a study on the periodic precipitation phenomena and their ... · a study on the periodic...

A Study on the Periodic Precipitation Phenomena and Their Application to Drug Delivery Systems

by

Beibei Qu

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Pharmaceutical Sciences University of Toronto

© Copyright by Beibei Qu (2012)

ii

A Study on the Periodic Precipitation Phenomena and Their

Application to Drug Delivery Systems

Beibei Qu

Doctor of Philosophy

Pharmaceutical Sciences University of Toronto

2012

Abstract

The main objective of this research was to better understand, predict and control of the periodic

precipitation process and to apply such programmed periodic precipitation to the design of a

pulsatile delivery system.

In the first part of this study, a generalized model taking into account both nucleation, particle

growth, and ripening process was refined and solved under various new concentration boundary

conditions not previously investigated. The results clearly delineate the key differences between

boundary conditions of infinite versus finite supply of inner electrolyte. When the inner

electrolyte boundary concentration was allowed to increase exponentially with time, equidistant

periodic precipitation was predicted and subsequently confirmed experimentally. In addition, the

effects of product solubility and reaction rate constant were also shown to be important in

determining the band number and band spacing.

In the second part of this study, the effects of gel crosslinking and gel charge density on the

periodic precipitation were investigated. The results indicate that increasing either the gel

crosslinking or decreasing the gel charge density will reduce the diffusion rate of the reactants

resulting in closely spaced bands. In addition, a new and improved rotating disk method for

iii

characterizing polyelectrolyte gels with ion-penetrable soft surfaces has been established by

taking into account the effect of surface conductivity which is usually ignored for ion-

impenetrable hard surfaces.

In the third part of this work, periodic precipitation formed in multi-component systems has been

shown to be governed by a heterogeneous nucleation mechanism. Using this approach, periodic

precipitation of an insulin mimetic compound VO2+ in gelatin gel, which cannot form alone in a

single reaction system, was induced by the periodic precipitation of Mg(OH)2 in a multi-

component system. Pulsatile release of VO2+ from the resulting multi-layered structure of

VO(OH)2 via a surface erosion mechanism was subsequently demonstrated.

iv

Acknowledgments

I would like to express my deep gratitude to my supervisor Prof. Ping I. Lee. During my graduate

study, Prof. Lee has been giving me enormous guidance and support. Looking back over the past

years, I realize how fast time flies and how much I have learned from Prof. Lee and his group. I

still remember his warm welcome email replying to my email introducing myself and asking for

an opportunity to join his group. As a student from abroad, I was not able to come for an

interview but Prof. Lee still offered me to be his graduate student. His research provides me a lot

of opportunities to learn new skills and techniques. At the beginning, the progress was quite slow

because I had to pick up the puzzles and put them together one by one. However, Prof. Lee is

always positive when I am struggling in the problem. He always has great suggestions and

advices when I have questions, and he always points to the right direction to improve the

research to a higher level when I have some results. Without his support, it would have ended up

nowhere. I really enjoy the time in Prof. Lee’s group. It is my great honor to have him as my

supervisor.

I also thank my committee members, Prof. Edgar. J. Acosta, Prof. Rob B. Macgregor and Prof.

Shirley X. Y. Wu. They have provided lots of valuable feedback on my research.

I thank my colleagues, Dr. Hui Zhao, Dajun Sun, Dr. Yan Li, Hong Liu and others. Working

together with them is always a nice experience.

Last but not least, I would like to thank my family for their support, emotionally and

intellectually.

v

Table of Contents

Abstract…………………………………………………………….……………………………..ii

Acknowledgement…………………………………………………………………………….….iv

Table of contents…………………………………………………………………………………..v

List of tables…………………………………………………………………………………....…ix

List of figures……………………………………………………………………………..…….…x

List of symbols…………………………………………………………………………………...xv

Chapter 1 Introduction

1.1. Background . ………………………………………………………………………………….1

1.2. Existing mechanisms and models of periodic precipitation. …………………………………3

1.3. The effects of gel phase property on the periodic precipitation …………………………….12

1.3.1 Gel mesh size..................................………………………………………..….…........ ….12

1.3.2 Gel charge property ...................................... ……………………………..………………13

1.4. Characterization of the density of gel charged groups. .…………………………………….15

1.5. Pulsatile drug delivery system and the application of periodic precipitation. ...................... 17

1.6. Hypothesis......................................................................................................................... 20

1.7. Research objectives ........................................................................................................... 20

Chapter 2 Programmed periodic precipitation

2.1. Introduction ....................................................................................................................... 21

2.2. Model ................................................................................................................................ 24

2.3. Methods and materials ....................................................................................................... 29

2.4. Results and discussion ....................................................................................................... 30

2.4.1 Experimental evidence of the periodic precipitation model .............................................. 30

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2.4.2 Concentration effect of the outer electrolyte..................................................................... 33

2.4.3 The finite reservoir effect................................................................................................. 37

2.4.4 Equidistant periodic precipitation..................................................................................... 42

2.4.5 Concentration effect of the inner electrolyte..................................................................... 47

2.4.6 The effects of other factors .............................................................................................. 50

2.4.7 The width of precipitate band........................................................................................... 54

2.5. Conclusions ....................................................................................................................... 56

Chapter 3 Effects of gel phase properties on periodic precipitation

3.1. Introduction ....................................................................................................................... 58

3.2. Materials and methods ....................................................................................................... 59

3.2.1 Gel preparation ................................................................................................................ 59

3.2.2 Preparation of periodic precipitation ................................................................................ 60

3.2.3 Gel mesh size analysis ..................................................................................................... 60

3.2.4 Quantifitation of gel charge property................................................................................ 60


3.3.1 Effects of the gel mesh size.............................................................................................. 60

3.3.2 Effects of the gel charge property..................................................................................... 65

3.4. Conclusions ....................................................................................................................... 71

Chapter 4 A rotating disk electrokinetic method for characterizing polyelectrolyte gels

4.1. Introduction ....................................................................................................................... 72


4.2.1 Materials.......................................................................................................................... 73

4.2.2 Preparation of gel samples ............................................................................................... 73

4.2.3 Rotating disk experiment ................................................................................................. 74

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4.3. Theory............................................................................................................................... 75

4.3.1 Improved rotating disk model .......................................................................................... 75

4.3.2 The density of gel charged groups.................................................................................... 82


4.4.1 Evaluation of ψ0 and 0 of PVA/PAA gel ....................................................................... 83

4.4.2 Evaluation of ψ0 and 0 of gelatin gel ............................................................................. 87

4.4.3 Evaluation of ψ0 and 0 of gelatin/PAA gel..................................................................... 89

4.5. Conclusions ....................................................................................................................... 91

Chapter 5 Periodic precipitation in multi-component systems

5.1. Introduction ....................................................................................................................... 92


5.2.1 Preparation of periodic precipitation in multi-component systems.................................... 93

5.2.2 Assay of precipitate composition ..................................................................................... 93


5.3.1 Periodic precipitation phenomena in multi-component systems........................................ 94

5.3.2 Composition analysis of bands.......................................................................................100

5.4. Conclusions ......................................................................................................................102

Chapter 6 Pulsatile drug release from periodic precipitation system

6.1. Introduction ......................................................................................................................104

6.2. Materials and methods ......................................................................................................106

6.2.1 Preparation of release medium ........................................................................................106

6.2.2 Drug release analysis ......................................................................................................106

6.2.3 Drug release mechanism ................................................................................................106

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6.3. Results and discussion ......................................................................................................107

6.3.1 Incorporation of model drug into the drug carrier ............................................................107

6.3.2 Pulsatile drug release ......................................................................................................107

6.4. Conclusions ......................................................................................................................112

Chapter 7 Summary and future directions

7.1. Summary ..........................................................................................................................114

7.2. Future directions...............................................................................................................116

References................................................................................................................... ............118

Appendix..................................................................................................................... ............128

ix

List of Tables

Chapter 2

Table 2.1 Effects of constant parameters on the formation of periodic precipitation……….…...53

Table 2.2 Effects of variable parameters on the formation of periodic precipitation…………....54

Chapter 4

Table 4.1 Calculated ψ0 and values of tested polyelectrolyte gel samples……………..…….90

Chapter 5

Table 5.1 Characteristics of insoluble salts…………………………………………………….. .97

Appendix

Table A-1 Parameter input in Comsol 3.5a (Example: Figure 2.4 in Chapter 2, Finite reservoir

boundary condition) ……………………..…………………………………………………......130

Table A-2 Parameter input in Comsol 3.5a (Example: Figure 2.10 in Chapter 2, Infinite reservoir

boundary condition) .……………………..………………………………………………….....133

Table A-3 Parameter input in Comsol 3.5a (Example: Figure 2.14 in Chapter 2, equidistant

periodic precipitation) .……………………..………………………………………………......135

Table A-4 Parameter input in Comsol 3.5a (Example: Figure 3.4 in Chapter 3, with real

parameters) .…………………………………………………………………………………….137

x

List of Figures

Chapter 1

Figure 1.1 Liesegang ring phenomena…………………………………………………………….2

Figure 1.2 Periodic precipitation in agates and in inflammatory breast lesion……...………….…2

Figure 1.3 Microfabrication of microlenses and molds for passive microfluidic mixers …….......3

Figure 1.4 Schematic diagram of Liesegang ring phenomenon…………………...………………4

Figure 1.5 The concentration change of reactant A and B in the gel phase based on the

prenucleation model……………………………………………………………………………….6

Figure 1.6 Schematic illustration of the particle growth process………………………………….8

Figure 1.7 Relation between Ceq(r) and r……………….………………………………...……….9

Figure 1.8 Experimental setup for producing equidistant periodic precipitation bands by

imposing time-dependent electric current………………………………………………..……....11

Figure 1.9 Ring and the tree like structures formed during precipitation.…………..…………...13

Figure 1.10 Surface potentials on “hard” and “soft” surfaces in contact with an electrolyte

solution…………………………………………………………………………………………...17

Figure 1.11 Periodic precipitation of AgI in PVA film………………………………………….19

Chapter 2

Figure 2.1 Schematic illustration of the periodic precipitation system………………………..…22

Figure 2.2 Schematic illustration of a model periodic precipitation system……………………..26

Figure 2.3 Exponential increase of CA0 in 24hr by the gradient HPLC solvent delivery unit …..30

Figure 2.4 Simulation results of periodic precipitation at two different dimensionless times.… .32

xi

Figure 2.5 Time evolution of periodic precipitation of Mg(OH)2 in gelatin gel under finite

reservoir boundary condition…………………………………………………………………….33

Figure 2.6 Concentration effect of outer electrolyte.…………….………………………………34

Figure 2.7 Concentration effect of outer electrolyte.…………….………………………………35

Figure 2.8 Concentration effect of outer electrolyte…………………………..………..………..36

Figure 2.9 Direct comparison between simulation results and experimental results………….…37

Figure 2.10 Time evolution of concentration profiles of outer electrolyte above the gel surface (x

< 0) at a = 15………………………………………………………………………………...…...38

Figure 2.11 Simulation results of periodic precipitation density F under finite and infinite

reservoir boundary conditions.…………………………………………………..………………39

Figure 2.12 Experimental results of periodic precipitation at 36 hr under finite (a1, b1, and c1)

and infinite (a2, b2, and c2) reservoir boundary conditions, respectively.……………………...40

Figure 2.13 Simulated concentration profiles of outer electrolyte above the gel surface, X < 0, at

T = 172.8 (CA0= 14.8 M and CB0 = 0.2 M) and different outer electrolyte volume (V1) to inner

electrolyte gel phase volume (V2) ratios.…………………………...…………………………...41

Figure 2.14 Simulation results of scaled density of precipitate…………………...……………..41

Figure 2.15 Simulation results of periodic precipitation density F (CB0 = 0.2 M) under the

following concentration boundary conditions in the outer electrolyte reservoir:………….…….43

Figure 2.16 Dimensionless outer electrolyte reservoir concentration profiles having different

rates of approach to the maximum concentration followed by maining the maximum

concentration for a fixed period of time…………………………………………………..…..…44

Figure 2.17 Experimental evidence of equidistant periodic precipitation…………..…………...45

Figure 2.18 Comparison of band spacing ΔX simulated under different outer electrolyte reservoir

concentration boundary conditions…………..………………………………………...………...46

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Figure 2.19 Typical secondary bands formed during the periodic precipitation process.……….47

Figure 2.20 Concentration effect of inner electrolyte.…………………………..…..…………...49

Figure 2.21 Simulation results of scaled density of precipitate, F, at T = 345.6 (24 hr) for

different CB0……………………………………………………………………………………...50

Figure 2.22 The effect of diffusion coefficient ratio, D1 = DA/DB.…………..….………..……...51

Figure 2.23 Simulation results of periodic precipitation density F with distance dependent

diffusion coefficient and a = 37………………………………..…………………....…………...52

Figure 2.24 The effect of product solubility, C0………………...…………………....…..……...52

Figure 2.25 The effect of reaction rate constant, k…………………….…………....…………...53

Figure 2.26 Schematic illustration of band width and band spacing…………………......……...54

Figure 2.27 Simulation results of periodic precipitation density F with a = 15……..…………...56

Chapter 3

Figure 3.1 Comparison of gel swelling ratio of PVA and gelatin gel as a function of salt

concentration at room temperature……………………………………………………………....61

Figure 3.2 Periodic precipitation of Mg(OH)2 in gelatin gel and PVA gel at 24 hr.………...…..62

Figure 3.3 Schematic diagram showing the effect of gel mesh size on periodic precipitation…..63

Figure 3.4 Simulation results of particle radius r (m) at t = 24 hr with CA0 =14.8 M and CB0 = 0.2

M………………………………………………..……………………………….....…………….64

Figure 3.5 Simulation results of precipitation density f (mol/m3) at t = 24 hr with CA0 =14.8 M

and CB0 = 0.2 M…………………………………………………………………………..……...65

Figure 3.6 Determination of pI value of gelatin gel from the pH dependence of UV absorbance

profiles……………………………………………………………………………………….…..67

Figure 3.7 Determination of pKa values of gelatin gel by pH titration.…………………...…….67

xiii

Figure 3.8 Determination of pKa values of PAA-gelatin gel by pH titration……….…………...68

Figure 3.9 Periodic precipitation in gelatin and PAA-gelatin gels..….…….……………...…….69

Figure 3.10 Schematic diagram showing the effect of gel surface curvature on the progression of

diffusion front.…………………...……………………………………………………………....70

Chapter 4

Figure 4.1 Rotating disc experimental setup……………………………………………………..75

Figure 4.2 Current flow on “hard” and “soft” surfaces in the rotating disc system…………..….77

Figure 4.3 The relation between ϕ(r/a,0)/(iza/πKL) and (r/a) in rotating disc system………........79

Figure 4.4 Typical plots of streaming potential as a function of disk rotation speed Ω1.5 for

PVA/PAA(100/1) gel disks in 0.04 mM NaCl...……………………………………….…..……86


PVA/PAA(200/1) gel disks in 0.04 mM NaCl...……………………………………….…..……86

Figure 4.6 Typical plots of streaming potential as a function of disk rotation speed Ω1.5 for 10%

gelatin gel disks in 0.04 mM NaCl...………………………..………………………….…..……88


gelatin/PAA gel disks in 0.04 mM NaCl...…………….....…………………………….…..……89

Chapter 5

Figure 5.1 Experimental results of precipitate in single and double reaction systems..…………95

Figure 5.2 Simulation results of periodic precipitation density F, as a function of solubility of

reaction product………………………………………………. ……….. …………………….…96

Figure 5.3 Schematic illustration of heterogeneous nucleation in Mg(OH)2- Ca(OH)2 system…98

Figure 5.4 Schematic illustration of heterogeneous nucleation in Mg(OH)2- VO(OH)2 system...99

Figure 5.5 Schematic illustration of sample positions.…………………………...………….…101

xiv

Figure 5.6 Results of ICP analyses of Mg2+ and VO2+ in sample 1-3, respectively.……... ...…102

Figure 5.7 Results of ICP analyses of Mg2+ and Ca2+ in sample 1-3, respectively.…. …...…...102

Chapter 6

Figure 6.1 Eroding front position as a function of time during the release of Mg2+ from periodic

precipitation structure in gelatin gel…………………………………………………………….108

Figure 6.2 The overall gel length S(t) as a function of time in Mg release system………...…..109

Figure 6.3 Concentration of Mg2+ in the release medium as a function of time analyzed by

ICP……………………………………………………………………………………………...109

Figure 6.4 Eroding front position as a function of time during the release of VO2+ and Mg2+ from

periodic precipitation structure in gelatin gel.……………………….…………………………111

Figure 6.5 The overall gel length S(t) as a function of time in Mg/VO release system………..112

Figure 6.6 Concentrations of VO2+ and Mg2+ in the release medium as a function of time

analyzed by ICP………………………………………………………………………………...112

xv

List of Symbols

A, outer electrolyte

A0, initial concentration of A at the gel surface a, dimensionless concentration of outer electrolyte A

a0, dimensionless concentration of outer electrolyte A at the gel surface B, inner electrolyte

b, dimensionless concentration of inner electrolyte C, reaction product

C0, solubility of product C CA, concentration of outer electrolyte A

CA0, initial concentration of A CB, concentration of inner electrolyte

CB0, initial concentration of B inside of gel phase CC, concentration of reaction product

Ceq(r), concentration of product C in equilibrium with a particle of radius r DA, DB, DC, diffusion coefficient of A,B and C, respectively

D1,D2, dimensionless diffusion coefficient of A and B, respectively d, typical molecular size taken as twice the diameter of the C molecule

E, precipitation from the reaction between A and B E(r), complete elliptic integral of the second kind

Er, radial component of the electric field close to the disk surface F, dimensionless concentration of precipitate

f, molar concentration of precipitate G, particle growth rate constant

ΔG*, free energy barrier in homogenous nucleation system ΔG*’, free energy barrier in heterogeneous nucleation system

H , A , charged groups in the gel phase H+ ,A+ ,Y- , soluble ions in solution i, valence of the charged groups

iz, normal current density J, nucleation rate

xvi

J(x,t), nucleation rate at position x and time t j, valence of the symmetrical electrolyte

J0(p) and J1(p), Bessel Functions of order 0 and 1, respectively jr, , total radial surface current

js, surface current K, dimensionless reaction rate constant

K(m), complete elliptic integrals of the first kind k, reaction rate constant

kB, Boltzmann’s constant KL, solution conductivity

Kσ, surface conductivity ksp, solubility product

L, length of gel L’, length of finite reservoir of outer electrolyte

N, dimensionless average particle number density n, average particle number density

Q, gel swelling ratio R, dimensionless particle radius

r, radial position r(x,t,t’), particle radius at time t which was nucleated at t’

rn (x,t) , critical radius of particle r (m), particle radius

s, dimensionless relation (Cc- C0)/ C0 T, dimensionless time

t, reaction time tn, band formation time of nth precipitate band

U, precipitation rate vr , local radial velocity

vm, molar volume of precipitate WS, gel swollen weight

W0, dry gel weight Wn, band width of nth precipitate band

w, capillary length

xvii

X, dimensionless distance x Xn, band position of nth precipitate band

ΔX, spacing between bands z, axial position

δ, thickness of the Gibbs surface Δ, dimensionless thickness of the Gibbs surface

ɛ0, permittivity of free space ɛ, dielectric constant of the solution.

ν, electrokinetic viscosity of the NaCl solution ρ, density of precipitate

ρ0, volume density of fixed charged groups in the gel ρe, density of charges at any space location in solution

σ, surface tension of the precipitate particle φ(r), radius dependence of surface tension

ψ0, surface (or Zeta) potential ψD, Donnan potential

Ω, rotation rate

λ, constant Gw /

θ, constant

),0( z or ψstr, measured streaming potential ),( zrd , potential that arises from the uniform flow of current to the disk surface

r , electric potential in the radial direction ),( zrr , potential responsible for the radial current

1

Chapter 1 Introduction

1.1 Background Periodic precipitation is the basis of many pattern formations in nature such as the band and ring

structures found in minerals, rocks, gall stones or even cystic or inflamed tissue. This

phenomenon was first investigaged by Liesegang in 1896, when he added a drop of silver nitrate

onto the surface of a gelatin gel containing potassium chromate and found a series of concentric

rings of silver chromate precipitate formed in the gel medium (Liesegang, 1896; Henisch, 1970),

as illustrated in Figure 1.1. These are the result of interdiffusion and reaction of two co-

precipitating reactants in the gel. Many minerals and rocks exhibit such Liesegang ring patterns

due to similar interplay between diffusion, reaction and precipitation processes as shown in

Figure 1.2a (Grzybowski et al.,2007). Similar phenomenon is also found in vivo including gall

stones (Xie et al., 1999) and inflamed tissue such as breast lesion and renal cyst as shown in

Figure 1.2b (Gavin et al., 2005). As a result, much effort has been made in modeling the band

formation and simulating the development of periodic precipitation in geoscience (Sultan et al.,

1990; Ortoleva, 1994). Recently, this phenomena has been applied in nanofabrication through

wet stamping, for instance, microscopic structures were formed when AgNO3 was released from

an agarose stamp onto a dried gelatin film pre-loaded with K2Cr2O7 (Grzybowski et al., 2007).

These fabricated microscale and complex structures have been applied to microlens arrays and

microfluidic architectures (Figure 1.3). In our present study, we are interested in an improved

understanding and more accurate theoretical description of the physical processes involved in

such a periodic precipitation phenomena. Meanwhile, new methodologies to manipulate and

control the spacing between the precipitate bands are provided. Ultimately, we envision that the

refined theories of this study can help predict the formation of periodic precipitate structures in

many of the above mentioned pattern formations in nature. Furthermore, we are interested in

exploring the application of the resulting periodic precipitation in multi-pulse drug delivery. By

loading the drug into such microstructured device through the periodic precipitation process, we

hope to provide a novel method of micro- and/or nanofabrication which can overcome the

limitations in the fabrication of existing pulsatile drug delivery systems.

2

Figure 1.1 Liesegang ring phenomena (Liesegang, 1896; Reprinted with permission from Müller

et al., 2003).

Figure 1.2 Periodic precipitation (a) in agates and (b) in inflammatory breast lesion. (Reprinted

with permission from Gavin et al., 2005; Grzybowski et al., 2007).

（a）（b）

3

Figure 1.3 Microfabrication of microlenses (a) and molds for passive microfluidic mixers (b).

Scale bar is 150 m (Reprinted with permission from Grzybowski et al., 2007；Campbell et al.,

2005).

1.2 Existing mechanisms and models of periodic precipitation Earlier investigators established three comprehensive laws to describe the precipitate band

position, band width and band spacing in the periodic precipitation phenomenon (Grzybowski,

2009): the time law (Morse and Pierce, 1903), the spacing law (Jablczynski, 1923) and the width

law (Pillai et al., 1980; Grzybowski, 2009). These phenomenological models are based on

simplified assumptions and therefore they may not describe the real system accurately.

For the time law, it is assumed that when the outer electrolyte, A, diffuses into the gel and reacts

with the inner electrolyte, B, the process is described by Eq. (1.1):

tC

xC

D AAA

2

2

Eq. (1.1)

4

Where t is the reaction time, CA and DA are the concentration and diffusion coefficient of outer

electrolyte A in the gel phase, respectively. In this case, it is assumed that the concentration of

inner electrolyte B, CB, is always constant in the gel phase (no reaction term in Eq. (1.1)).

Precipitate band forms when CA = CB, as illustrated in Figure 1.4.

Figure 1.4 Schematic diagram of Liesegang ring phenomenon.

The equation above can be solved with the initial concentration of A at the gel surface, CA0 . The

following relation between the band position and band formation time is then obtained:

Kt

xn

n Eq. (1.2) The Time Law,

Where Xn and tn are the band position and band formation time of nth precipitate band,

respectively. The ratio between Xn and tn approaches a constant K. By knowing the reaction time,

the precipitate band position can be predicted based on the time law.

For the width law, if it is assumed that the nucleation starts when CA equals to CB at Xn and ends

when CA is close to zero. The following relation between band width is achieved:

QW

Wn

n 1 Eq. (1.3) The Width Law

Where Wn and Wn+1 are the band width of the nth and n+1th precipitate band, respectively. The

ratio of which reaches a constant Q. Generally, by increasing the band number n, the band

becomes wider.

5

For the space law, if we further assume that the amount of reactant in the band is related to the

movement of A from the band gap to the band position, a space law can be applied to describe

the spacing between bands.

nn xx 1 Eq. (1.4) The Space Law,

Where Xn and Xn+1 are the band position of the nth and n+1th precipitate band, respectively. The

ratio of which reaches constant , which in most systems is larger than 1 and smaller than 1.5

(Grzybowski, 2009). In other words, the band spacing increases by increasing band number n

and the band becomes more separate following the diffusion direction.

Although these common laws can describe the general trends of the periodic precipitation

phenomena, they cannot make accurate predictions in the real systems because the assumptions

of these models are not precise in most cases. For instance, the inner electrolyte B in the finite

volume of gel is consumed by its reaction with the outer electrolyte A and its concentration CB is

not constant but decreasing with time. Therefore, the model equation Eq. (1.1) should be

corrected by adding one more reaction term regarding reactant B. Furthermore, these models are

not established according to the rigorous mechanism of this periodic precipitation phenomenon

and therefore these will not be employed in our study.

The mechanism of the periodic precipitation phenomenon has been widely investigated over the

years and different theories have been proposed, which can be divided into two general

categories (Stern, 1954, 1967; Henisch, 1988; Grzybowski, 2009): pre-nucleation and post-

nucleation models. The prenucleation model was first proposed by Ostwald in 1897 (Ostwald,

1897) and extended later by others (Prager, 1956; Smith, 1984; Le Van and Ross, 1987). Based

on this model, as the outer electrolyte diffuses into the gel phase and react with the inner

electrolyte, the precipitate rings are the result of nucleation and crystallization triggered by the

supersaturation of the reaction product, where the nucleation occurs when the reaction product

exceeds a certain threshold of supersaturation value. No further nucleation will occur at the ring

location due to the depletion of reactant in the immediate surrounding area of the nuclei and the

concomitant drop of the local level of product concentration (Figure 1.5) as the reaction zone

moves away (Prager, 1956; Smith ,1984; Le Van and Ross, 1987). When the supersaturation

level of the product achieves the critical level again at the moving front, the next precipitation

6

occurs and the repetition of these events continues. In this model, diffusion is a key factor in

determining the availability of the reactant product. Thus, periodic precipitation patterns can be

observed after repeated precipitate formation in the gel matrix with some spacing in-between.

Figure 1.5 The concentration change of reactant A and B in the gel phase based on the

prenucleation model (Figure adapted from Henisch. 1988).

This prenucleation model can be treated as an inter-diffusion and reaction process of reactants A

and B, and the fast reaction results in a slightly soluble intermediate reaction product C. After C

reaches a certain threshold value, it becomes the precipitate E. This process is typically described

by following equations:

tCCkC

xCD A

BAA

A

2

2

Eq. (1.5)

tCCkC

xCD B

BAB

B

2

2

Eq. (1.6)

tC

UCkCxC

D CBA

CC

2

2

Eq. (1.7)

UtE

Eq. (1.8)

7

Where DA, DB, DC are defined as the diffusion coefficients of reactant A, B and product C,

respectively, ρE represents the density of precipitation, k stands for the reaction rate constant,

and U denotes the precipitation rate, describing the removal of product from the reaction system

by nucleation and crystal growth process. When the product concentration is above a critical

threshold value (CC > Ccritical), then the nucleation occurs (or U >0).

The equations above for the prenucleation model can be numerically solved and the trend of

periodic precipitation obtained under appropriate boundary conditions. However, this model

neglects the kinetic characteristics of nucleation, particle growth and the time evolution of bands.

An extended prenucleation model incorporating the nucleation and growth kinetics was later

developed by Dee (Dee, 1986). In general, prenucleation models can generate periodic

precipitation bands satisfying the scaling law but are not able to describe other experimental

observations such as the formation of the secondary precipitation bands between two major

bands. Aiming to solve this problem, the postnucleation model was established taking into

account the competitive particle growth and ripening effects.

In the postnucleation model (Flicker and Ross, 1974; Feinn et al., 1978; Lovett et al., 1978;

Venzl and Ross, 1982; Feeney et al., 1983), A reacts with B and generates intermediate

compound C. In the later stage of the first-order phase separation, the existence of nuclei

particles with heterogeneous sizes leads to the growth of large particles by depleting the smaller

ones (Figure 1.6). The particle growth rate is related to the supersaturation level and particle size.

When particle size is larger than the critical radius rcritial, it grows by consuming the smaller

particles (r < rcritial) due to the lower surface energy of large particles. This model considers the

fact that the existence of nucleated particles with different sizes triggers the growth of larger

particles by depleting the surrounding smaller particles (Flicker and Ross, 1974; Feinn et al.,

1978; Lovett et al., 1978; Venzl and Ross, 1982; Feeney et al., 1983). It can be applied to

explain the occurrence of particle growth, the time evolution of bands and the existence of

secondary precipitations. However, most of the reported expressions between particle growth

and particle size are mostly based on qualitative assumptions and a rigorous theory was lacking.

8

Figure 1.6 Schematic illustration of the particle growth process

More recently, a generalized model has been developed by Chacron and L’Heureux (Chacron

and L’Heureux, 1999) which combines Dee’s extended prenucleation model incorporating both

nucleation and growth kinetics (Dee, 1986) with the postnucleation competitive particle growth

model pertaining to ripening (Feeney et al., 1983) and considers the role of supersaturation in

both nucleation and particle growth in the reaction process. This model bridges the gap between

the previous two models. The nucleation rate is related to the local supersaturation level, which

is still considered as a key factor in generating the bands pattern of periodic precipitation. A

rigorous theoretical relationship between particle growth and particle size has been established.

Accordingly, if the particle growth process is interface controlled, the particle growth rate can be

described by Eq. (1.9) (Le Van and Ross, 1987; Chacron and L’Heureux, 1999):

])(

[0C

rCCG

tr eq

Eq.(1.9)

Where, G is the particle growth rate constant and Ceq(r) is the concentration of product C in

equilibrium with a particle of radius r. C0 is the solubility of product C. The relation between

Ceq(r) and r can be described by the Gibbs-Thomson relation (Chacron and L’Heureux, 1999):

])(exp[)( 0 rrwCrCeq

Eq.(1.10)

where φ(r) represents the radius dependence of surface tension expressed as:

qrrrrr

/3)( 22

2

with q = 0.304359 and δ representing the thickness of the Gibbs

9

surface; this expression for φ(r) is simplified from a thermodynamically based expression of

Koenig (Koenig, 1950; Chacron and L’Heureux, 1999). w denotes the capillary length and has

the expression TkN

vwB03

2 , As a result, Eq.(1.10) is more accurate than the intuitive radius

dependence of equilibrium concentration employed in the previous postnucleation competitive

particle growth model (Feeney et al., 1983).

The particle with radius r is in equilibrium with the product concentration Ceq(r) as dictated by

its surface free energy. When r is larger than the critical radius, Ceq(r) decreases with r. In

contrast, when r is small, Ceq(r) increases with r (Figure 1.7). This suggests that the large

particles will become larger by consuming the surrounding small particles because of their small

surface energy. Therefore, the kinetic characteristics of nucleation, particle growth and the time

evolution of bands are all considered in this model. Meanwhile, when φ = 0, Ceq(r) is constant

and the generalized model of Chacron and L’Heureux reduces to Dee’s prenucleation model

(Chacron and L’Heureux, 1999). In this case, nucleation occurs when supersaturation reaches a

threshold, C0.

Ceq(r)

r

Figure 1.7 Relation between Ceq(r) and r.

To the best of our knowledge, this generalized model of Chacron and L’Heureux is the most

comprehensive one in describing the periodic precipitation process as it takes into account the

interplay between diffusion, nucleation, growth and ripening processes thereby bridging the gap

between the prenucleation and postnucleation models. This is supported by the fact that it

10

reduces to the Dee’s prenucleation model when nucleation and growth are dominant and it

reduces to the postnucleation competitive growth model when the nucleation phase terminates

and the ripening becomes important. Therefore, this generalized model is employed to analyze

the periodic precipitation results in our study.

Modeling the periodic precipitation phenomena more precisely can play a critical role in various

applications. Recently, the fabrication of microlenses has been demonstrated via the periodic

precipitation of silver nitrate and potassium hexacyanoferrate in a gel matrix (Grzybowski, 2009).

Thus, for the design and control of the periodic precipitation process in order to build a complex

and predesigned microstructure, it becomes increasingly important to be able to design and

control the precipitate band location, spacing and width. Most previous studies applied a

constant concentration (equivalent to a fixed concentration or an infinite reservoir) boundary

condition for the outer electrolyte in their modeling analysis while conducting the experiments

with a finite volume of outer electrolyte (Zrhyi et al. 1991; Carotenuto et al., 2002; Fiałkowski et

al. 2005; Lagzi, et al. 2007; Izsk and Lagzi, 2005). This constant concentration boundary

condition employed is in conflict with the experimental fact that the outer electrolyte

concentration at the solution/gel interface decreases with time in a finite volume. Such

inconsistency can generate considerable discrepancy between the predicted and experimental

band position and band spacing in periodic precipitation. Therefore, one major objective of the

present study is to examine and delineate the effect of concentration boundary conditions (for

both infinite and finite volume of the outer electrolyte) on the resulting characteristics of the

periodic precipitation.

As it is known, the band position Xn and spacing between bands ΔX can be adjusted only to a

limited extent by altering the initial concentration of the outer or inner electrolyte based on

existing approaches. In general, the spacing (ΔX) between consecutive precipitation bands

increases with distance X as a result of the inherently diffusion controlled transport process

which tends to slow down with distance. On the other hand equidistant banding pattern of a

reaction product is of significant technological importance in several fields ranging from micro-

and nanofabrication to drug delivery (Xu and Lee, 1993; Grzybowski, 2009). However, under

classical concentration boundary conditions (e.g. infinite or finite volume of outer electrolyte),

equidistant periodic precipitation will not result. Thus, additional driving forces need to be

introduced in order to alter the periodic precipitation pattern. Recently, Bena and coworkers

11

(Bena et al., 2005; Bena et al., 2008) reported the formation of equidistant periodic precipitation

by imposing a time-dependent electric current in the reaction-diffusion system, where the

concentration of reactants was varied by adjusting the electric field strength. Their experimental

setup was similar to that in Figure 1.8, where quasiperiodic time dependent electric current was

imposed to generate the desired equidistant precipitation. This approach is reasonable, because

the diffusion flux of reactants in the reaction zone is the governing factor controlling the

precipitate band position (Xn). However, the experimental setup of this method is quit

cumbersome and impractical for routine use. Furthermore, without considering the gel

deformation in an electric field (Yamaue et al., 2005), the current theory on the control of

periodic precipitation through electric current is inadequate for polyelectrolyte gel systems. More

recently, a continuous large-scale modification of the inhomogeneity of substrate and/or the

nucleation threshold has been proposed to obtain equidistantly spaced bands in the reaction-

diffusion system (Jahnke and Kantelhardt, 2010). However, this concept has only been shown in

the extended Monte-Carlo lattice-gas simulations but has yet to be demonstrated experimentally.

Judging from the experimental complexities involved in modifying the inhomogeneity of the

substrate and in changing the nucleation threshold, the applicability of this proposed approach

may be quite limited.

Figure 1.8 Experimental setup for producing periodic precipitation bands by imposing time-

dependent electric current (Reprinted with permission from Bena et al., 2005).

Other approaches have also been applied to control the pattern of periodic precipitation, such as

by varying the initial concentration of the reactants (Zrhyi et al., 1991; Attieh et al., 1998),

solubility of the reaction product (Msharrafieh et al., 2007), and gel thickness perpendicular to

12

the diffusion direction (Fialkowski et al., 2005). However, these approaches are not widely

applicable and the corresponding models describing the processes are complicated. Moreover, to

the best of our knowledge, equidistant periodic precipitation cannot be obtained by these

proposed methods. For practical purposes, there is a major need for a more flexible method to

control the band formation in the periodic precipitation process and to generate equidistant

periodic precipitation patterns. Therefore, as another objective of the present study, a novel and

practical method is introduced to manipulate the band position and band spacing ΔX and the

existing theories are further refined to achieve a better quantitative description of the phenomena.

In addition, some reactants, such as Ca(OH)2 and VO(OH)2, do not form periodic precipitation in

a single reaction system, regardless of the reactant concentration and reaction conditions.

Furthermore, it was reported that under certain conditions, periodic precipitation may be

facilitated by the impurity in the gel phase (Henisch, 1988). However, the mechanisms involved

are not clear and full explanations for these observations are not available. In our present study,

we propose to employ a multi-component system to assist the formation of periodic precipitation

of these problem reactants. To date, periodic precipitation phenomenon in multi-component

systems is a new topic and has only been reported in a limited number of cases (Shreif, et al.,

2002; Klajn, et al., 2004). However, existing studies focus mainly on the analysis of band

composition and the effect of reactant concentration on band position. This phenomenon does

not obey existing mechanisms or theories and more rigorous interpretations of these observations

are still lacking. Therefore, in our current study a potential mechanism for this phenomenon is

proposed to illustrate the formation of periodic precipitation in multi-component precipitate

system. Meanwhile, by applying multi-component periodic precipitation strategy, we want to

provide a guideline to facilitate reactants to form periodic precipitation in multi-component

systems, even for reagents which fail to form periodic precipitation in a single reaction system.

1.3 The effects of gel phase property on the periodic precipitation

1.3.1 Gel mesh size The three dimensional structures of a gel can maintain a stable concentration gradient of the

reactant solution and inhibit the occurrence of convection. The diffusion of reactants in the gel

phase depends on many factors, such as the mesh size, molecular size, valence, chemical nature

13

of diffusing reactants, swelling degree, charge density and the chemical nature of the gel matrix.

The gel mesh size is one of the most important factors in controlling the diffusion process and is

governed by the gel swelling degree and crosslinking density. The diffusion of reactants can be

retarded by reducing the gel mesh size. Besides the gel effect on diffusion, the gel structure can

also support the formed crystallites without exerting external forces on it before the crystal size

exceeds the mesh size of the gel. When the crystal size surpasses the mesh size, the gel starts to

suppress further crystal growth, which is one of the most important functions of the gel phase in

the periodic precipitation process. During the precipitation process, the initially formed small

crystallites undergo a dynamic evolution. The larger crystals, which have a lower solubility than

smaller ones due to surface energy effect, grow at the expense of smaller ones. Thus the system

will contain fewer but larger crystals at the end, known as the Ostwald ripening. However, in a

loose gel where adequate restriction by the gel phase on the nucleation step is lacking, the

crystals may evolve into other patterns of precipitation, such as tree like, large platelets or

needle-like precipitation patterns (Toramaru et al., 2003) (Figure 1.9). Therefore, the gel

structure is important to the initiation of nucleation and the evolution of crystal growth.

Figure 1.9 Ring and the tree like structures formed during precipitation (Reprinted with

permission from Toramaru et al., 2003).

1.3.2 Gel charge property

Physically crosslinked gelatin gel is a cationic polyelectrolyte gel, which has been widely applied

in periodic precipitation systems (Henisch, 1988; Msharrafieh and Sultan, 2005; Lagzi and

14

Ueyama, 2009). Conceptually, the ionic groups in the gel phase would interact (electrostatic

interaction or ion exchange) with diffusing reactant or reaction product and therefore the gel

charge property plays an important role affecting the periodic precipitation process.

The polyelectrolyte gel contains polymer network and pendent charged groups. The diffusion of

ionic species in polyelectrolyte gels and the electrostatic effects of charged groups have been

widely studied (Kim and Lee, 1992; Narita et al., 1998; Hyk and Ciszkowska, 1999; Baek and

Srinivasa, 2004; Darwish et al., 2004; Ogawa and Kokufuta, 2004; Yamaue et al., 2005; Masiak

et al., 2007). In response to the various pH stimuli in the environment, polyelectrolyte gels may

exhibit different swelling behavior. Similarly, with different densities of fixed charge groups,

polyelectrolyte gels may also exhibit different swelling properties under the same pH condition.

For cationic gels, gel shrinkage may occur during the diffusion and reaction of OH- ion which

can result in a large curvature on the gel surface confined in a glass tube. Consequently, the

reactant in the diffusion front cannot distribute evenly in this case, thereby affecting the radial

pattern of periodic precipitation downstream.

Meanwhile, the equilibrium between the precipitate and dissolved ions is affected by interactions

between the dissolved ions and the charged groups in polyelectrolyte gel. The ions may be

continuously removed from the precipitate until it is completed dissolved or reached an

equilibrium between the ionic gel and the precipitate. For example, when the 1,1-valent

precipitate AY (for instance, AgCl) is in a cationic gel with H+ dissociable groups, it will be in

equilibrium with its dissociated ions. The H+ on the cationic gel can exchange with the

dissociated A+ until it reaches equilibrium and the amount of dissolved precipitate depends on

the gel charge density (Helfferich, 1995):

YAAY

AYHHAY Where H and A are the charged groups on the gel, A+ , H+

and Y- are the soluble ions in solution.

Therefore, the physicochemical properties of the gel phase such as the gel mesh size or gel

charge property, and their effects on the periodic precipitation are critical. However, to the best

of our knowledge, these factors have not been investigated and quantified in relation to periodic

15

precipitation. Thus, in our current study the gel phase properties and their effects on the periodic

precipitation will be studied in order to be able to program the formation of desired periodic

precipitation patterns.

1.4 Characterization of the density of fixed charge groups As discussed above, the gel phase properties play an important role during periodic precipitation

process, especially the charge effect. Therefore, an accurate knowledge of gel charge properties

is critical to the understanding and control of periodic precipitation.

Experimental assessment of surface charge generally involves the measurement of surface (or

Zeta) potential ψ0, defined as the electric potential at the slipping plane or the plane of shear in

the electrical double layer relative to the electrically neutral bulk solution. This is typically

accomplished by measuring either the electrophoretic mobility of particles based on the laser

Doppler velocimetry or the streaming current and streaming potential of flat surfaces via either

the electrokinetic microslit method or the recently developed rotating disk method (Johnson and

Thornton, 1969; Spanos and Koutsoukos, 1999; Kushibiki et al., 2003; Sides and Hoggard, 2004;

Hoggard et al., 2005; Sides et al., 2006; Delgado et al., 2007; Lameiras and Nunes, 2008;

Tandon et al., 2008). The electrokinetic density of surface charge groups is then calculated from

the classical Gouy-Chapman theory for the screening of charges on a rigid ion-impenetrable

surface by counterions in the diffuse part of the electrical double layer (see Figure 1.10 a ), with

the contribution from adsorbed ions in the hydrodynamically stagnant layer ignored. This

approach is reasonable for the measurement of the electrokinetic potential of ion-impenetrable

materials having a “hard” surface such as silicon and mica (Sides and Hoggard, 2004; Hoggard

et al., 2005; Sides et al., 2006), where such contribution from adsorbed ions would indeed be

very small.

However, this classical electrokinetic model would be inaccurate for colloidal or bio-colloidal

systems involving ion-penetrable charged gel layers or “soft” surfaces (Ohshima, 1995; Dukhin

et al., 2004, 2006; Duval and van Leeuwen, 2004). This generally occurs in crosslinked

hydrogels, self-assembled bilayers, or adsorbed polyelectrolytes where ion penetration and

limited electro-osmotic solvent flow can exist within the substrate layer having a three-

dimensional charge distribution. This feature is different from the two-dimensional charge

distribution on a “hard” surface considered in classical electrokinetic theory. In other words, a

16

soft and ion-penetrable surface is normally present in polyelectrolyte gels, where the counterions

are distributed both inside the gel phase and in the diffuse layer external to the gel surface as

depicted in Figure 1.10 b. Depending on the gel structure and charge properties, the screening of

the three-dimensional surface charge by the counter ions can occur appreciably in the

polyelectrolyte gel layer with only a limited portion of the screening counter ions in the diffuse

layer. This is usually manifested in an increased surface conductivity. Theories on the

electrokinetic phenomena involving ion penetrable soft surfaces have been developed by several

groups (Ohshima, 1995; Dukhin et al., 2004, 2006; Duval and van Leeuwen, 2004) and have

been applied experimentally mostly to the electrophoresis of soft colloidal particles. Recently,

the electrokinetic microslit method has been employed experimentally to characterize the

electrokinetic properties of thin soft gel surfaces (grafted polyelectrolyte surface layer), and the

results confirmed the importance of surface conductivity in systems involving soft surfaces

(Duval and van Leeuwen, 2004; Duval, 2005; Zimmermann and Osaki, 2006). However, this

microslit setup is quite cumbersome for routine use and not readily adaptable to thicker gel films

of pharmaceutical interest. Meanwhile, a recently developed rotating disk electrokinetic method

for measuring streaming potential is much simpler to set up and operate, but the associated

electrokinetic analysis neglects the effect of surface conductivity K and is only applicable to

“hard” ion-impenetrable surfaces such as silicon and mica (Sides and Hoggard, 2004; Hoggard

et al., 2005; Sides et al., 2006). To the best of our knowledge, characterization of charge

properties of soft and ion-penetrable surfaces of polyelectrolyte gels using the rotating disk

electrokinetic approach has not been investigated in the literature either theoretically or

experimentally.

Therefore, a more accurate electrokinetic model taking into account the effect of surface

conductivity K needs to be developed to better characterize the charged “soft” polyelectrolyte

gels.

17

Figure 1.10 Surface potentials on “hard” and “soft” surfaces in contact with an electrolyte

solution. (a) Ion-impermeable surface - counter ions only move in the diffuse layer external to

the hard surface, forming an electric double layer. A surface potential between the surface and

the bulk solution is established; (b) ion–permeable surface - charges are distributed throughout

the gel phase and counter ions are distributed both inside the gel phase and in the diffuse layer

external to the gel surface. An equilibrium electrical potential called Donnan potential (ϕD)

forms across the phase boundary.

1.5 Pulsatile drug delivery system and the application of periodic precipitation Certain drugs can develop tolerance from constant drug administration (Wolff and Bonn, 1989;

Dighe et al., 2009), leading to sub-therapeutic effect. In order to avoid such tolerance effect,

these drugs can be loaded in a pulsatile release device, which delivers the drug intermittently in

order to avoid tolerance and to enhance the therapeutic effect. A biological example of this

effect is shown by gonadotropin-releasing hormone (GnRH), which is released in pulses

endogenously for the control of reproductive function (Woller et al., 2004). Similarly, a number

of other hormones, such as insulin, also exhibit circadian rhythm in plasma (Haus et al., 2001).

Therefore, there is a therapeutic rationale to produce appropriate repetitive pulses of drug release

in order to mimic physiological patterns of hormone release so as to enhance the effect of drug

therapy.

18

The multi-pulse drug delivery systems are named by their ability to produce more than one pulse

of drug release with definite time interval between the pulses as required by some special

physiological needs. Currently available multi-pulse drug delivery devices include laminated

delivery systems (Lee, 1986; Xu and Lee, 1993; Hassan et al., 2000), pellets-matrix osmotic

pressure systems (Ghosh and Ghosh, 2011), biodegradable microchip devices investigated

recently by Langer and colleagues (Grayson et al., 2003), and antigen responsive drug delivery

polymer proposed by Miyata and coworkers (Miyata et al., 1999). However, most pulsatile drug

delivery systems are only at the stage of academic interest due to the large device size, difficulty

in large scale manufacturing, and biocompatibility as well as toxicity issues with the new

material. For most stimuli-induced pulsatile drug delivery systems, the response time is still too

long and the release rate is too slow to be of practical value.

The Liesegang ring phenomena have been demonstrated in many pharmaceutically acceptable

polymers such as gelatin, PVA and silica gel. Among them gelatin gel is more suitable as a drug

loaded carrier for oral delivery as the gel erosion can be regulated by its reaction with pepsin

(enzymatic cleavage), which is usually present in the stomach. Based on the surface erosion

mechanism, drug loaded into a multi-layer structure in the gelatin gel can be released

periodically following oral administration and the drug release rate and release profile can be

manipulated by varying the spatial distribution of the drug loading in the gel matrix. In addition,

it is known that under proper conditions, periodic precipitation can be generated automatically

and applications of such periodic precipitation have been demonstrated on a microscopic scale

(micron to sub-micron) as shown in Figure 1.11 (Mueller, 1984; Grzybowski, 2009). Therefore,

by loading the drug into a multi-layer structure through periodic precipitation such that the

delivery pulses can be generated from the resulting multi layer micro-structure, it potentially can

serve as a pulsatile drug delivery device to overcome the limitations of existing delivery systems.

19

Figure 1.11 Periodic precipitation of AgI in PVA film. Scale bar 0.05mm (Reprinted with

permission from Mueller, 1984).

Recent studies show that several vanadium compounds exhibit insulin-mimetic effect (Sakurai et

al., 1999) due to its apparent improvement of metabolic disorders in type I and II diabetes,

resulting from its ability to normalize hyperglycemia and enhance insulin sensitivity (Marzban et

al., 2003). Vanadyl (VO2+) is also reported to be effective via oral administration in experimental

animals (Detata, et al., 1993; Cam et al., 1993). Therefore, vanadyl compounds may be

potentially applied in the oral therapy for diabetes (Marzban, 2003). In addition, it has been

recently reported that vanadyl sulfate is absorbed more thoroughly in the ileum (Fugono, 2001)

and the oral bioavailability of enteric-coated vanadyl sulfate (9.8%) is approximately doubled

compared to that taken directly (4.8%) (Fugono, 2002). Therefore, controlling the release pattern

of vanadyl sulfate in the GI tract would be potentially beneficial in enhancing its bioavailability

and reducing the insulin uptake by the diabetic patients, and the periodic precipitation of vanadyl

salt will be explored as a potential pulsatile drug delivery system.

In our study, vanadyl sulfate will be used as a model drug to form periodic precipitation in

gelatin gel and its release rate will be controlled by the spatial distribution of the precipitated

compound. Under appropriate conditions, such periodic distribution of vanadyl sulfate can

potentially offer elaborate rate control by providing a pulsatile release pattern of vanadyl sulfate

from a bioerodible gelatin gel.

20

1.6 Hypothesis Understanding the gel phase properties and reactant concentration effect on the periodic

precipitation process will allow a better prediction and control of the periodic precipitation. The

application of such programmed periodic precipitation will allow the design of a pulsatile

delivery system for insulin mimetics such as vanadyl salts.

1.7 Research objectives a) To refine existing theories on periodic precipitation by employing more precise boundary

conditions and to develop a novel and practical method to manipulate the band position and band

spacing beyond what is achievable at present. The eventual goal is to program the formation of

periodic precipitation.

b) To analyze the effect of gel phase properties on the periodic precipitation process and to

develop an improved rotating disk method for characterizing the charge density of

polyelectrolyte gels. A more accurate electrokinetic model will also be established taking into

account the presence of a soft gel matrix.

c) To show for the first time that periodic precipitation can be accomplished in the vanadyl

sulfate system based on the mechanistic study of periodic precipitation in multi-component

systems. Also, to explore the periodic precipitation of vanadyl salt as a potential pulsatile drug

delivery system.

21

Chapter 2 Programmed periodic precipitation*

2.1 Introduction Periodic precipitation is the result of inter-diffusion and reaction of two or more co-precipitating

reactants in a diffusion medium such as a gel. Under suitable conditions, a series of distinctly

spaced concentric rings or parallel bands of precipitate will form in the diffusion matrix as

depicted in Figure 2.1. The periodic bands so formed have been reported to obey the spacing

law (Grzybowski, 2009), which states that the ratio of consecutive band positions approaches a

constant nn xx 1 at large n, where Xn is the position of the nth band and the spacing

coefficient with >1 (Figure 2.1). In this case, the spacing between bands (ΔX) always increases

with distance from the interface between outer and inner electrolytes. The band position Xn and

spacing between bands ΔX of periodic precipitation can be varied only to a limited extent by

altering the process parameters such as the initial concentration of the reactants (Zrhyi et al.,

1991; Attieh et al., 1998), solubility of the product (Msharrafieh et al., 2007), and gel thickness

perpendicular to the diffusion direction (Fialkowski et al., 2005). In general, the spacing (ΔX)

between consecutive precipitation bands increases with distance X as a result of the inherently

diffusion controlled transport process which tends to slow down with distance.

Periodic precipitation experiments are generally conducted with an initial concentration of the

outer electrolyte in the solution phase higher than that of the inner electrolyte in the gel phase.

Most previous studies applied a constant concentration (equivalent to a fixed concentration or an

infinite reservoir) boundary condition for the outer electrolyte in their modeling analysis while

conducting the experiments with a finite volume of outer electrolyte (Zrhyi et al. 1991;

Carotenuto et al., 2002; Fiałkowski et al. 2005; Lagzi, et al. 2007; Izsk and Lagzi, 2005). This

constant concentration boundary condition employed mathematically is in conflict with the

experimental fact that the outer electrolyte concentration at the solution/gel interface decreases

with time in a finite volume. Such inconsistency can generate considerable discrepancy between

the predicted and experimental band position and band spacing in the periodic precipitation.

* The work presented in this chapter was all performed by Beibei Qu under supervision of Dr. Ping I. Lee.

22

Therefore, one major objective of the present study is to examine and delineate the effect of

concentration boundary conditions (for both infinite and finite volume of the outer electrolyte)

on the resulting characteristics of the periodic precipitation.

Xn-1 Xn+1Xngel surface

Figure 2.1 Schematic illustration of the periodic precipitation system. The n-th band formed at

position Xn (n is the band number, 1, 2, 3, …). In general, ΔX increases with n.

Due to the complexity of the periodic precipitation phenomena, many mechanisms and models

have been proposed (Stern, 1954, 1967; Henisch, 1988; Grzybowski, 2009), which generally fall

into two categories, the prenucleation mechanism and the postnucleation mechanism. The

prenucleation mechanism was first introduced by Ostwald (Ostwald. 1897) and later modeled by

Prager, Smith and Le Van et al. (Prager, 1956; Smith, 1984; Le Van and Rose, 1987). It assumes

that nucleation occurs only when the reaction products exceed a certain supersaturation threshold

level. However, this model often neglects the nucleation kinetics and parameters affecting

particle growth and evolution. An extended prenucleation model incorporating the nucleation

and growth kinetics was later developed by Dee (Dee, 1986). In general, prenucleation models

can generate periodic precipitation bands satisfying the scaling law but are not able to describe

other experimental observations such as the formation of the secondary precipitation bands

between two major bands. Aiming to solve this problem, the postnucleation model was

established taking into account the competitive particle growth and ripening effects (Feeney et

al., 1983). It considers the fact that the existence of nucleated particles with different sizes

triggers the growth of larger particles by depleting the surrounding smaller particles (Flicker and

Ross, 1974; Feinn et al., 1978; Lovett et al., 1978; Venzl and Ross, 1982; Feeney et al., 1983).

Recently, a generalized model was developed by Chacron and L’Heureux (Chacron and

23

L’Heureux, 1999; L’Heureux, 2008) which combines Dee’s extended prenucleation model (Dee,

1986) with the postnucleation competitive particle growth model pertaining to ripening (Feeney

et al., 1983), taking into consideration the role of supersaturation in both nucleation and particle

growth during the reaction process. This generalized model delineates the interplay between

diffusion, nucleation, growth and ripening processes thereby bridging the gap between the

prenucleation and postnucleation models, as discussed in Chapter 1. In fact, it reduces to Dee’s

prenucleation model when the nucleation and growth processes are dominant and becomes the

postnucleation competitive growth model when the nucleation phase terminates and the ripening

process dominates. This is by far the most comprehensive mechanistic model comparing with

other existing models on periodic precipitation. Therefore, this generalized model is employed

to analyze the periodic precipitation results in our study. Furthermore, using this model, we also

examine and delineate the effect of concentration boundary conditions (e.g. infinite or finite

volume of outer electrolyte) on the characteristics of the resulting periodic precipitation patterns,

particularly in achieving a programmed periodic precipitation from an accelerated concentration

boundary condition.

One of the most interesting periodic precipitation patterns is the equidistant periodic precipitation.

Equidistant banding pattern of a reaction product is of significant technological importance in

several fields ranging from micro- and nanofabrication to drug delivery (Xu and Lee, 1993;

Grzybowski, 2009). In the drug delivery field, there is a growing interest in creating laminated

delivery systems with equidistant alternating drug-containing and drug-free layers in order to

achieve pulsatile drug delivery upon dissolution of the delivery systems (Lee, 1986; Xu and Lee,

1993; Hassan et al., 2000). However, under classical concentration boundary conditions, e.g.

infinite or finite volume (or constant or time-varying concentration) of outer electrolyte,

equidistant periodic precipitation will not form. Thus, additional driving forces need to be

introduced in order to alter the periodic precipitation pattern. Recently, Bena and colleagues

(Bena et al., 2008) reported the formation of equidistant periodic precipitation by imposing a

time-dependent electric current in the reaction-diffusion system, where the concentration of

reactants was varied by adjusting the electric field strength. This approach is feasible since the

diffusion flux of reactants in the reaction zone is controlling the precipitate band position (Xn).

However, the experimental setup of this method is quite cumbersome and impractical for routine

use. Furthermore, without considering the gel deformation in an electric field (Yamaue et al.,

24

2005), the current theory on the control of periodic precipitation through electric current is

inadequate for polyelectrolyte gel systems. More recently, a continuous large-scale modification

of the substrate inhomogeneity and/or the nucleation threshold has been proposed to obtain

equidistantly spaced bands in a reaction-diffusion system (Jahnke and Kantelhardt, 2010).

However, this concept has only been shown in the extended Monte-Carlo lattice-gas simulations

but has yet to be demonstrated experimentally. Judging from the experimental complexities

involved in modifying the inhomogeneity of the substrate and in changing the nucleation

threshold, the applicability of this proposed approach would be limited. For practical purposes,

there is a major need for a more flexible method to control the band formation in the periodic

precipitation process and to generate equidistant periodic precipitation patterns. Therefore, the

second objective of this study is to develop a novel and practical approach to manipulate the

band position and band spacing ΔX and to refine the existing theories to achieve a better

quantitative description of the phenomena.

Additionally, from a practical perspective, microstructures of technological importance can be

constructed by applying the periodic precipitation phenomena. For example, the fabrication of

microlenses has been demonstrated via the periodic precipitation of silver nitrate and potassium

hexacyanoferrate in a gel matrix (Grzybowski, 2009). Thus, in order to fabricate complex and

predesigned microstructures using the periodic precipitation process, it becomes increasingly

important to design and control the precipitate band location, band spacing and band width.

Therefore, another major objective of the present study is to investigate factors determining the

formation of periodic precipitation and the location of precipitate band position. The eventual

goal of this work is to provide a novel and practical way of manipulating precipitate band

position and band spacing ΔX through proper programming of reaction conditions such as the

concentration of reactants and gel phase properties. Accordingly, periodic precipitation patterns

with predetermined band position and band spacing can be designed and generated.

2.2 Model For the rational design of a periodic precipitation system, a sound theoretical basis for the

associated diffusion and reaction phenomena needs to be established in order to investigate the

periodic precipitation process. In this section, the generalized model of Chacron and L’Heureux

25

is adopted and solved under various new boundary conditions specific to our objectives but not

previously investigated.

Gelatin gel formed in a glass tube was used as the model diffusion matrix in our study. The

length of the diffusion matrix was 10 cm and the diameter of the diffusion matrix was 2.4 mm

which was relatively small compared with the diffusion length (2.4 mm/10 cm = 0.024).

Therefore, for modeling purposes, it is appropriate to consider it as a one-dimensional reaction-

diffusion system (Figure 2.2). The gel phase occupies the space 0 ≤ x ≤ L with a uniform initial

concentration of a divalent inner electrolyte B (CB0). On top of the gel surface (x < 0 ) , a certain

amount of a monovalent outer electrolyte A with an initial concentration CA0 (>>CB0) is loaded.

When diffusing into the gel phase, A reacts with B to form a colloidal product C at a

concentration CC, which subsequently precipitates with a precipitation rate denoted by U. The

governing equations of this diffusion and reaction process are given as Eqs. (2.1) - (2.3).

The reaction is : B2+ + 2A- → C

tCCkC

xCD A

BAA

A

2

2

2

Eq. (2.1)

tCCkC

xCD B

BAB

B

2

2

2

Eq. (2.2)

tCUCkC

xCD C

BAC

C

2

2

2

Eq. (2.3)

26

Gelatin gel

Inner electrolyte B

Outer electrolyte A

End of gel,

Gel surface,

Gelatin gel

Inner electrolyte B

Outer electrolyte A

End of gel,

Gel surface,

Gelatin gel

Inner electrolyte B

Outer electrolyte A

End of gel,

Gel surface,

x>0

x=0

x<0

x=L

Figure 2.2 Schematic illustration of a model periodic precipitation system. Gel surface is located

at x = 0 and the end of gel at x = L. A certain volume of outer electrolyte A remains on top of the

gel surface, which is filled by a HPLC gradient pump unit.

Taking into consideration of the classical nucleation mechanism and assuming the precipitate

particles are spherical and stationary, the molar concentration of precipitate f and the

precipitation rate U can be presented by Eqs. (2.4) and (2.5), respectively (Dee, 1986; Le Van

and Ross, 1987; Chacron and L’Heureux, 1999).

')',,()',,()34(

0

3 dtttxrttxJv

ft

m

Eq. (2.4)

'''

0

2'3 ),,(),,(),()4(),(),()34( dtttx

trttxrtxJtxrtxJ

tfU

t

mn

m

Eq. (2.5)

Where, J(x,t) is the nucleation rate at position x and time t, r(x,t,t’) is the particle radius at time t

which was nucleated at t’, mv is the molar volume of precipitate, and rn (x,t) is the critical radius

of particle (Dee, 1986; Le Van and Ross, 1987; Chacron and L’Heureux, 1999).

27

Again, based on classical nucleation theory, the nucleation rate J can be obtained from the

following expressions (Dee, 1986; Chacron and L’Heureux, 1999):

)(sFJJ C and )(swgrn Where 0

0

CCCs C

,

0)( else,or 0 s if ]))((exp[)1()( 22 sFsgssF and )1ln(

1)(s

sg

Here, the constant dCwDJ cc /4 20

2 , 5.02

)3

4(

Tkw

B

, capillary length

TkNvw

B

m

032

, σ the

surface tension of the precipitate particle, kB Boltzmann’s constant, T the temperature, C0 the

solubility of product C, Dc the diffusion coefficient of product, d the typical molecular size taken

as twice the diameter of the C molecule (Chacron and L’Heureux, 1999), and N0 the Avogadro’s

number.

If the particle growth process is interface controlled, the particle growth rate can be described by

Eq. (2.6) (Le Van and Ross, 1987; Chacron and L’Heureux, 1999):

])(

[0C

rCCG

tr eq

Eq.(2.6)

Where, G is the particle growth rate constant and Ceq(r) is the concentration of product C in

equilibrium with a particle of radius r. The relation between Ceq(r) and r can be described by the

Gibbs-Thomson relation (Chacron and L’Heureux, 1999):

])(exp[)( 0 rrwCrCeq

Eq.(2.7)

where φ(r) represents the radius dependence of surface tension expressed as:

qrrrrr

/3)( 22

2

with q = 0.304359 and δ representing the thickness of the Gibbs

surface. This expression for φ(r) is simplified from a thermodynamically based expression of

Koenig (Koenig, 1950; Chacron and L’Heureux, 1999). As a result, Eq.(2.7) is more accurate

than the intuitive radius dependence of equilibrium concentration employed in the previous

postnucleation competitive particle growth model (Feeney et al., 1983).

28

For modeling purposes, the average particle number density, n, is defined by Eq. (2.8):

')',,(0

dtttxJnt

Eq. (2.8)

The molar concentration of precipitate as described in Eq. (2.4) can also be simplified to Eq. (2.9)

if the nucleation phase is assumed terminated while only the reaction, particle growth and

ripening steps occur (Chacron and L’Heureux, 1999):

),(34 3 txrv

nfm

Eq. (2.9)

This generalized model of Chacron and L’Heureux bridges the gap between the prenucleation

and postnucleation models. Therefore, by applying the model of Chacron and L’Heureux

(Chacron and L’Heureux, 1999; L’Heureux, 2008), the nucleation, particle growth and ripening

process can all be taken into account as described earlier. The validity of this model is later

tested through our experiments.

Similar to published analyses (Le Van and Ross, 1987; Chacron and L’Heureux, 1999), the

concentration of precipitate, particle radius and particle number density at different time t and

position X were numerically simulated in our study by the finite element method using Comsol

Multiphysics 3.5a. This is an engineering software based on finite element analysis which has

wide applications in the modeling and simulation of various engineering and physics problems.

For simulation purposes, the following realistic physical parameters were employed: CA0 = 1 -

14.8 M (NH4OH), CB0= 0.05-0.2 M (MgCl2) , C0 = 1.65*10-4 M (solubility of Mg(OH)2 ,

estimated from Ksp value) (Shreif et al., 2002), L = 5 -10 cm, L’/L=1/5, t = 9 – 48 hr, M = 58

g/mol (molecular weight of Mg(OH)2), density ρ = 2.3446 g/ cm3 /Mvm = 24.74 cm3/mol,

surface tension σ = 120 mJ/m2 (Mullin, 1992) , capillary length cmTkN

vwB

m 8

0

10*1.83

2

(Chacron and L’Heureux, 1999) , d the typical molecular size taken as twice the diameter of the

Mg(OH)2 molecule (Chacron and L’Heureux, 1999) and d = 8.56*10-8 cm , k = 10-6 L2/mol2.s, G

= 3.24*10-10 cm/s , DA= 1*10-5 cm2/s , DB = DC = 0.5*10-5 cm2/s. For prediction purposes, the

simulations were conducted with dimensionless parameters and they are scaled as follows:

29

twGT ， x

wDGXA

， 0B

A

CCa ,

0B

B

CCb ,

0

0

CCCs C

,G

kwCK B2

0 ,

A

B

DDD 1 ,

A

C

DDD 2 , n

CvwN

m 0

34 ,

wrR , 3NRF .

By inputting our model equations with appropriate boundary conditions and system parameters,

the coupled partial differential equations were numerically solved. The simulation details and the

parameters employed in our study are summarized and presented in the Appendix.

2.3 Methods and materials Typically, a gelatin gel containing inner electrolyte MgCl2 was prepared by dissolving 1.5 g

gelatin powder (type A from porcine skin, 300 Bloom, Sigma, USA) and 1.218 g MgCl2 in 30 ml

Milli-Q water at 50 oC. The solution was then filled into a glass tube (inner diameter 2.4 mm),

with one end sealed by parafilm, and placed in a refrigerator (4 oC) overnight. The resulting

physically crosslinked gelatin gel in glass tubing was cut into segments of 12 cm (gel length L =

10 cm + reservoir length L’ = 2 cm ). The periodic precipitation experiments were performed by

filling an outer electrolyte NH4OH at a prescribed concentration CA0 into the reservoir above the

gel surface while holding the tubing vertically. The unique approach taken here was to vary, CA0

programmatically using a gradient HPLC solvent delivery unit (L-2130 HTA Pump, Hitachi

High Technologies America, Inc.). The gradient concentration profile of CA0 was achieved by

stepwise manipulation of the volume flow ratio of two feed streams, 14.8 M NH4OH and

deionized water, throughout the duration of the experiment. A typical exponentially increasing

CA0 over 24 hr via a stepwise change of solvent gradient is depicted in Figure 2.3 below.

30

0 400 800 1200 16000

4

8

12

16

Time t (min)

CA0 (M)

Figure 2.3 Exponential increase of CA0 in 24 hr by the gradient HPLC solvent delivery unit

As the outer electrolytes diffuses into the gel, the following reaction occurs and bands of white

precipitate of Mg(OH)2 are generated.

Mg2++ 2OH- Mg(OH)2 (S)

The experiments were run for 9 - 48 hr, and bands formed were photographed with a digital

camera (Nikon D3100 or Canon 350D).

2.4 Results and discussion

2.4.1 Experimental evidence of the periodic precipitation model

The above diffusion and reaction model equations Eqs. (2.1)-(2.9) were numerically solved for

the Mg(OH)2 periodic precipitation system. The scaled particle radius R, particle number density

N and density of precipitate F at two different times (T1 and T2) were computed, respectively and

shown in Figure 2.4 where a finite reservoir boundary condition, CA0 = 7.4 M and CB0 = 0.2 M

31

was employed in the simulation. In all cases, periodic precipitation patterns are clearly seen.

From the simulation results, the particle number density N is seen to decrease with increasing

distance in the gel column due to a concomitant decrease of the degree of supersaturation of

product C (Figure 2.4a). Since classical nucleation theory predicts a larger particle number

density N with a smaller average particle size R at a high degree of supersaturation, it is

reasonable that when N is large in the initial bands at short distance, particles with smaller radii

are favored (Figure 2.4b). During the precipitation process, the density of precipitate F, which is

related to the product of N with the particle volume (or R3), continues to increase with time after

the formation of bands and the number of precipitate bands also increases with time (Figure 2.4c).

Meanwhile, particles with smaller radii evolve to become larger particles in the ripening process

(Figure 2.4b) and the precipitate bands tend to become dense and clearly spaced (Figure 2.4c).

0 10 20 30 40

0.0005

0.0010

0.0015

T1=172.8 T2=345.6

N

X(a)

32

0 10 20 30 400

100

200

300

400 T1=172.8 T2=345.6

R

X(b)

0 10 20 30 400

2000

4000

6000 T1=172.8 T2=345.6

F

X(c)

Figure 2.4 Simulation results of periodic precipitation at two different dimensionless times. T1 =

172.8 (black line) and T2 = 345.6 (black line), respectively with a = 37 (CA0 = 7.4 M and CB0 =

0.2 M) at T = 0. (a) scaled average particle number density, N. (b) scaled particle radius, R. (c)

scaled density of precipitate, F. Finite reservoir boundary condition was employed in the

simulation.

To elucidate the experimental evidence of periodic precipitation, one set of initial condition was

chosen for the dynamic study: 0.2 M of MgCl2 as the inner electrolyte and 7.4 M of NH4OH as

the outer electrolyte. A finite reservoir boundary condition was employed here to show the

phenomena of periodic precipitation. A typical periodic precipitation pattern of Mg(OH)2 was

observed and a series of pictures recording the periodic precipitation at different times are shown

in Figure 2.5. It can be seen that at small times (or small distances from the gel surface) there is

a turbid zone, where a large density of nuclei were produced due to the high degree of

supersatuation. At a larger distance down the gel column, this turbid zone is extended into a

region of closely-spaced bands followed by an extended cloudy colloidal zone. This colloidal

zone gradually evolves into a series of closely spaced but denser and sharper precipitate bands as

time progresses. Therefore, our model prediction strongly agrees with our experimental

observations. Furthermore, Figure 2.5 provides clear evidence of growing colloidal particles in

33

the region of a band prior to the band formation, consistent with the findings of Muller and Ross

(Muller and Ross, 2003; Kai, Muller and Ross, 1982).

Figure 2.5 Time evolution of periodic precipitation of Mg(OH)2 in gelatin gel under finite

reservoir boundary condition. (a) - (o) represent snapshots of periodic precipitation taken at time

t = 0.25 hr , 0.5 hr, 0.75 hr, 1 hr, 1.25 hr, 1.5 hr, 1.75 hr, 2 hr, 2.25 hr, 2.5 hr, 2.75 hr, 3 hr, 3.25

hr, 3.5 hr, 4 hr, respectively.

2.4.2 Concentration effect of the outer electrolyte

It is known that the initial concentration of outer electrolyte A on top of the gel surface, CA0, can

affect the spacing between bands (ΔX) during periodic precipitation (Attieh et al., 1998).

However, a critical examination of such effect through a comparison of the experimental and

simulation results is still lacking. Therefore, a series of measurements on the periodic

precipitation of Mg(OH)2 with different outer electrolyte concentrations were conducted to

investigate the effect of CA0 on band spacing, ΔX. In all cases, CB0 was kept at 0.2 M while CA0

was varied between 1 M and 14.8 M. An infinite reservoir boundary condition was employed

here (i.e. constant CA0 above the gel surface). We first tested our model for the concentration

effect of the outer electrolyte. The computed density profiles of F for different CA0 at T = 172.8

are presented in Figure 2.6. At a higher CA0, both ΔX and the corresponding band position are

predicted to decrease while the number of precipitate bands increases. There are substantial

similarities between the simulation and the experimental results (compare Figures 2.6 and 2.7).

34

0 10 20 30 40 500

2000

4000

6000

8000

10000 (1) (2) (3)

F

X

Figure 2.6 Concentration effect of outer electrolyte. Scaled density of precipitate, F simulated at

T = 172.8 (12 hr) and CB0 = 0.2 M for different CA0 , (1) CA0= 1 M (2) CA0= 3 M (3) CA0= 7.4

M , respectively. Infinite reservoir boundary condition was employed here (CA0 was constant

above the gel surface). Here, BD

DD Cor B A, , x

wDGXB

, diffusion coefficient of outer

electrolyte was assumed to be constant.

35

Figure 2.7 Concentration effect of outer electrolyte. Experimental results of periodic

precipitation with different CA0 . Infinite reservoir boundary condition was employed here (CA0

was constant above the gel surface). (a) - (e) demonstrate the periodic precipitation observed at

36 hr with CB0 = 0.2 M and CA0= 1 M, 3 M,7.4 M, 11 M and 14.8 M, respectively

In our present studies, the concentration of the inner electrolyte was fixed at 0.2 M. Therefore, it

reasonable to assume that the gel properties remain unchanged including the diffusion coefficient

of the outer electrolyte. Simulation results of periodic precipitation involving constant diffusion

coefficient for the reactants and product are illustrated in Figure 2.6. Even if the diffusion

coefficient of the outer electrolyte is concentration dependent, we can show that the predicted

trend will still be the same. To illustrate this point, periodic precipitation was simulated with

concentration dependent diffusion coefficients at different concentrations of outer electrolyte (i.e.

a larger diffusion coefficient at higher initial concentration). As shown in the simulation results

of Figure 2.8, both ΔX and the corresponding band position decrease while the number of

precipitate bands increases at a higher CA0, identical to the trends predicted for a constant

diffusion coefficient (compare Figures 2.6 and 2.8). Therefore, unless otherwise noted, all

subsequent simulations in the remaining thesis will be conducted with a constant diffusion

36

coefficient of the outer electrolyte for the sake of simplicity. Overall, the band number and band

spacing ΔX can be varied by changing the fixed outer electrolyte concentration CA0,. However,

under a specific CA0, ΔX always increases with increasing band distance X so that equidistant

bands cannot be generated by this approach.

0 10 20 30 40 500

2000

4000

6000

8000

10000

(1) (2)

F

X

Figure 2.8 Concentration effect of outer electrolyte. Scaled density of precipitate, F simulated at

T = 172.8 (12 hr) and CB0 = 0.2 M for different CA0 , (1) CA0= 3 M , DA= 1*10-5 cm2/s (2) CA0=

7.4 M, DA= 1.5*10-5 cm2/s. Infinite reservoir boundary condition was employed here (CA0 was

constant above the gel surface). Here,BD

DD Cor B A, , x

wDGXB

, diffusion coefficient of

outer electrolyte was assumed to be larger at a higher concentration of outer electrolyte.

For the sake of generality, simulations conducted so far have all been based on dimensionless

variables and parameters. However, for practical applications, a dimensionless simulation needs

to be converted to one based on real physical scales either from knowing the appropriate

conversion parameters involved from separate measurements or by estimating the conversion

parameters by matching the simulations with experimental data. To demonstrate this, the

simulation curve 3 of Fig. 2.6 based on the dimensionless F and X has been converted to one

based on measurable density of precipitation f (mol/m3) and the true distance x (m) by estimating

37

the appropriate parameters G/w and DB as shown in Fig. 2.9. These dimensionless simulation

results would be equally useful even if the periodic precipitation occurs in the microscopic scale

(e.g. Figure 1.11). In this case, our dimensionless simulation results can still be employed to

predict the formation of periodic precipitations by combining appropriate conversion parameters.

10-2 m

(a) (b)

0.0 1.0x10-2 2.0x10-2 3.0x10-20.00

1.50x102

3.00x102

4.50x102

x (m)

f

(mol/m3)

Figure 2.9 Direct comparison between simulation results (a) and experimental results (b). Infinite

reservoir boundary condition was employed here. the periodic precipitation observed at 12 hr

with CB0 = 0.2 M, CA0= 7.4 M, G/w =0.004 s-1 and DB=1.92*10-5 cm2/s

2.4.3 The finite reservoir effect

As discussed above, the magnitude of initial concentration of outer electrolyte CA0 is crucial in

determining the band position of periodic precipitation. A precise definition of CA0 in the model

is therefore very important to the understanding and design of periodic precipitation patterns. In

all previous experiments reported in the literature, a finite reservoir of outer electrolyte was

employed. In this case, CA0 at the gel surface (X = 0) is not constant but decreases with time due

to the diffusion and reaction processes in the gel phase (see Figure 2.10). However, to the best of

our knowledge, all existing models employed the assumption of an infinite reservoir of outer

electrolyte or a constant concentration boundary condition (Eq. 2.11) to describe the periodic

precipitation, equivalent to assuming a constant CA0 at all times. The validity of assuming an

infinite reservoir boundary condition is questionable since CA0 will be changing with time during

38

the precipitation experiments supplied with a finite volume of outer electrolyte. Thus, the infinite

reservoir assumption will significantly overestimate the local concentration in a finite reservoir

(Figure 2.10). This can generate considerable discrepancy between the predicted and

experimental band position and band spacing in the periodic precipitation. We will show below

the difference in predicted results between these two concentration boundary conditions and

verify them experimentally.

Figure 2.10 Time evolution of concentration profiles of outer electrolyte above the gel surface (x

< 0) at a = 15 (CA0 = 3 M and CB0 = 0.2 M). In a finite reservoir of outer electrolyte, the reduced

concentration a on the gel surface (x = 0) is much lower than that of infinite reservoir condition.

To assess the sensitivity and the difference between the infinite and finite reservoir boundary

conditions on the formation of periodic precipitation, boundary conditions defined by Eqs. (2.10)

and (2.11), respectively, were compared in our model simulations.

Infinite reservoir boundary condition: 00 AxA CC at all t Eq. (2.10)

Finite reservoir boundary condition: 00 at t ,0' AxLA CC Eq. (2.11)

39

Meanwhile, at a given CA0 (3 M, 7.4 M or 14.8 M), a set of experiments with finite and infinite

reservoir boundary conditions were conducted for 36 hr. Under finite reservoir conditions, the

volume ratio of outer electrolyte reservoir to that of gel phase was kept at 1 to 5 in both the

experiments and simulations. The computed profiles of precipitate density F with a = 15 under

finite and infinite reservoir boundary conditions at T = 345.6 exhibit similar behavior to that of

the experimental observations (compare Figure 2.11 and 2.12). In all cases, when the finite

reservoir condition was employed, fewer bands were generated and the band spacing ΔX

increased.

Figure 2.11 Simulation results of periodic precipitation density F under finite and infinite

reservoir boundary conditions. a = 15 (CA0 = 3 M and CB0 = 0.2 M) and T = 345.6

F

40

Figure 2.12 Experimental results of periodic precipitation at 36 hr under finite (a1, b1, and c1)

and infinite (a2, b2, and c2) reservoir boundary conditions, respectively. (a), (b), and (c) indicate

the initial concentration of CA0 = 3 M, 7.4 M, and 14.8 M, respectively, and CB0 =0.2 M.

Similarly, finite reservoir periodic precipitation systems with different volumes of outer

electrolyte were also simulated. The results of Figure 2.13 and Figure 2.14 show that, at T =

172.8, the reduced concentration a at the gel surface (X/L1 = 0) is lower at smaller outer

electrolyte volume (V1) to inner electrolyte gel phase volume (V2) ratios because of the faster

depletion of reactant from the outer electrolyte and the bands become closer with increasing

volume of the outer electrolyte. When the volume of outer electrolyte is twice that of the gel

phase, the band spacing is still much larger than that in the infinite system. Thus, the assumption

of an infinite reservoir of outer electrolyte employed by previous models would be inaccurate in

describing experimental results obtained from a finite reservoir of outer electrolyte.

41

Figure 2.13 Simulated concentration profiles of outer electrolyte above the gel surface, X < 0, at

T = 172.8 (CA0= 14.8 M and CB0 = 0.2 M) and different outer electrolyte volume (V1) to inner

electrolyte gel phase volume (V2) ratios. The position in the outer electrolyte is normalized to its

overall length L1 and the concentration “a” is normalized to the initial inner electrolyte

concentration.

Figure 2.14 Simulation results of scaled density of precipitate, F, at T = 172.8 (CA0= 14.8 M and

CB0 = 0.2 M) and different outer electrolyte volume (V1) to inner electrolyte gel phase volume

(V2) ratios.

F F

42

Therefore, the band position can only be predicted accurately by applying the appropriate

concentration boundary conditions. Our modified boundary condition, Eq. (2.11) which

corresponds to a finite reservoir is suitable for most experimental setups, where CA0 is decreasing

over time. On the other hand, a constant CA0 at the gel interface can only be achieved by

continually replenishing the outer electrolyte (e.g. by pumping fresh outer electrolyte into the

reservoir).

2.4.4 Equidistant periodic precipitation

In all cases studied above, the effect of the inner electrolyte concentration is not significant due

to the constant initial CB0 and the fact that CA0 >> CB0 . Thus, the diffusion of outer electrolyte

becomes a predominant factor in determining the precipitate band position Xn and band spacing

ΔX. In general, at any given outer electrolyte concentration CA0, the corresponding outer

electrolyte concentration in the gel phase, CA, decreases gradually with increased diffusion

distance. Therefore, the band spacing ΔX will always increase at larger distance due to the lower

rate of change of CA with distance. As a result, equidistant periodic precipitations cannot be

achieved naturally due to this inherent diffusion limitation. One important goal of our study is to

develop an approach to generate equidistant periodic precipitation. As mentioned in the

introduction section, there are some existing methods of varying CA in the gel phase to adjust the

precipitate band position, such as by imposing an electric field in the system. However, these

approaches are not widely applicable and the corresponding models describing the processes are

complicated. In our present study, a programmed increase of CA0 in an infinite reservoir is

imposed on the gel interface by the use of a HPLC gradient solvent pump which counteracts the

natural decrease of CA in the gel phase due to diffusion. Equidistant precipitation bands can be

expected through this approach.

To confirm this idea, the profile of precipitate density F is computed under an infinite reservoir

boundary condition with CA0 = A0Exp(θ*t) at t ≤ m hr, where A0 ,θ and m are selected such that

CA0 is exponentially increasing with time t, and CB0 = 0.2 M. The simulation results shown in

Figure 2.15 illustrate that equidistant periodic precipitation can be generated under such

boundary conditions beyond an initially turbid region of continuous density. Due to the

experimental limitation that the most concentrated NH4OH available is 14.8 M, a modified

boundary condition providing an exponential increase of the outer electrolyte reservoir was

43

employed in our study: CA0 = A0Exp(θ*t) at t ≤ m hr and CA0 = 14.8 M at m ≤ t ≤ m+q hr, where

q ≤ 12hr,. This reduces to a dimensionless form )*(Exp0 Taa , where a and T are the

dimensionless forms of CA0 and t, a0 = A0/CB0 and Gw / (see definitions in Section 2.2).

Such outer electrolyte reservoir concentration profiles having different rates of approach to the

maximum concentration of 14.8 M followed by maintaining CA0 at the maximum for a fixed

period of time are presented in Figure 2.16.

0 10 20 30 40 500

30000

60000

90000 (1) (2)

F

X

Figure 2.15 Simulation results of periodic precipitation density F (CB0 = 0.2 M) under the

following concentration boundary conditions in the outer electrolyte reservoir:

1) a = 5*exp(0.0078*T) at T ≤ 345.6 (t ≤ 24 hr) and a = 74 at 345.6< T ≤ 518.4 (24 hr < t ≤ 36hr)

2) a = 5*exp(0.0078*T) at T ≤ 518.4 (t ≤ 36 hr)

F

44

0 200 400 600 8000

20

40

60

80

(1) (2) (3) (4) (5) (6)

T

a

Figure 2.16 Dimensionless outer electrolyte reservoir concentration profiles having different

rates of approach to the maximum concentration followed by maintaining the maximum

concentration for a fixed period of time (CB0 = 0.2 M) :

1) a = 74 ，2) a = 5*exp(0.0312*T)*(T<=86.4)+74*(T>86.4)，

3) a = 5*exp(0.0208*T)*(T<=129.6)+74*(T>129.6)，

4) a = 5*exp(0.0156*T)*(T<=172.8)+74*(T>172.8)，

5) a = 5*exp(0.0078*T)*(T<=345.6)+74*(T>345.6)，

6) a = 5*exp(0.0051*T)*(T<=532.8)+74*(T>532.8)

In order to test the accuracy of our simulation results, a set of experiments with

programmatically changing CA0 were conducted. The corresponding experimental results were

recorded by a camera and shown in Figure 2.17. It is clearly seen that nearly equidistant periodic

precipitation bands can be developed from an exponentially increasing outer electrolyte reservoir

concentration beyond a transition region consisting of an initially turbid zone and some closely

spaced bands in the gel phase. Furthermore, Figure 2.18 illustrates the band spacing ΔX

predicted from our simulations under different outer electrolyte reservoir concentration boundary

conditions. The simulation results confirm that an exponential increase in CA0 can generate

45

equidistant bands during periodic precipitation. Moreover, ΔX can be adjusted by modifying the

parameter θ which directly affects the rate of the exponential increase in the outer electrolyte

reservoir concentration.

Figure 2.17 Experimental evidence of equidistant periodic precipitation: (a) – (e) formed under

outer electrolyte reservoir concentration profiles (2) - (6) of Figure 2.16.

(a) (b) (c) (d) (e)

46

0 3 6 9 12 150

2

4

6

8 (1) (2) (3) (4) (5) (6) (7)

Band number

ΔX

Figure 2.18 Comparison of band spacing ΔX simulated under different outer electrolyte reservoir

concentration boundary conditions: (1) a =15 , (2) a = 37 , (3) a = 74, (4) a = 5*exp(0.0312*T),

(5) a = 5*exp(0.0208*T), (6) a = 5*exp(0.0156*T), (7) a = 5*exp(0.0078*T). (and CB0 = 0.2 M)

Here, a and T are dimensionless forms of CA0 and t; see definitions in Section 2.2

It is to be noted that the density of precipitate F is determined by a combination of nucleated

particle number N and particle radius R (more precisely R3, the particle volume; see Figure 2.4).

Thus, in domains with large particle size but fewer particles, or with small particle size but a

larger number of particles, secondary bands may exist as a result of low to intermediate levels of

precipitate density (Figure 2.19). In this case, a hazy colloidal region below each formed

precipitate band is clearly visible.

47

Figure 2.19 Typical secondary bands formed during the periodic precipitation process.

2.4.5 Concentration effect of the inner electrolyte The concentration effect of the outer electrolyte has been discussed in Sec. 2.4.2 above. Similarly,

the concentration effect of the inner electrolyte on the precipitate band position is also important

(Zrhyi et al., 1991; Msharrafieh and Sultan, 2005), which is examined here. In our studies

described so far, CB0 was maintained at 0.2 M because varying the concentration of inner

electrolyte could affect the resulting properties of the physically crosslinked gelatin gel (through

variations in polymer volume fraction) such as the diffusion coefficients in the periodic

precipitation system. Therefore, the patterns of periodic precipitation can be influenced by such

variations in the inner electrolyte concentration. To understand these effects more quantitatively,

a set of experiments with varying CB0 (0.05 M , 0.1 M, 0.2 M and 0.4 M, respectively) but at a

fixed CA0 (14.8 M, finite reservoir boundary condition) was conducted for 36 hr. As shown in

Figure 2.20, periodic precipitation with CB0 at either 0.05 M or 0.1 M shows progressively

narrowing bands (radially) near the centre of the gel. Such deviation from regular periodic

precipitation patterns can be attributed to the shrinkage of the cationic gelatin gel upon its

reaction with OH- of the outer electrolyte resulting in the development of a larger curvature at the

gel surface as well at the diffusion front (see more detailed discussion in Chapter 3). On the

other hand, at a higher inner electrolyte concentration (0.2 M or 0.4 M), the periodic

precipitation patterns appear to be normal without any narrowing of bands. In this case, the

penetrating outer electrolyte is largely consumed by the inner electrolyte without any significant

gel shrinkage or curvature effect. However, a higher concentration of the inner electrolyte in the

gel phase decreases the polymer volume fraction and increases the free volume in the gel. Since

the reaction of both inner and outer electrolytes takes place in the gel phase, the diffusion

48

coefficients of the reactants and product are expected to increase with such an increase in free

volume. As shown in Sec. 2.4.2, at a fixed inner electrolyte concentration, the predicted trends

for band spacing and band location at different outer electrolyte concentrations are similar

irrespective of whether the diffusion coefficient of the outer electrolyte is assumed constant or

not. Based on this consideration, periodic precipitation patterns were simulated with

concentration dependent diffusion coefficients of the inner electrolyte CB0 (i.e. diffusion

coefficient increases with initial concentration) while keeping the diffusion coefficient of the

outer electrolyte constant to simplify the analysis. The computed precipitate density profiles F at

different CB0 at T = 345.6 are presented in Figure 2.21 where the band spacing ΔX is predicted to

increase while the number of precipitate bands decrease at a higher inner electrolyte

concentration, CB0. There are substantial similarities between the simulated and the experimental

results (compare Figures 2.20 and 2.21) thus indirectly validating the concentration dependency

for the inner electrolyte. Thus, the model employed here is reasonable in describing the

observed periodic precipitation patterns and in predicting the trend with increasing reactant

concentrations. However, since the form of the concentration dependency of diffusion coefficient

for the inner electrolyte is unknown, all subsequent periodic precipitation experiments in this

thesis will be conducted with a fixed inner electrolyte concentration of CB0 = 0.2 M in order to

ensure that the constant diffusion coefficient assumption is valid.

49

Figure 2.20 Concentration effect of inner electrolyte. Experimental results of periodic

precipitation with different CB0 at a fixed CA0 = 14.8 M. Finite reservoir boundary condition was

employed here. (a) - (d) demonstrate the periodic precipitation observed at 36 hr with CB0= 0.05

M, 0.1 M, 0.2 M and 0.4 M, respectively.

(a) (b) (c) (d)

50

0 10 20 30 40

5000

10000

15000

20000

(1) (2)

X

F

Figure 2.21 Simulation results of scaled density of precipitate, F, at T = 345.6 (24 hr) for

different CB0. (1) CB0 = 0.4 M , DB = DC = 1.5*10-5 cm2/s. (2) CB0 = 0.2 M, DB = DC = 0.5*10-5

cm2/s. Finite reservoir boundary condition was employed here.

2.4.6 The effects of other factors The formation of periodic precipitation depends not only on the concentration and the amount of

reactants (volume) in the system, but also on its physiochemical properties, such as the

diffusivity of reactants and product (DA, DB and DC), the solubility of the reaction product, C0,

and the reaction rate constant, k. Thus, the effects of these parameters on the periodic

precipitation process were also investigated in our simulation. For example, under finite reservoir

boundary conditions, with an increase in the diffusion coefficient ratio (DA/DB) or equivalently

an increase in DA, the periodic precipitation bands are more separated (larger band spacing) but

with more formed bands (see Figure 2.22). During the formation of periodic precipitation bands

in the gel matrix, the diffusivity of the diffusing species may be reduced successively by each

formed band along the diffusion direction due to the obstruction effect. This is simulated in

Figure 2.23 by comparing the effect of a constant diffusion coefficient ratio DA/DB=D1=3

51

(equivalently, a constant DA) with the case of a decreasing diffusion coefficient along the

diffusion direction, e.g. DA/DB = D1 = 3 – 0.05X. It is clear that the observed band spacing in

Figure 2.23 decreases with spatially decreasing diffusion coefficient, consistent with the trend of

a reduced diffusion coefficient shown in Figure 2.22. Similarly, simulations with increased

product solubility C0 and reaction rate constant are shown in Figure 2.24 and Figure 2.25,

respectively. With higher product solubility C0 fewer bands are produced and band spacing ΔX

increased, while at a higher reaction rate constant, more bands are generated and band spacing

ΔX decreased. Therefore, by employing reactants with a lower solubility of product or a higher

reaction rate constant, more and closer precipitate bands should appear.

0 10 20 30 40 500

1000

2000

3000

4000 D1=1 D1'=3

X

F

Figure 2.22 The effect of diffusion coefficient ratio, D1 = DA/DB. Simulation results of periodic

precipitation density F with a = 37 (CA0 = 7.4 M and CB0 = 0.2 M), T = 345.6, D1 = 1 and D1’ =

3D1 = 3 where D1 = DA/DB, X = (G/DBw)0.5x under finite reservoir condition.

52

0 10 20 30 40 500

2000

4000

6000

D1=3 D1'=3- 0.05X

X

F

Figure 2.23 Simulation results of periodic precipitation density F with distance dependent

diffusion coefficient and a = 37 (CA0 = 7.4 M and CB0 = 0.2 M), T = 345.6, D1 = 3 and D1’ = 3 -

0.05X where D1 = DA/DB , X = (G/DBw)0.5x under finite reservoir boundary condition.

0 10 20 30 40 500

2000

4000

6000

8000 (1) (2)

F

X

Figure 2.24 The effect of product solubility, C0. Simulation results of periodic precipitation

density F with a = 15 (CA0 = 3 M and CB0 = 0.2 M), T = 345.6, under finite reservoir boundary

condition (1) C0 = 1.65*10-4 M (2) C0 = 1.65*10-5 M

53

0 10 20 30 40 500

2000

4000

6000

8000

(1) (2)

F

X

Figure 2.25 The effect of reaction rate constant, k. Simulation results of periodic precipitation

density F with a = 15 (CA0 = 3 M and CB0 = 0.2 M), T = 345.6 under finite reservoir boundary

condition. Here,G

kwCK B0 , (1) K = 10-5 (2) K = 10-4

The effects of these parameters on the formation of periodic precipitation, both simulated and

experimentally observed, are summarize in Table 2.1 and Table 2.2:

Table 2.1 Effects of constant parameters on the formation of periodic precipitation

Parameter Effects

CA When CA↑, Band spacing ↓ . Band number ↑ as shown in Figure 2.6 & 2.7

DA When DA ↓ , Band spacing ↓ as shown in Figure 2.22

C0 When C0↑, Band spacing↑ . Band number ↓as shown in Figure 2.24

K When K↑, Band spacing ↓ . Band number ↑ as shown in Figure 2.25

54

Table 2.2 Effects of variable parameters on the formation of periodic precipitation

Parameter Effects

CA Finite/Infinite

reservoir boundary

condition

Going from infinite to finite reservoir boundary condition, CA decrease with time.

Band spacing ↑. Band number ↓as shown in Figure 2.11 & 2.12

DA When DA decrease with distance x, Band spacing ↓ as shown in Figure 2.23

2.4.7 The width of precipitate band In addition to investigating the trend in band spacing and band position in periodic precipitation,

the band width is another important factor to be considered. Based on mass balance in the

reaction and diffusion domain, the following relationship can be obtained (George and Varghese,

2005):

Cnnbandngapn CWXCWCX )(

where, Cc is the uniform initial concentration of the reaction product C. After the crystal growth

and phase separation, colloidal C aggregates at the band position to form observed band. The

concentration of C at the band position is Cband and at the gap between two bands is Cgap, as

shown in the schematic illustration of Figure 2.26.

Figure 2.26 Schematic illustration of band width and band spacing for Eq. (2.12)

ΔXn Wn

Cband Cgap

Eq. (2.12)

55

For the ease of data analysis, Eq.(2.12) can be rearranged to :

nnCbandgapCn XCFXCCCCW )()]/()[(

where, F(C) is a function of the concentration of reaction product C.

According to this simple mass balance relationship, an increase in either the reaction product

concentration or the band spacing will increase the band width. As shown in the photographs of

Figures 2.7 and 2.12, in any given gel tube, the observed band width becomes larger with

increasing band spacing, consistent with the prediction of Eq. (2.13). This can also be illustrated

in our simulation in Figure 2.27, where we have assumed that the precipitate particles become

large enough to be observed by the naked eye above an arbitrary precipitate density (F = 500). In

this case, the band width can be measured at each band position as shown in Figure 2.27. It can

be seen that for well developed bands, the bandwidth increases with increasing band spacing.

However, for the last few bands, this trend is not observed because these bands are still growing

and the band width increasing with time consistent with the simulation of Figure 2.4c and the

experimental observations of Figure 2.5.

Eq. (2.13)

56

0 10 20 30 40 500

2000

4000

6000

2.47 3.07 3.36 3.80 3.62 2.37

X

F

Figure 2.27 Simulation results of periodic precipitation density F with a = 15 (CA0 = 3 M and CB0

= 0.2 M). Infinite reservoir boundary condition, at T = 345.6. The band width is measured and

labelled at each band position.

2.5 Conclusions We have shown that the concentration profile of outer electrolyte plays a key role in controlling

the precipitate band’s position Xn and band spacing ΔX in the periodic precipitation system. We

have delineated for the first time the effect of finite versus infinite reservoir concentration

boundary conditions on the resulting periodic precipitation both in simulations and in

experiments. Our results show that in all cases when the finite reservoir condition is employed,

fewer bands with increasing band spacing ΔX will be generated. We have also shown that

equidistant precipitate bands can be simulated from the generalized model of Chacron and

L’Heureux and generated equally spaced bands experimentally by imposing a programmed and

57

exponentially increasing outer electrolyte reservoir concentration (CA0). To the best of our

knowledge, this unique approach has never been reported before. The general sensitivity of

periodic precipitation to concentration boundary conditions and the effect of various

physiochemical properties have been examined in detail in order to identify parameters important

to the design and control of the precipitate band position, band spacing and band width for the

application of these intriguing phenomena. In this regard, the effects of diffusion coefficient of

soluble species, reaction product solubility and reaction rate constant on the periodic

precipitation have been investigated to provide additional flexibility in manipulating the bands

position and band spacing in periodic precipitation. Furthermore, our experimental observations

show similar behavior as that predicted from our modeling simulations thus providing a sound

basis for the further application of periodic precipitation in pattern design and microfabrication

based on our current approaches.

58

Chapter 3 Effects of gel phase properties on periodic precipitation*

3.1 Introduction Various patterns of precipitation resulting from inter-diffusion and reaction in a gel medium such

as continuous, tree-like and periodic precipitation have been reported (Henisch, 1988; Toramaru

et al., 2003; Lagzi and Ueyama, 2009; Barge et al., 2010). However, other than varying the

reactant concentrations and reaction conditions, the importance of physicochemical properties of

the gel phase and their effects on the resulting periodic precipitation have not been sufficiently

emphasized or investigated. During periodic precipitation, the three dimensional structure of the

gel plays an important role in maintaining a stable concentration gradient of the reactants and

inhibiting the occurrence of convection. It not only supports the formed crystallites, but also

suppresses the nucleation and crystal growth process by limiting the gel mesh size. When the gel

mesh size is very large (or the gel concentration is very low), instead of generating periodic

precipitation patterns, the crystallites formed in the gel may evolve into other patterns, such as

tree-like or platelet-like precipitation (Toramaru et al., 2003). In addition, ionic groups in the gel

phase can interact with dissociated ions of the precipitate (e.g. through ion exchange), and affect

the diffusion of ions through electrostatic interactions in the gel phase (Kim and Lee, 1992;

Narita et al., 1998; Hyk and Ciszkowska, 1999; Baek and Srinivasa, 2004; Darwish et al., 2004;

Ogawa and Kokufuta, 2004; Yamaue et al., 2005; Masiak et al., 2007). Furthermore, the ionic

gel may shrink or swell during the diffusion of ionic species due to the electrostatic effect (Baek

and Srinivasa, 2004). When the gel shrinks in a glass tube, a large curvature on the gel surface

may develop which can propagate with the diffusion front and generate an uneven radial

distribution of reactants in the system thereby affecting the radial pattern of periodic

precipitation. Therefore, it is critical to understand the effect of gel phase properties in order to

design and control the desired patterns of periodic precipitation.

Previous studies have only investigated the effect of gel concentration or gel type on the periodic

precipitation patterns (Henisch, 1988; Toramaru et al., 2003; Lagzi and Ueyama, 2009).

However, studies on the gel physicochemical properties on the periodic precipitation are still

* The work presented in this chapter was all performed by Beibei Qu under supervision of Dr. Ping I. Lee.

59

lacking. To further our understanding of this phenomenon, we investigate in the present study the

effect of gel mesh size and gel charge property on the formation of periodic precipitation. To the

best of our knowledge, the influences of these parameters on periodic precipitation have not been

investigated previously. The eventual goal of our study is to generate the needed pattern of

periodic precipitation by controlling the parameters of the diffusion matrix, and to provide a

guideline for selecting appropriate gel properties, such as the gel mesh size or the density of

fixed charge groups. It is anticipated that, based on the results obtained here and those from

Chapter 2, periodic precipitation with predetermined band position, band spacing, and pattern

can be better programmed.

3.2 Materials and methods

3.2.1 Gel preparation

Gelatin powder (type A from porcine skin, 300 Bloom) and NaCl were obtained from Sigma-

Aldrich (USA). MgCl2 ， NH4OH ， HCl， polyacrylic acid (PAA; MW～ 1,250,000) and

glutaraldehyde were purchased from VWR (USA). Polyvinyl alcohol (PVA; DuPont™ Elvanol®

71-30) were kindly provided by DuPont (USA).

Gelatin gel samples were prepared by the physical crosslinking method as described in Chapter 2.

The PAA containing gelatin (PAA-Gelatin) gel samples were prepared similarly by dissolving

1.5 g gelatin powder, 0.006 g PAA and a predetermined amount of MgCl2 in 30 ml Milli-Q water

at 50 oC. The solution was loaded into glass tubing and stored at room temperature overnight to

form the PAA-Gelatin gel.

PVA gel samples were prepared by the chemical crosslinking method. The PVA powder was

completely dissolved in water at 90 oC. A predetermined amount of inner electrolyte and

glutaraldehyde (1 mL of 25% glutaraldehyde) were added into 100 mL PVA solution. A small

amount of HCl (300 ul of 1 M HCl) as also added as a catalyst to facilitate the formation of

chemically crosslinked PVA gel. Subsequently, the resulting solution was loaded into glass

tubing (inner diameter 2.4 mm) and stored at room temperature overnight to form PVA gel.

60

3.2.2 Preparation of periodic precipitation

The gel in glass tubing was cut into segments of 12 cm ( gel length L = 10 cm + reservoir length

L’ = 2 cm ). The periodic precipitation experiments were performed by placing the tubing

vertically and filling an outer electrolyte NH4OH at a concentration CA0 into the space above the

gel surface (finite reservoir system, see Chapter 2). The concentration of the inner electrolyte

MgCl2 was set in the range of 0.05 M and 0.2 M. The experiments were run for 24-36 hr and

photographed with a digital camera (Nikon, D3100 or Canon, 350D).

3.2.3 Gel mesh size analysis

A gel sample (about 2-3 g/sample) was immersed in Milli-Q deionized water at room

temperature for 48 h to achieve swelling equilibrium with a swollen weight of WS. Subsequently,

it was completely dried in a vacuum oven at ~ 40 oC until reaching a constant dry weight of Wd.

According to Yasuda’s model, there is a direct relation between gel swelling ratio Q and solute

diffusivity in the gel based on the free volume theory (Amidon et al., 2005; Yasuda and

Lamaze,1971). In this case, the free volume within the gel (or alternatively, the gel mesh size)

can be evaluated by measuring the gel swelling ratio Q, where d

s

WWQ . In our present study,

the gel swelling ratio Q is employed to obtain a qualitative comparison of gel mesh size between

different samples.

3.2.4 Quantification of gel charge property

The density of fixed charge groups was characterized by the improved rotating disc method, see

description in Chapter 4.


3.3.1 Effects of the gel mesh size Various gels have been applied in generating periodic precipitation in previous studies (Henisch,

1988; Lagzi and Ueyama, 2009). Conceptually, the gel mesh size represents the maximum size

of a drug molecule that can pass through the gel network and accounts for the screening effect of

the gel network on drug diffusion. Therefore, it is an important parameter which can affect the

reactant diffusion, product nucleation, and crystal growth during the periodic precipitation. In

61

our study, both physically crosslinked gelatin gel and chemically crosslinked PVA gel were used,

which have significantly different gel swelling ratios, thereby allowing us to assess the effect of

the corresponding gel mesh size on the periodic precipitation process.

Figure 3.1 summarizes the values of gel swelling ratio Q of the physically crosslinked gelatin

and chemically crosslinked PVA gels at different salt concentrations. It is seen that the gel

swelling ratio Q of gelatin gels is consistently larger than that of the chemically crosslinked PVA

gels irrespective of the salt concentration. The results suggest that in present study, the gel mesh

size of the physically crosslinked gelatin gels is generally larger than that of the chemically

crosslinked PVA gels.

Figure 3.1 Comparison of gel swelling ratios of PVA and gelatin gels as a function of salt

concentration at room temperature.

Periodic precipitation experiments of Mg(OH)2 in gelatin and PVA gels under the same

experimental conditions were conducted. The results shown in the photographs of Figure 3.2

demonstrate that there is a larger continuous precipitation region extending from the surface of

the PVA gel without clear band separation and it is followed by much wider precipitate bands

0.1 0.2 0.3 0.4 0.50

10

20

30

40

Swel

ling

ratio

(Ws/

Wd)

Salt Con. (M)

gelatin gel PVA gel

62

than those in the gelatin gel. According to the Ostwald ripening theory, a larger crystal grows by

consuming smaller crystals around it due to the smaller surface energy of the larger crystal. At

the location of the precipitate band, the band may become sharper and denser over time due to

the crystal growth and ripening (Figure 2.4). However, if the gel has a smaller mesh size such as

in the present PVA gel, the particle growth may be suppressed due to the spatial constraint in the

gel network (Figure 3.3).

Figure 3.2 Periodic precipitation of Mg(OH)2 in (a) gelatin gel and (b) PVA gel at 24 hr. Finite

reservoir boundary condition was applied with L’/L = 1/5, outer electrolyte CA0 was 14.8 M

NH4OH and inner electrolyte CB0 was 0.2 M MgCl2

63

(a)

(b)

Figure 3.3 Schematic diagrams showing the effect of gel mesh size on periodic precipitation. a)

precipitation in gel with smaller mesh size, b) precipitation in gel with large mesh size. When the

gel has a smaller mesh size, the particle growth is suppressed due to spatial constraint in the gel

network. The arrows indicate the force imposed by the gel network.

Based on this consideration, periodic precipitation patterns were simulated with different particle

growth rate G while keeping the diffusion coefficient of reactant constant to simplify the analysis.

The computed particle radius r and precipitate density profiles f at 24 hr as a function of distance

from the gel surface are presented in real units in Figure 3.4 and 3.5, respectively. In the first

situation, the gel mesh size is sufficiently large that it does not suppress the particle growth. In

this case G is set to be a constant. Whereas in the second situation, the gel mesh size is

sufficiently small that the gel network will suppress the particle growth. In this case, G is set at a

lower value when the particle radius becomes larger than the gel mesh size. The simulation

results of Figure 3.4 indicate that by decreasing the particle growth rate G the average particle

64

radius decreases. In addition, in the simulated periodic precipitation of Figure 3.5, if we

assumed that the precipitate density becomes large enough to be observed by the naked eye

above an arbitrary precipitate density (f = 100 mol/m3), a larger continuous precipitation region

appears at short distance in the gel having a smaller mesh size and a larger band width results

from a correspondingly decreased particle growth rate. These are consistent with the

observations of Figure 3.2.

0.0 1.0x10-2 2.0x10-2 3.0x10-20

1x10-7

2x10-7

3x10-7 (1) (2)

x (m)

r (m)

Figure 3.4 Simulation results of particle radius r (m) at t = 24 hr with CA0 = 14.8 M and CB0 = 0.2

M. (1) G = 3.24*10-12 m/s (2) G = 6.48*10-13 m/s for particle size larger than 10-8 m, otherwise,

G = 3.24*10-12 m/s. Finite reservoir boundary condition was employed in the simulation.

65

0.0 1.0x10-2 2.0x10-2 3.0x10-20.0

2.0x102

4.0x102

6.0x102

(1) (2)

x (m)

f (mol/m3)

Figure 3.5 Simulation results of precipitation density f (mol/m3) at t = 24 hr with CA0 = 14.8 M

and CB0 = 0.2 M. (1) G = 3.24*10-12 m/s (2) G = 6.48*10-13 m/s for particle size larger than 10-8

m, otherwise, G = 3.24*10-12 m/s. Finite reservoir boundary condition was employed in the

simulation. The dotted line indicates f = 100 mol/m3 above which precipitate density become

large enough to be observed by the naked eye. The intercept of this dotted line with each periodic

peak defines the observed band width.

In summary, the gel mesh size as characterized by the gel swelling ratio is an important factor in

governing the formation of periodic precipitation as it modulates the band location, band spacing,

and band width by affecting the particle growth rate during the precipitate band formation.

3.3.2 Effects of the gel charge property In general, polyelectrolyte gels consist of physically or chemically crosslinked polymer

backbone, with fixed charge groups. The charge groups are dissociable under certain conditions.

For example, under basic conditions, the carboxyl groups dissociate, while under acidic

66

conditions, the amino groups are dissociable. Therefore, most of the polyelectrolyte gels exhibit

cationic, anionic or amphoteric characteristics. As reported, charge groups in polyelectrolyte

gels play a critical role in the gel swelling behavior and they also affect the diffusion rate of the

mobile species in the gel phase (Kim and Lee, 1992; Narita et al., 1998; Hyk and Ciszkowska,

1999; Baek and Srinivasa, 2004; Darwish et al., 2004; Ogawa and Kokufuta, 2004; Yamaue et

al., 2005; Masiak et al., 2007;). The underlying causes of these observations have often been

attributed to the electrostatic repulsion effect of the charge groups (Baek and Srinivasa, 2004).

Therefore, at high pH values, an anionic gel swells and the diffusion rate of the mobile species in

the gel increases, whereas a cationic gel shrinks resulting in lower permeability, and vice versa.

As discussed in Chapter 2, the effects of concentration as well as diffusion rate of the reactants

are critical to the formation of periodic precipitation. Thus, the gel charge property is another

important factor affecting periodic precipitation and it will be investigated herein.

Gelatin is an amphoteric material having both ionizable carboxyl groups and amino groups. The

pI of the purified pig gelatin solution used in our experiments was measured to be 9.0 (Figure

3.6), which is in agreement with published data (Ward and Courts, 1977). The pKa values of the

gelatin gel used were analyzed by the standard titration method (Figure 3.7). The results indicate

that the physically crosslinked gelatin gel exhibits two pKa values at 3.25 and 10.25, and the gel

remains positively charged in the electrolyte solution of the present study (0.04 mM NaCl; pH

5.2). The results from our rotating disk experiments also confirmed this conclusion, which will

be discussed in Chapter 4. The density of fixed charge groups in the physically crosslinked

gelatin gel used in our rotating disk experiments was calculated to be 0.630 mM (discussed in

Chapter 4), supporting the idea that physically crosslinked gelatin gel behaves like a cationic gel

in our test electrolyte solution (0.04 mM NaCl; pH 5.2). This concentration of the test electrolyte

solution was selected to provide a balance between the sensitivity and signal to noise ratio in the

rotating disk electrokinetic measurement (see details in Chapter 4).

67

5 6 7 8 9 100.0

0.2

0.4

0.6

300nm 350nm 400nm 450nm 500nm

pH

ABS

Figure 3.6 Determination of pI value of gelatin gel from the pH dependence of UV absorbance

profiles (pI identified at maximum light scattering ~pH 9.0).

-1.0 -0.5 0.0 0.5 1.0 1.50

2

4

6

8

10

12

14

H+ added (mmol)OH- added (mmol)

pKa1=10.25

pKa2=3.25

pH

Figure 3.7 Determination of pKa values of gelatin gel by pH titration.

68

Polyacrylic acid (PAA) is an anionic polymer and its carboxylic groups are ionizable (pKa ~ 4.3)

(Leaist, 1989). When PAA is blended with the gelatin gel, the density of fixed charge groups in

gelatin gel will be reduced due to the addition of negatively charged carboxyl groups. The pKa

values were analyzed by the standard titration method (Figure 3.8) and the density of fixed

charge groups in the PAA-gelatin gel by the rotating disk method (discussed in Chapter 4). The

two pKa values of the PAA-gelatin gel are determined to be 3.5 and 10.5, very close to that of

the gelatin gel (compare with Figure 3.7). Thus, this PAA-gelatin gel still retains the

characteristics of a cationic gel in the test electrolyte (0.04 mM NaCl; pH 5.2) based on the

positive streaming potential observed in the rotating disk experiments to be discussed in Chapter

4. Furthermore, based on this data, the density of fixed charge groups in the PAA- gelatin gel is

determined to be 0.0682 mM, which is lower than that of the gelatin gel (0.630 mM) obtained

under the same experimental conditions. This trend is reasonable because part of the ionized

positively charged amino groups would be neutralized by the additional carboxyl groups from

PAA, thereby reducing the overall density of fixed charge groups.

-0.004 0.000 0.0042

4

6

8

10

12

H+ added (mol)OH- added (mol)

pKa1=10.5

pKa2=3.5

pH

Figure 3.8 Determination of pKa values of PAA-gelatin gel by pH titration

69

To investigate the effect of gel charge property, Mg(OH)2 periodic precipitation experiments

were conducted in gelatin and PAA-gelatin gels. The results in PAA-gelatin gel are presented in

Figure 3.9 (b), (d) and (f), in comparison with that in gelatin gel without PAA (Figure 3.9 (a), (c)

and (e)) at 3 different inner electrolyte concentrations (0.05, 0.1 and 0.2 M). It is clear that the

periodic precipitation bands formed in PAA-gelatin gels are more clear and well developed

whereas that of the gelatin gels exhibit a larger continuous precipitation region originating from

the gel surface without clear band separation followed by progressively narrowing bands

(radially) near the centre of the gel at large distance at low inner electrolyte MgCl2

concentrations (0.05 and 0.1M).

(a) (b) (c) (d) (e) (f)

Figure 3.9 Periodic precipitation in gelatin and PAA-gelatin gels. Finite reservoir boundary

condition was applied here CA0 = 14.8 M : (a) gelatin gel, 0.05 M MgCl2 (b) PAA- gelatin gel,

0.05 M MgCl2 (c) gelatin gel, 0.1 M MgCl2 (d) PAA- gelatin gel, 0.1 M MgCl2 (e) gelatin gel,

0.2 M MgCl2 (f) PAA- gelatin gel, 0.2 M MgCl2

Since gelatin gel is cationic with more positive charges than the PAA-gelatin gel in our test

electrolyte solution, large gel shrinkage can occur when fixed positive charges are neutralized

70

and shielded by the negative hydroxyl ions diffusing into the gelatin matrix from the outer

electrolyte compartment. As a result of this shrinkage, a larger curvature develops at the

diffusion front, as shown in the photographs of Figure 3.9 (a), (c) and (e), and depicted in the

schematic drawing of Figure 3.10. As a result of this curvature, the diffusion front in the centre

where more reaction product is formed will move ahead of that near the edge. Therefore, at any

radial direction, there will be more product (or denser band) formed at the center than that near

the wall. This situation can become exacerbated by the low initial concentration of inner

electrolyte where the product concentration near the wall will eventually become lower than the

critical supersaturation to form precipitate band. Thus, progressively narrowing bands (radially)

are observed near the centre of the gel as shown in Figure 3.9. In contrast, in the PAA-gelatin

gel, the net positive charge of the gel decreases with the addition of anionic PAA thereby

resulting in less gel shrinkage or less change of curvature at the gel surface upon diffusion of

OH- from the inner electrolyte into the gel phase. This will result in less curvature at the moving

diffusion front, and the reaction product will be more evenly distributed at any radial direction

than that in the gelatin gel. As a result, distinct and more evenly distributed precipitate bands are

formed in the radial direction in the PAA-gelatin gel. Therefore, the gel charge property is also

an important factor in affecting the periodic precipitation as it can influence the gel swelling (or

shrinking) and the diffusion of mobile ions in the gel phase.

(a) (b)

Figure 3.10 Schematic diagram showing the effect of gel surface curvature on the progression of

diffusion front in: (a) gel surface with large shrinkage; and (b) gel surface with small shrinkage.

71

3.4 Conclusions In this chapter, by characterizing the gel swelling ratio (or gel mesh size) and the gel charge

property, the effects of gel phase properties on the periodic precipitation have been investigated.

Our simulation and experimental results indicate that by decreasing the gel mesh size and the

particle growth rate as in the PVA gel, the width of the periodic precipitation bands increases. In

addition, as a result of a larger extent of gel shrinkage at the gelatin gel surface upon reacting

with the diffusing outer electrolyte, a larger curvature develops at the gel surface and at the

diffusion front. As a result, there will be more product (or denser band) formed at the center than

that near the wall at any radial direction. In the case of low initial concentration of inner

electrolyte, the product concentration near the wall will eventually become lower than the critical

supersaturation needed to form precipitate band. Indeed, such progressively narrowing bands

(radially) are observed near the centre of the gelatin gel at low inner electrolyte concentration.

Therefore, by selecting appropriate gel properties such as the gel swelling ratio or mesh size and

the gel charge property, the formation of desired periodic precipitation patterns can be better

controlled.

72

Chapter 4 A rotating disk electrokinetic method for characterizing

polyelectrolyte gels*

4.1 Introduction Polyelectrolyte gels have been widely employed in periodic precipitation systems. Charge

groups in the bulk of such polyelectrolyte gels can interact with entrapped ionic diffusion species

(Baek and Srinivasa, 2004; Darwish et al., 2004; Hyk and Ciszkowska, 1999; Kim and Lee, 1992;

Masiak et al., 2007; Narita et al., 1998; Ogawa and Kokufuta, 2004; Yamaue et al., 2005)

thereby affecting the periodic precipitation. An accurate knowledge of gel charge properties is

therefore important to the understanding and design of periodic precipitation in polyelectrolyte

gels systems. The existing rotating disk method for quantifying the surface potential of flat

surfaces is based on the classical electrokinetic model which neglects the effect of surface

conductivity and therefore is only applicable to ion-impenetrable charged surfaces or “hard”

surfaces (Hoggard et al., 2005; Sides and Hoggard, 2004; Sides et al., 2006). This classical

electrokinetic model would be inaccurate for polyelectrolyte gel systems involving ion-

penetrable charged layers or “soft” surfaces. In this study, we present a new rotating disk model

for characterizing charge properties of ion penetrable soft surfaces using gelatin as a model

polyelectrolyte. A new and more accurate electrokinetic model taking into account the effect of

surface conductivity K is developed to better characterize charged “soft” gel surfaces. This

method is much simpler to use and more applicable to polyelectrolyte gels of pharmaceutical

interest than the microslit electrokinetic method.

In addition to electrokinetic parameters already considered in the existing rotating disk model,

the contribution of surface conductivity known to be very significant for soft and ion-penetrable

gel surfaces has been taken into account. Based on this new approach, two rotating gel disks of

different radius but with identical gel composition and preparation procedures were employed for

determining the surface potential and charge density. Gelatin was selected as a model

polyelectrolyte gel because it has been widely applied in various periodic precipitation systems

* The work presented in this chapter was all performed by Beibei Qu under the supervision of Dr. Ping I. Lee.

73

(Henisch, 1988; Msharrafieh and Sultan, 2005; Lagzi and Ueyama, 2009). PAA containing PVA

gel and PAA containing gelatin gel were also selected to test the proposed new rotating disk

model because of their wide application as pharmaceutical gels and their ease of achieving

different charge densities through polymer blending or crosslinking. Our results confirm that the

contribution from surface conductivity is significant in polyelectrolyte gels. For example, the

surface potential and charge density of 10% physically crosslinked gelatin has been determined

to be 47.79 mV and 0.630 mM, respectively based on the present model for ion penetrable soft

surfaces. In contrast, the existing rotating disk model ignoring the surface conductivity

contribution would have resulted in 6.79 mV and 0.0438 mM, respectively for the large disk, a

significant underestimation.


4.2.1 Materials Polyvinyl alcohol (PVA; DuPont™ Elvanol® 71-30) were kindly provided by DuPont (USA).

Gelatin powder (type A from porcine skin, 300 bloom) and NaCl were obtained from Sigma-

Aldrich (USA). Polyacrylic acid (PAA; MW～1250000) and glutaraldehyde were purchased

from VWR (USA). Gel sample dishes constructed from PEEK (polyether ether ketone) were

custom made and supplied by Pine Research Instrument Company (USA). Silver wires (diameter

2.0 mm, 99.9% pure) were purchased from Alfa Aesar (USA).

4.2.2 Preparation of gel samples PVA/PAA gel samples were prepared by the chemically crosslinking method. PVA powder was

completely dissolved in water at 90 oC to form a 10% w/v PVA solution. A predetermined

amount (0.1 g or 0.05 g) of PAA and glutaraldehyde (1 mL of 25% glutaraldehyde) were added

to 100 mL of PVA solution for the preparation of PVA/PAA (100/1) and PVA/PAA (200/1) gels.

A small amount of HCl (300 ul of 1 M HCl) was also added as a catalyst to facilitate the

chemically crosslinking of PVA. Subsequently, the resulting solution mixture was loaded into

sample dishes of internal radius 1.1 cm and 2.0 cm, respectively and heated in an oven at 60 oC

for 45 min to achieve crosslinking.

Gelatin containing gel samples were prepared by the physically crosslinking method. A suitable

amount of gelatin powder was first dissolved in water at 50 oC to form a 10% w/v solution. A

74

small amount (0.02 g) of PAA was dissolved into 100 mL of gelatin solution at 50 oC for the

preparation of gelatin/PAA (500/1) gels. These gel forming solutions were immediately

dispensed into sample dishes of internal radius 1.1 cm and 2.0 cm, respectively. Subsequent

cooling to room temperature for 24 hours produced physically crosslinked gelatin containing gel

disks.

Prior to a rotating disk experiment, the gel disk samples were equilibrated in 0.04 mM NaCl

solution (pH 5.2). The pKa values of gel samples were determined by the standard pH titration

of a suspension of finely divided gel particles.

4.2.3 Rotating disk experiment

The gel disk sample was attached to the spindle on a modulated speed rotator (Pine Research

Instrumentation) with the gel surface facing downward. The sample surface was fully immersed

in a 0.04 mM NaCl solution in a 2 L beaker and rotated at a fixed speed (up to 2600 rpm). The

concentration of NaCl (0.04 mM) solution with a corresponding pH of 5.2 (most likely from

dissolved carbon dioxide) was chosen to achieve a good sensitivity in this electrokinetic

measurement while at the same time maintaining a stable streaming potential reading. In this

case, at ion concentrations much larger than 0.04 mM (or higher ionic strength), the electrical

double layer on the charged gel surface becomes too thin to produce a measurable streaming

potential, whereas at ion concentrations much lower than 0.04 mM,(or lower ionic strength) the

electrical double layer on the charged gel surface becomes too thick to produce stable streaming

potential readings. Two Ag/AgCl electrodes connected to a Keithley Electrometer (Model 614)

were placed with the positive electrode at the center of the dish at a distance of 0.5 mm from the

sample surface and the negative electrode far away from the sample (Figure 4.1). The Ag/AgCl

electrodes were prepared by an established method (Hoggard et al., 2005). To ensure the

measurements were conducted under laminar flow conditions (Reynolds number, a2Ω/ν <

2x105 )(Newman and Thomas-Alyea, 2004), the gel disks were kept at a rotating speed range of

1600-2600 rpm for the small sample disk and 600-1200 rpm for the large sample disk,

respectively. Here, a is the radius of gel disk, ν is the electrokinetic viscosity of the NaCl solution

and Ω is the rotation rate. The surface potential ψ0 and charge density of the polyelectrolyte gel

were calculated from the measured streaming potential ),0( z (or ψstr) based on our rotating disk

75

model. The measured values of 5.1/str are expressed as their mean and standard error (x ±

S.E.) based on 35-50 sets of measurements.

Ag/AgCl electrodesAg/AgCl electrodes

Gel

-

+

Ag/AgCl electrodesAg/AgCl electrodes

Gel

-

+

Figure 4.1 Rotating disc experimental setup. The gel was loaded into the dish attached to a

rotating spindle and immersed in 0.04 mM NaCl solution. The streaming potential in the system

was measured by two Ag/AgCl electrodes, which were connected to the electrometer.

4.3 Theory

4.3.1 Improved rotating disk model

In a rotating disk system, the complete description of the surface current, js, consisting of

components originated from the outward radial convective flow of ions in the diffuse layer as

well as that originated from surface conductivity K due to the movement of adsorbed surface

charges can be expressed by Eq. (4.1) (Newman and Thomas-Alyea, 2004) :

dzvEKj errs

0

Eq. (4.1)

76

where e is the charge density at any space location in solution, vr the local radial velocity, r the

radial position, z the axial position, and rE the radial component of the electric field close to the

disk surface which can be evaluated from:

0

z

rr dr

dE Eq. (4.2)

where z = 0 is at the disk surface and r the electric potential in the radial direction. This radial

electric field is produced by the radial movement of adsorbed surface charges. Since the gel

charge density is relatively low, the generated radial electrical field and the associated

polarization of electric double layer would be quite small. Therefore, the effect of double layer

polarization is not considered in the present model.

When the rotating disk method is applied to a charged but ion-impenetrable hard surface, the

contribution from surface conductivity K is generally negligible. This has been shown to be a

valid assumption for disks with radius greater than 1 cm (Sides et al. 2006). Consequently, by

neglecting the first term of Eq. (4.1) as generally done in the so-called “hard” surface model

(Hoggard et al., 2005; Sides and Hoggard, 2004; Sides et al., 2006; Lameiras and Nunes, 2008),

the surface current can be simplified to:

rrdzvj ers 005.0

5.1

000

51023.0

Eq (4.3)

where 5.0

5.151023.0

, Ω is the rotating speed in radians per second, ψ0 the surface (or Zeta)

potential, the kinematic viscosity of the solution, 0 the permittivity of free space, and the

dielectric constant of the solution. In contrast, for ion-penetrable “soft” surfaces, the charged

groups are confined in the gel phase whereas the counter ions are distributed both in the gel

phase and in the external diffuse layer. In this case, the first term of Eq. (4.1) containing the

surface conductivity K generally cannot be ignored (Dukhin et al., 2004, 2006; Duval and van

Leeuwen, 2004; Duval, 2005; Ohshima, 1995; Zimmermann and Osaki, 2006). Therefore, the

77

general expression of Eq. (4.1) including the surface conductivity term will be used as the basis

for subsequent derivations.

Given the configuration of the present experimental setup as shown in Figure 4.2b where the

rotating gel disk is insulated on all sides except the gel surface and a thin exposed edge

(negligible thickness) contacting the bulk electrolyte solution, the fluid velocity inside a porous

substrate such as the ion-penetrable gel phase will generally be much smaller than that of the

local radial convective velocity at the disk surface in the solution phase (Nam and Bonnecaze,

2007; Sides et al., 2006). It is therefore reasonable to assume that the surface electric current in

our rotating gel disk system follows a similar flow pattern as that of the “hard” surface rotating

disk system previously studied by Sides et al. (Sides et al., 2006 ) (comparing Figure 4.2a and

Figure 4.2b). In both cases, the surface current emerging at the periphery of the disk will return

as bulk current through the electrolyte to the surface of the disk to complete the circuit. Except

that the surface current on the charged gel surface will have more significant contribution from

the surface conductivity.

Figure 4.2 Current flow on “hard” and “soft” surfaces in the rotating disc system. (a) Ion-

impermeable surface - surface current flows radially along the hard surface and returns back to

the surface from the bulk electrolyte solution; (b) ion–permeable surface - the “apparent” current

flow on the “soft” surface follows the similar pattern of that on the “hard” surface, as gel disc is

insulated on all sides except for the bottom surface. The fluid velocity inside of the gel phase is

much smaller than that of the local radial convective velocity on the gel surface.

78

As shown in Figure. 4.2a and 4.2b, the radial surface current has to be compensated by a bulk

current normal to the surface of the rotating disk at each r as required by the conservation of

charge. It has been shown that this normal current density zi returning to the diffuse layer of the

“hard” surface on a rotating disk is uniform and independent of r (Lameiras and Nunes, 2008;

Sides et al., 2006). We will assume here that in the case of rotating disk with a “soft” surface, zi

is also constant and independent of r; this assumption will later be shown to be valid. The electric

potential required to drive these currents through the electrolyte is expressed

as: ),(),(),( zrzrzr dr , where ),( zrr is the potential responsible for the radial current

and ),( zrd the potential that arises from the uniform flow of current to the disk surface. Based

on known solutions for disk electrode systems, the overall potential distribution in the present

case can be described by Eq. (4.4) (Nanis and Kesselman, 1971; Sides et al., 2006):

5.022001

])1()[()()()(),(

rzmKdperpJ

ppJ

aiKzr zp

z

L

Eq. (4.4),

where the coordinates are normalized by the disk radius a such that arr , a

zz and

22 )1()(4

rzrm , and zi is the returning current normal to the disk surface, KL the solution

conductivity, J0(p) and J1(p) Bessel Functions of order 0 and 1, respectively, and K(m) the

complete elliptic integrals of the first kind .

When the streaming potential ),0( z (or ψstr ) is measured at the center of the rotating gel disk,

the following approximate solution to Eq. (4.4) can be obtained (Lameiras and Nunes, 2008;

Sides et al., 2006):

2

22

)(1

)(12)(2121),0(

z

zzzai

Kz

z

L

Eq. (4.5)

Correspondingly, the potential near the surface of the rotating gel disk at any radial position can

be derived as (Nanis and Kesselman, 1971; Sides et al., 2006):

79

)1()(1)(2)0,(

rmKrE

aiKr

z

L

Eq. (4.6)

where )(rE is the complete elliptic integral of the second kind. The right hand side of Eq. (4.6)

evaluated from numerical results of the elliptic integrals (Abramowitz and Stegun, 1970) can be

approximated by a second degree polynomial of (see Figure 4.3) to obtain a simplified

expression for )0,(r :

]477.1)(774.1[)0,( 2 rKair

L

z

Eq. (4.7)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Original Simulated

y = y = --1.774x1.774x22+1.477+1.477RR2 2 = 0.994= 0.994

r/a

ϕ(r/a

,0)/(

i za/π

KL)

Figure 4.3 Relationship between ϕ(r/a,0)/(iza/πKL) and (r/a) in a rotating disc system. The

numerical results of Eq. (4.6) were fitted to a second degree polynomial of r/a to obtain Eq. (4.7),

which was used to calculate Er and js based on Eq. (4.1) and (4.2).

Combining Eq. (4.7) with Eqs (4.1) - (4.3), we obtain:

80

aKrirE

L

z

548.3

)( Eq. (4.8)

and

rrKa

KijL

zs 00

548.3 Eq. (4.9)

It should be stressed here that Eq. (4.9) is a general expression including the surface conductivity

term applicable to ion-penetrable soft surfaces.

Considering a total radial surface current defined as sr rjj 2 , the increment of such total

radial surface current in a ring of thickness Δr can be obtained as (Lameiras and Nunes, 2008):

rrrraK

Kij

L

zr 004

192.14

Eq. (4.10)

As mentioned above, this radial surface current increment has to be compensated by a bulk

current normal to the surface of the rotating disk as required by the conservation of charge.

Therefore, the current density zi returning to the diffuse layer of the rotating disk can be shown

to be:

002096.72

L

zrz aK

Kirr

ji Eq. (4.11)

Similar expression without the surface conductivity term has been derived for ion-impenetrable

hard surface (Lameiras and Nunes, 2008; Sides et al., 2006). Since every parameter on the right

hand side of Eq. (4.11) is a constant independent of r for a given system, the surface current

density zi is therefore also constant and independent of r thus validating our previous

assumption.

Rearrangement of Eq. (4.11) results in a working equation for determining the current density zi

returning to the surface of the rotating disk:

81

]548.3

5.0[

00

aKKi

L

z

Eq. (4.12)

Substituting Eq. (4.12) into Eq. (4.5), an expression for the streaming potential ),0( z (or ψstr)

on the soft surface is given by:

2

22

00 )(1

)(12)(21)

096.71(),0(

z

zzza

aKK

KzL

L

Eq.(4.13)

For the ease of data analysis, Eq.(4.13) can be rearranged to:

2

22

5.0

005.1

)(1

)(12)(21

)096.7

1(

51023.0),0(

z

zzz

aKKK

az

LL

Eq.(4.13a)

As a comparison, the corresponding expression for the streaming potential of a hard surface

(Sides et al., 2006) and its rearranged form are given below:

2

22

00 )(1

)(12)(21),0(

z

zzza

Kz L

Eq.(4.14)

2

22

5.000

5.1)(1

)(12)(2151023.0),0(

z

zzzK

az

L

Eq.(4.14a)

It is clear that Eq.(4.13) differs from Eq.(4.14) by an extra term containing the ratio of surface

conductivity K to solution conductivity KL. When the contribution of surface conductivity is

large for ion-penetrable soft surfaces, e.g. when 1096.7

LaKK

, the effect of surface conductivity

K will not be negligible. This is supported by published reports suggesting that the magnitude

of measured K for ion-penetrable soft surfaces is generally very large and not negligible in

comparison with that of ion-impenetrable hard surfaces (Dukhin et al., 2004, 2006; Duval and

82

van Leeuwen, 2004; Duval, 2005; Ohshima, 1995; Zimmermann and Osaki, 2006). It is known

that the gel physicochemical properties such as cross-linking density, porosity, or permeability

can alter the ion flow profile near the gel surface and affect the surface current (Yaroshchuk and

Luxbacher, 2010). However, for all practical purposes, this effect has been lumped into the

surface conductivity term, K , in the present analysis. It should be noted that Eq.(4.13) for soft

surfaces reduces to Eq.(4.14) for hard surfaces when surface conductivity K is negligibly small

(≈ 0). As will be shown later, if one ignores the contribution of surface conductivity in soft ion-

penetrable surfaces and only uses Eq.(4.14) to analyze results, the surface potential ψ0 and the

density of fixed charge groups of polyelectrolyte gels will be significantly underestimated.

Therefore, this extra term in Eq.(4.13) is critical for quantifying ψ0 of ion-penetrable soft

surfaces using the present rotating disk system.

For any given rotating disk gel sample of radius a , there are two unknowns in Eq. (4.13), ψ0 and

K . In practice, one may employ two rotating gel disks of different radius but identical gel

composition and preparation procedures (therefore identical ψ0 and K ). Plotting the measured

streaming potential ),0( z (or ψstr) of these two gel disks as a function of Ω1.5 under identical

experimental conditions and substituting the resulting slope of regression of such plot into Eq.

(4.13a) yields two algebraic equations containing ψ0 and K as unknowns. Subsequent solution

of these two simultaneous equations results in the determination of both ψ0 and K for the

specific charged gel.

4.3.2 The density of gel charged groups

Based on the work of Ohshima and Kondo (Ohshima and Kondo, 1990a, 1990b), a mathematical

equation relating the surface potential ψ0 to the Donnan potential ψD for soft ion-penetrable

material can be written as:

)2

tanh(0 Tkje

jeTk

B

DBD

Eq.(4.15)

83

where Bk is the Boltzmann constant taken as 1.38 x 10-23 J/K; j the valence of the symmetrical

electrolyte, i.e. 1 for NaCl; e the elementary electric charge taken as 1.602 x 10-19 C; and T the

absolute temperature.

Similarly, ψD can be directly related to the gel charge density and the concentration of the bulk

electrolyte solution according to the following Equation (Ohshima and Kondo, 1990a, 1990b):

5.02

0

0

0

0

0

0 1)2

(2

ln)2

(arcsinjC

ijC

ijeTk

jCih

jeTk BB

D

where ρ0 is the density of charged groups due to fixed charges in the gel, i the valence of the

charged groups which is 1 in this case and C0 the electrolyte concentration. Therefore, by

measuring the streaming potential ),0( z (or ψstr) of a gel disk sample using our rotating disk

system, the Donnan potential ψD of the gel phase and ρ0 can readily be determined from Eqs.

(4.15) and (4.16).

4.4 Results and discussion From Eqs (4.13) and (4.14), the measured streaming potential is expected to be proportional to

Ω1.5. The mean slopes of linear regression of such data measured in a 0.04 mM NaCl solution

from multiple runs are determined for the small and large gel disks, respectively. Since two

rotating gel disks of different radius but identical gel composition and preparation procedures are

expected to have identical ψ0 and K , substitution of the determined mean slopes of regression

for the two gel disks into Eq. (4.13) results in two algebraic equations containing ψ0 and K as

unknowns. Other known parameters in Eq. (4.13) such as ε, , and KL are taken to be 79.63,

9.57x10-7 m2/s and 0.000475 S/m at room temperature based on literature data (Conway, 1952;

Schmidt, 1927; Weast et al., 1988).

4.4.1 Evaluation of ψ0 and ρ0 of PVA/PAA gel

Polyvinyl alcohol contains hydroxyl functional groups, which can form acetal linkages with

aldehyde groups in the presence of HCl. Pure PVA gel does not bear dissociable groups,

however by adding polyacrylic acid (PAA) into PVA, the resulting gel matrix exhibits negative

Eq. (4.16)

84

charges in 0.04 mM NaCl solution due to the introduction of carboxyl groups. The pKa value of

the PAA containing PVA gel used in the present study was analyzed by the standard titration

method. The result indicates that the pKa value of this chemically crosslinked PAA/PVA gel is

approximately 5.6 and the gel is negatively charged in the electrolyte used in the present study

(0.04 mM NaCl; pH 5.2).

For the PVA/PAA (100/1) gels, the mean slopes of measured streaming potential versus Ω1.5 data

in a 0.04 mM NaCl solution from multiple runs (n = 35) are determined to be -3.85x10-4 mV.s1.5

and -8.64x10-4 mV.s1.5 for the small and large gel disks, respectively (see insets of Figs. 4.4a &

4.4b). Solution of these simultaneous algebraic equations results in the determination of ψ0 and

K for the present PVA/PAA(100/1) gel to be -75.71 mV and 1.22x10-6 S, respectively.

To appreciate the importance of surface conductivity K , corresponding values of LaKK

096.7 in

Eq. (4.13) are calculated to be 0.53 and 0.29 for the small and large disk, respectively. In this

case, the effect of surface conductivity K for the present soft gel surfaces should not be ignored

in Eq. (4.13) because the term LaKK

096.7 is of the same order of magnitude as 1. If the existing

“hard” surface rotating disk model without any contribution from surface conductivity (Eq.

(4.14)) is applied here, the calculated surface potential ψ0 would have been, for example, -49.57

mV and -58.68 mV for the small and large gel disk, respectively. It should be noted here that the

surface potential data evaluated from the hard disk model seem very different for the small and

large gel disks of the same gel composition under otherwise identical experimental conditions

thus highlighting its inconsistency with the physical situation. In addition, these surface potential

values are much smaller in magnitude than the -75.71 mV calculated from the present rotating

disk model for ion-penetrable soft surfaces (Eq. (4.13)). Similar discrepancies also exist in the

density of fixed charge groups determined from the surface potential data (see Table 4.1). These

observations further demonstrate that the contribution of surface conductivity is not negligible in

evaluating the eletrokinetic properties of these charged gel samples.

Substituting the surface potential ψ0 determined above into Eqs. (4.15) & (4.16), the density of

fixed charge groups is determined to be 2.06 mM for the PVA/PAA(100/1) gel. In contrast, if

the surface potential obtained from the hard surface rotating disk model (Eq (4.14)) is employed,

85

the calculated density of fixed charged groups will be, for example, 0.683 mM for the small disk

much smaller than the value determined by the present new rotating disk model for soft charged

gel surfaces. This observation is consistent with the data shown by Dukhin et al. (Dukhin et al.,

2005) which indicate that only a very small fraction of the surface charge in ion-penetrable soft

materials is detectable by the classical electrokinetic method neglecting the surface conductivity.

Since there are no reported data available for comparison, the consistency of our model

prediction is checked as follows. Based on the gel titration curve, the pKa of the gel is in the

range of 5.6. As a result, in our 0.04 mM NaCl solution (pH 5.2), the total density of dissociated

gel charge groups should be about 3.96 mM, which is reasonably close to the result of 2.06 mM

determined from our new rotating disk model. The observed deviation may be attributed to the

leaching loss of PAA from this PAA/PVA gel during sample preparation (equilibrated in water

for 4 days). Thus, the new rotating disk method presented here should provide a more realistic

and reasonably accurate charge density data for polyelectrolyte gels.

Similarly, the surface potential and gel charge density of PVA/PAA(200/1) gel were also

determined the same way. The mean slopes of linear regression of streaming potential versus

Ω1.5 data measured in a 0.04 mM NaCl solution from multiple runs (n = 35) are determined to be

-3.40x10-4 mV.s1.5 and -7.14x10-4 mV.s1.5 for the small and large gel disks, respectively (see

insets of Figs.4.5a & 4.5b). The corresponding gel surface potential and density of fixed charge

groups are determined to be -55.87 mV and 0.90 mM, respectively. Thus, the calculated ρ0 of this

PVA/PAA(200/1) gel is approximately 50% of that of the PVA/PAA(100/1) gel. This is very

reasonable as only 50% of carboxyl groups is present in the PVA/PAA(200/1) gel.

86

ψst

r(m

V)

Ω1.52500 3000 3500 4000

-2.6

-2.4

-2.2

-2.0

ψst

r(m

V)

Ω1.5600 800 1000 1200 1400

-2.2

-2.0

-1.8

-1.6

-1.4

0 10 20 30 40-6

-4

-2

0

0 10 20 30 40-12

-10

-8

-6

-4

Group number

ψstr/Ω1.5 = (-8.64±0.0508)X10-4 mV.s1.5 ψstr/Ω1.5 = (-3.85±0.0235)X10-4 mV.s1.5

Group number

ψst

r/Ω1.

5 ×10

4

ψst

r/Ω1.

5 ×10

4

(a) (a) (b)(b) Figure 4.4 Typical plots of streaming potential as a function of disk rotation speed Ω1.5 for

PVA/PAA(100/1) gel disks in 0.04 mM NaCl: (a) small disk (a = 1.1cm); and (b) large disk (a =

2 cm). The slopes of linear regression from multiple runs (n = 35) are shown in the insets and

noted in each figure as mean with standard error (x ± S.E.).

ψst

r(m

V)

Ω1.52500 3000 3500 4000

-2.2

-2.0

-1.8

-1.6

ψst

r(m

V)

Ω1.5600 800 1000 1200 1400

-2.0

-1.8

-1.6

-1.4

0 10 20 30 40-6

-4

-2

0

0 10 20 30 40-12

-10

-8

-6

-4

Group number

ψstr/Ω1.5 = (-7.14±0.0396) X10-4 mV.s1.5 ψstr/Ω1.5 = (-3.40±0.0310) X10-4 mV.s1.5

Group number

ψst

r/Ω1.

5 ×10

4

ψst

r/Ω1.

5 ×10

4

(a) (b)(a) (b)


PVA/PAA(200/1) gel disks in 0.04 mM NaCl: (a) small disk (a = 1.1 cm); and (b) large disk (a

= 2 cm). The slopes of linear regression from multiple runs (n = 35) are shown in the insets and

noted in each figure as mean with standard error (x ± S.E.).

87

4.4.2 Evaluation of ψ0 and ρ0 of gelatin gel

Gelatin is an amphoteric material having both ionisable carboxyl groups and amino groups

which can be crosslinked physically through hydrogen bonding. The physically crosslinked

gelatin gel of the present study exhibits two apparent pKa values (3.25 and 10.25) and the gel

remains positively charged in the electrolyte solution used in the present study (0.04 mM NaCl;

pH 5.2),

The mean slopes of streaming potential versus Ω1.5 data measured in a 0.04 mM NaCl solution

from multiple runs (n = 50) are determined to be 3.10x10-5mV.s1.5 and 1.00x10-4 mV.s1.5 for the

small and large gel disks, respectively (see insets of Figs. 4.6a & 4.6b). ψ0 and K for the

present 10% gelatin gel are calculated to be 47.79 mV and 2.54x10-5 S, respectively.

To demonstrate the importance of surface conductivity K , corresponding values of LaKK

096.7

in Eq. (4.13) are calculated to be 10.97 and 6.04 for the small and large gelatin disks,

respectively. It is clear that 1096.7

LaKK

for both gel disks and therefore the effect of surface

conductivity K for the present soft gelatin gel surfaces is quite significant and cannot be ignored

in Eq. (4.13). If the existing “hard” surface rotating disk model without any contribution from

surface conductivity (Eq. (4.14)) is applied here, the calculated surface potential ψ0 would have

been, for example, 3.99 mV and 6.79 mV for the small and large disk, respectively., much

smaller than the 47.79 mV calculated from the present rotating disk model for ion-penetrable soft

surfaces (Eq. (4.13)). Again, as discussed in the previous section, this discrepancy highlights the

importance that the contribution of surface conductivity is not negligible in evaluating the

eletrokinetic properties of these charged gel samples

It would be desirable at this point to compare the results of this study with available literature

data. However, electrokinetic data on pure gelatin gels are very scarce. In one previous study

based on the electrophoretic light scattering (ELS) method (Kushibiki et al., 2003), the surface

potential ψ0 of a 2% physically crosslinked gelatin gel (also type A from porcine skin) was

reported to be approximately 4.6 mV. From another previous work based on an electro-osmotic

method (Johnson and Thornton, 1969), ψ0 of 2% and 8% physically crosslinked gelatin gel

88

samples (also type A from porcine skin) obtained from the reported streaming potential versus

pressure data are 2.33 mV and 1.46 mV, respectively. It should be noted however that both these

previous studies evaluated ψ0 by the so-called Smoluchouski equation (Johnson and Thornton,

1969; Kushibiki et al., 2003; Tandon et al. 2008), which also neglects the effect of surface

conductivity and is in essence equivalent to the classical electrokinetic model for ion-

impenetrable hard surfaces. As such, it is not surprising that these literature values of ψ0 for

gelatin appear to be lower than our present result of 47.79 mV because neglecting the surface

conductivity has been shown to result in an underestimation of ψ0 for gel samples having “soft”

ion-penetrable surfaces.

Substituting the surface potential ψ0 determined here into Eqs. (4.15) & (4.16), is calculated to

be 0.630 mM for our 10% gelatin gel. In contrast, if the surface potential obtained from the hard

surface rotating disk model (Eq (4.14)) is employed, the calculated ρ0 will be, for example,

0.0253 mM and 0.0438 mM for the small and large disk, respectively, less than 10% of the

charged density determined by the present new rotating disk model for soft gelatin gel surfaces.

This is again in agreement with data reported by Dukhin et al. (Dukhin et al., 2005) which show

that only a very small fraction of the surface charge in ion-penetrable soft materials is detectable

by the classical electrokinetic method neglecting the surface conductivity.

0 10 20 30 40 5002468

0 10 20 30 40 500123

2000 2500 3000 3500 4000 4500

0.34

0.36

0.38

0.40

0.42

st

r (m

V)

1.5

ψψstrstr//ΩΩ1.5 1.5 = = (3.10(3.10±±0.0517)X100.0517)X10--5 5 mV.smV.s1.5 1.5

500 700 900 1100 1300 1500

0.98

1.00

1.02

1.04

1.06

1.08

st

r (m

V)

1.5

ψψstrstr//ΩΩ1.5 1.5 == (1.00(1.00±± 0.0147 ) X100.0147 ) X10--4 4 mV.smV.s1.5 1.5

ψψst

rst

r(( m

Vm

V ))

ψψst

rst

r((m

Vm

V ))

2000 2500 3000 3500 4000 4500

0.34

0.36

0.38

0.40

0.42

st

r (m

V)

1.5

ψψstrstr//ΩΩ1.5 1.5 = = (3.10(3.10±±0.0517)X100.0517)X10--5 5 mV.smV.s1.5 1.5

500 700 900 1100 1300 1500

0.98

1.00

1.02

1.04

1.06

1.08

st

r (m

V)

1.5

ψψstrstr//ΩΩ1.5 1.5 == (1.00(1.00±± 0.0147 ) X100.0147 ) X10--4 4 mV.smV.s1.5 1.5

ψψst

rst

r(( m

Vm

V ))

ψψst

rst

r((m

Vm

V ))Group number Group number

ψψst

rst

r//ΩΩ

1.5

1.5 ××

101055

ψψst

rst

r//ΩΩ

1.5

1.5 ××

101044

(a) (b)

Figure 4.6 Typical plots of streaming potential as a function of disk rotation speed Ω1.5 for 10%

gelatin gel disks in 0.04 mM NaCl: (a) small disk (a = 1.1 cm); and (b) large disk (a = 2 cm).

89

The slopes of linear regression from multiple runs (n = 50) are shown in the insets and noted in

each figure as mean with standard error (x ± S.E.).

4.4.3 Evaluation of ψ0 and ρ0 of gelatin/PAA gel Gelatin gels containing 0.02 % (w/v) polyacrylic acid (PAA) to modify the charge properties

were also investigated by the present rotating disc method. Polyacrylic acid is an anionic

polymer and its carboxylic groups are ionizable. The mean slopes of linear regression of

streaming potential versus Ω1.5 data measured in a 0.04 mM NaCl solution from multiple runs (n

= 50) are determined to be 1.68x10-5 mV.s1.5 and 4.94x10-5 mV.s1.5 for the small and large gel

disks, respectively (see insets of Figs. 4.7a & 4.7b). Correspondingly, the ψ0 and ρ0 of the PAA-

gelatin gel are calculated to be 10.29 mV and 0.0682 mM, which are lower than that of the

gelatin gel samples described in the previous section due to the neutralization of part of the

positively charged amino groups on the gelatin by the carboxyl groups of added PAA.

2000 2500 3000 3500 4000 4500

3.36

3.38

3.40

ψ

str(m

V)

Ω1.5

400 600 800 1000 1200 1400

1.94

1.96

1.98

2.00

2.02

ψst

r(m

V)

Ω1.5

0 10 20 30 40 500

1

2

3

4

0 10 20 30 40 500

2

4

6

8

10

Group number

ψstr/Ω1.5 = (1.68±0.0203)X10-5 mV.s1.5

Group number

ψstr/Ω1.5 = (4.94±0.0770)X10-5 mV.s1.5

ψst

r/Ω1.

5 ×10

5

ψst

r/Ω1.

5 ×10

5

(a) (b)(a) (b)


gelatin/PAA gel disks in 0.04 mM NaCl: (a) small disk (a = 1.1 cm); and (b) large disk (a = 2

cm). The slopes of regression of such plot from multiple runs (n = 50) are shown in the insets

and noted in each figure as mean with standard error (x ± S.E.).

90

The calculated gel surface potential ψ0 and the density of fixed charge groups ρ0 are summarized

in Table 4.1. The results indicate that by employing the hard disk model (Eq.(14)) and ignoring

the surface conductivity contribution, the calculated ψ0 and ρ0 will be significantly

underestimated. In addition, different values of ρ0 are obtained for the large and small disks

based on the hard disk model inconsistent with the physical situation, whereas only one value of

ρ0 is obtained for both the large and small disks based on our model taking into consideration of

the surface conductivity contribution.

Table 4.1 Calculated ψ0 and ρ0 values of tested polyelectrolyte gel samples

Gel Sample ψ0

(our model)

K ρ0 (Large & Small Disks )

(our model)

ρ0 (Large Disk)

(hard model)

ρ0 (Small Disk)

(hard model)

PVA/PAA(100/1) -75.71 mV 1.22x10-6 S 2.06 mM 1.01 mM 0.683 mM

PVA/PAA(200/1) -55.87 mV 6.39x10-7S 0.900 mM, 0.651 mM 0.526 mM

Gelatin 47.79 mV 2.54x10-5 S 0.630 mM 0.0438 mM 0.0253 mM

Gelatin/PAA

(500/1)

10.29 mV 8.69x10-6 S 0.0682 mM 0.0212 mM 0.0136 mM

The density of fixed charge groups under any pH conditions can be predicted for a synthetic

cationic or anionic gel, such as the PVA/PAA gel of Sec. 4.4.1, based on its known pKa.

However, for amphoteric gels based on polypeptides such as gelatin, the density of fixed charge

groups under different pH conditions cannot be accurately determined unless the quantity and

composition of dissociable peptide groups in the gel phase are known.

91

4.5 Conclusions A new and improved rotating disk model for characterizing polyelectrolyte gels has been

established. In addition to electrokinetic parameters considered in the existing rotating disk

model for ion-impenetrable hard surfaces, the effect of surface conductivity in soft and ion-

penetrable surfaces has been taken into account in the present model. Our rotating disk data show

a significant contribution of surface conductivity to the electrokinetic results in polyelectrolyte

gels with ion-penetrable soft surfaces which is usually ignored in the existing rotating disk model

for ion-impenetrable hard surfaces. Moreover, the improved rotating disk method can be

conveniently set up and applied to quantify the charge density and surface potential of other

polyelectrolyte gels of pharmaceutical interest.

92

Chapter 5 Periodic precipitation in multi-component systems*

5.1 Introduction As one of the objectives of this thesis, we propose to explore the periodic precipitation of

vanadyl salt in an erodible gel matrix and the use of the resulting laminated structure of

VO(OH)2 precipitate to achieve pulstatile delivery of vanadyl compounds. This is motivated by

the fact that vanadyl comounds are known to exhibit insulin-mimetic effect and a number of

hormones including insulin all exhibit circadian rhythm in plasma (Bussemer, et al., 2001).

Periodic precipitation phenomena involving a single-component precipitate have been

investigated for over a century (Liesegang, 1896; Stern, 1954 ， 1967; Henisch, 1988;

Grzybowski, 2009). However, certain insoluble precipitates such as Ca(OH)2 and VO(OH)2 do

not form periodic precipitation in a single-component precipitate system regardless of the

reactant concentration and reaction conditions.

In Chapter 2, we showed that the band position, band spacing and band width can be

manipulated by programming the concentration profile of reactants and reaction conditions. The

effects of process parameters such as the reaction rate constant, product solubility and diffusion

coefficient of reactant were also investigated, which provided trends and guidelines for the

design and control of the formation of periodic precipitation. Meanwhile, the effects of gel phase

properties on the periodic precipitation were studied in Chapter 3, where the gel mesh size and

gel charge density were shown to have a direct effect on the band width and patterns of periodic

precipitation. Herein, we propose to employ a multi-component precipitate system to assist the

formation of periodic precipitation of our model drug, VO2+. Up to date, periodic precipitation

phenomenon in multi-component systems is a new topic and has only been reported in a limited

number of cases (Shreif, et al., 2002; Klajn, et al., 2004). However, these existing studies

* The work presented in this chapter was all performed by Beibei Qu under supervision of Dr. Ping I. Lee, except

for the sample testing with Inductively Coupled Plasma Atomic Emission Spectrometry (ICP), which was conducted

by the Analytical Laboratory for Environmental Science Research and Training in the Chemistry Department,

University of Toronto.

93

focused mainly on the analysis of band compositions and the effect of reactant concentration on

band position. As a result, the underlying mechanisms are not clear and a more rigorous model

able to explain these observations is still lacking. Therefore, in our current study a potential

mechanism of this phenomenon is also proposed. In the proposed Mg(OH)2- VO(OH)2 multi-

component precipitate system, foreign nuclei of Mg(OH)2 were generated in situ to facilitate the

aggregation and crystal growth of VO(OH)2 at the band position. The composition of the

precipitate bands was further examined by Inductively Coupled Plasma Atomic Emission

Spectrometry (ICP). The results confirmed that VO(OH)2 is successfully precipitated with the

laminated structure (bands) formed by periodic precipitation of Mg(OH)2. Therefore, this system

demonstrates the feasibility of employing the heterogeneous nucleation mechanism to facilitate

the periodic precipitation of our model drug VO2+ which does not form periodic precipitation on

its own in a single reaction system.


5.2.1 Preparation of periodic precipitation in multi-component systems

Typically, a gelatin gel containing inner electrolytes was prepared by dissolving 1.5 g gelatin

powder (type A from porcine skin, 300 Bloom, Sigma, USA), 1.218 g MgCl2 and a

predetermined amount of CaCl2, CoCl2 or VOSO4 into 30 ml Milli-Q water at 50 oC. The

solution was then filled into a glass tube (inner diameter 2.4 mm), with one end sealed by

parafilm, and placed in a refrigerator (4 oC) overnight. The resulting physically crosslinked

gelatin gel in glass tubing was cut into segments of 12 cm (gel length L = 10 cm + reservoir

length L’ = 2 cm). The periodic precipitation experiments were performed by filling the outer

electrolyte NH4OH (14.8 M) into the reservoir above the gel surface (finite reservoir system, see

Chapter 2) while holding the tubing vertically. The experiments were run for 24-36 hr, and bands

formed were photographed with a digital camera (Nikon D3100 or Canon 350D).

5.2.2 Assay of precipitate composition

The gel containing periodic precipitation was frozen and then carefully removed from the tubing.

The precipitate bands were individually cut and completely dissolved in 10 mL concentrated

nitric acid. The ion concentrations of Mg2+, Ca2+, Co2+, or VO2+ in samples were analyzed by

94

Inductively Coupled Plasma Atomic Emission Spectrometry (ICP), which is employed to detect

trace metal element (detection limit of 0.01 μg/mL). Eletromagnetic radiation of target atom in

the sample can be generated by this emission spectroscopy and its intensity indicates the

concentration of the target element. The ICP tests were conducted by the Analytical Laboratory

for Environmental Science Research and Training in the Chemistry Department at the University

of Toronto.


5.3.1 Periodic precipitation phenomena in multi-component systems The formation of periodic precipitation in a multi-component system can be considered as a

heterogeneous nucleation process where multiple types of nuclei co-exist. The nucleation

process plays an important role in controlling the crystalline aggregation and growth and it

generally involves the following steps: the molecules collide into clusters to form embryos; once

the free energy barrier ΔG* is overcome, the embryo reaches a critical radius and starts to grow.

In general, if the embryo contains only one species, homogenous nucleation occurs. If the

embryo contains multiple species, heterogeneous nucleation takes place (Gorbunov, 1999; Liu,

1999, 2000; Djikaev and Donaldson, 2000; Cacciuto et al. 2004).

During heterogeneous nucleation, the existence of a foreign nucleus lowers the nucleation energy

barrier and reduces the ΔG* to ΔG*’, where ΔG*’= f(m,x) ΔG* and f(m,x) varies between 0 and

1. f(m,x) is determined by the interfacial interaction parameter, m, and the relative particle size

of the foreign nucleus, x (Liu,1999, 2000). This relation indicates that heterogeneous nucleation

is kinetically more favored. As a result, the nucleation rate and nuclei density increase

(Gorbunov, 1999; Liu, 1999, 2000; Djikaev and Donaldson, 2000; Cacciuto et al. 2004).

Photographs shown in Figure 5.1 illustrate that periodic precipitation occurs in single reaction

systems of Mg(OH)2 and Co(OH)2, but only continuous precipitation is formed in VO(OH)2

single reaction system and no precipitation is observed in Ca(OH)2 single reaction system. In

this case, the solubility of reaction product is the dominant factor in determining the occurrence

of periodic precipitation. If the solubility of reaction product is very high, little or no

precipitation will be generated and periodic precipitation will not be observed, as supported by

the simulation result of Figure 5.2a. On the other hand, if the solubility of reaction product is

95

very low, very close bands (as shown in Figure 2.24) or continuous precipitation will occur

(Figure 5.2b).

(a) (b) (c) (d) (e) (f) (g)

Figure 5.1 Experimental results of precipitation in single reaction, (a)-(d), and double reaction,

(e)-(g), systems after 36 hr under finite reservoir boundary conditions. (a) 0.2 M Mg2+ , (b) 0.2

M Ca2+ , (c) 0.002 M VO2+, (d) 0.2 M Co2+, (e) 0.2 M Mg2+ + 0.2 M Ca2+, (f）0.2 M Mg2+ +

0.002 M VO2+, (g) 0.2 M Mg2+ + 0.2 M Co2+

96

0 10 20 30 400

2000

4000

6000

8000

(1) (2)

X

F

(a)

0 10 20 30 400.00E+000

2.00E+007

4.00E+007

6.00E+007

(3)

X

F

(b)

Figure 5.2 Simulation results of periodic precipitation density F with a = 74 (CA0 = 14.8 M and

CB0 = 0.2 M) at T = 172.8, as a function of solubility of reaction product: (a) (1) C0 = 1.65*10-4

M; (2) C0 = 1.65*10-2 M (b) (very high solubility); (3) C0 = 1.65*10-8 M (very low solubility).

Simulated under finite reservoir boundary condition with density ρ = 2.3446 g/ cm3

/Mvm = 24.74 cm3/mol, surface tension σ = 120 mJ/m2 (Mullin, 1992) , capillary

97

length cmTkN

vwB

m 8

0

10*1.83

2

(Chacron et al., 1999) , the typical molecular size d =

8.56*10-8 cm , k = 10-6 L2/mol2.s, G = 3.24*10-10 cm/s , DA= 1*10-5 cm2/s , DB = DC = 0.5*10-5

cm2/s.

The solubility of the insoluble product X(OH)2 can be calculated from its ksp value based on the

following equation (Mullin , 1992):

3/10 )

4( spk

C Eq. (5.1)

where C0 is the solubility and ksp is the solubility product.

Mg(OH)2, with its solubility in the intermediate range, is a well-suited reagent for periodic

precipitation. As shown in Table 5.1, Ca(OH)2 exhibits a much higher solubility (2 orders of

magnitude higher) than Mg(OH)2. Thus, using the same amount of calcium as the starting

reactant, less precipitate is formed. In contrast, the solubility of VO(OH)2 is 4 orders of

magnitude lower than that of Mg(OH)2. So a small amount of vanadyl ions generates a lot more

precipitation in a short time and pattern of precipitation appears continuous. The solubility of

Co(OH)2 is relatively closer to that of Mg(OH)2, and can also form periodic precipitation similar

to Mg2+. Therefore, when the solubility of X(OH)2 is outside a certain range, it cannot form

periodic precipitation in the single reaction system.

Table 5.1 Characteristics of insoluble salts (Shreif et al. 2002; Chasteen, 1981)

Salt Ksp Solubility (M)

Mg2+ 1.8*10-11 2.0*10-4 Ca2+ 6.0*10-6 1.1*10-2 Co2+ 3.0*10-16 4.0*10-6 VO2+ 1.1 *10-22 3.0*10-8

98

Therefore, in order to assist X(OH)2 to overcome the limitation of its too low or too high

solubility and form periodic precipitation, foreign nuclei of Mg(OH)2 were introduced into the

X(OH)2 system through in situ generation. X2+ represents Ca2+, Co2+ or VO2+ in our current study.

Since Co(OH)2 and VO(OH)2 tend to re-dissolve in excess outer electrolyte NH4OH, a finite

reservoir system (see Chapter 2) was employed here such that the reaction between Co(OH)2 or

VO(OH)2 with excess NH4OH can be minimized. When the outer electrolyte, NH4OH, diffuses

into the gelatin gel containing Mg2+ and X2+, it was observed that periodic precipitation indeed

occurred in this system. The results of Figure 5.1 indicate that, compared to the single precipitate

Mg(OH)2 system, the band positions of precipitate in the multi-component precipitate system of

Mg(OH)2/Ca(OH)2 do not change significantly, except the bands become thicker. The

hypothesis is that Mg(OH)2, having a lower solubility than Ca(OH)2, nucleates quickly in the

system. The nuclei aggregate and start to grow after reaching a critical size. When they grow to

a certain size they become “seed” particles, which can catalyze the formation of more Ca(OH)2

nuclei based on the heterogeneous nucleation mechanism. With the same concentration of Ca2+

and Mg2+ as the starting reactants, the particles of Ca(OH)2 grow slower than the Mg(OH)2 due

to the relatively high solubility of Ca(OH)2, so it does not promote the nucleation rate of

Mg(OH)2. Therefore, the band position of Mg(OH)2 does not change significantly and the bands

only become thicker due to the aggregation of crystallite of Ca(OH)2 at the same band location of

Mg(OH)2 (Figure 5.3). The presence of Ca(OH)2 precipitate in the same band as crystal

aggregates with Mg(OH)2 will be verified by the ICP analysis later in this section.

Ca(OH)2

Mg(OH)2

Figure 5.3 Schematic illustration of heterogeneous nucleation in Mg(OH)2- Ca(OH)2 system

99

Similarly, in the Mg(OH)2-VO(OH)2 system, periodic precipitation was also observed. As shown

in Figure 5.1, more precipitate bands were formed in this multi-component system with smaller

band spacing than that of the Mg(OH)2 single precipitate system. In this case, VO(OH)2 has a

much lower solubility than Mg(OH)2 and therefore would produce nuclei first in the system,

which then promote the nucleation of Mg(OH)2 in the system based on the heterogeneous

nucleation mechanism. Note that the Mg2+ concentration is 100 times higher than VO2+, the

formation rate of Mg(OH)2 exceeds that of the VO(OH)2, so the particles of Mg(OH)2 continue to

grow in the multi-component system. Once reaching a certain size larger than the particles of

VO(OH)2, they become “seed” particles which catalyze the growth of crystalline VO(OH)2 near

the precipitate band position of Mg(OH)2. Therefore, VO(OH)2 serves as the seed nucleus at the

beginning of the process before Mg(OH)2 takes over, resulting in the co-existence of VO(OH)2

and Mg(OH)2 at the band position, as shown in Figure 5.4. The presence of VO(OH)2 precipitate

in the same band as crystal aggregates with Mg(OH)2 will also be verified by the ICP analysis

later in this section.

VO(OH)2

Mg(OH)2

Figure 5.4 Schematic illustration of heterogeneous nucleation in Mg(OH)2- VO(OH)2 system

On the other hand, in the Mg(OH)2-Co(OH)2 system, the bands contain white Mg(OH)2

precipitate in the upper region displaying smaller band space ΔX comparing with the Mg(OH)2

single precipitate system (Figure 5.1). Due to the lower solubility of Co(OH)2 than that of

Mg(OH)2, the nuclei of Co(OH)2 are generated first, which then facilitate the nucleation of

100

Mg(OH)2 in the system and thus the band position of Mg(OH)2 is affected accordingly (e.g.

smaller ΔX) as shown in Figure 5.1. Therefore, the mechanism of periodic precipitation in the

Mg(OH)2-Co(OH)2 system is similar to that of the Mg(OH)2-VO(OH)2 system. The nuclei of

Co(OH)2 promote the nucleation of Mg(OH)2 in the system initially and after the Mg(OH)2

particle size becomes large enough, they start to catalyze the crystallization of Co(OH)2 in the

system. Because Co(OH)2 can re-dissolve in excess of outer electrolyte NH4OH, the bands of

Co(OH)2 then re-dissolve gradually from the gel surface where the concentration of NH4OH is

the highest. This re-dissolution phenomenon is also observed in the Co(OH)2 single precipitate

system (see Figure 5.1(d)). Therefore, in the Mg(OH)2-Co(OH)2 system the white precipitate of

Mg(OH)2 only exists in the upper part of the glass tubing, the undissolved Co(OH)2 bands stay

in the lower part of the tubing as shown in Figure 5.1(g).

In the Mg(OH)2-VO(OH)2 or Mg(OH)2-Co(OH)2 system, the existence of VO(OH)2 or Co(OH)2

will accelerate the nucleation rate of Mg(OH)2 and the precipitate bands of Mg(OH)2 become

closer (smaller ΔX), as shown in Figure 5.1(f) and(g). It is generally known that when multi-

species nuclei coexist in the system, heterogeneous nucleation occurs. The nucleation rate and

the nuclei density are increased by the lower nucleation free energy (Liu, 1999, 2000; Djikaev

and Donaldson, 2000; Gorbunov, 1999; Cacciuto et al. 2004). Therefore we propose that the

heterogeneous nucleation caused by the foreign nuclei is the possible mechanism of these

experimental observations.

In summary, by introducing the foreign nuclei in situ, we successfully generated periodic

precipitation with Ca2+ and VO2+, neither of which can produce periodic precipitation alone in

their individual single reaction systems. The phenomena of periodic precipitations in multi-

component systems can be explained by a heterogeneous nucleation mechanism. Periodic

precipitation of VO2+ so produced will be tested in vitro in the next chapter to demonstrate its

applicability as a novel delivery system in achieving pulsatile drug release.

5.3.2 Composition analysis of bands

In order to examine the precipitate composition at various band positions, sections of the gel

were cut at different positions (at the band or between bands) and analyzed by ICP (see Figure

5.5). As a control analysis, the concentration of Mg2+ was also measured. Ordinarily，VO2+ and

Mg2+ can be analyzed by the UV and EBT-EDTA titration method, respectively. However, the

101

concentrations of our gel samples were below the detection limit of these classical methods, so

the ICP method was selected which has the optimal detection limit of 0.01 μg/mL. The results

so obtained are presented in Figure 5.6, which indicate the existence of VO2+ and Mg2+ at both

bands 1 and 3 in the Mg(OH)2-VO(OH)2 periodic precipitation system. It is clear that the

concentrations of VO2+ and Mg2+ at these band positions (sample 1 and 3) are significantly

higher than that in the space between bands (sample 2). To further confirm the precipitate state

of VO2+ and Mg2+ in the bands, samples taken at position 1 and 2 were washed with deionized

water for 24 hours and analyzed again. Dissolved VO2+ and Mg2+ in the gel phase between bands

(sample 2) were washed out, but the crystalline precipitate of VO2+ and Mg2+ still remained at the

band position in high concentrations similar to that before washing (Figure 5.6). Similarly, in the

Mg(OH)2-Ca(OH)2 periodic precipitation system, the concentration of Ca2+ and Mg2+ at the band

positions (sample 1 and 3) are also higher than that in the gel space between bands (sample 2), as

shown in Figure 5.7. However, since the solubility of Ca(OH)2 is much larger than Mg(OH)2,

part of the crystalline Ca(OH)2 at the band position can still be removed by washing.

Nevertheless, the concentrations of Ca(OH)2 and Mg(OH)2 remained at the band position are still

higher after washing as compared with that of sample 2. Therefore, we can confirm that

VO(OH)2 and Ca(OH)2 precipitated at the same band positions of Mg(OH)2 in the multi-

component periodic precipitation systems studied here.

Figure 5.5 Schematic illustration of sample positions. Samples were taken at position 1-3 for

composition analysis: Samples 1 and 3 at the precipitate bands; sample 2 between bands.

gel surface

1 2 3

102

1 2 30

5

10

15

20

25

not wash wash

1 2 30.0

0.1

0.2

0.3

0.4

0.5

not wash wash

Con.

of M

g2+(u

g/m

L)

Con

. of V

O2+

(ug/

mL)

Figure 5.6 Results of ICP analyses of Mg2+ and VO2+ in samples 1-3, respectively. The

experiments were repeated by washing samples 1 and 2 with deionized water.

1 2 30

15

30 not wash wash

1 2 30

2

4

6 not wash wash

Con

. of M

g2+(u

g/m

L)

Con

. of C

a2+(u

g/m

L)

Close to 0

Figure 5.7 Results of ICP analyses of Mg2+ and Ca2+ in samples 1-3, respectively. The

experiments were repeated by washing samples 1 and 2 with deionized water.

5.4 Conclusions Heterogeneous nucleation has been proposed to be the major mechanism controlling the band

formation in multi-component periodic precipitation system. This aspect has been investigated

Mg2+ VO2+

Mg2+ Ca2+

103

experimentally and mechanistically in the present study. Our results show that periodic

precipitation can be facilitated by adding foreign nuclei to facilitate the periodic precipitation of

compounds which do not form periodic precipitation in a single-component precipitate system

regardless of the reactant concentration and reaction conditions. In this case, the solubility of the

reaction product is the dominant factor in determining the occurrence of periodic precipitation. If

the solubility of reaction product is very low such as VO(OH)2, very close bands to continuous

precipitation will occur. On the other hand, if the solubility of reaction product is very high such

as Ca(OH)2, less or no precipitation will be generated and periodic precipitation cannot be

observed. Our results show that by introducing foreign nuclei of Mg(OH)2 into the X(OH)2

system through in situ generation, periodic precipitation of VO(OH)2 or Ca(OH)2 can be induced

to occur as crystal aggregates in the same band with Mg(OH)2. The existence of precipitates of

VO(OH)2 or Ca(OH)2 in the periodic bands has been confirmed by ICP test which has a very low

detection limit of 0.01 μg/mL. The work reported herein provides a novel method to facilitate

reactants to form periodic precipitation in multi-component precipitate systems, even for

reagents which fail to form periodic precipitation in a single precipitate system. Furthermore, it is

the first time that periodic precipitation is accomplished with our model drug VO2+ and this

system can serve as a potential pulsatile drug delivery system, which will be further explored and

discussed in Chapter 6.

104

Chapter 6 Pulsatile drug release from periodic precipitation system*

6.1 Introduction It is well known that clinically certain drugs (such as nitroglycerine) can develop tolerance from

continuous constant drug administration (Wolff and Bonn, 1989; Dighe et al.,2009), causing the

therapeutic effect to be reduced or even lost completely. In order to avoid this tolerance effect,

delivery systems or therapeutic regimens have been designed to deliver the drug in a pulsatile or

other non-constant fashion. Examples of bioactive agents that can benefit from non-constant

drug delivery include gonadotropin-releasing hormone (GnRH) which is secreted in pulses

endogenously (Woller, et al., 2004) and a number of other hormones, such as insulin, all

exhibiting circadian rhythm in plasma (Haus et al., 2001). Therefore, there is a therapeutic

rationale for developing pulsatile drug delivery systems in order to mimic physiological patterns

of hormone release so as to enhance the effect of drug therapy. In designing such pulsatile

delivery systems, there has been a significant interest in creating laminated delivery systems with

equidistant alternating drug-containing and drug-free layers in order to achieve a pulsatile drug

release upon dissolution of the delivery system (Lee, 1986; Xu and Lee, 1993; Hassan et al.,

2000). However, such multilayer polymer matrices have limitations in their large scale

manufacturability and the labor intensive nature of the manufacturing process. To overcome

these drawbacks, we propose that such alternating structure of a precipitated drug can be

generated automatically and more effectively by the nanofabrication process of periodic

precipitation. This idea is supported by the fact that multilayer structures at the micro- or nano-

scale (micron to sub-micron) have been constructed through periodic precipitation (Mueller,1984)

and applied to microlens arrays and microfluidic architectures (Grzybowski, 2009). Therefore,

by loading a selected drug into this multi-layer alternating structure through periodic

precipitation, it provides a convenient and novel way to produce pulsatile drug delivery systems.

* The work presented in this chapter was all performed by Beibei Qu under supervision of Dr. Ping I. Lee, except

for the sample testing with Inductively Coupled Plasma Atomic Emission Spectrometry (ICP), which was conducted

by the Analytical Laboratory for Environmental Science Research and Training in the Chemistry Department,

University of Toronto.

105

To the best of our knowledge, this new concept has never before been applied to the design of

pulsatile drug delivery systems.

Vanadium (V) compounds including vanadium salts and vanadium complexes exhibit insulin-

mimetic effect due to their ability to normalize hyperglycemia and enhance insulin sensitivity.

Vanadyl (VO2+) compounds are potentially useful in the oral therapy for diabetes because of its

good oral absorption properties (Marzban and McNeill, 2003; Orvig et al., 1995; Nriagu et al.,

1998). Therefore, vanadyl sulfate (VOSO4) was selected as the model drug in our present study.

As shown in Chapter 5, despite the fact that VO2+ does not form periodic precipitation on its own

in a single reaction system, periodic precipitation of vanadyl salt can be induced by employing

the approach of a multi-component precipitate system, where the existence of high concentration

of VO(OH)2 precipitate in the resulting laminated structure can be confirmed by the ICP test.

Thus, periodic precipitation of VO(OH)2 will be explored in this study as a potential pulsatile

drug delivery system.

Mechanistically, pulsatile drug delivery can be achieved in bioerodible polymers containing non-

uniform alternating drug containing and drug free layers through the surface erosion process (Xu

and Lee, 1993). In the present study, our model drug VO2+ will be incorporated into the

laminated structure via periodic precipitation and its subsequent release behavior evaluated in

simulated gastric fluid (pepsin solution with pH ~ 1.2). In this case, the gel erosion rate can be

controlled by the pepsin concentration and solution pH, both of which affect the activity of the

pepsin solution. In principle, the drug release rate and release profile from such a system can be

further programmed by varying the spatial distribution of the drug precipitate in the gel matrix.

As discussed in Chapter 2 and Chapter 3, this potentially can be achieved by varying the gel

matrix and reaction conditions.

In this chapter, we will demonstrate that pulsatile vanadyl release profiles can be successfully

achieved from a surface erodible gelatin gel matrix containing VO(OH)2 loaded in laminated

structure formed by the periodic precipitation process.

106


6.2.1 Preparation of release medium

2.0 g NaCl and 9.6 - 15 g purified pepsin were dissolved in 1 L Milli-Q water at room

temperature (USP, 2007). 7.0 ml HCl was added (pH ~ 1.2). Purified pepsin (from porcine

stomach mucosa, activity of 800 to 2500 units per mg of protein), NaCl and HCl were purchased

from VWR (USA). During the release study, the precipitate dissolved completely in the release

medium, which was sampled periodically.

6.2.2 Drug release analysis

Gelatin gel in glass tubing containing periodic precipitate of both Mg(OH)2 and VO(OH)2 was

cut into a small segment retaining only three consecutive bands and with one end sealed with

parafilm. In the release medium, the gel erosion and drug release occurred only through the open

end of the glass tubing segment. Release studies on gel samples containing only Mg(OH)2

precipitate bands were conducted in 20-ml vials each containing 20 ml of release medium.

Release studies on gel samples containing both Mg(OH)2 and VO(OH)2 precipitate bands were

conducted in 7.4-ml vials each containing 7 ml of release medium. These were agitated on a vial

rotator at room temperature. The release medium was replaced every 8 hr and 12 hr respectively

and the drug concentration in the sampled release medium was analyzed by ICP, which were

conducted by the Analytical Laboratory for Environmental Science Research and Training in the

Chemistry Department, University of Toronto.

6.2.3 Drug release mechanism

For a surface erodible system with contributions from diffusion, the mechanism of drug release

can be analyzed by evaluating the dimensionless parameter Ba/D, where B is the surface erosion

rate, a denotes the initial gel length and D is the diffusion coefficient of drug in the gel phase.

When the parameter Ba/D is greater than 1, the drug release becomes mostly controlled by the

surface erosion mechanism (Lee, 1980). To apply this analysis to the present system, the

diffusion coefficient of the model drug in our gel phase is taken to be 1 x 10-5 cm/s2 which is

quite reasonable, and the surface erosion rate B can be calculated from the measured changes in

the overall gel length S(t) based the relationship S(t) = a-Bt.

107


6.3.1 Incorporation of model drug into the drug carrier

As discussed in Chapter 5, periodic precipitation of VO2+ cannot be generated alone in a single

precipitate system (Figure 5.1c), but can be induced by the periodic precipitation of Mg2+ in a

multi-component precipitate system based on a heterogeneous nucleation mechanism (Figure

5.1f). The existence of VO2+ and Mg2+ in the band position has been confirmed by the ICP test

(Figure 5.6). The results show that the concentrations of VO2+ and Mg2+ at the band position are

much higher than that in the clear space between bands. Therefore, our model drug VO2+ can be

successfully loaded into a laminated structure through the periodic precipitation process.

6.3.2 Pulsatile drug release To demonstrate the feasibility of pulsatile drug release from such laminated structure generated

by periodic precipitation, the release of Mg2+ from a gel sample containing periodic precipitation

of only Mg(OH)2 was first investigated as a control. In the release medium, the gelatin gel was

eroded from the open end due to its digestion by pepsin. The eroding front position at the gel

surface was monitored by the addition of a pH indicator (0.2% Thymol Blue) in the release

medium (see Figure 6.1). The gel surface erosion rate constant B was calculated from time

dependent changes in the overall gel length to be 0.0396 cm/ hr with an initial gel length a = 3.6

cm in this case (Figure 6.2). Accordingly, the parameter Ba/D was determined to be 3.96, much

larger than 1. Therefore, it can be concluded that under the present experimental conditions, the

Mg2+ release from our gel sample was mainly controlled by the surface erosion mechanism. The

precipitate bands are observed to dissolve rapidly as the eroding front sweeps over the band

position with time as shown in images 2, 5 and 7 of Figure 6.1. The corresponding

concentrations of Mg2+ in the release medium at the moment of band dissolution are seen to be

much higher than in other samples when the eroding front is moving between bands as shown in

Figure 6.3. In this case, lower concentrations of Mg2+ are detected at 12, 36, 48 and 72 hr

corresponding to the presence of dissolved Mg2+ between precipitate bands in the gel phase.

This confirms that the drug release from our current laminated drug carrier in gelatin gel is

mainly controlled by the surface erosion mechanism. In addition, we observe that by decreasing

the band spacing ΔX (ΔX2 < ΔX1, seen in Figure 6.1), the compound release interval is decreased

(Figure 6.3). Therefore, by manipulating the reaction conditions and gel properties as described

108

in previous chapters, predetermined periodic precipitate bands can be generated and the pulsatile

frequency and compound release rate can be adjusted. From this result, it is clear that the

periodic precipitation system is very suitable as a pulsatile drug release platform and appropriate

compound release rate based on application requirement can be achieved by programming the

formation of periodic precipitation.

Sample

1

2

3

4

5

6

7

12hr

24hr

36hr

48hr

60hr

72hr

84hr

image

indicates the moving front

Release direction

ΔX1 ΔX2

Figure 6.1 Eroding front position as a function of time during the release of Mg2+ from periodic

precipitation structure in gelatin gel. Here, a pH indicator was added to enhance the observation

of gel surface movement.

Indicates the position of moving front

109

20 40 60 80

0

1

2

3

S(t)

(cm

)

t (h)

Figure 6.2 The overall gel length S(t) as a function of time in the Mg release system

1 2 3 4 5 6 70

5

10

15

Con

. of M

g2+(u

g/m

L)

12hr 24hr 36hr 48hr 60hr 72hr 84hr

Figure 6.3 Concentration of Mg2+ in the release medium as a function of time analyzed by ICP.

Mg2+

110

Having confirmed that pulsatile Mg2+ release can be achieved from the current periodic

precipitation system, we proceed to investigate the release behavior of the periodic precipitation

system with a model drug. When the model drug VO2+ was loaded through the multicomponent

periodic precipitation approach discussed in Chapter 5 by adding Mg2+ (0.2 M) into the system,

the resulting precipitate bands were much closer than those in pure Mg2+ (0.2 M) periodic

precipitation system (Figure 5.1). This smaller band spacing will result in faster drug release

from this VO2+ /Mg2+ system than that from the pure Mg2+ precipitate system, which would have

made it more difficult to synchronize the sampling time with the dissolution of the bands. To

reduce the drug release rate in this VO2+/Mg2+ system, the release medium with a lower

concentration of pepsin (9.6 g/L) was used in this study which allowed a reasonable sampling

time of every 8 hr. The eroding front position at the gel surface was also monitored by the

addition of a pH indicator (0.2% Thymol Blue) in the release medium. The precipitate bands

containing both VO2+ and Mg2+ are observed to dissolve rapidly as the eroding front sweeps over

the band position with time as shown in images 2, 4 and 6 of Figure 6.4. The gel surface erosion

rate constant B was calculated to be 0.0404 cm/ hr with an initial gel length a = 2.1 cm in this

case (Figure 6.5). Accordingly, the parameter Ba/D was determined to be 2.36 also larger than 1

in this case. Therefore, it can be concluded that under the present experimental conditions, the

release of VO2+ and Mg2+ from our gel sample was mainly controlled by the surface erosion

mechanism. The corresponding concentrations of VO2+ and Mg2+ in the release medium at the

moment of band dissolution are seen to be much higher than in other samples when the eroding

front is moving between bands as shown in Figure 6.6. In this case, lower concentrations of

VO2+ and Mg2+ are detected at 8, 24 and 40 hr corresponding to the presence of dissolved VO2+

and Mg2+ between precipitate bands in the gel phase. Results in Figure 6.6 confirm that

simultaneous pulsatile release of VO2+ and Mg2+ can be achieved from the present gelatin gel

through a surface erosion mechanism. Therefore, the pulsatile release of our model drug VO2+

has been successfully demonstrated using our proposed periodic precipitation platform.

111

Sample

0

1

2

3

4

5

6

0hr

8hr

16hr

24hr

32hr

40hr

48hr

image

indicates the moving front

Release direction

ΔX1 ΔX2

Figure 6.4 Eroding front position as a function of time during the release of VO2+ and Mg2+ from

periodic precipitation structure in gelatin gel. Here, a pH indicator was added to enhance the

observation of gel surface movement.

Indicates the position of moving front

112

20 40 60

0

1

2

S(t)

(cm

)

t (h)

Figure 6.5 The overall gel length S(t) as a function of time in the Mg/VO drug release system

1 2 3 4 5 60

2

4

6

8

1 2 3 4 5 60.00

0.05

0.10

0.15

0.20

Con

. of M

g2+(u

g/m

L)

Con

. of V

O2+

(ug/

mL)

8hr 16hr 24hr 32hr 40hr 48hr 8hr 16hr 24hr 32hr 40hr 48hr

Figure 6.6 Concentrations of VO2+ and Mg2+ in the release medium as a function of time

analyzed by ICP.

6.4 Conclusions In our present study, the potential of using a laminated structure generated by the periodic

precipitation as a pulsatile drug delivery device has been demonstrated. The insulin mimetic

Mg2+ VO2+

113

drug VO2+ was loaded into the laminated structure in a gelatin gel matrix through the multi-

component periodic precipitation approach. Based on the surface erosion mechanism, VO2+ was

released from the laminated structure layer by layer and as a result a pulsatile drug release profile

was achieved. As a reference, the release of Mg2+ located at the band position was similarly

analyzed which also exhibited the pulsatile release pattern. Therefore, by controlling the spatial

distribution of the precipitated compound, the release rate and release interval between pulses

can be manipulated based on different application requirements. Our results from previous

chapters indicate that the spatial distribution of the precipitated compound can be programmed

by manipulating the reaction conditions and properties of the gel matrix. Therefore, under

appropriate conditions, such periodic distribution of vanadyl compound can be formed to provide

a desired pulsatile drug release pattern and it potentially can offer a wide range of drug release

rates by utilizing the characteristics of periodic precipitation phenomena.

114

Chapter 7 Summary and future directions

7.1 Summary The primary objective of this research was to better understand, predict and control of the

periodic precipitation process and to apply such programmed periodic precipitation to the design

of a pulsatile delivery system.

In the first part of this study, we delineated for the first time the effect of finite versus infinite

reservoir concentration boundary conditions on the resulting periodic precipitation both in

simulations and in experiments. Prior to this, there was a major inconsistency between the

boundary conditions employed in the modeling (infinite reservoir of outer electrolyte) and

experimental (finite reservoir of outer electrolyte) approaches, which produced considerable

discrepancies between the predicted and experimental results. In addition, most of the modeling

analyses were based on the prenucleation model which does not take into account the

competitive particle growth and ripening effects. Therefore, we employed a generalized model of

Chacron and L’Heureux which combines the essence of both the prenucleation and

postnucleation models, and refined and solved it under various new concentration boundary

conditions not previously investigated. We have shown that the concentration profile of outer

electrolyte plays a key role in controlling the precipitate band’s position Xn and band spacing ΔX

in the periodic precipitation system. Our results show that in all cases when the finite reservoir

condition is employed, fewer bands with increasing band spacing ΔX will be generated. We have

also shown that equidistant precipitate bands can be simulated mathematically and generated

experimentally by imposing a programmed and exponentially increasing outer electrolyte

reservoir concentration. To the best of our knowledge, this unique approach has never been

reported in the literature before. The general sensitivity of periodic precipitation to concentration

boundary conditions and the effect of various physiochemical properties such as the diffusion

coefficient of the soluble species, the solubility of the reaction product and the reaction rate

constant were examined in detail to identify parameters important to the design and control of

the precipitate band position, band spacing and band width for the application of these intriguing

phenomena. Furthermore, our experimental observations show behavior similar to that predicted

115

from our modeling simulations thus providing a sound basis for the further application of

periodic precipitation to generate desired band position and band spacing.

The physicochemical properties of the gel phase such as the gel mesh size or gel charge property,

and their effects on the periodic precipitation are critical. However, they have not been

sufficiently emphasized or investigated previously. Therefore, in the second part of our study, the

gel mesh size was related to the free volume in the gel through measured gel swelling ratio. It is

clear that by decreasing the gel mesh size and the particle growth rate, the width of the periodic

precipitation bands increases but with less number of bands and smaller band spacing. In

addition, we have characterized the gel charge property by an improved rotating disc method.

There were indications that by increasing the density of charged groups in a cationic gel, a larger

curvature could develop at the gel surface and at the diffusion front of the outer electrolyte,

which produced an uneven distribution of reaction product in the radial direction with

progressively narrowing bands when the initial concentration of inner electrolyte was low.

Therefore, by selecting appropriate gel properties such as the gel swelling ratio or the density of

gel fixed charge groups, the formation of desired periodic precipitation patterns can be better

controlled.

The improved rotating disk approach was developed in our study to characterize the charge

properties of polyelectrolyte gels. The existing rotating disk method for quantifying the charge

property of flat surfaces is based on the classical electrokinetic model which neglects the effect

of surface conductivity and therefore is only applicable to ion-impenetrable hard surfaces.

However, this classical electrokinetic model would be inaccurate for polyelectrolyte gel systems

involving ion-penetrable charged surfaces such as gelatin gel surface. In addition to

electrokinetic parameters already considered in the existing rotating disk model, the contribution

of surface conductivity, known to be very significant for soft and ion-penetrable gel surfaces, has

been taken into account in our current analysis. Based on this new approach, two rotating gel

disks of different radius but with identical gel composition and preparation procedures were

employed for determining the surface potential and density of fixed charge groups in different

gel samples. Our results confirm that the contribution from surface conductivity is significant in

polyelectrolyte gels and the densities of fixed charge groups in PVA/PAA, gelatin and

gelatin/PAA gels have been determined based on our improved rotating disc model.

116

The ultimate goal of our work is to apply the periodic precipitation process to fabricate a

pulsatile drug delivery device. In the third part of our study, vanadyl sulfate was chosen as a

model compound, which is an insulin mimetic drug and is effective through oral administration.

However, it does not form periodic precipitation in a single-component precipitate system

regardless of the reactant concentration and reaction conditions. Because the solubility of

VO(OH)2 is extremely low, only continuous bands are generated. Our results indicate that by

introducing foreign nuclei of Mg(OH)2 into the system, periodic precipitation of VO(OH)2 is

facilitated. The existence of precipitates of VO(OH)2 in the periodic bands has been confirmed

by the ICP test. This is the first time that periodic precipitation is accomplished with VO2+ and

this system would be useful as a potential pulsatile drug delivery system. The mechanism of such

multi-component periodic precipitation is not clear and had not been studied previously. Based

on our study, heterogeneous nucleation has been proposed to be the major mechanism

controlling the band formation in multi-component periodic precipitation system. Our work

herein provides a novel method to facilitate reactants to form periodic precipitation in multi-

component precipitate systems, even for reagents which fail to form periodic precipitation in a

single precipitate system.

Finally, we have also demonstrated the potential of using laminated structure generated by the

periodic precipitation as a pulsatile drug delivery device. The insulin mimetic drug VO2+ was

loaded into the laminated structure in a gelatin gel matrix through the multi-component periodic

precipitation approach. Based on the surface erosion mechanism, VO2+ was released from the

laminated structure layer by layer and, as a result, a pulsatile VO2+ release profile was

successfully demonstrated. Therefore, by controlling the spatial distribution of the precipitated

compound, the release rate and release interval between pulses can be manipulated based on

different application requirements.

7.2 Future directions From our improved understanding of the periodic precipitation process based on the present

thesis work, we can design and program the formation of periodic precipitation in most cases.

But there are still many hurdles to overcome in order to be of practical utility. For instance, in

our current simulation, some process parameters, such as particle growth rate constant, are

estimated, as appropriate approaches for their evaluation are not available for the periodic

117

precipitation system. Meanwhile, in real systems involving diffusion and reaction, non-constant

process parameters rather than the constant ones employed here such as position dependent

diffusion coefficient or concentration dependent reaction rate constant may have to be introduced

in order to better simulate the periodic precipitation process. Furthermore, our current

simulations were based on the finite element method. Its accuracy is determined by the number

of elements (or mesh size) one can accommodate in the simulation, which is currently limited by

the power of the available desktop computer. Therefore, in the future as more process

parameters are evaluated and more powerful computers become available, more accurate band

position and other band characteristics can be predicted for real systems.

In addition, some abnormal periodic precipitation phenomena, such as the reverse Liesegang

rings, have been reported. They cannot be predicted by our current model because the underlying

mechanisms of these phenomena are still unclear. Therefore, in the future, an improved

understanding on these phenomena and a further refinement on the current model would be

needed.

Furthermore, most of the drugs on the market today are organic or bio-organic molecules

(antibody, enzyme or nucleic acid) which do not precipitate easily as inorganic salts. The

periodic precipitation technique will be more powerful if these drugs can be loaded into desired

laminate structures to achieve specific release patterns. It is known that some organic molecules

may form precipitate under basic conditions or with a chelating agent. This may offer the

potential to employ the present multi-component periodic precipitation approach to induce the

periodic precipitation of a selected organic molecule. This aspect needs to be further investigated.

Therefore, loading these organic drugs into periodic precipitation structures becomes one of the

key tasks for future exploration.

Finally, the successful application of periodic precipitation in the design of a pulstaile drug

delivery device in the present study and the reported fabrication of complex structures by wet

stamping all demonstrate the great potential of applying periodic precipitation as a

microfabriction technique to achieve complex but useful microstructure in a very simple way.

Therefore, in the future this technique potentially can find additional applications in other fields

for fabricating useful microstructures and for controlling the time varying patterns of liberation

of drugs or reactants at the desired time and spatial location.

118

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128

Appendix

Dimensionless model equations transformed from Eq. (2.1)-Eq.(2.9) were employed in our study

(Chacron and L’Heureux, 1999) :

TabKa

Xa

2

2

2

Eq. (A.1)

TbbKa

XbD

2

2

2

1 Eq. (A.2)

TssFsggalfaesbKKa

XsD R

)()(

3)1( 332

2

2

2

Eq. (A.3)

TsFsggalfaes R

)()()1(2 2

Eq. (A.4)

TsFsggalfaNes R

)()()1( Eq. (A.5)

TNsFalfa

)( Eq. (A.6)

TRes R

1 Eq. (A.7)

0)( else, if ,0),)]([exp()1()( 22 sFssgbetassF Eq. (A.8)

)1ln(1)(s

sgg

Eq. (A.9)

304359.0/3 22

2

AARAARRAAR

Eq. (A.10)

129

Where twGT , x

wDGXA

,0B

A

CCa ,

0B

B

CCb ,

0

0

CCCs C ,

GkwCK B0

2

, 0

0

CCKKK B ,

A

B

DDD 1 ,

A

C

DDD 2 ,

wAA

, nCvwN

Nm 0

304

, wrR ,

dGvCDw

alfam

C 06216

,

2/12

).3

4(Tempkwbeta

B

,

dCDwJ C

C0

224 .

List of symbols employed in Comsol files

u1, dimensionless parameter a

u2, dimensionless parameter b u3, dimensionless parameter s

u4, dimensionless parameter

u5, dimensionless parameter

u6, dimensionless parameter N u7, dimensionless parameter R

OO, dimensionless expression Eq.(A.10) SS, dimensionless expression Eq.(A.7)

P, constant equals to 0 AA, dimensionless thickness of the Gibbs surface

130

Table A-1 Parameter input in Comsol 3.5a (Example: Figure 2.4 in Chapter 2, Finite reservoir

boundary condition)

Model Navigator One dimension

PDE Modes, Coefficient Form, Time dependent analysis

Draw Domain one: line (-10,0)

Domain two: line (0,50)

Options: Constants alfa = 3.83*0.5*10^10 beta = 9.0

K= (10^-5) KK = 0.012

P =0 AA = 4.61

Options: Scalar

expressions

F = ((1+u3)^2)*exp(-(beta*gg)^2)*(u3>0)+P*(u3<=0) gg= 1/(log(abs(1+u3)))*(u3>-1)*(u3<0)

+1/(log(abs(1+u3)))*(u3>0)+P*(u3<=-1)+(10^20)*(u3==0) OO=(u7^2+AA*u7)/(u7^2+3*AA*u7+AA*AA/0.304359)

SS=1+u3-exp(OO/u7)

Physics:

Subdomain

settings

PDE 1:

Domain (1):diffusion coefficient =1, mass coefficient = 1, u(t0)= 5*7.4 Domain (2):diffusion coefficient =1, mass coefficient = 1,

source term = -K*u*u*u2, u(t0)= 0 PDE 2:

Domain (1):diffusion coefficient =0, mass coefficient = 1, u2(t0)= 0 Domain (2):diffusion coefficient = 0.5, mass coefficient = 1,

source term = -K*u*u*u2, u2(t0)= 1 PDE 3:

Domain (1):diffusion coefficient =0, mass coefficient = 1, u3(t0)= -0.99 Domain (2):diffusion coefficient = 0.5, mass coefficient = 1,

source term = KK*u*u*u2-SS*u4-alfa*gg*gg*gg*F/3, u3(t0)= -0.99 PDE 4:

131

Domain (1):diffusion coefficient =0, mass coefficient = 1, u4(t0)= 0 Domain (2):diffusion coefficient = 0, mass coefficient = 1,

source term = alfa*gg*gg*F+2*SS*u5, u4(t0)= 0 PDE 5:


source term = alfa*gg*F+SS*u6, u5(t0)= 0 PDE 6:


source term = alfa*F, u6(t0)= 0 PDE 7:

Domain (1):diffusion coefficient =0, mass coefficient = 1, u7(t0)= 10^-20 Domain (2):diffusion coefficient = 0, mass coefficient = 1,

source term = SS, u7(t0)= 10^-20

Physics: Boundary

settings

PDE 1: Boundary (1) = Newmann boundary condition q=0, g=0

Boundary (3) = Newmann boundary condition q=0, g=0 PDE 2: Boundary (1) = Newmann boundary condition q=0, g=0

Boundary (3) = Newmann boundary condition q=0, g=0 PDE 3:

Boundary (1) = Newmann boundary condition q=0, g=0 Boundary (3) = Newmann boundary condition q=0, g=0





Boundary (3) = Newmann boundary condition q=0, g=0

132


Boundary (3) = Newmann boundary condition q=0, g=0

Time stepping range(0.01,0.01,345.6)

Solver: Time dependant

Linear system solver: Direct(UMFPACK)

Solve Press “=”

133

Table A-2 Parameter input in Comsol 3.5a (Example: Figure 2.10 in Chapter 2, Infinite reservoir

boundary condition)



Draw Domain one: line (0,50)

Options: Constants alfa = 3.83*0.5*10^10

beta = 9.0 K= (10^-5)

KK = 0.012 P =0

AA = 4.61

Options: Scalar

expressions

F = ((1+u3)^2)*exp(-(beta*gg)^2)*(u3>0)+P*(u3<=0)

gg= 1/(log(abs(1+u3)))*(u3>-1)*(u3<0) +1/(log(abs(1+u3)))*(u3>0)+P*(u3<=-1)+(10^20)*(u3==0)

OO=(u7^2+AA*u7)/(u7^2+3*AA*u7+AA*AA/0.304359) SS=1+u3-exp(OO/u7)

Physics:

Subdomain

settings

PDE 1: diffusion coefficient =1, mass coefficient = 1, u(t0)= 0

source term = -K*u*u*u2 PDE 2:

diffusion coefficient = 0.5, mass coefficient = 1, source term = -K*u*u*u2, u2(t0)= 1

PDE 3: diffusion coefficient = 0.5, mass coefficient = 1,


diffusion coefficient = 0, mass coefficient = 1, source term = alfa*gg*gg*F+2*SS*u5, u4(t0)= 0

PDE 5: diffusion coefficient = 0, mass coefficient = 1,

source term = alfa*gg*F+SS*u6, u5(t0)= 0

134



diffusion coefficient = 0, mass coefficient = 1, source term = SS, u7(t0)= 10^-20

Physics: Boundary

settings

PDE 1:

Boundary (1) = Dirichlet boundary condition q=0, g=0, h=1, r = 15*flc2hs(t-1,1)





Boundary (1) = Newmann boundary condition q=0, g=0 Boundary (2) = Newmann boundary condition q=0, g=0 PDE 5:






Solver: Time dependant Linear system solver: Direct(UMFPACK)

Solve Press “=”

135

Table A-3 Parameter input in Comsol 3.5a (Example: Figure 2.14 in Chapter 2, equidistant

periodic precipitation)



Draw Domain one: line (0,50)

Options: Constants alfa = 3.83*0.5*10^10

beta = 9.0 K= (10^-5)

KK = 0.012 P =0

AA = 4.61

Options: Scalar

expressions

F = ((1+u3)^2)*exp(-(beta*gg)^2)*(u3>0)+P*(u3<=0)

gg= 1/(log(abs(1+u3)))*(u3>-1)*(u3<0) +1/(log(abs(1+u3)))*(u3>0)+P*(u3<=-1)+(10^20)*(u3==0)

OO=(u7^2+AA*u7)/(u7^2+3*AA*u7+AA*AA/0.304359) SS=1+u3-exp(OO/u7)

Physics:

Subdomain

settings

PDE 1: diffusion coefficient =1, mass coefficient = 1, u(t0)= 0

source term = -K*u*u*u2 PDE 2:

diffusion coefficient = 0.5, mass coefficient = 1, source term = -K*u*u*u2, u2(t0)= 1

PDE 3: diffusion coefficient = 0.5, mass coefficient = 1,


diffusion coefficient = 0, mass coefficient = 1, source term = alfa*gg*gg*F+2*SS*u5, u4(t0)= 0


source term = alfa*gg*F+SS*u6, u5(t0)= 0

136



diffusion coefficient = 0, mass coefficient = 1, source term = SS, u7(t0)= 10^-20

Physics: Boundary

settings

PDE 1:

Boundary (1) = Dirichlet boundary condition q=0, g=0, h=1, r = A0*flc2hs(t-1,1)





Boundary (1) = Newmann boundary condition q=0, g=0 Boundary (2) = Newmann boundary condition q=0, g=0 PDE 5:







Solve Press “=”

137

Table A-4 Parameter input in Comsol 3.5a (Example: Figure 3.4 in Chapter 3, with real

parameters)



Draw Domain one: line (-0.01,0)

Domain two: line (0,0.05)

Options: Constants Jc= 9.498*10^28 P =0

AA = 4.61*8.1*10^-10 K= (10^-12)

vv= 24.74*10^-6 w= 8.1*10^-10

C0= 0.165 GG= 3.24*10^-12

beta= 9.0

Options: Scalar

expressions

F = ((1+uu)^2)*exp(-(beta*gg)^2)*(uu>0)+P*(uu<=0)

gg= 1/(log(abs(1+uu)))*(uu>-1)*(uu<0) +1/(log(abs(1+uu)))*(uu>0)+P*(uu<=-1)+(10^20)*(uu==0)

OO=(u7^2+AA*u7)/(u7^2+3*AA*u7+AA*AA/0.304359) SS= GG*(u3-Ceq)/C0

A0= 14800 Ceq= C0*exp(w*OO/u7)

uu= (u3-C0)/C0

Physics:

Subdomain

settings

PDE 1:

Domain (1):diffusion coefficient = 10^-9, mass coefficient = 1, u(t0)= 14800

Domain (2):diffusion coefficient = 10^-9, mass coefficient = 1, source term = -K*u*u*u2, u(t0)= 0

PDE 2: Domain (1):diffusion coefficient =0, mass coefficient = 1, u2(t0)= 0

Domain (2):diffusion coefficient = 0.5*10^-9, mass coefficient = 1,

138

source term = -K*u*u*u2, u2(t0)= 200 PDE 3:

Domain (1):diffusion coefficient =0, mass coefficient = 1, u3(t0)= 0.0001*C0

Domain (2):diffusion coefficient = 0.5*10^-9, mass coefficient = 1, source term = K*u*u*u2-SS*u4/vv-4*3.14*Jc*w*w*w*gg*gg*gg*F/(3*vv), u3(t0)= 0.0001*C0 PDE 4:


source term = 4*3.14*Jc*w*w*gg*gg*F+8*3.14*SS*u5, u4(t0)= 0 PDE 5:


source term = w*Jc*gg*F+SS*u6, u5(t0)= 0 PDE 6:


source term = Jc*F, u6(t0)= 0 PDE 7:

Domain (1):diffusion coefficient =0, mass coefficient = 1, u7(t0)= 10^-20 Domain (2):diffusion coefficient = 0, mass coefficient = 1,

source term = SS, u7(t0)= 10^-20

Physics: Boundary

settings

PDE 1:






139






Time stepping range(0.01,1,86400)


Solve Press “=”

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