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Guest–host van der Waals interactions decisively affect the moleculartransport in mesoporous media†

Tina Ukmar,a Uro�s Maver,a Odon Planin�sek,b Albin Pintar,a Ven�ceslav Kau�ci�c,ac Alja�z Godec*a

and Miran Gaber�s�cek*ac

Received 22nd July 2011, Accepted 18th October 2011

DOI: 10.1039/c1jm13493k

We present clear evidence that the global (macroscopic) transport from/to mesoporous materials is

significantly affected by the interactions between the mesoporous host and the guest molecules. The

problem is considered in a most general way so the solutions apply for a variety of cases such as the

release of a guest from porous matrices, catalysis occurring in porous materials or processes taking

place in separation techniques. The concept is proved on the experimentally determined release profiles

of a model drug (indomethacin) from accurately designed SBA-15 and MCM-41 mesoporous silicates.

In order to allow for a full quantitative analysis, a very high frequency of sampling was carried out at

short release times. The agreement between the experimentally determined and the theoretically

predicted curves is excellent not only in shape but also in all major trends. In the broadest sense, one

might say that the host–guest interactions change the effective cross-section of pores through which the

transport of guest occurs. In addition, the interactions lower the efficiency of utilization of the guest. In

drug release this is observed as a decrease of released matter at long times, in catalysis this would

correspond to a decrease of global turnover efficiency etc.However, it is not only the final outcome that

is affected but also the transport pattern (e.g. the shape of release curves) during a wide range of

timescales. Our finding might have a profound influence on the design of various devices based on

meso- or macroporous materials.

1 Introduction

Molecular transport in or across porous media plays a key role in

very diverse fields such as rheology, composite materials, poly-

mer and colloidal science, geology, catalysis, separation science,

biophysics and drug delivery. As with regards to the molecular

motion inside the pores, many interesting phenomena have been

reported.1–6 For example, in the case of pore sizes up to 1 nm,

such as typically found in zeolites, the molecular dynamics are

characterized by single file diffusion,7 incommensurate diffu-

sion8,9 and loading dependent diffusion path control.10 In such

cases, the transport is determined by a complex interplay

between adsorbent–adsorbate and adsorbate–adsorbate inter-

actions. In the case of zeolites the role of guest–host interactions

is conventionally termed the ‘thermodynamic factor’11 and was

also found to be important for transport in molecular framework

aNational Institute of Chemistry, Hajdrihova 19, Ljubljana, Slovenia.E-mail: aljaz.godec@ki.si; miran.gaberscek@ki.si; Fax: +386-1-4760-300; Tel: +386-1-4760-316bFaculty of Pharmacy, University of Ljubljana, Ljubljana, SloveniacFaculty of Chemistry and Chemical Technology, University of Ljubljana,Ljubljana, Slovenia

† Electronic Supplementary Information (ESI) available: See DOI:10.1039/c1jm13493k/

1112 | J. Mater. Chem., 2012, 22, 1112–1120

microporous materials.12 When the pore size is increased up to

2 nm, features of classical Brownian dynamics appear, but the

conventional mobility-dependent free volume relationship still

does not apply.3 Meanwhile, as we enter the domain of meso-

porous materials (with pore sizes between 2.5 and 30 nm), the

transport occurs in terms of regular Brownian motion, man-

ifested as Knudsen diffusion established via collisions with pore

walls in the case of gases13 and regular diffusion at liquid

densities. Recently it was shown that transport of gases in the

case of strong guest–host attractions does not follow the

Knudsen prediction because strong adsorptions creates highly

conducting paths in small pores that compete with large pores

and leads to a reduction in tortuosity with molecular size.14 Also,

the attractions to host walls were shown to intimately determine

the nature of the mechanism of wall reflections and in particular

the momentum exchange at a collision (see ref. 15 and 16 for an

in-depth review of the subject).

It remains questionable, however, to what extent all these

interesting phenomena that occur inside micro, meso and macro

pores, affect the macroscopic (also termed large scale) transport.

Namely, it is the latter that is actually probed in basically all

applications mentioned above. For example, in separation,17

catalysis,18–21 drug delivery22–32 and other applications any

changes (e.g. in concentration of active species) are probed ‘far

This journal is ª The Royal Society of Chemistry 2012

Fig. 1 Schematic of the scales involved in typical experiments: the

measurements are performed on a macroscopic scale (left) while the

transport phenomenon involves the microscopic scale (right).

Fig. 2 (a) Top panel: Schematics of 2-dimensional model porous

matrices. The pore volume, Vp ¼ L(L � na) ¼ const., and pore entrance

area, Ap ¼ 2(n� 1)d¼ const. are held constant while varying n, d and a in

a way that preserves L ¼ na + (n � 1)d. Specifically we take the following

values of d (a): d0 (18D), 4d0/5 (15D), d0/2 (10D) and 2d0/5 (9D) with d0 ¼10D, L¼ 130D and LB ¼ 175D. The thickness of the outermost two walls

in the last parameter set is 1D larger than those of inner walls (that is 10D

instead of 9D), which is necessary to preserve constant L, Vp and Ap in all

cases. Bottom panel: Types of initial conditions used in the calculations.

Type I correspond to a homogeneous concentration inside the pores,

Type II to an internal concentration proportional to the local Boltzmann

factor and Type III to a thin homogeneous adlayer at the internal pore

walls. To assess its influence on release kinetics an additional thin

homogeneous surface adlayer was added on the external matrix surface.

(b) Various effective pair interaction potentials (top panel) and (bottom

panel) resulting average contact potentials, hVconti (dotted lines), and

average potentials in the center of pores, hVcenti (full lines).

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away’ from the porous media (Fig. 1). Thus, even if inside the

pores very complex phenomena may occur, does the probe feel

these local changes at distances that are many orders larger than

the typical dimensions of pores? Similarly, can we probe changes

inside the pores that, by nature, are much faster (adsorption,

chemical or electrochemical reaction etc.) than the macroscopic

transport towards the probe?What one needs, therefore, is a clear

coupling between the local processes (i.e. processes inside the

pores) and large-scale transport, that is, the transport from the

porousmedium to the probe positioned in bulk system (seeFig. 1).

An attempt of such a quantitative coupling has been made in our

previous work33 where it was found that a decrease in pore size

leads to significant enhancement ofmacroscopic transport, within

a certain timescale. However, in that preliminary work we

neglected the interactions between the porous system and the

guest which is a rather non-realistic situation, so the results were

mainly interesting from the conceptual point of view.

In the present work we demonstrate, for the first time on

a basis of realistic experiment, a quantitative coupling between

the local dynamics in a porous system and the large scale

transport. In particular, we show how a general interaction

between the porous matrix and the guest matter affects the

macroscopic transport of the guest to a probe positioned outside

the porous matrix at relatively large distances, as found in most

of the applications. As a model system we chose a drug release

system, although the results are valid for any other porous

system with considerable guest–matrix interaction (most systems

of practical interest). The drug system is chosen mainly due to the

fact that during the recent years very precise nanoarchitecturing

of certain porous matrixes has been developed. Thus, not only

the pore composition, but also its size, distribution of sizes, shape

etc. can effectively be tailored. In the present paper we have been

able to control, very accurately, another crucial parameter

needed for quantitative treatment—the particle size of different

mesoporous hosts. In particular we preparedMCM-41 and SBA-

15, which have an equal particle size and show no intergrowth

aggregation. For an easier comparison with our related work34

we have selected indomethacin as a model guest (solute).

Considering the fact that the large scale molecular transport in

most systems studied experimentally occurs on the time scale of

several (tens of) hours while the transport inside the pores is

several orders of magnitude faster, any theoretical approach

must inherently be able to capture the features of the system

dynamics on such a wide span of time and length scales.

Unfortunately, most molecular techniques such as classical or ab

initioMolecular Dynamics, Langevin Dynamics simulations, etc.

are inapplicable due to the fact that a simulation of the system

This journal is ª The Royal Society of Chemistry 2012

with several pores on an experimentally relevant time scale is

simply infeasible. Therefore, we chose a mesoscopic continuum

description which describes molecular transport in terms of the

dynamics of the dimensionless solute concentration field. As we

show in continuation, such a model is able to capture and explain

the most essential features observed experimentally.

Hexagonal porous matrices of size L are modeled as

2-dimensional arrays of channels with diameter d and pore wall

thickness a (see Fig. 2(a) top panel). The matrix is surrounded by

a buffer zone of thickness LB, which is chosen to be large enough

to assure that the interactions with pore walls vanish and that the

concentrations never reach appreciable values at large distances

from the matrix. While varying the pore size and attractions to

pore walls the total pore volume, pore entrance area along with L

and LB are kept constant in all cases (see Fig. 2(a) and the

Methods section for further details).

When guest molecules enter a mesoporous host, one expects at

least the development of van der Waals (vdW) type interactions

between both phases. The essential and fundamental questions

that we want to address in the present paper are the following:

Do local interactions (of vdW type) between the porous host and

guest molecules affect the macroscopic transport from the pores

to a large distant probe? If yes, how important is the strength of

these local interactions? And finally, does the pore size matter?

J. Mater. Chem., 2012, 22, 1112–1120 | 1113

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Needless to say, any progress in understanding these correlations

could lead to a significant optimization of devices and materials

used in drug release, separation techniques, catalysis, etc.

2 Methods

2.1 Synthesis of SBA-15 matrix

SBA-15 with the average pore size of 9 nm was prepared with

a hydrothermal synthesis using the structure directing agent

Pluronic P123 (a PEG-PPG-PEG block copolymer, Aldrich) and

tetraethyl orthosilicate (98% TEOS, Aldrich) as a silica

precursor. The material was prepared closely along the lines

reported in ref. 35 but with slight modifications to achieve the

desired particle size and morphology. First, 4.0 g of triblock

copolymer Pluronic P123 was added to a mixture of 30 g of water

and 120 g of 2 M HCl aqueous solution, which was stirred at

room temperature for 3–4 h until Pluronic P123 was dissolved.

Then, 8.50 g of TEOS was added to this solution under vigorous

stirring. After 10 min, the mixture was kept under static condi-

tions at 312 K (instead of 308 K) for 20 h, followed by a hydro-

thermal synthesis in a Teflon-lined autoclave for 24 h at 373 K.

The solid products were collected by filtration, washed with

water, dried, and calcined at 823 K in flowing air. A comparison

of material characteristics with those reported in the literature35

confirmed that the prepared material was in fact of SBA-15 type,

thus the sample was denoted SBA-15.

2.2 Synthesis of MCM-41 matrix

MCM-41 with a mean pore diameter of 3 nm was synthesized

using the structure directing agent hexadecyl-

trimethylammonium bromide (CTAB, Aldrich) and tetraethyl

orthosilicate (98% TEOS, Aldrich) as a silica source. The mate-

rial was prepared along the lines reported in ref. 36 but with

slight modifications of the synthesis conditions. First, 26.7 g of

ammonia solution (25% NH4OH, Merck) was mixed with 105 g

of distilled water and then 0.5 g of CTAB was added to the

solution. The mixture was placed in an oil bath under stirring and

heated to a homogeneous solution. When the solution had

become clear, 2.35 g of TEOS was added dropwise. After 3 h of

stirring and heating at 348 K, the obtained white powder was

washed on a filter and calcined at 823 K for 6 h in an air flow to

remove the CTAB from the pores. A comparison of material

characteristics with those reported in the literature36 confirmed

that the prepared material was in fact of MCM-41 type, thus the

sample was denoted MCM-41.

2.3 Guest molecule loading

Loading with the model guest molecule indomethacin was per-

formed as follows. First a solution of indomethacin (g-IMC,

Sigma) in tetrahydrofuran (THF, Sigma-Aldrich) was prepared

with the IMC solution at a concentration of 75 mg g�1. THF was

added drop-wise to a fine layer of calcined samples of SBA-15

andMCM-41, allowing the powder to soak the added drops. The

prepared samples were then dried, first at 313 K for 24 h in

a ventilation dryer and then, additionally, for 24 h at 313 K in

a vacuum dryer. The first step is supposed to remove the majority

of the solvent while the second step removes the residual solvent

1114 | J. Mater. Chem., 2012, 22, 1112–1120

which is adsorbed more strongly to the pore walls or capillary

condensed. Immediately after drying, the samples were placed in

an desiccator under an inert Ar atmosphere to avoid exposure to

moisture and kept under such conditions until right before the

measurements. It was found that such pretreatment decisively

improved the repeatability and enabled one to avoid the other-

wise very prominent effects of residual loading of solvent and/or

water.

2.4 Scanning (SEM) and transmission (TEM) electron

microscopy

SEM micrographs were obtained by a Zeiss SupraTM 35VP

Scanning Electron Microscope (SEM) operated at 1 keV. TEM

analysis was performed on a 200-kV field-emission gun (FEG)

microscope JEOL JEM 2100. For TEM studies, a drop of an

ethanol-diluted nanoparticle solution was placed on a copper

grid and dried at room temperature. The specimens were addi-

tionally coated with carbon in order to prevent excessive

charging and decomposition of the sample under the electron

beam.

2.5 X-Ray powder diffraction (XRPD)

XRPD patterns were recorded on a PANalytical X-Pert PRO

high-resolution diffractometer with a1 configuration using

CuKa1 radiation (1.5406 �A) in the range from 0.5 to 10� 2q, usinga step of 0.017� per 100 s and separately in the range from

5 to 35� 2q, using a step of 0.033� per 100 s under air conditions.

2.6 N2 adsorption–desorption analysis

Adsorption and desorption isotherms of nitrogen were measured

on a Micromeritics ASAP 2020 volumetric adsorption analyzer

at 77 K. Before the sorption analysis, the calcined samples were

outgassed under vacuum for 2 h at 473 K and the loaded samples

for 2 h at 333 K. The specific surface area (SBET) was determined

using the BET (Brunauer–Emmett–Teller) equation. The total

pore volume (Vt) was estimated from the amount adsorbed at

a relative pressure of 0.989, converting it to the volume of liquid

nitrogen at 77 K. The microporosity of SBA-15 (Vmi) was

calculated by the t-plot method from the adsorption data. The

calculation of pore size distribution (PSD) was performed by

analyzing the adsorption data of the N2 isotherm using the

Barrett–Joyner–Halenda (BJH) method with a Halsey–Faas

correction for bare calcined and loaded SBA-15, and the BJH

method with a Halsey–Kruk–Jaroniec–Sayari correction for

bare calcined and loaded MCM-41. The average pore diameter,�dme corresponds to the peak value of the BJH distribution.

2.7 Quantification of loading fraction

The quantification of the relative amount of the loaded guest

molecules was carried out by means of thermogravimetric anal-

ysis (TGA) coupled with differential scanning calorimetry

(DSC). Measurements were carried out between 303 and 1073 K

with a heating rate of 20 K min�1 using a Mettler Toledo

(Schwerzenbach, Switzerland) thermogravimetric analyzer

model TGA/DSC 1 under a constant gas flow rate (oxygen,

This journal is ª The Royal Society of Chemistry 2012

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50 mLmin�1). The initial sample mass ranged from 5–10 mg. The

measurements were repeated three times.

The weight loss accompanying the combustion of IMC was

used to determine the fraction of loaded IMC (for details see the

ESI).† The temperature interval of the IMC combustion was

determined from the first derivative of the weight loss with

respect to temperature. The absence of IMC particles located

outside the pores was confirmed with a combination of X-ray

powder diffraction and solid-state NMR, both performed at

several temperatures. Further details of the characterization

procedures are given elsewhere.34

2.8 Guest release experiments

Release experiments were performed in highly purified water

using a standard dissolution apparatus (USP Dissolution

Apparatus 1, Vankel Dissolution apparatus, model VK 7000,

USA) equipped with a wire-mesh basked attached to a rotation

shaft adjusted to 50 rpm at 298 K. The approach (Apparatus 1

and slow rotation) was chosen specifically to avoid the intrinsic

hydrodynamic issues arising in the rotating paddle method (USP

Dissolution Apparatus 2),37 which is usually used for these sorts

of experiments, and to minimize all hydrodynamic effects in

general. Samples were collected in triplet replicas at frequent,

predetermined time intervals and filtered through 0.45 mm filters.

The concentration was determined spectrophotometrically at 268

nm using a HP diode array UV-VIS spectrophotometer (8453,

Germany). At this wavelength the THF absorption does not

interfere with the IMC absorption (for details see the ESI).† The

collected volumes were replaced by equal volumes of the disso-

lution medium (highly purified water) to maintain a constant

volume. These replacements were considered when determining

actual concentrations at different time intervals. The release

experiments were carried out in five-fold replicas to minimize the

statistical error.

2.9 Model and computational methods

Porous matrices were modeled as 2-dimensional arrays of

channels with diameter d and wall thickness a (see Fig. 2(a) top

panel). This structure should describe well the geometry of the

porous network in mesoporous silicates with an hexagonal pore

arrangement, such as SBA-15, MCM-41 and related materials.

For a discussion of the geometrical relations between these 2D

models and real matrices see ref. 33. We denote the size of the

porous matrix with L and the thickness of the surrounding buffer

zone by LB. LB is chosen to be large enough to assure that the

interactions with pore walls vanish and that the concentrations

never reach appreciable values at large distances from the matrix.

L and LB are kept constant in all cases. We fix the total pore

volume and the total pore entrance area and consider various

pore sizes (see Fig. 2(a)).

As they move through the porous matrix, solute molecules

experience an external force due to interactions with pore walls.

The total external interaction potential experienced by

a diffusing solute molecule at a given position r is equal to the

sum of interactions with all molecules constituting pore walls.

We describe the attractive part of these effective intermolecular

interactions using the attractive part of a pairwise additive

This journal is ª The Royal Society of Chemistry 2012

interaction potential of the Morse type and the repulsive part

simply with a hard repulsive core (see Fig. 2(b)) Thus, the local

external potential is defined as

VðrÞ ¼ðR

rðr0Þupðjr� r0jÞd2r0; (1)

where the pair potential is defined as

upðRÞ ¼qmin

�e2sðz�RÞ � 2esðz�RÞ� if R$D

N otherwise:

((2)

where r(r) is the local density of the matrix at position r ¼ (x, y)

and is equal to 1 if r lies inside a pore wall and zero otherwise.

qmin and z are the depth and position of the pair potential

minimum, s describes the width of the potential well and thereby

also its range and D is a lower cut-off distance, which will be

explicitly defined later.

The dynamics of the dimensionless solute concentration fieldC

(r) are assumed to be governed by drift due to the local forces

exerted on diffusing molecules by pore walls and diffusion due to

the presence of local concentration gradients. We consider a low

Reynolds number environment (overdamped regime) and

isotropic mobility, m ¼ D/kBT, and thus we assume a linear

relationship between force and velocity, v ¼ mF. The system

evolves according to the 2-dimensional Fokker–Planck equation

(FPE):

vCðrÞvt

¼ DVxy$

�CðrÞkBT

VxyVðrÞ þ VxyCðrÞ�h� Vxy$j (3)

where Vxy represents the 2-dimensional gradient operator, Vxy ¼v/vxx + v/vyy, D is the diffusion coefficient, kB is the Boltzmann

constant, T is the temperature, and j the generalized flux. Since

the impenetrability conditions simply reflect the absence of an

effective driving force for solute transport into the pore walls, we

can replace the repulsive part of the external potential by

reflecting boundary conditions (BCs) for C and V at the walls,

VxyC$n¼ 0 and VxyV$n¼ 0, where n is the unit surface normal of

pore walls. Such a choice of BCs is accurate in the overdamped

regime while this would not be the case if inertial effects would be

important. In the latter case the appropriate choice would

instead be the point of zero velocity in the normal direction of

pore walls. In the present case, however, the temporal resolution

of the model is longer than the typical time scale of momentum

relaxation. Note that the assumption of the mobility being

isotropic is likely not to hold rigorously (see for example ref. 16)

but is expected to differ in the axial and normal directions mainly

due to molecular packing and boundary-layer hydrodynamic

effects. A self-consistent inclusion of the pore size dependence of

the anisotropic diffusion tensor is not manageable within the

present theoretical framework and is therefore omitted. Such

a simplification is not expected to affect the qualitative nature of

the dynamics on the scales of interest.

All calculations are performed numerically on a square grid

with spacing D. First we evaluate the integral in 1 using the

attractive part of up with z¼ D and s¼ 1 and then solve the FPE

under reflecting boundary conditions for C and V at pore walls

and external borders of the grid. Further computational details

and formal definitions of average potential characteristics hVcentiand hVconti are presented elsewhere.6

J. Mater. Chem., 2012, 22, 1112–1120 | 1115

Fig. 3 SEM and TEM images of prepared (a) SBA-15 and (b) MCM-41.

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We consider different types of initial conditions (IC), which

are most likely to be met in experimental studies (see Fig. 2(a)

bottom panel). The basic types of IC (I–III) represent situations

where no guest molecules are initially located on the external

surface. Specifically, type I corresponds to a homogeneous

concentration inside the pores, type II to a concentration

proportional to the local Boltzmann factor, exp(�V(r)/kBT), and

type III to a thin adlayer of equal concentration as in type I. In

a separate set of calculations we additionally introduce a thin

adlayer of various concentrations on the external surface of the

matrix.

Along with complete absence of solute–wall attractions

(qmin ¼ 0) we consider several strengths of attraction with qmin

between 0.05 and 0.5 kBT (see Fig. 2(b)). We perform 280 � 103

integration steps with increment Dt taking the dimensionless

parameter a h DDt/D2 ¼ 0.2.

The large-scale transport kinetics are quantified by means of

two effective parameters. One is the mean release time (or mean

transport time in the case of separation and catalysis), t1/2,

defined as the time in which half of guest molecules are trans-

ported into the bulk solution (for details see the ESI)† and is

obtained simply by integration of the total flux across the outer

surface of the so-called diffusion layer, O:ðt1=20

ds#Ojðr; sÞ$dS ¼ 1

2

ðR

Cðr; s ¼ 0Þd2r; (4)

where dS is the differential of the surface normal of O. The other

is the effective large-scale diffusion coefficient, Deff, describing

intrinsic large-scale transport behavior on large (‘macroscopic’)

time scales. The calculation of the latter is based on fitting the

numerical solution for the time evolution of the guest fraction

outside the porous host to an effective 1-dimensional model,

which is somewhat tedious and details are given in the ESI.†

Fig. 4 Top panel: N2 sorption isotherms of empty and loaded meso-

porous (a) SBA-15 and (b) MCM-41. Bottom panel: the corresponding

XRPD diffraction patterns. The inset shows corresponding pore size

distributions. The black lines correspond to bare silicates and red lines to

drug-loaded silicates.

3 Results and discussion

As mentioned in the Introduction section, a precondition for any

quantitative treatment of experimental data is that the samples

are extremely well defined. This was the reason that we selected

release porous systems based on silicates which can be tailored

very precisely on the nanoscale. Ideally, if different samples are

probed, it is desired that only one experimental variable is

changed. For example, SBA-15 andMCM-41 can be prepared so

that the only significant difference is in pore sizes, as shown in

continuation.

SEM images of the obtained calcined mesoporous silicates

show well defined polyhedral particles without intergrowth

aggregation for both SBA-15 and MCM-41. For both samples

the sizes of mesoporous silica particles range from 400 nm to

1000 nm with an average size around 600 nm (see Fig. 3). In both

cases TEM images clearly show hexagonal arrangements of the

porous system and the uniformity of the cylindrical pores (Fig. 3

inset). The average pore diameter of the silicates is in good

agreement with XRPD patterns and N2 sorption measurements.

In conjunction with TEM images, a high degree of order in the

pore arrangement is also observed from XRPD patterns exhib-

iting three well resolved peaks, which are assigned to reflections

from the (100), (110) and (200) planes of a hexagonal well

ordered structure with p6mm space group. Equal particle size is

1116 | J. Mater. Chem., 2012, 22, 1112–1120

particularly important, since it eliminates particle-size effects on

large-scale transport kinetics and at the same time allows a direct

comparison of experimentally determined release kinetics with

theoretical predictions. The N2 adsorption–desorption isotherms

(shown in Fig. 4 top panel) are found to be of type IV sorption

isotherms according to the IUPAC classification and exhibit

well-defined H1 hysteresis loops, typical for SBA-15 and MCM-

41 silicates. The presence of H1 hysteresis type confirms that they

have open-ended cylindrical mesopores. Both isotherms have

hysteresis loops with sharp adsorption–desorption branches,

which indicates a narrow pore size distribution in both silicates.

The corresponding pore size distributions are shown in Fig. 4

(inset in bottom panel). The specific surface areas, pore volumes

and average pore diameters are given Table 1. Note that the

discrepancy in the total pore volume between SBA-15 and

MCM-41 is only around 28%, which in principle should allow

a direct qualitative comparison with theoretical predictions. As

expected, upon guest loading the specific surface area and the

total pore volume decrease significantly. This suggests that most

of the nanometre channels of the silicates are still open, although

a portion of the voids are filled with guest molecules. The guest

molecules embedded inside the pores do not fully occupy the

This journal is ª The Royal Society of Chemistry 2012

Table 1 Specific surface area, total and micropore volumes and porediameter of prepared mesoporous silicates

sample S [m2 g�1] Vt [cm3 g�1] Vmi [cm

3 g�1]

�dme

[nm]

SBA-15 776 1.075 0.058 9.2MCM-41 1036 0.776 — 3.1SBA-15 + IMC 320 0.5929 0.002 8.4MCM-41 + IMC 224 0.2024 — 2.9

Fig. 5 Experimentally determined release profiles from SBA-15 (red

symbols) and MCM-41 (blue symbols) with average pore diameters 9.2

and 3.1 nm, respectively. Inset: Theoretically predicted release profiles for

type I initial conditions in the case of d ¼ d0 ( ) and d ¼ d0/2( ):

diffusion in the absence of guest–wall attractions (full lines) and in the

case of attractions with qmin ¼ 0.4kBT (dashed lines). Note that the

theoretical results are given in numerical time units Dt ¼ aD2/D.

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available space, such that there is still some space for N2

adsorption. The effect of loading is also nicely seen in the pore

size distributions (Fig. 4 red lines). Note that in the case of SBA-

15, the average pore size decreases upon loading, which suggests

that besides fully loaded pore segments, one also finds regions of

thin adlayers. From the diffraction patterns one observes that the

hexagonal pore arrangement is not disrupted upon drug loading.

In agreement with the change in the average pore diameter, the

positions of the (110) and (200) peaks are shifted slightly towards

larger 2q values upon drug loading in the case of SBA-15.

Note also that in the case of SBA-15 at small pore diameters,

the tail in the pore size distribution corresponding to traces of

micropores vanishes, which suggests either filling of micropores

or an extensively obstructed access to the micropores, for

example, due to the presence of a larger guest particle. The

various effects of loading are quantified in Table 1. The relative

fraction of loaded IMC, as determined from TGA/DSC (see

ESI),† is 32% in SBA-15 and 33% inMCM-41. Thus, both model

systems are sufficiently similar and comply with the assumptions

made in the theoretical approach, which allows a direct quali-

tative comparison between the experimental results of the release

kinetics with the theoretical predictions. Note that the well

defined PFG NMR self-diffusion measurements in MCM-41

type materials have revealed that these materials can exhibit

significant deviations from an ideal, uniform structure.38 In other

cases they were also found to form homogeneous ‘single crys-

tals’,39 which is highly likely to hold for materials prepared and

investigated in the present study.

The results of release experiments, presented in terms of the

fraction of released guest as a function of time, are given in

Fig. 5, where one observes two important facts. First, the overall

release is faster from SBA-15, the matrix with larger pores, and

second, the total fraction of released guest is also larger in the

case of SBA-15. This behavior, as well as similar observations in

the literature,23 are unexpected on the basis of the conventional

picture about the molecular release from pores of matrices. In

particular, the difference in total fraction released obtained from

measurements after 5 days (0.44 vs. 0.39 for SBA-15 and MCM-

41, respectively) cannot be explained by the geometrical effect of

the difference in pore size of both samples; the only apparent

difference as inferred from the characterization above. Namely,

it is easy to show that in conventional experimental setups such

as those used here (the volume of the release medium is constant

and the volume of the porous particles is negligible with respect

to the volume of the release medium), the curves for both samples

should converge to the same point (as calculated in inset of

Fig. 5, solid line). Even more, the conventional (diffusion

controlled release) picture predicts that the kinetics of release

from porous matrices are in fact faster in the case of smaller

This journal is ª The Royal Society of Chemistry 2012

pores.33 In other words, the curve for MCM-41 with smaller

pores is expected to have higher values than that for SBA-15 at

all times, as indeed shown in the inset of Fig. 5 (solid curves).

This feature is a direct consequence of the coupling of the

transport in/from several nearby channels which renders single-

channel models inappropriate. In the case of smaller pores, the

lateral volume per unit pore exit area into which the molecules

can diffuse is larger. As a result, the coupling of fluxes from

nearby pores is less obstructive and the flux from the pores is

therefore larger, which can be formulated approximately as more

efficient draining. One may conclude that the classical predic-

tions fail even to reproduce the qualitative trends observed in real

experiments (Fig. 5).

Analyzing many similar measurements one inevitably comes to

a conclusion that in the conventional treatment a crucial ingre-

dient is missing, which leads to systematic errors when one tries

to replicate experiments in a quantitative way. Careful inspection

of literature data shows, quite surprisingly, that one such ingre-

dient that would be expected but is obviously missing is the

interaction between the host matrix and the guest molecules.

Namely, one intuitively expects at least a vdW type interaction

between the two phases that otherwise do not interact via

chemical bonding. Aside from the fact that the inclusion of such

an interaction introduces significant complications in the calcu-

lations, it was also shown that the role of local interactions in the

effective transport on large scales (in particular the coupling of

local and macroscopic fluxes) is far from trivial,6 such that

intuitive ideas about molecular binding to the surface of pore

walls are by far not sufficient to rationalize experimental obser-

vations. These are probably the main reasons for the lack of such

a description in the literature. In an ideal case the role of guest–

wall interactions in transport on all length scales would be best

described with some method with molecular resolution, which is

unfortunately infeasible due to obvious numerical limitations.

Even on a continuum level there arise several technical issues

which we solved by developing a hybrid 4-step operator-splitting

discretization scheme, where we differentiated the diffusive and

drift terms of the FPE on separate grids each with the

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‘alternating-direction fully implicit method’ (for details refer to

ref. 6). Furthermore, an unambiguous evaluation of the effective

large-scale transport (intrinsic and cumulative) with respect to

matrix parameters is difficult by itself, because of the highly non-

trivial relation between local (in pores) and large-scale transport,

whereby a 1-dimensional effective diffusion process is observed

only for asymptotic times.6 With regards to the intrinsic large-

scale transport, i.e. the effective diffusion coefficient, Deff, one

must additionally assure the absence of finite size-effects, either

by achieving a separation of scales or by assuming a sufficient

symmetry of the porous matrix.40

Quite surprisingly, such an inclusion of a vdW interaction into

the classical transport equations for porous media, immediately

removes all the inconsistencies discussed above. A comparison of

experimental and simulation data (Fig. 5 and Fig. 5, inset,

dashed lines) shows complete reproduction of both important

features, that is, the significant difference in the total fraction and

the faster release from bigger pores (red curves have higher

values at all times). In continuation we explain why the inclusion

of the vdW interaction completely changes the release patterns

and their dependence on selected materials parameters.

Attractions of guest molecules to host walls cause accumula-

tion of the guest at the pore walls, which can be understood as

a multilayer adsorption process (see Fig. 6). Here it should be

stressed that the relaxation towards this transient quasi-equilib-

rium state is very fast and leads to a local distribution, which is

proportional to the Boltzmann factor, exp(�V(r)/kBT) (see ref.

6). As a consequence, the time scales of local (in pores) and large

scale transport are decoupled. To understand this fast local

quasi-equilibration first recall that the first moment of the

incremental molecular motion (the average displacement), hDx(r)i, is zero for diffusion and hDx(r)if �mVV(r)Dt in the case of

drift due to the presence of an external potential V(r)41 (recall

that low Reynolds numbers were assumed). Being a random

motion, the first non-vanishing moment of a diffusive motion is

the variance related to the diffusion coefficient via h(Dx)2i ¼DDt.

In other words, the average displacement of a molecule is influ-

enced only by the local drift whereas its mean square displace-

ment is due to both drift and diffusion. Hence, on a continuum

Fig. 6 Simulation snap-shots in the case of different strengths of

attractions to pore walls.

1118 | J. Mater. Chem., 2012, 22, 1112–1120

scale the concentration field will be displaced by the local drift

and at the same time disperse due to diffusion in a concentration

gradient. Due to the geometry of the porous network V(r) inside

each pore is almost axially symmetric. Thus, to a very good

approximation, the only non-vanishing component of the local

drift will be in the direction normal to pore walls. On the

contrary, due to the large scale concentration gradient between

the interior of the porous matrix and the surrounding bulk

solution the large scale transport will be determined only by the

magnitude of the axial flux component, which, as we show in

continuation, depends greatly on the pores size and strength of

guest–wall attraction. Consequently, the drift displacement,

which depends on the variation of the external potential across

the cross section of pores, causes a quasi-equilibration in the

direction normal to pore walls, the result of which is readily

observed in Fig. 6.

In turn, the transient concentration difference across the cross

section of pores, more specifically the concentration difference

between the center of the pores and the adlayer at pore walls,

depends on the corresponding potential difference. Naturally,

since the interactions between guest and host molecules are short

ranged, the aforementioned potential difference is larger in the

case of larger qmin. The external potential directly at pore walls is

almost independent of pore size, while the potential in the center

of pores is less favorable in larger pores (see Fig. 2(b)), which

means that the aforementioned concentration difference is larger

in larger pores.

The local dynamics depend on the local forces (the drift

component mF(r) ¼ �mVV(r)) and the local concentration

gradient. The net effect is a larger total local flux, but with

a smaller axial component (i.e. a larger component towards the

pore walls), which is readily observed in Fig. 7. Since only the

axial component contributes to large scale release, the effective

release kinetics become depleted. This is shown in Fig. 7.

One can see that the relative cross section of pores, where the

local flux has a non-zero axial component, is smaller in the case

of stronger solute–matrix attractions. Because of the pore size

Fig. 7 Calculated local flux fields in the case of: qmin ¼ 0 (left), qmin ¼0.2kBT (middle) and qmin ¼ 0.4kBT (right). For visual convenience the

magnitude of vectors was scaled up to the ratio (left):(middle):(right) ¼7 : 6 : 1.

This journal is ª The Royal Society of Chemistry 2012

Fig. 9 Release profiles on a logarithmic time scale: (a) experimental

results for SBA-15 + IMC (red symbols) and MCM-41 + IMC (blue

symbols); (b) theoretical profiles for matrices with qmin ¼ 0.4kBT and

pore sizes d¼ d0 ( ) and d¼ d0/2 ( ). Dashed lines correspond to type I

initial conditions and full lines to type I with an additional surface adlayer

(see Fig. 2(a)). For comparison the inset shows corresponding curves in

the case of no guest–host attractions (qmin ¼ 0) for pore sizes d ¼ d0 (——)

and d ¼ d0/2 ( ). Note that the theoretical results are given in numerical

time units Dt ¼ aD2/D.

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dependence of the external potential, the relative cross section of

pores where the local flux has a non-zero axial component for

given qmin is also smaller in smaller pores. It is this decrease of the

relative cross section that leads to a reverse dependence of release

kinetics shown in the experiments and in simulation (Fig. 5,

dashed line). The effects of pore size and strength of solute–host

attraction are quantified in Fig. 8 in terms of the mean release

time and effective diffusion coefficient. For sufficiently weak

guest–host attractions (qmin < 0.15 kBT) the transport is faster

from matrices with smaller pore size, which can bee seen by

comparing the results of t1/2 and Deff corresponding to different

pore sizes (i.e. on going from the black line to the red one in the

insets of Fig. 8(a) and (b) respectively). In this weak interaction

region the effect of the pore size is of purely geometrical origin,

hence we term this situation the ‘geometry controlled regime’.

Details of the geometrical release enhancement mechanism can

be found in ref. 33. In the strong attraction regime (qmin > 0.25

kBT) the release is always faster from larger pores, which can be

seen from Fig. 8 by noting that the pore size dependences of t1/2and Deff become reversed when compared to the dependence in

the weak interaction region. Obviously, in this guest–host

attraction region the effect of interactions prevails over the

geometrical effect. Conveniently, such a situation can be termed

‘the interaction controlled regime’. For intermediate guest–host

attractions (0.15 < qmin < 0.25 kBT) a small reduction of pore size

accelerates the transport kinetics (compare black, green and blue

lines in the bottom inset of Fig. 8(a)), while a further reduction

results in a slower effective transport (compare blue and red lines

in the bottom inset of Fig. 8(a)). Note that the release kinetics

(the rate of change of the fraction of released solute) are insen-

sitive to the degree of local concentration relaxation (compare

full, dashed and dash-dotted lines in Fig. 8). We have shown

earlier (see Fig. 7) that the dynamics inside the pores may be

much faster than the transport on large scales which is actually

probed and exploited in almost every application involving

porous materials. In order to study the coupling between the fast

local processes and the much slower large scale transport it is

convenient to plot the release (or, generally, the transport) curves

on a logarithmic time scale. However, in terms of measurements,

this requires a very high sampling frequency at short times which

is rarely reported in the literature. The results for the present

systems are shown in Fig. 9. We see that on various timescales

different features appear. For example, in the absence of a guest–

host interaction (inset in Fig. 9(b)), the curves corresponding to

Fig. 8 (a) Mean release times, t1/2, and (b) effective diffusion coefficient

Deff for various pore sizes as a function of the strength of attraction to

host walls: d0(——), 4d0/5 ( ), d0/2 ( ) and 2d0/5 ( ) as a function of

qmin. Full lines correspond to type I IC, dash-dotted lines to type II and

dashed to type III IC.

This journal is ª The Royal Society of Chemistry 2012

different pore sizes first diverge at short times and afterwards

converge at long times; however, they do not become parallel at

any time interval. On the other hand, the experimental profiles

(Fig. 9(a)) for different pore sizes are parallel (but not linear) at

long times. This parallelity is completely reproduced in the

theoretical curves that take account of the guest–host interaction

(Fig. 9(b), red and blue curves). Note that the shape of the

profiles is not changed significantly, if initially an external

adlayer is deposited on the surface of porous matrix (compare

solid and dashed curves in Fig. 9).

4 Conclusions

Wemay conclude based on both the experimental and theoretical

evidence given above that the release profiles in the presence of

guest–host interactions are much different than expected in the

absence of such interactions. Even more, if interactions are not

taken into account, completely wrong trends might be predicted,

for example, enhancement/depletion of transport from different

pore sizes may be reversed in the case of significant attractions,

the efficiency of guest release may be significantly decreased due

to interactions etc.

Our results have shown that intuitive speculations about the

effect of local pore–molecule interactions on the large scale

transport are utterly insufficient to rationalize the features of the

system dynamics on several length scales. We established, for the

first time, the qualitative and the quantitative effects of such

interactions on the coupling of the microscopic (inside the pores)

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and macroscopic (experimental scale) transport in the presence

of a macroscopic concentration gradient.

It is worth stressing that the present theoretical treatment is

completely general and does not only apply for release systems

(such as drug release, dissolution from matrices etc.) but for any

transport involving porous systems (down to the lower limit of

mesoporosity), involving various degrees of interactions between

host and guest substances. As regards the drug release, our

results clearly suggest that maximization of the specific surface

area of host (minimization of pore size) will lead not only to

slower release kinetics but also to a decrease of drug utilization,

that is, a significant amount of drug will remain captured inside

the pores due to an attractive vdW interaction.

Thus, in order to efficiently maximize the transport in devices

exploiting meso- or macroporous materials, it is of primary

significance that the interactions between the host and guest be

appropriately controlled (e.g. by selecting appropriate compo-

sitions of both components, by surface modification of host or by

appropriately optimizing the size of pores etc.).

Acknowledgements

The authors acknowledge support from the National programs

P2-0148 and P1-0021.

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