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1. Introduction

Structure formation as a result of phase separation in aqueous formulations of polymers is an important aspect of the func-tionality of these polymers. Examples are water-based paint, encapsulation of food ingredients and pharmaceuticals, and the texture and feel in the mouth of food. Although struc-ture formation plays a role in the industrial processing of most polymer mixtures, in recent years water-based systems have received special attention because of their (generally) environmentally friendly, biocompatible or even food-grade qualities. With respect to food, structured water-based sys-tems may offer the additional advantage of replacing fat and oil by aqueous solutions of protein and polysaccharides. Control over the thermodynamics of phase separating aque-ous polymer systems, leading for example to stable water-in-water emulsions, deserves therefore thorough theoretical and experimental scientific study. The work presented here is intended to be part of such a study, as well as a means of highlighting some possibly intriguing scientific issues.

In general, mixtures of chemically distinct polymers are thermodynamically unstable. When compared to the same mix-ture of unconnected monomers, the entropy of a polymer mix-ture is—in the case of flexible, random coil polymers—lower

by a factor roughly equal to the average degree of polymeri-zation, whereas the (positive) mixing enthalpy is the same. Therefore, a mixture will lower its free energy of mixing by forming homogeneous phase regions. In the case of two poly-mers differing in stiffness or shape (with the extremes of stiff rods and dense spheres), an unfavorable entropy of mixing may add to the driving force for phase separation.

Solutions of two chemically distinct polymers will be unsta-ble for the same reasons as polymer blends, but the instability will be weakened because the positive enthalpy of mixing is reduced by the presence of solvent [1, 2]. In practice, above vol-ume fractions of 5–10%, polymer solutions will phase separate. Between the phase regions an interface is formed, which sepa-rates two polymer solutions, each containing up to 95% solvent. These interfaces will be called solvent–solvent interfaces, or, in the case of water as a solvent, water–water interfaces.

Water–water interfaces are formed in phase-separated solu-tions of proteins and polysaccharides [3], but also in mixed solutions of synthetic polymers, such as poly(ethylene oxide), poly(vinylpyrrolidone), (PVP) or poly(acrylamide) (PAM) and sodium poly(acrylate) (NaPAA) [4]. Solvent–solvent interfaces have properties which make them fundamentally different from interfaces in binary melts in two ways. At first, the presence of an excess of solvent at the interface, due to

Journal of Physics: Condensed Matter

Composition, concentration and charge profiles of water–water interfaces

R Hans Tromp1,2, M Vis2, B H Erné2 and E M Blokhuis3

1 Department of Flavour and Texture, NIZO food research, Kernhemseweg 2, 6718 ZB Ede, The Netherlands2 University of Utrecht, Van ’t Hoff laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials Science, Padualaan 8, 3584 CH Utrecht, The Netherlands3 Colloid and Interface Science, Leiden Institute of Chemistry, Gorlaeus Laboratories, PO Box 9502, 2300 RA Leiden, The Netherlands

E-mail: Hans.Tromp@NIZO.com

Received 10 February 2014, revised 21 March 2014Accepted for publication 24 March 2014Published 27 October 2014

AbstractThe properties of interfaces are discussed between coexisting phases in phase separated aqueous solutions of polymers. Such interfaces are found in food, where protein-rich and polysaccharide-rich phases coexist. Three aspects of such interfaces are highlighted: the interfacial profiles in terms of polymer composition and polymer concentration, the curvature dependence of the interfacial tension, and the interfacial potential, arising when one of the separated polymers is charged. In all three cases a theoretical approach and methods for experimental verification are presented.

Keywords: interface, aqueous, biopolymer, food

(Some figures may appear in colour only in the online journal)

Printed in the UK

464101

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© 2014 IOP Publishing Ltd

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10.1088/0953-8984/26/46/464101

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osmotic compressibility, can relax to some extent the energy involved in the interpenetrating of incompatible solutions at the interface. Secondly, the continuity of the aqueous back-ground imparts permeability for solvent and small solutes. Thirdly, specifically for water–water interfaces is the possibil-ity of an electric potential across the interface in the case of charged polymers.

In this paper three topics of water–water interfaces will be discussed:

composition. These profiles differ because water is able to reduce the composition gradient by accumulating at the interface.

case of phase separated polymers of different sizes and Kuhn lengths. A preferential curvature, and to a lesser extent the stiffness affects the coarsening kinetics of a separating system. A preferential curvature is not specific for interfaces between polymer solution.

(Donnan potential) across the interface.

The theory of each topic will be presented first. After that for each topic the experimental consequences and available experimental data will be discussed and an approach for experimental validation will be presented.

2. Theory

2.1. The interfacial profile and tension in the presence of solvent

Profiles of interfaces between polymer melts and solutions were first calculated by Helfand et al [7] in a mean self-consis-tent field approach. Here, we follow the approach of Broseta et al [6], who go one step further by using the so-called ‘blob’ model [1], which takes excluded volume interaction between monomers into account. Excluded volume interactions are important because phase separation takes place at concentra-tions that are too low for these interactions to be screened. The description of the water–water interface between two coexist-ing phases of aqueous polymer solutions begins [5] with the Flory-Huggins free energy density of mixing of the melts of polymers A and B of an equal degree of polymerization N

ϕ ϕ ϕ ϕ χ ϕ ϕ= + − − + −⎡⎣⎢

⎤⎦⎥

Fk TV a N N

1log

1log(1 ) (1 )FH

B3 FH (1)

(ϕ is the volume fraction of, say, polymer A, χFH Flory-Huggins interaction parameter, V the volume and kB Boltzmann’s con-stant and T the temperature). Phase separation in polymer solutions takes place above the concentration of overlapping polymer coils, approximated by (in monomers per unit volume)

π* ≈cN

R3

4 g3 (2)

which is typically less than 1% (Rg is the radius of gyration of the coil in dilute (non-overlapping) conditions. Therefore, the coexisting polymer solutions are semi-dilute, and the blob

model can be used [1]. In the blob model the polymers, dis-solved in what is assumed to be a good solvent, are divided into ‘blobs’, which are stretches of monomers along the chain which interact by excluded volume. Beyond a certain distance ξ, the entanglement distance or blob size, monomers do not interact. ξ can therefore be considered a correlation length and be expected to set the interfacial width between coexisting or meta-stable phase regions. Because monomer interactions are absent beyond ξ, a chain of blobs is an ideal Gaussian chain, and the collection of chains of blobs is analogous to a melt. The expression for the free energy density of mixing is therefore a rescaled version of equation (1) with the monomer length replaced by the blob size ξ and the degree of polymer-ization by the number of blobs per chain Nb [6]

ξϕ ϕ ϕ ϕ ϕ ϕ= + − − + − +

⎡⎣⎢

⎤⎦⎥

FVk T N N

u K1

log1

log(1 ) (1 ) .blob

B3

b b (3)

Because the distance between entanglements is dependent on the polymer concentration, ξ, Nb and u are concentration dependent:

⎜ ⎟⎛⎝

⎞⎠ξ = *

νν−

c Rc

c( ) 0.43 g

1 3 (4)

ξ=N cN

c( )b 3 (5)

and

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟= =

χν

χν− −

u c uc

c Nc

c( )

2crit

crit

3 1

b,crit crit

3 1 (6)

where Nb,crit is the number of blobs per chain at the critical concentration of mixing ccrit and χ ≅ 0.22 [8]. For a good solvent

≅ νR bNg (7)

in which b is the Kuhn length and ν = 3/5. K accounts for the free energy of mixing of the monomers inside a blob with solvent. Because the solvent is good, K is set by the properties of a self-avoiding walk and therefore a constant in the con-centration [9]. The specific effects of polymer-solvent interac-tions are in the Rg, which, together with c* and the monomer concentration c, determines the blob size. Equation (6) pro-vides experimental access to the theory through N, Rg and ccrit, which can be measured.

In order to describe the profile of the interface the expres-sion of the free energy density is extended by two energy gradient terms, one accounting for the composition gradient, and the other for the concentration gradient (for equal values of the degree of polymerization N for the two polymers the polymer-solvent interaction energy has no gradient)

∫ϕ ϕξ ϕ ϕ

ξ ϕ

ξ ϕ ϕ ξϕ

ξϕ ϕ ξ

= + − −

+ − + + ∇− + ∇

⎡⎣⎢⎢

⎤⎦⎥⎥

F ck T

rN N

u K cc

[ , ] d log1

log(1 )

(1 )24 (1 ) 24

VB b

3b

3

3 3

2 2

2

(8)

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where r is the position variable. The interfacial tension is the excess grand potential per unit area

∫ ϕξ ϕ ϕ

ξ ϕ

ξ ϕ ϕ ξη

ξ η

εξ ε μ η μ ε

= + − −

+ − + + −

+ − − − +η ε

⎡

⎣⎢⎢

⎤

⎦⎥⎥⎥

Ak Tz

N N

u K

uu

p

Ωd log

1log(1 )

(1 )˙

24 (1 )

˙24 (1 )

V

ex

B b3

b3

3 3

2

2

2 2

2

(9)

due to the gradients at the interface, with the composition variable

η ϕ= −2 1 (10)

and the concentration variable

ε ≡ −z

c z cuc

( )( )

. (11)

The distance to the interface is z. Symbols with a bar are bulk values, far from the interface. μη is the coexistence value of the exchange chemical potential of the polymers, which is zero for equal N and equal random chain statistics for both polymers, με is the coexistence value of the exchange chemi-cal potential of the solvent, and p is the pressure far from the interface, where gradients are zero.

In order to calculate the equilibrium interfacial tension, the compositional and concentration profiles have to be found for which Ω is minimal, using the Euler–Lagrange equation. For infinite N, and in the absence of solvent, the Euler–Lagrange equation becomes

ξ η η η ηηη= − + −u

z6

¨( ) (1 )˙

1.

22

2

2 (12)

When Nb = ∞, boundary conditions are η(±∞) = ±1. The solu-tion therefore is

ηξ

=z zu

( ) tanh6

.2 (13)

The width of this interface

ηη

ξ˙ = ≡

=∞

uD

6z 0

2

(14)

will be used as a length unit in the following. Because the two incompatible polymers try to minimize mixing, it is expected that at the interface the solvent will accumulate to some extent, weakening the interaction between the polymers [6, 7]. If it is assumed that this accumulation is small, or in other words the deviation from the bulk conditions is small and ε<<u 1 (see equation (11)), the compositional profile is not affected by the presence of solvent and can be calculated by minimizing equation (9) for ε = 0.

Using equations (4)–(6), expressing the rescaled interac-tion parameter by ω = uNb and expanding the concentration-dependent quantities to second order in εu expressions can be

obtained for the composition and concentration profiles (for a detailed treatment see [6] and [10])

ηη

ηω η η

ω η η′− = + + + − − − + p

4(1 )

12

log(1 )12

log(1 )4

2

2

2

(15)

ε ηω η η

ω η

νν ε ν χ

ν η μ

′′ = + + + − −

+ − − +− − ε

u

K2

12

log(1 )12

log(1 )

3(3 1)

34(3 1)2

2 (16)

with boundary conditions η (±∞) = ±1 and ε (±∞) = 0 and

μ ηω η η

ω η ν χν η= + + + − − − +

−ε12

log(1 )12

log(1 )3

4(3 1)2 (17)

ηω η η

ω η η= − + + − − − +p12

log(1 )12

log(1 )4

.2

(18)

2.2. The curved interface

When the polymer solutions separated by the interface have different degrees of polymerization, or different blob sizes, the phase diagram (figure 1) is non-symmetric and therefore it is expected that the interface profile is also non-symmetric [11, 12]. This means that the gradient energy is not equally distributed, which causes a curved state to have a lower energy than a flat state. The extent to which this leads to a curved interface depends not only on the asymmetry of the profile, but also on the interfacial stiffness. Assuming a curvature radius much larger than the interfacial width, the interfacial tension can be expressed by

σ σ δσ= − + + +RR

k kR

( )2 2

...00 G

2 (19)

σ0 is the interfacial tension of a flat interface, δ is the Tolman length [13], which is a measure for the spontaneous curvature

Figure 1. Phase diagram of polymer (blob) blends with different ratios α of the degree of polymerization. ϕA is the volume fraction of the polymer with the highest N.

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and differs from zero only for asymmetric interfacial profiles, and k and kG are the rigidities of bending and Gaussian curva-ture, respectively.

The starting point of a derivation, given in detail in [10] of expressions for δ and the bending rigidities, is equation (8), now with different numbers of blobs per chain and different blob sizes for the two polymers. It is assumed that the pres-ence of a solvent profile will not change the essentials of the expressions for curvature and stiffness. Therefore, the gradi-ent in the solvent concentration will be ignored. This means that, except for the polymer composition, expressed by ϕ r( ), all space-dependent quantities can be expressed by their bulk values, denoted by a bar. We assume a meta-stable droplet of minority phase. The free energy of mixing of the solution in this droplet is

∫ϕ ϕξ

ϕ ϕξ

ϕ

ξϕ ϕ

ξϕξ ϕ

ϕξ ϕ

= + − −

+ − + + ∇ + ∇−

⎡⎣⎢⎢ ⎤

⎦⎥⎥

F ck T

rN N

u K

[ , ] d log1

log(1 )

(1 )24 24 (1 )

VB b,A A

3b,B B

3

eff3

eff3

2

A

2

B

(20)

in which subscripts A and B refer to the two phase separated polymers. From now on, the bars will be omitted. We assume, as is commonly done in polymer melts that the interaction between monomers and monomers and solvent is not depen-dent on the degree of polymerization and therefore we have defined an effective correlation length as

⎛⎝⎜

⎞⎠⎟ξ ξ ξ= +1 1

21 1

.eff A B

(21)

We also define, in terms of the bulk blob sizes, two asymmetry parameters α and r0

α ξξ= =N

NNN

b,A A3

b,B B3

A

B (22)

and

ξξ=r0

A

B (23)

where α represents the difference in degree of polymeriza-tion of the two polymers, while r0 reflects the difference in size of the statistical monomers; different due to, for exam-ple, a difference in solvent quality or chain stiffness. The excess interfacial grand potential is for the radial symmetry of the droplet

∫ ϕ μϕ ϕ ϕ= − + ∇⎡⎣ ⎤⎦r f mΩ d ( ) ( )V

ex2

(24)

with

ϕ ξϕα ω ϕ

α ϕα ω ϕ ϕ

= +

+ −+ − −

⎡⎣⎢

⎤⎦⎥

fuk T

( )2

(1 )log

2 (1 )(1 )

log(1 )

B3

2 (25)

and

ϕ ϕξ ϕ ϕ= + −

+ −mk T r

r( )

[1 ( 1) ]12 (1 ) (1 )

.B 0

0 (26)

ω is now defined by

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ω ξ ξ ξ

αα= + = +

+

χν− − −

−uN N

c

c2 1 1 (1 )

(1 )eff3

b,A A3

b,B B3

1 2p

crit

11 3

(27)

and μ is the exchange chemical potential of the meta-stable curved state of the interface of a droplet of one phase in an infinite volume of the other. This corresponds to a pressure difference, for which the phase diagram can be calculated. It should be noted that the blob size ratio r0 occurs only in the gradient term. Therefore, the phase diagram, shown in figure  1, is asymmetric only because α ≠ 1, not because r0 ≠ 1. This is the consequence of the introduction of an effec-tive blob size governing the interaction between the blobs. As mentioned above, this assumption is justified by the indepen-dence of the interaction on N.

After finding the composition profile ϕ(r) which minimizes Ωex, the radius R of the metastable droplet—corresponding to the chosen value of the non-coexistence value of μ—can be found from

∫π ϕ ϕ π ϕ ϕ− = −∞

[ ] ( )r r r R4 ( ) d43v l v

0

2 3 (28)

in which ϕl and ϕv are the compositions of the droplet and the bulk phase (we assumed the droplet to be denser than the bulk). The full expressions for k, kG and δ can be found in [10].

2.3. The effect of charge

In general, a difference in charge of incompatible aqueous polymer solutions suppresses the drive to phase separate, because phase separation confines counter-ions to only a part of the volume [14]. The corresponding difference in osmotic pressure has to be balanced by a larger polymer incompat-ibility for separation to occur. However, when the difference in charge of the polymers is only small, phase separation may still occur and an electrical potential across the interface is expected, i.e. a Donnan potential [15, 16].

We assume an interface between phases, one rich in a neu-tral polymer and the other rich in a weakly positively charged polymer (concentration c). Small ions of omnipresent low molar weight salt (concentration cs, assumed to be monova-lent), and the negative counter ions will maximize their entropy by partially diffusing across the interface (which is continuous in water). This causes a departure from charge neutrality at the interface, balanced by the increase in small ion entropy. The electrical potential difference between points, one in each phase, far from the interface is then

ψ = − =α

β

α

β

+

+

−

−N k T

Fcc

N k TF

cc

log logDav B av B

(29)

where c±i are concentrations of small ions, and the index α

refers to the phase containing the charged polymer, and β to

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the phase containing the neutral polymer. F is Faraday’s con-stant and Nav is Avogadro’s number. Far from the interface charge neutrality is maintained:

+ =α α+ −c zc c (30)

and

= =β β+ −c c cs (31)

in which z is the number of positive charges on a polymer chain. Therefore, the interfacial or Donnan potential can be expressed by

ψ = ≈N k TF

zcc

N k TF

zcc

arcsinh2 2

.Dav B

s

av B

s (32)

The sign of this potential is determined by the sign of the charge of the polyelectrolyte.

3. Results and discussion

3.1. The interface profile

The experimentally accessible quantities N, Rg and ccrit can be introduced in the expression for the interface profile equa-tion (16). The values chosen were 1000 for N, 18 nm for Rg and 0.5, 1.5 and 3.5% (w/w) for ccrit. These values are rep-resentative for gelatin and dextran or pullulan, mixtures of which are extensively studied in the context of phase sepa-ration [17]. ccrit is the measure for incompatibility. Together these parameters determine the blob size ξ, the number of blobs per chain Nb and the interaction parameter u. The result is shown in figure 2A. The polymer depletion at equal distances from the critical concentration (c/ccrit = 2.85, i.e. ω = 10) is strongly dependent on the incompatibility. For the highest degree of incompatibility, ccrit = 0.5%, the depletion is nearly 30%. In this case, ε ≈u 0.2, so at a stronger incompat-ibility the theory becomes invalid. The depletion profile corre-sponds to a profile of the blob size ξ which has a maximum at the interface as is shown in figure 2B. Because the distance in concentration to ccrit is the same for the three cases, the overall concentration for a low ccrit is lower, and therefore the overall blob size ξ larger. Consistent with the assumption made at the

outset that ξ is the relevant length scale, the width of the pro-file is found to be of the order of 4ξ, i.e. 8–20 nm.Measurements of the interfacial profile are not yet available. Ellipsometry [19] may be able to give information, although the refractive index difference at the interface in practical sys-tems is usually less than 0.001, and therefore poses a serious challenge. Neutron reflectivity may be another possibility [18], in particular in combination with index matching one of the polymers with the solvent, using partially deuterated polymers. The excess of the other polymer will in that case dominate the reflectivity. A difficulty will be the fact that the air-solution interface, which has to be passed by the neutron beam, has a much stronger reflectivity. By matching the sol-vent of the top phase with air this problem may be avoided. Still, there may be a polymer depletion or absorption layer at the air-solution interface which has a similar, or larger, effect on the reflectivity as the interface between the phases.

An indirect consequence of the polymer depletion at the interface between the phases may be the redistribution of water during coarsening. This effect can be estimated by com-paring the total excess of water at the interface per unit area,

∫ ε= − ∞−∞

∞

C c c cuD x x( , ) ( ) dex crit (33)

with the total amount of water present in the macroscopic phase separated system. Some examples of Cex are plotted in figure 3. The typical values of this excess are between 2·10−5 and 10·10−5 g m−2, and rather independent on concentration at concentrations higher than two times the critical concen-tration. A monolayer of pure water would be of the order of 1·10−4 g m−2 , so at the highest incompatibility of practical rel-evance the interfacial layer is, in terms of mass, comparable to a monolayer of water.

The amount of water accumulated in the interface is dependent on the interface area per unit volume of macro-scopic system. For a volume fraction ϕd of phase droplets of size R, immersed in a continuous phase, the total mass of excess water at the interface is

ϕ=MR

C3

.ex ex (34)

Figure 2. Interface profiles, calculated by solving equation (16) for different values of ccrit, i.e. degree of incompatibility for polymers of equal degree of polymerization N = 1000 and c/ccrit = 2.85 (ω = 10). A: the polymer concentration profile in units of the bulk concentration; B: the blob size profile. Rg = 18 nm, monomer molar mass 120 g.

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Mex is in the order of 100–240 g m−3 for R = 1 μm and ϕ = 0.5. Therefore, at most 0.03% of all water present is accumulated at the interface, which will be difficult to detect in a direct way by any technique.

The accumulation of water at the interface will reduce the interfacial tension in excess of what is expected in the case of a homogeneous distribution of water. The interfacial tension can be measured accurately by interpreting the shape of the menis-cus curving up or down against a vertical glass wall [20]. In the case of gelatin/polysaccharide interfaces, the gelatin phase fully wets the wall, so the contact angle is zero. An example is shown in figure 4. The fitting results in a capillary length

= σρΔL

gcap2

(35)

in which g is the acceleration due to gravity and ∆ρ the dif-ference in density between the phases. The latter can be mea-sured by standard methods. Another method to measure the interfacial tension is by the spinning drop technique [21, 22], or by deformation of the interface by radiation pressure from an intense laser beam [23].

Although the interfacial tension between coexisting polymer solution interfaces has been reported several times [22–27] the specific effect of the excess of solvent was never studied. This could be done by taking monodisperse phase-separating polymers, of which the coexisting phase composi-tion and ccrit is accurately known.An expression for the interfacial tension is obtained from mini-mizing Ω with respect to the solvent and polymer profiles [6, 10]. The result contains three terms:

σξ

σ= −Δ − Δ = −Δ − Δ∞⎜ ⎟⎛⎝

⎞⎠

k T uu u

6(1 ) (1 )B

2

1/2

1 2 1 2 (36)

with

∫ η ηηΔ = − ′

−

ηx

1 d( )

11

02 (37)

and

∫ ηη

χν η η εΔ = − − ′

− + +− −

−∞

∞ ⎡⎣⎢

⎤⎦⎥x

xx

18

d( )

(1 )

1(3 1)

( ) ( ).2

2

22 2

(38)

A measure for the effect of excess of water at the interface is

Ξ = −Δ − Δ−Δ

u11

.1 2

1 (39)

The first term in equation (36) is σ∞, the interfacial tension for infinite N and no solvent. σ∞ can be calculated from c, ccrit and Rg. The second term accounts for the decrease in σ when N is finite, in absence of a solvent gradient. The third term accounts for the weakening of the gradient energy due to the excess of solvent at the interface. Figure 5 shows calculated examples of the relative effect of excess water at the interface. It turns out that the effect is most significant at high incompat-ibility close to ccrit. The interfacial tension measured in the concentration range should be 10–15% lower than expected from a calculation which takes into account only the compo-sitional gradient. It should be noted that the polymer material used must be monodisperse, because low molar mass material will weaken the concentration gradient and obscure the effect of excess water.

Figure 3. Excess water at the interface as a function of polymer concentration, for different incompatibilities. For parameter values see caption to figure 2.

Figure 4. Meniscus between phases of phase separated 3.6% fish gelatin/2.4% dextran water–water interface. The density difference is 0.25 kg m−3 and Lcap, indicated by the arrow, is 80 μm. The interfacial tension is therefore 0.008 μN m−1. The two polymers have roughly the same degrees of polymerization.

Figure 5. Calculated effect of excess of water at the interface for ccrit = 0.5%.

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3.2. The interfacial rigidity

The theoretical predictions for the interfacial rigidity and asymmetry are shown in figure 6. It turns out that the factor determining the rigidity, 2k + kG, is negative, indicating that the interface decreases its energy by being curved. The effect becomes weaker further away from the critical point, because the interfacial width decreases. There turns out to be a mini-mum between 1.5 and 2 c/ccrit where the decrease in interfa-cial tension closer to the critical point cancels the thinning of the interface with increasing concentration.

Asymmetry in blob size, i.e. r0 ≠ 1, has an insignificant effect on the rigidity (result not shown) but it has a signifi-cant effect on the Tolman length δ, shown in figure 7, shifting it by about 0.5 nm, quite independent in concentration or α. The asymmetries in blob size and degree of polymerization

N appear to have opposite effects. Increasing N on one side of the interface causes it to curve to the other side, whereas increasing the blob size on one side causes it to curve towards that same side. The reason for this may be the fact that the blob size sets the interfacial width and therefore the gradient energy (see equations (24)–(26)). This energy is higher on the side of the smaller blobs, and is increased more by a curvature towards the side of the smaller blobs.

The asymmetry in N affects the composition of the drop-let. A droplet rich in high N polymer is further from coexist-ence than a droplet of the same size rich in low N, and has therefore a high internal pressure, corresponding to a higher interfacial tension.In the procedure to calculate the curvature dependence of the interfacial tension using the model of a meta-stable droplet of minority phase, it was assumed that the droplet consisted of the phase rich in the higher degree of polymerization. In that case, it turned out that the Tolman length is negative: or, in other words, the interfacial tension increases, relative to the flat state, when the interface gets shaped into a droplet. A droplet of the phase rich in the smaller polymer immersed in the phase with the larger polymer would have a lower inter-facial tension compared to the flat state of its interface. As a consequence, droplets in the meta-stable phase-separating mixture have a higher (meta)stability when they consist of the smaller polymer, and vice versa. To a lesser extent, in the case of equal values of N but a difference in blob size, droplets con-taining the smaller blob are less stable, and vice versa.

This asymmetry in the stability of droplets corresponds to a difference in interfacial energy between a certain volume V = (4π/3)Nd of phase A or B distributed over a number of droplets Nd. The excess interfacial energy relative to the flat state is

Figure 6. Interfacial rigidity 2k + kG of a flat interface as a function of concentration, for different values of the ratio of degrees of polymerization of the separated polymers.

Figure 7. The Tolman length, a measure for the interfacial profile asymmetry, as a function of polymer concentration, for two values of the ratio of degrees of polymerization, and three values of the blob size ratio. Ratios of 1 correspond to symmetric cases.

Figure 8. Excess interfacial energy of a collection of monodisperse phase droplets relative to a flat interface of the same size, as a function of droplet curvature, with asymmetric (α = 8) and symmetric interfaces (α = 1). The total volume of the droplets is fixed. ucrit = 0.0061, ξcrit = 2.6 nm, c/ccrit = 1.8. The sizes at the maxima are Rpref. Dashed line: droplets rich in large polymers in a continuous phase rich in small polymers; Dash-dotted line: vice versa.

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π σ δσ= − + +⎡⎣⎢

⎤⎦⎥F R N R N

Rk k

R( , ) 4

2 2.d

2d 0

0 G2 (40)

For α = 1 and α = 8 F is plotted in figure 8. When minimiz-ing with respect to R at constant volume a preferred radius is obtained

δ δ σ= + − +( )R k k2 4 3 (2 ) .pref2

0 G (41)

Rpref is in the order of 20–40 nm. It is smaller for droplets of high N phase in low N phase than for the opposite case, because in the former case the interface energy per unit area of a droplet state is higher. Rpref is the size of smallest droplet that is meta-stable and will lower its free energy by growing. In may, therefore, be expected that after equal coarsening times and at equal distance from the critical point droplets in low N phase will be larger than droplets in high N, after correcting for the difference in diffusion constants.

Experimentally, this effect may be accessible by studying the coarsening rate and size distribution of droplets, formed after nucleation or by shaking or stirring. Assuming the sim-plest case of isolated droplets of one phase, growing by dif-fusion in a supersaturated, infinitely large majority phase, the increase in size is expressed by

π σR

Rt

DRV R

kTR R4

dd

~2 ( ) 1

.2 2 p (42)

For a diffusion constant D of 10−12 m2s−1 and a molar volume of 0.075 m3 the development during an hour of a log-normal distribution of droplets centred at 1.2 Rpref is shown in figure 9. In the experimental case the initial size will be slightly larger than Rpref, so the positions of the distributions will differ accordingly. Growing according to equation (42) the effect of a size dependent σ can be better seen when an identical initial size and size distribution are chosen (figure 9B). It is clear that there is a small but significant effect.

3.3. The effect of charge

In the case of phase separation between a neutral polymer and a weakly charged polyelectrolyte, the interface is pre-dicted to not only contain composition and concentration profiles, but also a charge density profile, which gives rise to a Donnan or membrane potential. Our recent experiments have shown that such a potential is indeed measurable. The method was a measurement of the interface potential dif-ference using two Ag/AgCl reference electrodes on both sides of the interface [28–30]. The details of the method will be published in a separate paper. Some of the results are in table 1.

The system was a phase-separated aqueous solution of dextran and cold-water fish gelatin (non-gelling at room tem-perature) with approximately equal phase volumes. The con-centrations were chosen in such a way that the differences in

Table 1. Preliminary results of the interface potential (Donnan potential) across an equilibrium water–water interface between phase sepa-rated fish gelatin and dextran solutions.

pH

Concentration (%w/w)Salt concentration cs (mM)

Donnan emf (mV)

Charge per chain zdextran gelatin ∆ gelatin

4.8 6.5 6.5 5.8 ± 0.6 9.4 +3.8 ± 0.4 +4.86.2 4.0 4.0 5.65 ± 0.20 4.5 +0.55 ± 0.04 +0.349.2 4.3 4.3 5.80 ± 0.09 6.9 −2.12 ± 0.17 −2.0

Figure 9. Development after 1 h due to diffusive growth of the number weighted size distribution P(R) of droplets of phase rich in polymer larger than the polymer in the continuous phase (δ = −2.7 nm, α = 8), droplets of phase rich in polymer smaller than the polymer in the continuous phase (δ = 2.7 nm, α = 8) and droplets in the case of equal polymer sizes (δ = 0, α = 1). A: starting from a distribution centered on 1.2Rpref; B: starting from a distribution centered on the same value for all three cases. c/ccrit = 1.8, D = 10−12 m2 s−1, T = 298 K, Vp = 0.075 m3.

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gelatin concentration between the phases are about the same. Therefore, the Donnan potentials do not have to be corrected for gelatin concentration and are directly comparable. The fish gelatin was acid-extracted and is therefore expected to have an isoelectric point (IEP) near pH 8. The absolute value of the Donnan potential was found to decrease at pH values closer to the IEP, and the sign changed when crossing the IEP. From the Donnan potential and the degree of polymerization (about 1000) the valency z of the gelatin chains can be calcu-lated using equation (32). The values obtained are in the range expected on the basis of the number of charged amino acid residues in the chain. It can therefore be concluded that an interfacial potential exists and behaves according to a Donnan potential. This interfacial potential differs from the well-established membrane potential in the fact that this potential arises spontaneously across an interface without quenched degrees of freedom or actively maintained gradients, such as in a living cell.

The implications of this potential for the interfacial proper-ties which are relevant for phase separation kinetics and meta-stable drop size distributions are unknown. To our knowledge no theoretical predictions are available for the effect of inter-facial charge on interfacial tension.

An experimental consequence which may be of practi-cal importance is shown in figure 10. A confocal microscope image [31] is shown of a fully phase separated aqueous gela-tin/dextran system, which was broken up into droplets by stirring. A part of the dextran was labeled with fluorescent labels (fluorescein, labeling density about 1 per 1000 mono-mers). The labels are negatively charged, and therefore the labeled dextran was a weakly charged polyelectrolyte. The labeled dextran turned out to accumulate in the interface of the droplets. A probable explanation is an association between the negatively charged dextran and the outside of the gelatin-rich droplets, which are positively charged due to the Donnan effect.

4. Conclusions

Interfaces between phase-separated aqueous polymer solu-tions are, on the basis of theory, expected to be characterized not only by a composition profile, but also by a concentra-tion profile, a preferential curvature and an interfacial electric potential. Experimental evidence of the interfacial charge has been established. Experimental verification of a (negative) contribution to the interfacial tension from accumulation of water at the interface might be achieved by measuring the interfacial tension as a function of the water concentration at constant critical point of mixing. Proof of a preferential cur-vature, expected in the case of different degrees of polymer-ization or chain statistical lengths of the two phase separated polymers is the most difficult to obtain. This might be done by careful comparison of coarsening rates after nucleation of high and low molar mass phase droplets.

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