Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 26 (2014) 464101 (10pp)
doi:10.1088/0953-8984/26/46/464101
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 Netherlands
University of Utrecht, Van ’t Hoff laboratory for Physical and Colloid Chemistry, Debye Institute for
Nanomaterials Science, Padualaan 8, 3584 CH Utrecht, The Netherlands
3
Colloid and Interface Science, Leiden Institute of Chemistry, Gorlaeus Laboratories, PO Box 9502,
2300 RA Leiden, The Netherlands
2
E-mail: [email protected]
Received 10 February 2014, revised 21 March 2014
Accepted for publication 24 March 2014
Published 27 October 2014
Abstract
The 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 polysacchariderich 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)
1. Introduction
by a factor roughly equal to the average degree of polymerization, 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 polymers 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 unstable 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 volume fractions of 5–10%, polymer solutions will phase separate.
Between the phase regions an interface is formed, which separates 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 solutions 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
Structure formation as a result of phase separation in aqueous
formulations of polymers is an important aspect of the functionality 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 structure 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 systems may offer the additional advantage of replacing fat and
oil by aqueous solutions of protein and polysaccharides.
Control over the thermodynamics of phase separating aqueous polymer systems, leading for example to stable waterin-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 mixture of unconnected monomers, the entropy of a polymer mixture is—in the case of flexible, random coil polymers—lower
0953-8984/14/464101+10$33.00
1
© 2014 IOP Publishing Ltd
Printed in the UK
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
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 background imparts permeability for solvent and small solutes.
Thirdly, specifically for water–water interfaces is the possibility 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:
model can be used [1]. In the blob model the polymers, dissolved 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
r5IFQSPàMFPGUIFJOUFSGBDFJOUFSNTPGDPODFOUSBUJPOBOE
melt. The expression for the free energy density of mixing is
composition. These profiles differ because water is able
therefore a rescaled version of equation (1) with the monomer
to reduce the composition gradient by accumulating at
length replaced by the blob size ξ and the degree of polymerthe interface.
ization by the number of blobs per chain Nb [6]
r5IFQSFGFSFOUJBMDVSWBUVSFBOEJOUFSGBDFTUJGGOFTTGPSUIF
⎤
case of phase separated polymers of different sizes and Fblob
1−ϕ
1⎡ ϕ
(3)
Kuhn lengths. A preferential curvature, and to a lesser VkBT = ξ3 ⎢⎣ Nb log ϕ + Nb log ( 1 − ϕ ) + uϕ ( 1 − ϕ ) + K ⎥⎦ .
extent the stiffness affects the coarsening kinetics of a
separating system. A preferential curvature is not specific Because the distance between entanglements is dependent
on the polymer concentration, ξ, Nb and u are concentration
for interfaces between polymer solution.
r5IF QSFTFODF PG BO FMFDUSPTUBUJD QPUFOUJBM EJGGFSFODF dependent:
ν
(Donnan potential) across the interface.
⎛ c ⎞ 1 − 3ν
⎟
ξ ( c ) = 0.43Rg⎜
(4)
⎝ c* ⎠
The theory of each topic will be presented first. After that
for each topic the experimental consequences and available
N
experimental data will be discussed and an approach for
Nb ( c ) = 3
(5)
ξc
experimental validation will be presented.
and
2. Theory
⎛ c ⎞ 3ν − 1
2 ⎛ c ⎞ 3ν − 1
=
u ( c ) = ucrit ⎜
⎟
⎜
⎟
⎝ ccrit ⎠
Nb,crit ⎝ ccrit ⎠
χ
2.1. The interfacial profile and tension in the presence
of solvent
Rg ≅ bN ν
(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 concentration [9]. The specific effects of polymer-solvent interactions are in the Rg, which, together with c* and the monomer
concentration c, determines the blob size. Equation (6) provides experimental access to the theory through N, Rg and ccrit,
which can be measured.
In order to describe the profile of the interface the expression 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)
(1)
(ϕ is the volume fraction of, say, polymer A, χFH Flory-Huggins
interaction parameter, V the volume and kB Boltzmann’s constant 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)
3 N
c*≈
4π Rg 3
(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
Profiles of interfaces between polymer melts and solutions
were first calculated by Helfand et al [7] in a mean self-consistent 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 concentrations that are too low for these interactions to be screened. The
description of the water–water interface between two coexisting 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
⎤
1−ϕ
FFH
1 ⎡ϕ
= 3 ⎢ log ϕ +
log ( 1 − ϕ ) + χFH ϕ ( 1 − ϕ ) ⎥
⎦
kBTV a ⎣ N
N
χ
F [ ϕ, c ]
=
kBT
(2)
∫
V
⎡
1−ϕ
ϕ
d r ⃗⎢
log ϕ +
log ( 1 − ϕ )
⎢⎣ Nbξ3
Nbξ3
2
2 ⎤
∇ϕ⃗
∇c⃗ ⎥
u
K
+
+ 3 ϕ(1−ϕ) + 3 +
(8)
ξ
ξ
24ξϕ ( 1 − ϕ ) 24ξc 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
2
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
⃗ the position variable. The interfacial tension is the
where r is
excess grand potential per unit area
⎡
Ωex
=⎢
AkBT ⎢⎣
∫ dz Nϕbξ3 log ϕ + 1N−bξϕ3 log ( 1 − ϕ )
V
η˙ 2
u
K
ϕ 1−ϕ) + 3 +
3 (
ξ
ξ
24ξ ( 1 − η2 )
⎤
u 2 ε˙ 2
⎥
+
− μη η − με ε + p ⎥
2
24ξ(1 − u ε )
⎥⎦
+
(9)
due to the gradients at the interface, with the composition
variable
η = 2ϕ − 1
(10)
and the concentration variable
ε(z) ≡
c(z) −c
.
uc
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.
(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 chemical 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
ηη˙2
ξ2
.
η¨ ( z ) = η ( 1 − η2 ) +
1 − η2
6u
obtained for the composition and concentration profiles (for a
detailed treatment see [6] and [10])
1−η
1+η
η ′2
η2
log ( 1 − η ) − + p (15)
log ( 1 + η ) +
=
2
2ω
4
2ω
4(1−η )
1+η
1−η
u
log ( 1 + η ) +
log ( 1 − η )
ε ′′ =
2ω
2ω
2
3ν + χ 2
3νK
ε−
η − με
+
2
(3ν − 1)
4(3ν − 1)
with boundary conditions η (±∞) = ±1 and ε (±∞) = 0 and
με =
(12)
When Nb = ∞, boundary conditions are η(±∞) = ±1. The solution therefore is
η ( z ) = tanh z
6u
ξ2
.
ξ2
≡ D∞
6u
1+η
1−η
3ν + χ
log ( 1 + η ) +
log ( 1 − η ) −
η 2 (17)
2ω
2ω
4(3ν − 1)
p =−
(13)
1+η
1−η
η2
log ( 1 + η ) −
log ( 1 − η ) + .
2ω
2ω
4
(18)
2.2. The curved interface
The width of this interface
η
=
η˙ z = 0
(16)
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
(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 u ε<< 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 interaction parameter by ω = uNb and expanding the concentrationdependent quantities to second order in u ε expressions can be
σ ( R ) = σ0 −
2δσ0 2k + k G
+
+ ...
R
R2
(19)
σ0 is the interfacial tension of a flat interface, δ is the Tolman
length [13], which is a measure for the spontaneous curvature
3
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
and
and differs from zero only for asymmetric interfacial profiles,
and k and kG are the rigidities of bending and Gaussian curvature, 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 presence of a solvent profile will not change the essentials of the
expressions for curvature and stiffness. Therefore, the gradient 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
F [ ϕ, c ]
=
kBT
m(ϕ) =
−1
2u ⎛ 1
1 ⎞
(1 + α )2 ⎛ cp ⎞ 1 − 3ν
ω=
+
=
⎜
⎟
⎜
⎟
3
3
3
(1 + α ) ⎝ ccrit ⎠
ξeff ⎝ Nb,AξA Nb,BξB ⎠
−χ − 1
+
u
3
ξ eff
b,B B
ϕ(1−ϕ) +
K
3
ξ eff
⎤
2
2
∇ϕ⃗
∇ϕ⃗
⎥
+
+
24ξAϕ 24ξB ( 1 − ϕ ) ⎥⎦
(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 dependent on the degree of polymerization and therefore we have
defined an effective correlation length as
1
1⎛ 1
1⎞
= ⎜ + ⎟.
ξeff 2 ⎝ ξA ξB ⎠
∞
4π
N ξ
N
= A
3
Nb,BξB NB
(22)
ξA
ξB
(23)
∫ d r ⃗ ⎡⎣ f ( ϕ ) − μϕ + m ( ϕ )
2 ⎤
∇⃗ ϕ ⎦
(24)
V
with
f (ϕ) =
2ϕ
ukBT ⎡
log ϕ
⎢
ξ3 ⎣ (1 + α ) ω
+
⎤
2α ( 1 − ϕ )
log ( 1 − ϕ ) − ϕ2 ⎥
⎦
(1 + α ) ω
dr =
4π 3
R ( ϕl − ϕv )
3
(28)
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 incompatibility 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 neutral 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 monovalent), 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
where α represents the difference in degree of polymerization of the two polymers, while r0 reflects the difference in
size of the statistical monomers; different due to, for example, a difference in solvent quality or chain stiffness. The
excess interfacial grand potential is for the radial symmetry
of the droplet
Ωex =
2
2.3. The effect of charge
and
r0 =
v
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].
We also define, in terms of the bulk blob sizes, two asymmetry
parameters α and r0
α=
∫ [ϕ(r) −ϕ ] r
0
(21)
3
b,A A
(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 effective blob size governing the interaction between the blobs. As
mentioned above, this assumption is justified by the independence 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
⎡
b,A A
(26)
ω is now defined by
∫ d r ⃗ ⎢⎢⎣ N ϕξ 3 log ϕ + N1 − ϕξ3 log ( 1 − ϕ )
V
kBT [ 1 + ( r0 − 1) ϕ ]
.
12ξ (1 + r0 ) ϕ ( 1 − ϕ )
ψD = −
NavkBT
c+ N k T
c−
log α+ = av B log α−
F
cβ
F
cβ
(29)
where c±i are concentrations of small ions, and the index α
refers to the phase containing the charged polymer, and β to
(25)
4
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
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.
outset that ξ is the relevant length scale, the width of the profile 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 systems 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 solvent 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 comparing the total excess of water at the interface per unit area,
the phase containing the neutral polymer. F is Faraday’s constant and Nav is Avogadro’s number. Far from the interface
charge neutrality is maintained:
cα+ + zc = cα−
(30)
cβ+ = cβ− = cs
(31)
and
in which z is the number of positive charges on a polymer
chain. Therefore, the interfacial or Donnan potential can be
expressed by
ψD =
NavkBT
zc NavkBT zc
arcsinh
≈
.
F
2cs
F 2cs
(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 equation (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 representative for gelatin and dextran or pullulan, mixtures of
which are extensively studied in the context of phase separation [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, u ε ≈ 0.2, so at a stronger incompatibility the theory becomes invalid. The depletion profile corresponds 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
∞
Cex ( c , ccrit ) = − cuD∞
∫ ε ( x ) dx
(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 concentration. A monolayer of pure water would be of the order of
1·10−4 g m−2 , so at the highest incompatibility of practical relevance 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 macroscopic 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
Mex =
5
3ϕ
Cex .
R
(34)
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
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.
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 meniscus 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
L cap =
2σ
gΔρ
(35)
in which g is the acceleration due to gravity and ∆ρ the difference in density between the phases. The latter can be measured 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 phaseseparating polymers, of which the coexisting phase composition and ccrit is accurately known.
An expression for the interfacial tension is obtained from minimizing Ω with respect to the solvent and polymer profiles [6, 10].
The result contains three terms:
σ=
Figure 5. Calculated effect of excess of water at the interface for
ccrit = 0.5%.
A measure for the effect of excess of water at the interface is
Ξ=
with
η
∫ dη 1η−′ (ηx2)
(37)
0
and
Δ2 = −
1
8
∞
⎡
∫ dx⎢⎣ (−1η−′(xη)2)
−∞
2
+
⎤
1+χ
2
2
(η − η )⎥ε(x ).
⎦
(3ν − 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 incompatibility 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 compositional 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.
kBT ⎛⎜ u ⎟⎞1/2
(1 −Δ1 − uΔ2 ) = σ∞(1 −Δ1 − uΔ2 ) (36)
ξ2 ⎝ 6 ⎠
Δ1 = 1 −
1 −Δ1 − uΔ2
.
1 −Δ1
(38)
6
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
Figure 6. Interfacial rigidity 2k + kG of a flat interface as a function
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; Dashdotted line: vice versa.
of concentration, for different values of the ratio of degrees of
polymerization of the separated polymers.
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 droplet. A droplet rich in high N polymer is further from coexistence 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 interfacial 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 containing 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 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.
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 minimum between 1.5 and 2 c/ccrit where the decrease in interfacial 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 significant 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
7
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
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.
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.
Concentration (%w/w)
pH
dextran
gelatin
∆ gelatin
Salt concentration
cs (mM)
Donnan
emf (mV)
Charge per
chain z
4.8
6.2
9.2
6.5
4.0
4.3
6.5
4.0
4.3
5.8 ± 0.6
5.65 ± 0.20
5.80 ± 0.09
9.4
4.5
6.9
+3.8 ± 0.4
+0.55 ± 0.04
−2.12 ± 0.17
+4.8
+0.34
−2.0
⎡
2δσ0 2k + k G ⎤
+
F ( R, Nd ) = 4πR Nd ⎢ σ0 −
⎥.
⎣
R
R2 ⎦
2
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.
(40)
For α = 1 and α = 8 F is plotted in figure 8. When minimizing with respect to R at constant volume a preferred radius is
obtained
( )
Rpref = 2δ + 4δ 2 − 3 σ0 (2k + k 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 simplest case of isolated droplets of one phase, growing by diffusion in a supersaturated, infinitely large majority phase, the
increase in size is expressed by
4πR2
2Vpσ ( R ) 1
dR
~ DR2
.
dt
kTR R
3.3. The effect of charge
In the case of phase separation between a neutral polymer
and a weakly charged polyelectrolyte, the interface is predicted 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 difference 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 temperature) with approximately equal phase volumes. The concentrations were chosen in such a way that the differences in
(42)
8
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
Figure 10. CLSM-images of phase separated mixtures of 10% (w/w) gelatin and 10% (w/w) dextran. The molar mass of dextran is (a)
2000 kDa and (b) 464 kDa. In both figures 5% of the dextran fraction is covalently labelled with FITC. The degree of labelling is 30 per
1000 monomers.
4. Conclusions
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 calculated 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 wellestablished 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 properties which are relevant for phase separation kinetics and metastable drop size distributions are unknown. To our knowledge
no theoretical predictions are available for the effect of interfacial charge on interfacial tension.
An experimental consequence which may be of practical importance is shown in figure 10. A confocal microscope
image [31] is shown of a fully phase separated aqueous gelatin/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 monomers). 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 gelatinrich droplets, which are positively charged due to the Donnan
effect.
Interfaces between phase-separated aqueous polymer solutions are, on the basis of theory, expected to be characterized
not only by a composition profile, but also by a concentration 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 curvature, expected in the case of different degrees of polymerization 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.
References
[1] de Gennes P G 1979 Scaling Concepts in Polymer Physics
(Ithaca, NY: Cornell University Press)
[2] Scott R L 1949 J. Chem. Phys. 17 279–84
[3] Grinberg V Y and Tolstoguzov V B 1997 Food Hydrocoll.
11 145–58
[4] Perrau M B, Iliopoulos I and Audebert R 1989 Polymer
30 2112–217
[5] Flory P F 1953 Principles of Polymer Chemistry (Ithaca, NY:
Cornell University Press)
[6] Broseta D, Leibler L, Kaddour O and Strazielle C 1987
J. Chem. Phys. 87 7248
[7] Helfand E and Tagami Y 1972 J. Chem. Phys. 56 3592
[8] Joanny J F, Leibler L and Ball R 1984 J. Chem. Phys. 81 4640
[9] Broseta D, Leibler L and Joanny J F 1987 Macromolecules
20 1935
[10] Tromp R H and Blokhuis E M 2013 Macromolecules
46 3639–47
[11] Helfand E and Sapse A M 1975 J. Chem. Phys. 62 1327
9
R H Tromp et al
J. Phys.: Condens. Matter 26 (2014) 464101
[22] Scholten E, Tuinier R, Tromp R H and Lekkerkerker H N W
2002 Langmuir 18 2234–8
[23] Mitani S and Sakai K 2002 Phys. Rev. E 66 031604
[24] Heinrich M and Wolf B A 1992 Polymer 33 1926–31
[25] Simeone M, Alfani A and Guido S 2004 Food Hydrocoll. 18 463
[26] van Puyvelde P, Antonov Y A and Moldenaers P 2002 Food
Hydrocoll. 16 395
[27] Antonov Y A, van Puyvelde P and Moldenaers P 2004 Int. J.
Biol. Macromol. 34 29
[28] Overbeek J T G 1953 J. Colloid Sci. 8 593
[29] Overbeek J T G 1956 Prog. Biophys. Biophys. Chem. 6 58
[30] Rasa M, Erné, B H, Zoetekouw B, van Roij R and Philipse A P
2005 J. Phys.: Condens. Matter 17 2293
[31] Edelman M W 2003 Segregative phase separation in aqueous
mixtures of polydisperse biopolymers Thesis Wageningen
University
[12] Bates F S and Fredrickson G H 1994 Macromolecules
27 1065–7
[13] Tolman R C 1949 J. Chem. Phys. 17 333
[14] Bergfeldt K, Piculell L and Linse P 1996 J. Phys. Chem.
100 3680–7
[15] Donnan F G 1911 Z. Elektrochem. Angew. Phys. Chem.
17 572–81
[16] Donnan F G 1924 Chem. Rev. 1 73–90
[17] Edelman M W, Tromp R H and van der Linden E 2003
Phys. Rev. E 67 021404
[18] Penfold J 2002 Curr. Opin. Colloid Interface Sci. 7 139–47
[19] Meunier J 1987 J. Phys. 48 1819–31
[20] Aarts D G A L, van der Wiel J H and Lekkerkerker H N W
2003 J. Phys.: Condens. Matter 15 S245–50
[21] De Hoog E H A and Lekkerkerker H N W 1999 J. Phys.
Chem. B 103 5274
10