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Capillary-Induced Phase Separation in Binary and Quasi-Binary Polymer Solutions. A Mean-Field Lattice Study Martin Olsson, Per Linse,* and Lennart Piculell Division of Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden Received October 15, 2003 A capillary-induced phase separation (CIPS) can arise in a polymer solution enclosed between two walls, provided the solution is close to phase separation. Here the CIPS phenomenon is investigated in binary and quasi-binary polymer solutions by a mean-field lattice theory. A quasi-binary polymer solution represents a binary polymer solution with a polydisperse polymer. The CIPS involves a new phase between the two walls with a different solute composition than the surrounding solution. The molecular length of the polymer, the interaction between the polymer and the walls, and the separation between the two walls are factors that all influence the CIPS. A polydisperse polymer, as in the quasi-binary polymer solutions, will give a slightly increased tendency for CIPS at short distances between the walls, while the effect decays at larger separations.
Introduction An added soluble polymer generally affects the forces between two surfaces immersed in a solution. Such forces can be observed macroscopically, e.g., in a surface force measurement. They are also manifested in the stability of particles dispersed in a polymer solution. The effects of polymers on the forces are of different types. A nonadsorbing polymer induces a depletion attraction between dispersed particles,1-7 while an adsorbing polymer gives rise to either an attraction due to bridging or a repulsion caused by excluded-volume effects between the particles.8-13 Recently, a new type of polymer-induced attraction has been observed between surfaces immersed in polymer solutions that are close to phase separation. This attraction has been ascribed to a capillary-induced phase separation (CIPS) in the gap between the surfaces.14-17 A long-range force is associated with the CIPS, where the range may extend a few hundred nanometers. * Corresponding author. E-mail:
[email protected]. Fax: +46 46 222 44 13. (1) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255-1256. (2) Vrij, A. Pure Appl. Chem. 1976, 48, 471-483. (3) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251-267. (4) Lekkerkerker, H. N. W.; Poon, W. C. K.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Europhys. Lett. 1992, 20, 559-564. (5) Meijer, E. J.; Frenkel, D. J. Chem. Phys. 1994, 100, 6873-6887. (6) Sear, R. P.; Frenkel, D. Phys. Rev. E 1997, 55, 1677-1681. (7) Warren, P. B. Langmuir 1997, 13, 4588-4594. (8) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press Inc.: London, 1983. (9) Klein, J.; Luckham, P. F. Nature 1984, 308, 836-837. (10) Granick, S.; Patel, S.; Tirrell, M. J. Chem. Phys. 1986, 85, 53705371. (11) Dickinson, E.; Eriksson, L. Adv. Colloid Interface Sci. 1991, 34, 1-29. (12) Lafuma, F.; Wong, K.; Cabane, B. J. Colloid Interface Sci. 1991, 143, 9-21. (13) Liu, S. F.; Legrand, V.; Gourmand, M.; Lafuma, F.; Audebert, R. Colloid Surf., A 1996, 111, 139-145. (14) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain. Where Physics, Chemistry, Biology and Technology Meet, 2nd ed.; Wiley: New York, 1999. (15) Freyssingeas, E.; Thuresson, K.; Nylander, T.; Joabsson, F.; Lindman, B. Langmuir 1998, 14, 5877-5889. (16) Wennerstro¨m, H.; Thuresson, K.; Linse, P.; Freyssingeas, E. Langmuir 1998, 14, 5664-5666. (17) Joabsson, F.; Linse, P. J. Phys. Chem. B 2002, 106, 3827-3834.
Figure 1. Illustration of CIPS in the gap between two particles, where schematic polymer molecules are indicated in the figure.
To predict the occurrence of a CIPS, the mechanism behind it has to be understood. A capillary phase separation can take place between two surfaces, if the surfaces are brought close enough to each other. Such a phase separation gives a new phase, referred to as a capillary phase, between the surfaces. The composition of the capillary phase differs from that of the surrounding solution, which is referred to as the reservoir phase. A schematic picture of a CIPS between two surfaces is presented in Figure 1. The capillary phase separation is a very general phenomenon and can occur in water, oil, air, etc. A necessary requirement for a CIPS is that the interfacial tension between the capillary phase and the surface is lower than that between the reservoir phase and the surface. This is needed to counteract the higher free energy density of the capillary phase. At some distance, the two free energy contributions will balance. This is the largest separation at which a CIPS can occur. Because of the higher free energy density of the capillary phase, the system will tend to minimize the volume of this phase, leading to an attraction between the surfaces. The aim of this study is to theoretically investigate CIPS in binary solutions, binary polymer solutions, and quasibinary polymer solutions. The study is motivated by surface force measurements15,16 and experimental findings of particle-induced phase separation in various types of
10.1021/la035931d CCC: $27.50 © 2004 American Chemical Society Published on Web 01/22/2004
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solutions. For example, such phase separations have been observed in as simple systems as the binary lutidinewater system with dispersed silica18-21 or polystyrene latex22,23 particles. Solutions with dispersed particles displayed a more extended two-phase region as compared to corresponding particle-free systems. In an accompanying paper to this present one, particle-induced phase separation in quasi-binary polymer solutions of aqueous ethyl(hydroxyethyl)cellulose (EHEC) has been documented.24 Moreover, particle-induced phase separation has also been seen in a ternary aqueous polymer solution of poly(ethylene oxide) and dextran with dispersed silica particles.25 In the present contribution, CIPS and the associated attractive force between the particles are investigated for binary and quasi-binary solutions using a mean-field lattice theory with two impenetrable walls representing the particles. We are employing the same approach as in a previous modeling of the CIPS phenomenon appearing in ternary polymer solutions.17 For binary polymer solutions, an attractive polymer-wall interaction is necessary to promote the polymer-rich capillary phase experimentally observed. Such an attractive interaction establishes a clear border between CIPS and depletion attraction. Previously, CIPS has been demonstrated by different theoretical approaches to occur in binary,26,27 quasibinary,28 and ternary17 polymer solutions. Here, we provide a more detailed examination of CIPS and, in particular, on the associated long-range attractive interaction. Moreover, conditions are selected to reasonably well compare with experimental data on binary solutions18-23 and binary and quasi-binary polymer solutions.24 Theoretical Approach Capillary-Induced Phase Separation. A CIPS between surfaces in a solution requires that the solution itself is close to phase separation. As mentioned above, the CIPS is driven by a decrease of the surface tension, and hence, the surface tension between the capillary phase and the surface has to be lower than the surface tension between the reservoir phase and the surface. In the case considered here, the polymer adsorbs to the walls. If the gap between the walls is much larger than the decay length of the concentration profile associated with the adsorption, there will be a region of constant composition in the center of the gap. This constant composition is referred to as the bulk composition of the phase in the gap. Another property of the capillary phase is that its excess free energy will be lower if the surface separation decreases. This is a consequence of a decreasing volume of the capillary phase. Under given conditions, the difference in excess surface (18) Beysens, D.; Este`ve, D. Phys. Rev. Lett. 1985, 54, 2123-2126. (19) Gurfein, V.; Beysens, D.; Perrot, F. Phys. Rev. A 1989, 40, 25432546. (20) Van Duijneveldt, J. S.; Beysens, D. J. Chem. Phys. 1991, 94, 5222-5225. (21) Beysens, D.; Narayanan, T. J. Stat. Phys. 1999, 95, 997-1008. (22) Gallagher, P. D.; Kurnaz, M. L.; Maher, J. V. Phys. Rev. A 1992, 46, 7750-7755. (23) Gallagher, P. D.; Maher, J. V. Phys. Rev. A 1992, 46, 20122021. (24) Olsson, M.; Joabsson, F.; Piculell, L. Langmuir 2004, 20, 16051610. (25) Joabsson, F.; Calatayu, A.; Thuresson, K.; Piculell, L. Submitted for publication in J. Phys. Chem. B. (26) Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1985, 18, 1882-1900. (27) Forsman, J.; Woodward, C. E.; Freasier, B. C. J. Chem. Phys. 2002, 117, 1915-1926. (28) Chhajer, M.; Gujrati, P. D. J. Chem. Phys. 1998, 109, 1101811026.
Olsson et al.
Figure 2. Typical reduced excess surface free energies (βAσ) for the reservoir and the capillary phases in a gap between two walls, vs the wall separation (D). A stable capillary phase appears for D < Donset.
free energy between the capillary (cap) and the reservoir (res) phases can be expressed as14
∆Aσ(D) ) area‚[(γcap - γres) + D(gcap - gres)] (1) where g is the free energy per unit volume for the bulk composition of a phase and γ is the surface tension of the phase. From eq 1, the CIPS force then becomes
F(D) ) -
d∆Aσ(D) ) -area‚(gcap - gres) dD
(2)
When ∆Aσ(D) ) 0, the separation D is called Donset and can be expressed14 from eq 1 as
Donset ) -
γcap - γres gcap - gres
(3)
Donset is therefore the largest separation at which the capillary phase is thermodynamically stable. ∆Aσ(D) can also be viewed as the difference in the excess free energy between the reservoir and the capillary phases as
∆Aσ(D) ) Aσcap(D) - Aσres(D)
(4)
where Aσ(D) is the excess surface free energy of the phase at a separation D between the surfaces. Note that both Aσcap(D) and Aσres(D) are excess free energies with respect to the bulk of the reservoir. Equation 4 constitutes a convenient way to determine the excess free energies for the solutions in our numerical approach. Figure 2 illustrates the normal influence of D on Aσcap and Aσres. The figure shows (cf. eq 4) that the stable phase will be the capillary phase at D < Donset and the reservoir phase at D > Donset. Mean-Field Lattice Theory. The Flory-Huggins lattice theory29 was used to investigate the binary and the quasi-binary polymer solutions theoretically. Scheutjens and Fleer30,31 have extended this theory to describe specific properties such as adsorption and interfacial (29) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (30) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635.
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properties of polymers at solid surfaces. In the theoretical model concerned in this study, two parallel impenetrable walls are immersed in a polymer or a monomer solution. The monomer solution is considered as a limiting case of the polymer solution. The distance between the walls is divided into layers parallel to the surface of the walls. The number of layers, numbered with i ) 1, 2, ..., D, between two surfaces represents the distance between the walls. Each layer is divided into lattice sites, filled by solvent and solute (polymer or monomer). This theory has been applied earlier to the study of CIPS in a ternary polymer solution.17 Some of the expressions from the latter study are needed here in order to follow the result and discussion parts. The first expression gives the volume fraction, φA,i, of the segments of type A in layer i between the two walls as
φA,i )
1 L
rs
∑x s)1 ∑ δA,t(x,s)nxsi
(5)
where L is the number of lattice sites in a layer, nxsi denotes the segment distribution, and the Kronecker δ only selects segments of rank s of component x if they are of type A. An expression for nxsi, the number of sites in layer i occupied by segments of rank s belonging to component x, is given by using a matrix method as32 s+1
s
(Wt(x,s′))T]‚s}{∆iT‚[∏Wt(x,s′)]‚p(x,1)} ∏ s′)r s′)2
nxsi ) Cx{∆iT‚[
x
(6)
where Cx is a normalization factor, Wt(x,s) is a tridiagonal matrix that describes the lattice topology and weighting factors for the segment of rank s belonging to component x, and p(x,1) is a vector that includes the distribution of the first segment of component x in the layers, with ∆ and s being elementary column vectors. By knowing nxsi, the volume fraction φA,i can be determined from eq 5. Finally, the force operating between the walls can be expressed as17
F(D) ) -
dAσ(D) 1 ≈ - [Aσ(D + 1) - Aσ(D - 1)] (7) dD 2
where Aσ is either Aσcap or Aσres. A few extensions and modifications have been done to the earlier work17 concerning the number of components and polydispersity in the studied solutions. Previously, the model treated two different solutes in a solution in contrast to this study where only one polymeric solute is considered. However, in the investigation of a quasi-binary polymer solution, the polydisperse polymer is described by using several components of different length. Solutions. The investigated solutions are (1) binary and (2) quasi-binary polymer solutions. In both kinds of solution, the solvent has a length of rs ) 1 segment. For the binary solutions, two different kinds of solutes are studied, namely, a monomer with rp ) 1 segment and a polymer with rp ) 100 segments. The two binary solutions are denoted as (1a) the monomer and (1b) the binary polymer solution, respectively. The phase behavior of a binary solution is characterized by the critical interaction (31) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178-190. (32) Linse, P.; Bjo¨rling, M. Macromolecules 1991, 24, 6700-6711.
parameter, χp,c, and the critical concentration, φp,c, given by29
χp,c )
(1 + rp1/2)2 2rp
(8)
φp,c )
1 (1 + rp1/2)
(9)
The upper consolute point of the solution is given by
Tp,c ) w/Rχp,c
(10)
where w is the interaction parameter in units of kJ/mol. For the quasi-binary polymer solution, the length polydispersity is represented by a discrete Schulz distribution if nothing else is stated. The Schulz distribution is asymmetric with a larger weight for high-molecularweight components compared to low-molecular ones and gives, therefore, a reasonable description of a real polymer solution. The Schulz distribution, f(σ), of quantity σ is given by33,34
f(σ) )
( ) ( t+1 〈σ〉
t+1
‚
) (
)
(t + 1)σ σt exp Γ(t + 1) 〈σ〉
(11)
t>0 where t is the width of the distribution, Γ(t) is the gamma function and 〈σ〉 is the mean value of the distribution. The normalized moments of the Schulz distribution are
〈σn〉 )
(n + t)!
∫0∞ σnf(σ) dσ ) 〈σn〉‚t!(t + 1)n
(12)
n ) 0, 1, 2, ... For the quasi-binary polymer solution, the mean value of the solute length was set to 100 segments and the polydispersity index (PI) was 1.4. The length and the relative amount of the solute fractions of the discrete representation involving four components used in the calculations were obtained by requiring their first eight moments to fulfill eq 12.33,34 For one specific study (see below) a quasi-binary polymer solution representing the simplest type of polydispersity is studied, where the polymer solute contains only two fractions. This solution is denoted as a bidisperse polymer solution to distinguish it from the quasi-binary polymer solution represented by a Schulz distribution. The two solute fractions of the bidisperse polymer solution differ in length with a corresponding (cf. eq 8 and 10) difference in upper consolute point, ∆Tp,c, between the pure fractions in their respective mixtures with the solvent. In the bidisperse polymer solution, the two solute fractions were chosen to have rp1 ) 100 and rp2 ) 988 segments, respectively, yielding a ∆Tp,c of 40 K. The volume fraction of the shorter fraction was nine times larger than that of the longer fraction in the solution. This corresponds to a number average molecular weight of 189 segments for the bidisperse solute and a PI of 2.99. The interactions between the solute and the solvent were chosen to get similar upper consolute points in the different solutions. This can be seen from the studies of (33) D’Aguanno, B.; Klein, R. J. Chem. Soc., Faraday Trans. 1991, 87, 379-390. (34) Phalakornkul, J. K.; Gast, A. P.; Pecora, R.; Naegele, G.; Ferrante, A.; Mandl-Steininger, B.; Klein, R. Phys. Rev. E 1996, 54, 661-675.
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Olsson et al. Table 1. Parameters for the Solutions
solution monomer binary polymer quasi-binary polymer
bidisperse polymer
degree of polymerization
solution interaction parameter (kJ/mol)
wall interaction parametera (kJ/mol)
rp ) 1 rp ) 100 rp1 ) 39b rp2 ) 120 rp3 ) 253 rp4 ) 468 rp1 ) 100c rp2 ) 988
wp-s ) 5.54 wp-s ) 1.675 wp-s ) 1.675 wpi-pj ) 0d
∆wp ) -1.0; -2.0; -3.0; -5.0; -10.0 ∆wp ) -2.0; -2.5; -3.0 ∆wp1 ) ∆wp2 ) ∆wp3 ) ∆wp4 ) -2.0; -2.5; -3.0
wp-s ) 1.474 wpi-pj ) 0e
∆wp1 ) ∆wp2 ) -2.5
b a ∆w ≡ w p p-w - ws-w, ∆wp1 ) wp1-w - ws-w, ∆wp2 ) wp2-w - ws-w, ∆wp3 ) wp3-w - ws-w, and ∆wp4 ) wp4-w - ws-w. The relative volume fractions of the four solute fractions are 0.39935 for p1, 0.50943 for p2, 0.08948 for p3, and 0.00174 for p4. c The relative volume fractions of the two solute fractions are 0.90 for p1 and 0.10 for p2. d i ) {1,2,3,4}; j ) {1,2,3,4}; i * j. e i ) {1,2}; j ) {1,2}; i * j.
Figure 3. Cloud point curve (solid curve) and iso-Donset curve at indicated Donset (dashed curve with filled symbols) for the monomer solution at ∆wp ) -5.0 kJ/mol. The open circle refer to the change in phase behavior created by the walls for the solvent-rich phase at T ) 301.50 K.
Figure 4. Cloud point curve (solid curve) and iso-Donset curve at indicated Donset (dashed curve with filled symbols) at low polymer volume fractions for the binary polymer solution at ∆wp ) -2.5 kJ/mol.
the phase behavior of the monomer, the binary polymer, and the quasi-binary polymer solutions presented in the result part, where the upper consolute point is ca. 333 K in all solutions. To isolate the effect of the solvent interaction, the interaction between all different solute fractions is athermal. A specification of the interaction of the solute or the solvent with the walls (w) is also needed. Only the difference between the wall-solute interaction parameter, wp-w, and the wall-solvent interaction parameter, ws-w, is relevant. The difference ∆wp ≡ wp-w - ws-w ) -2.5 kJ/mol is used in most cases, giving an enthalpic preference for the solute to the surface. In the quasi-binary polymer and the bidisperse polymer solutions, all solute fractions have the same wall interaction parameter. Parameters for all solutions are collected in Table 1. Results Cloud Point Curves. The phase behaviors of the monomer, the binary polymer, and the quasi-binary polymer solutions have been mapped as cloud point curves (Figures 3-5) using the Flory-Huggins theory. The cloud point temperatures were determined at different solute concentrations, and the solutions separate into two liquid phases below the cloud point curves. For the monomer and the binary polymer solutions, the cloud point curve is identical to the binodal curve of the corresponding system. The cloud point curve of the monomer solution is presented in Figure 3. This solution represents a system
Figure 5. Cloud point curve (solid curve) and iso-Donset curve at indicated Donset (dashed curve with filled symbols) at low polymer volume fractions for the quasi-binary polymer solution at ∆wp ) -2.5 kJ/mol.
of the same type as the previously studied system of lutidine-water.18-23 For the monomer solution, the upper consolute point, Tp,c, was 333 K and the critical volume fraction, φp,c, was 0.5 according to eqs 9 and 10. In Figure 4, the cloud point curve of the binary polymer solution is shown. For this solution, Tp,c was 333 K and φp,c was 0.0909 according to eqs 9 and 10. The cloud point curve for the quasi-binary polymer solution is illustrated in Figure 5.
Capillary-Induced Phase Separation
The polydispersity in the quasi-binary polymer solution was leading to a widening of the two-phase region compared to the phase behavior seen for the binary polymer solution. The influences of the introduced walls on the phase behavior at a fixed separation, Donset, between the walls are also shown in Figures 3-5. The interaction between the solute and the walls, ∆wp, was chosen to be -5.0 kJ/ mol for the monomer solution and -2.5 kJ/mol for the two polymer solutions. This gives a new phase boundary referred to as an iso-Donset curve. The strict interpretation of such a curve is as follows. The model involves two infinite planar surfaces at a fixed separation in equilibrium with an infinite reservoir phase. Far into the one-phase region, the gap between the surfaces is filled with the reservoir phase. The iso-Donset curve indicates the conditions (composition and temperature) when a new phase, the capillary phase, enriched in solute, replaces the reservoir phase between the surfaces. Donset was 25 layers in the study of the monomer solution and 50 layers for the binary and the quasi-binary polymer solutions. For the polymer solute, the radius of gyration, Rg, corresponds to ca. 10 layers. The different choices of Donset for the solutions is motivated by the different concentration profiles of the solute close to the walls (see below). Donset is here chosen to be sufficiently large that the concentration profile of the reservoir phase has decayed to the bulk composition in the middle of the gap between the walls. It has to be pointed out that the choice of wall separation is otherwise arbitrary, but with a fixed separation, trends in the displacement of the capillary induced phase separation, relative to the bulk phase separation, can be monitored. Iso-Donset curves can, in principle, be obtained from surface force measurements. By contrast, there exists no straightforward mapping of the capillary system modeled here onto the particle dispersions that have been studied experimentally. However, the shift in the cloud point curves induced by dispersed colloidal particles should follow the same trends as the model system with infinite surfaces, if the two systems have the same surface characteristics. The corresponding iso-Donset curves to the cloud point curves in Figures 3-5 show that the two-phase regions for the solutions are increased on the solute-poor side of the biphasic region in the cloud point curves when the two walls are introduced. The effect of the walls on the solute-rich side of the upper consolute point has been examined for the monomer solution at a single temperature. As might be expected, the two-phase region is decreasing on the solute-rich side as a consequence of the two walls. From now on, we will limit our investigation to the solvent-poor side of the upper consolute point where the CIPS can occur and give rise to the attraction that is the focus of our study. For all the three different types of solution, it is showed that the introduction of the walls brought about similar changes in cloud point temperature at different investigated solute concentrations. The effects of the walls are relatively small for all the studied solutions, and hence, the corresponding reservoir solutions have to be very close to phase separation, or the capillary has to be quite thin, in order for CIPS to occur. Dependence on Surface Separation. Clearly (cf. eq 1), there is a relation between the surface separation and the shift in the phase boundary induced by CIPS. The smaller the surface separation, the larger is the shift in the phase boundary. In Figure 6, Donset is determined for some distances to the cloud point curve for the monomer, the binary polymer, the quasi-binary polymer, and the bidisperse polymer solutions. To relate the investigation
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conditions to the cloud point curves of the solutions, a new variable describing the shift in cloud point temperature by introducing the walls is used. This variable is defined as ∆Tp ) Tp - Tp*, where Tp is the temperature onset of CIPS and Tp* is the cloud point temperature of the solution in the absence of the walls. Donset as a function of ∆Tp is given in Figure 6 at solute concentration φp ) 0.0050 for the three polymer solutions. However, φp ) 0.0050 will correspond to an unreasonable low cloud point for the monomer solution. Therefore, the study of Donset in the monomer solution was done at the same temperature as for the binary polymer solution. This corresponds to a value of φp ) 0.2433 for the monomer solution. For all studied solutions, the dependences on surface separation for CIPS have also been studied at different φp than mentioned above. This showed that the results presented in Figure 6 are representative, i.e., the curves depended only weakly on the solute concentration. A comparison between the monomer and the binary polymer solutions in Figure 6a gives that an increase in the number of segments for the solute leads to an occurrence of CIPS at larger wall separation. Expressed alternatively, ∆Tp will be larger for the binary polymer solution than for the monomer solution if the same surface separation is chosen. Figure 6a also shows how Donset changes if ∆wp is varied. For the monomer solution, a negative ∆wp is enough to give rise to a CIPS at small Donset or ∆Tp. An increase in the magnitude of ∆wp raises Donset at a given ∆Tp. However, the effect becomes saturated at ∆wp ≈ -5.0 kJ/mol for the monomer solution. In Figure 6b, the occurrences of CIPS for the binary and the quasi-binary polymer solutions are compared. It is necessary to have a net attraction of the solute to the wall of at least ca. ∆wp ) -2.0 kJ/mol to have a CIPS in the binary and the quasi-binary polymer solutions. This requirement of a finite net attraction differs from the case of the monomer solution, where only a negative ∆wp was necessary. For the polymer solutions, increase of the magnitude of ∆wp from -2.0 to -2.5 kJ/mol will affect Donset strongly at a given ∆Tp, but a further increase gives only small changes in Donset. It is found that Donset at a given ∆Tp is slightly larger for the quasi-binary polymer solution at ∆wp < -2.5 kJ/mol, especially at small Donset. However, at the smallest investigated wall preference of ∆wp ) -2.0 kJ/mol, the value of Donset is instead larger for the binary polymer solution. These variations for the two solutions can be explained by differences in the adsorption of the solute to the walls at the studied ∆wp. This will be more treated in the discussion section. Parts a and b of Figure 6 also motivate our choices of ∆wp to obtain the iso-Donset curves in Figures 3-5. ∆wp values in Figures 3-5 were chosen to be sufficiently large that a stronger net attraction for the solute to the walls will have a minor effect on the phase behavior. To gain a deeper understanding of the differences between the binary and the quasi-binary polymer solutions, the bidisperse polymer solution was studied. In Figure 6c, the corresponding data for the bidisperse polymer solution together with the binary and the quasibinary polymer solutions are displayed at ∆wp ) -2.5 kJ/mol. φp was chosen to be 0.0050 for all examined solutions. Figure 6c shows that Donset increases for a given ∆Tp, at least for ∆Tp > 1 K, for the quasi-binary polymer and the bidisperse polymer solutions compared to the binary polymer solution. This indicates that an increase in the polydispersity and in length of the longest solute fraction give an increase in the range of the CIPS force. Strength of the CIPS Force. An attractive force operating between the walls is associated with the CIPS.
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Figure 7. Strength of the CIPS force vs shift in cloud point temperature (∆Tp) for the binary polymer solution at φp ) 0.0050 and 0.040. In both cases, results for ∆wp ) -2.0 and -2.5 kJ/ mol overlap. Tp* was 301.50 K for φp ) 0.0050 and 327.70 K for φp ) 0.040.
Figure 6. Relation between the range of the capillary force (Donset) and the shift in cloud point temperature (∆Tp) for (a) the monomer (filled circles) and the binary polymer (filled squares) solutions at indicated ∆wp, (b) the binary polymer (filled squares) and the quasi-binary polymer (filled triangles) solutions at indicated ∆wp, and (c) the binary polymer (filled squares), the quasi-binary polymer (filled triangles), and the bidisperse polymer (filled diamonds) solutions at ∆wp ) -2.5 kJ/mol. φp ) 0.2433 for the monomer solution and 0.0050 for the three polymer solutions. Tp* was 301.50 K for the monomer and the binary polymer solutions, 324.85 K for the quasi-binary polymer solution, and 316.15 K for the bidisperse polymer solution.
Over most of the range where a force exists (D e Donset), the force is nearly independent of D since the excess surface free energy of the capillary phase, Aσcap(D), is nearly linearly dependent on the wall separation. This is true except at short distances between the walls where the concentration profiles of the solute near the walls become perturbed, making the force dependent on D. We have used eq 7 to calculate the CIPS force at D ) Donset as a function of the shift in temperature of the cloud point, ∆Tp. The results at φp ) 0.0050 and 0.040 for the binary polymer solution are presented in Figure 7 for ∆wp ) -2.0 and -2.5 kJ/mol. For reasons explained above, the attractive force is stronger as the distance to the cloud point increases, and the attractive force seems to depend linearly on ∆Tp. Moreover, a higher solute concentration, closer to the upper consolute point, reduces the strength of the CIPS force. The strength of the CIPS force is governed by the difference in composition between the reservoir and capillary phases. Figure 7 also shows that the strength of the CIPS force is in practice independent of ∆wp. The same trends with dependence on solute concentration and independence of ∆wp for the strength of the CIPS force are seen for the monomer and the quasibinary polymer solutions. Concentration Profiles. Concentration profiles, given as a variation in the volume fractions of the solute components near the walls, have also been investigated. In Figure 8, the concentration profiles for the different solutions are presented. The results are for D ) Donset and show the compositions of both the reservoir and the capillary phases. The concentration profiles refer to φp ) 0.2433 for the monomer solution and φp ) 0.0050 for the two polymer solutions. Close to the surfaces, an enrichment of the solutes can be seen in both the reservoir and the capillary phases. In the monomer solution the solute is enriched in the first ca. 8 layers and in the binary and the quasi-binary polymer solutions the concentration of the solute is enhanced up to ca. 20 layers out from the surface. Both these results refer to the reservoir phases. This solute enrichment close to the surface in the reservoir phase is often referred to as a wetting layer. In our calculations, the gap between the walls has been sufficiently large that the composition in the middle of the gap is equal to the bulk composition of the reservoir or the capillary phases. CIPS is therefore
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Table 2. Compositions of the Coexisting Phases for the Conditions in Figure 8b phase
φp
reservoir phase (bulk) capillary phase (bulk) coexisting phase at the bulk cloud point
0.0041 0.30 0.32
seen to arise at separation distances between the walls that are considerably larger than the thickness of the wetting layers. Similar calculations of concentration profiles for a ternary polymer solution confirm this conclusion.17 Figure 8 also shows that the bulk compositions of the reservoir and the capillary phases differ, where the capillary phase is much richer in the solute. The composition of the bulk capillary phase is typically similar to that of the phase that coexists with the bulk reservoir phase at its clouding temperature. This is demonstrated, for the case shown in Figure 8b, in Table 2. Similar differences of a few percent in the solute fractions were observed for the other solutions studied. The compositions of the different solute fractions in the reservoir and the capillary phases for the quasi-binary polymer solution are shown in Figure 9. First, the concentration profiles of the reservoir phase show that the longer solute fractions will be more enriched at the surface than the shorter ones. Second, these components are also strongly enriched in the capillary phase. On comparison of the bulk concentrations in Figure 9, the longest solute fraction will have a bulk concentration in the capillary phase that is 33 000 times larger than that in the reservoir phase. By contrast, the bulk composition of the shortest polymer fraction is only 2.4 times larger in the capillary phase compared to the reservoir phase. Discussion Effect of Wall Interaction and Polydispersity. For CIPS to occur on the solute-poor side of the biphasic regions in Figures 3-5, an adsorption of the solute is needed. To obtain an adsorption of a polymeric solute to the surface, the enthalpic attraction to the surface has to be large enough to overcome the loss in conformational entropy of the solute close to the surface. For a single polymer of infinite length at athermal conditions, the required attraction is given by a dimensionless critical adsorption parameter according to35
χscrit ) -ln(1 - λ1)
(13)
where λ1 is the a priori probability to cross to an adjacent layer. For a hexagonal lattice used here, λ1 ) 1/4. The critical adsorption parameter can be related to ∆wp according to
(
∆wp ≡ wp-w - ws-w ) kT -
)
1 crit kT χ ) ln(1 - λ1) λ1 s λ1 (14)
and becomes -1.15kT ≈ -3.0 kJ/mol. For chains of finite length, the critical adsorption parameter will be smaller. Furthermore, the adsorption process becomes gradual for chains of finite length, and a finite density of the chains further affects the adsorption behavior.30 Figure 10 shows the excess adsorbed amount, Γex, as a function of the absolute value of ∆wp for a wall in contact with the reservoir phases of the monomer and the binary polymer solutions (conditions as in parts a and b of Figure (35) Silberberg, A. J. Phys. Chem. 1962, 48, 1872-1884.
Figure 8. Solute concentration profiles for the reservoir (solid curves) and the capillary (dashed curves) phases at D ) Donset for (a) the monomer solution (∆wp ) -5.0 kJ/mol, Tp* ) 301.50 K, φp ) 0.2433, and Donset ) 25), (b) the binary polymer solution (∆wp ) -2.5 kJ/mol, Tp* ) 301.50 K, φp ) 0.0050, and Donset ) 50), and (c) the quasi-binary polymer solution (∆wp ) -2.5 kJ/mol, Tp* ) 324.85 K, φp ) 0.0050, and Donset ) 50).
8). The excess adsorbed amount is the integrated excess amount of solute with the bulk concentration as reference. For the binary polymer solution, there is a reminiscence of a criticality at ∆wp ≈ -2.0 kJ/mol where Γex rapidly increases from zero. This corresponds to ca. 2/3 of ∆wp for an infinite chain determined from eq 14. At ∆wp > -2.0
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Langmuir, Vol. 20, No. 5, 2004
Olsson et al. Table 3. Excess Adsorbed Amount, Γex, for the Binary Polymer and the Quasi-Binary Polymer Solutions at Op ) 0.0041a ∆wp (kJ/mol) binary polymer quasi-binary polymer
-2.0
-2.5
0.14 0.052
1.81 2.16
a The conditions of the solutions are the same as in parts b and c of Figure 8.
Figure 9. Concentration profiles of the solute fractions (p1p4) in the reservoir (solid curves) and the capillary (dashed curves) phases at D ) Donset for the quasi-binary polymer solution with ∆wp ) -2.5 kJ/mol at Tp* ) 324.85 K, φp ) 0.0050, and Donset ) 50. See Table 1 for the relative bulk concentrations of the solute fractions in the reservoir phase.
Figure 10. Excess adsorbed amount to a wall for the reservoir phase vs absolute value of ∆wp for the monomer (filled circles) and the binary polymer (filled squares) solutions. Tp* was 301.50 K for both solutions and φp ) 0.2433 and 0.0050 for the monomer and the binary polymer solutions, respectively.
kJ/mol for the binary polymer solution, the enthalpy gain is too small for the chains to be adsorbed at the wall. For the monomer solution, the cooperativity is lost. Even at ∆wp ) 0.0, Γex seems to be positive due to poor solvent conditions. We believe that the much more prominent saturation effect seen in Figure 6a for the binary polymer solution is related to the adsorption behavior just discussed. For a polymeric solute, there is only a small ∆wp window, where the stability of the capillary phase depends on ∆wp, while for a monomeric solute the onset distance and the phase behavior for the CIPS phenomenon can be varied over larger ranges with the surface attraction, as demonstrated in Figure 6a. In Figure 6b, CIPS was found at a larger Donset at a given ∆Tp for the binary polymer solution than for the quasi-binary polymer solution at ∆wp ) -2.0 kJ/mol while the opposite occurred at ∆wp < -2.5 kJ/mol. This is also believed to be a consequence of different adsorption of the solute to the walls, where an increased adsorption to the wall gives a larger Donset. In Table 3, Γex is presented for
the binary polymer and the quasi-binary polymer solutions at ∆wp ) -2.0 and -2.5 kJ/mol. Γex is determined for both solutions (conditions as in parts b and c of Figure 8) at a solute concentration of φp ) 0.0041. φp corresponds to the bulk concentration of the reservoir phase in the binary polymer solution for the case shown in Figure 8b. The lower Γex for the quasi-binary polymer solution at ∆wp ) -2.0 kJ/mol probably indicates that a larger preference for the wall is necessary for an adsorption of the shortest solute fraction of the quasi-binary polymer solution. The larger Γex for the quasi-binary polymer solution at ∆wp ) -2.5 kJ/mol is an effect of a stronger adsorption of the longer solute fractions present in low amounts in the polydisperse solution. The adsorption dependence on the occurrence of CIPS is further supported by the adsorption seen in the bidisperse polymer solution (data not shown). The better adsorption of the longer solute fractions to the walls can also explain the behavior seen in Figure 6c, where a polydisperse or a bidisperse solution gave CIPS at larger Donset at ∆Tp > 1 K than a binary polymer solution. An additional explanation to Figure 6c is that the longer solute fractions in the polydisperse or the bidisperse solution give adsorption profiles that extend further out from the walls. At a small Donset, the composition of the reservoir phase between the walls does not completely reach the true bulk composition for the longest solute fractions. As a consequence, the polydispersity affects the studied solutions more heavily at small wall separations. Comparison with Experiments. The phase behavior of quasi-binary polymer solutions of different batches of aqueous ethyl(hydroxyethyl)cellulose (EHEC) were affected by dispersion of particles, shown in the accompanying study.24 The dispersed particles induce a decrease in Tp for solutions on the solute-poor side of the biphasic region. Note that EHEC itself shows reverse phase behavior; i.e., the solute will be less soluble in water as the temperature is increased. The magnitude of the decrease in cloud point temperatures for the EHEC solutions induced by the dispersed particles was believed to depend on the polydispersity of the solution. For binary mixtures of lutidine in water, dispersed particles also affect the phase behavior, first seen by Beysens and Este`ve.18 If lutidine has an affinity for the particles, a change in the phase behavior will be seen on the lutidine-poor side of the biphasic region, like for the study with dispersed particles in EHEC solutions. The theoretical results presented in this work predict, qualitatively, the changes in phase behavior seen experimentally at the solute-poor side of the biphasic region by the dispersed colloidal particles both for solutions of lutidine in water and for quasi-binary polymer solutions. A more quantitative comparison between the theoretical predictions and the experimental results requires a more advanced theoretical model that treats the particles as a component in the solutions. Comparison with Earlier Theoretical Studies. Similar to the present study, previous investigations on
Capillary-Induced Phase Separation
binary polymer26,27 and quasi-binary polymer28 solutions showed that a polymer-rich phase could be established when two surfaces were brought to a short separation from each other. Scheutjens and Fleer,26 using a meanfield theory, and Forsman et al.,27 employing a density functional theory, found that the bulk concentration of the capillary phase in a binary polymer solution was similar to that of the phase which would coexist with the reservoir phase at its clouding temperature. The meanfield lattice modeling in our study confirmed those predictions. Moreover, our results showed that these observations also are valid for monomer solutions and quasi-binary polymer solutions. Noticeably was an enrichment of the longer solute fraction in the capillary phase for the quasi-binary polymer solution. The present study also agreed with Chhajer and Gujrati’s28 observation of a critical polymer-surface interaction for the appearance of CIPS. Finally, the presence of a long-range attractive force, independent of the surface separation, associated with CIPS as described by Scheutjens and Fleer26 was here supported, and the strength of the attractive force has been examined in more detail. Conclusions In binary and quasi-binary polymer solutions, a capillary-induced phase separation (CIPS) has been shown to
Langmuir, Vol. 20, No. 5, 2004 1619
occur in a gap between two walls when the solute has a net affinity to the walls. To demonstrate this, mean-field lattice modeling has been applied. For a CIPS to occur, the solution needs to be close to phase separation. For a polymeric solute, a threshold attraction of the solute to the wall must be exceeded, while no threshold attraction is required for a monomeric solute. CIPS can be promoted by changing the length of the polymeric solute. An increase in the length will give a CIPS further away from the bulk cloud point curve of the solution. In the CIPS, the composition of the bulk capillary phase is closely similar to that of the phase that coexists with the bulk reservoir phase at its clouding temperature. A polydispersity of the binary polymer solution was shown to give a slightly larger influence of CIPS in the solution, but the effect is still small compared to the effects of CIPS found in a ternary polymer solution.17 The theoretical results are in qualitative agreement with available experiments on particles dispersed in binary or quasi-binary solutions. Acknowledgment. This work was financed by the Center for Amphiphilic Polymers from Renewable Resources (CAP) and the Swedish National Research Council (NFR). LA035931D
