Influence of Added Particles on the Phase Behavior of Polymer

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Influence of Added Particles on the Phase Behavior of Polymer Solutions. Analysis by Mean-Field Lattice Theory Martin Olsson, Per Linse,* and Lennart Piculell Division of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden Received May 20, 2005. In Final Form: August 17, 2005 The influence of added colloidal particles on the phase stability of polymer solutions is investigated theoretically. The polymer has an affinity to the particle surface. A mean-field lattice theory based on the Flory-Huggins theory is used to calculate the phase behavior in solutions containing a single polymer component and particles. The particles are described in two different ways. The first approach considers the surface free energy associated with added solid particles and the mixing entropy of the particles. In the second approach, the particles are simply modeled as large polymers. Both ways of describing the added particles show that the added particles decrease the stability of the polymer solution when the polymer-particle attraction is strong. A higher particle concentration enhances the effect. Experiments where polystyrene latex particles are added at different concentrations to aqueous dispersions of ethyl(hydroxyethyl)cellulose (EHEC) support the theoretical findings.

Introduction Mixed solutions of polymers and colloidal particles have been extensively investigated. Two different classes of such mixtures concern nonadsorbing and adsorbing polymers, respectively, where different types of stability/ instability effects are found. For a nonadsorbing polymer, attraction between the particles due to depletion can occur, leading to a destabilization of the mixture.1,2 An adsorbing polymer, on the other hand, may stabilize the mixture by “steric stabilization”, since the adsorbed polymer chains prevent particles from coming close to each other for entropic reasons. However, it may also give rise to destabilization through bridging flocculation, which usually occurs in mixtures where the particle surfaces are not saturated with the adsorbing polymer.3,4 A third possible scenario in adsorbing polymer-particle mixtures is a capillary-induced phase separation (CIPS).5-9 CIPS has recently been discovered in polymer solutions that are close to phase separation even in the absence of particles. The basic physics of a CIPS is that a new phase, the “capillary phase”, can be formed in the gap between the colloidal particles, if the interfacial free energy between the capillary phase and the particle surface is lower than between the original “reservoir phase” and the surface.10 CIPS may occur in any fluid enclosed by surfaces; it is by no means limited to polymer systems.6 The aim of this work is to model how the phase behavior of a polymer solution is affected by added particles to which * Corresponding author. E-mail: [email protected]. (1) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255-1256. (2) Vrij, A. Pure Appl. Chem. 1976, 48, 471-483. (3) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (4) Dickinson, E.; Eriksson, L. Adv. Colloid Interface Sci. 1991, 34, 1-29. (5) Freyssingeas, E.; Thuresson, K.; Nylander, T.; Joabsson, F.; Lindman, B. Langmuir 1998, 14, 5877-5889. (6) Wennerstro¨m, H.; Thuresson, K.; Linse, P.; Freyssingeas, E. Langmuir 1998, 14, 5664-5666. (7) Joabsson, F.; Linse, P. J. Phys. Chem. B 2002, 106, 3827-3834. (8) Olsson, M.; Joabsson, F.; Piculell, L. Langmuir 2004, 20, 16051610. (9) Olsson, M.; Linse, P.; Piculell, L. Langmuir 2004, 20, 1611-1619. (10) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain. Where Physics, Chemistry, Biology and Technology Meet, 2nd ed.; Wiley: New York, 1999.

the polymer adsorbs. Previously, an experimental study from our group has shown that added particles enhance the phase separation in aqueous solutions of ethyl(hydroxyethyl)cellulose (EHEC).8 In the latter experiments, EHEC adsorbed to the particles but the polymerto-particle ratio was high, making bridging unlikely as the destabilizing mechanism. Figure 1a illustrates schematically that added particles gave phase separation in solutions that were close to phase separation in the absence of particles. This particle-induced phase separation was seen at the polymer-poor side of the critical point. The magnitude of the effect was influenced by parameters such as the affinity of the polymer to the particle surfaces, the polymer polydispersity, and the salt content.8,9 Added particles have previously been shown to affect in a similar manner the phase behavior of binary solutions of lutidine and water,11-16 illustrated schematically in Figure 1b. Both systems in Figure 1 feature a lower critical solution temperature (LCST), in contrast to the more common situation with an upper critical solution temperature (UCST). Upon heating, an aqueous LCST solution phase separates at a (concentration dependent) temperature, Tcp, referred to as the cloud-point. The phenomenon of particle-induced phase separation is of fundamental interest, and it is expected to be important in technical applications involving mixed polymer-particle dispersions. Our initial attempts at theoretical modeling of the particle-induced phase separation described above were to describe the effects in terms of a CIPS.7,9 We used a mean-field lattice theory to calculate the phase separation in a gap between two walls for polymer-solvent and polymer 1-polymer 2-solvent solutions. The obtained theoretical results from these investigations predicted, qualitatively, the changes in (11) Beysens, D.; Este`ve, D. Phys. Rev. Lett. 1985, 54, 2123-2126. (12) Gurfein, V.; Beysens, D.; Perrot, F. Phys. Rev. A 1989, 40, 25432546. (13) Van Duijneveldt, J. S.; Beysens, D. J. Chem. Phys. 1991, 94, 5222-5225. (14) Gallagher, P. D.; Kurnaz, M. L.; Maher, J. V. Phys. Rev. A 1992, 46, 7750-7755. (15) Gallagher, P. D.; Maher, J. V. Phys. Rev. A 1992, 46, 20122021. (16) Beysens, D.; Narayanan, T. J. Stat. Phys. 1999, 95, 997-1008.

10.1021/la051343j CCC: $30.25 © 2005 American Chemical Society Published on Web 09/27/2005

Analysis by Mean-Field Lattice Theory

Langmuir, Vol. 21, No. 23, 2005 10863 Table 1. Polymer Data polymer N-EHEC HM-EHEC

DSethyl a 0.8 0.8

MSEOb 2.1 2.1

MShydrophobc

Mw (Da)d

Mw/Mne

0.01

5.4 × 5.1 × 105

6 7

105

a Average total number of ethyl groups per anhydroglucose unit. Average total number of ethylene oxide groups per anhydroglucose unit. c Average total number of nonylphenol groups per anhydroglucose unit. d Weight-averaged molecular mass. e Polydispersity index.

b

component mixture of two polymers in a solvent may be applied. In the present context, an attractive polymerpolymer interaction parameter is employed, giving rise to an associative phase behavior. Some earlier FloryHuggins calculations of the phase behavior in mixed polymer solutions also treat the associative case, but for different regions of the parameter space.21-23 Some additional experiments on aqueous dispersions containing EHEC and polystyrene latex particles have been performed as a comparison to the theoretical results and to complement the picture from the earlier study.8 One set of experiments focuses on the effect of an increasing particle concentration. Further, experiments have also been performed to study the particle distribution in phase-separated EHEC solutions and to investigate the sensitivity of the determined phase behavior to the experimental procedures. Figure 1. Schematic illustration of cloud-point curves for particle-free (solid curves) and particle-containing (dashed curves) for (a) EHEC-water mixtures and (b) lutidine-water mixtures, where the EHEC(lutidine)-particle interaction is attractive. The concentration of the added particles is low in both cases. The temperature interval on the y axis is ca. 25 °C.

phase behavior by the colloidal particles obtained for the two different types of polymer solutions. The limitation of these analyses was that the phase separation in a gap could not be translated to the behavior of a particle dispersion at a certain particle concentration. However, it was found in the model that a decrease in the separation distance between the two walls promoted phase separation. In this present study, the aim is to obtain a model that includes the particle concentration explicitly. As in the previous studies,7,9 we calculate the free energy in the system by using a mean-field lattice theory based on the Flory-Huggins theory. A binary polymer-solvent mixture or a binary “monomer”-solvent mixture is used as reference solution. Two approaches are used to investigate the effect of the added particles. In the first approach, two extra terms are added to the free energy expression to describe the surface free energy and the mixing entropy associated with the particles. We will here only consider solutions of an adsorbing polymer and colloidal particles, although the model should have a more general applicability. Some theoretical attempts to model the effect of added particles found in binary solutions of lutidine and water use similar approaches to account for the perturbation in free energy that the particles create.17-20 A limitation with this approach is, however, that high particle concentrations are not treated properly. Therefore, in the second approach, the particle is described in a cruder way, simply as another polymer. Hence, the classical Flory-Huggins theory adapted to a three(17) Sluckin, T. J. Phys. Rev. A 1990, 41, 960-964. (18) Lo¨wen, H. Phys. Rev. Lett. 1995, 74, 1028-1031. (19) Netz, R. R. Phys. Rev. Lett. 1996, 76, 3646-3649. (20) Gil, T.; Ipsen, J. H.; Tejero, C. F. Phys. Rev. E 1998, 57, 31233133.

Experimental Section Materials. Ethyl(hydroxyethyl)cellulose (EHEC) was a kind gift from Akzo Nobel Surface Chemistry AB, Stenungsund, Sweden. Two different batches of EHEC were used, referred to as nonmodified EHEC (N-EHEC) and hydrophobically modified EHEC (HM-EHEC). The difference between the batches is that HM-EHEC contains a few nonylphenol groups grafted onto the EHEC backbone. The EHECs were purified before use as previously described.24 The degrees of substitution of ethyl, hydroxyethyl, and nonylphenol groups on the cellulose backbone in N-EHEC and HM-EHEC given by the manufacturer and the molecular weight and the polydispersity index of N-EHEC and HM-EHEC determined by size exclusion chromatography (SEC)8 are compiled in Table 1. Polystyrene latex particles with a mean diameter of 350 nm were purchased from Polyscience Inc., Warrington, USA, and obtained as a stock dispersion of 2.6 wt % particles in water. The water used in the experiments was of MilliPore quality. Methods. Samples were prepared by weighing the desired amounts of EHEC, polystyrene latex particles, and water directly into test tubes, which were sealed with Teflon screw caps. The concentrations of polymer and particles were 0-1.0 wt %. All samples were equilibrated on a tilting board for at least 12 h at room temperature. The conventional method to determine the cloud-point of a polymer solution is to observe the change in solution turbidity at Tcp. This was done by visual inspection in this work for samples with low particle concentration. The cloud-point was measured both by increasing and by decreasing the temperature at a rate of 0.5 °C/min for mixtures of N-EHEC and polystyrene latex particles for a particle concentration of 0.01 wt % (corresponding to a particle volume fraction of ca. 0.0001). In these experiments, Tcp was simultaneously determined for a particle-containing sample and for its particle-free reference sample. This procedure minimizes the experimental error. The reproducibility in the determination of Tcp was within (0.5 °C. (21) Malmsten, M.; Linse, P.; Zhang, K. W. Macromolecules 1993, 26, 2905-2910. (22) Bergfeldt, K.; Piculell, L.; Tjerneld, F. Macromolecules 1995, 28, 3360-3370. (23) Bergfeldt, K.; Piculell, L.; Linse, P. J. Phys. Chem. 1996, 100, 3680-3687. (24) Thuresson, K.; Karlstro¨m, G.; Lindman, B. J. Phys. Chem. 1995, 99, 3823-3831.

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T the temperature. At sufficiently large χ1,2 (high w1,2 and/ or low T), the solution becomes thermodynamically unstable and separates into two phases; that is, this simple model gives an UCST behavior. More complex polymer models, including internal degrees of freedom, may generate LCST behavior as found in, for example, EHEC solutions. The type of phase behavior (UCST or LCST) is, however, not our primary concern here, since we are interested in the effect of added particles. Therefore, we will be content with the simplest (UCST) type of temperature dependence. The critical point of the phase instability in a solution with UCST behavior is given by

Figure 2. Viscosity as a function of the temperature for a 0.80 wt % HM-EHEC solution without added particles. Tcp is determined to 50.7 ( 0.5 °C from the point of intersection of the two lines indicating the temperature dependence on the viscosity below and above Tcp. However, at high particle concentrations, the particles gave a strong background turbidity in the solution and the change in the turbidity at Tcp was difficult to detect accurately even in a spectrophotometer. Therefore, a rheological method was used to determine Tcp. This method is based on the observation that the value of |dη/dT|, where η is the solution viscosity, changes discontinuously at Tcp. Essentially, the volumes of the formed phases determine whether |dη/dT| decreases or increases at Tcp. Such a determination of Tcp from rheological data is illustrated in Figure 2. Here, the larger dilute phase constitutes the continuous phase at the onset of phase separation leading to an increase in |dη/dT|. In the rheological measurements, Tcp was determined in HMEHEC-polystyrene latex particle mixtures with polymer and particle concentrations of 0.80 wt % and 0-1.0 wt %, respectively. The determination of Tcp was carried out on a StressTech rheometer from Rheologica, Sweden. A 4 cm plate and plate geometry was used and a shear rate of 100 Hz was constantly applied during the measurements. The temperature in the system was controlled by an external water bath, where the temperature was raised continuously by 1.0 °C/min.

(2)

φ2,c ) [1 + (r2/r1)1/2]-1

(3)

Obviously, the upper critical temperature can be expressed as Tc ) w1,2/Rχc. In the following, component 1 will be considered to be the solvent, hence r1 ) 1, and component 2 to be the solute. Two different types of solute have been studied, viz. (i) a monomer with r2 ) 1 and (ii) a polymer with r2 ) 100. According to eq 3, the critical solute volume fractions of the binary monomer-solvent and polymer-solvent solutions become φ2,c ) 1/2 and 0.091, respectively. For both kinds of system, w1,2 is chosen to give an upper critical temperature Tc ) 333 K. Parameters characterizing the systems investigated are collected in Table 2. Particle Approach. In this approach, the particles added to the binary solution are assumed to be spherical with the radius Rp (in lattice units), and where each particle possesses an area Ap ) 4πRp2 and a volume Vp ) (4π/3)Rp3. The overall particle volume fraction φ3 is defined according to

φ3 ≡

n3Vp n1r1 + n2r2 + n3Vp

On the basis of a lattice mean-field description, we use two different approaches to model how added particles influence the phase behavior of the polymer solution. The two approaches differ with respect to how the particles are treated, and they are referred to as the particle and the polymer approach, respectively. Both approaches start from the Flory-Huggins theory of a binary solution composed of a solvent and a homopolymer.25 Binary Solution. According to the Flory-Huggins theory, the free energy of mixing two components from their pure noncrystalline reference states is given by

(1)

where nx denotes the number of molecules of component x, rx the number of lattice cells occupied by a molecule of component x, and φx ≡ nxrx/(n1r1 + n2r2) the volume fraction of component x, with x ) 1 or 2. Moreover, χ1,2 is the dimensionless Flory interaction parameter related to the effective interaction parameter w1,2 according to χ1,2 ) w1,2/RT, and, as conventionally, R is the gas constant and (25) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

(4)

and the total particle area can be expressed as

S ) n3Ap )

Theoretical Approach

n1 n2 A({nx})/RT ) ln φ1 + ln φ2 + n1φ2χ1,2 r1 r2

χc ) (r1-1/2 + r2-1/2)2/2

3φ3 (n r + n2r2 + n3Vp) Rp 1 1

(5)

In the presence of particles, the free energy expression, given by eq 1, is augmented to account for (i) the surface free energy associated with the particles and (ii) the mixing entropy of the particles according to

A({ nx})/RT )

n1 n2 ln φ1,s + ln φ2,s + n1φ2,sχ1,2 + r1 r2 mix Asurf part/RT - Spart/R (6)

where φx,s ≡ nxrx/(n1r1 + n2r2) represents the volume fraction excluding the particles. The surface free energy term is approximated to

Asurf part ) Sγ(φ1,s,φ2,s)

(7)

where S is the total particle area and γ denotes the surface tension of the particle in contact with a solution with a composition given by φ1,s and φ2,s ) 1 - φ1,s. The surface tension γ was calculated using an extension of the FloryHuggins theory to heterogeneous polymer solutions as formulated by Scheutjens and Fleer.26 In brief, a potential gradient perpendicular to a surface is introduced, which allows volume fraction gradients to be established as the


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Table 2. Parameters Characterizing the Systems Investigateda approach particle particle polymer polymer

c

degree of polymerizationb

solute-solvent interaction parameter (kJ/mol)

particle interaction parameters (kJ/mol) c

particle radius (lattice units)

r2 ) 1 r2 ) 100 r2 ) 1 r3 ) 1000 r2 ) 100 r3 ) 1000

w1,2 ) 5.54 w1,2 ) 1.675 w1,2 ) 5.54

∆w2 ) -4.0, -1.0 ∆w2 ) -2.5, -2.0 w1,3 ) 0 w2,3 ) -4.0, -1.0 w1,3 ) 0 w2,3 ) -2.5, -2.0

Rp ) 20 Rp ) 20

w1,2 ) 1.675

a Index 1 denotes solvent, 2 solute, and 3 particle. Positive interaction parameter implies repulsion and negative attraction. b r ) 1. 1 ∆w2 ≡ w2,3 - w1,3.

free energy of the system is minimized. The Gibbs surface free energy per unit area of the heterogeneous system was taken as the surface tension. For simplicity, a planar surface was used. The mixing entropy of the particles also entering in eq 6 was evaluated according to

Smix part/R ) - n3 ln

n3 n1r1 + n2r2 + n3Vp 1 2 + - 3 (8) n3 1 - φ3 (1 - φ )2 3

[

]

where the first term represents the ideal mixing based on the particle number density and the second term the hardsphere contribution as given by Carnahan and Starling. The particle approach has limitations originating from its pertubative nature. It originates from the binary solute-solvent solution and should therefore only be valid for limited addition of particles. First, surface tensions cannot be calculated for conditions at which the binary solute-solvent solution is unstable and in practice not for a part of the metastable region. Second, an inconsistency occurs when the adsorbed amount of the solute becomes comparable to the total amount of solute in that phase. Since the calculations of the surface tension entering in eq 7 are based on the average volume fractions of that phase, the chemical potential of the solute is exaggerated when the adsorbed amount of solute is large. In the particle approach, the particle radius Rp ) 20 lattice units, corresponding to Ap ) 5.0 × 103, is used throughout. We will consider up to n3 ) 1.2 × 10-6 particles added to a unit volume of the binary solution (n1r1 + n2r2 ) 1), hence corresponding up to φ3 ) 0.04. Only the difference of the effective solvent-particle interaction parameter w1,3 and the effective solute-particle interaction parameter w2,3 is of importance. Here, we define ∆w2 ≡ w2,3 - w1,3 and ∆w2 ) -4.0 to -1.0 kJ/mol, giving an adsorption of the solute to the particle surface, is used. Again, we refer to Table 2 for full details. When connecting to experiments, the typical lattice spacing d ) 0.5 nm gives a particle radius dRp ) 10 nm. Polymer Approach. In the polymer approach, the particles are represented as long polymers. The free energy of the solute-solvent-“particle” mixture is obtained by a standard generalization of eq 1 to three components according to

interaction parameters, respectively. Hence, in this approach, the particles are less realistically described, but the free energy expression is valid at all compositions. The polymer representing the particle is characterized by r3 ) 1000. Here, w1,3 ) 0 and w2,3 ) -4.0 to -1.0 kJ/mol have been used. Determination of the Composition of Coexisting Phases. Examination of phase stability can be achieved by using different techniques. Here, given the overall composition of the system {nx} (x ) 1, 2, and 3), equilibrium is achieved when {nx} is distributed among the coexisting phases {R} such that the total free energy Atot ) ∑RAR({nRx }) is minimized, where AR({nRx }) for phase R is given by eqs 6-8 in the particle approach and by eq 9 in the polymer approach. From {nRx }, the volume and the volume fractions of the components in the coexisting phases are readily available. This protocol is obviously equivalent to procedures originating from imposing equivalence among the chemical potentials of each component in all coexisting phases. Experimental Results

where r3 is the length and φ3 the volume fraction of the polymer representing the particle. Moreover χ1,3 and χ2,3 are the solvent-“particle” and the solute-“particle”

Cloud-Point Curves. The cloud-point curves for NEHEC with and without added polystyrene latex particles at a particle volume fraction of 0.0001 are given in Figure 3. Figure 3a shows experiments obtained by increasing the temperature, similar to those published previously.8 To check whether the effect of the particles could be a kinetic, rather than an equilibrium effect, we here also made additional experiments where the temperature was lowered, Figure 3b. The Tcp of the particle-free solutions obtained upon increasing the temperature was ca. 2 °C higher than that obtained upon decreasing the temperature, in agreement with previous studies.24,27 Figure 3 shows that the added particles lower the Tcp independently of the procedure used to determine Tcp, which indicates that it is not a kinetic effect. The magnitude of the lowering in Tcp by the added particles is somewhat larger, at least at the highest polymer concentrations, when Tcp was determined by decreasing the temperature. Some selected polymer-particle samples used in Figure 3 were phase separated macroscopically at temperatures close to Tcp by keeping the solutions at the chosen temperature during ca. 12 h. When the phase separation temperature was more than 1 °C above the Tcp obtained upon increasing the temperature, virtually all particles accumulated in the phase rich in N-EHEC, as determined by visual inspection. However, both of the separating phases contained visible amounts of particles when Tcp e T e Tcp + ∼1 °C. Increasing the Particle Volume Fraction. The dependence of Tcp of HM-EHEC on the particle volume fraction is illustrated in Figure 4. The extension of the two-phase area increased with increasing particle volume

(26) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635.

(27) Karlstro¨m, G.; Carlsson, A.; Lindman, B. J. Phys. Chem. 1990, 94, 5005-5015.

A({ nx})/RT )

n1 n2 n3 ln φ1 + ln φ2 + ln φ3 + r1 r2 r3 n1φ2χ1,2 + n1φ3χ1,3 + n2φ3χ2,3 (9)

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Figure 3. Cloud-point diagram of N-EHEC with (filled circles) and without (open circles) dispersed polystyrene latex particles at a particle volume fraction of 0.0001 determined with (a) a continuous increase and (b) a continuous decrease in temperature of 0.5 °C/min.

Figure 4. Clouding temperature, as determined by rheology, of 0.80 wt % aqueous HM-EHEC solutions as a function of the volume fraction of added polystyrene latex particles.

fraction. For the particle-free polymer solution, Tcp was ca. 50 °C. At the highest particle volume fraction of 0.01, the decrease in Tcp was ca. 8 °C. Theoretical Results Binary Solutions. Most calculations have been performed at T ) 320 K, which is below the critical temperature of the binary solution, Tc ) 333 K. Conse-

Olsson et al.

Figure 5. Ternary solute-solvent-particle phase diagram at T ) 320 K for (a) r2 ) 1 and ∆w2 ) -4.0 kJ/mol and (b) r2 ) 100 and ∆w2 ) -2.5 kJ/mol, as predicted by the particle approach, displaying compositions of coexisting phases (filled circles), tie-lines (solid lines), and dilution lines connecting φconc on the binary solute-solvent axis with the particle corner 2 (dotted lines). Other parameters are given in Table 2.

quently, one of the two phases formed is diluted and one concentrated with respect to the overall solute volume fraction. In the following, these phases are denoted as “dil” and “conc”, respectively. The solute volume fractions in the two coexisting phases at T ) 320 K were φdil 2 ) 0.331 ) 0.669 in the monomer-solvent solution and and φconc 2 conc φdil ) 0.218 in the polymer-solvent 2 ) 0.0210 and φ2 solution. Below, we will consider the effects on the phase behavior of the monomer-solvent and polymer-solvent solutions by added particles, using the two approaches described above. Particle Approach. Figure 5 shows ternary phase diagrams based on the solute, solvent, and particle volume fractions for the more attractive solute-particle interaction parameters at T ) 320 K. The diagrams display the compositions of coexisting dilute and concentrated phases, on the binary tie-lines, and dilution lines connecting φconc 2 solute-solvent axis with the particle corner. The latter lines represent systems where the particle content is varied at a fixed ratio of solute to solvent. A magnification of an area containing the dilute branch of the binodal curve at low particle volume fraction is presented in Figure 6.


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Figure 6. Magnification of the ternary polymer-solventparticle phase diagram shown in Figure 5b displaying the dilute branch of the binodal curve at low particle volume fractions in a Cartesian representation. The slope kdil (see text) is also shown (dotted line).

Our main interest lies in how the compositions of the coexisting dilute and concentrated phases are affected by the particle addition. For that purpose, we use the solute volume fraction excluding the volume occupied by the particles, φ2,s ≡ φ2/(φ1 + φ2). Along the dilution lines given conc in Figure 5, φconc 2,s ) φ2,s (φ3 ) 0) at all φ3. Moreover, to dil quantify how φ2,s and φconc 2,s deviate from their initial conc values φdil 2,s (φ3 ) 0) and φ2,s (φ3 ) 0), respectively, we dil ≡ (dφ /dφ )| dil introduce the slopes k 2,s 3 φ2,s ) φ2,s , φ3 ) 0 and conc kconc ≡ (dφ2,s/dφ3) |φ2,s ) φ2,s , φ3 ) 0, which describe how the equilibrium solute concentration in a phase is initially affected by adding particles. Figure 6 exemplifies how the value of kdil was determined (dotted line). As particles are added to the solute-solvent solution, Figure 5b shows that (I) the polymer volume fraction in decreases weakly from 0.218 the concentrated phase φconc 2 at φ3 ) 0 to, e.g., 0.20 at φ3 ) 0.10; (II) the polymer volume fraction in the concentrated phase disregarding the volume occupied by the particles φconc 2,s increases, i.e., the polymer-solvent ratio changes toward a larger fraction of polymer; and (III) the particles are almost exclusively partitioned to the concentrated phase. The qualitative behavior is the same for the monomer system (Figure 5a) as for the polymer system. The enlargement given in Figure 6 shows that for the polymer system φdil 2 and consequently also φdil 2,s decreases as particles are added. A more quantitative analysis gives the slopes kdil , -1‚102 and kconc ) 0.03 for the monomer system, and kdil ) -60 and kconc ) 0.04 for the polymer system. For the monomer system, no exact value of kdil is given, since the particle concentration in the dilute phase was extremely small and could not be determined accurately at low φ3. The same holds for some other cases presented below. Hence, the dilute phase becomes diluted and the concentrated phase concentrated by adding particles, and the composition of the dilute phase is much more strongly affected than the composition of the concentrated phase. We also examined how the effect of the added particles depended on the strength of the solute-particle interaction ∆w2. Ternary phase diagrams (not shown) for the less attractive solute-particle interactions (∆w2 ) -1.0 kJ/ mol for the monomer system and ∆w2 ) -2.0 kJ/mol for the polymer system) showed large similarities to those given in Figure 5. For the monomer system, kdil was again strongly negative and kconc ) -0.02, whereas for the

Figure 7. Ternary monomer-solvent-particle phase diagram at T ) 320 K for r2 ) 1 with (a) ∆w2 ) -1.0 kJ/mol and (b) ∆w2 ) -4.0 kJ/mol, as predicted by the polymer approach, displaying compositions of coexisting phases (filled circles), tie-lines (solid lines), and dilution lines connecting φconc and φdil 2 2 on the binary monomer-solvent axis with the particle corner (dotted lines). Other parameters are given in Table 2.

polymer system kdil ) -55 and kconc ) -0.004, respectively. Thus, for the dilute phase the initial influence on the solute concentration was similar, although for the concentrated phase the change in φconc 2,s had altered sign. The depenon φ was still small, however. dence of φconc 3 2,s Polymer Approach. Two ternary monomer-solventparticle phase diagrams at T ) 320 K for two different values of ∆w2 are shown in Figure 7. Magnifications of areas containing the dilute branch of the binodal curve at low particle volume fraction are provided in Figure 8. As for the particle approach, the volume fraction of particles in the dilute phase was too small to be numerically quantified at very low φ3. Again with a focus on small additions of particles, from Figure 7 we conclude that (I) the monomer volume fraction and the monomer volume in the concentrated phase φconc 2 fraction excluding the particles φconc 2,s decrease for ∆w2 ) -1.0 kJ/mol (Figure 7a) and increase for ∆w2 ) -4.0 kJ/ dil mol (Figure 7b); (II) φdil 2 and φ2,s strongly decrease, the largest effect appearing for ∆w2 ) -4.0 kJ/mol; and (III) the particles strongly partition to the concentrated phase. In addition, we notice that the two-phase region is closed; that is, at sufficiently high particle concentrations, the

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Figure 8. Magnification of the ternary monomer-solventparticle phase diagram shown in (a) Figure 7a and (b) Figure 7b, displaying an area containing the dilute branch of the binodal curve at low particle volume fractions in a Cartesian representation.

system becomes monophasic at all monomer-solvent compositions. In more detail, the quantitative analysis gave kconc ) -2 and 1.5 for ∆w2 ) -1.0 and -4.0 kJ/mol, respectively, and again only kdil , -1 × 102 could be concluded for kdil. Moreover, Figure 8 displays how the monomer volume fraction in the dilute phase initially decreases and then increases upon an increased particle volume fraction. The lowest monomer volume fraction occurred at the particle -3 volume fraction φdil 3 ≈ 10 , independently of the value of ∆w2. Figure 9 provides the corresponding ternary polymersolvent-particle phase diagram at T ) 320 K, also for two different values of ∆w2, whereas Figure 10 displays magnifications of areas containing the dilute branch of the binodal curve at low particle volume fractions. The qualitative aspects of the predicted phase behavior of the polymer-solvent-particle system are the same as for the monomer-solvent-particle system. The two main differences, however, are (I) the shift of the two-phase region to lower solute volume fractions, which of course is related to the lower critical volume fraction for the polymersolvent solution and (II) a stronger tendency for φconc 2,s to increase upon addition of the particles. Numerically, kconc ) 1.5 and 4 for ∆w2 ) -2.0 and -2.5 kJ/mol, respectively, were obtained. Important for comparison with experimental data is the still very strong reduction of φdil 2,s upon addition of particles. For example φdil 2,s ) 0.025 at φ3 ) 0 dil -3 (Figure 10b). is reduced 8-fold to φdil 2,s ≈ 0.003 at φ3 ) 10

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Figure 9. Ternary polymer-solvent-particle phase diagram at T ) 320 K for r2 ) 100 with (a) ∆w2 ) -2.0 kJ/mol and (b) ∆w2 ) -2.5 kJ/mol as predicted by the polymer approach, displaying compositions of coexisting phases (filled circles), tielines (solid lines), and dilution lines connecting φconc and φdil 2 2 on the binary polymer-solvent axis with the particle corner (dotted lines). Other parameters are given in Table 2.

Experimentally, effects of an additive, like particles, on the phase behavior of a solute-solvent mixture are often presented in a Cartesian representation, e.g., with Tcp as a function of the concentration of additive at a specific φ2 (cf. Figure 4). In Figures 11 and 12, theoretical results, at the dilute branch of the binodal curve, are presented in this way. The figures show that Tcp of the two mixtures are strongly influenced by added particles, also at very small particle concentrations. Compared to the particlefree situation, the two mixtures in Figures 11 and 12 gave increases in Tcp by ca. 10 °C at the highest φ3. For mixtures with more attractive interactions between the solute and the particles, the increase in Tcp by the particle addition became larger. Discussion General. The particle approach predicts that the compositions of the two coexisting solutions are only weakly affected by the addition of particles. Moreover, virtually all particles partition to the concentrated phase, even at rather high particle concentrations in the concentrated phase. This partitioning of particles to one phase indicates that the mixing entropy was insignificant in comparison to the other free energy terms. At the highest


Langmuir, Vol. 21, No. 23, 2005 10869

Figure 12. Clouding temperature of the polymer-solventparticle mixture for r2 ) 100 with ∆w2 ) -2.0 kJ/mol at φ2 ) 0.0116 as a function of the particle volume fraction as predicted by the polymer approach. Other parameters are given in Table 2.

Figure 10. Magnification of the ternary polymer-solventparticle phase diagram shown in (a) Figure 9a and (b) Figure 9b, displaying an area containing the dilute branch of the binodal curve at low particle volume fractions in a Cartesian representation.

Figure 11. Clouding temperature of the monomer-solventparticle mixture for r2 ) 1 with ∆w2 ) -1.0 kJ/mol at φ2 ) 0.281 as a function of the particle volume fraction as predicted by the polymer approach. Other parameters are given in Table 2.

particle volume fractions shown in Figure 5 (10%), the validity of the assumptions made in the approach is already doubtful. Calculations at still higher particle contents thus become meaningless. The depletion of the bulk solution with respect to the solute, due to adsorption to the particle, could, in principle, be handled but would

be technically demanding. More seriously, at high particle concentrations the limitation is that the surface tension cannot be calculated in the region where the binary solutesolvent solution is unstable. Nevertheless, regardless of these limitations, the model was able to predict the initial effects on the phase boundaries of added particles. Because of the limitations of the particle approach, we also found it useful to investigate the predictions of the polymer approach. The advantage of this approach is that all three components, the solute, the solvent, and the particle, are treated on the same level, and the entire two-phase region may thus be calculated. The question is to what extent the polymer approach gives predictions that are valid also for a solid particle component. The free energy of mixing, as given by the Flory-Huggins theory, contains conceptually three different contributions.10 (i) Translational entropy of mixing. This should not be a problem, since by changing the degree of polymerization this entropy can be adjusted. (ii) Configurational entropy of mixing. Flexible polymers resist being concentrated not primarily due to the translational entropy, but because of the fact that a flexible polymer molecule has fewer possibilities to fold at each flexible bond when other polymer molecules are present and already occupy part of the available space. This contribution due to the internal degrees of freedom of the polymer molecules is absent from the entropy of mixing of solid particles. Hence, the polymer approach represents an overestimation of the entropy of mixing of a particle. (iii) Enthalpy of mixing. The polymer approach also overestimates the enthalpy of mixing, since a polymer molecule has a much larger surface area that interacts with the surrounding solution than a particle. The overestimated configurational entropy and the overestimated mixing enthalpy partially cancel each other, but to what extent is not clear. We will now proceed to discuss some of the features of the phase diagrams predicted by the two approaches and try to interpret these physically. Just like the particle approach, the polymer approach predicts a strong partitioning of the particle component to the concentrated phase. This is because the large enthalpic preference for the solute outweighs the loss in translational entropy; a decrease in r3 gives a less strong partitioning. A further similarity between the approaches is that they predict a strong depletion of the dilute phase with respect to the solute, when particles are added (kdil is strongly negative).

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We interpret this in terms of the attraction between the particles and the solute; as the particles partition to the concentrated phase, they bring some solute molecules along. This effect is obviously facilitated by the unfavorable interactions between the solute and the solvent. One may suspect that this effect is exaggerated for a particle modeled as a polymer, owing to the large surface area (many interaction points) of a polymer molecule. When the particle-solute attraction is made stronger (cf. Figure 7, panels a and b, and Figure 9, panels a and b, respectively), the depletion of the dilute phase also becomes stronger. Another result of a stronger particle-solute attraction, which also is apparent by looking at the same pairs of phase diagrams, is that the concentrated phase becomes more concentrated. This phase increases the number of favorable particle-solute interactions by expelling solvent. We can see, furthermore, that the extent of this effect varies with the degree of polymerization of the solute. When the solute is a monomer, the osmotic pressure of the concentrated phase is still high, owing to the translational entropy of mixing. Hence (Figure 7a), conc φ2,s actually decreases with increasing particle concentration (kconc is negative) when the effective solute-particle interaction is only modestly attractive (∆w2 ) - 1.0 kJ/ mol). When the attraction is sufficiently strong, as in the conc example drawn in Figure 7b (∆w2 ) - 4.0 kJ/mol), φ2,s initially increases with increasing particle concentration (kconc is positive). Comparison between Experiments and Modeling. Figure 4 (experiment) and Figures 11 and 12 (theoretical modeling) show the variation of the Tcp of a polymer solution as a function of the concentration of added particles. Only the dependence at the dilute side of the two-phase region is shown, since the large solution viscosity make experiments on the concentrated side difficult. As previously mentioned, the experiment concerns a LCST mixture and the theory an UCST mixture, implying that a given physical response is related to opposite temperature changes. This is indeed the situation, and in both cases, the shift in Tcp shows that the twophase region of the polymer solution increases in extension as the particles are added. Hence, the predictions of the particle and polymer approaches are here consistent with the experimental results. Moreover, the experimental analyses of the particle distribution in phase-separated polymer-particle solutions showed that the particles had a strong preference for the concentrated phase, in agreement with the phase diagrams obtained in the particle and polymer approaches (see Figures 5, 7, and 9). Relevance to other Mixtures. Added macromolecules or association colloids may, like added solid particles, influence the phase behavior of a polymer solution, when the polymer molecule and the macromolecule associate. One example concerns mixtures of polymers and surfactants that form mixed micelles at the critical association concentration (cac) of the surfactant. When an ionic surfactant is added to a solution of an LCST polymer, Tcp decreases at surfactant concentrations close to the cac.28-32 (28) Carlsson, A.; Karlstro¨m, G.; Lindman, B. Langmuir 1986, 2, 536-537. (29) Wormuth, K. R. Langmuir 1991, 7, 1622-1626. (30) Pandit, N. K.; Kanjia, J.; Patel, K.; Pontikes, D. G. Intern. J. Pharm. 1995, 122, 27-33. (31) Thuresson, K.; Nystro¨m, B.; Wang, G.; Lindman, B. Langmuir 1995, 11, 3730-3736.

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At higher surfactant concentrations, the properties of the solutions are mainly determined by the high solubility of the ionic surfactant, and Tcp of the LCST polymer solution increases. These changes in Tcp at different concentrations of added surfactant were also found in a modeling study by Karlstro¨m et al. focusing on the phase behavior of polymer-surfactant mixtures.27 The effects of added ionic surfactant on a polymer mixture are thus in close analogy to what is shown with the polymer approach for the polymer-particle mixtures having a large polymerparticle attraction. A further similarity to the polymerparticle mixtures is that a partitioning of surfactant to the polymer-rich phase is found in the phase-separated polymer-ionic surfactant solutions.33,34 Another class of macromolecules that are well-known to associate with polymer molecules are proteins. We are aware of only one group who has studied the effect on Tcp of a polymer solution by added protein. They found that Tcp of a cationic hydrophobically modified ethylene oxide polymer (HM-EO) decreases by 1-3 °C (depending on the polymer concentration) by addition of an oppositely charged protein, BSA.35,36 However, the opposite charge of the protein and the polymer makes these mixtures different from the polymer-particle systems considered here. Conclusions In a mean-field lattice theory, added particles were seen to influence the phase behavior of a monomer or a polymer solution. The added particles increased the two-phase region at the dilute side of the critical point. At the concentrated side of the critical point, the effect of the particles depended on the strength of the attraction between the solute and the particles: with a strong attraction, a small addition of particles increased the twophase region, whereas the opposite trend occurred with a weak attraction. The added particles influenced the phase boundary to the dilute phase much more strongly than the phase boundary to the concentrated phase. Furthermore, a partitioning of the added particles to the concentrated phase in phase-separated polymer solutions was observed. In agreement with the theoretical predictions, experimental results from aqueous dispersions of EHEC including different concentrations of polystyrene latex particles showed that the influence on the two-phase region was stronger at a higher particle concentration and that a partitioning of the particles to the concentrated phase occurred. Acknowledgment. We are grateful to Håkan Wennerstro¨m for stimulating discussions and encouragement and to Leif Karlson at Akzo Nobel Surface Chemistry AB, Stenungsund, Sweden, for the rheology measurements. This work was financed by the Centre for Amphiphilic Polymers from Renewable Resources (CAP) and the Swedish National Research Council (VR). LA051343J (32) Thuresson, K.; Lindman, B. J. Phys. Chem. B 1997, 101, 64606468. (33) Piculell, L.; Lindman, B.; Karlstro¨m, G. In Polymer-Surfactant Systems; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998 (34) Piculell, L.; Thuresson, K.; Lindman, B. Polymer Adv. Technol. 2001, 12, 44-69. (35) Jo¨nsson, M.; Johansson, H.-O. J. Chromatogr. A 2003, 983, 133144. (36) Jo¨nsson, M.; Tjerneld, F. Private communications.

Influence of Added Particles on the Phase Behavior of Polymer

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