Specific Ion-Dependent Attraction and Phase Behavior of Polymer

Langmuir 2004, 20, 11393-11401

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Specific Ion-Dependent Attraction and Phase Behavior of Polymer-Coated Colloids Kildong Hwang, Hung-Jen Wu, and Michael A. Bevan* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122 Received August 25, 2004. In Final Form: October 1, 2004 This paper presents results demonstrating the role of temperature and specific ions in mediating attraction between polymer-coated colloids and determining their equilibrium phase behavior. In particular, theoretical predictions of continuum van der Waals attraction between poly(ethylene oxide)-poly(propylene oxide)poly(ethylene oxide) (PEO-PPO-PEO)-coated polystyrene colloids are used to explain measured temperature and specific ion-dependent fluid-gel transitions in dispersions of these particles. Building on previous studies of PEO-PPO-PEO-coated polystyrene colloids dispersed in aqueous NaCl media, this work reports rheologically measured fluid-gel transitions as a function of temperature and NaCl/MgSO4 composition. Adhesive-sphere predictions of percolation thresholds are fit to measured fluid-gel data by allowing the adsorbed copolymer layer thickness as a single adjustable parameter. This allows the attraction between the PEO-PPO-PEO layers to be interpreted as a function of temperature and NaCl/MgSO4 composition. Quantitative predictions of a polymeric van der Waals attraction associated with the layer collapse in diminishing solvent conditions provides a simple mechanism for explaining the measured fluid-gel data as a dynamic percolation transition. Ultimately, this work identifies the importance of continuum polymeric van der Waals attraction for explaining specific ion-dependent phenomena.

Introduction In this paper, we compare theory and experiments to understand how temperature and specific ions together mediate attraction between colloids bearing adsorbed polymer layers and their resulting structural transitions. We show that measured fluid-gel transitions of polymercoated particles (see Figure 1), which are highly dependent on temperature, particle concentration, and electrolyte composition, can be explained by a net interparticle attraction dominated by continuum van der Waals contributions due to the adsorbed polymer layers. In particular, increasing temperature and specific ion concentration produce a continuous dimensional collapse of the adsorbed polymer layers, which alters their nonuniform dielectric properties. This solvent quality-dependent polymer layer collapse produces significantly stronger van der Waals attractions on the kT energy scale, which ultimately drives the observed fluid-gel transitions. The combined contributions of polymeric van der Waals and specific ion effects together have not been previously considered but appear to be significant for describing specific ion-dependent phenomena in a broad range of natural and synthetic systems relevant to numerous fundamental and technological problems. Total internal reflection microscopy has previously been used to perform the first direct measurements of continuum van der Waals attraction between a colloidal particle and a surface bearing adsorbed polymer in good solvent conditions.1 The continuum van der Waals contribution from highly solvated polymer layers was found to be relatively insignificant due to the similarity of the layer and solvent dielectric properties. In a subsequent study, it was shown that densification of the polymer layer with decreasing solvent conditions produced sufficient van der Waals attraction to drive rheologically measured * Author to whom correspondence should be addressed. E-mail: [email protected]. (1) Bevan, M. A.; Prieve, D. C. Langmuir 2000, 16, 9274.

Figure 1. Schematic illustration of relevant separations defined for the interaction of colloidal particles with adsorbed polymer as a function of temperature and specific ion concentration (not to scale).

fluid-gel transitions.2,3 These results could be predicted in an a priori fashion using existing theories for the van der Waals attraction between polymer-coated colloids4 with no adjustable parameters. These initial studies were conducted in 0.5 M NaCl, and temperature was used to manipulate solvency of poly(ethylene oxide) moieties contained in PEO-PPO-PEO triblock copolymers. This previous work demonstrated how temperature mediates adsorbed polymer layer density profiles and their associated nonuniform dielectric properties to influence the net van der Waals attraction between polymer-coated colloids. In this paper, we are concerned with the role of specific ion effects (SIE) in addition to temperature for controlling attraction between polymer-coated colloids. SIE are critical to a broad range of well-established phenomena ranging from solvency effects in aqueous polymer solutions5 to (2) Bevan, M. A.; Scales, P. J. Langmuir 2002, 18, 1474. (3) Bevan, M. A.; Petris, S. N.; Chan, D. Y. C. Langmuir 2002, 18, 7845. (4) Parsegian, V. A. In Physical Chemistry: Enriching Topics from Colloid and Surface Science; Mysels, K. J., Ed.; Theorex: La Jolla, CA, 1975; p 27. (5) Boucher, E. A.; Hines, P. M. J. Polym. Sci. 1976, 14, 2241.

10.1021/la0478752 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/19/2004

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complex biological phenomena.6,7 Although SIE have been well documented in numerous experiments, data are generally interpreted simply by ranking the degree to which specific ions affect some observable phenomenon and then comparing with the classic Hofmeister series.8 Theoretical efforts to describe the role of specific ions on the solubility of nonionic macromolecules in aqueous media have focused on a competition between ion and segment solubility related to water structure-solvation effects,9 although the presence of perturbed water structures have been difficult to verify experimentally.10 Recent efforts have focused on how specific ions with different polarizabilities affect the interaction between colloidal and macromolecular species via their influence on dispersion forces,11 although these studies have not dealt directly with nonionic macromolecules. Despite considerable theoretical effort, SIE continue to defy description by a universal theory. Colloids composed of “solvent quality” sensitive polymeric coatings are of technological interest because of their use as controlled-release vehicles in drug delivery.12 Thermo-responsive polymer-coated colloids are also used as model systems to understand equilibrium phase behavior of particles with potentials that can be sensitively tuned from hard spheres to adhesive spheres.13 The dominant rheological transition displayed by such particles with increasing attraction is generally from a viscous fluid to an elastic solid.14 Despite numerous studies on thermo-reversible colloids, some uncertainty concerning the exact nature of these transitions remains; measured transitions have been attributed to equilibrium dynamic percolation,14 irreversible static percolation, or dynamically arrested attractive glass formation.15-17 Understanding these transitions requires independent measurements of static and dynamic structure and the interparticle potential. Although scattering and rheological methods are often used to detect structural transitions in colloidal systems, in general, interparticle attraction on the kT energy scale is not known. In this work, we show that continuum van der Waals attraction between PEO-PPO-PEO-coated polystyrene colloids can be used to explain their fluid-gel transition behavior as a function of temperature, electrolyte composition, and particle concentration. Solvent quality is diminished in each experiment using a combination of increasing temperature and specific ion concentration, which is limited to NaCl and MgSO4 in this study. Because continuum van der Waals attraction between the PEOPPO-PEO layers in aqueous NaCl media is well understood from previous TIRM measurements,1 in this work, the PEO-PPO-PEO layer thickness as a function of MgSO4 concentration is allowed as a single adjustable (6) Collins, K. D.; Washabaugh, M. W. Quart. Rev. Biophys. 1985, 18, 323. (7) Cacace, M. G.; Landau, E. M.; Ramsden, J. J. Quart. Rev. Biophys. 1997, 30, 241. (8) Hofmeister, F. Arch. Exp. Pathol. Pharmakol. 1888, 24, 247. (9) Florin, E.; Kjellander, R.; Erikson, J. C. J. Chem. Soc., Faraday Trans. 1 1984, 80, 2889. (10) Omta, A. W.; Kropman, M. F.; Woutersen, S.; Bakker, H. J. Science 2003, 301, 347. (11) Bostrom, M.; Williams, D. R. M.; Ninham, B. W. Phys. Rev. Lett. 2001, 87, 168103. (12) Langer, R.; Peppas, N. A. AIChE J. 2003, 49, 2990. (13) Gast, A. P.; Russel, W. B. Phys. Today 1998, 51, 24. (14) Grant, M. C.; Russel, W. B. Phys. Rev. E 1993, 47, 2606. (15) Mallamace, F.; Gambadauro, P.; Micali, N.; Tartaglia, P.; Liao, C.; Chen, S.-H. Phys. Rev. Lett. 2000, 84, 5431. (16) Chen, S.-H.; Chen, W.-R.; Mallamace, F. Science 2003, 300, 619. (17) Pham, K. N.; Puertas, A. M.; Bergenholtz, J.; Egelhaaf, S. U.; Moussaid, A.; Pusey, P. N.; Schofield, A. B.; Cates, M. E.; Fuchs, M.; Poon, W. C. K. Science 2002, 296, 104.

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parameter to fit equilibrium percolation threshold predictions to measured fluid-gel data. This yields the polymer thickness and interparticle attraction as a function of temperature and NaCl/MgSO4 composition. This provides a mechanism to explain the continuously reversible interparticle attraction in terms of the PEO-PPO-PEO layer contribution mediated by temperature and specific ion-dependent solvent quality effects. Ultimately, our results indicate new fundamental insights into specific ion-dependent polymeric forces and colloidal phase behavior, which provides a basis to interpret and manipulate a broad range of phenomena in both synthetic and biological systems. Theory Attraction between Colloids Bearing Nonuniform Adsorbed Polymer Layers. To rigorously compute the continuum attraction between polymer-coated colloids as depicted in Figure 1, it is necessary to consider both the uniform core particle and nonuniform adsorbed polymer layer dielectric properties. The 1975 theory of Parsegian18 is one of the few attempts to include nonuniform adsorbed polymer properties when calculating the net van der Waals attraction between a pair of polymer-coated colloids. The total separation-dependent interaction potential between polymerically stabilized particles (see Figure 1) can generally be modeled using a repulsive hard-wall term and Parsegian’s attractive composite particle van der Waals term as18

{

E(r) ) ∞, 0 < r < 2(a + δ) -

dln (r1,ω) × r1

∫0∞dω∫0a + δdr1

p 32π

dln (r2,ω) G(r1,r2), 2(a + δ) < r r2

∫0a + δ dr2

(1)

where G(r1,r2) is a geometric factor given by

G(r1,r2) )

[

2r1r2 2

(a + δ) - (r1 + r2)

2

+

ln

2r1r2 2

(a + δ) - (r1 - r2)2

(

+

)]

(a + δ)2 - (r1 + r2)2

(a + δ)2 - (r1 - r2)2

(2)

where r1 and r2 are the radial coordinates for particles 1 and 2. The radially dependent dielectric function, ln (r,ω), describes the composite particle material properties, including the uniform core particle and nonuniform adsorbed polymer density profile. For uniform core particles of material 1 with nonuniform layers of material 2 in a uniform medium of material 3 as shown in Figure 1, ln (r,ω) in eq 1 is given by (18) Parsegian, V. A. J. Colloid Interface Sci. 1975, 51, 543.

Attraction and Behavior of Polymer-Coated Colloids

Langmuir, Vol. 20, No. 26, 2004 11395

ln (r,ω) )

{

ln 1(ω)H(a - r)

}core particle

+ ln 2(r,ω)[H(a + δ - r) - H(a - r)] }adlayer + ln 3(ω)H(a + δ - r)

analytical PY results

}medium (3)

where the polymer layer thickness, δ, is the radial distance from the core-particle surface to the location at which the adsorbed polymer density profile vanishes. The Heaviside step function, H, is used to indicate how material properties vary as a function of the radial particle coordinates. Adhesive Sphere Potential and Phase Behavior. The “adhesive sphere” (AS) potential due to Baxter19 is convenient for modeling equilibrium phase behavior of colloids with short-range attractive interactions. A number of analytical expressions19-29 and Monte Carlo (MC) simulation results30-33 are readily available in the literature for comparison with experimental results. The AS potential can be written in a similar form to eq 1 with a repulsive hard-wall term, an attractive surface adhesion term, and a noninteraction term for separations beyond the range of the surface adhesion as19

{

EAS(r) )

∞, 0 < r < 2(a + δ)

lim

σf2(a + δ)

-ln

(

σ 12τ(σ - 2(a + δ))

)

2(a + δ) < r < σ (4)

0, σ < r

where τ characterizes the magnitude of the contact adhesion and all remaining variables are defined in Figure 1. An arbitrary potential, E(r), can be mapped onto the AS potential using the “B2 device”,20,23 which involves equating the second virial coefficients, B2, of the two potentials. This allows an equivalent value of τ to be specified for any potential, E(r), using

8(a + δ)3 ) 2πτ 8(a + δ)3 + 3

∫0∞ r2{exp[-

Table 1. Percolation Threshold in Terms of Adhesive Sphere Parameter, τ, and Particle Volume Fraction, O, Used to Construct Phase Diagram from Analytical Predictions25 (Eq 6) Using the PY Closure and MC Simulations31

] }

E(r) - 1 dr (5) kT

Using eq 5, values of the adhesion parameter, τ, can be determined for the potential in eq 1, which allows for prediction of phase behavior for polymer-coated colloids from tabulated τ, φ data. Of relevance to the current work, values for the percolation threshold of adhesive spheres are reported in Table 1 from the MC simulation results (19) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. (20) Regnaut, C.; Ravey, J. C. J. Chem. Phys. 1989, 91, 1211. (21) Seaton, N. A.; Glandt, E. D. J. Chem. Phys. 1987, 86, 4668. (22) Cummings, P. T.; Perram, J. W.; Smith, E. R. Mol. Phys. 1976, 31, 535. (23) Stell, G. J. Stat. Phys. 1991, 63, 1203. (24) Chiew, Y. C. J. Chem. Phys. 1999, 110, 10482. (25) Chiew, Y. C.; Glandt, E. D. J. Phys. A 1983, 16, 2599. (26) Smithline, S. J.; Haymet, A. D. J. J. Chem. Phys. 1985, 83, 4103. (27) Barboy J. Chem. Phys. 1974, 61, 3194. (28) Coniglio, A.; Angelis, U. D.; Forlani, A.; Lauro, G. J. Phys. A: Math. Gen. 1977, 10, 219. (29) Marr, D. W.; Gast, A. P. J. Chem. Phys. 1993, 99, 2024. (30) Kranendonk, W. G. T.; Frenkel, D. Mol. Phys. 1988, 64, 403. (31) Lee, S. B. J. Chem. Phys. 2001, 114, 2304. (32) Seaton, N. A.; Glandt, E. D. J. Chem. Phys. 1987, 87, 1785. (33) Seaton, N. A.; Glandt, E. D. J. Chem. Phys. 1986, 84, 4595.

MC simulation

τ

φ

τ

φ

0.10 0.15 0.20 0.35 0.50 0.70

0.095 0.174 0.218 0.297 0.345 0.391

0.10 0.15 0.20 0.35 0.50 0.70

0.133 0.177 0.210 0.270 0.311 0.352

of Lee31 and analytical results of Chiew and Glandt given by25

τ)

19φ2 - 2φ + 1 12(1 - φ)2

(6)

Experimental Section Materials. Surfactant-free polystyrene (PS) latex particles were prepared using the emulsion polymerization method of Goodwin, et al.34 These particles had an average diameter of 410 nm and polydispersity index of 1.09 as characterized by static and dynamic light scattering measurements using a Brookhaven Instruments Corporation BI-200SM goniometer setup. The adsorbed polymer used in this work is poly(ethylene oxide)poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) copolymer supplied by BASF Wyandotte Corporation, which is sold under the commercial name “F108 Pluronic”. The average total molecular weight of the copolymer used in this study is 14 000 g mol-1 with a PEO molecular weight of 10 750 g mol-1 and a PPO molecular weight of 3250 g mol-1.35 These molecular weights correspond to an average anchor block of 56 PPO repeat units and two buoy blocks, each with 122 PEO repeat units. Since PEO is known to degrade by oxidation in aqueous solutions,36 experiments were performed within 2 weeks of dispersion preparation. Sodium chloride (NaCl) and magnesium sulfate (MgSO4) salts used to control ionic strength in these experiments were purchased from Aldrich and were used without further purification. All solutions were prepared using doubledeionized water. Methods. High-volume-fraction PS particle dispersions were polymerically stabilized by slow addition of 1000 ppm PEOPPO-PEO solution to dispersions of less than 10% volume fraction over a 24 h period while stirring. Centrifugation was used to remove excess bulk polymer and to replace the supernatant with various compositions of NaCl and MgSO4 with a total concentration of 0.5 M to screen electrostatic interactions. These washing steps were repeated until dispersions displayed an unchanging shear-dependent viscosity. In each wash step, the total centrifugation time was 75 min at a rate of 5000 rpm. Once the PEO-PPO-PEO-stabilized PS particle dispersions were adequately cleaned, centrifugation was used to prepare two stock dispersions at the lowest and highest volume fractions. All other dispersions used in this study were prepared by mixing these two stock batches in different proportions. The core-particle volume fraction in the two stock batches was determined gravimetrically using the density of PS as 1.055 kg m-3, which resulted in the same volume fraction whether the adsorbed polymer layer density was included or not. Viscosity and shear storage modulus were measured as a function of temperature for a number of particle concentrations and electrolyte compositions using a Paar-Physica MCR 300 rheometer. The cone-and-plate geometry was used with a 75 mm diameter cone having an angle of 1°. A temperature-controlled plate was used with the cone and plate geometry, and the sample (34) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464. (35) Baker, J. A.; Berg, J. C. Langmuir 1988, 4, 1055. (36) Aklyene Oxides and their Polymers; Bailey, F. E., Koleske, J. V., Eds.; Marcel Dekker: New York, 1990; Vol. 35, p 261.

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Figure 2. Low shear viscosity, η (Pa s) (open symbols), and shear storage modulus, G′ (Pa) (closed symbols), for a polystyrene core-particle volume fraction of φcore ) 0.14 as a function of temperature T (°C) and NaCl/MgSO4 concentration. Electrolyte conditions are designated by circles (O) for 0.5 M NaCl, squares (0) for 0.45 M NaCl/0.05 M MgSO4, triangles (4) for 0.4 M NaCl/0.1 M MgSO4, and inverted triangles (3) for 0.3 M NaCl/0.2 M MgSO4. The horizontal dashed line (- - -) indicates critical viscosity and storage modulus, and vertical lines (- - -) indicate liquid-gel transition temperatures for each electrolyte composition. was protected with a cover, solvent trap, and low-viscosity silicone oil to protect samples from evaporation. Various heating rates were checked in each experiment to check for thermal equilibration at each temperature. Steady shear experiments were performed in an equilibrium mode to obtain a steady rate at each applied stress. Oscillatory shear measurements were performed with a strain amplitude of 0.005 at a frequency of 3 Hz to generally obtain measurements of linear viscoelastic behavior for fluid and gelled structures at all temperatures and particle concentrations.

Results and Discussion Rheological Measurements of Temperature-Specific Ion Liquid-Gel Transitions. In the following, we report effects of specific ions on temperature-dependent rheological properties of latex colloids bearing adsorbed polymer layers, which we explain as an equilibrium percolation phenomenon mediated by interparticle van der Waals attraction. Figure 2 reports low shear viscosity, η (Pa s), and shear storage modulus, G′ (Pa), for a temperature- and MgSO4-dependent liquid-gel transition of 410 nm polystyrene particles with adsorbed PEOPPO-PEO copolymer. Viscosity and modulus data are designated by open and closed symbols. The core-particle volume fraction is φcore ) 0.14, and the effective particle volume fraction is φeff ) 0.19 due to the additional volume occupied by the 20 nm thick PEO-PPO-PEO layer at 25 °C in 0.5 M NaCl.1 Because the adsorbed PEO-PPOPEO layer collapses with increasing temperature and MgSO4 concentration, the effective volume fraction decreases for T > 25 °C. Four different NaCl/MgSO4 solutions were investigated with a fixed 0.5 M concentration but variable ionic strength due to a valence of two for both Mg and SO4 ions. Specific ion effects (SIE) generally depend on ion concentration rather than ionic strength.37 The electrolyte compositions investigated in Figure 2 are 0.5 M NaCl, 0.45 M NaCl/0.05 M MgSO4, 0.4 M NaCl/0.1 M MgSO4, and 0.3 M NaCl/0.2 M MgSO4. The temperature was varied between T ) 20 and 95 °C in each experiment. (37) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983.

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Viscosity measurements in Figure 2 initially correspond to hard-sphere values at 25 °C based on the effective particle volume fraction. With increasing temperature, the viscosity displays a small initial decrease before increasing by orders of magnitude in the vicinity of a unique temperature range for each electrolyte composition. The initial decrease in viscosity with increasing temperature occurs due to both a decreasing continuous medium viscosity and decreasing effective hydrodynamic particle size, while the abrupt increase results from aggregation and subsequent network formation. The shear modulus in Figure 2 is initially undetectable for temperatures near 25 °C, but becomes finite and rapidly increases at elevated temperatures within some critical range specific to each electrolyte composition. Initially imperceptible values of the modulus at low temperatures are consistent with a purely viscous, dissipative fluid, whereas the increased modulus at elevated temperatures is taken as evidence of elastic network formation, or a gelation process. Each sample in Figure 2 is considered to be thermally equilibrated since slower heating rates produced no discernible differences from the reported curves. The horizontal dashed line at η ) 0.02 Pa s and G′ ) 1 Pa in Figure 2 is taken to indicate the transition from a viscous fluid to an elastic gel network, or a fluid-gel transition of the PEO-PPO-PEO-coated PS colloids. Although associating specific values of viscosity and storage modulus with a fluid-gel transition is somewhat arbitrary, choosing a temperature in the middle of the steep transition from a viscous fluid to an elastic solid is reasonable without a more direct detection method.38 The measured fluid-gel transition can be associated with an equilibrium, dynamic percolation threshold, which is defined from a statistical mechanical viewpoint to occur when half of all independent particle configurations for given thermodynamic conditions have at least one cluster spanning the system.21 Although it is not generally possible to precisely identify the presence of single spanning clusters from rheological measurements due to their indirect nature and limited sensitivity, detection of a divergent viscosity and finite modulus together38 can be expected to consistently identify fluid-gel transitions correlated with percolation threshold predictions. The viscosity and modulus measurements in Figure 2 are expected to bound the percolation threshold; the initial increase in viscosity may be due to changing cluster size, shape, and attraction that occur short of percolation, while detection of a finite storage modulus probably occurs beyond the percolation threshold for stresses exceeding that for linear elastic deformation of a single percolating cluster. In any case, the approaches employed in Figure 2 are consistent with typical literature rheological techniques and static14 and dynamic15 light-scattering methods for experimentally estimating percolation thresholds. Vertical dashed lines in Figure 2 indicate the fluid-gel transition temperature, Tgel, for each of the four ionicstrength conditions investigated in Figure 2. A trend of decreasing Tgel with an increasing proportion of MgSO4 is observed for the data set in Figure 2. Additional observations revealed the measured fluid-gel transitions to be thermo-reversible in each case, which has also been demonstrated in previous light-scattering measurements of PEO-PPO-PEO-coated PS colloids.39,40 Thermoreversibility suggests the equilibrium nature of the (38) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1998, 42, 1451. (39) Bevan, M. A.; Prieve, D. C. In Polymers in Particulate Systems: Properties and Applications; Hackley, V. A., Somasundran, P., Lewis, J. A., Eds.; Marcel Dekker: New York, 2001; Vol. 104. (40) Bevan, M. PhD Dissertation, Carnegie Mellon University, 1999.


Figure 3. Summary phase diagram of experimental liquidgel transition temperatures from rheological measurements with theoretical percolation threshold fits from analytical25 and Monte Carlo simulation31 results (see text for details). Experimental data (open symbols) for polystyrene core particle volume fractions of φcore ) 0.11, 0.14, 0.21, 0.32, and electrolyte conditions designated by circles (O) for 0.5 M NaCl, inverted triangles (3) for 0.45 M NaCl/0.05 M MgSO4, squares (0) for 0.4 M NaCl/0.1 M MgSO4, diamonds (]) for 0.3 M NaCl/0.2 M MgSO4, and triangles (4) for 0.2 M NaCl/0.3 M MgSO4. Filled symbols indicate the critical point (TC,φC) for each electrolyte composition.

measured transitions in Figure 2. This data set clearly demonstrates the importance of SIE and temperature together in controlling attractive interactions and phase behavior of polymer-coated colloids in aqueous media. With evidence of synergistic SIE and temperature-dependent liquid-gel behavior at one core-particle volume fraction, a number of rheology experiments similar to those reported in Figure 2 were conducted for a range of particle volume fractions. These results are reported and discussed in the following section. Phase Diagram for Temperature-Specific IonDependent Fluid-Gel Transition. By performing several experiments similar to the one reported in Figure 2 for a range of volume fractions, a summary phase diagram is constructed in Figure 3 showing liquid-gel transition temperatures, Tgel, as a function of core polystyrene particle concentration, φcore, and NaCl/MgSO4 composition. Four core polystyrene particle volume fractions were investigated including φcore ) 0.11, 0.14, 0.21, and 0.32 (with effective volume fractions of φeff ) 0.15, 0.19, 0.28, and 0.42). The data presented in Figure 3 represent the fluid-gel transition for each electrolyte composition, which can immediately be observed to display a strong dependence on MgSO4 concentration. Leastsquares fit lines initially provide empirical representations of the fluid-gel transition locus for each electrolyte composition in Figure 3 by assuming the simplest linear dependence on particle concentration. The atypical, inverted phase behavior for each electrolyte composition in Figure 3 is characterized by each fluid-gel transition occurring at elevated rather than decreased temperatures. This inverted behavior occurs due to the lower critical solution temperature behavior of the PEO moieties in the adsorbed PEO-PPO-PEO layer.3,40 To explain the fluid-gel transitions in Figure 3, a number of candidate attractive interactions could be considered including (1) continuum van der Waals at-

Langmuir, Vol. 20, No. 26, 2004 11397

traction between PS core particles,41 (2) a net attraction due to favorable “mixing” of PEO-PPO-PEO layers in poor solvent conditions,37,42 and (3) continuum van der Waals attraction between adsorbed PEO-PPO-PEO layers.3,18,40 With regard to the relative core PS particle and PEO-PPO-PEO layer van der Waals contributions, eq 1 reveals the dominant contribution for all SIE and temperature conditions is due to the nonuniform dielectric properties of the adsorbed PEO-PPO-PEO. This results from the relatively large thickness of the PEO-PPOPEO layer compared with the particle radius at all solvent conditions (a/δ ≈ 10 at 25 °C), which prevents significant van der Waals attraction between the 410 nm PS core particles. The surprising aspect of this finding is that the PEO-PPO-PEO layer by itself can generate significant continuum van der Waals attraction to produce equilibrium phase transitions of the polymer-coated colloids. This conclusion has also been discussed in previous work involving PEO-PPO-PEO-coated PS particles in 0.5 M NaCl aqueous media.1-3,39-41 Curve Fits to Fluid-Gel Transitions Using a Continuum van der Waals Potential. Although the dashed lines in Figure 3 were initially provided as empirical representations of each liquid-gel transition, these lines ultimately correspond to theoretical interpretations of the data based on a subsequent fitting procedure using anaytical25 and MC simulation results31 and the van der Waals potential in eq 1. To obtain the “curve fits” in Figure 3, the only information required is the spatially varying PEO-PPO-PEO layer dielectric properties, 2(r,ω), in eq 1 as a function of temperature and electrolyte composition. Previous measurements provide all the information necessary to fully specify the spatially varying layer dielectric properties, 2(r,ω), as a function of temperature in 0.5 M NaCl as3

where Cuv(r,T*) ) 0.762 + 0.417E-3‚ν(r,T*) and ωuv ) 1.90E16 rad s-1 are used to model the dielectric properties of the PEO-PPO-PEO layers as a function of polymer concentration. These parameters were obtained from “Cauchy plots”43 for a range of PEO-PPO-PEO solution concentrations.3 The reduced layer thickness as a function of temperature, δ*(T*), was measured directly using total internal reflection microscopy and indirectly using smallangle light scattering,1,2,39 The adsorbed amount, which is the integral of eq 8 over the layer thickness, was measured using reflectometry to be 2 mg m-2.40 A reduced layer thickness, δ*, is defined in eq 9 to quantify the (41) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (42) Fleer, G. J.; Stuart, M. A. C.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: New York, 1993. (43) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3.

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dimensional collapse of the PEO-PPO-PEO layers relative to their initial and final thicknesses, which have previously been used to generate universal curves for several PEO-PPO-PEO molecular weights adsorbed to PS particles.2,3 A reduced temperature, T*, is defined in eq 10 to capture the temperature dependence of the layer collapse relative to the lower critical solution temperature of ΘL ) 92 °C for PEO in 0.5 M NaCl, which corresponds to the phase separation temperature for bulk PEO solutions. A single assumption contained in eq 8 is that the PEO-PPO-PEO layer has a parabolic density profile for all solvent conditions, which is reasonable on the basis of theory and measurements of solvent quality effects in other material systems.42,44 In previous work, eqs 7-10, along with PS and water dielectric properties,45 were used to calculate the potential in eq 1 and phase behavior in good agreement with rheological data for 0.5 M NaCl conditions using no adjustable parameters. In this work, the layer thickness is allowed as a single adjustable parameter, δ(T,CSI), which, in addition to having a temperature dependence, is also a function of specific ion concentration, CSI. To simplify the process of constructing phase diagrams, the net potential at any temperature can be specified using an empirical potential fit to eq 1 given by

{

E(r,T,C) ) ∞, 0 < r < 2(a + δ(T,C)) 28.97 - 39.02h + 0.4759h2 - (1.600 × 10-3)h3 a , 180 1 - 0.9158h + 0.07360h2 + 0.05770h3

( )

r > 2(a + δ(T,C)) (11)

where the units on all constants are in nanometers raised to the appropriate power, and the fitting procedure is described in detail elsewhere.3 This potential can then be mapped onto the AS potential (eq 4) using the “B2 device” in eq 520,23 to construct a theoretical T, φcore phase diagram using literature τ, φ data from the analytical result in eq 625 and the MC results31 summarized in Table 1. The “B2 device” has previously been shown to produce accurate phase-behavior predictions for attractive particles, provided the range of attraction does not exceed 0.10a,20 which is the case for the potential in eq 1 (and eq 11). By adjusting δ(T,CSI) in eq 11 to produce a locus of T, φcore points identical to each of the empirically fit lines in Figure 3, phasebehavior predictions are obtained in excellent agreement with the measured data. An additional assumption in adjusting only δ(T,CSI) in the potential in eq 11 is that specific ions only affect the van der Waals forces by determining the PEO-PPO-PEO density profile via solvent quality effects9 but not via ion specific dispersion forces.11 This issue will be revisited following a morethorough analysis of the data in Figure 3. In any case, the curve fits based on adjusting only δ(T,CSI) in the potential in eq 11 demonstrate that the phase behavior in Figure 3 can be fully explained by temperature and SIE-mediated polymeric van der Waals attraction. By fitting δ(T,CSI) over the temperatures and specific ion compositions in Figure 3, fluid-fluid3 and fluidsolid26,29 coexistence can also be predicted using eq 11 with tabulated AS results. This allows the measured fluid(44) Karim, A.; Satija, S. K.; Douglas, J. F.; Ankner, J. F.; Fetters, L. J. Phys. Rev. Lett. 1994, 73, 3407. (45) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285.

gel data and equilibrium percolation predictions to be compared with other equilibrium transitions at low- and high-concentration limits on the phase diagram in Figure 3. The spinodal is not shown for each data set in Figure 3 for clarity; however, filled symbols indicate the critical point (TC,φC) for each electrolyte composition. The lowest volume-fraction data points measured for each data set nearly coincide with the critical points and percolation threshold predictions similar to other studies.14 The critical core-particle volume fraction for every electrolyte composition is φcore ) 0.113 (φeff ) 0.15), and the critical temperatures from fits to MC results31 are TC ) 91, 85, 76, 69, and 56 °C for CMgSO4 ) 0, 0.05, 0.10, 0.20, and 0.30 M. The percolation locus for each electrolyte composition cuts across the phase diagram from near the critical point to conditions at lower temperatures and higher coreparticle volume fractions at φcore ) 0.32 (φeff ) 0.42). Fluidsolid coexistence is predicted to occur in all cases for coreparticle volume fractions around φcore ≈ 0.40-0.45 (φeff ) 0.53-0.59); however, samples more concentrated than φcore ) 0.32 (φeff ) 0.42) could not be reproducibly prepared. This presumably results from dynamical factors, possibly due to glassy states driven by both higher particle concentrations and attractive interactions.15-17 Although predicted freezing conditions extrapolate almost linearly from each percolation locus to higher particle concentrations, fluid-solid predictions are not shown in Figure 3 because they are not easily attained thermodynamic states, as evident in our results. Excellent agreement is observed between fluid-gel measurements and equilibrium percolation threshold predictions. This illustrates the straightforward prediction of the dynamic percolation of attractive polymer-coated colloids using a continuum polymeric van der Waals potential mediated by SIE and temperature-dependent solvent quality effects. In the following section, we examine more closely the temperature and MgSO4-dependent PEO-PPO-PEO thickness, δ(T,CSI), and resultant attraction interpreted from the curve fits in Figure 3. Temperature-Specific Ion-Dependent Polymeric Layer Thickness. Figure 4a reports fit values of δ(T,CSI) from Figure 3 as a function of temperature for each electrolyte composition. Data are limited to the fluid-gel transition temperatures reported in Figure 3. The symbol convention is the same as in Figure 3, with open symbols corresponding to curve fits from MC simulation results (Table 1) and closed symbols from analytical percolation results (eq 6, Table 1). Figure 4b reports the same data shown in Figure 4a, except that the layer thickness and temperature scale for each experiment have been normalized using eqs 9 and 10. To define δ* for each electrolyte composition in Figure 4b, values of the initial layer thicknesses, δ0, were estimated by linearly extrapolating the data in Figure 4a to T ) 25 °C. The final layer thicknesses, δf, were taken as the fit values at the highest measured temperature. To define T* for each electrolyte composition in Figure 4b, lower critical solution temperatures, ΘL, were taken as the highest measured temperature for each set, which are indicated by vertical dashed lines in Figure 4a. For comparison, Figure 4b also reports the normalized PEO-PPO-PEO layer collapse from previous direct measurements in 0.5 M NaCl conditions.3 The results in Figure 4a and b represent an interpretation of the results in Figure 3 that is consistent with molecular-scale polymer solution theory37,42,46 and predic(46) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithica, NY, 1953.


Langmuir, Vol. 20, No. 26, 2004 11399

Figure 4. (a) Polymer layer collapse inferred from curve fits to percolation loci at each electrolyte composition in Figure 3. Symbol shapes indicate electrolyte composition as defined in Figure 3. Closed symbols correspond to analytical25 fit results, and open symbols correspond to MC simulation31 results (see text for details). Vertical dashed lines (- - -) indicate temperature for the total dimensional collapse of PEO-PPO-PEO layer onto the PS particle surface. Vertical dashed lines (- - -) also correspond to bulk PEO solution θ temperatures as a function of MgSO4 (without NaCl) as measured by Boucher and Hines.5 (b) Polymer layer collapses reported in Figure 4a normalized using reduced layer thickness and temperature as defined in eqs 9 and 10. Data are also shown for the PEO-PPO-PEO temperature-dependent layer collapse in 0.5 M NaCl from previous direct measurements.3 Symbols are the same as those used in Figure 3, except for hexagons representing directly measured collapse in 0.5 M NaCl.

tions of continuum polymeric attraction.18,47-51 The curve fits in Figure 3 were obtained considering only the continuum van der Waals attraction that arises between polymer-coated colloids as a function of the polymer layer thickness at various solvent conditions, δ(T,CSI). Because the observed fluid-gel transitions in Figure 3 all occur for T < ΘL, where net intermolecular attractive interactions are not expected, these data represent a case where continuum polymeric van der Waals attraction is the dominant factor controlling the observed transitions. In fact, because the PS core-particle contribution is small compared the PEO-PPO-PEO contribution, the potential in eq 11 should also be applicable to predicting/interpreting temperature and specific ion-dependent phase behavior of PEO solutions52 and PEO-PPO-PEO micelles.15 The PEO-PPO-PEO layers display a continuous dimensional collapse with increasing temperature and specific ion concentration that is also consistent with molecular descriptions of how polymer brush density profiles evolve with diminishing solvent conditions.53 Although interfacial confinement of adsorbed polymers can be expected to produce different thermodynamic behavior from unadsorbed polymers,42 each dimensional collapse in Figure 4a terminates within (2 °C of well-established ΘL values for bulk PEO measured by Boucher and Hines.5 Values of ΘL from Boucher and Hines and the dashed lines in Figure 4a are shown in Figure 5a to have a linear dependence on MgSO4 concentration well described by

θL(CSI) ) 93.3-132CSI

where the constants have units of degrees Celsius (°C) and degrees Celsius/molarity (°C M-1). For comparison, Figure 5a also reports critical points, TC, determined from fits to MC simulation results31 in Figure 3. From extrapolated values of the initial PEO-PPOPEO layer thickness for each electrolyte composition in Figure 4a, the layer collapse at 25 °C as a function of MgSO4 concentration is constructed in Figure 5b. The initial layer thickness shown in Figure 5 is close to 20 nm, in agreement with previous direct measurements in 0.5 M NaCl at 25 °C,1 and the final thickness is consistent with the nearly full collapse of the layer also measured directly for T > ΘL in 0.5 M NaCl.1 The dependence of δ0 on CMgSO4 is well described by the empirical curve fit

δ0(CSI) ) 5.54 + 15.9 exp(-10.1CSI)

where the fit constants have units of nanometers (nm) and 1/molarity (M-1). To the authors' knowledge, neither direct nor indirect measurements of the dimensional collapse of an adsorbed polymer layer due to SIE have been previously reported. By compiling fit results as a function of temperature for each electrolyte composition in Figure 3, it is possible to specify the layer thickness for all solvent conditions investigated here. The polymer layer thickness as a function of temperature and MgSO4 concentration can be obtained by fitting eqs 10, 12, and 13 to the normalized data in Figure 4b to give

(12) δ*(T*,CSI) ≡

(47) Dzaloshinskii, I. E.; Lifshitz, E. M.; Pitaerskii, L. P. Adv. Phys. 1961, 10, 165. (48) Langbein, D. van der Waals Attraction; Springer-Verlag: Berlin, 1974; Vol. 72. (49) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976. (50) Vold, M. J. J. Colloid Sci. 1961, 16, 1. (51) Vincent, B. J Colloid Interface Sci. 1973, 42, 270. (52) Polik, W. F.; Burchard, W. Macromolecules 1983, 16, 978. (53) Zhulina, E. B.; Borisov, O. V.; Pryamitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140.

(13)

δ(T*) - δf

) δ0(CSI) - δf 1.56 exp[-exp(4.05T* + 1.59)] (14)

where δf ) 4.5 nm. Equation 14 allows specification of the PEO-PPO-PEO layer thickness in the range T ) 20-95 °C and CMgSO4 ) 0-0.3 M (with complementary NaCl concentration, CNaCl, to give a total 0.5 M concentration in each case). Equation 14 is summarized by the contour plot shown in Figure 6a, with dashed lines indicating the

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Hwang et al.

Figure 5. (a) Linear regression to bulk PEO solution θ temperatures (closed circles) as a function of MgSO4 (without NaCl) as measured by Boucher and Hines.5 Open circles indicate transition temperatures shown by dashed lines in Figure 4a, and open squares indicate TC values determined from fits to MC simulation results.31 (b) Initial layer thickness, δ0, as a function of MgSO4 concentration estimated from curve fits in Figure 4a using procedure described in text.

Figure 6. (a) Layer thickness, δ, as a functions of MgSO4 concentration and temperature estimated from curve fits in Figure 3a. (b) Depth of van der Waals minimum determined by polymer layer thickness in Figure 6a and polymeric van der Waals as a function of polymer layer thickness given by eq 11. Solid lines are constant contour lines, and dashed lines indicate the T-CMgSO4 region in which data were fit in Figure 3.

T-CMgSO4 region in which data were fit in Figure 3. The low-T-low-CMgSO4 region in Figure 6a represents conditions for which the polymer layer collapse was extrapolated to room-temperature values for each data set in Figure 4. Although the extrapolated region is somewhat speculative, the result in eq 14 is qualitatively similar to the directly measured collapse in 0.5 M NaCl (CMgSO4 ) 0) shown in Figure 4b, which provides some confidence in the inferred collapse. The high-T-high-CMgSO4 region in Figure 6a is also extrapolated from the fitted data in Figure 3; however, the layer collapse is already saturated at these conditions so that no additional significant collapse is expected. With knowledge of the PEO-PPO-PEO layer thickness for all conditions between T ) 20-95 °C and CMgSO4 ) 0-0.3 M, it is possible to make predictions of the continuum polymeric attraction and ultimately phase behavior for the same conditions.

Temperature-Specific Ion-Dependent Attraction and Nature of Observed Transitions. Figure 6b shows the depth of the van der Waals minimum for the same range of conditions as in Figure 6a by evaluating the potential in eq 11 at r ) 2δ(T*,CSI) (from eq 14). Although the complete separation-dependent potential is known for all conditions in Figure 6b, only the magnitude of the attractive minimum is reported to simply demonstrate the variation in attraction with changing conditions. Because all fluid-gel transition data reported in Figure 3 are contained in the dashed-line region in Figure 6b, it appears that all transitions occur for attractive particle interactions between 2 and 6kT. This suggests that each transition corresponds to equilibrium dynamic percolation rather than static percolation that would occur for >6kT of attraction. The attraction predicted by eq 1 is expected as an upper bound because ignoring the polymeric


contribution or including nonidealities such as roughness41 would only predict weaker attraction. The energy scale in Figure 6b that appears to easily explain the measured transitions in Figure 3 is consistent with dynamic percolation behavior. A factor that could complicate interpretation of the fluid-gel data in Figure 3 is the possibility of “attractive glass” formation that could contribute to the observed rheological transitions. When the data in Figure 3 is fit and the plots in Figure 6 are constructed, equilibrium percolation behavior is assumed as the source of the rheologically measured fluid-gel transitions. If the transitions instead resulted from attractive glass formation, it would not be appropriate to “fit” equilibrium percolation predictions to the data in Figure 3. However, attractive glasses have not been previously observed for the relatively low volume fractions in Figure 3,15,17 which we take as evidence that our measurements correspond to dynamic percolation. In any case, unambiguous knowledge of the potential from TIRM measurements and theoretical predictions via eq 1 indicate the importance of continuum polymeric van der Waals attraction for the measured transitions in Figure 3. A final issue to consider is whether specific ions appreciably affect the net particle attraction by any other mechanism than simply causing the PEO-PPO-PEO layer to collapse more rapidly with increasing temperature. In using eq 1 to interpret our results, we assume specific ions only affect van der Waals interactions by determining the PEO-PPO-PEO density profile via solvent quality effects.9 A number of recent literature articles mention the importance of the polarizability of specific ions in altering DLVO forces.11 Because our work is concerned with nonionic polymers, the applicability of these findings to our work is not immediately obvious. Although this issue is not easily addressed, our results are explained in a straightforward manner using the continuum potential in eq 1, which produces consistent estimates of molecular-scale parameters, such as the ΘL values in Figure 5a. Precise knowledge of the PEO-PPO-PEO thickness and its attractive contribution as a function of temperature and electrolyte composition in Figure 6 provides a powerful method to tune the interparticle potential. From a priori knowledge of kT attractive interactions, phase diagrams

Langmuir, Vol. 20, No. 26, 2004 11401

can be predicted, and equilibrium self-assembly processes can be manipulated to perform “phase engineering”. Because SIE have been widely documented in a variety of biological phenomena involving macromolecules,6,7 it is possible that the continuum van der Waals contributions discussed in this work are also relevant to these systems. Ultimately, we have demonstrated the importance of continuum polymeric van der Waals contributions in determining the phase behavior of attractive polymercoated colloids. Conclusion This paper shows that continuum van der Waals attraction between PEO-PPO-PEO layers adsorbed to polystyrene colloids can explain observations of temperature and specific ion-dependent fluid-gel transitions in dispersions containing these particles. Because the continuum van der Waals attraction was directly measured and used to explain phase behavior in previous studies using only NaCl, in this work, the PEO-PPO-PEO thickness as a function of MgSO4 concentration is allowed as a single adjustable parameter to fit percolation predictions to the measured fluid-gel transition data. This yields the PEO-PPO-PEO thickness and interparticle attraction on the kT energy scale as a function of both temperature and NaCl/MgSO4 composition. The evolution of van der Waals attraction between adsorbed polymer layers with diminishing solvent conditions provides a simple mechanism for explaining measured temperature and specific ion-dependent fluid-gel transitions of polymercoated colloids. This mechanism has not previously been considered in the explanation of temperature and specific ion-dependent phenomena in either synthetic or biological systems. Understanding effects of specific ions on continuum van der Waals interactions between polymercoated colloids and solvated macromolecules provides a basis to interpret and manipulate a broad range of phenomena. Acknowledgment. We acknowledge financial support for this work by the National Science Foundation (CTS0346473), the donors of the ACS Petroleum Research Fund (41289-G5), and the Robert A. Welch Foundation (A-1567). LA0478752

Specific Ion-Dependent Attraction and Phase Behavior of Polymer

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