Interactions and Phase Behavior of Nanosized Particles - Langmuir

Interactions and Phase Behavior of Nanosized Particles. P. C. Zamora andC. F. Zukoski*. Department of Chemical Engineering and Beckman Institute, Univ...
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Articles Interactions and Phase Behavior of Nanosized Particles P. C. Zamora and C. F. Zukoski* Department of Chemical Engineering and Beckman Institute, University of Illinois, Urbana, Illinois 61801 Received October 10, 1995. In Final Form: March 18, 1996X As the concentration of silicotungstate particles (SiW12O404-) is increased at fixed background electrolyte concentration, the suspension separates into a dense, crystalline colloidal phase in equilibrium with a dilute colloidal liquid. The solubility of the particles ranges from 2.67 to 0.2 g/mL as the background 1:1 electrolyte concentration increases from 0 to 5 M. The second virial coefficient of suspensions of silicotungstate particles is a decreasing function of electrolyte concentration and, if interpreted in terms of square well or adhesive hard sphere pair potentials, suggests the particles’ feel attractions are on the order of 0-2kT at the phase boundary. Despite attractions of sufficient strength, colloidal gas/colloidal liquid phase transitions are not observed. These results are interpreted as indicating particles feel an attraction with an extent which is much smaller than the core particle diameter.

I. Introduction The phase behavior of suspensions of uniform colloidal particles has seen considerable attention particularly for systems where the particles feel purely repulsive pair potentials. For these systems, as the particle density is increased, order/disorder phase transitions are driven by the reduced entropy in the crystalline state.1 Considerably less attention has been paid to the phase behavior of colloidal systems with attractive potentials primarily due to difficulties in reducing the strength of attractions to the point that aggregation is reversible. Two examples where this has been accomplished can be found in depletion flocculated suspensions where the range and strength of the attractive potential are controlled by the radius of gyration, RG, and the concentration of added polymer1-4 and the octadecylsilica suspensions extensively studied by Vrij and co-workers.5-8 For the depletion flocculated systems experimental and modeling studies suggest that when RG > 0.3σ, gas/liquid and fluid/crystal phase transitions will be seen.1-4 Here σ is the particle diameter. As RG shrinks below 0.3σ, the triple and critical points move together such that no gas/ liquid transitions are seen despite attractions on the order * To whom correspondence should be addressed. Email: [email protected]. Phone: (217) 333-7379. Fax: (217) 333-5052. X Abstract published in Advance ACS Abstracts, June 15, 1996. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, Great Britain, 1989; p 343. (2) Gast, A. P.; Hall, C. K.; Russel, W. B. Polymer Induced Phase Separations in Nonaqueous Colloidal Suspensions. J. Colloid Interface Sci. 1983, 96, 251. (3) Gast, A. P.; Russel, W. B.; Hall, C. K. An Experimental and Theoretical Study of Phase Transitions in the Polystyrene Latex and Hydroxyethylcellulose System. J. Colloid Interface Sci. 1986, 109, 161. (4) Lekerkerker, H. N. W.; Poon, W. C.-K.; Pusey, P. W.; Stoobanb, A.; Waiven, P. B. Europhys. Lett. 1992, 20, 559. (5) de Kruif, C. G.; Jansen, J. W.; Vrij, A. Sterically Stabilized Silica Colloid as a Model Supramolecular Fluid. In Physics of Complex and Supramolecular Fluids; Safron, S. A., Clark, N. A., Eds.; 1987; pp 31543. (6) Vrij, A.; Jansen, J. W.; et al. Light Scattering of Colloidal Dispersions in Non-polar Solvents at Finite Concentrations: Silica Spheres as Model Particles for Hardsphere Interactions. Faraday Discuss. Chem. Soc. 1983, 76, 19. (7) Van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354. (8) Jansen, J. W.; de Kruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 481, 492, 501.

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of small fractions to many multiples of kT. For the octadecylsilica systems, on the basis of chemical and physical considerations, pair potentials are expected to consist of a hard core with a short range, temperature sensitive, attractive well.5-7 As the suspension temperature is lowered, the attractions become stronger. For this system, depending on the solvent used, space-filling gels or amorphous separated phases are observed.8 From experimental and modeling studies of both systems one reaches the startling conclusion that as the range of attractions becomes significantly smaller than the particle core diameter, the solid phase is always at a lower energy than the liquid phase and, despite considerable attractions, colloidal suspensions will exhibit no critical point.4,9-11 For conventional colloidal systems, attractions are often ascribed to van der Waals attractions which do not become significant until surfaces are separated by a distance on the order of 0.2σ-0.1σ. As a consequence one might conclude that gas/liquid transitions will only occur in carefully designed systems where the range of the attractive well is expanded as in depletion flocculation. However, this hypothesis is difficult to test. For particles 100 nm and larger, attractions are rarely sufficiently weak for phase separations to be seen (i.e., rather than undergoing phase separation, particles flocculate, often in an irreversible manner). However as the attractive energy at contact can be approximated as Aσ/24δ′, where A is the Hamaker constant, for a fixed value of δ′, as σ shrinks the maximum attractive energy will become comparable to kT. The minimum surface to surface separation δ′ is estimated as being on the order of an atomic diameter.11 Thus, even for systems with large Hamaker coefficients, with δ′ fixed, for particle diameters on the order of 1-5 nm, the attractive well depth will diminish in magnitude to the point where reversible aggregation, and thus phase transitions, will be observed. Here we use this phenomenon to our advantage in characterizing the interactions and phase behavior of (9) Lomba, E.; Almarza, N. G. Role of the Interaction Range in the Shaping of Phase Diagrams in Simple Fluids. The Hard Sphere Yukawa Fluid as a Case Study. J. Chem. Phys. 1994, 100, 8367. (10) Hagen, M. J.; Frenkel, D. Determination of phase diagrams for hard-core attractive systems. J. Chem. Phys. 1994, 101, 4093. (11) Tejero, C. F.; Baus, M. Freezing of Adhesive Hard Spheres. Phys. Rev. E 1993, 48 (5), 3793.

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attraction is small when compared to the particle diameter. In section VI, we draw conclusions. II. Experimental Section

Figure 1. Ball and stick model for the SiW12O40 anion. The size of the atoms has been decreased to provide a picture of the internal structure of the anion. If drawn with van der Waals atomic diameters, the oxygens are close packed.

nanometer-sized particles. As discussed below, these particles are sufficiently small that van der Waals forces give a maximum attractive energy of =0.5kT, making them ideal candidates for investigating the presence or lack of existence of gas/liquid transitions. We find that even for particles at the lower end of the colloidal size range, in the absence of additives such as polymers, pair potentials are sufficiently short range that equilibrium liquid phases are not seen. In this study we characterize the interactions and phase behavior of suspensions of the silicotungstate anion {SiW12O404-}, which has a Keggin structure12,13 composed of a central SiO4 tetrahedron surrounded by a cage of six WO6 octahedra having common vertices and edges (Figure 1). Silicotungstate (STA) retains the Keggin structure in solution and is highly soluble in water with an effective, spherical hydrodynamic diameter of 1.1-1.2 nm.15,16 When the STA concentration is sufficiently large (solution densities of 2.7 g/mL), heavily hydrated crystals are formed. Detailed studies of the dehydration of these crystals suggest the particles feel oscillatory pair potentials of a hydration or structural origin.17 The interactions between STA molecules are characterized here using static light scattering to determine the second virial coefficient of the suspension osmotic pressure, B2. Further progress requires a model of the pair potential be chosen. First we compare with predictions made using classical colloid theory, where particles interact with van der Waals and electrostatic energies. Due to discrepancies between the ionic strength dependencies of measured and predicted values, we then generalize the model for interactions to the square well and adhesive hard sphere pair potentials. Below in section II we discuss the experimental system while in section III comparison is made of measured values of B2 with those predicted from conventional colloidal interaction models. The attractions are larger than can be predicted, suggesting an additional attraction. In section IV, links between general pair potentials and B2 are made and predictions of phase behavior are discussed. Section V contains a comparison of predicted and measured phase behavior which confirms the existence of attractive interactions and provides evidence that the range of the (12) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press Inc.: San Diego, CA, 1992; p 203. (13) Signer, R.; Gross, H. U ¨ ber den Bau einiger Heteropolysa¨uren. Helv. Chim. Acta 1934, 17, 1076. (14) Keggin, J. F. The Structure and Formula of 12-Phosphotungstic Acid. Proc. R. Soc. London 1934, 144A, 75. (15) Baker, M. C.; Lyons, P. A.; Singer, S. J. Velocity Ultracentrifugation and Diffusion of Silicotungstic Acid. J. Am. Chem. Soc. 1955, 77, 2011. (16) Kurucsev, T.; Sargeson, A. M.; West, B. O. Size and Hydration of Inorganic Ions from Viscosity and Density Measurements. J. Phys. Chem. 1957, 61, 1567. (17) Zamora, P. C. Ph.D Dissertation, University of Illinois, 1995.

Experiments were performed using three different silicotungstate compounds (H4SiW12O40, Li4SiW12O40, and Na4SiW12O40) with different background salt concentrations 0.3, 1.0, 3.0, 4.0, and 5.0 mol/L of HCl, LiCl, and NaCl, respectively. For the lightscattering experiments, each experimental set was comprised of five dilutions of the silicotungstate solutions (concentrations varying from 0.43 to 0.04 g/mL) and a particular background salt concentration of interest. Silicotungstic acid was purchased from Fisher Chemicals (H4SiW12O40‚24H2O) and recrystallized in distilled water a minimum of three times. Solutions were covered with a Kimwipe (a delicate, low-lint paper) containing small holes punctured by the tip of a needle in order to minimize dust from contaminating the solution. The resulting crystals were stored in their saturated solution. The silicotungstate salts were prepared by procedures outlined by Krotov and Rode.18 A stoichiometric amount of the carbonate corresponding to the desired salt was added to a solution of silicotungstic acid. The mixture was initially heated to approximately 70 °C until the evolution of CO2 bubbles ceased, and was allowed to cool at room temperature, where it subsequently crystallized. The lithium salt (Li4SiW12O40) was prepared by crystallization from a mixture of solutions of H4SiW12O40 and Li2CO3 in a 1:2 stoichiometric molar ratio. The sodium salt (Na4SiW12O40) was prepared by crystallization from a mixture of solutions of H4SiW12O40 and Na2CO3 in a 1:2 stoichiometric molar ratio. Resulting crystals were stored in their saturated solutions. Elemental analyses were performed on all three samples in order to verify elemental ratios. Silicon nuclear magnetic resonance (General Electric GN300NB) experiments using saturated or nearly saturated aqueous solutions of the three different samples verified the presence of the silicotungstate anion. All static light-scattering measurements were performed using a DAWN DSP Model F laser photometer from Wyatt Technology Corporation (Santa Barbara, CA). The light source for the system is a 5 mW linearly and vertically polarized helium neon laser with a wavelength of λ ) 632.8 nm. Meticulous procedures were followed in preparing the lightscattering samples in order to prevent contamination by dust and other extraneous materials. All water used for dilutions and in making LiCl and NaCl solutions was filtered through 0.02 µm Whatman brand inorganic membrane Anodisc 47 filters. Nalgene 115-mL filter units with a pore size of 0.2 µm were used to filter solutions of 10 M LiCl, 5 M NaCl, saturated Li4SiW12O40 (2.15 g/mL), and saturated Na4SiW12O40 (2.00 g/mL). Nalgene 115-mL filter units with a pore size of 0.45 µm were used to filter stock solutions of H4SiW12O40 (1.35 g/mL) and 10 M HCl. Clean, 20-mL scintillation vials were used for each dilution. Calibration of the 90° detector was performed using HPLCgrade toluene. The remaining detectors were calibrated by normalization to the 90° detector using the “MagicGlass” cell (a highly polished cylinder of glass that scatters isotopically). Values for dn/dc (the change in suspension refractive index, n, with solute concentration (mass/volume), c, could not be obtained at high electrolyte strengths because the silicotungstate solution reacted with the metal tubing of the refractometer. Thus, an appropriate dn/dc value was chosen that would result in the correct molecular weight value of the silicotungstate particle (2.87 × 103 g/mol). In order to maintain consistency, fitted values were used in all second virial coefficient calculations. At low ionic strengths, there was an excellent agreement of values of dn/dc estimated in this manner and those measured using a differential refractometer.17 Intensities were found to be independent of scattering angle at all concentrations, indicating that, as expected, the STA particles are in the Raleigh region where scattering intensity can be related to concentration and molecular weight,22 Mw: (18) Krotov, N. A.; Rode, E. Ya. Lithium Acid Salts of Hydrogen Dodecatungstosilicate. Russ. J. Inorg. Chem. 1963, 8, 895. (19) Kerker, M.; Kratohvil, J. P.; Ottewill, R. H.; Matijevic, E. Correlation of Turbidity and Activity Data. II. Tungstophosphoric and Tungstosilicic Acids. J. Chem. Phys. 1963, 67, 1097.

Interactions and Phase Behavior of Nanosized Particles 1 1 dπ K*c ) + 2A2c ) R0 RT dc Mw

(1)

where π is the suspension osmotic pressure, R0 is the excess Raleigh ratio, RT is the product of the gas constant and the absolute temperature, c is the silicotungstate concentration {mass/volume}, K* ) 4π2n02(dn/dc)2/λ4NA, n0 is the solvent refractive index, and A2 is the second virial coefficient (with units of (mole volume)/mass2). As indicated in Figure 2, K*c/R0 was a linear fraction of c from which we extracted values of A2. The measured values of A2 are in agreement with previous work.19-21 In order to calculate the solubilities of the silicotungstate compounds at different electrolyte background concentrations, a thermogravimetric analyzer (Thermal Analysis Instruments model 2950, balance resolution 0.1 µg, balance accuracy (0.1%, noise +0.2 µg) was used to heat a known volume (10 µL) of silicotungstate solution to 700 °C. This temperature was sufficiently high enough to remove all water and all hydrates of the silicotungstate compounds. The remaining residue was comprised of the anhydrous silicotungstate compound and the added background salts of either LiCl or NaCl. The maximum weight contribution of the background salt added relative to the weight of the silicotungstate was insignificant. Thus all of the remaining residue was assumed to be the anhydrous silicotungstate compound. The stable hydrates formed at the crystallization boundary were H4STA‚31H2O, Li4STA‚21H2O, and Na4STA‚18H2O.

III. Interactions between STA Particles The osmotic pressure π of colloidal suspensions can be written in terms of the pair interaction energy and the pair distribution fraction, g(r), and pair interaction energy V(r) as1,23

π ) FkT -

g(r) dr ∫0∞r3 dV dr

2π 2 F 3

(2)

At low particle number densities, F, this may be written as an expansion in density where

π ) 1 + B2F + O(F3) FkT B2 ) -2π

(3)

∫0∞(e[-V(r)/kT] - 1)r2 dr

(4)

is the second virial coefficient related to A2 by B2 ) A2 Mw2/NA, where NA is Arogadro’s number. Measured values of A2, converted to B2, are given in Table 1. As indicated in eq 4, B2 provides indirect information about the pair interaction energy. If the assumption is made that STA particles interact like small charge-stabilized colloidal particles, B2 can be calculated assuming a hard core with van der Waals attractive and electrostatic repulsive interactions. Using classical colloid interaction models, van der Waals attractions are written1,12

Va(r) ) -

{

[

Langmuir, Vol. 12, No. 15, 1996 3543

]}

(r - σ)2 A σ2 σ2 + + 2 ln 12 r2 - σ2 r2 r2

(5)

r > σ + δ′ (20) Kratohvil, J. P.; Oppenheimer, L. E.; Kerker, M. Correlation of Turbidity and Activity Data. III. The System Tungstosilicic Acid-Sodium Chloride-Water. J. Phys. Chem. 1966, 70, 2834. (21) Kronman, M. J.; Timasheff, S. N. Light Scattering Investigation of Ordering Effects in Silicotungstic Acid Solutions. J. Phys. Chem. 1959, 63, 629. (22) Zimm, B. H. The Scattering of Light and the Radial Distribution Function of High Polymer Solutions. J. Chem. Phys. 1948, 16, 1093. (23) Brady, J. F. Brownian Motion, Hydrodynamics, and the Osmotic Pressure. J. Chem. Phys. 1993, 98, 3335.

Figure 2. Typical light-scattering data for H4STA suspended in HCl at various concentrations. Table 1. Second Virial Coefficients of Silicotungstate Suspensions B2b (nm3)

XCla (M)

H4STA

Li4STA

Na4STA

0.3 1.0 3.0 4.0 5.0

10.5 4.7 1.1 -1.8 -1.2

10.3 6.1 1.3 -0.3 -2.0

10.2 4.1 .08 -1.3

ψ0c (mV)

B2DLVO d (nm3)

-60.7 -43.5 -30.1 -27.1 -24.9

13.6 5.8 4.0 3.8 3.7

a Concentration of supporting electrolyte. X is the appropriate counterion for the silicotungstate particle indicated. b Second virial coefficients measured at 25 °C by static light scattering. Uncertainties are estimated at (0.2 nm3. For purposes of determining B2 at the total small ion concentration at the solubility boundary, curve fits to the values given below were developed. For HCl, B2 (nm3) ) 8.15-9.91 log(2[HCl]). For LiCl, B2 (nm3) ) 8.52-9.841 log(2[LiCl]). For NaCl, B2 (nm3) ) 7.75-9.78 log(2[NaCl]). c Surface potential of the STA particle calculated assuming a surface charge of four elemental charges at the indicated background ionic strength. d Second virial coefficient calculated from a summation of electrostatic repulsion and van der Waals attraction with a Hamaker coefficient of 5kT and a minimum separation distance of 0.165 nm.

where δ′ represents the cutoff of the van der Waals attractions, which is introduced to keep B2 finite. δ′ is expected to be on the order of an atomic diameter and has been investigated for a number of systems from organic liquids to mica. Adhesive energies are found to be well predicted when δ′ ) 0.165 nm.12 The Hamaker constant for STA, A, is expected to lie near 5kT (2.05 × 10-20 J), which is characteristic of mica and fused quartz interacting across water at 25 °C. Electrostatic repulsions are evaluated from the linear superposition approximation where the surface potential, ψ0, is calculated at each ionic strength assuming a constant of four primary charges uniformly distributed over the particle surface. Conductivity measurements made with increasing STA concentration at constant background electrolyte concentration indicate that, up to 1 M HCl, LiCl, and NaCl, STA retains four primary charges. (As [STA] increases, conductivities increase at the limiting conductivity of the counterion times 4[STA]. The silicotungstate anion contributes a negligible level to the suspension conductivity, as confirmed by the exact solutions to the governing equations as solved using the method of O’Brien and White.24). We assume that while ψ0 varies to reflect the background ionic strength, particles interact with constant ψ0, yielding an electrostatic repulsive potential of (24) O’Brien, R. W.; White, L. R. Electrophoretic Mobility of a Sherical Colloidal Particle. J. Chem. Soc., Faraday Trans. II 1978, 74, 1607-26.

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Ve(r) ) π0σψ20 exp{σκ(r/σ - 1)}/(r/σ)

Zamora and Zukoski

(6)

where 0 is the product of the relative permittivity of thesolution and the permittivity of free space and κ is the Debye-Huckel screening parameter

κ2 )

e22nb 0kT

(7)

where we have assumed that, at STA concentrations where B2 is measured, the counterions balancing the particle charge yield a negligible contribution to κ. Here e is the charge on a proton and nb is the bulk electrolyte concentration. As seen in Figure 3, B2 is weakly dependent on counterion type and is a decreasing fraction of background ionic strength, indicating decreasing repulsions or increasing attractions as the ionic strength increases. Indicated on Figure 3 are calculations of B2 from V(r) ) Va(r) + Ve(r) (for two values of A and δ′). In these calculations σ has been taken as 1.12 nm. Note that the predicted values decrease to a plateau value where electrostatic forces are completely screened. For nominal values of δ′ ) 0.165 nm and A ) 5kT, classical potentials closely predict B2 up to ionic strengths of 1 M, above which values of B2 are severely overpredicted. As indicated in Figure 3, extreme values of A ) 20kT continue to overpredict at the highest ionic strengths. B2 values calculated from the classical potentials are very sensitive to δ′. However, with A ) 5kT even with δ′ ) 0.1 nm (less than the van der Waals diameter of an oxygen atom) the conventional models fail to predict B2 at high ionic strengths. The colloidal interaction potentials used above have been developed to predict interactions of large particles where the solvent is treated in the continuum limit. These approximations are expected to fail at high ionic strengths. Nevertheless, applying the conventional models with physically reasonable Hamaker constants and distances of closest approach suggests van der Waals attractions remain weak, yielding maximum attractions of approximately 0.5kT. The measured B2 values indicate that, at high background electrolyte concentrations, the STA particles feel larger attractions. IV. General Pair Potentials and Phase Behavior Due to a lack of knowledge about the origin of the high ionic strength interactions of STA particles, we have turned to simple forms of the pair potential to gain insight into the strength of interparticle attractions. The first model we use is that of a square well potential:25-28

{

)∞ V(r) ) - )o

r 3, σ/l > 6), thermodynamic properties of fluids are well matched by those predicted for adhesive hard spheres30 if a mapping is done at constant B2. The adhesive hard sphere potential has the advantage of requiring only two parameters, σ (29) Ashcroft, N. W. Elusive Diffusive Liquids. Nature 1993, 365, 387. (30) Baxter, R. J. Percus-Yevick Equation for Hard Spheres with Surface Adhesion. J. Chem. Phys. 1968, 49 2770.

Interactions and Phase Behavior of Nanosized Particles

Langmuir, Vol. 12, No. 15, 1996 3545 Table 2. Solubility of Silicotungstate Compounds at Different Background Electrolyte Concentrations XCla M

[H4STA]eq (g/mL)

[Li4STA]eq (g/mL)

[Na4STA]eq (g/mL)

0.0 0.3 1.0 2.0 3.0 4.0 5.0

2.18 ( 0.01 2.10 ( 0.08 1.87 ( 0.06 1.67 ( 0.01 1.41 ( 0.09 1.18 ( 0.03 0.61 ( 0.02

2.15 ( 0.01 2.56 ( 0.09 2.14 ( 0.08 1.86 ( 0.13 1.37 ( 0.04 0.85 ( 0.02 0.35 ( 0.01

1.99(0.02 2.01 ( 0.11 1.88 ( 0.09 1.71 ( 0.17 1.31 ( 0.01 0.76 ( 0.02 0.50 ( 0.01

a Concentration of background electrolyte as indicated: X ) H, Li, or Na.

virial coefficient for the adhesive hard sphere potential is related to τ through

τ ) {4[1 - 3B2/2πσ3]}-1

Figure 4. (A, top) Dimensionless temperature as a function of scaled particle density. kT/ is calculated from eq 9 using measured values of B2 for H4STA, σ ) 1.12 nm, and various values of λ ) 1+∆/σ: (b) λ ) 1.5; (9) λ ) 1.375; (2) λ ) 1.25. Densities are calculated from solubilities of H4STA. Lines are spinodals calculated by Heyes and Aston26 for square well fluids: (solid line) λ ) 1.5; (dashed line) λ ) 1.375; (dotted line) λ ) 1.25. The vertical line is drawn to estimate the location of the fluid/crystal phase boundary at Fσ3 ) 0.945. Similar behavior is seen for Li4STA and Na4 STA.17 (B, bottom) Phase behavior predicted for H4STA using the adhesive hard sphere model for converting measured values of B2 to the stickiness parameter τ. Solid circles and squares are experimental points. Conversion of B2 and [STA]eq to τ and Fσ3 was made with σ ) 1.12 nm (circles) and σ ) 1.25 nm (squares). The solid and dashed lines at high τ are calculations of the fluid/crystal coexistence given by Tejero and Baus11 and Smithline and Hayment.31 The longdashed curve is the spinodal predicted by Baxter.30 The heavy solid lines are predictions of the equilibrium fluid/solid phase boundary and the metastable spinodal as predicted by Hagen and Frenkel10 for attractive Yukawa potentials with σ/l ) 9. Conversion to adhesive hard sphere phase behavior made at constant B2. Similar behavior is seen for Li4STA and Na4STA.17

and the stickiness parameter τ, which plays the role of temperature. For large τ, the hard sphere or hightemperature limit is approached while attractions become increasingly important as τ f 0. The adhesive hard sphere potential is written as

{

∞ r σ′

(11)

where V(r) is evaluated in the limit as σ′ f σ. The second

(12)

The gas/liquid spinodal30 and high-temperature fluid/solid phase11,31,32 behavior have been predicted for adhesive hard spheres. However, for 0.13 < τ < 0.5, the phase behavior is poorly understood. To overcome the lack of phase information, we follow previous studies which suggest that, for potentials with limited extents of attraction, structure factors and equations of state are well described by adhesive hard sphere interactions if comparisons are made at constant B2.27,28 On the basis of these observations we equate solubilities of particles interacting with Yukawa potentials with σ/l ) 9 to adhesive hard spheres when the two systems have the same second virial coefficient. This process is accomplished by using Hagen and Frenkel’s results to determine the solubility (written in terms of a dimensionless number density, Fσ3, in Figure 4) at a given value of /kT. The second virial coefficient of Yukawa particles interacting with this value of /kT is then calculated from eq 4, and this value of B2 is converted to τ through eq 12. At this point we have a solubility and a value of τ. Repeating this process for many solubilities yields the heavy solid line in Figure 4B. The same process can be carried out for the spinodal of Hagen and Frankel and the melting curve. Results of these calculations and a comparison with extant predictions for adhesive hard spheres are given in Figure 4B. Note that the critical densities are not well matched. However, in terms of ranges of τ, the two spinodals are not dramatically different. Of greater significance is that the fluid/solid phase boundary for the Yukawa potential lies above both spinodals, suggesting that stable liquid phases will not be observed in particles interacting with attractions of limited extent. V. Phase Behavior of STA Suspensions As conventional colloid interaction potentials fail to predict B2 at elevated ionic strengths, a natural question arises as to the magnitude of the attractions required to yield measured values of B2. As discussed above, by choosing square well, Yukawa, or adhesive hard sphere potentials, the strength of the attraction can be determined. These attractions are expected to be on the order of 0.5kT, which is sufficient to induce phase separation. Thus a test of the assumed interaction potentials comes in comparing measured and predicted phase behavior. For this purpose the solubility of STA was measured with increasing background ionic strengths (Table 2). The saturated solution concentration of silicotungstate, [STA]eq, (31) Smithline, S. J.; Haymet, D. J. Solid-Liquid Coextistence in the Adhesive Hard Sphere System. J. Chem. Phys. 1985, 83, 4103. (32) Marr, D. W.; Gast, A. P. On the solid-fluid interface of adhesive hard spheres. J. Chem. Phys. 1993, 99, 2024.

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is a decreasing function of electrolyte concentration and a weak function of counterion type. Second virial coefficients were measured at STA concentrations sufficiently low that the counterions which balance the STA charge made a negligible contribution to the overall ionic strength. This is not true at the solubility boundary where counterions can have a dramatic effect on the small ion concentration. Following BeresfordSmith and Chan,33 as a first approximation for accounting for the increased small ion activity (decrease in solvent activity) at the solubility limit, effective values of B2 were estimated at equivalent small ion concentrations. This was accomplished by curve fitting measured values of B2 as a function of total background small ion concentration, 2[salt], and then evaluating these functions at small ion concentrations equivalent to that occurring in solution at [STA]eq (i.e., at (2[salt] + 4[STA]eq)/(1 - φ)). For example, the solubility of STA in 1 M HCl is 0.65 M (φ ) 0.4). Thus, at the solubility boundary, the total small ion concentration is (2[salt] + 4[STA]eq)/(1 - φ) ) 7.7 M. The effective value of B2 at the solubility limit is then evaluated from the B2 measured in dilute STA conditions at a total small ion concentration of 7.7 M rather than 2 M. Determination of /kT and τ requires knowledge of σ and ∆. Assuming the STA diameter is 1.12 nm (as determined from viscosity measurements) results in phase behavior as indicated in Figure 4A and B. In Figure 4A we give solubilities of H4STA in terms of dimensionless temperatures determined from B2 using eq 9 for different values of ∆. In Figure 4B, B2 has been converted to τ using eq 12. Two points should be made about STA phase behavior with this choice of σ. First, the phase behavior shows a rapid increase in kT/ or τ near Fσ3 ) 0.6 as the apparent high-temperature limit is approached. This density is substantially smaller than that predicted for the high-temperature limiting solubilities of square well fluids or adhesive hard spheres (Fσ3 ) 0.945). The poor comparisons of limiting behavior may be due to the failure of the model to account for specific interactions or, as discussed below, to an underestimation of the effective hard core diameter. Second, no gas/liquid phase transition is observed. For the sake of comparison we have included in Figure 4A and B predictions of phase boundaries from simulation and analytical studies. Note that, for square well potentials, phase behavior is best matched with ∆/σ ) 0.25 (the lowest value for ∆/σ for which we could find phase behavior predictions.) An alternative method of estimating σ is to force a fit between predicted and measured solubilities at a single value of τ. Figure 5 gives a comparison where the experimental solubility at the largest value of τ lies on the theoretical curve. With this procedure σ is 1.25, 1.26, and 1.29 nm for H4STA, Li4STA, and Na4STA, respectively. Two points can be made about Figure 5. First, within experimental uncertainty the model and experimental phase diagrams are the same, providing strong evidence that the STA particles interact with short range attractions which are sensitive to background electrolyte concentration. Second, the use of Hagen and Frenkel’s10 data for attractive Yukawa potentials to describe the phase behavior of adhesive hard spheres obscures the reentrant fluid phase predicted for adhesive hard spheres by a variety of techniques.11,31,32 The coexistence fluid density increases above a Fσ3 value of 0.945 at intermediate values of τ (Figure 4B). This behavior is not observed for the Yukawa systems. Experimentally we find evidence for

such reentrant behavior for Li4STA and Na4STA. However, the effect is not large and lies at the edge of our experimental uncertainty (Table 2). Using hard core diameters yielding the phase behavior in Figure 5, the strength of the interparticle attractions along the phase boundary can be extracted from τ by assuming an equivalence between sticky hard spheres and very narrow square wells. For a well width of 0.165 nm, the strength of interparticle attractions for 0.05 < Fσ3 < 0.6 corresponds to 1.0 < /kT < 1.5. If we use the nominal Hamaker coefficient of 5kT and δ′ ) 0.165 nm, the maximum van der Waals attractive strength is 0.45kT. Thus for the van der Waals attractions to match the experimental value, the Hamaker coefficient would have to be double or triple to absurdly high values. These results suggest there is an attraction of non-van der Waals origin giving rise to the observed phase behavior of STA particles. Frink and van Swol34 have recently shown that as the chemical potential of the solvent is decreased (in our case, this corresponds to increasing background electrolyte concentrations), hydrophilic particles begin to feel an attraction. The attraction is a manifestation of solvation or structural forces and arises from a competition for the solvent between the surfaces and the bulk. Detailed studies of the crystalline STA phases indicate that STA particles are highly hydrated and dehydration occurs in a series of discrete steps as the solvent chemical potential is lowered.17 The dehydration steps provide clear evidence for oscillations in the particle interaction energies and are readily interpreted in terms of solvation interactions. Frink and van Swol demonstrate that the solvation minimum grows as solvent chemical potential is lowered and may be large compared to van der Waals forces at similar surface separations. Our results are consistent with the interpretation that the attraction giving rise to measured values of B2 and phase behavior for STA particles is not van der Waals in nature but arises from a “solvation attraction”. Note however that this interpretation will have to be modified for charged surfaces in the presence of large electrolyte concentrations where charge/charge correlations may become significant.

(33) Beresford-Smith, B.; Chan, D. Y. C. Highly Asymmetric Electrolytes: A Model for Strongly Interacting Colloidal Systems. Chem. Phys. Lett. 1982, 92 (5), 474.

(34) Frink, L. J. D.; van Swol, F. Solvation forces and colloidal stability: A combined Monte Carlo and density functional approach. J. Chem. Phys. 1994, 100, 9106.

Figure 5. Adhesive hard sphere phase diagram: (b) H4STA; (9) Li4STA; (2) Na4STA. τ and Fσ3 were calculated from the B2 values in Table 3 and the solubilities in Table 2 assuming σ ) 1.25, 1.26, and 1.29 nm for H4STA, Li4STA, and Na4STA, respectively. The lines drawn are predictions of phase behavior as indicated in Figure 4.

Interactions and Phase Behavior of Nanosized Particles

VI. Conclusion Silicotungstate particles lie at the lower end of the colloidal size range. However, they are sufficiently large that they dominate the light scattered from STA suspensions, making static light scattering a useful tool to probe particle interactions. Second virial coefficients measured in HC1, LiCl, and NaCl are decreasing functions of electrolyte concentration. For electrolyte concentrations below 1 M, particle interactions are dominated by electrostatic repulsions and use of the linear superposition approximation for electrostatic repulsions with four charges per particle provides good agreement between predicted and measured values of B2. At higher salt concentrations, B2 decreases through zero and a combination of electrostatic and van der Waals attractive interactions fails to provide agreement with measured B2 values. The STA solubility boundary decreases as background electrolyte concentration increases. By assuming the particles at the solubility boundary interact as dilute STA particles at the same total small ion concentration, a phase diagram can be constructed where B2 is used as a measure of effective temperature. The experimental phase diagram corresponds well with predictions for particles with narrow attractive well widths, suggesting STA particles may be treated as adhesive hard spheres. The agreement of predicted and measured phase behavior suggests the magnitude of the interparticle attractions continues to grow as electrolyte concentration is raised. The strength of the attractions is larger than can be predicted from van der Waals forces, suggesting an additional attractive potential is acting. The high electrolyte concentrations and small interparticle spacings at the solubility boundary give one pause in applying conventional electrostatic and van der Waals colloid interaction models. Nevertheless, these models provide

Langmuir, Vol. 12, No. 15, 1996 3547

good agreement for B2 values at ionic strengths up to 1 M. In addition, attractive energies are estimated from B2 which is measured in dilute STA solutions where average particle spacings are large. Thus we conclude conventional models will not explain the observed dilute and concentrated suspension behavior of STA particles and predictions of the magnitude of B2 will require more detailed understanding of solvent and counterion size effects. Finally, we note that, when compared on an equal footing, solubilities are very similar for the three compounds investigated. However, the crystals formed have substantially different hydrate states. The acid compound crystallizes with 31 waters of hydration per STA while Li4STA and Na4STA crystallize with 24 and 18 waters per STA, respectively. Recently, we have found that a globular protein, lysozyme (σ ) 3.4 nm), has solubility behavior which superimposes on the STA data in Figure 5.35 From this we conclude that solubility boundary is sensitive to an average strength of attraction while packing in the crystal is dependent on specific interactions. Acknowledgment. P.C.Z. was supported by an NSF Fellowship in the early stages of this work and in the later stages by the Materials Research Laboratory at the University of Illinois through Grant DEFG02-91.ER45439. The authors would like to thank L. Frink, D. Rosenbaum, and F. van Swol for stimulating discussions on the origins of interaction potentials and interpretation of experimental results. W. G. Klemperer is thanked for his help in identifying the experimental system and in the handling of the silicotungstate anion. LA950854A (35) Rosenbaum, D.; Zamora, P. C.; Zukoski, C. F. Phase Behavior of Small Attractive Colloidal Particles. Phys. Rev. Lett. 1996, 76, 150.