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J. Phys. Chem. B 2003, 107, 5294-5299
Freezing of Charge-Stabilized Colloidal Dispersions Shiqi Zhou* and Xiaoqi Zhang Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Wenhua Road, Zhuzhou City 412008, P. R. China ReceiVed: October 26, 2002; In Final Form: January 2, 2003
The Rogers-Young approximation for the Ornstein-Zernike integral equation is combined with the HansenVerlet one-phase criterion for freezing to predict freezing of a hard core repulsive Yukawa model (HCRYM) fluid. Comparison of theoretical predictions with corresponding computer simulation data discloses the superiority of the Rogers-Young approximation over the hypernetted chain approximation and the rescaled mean spherical approximation for freezing. Then, the Rogers-Young approximation combined with the Hansen-Verlet one-phase criterion is employed for the freezing of many-component charge-stabilized colloidal dispersions, which consist of colloidal macroions, electrolyte small ions, and solvent molecules and are modeled as a single-component charged hard core macroion interacting through a screened Coulomb potential. The theoretically predicted freezing line with the macroion surface charge number being assumed as an adjustable parameter is in very good agreement with the corresponding experimental data. The reason why, by the empirical Hansen-Verlet structure function approach, the single-component coarse-grained effective potential is valid for the freezing description of the many-component charge-stabilized colloidal solutions but not valid for the case of asymmetric binary hard sphere mixtures is discussed.
I. Introduction Charged colloidal solutions are presently a subject of intense experimental and theoretical interest. Unlike the case of simple electrolytes,1-4 the thermodynamic properties and phase structures of which are reasonably well understood, it is fair to say that our understanding of the charge-stabilized colloidal dispersions is far from complete. Even the fundamental problem of what is the form of the interaction potential between strongly charged colloidal particles still remains controversial.5-8 The above situation is due to the challenging aspects encountered by the theoretical studies of highly asymmetric charged colloidal mixtures: one, the existence of spatial correlations among the charged species on different length scales; the other, the longrange nature of the electrostatic Coulomb interaction. However, the experimental studies and various computer simulation investigations of the charge-stabilized colloidal dispersions have shown that the experimental behavior can be reproduced (when the concentration of the electrolytes is high) at least qualitatively by assuming that the colloidal particles interact via a spherically symmetrical hard core Yukawa model (HCYM) potential. Here, the hard core term represents the finite size of the colloidal particles, while the Yukawa term represents the screened repulsive or attractive electrostatic interaction. In the HCYM, only the macroions enter the model explicitly, now often interacting via a screened Coulomb potential, while the solvent is treated as a dielectric medium, with the small counterions and coions contributing only to screening the electrostatic Coulomb interaction. A comprehensive review of the theories of electrostatic screening was recently made by Vlachy.9 Although this represents an oversimplification of the original very complex interactions between the colloidal particles suspended in electrolyte solution, this dramatic simplification can be built on the theoretically derived Derjaguin-LandauVerwey-Overbeek (DLVO) interaction between the spherical
electrostatic double layers surrounding the colloidal particles.10 Among the thermodynamic behaviors of charged colloidal solutions, one thing that both experimentalists and theorists accept is that when the concentration of colloidal particles is sufficiently large, it freezes into a crystal.11,12 Traditionally, the theories of liquid-solid phase transition can be divided into two classes. One is based on the description of the system free energy;13-16 the other is based on the system structure function description. The examples of the latter include the HansenVerlet empirical rule for freezing,17 which declares that at freezing the maximum in the static structure factor Sˆ max ∼ 2.85 is found to hold remarkably for systems interacting with a 1/rn potential. The Raveche-Mountain-Street empirical rule governing the freezing18 of a Lennard-Jones system is based on the examination of the pair correlation function and states that the liquid freezes when the ratio R ) g(rmin)/g(rmax), where rmax is the distance corresponding to the maximum in the radial distribution function and rmin is the distance corresponding to the subsequent minimum in g(r), is approximately 0.2. Finally, the Lindemann criterion19 states that the solid should melt when the ratio of the square root of the mean-square distance, 〈r2〉1/2, to the lattice spacing d exceeds 0.1. It is the aim of the present paper to investigate the validity of the Hansen-Verlet one-phase criterion for the freezing properties of the HCYM fluid with a repulsive screening interaction by obtaining the static structure factor from solving numerically the Ornstein-Zernike (OZ) integral equation under several well-known bridge function approximations. Also, we will employ the combination of the Hansen-Verlet one-phase rule and the Rogers-Young approximation to investigate the freezing of the colloidal suspensions of charged polystyrene spheres and compare the theoretical predictions with the existing experimental data. Finally, we discuss the validity of the socalled single-component coarse-grained effective potential for the phase transition description of the many-component system.
10.1021/jp027319l CCC: $25.00 © 2003 American Chemical Society Published on Web 05/03/2003
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J. Phys. Chem. B, Vol. 107, No. 22, 2003 5295
II. Theory The HCYM potential of charged colloidal particles consists of a hard core part and a pure Yukawa part
exp(-κ(r - σ)) βu(r) ) βλ r/σ
r/σ < 1
(1)
where σ is the hard core diameter of the colloidal particles, κ and λ are the inverse screening length and the interaction strength, respectively, and β ) 1/kBT, is the inverse temperature. Given the HCYM potential between the charged colloidal particles, all equilibrium properties follow from the basic principles of statistical mechanics.20 The pressure P of a fluid confined in a volume V at a temperature T is given by
1
∫dr1...drNzN exp(-β∑u(ri - rj)) ∑ N! Ng0 i 0), which was studied much less
5296 J. Phys. Chem. B, Vol. 107, No. 22, 2003
Figure 1. Phase diagram of the repulsive HCYM system with σκ fixed at 5 in a βλ versus Fσ3 plot. The implications of the symbols are given in the inset.
Figure 2. Phase diagram of the repulsive HCYM system with σκ fixed at 5 in a βPσ3 versus log 10[βλ] plot. The points stand for the prediction from the RY approximation combined with the Sˆ max ) 3-Hansen-Verlet one-phase criterion. The line stands for the result based on the computer simulation data fit26 for the same system parameters.
compared with its attractive counterpart. To test whether the Hansen-Verlet one-phase criterion for freezing is applicable to the repulsive HCYM fluid, in Figure 1 we compare the predictions from the OZ-RY formalism and the OZ-RMSA formalism about the freezing point with the corresponding computer simulation data and those from ref 25, which are based on the OZ-HNC formalism. By comparing with the Monte Carlo simulation data,26 in ref 25 it was assumed that the freezing transition appears when the height of the first peak of Sˆ (q) reaches the value Sˆ max ) 3; then we extract in the formalism of the HNC approximation the freezing point based on this criterion. In the present study, we found that, in the formalism of the RY approximation, the Hansen-Verlet one-phase criterion based on Sˆ max ) 3 also produces the best agreement with the corresponding computer simulation data, far superior to the HNC approximation. In the same Figure 1, we also plot the predictions from the RMSA combined with the Hansen-Verlet one-phase criteria based on Sˆ max ) 3 and Sˆ max ) 2.65, respectively. It can be found that the RY approximation is superior to the HNC approximation and the RMSA; even when Smax is adjusted to be 2.65, the RMSA is also inferior to the HNC approximation. In Figures 2 and 3, we also give the phase diagram in a βPσ3 versus log 10[βλ] plot and a log 10[βλ] versus Fσ3 plot in the formalism of the RY approximation. The same good agreement with the computer simulation data is obtained.
Zhou and Zhang
Figure 3. Phase diagram of the repulsive HCYM system with σκ fixed at 5 in a log 10[βλ] versus Fσ3 plot. The points stand for the prediction from the RY approximation combined with the Sˆ max ) 3-Hansen-Verlet one phase criterion. The line stands for the result based on the computer simulation data fit26 for the same system parameters.
Thus, we can conclude that, for the freezing of the repulsive HCYM fluid, the Hansen-Verlet one-phase criterion based on Sˆ max ) 3 can give the best predictions in the formalism of the RY approximation, so we will employ the RY approximation and the Sˆ max ) 3-Hansen-Verlet one-phase criterion for the study of the freezing of the charge-stabilized suspension of spherically shaped colloidal particles immersed in the electrolyte solution. As opposed to simple liquids, the colloidal solutions are intrinsically complex systems containing mesoscopic particles, the colloids, in the nanometer to micrometer size range and small solvent (water or organic molecule) and solute molecules (electrolytes). The presence of the solute component makes it possible to modify at will and on a large scale the static and dynamic macroscopic properties of the whole solution. This explains the variety of phase transitions displayed by the chargestabilized colloidal suspension when the density of the colloidal particles and the concentration of the added electrolytes are varied. To treat the colloidal mixture theoretically on the same footing for every species in the mixture is obviously a numerical challenge. Fortunately, due to the large asymmetry in size, mass, and time scale between the colloidal particles and solvent/solute molecules, most of the experimental techniques are directly sensitive only to the former. Thus, this allows for the application of the single-component coarse-grained effective potential, that is, considering the colloidal solution not as a mixture but as a monodisperse system formally equivalent to a simple liquid, with the colloids playing the role of atoms and interacting by a solvent and solute-averaged effective pair-potential. There are various effective pair-potentials, for example, the DerjaguinLandau-Verwey-Overbeek (DLVO)10,27 potential in the case of the charge-stabilized colloidal suspension and the depletion potential28 in the case of colloid-polymer or colloid-colloid mixtures and mixtures of micelles and emulsion droplets. The repulsive part of the DLVO potential corresponds to the screened Coulomb interaction resulting from the linearized PoissonBoltzmann (PB) theory, that is, the Debye-Huckel approximation. In real systems, the Debye-Huckel approximation may not be valid, since the surface electrical potential is often not less than one. Alexander et al.29 have shown, however, that the linearized form of the PB equation still can be used with the actual surface charge number being replaced by a renormalized effective one. The interaction potential is given by
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Figure 4. Static structure factor of a monodisperse suspension of charged silica spheres. The symbols stand for the experimental results of Philipse and Vrij;30 the full line is the RY fit. The bulk parameters used in the RY calculation are written out in the figure.
Figure 5. Same as in Figure 4 but with η ) 0.079. From Nagele et al.31
u(r) ) (Z2e2/)(1 + κσ/2)-2
exp(-κ(r - σ)) r/σ > 1 r
)∝
r/σ < 1 (12)
Here the inverse screening length κ is given by
κ)
x
(4πβ/)[FZe2 +
∑R nRZ2Re2]
(13)
where F is the number density of the colloidal particle of charge Ze with e as the electronic charge, nR is the number density of an ion of type R with charge ZRe, and is the dielectric constant of the medium. To apply the single-component Hansen-Verlet one-phase criterion and the repulsive part of the DLVO potential to colloidal mixtures, first one has to ascertain whether the repulsive part of the DLVO potential can predict the structural properties of the colloidal solution in good agreement with the experimental data, not with the computer simulation data based on the same model potential. In Figures 4 and 5 we show a comparison of the static light scattering data for the static structure factor Sˆ (q)30,31 for the most concentrated samples with η ) 0.101 and µ ) 0.079 and those calculated from the RY approximation; the parameters are chosen to be Z ) 97, σ ) 160 nm, ) 10, and κσ ) 3.4 for the calculation of the RY approximation. One can see clearly that the predictions from
Figure 6. Phase diagram of a charge-stabilized colloidal dispersion of a polystyrene in water. The line + symbols stands for the prediction from the RY approximation combined with the Sˆ max ) 3-Hansen-Verlet one-phase criterion. The implications of the other symbols, which are for the experimental data from ref 32, are written out in the figure.
the RY approximation based on the single-component effective screened Coulomb interaction potential, when choosing appropriate model parameters, can reproduce well the static structure factor Sˆ (q) of the colloidal mixture even in the concentrated region of the phase diagram. This fact combined with the conclusion that the Sˆ max ) 3-Hansen-Verlet one-phase criterion can predict accurately the freezing of a repulsive HCYM fluid encourages us to employ the Sˆ max ) 3-HansenVerlet one-phase criterion for the study of the charged colloidal mixture freezing point in the formalism of the RY approximation. Two recent experimental studies on the colloidal phase diagrams are due to Monovoukas and Gast32 for a colloidal suspension of charged polystyrene spheres in an aqueous medium and due to Sirota et al.33 for the same system in 90% methanol solvent using synchrotron X-ray scattering. As mentioned above, the colloidal particle surface charge number Z is usually used as an adjustable parameter. For the former study (system I), σ ) 1334A and ) 78, while, for the latter (system II), σ ) 910A and ) 38. We found that when the surface charge number Z for the systems I and II is adjusted to be 692 and 350, respectively, the predicted freezing line can be in very good agreement with experimental data (see Figures 6 and 7). Compared with the DFT for the phase diagram of charge-stabilized colloidal suspensions, the present integral equation (IE) approach is computationally much simpler, but it can only predict the freezing point and cannot predict which crystal lattice (BCC or FCC etc.) will be formed. This limitation, however, does not devaluate the IE approach. In the biology studies, protein crystals are needed for diffraction studies34 or as a means of batch purification and as organic zeolites for enzymatic reactions.35 In current practice, the conditions for crystallization are determined by trial and error. This method is time-consuming; furthermore, trial and error is wasteful, especially since new proteins are often available only in very small quantities. If there exists a simple approach for the determination of the freezing point of the protein solution, the crystallization process can be speeded greatly and the quantity of the sample protein needed can be reduced largely. The present IE approach obviously satisfies this requirement. IV. Concluding Remarks and Discussion The thermodynamically self-consistent RY approximation for the OZ integral equation combined with the Hansen-Verlet one-
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Figure 7. Phase diagram of a charge-stabilized colloidal dispersion of a polystyrene in 90% methanol. The line + symbols stands for the prediction from the RY approximation combined with the Sˆ max ) 3-Hansen-Verlet one-phase criterion. The implications of other symbols, which are for the experimental data from ref 33, are written out in the figure.
phase criterion with Smax ) 3 can predict very accurately the freezing point of the repulsive HCYM fluid; the prediction accuracy is far better than that due to the HNC approximation and the RMSA. By employing the colloidal particle surface charge number as an adjustable parameter, the effective screened Coulomb interaction potential can predict the structural properties of the colloidal solution, in good agreement with the experimental static structure factor in the formalism of the RY approximation. The above two successes naturally lead to the successful prediction of the freezing point of the chargestabilized colloidal solution in the formalism of the RY approximation and the Hansen-Verlet one-phase criterion. The extension of the present approach to the two-dimensional chargestabilized colloidal solution is straightforward with reference to ref 36, which extends the approach of ref 25 based on the HNC approximation to a two-dimensional charge-stabilized colloid system. The screened effective Coulomb interaction potential in the case of charge-stabilized colloidal suspensions is actually an effective pair potential obtained by eliminating the degrees of freedom of the smaller particle and solvent; the resulting onecomponent system is then described by the effective pair potential. But an important question is whether the equilibrium structure and various thermodynamic properties of the original mixture system can be described well by the effective pair potential. Some recent studies give a negative answer to the above question. For example, a recent experiment37 on a chargestabilized colloidal crystal shows that the bulk modulus, the inverse of the compressibilitysas determined from the colloidal structure factor at long wavelengthssis three times smaller than that calculated from the DLVO theory. Another example38 is the Asakura-Oosawa model for colloid-polymer mixtures; the iosthermal compressibility χT obtained from the k f 0 limit of the partial structure factors in the two-component AsakuraOosawa model is smaller by an order of magnitude than the effective osmotic compressibility, obtained from the k f 0 limit of the colloidal structure factor of the corresponding effective one-component system described by the depletion potential, which is also an effective potential, by integrating out the freedom degree of the polymer. Finally, a rather recent study of a highly asymmetric binary hard sphere mixture was carried out.39 In this study, a formal expression for the effective Hamiltonian of the large sphere was derived by first integrating out the degrees of freedom of the small sphere in the partition
Zhou and Zhang function; then an explicit pairwise depletion potential was employed in this effective one-component large sphere Hamiltonian system to determine the fluid-solid coexistence for size ratios of small to large sphere q ) 0.033, 0.05, 0.2, and 0.1. However, in agreement with the above two negative answers, the structure factor S(q) of the effective one-component system shows no sharp signature of the onset of the freezing transition, and at most points on the fluid-solid boundary, the value of S(q) at its first peak is much lower than the value given by the Hansen-Verlet freezing criterion. For example, the height of the first peak is near 1.04 for large sphere packing fraction 0.05 and near 1.52 for large sphere packing fraction 0.3 in the case of the size ratio q ) 0.1. It should be noted, however, that the single-component depletion potential approach is successful for the prediction of the mixture phase transition by the free energy approach,39 even if it fails for the description of the mixture phase transition by the correlation function approach. However, the present study gives a counter-example to the above three negative answers. We think that there are mainly three explanations for the above contradiction. In the present study of the charge-stabilized colloidal suspensions, the value of surface charge number is used as an adjustable parameter; in our investigation, we found that, with all other bulk parameters fixed, the height of the first peak of the structure factor increases as the value of surface charge number increases. So by adjusting the surface charge number, one can always adjust the value of the first peak of the structure factor to 3.0. Second, unlike the size ratios of small to large sphere q in ref 39, in the present investigation, the size ratios q are near 4/1000 ) 0.004, an order smaller than that in ref 39. One can intuitively conclude that the single-component effective potential approach will become more accurate as the size ratio q becomes smaller. Finally, in the present charge-stabilized colloidal suspension, the concentration of the electrolyte is low, so the hard core steric interaction of the electrolyte small ions is dominated by the electrostatic interaction, which is successfully incorporated into the effective screened Coulomb interaction potential by the screening parameter κ. It is well-known that the single-component DLVO potential is suitable for the description of the colloidal solution phase diagram by the free energy approach.40-42 However, by the present investigation, we conclude that the effective one-component DLVO potential is also suitable for the phase diagram description of a charged colloidal mixture suspended in an electrolyte solution by the correlation function approach; this conclusion is in contradiction with that of the case of asymmetric binary hard sphere mixtures.39 The key conditions for the present success include, first, the size of the colloidal particle being far larger than that of the electrolyte ions or counterion and solvent molecule; second, the presence of the electrostatic interaction, which dominates over the small ions and solvent molecule steric interaction, which is the origin for the failure of the singlecomponent depletion potential approach for the asymmetric binary hard sphere mixtures by the correlation function approach;39 and third, the surface charge being regarded as an adjustable parameter. It should be noted that there exist some models in the literature that could be used to specify the renormalization effective charge, but we have not employed these models in the present prediction or tested these models by the present work. The reasons are as follows. The renormalization effective charges specified by these models are only accessible by titration experiment, while the present adjustable surface charge is not accessible by any experiments; they are two different concepts.
Freezing of Charge-Stabilized Colloidal Dispersions The final numerical value of the present adjustable surface charge results from two effects: One is counterion condensation effect, which can be reflected in the renormalization effective charge, whose numerical value can be predicted by the abovementioned models. The other is closely related employing the single-component effective potential for the description of the original mixture. As shown in the above-mentioned example of a highly asymmetric binary hard sphere mixture,39 at most points on the fluid-solid boundary, the value of S(q) at its first peak, based on the single-component depletion potential (it should be noted that the DLVO potential and the depletion potential are all the solvent and/or solute-averaged effective potentials), is much lower than the value given by the HansenVerlet freezing criterion. Thus, to make the value of S(q) at its first peak reach the Sˆ max ) 3-Hansen-Verlet one-phase criterion, one has to adjust the density to a larger numerical value. However, the increasing density has the same effect to raise the value of S(q) at its first peak, as the increasing surface charge does in the present case. But the latter effect cannot be covered by these charge renormalization models. So one cannot employ these models to predict the present adjustable surface charge and/or test these models by comparing the result with the experimental data. However, the breakdown of these charge renormalization models in the present case does not devalue the predictive power of the present approach, since only if one employs the experimental data of one liquid-solid transition point can one determine the adjustable surface charge for the calculation of all of the other state points and predict the freezing line with good accuracy, as shown in the present Figures 6 and 7. Acknowledgment. This project was supported by the National Natural Science Foundation of China (Grant No. 20206033) and the Scientific Research Fund of the Hunan Provincial Education Department (Grant No. 02B033). It is a pleasure for the authors to thank two reviewers for their comments, on which the revised version of the present paper is based. References and Notes (1) Romero-Enrique, J. M.; Orkoulas, G.; Panagiotopoulos, A. Z.; Fisher, M. E. Phys. ReV. Lett. 2000, 85, 4558. (2) Bekiranov, S.; Fisher, M. E. Phys. ReV. E 1999, 59, 492. (3) Fisher, M. E.; Levin, Y. Phys. ReV. Lett. 1993, 71, 3826. (4) Zuckerman, D. M.; Fisher, M. E.; Lee, B. P. Phys. ReV. E 1997, 56, 6569. (5) Palberg, T.; Wurth, M. Phys. ReV. Lett. 1994, 72, 786. (6) Tehver, R.; Ancilotto, F.; Toigo, F.; Koplik, J.; Banavar, J. R. Phys. ReV. E 1999, 59, R1335.
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