Consistent Surface Potentials from Bulk Suspension Properties

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

195, 32–41 (1997)

CS975132

Consistent Surface Potentials from Bulk Suspension Properties William J. Hunt and Charles F. Zukoski 1 Department of Chemical Engineering, University of Illinois at Urbana—Champaign, 114 Roger Adams Laboratory, 600 South Mathews Avenue, Urbana, Illinois 61801 Received September 24, 1996; accepted August 19, 1997

tion, electrophoresis, conductivity, or acoustic measurements are used to infer the surface potential or surface charge of a colloidal particle. Each method can provide reasonable values of the charge or zeta potential, but the assumptions that go into the models for converting electrokinetic measurements to a surface potential often fail to capture the complexities of the solid–liquid interface, leading to inaccurate estimates of the interaction potential and thus to poor predictions of the suspension’s modulus or osmotic pressure. For example, Zukoski and Saville (5) showed that independent measurements of the electrophoretic mobility and the conductivity of the same system did not yield the same zeta potential. Variations could be as large as a twofold shift in zeta potential on the same particle by the two methods. The difference was attributed to migration of ions in the Stern layer. Similar discrepancies have been reported by Dukhin and Derjaguin (6), O’Brien and Perrins (7), and van der Put and Bijsterbosch (8). Electrokinetic measures of the surface potential are extremely sensitive to the details of the boundary conditions used to describe the hydrodynamics and the electrodynamics of the double layer. The model must appropriately describe the Stern layer, static or dynamic, and correctly identify the drag function. The drag function becomes very complicated if the particle is not spherical or if the particle does not have a smooth, hard surface over which the charge is uniformly distributed, thereby particles with ‘‘hairy layers’’ are difficult to treat. Assumptions about surface conduction and adsorbtion must also be included in an accurate model of electrokinetic mobility or conductivity. On the other hand, bulk suspension mechanical properties, such as osmotic pressure or elastic modulus, are relatively insensitive to the details of the doublelayer structure and its response under an imposed field. Setting aside, for the moment, the difficulties in obtaining surface potentials by standard techniques, several models for calculating elastic constants and osmotic pressures have been developed. Numerous means of calculating osmotic pressure have been developed, and current models derived from statistical mechanics appear to capture both quantitative and qualitative aspects of these measures of the suspension’s thermodynamic state. Local neighborhood models for elas-

The equilibrium mechanical properties of face centered cubic (FCC) colloidal crystals interacting through the Yukawa potential are investigated. We investigate the use of a screened Coulomb interaction potential to define a single, electrostatic surface potential capable of predicting both the elastic modulus and osmotic pressure of colloidal crystals. Correlations are developed which capture the volume fraction, particle size, and ionic strength dependencies of these properties. q 1997 Academic Press Key Words: suspensions; surface potential; elasticity; osmotic pressure.

1. INTRODUCTION

When working with colloidal suspensions, one often needs engineering estimates of the physical properties. The presence of a yield stress or how much work must be done to concentrate the sample or transport it plays a crucial role in the process economics (1). Estimations of the magnitudes of these quantities rely on the development of sound models that relate the particle interaction energy to the suspension elasticity and osmotic pressure. Much of the behavior of charge stabilized colloidal suspensions can be described using the Yukawa potential (2–4). U(r) Å peeoc 2o s 2

exp( 0 k(r 0 s )) , r

[1]

where r is the center to center separation of the particles, e and eo are the relative dielectric permittivity of the suspending medium and the dielectric permittivity of free space, k is the Debye screening length, s is the particle diameter, and co is the surface potential. Numerous models to describe osmotic pressure and elasticity behavior of colloidal suspensions have been developed using this form of the interaction energy. The difficulty faced in application of these models lies in the accurate estimation of the surface or diffuse double-layer potential. By application of models with various levels of sophistica1

To whom correspondence should be addressed. 32

0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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CONSISTENT SURFACE POTENTIALS FROM SUSPENSIONS

ticity qualitatively capture the volume fraction dependencies of osmotic pressure and elastic modulus (3, 9–12). One of the most complete treatments is that of Robbins et al. (9), who calculate osmotic pressures, phase behavior, and elastic constants for charged particles in FCC and BCC lattices interacting through the Yukawa potential. Beyond the testing of understanding of the physical chemistry of dense suspensions, a means of predicting suspension elastic moduli would be helpful in predicting the suspension’s rheological properties. Chow and Zukoski (13) and Fagan (14) have shown that dynamic suspension properties, such as yield stress or the shear stresses at which microstructural transitions occur, scale on the elastic modulus of the suspension over a wide range of loadings and particle properties. As a consequence, given sufficiently accurate methods of predicting the surface potentials, models such as those developed by Buscall et al. (12), Russel et al. (3), or Robbins et al. (9), which provide a consistent description of suspension equilibrium properties, can be employed to make estimations of dynamic properties useful in the processing of these suspensions. In this work, we provide a method of inferring a surface potential from measured elastic moduli or osmotic pressures for dense suspensions of monodisperse, charge-stabilized colloidal particles. We assume the particles form a colloidal crystal with a FCC lattice as is appropriate for particles where ks ú 1 and typical reduced temperatures (9, 15). Under these conditions, except in extremely dense suspensions where the particle separations are a few percent of the particle diameter, electrostatic forces will dominate the pair potential and van der Waals forces may be neglected. The correlations provided estimate suspension equilibrium properties to a high degree of accuracy for values of ak from 1 to 40. Correlations are developed based on a single interaction energy for a given suspension that characterizes the volume fraction dependence of both the osmotic pressure and the elastic modulus. Osmotic pressures are calculated using standard statistical mechanical theories, lattice calculations and molecular dynamics techniques. The elastic constants are calculated by a method commonly used in crystal mechanics, but not usually seen in discussions of colloidal suspensions. The first attempts to test models of suspension properties in this manner were carried out by Russel and Benzing (16, 17), whose data have withstood the test of time. The use of the models and calculations made there are limited to the system under investigation. We provide general expressions that can be used for a wide variety of systems. We note that the Yukawa potential has a limited range of applicability and that more exact descriptions are available (18), however, except at volume fractions where k(r 0 s ) ! 1, the Yukawa potential provides an excellent approximation for true interaction between two charged particles in a screening continuous phase. We find the self-consistent surface potential ascribed provides an excellent approximation for many

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experimental systems. The purpose of this paper is to describe the development of the correlations for elastic modulus and osmotic pressure and demonstrate the applicability of measurements of these properties for establishing a means of predicting other suspension properties. 2. MATERIALS AND METHODS

The osmotic pressure computations used in this work employ two simulation techniques. First, a molecular dynamics (MD) code using the constant pressure techniques of Brown and Clarke (19) and the leap-frog Verlet algorithm (20) was used to investigate the behavior of bulk, monodisperse, charge-stabilized colloidal suspensions. The simulation is carried out with full periodic boundary conditions at constant temperature, particle number, and applied pressure (i.e., the NPT ensemble). Particles interact through the screened coulomb (Yukawa) potential in a pairwise additive manner through the minimum image convention (20). This technique sets the osmotic pressure, and the particle density is then equilibrated for a given set of parameters. Thermal equilibrium is maintained by ad hoc rescaling of velocities such that the total kinetic energy of the system matches that for the number of Brownian particles in the simulation. The second method used was a lattice method. An FCC lattice was constructed, and the contributions of the interparticle forces to the osmotic pressure were calculated at zero temperature according to N0 1

PÅ

N

1 ∑ ∑ rijrFij , 3V iÅ1 j Åi/1

[2]

where V is the volume of the simulation cell containing the N particles, and rij and Fij are the separation and force vectors between pair ij. In the lattice method osmotic pressure calculation we have ignored the ideal gas contribution. This is a constant factor added to the dimensionless osmotic pressure computed above. The correction that must be added is P* HS Å

6f kT exp( 0 ks ), p Uo

[3]

where f is the suspension volume fraction. The lattice method and the full MD calculations were compared at several values of ks and found to be in good agreement for the same value of the Yukawa interaction energy (surface potential). The correlations were developed using the results of the lattice method because results could be generated much more rapidly. The method of Catlow and Mackrodt (21) was used to calculate the elastic constants of the suspensions. This calculation uses the same suspension parameters, pair potential,

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and particle volume fraction determined by the osmotic pressure calculation to compute the elastic constants for a defectfree FCC lattice. The output is a rank 4 tensor describing how the crystal reacts to an applied strain at zero frequency. The components of the tensor can be transformed to yield an estimate of the elastic constant a particular crystal face sheared in a particular direction. In this case, the densepacked 111 planes of the FCC lattice were used to estimate the modulus with the 110 direction, i.e., the direction of shear and the stress transmitted normal to the 111 planes (22, 23).

TABLE 1 Dimensionless Groups Used in MD, Osmotic Pressure Lattice Method, and Suspension Elasticity Calculations Debye length

k* Å ks

Temperature

T* Å

kT Uo f f* Å fmax r r* Å s Fs F* Å Uo Ps3 P* Å exp(0ks) Uo 3 Ga G* Å exp(0ks) Uo

Volume fraction Separation Force

2.1. Pair Potential

Pressure

For colloidal particles where van der Waals forces are small compared to the electrostatic forces, the Yukawa potential has been shown to work well (2–4, 9). To approximate the hard core, a steep short-range repulsion was used for separations of s or smaller (20).

U Å Uo

exp{ 0 ks((r/ s ) 0 1)} r/ s

for r ú s,

exp{ 0100(r/ s )} r/ s

for r ° s,

[5]

where e is the relative dielectric permittivity of the suspending medium, eo is the permittivity of free space, e is the charge on an electron, zi is the number of charges per ion, and nbi is the bulk solution concentration of ion type i. The strength of the interaction is characterized by Uo . This value is the primary variable of manipulation in the simulations, but it is a masked version of the surface potential and the particle diameter: Uo Å peeoc 2o s.

[6]

In the MD simulations, the ‘‘cut and shift’’ versions of the potential were used to speed the calculations (20). Dimensionless variables used in these calculations are summarized in Table 1. The mean separation is used to nondimensionalize the elasticity. It is defined as

S D

aÅs

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fmax f

where fmax is the q maximum volume fraction of the system, assumed to be 2 p /6 Å 0.74 in this work. 2.2. Elasticity Calculation

[4]

where the constant, k, is the Debye screening length (Eq. [5]). eeokT 1 Å , 2 2 k 2e ( z 2i nbi

Modulus

Many models of the shear modulus of dense suspensions of charge-stabilized particles have been developed and have met with varying degrees of success (3, 10–12). Direct computation of a modulus is difficult and requires assumptions about how the pairs contribute and the suspension microstructure. These models typically only account for harmonic contributions from nearest neighbors moving in one dimension along the line connecting their centers. Calculation of the stiffness constants, Cij , while still a complicated procedure, requires only an interaction potential and the particle positions. All interactions are assumed to add pairwise, but the stiffness constants are calculated for a particle in a three-dimensional potential field rather than the one-dimensional models commonly referred to in the literature. The stiffness constants also provide more information about the system than the direct modulus calculations based on assumed models. When combined in the proper relation, the stiffness constants yield moduli of the crystal when sheared in any direction, the bulk modulus, estimations of polycrystalline moduli, and information about symmetries of the suspension microstructure (22). A pairwise additive Yukawa potential is assumed, and an FCC lattice at the appropriate volume fraction is constructed. The lattice energy is then defined as the sum over all pairs of the potential evaluated at their separation (21). N0 1

UL Å ∑

N

∑ U(rij ).

[8]

iÅ1 j Åi/1 1/3

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[7]

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The stiffness constants are defined as the second derivative

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FIG. 1. The comparison of the C * 11 elastic constant for an FCC lattice computed by the method presented here with that of Robbins et al. Note the effect of volume fraction when a finite-sized core is used.

of the lattice energy with respect to the strain elements ( ek ) per unit of volume. Ckl Å

1 Ì 2UL , V Ìek Ìel

[9]

where V is the volume of the simulation cell that contains the particles. If one employs a potential that is a function of separation (i.e., r 2 ), the lattice energy can be calculated by the variation of the positions with strain at constant pair separations. The stiffness constants are expressed in terms of spatial derivatives of the pair interaction energy and strain derivatives of the separation between the pairs in the limit of zero strain. Now the chain rule of differentiation can be used to obtain the elastic constants. Completion of this algebra, detailed in the Appendix, yields Ckl Å

1 V

S

D 1kl

ÌUL Ì 2UL / D 2kl Ìr Ìr 2

D

,

35

sults of Robbins et al. ( 9 ) , who consider the dynamics of charged particles in terms of three parameters, k, Uo , and ( 1 / r ) 1 / 3 , where r is the particle number density ( Figs. 1 – 3 ) . When calculating elastic constants, this approach yields ( 1 / the number of independent elastic constants ) dimensionless groups, l Å ak, and Cij a 3 / Uo . What this work does not include is the volume fraction effects of using a finite core size. That is, specifying ks and r is not sufficient to uniquely determine volume fraction, f. The simulations presented here account for finite particle size yielding the fundamental parameters k, s, f, and Uo which yields one additional dimensionless group, f / fmax . Both the results reported here and those reported by Robbins et al. use an FCC lattice and the Yukawa potential with a finite core size. Elastic constants calculated by Robbins et al. were obtained by measuring phonon velocities through particular directions of the lattice, which, along with the crystal density in that direction, yields the elastic constants. At a volume fraction of 0.52, the scalings used here and that used by Robbins et al. are identical. Under these conditions, the two methods are in agreement. The elastic constants calculated here show good agreement with the work of Robbins et al. for a wide range of ak values with differences becoming apparent in the limit as ak goes to zero ( point particle limit ) . As an additional check, we find the correct symmetries and zero values for the elastic constant matrix are obtained from the calculations. For materials with cubic symmetry, there are three unique, nonzero elastic constants: C11 , C12 , and C44 ( 22 ) . These three constants can then be used to obtain elastic moduli for the deformation of any plane of the crystal.

[10]

where D 1kl and D 2kl are complicated expressions involving products of the x, y, and z components of the separation. Given the particle positions and their interaction potential, the stiffness constants, Cij , are easily calculated. 3. RESULTS

3.1. Analysis of Previous Results Elastic constants of particles in an FCC lattice interacting with Yukawa potentials are compared with the re-

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FIG. 2. The comparison of the C * 12 elastic constant for an FCC lattice computed by the method presented here with that of Robbins et al. Note the effect of volume fraction when a finite-sized core is used.

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the correlation for predicting osmotic pressures that results is P* Å

Ps 3 exp( 0 ks ) Uo

S

Å 15.6

D S

S D D

f fmax 0 0.899 exp 0 ks0.937 fmax f

0.347

.

[11]

FIG. 3. The comparison of the C * 44 elastic constant for an FCC lattice computed by the method presented here with that of Robbins et al. Note the effect of volume fraction when a finite-sized core is used. Also note that the method outlined here does not capture the plateau observed by Robbins et al. at low values of ak.

In the following, all of the elastic moduli compared to experimental data assume that ( 111 ) planes of particles are stacked and the stress is transferred in the direction normal to the planes resulting from a strain along the dense packed direction of the ( 111 ) plane. This is consistent with the observations of Chen et al. ( 24 ) and Chow and Zukoski ( 13 ) who found using SANS that ( 111 ) planes of particles align parallel to the walls of a rheological geometry with the dense-packed direction of the ( 111 ) plane aligned with the velocity vector after shearing the suspension. 4. DISCUSSION

4.1. Development of Osmotic Pressure and Elastic Modulus Correlations The osmotic pressure and elastic modulus correlations were developed in a straightforward manner exploring a wide range of the available ks to 0 f space. A plot of dimensionless osmotic pressure versus ks was made at a constant value of f / fmax . This curve was then fitted with an exponential curve, and the resulting prefactor and argument were extracted. This process was repeated for the other values of f / fmax . The prefactors and arguments of the exponential fit were then plotted as a function of f / fmax . The prefactors are well described by a linear function of f / fmax , while the argument is a power law function of f / fmax . When all of the aforementioned fit functions are compiled,

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The elastic constants are computed for an FCC lattice of given volume fraction and are then rotated to provide the modulus of stacked 111 planes sheared in the 110 direction with stress transferred normal to the planes. The same procedure of mapping out a series of dimensionless moduli for various values of ks and f / fmax was used, fitting exponential curves to these data and then fitting functions to the resulting prefactor and argument. The prefactor was well described by a linear function, and power law dependence of the argument was observed. The resulting function is G* Å

Ga 3 exp( 0 ks ) Uo

S

Å 7.44

D S

S D D

f fmax / 0.332 exp 0 ks0.863 fmax f

0.366

.

[12] When working with highly de-ionized systems, one is often concerned with the influence of the particle counterions on the Debye length. This leads to a volume fraction dependent Debye length that can be accounted for using several well-extablished models (3, 16, 17) when calculating and redimensioning the output of the correlations. The calculations made here fix the Debye length but do not assign its origin. Thus, if the counterions from the particles contribute substantially to the electrostatic screening length, this can be accounted for by altering k appropriately as a function of volume fraction and ascribing the surface potential becomes an iterative approach. 4.2. Comparison of Elastic Modulus Models Models of the elastic modulus have been considered by several authors (3, 10–12). Most are based on the harmonic component of the interaction energy ( Ì 2U(r)/ Ìr 2 ) of nearest neighbors. A major assumption in these models is the influence of a geometric constant for a given structure to describe the contributions from nearest neighbors. The results of the calculations reported here will be compared with three similar models proposed by van der Vorst et al. (11), Buscall et al. (12), and Russel et al. (3). The latter two models assume a Yukawa interaction potential and a harmonic oscillation for small strains

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between nearest neighbors only. They differ in their treatment of the geometric factor. The model of van der Vorst et al. includes an anharmonicity term in the particle interactions. All models, however, consider the particles to be strained along a line connecting their centers. The model proposed by Buscall et al. considers the change in separation when two particles initially at equilibrium are strained a small amount. The force is based on the new positions and the area through which particles interact. This is then averaged over all orientations and multiplied by the number of nearest neighbors to the origin particle. The resulting expression for FCC lattices is as follows.

S

G Å apeeoc 2o s 2

D S S DD

k 2a 2 / 2ka / 2 a exp 0 ks 01 s a4

,

[13] where a Å (3/32) fm N Å 0.833 for an FCC lattice ( fm Å 0.74 and N Å 12 neighbors) and a is the mean separation of the particles. Russel et al. treat the geometric factor in a much simpler manner by assuming potential energy is stored as a Hookean spring. The model uses the fact that displacing a pair from its equilibrium position by a small amount, g, increases the pair energy. This small increase in energy is summed over all pairs, and, if plotted vs g 2 , the slope of the resulting curve is G/2. Using the Yukawa potential, results in

S

Nr k 2a 2 / 2ka / 2 GÅ peeoc 2o s 2 2 a

D a 01 s

,

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S

fmax N k 2a 2 0 2ka 0 2 peeoc 2o s 2 5p a4

D

S S DD

1 exp 0 ks

[14]

where N is the number of nearest neighbors to a given particle and r is the number density of particles. Comparison demonstrates that these two models differ by a constant factor, numerically equal to 10.18 for FCC lattices, that arises from the different treatments of the geometry of the interaction. Consequently, for given values of particle size, Debye length, modulus, and volume fraction the two models will predict surface potentials that differ by roughly a factor of 3. In recent work by van der Vorst et al., a similar calculation was performed in which the resulting expression for the elastic modulus includes a first-derivative term providing anharmonicity to the potential well. This approach takes the statistical mechanical expression for the stress on a system, expands it in a Taylor series for small strains, and averages it over all conformations. Again, the assumptions are made that the single particle neighborhood resembles an FCC lattice, pairwise additivity of the interactions is valid, and the

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particle interaction is described by the Yukawa potential. The resulting expression for the elastic modulus is

GÅ

S S DD

1 exp 0 ks

FIG. 4. Comparison of the van der Vorst et al., Buscall et al., and Russel et al. models and the predictions of the work presented here for the dimensionless modulus for values of ks Å 20 and 5. The parameters used were 1/ k Å 9.6 nm, s Å 50 or 200 nm, and co Å 50 mV.

a 01 s

,

[15]

where N is the number of nearest neighbors to a given particle (assumed to be 12 for an FCC lattice), fmax is the maximum packing density (0.74 for FCC), and a is the mean separation of the particles. Note this differs slightly from the purely harmonic models proposed by Buscall et al. and Russel et al., but it offers only a small variation from the surface potential predicted by the model of Buscall et al. In Fig. 4, these models are compared with the elasticity calculation described in this work. A range of volume fractions are examined for typical colloidal particles of 200 and 50 nm diameters with a 50 mV surface potential suspended in a 10 03 M 1:1 electrolyte ( k 01 Å 9.6 nm). Models proposed by van der Vorst et al., Buscall et al., and Russel et al. capture the correct volume fraction dependence, but if forced to fit experimental data, assumptions about the geometry of the interaction lead to surface potentials that can differ by up to an order of magnitude from the full calculation model presented here.

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4.3. Comparisons with Experimental Data In this section, we compare our predictions to several sets of published experimental data for charge-stabilized colloidal suspensions. The data cover a wide range of ks and a large range of measured osmotic pressures and elastic moduli (Figs. 5 and 6). The systems also vary dramatically. The data from Chen, Benzing, Chow, and Voegtli are for charge-stabilized polymeric latices. The data from Fagan and Chang are chargestabilized silica particles. Particle sizes range from 30 to 700 nm in diameter, ionic conditions are from 10 06 to 10 02 M, and solid volume fractions are from 0.1 to 0.6. The microstructure of the data from Chen and Chow is known to be (111) planes of an FCC lattice parallel to the walls of the rheometer as confirmed by SANS (24). For the other data sources, the suspension microstructure is not known explicitly, but the suspension is assumed to be in an FCC lattice. In cases where both osmotic pressure and elasticity data exist, the same prefactor of the Yukawa energy (surface potential) is used to fit both sets of data. The pertinent parameters for the data presented here are summarized in Table 2. The surface potentials are indeed found to provide a consistent description of the osmotic pressure and elastic modulus behavior of charge stabilized suspensions (Figs. 7 and 8). For those systems where z has been determined, we find co / z scatters between 0.81 and 1.67 (see Table 3). Uncertainties in co used to fit the P and G * data are on the order of {10%. Uncertainties in z are difficult to determine due to the nonlinear relationship between z and the electro-

FIG. 6. Comparison of the predictions of the correlation presented here with the measured elastic moduli. The solid line bisecting the graph is G e*xperimental Å G c*orrelation , depicting perfect agreement between the correlation and experiment. The dashed lines indicate the estimated error of {20% of the predicted value. Also note the data presented here cover nearly four orders of magnitude in elastic modulus and range in ak from 1 to 60.

phoretic mobility. A lower bound on the uncertainty in z can be estimated from the uncertainty in the electrophoretic mobility, which is typically {5–10%. Thus, if the electrokinetic surface potential and the self-consistent surface potenTABLE 2 Parameters Used in Fitting the Experimental Data

FIG. 5. Comparison of the predictions of the correlation presented here with the measured osmotic pressures. The solid line bisecting the graph is Pexperimental Å Pcorrelation , depicting perfect agreement between the correlation and experiment. The dashed lines indicate the estimated error of {20% of the predicted value. Also note the data presented here cover nearly four orders of magnitude in osmotic pressure and range in ak from 1 to 60.

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Data source

Reference

Diameter (nm)

1/k (nm)

ks

co (mV)

Chen 34 Chen 146 Chen 229 Benzing A Benzing A Benzing A Benzing B Benzing B Voegtli A Voegtli B Fagan 125 Fagan 500 Fagan 800 Chow Chang 95 Chang 370 van der Vorst

(25) (25) (25) (17) (17) (17) (17) (17) (26) (26) (14) (14) (14) (13) (27) (27) (11)

34 146 229 105 105 105 366 366 176 150 124 465 778 255 95 368 490

9.6 9.6 9.6 9.6 304.4 96.1 96.1 960 30.4 30.4 7.84 7.84 7.84 9.6 9.6 9.6 304

3.54 15.21 23.85 10.94 3.45 1.09 3.81 0.38 5.75 4.93 15.82 59.3 99 26.6 15.6 38.3 1.61

63 83 42 66 58 46 52 34 87 78 148 262 153 38 98 111 53

Note. Particle sizes and Debye lengths were provided by the authors. Surface potentials were adjusted to fit the experimental elasticity data and osmotic pressure data if available.

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However, there is good agreement with the potential fit here and the potential found by Fagan when comparing predictions and measures of force–distance curves for silica spheres suspended in electrolyte solutions using an atomic force microscope (AFM). This observation, perhaps, reflects the similarity of AFM, G *, and P measurements in probing the overlap of double layers, while electrokinetic measurements are sensitive to a variety of other double-layer properties. 5. SUMMARY

FIG. 7. Model predictions for the 146 nm particles of Chen. ( l ) Experimental osmotic pressures. ( n ) Experimental elastic moduli. The lines are the predictions of the correlations presented here. The parameters used were s Å 146 nm, 1/ k Å 9.6 nm, and co Å 83 mV.

tial ascribed here arise from the same location in the double layer, we expect co / z to lie within {15–20% of unity. This expectation is reasonably well matched; unfortunately an error of {10% in co results in an error in P or G * of {20%. In addition, there are uncertainties due to the unknown suspension microstructure in most cases.

The methods described here provide a simple and accurate means of assigning a self-consistent surface potential for charge-stabilized colloidal suspensions on the basis of osmotic pressure or elasticity data. The correlations presented in this work provide a means of predicting suspension properties from a knowledge of fundamental particle and suspension parameters and a single adjustable parameter. This can be useful in processes where predictions are necessary for the pressures to consolidate a ceramic green body or to obtain an order of magnitude estimate of a suspension’s rheological behavior. For example, it has been shown by Chow and Zukoski ( 13 ) that the dynamic yield stress, t yd , when scaled by the modulus of charge stabilized suspensions over a large range of volume fraction, has a constant value of about 0.03. Similar scalings have also been shown to work for the dynamic properties of charge-stabilized silica suspensions ( 14 ) . Models of suspension elasticity proposed by van der Vorst et al., Buscall et al., and Russel et al. capture the correct form of the volume fraction dependence of the elastic modulus, but the assumptions made in treating the geometry of the interacting system lead to an estimation of surface potentials that can differ by up to an order of magnitude from the full calculation model presented here. As a consequence, if one estimates the surface TABLE 3 Comparison of the Potentials Fit Using the Correlations Presented in This Work and the Measured z Potentials Reported by the Authors

FIG. 8. Model predictions for the 105 nm particles of Benzing in 10 04 M KCl. ( l ) Experimental osmotic pressures. ( n ) Experimental elastic moduli. The lines are the predictions of the correlations presented here. The parameters used were s Å 105 nm, 1/ k Å 30.4 nm, and co Å 58 mV.

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Data source

Reference

co (mV)

z (mV)

Voegtli A Voegtli B Benzing A in 1003 Benzing A in 1004 Benzing A in 1005 Benzing B in 1005 Benzing B in 1007 Fagan 125

(26) (26) (17) (17) (17) (17) (17) (14)

87 78 66 58 46 52 34 148

72 90 47 52 54 39 42 81

co by AFM (mV)

co /z

120

1.21 0.87 1.40 1.12 0.85 1.33 0.81 1.83 (1.23)

Note. Estimated error in co and AFM fits is {10%. Uncertainties in z potentials are difficult to ascertain. Typical uncertainties in electrophoretic mobilities are of the order of 5–10%.

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potential from elasticity data, this potential will fail to correctly predict the suspension’s osmotic pressure. Since Eqs. [11] and [12] are based on the same form of particle interaction energy and suspension microstructure, surface potentials estimated by fitting the measurement of one parameter (say P ) will accurately predict the second parameter (G). This correspondence provides the basis for calling the fit surface potential self-consistent. Due to different models for double-layer structure used in developing links between electrokinetic measurements and surface potential and those used in the model developed here and uncertainties inherent in measuring suspension properties, the lack of agreement shown in Table 3 is not surprising. However, we emphasize that the purpose of estimating co from P or G is not to study the chemistry of the surface but to provide an accurate estimate of Uo for use in predicting thermodynamic or mechanical properties of particles interacting with a Yukawa potential. As a consequence, the success of the correlations presented here is best tested in the agreement between predicted and measured values of P and G with the one adjustable parameter Uo .

V (1) Å V (2) Å

In order to calculate the elastic constants, the positions of the particles and a pair potential dependent on the magnitude q of the separation ( r 2 ) are required. This derivation is general but will be adapted for use with the Yukawa potential here. There are no assumptions made about the arrangement of the particles. Any information about the symmetry of the system will be inferred from the symmetries, equalities, and zero values of the elastic constant matrix. The derivations will also be made using the contracted Voigt notation. The symmetric strain tensor is defined as

F

G

d 2U(r) 1 0 V (1) . dr 2 ÉrÉ2

UL Å ∑ U(r).

The elastic constants are defined as the second derivatives of the lattice energy with respect to strain, with the lattice energy normalized to a unit volume: Ckl Å

1 Ì 2UL . Volume ÌekÌel

Å

∑ pairs

S

D

1 Ì 2r 2 1 Ìr 2 1 Ìr 2 ( 2 ) V (1) / V 2 ÌekÌel 2 Ìek 2 Ìel

∑ D 1klV ( 1 ) / D 2klV ( 2 ) .

1

1

x2

x4

1

2

0

x2y2

1

3

0

x 2z 2

1

4

0

x 2 yz

1

5

1 xz 2

0

1

6

1 xy 2

0

The necessary expressions for the code are more simply derived as the square of the strained positions. The expressions will also be computed in the zero strain limit.

2

2

y2

y4

2

3

0

y 2z 2

r * 2 Å r * T r r * Å r 2 / 2r T rer r / r T rerer r. [18]

2

4

1 yz 2

0

2

5

0

y 2 xz

2

6

1 xy 2

0

S

D

[16]

The strained position vector between particles i and j is then defined as using r Å ri 0 rj and the identity tensor, d . r * Å ( d / e )r r

[17]

When the derivatives of the lattice energy are taken, the chain rule is used. Two expressions arise that, when grouped appropriately, slightly simplify algebraic manipulations:

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[23]

The expressions for D 1kl and D 2kl involve only the x, y, and z components of the separation, detailed in Eq. [24]. The values of the coefficients are symmetric (i.e., kl } l k ). D 2kl

.

[22]

pairs

D 1kl

2e1 e6 e5 e6 2e2 e4 e5 e 4 2e 3

[21]

Using the chain rule and taking the appropriate derivatives of the strained position vector, the final expressions for calculating the elastic constants are

l

1 2

[20]

pairs

k

eÅ

[19]

To begin the calculation of the elastic constants, the lattice energy is defined as the sum of the interaction energy for all pairs evaluated at the equilibrium positions:

Ì 2UL Å ÌekÌel

6. APPENDIX

dU(r) 1 , dr ÉrÉ

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CONSISTENT SURFACE POTENTIALS FROM SUSPENSIONS

3 3

z2

z4

3 4

1 yz 2

0

3 5

1 xz 2

0

3 6

0

z 2 xy

4 4

1 2 (y / z 2 ) 4

y 2z 2

4 5

1 xy 4

0

4 6

1 xz 4

0

5 5

1 2 (x / z 2 ) 4

x 2z 2

5 6

1 yz 4

0

6 6

1 2 (x / y 2 ) 4

x2y2

REFERENCES

[24]

For the Yukawa potential, the expressions for V ( 1 ) and V are (2)

U(r) Å

exp( 0 k(r 0 s )) , r

V (1) Å 0

U(r) r

V (2) Å 0

U(r) r2

[25]

S D S D k/

1 r

k2 /

,

3k 3 / 2 r r

[26] .

[27]

ACKNOWLEDGMENT This work is funded by the Department of Energy through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana—Champaign. We also thank Dr. F. B. van Swol for many helpful discussions.

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