Calculation of the collective diffusion coefficient of electrostatically stabilised colloidal particles

Colloids and Surfaces A: Physicochemical and Engineering Aspects 138 (1998) 161–172

Calculation of the collective diffusion coefficient of electrostatically stabilised colloidal particles W. Richard Bowen *, Anne Mongruel 1 Biochemical Engineering Group, Department of Chemical Engineering, University of Wales Swansea, Swansea, SA2 8PP, UK

Abstract The collective diffusion of concentrated suspensions of charged spheres is theoretically investigated. The analysis is based on a fundamental calculation of colloidal interactions between particles expressed in terms of osmotic pressure. The osmotic pressure accounts for multiparticle electrostatic interactions, dispersion forces, and configurational entropy effects. This osmotic pressure is used in the generalised Stokes–Einstein equation to derive the collective diffusion coefficient. This allows the prediction of the concentration dependence of the collective diffusion coefficient for various physicochemical conditions, such as particle size, surface charge or potential and electrolyte concentration. Results are compared with existing experimental data on the collective diffusion coefficient of bovine serum albumin, and to other dilute theories. Very good quantitative agreement is found. © 1998 Elsevier Science B.V. Keywords: Colloid; Electrostatically stabilised; Gradient diffusion coefficient; Osmotic pressure; Protein

1. Introduction The well-known Stokes–Einstein equation describes the Brownian motion of a single sphere in a liquid medium, due solely to the thermal fluctuations of the molecular movements around the particle. This equation yields the Brownian diffusion coefficient k T D = B 0 6pma

(1)

where a is the radius of the sphere, m the viscosity of the fluid, and k T the thermal energy. When B considering an ensemble of particles, the diffusion of one particle is influenced by the presence of its * Corresponding author. 1 Present address: GEMICO, ENSIC, 1 rue Grandville- BP 451, 54001 Nancy ce´dex, France. 0927-7757/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 6 ) 0 3 95 4 - 4

neighbours. Moreover, one has to distinguish between two diffusion processes. The self-diffusion coefficient D describes the fluctuating trajectory s of a tracer particle among others, and is obtained by a time average of the position correlation function. By contrast, the collective diffusion coefficient D , also termed the gradient or mutual c diffusion coefficient, describes a collective process, the relaxation of a concentration gradient. It is the coefficient appearing in Fick’s law of diffusion. D and D coincide with D at infinite dilution, s c 0 but differ for more concentrated suspensions. Both self and collective diffusion coefficients depend on the interactions between particles: hydrodynamic interactions mediated by the solvent, and direct colloidal interactions. In the present paper, the concentration dependence of the collective diffusion coefficient D of c colloidal particles as a function of the governing

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physicochemical parameters is investigated. This problem is of great practical interest because the collective diffusion coefficient is a key parameter in a large number of processes involving colloidal particles, such as membrane filtration or sedimentation. Moreover, in such operations where particles are concentrated, it is crucial to have some knowledge of the concentration dependence of the collective diffusion coefficient for a large range of volume fraction W. Theoretically, the calculation of the collective diffusion coefficient is a difficult task, because the calculation of the multiparticle interactions, hydrodynamic and colloidal, is a complicated and still not entirely resolved problem. At low volume fraction, where only pair particle interactions are important, it is possible to derive relatively simple expressions for the diffusion coefficient [1–3]. Some theories are also available for intermediate volume fractions, but only up to W=0.2 [4,5]. No theory at present is able to derive the concentration dependence of the collective diffusion coefficient for the whole range of volume fractions. The same problem occurs with the experimental measurement of the collective diffusion coefficient: measurements, usually performed by dynamic light scattering (DLS ) methods, fail at high volume fractions. Despite the considerable amount of experimental data in the literature on diffusion coefficients of various colloidal systems [2,3,6–12], they are mostly limited, with some exceptions, to the dilute limit. Moreover, in many of these publications there is a lack of detail concerning the exact physicochemical characterisation of the colloidal systems under investigation, which makes the comparison with theory particularly difficult. In the present work, the collective diffusion coefficient is derived from an ab initio calculation of the osmotic pressure of the suspension, P(W). The calculation of the osmotic pressure is based on a fundamental modelling of the colloidal interactions. This osmotic pressure model was originally developed for studying the ultrafiltration of colloidal particles, and proved to give excellent results when compared with ultrafiltration data of colloidal systems [13,14]. The generalised Stokes–Einstein equation [15] is used to derive the

diffusion coefficient from the osmotic pressure:

C

D

4pa3 ∂P (W) D (W)=D K(W) (2) c 0 3k T ∂W B where K(W) is the sedimentation coefficient. Eq. (2) is valid for the whole range of volume fractions, and any type of interparticle potential. The present study is concerned, in particular, with electrostatically stabilised colloidal dispersions. The aim of the study is to obtain an a priori prediction for the concentration dependence of the collective diffusion coefficient as a function of the main physicochemical parameters (particle size, surface charge, surface potential or zeta potential, ionic strength). This work was originally motivated by the necessity of having information on the collective diffusion coefficient to describe the rate of cross-flow ultrafiltration of colloidal particles. The paper is organised as follows. Section 2 gives a brief summary of the theoretical background for the calculation of the collective diffusion coefficient, and a review of the theoretical results available in the literature. In Section 3, experimental techniques for determining the diffusion coefficient are reviewed briefly. Section 4 describes the derivation of the equations for the calculation of the osmotic pressure and diffusion coefficient used in the present work. Finally, Section 5 presents a comparison of the calculated diffusion coefficient with results for the protein bovine serum albumin (BSA), available in the literature in the dilute regime, and with existing dilute theories.

2. Theoretical aspects In this section, some theoretical aspects of the collective diffusion of monodisperse, charged, colloidal spheres are reviewed. 2.1. Generalised Stokes–Einstein equation The generalised Stokes–Einstein equation describes the concentration dependence of D for c the whole range of volume fractions, and for any interparticle potential. Its basis is that the driving

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force underlying diffusion is thermodynamic in origin [16,17]. In the dispersion, there exists a thermodynamic force acting on the particles, which can be expressed in terms of the osmotic pressure and the number n of particles per unit volume [15,17]: F=−

1 n

VP=−

4pa3 ∂P 3W ∂W

VW

(3)

Under the action of this force, particles acquire a velocity V equal to V=MF, where M is the hydrodynamic mobility of the particles. For an isolated sphere, the mobility is given by Stokes law: M =1/(6pma). For concentrated suspensions, 0 the mobility deviates from this value as hydrodynamic interactions between particles become more important, and is then concentration dependent. It is given by M(W)=K(W)M , where K(W) is the 0 sedimentation coefficient. The flux of particles may be written as (4)

J=WMF

This can also be written in the form of Fick’s law: F=−D VW (5) c Combining Eqs. (3)–(5), together with the expression for the mobility, yields the generalised Stokes–Einstein equation in the form of Eq. (2) which can be rewritten as: D (W) K(W) c = D S(W) 0 with 1 S(W)

=

C

(6)

4pa3 ∂P (W)

3k T B

∂W

D

This relation is valid for any volume fraction, provided that the concentration gradient is not too great [16 ]. In Eq. (6), S(W) coincides with the structure factor of the suspension and accounts for its thermodynamic properties. The sedimentation coefficient K(W) accounts for the hydrodynamic properties. For systems of charged spheres, where the electrostatic repulsion is the dominant colloidal interaction, both S(W) and K(W) decrease

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with volume fraction. However, instead of their having cancelling effects on the diffusion coefficient, the resulting concentration dependence of D (W) can be significant in many cases. c 2.2. Equilibrium properties of a suspension: osmotic pressure The thermodynamic term S(W) in Eq. (6) directly involves the colloidal interactions through the osmotic pressure P(W). The osmotic pressure of colloidal suspensions is a function of particle concentration. This is due to the combined effects of electrostatic double layer repulsion, dispersion forces, and configurational entropy. Dispersion forces, which are attractive, give rise to a diminution of osmotic pressure with volume fraction, but as their contribution is usually less than the electrostatic and entropic contributions in the case of charged spheres, the net effect is always an increase of P(W) with W. In the dilute limit, it is possible to derive analytical expressions for the W dependence of S(W), using pair interaction functions [1–3,7]. This yields the first-order correction to S(W): S(W)=1−S W+O(W2) (7) 2 with S >0. For hard spheres S =8 [15], whereas 2 2 for charged spheres, the osmotic pressure increase due to the repulsion considerably increases the value of S . It can be difficult to predict the 2 behaviour of the osmotic pressure for the whole range of concentration and for a given set of physicochemical conditions. Some models provide good approximations at high volume fraction, but do not describe the complete concentration range. Other models are available for moderate volume fractions, up to W=0.2 only [4,5]. 2.3. Dynamic properties of a suspension: sedimentation and diffusion The sedimentation coefficient of a suspension of charged spheres depends on the influence of colloidal forces on hydrodynamic interactions. For a given volume fraction, hydrodynamic interactions can either enhance or slow down the sedimentation, in the presence of attractive or repulsive

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colloidal interactions respectively [18]. In the case of electrostatically stabilised spheres, which are the subject of this study, the sedimentation coefficient always decreases substantially with the volume fraction. Many theoretical works have been devoted to the determination of the sedimentation coefficient of colloidal suspensions. Analytical expressions are available at low volume fraction, and are limited to the first-order dependence of K with W [1]: K(W)=1−K (W)+O(W2) 2

(8)

with K >0. For charged spheres, the value of K 2 2 can be significantly larger than Batchelor’s result [16 ] for Brownian hard spheres (K =6.55). 2 At finite concentration, the situation is less clear. For Brownian hard spheres, many experimental data can be correctly described up to W=0.5 by the empirical correlation [1] K(W)=(1−W)6.55

(9)

which coincides with Batchelor’s result in the dilute limit. This correlation gives a satisfactory description of the sedimentation of a disordered suspension of hard spheres. Calculations of hydrodynamic interactions for periodic arrays of hard spheres have been made using either integral equations methods [19] or Stokesian dynamic computations [20]. Both methods give comparable results up to W=0.2, regardless of the geometry of the array (simple cubic, body centred, faced centred). An earlier model was developed by Happel and Brenner [21] using a cell model for ordered suspensions of spheres, and yielded the following analytical expression for K(W): K(W)=

1−3W1/3+3W5/3−W2 2 2 1+2W5/3 3

(10)

For suspensions of charged spheres, many empirical correlations of the form K(W)=[1−(W/W )]c m are available, W and c depending on the system m under consideration [1]. So far, a few theories have attempted to describe the W dependence of K(W) up to W=0.2 [4], using sophisticated descriptions of the hydrodynamic interactions (RMSA closure or Roger–Young system), but with crude

approximations for the colloidal interparticle potential (reduced to a pair potential ). It follows from Eqs. (7) and (8) that the W dependence of the collective diffusion coefficient can be written in the dilute limit as D (W)=D (1+lW) (11) c 0 with l=S −K . For hard spheres, l=1.45 and 2 2 the concentration dependence of D is weak, even c at moderate volume fractions. This is due to a nearly cancelling effect of the osmotic pressure contribution and of the sedimentation coefficient contribution in that case. If the interaction energy between particles is repulsive (charged spheres), then l is positive and D increases with concenc tration in the limit of low volume fraction. This increase can be very significant, particularly as the range of the repulsion increases [7].

3. Experimental aspects Experimental measurement of diffusion coefficients is usually performed by DLS. This technique provides the overwhelming part of the diffusion data to be found in the literature. Unfortunately, problems arise as the volume fraction increases due to the opacity of the concentrated suspension, and of multiple light scattering effects. In some cases, those effects can be minimised by using an optically matching solvent [9,11], but, generally, the volume fraction explored does not exceed W= 0.2. Alternative methods to DLS are not numerous, and are more rarely used: capillary methods [7,22,23], porous barrier (membranes) methods (cited in Ref. [7]), or Gouy interferometric methods [8]. They are generally also limited to diluted samples. In the present study, we wished to compare our modelling of diffusion coefficients to different existing sets of data, chosen because they provided a good physicochemical description of the colloidal system under consideration and, hence, provided all the useful parameters for the calculations. The choice proved to be very limited. The most complete set of data is formed by the experiments of Anderson et al. [7] on the diffusion of the protein BSA, and performed with a capillary method. This

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165

is the set of data which has been most widely used by other workers who have developed theoretical approaches to the calculation of diffusion data [1,2]. It is also the data with which we have chosen to test the theoretical approach developed in the present paper. The capillary method used by Anderson et al. [7] is based on the principle of hydrodynamic stability in a vertical tube. A fluid of higher density (the suspension) is placed above a lighter fluid (the suspending medium), thus creating an inverse concentration gradient. A circulatory flow then occurs, the lighter fluid rising and the denser fluid falling. This process stops when the concentration gradient is balanced by viscous and Brownian forces. This occurs for a given value of the diffusion coefficient D and suspension viscosity m. The same condition c holds even for more concentrated samples, for which D and m are concentration dependent. At c the end of the experiment, when the concentration equilibrium is reached, the mean concentration in the capillary is determined. This mean concentration gives access to the value of D , provided c that the concentration dependence of the viscosity is known, and that a linear dependence of D with c volume fraction is assumed. This is, therefore, only valid for dilute samples. The coefficient l of Eq. (11) can then be determined for different concentrations and physicochemical conditions. The experimental technique has been subsequently refined [22,23]. The experiments of Anderson et al. proved to give excellent reproducibility for the coefficient l.

of physicochemical conditions such as particle size, surface charge, surface potential or zeta potential and ionic strength. A given local geometry of the suspension has to be assumed. There is a considerable body of evidence that concentrated electrostatically stabilised dispersions exist in a regular packing form, typically hexagonal packing.2 Such a hexagonal packing is assumed throughout the range of volume fraction, allowing a calculation of the osmotic pressure at any point of the suspension, from [13,24]

4. Calculation of osmotic pressure, sedimentation coefficient and diffusion coefficient

4.1.1. Electrostatic interactions For concentrated suspensions where the interparticle distance is comparable to the double layer thickness, multiparticle electrostatic interactions have to be taken into account. This is done by means of a cell model [13,25], where the net effect of multiparticle interactions on one particle is equivalent to the enclosure of the particle in a spherical cell. The radius b of the cell is determined by the volume fraction: W=(a/b)3.

4.1. Osmotic pressure In the present work, we use a calculation of osmotic pressure based on the addition of the different contributions to the interaction energy, in accordance with extended DLVO theory. This calculation was originally developed to predict the rate of dead-end filtration of colloids [13,14]. It allows the prediction of osmotic pressure for the whole range of concentration and for a given set

P (D)=

E6 A

[F(D)+F (D)]+p A ent

(12)

h where p is the entropic pressure term, f(D) and ent F (D) the interparticle forces due to the double A layer electrostatic forces and dispersion forces respectively, between particles at a distance D apart. A is the surface of interaction, in a hexagoh nal array of volume fraction W. Only nearestneighbour interactions are included for the dispersion forces. The electrostatic force, on the contrary, results from an electrostatic cell model (see below) which accounts for multiparticle electrostatic interactions, the latter being very important when the repulsion is long range. The contribution of entropic pressure p is then added. The contribuent tion of the attractive dispersion forces to Eq. (12) will be generally small compared with the electrostatic repulsion, except when the distance between particles is small. The entropic pressure term becomes more important as colloids decrease in size, and, hence, is very significant for proteins such as BSA.

2 Discussed and cited in Ref. [13].

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The Poisson–Boltzmann equation (PBE ), which describes the electrostatic potential y as a function of the distance r from the particle centre, is written in spherical coordinates: ∂2y

+

2 ∂y

=

2n0ze

sinh

A B ezy

(13) e e k T 0 r B where e is the permittivity of vacuum, e the 0 r dielectric constant, e the elementary charge, n0 the number of ions per unit volume of bulk electrolyte, and z the valency of the electrolyte. Eq. (13) is solved numerically inside the cell with appropriate boundary conditions: a condition of constant charge, or constant potential, or charge regulation at the particle surface, and the electroneutrality of the cell for the outer boundary condition (r=b). Knowing the potential at the cell surface, y , gives b the electrostatic potential per particle and hence the configurational electrostatic force between particles used in Eq. (12): ∂r2

r ∂r

C A B D

1

zey

b −1 (14) k T B where S is the surface area of the spherical cell. b

f (D)=

S n0k T cosh B 3 b

4.1.2. Dispersion forces The van der Waals attractive interaction energy between two spheres of radius a at a distance D apart is A V (D)=− H A 6 +

G

2a2 D2+4aD

2a2 (D+2a)2

C

+ln 1−

4a2 (D+2a2)

DH

dV (D) A dD

p

ent

A

=

A B 3k T B 4pa3

W+W2+W3−0.67825W4−W5−0.5W6−XW7 1−3W+3W2−1.04305W3

B

(17) where X=6.028 exp[Y(7.9−3.9Y )] and Y= (pE2/6)−W. A detailed examination of this approach to the calculation of osmotic pressure [28] shows that it gives excellent agreement with the best available experimental osmotic pressure data for BSA [29]. However, those experimental osmotic pressure data were for physicochemical conditions quite different from those for the best available collective diffusion data for BSA [7] against which calculations of diffusion coefficients will be made in the present work. The present calculations, therefore, also provide a further independent assessment of the validity of the proposed description of protein–protein interactions.

(15) 4.2. Sedimentation coefficient

The Hamaker constant A depends on the properH ties of the particles and of the suspending medium [26 ]. For BSA particles, a detailed knowledge of the dielectric constants and frequency of electromagnetic radiation allows the calculation of a screened, retarded, Hamaker constant [14]. The force between two spheres can be calculated from the interaction energy expression as F (D)= A

4.1.3. Entropic pressure The osmotic pressure of colloidal suspensions undergoes a steep rise as the volume available to the particles decreases. For this reason, virial equations of state valid at low volume fraction become very rapidly limited as W increases and recourse has to be made to molecular dynamic computer simulations. In the present work, an equation of state is used which approximates closely the results of such simulations for the whole range of volume fraction [27]:

(16)

In Fig. 1, the sedimentation coefficients K(W) obtained by several authors (see Section 2) have been drawn for comparison. The open circles, squares and triangles correspond to the exact solution of Zick and Homsy [19] for face-centred, hexagonal-centred and simple cubic arrays respectively. The continuous line depicts Happel’s cell model (Eq. (10), and the dashed line the correlation for a disordered dispersion of hard spheres ( Eq. (9). It can be seen that Happel’s cell model

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167

Moreover, these two equations are convenient analytical expressions for K(W). Consequently, they will be used in the generalised Stokes–Einstein equation to derive the diffusion coefficient in Section 5. 4.3. Electroviscous effect

Fig. 1. Sedimentation coefficient K(W) vs. W for different assemblages of uncharged spheres: Happel’s cell model of Eq. (10) (———); correlation for a disordered hard-sphere dispersion of Eq. (9) (– – –); the exact solution of Zick and Homsy [19] for a face-centred cubic array, #; body-centred cubic array, %; and simple cubic array, 6 (symbols are superimposed when not all are visible).

is close to the results of Zick and Homsy, in agreement with the order assumed in these two models. The correlation for disordered hard-sphere dispersions lies well above the two other models, indicating that the sedimentation is less hindered in that case because the disordered configuration restricts the back-flow less. As mentioned previously, it is reasonable to assume that in the case of charged spheres the electrostatic repulsion induces an ordering of the suspension even at moderate volume fractions. This can be confirmed by comparing the results of the Roger–Young system calculations [4] for the sedimentation coefficient with the curves in Fig. 1: one finds that they lie very close to Happel’s results in the case of strong and long-range repulsion, and close to the hard-sphere correlation in the case of a low repulsion. We conclude that Eqs. (9) and (10) give a good description of the sedimentation coefficient in two limiting cases: a disordered dispersion of uncharged spheres, and ordered dispersion of charged spheres with long-range repulsion.

Two electrokinetic effects are likely to affect the diffusion coefficient of colloidal particles. The primary electroviscous effect, due to the deformation of the electrical double layer during the particle motion is negligible compared to the effects of interparticle interactions. However, it can modify slightly the dilute limit value D of the diffusion 0 coefficient, depending on the physicochemical conditions. This effect explains the scattered results sometimes encountered experimentally for D [2]. 0 The secondary electroviscous effect is due to the streaming potential which is set up during the motion of an electrolyte relative to a suspension of charged particles. It results in a diminution of the flow of liquid, or, equivalently, to an increase of the apparent viscosity of the suspension [30]. This leads to a modification to Eq. (2) to account for the effect of the secondary electroviscous effect on K(W):

C

4pa3 ∂P (W) m K(W) m K(W) D =D =D c 0 m S(W) 0m k T ∂P a a B

D (18)

This effect can be accounted for mathematically, and the ratio m/m calculated [31]. For low ionic a strengths and large zeta-potential values, this effect can become significant and lead to a maximum viscosity increase of about 20%. It is included in the following calculations of the diffusion coefficient.

5. Theoretical predictions for the diffusion coefficient of BSA In this section, the method described in Section 4 for calculating the collective diffusion coefficient ( Eq. (18), Eq. (9) or Eq. (10) and Eq. (12) is applied to the case of BSA solutions.

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5.1. Physicochemical characteristics of BSA in solution In the experiments of Anderson et al. [7], with which our calculations will be compared, the pH of the BSA solution was kept constant at 6.5 and the ionic strength of the KCl electrolyte solution was varied from 10−1 to 5×10−4 M. The temperature was 4°C. 5.1.1. Effective size of BSA in solution Anderson et al. give the hydrodynamic radius a of BSA as 3.6 nm (calculated from D ), a value 0 which has also been used by subsequent workers in comparing theoretical approaches with the data of Anderson et al. We will, therefore, use this value in our calculations. However, we wish to distinguish in our calculations between the surface of the protein and the outer Helmholtz plane (OHP), that is we will describe the electrical double layer in terms of a modified Gouy–Chapman model [30] so that the surface potential y and surface charge 0 s are distinguished from the potential f at the 0 OHP and the diffuse layer charge s . It will be d assumed that the OHP coincides with the hydrodynamic radius and the zeta potential (if this were to be experimentally determined ). The distance d of the OHP from the surface of the protein is taken to be the ionic radius of a hydrated counterion, 0.23 nm [14]. Electrostatic interactions and volume fractions are calculated using the hydrodynamic radius. Dispersion forces are calculated using the effective solution radius of the protein (a=3.6−0.23=3.37 nm). It is assumed that the hydrodynamic radius does not vary with ionic strength. 5.1.2. Charge properties of BSA in solution Solution of the Poisson–Boltzmann equation, Eq. (13), requires an appropriate boundary condition at the BSA surface. Possibilities include constant surface charge, constant surface potential or constant zeta potential [13]. However, a more complete description is given by the charge regulation boundary condition [30] which accounts explicitly for the way in which the charge at the surface is generated. In the case of BSA the charge

originates from the ionisation of the charged sidechains of the amino acids comprising the protein [32] and from specific binding of chloride ions [33]. Here, we use the data of Tanford et al. [32] for the numbers of charged amino acid side chains and their pK values, and of Scatchard and Black [33] for a quantitative description of chloride-ion binding. When these are taken in conjunction with the modified Gouy–Chapman description of the electrical double layer they allow a prediction of the charge properties, and hence interactions of BSA molecules, for any values of pH or ionic strength. For illustration, the f values calculated in this way for an isolated BSA molecule (in the limit of vanishing BSA concentration) are listed in Table 1 for the different ionic strengths used by Anderson et al. The values of surface potential y calculated 0 by Anderson et al. from the charge z are also 0 shown for comparison. It should be noted that, in the paper of Anderson et al., these potential values were calculated at the distance r=a and so correspond to our zeta potentials. There is good agreement between the two sets of potentials. 5.1.3. Hamaker constant For evaluating the dispersion forces (Eq. (15), a detailed expression for the screened, unretarded Hamaker constant A of BSA in water is used H [26 ]. The latter is obtained from knowledge of the dielectric constant (e ) and frequency of electror magnetic variation (v ) of water and protein. i For water, the values e =78.55 and v = r i Table 1 Surface charge z , surface potential y and zeta potential f of 0 0 BSA at pH 6.5 and T=4°C, for different KCl concentrations [ KCl ] (M )

−z a 0 (e)

−y b 0 (mV )

−fc (mV )

−y c 0 (mV )

5×10−4 10−3 2.5×10−3 10−2 10−1

6.0 6.3 6.8 9.6 15.1

22.20 21.48 20.05 20.53 15.04

22.24 21.10 19.59 17.59 14.66

23.90 22.83 21.46 19.89 18.82

a Data of Ref. [7]. b Calculations of Ref. [7]. c Calculated from present charge regulation model.

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1.899×1016 rad s−1 are used [34], for the protein e =1.9, v =1.350×1016 rad s−1 [14]. For referr i ence, the resulting unscreened and unretarded Hamaker constant is then A =0.753×10−20 J. H 5.2. Results and discussion Calculations of the dependence of D on W are c presented in Fig. 2, for the different ionic strengths used in the experiments of Anderson et al. The sedimentation coefficient K(W) used is Happel’s ( Eq. (10), except for the dotted line drawn for the highest ionic strength, 0.1 M, where the correlation for a disordered hard-sphere dispersion ( Eq. (9) has been used. The variations of the diffusion coefficient exhibit the same trends for all values of ionic strength: an initial increase up to a maximum value, and then a decrease. An important feature is that this maximum value increases with decreasing ionic strength, that is with the range of the electrostatic repulsion. Experimental data [7] are also shown in Fig. 2.

Fig. 2. Adimensional diffusion coefficient D (W)/D vs. volume c 0 fraction W for BSA solutions at different ionic strengths, pH 6.5 and T=4°C. Theoretical lines calculated using Eq. (18) (for KCl solutions): (———) using Happel’s sedimentation coefficient: (a) 5×10−4 M, (b) 10−3 M, (c) 10−2 M, (d) 10−1 M; (,) using a disordered hard-sphere sedimentation coefficient, 10−1 M. Experimental data: &, Ref. [7], 10−3 M KCl; $, Ref. [7], 10−1 M KCl; #, Ref. [6 ], 0.15 M, pH 7 and T=25°C.

169

Also shown on the same graph are the data of Phillies et al. [6 ]. The physicochemical conditions in the latter (pH 7, I=0.15, T=25°C ) do not match exactly those of Ref. [7]. However, it seems that at high ionic strength the protein diffusion behaviour approaches that of hard spheres and is neither very sensitive to the exact physicochemical conditions nor does it show a great variation with volume fraction. There is, nevertheless, a good agreement between the experimental data and the present calculations when using the hard-sphere correlation for the sedimentation coefficient ( Eq. (9). The agreement between theory and experiment appears better than that presented in Ref. [1], which gives a comparison with an approximate hard-sphere theory, and especially so at higher values of W where the theoretical line in Ref. [1] lies below the data. The latter treatment is based on a Carnahan–Starling result for the osmotic pressure together with Eq. (9) for the sedimentation coefficient, both of which are hardsphere expressions which do not take electrostatic interactions into account. The better agreement in the present case may, therefore, be evidence for the influence of electrostatic interactions on D (W), as W increases even for cases where such c interactions are relatively weak (here due to the high ionic strength). At such high ionic strength there is not sufficient long-range order for Happel’s expression for the sedimentation coefficient ( Eq. (10) to be appropriate (curve (d)). A much better test of the present calculations is the comparison between the experimental data in 10−3 M solution and our theoretical curve (b). Under such conditions long-range order is established, and so our treatment of the calculation of osmotic pressure as well as the use of Happel’s expression are valid. The good agreement between experimental data and theoretical calculation is very encouraging. A more quantitative validation of our calculations can be performed in the dilute limit. Calculating the initial slopes of curves (a), (b) and (c) of Fig. 2 yields the coefficient l of Eq. (11), which can be compared directly to the same coefficient obtained either experimentally by Anderson et al., or from dilute theories by other authors [2,15,35]. Numerical values for l are given

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in Table 2. There is good agreement between the present theoretical calculations and the experimental data of Anderson et al. [7]. Our calculations also compare well with those presented in Ref. [7] based on a treatment by Batchelor [15] and those in Ref. [7] based on a treatment by Anderson and Reed [35]. There is also good agreement between the present calculations and the recent calculations of Petsev and Denkov [2] who used Felderhof ’s theory. Comparisons of two theoretical treatments (an adhesive excluded-shell model and a detailed model ) with the experimental data of Ref. [7] are also given in Ref. [1]. The adhesive excluded-shell model gives results close to those based on the treatment of Batchelor given in Ref. [7], which is to be expected as the equations developed are very similar. The more detailed theory gives comparable agreement. It should be noted, however, that, unlike the cell model developed in the present work, all of these previous treatments use pairwise calculation of electrostatic interactions and will be less applicable to higher volume fractions.

6. Conclusions A method for calculating the collective diffusion coefficient of a colloidal dispersion from the Table 2 Coefficient l in D=D (1+lW) for different KCl concentrations 0 [ KCl ](M )

la

lb

lc

ld

le

5×10−4 10−3 2.5×10−3 10−2 10−1

236 115 15 19 0

200 95 37 12 4

323 156 65 21 0

224 (242) 122 (123) 46 (50) 17 (19) 2 (3.4)

269 135 57 22 1.2

a Experimental results taken from graph of Ref. [7]. b Theory of Batchelor [15], taken from graph of Ref. [7]. c Theory of Anderson and Reed [35], taken from graph of Ref. [7]. d Dilute theory of Petsev and Denkov. The first values are those quoted in Ref. [2] and the values in parentheses are our calculations using eq. (2.34) in Ref. Eq. (2). e Present model with charge regulation, calculated from the initial slopes of lines in Fig. 2 (with an additional value for 2.5×10−3 M ). The value for 10−1 M is for the dotted line in the figure.

osmotic pressure and a sedimentation coefficient has been developed. The osmotic pressure is calculated from a fundamental modelling of the colloidal interparticle interactions. This takes into account electrostatic double layer interactions, dispersion interactions and entropic forces. Electroviscous effects are also taken into consideration. The electrostatic interactions may be calculated directly from a knowledge of the surface chemistry of the colloidal material through the use of a charge regulation description of the surface. The calculations assume long-range electrostatic ordering of the dispersion. Two simple expressions have been used for the sedimentation coefficient: Happel’s cell model for an ordered dispersion and a correlation valid for a disordered hard-sphere dispersion. A comparison of our calculations with existing data for BSA shows good agreement in the dilute limit at low ionic strengths where long-range ordering is expected and the diffusion coefficient increases markedly with colloid concentration. This indicates the validity of our osmotic pressure model for these biocolloids, whose physicochemistry is fairly well understood, and of our subsequent use of such osmotic pressure values in the calculation of collective diffusion coefficients. Good agreement is also obtained with data at high ionic strength if a sedimentation coefficient correlation for a disordered hard-sphere dispersion is used. The approach developed is, in principle, valid for a wide range of volume fraction. However, there do not at present exist collective diffusion coefficient data with fully known physicochemical conditions for proteins or inorganic colloids for even moderate values of the volume fraction. More extensive testing of the calculations awaits such data. For an engineer designing membrane processes such data and predictions are important, as volume fractions up to close packing may occur at the membrane surface during cross-flow filtration.

Acknowledgment This work forms part of a programme of research funded by the UK Engineering and

W.R. Bowen, A. Mongruel / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 161–172

Physical Sciences Research Council. Dr. A. Mongruel was funded through an EC Human Capital and Mobility programme.

7.1.1. Greek letters a b e 0

Appendix

e r f l m

7.1. Nomenclature

m

a

P(W) a A h A H d D D (W) c D S D 0 e f(D) F F (D) A J k B K(W) M(W) M 0 n n0 p ent r S(W) S b T V V (D) A z

effective hard sphere radius in solution (m) area of a hexagon (m2) Hamaker constant (J ) distance of OHP from particle surface (m) interparticle distance (m) gradient or mutual diffusion coefficient (m2 s−1) self-diffusion coefficient (m2 s−1) Brownian diffusion coefficient in the dilute limit (m2 s−1) elementary charge (1.6×10−19 C ) configurational force (N ) thermodynamic force (N ) attractive interparticle force (N ) flux of particles (m s−1) Boltzmann constant (1.38×10−23 J K−1) sedimentation coefficient particle hydrodynamic mobility (N s m−1) Stokes hydrodynamic mobility (N s m−1) number of particles per unit volume (m−3) number of ions per unit volume in the bulk electrolyte (m−3) entropic pressure (N m−2) radial coordinate (m) structure factor surface area of spherical cell (m3) absolute temperature ( K ) velocity (m s−1) attractive interaction energy (J ) valency

171

s d s 0 W y y b y

0

v

i

hydrodynamic sphere radius (m) spherical cell radius (m) permittivity of vacuum (8.854×10−12 C V−1 m−1) dielectric constant zeta potential ( V ) coefficient in Eq. (11) dynamic viscosity of the solvent (N s m−2) apparent dynamic viscosity of the solvent (N s m−2) osmotic pressure of the suspension (N m−2) diffuse layer charge (C ) particle surface charge density (C m−2) colloid volume fraction electrostatic potential ( V ) electrostatic potential at the outer spherical cell boundary ( V ) electrostatic potential at the particle surface ( V ) characteristic frequency of electromagnetic radiation (rad s−1)

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