Electrophoretic Mobility of Colloidal Particles in Weak Electrolyte Solutions

Journal of Colloid and Interface Science 211, 160 –170 (1999) Article ID jcis.1998.5984, available online at http://www.idealibrary.com on

Electrophoretic Mobility of Colloidal Particles in Weak Electrolyte Solutions Constantino Grosse*,1 and Vladimir Nikolaievich Shilov† *Instituto de Fı´sica, Universidad Nacional de Tucuma´n, Av. Independencia 1800, 4000 Tucuma´n, Argentina, and Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Argentina; and †Institute of Biocolloid Chemistry of the National Academy of Sciences, Kiev, Ukraine E-mail: [email protected], [email protected] Received October 14, 1998; accepted November 9, 1998

which makes it impossible to assign any meaningful z potential value to the suspended particles. Many attempts have been made to account for this incompatibility, the main approaches being refinements of the treatment of the thin stagnant layer behind the slip surface (25, 27–31). The inclusion in the standard model of an anomalous conductivity due to the movement of ions inside this layer does increase the dielectric increment but it also decreases the electrophoretic mobility (29, 31). The problem remains, therefore, essentially unsolved. A generalization of the theory to the case of weak electrolytes was first presented by Baygents and Saville (32). These authors developed a numerical theory of electrophoresis considering arbitrary values of the equilibrium dissociation–recombination constant but for the limiting case of very fast kinetics, i.e., for infinitely large values of the dissociation– recombination rate constant. They also took into account the possible conductivity and hydrodynamic movement inside the disperse phase. In this work we consider, in the framework of the standard model, the problem of a rigid nonconducting colloidal particle in a weak electrolyte solution with arbitrary values of both the equilibrium and the rate constants of dissociation–recombination. We solve analytically the electrodiffusion equations in the thin double layer approximation. Our results extend the applicability of the analytical theory to systems for which the strong electrolyte hypothesis does not apply and the finite values of the equilibrium and rate constants are important. They also show that even for moderate deviations from this hypothesis, the presence of ion pairs and of the corresponding recombination and dissociation processes can substantially increase the electrophoretic mobility value of colloidal particles. A comparison of the new analytical theory with the numerical one (32) shows full agreement for the cases when the restrictions of both theories are simultaneously fulfilled.

The analytical theory of the thin double layer concentration polarization in suspensions of colloidal particles is generalized to the case of weak electrolyte solutions, i.e., when the dissociation– recombination equilibrium and rate constants have both finite values. It is shown that under the action of a static applied field, regions near the particle appear where there is departure from the dissociation–recombination equilibrium. The resulting ion and ion-pair sources have a strong bearing on their flows, leading to a change of the electrolyte concentration gradients around the particle. This phenomenon also modifies the value of the particle electrophoretic mobility, which is dependent on the concentration polarization. At constant ionic strength, the theoretical maximum of the electrophoretic mobility versus z potential curve can substantially surpass in weak electrolyte solutions the corresponding value attained in strong electrolytes. © 1999 Academic Press Key Words: weak electrolytes; colloidal particles; concentration polarization; dissociation–recombination; electrophoretic mobility.

INTRODUCTION

All the theoretical treatments of the electrokinetic and dielectric properties of colloidal suspensions, based on the standard model (1–13), consider that the electrolyte is strong. This means that no ion pairs are present in the electrolyte solution and that no dissociation or recombination processes take place. The theoretical results show that the electrophoretic mobility is strongly dependent and that the dielectric increment is mainly determined by the field-induced changes of the ion densities outside the double layer. This suggests that even a small amount of dissociation and recombination could have an appreciable bearing on these parameters. Experimental results show that electrokinetic and dielectric measurements are usually incompatible: z potential values calculated from the electrophoretic mobility lead to theoretical predictions for the dielectric increment that are far too low (14 –22). Worse still, the observed electrophoretic mobility is often higher than the theoretical maximum (19, 20, 23–26), 1

THEORETICAL FORMULATION

We consider a weak electrolyte solution made of the solvent and three types of solutes: positive ions (upper index 1),

To whom correspondence should be addressed.

0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

160

161

ELECTROPHORETIC MOBILITY IN WEAK ELECTROLYTES

negative ions (upper index 2), and ion pairs (upper index n). In equilibrium (lower index 0), their concentrations C (number per unit volume) are related by 2 KC n0 5 C 1 0 C0

e 6 6 D C 0 ¹ f 0 2 D 6¹C 6 0 5 0 kT

j n0 5 2D n¹C n0 5 0,

6 C6 5 C6 0 1 dC

[1]

where K is the dissociation constant (in the limit of a strong electrolyte K 3 `). In the absence of an applied field, the ion and the ion-pair flows j around a charged spherical particle suspended in the electrolyte solution vanish, j6 0 5 7

Writing each field-dependent quantity as an equilibrium value plus a term proportional to the field,

C n 5 C n0 1 d C n

f 5 f 0 1 df , keeping only terms that are linear in the applied field, and using Eqs. [2] and [3], transforms Eqs. [6], [7], and [9] into e 6 6 D C 0 ¹ df 2 D 6¹ d C 6 kT

[2]

j6 5 7

[3]

j n 5 2D n¹ d C n

where f is the electric potential, D are the diffusion coefficients, and it was assumed that the electrolyte is univalent ( z 1 5 z 2 5 1). The solutions of these equations are the equilibrium ion concentrations around the particle,

C 0e 2f0e/kTC 0e 1f0e/kT C 20 C 5 5 . K K

Outside the thin double layer, in the electroneutral solution,

[4]

e 6 6 D C ¹ f 2 D 6¹C 6 kT

j 5 2D ¹C . n

n

n

d C 1 5 d C 2 5 d C. Equations [10] and [12] further simplify to j6 5 7

[5]

In the presence of a homogeneous static electric field, ion and ion-pair flows appear, which obey the equations j6 5 7

[12]

C6 0 5 C0

and the equilibrium concentration of ion pairs, which is a constant independent of the distance to the particle: n 0

[11]

2 2 1 ¹ z j n 5 2S~K d C n 2 C 1 0 d C 2 C 0 d C !.

7f0e/kT C6 0 5 C 0e

C0 5 C6 0 ~`!,

[10]

[6]

e 6 D C 0¹ df 2 D 6¹ d C kT

¹ z j n 5 2S~K d C n 2 2C 0d C!,

[13]

which, combined with Eqs. [8], [9], and [11], lead to 7

e 6 D C 0¹ 2df 2 D 6¹ 2d C 5 S~K d C n 2 2C 0d C! kT

[14]

[7] ¹ 2d C n 5

These flows are related by the continuity equation ¹ z j 6 5 2¹ z j n,

[8]

¹ z j n 5 2S~KC n 2 C 1C 2!.

[9]

where

In this expression, which out of equilibrium replaces Eq. [1], S is the dissociation–recombination rate constant. The first addend in the right-hand side is equal to the rate of dissociation of ion pairs per unit volume, while the second corresponds to the recombination rate of ions.

S ~K d C n 2 2C 0d C!. Dn

[15]

In order to solve this system of coupled equations for the unknowns dC, dC n, and df, it is first transformed to the form ¹ 2S 5 L 22S

[16]

¹ 2d C 5 2

S S D ef

[17]

¹ 2df 5 2

kT SD S, e C 0D ef

[18]

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GROSSE AND SHILOV

where S 5 ~K d C n 2 2C 0d C!

[19]

SK C 0D n 1 1 2 Dn KD ef

D

[20]

D2 2 D1 . D1 1 D2

[21]

L 22 5 D ef 5

S

2D 1D 2 D1 1 D2

D5

Expression [16] has the form of the Helmholtz equation. The part of its general solution that tends to zero at infinite distance is S 5 Pf~r, L!cos u ,

[22]

shows the following differences. The electric potential change includes, in addition to the harmonic Laplace solution term, a new anharmonic term that vanishes when the diffusion coefficients of counterions and co-ions are the same, Eq. [21]. The ion and ion-pair concentration changes also include new anharmonic terms but they do not vanish with D. The farreaching harmonic terms are proportional to one another, their ratio being determined by the equilibrium condition K d C n 5 2C 0 d C. This means that, far from the particle, the concentrations are modified in such a way that there is no net dissociation or recombination. While the ratio of the anharmonic terms is also constant, they have different signs: d C n D n 5 2 d CD ef . This means that, close to the particle (r 2 a # L), there is predominantly recombination of ions on one side of the particle and dissociation of ion pairs on the other.

where e 2~r2a!/L 1 1 r/L f~r, L! 5 2 2 r /a 1 1 a/L

BOUNDARY CONDITIONS

[23]

and P is an integration constant. Its value is a measure of the deviation from equilibrium, so that the product SP is proportional to the rate of net dissociation of ion pairs, Eqs. [13], [19], and [22]: ¹ z j n 5 2SPf~r, L!cos u .

dc L SP cos u 2 f~r, L!cos u r2 D ef 2

df 5 2Er cos u 1 2

[24]

de cos u r2

D kT L 2SP f~r, L!cos u C 0 e D ef

[25]

in which the first addends in the right-hand side are solutions of the Laplace equation. The expression for the field-induced change of the distribution of ion pairs is finally obtained combining Eqs. [19], [20], [22], and [24]:

dCn 5

2C 0 d c L 2SP cos u 1 f~r, L!cos u . K r2 Dn

x a @ 1,

[27]

L @ x 21.

[28]

in local equilibrium,

Equation [23] shows that the value of L characterizes the thickness of the region surrounding the particle inside which this net dissociation or recombination takes place. Expressions [17] and [18] have the form of the Poisson equation. Using Eq. [22], the parts of their general solutions that tend to zero at infinite distance (except for a uniform field term) can be written as

dC 5

The boundary conditions required to determine the constants P, d c , and d e can be obtained in the case of a thin quasiflat double layer,

[26]

A comparison of these general solutions with the corresponding results for strong electrolytes in a weak static field

In these expressions x is the reciprocal of the Debye screening length,

x5

Î

2e 2C 0 , kT e

[29]

and e is the absolute permittivity of the electrolyte solution. The local equilibrium of the double layer arises because the characteristic thickness of the region where the field-induced electrolyte concentration gradients expand from the surface of the particle toward the surrounding volume is far greater than the characteristic thickness of the double layer. This allows us to consider that, in spite of the space dependence of the field induced changes of the electrolyte concentration, the double layer near every small area of the surface is in contact with a solution characterized by a certain value of the electrolyte concentration. In the present case, the characteristic thickness is given by the least among a and L, Eq. [24], in the same way as the least among a and =2D/ v determines this thickness in the case of the low-frequency dispersion (see, for example, (2) or (4)). Therefore, the background of the validity conditions [27] and [28] for the local equilibrium of a thin polarized double layer in a weak electrolyte solution and a static field is fully similar to that of the local equilibrium of a thin polarized

163

ELECTROPHORETIC MOBILITY IN WEAK ELECTROLYTES

double layer in a strong electrolyte solution and an alternating low-frequency field, which was first established in (2). Combining Eqs. [20], [28], and [29] leads to the requirement

S!

2e 2C 0 D n kT e K

1

n5

C 0D 112 KD ef

2s Dn e ~D 1 1 D 2! K

5

1 C 0D n 112 KD ef

s ~1 2 D 2! C 0D n eC0 KD ef

1 112

C 0D n KD ef

,

where

s 5 e 2u 1C 0 1 e 2u 2C 0

which, combined with Eq. [26], gives L 2SP 4C 0 d c f9~a, L! 2 5 0, Dn K a3

[31]

where f9(a, L) is the derivative of f(r, L) with respect to r, calculated at the boundary r 5 a. The other two boundary conditions are obtained from the continuity equation for the normal components of the ion flows. For a quasiflat thin double layer in local equilibrium, they have the form (see, for example, (34))

S

7

e 6 D C 0¹ ndf 2 D 6¹ nd C kT

DU

a

5 2¹ s z I 6 s ,

[32]

where I6 s are the surface flows,

is the conductivity of the electrolyte solution, u 6 are the ion mobilities, and Einstein’s equation

I6 s 5 7

G6 e 6 6 0 D G 0 ¹ udf 2 D 6 ¹ dC kT C0 u

D 6 5 u 6kT

1

E

`

~C 6 0 2 C 0!Vdr,

[33]

a

was used. The condition obtained for S is equivalent to

S!

V is the velocity of the fluid, and G 6 0 are the adsorption coefficients

s 1 1/~ t C 0! S max 5 5 , e K 1 2C 0 K/C 0 1 2 K/C 0 1 2

where t is the electrolyte solution relaxation time and Smax is the maximum value which the recombination constant can attain according to Onsager’s theory (33) (for an aqueous KCl electrolyte solution with C0 5 1024 m23, the conductivity at 300 °K is s 5 0.025 S/m, so that Smax 5 3.5 10217 m3/s). Considering that Eqs. [27] and [28] are fulfilled, the electrolyte concentration on the surface of the particle can be obtained as usual (4), equating the normal component of the ion or ion-pair flows leaving the surface to minus the surface divergence of the surface ion or ion-pair flows. The first boundary condition is obtained from Eq. [5], which shows that in equilibrium there is no excess of ion pairs near the surface of the particle. Therefore, the surface flow of ion pairs and, consequently, their normal flow leaving the surface must vanish, j nnu a 5 0,

E

`

~C 6 0 2 C 0!dr.

[30]

[34]

a

These coefficients can be calculated in the case of a thin double layer, i.e., when the interface can be considered as locally flat, G6 0 5

2C 0 7ez/ 2kT ~e 2 1!, x

[35]

where z is the potential value on the surface (r 5 a). The velocity of the fluid tangential to the particle surface can also be calculated in the case of a quasiflat interface, considering the contributions of the electroosmotic (due to the tangential gradient of the electric potential) and the capillary osmotic (due to the tangential gradient of the concentration) terms (34):

1 2

ef0 4kT e e ~ z 2 f 0! 4 e kT V5 ¹ udf 1 ln ¹ ud C. ez kT h C 0h e cosh 4kT

S D

so that, Eq. [11], ¹ nd C nu a 5 0

G6 0 5

2

cosh

164

GROSSE AND SHILOV

This result, together with Eq. [35], transforms Eqs. [32] and [33] to the final form, 6

e de 6 1 dc 6 ~R 1 2 2 U 6! 3 ~R 1 2! 1 kT a C0 a3 2

L 2SP $~1 6 D!@R 6 2 af9~a, L!# 2 U 6% D ef C 0a 7

e E~R 6 2 1! 5 0, kT

[36]

where R6 5

F

ez 2 xG6 0 ~1 1 3m 6! 6 3m 6 xa C0 kT

U 6 5 48 m6 5

S

m6 ez ln cosh xa 4kT

S D

D

G

2e kT 2 . 6 3hD e

[37]

FIG. 1. Field induced ion concentration changes outside the thin double layer on the right-hand side of a negatively charged particle with field applied from left to right (cos u 5 1). Filled squares, strong electrolyte; squares, weak electrolyte with D2 5 D1; circles, weak electrolyte with D2 5 D1/5; diamonds, weak electrolyte with D2 5 5D1 (the dependence chosen for D2 maintains constant the value of R). Lines, harmonic part of the solutions. Values used: a 5 0.2 mm, z 5 2200 mV, C0 5 1024 m23, K 5 1024 m23, S 5 10218 m3/s, D1 5 Dn 5 2 1029 m2/s.

[38] The solutions of these equations and Eq. [31] are

The coefficients P, dc, and de can be now obtained solving the system of three equations [31] and [36] with three unknowns. Equation [36] can be substantially simplified in two aspects. The first is to keep only the main term related to convection, i.e., the term in m6 which multiplies G6 0 in Eq. [37], and neglect the term linear in z and the logarithmic term. The second is to set to zero the value of G2 0 (assuming a negatively charged particle) since the surface flow of co-ions is negligible as compared to the corresponding flow of counterions, Eq. [35]. With these simplifications, Eqs. [37] and [38] transform to R1 5

2G1 4 0 ~1 1 3m1! 5 ~e2ez/ 2kT 2 1!~1 1 3m1! 5 R aC0 xa

[39]

R2 5 0

[40]

U6 5 0

[41]

so that Eq. [36] written for counterions and co-ions becomes e de 1 dc ~R 1 2! 1 ~R 1 2! kT a 3 C0 a3 2 22

L 2SP e @R 2 af9~a, L!#~1 1 D! 2 E~R 2 1! 5 0 D ef C 0a kT

1 dc e de 12 kT a 3 C0 a3 1

L 2SP 5 2

C 0eEa 3R kT R 1 2

3 112 dc 5

C 0D n KD ef

G

[42]

F

1 R~1 1 D!~1 2 2h! 12 2~R 1 2!

G

[43]

C 0eEa 3 3R 4kT R 1 2 3

n

C 0D 112 KD ef

Ea 3 R 2 4 de 5 4 R12

3

F

G

~1 1 D!~1 2 2h! 2 3D C 0D n 11R KD ef R24 , n C 0D R~1 1 D!~1 2 2h! 112 12 KD ef 2~R 1 2!

112

F

G

[44]

where, Eq. [23], h52

L 2SP e af9~a, L!~1 2 D! 2 E 5 0. D ef C 0a kT

F

C 0D n KD ef R~1 1 D!~1 2 2h! 12 2~R 1 2! hD ef

1 1 a/L 1 5 . af9~a, L! 2 1 2a/L 1 a 2/L 2

This parameter only depends on the quotient L/a and varies

ELECTROPHORETIC MOBILITY IN WEAK ELECTROLYTES

FIG. 2. Same as Fig. 1 but for the ion-pair concentration changes.

between the limits 0 # h # 21 , which correspond to L/a 3 0 (S 3 ` or K 3 `) and L/a 3 ` (S 3 0), respectively. In the limit of strong electrolytes (K 3 ` so that L 2 S 3 0 and h 3 0), Eqs. [42] to [44] reduce to P30 dc 3

C 0eEa 3 3R 4kT R 1 2

[45]

de 3

Ea 3 R 2 4 4 R12

[46]

while, in the opposite limit of very weak electrolytes (K 3 0 so that L 2 S 3 D ef / 2C 0 ), P32

C 20eEa 3R kT R 1 2

h R~1 1 D!~1 2 2h! 12 2~R 1 2!

concentration change has a similar behavior, Fig. 2, except that close to the surface the concentration decreases with respect to the far reaching harmonic term (this makes it possible for the dC n versus distance curve to reach the surface with zero slope, Eq. [30]). Because of this difference of behavior, there is predominantly recombination of ions close to the surface on the right side of the particle and dissociation of ion pairs on its left side. These processes constitute the causes of the increase (decrease) of the ion-pair concentration on the right (left) side of the particle, which are responsible for their movement. Ion pairs move toward the particle from the left because of the decreasing concentration, dissociate forming negative co-ions which return toward the left, and positive counterions which enter the double layer and are transferred to the right side. Counterions leaving the double layer on the right side recombine with co-ions approaching from the right to form ion pairs that move toward the right because of the decreasing concentration. As for the dipolar coefficient, Eq. [44], it has a more complicated behavior, Fig. 3. This coefficient is determined by its high frequency value (function of the local conductivity distribution) modified by three factors. The first two, which are also present in strong electrolytes, always contribute to lower the dipolar coefficient and are both caused by the ion concentration changes. The first effect is a redistribution of counterions in the double layer, which tend to move due to diffusion along the particle surface toward its left side. The second is the compression of the double layer on the right side of the particle and its expansion on the left, which is due to the dependence of the Debye screening length with the ion concentration. This deformation, which is equivalent to a displacement of the positive counterion charge toward the left with respect to the negatively charged particle, produces a dipole moment in the opposite direction to the applied field. Only these two effects are present when D1 5 D2, (D 5 0), in which case the dipolar coefficient in weak electrolytes becomes bigger than in strong ones (at constant ionic strength), Fig. 3, due to the decrement of the ion concentration changes, Fig. 1.

dc 3 0 Ea 3 R~1 2 3D! 2 4 de 3 4 R12

R~1 1 D!~1 2 2h! R~1 2 3D! 2 4 . R~1 1 D!~1 2 2h! 12 2~R 1 2!

11

A comparison of Eqs. [42] to [43] with the results corresponding to strong electrolytes shows the following differences (which we will discuss considering that the particle is negatively charged and the field is applied from left to right). The far-reaching harmonic ion concentration changes around the particle, Eq. [43], are always diminished, Fig. 1. On the right side of the particle where the ion concentration increases, there is an additional increase close to the surface due to the anharmonic term of Eq. [24] (in view of the negative sign of the coefficient P). The ion-pair

165

FIG. 3. Same as Fig. 1 but for the electric potential changes.

166

GROSSE AND SHILOV

The third effect can be either a decrement or an increment of the dipolar coefficient depending on the difference of the diffusion coefficients of counterions and co-ions. It only exists for static fields in weak electrolytes and is due to the field-induced charge density that builds up outside the double layer. Its origin is in the dependence with distance of the anharmonic ion concentration change term, Eq. [24]. The function f(r, L) varies with distance faster than 1/r 2, Eq. [23], which means that on the right-hand side of the particle, more ions enter by diffusion any given volume from the left than leave it from the right. This process, which always causes an increase of the ion concentration with respect to the harmonic term, Fig. 1, also leads to the appearance of a charge density which has the sign of the ion of highest mobility. If the diffusion coefficient of counterions is greater than that of co-ions D 1 . D 2 , (D , 0), positive (negative) charge builds up close to the surface on the right (left) side of the particle, as can be seen from Eq. [18] (its right-hand side is minus the charge density and both D and S are negative). This results in a further increment of the dipolar coefficient, Fig. 3. In the opposite case when the mobility of co-ions is greater than that of counterions D 1 , D 2 , (D . 0), negative (positive) charge builds up on the right (left) side of the particle. The resulting dipole decreases the total dipolar coefficient, which for sufficiently large values of D, can become smaller than in the case of a strong electrolyte, Fig. 3. The incidence of the value of the dissociation–recombination rate constant S on the obtained results can be appreciated considering the limit S 3 0 (h 3 21 ): L 2SP 5 2

C 0eEa 3R C 0D n 2kT R 1 2 KD ef

C 0eEa 3 3R dc 5 4kT R 1 2

C 0D n KD ef

S

Vt 5

S

L 2SP C 0eEa 3 3R f~r, L! 3 D ef 4kT R 1 2

Ea 3 R 2 4 D kT L 2SP f~r, L! 3 C 0 e D ef 4 R12

2C 0 L 2SP dc 1 r2 f~r, L! 3 0. K Dn

D

ez 4 e kT kT ez ln cosh ¹f2 ¹ d C. h t h eC 0 e 4kT t

[50]

For a spherical particle with a thin double layer, the electrophoretic velocity is related to the tangential fluid velocity along the outer boundary of its double layer by 2 V t~a, u ! . 3 sin u

Combining this expression with Eq. [50], and using Eqs. [24] and [25], the following result is obtained,

D

V eph 5

H F

2 de kT L 2SPD ez E 12 31 3 h Ea eE C 0aD ef 2

These expressions do not reflect the values of the harmonic and anharmonic terms in Eqs. [24] to [26] since, in the considered limit, L 3 `. This causes the function f(r, L), Eq. [23], to reduce to a2/r 2 so that the anharmonic terms in Eqs. [24] to [26] transform to harmonic. Therefore, the resulting harmonic coefficients become

de 2 r2

The total field induced velocity of a fluid along a charged flat surface is determined by two contributions (34, 35): the electroosmotic term which is proportional to the tangential gradient of the electric potential and the capillary osmotic term which is proportional to the tangential gradient of the electrolyte concentration:

V eph 5 2

C 0D n 3DR 1 1 2 12 3 KD ef R24 Ea R 2 4 de 5 . 4 R12 C 0D n 112 KD ef

dc 2 r2

ELECTROPHORETIC MOBILITY

D ef C 0D n 112 KD ef

1 112

Comparing Eqs. [47] and [48] with the corresponding results obtained in the limit of strong electrolytes, Eqs. [45] and [46], shows that they are identical. This means that, for extremely slow rate constants, no dissociation or recombination processes take place around the particle. The weak electrolyte behaves then just as a strong one. The only difference is the presence of a distribution of ion pairs that is uniform, Eqs. [26] and [49], and does not have any influence on the concentration changes of counterions or co-ions.

[47]

S

4 e kT ez ln cosh he 4kT

2

GJ

kT d c kT L 2SP 31 eE C 0a eE C 0aD ef

F

G

C 0D n R~1 2 2h! 12 KD ef R14 V eph ez R 1 4 5 n E 2h R 1 2 C 0D R~1 1 D!~1 2 2h! 112 12 KD ef 2~R 1 2! 112

F

1

S

G

D

R 2 e kT ez ln cosh he 4kT R 1 2 C 0D n KD ef , R~1 1 D!~1 2 2h! 12 2~R 1 2!

1 1 4h

[48] 3 [49]

DF

G

112

C 0D n KD ef

F

G

@51#

167

ELECTROPHORETIC MOBILITY IN WEAK ELECTROLYTES

or, separating the Smoluchowski (36) term,

This transforms the general expression [51] into

V eph ez e kT 5 1 R E h 2he

S H

F

DGS

D

ez ez C 0D n 1 4 ln cosh 1 1 4h kT 4kT KD ef 2e z C 0D n 2 1 1 @2 2 ~1 1 D!~1 2 2h!# kT KD ef 3 . n C 0D R 1 1 @2 2 ~1 1 D!~1 2 2h!# KD ef C 0D n 12 112 KD ef

H

S

J

J

V eph 2 e kT 32 ln~2! E he

1 1 4h

C 0D n KD ef

1 1 @2 2 ~1 1 D!~1 2 2h!#

C 0D n KD ef

which, for strong electrolytes reduces to V eph 2 e kT 32 ln~2! E he

D

[53]

and for very weak electrolytes becomes In the limit of strong electrolytes (K 3 `), Eq. [51] reduces to

S

H

D

V eph e kT e z R 1 4 ez 2R 3 1 ln cosh E h e 2kT R 1 2 4kT R 1 2

J

V eph 3 0. E [52]

while, for very weak electrolytes (K 3 0), it becomes V eph ez 4 3 . E h R~1 2 D! 1 4 The first limiting expression coincides with the well-known result for strong electrolytes and negatively charged particles, when R is expressed as

For high, but not infinite values of the dissociation constant K, the preceding results can be expressed as a term corresponding to a strong electrolyte plus an additional term.

S D

V eph V eph 5 E E

2 str

3

H S

e kT R~1 2 2h! C 0D n @R~1 2 D! 2 4D# 2 h e ~R 1 2! 2 KD ef

ez ez 1 4@R~1 2 D! 1 4#ln cosh kT 4kT

DJ

[54]

which, for z 3 2`, tends to R5

2l D1 1 D2 2l 2 , 5 as D1 as 1 2 D

where l is the surface conductivity of the particle. The limiting result [52] is also obtained for very low values of the dissociation–recombination rate constant (S 3 0), independent of the value of the equilibrium constant K. In the limit z 3 2`,

S

ln cosh

D

ez ez 32 2 ln~2!, 4kT 4kT

S D

V eph V eph 3 E E

1 str

2 e kT C 0D n ~1 2 D!~1 2 2h!ln~2!. h e KD ef

The first term in the right-hand side of this last expression is negative, Eq. [53], while the second is positive. Therefore, a finite value of the dissociation constant always leads to a smaller absolute value of the high z potential electrophoretic mobility limit. On the contrary, the term inside the curly brackets in Eq. [54] can be either positive or negative, depending on the sign of D. This occurs because for small values of the z potential,

while, from Eqs. [35], [37], and [39],

R3

4 2ez/ 2kT ~1 1 3m 1!. e xa

S D

ez 4kT ez 3 ln cosh 4kT 2!

S

D

2

168

GROSSE AND SHILOV

FIG. 4. Electrophoretic mobility of a spherical particle with thin double layer in a weak electrolyte solution with D 2 5 D 1 , relative to the corresponding value in a strong electrolyte with the same ion content. Circles, K 5 10 22 m23; diamonds, K 5 10 23 m23; filled squares, K 5 10 24 m23; filled diamonds, K 5 10 25 m23; filled circles, K 5 10 26 m23. Remaining values as in Fig. 1.

so that this term tends to

H

@R~1 2 D! 2 4D#

ez kT

S

1 4@R~1 2 D! 1 4#ln cosh

ez 4kT

DJ

3 24D

ez . kT

This means that for D . 0 (co-ions with higher mobility than counterions), a finite value of the dissociation constant could lead to a higher absolute value of the maximum of the electrophoretic mobility versus z potential curve. It is worth noting that Eqs. [42] to [44], as well as Eq. [51] and following equations, correspond to the case of negatively charged particles. The transformation of Eqs. [42] to [44] to the case of positively charged particles (z . 0) consists in changing the signs of the right-hand sides of these equations, in substituting D 2 for D 1 and 2D for D, and in replacing Eq. [39] by R2 5

FIG. 5. Same as Fig. 4, but for D 2 5 D 1 /5.

that of counterions, the electrophoretic mobility for low and medium values of the z potential is increased by the existence of the dissociation–recombination processes. This z potential range includes the values for which the calculations based on the standard model predict a maximum of the absolute value of the electrophoretic mobility. Therefore, the presence of ion pairs increases the absolute value of the mobility maximum, extending the possibilities of the interpretation of electrophoretic measurements. The physical mechanism responsible for the increase of the mobility maximum can be understood examining the influence of the field-induced electrolyte concentration changes outside the double layer on the particle movement. For a negative particle immersed in an electric field directed from left to right, the electrolyte concentration increases along its surface in this same direction. The influence of this concentration change can be evaluated, at least qualitatively, considering the diffusiophoretic movement of the particle in an electrolyte concentration gradient oriented from left to right.

2G 2 4 0 ~1 1 3m 2! 5 ~e ez/ 2kT 2 1!~1 1 3m 2! 5 R. aC 0 xa

Besides these changes, the transformation of Eq. [51] to the case of positively charged particles consists in substituting 2ln(cosh e z /4kT) for ln(cosh e z /4kT). DISCUSSION

Figures 4 – 6 show that for D 1 5 D 2 , and especially for D 1 , D 2 when the diffusion coefficient of co-ions is larger than

FIG. 6. Same as Fig. 4, but for D 2 5 5D 1 .

ELECTROPHORETIC MOBILITY IN WEAK ELECTROLYTES

It is well known from the theory of diffusiophoresis (37, 38) of particles with a thin double layer, that for small or moderate values of the surface conductivity l, the diffusiophoretic movement is in the same direction as the electrolyte concentration gradient. On the contrary, for high z potential values (surface conductivities such that l /( x a) ' e 2e z / 2kT /( x a) @ 2), the surface diffusion of counterions leads to a surface current which gives rise to a positive (negative) charge density inside the double layer on the left (right) side of the particle. This produces an additional electric field (oriented in the same direction as the applied field) acting on the particle, so that the diffusiophoretic contribution to the movement is in a direction opposite to the concentration gradient. It is worth noting that because of this same field, the high z potential limit of the electrophoretic mobility is not zero (despite the exponential increase of l with z) but has a finite value which, for x a @ 1, is

e kT 2 ln~2!. he The presence of ion pairs always lowers the field-induced concentration changes along the outer boundary of the double layer, Fig. 1, diminishing the contribution of these changes to the particle movement. Therefore, for relatively low z potential values (for which the contribution of the concentration changes is in the direction opposite to the electrophoretic velocity), the presence of ion pairs increases the electrophoretic mobility. On the contrary, for large values of the z potential (for which the contribution of the concentration changes is in the same direction of the electrophoretic velocity), the presence of ion pairs decreases the electrophoretic mobility. The above-described mechanism, which might be designated as the decrease of the diffusiophoretic contribution to electrophoresis, constitutes all the contribution of the ion pairs to electrophoresis when D 1 5 D 2 . On the contrary, when the diffusion coefficients of counterions and co-ions differ, an additional mechanism related to the volume charge density that builds up outside the thin double layer when D 1 Þ D 2 appears. As it was already shown above, the sign of this volume charge on the side of the particle where the field induced concentration change is positive (negative) coincides with the sign of the ion of highest (lowest) mobility, Fig. 3. Therefore, when the mobility of co-ions is higher than that of counterions, the additional field generated by the fieldinduced charge density outside the double layer is in the same direction as the applied field, contributing to increase the electrophoretic mobility. On the contrary, when the counterions have the highest mobility, the additional field diminishes the electrophoretic mobility to the point that, for a sufficiently large difference between diffusion coefficients, V eph /E becomes lower in a weak electrolyte solution than when the electrolyte is strong (at equal ionic strength and z potential values).

169

FIG. 7. Comparison of the numerical theory of Baygents–Saville (32) with the present analytical theory. Values used: a 5 0.2 mm, C 0 5 1.4314 10 25 m23 (corresponding to x a 5 100), D 1 5 D 2 5 D n 5 2 10 29 m2/s. Strong electrolyte (complete ionization): squares, numerical results; lower thick line, analytical results. Weak electrolyte with K 5 C 0 /198 (1% ionisation) and S 3 `: diamonds, numerical results; upper thick line, analytical results. Thin lines, analytical results for a weak electrolyte with K 5 C 0 /198 and, from top to bottom, S 5 10 218 , 10 219 , 10 220 , and 10221 m3/s.

There is a full qualitative agreement between the results of the present theory and those of Baygents and Saville (32), which were obtained for the limiting case of infinitely fast dissociation–recombination processes (S 3 `). For a quantitative comparison, a sufficiently large value of S should be substituted in Eq. [51] (for which the dependence of V eph /E on S vanishes). Also, only those results from (32) that correspond to x a @ 1 should be used, in order to respect the restriction [27] of the theory. An example that fulfills these requirements is given in Fig. 10 from (32), namely the two curves for rigid particles. Data points taken from this figure, together with the corresponding theoretical results, are represented in Fig. 7. As can be seen, the numerical and the analytical results coincide both for small and large degrees of dissociation. This shows the full agreement between the new analytical theory, available for arbitrary values of the dissociation–recombination rate constant, but only for very thin double layers, and the existing numerical theory, available for arbitrary double layer thicknesses, but only for very large dissociation–recombination rate constants. This agreement takes place in the common range of applicability of both theories. Figure 7 also shows the influence of the value of the dissociation–recombination rate constant on the electrophoretic mobility. For decreasing values of S, the mobility curve lowers until it reduces to the curve corresponding to strong electrolytes. CONCLUSION

We presented a generalization of the analytical theory of the concentration polarization in suspensions of colloidal particles

170

GROSSE AND SHILOV

with a thin double layer to the case of weak electrolyte solutions. The presence of ion pairs and of the dissociation–recombination processes always decreases (at constant ionic strength) the field-induced ion concentration changes which build up around a particle. On the contrary, the dipolar coefficient usually increases, except when the mobility of co-ions is much higher than that of counterions. These effects have a strong bearing on the electrophoretic mobility of colloidal particles. We showed that the theoretical maximum of the electrophoretic mobility in weak electrolyte solutions can substantially surpass (at constant ionic strength) the corresponding value in strong electrolytes, extending the possibilities of the interpretation of electrophoretic measurements. We also showed that the effect of electrolyte weakness on the electrophoretic mobility decreases with decreasing values of the dissociation–recombination rate constant S. In a forthcoming work we shall use the obtained results for the study of the conductivity dispersion amplitude, the permittivity increment, and the characteristic time of the low frequency dielectric dispersion of colloidal suspensions in weak electrolyte solutions. ACKNOWLEDGMENTS This work was partially supported by a grant of the Consejo de Investigaciones de la Universidad Nacional de Tucuma´n, by INTAS Project 93-3372 (extension) and by INTAS/UA Project 95-165.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

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