Generalization of a Classic Theory of the Low Frequency Dielectric

J. Phys. Chem. B 2009, 113, 11201–11215

11201

Generalization of a Classic Theory of the Low Frequency Dielectric Dispersion of Colloidal Suspensions to Electrolyte Solutions with Different Ion Valences Constantino Grosse Departamento de Fı´sica, UniVersidad Nacional de Tucumań, AV. Independencia 1800, (4000) San Miguel de Tucumań, Argentina, and Consejo Nacional de InVestigaciones Cientı´ficas y Tećnicas, Argentina ReceiVed: May 20, 2009; ReVised Manuscript ReceiVed: June 9, 2009

The classic Shilov-Dukhin theory of the low frequency dielectric dispersion of colloidal suspensions in binary electrolyte solutions was developed for symmetric electrolytes: equal counterion and co-ion valences. A rigorous generalization of this theory to asymmetric electrolytes, such that the valence of counterions is double or half the valence of co-ions, is presented. This generalization is possible because analytical solutions of the intervening integrals exist for these two particular cases but do not exist in the general case of different counterion and co-ion valences, a result apparently overlooked or ignored in the past. Introduction The classic Shilov-Dukhin theory of the low frequency dielectric dispersion (LFDD) of colloidal suspensions in binary electrolyte solutions was developed considering that both ion species have the same valence.1,2 This requirement has a purely mathematical nature: analytic expressions for the intervening integrals do not exist for the general case z+ * z- (z( are the unsigned ion valences). Nevertheless, a generalization has been made,3 but it is based on approximate solutions for the equilibrium potential in the double layer. In a recent work4 we presented a result apparently overlooked since the development of the theory in the early 1970s: while it is true that there is no analytic solution in the general case z+ * z-, analytic results do exist in the all-important cases z+ ) 2z- and z- ) 2z+ (this was already shown in 1953,5 mentioned in 1961,6 but apparently ignored in later works). In ref 4 we used these solutions to generalize the DC thin double layer polarization theory and obtain new expressions for the dipolar coefficient, conductivity increment, and electrophoretic mobility. Now we extend these results to the frequency domain and generalize the LFDD theory to obtain new expressions for the dipolar coefficient and its extension including the high frequency relaxation. We then deduce analytic expressions for the dielectric increment, the conductivity increment, and the dynamic electrophoretic mobility valid in a broad frequency range. In doing this, and in view of the general lack of detail in the existing presentations of the Shilov-Dukhin theory2,7,8 we provide, just as in ref 4, a full deduction including all the intermediate steps, assumptions, and justifications, that lead to the final solution. We also include a full proof of the hypothesis of approximate electroneutrality that is used in the theory. To avoid useless repetitions, we base the presentation on ref 4 and only add the modifications required for the AC case.

According to the standard model,9 the suspended particle is represented by an insulating sphere of radius a, with a uniform fixed surface charge density σ0. The surrounding electrolyte solution is characterized by its viscosity ηe, absolute permittivity εe, the unsigned valences of its ions z(, their diffusion coefficients D(, and their concentrations far from the particle C( (∞). b,t), electric potential Φ(r b,t), fluid The ion concentrations C( (r velocity b V(r b,t), and pressure P(r b,t), are determined by the usual set of the Nernst-Planck, continuity, Poisson, Navier-Stokes, and incompressibility equations:

bj ( ) -D(∇C( - C(z(D(∇Φ ˜ + C(b V ∇ · bj ( ) -

∂C( ∂t

∇2Φ ) -(z+C+ - z-C-)

e εe

(3)

(4)

∇·b V)0

(5)

where the symbol ∼ denotes a dimensionless magnitude ˜ ) Φe/(kT). These equations are identical to the DC case, Φ eqs 1-5 in ref 4 except for the time dependences of all the magnitudes and of the time derivative appearing in eq 2. Just as in the DC case, the equation set is first solved in equilibrium (lower index 0). This leads to the equilibrium ion concentrations (Φ ˜

The classic Shilov-Dukhin LFDD theory is based, just as the DC formulation, on the standard electrokinetic model, the thin double layer approximation, and the principle of local equilibrium. The extension to AC fields requires, furthermore, the use of the hypothesis of approximate electroneutrality.

(2)

ηe∇2b V - ∇P ) (z+C+ - z-C-)e∇Φ

-z C( 0 ) z Ne

Equation Set

(1)

0

where the concentration N is defined as

N)

C((∞) z-

10.1021/jp904742v CCC: $40.75  2009 American Chemical Society Published on Web 07/21/2009

(6)

11202

J. Phys. Chem. B, Vol. 113, No. 32, 2009

Grosse

µ˜ *( ) δµ˜ *( + ...

to the Poisson-Boltzmann equation

˜0 ) ∇2Φ

-˜ z+z-e2N -z+Φ˜ 0 - ez Φ0) (e εekT

where all the magnitudes preceded by the δ character are linear in the applied field. Combining these expansions with the original equations, dropping all the higher than first order terms, and using the equilibrium expressions, leads to

and to the expression for the equilibrium pressure ˜0 - -z+Φ

P0 - P(∞) ) NkT(z e

˜0 + z-Φ

+z e

+

( b* δbj *( ) -C( ˜ *( + C( 0 D ∇δµ 0 δV

(8)

˜* ∇ · δbj *( ) -iωC( ˜ * ( iωz(C( ˜ *) 0 δn 0 (δΦ - δφ

(9)

-

-z -z )

(7) where P(∞) is the pressure far from the particle. When a macroscopic AC electric field Eeiωt is applied to the system, eqs 1 can be transformed writing the ion flows as

- - ˜ * ) [(z+z+C+ ˜* ∇2δΦ ˜ *) 0 + z z C0 )(δΦ - δφ - ˜ *] (z+C+ 0 - z C0 )δn

e2 εekT

- b* - ∇δp* ) kT(z+C+ ˜* + ηe∇2δV 0 - z C0 )∇δφ

bj *( ) -C*(D(∇µ˜ *( + C*(b V*

+ ˜* kT(C+ 0 - z N + C0 - z N)∇δn

where the asterisk denotes a complex frequency dependent magnitude and µ˜ *( ) µ*(/(kT) are the dimensionless electrochemical potentials:

b* ) 0 ∇ · δV

(10)

where

µ˜ *( ) ln

*(

C ˜* ( z(Φ z-N

n˜* ) n* /N

The equations are then simplified using the principle of local equilibrium: each sufficiently small volume element of the system is considered to be in a state of equilibrium, even when different volume elements are not in equilibrium with one another. This condition is expressed in terms of a Wirtual system that is defined by the following conditions. (i) it is electroneutral in its entire volume:

c*( ) z-n* (ii) each of its volume elements is in equilibrium with the corresponding element of the real system

µ˜

*(

δn˜* ) δn* /N δµ˜ *( ) δn˜* ( z(δφ˜ *

These equations are identical to the DC case, eqs 25-29 in ref 4, except for the right-hand side in eq 9 and the presence of the asterisks. Combining the divergence of eq 8 with eq 1 written for equilibrium and with eqs 10 and 11, and equating the righthand side of the resulting expression to the right-hand side of eq 9, leads to two equations, one for each ion sign. Solving these equations for ∇2δn˜* and ∇2δφ˜ * transforms the equation set into

(

∇2δn˜* ) z+z- ∇δφ˜ * -

n* ) ln ( z(φ˜ * N

where c*( and φ˜ * are the ion concentrations and the potential of the virtual system. Accordingly, the local equilibrium values of the real system parameters are related to the corresponding parameters of the virtual system by means of expressions that are analogous to eqs 6 and 7:

[

((Φ ˜ *-φ ˜ *)

˜ *-φ - -z+(Φ ˜ *)

P - p ) n kT[z e *

*

˜ *-φ + z-(Φ ˜ *)

+z e

+

-

-z -z ]

where p* is the pressure of the virtual system. The equation set is then linearized, writing all the fielddependent magnitudes as an expansion in successive powers of the applied field strength, for example: *( C*( ) C( + ... 0 + δC

)

∆ b* ˜0 + δV · ∇Φ Def iω [δn˜* - (δΦ* - δφ˜ *)z+z-∆] Def

∇2δφ˜ * ) ∇δn˜* + (z+ - z-)∇δφ˜ * -

C*( ) z-n*e-z *

(11)

]

Q b* ˜0 + δV · ∇Φ Def

iω ˜ * - δφ˜ *)] [δn˜*∆ - Q(δΦ Def

- - ˜ * ) [(z+z+C+ ˜* ∇2δΦ ˜ *) 0 + z z C0 )(δΦ - δφ - (z+C+ ˜ *] 0 - z C0 )δn

e2 εekT

- b* - ∇δp* ) kT(z+C+ ˜* + ηe∇2δV 0 - z C0 )∇δφ + ˜* kT(C+ 0 - z N + C0 - z N)∇δn

where

Generalization of a Classic Theory of LFDD

∆)

D- - D+ z+D+ + z-D-

√ξ*κ(a-r) a 2 1 + √ξ κr eEa

z-D+ + z+D) 1 + (z+ - z-)∆ z+D+ + z-D-

*

( r ) 1 + √ξ κa kT cos θ

δn˜* ) K*c e

(z+ + z-)D+Dz+D+ + z-D-

Def ) Q)

J. Phys. Chem. B, Vol. 113, No. 32, 2009 11203

(12)

These equations greatly simplify in the electrolyte solution outside the boundary of the equilibrium double layer, where ˜ 0 ) 0, the equilibrium potential and charge density vanish (Φ C 0( ) z- N), leading to the final set to be solved:

∇2δn˜* ) κ2ξ*(δn˜* + δF˜ *z+z-∆)

(13)

∇2δφ˜ * ) κ2ξ*(δn˜*∆ + δF˜ *Q)

(14)

∇2δF˜ * ) κ2δF˜ * + κ2ξ*(δn˜*∆ + δF˜ *Q)

(15)

where K*c is a complex frequency dependent integration coefficient. This result shows that in the AC case δn˜* decays exponentially, while it is proportional to r-2 in the DC case, eq 30 in ref 4. While the coefficient multiplying r in the exponent decreases when the frequency decreases, eq 18 does not reduce to the DC solution even in the limit ω f 0. The third expression is also a Helmholtz type equation so that √1+ξ*κ(a-r) a

δF˜ * ) KF*e

b* ) ∇δp* ηe∇ δV

2

(r)

(

[

K*da2 2

r

-

a r √ * + KF*e 1+ξ κ(a-r) a r

(

where

κ)

z+z-(z+ + z-)e2N εekT

˜ * ) δφ˜ * - δF˜ * ) δΦ iω κ Def 2

and

δF˜ * )

e ˜ * - δφ˜ *) δF* ) -(δΦ κ2εekT

()

2

(16)

is the dimensionless field induced charge density. Unlike the DC case, the variables δn˜* , δφ˜ * , and δF˜ * are coupled in these equations and only uncouple when the diffusion coefficients of counterions and co-ions have the same value: ∆ ) 0 and, in view of eq 12, Q ) 1. In this particular case, eqs 13-15 reduce to

∇ δn˜ ) κ ξ δn˜ *

2 *

*

)]

1 + √1 + ξ*κr eEa cos θ kT 1 + √1 + ξ*κa

[

K*da2 2

-

r

(17)

∇2δφ˜ * ) κ2ξ*δF˜ * ∇2δF˜ * ) κ2(1 + ξ*)δF˜ * The first expression has the form of a Helmholtz equation, so that the appropriate solution can be written as

]

r eEa cos θ a kT

In these expressions K*d is another complex frequency dependent integration coefficient. In the general ∆ * 0 case, the equation set is classically solved using the hypothesis of approximate electroneutrality, which states that the electrolyte solution outside the double layer that is electroneutral in equilibrium, remains electroneutral when an AC field is applied:

δF˜ * ) 0

2

×

while the field-induced potential change becomes

is the reciprocal Debye length,

ξ* )

)

1 + √1 + ξ*κr eEa cos θ kT 1 + √1 + ξ*κa (19)

where KF* is another complex frequency dependent integration coefficient. The presence of the reciprocal Debye length in the exponent shows that δF˜ * decays on a length scale of the double layer thickness. Finally, the second expression has the form of a Poisson equation, so that its appropriate solution is

δφ˜ * )

2

(18)

*

(20)

This transforms eqs 13 and 14 into

∇2δn˜* ) κ2ξ*δn˜*

(21)

∇2δφ˜ * ) κ2ξ*δn˜*∆

(22)

The justification of this hypothesis, which is rigorous (except for a rapidly decaying exponential term) for ∆ ) 0, eq 19, is that the field-induced counterion and co-ion concentration changes outside the double layer have almost exactly the same value, so that the virtual system parameters can be calculated neglecting the relatively small difference between these concentrations. It should be noted that this hypothesis does not imply that the final solution of the problem corresponds to a system that is electroneutral outside the double layer: eq 22

11204


Grosse

is a Poisson rather than a Laplace-type equation. A proof of the validity of this hypothesis is presented in Appendix 1. Equation 21 for δn˜* is identical to eq 17 so that its solution is given in eq 18. The appropriate solution of the Poisson type eq 22 is

δφ˜ * )

(

K*da2 r2

-

contribute to change the ion density values (last term in the right-hand side). The integrals can only be analytically evaluated when the double layer is thin as compared to the radius of the particle:

κa . 1

)

r eEa cos θ + δn˜*∆ a kT

(23)

assuming that the tangential gradient of the electrochemical potential does not change across the double layer:

Finally, using eqs 16 and 20

-z-ND(∇rδµ˜ *( | a ) ˜ * ) δφ˜ * δΦ

(24)

while the field-induced charge density, obtained combining eqs 12, 15, and 21 is

-ξ*∆ δn˜* 1 + (z+ - z-)ξ*∆

δF˜ * )

(25)

This expression shows that while δF˜ * is confined inside the double layer for ∆ ) 0, eq 19, it extends to much longer distances in the general case. Boundary Conditions The coefficients K*c and Kd* can be determined integrating the continuity equations written for the differences between the actual ion flows δbj *(, eq 8, and the flows δbj *( l calculated using expressions that are only valid outside the double layer. The only difference with respect to the DC case is that the time derivatives do not vanish so that eq 33 in ref 4 becomes

∫{ ∞

a

1 ∂ 2 *( 1 [r (δjr - δjlr*()] + × 2 ∂r r sin θ r ∞ ∂ ∂ *( )] dr ) - a [sin θ(δjθ*( - δjlθ (δC*( ∂θ ∂t

}

∫

z-Nδn˜*) dr (26)

where the long-range (lower index l) ion flows are

b* δbj l*( ) -z-ND(∇δµ˜ *( + z-NδV Proceeding exactly as in the DC case transforms eq 26 into

-z-ND(∇rδµ˜ *( | a )

1 × a sin θ

∫

∞ ∂ ( ˜ *( dr] [sin θ a (C( 0 - z N)D ∇θδµ ∂θ ∞ 1 ∂ * [sin θ a (C( 0 - z N)δVθ dr] a sin θ ∂θ ∂ ∞ (δC*( - z-Nδn˜*) dr ∂t a

∫

∫

The left-hand side of this expression represents the electrodiffusive flows of ions arriving to the outer boundary of the double layer. These flows spread out inside this layer in the form of surface ion flows (first two terms in the right-hand side) or

(27)

D( × a sin θ

∫

∞ ∂ 1 (sin θ∇θδµ˜ *( | a) a (C( × 0 - z N) dr ∂θ a sin θ ∞ ∞ ∂ * *( [sin θ a (C( 0 - z N)δVθ dr] - iω a (δC ∂θ z-Nδn˜*) dr (28)

∫

∫

The first integral in this expression represents the nonspecific adsorption coefficients G( 0 (excess equilibrium surface densities of counterions or co-ions). Their values are given in ref 4 for the only three cases that can be analytically solved: (11) z+ ) z- ) z, (21) z+ ) 2z- ) 2z, and (12) 2z+ ) z- ) 2z. The second integral represents the convective flow of ions along b is a superposithe particle surface, where the fluid velocity δV tion of electroosmotic and capillary osmotic contributions:

εN

∫a∞ (C(0 - z-N)δVθ* dr ) - κηe e ( kTe ) (∇θδφ˜ *I(eo + 2

∇θδn˜*I( co) ( ( and Ico , calculated for Expressions for the coefficients Ieo the same 11, 21, and 12 cases, are also given in ref 4. Finally, the last addend is neglected because it is proportional to the frequency of the applied field, while the theory of the LFDD is formulated for low values of ω. Except for the presence of the asterisks, eq 28 so reduces to exactly the same form as in the DC case:

(

G( 0

)

1 z(z( ( ( 3m Ico × a sin θ z N 2z-κ z(G( 0 z(z( ∂ (sin θ∇θδn˜* | a) + ( - + - 3m(I( eo × ∂θ z N 2z κ ∂ 1 (sin θ∇θδφ˜ * | a) a sin θ ∂θ

-∇rδµ˜ *( | a )

+

-

(

)

where

m( )

2εe (

3ηeD

( ) kT z(e

2

However, the expressions for the derivatives now obtained using eqs 18 and 23 are quite different:

∂ 1 2 eEa (sin θ∇θδn˜* | a) ) - 2 K*c cos θ a sin θ ∂θ kT a



∂ 1 2 (sin θ∇θδφ˜ * | a) ) - 2 (K*d - 1 + K*c ∆) × a sin θ ∂θ a eEa cos θ kT

(

[

-1 (1 ( z(∆)K*c 2 + a

ξ*κ2a2 1 + √ξ κa *

)

(

]

z((2K*d + 1)

eEa cos θ kT

K*c [(1 ( z(∆)(R( + f* ) - U(] ( z(K*d(R( + 2) ) (z((R( - 1) (29) where the coefficients

2z((z+ + z-)e2 ( 3m(z( G0 ( - I( eo 2 κ aεekT z κa

U( )

(1 - z-∆)U+ + (1 + z+∆)UA

2 + z + z-

2z+ 2zR+(1 - z-∆) + + R-(1 + z+∆) + 4 z +z z +z +

B ) (R+ + 2)(R- + 2) -

Combining these results and factoring with respect to the coefficients, leads to the final equations:

R( )

H)

A)

while, using eq 11 together with eqs 18 and 23

∇rδµ˜ *( | a )

(1 - z-∆)(1 + z+∆)(R+ - R-) -

(30)

3m(z( ( ( ((I( eo - z Ico) z-κa

z+(R+ + 2)U- + z-(R- + 2)U+ z+ + z-

while the complex coefficient h* , eq 31, contains all the frequency dependence. The first addend on the right-hand side of eq 32 is proportional to the “fast” part of the dipole coefficient, which is always in phase with the applied field. The second addend is proportional to the “slow” part of this coefficient, which is always in phase with the field-induced concentration change around the particle. Figures 1 and 2 show the spectra of the real parts of the dipolar and the concentration coefficients, as well as their dependences on the ion valences. The calculations were performed using the parameter values given in Table 1. Note that these parameters coincide with those used in the DC study4 except for κa ) 30 instead of 100 and ζ˜ ) -4 instead of -2. These changes were necessary in order to make all the frequency dependences visible in the different figures. As can be seen, both the monovalent counterion (cases 11 and 12) and divalent counterion (cases 21 and 22) curves strongly differ from one another, despite the relatively high surface potential value. This means that the co-ion valence has a strong bearing on the LFDD phenomenon. For ω ) 0, or in view of eq 31 h* ) 0, the concentration coefficient Kc(0) reduces to the corresponding stationary result, eq 51 in ref 4 as expected. On the contrary, the dipolar coefficient

are the same as in the DC case,4 and

z+(R+ - 1)(R- + 2 - U-) + f* ) 2 +

* 2 2

ξκa

1 + √ξ κa

) 2 + 2h*

(31)

*

2z+z- R+ - R3 + z + z- 2B K*c ) 1 + h*A/B

(32)

does not reduce to eq 50 in 4. This behavior arises because of the difference between the AC and DC equations for the electric potential outside the double layer: eq 23 and eq 31 in 4, respectively. In the stationary limit these potentials must be equal, which requires that the following condition should be fulfilled:

Kd(0) + Kc(0)∆ ) KdDC

(33)

where Kd∞, H, A, and B are real frequency independent coefficients:

Kd∞ ) lim Kd ωf∞

)

z-(R- - 1)(R+ + 2 - U+) - z+z-(R+ - R-)∆ z+(R+ + 2)(R- + 2 - U-) + z-(R- + 2)(R+ + 2 - U+)

The final results for the dipole and the concentration coefficients, obtained solving the equation system 29, can be written in the following form:7

K*d ) Kd∞ - K*c H

Kd(0) )

z+R+(1 - z-∆) + z-R-(1 + z+∆) - (z+ + z-) z+R+(1 - z-∆) + z-R-(1 + z+∆) + 2(z+ + z-) (34)

(35)

which is indeed satisfied. TABLE 1: System Parameters Used in All of the Figures, Unless Specified Otherwise particle radius particle absolute permittivity particle mass density electrolyte solution viscosity electrolyte solution absolute permittivity electrolyte solution mass density ion diffusion coefficients ion concentrations such that dimensionless surface potential temperature

a ) 100 × 10-9 m εi ) 2.0 × ε0 Fi ) 1000 kg/m3 ηe ) 8.90 × 10-4 poise εe ) 78.54 × ε0 Fe ) 1000 kg/m3 D+ ) D- ) 2 × 10-8 m2/s κa ) 30 ζ˜ ) -4 298 K

11206


Grosse

Figure 1. Spectra of the real part of the dipolar coefficient calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Dot lines correspond to the single time constant relaxation, eq 37. Remaining parameters are given in Table 1.

Figure 2. Spectra of the real part of the concentration coefficient calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

The high frequency limit of the dipolar coefficient can be written in terms of the Dukhin number Du:10,11

Kd∞ )

2Du - 1 2Du + 2

( Using eqs 30 and 34 and writing Ieo in terms of G0(, leads to the following generalized expression:

the values obtained for divalent counterions and monovalent co-ions are slightly higher than for divalent counterions and co-ions. Analogously, values obtained for monovalent counterions and divalent co-ions are slightly lower than for monovalent counterions and co-ions. Writing the Dukhin number in terms of the surface conductivity λ,

Du ) +

Du )

-

(z + z )e + [z+z+D+G+ 0 (1 + 3m ) + (z+D+ + z-D-)κ2aεekT 2

z-z-D-G0 (1 + 3m )]

Figure 3 shows the dependence of the Dukhin number on the surface potential for different ion valences. Unlike Figures 1 and 2, the Dukhin number depends almost exclusively on the counterion valence while the co-ion valence only contributes with a small correction. Since the co-ion contribution is negative,

λ aKe

where

Ke )

z+D+ + z-Dz+z-e2N(z+D+ + z-D-) ) ) εeκ2 kT z+ + zD+D(36) εeκ2 Def



Figure 3. Dukhin number as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

is the conductivity of the electrolyte solution, makes it possible to generalize the Bikerman surface conductivity expression12 to different ion valences:

so that the dipolar coefficient can be written as

K*d ) Kd∞ + λ)

2

e ++ + + [z z D G0 (1 + 3m+) + z-z-D-G0 (1 + 3m )] kT

It should be noted that the two addends in the right-hand side of this expression do not correspond to the full counterion and co-ion contributions to the surface conductivity, since part of the electroosmotic contributions, eq 30, cancel out. The frequency dependence of the dipolar coefficient, eqs 31, 32 and 33, is much broader than a single time constant relaxation, dotted lines in Figure 1, which would have the following form:

K*d ) Kd∞ +

Kd(0) - Kd∞ 1 + iωτR

(37)

Kd(0) - Kd∞ iωτR 1+ 1 + (1 + i)W

(39)

where

W ) √ωa2 /(2Def) Figure 4 shows the dependence of the dipolar coefficient relaxation amplitude on the surface potential for different ion valences. This rather complicated behavior can be explained taking into account that for very high (in modulus) surface potential values, the high frequency dipolar coefficient tends, in all cases, to a value that is close to unity:

lim Kd∞ ≈ 1 Because of this, it is not possible to define a single relaxation time for this low-frequency process. However, a characteristic time can be defined, identifying the low-frequency asymptotic form of eq 32,

Kd∞ - K*c H f Kd∞ - 3

2z+z- R+ - R× z+ + z- 2B

(

1 - iω

)

a2 A H 2Def B

|ζ˜|f∞

(the equality would hold for κa f ∞). As for the low frequency limit of this coefficient, it reduces to the corresponding stationary value, eq 35, since the curves in Figure 4 correspond to ∆ ) 0, Table 1. Therefore, for the considered cases, the high surface potential limits coincide with the values given in ref 4: 22 lim K11 d (0) ) lim Kd (0) ≈ |ζ˜|f∞

˜|f∞ |ζ

to the low frequency asymptote of eq 37,

Kd∞ +

Kd(0) - Kd∞ f Kd∞ + [Kd(0) - Kd∞](1 - iωτR) 1 + iωτR

12 lim K21 d (0) ) lim Kd (0) ≈ ζ˜f-∞

ζ˜f∞

1 4 1 2

12 lim K21 d (0) ) lim Kd (0) ≈ 0 ζ˜f∞

This leads to

a2 A τR ) 2Def B

(38)

˜ ζf-∞

so that the relaxation amplitude tends approximately to 1/2, 3/4, or 1, when the counterion valence is double, equal, or half the co-ion valence.

11208


Grosse

Figure 4. Low frequency dipolar coefficient relaxation amplitude as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

Figure 5. Low frequency characteristic time as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

Figure 5 shows the dependence of the low frequency characteristic time on the surface potential for different ion valences. This dependence is determined by the corresponding dependence of the quotient A/B, eq 38. For low (in modulus) surface potential values this quotient tends to one. For large negative (positive) surface potential values, R+ (R-) becomes large and unbounded so that

lim

A z+ (1 - z-∆) ≈ + B z + z-

lim

A z(1 + z+∆) ≈ + B z + z-

ζ˜f-∞

˜ ζf∞

(the equalities would hold for κa f ∞). Therefore, for |ζ˜ | f ∞, the characteristic time value corresponding to low surface potentials, τR ) a2/(2Def), is multiplied by approximately 2/3, 1 /2, or 1/3, when the counterion valence is double, equal, or half the co-ion valence. Furthermore, the transition from the low to

the high surface potential limit occurs at lower surface potentials for divalent counterions since, in this case, the corresponding R increases much faster with ζ˜ . Complex Conductivity Increment The complex conductivity increment of a dilute suspension has the general form

K* - K*e ) 3K*e K*d φ

(40)

where

K* ) K(ω) + iωε(ω) K*e ) Ke + iωεe are the complex conductivities of the suspension and the electrolyte solution, respectively, and φ is the volume fraction



Figure 6. Dielectric increment amplitude as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

of particles in the suspension. Separating the real and imaginary parts of this equation and using eqs 36, 38, and 39, leads to the following expressions for the conductivity and permittivity increments:

{

K(ω) - Ke ) 3Ke Kd∞ - [Kd(0) - Kd∞] × φ

[

]}

2BD2ef 1 + W (ωτR)2 AD+D- (κa)2 (1 + W)2 + (W + ωτR)2 (41)

(1 + W)2 + W(W + ωτR) +

{

ε(ω) - εe ) 3εe Kd∞ - [Kd(0) - Kd∞] × φ

[

(1 + W)2 + W(W + ωτR) - (κa)2

AD+D(1 + W) 2BD2ef

(1 + W)2 + (W + ωτR)2

]}

(42)

The first of these expressions shows that in the frequency range of the LFDD, the last addend in the square bracket is negligible as compared to the first, eq 27. Analogously, the first addends in the square bracket of the second expression are negligible as compared to the last. Dropping these terms and combining the resulting expressions leads to the following modified form of the dipolar coefficient:

K*d ) Kd∞ +

() Ke

K*e

Kd(0) - Kd∞ iωτR 1+ 1 + (1 + i)W

(43)

Aside from being consistent with eq 27, this modified expression has the important advantage of being well behaved in the whole

frequency range. On the contrary, the original eq 39 leads to a conductivity increment expression, eq 41, that diverges for ω f ∞ instead of attaining a finite value. Using eqs 40 and 43 makes it possible to determine the conductivity and permittivity increment amplitudes corresponding to the LFDD:

K∞ - K(0) ) 3Ke[Kd(0) - Kd∞] φ ε(0) - ε∞ AD+D) 3εe[Kd(0) - Kd∞](κa)2 φ 2BD2 ef

These expressions show that the amplitude of the conductivity increment is proportional to the relaxation amplitude of the dipolar coefficient, Figure 4. On the contrary, the dielectric increment amplitude represented in Figure 6, is proportional to this same relaxation amplitude multiplied by A/B, which is proportional to the characteristic time τR, eq 38 and Figure 5 (the remaining coefficient only depends on the ion valences when ∆ * 0). The proper limiting behavior of eq 43 makes it possible to add to this expression a high frequency Maxwell-WagnerO’Konski relaxation term in order to extend its applicability to a broad frequency range:13,14

K*d ) Kdδ +

()

Kd∞ - Kdδ Ke + * 1 + iωτδ Ke

Kd(0) - Kd∞ iωτR 1+ 1 + (1 + i)W (44)

where15-17

Kdδ )

εi - εe εi + 2εe

11210


τδ )

εi + 2εe 2Ke(Du + 1)

Grosse

(45)

Figure 7 shows the spectra of the real part of the resulting dipolar coefficient, eq 44, and its dependence on the ion valences. Also shown are curves corresponding to the original expression 39 and the modified eq 43. Figure 8 shows the high frequency dipole coefficient relaxation amplitude as a function of the surface potential and its dependence on the ion valences. Note that for monovalent counterions and divalent co-ions, the high frequency relaxation amplitude becomes negative for low values (in modulus) of the surface potential. This behavior can be appreciated for case 12 in Figure 7: an increase rather than a decrease of the dipolar coefficient at high frequencies. Figure 9 shows the high frequency relaxation time (in log scale) as a function of the surface potential and its dependence on the ion valences. Note the very strong dependence on the ion valences specially at high (in modulus) surface potentials:

much lower high frequency relaxation time values for divalent than for monovalent counterions, and a very weak dependence on the valence of co-ions. This behavior, which can be clearly seen in Figure 7 (a much higher relaxation frequency for cases 22 and 21 as compared to 11 and 12), reflects the dependence of the Dukhin number on the ion valences, eq 45 and Figure 3, since the electrolyte solution conductivity does not depend on the ion valences at constant κ, eq 36. The final broad frequency conductivity and permittivity increment spectra, obtained combining eqs 40 and 44 are represented in the Figures 10 and 11. As can be seen, the dependences on the ion valences are very strong, both at low and high frequencies. However, while the low-frequency properties depend on both the counterion and the co-ion valences, the high frequency properties are mostly determined by the counterion valence. Dynamic Electrophoretic Mobility In the frequency range of the LFDD, the electrophoretic mobility can be calculated as in the DC case from the sum of

Figure 7. Spectra of the real part of the dipolar coefficient, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Dot lines correspond to the modified eq 43 while the thin black lines represent the extended eq 44. Remaining parameters are given in Table 1.

Figure 8. High frequency dipolar coefficient relaxation amplitude as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.



Figure 9. High frequency relaxation time as a function of the surface potential, calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

Figure 10. Conductivity increment spectra calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

the electroosmotic and of the capillary osmotic velocities on the particle equator just outside the double layer.18 Accordingly, the results obtained for the dimensionless dynamic electrophoretic mobility,

u˜* )

(21) For z+ ) 2z- ) 2z 21 u˜* ) ζ˜ (1 - K*21 - K*21 d c ∆) +

[

]

ezζ/3 √2 + e-zζ + √3e-zζ *21 3 ln Kc 2 z 2√3

3eηeV*p 2εekTE

are similar to the DC expressions,4 except for the asterisks and the inclusion of the concentration coefficient in the expression of the tangential electric field on the particle equator, eq 35: (11) For z+ ) z- ) z

(

˜

˜

)

11 ezζ/4 + e-zζ/4 *11 4 u˜* ) ζ˜ (1 - K*11 - K*11 Kc d c ∆) + 2 ln 2 z

˜

(

˜

˜

)

(12) For 2z+ ) z- ) 2z 12 u˜* ) ζ˜ (1 - K*12 - K*12 d c ∆) +

[

]

e-zζ/3 √2 + ezζ + √3ezζ *12 3 ln Kc 2 z 2√3 ˜

(

˜

˜

)

where Kd* is given in eq 43. In the high frequency limit, the concentration coefficient tends to zero, Figure 2, while the

11212


Grosse

Figure 11. Relative permittivity increment spectra calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). The amplified high frequency part of the spectra appear in the inset. Remaining parameters are given in Table 1.

dipolar coefficient tends to its limiting high frequency value, eq 34, so that these expressions tend to

lim u˜* ) ζ˜ (1 - Kd∞)

(46)

ωf∞

(21) For z+ ) 2z- ) 2z * u˜*21 ) ζ˜ (1 - K*21 - K*21 d c ∆)G +

[

For frequencies above the LFDD, the dynamic electrophoretic mobility is determined by:19,20 * * u˜HF ) ζ˜ (1 - KdHF )G*

(47)

* where KdHF is the dipolar coefficient corresponding to the Maxwell-Wagner-O’Konski relaxation (first two addends in the right-hand side of eq 44), while G* is a factor related to the particle and fluid inertia:

1+ G* ) 1+

iωa2Fe ηe

(

iωa2Fe Fi - Fe iωa Fe + 3+2 ηe 9ηe Fe 2

)

where Fe and Fi are the mass densities of the electrolyte solution and particle, respectively. * f Kd∞, so that In the low frequency limit, G* f 1 and KdHF the high frequency mobility converges to the value

lim

ωf0

* u˜HF

) ζ˜ (1 - Kd∞)

which coincides with eq 46. This behavior makes it possible to combine the low and high frequency dynamic mobility expressions, which leads to the final results valid over a broad frequency range: (11) For z+ ) z- ) z

(

˜

˜

)

ezζ/4 + e-zζ/4 *11 4 Kc u˜*11 ) ζ˜ (1 - Kd*11 - Kc*11∆)G* + 2 ln 2 z

]

ezζ/3 √2 + e-zζ + √3e-zζ *21 3 ln Kc z2 2√3 ˜

(

˜

˜

)

(12) For 2z+ ) z- ) 2z * u˜*12 ) ζ˜ (1 - K*12 - K*12 d c ∆)G +

[

]

e-zζ/3 √2 + ezζ + √3ezζ *12 3 ln Kc z2 2√3 ˜

(

˜

˜

)

where Kd* is now given in eq 44. The dynamic electrophoretic mobility modulus and phase spectra and their dependence on the ion valences are represented in Figures 12 and 13. As can be seen, these dependences are very strong in the whole frequency range and mostly determined by the valence of counterions.

Conclusion In this work we present a generalization of the classic Shilov-Dukhin theory of the low frequency dielectric dispersion of colloidal suspensions, originally developed for symmetrical electrolytes, to the cases when the valence of counterions is double or half the valence of co-ions. This completes our previous study,4 where we generalized the Dukhin-Shilov thin double layer polarization theory to these same cases. These rigorous and purely analytic generalizations were possible because the intervening expressions can be integrated in these two particular cases, a fact apparently overlooked in the past. In view of the general lack of detail in the existing outlines of the theory, the presentation was made providing all the



Figure 12. Spectra of the dimensionless dynamic mobility modulus calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z- ) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

Figure 13. Spectra of the dimensionless dynamic mobility phase angle calculated for z+ ) z- ) 1 (black line), z+ ) 2z- ) 2 (squares), 2z+ ) z) 2 (diamonds), and z+ ) z- ) 2 (gray line with dots). Remaining parameters are given in Table 1.

intermediate steps, assumptions, and justifications that lead to the final solution. This way of doing required the inclusion of a full proof of the hypothesis of approximate electroneutrality used by the theory, which is presently unavailable in the literature to the best of our knowledge. We hope that the presented results will be useful for the interpretation of a wealth of experimental dielectric and electrokinetic spectroscopy data, for which analytical expressions were unavailable. We also hope that the detailed presentation of the Shilov-Dukhin LFDD theory might lead to future theoretical extensions, intended to generalize the standard electrokinetic model. Acknowledgment. Financial support for this work by CIUNT (projects 26/E312 and 26/E419) and by CONICET (PIP 4656) is gratefully acknowledged.

Appendix: Proof of the Hypothesis of Approximate Electroneutrality We seek a solution of the equation set 13-15. Expressions 13 and 15 can be solved using the following change of variables:

δn˜* ) m* + R*n* δF˜ * ) β*m* + n* where the coefficients R* and β* are chosen in such a way that the equations for m* and n* become independent of one another. This leads to two solutions for R* and two for β* . Two of the resulting combinations must be discarded because they lead to coefficients multiplying ∇2m* and ∇2n* that are equal to zero. One of the remaining combinations must be discarded because

11214


Grosse

the corresponding R* and β* expressions diverge when ∆ f 0. The last combination is

R* ) -

β* )

1 + ξ*(z+ - z-)∆ - Z* 2ξ*∆

1 + ξ*(z+ - z-)∆ - Z* 2ξ*z+z-∆ *

+

-

*2

+

- 2

2

ξ*(Q - z+z-∆2) 1 + ξ*(Q - z+z-∆2)

[

H*nn*

()

R*Kne√Hn(a-r) *

* * ∇2m* ) Hm m

∇n )

Y* )

* * a 2 1 + √Hmr * √Hm + δn˜* ) Km e (a-r) r * 1 + √Hm a

so that R* and β* vanish for ∆ f 0. These coefficients lead to the following Helmholtz equations

2 *

∆ 1 + ξ (Q - z+z-∆2) *

Combining these results leads to the final expressions:

Z ) √1 + 2ξ (z - z )∆ + ξ (z + z ) ∆ *

X* )

[

δF˜ * ) β*Km* e√Hm(a-r) *

()

Kne√Hn(a-r) *

* Hm

H*n

2

κ ) {1 + ξ*[2 + (z+ - z-)∆] - Z*} 2 κ2 ) {1 + ξ*[2 + (z+ - z-)∆] + Z*} 2 *

[

δφ˜ * )

K*da2

* * √Hm Km e (a-r)

m )

n ) *

* K*ne√Hn(a-r)

δφ ˜* )

(

r2

-

)

r eEa cos θ + δn˜*X* + δF˜ *Y* a kT

so that

∇2δφ˜ * ) δn˜*κ2ξ*(X* + Y*∆) + δF˜ *κ2[X*ξ*z+z-∆ + Y*(1 + ξ*Q)] This leads to the following equation set

r + a

* * a 2 1 + √Hmr * √Hm + e (a-r) (X* + Y*β*)Km r * 1 + √Hm a

()

()

The Poisson eq 14 can now be solved writing the solution in the general form:

()

]

* * a 2 1 + √Hnr eEa (X*R* + Y*)K*ne√Hn(a-r) cos θ r kT 1 + √H*na

and, using eq 16:

()

K*da2

r

* a 2 1 + √Hmr eEa cos θ r kT * 1 + √Hm a

* a 2 1 + √Hnr eEa cos θ r kT 1 + √H*na

-

]


()

*

The solutions for m and n can be written as

*

2

()

* a 2 1 + √Hmr + r * 1 + √Hm a

where

]


˜* ) δΦ

[

K*da2 2

-

r

r + a

[X + (Y - 1)β *

*

*

* * √Hm ]Km e (a-r)

* a 2 1 + √Hmr + r * 1 + √Hm a

()

]

* * a 2 1 + √Hnr eEa (X*R* + Y* - 1)K*ne√Hn(a-r) cos θ r kT 1 + √H*na

()

In the frequency range of the LFDD, and in view of eqs 27 and 38,

ξ* )

ωa2 /Def iω ) i ,1 κ2Def κ2a2

X* + Y*∆ ) ∆ X*ξ*z+z-∆ + Y*(1 + ξ*Q) ) ξ*Q that can be solved for X* and Y*

which shows that it is possible to simplify the above results using their series expansions. This leads to

Z* f 1 + ξ*(z+ - z-)∆ + 2ξ*2z+z-∆2



R* f ξ*z+z-∆

References and Notes

β* f -ξ*∆ Hm* f κ2ξ* H*n f κ2 X* + Y*β* f ∆ X*R* + Y* f Qξ*

[

√ξ*κ(a-r) a 2 1 + √ξ κr *

( r ) 1 + √ξ κa +

* δn˜* ) Km e

*

ξ*z+z-K*neκ(a-r)

[

2

√ξ*κ(a-r) a 2 1 + √ξ κr

* e δF˜ * ) -ξ*Km

*

( r ) 1 + √ξ κa ∆ + *

K*neκ(a-r)

δφ ˜* )

[

K*da2 r2

-

[

K*da2 r

2

-

]

cos θ ( ar ) 11 ++ κaκr eEa kT 2

* 2 r * √ξ*κ(a-r) a 1 + √ξ κr ∆+ e + Km a r 1 + √ξ*κa

()

Qξ*K*neκ(a-r)

˜* ) δΦ

]

cos θ ( ar ) 11 ++ κaκr ∆ eEa kT

]


* 2 r * √ξ*κ(a-r) a 1 + √ξ κr ∆e + Km a r 1 + √ξ*κa

()

K*neκ(a-r)

]


These expressions coincide with eqs 18, 25, 23, and 24, which were obtained using the hypothesis of approximate electroneutrality, except for terms that decrease exponentially with κr.

(1) Shilov, V. N.; Dukhin, S. S. Theory of low-frequency dispersion of dielectric permittivity in suspensions of spherical colloidal particles due to double-layer polarization. Colloid J. 1970, 32, 245. (2) Dukhin S. S.; Shilov V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes; Wiley: New York, 1974. (3) Hinch, J. E.; Sherwood, J. D.; Chew, W. C.; Sen, P. N. Dielectric response of a dilute suspension of spheres with thin double layers in an asymmetric electrolyte. J. Chem. Soc., Faraday Trans. 2 1984, 80, 535. (4) Grosse, C., Generalization of a classic thin double layer polarization theory of colloidal suspensions to electrolyte solutions with different ion valences. J. Phys. Chem. B. 2009, 113, 8911-8924. (5) Grahame, D. C. Diffuse double layer theory for electrolytes of unsymmetrical valence types. J. Chem. Phys. 1953, 21, 1054. (6) Loeb, A. L.; Overbeek, J. Th. G.; Wieresma, P. H. The Electrical Double Layer around a Spherical Colloid Particle; M.I.T. Press: Cambridge, U.K., 1961. (7) Grosse, C.; Shilov, V. N. Theory of the low frequency electrorotation of polystyrene particles in electrolyte solution. J. Phys. Chem. 1996, 100, 1771. (8) Shilov, V.; Delgado, A. V.; Gonza´lez-Caballero, F.; Grosse, C.; Donath, E. Suspensions in an alternating electric field. Dielectric and electrorotation spectroscopies. Interfacial Electrokinetics and Electrophoresis; Delgado A., Ed.; Marcel Dekker: New York, 2001; p 329. (9) Overbeek, J. Th. G. Theorie der Elektrophorese. Der Relaxationseffekt. Koll. Beihefte 1942, 54, 287. (10) Dukhin, S. S. Nonequilibrium electric surface phenomena. AdV. Colloid Interface Sci. 1993, 44, 1. (11) Lyklema, J. Fundamentals of Interface and Colloid Science Vol. II: Solid-Liquid Interfaces; Academic Press: London, 1995. (12) Bikerman, J. J. Electrokinetic equations and surface conductance. A survey of the diffuse double layer theory of colloidal solutions. Trans. Faraday Soc. 1940, 35, 154. (13) Shilov, V. N.; Delgado, A. V.; Gonza´lez-Caballero, F.; Grosse, C. Thin double layer theory of the wide-frequency range dielectric dispersion of suspensions of non-conducting spherical particles including surface conductivity of the stagnant layer. Colloids Surf., A 2001, 192, 253. (14) O’Brien, R. W. The high-frequency dielectric dispersion of a colloid. J. Colloid Interface Sci. 1986, 113, 81. (15) O’Konski, C. T. Electric properties of macromolecules. V. Theory of ionic polarization in polyelectrolites. J. Phys. Chem. 1960, 64, 605. (16) Grosse, C. Relaxation mechanisms of homogeneous particles and cells suspended in aqueous electrolyte solutions. In Interfacial Electrokinetics and Electrophoresis, Delgado A. Ed.; Marcel Dekker, New York, 2001. (17) Grosse, C. Dielectric properties of suspensions of solid particles. In Encyclopedia of Surface and Colloid Science; Hubbard A., Ed.; Marcel Dekker: New York, 2002. (18) Lo´pez-Garcıá, J. J.; Grosse, C.; Horno, J. Influence of the counterion and co-ion diffusion coefficient values on some dielectric and electrokinetic properties of colloidal suspensions. J. Phys. Chem. B 2005, 109, 11907. (19) O’Brien, R. W. Electro-acoustic effects in a dilute suspension of spherical particles. J. Fluid Mech. 1988, 190, 71. (20) Ahualli, S.; Delgado, A. V.; Grosse, C. A simple model of the high-frequency dynamic mobility in concentrated suspensions. J. Colloid Interface Sci. 2006, 301, 660.

JP904742V

Generalization of a Classic Theory of the Low Frequency Dielectric

Recommend Documents