The viscoelastic properties of ordered latices: a self-consistent field theory

The Viscoelastic Properties of Ordered Latices: A Self-Consistent Field Theory W. B. RUSSEL AND D. W. BENZING 1 Department o f Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received October 10, 1980; accepted January 26, 1981 The self-consistent field model developed herein can predict both equilibrium and transport properties for ordered monodisperse latices. The osmotic pressure and shear modulus calculated from the multiparticle electrostatic interactions depend on the dimensionless surface charge density and electrolyte concentration as well as the volume fraction. The results for the dielectric permittivity and dynamic viscosity account only for the first-order effect of volume fraction. The theory reveals the critical role of the counterions in shielding the surface charge and illustrates the significant difference between systems in which the surface potential, as opposed to the surface charge, remains fixed with increasing concentration. INTRODUCTION

The thermodynamical and mechanical properties of ordered monodisperse latices have been examined rather extensively in the past 10 to 15 years. For example, Hiltner and Krieger (1) established that the characteristic irridescence arises from the Bragg diffraction of visible light by the crystalline array of suspended particles, which apparently changes from a body-centered to a face-centered cubic symmetry with increasing volume fraction (2). This ordering disappears with dilution (3), the addition of excess electrolyte (4), or an increase in temperature (5), each of which decreases the magnitude of the electrostatic repulsions immobilizing the charged spheres within the lattice, allowing thermal motion to disrupt the structure. Indeed, several theories ranging from semitheoretical correlations (6-8) to detailed Monte Carlo simulations (9) predict the order-disorder transition reasonably well. The mechanics of ordered suspensions are equally interesting because of the elas1 Present address: Signetics Corporation, P.O. Box 9052, Sunnyvale, Calif. 94086.

ticity imparted by the screened electrostatic forces between particles. Disordered suspensions of spheres generally behave as non-Newtonian fluids with a Newtonian low shear limit. With the onset of order, however, the low-shear viscosity becomes infinite, indicating that a finite stress must be applied to initiate steady flow (10). The suspension then responds to small-amplitude deformations as a linear viscoelastic solid with a nonzero static shear modulus and a low dynamic viscosity, which allow the suspension to propagate low-frequency shear waves (11), and a high osmotic pressure (12). The present study examines in detail these viscoelastic effects, theoretically in this paper and experimentally in the following. Our motivation derives from the markedly non-Newtonian rheology of aqueous colloidal suspensions in general, which is due in part to forces of electrostatic origin (13). Previously, the severe difficulties in characterizing simultaneously many-body interactions plus the suspension microstructure have restricted theoretical studies to the pair interaction, or dilute, limit. Ordered latices, however, provide a perfect model for studying multiparticle interactions within

163 Journal of Colloid and Interface Science, Vol. 83, No. 1, September1981

0021-9797/81/090163 - 15502.00/0 Copyright© 1981by AcademicPress,Inc. All rightsof reproductionin any formreserved.

164

RUSSEL

AND BENZING

the context of a well-characterized microstructure. Existing theories for both the equilibrium and mechanical properties of ordered latices generally assume pairwise additive electrostatic forces, (e.g., (14, 11)) and thereby suffer two shortcomings. First, no exact solution to the Poisson-Boltzmann equation, even in the linearized form, exists for two interacting spheres, leaving a choice between approximations for thin, or thick, double layers or weak interactions, none of which are appropriate over the full range of interest. A more serious problem at finite concentrations, however, is the omission of the role of counterions in shielding the surface charges. For example, Barnes et al. (6) estimated that at the order-disorder transition the counterion density in the fluid accounts for 25-35% of the total ionic strength. At lower concentrations of added salt the counterions dominate, rendering highly suspect pairwise additive theories which are not corrected for this multiparticle effect. An alternative approach devised by Ohtsuki et al. (7) considers an individual sphere positioned eccentrically within a spherical unit cell. Their numerical solutions to the nonlinear Poisson-Boltzmann equation account for the counterions and determine the potential energy as a function of eccentricity. Unfortunately, while the proposed relationship between the static shear modulus and the second derivative of this energy with respect to the sphere position appears plausible, its validity remains unclear. In this paper we develop a self-consistent field theory relating the bulk electrical and mechanical properties of ordered latices to the size, charge, and volume fraction of the suspended spheres and the ionic strength of the fluid. The general approach, originally proposed by Wigner and Seitz (15) for electron crystals and applied later to solid composites (16) and polymer solutions (17), Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

comprises an approximate, mean field treatment of multiparticle interactions. Our formulation is ad hoc but consistent with recent work by Hinch (18) which illustrates the nature of the approximations involved. The model represents the ordered array as a spherical unit cell, of radius a/61/8 (~b = volume fraction) and concentric with a rigid sphere o f radius a, surrounded by a homogeneous continuum with the properties of the bulk suspension. The sphere has electrical charges distributed uniformly over its surface and the Newtonian fluid contains an equal number of counterions plus any excess electrolyte. Any field imposed on the continuum is transmitted to the cell-suspension interface, thereby disturbing the spherically symmetric rest state within the unit cell. Once fields satisfying the governing equations and all boundary conditions are obtained, the unknown properties of the bulk follow from the selfconsistency condition, that averages within the unit cell must equal those in the surrounding continuum. While not rigorous, the approach appears well suited to ordered arrays. First, it avoids the assumption of pairwise additivity in a fashion similar to the eccentric cell model but without the ambiguity associated with this approach. Furthermore, all bulk properties, both equilibrium and nonequilibrium, are accessible. Finally, the ordered structure eliminates the near-field interactions which make self-consistent treatments poor models for random suspensions. Since order is assumed and characterized by the spherical unit cell, distinctions between different crystal symmetries are eliminated along with the possibility of predicting the orderdisorder transition. In the first few sections we formulate the complete governing equations with boundary conditions and then invoke scaling arguments to establish several reasonable simplifications. In subsequent sections solutions are presented for the rest state, an

165

VISCOELASTIC PROPERTIES OF ORDERED LATICES

applied electric field, and a low-frequency oscillatory deformation, thereby determining the osmotic pressure, the dielectric permittivity, and the shear modulus and dynamic viscosity, respectively. AVERAGED EQUATIONS AND S U S P E N S I O N PROPERTIES

The characterization of a multiphase material as an effective continuum has long been standard practice for both fluid suspensions (19) and solid composites (16). If the microscale of the suspension, either the characteristic separation or the particle size, remains small relative to the macroscopic length scales, the constitutive relations for the continuum will depend only on the microstructure of the material. Then one can focus theoretically on a representative volume, with dimensions intermediate between the macro- and microscales, in which the suspension and the deformation are statistically homogeneous. The equations governing the behavior of the continuum within this representative volume are obtained by ensemble averaging (indicated by ( )) the balances valid locally within the individual phases. For a suspension of neutrally buoyant particles subject to both mechanical and electrostatic forces the equilibrium equation p

o Ot

- V-(or + m)

[1]

includes the local inertia pv, the mechanical stress or, and the Maxwell stress m. Ordered suspensions respond to small-amplitude motions as incompressible, linear viscoelastic solids with the stresses related to the strain tensor e as (or + m ) = -(Tr +p0)8 + 2G'(e) +

\Ot/'

where e = (1/2)(Vu + (Vu)T)

[21

with u the local displacement from rest and P0 the ambient pressure. The osmotic pressure 7r, the shear modulus G', and the dynamic viscosity 7' remain to be determined from microstructural parameters. The statistical homogeneity within the representative volume permits the conversion from an ensemble to a volume average. Within an ordered system the representative volume comprises a large number of identical unit cells; hence, integration over a single cell suffices. With the divergence theorem (or + m) = ~

_

(or + m)d3x

1 ~ x(or + m).nd2x V Js

[3]

and, similarly, (e) =-~--~

(un + nu)d2x,

[41

where V and S denote the volume and surface of the cell, respectively, and n is the outward normal. A similar treatment of the electrostatics provides the averaged equation for the electric displacement vector D V'(I))

= (Pe)

[5]

with the averaged charge density (Pe) = 0 since the suspension must be electrically neutral on the macroscale. The constitutive relation (D) = -e'(Vt~) [6] defines the effective dielectric permittivity e' with ~b the electrostatic potential. These averages also represent integrals of the local fields over the unit cell. The basic premise of the self-consistent approach is that the fields (denoted below by *) within the effective continuum surrounding a particular unit cell obey the averaged equations for both the mechanics, Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

166

RUSSEL

OV* p - -

AND BENZING N

-_- - - ~ 7 p *

+

G ' V 2 I I * -{- . 0 ' V 2 v *

[7]

Pe = e ~ zkn k,

Ot

k=l

and the ion conservation equations

with

On k

V.u* = 0 v* =

0u* Ot

+ v. Vn k = oJk(kTV2nk+ ezkV.nkV~b). [11]

Ot

,

[81

and the electrostatics, V2tk* = 0.

[9]

Classical boundary conditions then relate these continuum variables to the fields applied to the bulk material, i.e., far from the unit cell, and the local fields within the unit cell, as discussed further in the following sections. Once the complete solutions to the appropriate boundary value problems are known, Eqs. [2] through [6] determine implicitly the unknown moduli appearing in both the constitutive equations and the solutions.

e is the electronic charge. The balance of convection against diffusion and conduction in [11] introduces a coupling with the momentum equation p

0v

- -Vp + /zV2v - peVtk.

The last term in [12] comprises the electrostatic body force on an ion in an electric field. Finally, incompressibility dictates that V.v = 0

with the space charge density Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

[13]

completing the set of N + 5 equations in a like number of unknowns. In the absence of solid-body translation or rotation

LOCAL EQUATIONS The properties of the individual phases enter through the governing equations at the microscopic level. The small latex spheres of interest are effectively rigid and the fluid is incompressible and Newtonian. The interesting phenomena arise from fixed charges on the sphere surfaces which interact electrically with mobile ions in the fluid. Here we adapt the simplest conventional model (20) which treats the former as a uniform surface charge of density q0 and the latter as point charges of N different species with valence zk, mobilities o~k, and number densities n k (k = 1 , . . . , N). This model should remain valid at the relatively low ionic strengths relevant to ordered latices. The potential and the distributions of mobile ions within the fluid obey the Poisson equation ~V2t~ = --Pc, [10]

[12]

Ot

v=0

at

r=a.

[14]

The exact electrostatic boundary conditions at r = a simplify to 00

-e

Or

= q

or

0 = t~0

[15]

since the dielectric constant for the fluid (water) greatly exceeds that for the solid (polystyrene). Thus, neither electrostatic nor velocity fields within the particle need be considered. At the interface between the fluid within the cell and the bulk suspension, denoted by r = r* with unit normal n, the velocity Or* V----V*

--

Ot

the tractions (or + m ) . n = or*.n, the electric displacement EVO'n = ~'V~b*'n, and the potential

[16]

VISCOELASTIC PROPERTIES OF ORDERED LATICES

tO = tO*

[17]

must be continuous. These conditions result from the basic premises of the self-consistent field approach; hence, none of the

167

ambiguities regarding boundary conditions in conventional cell models arise (21). In the form shown above this coupled set of nonlinear partial differential equations would be quite difficult to solve. Scaling with the characteristic quantities,

length

a

(~ 10-7 m)

frequency

__e(k_.T_T]2

(~ 105 sec -1)

pc \ a e z /

potential

kT

(~25 mV)

ez

concentrations tresses

EkT

(~1022 m -3 = 10-5 M)

(aez) 2 (~102 N/m 2 = 10-2 m H20),

[18]

\aez/

however, reveals the dominant role of electrical effects and indicates several physically reasonable and mathematically significant simplifications. First, at frequencies small relative to (e//x) × ( k T / a e z ) z the viscous stresses become lower order than the Maxwell stresses and pressure which thereby govern the deformations. Furthermore, the short viscous (a 2p/lx 10-8 sec) and diffusion (a2/oJkT ~ 10-5 sec) time scales make the time-dependent terms unimportant as well. As a result the fluid motion is pseudosteady and the charge cloud remains near equilibrium, but not spherical, as the cell boundary deforms. Coupled with the small amplitude of the motion (~a) these permit regular perturbation expansions of all the dependent variables about the rest state with the dynamic, or viscous, effects small relative to the electrostatics responsible for the elasticity. To obtain analytical solutions we also must linearize the conduction term in the ion conservation equation, or equivalently the Poisson-Boltzmann equation governing the equilibrium potential, for small potentials ezto/kT ~ 1. Fortunately, the resulting approximation remains numerically accurate for O(1) values (22).

The following sections address first the rest state to calculate the osmotic pressure, then the response to a small static deformation which reflects the shear modulus, and finally the effect of low-frequency oscilla. tions determining the dynamic viscosity. All variables are rendered dimensionless with the scale factors noted above but, for simplicity, are represented by the same symbol as before.

OSMOTIC PRESSURE

At equilibrium, with no applied mechanical or electric fields, the electrostatic repulsions responsible for the ordered structure of the suspension cause an internal pressure. When the suspension equilibrates with pure fluid through a semipermeable membrane, an osmotic pressure, which greatly exceeds that for a disordered suspension of hard spheres, can be measured. In this section we calculate the pressure as a function of the volume fraction of particles with dimensionless surface charge Q = e a z q / e k T for a binary z - z electrolyte at the dimensionless concentration N = e2a2z2no/ekT. Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

168

RUSSEL

AND

The Boltzmann distributions of positive and negative ions, obtained from the equilibrium ion conservation equations, n +- = N exp w- to,

BENZING

e<~>f = 3 Q ¢ 2 N 1 -4~

[19]

depend on the local potential to and the ion concentration N across the membrane, where to = 0. Within the suspension

and

tO = (to)f + to'

with

[20]

with the local variations in the potential to' remaining small providing the surface potential too ~ 1; the average potential in the fluid (to)f is also small for N - O(1) but diverges as N ~ 0. In the Poisson-Boltzmann equation, obtained from [10] and [19], linearization with respect to to' alone produces V2to' = K2to' - Po. [21] The dimensionless parameter K2 =

[26]

1-05 F(x) = 1 +

05

x2[1 + (1 + x-Z) 1/2]

The limiting forms of (to)f and K2 with respect to the key parameter 3 Q

05

(

counterions

]

2 N 1 -- 05 \exc--~ss ~ y t e / illustrate the physics of Concentrated suspensions rather well. At high ionic strengths

Q

N(e<¢)f + e-<~>9

determines the effective Debye length in the concentrated suspension. The last term

N1

05

N1-05 leaving K2 - 2N,

P0 = N(e -(~)f - e<~>r)

[22]

represents the excess counterions required by electroneutrality to compensate the particle charge. The average fluid potential (to)f follows from the electroneutrality condition and the definition of to'. The former dictates that

(o (n + _ n-)r2dr = - Q


3Q 2N

05 1-05

The Debye length then corresponds to that for a dilute suspension and the average potential remains small. With low ionic strengths,

Q

[23]

4,

m > > l

N1-05 or with [19] and [20] and spherical symmetry

dto'/dr = 0

at

r =r0=

05-1/3. [24]

and

05

K2 - 3Q - 1-05

,

Then since

[o to'r2dr = 0 by definition, [21] leads to P0 = - 3 Q - -

05

, [25] 1-05 exactly the density of counterions. With [22] and [25] we now find that Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

N1

05

Now only the counterions shield the electrostatic interactions and the potential increases without bound. These long-range interactions and elevated potentials generate the large electrostatic effects of interest. The solution to [21] satisfying [24] and

169

VISCOELASTIC PROPERTIES OF ORDERED LATICES

dtO'

-

Q

at

r =

1

[27]

dr

From [28] the average potential in the suspension (tO) equals

is

Q - -6 + tO' = - 3 Ks 1 -

(2• (tO) = (tO)f+ Q D

Q

6

3 0 ) . [ 3 0 K2 1 - 6

]

Dr

× ((Kr0 + 1)e -~('°-'~ + (Kr0 - 1)e ~r°-r~) [28] with D = (~cr0 - 1)(1 + K)e K~'°-I~ + (Kro + 1)(1

-

~)e -K(r°-l).

Note that with tOo ~< O(1), tO' ~< O(1) for all values of N. The force exerted by this nonuniform potential on the ions within the fluid generates a radial pressure gradient in the annulus; while in the bulk suspension the constant dimensionless pressure,

The surface potential corresponding to the charge density Q is

p* = 7r + Po,

[31]

exceeds the total pressure on the opposite

tOo = Q

(Kro + 1)e -~(r°-l) + (Kro - 1)e ~(r°-l) (Kro + 1)(1 -

K)e -K(r°-l) + (Kro -

1)(1 +

side of the membrane by the colloid osmotic pressure. From the momentum equation [12] without flow Vp = - ( n + - n-)Vto

[32]

which with [19] integrates to 2).

P = Po + N ( e * + e - * -

[33]

The subtraction of 2 N corrects for the osmotic pressure of the neutral electrolyte which freely permeates the membrane. At r = r0 the fluid pressure equals that in the bulk, determining the dimensionless osmotic pressure, after a careful expansion for to' ~ 1, as 7r=3Q

6 F(1 1- 6

+

\ D2

3

2N

6 -

6

[29]

Ks 1 - 0

10-2,

10 -3-

47r E a k T

for the latices of interest here. Numerical results from [34] for conditions relevant to ordered latices will be examined in the final section. Limiting values with respect to Q

6

N1-6 the ratio of counterions to added salt, are instructive, however. At high ionic strengths the predicted osmotic pressure vanishes exponentially as -

[35]

but the suspension also becomes disordered, rendering the model invalid. The low ionic strength limit with

1-6

Q F=F

(ez) ~

- -

Q

7r - 2Q26 ~/3 e x p ( - ( 8 N ) 1/2 1 (~1/3 (])1/3t] '

1 2F2

with 3Q

3

g ) e K(r°-l)

6 1 -6

6 -

) "

-

>> 1 but

+1/3 Q - -

N1-6

1-6

<1

produces The thermal energy of the particles has been ignored since it contributes little,

7r

3 -

6 - - ,

2Ql_

Journal of Colloid and Interface Science,

6

[36]

Vol. 83, No. 1, September 1981

170

RUSSEL

AND BENZING

corresponding to the thermal energy of the counterions present to insure electroneutrality. Clearly the exact concentration of free electrolyte becomes unimportant when

N
to = - E - r

(1 -

4,

4,r3

PERMITTIVITY

The mechanical fields considered in the next section deform the cell-suspension interface from its spherical equilibrium shape, causing a nonuniform potential within the surrounding suspension. The bulk dielectric permittivity then enters the interfacial conditions and affects the elastic modulus. In this section we calculate the low-frequency limit of e' (scaled on the value in the fluid) from the response of the suspension to a steady external electric field. The potential within a suspension of charged particles subjected to an electric field of dimensionless strength E ~ 1 can be expanded as the O(to0) equilibrium result, an O(E) effect from the interaction of the applied field with uncharged dielectric spheres, plus terms of O(Etoo) and smaller reflecting the polarization and convection of the equilibrium ion distributions by the external field (20). Fortunately, determination of the elastic modulus to leading order in too requires only O(1) knowledge of e'; hence, we do not attempt the difficult calculation of these last effects. The O(E) potential satisfies Laplace's equation in both the fluid and the suspension with the boundary conditions as

r--~,

Oto _ 0

at

r = 1,

Or

[37]

and at r = ro 0to

0to*

Or

Or

to = to*.

[38]

The solutions are Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

4,) + E'(2 + 4,)

tO* = - E - r

1-4'

O*~E'r

2 +

+e-7~ +- ~-~ ] .

[39]

The bulk permittivity e' then follows from the average electric field and electric displacement vector within the cell as 2(1 - 4,)

e' -

[40]

2+4, SHEAR

MODULUS

The low-frequency viscoelastic response of the suspension to a small-amplitude deformation consists of distinct in-phase and out-of-phase components. Here we examine the former in order to calculate the shear modulus G = (aez/kT)2(G'/E) characterizing the elastic stresses within the bulk suspension. For a static deformation at constant volume, i.e., u* = e . r

[41]

with lel = 0, the pressure and Maxwell stresses arising from the nonspherical, deformed charge distributions within the annulus balance the stresses in the surrounding suspension, determining the shape of the interface. The small amplitude of the deformation permits us to simplify the free surface problem by projecting the boundary conditions onto the original spherical interface and to linearize the equations. Terms of O(e:e), therefore, are omitted in both the solution of the boundary value problem and the subsequent calculation of the shear modulus. Within the bulk suspension the deformation and pressure fields satisfying [7] and [8] and the boundary condition [41] asr --> ~, u* = e . r

1+

p* = G

A r.e.r __ r3

r2

+r ,

r2

2r 3

' [42]

171

V I S C O E L A S T I C P R O P E R T I E S OF O R D E R E D L A T I C E S

generate the stresses o'*=G

2e 1 + 10B t B r.e.r 7 ] - 56 r ~ /.2

+ 2 r e( .~r/ . 2

5( ar 3

7B)rrr 5 /.2 r.e.r}__r e .

[43]

with that within the annulus will fix the unknown constants A and B characterizing the deformation. The ellipsoidal form of the interface suggests that r.e.r O(r) = (O>f + O'(r) + - -/.2 r.e.r 0 " ( r ) = ( 0 ) + - /.2

The perturbed interface at r* = ((ro + u*(ro))'(ro + u*(ro))) 1/2

( a 34))

r0"e'ro 1 + = ro 1 + rg 2ro3

[44]

0~(Kr),

[47]

O*(r) •

The potential within the fluid then follows from [21] in terms of modified spherical Bessel functions of the first and second kinds of order two, or

now has unit normal n-

go r0

2

( A 34)

0I(X) = C1

+

e'ro

x

ro r o ' e ' r o 1

r0

-/.~

/.o

.

x 2 cosh x

+ C2(- 3 sinhx

[45]

]

Subsequent matching of the O(e) traction a t r = r*,

~'nlr ....

sinh x

1 + 2/.2

2G

{(

_z

+ (~+l)

coshx) ;

[48]

from [9] the potential in the bulk varies as

1 + 2ro3

r~ / r0 02

(A - 2 ~o

5B l ro r0 "e'r0 l r~ /r0 r~ J '

r~

[46]

Expansion of these about r = r0 reduces the boundary conditions [17] to 01(t
=

0*(ro),

dO~ ro 1 + 2r---~

2r~] dr 2 (/.o) + T r (Kro) = e'

dO* dr (to)

[49]

while 2(ros:2Q/D)

d01 - -

(K)

=

0.

dr

Note that because of the deformation the O(1) charge distribution (cc d"O1/dr 2) appears as a surface charge in the O(e) boundary condition on the normal field. As a consequence of [49]

C1 = -

x

Hz(Kro) + (3E'/ro)H2(Kro) 1 + 2r3o

C2 H1

D2

r~H2(~:ro)

[50]

Journal of Colloid and Interface Science, Vol. 83, No. 1, September1981

172

RUSSEL AND BENZING

with

2 1

K(l+--9K2) H1 =

-(1

B

4

+--9K2) tanhK

4 ( 1 ) 1 + - - K ~ - K 1 + - - K S tanhK 9 9

3((1 = ~-~

H2(x)

+

[ QK 2 ~2 H2(Kro) H = Kro(---~-) Hz(Kro) + (3e'/Kro)H2(Kro)'

sinh x - x cosh x

Ha(x)=9(-(l+x2)sinhx

completing the solution to the boundary value problem. Calculation of the shear modulus requires the average strain [531

(e) = e 1 +

coshx

and the O(e) average stress

+H,(x(l+@)sinhx

(o'+m)

))

(,

Now the balance of tractions across the interface will determine A and B. Within the annulus the expansion about r = r0 leaves only the O(e) pressure from [33], ( a + m).nlr' = N(e(¢) _ e_<¢)) r0 ro. e. r0

ro

[521

5H l+--6G

with

- H @ sinhx - (1 + ~ ) c o s h x ) ) ,

+ x(1 +

r~

r~

~l(Kro),

N(e (~) - e -(~)) = 2 ~3Q

[51]

where

= ~ 3q~(Ir~ro (o" + m)dv

+ fro<_r<_r,(O" + m)rodv)

[541

within the cell. The latter integral would contribute at O(e) but equals zero since the deformation preserves volume. Application of the divergence theorem to the remaining integral and substitution of the appropriate solutions then determines (o" + m) -

4 QK a -5 D

~bl(Kro)e. [55]

Consequently, from [53] we find

D from [30]. The Maxwell stresses do not enter directly because of their quadratic nature and the condition dO'Mr (r0) = 0, but the form of the pressure indicates the essential role of electric fields within the annulus. Equating [51] and [46] produces two relations between A and B which lead to 1H 5 2G 3 5H l+--6G __ ___

A 2r03

Journal of CoUoid and Interface Science, V o l . 83, N o . 1, S e p t e m b e r 1981

G = (aez] 2 G' _ 1H. \kT] E 2

[561

The self-consistency of the complete treatment, that is, the formulation of the boundary value problem and the definition of bulk properties, now becomes clear. Since A = 0 problem and B =- 1/4, independent of Q, N, and ~b, the average strain and stress are (e) : e

and (o" + m) = 2G(e)

173

VISCOELASTIC PROPERTIES OF ORDERED LATICES

within both the unit cell and the surrounding suspension, as required. The limiting forms of the shear modulus are

G - ~ 1 Q261/3(2N)l/2 X exp ( 8_N ) 1 / 2 ,

suspension becomes evident once the displacements u*(r)e iat and pressure p*(r)e mr, with 1~ the dimensionless frequency, are substituted into the momentum equation producing - R e 1~2u*

1 -(~1~3(])113)

= - V p * + GV2u * A- /~-~T/V2n*,

,

where

(~1/3 N---~ 0 with

9 Q2 G - 1 6--

@

1-6

Re = /z2 \ ez /

~ 1,

[57]

In the former case the shear modulus exceeds the osmotic pressure, G (2N) 112 ---4~>~1, 77" (~1/3 although both asymptote to zero. At low ion densities, however, G -.-8

3

6 lj3 - - ~ 1 Q

-10 3

for water. Since the O(e) pressure (Eq. [42]) is zero, the O(O.e) equation for the out-ofphase component reduces to

q~4/3( 1 + 1 2 ~b513) 26)(1 + 3 (1 - q~)2(1 - ~ 6 _ 3 68/3 ) /

[58]

1 -

and the ratio is small. The full dependence of G on N, Q, and 4) will be explored later. DYNAMIC VISCOSITY

At finite frequencies dissipative processes, characterized by the dimensionless dynamic viscosity ~ = ~'//z, generate stresses which lag the deformation. For the small electrostatic potentials considered here, the first-order contribution is purely hydrodynamic, from the viscous stresses within the Newtonian annular fluid. Although the application of the self-consistent treatment to the steady shear problem, which is mathematically identical, has been discussed [16], the results are not readily available. We, therefore, present the full derivation below. The role of viscous processes in the bulk

0 = -Vp* + nV2v*

[59]

with v*~il~e.r=6.r

as

r~.

The velocity field must also satisfy the incompressibility condition. A similar decomposition of the velocity, pressure, and electric fields and the charge distribution within the annulus provides the dimensionless O(l~e) momentum equation 0 = -Vp2 + VZv2- p ' V O 2 - p2VO'

[601

with the primes indicating the isotropic rest state. The body force terms represent a relaxation effect due to the nonequilibrium deformation of the charge cloud by the flow, distinct from the in-phase equilibrium deformation driven by the motion of the cellsuspension interface. The velocity v2 therefore consists of the O(e) hydrodynamic portion caused by the imposed shear plus the O(+~)e) response to the electrical stresses. The latter, comprising the primary electroviscous effect analogous to that analyzed by Booth (23) and Sherwood (22) for dilute suspensions, we neglect within the small potential approximation. Furthermore, the O(e) deformation of the interface induces O(e:e) stresses and can be dropped in the small-amplitude limit. Determination of the dynamic viscosity thereby reduces to a purely hydrodynamic problem governed by the Stokes equations Journal of Colloid and Interface Science,

Vol. 83, No, 1, September1981

174

RUSSEL AND BENZING tinuity o f velocities at r = ro as 2 r05 ( -r~ + - 1 + 21 1 - r~ r~

C*-

(r ~ - ~ - r 0 ,2+22~

x

~llM.o

2

(~)-1)

2,

4r--=~ +

,

[63]

leaving the continuity of tractions to resolve the d y n a m i c viscosity, OIj

o

0!2

&

",

q~

o,

FIG. 1. Dimensionless dynamic viscosity as a function of volume fraction.

35

"q=l+

o+

32

in both regions. T h e general solution

2

1175

__

_

+ 7(/) 184 61o/a

+r

16 la/3 64 )u2, - ~-~ 6 + --49420/3

M2 = 1 - 15

8 61o/a,

[611

pertains to the annulus as written; in the bulk the constants are A*, B*, C*, and D*. T h e b o u n d a r y conditions [14] and [59] on the surface o f the sphere and at infinity, t o g e t h e r with the self-consistency conditions (o-) = 2~/(e_') and

62 + 4 8 6 +/a

49

- T

v2 = 26"r A + __C + __ r2B ro 42 2 r.~B + _ 2--1 277

M3

with M 1 - - (1

C r_.e_'.r( rz \-577-

Ms

M 1 -

(e_') = e_',

M3=

25 1-T6+T

21 65/a _ 25 67]z + ~blo/a" T

In the dilute limit 5 "0- 1+-6+ 2

5 42

+

....

102

determine 1 A:~

--

2

o

B* = D* = 0, rr

2

A =

1

1 +-('~5

2

1 ) ( 1 - r o ~)

1 - r~ ////

2

1 +-(7-

1)(1-r

5

B=-5

{

21

O = -2ra(r/-

z,,./

z// //

,.///

25 21 5~ -T roa + --~ro) o.I

1 - rg 1).

/6//

II

1 - r~ 1)~1-

z./

a) tO"

2 2 1 + - (5' 0 C _

G

[62]

T h e final constant t h e n follows from the conJournal of Colloid and Interface Science, Vol. 83; No. 1, September 1981

o.2

0.3

0.4

0.5

0.6

FIG. 2. Effect of mechanism controlling surface charges: ( ) ~b0= 1.0, (- - -) Q = 1.0, withN = 10-4.

VISCOELASTIC

PROPERTIES

The 0(62 ) term equals the contribution from far-field hydrodynamic interactions found by rigorous pair interaction theories (24, 25), indicating that self-consistent theories treat interactions in a mean-field sense. In the dilute theory this term emerges from the renormalization required to overcome nonconvergent integrals. The self-consistency constraint within the present theory has the same effect. Figure 1 illustrates the complete volume fraction dependence of ~9. The magnitude, e.g., ~ - 8 for 6 = 0.50, provides further insight into the mechanics of these systems. For example, for disordered suspensions of uncharged spheres, which behave as viscoelastic fluids, the low-shear viscosity and the low-frequency dynamic viscosity coincide at - 2 4 (relative to the fluid) for 6 = 0.50 (26). With ordered suspensions, on the other hand, relative viscosities in shear have been measured as high as 106 and apparently diverge in the limit (10), emphasizing the solid-like nature of the equilibrium structure. The small-amplitude oscillatory motion invokes weaker hydrodynamic interactions than the disordered suspensions, since the spheres maintain their separation, and recovers the energy required to deform the structure rather than dissipating it as in steady shear. Hence the significantly smaller values of ~1. DISCUSSION

Since the algebraic complexity of the explicit analytical forms for ~- and G obscures their full dependence on 6, N, and Q or tOo, we will examine a few specific cases to demonstrate the relevant trends. The first question of interest concerns the role of the charging mechanism for particles in concentrated suspensions. At fixed N variations in the volume fraction change the ionic environment of the surface charges, as reflected by K [26], and will thereby alter either Q or tOo,or both. For example, for an ideal, reversible interface the surface potential depends only on the concentration of

OF ORDERED

175

LATICES

14

0

(

I

I

I

I

I

OI

0.2

03

0.4

0.5

0.6

FIG. 3. V a r i a t i o n in charge w i t h v o l u m e fraction: )% = 1.0,(---)Q = 1.0, w i t h N = 10 -4 .

potential determining ions and not on 6. Particles possessing a fixed number of ionizable groups, on the other hand, may maintain a constant charge density, in the absence of specific ion adsorption, over the entire concentration range. The significant differences in 7r and G between these two limiting cases are illustrated in Fig. 2 for a low ionic strength so that Q = too = 1.0 at infinite dilution. The larger values for the fixed potential case arise from the variation in charge with increasing volume fraction shown in Fig. 3. Although Q becomes quite large at higher volume fractions the linearization of the Poisson-Boltzmann equation with respect to the potential remains valid. In the remaining discussion we assume the fixed potential condition but treat 00 and N as independent parameters. Similar trends would result with Q fixed, however. The suppression, by excess electrolyte, of the electrostatic forces responsible for the large osmotic and elastic stresses in ordered latices depends on the value of N relative to the counterion concentration 3 6

Q--

2

1-6

Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

176

RUSSEL

AND BENZING

102 !

ID

1.8

I0 ~

).6

N--O )g

\ \ I

N

". -

~

,,

1.0 0

3.16

-,

I0-~

10

I

I0

2 N I-~ 3Q¢ k

31.6 I00

F~G. 6. C o r r e l a t i o n o f l o s s o f e l a s t i c i t y G/(limN~0 G ) with the ratio of added electrolyte to counterions, ( 2 / 3 ) ( N / Q ) ( ( 1 - ~b)/O): ( ) 00 = 2 . 0 , q5 = 0 . 5 0 ; ( - . - ) Oo = 2.0,(;b = 0 . 2 0 ; ( . . . . . )~bo = 1.O, q5 = 0 . 2 0 ; ( - - - )

qJo =

io-2 0

0.1

0.2

&5

FIG. 4. E f f e c t o f a d d e d p r e s s u r e , qJo = 1.0.

0.4

electrolyte

0.5

0.5,

4~ =

0.20.

0.6

on osmotic

I0z

At low salt concentrations 7r (Fig. 4) and G (Fig. 5) become insensitive to N; however, a b o u t N - 1.0fortko = 1.0 both decay rapidly. In fact the precise value at which N becomes significant occurs about 3 Q

4)

(

counterions

]_

2 N 1 - ~b \excess electrolyte/

I0

1

(Fig. 6) whence the added electrolyte begins contributing to the shielding. The asymptotic curves for N < I depicted in Figs. 4 and 5 do not conform to the analytical limits [36] and [57] for

I

G

3Q

~b -->>1 2N 1-~b because

I0-'

(~I/3 (KroY' =

3Q

-

-

~> O ( 1 ) ,

1-4, IO-Z

0

0.1

0.2

03

0.4

0.5

0.6

FIG. 5. E f f e c t o f a d d e d e l e c t r o l y t e o n s h e a r m o d u l u s , Oo = 1.0. Journal of Colloid and Interface Science, Vol. 83, No. 1, September 1981

except at exceedingly low volume fractions, indicating substantial shielding by the counterions. Under extreme conditions this important phenomenon introduces a somewhat surprising maximum into the dependence of the shear modulus on the surface potential (Fig. 7). The linearization

VISCOELASTIC PROPERTIES OF ORDERED LATICES

177

25

othertheoriesare presentedinthefollowing paper.

20

REFERENCES

15 G

I0

i 2

i 3

i 4

5

FIG. 7. Shear modulus as function of surface potential showing effect of counterion shielding with N = 10-8 and ~b = 0.50.

of the Poisson-Boltzmann equation probably becomes a poor approximation in the vicinity of the peak; however, nonlinearities merely suppress the potential, thereby reducing the magnitudes but not altering the shape of the curve. Coupled with the scale factors [18] these results clearly indicate that electrostatic stresses within ordered polystyrene latices can produce quite significant osmotic pressures and shear moduli under reasonable conditions. For example, with a - 0.1 /xm and a modest surface potential, - 2 5 mV, a pressure of -0.1 m H20 and a modulus of -350 N/m 9 are predicted at 4) = 0.35 for ion concentrations below 10-5 M. The magnitudes, however, decrease markedly with increasing particle size; at a - I /zm the same volume fraction and potential produce stresses two orders of magnitude smaller which would just be measurable only in fully deionized water, ~10 -7 M. These predictions agree qualitatively with a variety of observations in the literature. Quantitative comparisons with experimental results and

1. Hiltner, P. A., and Krieger, I. M., J. Phys. Chem, 73, 2386 (1969). 2. Williams, R., and Crandall, R. S., Phys. Lett. Ser. A 48, 225 (1974). 3. Krieger, I. M., and Hiltner, P. A., in "Polymer Colloids" (R. Fitch, Ed.), p. 63, Plenum, New York, 1971. 4. Hachisu, S., and Kobayashi, Y,, J. Colloid Interface Sci. 46, 470 (1974). 5. Williams, R., Crandall, R. S., and Wojtowicz, P. J., Phys. Rev. Lett. 37, 348 (1976). 6. Barnes, C. J., Chan, D. Y. C., Everett, D. H., and Yates, D. E.,J. Chem. Soc. Faraday Trans. 2 74, 136 (1978). 7. Ohtsuki, T., Mitaku, S., and Okano, K., Japan J. Appl. Phys. 17, 627 (1978). 8. Forsyth, P. A., Marcelja, S., Mitchell, D. J., and Ninham, B. W. Adv. Colloid Interface Sci. 9, 37 (1978). 9. Snook, I., and van Megen, W., J. Chem. Soc. Faraday Trans. 2 72,216 (1976). 10. Krieger, I. M., and Eguiluz, M., Trans. Soc. Rheol. 20, 29 (1976). 11. Goodwin, J. W., and Khidher, A. M., in "Colloid and Surface Science" (M. Kerker, Ed.), Chap. IV, p. 529, Academic Press, New York, 1976. 12. Barclay, L., Harrington, A., and Ottewill, R. H., Kolloid Z. Z. Polym. 250,655 (1972). 13. Russel, W. B., J. Rheol. 24, 287 (1980). 14. Snook, I., and van Megen, W., J. Colloid Interface Sci. 57, 40, 48 (1976). 15. Wigner, E., and Seitz, F., Phys. Rev. 43, 804 (1933). 16. Hashin, Z., J. Compos. Mater. 2, 285 (1968). 17. Freed, K. F., and Metiu, H., J. Chem. Phys. 68, 4604 (1978), 18. Hinch, E. J., J. Fluid Mech. 83, 695 (1977). 19. Batchelor, G. K., J. Fluid Mech. 41, 545 (1970). 20. Saville, D. A., Annu. Rev. Fluid Mech. 9, 321 (1977). 21. Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics," Prentice-Hall, Englewood Cliffs, N. J., 1965. 22. Sherwood, J., J. Fluid Mech., in press. 23. Booth, F.,Proc. Roy. Soc. Ser. A 203, 533 (1950). 24. Batchelor, G. K., and Green, J. T., J. Fluid Mech. 41,401 (1972). 25. Batchelor, G. K., J. Fluid Mech. 83, 97 (1977). 26. Krieger, I. M,, Adv. Colloid Interface Sci. 3, 111 (1972).

Journal of Colloidand InterfaceScience. Vol.83, No. 1, September 1981