Non-Newtonian Stress in an Electrolyte - The Journal of Physical

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Non-Newtonian Stress in an Electrolyte J. D. Sherwood* Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K. ABSTRACT: Analogies are drawn between the dynamics of electrolyte solutions and those of dilute suspensions of charged colloidal particles. The viscosity of both electrolytes and suspensions is a function of the ionic concentration c and of the Peclet number Pe characterizing the ratio of applied shear rate that tends to deform the ionic charge clouds, to diffusion that allows them to relax back to equilibrium. In particular, previously published results on the rheology of colloidal suspensions (Lever, D. A. J. Fluid Mech. 1979, 92, 421-433) imply not only that the Falkenhagen O(c1/2) electrical contribution to viscosity is shear thinning, as shown by H. Wada (J. Stat. Mech.: Theory Exp. 2005, P01001), but also that this contribution to the stress is elastic, with normal stress differences appearing at O(Pe2). In practice, the shear rates required for substantial departure from a Newtonian rheology are large, typically 109 s-1.

1. INTRODUCTION The viscosity of an electrolyte solution1-3 depends upon its ionic concentration c, as do other dynamic properties of the electrolyte such as its electrical conductivity and the diffusivity of ions within the solution.2,3 Similarly, the electrokinetic properties of colloidal suspensions (e.g., electrophoresis, viscosity, and particle diffusivity) are modified by the ionic strength of the suspending electrolyte, since this controls the Debye length κ-1 characterizing the thickness of the electrical charge cloud around each particle. In the limit in which colloidal particles shrink to the size of ions, theories for the dynamic properties of a colloidal suspension ought to tend toward a limit that corresponds to the equivalent theory for electrolytes. Allison4 recently confirmed the correspondence between electrolyte conductivity and the electrophoresis of small colloidal particles. Here we shall consider a similar link between theories for the viscosity of electrolyte solutions and those for suspensions of charged colloidal particles that are small compared to the Debye length κ-1 in the surrounding electrolyte. In particular, we shall show that the non-Newtonian rheology of an electrolyte5 is to be expected from equivalent results for dilute suspensions of charged colloidal particles surrounded by ionic charge clouds.6 Booth7 considered the viscosity of a dilute suspension of charged spherical particles of radius a. The imposed shear flow causes the charge clouds around individual particles to deform, which in turn leads to an increase in the stress within a flowing suspension (the primary electroviscous effect). The magnitude of this deformation is controlled by the Peclet number Pe that characterizes the ratio of ionic convection deforming the charge cloud to ionic diffusivity that allows the cloud to relax back to its r 2011 American Chemical Society

equilibrium shape. Most theories of the primary electroviscous effect make the (usually reasonable) assumption that the Peclet number is small,7,8 and that the deformation of the charge cloud is proportional to this small Peclet number. Russel9 computed the O(Pe2) change to the stress in the limit of a thin charge cloud (κ-1 , a). Lever6 considered the opposite limit of thick charge clouds κ-1 . a, and not only found O(Pe2) results for arbitrary linear flows, but also for three particular flows (simple shear, axisymmetric straining, and two-dimensional straining) he obtained results at arbitrary Peclet numbers. The deformed charge clouds were obtained by a Fourier transform method very similar to that used by Onsager and Kim10 to study the nonlinear electrical conductivity of electrolytes at high field strength (the Wien effect), and later used by Wada5 to discuss the viscosity of an electrolyte in simple shear. The results of Lever6 and of Wada5 for the shear-thinning O(c1/2) electrical contribution to the viscosity appear to be almost identical. However, Lever6 also found that normal stress differences appear at O(Pe2). We show here how these results apply to an electrolyte. Wada5 appeals, in his conclusions, for further work on the viscoelastic properties of electrolytes, and the main message of the results presented here is that much can be learned from similar analyses of colloidal suspensions. In a dilute suspension of particles with volume fraction φ , 1, the viscosity μ is expected to vary as ð1Þ μ ¼ μ0 ð1 þ a1 φ þ a2 φ2 þ ... Þ Received: October 22, 2010 Revised: December 10, 2010 Published: January 10, 2011 1084

dx.doi.org/10.1021/jp110131g | J. Phys. Chem. B 2011, 115, 1084–1088

The Journal of Physical Chemistry B

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where μ0 is the viscosity of the suspending fluid and a1 = 2.5 for rigid spherical particles.11 The presence of the particles modifies the stress in a volume O(φ). When each particle is surrounded by an ionic charge cloud, imposed shear perturbs the cloud and modifies the electrical stresses between ions and the particle within the volume of the cloud. As a result, a1 increases and the viscosity of the suspension of charged particles is greater than the viscosity of a suspension of uncharged particles. At low ionic concentration c , 1, the viscosity μ of a dilute electrolyte is found experimentally to vary as3 μ ¼ μ0 ð1 þ Ac1=2 þ Bc þ ... Þ

ð2Þ

1/2

The term Ac is due to electrical interactions between ions: distortion of the spatial distribution of ions relative to one another causes the stress to be anisotropic. This is the effect that we shall discuss in this paper. The ions themselves play the role of the charged colloidal particles in eq 1. As the ionic concentration decreases, the number density of the ions decreases but the length κ-1 over which ion distributions are perturbed increases. Water molecules are not explicitly included in the computation of the stress. However, the solvent properties are important and enter the calculation via (i) the electrical permittivity of the solvent through which act the electrical forces between ions, (ii) the ionic mobilities that relate the velocity of an ion to the force imposed upon it, and (iii) the assumption that the volume of ions (and associated hydration shells) can be neglected, so that ions act as point particles. The term Bc in eq 2 is analogous to the term a1φ in eq 1. The presence of an ion changes the stress over a volume similar to that of the ion (together with any hydration shell). Thus B is typically large for ions that are strongly hydrated; conversely, B can be negative for ions that break the structure of the surrounding water. It should be borne in mind that an experimental fit such as eq 2 is typically only useful up to concentrations of order 0.1 N, and the electrical interaction term Ac1/2 dominates eq 2 only at much smaller ionic strengths. In essence, all the analysis we require is to be found in Lever's colloidal suspension paper.6 However, in electrolytes the anions and cations should be treated on an equal footing, with ionic densities expressed in terms of the probability of finding an ion of species j at r1 subject to there being an ion of species i at r2. We therefore initially follow the analysis of Onsager and Fuoss,2 as presented by Harned and Owen.3

2. ION DISTRIBUTIONS We consider an electrolyte containing m species of ions with number densities ni (i = 1, ..., m). The charge on an ion of the ith species is ezi, where e is the charge on a proton and zi the ionic valence. Electrical neutrality requires that the mean number P -1 0 is densities n0i satisfy m i=1 zini = 0. The Debye length κ defined by m X n0i ðezi Þ2 ð3Þ k2 ¼ εkT i¼1

where nji is the number density of i ions at r2 given that there is a j ion at r1. When r1 - r2 . κ-1 we expect the ionic distributions to be uncorrelated, so that fij f n0i n0j as r12 f ¥. Given that there is an ion j at r1, the charge density at r21 is X zi fji X nji zi e ¼ e ð5Þ F ¼ n0j i i Poisson's equation for the electric field -r2ψj(r1,r12) around the j ion is e X zk fjk ezj r22 ψj ¼ - δðr12 Þ ð6Þ ε k n0j ε where the term ezjδ(r12) represents the charge on the j ion itself. We assume that the ions behave as point ions; i.e., we neglect the possibility that there is a nonzero closest distance of approach between two ions. Ions are convected by the fluid velocity u and move relative to the fluid under the influence of electrical and thermal forces. We neglect hydrodynamic interactions between individual ions, and assume that the electrolyte is incompressible, so that r 3 u = 0. The velocity vji of an i ion in the vicinity of a j ion is vji ¼ uðr2 Þ - ωi r2 ðezi ψj þ kT ln fji Þ

ð7Þ

where ωi is the mobility of the i ion. The equation of continuity for ions is Dfji ðr1 , r12 Þ ¼ r1 ðfij vij Þ þ r2 ðfji vji Þ Dt ¼ -

Dfij ðr2 , r21 Þ Dt

ð8Þ

In steady state, on substituting the ionic velocities (7) into the equation of continuity (8), we obtain uðr1 Þ 3 r1 fij þ uðr2 Þ 3 r2 fji ¼ ωi r2 ð fji ezi r2 ψj þ kTr2 fji Þ þ ωj r1 ð fij ezj r1 ψi þ kTr1 fij Þ

ð9Þ

We follow previous work2,3,10 and assume that potentials are small, so that fij and fji in the electrical terms of (9) can be approximated by their unperturbed values n0i n0j . Hence ½uðr2 Þ - uðr1 Þ 3 r2 fji ¼ n0i n0j e½ωi zi r22 ψj ðr12 Þ þ ωj zj r21 ψi ðr21 Þ þ kTðωi þ ωj Þr22 fji

ð10Þ

We are interested in linear flows of the form uðr1 Þ - uðr2 Þ ¼ Γ 3 ðr1 - r2 Þ

ð11Þ

where ε is the electrical permittivity of the fluid and kT is the Boltzmann temperature. Consider the probability fji(r1, r12) of finding an ion of species j at r1 and an ion of species i at r2 = r1 þ r12. By symmetry

where Γ = (ru)T and E = (Γ þ ΓT)/2 is the rate of strain tensor. In the absence of any forcing term (i.e., zero flow), the charge clouds and potentials must be symmetric in r = r21. The velocity u is antisymmetric in r, as is the r operator. We conclude that the fij and ψi remain symmetric in r when the flow (11) is imposed. In consequence, fij(r) = fji(-r) = fji(r). The analysis is therefore simpler than that for conductivity (and the Wien effect), in which both symmetric and antisymmetric parts of the potentials and number densities have to be considered. We now restrict our analysis to a two-component electrolyte with valences z( and mean ionic number densities n0(, so that

fji ðr1 , r12 Þ ¼ n0j nji ¼ n0i nij ¼ fij ðr2 , r21 Þ

zþ n0þ þ z - n0- ¼ 0

ð4Þ 1085

ð12Þ

dx.doi.org/10.1021/jp110131g |J. Phys. Chem. B 2011, 115, 1084–1088


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and thus k2 ¼

e2 n0þ z þ εkT

ðz þ - z - Þ

The Poisson equation for the potentials (6) becomes e ez þ r2 ψ þ ¼ - 0 ½z þ fþþ þ z - fþ- δðrÞ εn þ ε r2 ψ -

e ez δðrÞ ¼ - 0 ½z þ f-þ þ z - f-- εn ε

ð13Þ

ð14Þ

^ 2 ð^f - ^f Þ þ r ^ 2ψ ^ ^f - ^f Þ ¼ r ^þ ^ T rð Pe ^r 3 Γ þþ þþþ þ3 ð22Þ where

ð15Þ

r 3 ΓT 3 rfþþ ¼ 2kTω þ r2 fþþ þ 2ðn0þ Þ2 eω þ z þ r2 ψ þ

ð16Þ

r 3 ΓT 3 rf-- ¼ 2kTω - r2 f-- þ 2ðn0- Þ2 eω - z - r2 ψ ð17Þ r 3 ΓT 3 rfþ- ¼ kTðω þ þ ω - Þr2 fþð18Þ

We have five unknown functions fþþ, f--, fþ-, ψþ, and ψrather than Lever's two (charge density and electric potential). When the deformation of the charge clouds is small (i.e., at low Peclet numbers Pe , 1, to be defined below) the analysis is (just about) tractable, and schemes for dealing with arbitrary electrolytes are proposed by Onsager and Kim.12 The low Peclet number viscosity of an electrolyte containing two (arbitrary) ionic species was obtained by Falkenhagen.1,3 An increase in the mobility of the more mobile species reduces the viscosity, which is controlled by the ion of lower mobility when the two mobilities are very different. However, at O(Pe2), where non-Newtonian effects start to appear, the analysis of an arbitrary electrolyte is more complicated than appropriate in a short note such as this. We therefore restrict our attention to a symmetric electrolyte zþ = -z- = z, n0þ = n0- = n0, with ionic mobilities ωþ = ω- = ω. The roles of the anions and cations are now completely symmetric, with ψþ = -ψ- and fþþ = f--. We scale all lengths by κ-1. We assume that potentials over the bulk of the charge cloud are small, and so scale potentials by ezκ/ ε (rather than by kT/e). The obvious scaling for probability distributions fij would be by n0i n0j , so that, after scaling, fij f 1 at infinity. However, it turns out to be easier to set ! 0 f n ij ^f ¼ ð19Þ ij n0i n0j k3 n εkT 1 ¼ ¼ 2kz2 e2 8πλB k k3

ð23Þ

is a Peclet number. Lever used a Peclet number γ = 2Pe. In an electrolyte the relaxation of the charge cloud is determined by the diffusivity of one ion relative to the other, rather than by the diffusivity of an ion relative to a fixed colloidal particle. In our symmetric electrolyte this relative diffusivity is twice that of a single ion. To achieve a given Peclet number and charge cloud deformation in an electrolyte, the shear rate required is twice that required to deform the ion cloud around a stationary charged colloidal particle. Both ψþ and ^f þþ - ^f þ- decay at infinity, so we can follow Onsager and Kim,10 Lever,6 and Wada5 and take 3-dimensional Fourier transforms, denoted by a tilde. The Poisson equation (21) becomes ~ ¼ - ½~f - ~f - 1 ð24Þ - k2 ψ þ

þþ

þ-

and the convection-diffusion equation (22) for the ionic probability distributions becomes 2 ~ 2 ~ ^ rð ~ ~f ~ ~ Pe k 3 Γ þþ - f þ- Þ ¼ k ðf þþ - f þ- Þ þ k ψ þ 3 ¼ ðk2 þ 1Þð~f - ~f Þ þ 1 ð25Þ þþ

þ-

Note that Peclet numbers tend to be small. If we take typical values for a 0.1 mol/L solution of KCl, with κ-1 = 1 nm, ω = 5 1011 m N-1 s-1, k = 1.38 10-23 J K-1, and T = 300 K, we find ωkTκ2 = 2 109 s-1. The limit of small Peclet numbers is therefore the limit of most practical interest (though Wada5 discusses the possibility of performing experiments in nearcritical ionic fluids in which lower shear rates might produce high Peclet numbers). The convection-diffusion equation (25) is (in essence) the equation solved by Lever (his eq 11) and by Wada (his eq 21). Rather than repeat their work in full, we assume Pe , 1 and look for solutions of the form ¥ X ð~f þþ - ~f þ- Þ ¼ Pen~f n ð26Þ n¼0

with 1 1 þ k2

ð27Þ

^ ~f ¼ 2k 3 Γ 3 k 1 ð1þk2 Þ3

ð28Þ

~f ¼ 0

where 0

εΓ Γ ¼ 2 2 2 0 4ωe z n 2k ωkT

Pe ¼

The convection-diffusion equations (9) for the ionic probability distributions become

þ n0þ n0- eðω þ z þ r2 ψ þ þ ω - z - r2 ψ - Þ

The convection-diffusion equations, eqs 16, 17, and 18, for the ionic probability distributions reduce to

ð20Þ

^ ^ ^T ^ ^ ~f ¼ 2k 3 Γ 3 ð Γ þ Γ Þ 3 k - 12k 3 Γ 3 kk 3 Γ 3 k 2 4 5 2ð1þk2 Þ ð1þk2 Þ

where λB = e2z2/(4πεkT) is the Bjerrum length for an ion of valence z. The Poisson equation (14) becomes ^ þ ¼ - ½^f ^ 2ψ ^ rÞ ð21Þ r þþ - f þ- - δð^

ð29Þ

The electric potential can similarly be expanded as ~ ¼ ψ þ

We scale all ionic mobilities by ω and scale fluid velocities by Γ*κ-1, where Γ* is a typical magnitude of the rate of shear. 1086

¥ X n¼1

~ Pen ψ n

ð30Þ



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with ~ ¼ 1 ψ 0 1 þ k2 ~ ¼ ~f k - 2 , ψ n n

ng1

ð31Þ ð32Þ

3. STRESS The direct contribution to the electrical stress is usually computed in the electrolyte literature from the force between pairs of ions on either side of a plane (though we note that recent work13,5 is based on a Green-Kubo formulation). In the colloidal literature, the Maxwell stress tensor at a point ri has usually been obtained as an integral over the position of the charged particle at rj and can conveniently be considered as an integral over the charge cloud surrounding each ion. After integration by parts, the stress in the cloud around each cation would be Z Z e xFrψ dV ¼ 0 x½z þ fþþ þ z - fþ- rψ þ dV ð33Þ nþ This should be multiplied by the number density n0þ of cations, with an equivalent result for the charge cloud around the anions. Hence, using the same scalings as previously, the electrical contribution to the stress is found to be Z e2 z2 kn0þ ^ψ ^ þ dV^ ^x½^f þþ - ^f þ- r σe ¼ ε Z e2 z2 kn0^ψ ^ - dV^ þ ^x½^f -þ - ^f -- r ð34Þ ε Lever shows that the integral (33) can be expressed in terms of an integral in Fourier transform space. For our simplified symmetric electrolyte, with ωþ = ω-, the two integrals in eq 34 can be combined to give Z 2e2 z2 kn0 e ~ ~f ~ d3 k ^ k r½ - ~f þ- ψ ð35Þ σ ¼ þ ð2πÞ3 ε þþ where the integral is over the full 3-dimensional k space. We scale σ by 2e2z2κn0/ε = κ3kT and look for an expansion of the stress in the form ¥ X σê ¼ Pen σen ð36Þ n¼1

which is determined by inserting the expansions (26) and (30) into eq 35. At O(Pe) we find Z 1 e ~ ~f Þ ψ ~ þ ðr ~ ~f Þ ψ ~ d3 k k½ð r σ1 ¼ 0 1 1 0 ð2πÞ3 " # Z 4 1 - 3k2 ^ ¼ d3 k 3 kkk 3 Γ 3 k 2 k ð1þk2 Þ5 ð2πÞ Z ^ ^ k Γ 3 k þ kk 3 Γ 2 d3 k þ 3 4 ð1þk2 Þ ð2πÞ ^ E ð37Þ ¼ 120π At O(Pe2), although we consider steady state we nevertheless use the upper convected time derivative r DE - Γ 3 E - E 3 ΓT ð38Þ E ¼ Dt

in order to put the resulting constitutive relation into the form of that of a second-order fluid. Thus Z 1 e ~ ~f Þ ψ ~ þ ðr ~ ~f Þ ψ ~ þ ðr ~ ~f Þ ψ ~ d3 k ^ k½ð r σ2 ¼ 0 1 2 2 1 0 ð2πÞ3 r

^ Þ þ βE ^ E ^ ¼ - 2Rð E 3

ð39Þ

with R ¼

1 , 640π

β ¼ -

1 224π

ð40Þ

in agreement with Lever. The normal stress differences in a second-order fluid with constitutive relation given by eq 39 and undergoing steady simple shear ux = γ· y are σê2xx - σê2yy ¼ 2R γ^_2

ð41Þ

σê2yy - σê2zz ¼ - β γ^_2=4

ð42Þ

again in agreement with Lever's results for normal stress differences. In dimensional form, the first two terms of the expansion of the electrical contribution to the stress (36) may be written as σe ¼

¼

r ^ E 2e2 z2 kn0 3 ^ þ βPe2 E ^ E ^ - 2RPe2 E ½Pe 3 þ OðPe Þ 120π ε ð43Þ

Ek R r β Eþ E E þ OðPe3 Þ 240ωπ 2kω2 kT 4kω2 kT 3

ð44Þ

and the electrical contribution to the normal stress differences in simple shear becomes R N1 ¼ σexx - σeyy ¼ ð45Þ γ_ 2 þ OðPe3 Þ 2kω2 kT N2 ¼ σeyy - σ ezz ¼ -

β γ_ 2 þ OðPe3 Þ 16kω2 kT

ð46Þ

At low Peclet number the electrical contribution to the shear stress (the first term on the right-hand side of eq 44) is proportional to κ. If the ionic strength is reduced, then so is _ However, a reduction in κ the shear stress (at fixed shear rate γ). increases the Peclet number, and we see from eqs 45 and 46 that normal stress differences should be enhanced if the ionic strength _ At higher Pe (considered of the electrolyte is reduced at fixed γ. by Lever) the normal stresses attain a maximum, so the κ-1 dependence predicted by eqs 45 and 46 eventually breaks down. Indeed, Lever predicts that for Pe J 1 nondimensional normal stresses increase more slowly than Pe3/2 so that any further increase in Pe caused by a reduction in ionic strength tends to reduce electric normal stress differences rather than enhance them. If we set Pe = 1 in eq 43, we conclude that the maximum stress that can be attained varies as κ3, and the limit c f 0 does not lead to an unphysically large elasticity. In a 10-3 mol/L solution of KCl at T = 300 K with κ-1 = 10 nm, we find R/(2κω2kT) ≈ 2 10-15 Pa s2, so exceedingly high shear rates would be needed to generate an appreciable normal stress difference. Lever6 found that the O(Pe2) expressions 41 and 42 for the normal stresses are valid up to a Peclet number ≈0.3 which for a 10-3 mol/L KCl solution corresponds to a shear rate γ_ ≈ 107 s-1, and results for this electrolyte are shown in dimensional form in Figure 1. At higher Peclet numbers Lever found that the nondimensional first normal stress has a maximum 1087



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Figure 1. Low Peclet number predictions of normal stress differences in a 10-3 mol/L KCl solution at temperature T = 300 K, as functions of shear rate _ (a, solid line) first normal stress difference N1 = σexx - σeyy (45); (b, dashed line) second normal stress difference -N2 = σezz - σeyy (46). γ:

value approximately 0.007 at a Peclet number 70, which for a 10-3 mol/L KCl solution corresponds to a first normal stress difference σxx - σyy ≈ 30 Pa at a shear rate γ_ ≈ 3 109 s-1. From eq 44 we see that the electrical contribution to the shear viscosity μ at O(Pe) is μe = κ/(480πω), as found by Falkenhagen1 and Onsager and Fuoss.2 This remains unchanged at O(Pe2), as found both by Lever6 and by Wada.5 At very high Peclet numbers Lever found that the electrical contribution to the stress decreases as γ_ -1/2, corresponding to a viscosity contribution that decreases as γ_ -3/2. This differs from the dependence γ_ -5/4 found by Wada, who did not present results for Peclet numbers higher than 100, much lower than the Peclet numbers at which Lever's asymptote holds. Wada's and Lever's results for the electrolyte viscosity in simple shear appear similar, but it is difficult to obtain precise values from the published graphs. Lever found a maximum shear stress at a Peclet number ≈10, so we expect a maximum shear stress at Pe ≈ 10 in an electrolyte. Wada found a maximum stress when his Peclet number x = 2Pe was somewhat greater than 10, but less than 20, so that the ratio of the Peclet numbers for maximum stress in the electrolyte and in the colloidal suspension is lower than the expected value 2. This discrepancy remains unresolved.

’ REFERENCES (1) Falkenhagen, H. Phys. Z. 1931, 32, 745–764. (2) Onsager, L.; Fuoss, R. J. Phys. Chem. 1932, 36, 2689–2778. (3) Harned, H. S.; Owen, B. B. Physical chemistry of electrolytic solutions, 2nd ed.; Reinhold: New York, 1950. (4) Allison, S.; Wu, H.; Twahir, U.; Pei, H. J. Colloid Interface Sci. 2010, 352, 1–10. (5) Wada, H. J. Stat. Mech.: Theory Exp. 2005, (01) P01001, 1–13. (6) Lever, D. A. J. Fluid Mech. 1979, 92, 421–433. (7) Booth, F. Proc. R. Soc. London Ser. A 1950, 203, 533–551. (8) Sherwood, J. D. J. Fluid Mech. 1980, 101, 609–629. (9) Russel, W. B. J. Fluid Mech. 1978, 85, 673–683. (10) Onsager, L.; Kim, S. K. J. Phys. Chem. 1957, 61, 198–215. (11) Einstein, A. Ann. Phys. 1906, 19, 289–306; Ann. Phys. 1911, 34, 591–592. (12) Onsager, L.; Kim, S. K. J. Phys. Chem. 1957, 61, 215–229. (13) Chandra, A.; Bagchi, B. J. Chem. Phys. 2000, 113, 3226–3232.

4. CONCLUSIONS Lever's computations6 of the electrical contribution to stress in a sheared colloidal suspension can be used with little modification to predict the O(c1/2) contribution to the stress in an electrolyte. In particular, elastic effects (e.g., normal stress differences), not evaluated by Wada,5 can be obtained from Lever's analysis. It is predicted that at the small Peclet numbers likely to be attained in experiments the elastic effects increase as the electrolyte concentration decreases. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel: þ44 (0)1223 760436. 1088


Non-Newtonian Stress in an Electrolyte - The Journal of Physical

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