J. Phys. Chem. C 2007, 111, 141-148
141
Electroviscous Effect of Concentrated Colloidal Suspensions in Salt-Free Solutions Emilio Ruiz-Reina*,† and Fe´ lix Carrique‡ Departamento de Fı´sica Aplicada II, UniVersidad de Ma´ laga, Campus de El Ejido, 29071, Ma´ laga, Spain, and Departamento de Fı´sica Aplicada I, UniVersidad de Ma´ laga, Campus de Teatinos, 29071, Ma´ laga, Spain ReceiVed: April 18, 2006; In Final Form: August 27, 2006
The electroviscous effect of salt-free suspensions is analyzed theoretically. Our system is a colloidal suspension that has no ions different than those arising from the charge process of the spherical colloidal particles. The theory is based on a cell model for the particle-particle electrohydrodynamic interactions, and is valid for arbitrary surface charge density and particle concentrations. We study the influence on the electroviscous coefficient of the surface charge density, the particle volume fraction and the drag coefficient and valence of the added counterions.
5 η ) η0 1 + φ(1 + p) 2
Introduction An increasing interest is being devoted to the study of the rheology of concentrated suspensions, not only for the advent of new available theoretical models, but mainly because most of the suspensions usually found in industrial applications are concentrated. The rheological behavior is a key factor regarding quality and control processes of these materials. However, very few theoretical results have been presented for the case of concentrated suspensions that have no ions different than those stemming from the charge process of the colloidal particles, i.e., the added counterions that counterbalance the particle surface charge. We will call these kind of systems suspensions in a salt-free solution or salt-free suspensions. The electrokinetic phenomena of such systems are now under investigation.1-7 An aqueous dispersion of colloidal particles can be treated as a salt-free suspension if the concentration of counterions dissociated from the particles are much higher than the H+ and OH- ions originated from water hydrolysis. Understanding the rheology and the electrokinetics of salt-free suspensions can be applied also for the study of suspensions in nonaqueous solutions. A central parameter in the study of the rheology of a colloidal suspension is the newtonian low-shear viscosity. The viscosity η of a dilute suspension of spherical, rigid and nonslipping colloidal particles was first calculated by Einstein,8
5 η ) η0 1 + φ 2
(
)
(1)
where η0 is the viscosity of the suspending medium and φ is the particle volume fraction. The increase in suspension viscosity is due to the distortion of the applied flow field in the neighborhood of the particles. When the particles are charged, there is an additional increment of the flow distortion and, consequently, of the viscosity, which is caused by the presence of electric double layers around the particles. This phenomenon is called the primary electroviscous effect and Einstein’s equation is modified to take it into account, * Corresponding author phone: 34 952131355; fax: 34 952131450; e-mail:
[email protected]. † Departamento de Fı´sica Aplicada II. ‡ Departamento de Fı´sica Aplicada I.
(
)
(2)
where p, the primary electroviscous coefficient, is a function of the charge on the particle and properties of the suspending medium. In the case of salt-free suspensions, the only ions in the liquid medium are the counterions that balance the surface charge. The thickness of the electric double layer is given by the Debye length κ-1, which is given, in the case of a salt-free suspension, by
-3ezc e2njczc2 φ ) σ κ ) 0rskBT 0rskBTa P1 - φ 2
(3)
where e is the elementary electric charge, kB is Boltzmann’s constant, T is the absolute temperature, rs is the relative permittivity of the suspending medium, 0 is the vacuum permittivity, a is the particle radius, σP is the particle surface charge density, and njc and zc are the mean number density in the liquid medium and valence of counterions. In contrast with the situation of added electrolyte, there is a singular relationship between the surface potential and the surface charge density in salt-free suspensions of spherical particles.3,4 While for low surface charge density this relation is roughly linear, above a critical value of the surface charge density the surface potential grows very slowly. This phenomenon is called counterion condensation. The primary electroviscous coefficient has been widely studied in the case of added electrolyte. It is of small magnitude and its experimental determination is difficult.9,10 Most of the experimental works11-18 on the primary electroviscous effect find that it is underestimated by the theoretical models.19-24 Various theories that try to predict the experimental results have been presented. Rubio-Herna´ndez et al.25-28 and Sherwood et al.29 included in the theoretical models the contribution of the ionic transport into the Stern-layer. They found an improvement of the predictions when the influence of a dynamic Stern-layer was taken into account in both the viscosity calculations and experimental determinations of the state of charge and properties of the suspension. Garcı´a-Salinas and de las Nieves10 studied the influence of the reduction of ionic diffusion in the neighbor-
10.1021/jp062372l CCC: $37.00 © 2007 American Chemical Society Published on Web 11/22/2006
142 J. Phys. Chem. C, Vol. 111, No. 1, 2007
Ruiz-Reina and Carrique particles, such as static electrophoresis and electrical conductivity,34-37 sedimentation velocity and potential,38,39 dynamic electrophoresis,40-42 complex conductivity and dielectric response,43 and electroacoustic phenomena,44,45 to mention just a few. Most of them are based on Kuwabara’s46 boundary conditions. According to this model (Figure 1), each spherical particle of radius a is surrounded by a concentric shell of an electrolyte solution, having an outer radius b such that the particle/cell volume ratio in the cell is equal to the particle volume fraction throughout the entire suspension, i.e.,
φ) Figure 1. Cell model.
hood of the colloidal particles and found a better theoryexperiment agreement. Recently, Ruiz-Reina et al.9 developed a theoretical model of the electroviscous effect (we eliminate the word “primary” because the suspension does not have to be necessarily dilute), which includes the possibility of hydrodynamic interactions and which can predict the electroviscous coefficient for moderately concentrated suspensions. Posteriorly, Ruiz-Reina et al.30 extended the range of validity of the theory by introducing the overlapping of electrical double layers, and Carrique et al.31 considered the influence of the dynamic Stern-layer in the moderately concentrated case. It is well-known that the primary electroviscous effect is more important at low electrolyte concentrations, so an advantage of using salt-free suspensions is that the electroviscous effect is maximum, because the extension of the electric double layer, which is characterized by the Debye length κ-1, is the greatest in comparison to the case of added salt for given particle radius and volume fraction values. In fact, according to eq 3, the ratio of the particle radius a to the double layer extension κ-1, which is known as the electrokinetic radius, is always very low for salt-free suspensions, even for very highly charged particles. For example, for a salt-free suspension with particle surface charge density of -20 µC/cm2, particle radius a ) 100 nm, and volume fraction 10-3, the electrokinetic radius κa is about 1.8, whereas typical values in added salt solutions are of the order of tens or hundreds. Consequently, the experimental study of the electroviscous effect is easier and more confident with salt-free suspensions. It is important that the measurements be carried out with a sample perfectly free of salt impurities and under inert atmosphere to avoid CO2 contamination. Apart from these requisites, the experiments have to be performed with the same procedure as in salt solutions.11-18 In this work, we develop a theory of the electroviscous effect of salt-free suspensions, valid for arbitrary surface charge density and very low to moderate particle concentrations, 0 < φ < 0.5. Also, we have analyzed the influence on the electroviscous coefficient of the surface charge density, the particle volume fraction, and the type of added counterions. Theory The Cell Model. To take into account the hydrodynamic particle-particle interactions, Happel’s cell model32 with Simha’s hydrodynamic boundary conditions33 at the outer surface of the cell will be used. The cell model concept has been successfully applied to develop theoretical models for different electrokinetic phenomena in moderately concentrated colloidal suspensions of charged
(ba)
3
(4)
The surface, r ) a, is usually called the “slipping-plane”. This is the plane outside which the continuum equations of hydrodynamics are assumed to hold. In this work we assume that the slipping-plane coincides with the true surface of the particle. The basic assumption of the cell model is that the suspension properties can be derived from the study of a unique cell. The disturbance due to the presence of each solid sphere is considered to be confined inside the cell. According to Simha, each cell-enclosing fluid is surrounded by a spherical wall and the perturbations of the different magnitudes disappear at the surface of the cell (r ) b). By its own nature, the cell model is only applicable when the suspension is homogeneous and isotropic. When a flow field is imposed, this is only true in the low shear region. Different boundary conditions on the surface of the cell can be found in the literature. Simha’s boundary condition33 arises from his hypothesis of no disturbance velocity at the cell wall, i.e., u′ ) 0. It is based on the hydrodynamic assumption that the perturbations of the flow caused by the presence of other particles outside the cell cannot influence the dilatational flow within it. It is accepted that the effect of all other particles in the suspension being sheared is such as to cause the disturbance velocity to the dilatational flow to vanish at the surface of the cell. This simplified model stresses the interaction between the central particle and its immediate neighbors.47 In an alternative treatment, Happel32 introduced a different hypothesis about the behavior of the disturbances at r ) b. He assumed that only the normal component of the perturbation velocity vanishes at the surface of the cell, supplemented with the condition of no friction on it due to the disturbance, which corresponds to the vanishing of the tangential components of the stress perturbation. In the limit of the highly dilute case, Simha’s condition resembles Einstein’s boundary condition, which states that the disturbance velocity vanishes at very large distances from the particle center, i.e., at infinity. The consequence is that Einstein’s limit is recovered when Simha’s condition is used. We want our present theory to cover the range of volume fractions from the very dilute limit to the moderately concentrated case and, in that sense, we think that Simha’s cell model is the natural way of extending the limiting treatment of Einstein to moderately concentrated suspensions. Although Happel’s boundary condition is also a plausible hypothesis for concentrated suspensions, and it may be more appropriate for volume fractions larger than in the case of Simha, it is not applicable when the volume fraction approaches zero. It is important to note that in the own nature of the cell models it is inherent the fact that the boundary conditions are, to some extent, ad hoc hypotheses, although they are based on hydro-
Electroviscous Effect Of Concentrated Salt-free Suspensions dynamic arguments. The different choices have to be validated posteriorly with the experiments. Governing Equations. Let us consider now a charged spherical particle of radius a immersed in an salt-free medium, with the only presence of the added counterions. These ions have valence zc and drag coefficient λc. The axes of the spherical coordinate system (r, θ, φ) are fixed at the center of the particle. In the absence of any flow field, the particle is surrounded by a spherically symmetrical charge distribution. We will consider the case when a flow field is applied to the suspension. If a linear shear field is applied to the system, the velocity field u(r) can be written as
u(r) ) R‚r + ∇ × ∇ × [R‚∇f(r)]
(5)
where R is a constant second-order tensor, and f(r) is a function that only depends on the radial coordinate. The second term corresponds to the perturbation in the flow field due to the presence of particles in the suspension.48 A complete solution of the problem would require the knowledge of the electric potential Ψ(r), the number density nc(r) and the drift velocity vc(r) of the ions, the fluid velocity u(r), and the pressure P(r) at every point r in the system. The governing equations for these fields are23
∇2Ψ(r) ) -
Fel(r) rs0
(6)
Fel(r) ) zcenc(r)
(7)
η0∇2u(r) - ∇P(r) ) Fel(r)∇Ψ(r)
(8)
∇‚u(r) ) 0
(9)
-λc[vc(r) - u(r)] - zce∇Ψ(r) - kBT∇ln nc(r) ) 0 ∇‚[nc(r)vc(r)] ) 0
(11)
(12)
Ψ(r) ) Ψ (r) + δΨ(r)
(13)
Fel(r) ) Fel0(r) + δFel(r)
(14)
0
where δnc(r), δΨ(r) and δFel(r) are perturbation terms. The equilibrium quantities (with superscript 0) are related by
(
)
0 Fel0(r) 1 d 2 dΨ (r) r ) dr rs0 r2 dr
Fel0(r) ) zcenc0(r)
[
(16)
]
zceΨ0(r) kBT
nc0(r) ) bc exp -
(17)
where bc is a constant representing the ionic concentration where the equilibrium electric potential is zero. It is an unknown coefficient arising by the integration of eq 10 in the absence of flow field, i.e., u(r) ) 0 and vc(r) ) 0. The combination of eqs 15, 16, and 17 leads to the well-known Poisson-Boltzmann equation for the equilibrium electric potential,
(
[
)
]
0 z ce zceΨ0(r) 1 d 2dΨ (r) r ) b exp dr rs0 c kBT r2 dr
(18)
The electroneutrality of the cell implies that
Q ) 4πa2σp ) -4π
∫ab r2Fel0(r)dr )
-4πezcbc
∫
b 2 r a
(
)
zceΨ0(r) dr (19) kBT
exp -
where Q is the total charge of the particle. The imposed linear shear field is low and the distortion of the system from equilibrium is small. It is well-known that the suspension has a Newtonian behavior in this case (linear relationship between the volume-averaged stress and shear rate tensors) as can be seen in typical flow curves.49 The concentration of ions nc(r) around the particles will change slightly, and we define the perturbation terms Φc(r) as23
[
nc(r) ) bc exp -
(10)
Equation 6 is Poisson’s equation, where Fel(r) is the electric charge density given by eq 7. Equations 8 and 9 are the NavierStokes equations appropriate to a steady incompressible fluid flow at low Reynolds number in the presence of an electrical body force. Equation 10 expresses the balance of hydrodynamic drag, electrostatic and thermodynamic forces on the counterions at position r. Finally, eq 11 is the continuity equation of counterions. For a low shear field, represented by a symmetric (only the symmetric part of the velocity gradient tensor contributes to dissipation of energy and, consequently, to the viscosity of the suspension) and traceless (because the liquid medium is incompressible) tensor R, the following perturbation scheme applies,
nc(r) ) nc0(r) + δnc(r)
J. Phys. Chem. C, Vol. 111, No. 1, 2007 143
zce [Ψ(r) + Φc(r)] kBT
]
(20)
which allows us to write for the perturbation of the ionic concentration, in combination with eq 12,
δnc(r) ) -
( )
zce 0 n (r)[δΨ(r) + Φc(r)] kBT c
(21)
Making use of the symmetry of the problem, it is possible to separate radial and angular dependences,
δΨ(r) ) ψ(r)(rˆ ‚R‚rˆ )
(22)
Φc(r) ) φc(r)(rˆ ‚R‚rˆ )
(23)
where rˆ is the unitary vector (rˆ ) r/r, being r the modulus of the position vector r). Substituting eqs 12-17, 20 and 21 into the differential eqs 6-11, neglecting products of small perturbation quantities, and applying the symmetry eqs 5, 22, and 23, the governing equations transform to
L4F(r) ) L2φc(r) )
( )
2e2 dΨ0 0 nc (r)zc2φc(r) r2η0kBT dr
( )[
ezc dΨ0 dφc λc + [r - 3F(r)] kBT dr dr z ce
zc2e2nc0(r) [ψ(r) + φc(r)] ) 0 L2ψ(r) rs0kBT
(15) The function F(r) is defined by
(24)
]
(25)
(26)
144 J. Phys. Chem. C, Vol. 111, No. 1, 2007
F(r) )
(
)
d 1 df(r) dr r dr
Ruiz-Reina and Carrique
(27)
and the differential operators L2 in eqs 25, 26, and L4 in eq 24 are defined by
d2 2 d 6 + dr2 r dr r2
(28)
d4 8 d3 24 d 24 + + dr4 r dr3 r3 dr r4
(29)
L2 ≡ L4 ≡
The three coupled differential equations 24-26 must be solved with specific boundary conditions. The boundary conditions here used are
|
dΨ0 dr
σP ) -rs0 dΨ0 dr
|
r)a
(30)
)0
(31)
u(r)|r)a ) 0
(32)
u(r)|r)b ) R‚r
(33)
nc(r)|r)b ) nc0(b)
(34)
rˆ ‚∇Φc(r)|r)a ) 0
(35)
rs(nˆ ‚∇δΨ)|r)a ) rp(nˆ ‚∇δΨp)|r)a
(36)
∇δΨ‚rˆ |r)b ) 0
(37)
r)b
In these equations, rp is the particle relative permittivity, and Ψp(r) is the electric potential inside the particle. The electric state of the particle surface is specified by condition 30. The condition in eq 31 implies the electroneutrality of the cell in equilibrium. Equation 32 reflects the no-slip condition at the surface of the particle. Condition 33 is stated by Simha33 which considers that the perturbation of the dilatational flow ∇ × ∇ × (R‚∇f(r)) is zero at r ) b. Condition 34 expresses that the alteration of the equilibrium ion distribution disappears on the cell boundary. Equation 35 arises from the impenetrability of the particle surface. Expression 36 follows from the discontinuity of the normal component of the electric displacement vector. Finally, eq 37 ensures that the electroneutrality of the cell is maintained when the flow field is imposed. According to the symmetry eqs 5, 22, and 23, the boundary conditions given by eqs 32-37 transform into
| |
a 3
dF 1 ) dr r)a 3
(38)
F(b) ) 0
dF )0 dr r)b
(39)
F(a) )
φc(b) ) -ψ(b)
(40)
dψ )0 dr r)b
(41)
| |
dφc dr
|
r)a
)0
2rp dψ ψ(a) ) 0 dr r)a rsa
As we will see, for the calculation of the electroviscous coefficient we must solve the coupled differential eqs 18, 2426 with boundary conditions 30, 31, and 38-43. We use the mathematical application MATLAB with its built-in routines for this purpose. The boundary value problem solver used is a finite difference code that implements the three-stage LobattoIlla formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fourth order accurate uniformly in the functions domain.50 Mesh selection and error control are based on the residual of the continuous solution. The relative tolerance, which applies to all components of the residual vector, has been taken equal to 10-6. Effective Viscosity. We will follow a similar formalism as that derived by Batchelor51 for dilute suspensions of uncharged spheres. However, in our case, use will be made of a cell model to calculate the viscosity of a moderately concentrated suspension. Hydrodynamic interactions between particles will be taken into account by means of Simha’s hydrodynamic boundary conditions at the outer surface of the cell.33 Let the velocity flow field and pressure for a pure straining motion (represented by a symmetrical and traceless tensor R) in the liquid medium be
ui ) Rijxj + ui′ ) ui0 + ui′ P ) P0 + P′
(44)
where i and j are Cartesian indexes in three dimensions (i, j ) 1, 2, 3) and xi are the Cartesian coordinates. The quantities with a prime represent deviations from the corresponding averaged quantities in the suspension (superscript 0). The stress tensor σij at any point in the liquid is
σij ) -P0δij + 2η0Rij + σij′
(45)
where δij is the Kronecker tensor and the additional perturbation stress tensor, σij′, is defined as
(
σij′ ) -P′δij + η0
)
∂uj′ ∂ui′ + ∂xi ∂xj
(46)
The rate at which the forces at the outer surface S of the cell do work is
dW ) dt
∫SuiσiknkdS ) ∫SRijxjσiknkdS )
Rij
[
∫S -P0δik + 2η0Rik - P′δik + η0
(
)]
∂ui′ ∂uk′ + x n dS ∂xk ∂xi j k (47)
being nk (k ) 1, 2, 3) the coordinates of a unit vector normal to the surface S. In the last expression use has been made of the condition that, at the outer surface of the cell, the perturbation velocity field is zero (just Simha’s),
ui′ ) 0 w ui ) ui0 ) Rijxj
(48)
and then
(42) (43)
2Rij )
∂ui0 ∂uj0 + ∂xj ∂xi
(49)
The effective viscosity is defined as that of a homogeneous fluid with the same viscosity of the suspension, η, for which the stress tensor is defined as
Electroviscous Effect Of Concentrated Salt-free Suspensions
σij* ) -P0δij + 2ηRij
J. Phys. Chem. C, Vol. 111, No. 1, 2007 145
(50)
and the rate of working at the external boundary will be
dW ) dt
∫S Rijxjσik*nkdS ) Rij∫S [-P0δik + 2ηRik]xjnkdS
(51)
Operating, we obtain
Rij
∫S 2ηRikxjnkdS ) 2ηRijRik ∫S xjnkdS ) 4 4 2ηRijRik πb3δkj ) 2ηRijRij πb3 (52) 3 3
Equating the two equations 47 and 51 representing the energy dissipation rates, the term involving P0 can be canceled because it is common in both expressions. Taking into account eq 52, the dissipation rate equality becomes
4 2ηRijRij πb3 ) Rij 3
[(
∫S
)
or defining S(φ) (first derived by Simha33) as
∂ui0 ∂uk0 η0 + - P′δik + ∂xk ∂xi
(
Figure 2. Relative viscosity increment against volume fraction for the uncharged case, eq 61. The dashed line, with slope 5/2, is Einstein’s result for very low volume fraction.
)]
∂ui′ ∂uk′ + x n dS (53) η0 ∂xk ∂xi j k
S(φ) )
4 2ηRijRij πb3 ) Rij 3
[
∫S
(
)]
We recall the expressions of the perturbation pressure function for overlapping conditions,30 and the velocity field in terms of the F(r) function,
(
)
0 r2 d3F d2F Fel (r)ψ(r) + 3r 2 + (rˆ ‚R‚rˆ ) P′(r) ) -η0 2 dr3 η0 dr
[
( )]
d F(r) u(r) ) (R‚r) r - 3F(r) - r2 dr r
( )
By evaluating them at the outer surface of the cell and regarding again the above-mentioned Simha’s condition, we obtain
27 dF 4 4 6 (b) + 2ηtr(R2) πb3 ) 2η0tr(R2) πb3 1 - F(b) 3 3 5b 5 dr
}
Fel0(b)ψ(b) 3 d2F b 2 d 3F b (b) + (b) + 10 dr2 10 dr2 5η0
(57)
The effective viscosity is thus calculated to give
{
η ) η0 1 -
}
Fel0(b)ψ(b) b 2 d 3F (b) + 10 dr3 5η0
(58)
When the particles are uncharged, the last expresions turns out to be30
[ {
5 η ) η0 1 + φS(φ) 2
[
]
(61)
When the suspension is very dilute (φ f 0), the Simha function S(φ) tends to 1, and eq 61 transforms into Einstein’s result, as can be observed in Figure 2, showing the relative viscosity increment against volume fraction in the uncharged case. Electroviscous Coefficient. If the particles are charged, we can define an electroviscous coefficient p by analogy with the dilute case,
5 η ) η0 1 + φS(φ)(1 + p) 2
[
}]
4(1 - φ7/3) 5 η ) η0 1 + φ 2 4(1 + φ10/3) - 25φ(1 + φ4/3) + 42φ5/3 (59)
]
(62)
Using eq 58, the electroviscous coefficient is given by
p)
[
27 dF 3 d2F 6 2 (b) + b 2 (b) + - F(b) 5 dr 10 dr 5φS(φ) 5b
]
Fel0(b)ψ(b) b2 d3F (b) + - 1 (63) 10 dr3 5η0
The data used in the generation of all figures can be found in Table 1. The drag coefficient λc in eq 10 are related with the limiting equivalent conductance Λc0 by
λc )
2
6 27 dF 3 dF F(b) (b) + b 2 (b) + 5b 5 dr 10 dr
(60)
(55)
d F(r) +r rˆ (rˆ ‚R‚rˆ ) dr r (56)
{
4(1 + φ10/3) - 25φ(1 + φ4/3) + 42φ5/3
the effective viscosity becomes
or also
∂ui ∂uk -P′δik + η0 + x n dS (54) ∂xk ∂xi j k
4(1 - φ7/3)
N Ae 2 Λc0
(64)
where NA is Avogadro’s number. The following figures show the numerical results for the electroviscous coefficient and its dependence with the electrokinetic parameters involved in the problem. Figures 3 and 4 show the variation of the electroviscous coefficient as a function of the particle surface charge density at various volume fraction values. Both figures indicate that the electroviscous effect is more important for low volume fractions. In Figure 3 it can be observed that p always rises with the surface charge density inside the range shown in the figure.
146 J. Phys. Chem. C, Vol. 111, No. 1, 2007
Figure 3. Electroviscous coefficient against particle surface charge density for various φ values ranging from 10-5 to 10-1, with Na+ counterions.
Figure 4. Electroviscous coefficient against particle surface charge density for various φ values ranging from 10-1 to 5 × 10-1, with Na+ counterions.
However, in Figure 4, this is true if the surface charge density is not very elevated, because p is approximately constant for larger values. The surface charge value, where the plateau starts, moves to higher values when the volume fraction is increased. When we increase the surface charge density at a fixed φ-value, two mechanisms appears: (a) the distortion of the flow field inside the electrical double layer augments because there are more ions in it and the electrical body force is greater. This provokes p to grow with σP. The increase is very rapid for low surface charge density, but decelerates when the surface charge density is high due to the counterion condensation effect,3,4 which causes the surface potential to rise very slowly with the surface charge density; (b) the concentration of counterions inside the cell increases, causing a contraction of the electric doble layer and, consequently, confining the distortion into a smaller region. This reduces p with σP. The interplay of these two mechanisms can explain the observed behavior.
Ruiz-Reina and Carrique
Figure 5. Electroviscous coefficient against volume fraction for various surface charge density values, with Na+ counterions.
Figure 6. Percentage of relative viscosity increment (η - ησp)0)/ησp)0 against volume fraction for various surface charge density values, with Na+ counterions.
The dependence of the electroviscous effect on the volume fraction is illustrated in Figure 5, for various fixed surface charge density values. As was noted before in Figures 3 and 4, the electroviscous coefficient diminishes with volume fraction for all charge density values. The explanation of this behavior is as follows: for a fixed σP-value, an increase of the volume fraction have two effects: (a) sets the cell surface nearer to the particle surface, confining the electrical double layer in a smaller region, and (b) augments the concentration of counterions inside the cell, causing an additional contraction of the electrical double layer. As a consequence, the distortion of the flow field takes place in a smaller region, resulting in a diminution of energy dissipation and, consequently, of the electroviscous coefficient p. In Figure 6, we show the percentage of relative viscosity increment (η - ησp)0)/ησp)0 against volume fraction for various surface charge density values. The symbol ησp)0 represents the viscosity of the suspension in the uncharged case, which is given by eq 59. If the surface charge density is high enough (see the
TABLE 1: Data Used in the Generation of the Figures particle radius: temperature: relative permittivity of suspending liquid: relative permittivity of solid particles: counterion Na+ limiting equivalent conductance: counterion H+ limiting equivalent conductance: counterion K+ limiting equivalent conductance: counterion Li+ limiting equivalent conductance: counterion Ba2+ limiting equivalent conductance: counterion La3+ limiting equivalent conductance:
a ) 100 nm T ) 298.16 K rs ) 80 rp ) 2 0 2 ΛNa + ) 50.08 cm /(ohm‚g.eq.) 0 ΛH+ ) 349.65 cm2/(ohm‚g.eq.) ΛK0 + ) 73.48 cm2/(ohm‚g.eq.) 0 2 ΛLi + ) 38.66 cm /(ohm‚g.eq.) 0 2 ΛBa ) 63.6 cm /(ohm‚g.eq.) 2+ 0 2 ΛLa ) 69.7 cm /(ohm‚g.eq.) 3+
Electroviscous Effect Of Concentrated Salt-free Suspensions
Figure 7. Electroviscous coefficient against particle surface charge density at φ ) 10-3, with Li+, Na+, K+, and H+ counterions.
J. Phys. Chem. C, Vol. 111, No. 1, 2007 147 The reduction of the region of flow distortion explains the decrease of p. In conclusion, we have developed a theoretical model for predicting the electroviscous effect of salt-free suspensions, valid for arbitrary surface charge density and for dilute to moderately concentrated cases. The electroviscous coefficient increases with the surface charge density up to a certain value where the counterion condensation and the contraction of the electrical double layer start reducing it. The electroviscous coefficient always diminishes with the augment of the volume fraction due to the contraction of the electrical double layer. It has also been shown that the magnitude of the electroviscous coefficient is lower the lower the drag coefficient or the higher the valence of the counterions. We stress the necessity of new experimental data, using saltfree suspensions, to test this model. The electroviscous effect is maximum in the case of salt-free suspensions, and we think this fact could mitigate the difficulties usually encountered in its experimental determination. Furthermore, we believe that it is necessary to reinterpret the experimental results of the electroviscous effect that were obtained before in the case of added electrolyte, because the disagreements between the existing theories and the experiments arise mainly under the condition of low added electrolyte concentration. In this situation, the influence of the counterions stemming from the surface charge of the particles can be very important, and it has not yet been taken into account in the theoretical analysis. Acknowledgment. Financial support for this work by Ministerio de Ciencia y Tecnologı´a, Spain (Project MAT200304688) is gratefully acknowledged.
Figure 8. Electroviscous coefficient against volume fraction with σP ) -1.0 µC/cm2, with K+, Ba2+, and La3+ counterions.
References and Notes
curves corresponding to σP ) -1.0 µC/cm2 and σP ) -0.5 µC/cm2), the relative viscosity increment becomes approximately insensitive to the surface charge density, mainly in the low volume fraction region. This can also be observed in figures 3, 4, and 5, and is a consequence of the counterion condensation: there is a reduction of the rate of increase of the electric potential as the surface charge density value increases. For the larger volume fractions, the viscosity ratio diminishes indicating that the electroviscous effect is relatively less important in comparison with pure hydrodynamic effects, which dominates in the case of a large concentration of particles. Figure 7 shows the effect of changing the drag coefficient of the counterions. It is observed that the higher the drag coefficient, the higher the electroviscous coefficient for all particle charge density values. The drag coefficient enters the problem through the force balance eq 10. If the drag coefficient is augmented, the counterions are more dragged by the fluid motion and the electrical double layer becomes more distorted from the equilibrium distribution and, consequently, also the flow field, leading to an increased energy dissipation. The influence of the valence of the counterions is illustrated in Figure 8, where data for mono-, di-, and trivalent ions are presented. The drag coefficients of K+, Ba2+, and La3+ counterions are very similar (see Table 1 and eq 64), so the differences between the curves appearing in Figure 8 can be mainly attributed to the change in valences. It is clear that the magnitude of the electroviscous coefficient is diminished if the valence is raised. The reason of this behavior is simple: it is well-known that the thickness of the electrical double layer is a strongly decreasing function of the valence of the counterions.
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