J. Phys. Chem. C 2008, 112, 12455–12462
12455
Diffusiophoresis of Concentrated Suspensions of Liquid Drops James Lou and Eric Lee* Department of Chemical Engineering, Institute of Polymer Science and Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed: January 30, 2008; ReVised Manuscript ReceiVed: May 5, 2008
The diffusiophoresis of concentrated suspensions of liquid drops subject to a small electrolyte gradient is analyzed theoretically at arbitrary levels of surface potential and double-layer thickness. The effect of doublelayer polarization and double-layer overlapping is taken into account. Kuwabara’s unit cell model is employed in modeling the suspension system, and a pseudospectral method based on Chebyshev polynomial is adopted to solve the resulting general electrokinetic equations. Key factors are examined such as the thickness of the electric double layer, the magnitude of the surface potential, the volume fraction of liquid drops, and the viscosity of the internal fluid. The results presented here add another dimension to the previous corresponding study of rigid particles by considering the internal flow of liquid drops, characterized by their viscosity. It is found, among other things, that the lower the viscosity of the internal fluid, the greater the diffusiophoretic velocity of liquid drops. In particular, the diffusiophoretic velocity of an invicid drop is about 3 times higher than that of a rigid one, whereas it is about 2 times higher for a liquid drop with a viscosity similar to that of the suspending medium. The surfactant that might be absorbed on the droplet surface is crucial to the overall observations made above, though. Introduction Diffusiophoresis, the motion of a charged particle in an electrolyte solution due to the concentration gradient of the electrolytes, is a very important and interesting fundamental electrokinetic phenomenon with great potential in practical industrial applications, such as the deposition of colloidal paint in the car industry.1,2 Because the concentration gradient is usually very large near the surface of a catalyst, it also has a great impact in catalytic reaction engineering. Academic studies of this phenomenon can be found on both experimental3–7 and theoretical fronts.8–16 Among them, Dukhin was the first to point out the electrokinetic origin of diffusiophoresis, and he compared it with the much more widely known electrophoresis phenomenon, the motion of a charged particle under the influence of an externally applied electric field. He concluded that the diffusion of the electrolytes will generate an induced electric field around the colloidal particle and set it in motion as a result.17 Prieve and coworkers1,4 studied a latex system in the car industry both experimentally and theoretically and pointed out that diffusiophoresis is the underlying driving force for a competitive coating process serving as a rival of the traditional electrodeposition of paint onto a metal surface. They used standard electrokinetic equations to describe the very dilute suspension involved and came up with a very satisfactory agreement with the experimental results. As an extension of the previous studies on dilute suspensions, Keh and coworkers18–21 considered the diffusiophoresis in a concentrated suspension on the basis of the concept of unit cell models proposed by Happel22 and Kuwabara.23 They found that the Kuwabara unit cell model was more suitable than the Happel model for the system under consideration. However, their results were restricted to low surface potential with no account of the polarization effect and the distortion of the electric double layer * Corresponding author. E-mail:
[email protected]. Tel: 886-223622530. Fax: 886-2-23622530.
as a result of the convection. This constraint was eliminated recently by Lee and workers,24,25 who considered the diffusiophoresis of concentrated suspensions of spherical particles based on Kuwabara’s cell model, taking into account the polarization effect. The effect of double-layer overlapping was considered as well. The previous studies reviewed so far were focused on rigid colloids. Corresponding suspensions of liquid drops are involved in many important processes in the colloid industry, especially in microfluidic and nanofluidic applications.26 The tiny drops in microfluidic flows, for instance, are almost-ideal chemical reactors characterized by fast thermal transfer, efficient mixing, narrow residence time, and an absence of a hydrodynamic dispersion.26 Therefore, these flows have been utilized in the emulsification,27 the encapsulation,28 the microreaction,29 and so forth. As for the diffusiophoresis of liquid drops in particular, Levich and Kuznetsov30 studied the motion of liquid drops in a nonelectrolyte solution due to the surface reaction near active substrates. In the meantime, Pilat and Prem31 proposed a device that makes use of diffusiophoresis to scrub the pollutant particles from the air. Their idea was further improved by Wang et al.32 Karpov and Oxtoby33 investigated theoretically the capillary flow of liquid drops induced by the concentration gradient, which was supported recently by experimental observations made by Molin et al.34 Molin et al.34 predicted that a 10-100 µm drop might move at a speed exceeding 10 µm/s. As for drops suspended in electrolyte solutions, Baygents and Saville35 studied numerically the diffusiophoresis of a single liquid drop or bubble with a finite element method; hence their results are applicable to only very dilute suspensions. The presence of surfactant- or surface-active substances may lead to a significant impact on the motion of a liquid drop. Usually, a tiny surfactant concentration, which is always present in tap water, natural water, and electrolyte samples, affects the rheology of fluid interfaces, for example, the surface viscosity
10.1021/jp8008749 CCC: $40.75 2008 American Chemical Society Published on Web 07/18/2008
12456 J. Phys. Chem. C, Vol. 112, No. 32, 2008 and the surface tension. In the case of ultrapure water and ultrapure salts, the droplet surface is mobile, and the so-called Rybczynski-Hadamard theory36 in hydrodynamics is applicable. However, in reality, the liquid drop is always covered with a film of surfactant in the solution, and the surfactant reduces the mobility of the droplet interface. As the liquid drop moves, some of the adsorbed surfactants are pulled down to the rear of the liquid drop, and a local surfactant concentration gradient arises along the liquid drop surface. Thus, the concentration of the adsorbed surfactant increases from the leading edge to the rear part of the liquid drop. This surfactant concentration gradient causes a surface tension gradient to retard the motion of the drop, and consequently the drop velocity is lowered. Levich and coworkers developed an adsorption theory to explain this phenomenon,37 and later on Dukhin et al.38 gave an excellent review about theoretical models applied to the effect of the surfactant on hydrodynamics of the bubble motion. Furthermore, an alternative kind of diffusiophoresis was proposed by them,30,39 namely, the diffusiophoresis caused by surfactant gradient. We present here the study of diffusiophoresis in concentrated suspensions of liquid drops dispersed in an electrolyte solution, and we take into account both the double-layer polarization and the double-layer overlapping effects. The effect of the internal flow inside the liquid drop, characterized by its viscosity, is investigated extensively and is absent in our previous studies of rigid particles.24,25 It turns out to have a great impact on the diffusiophoretic velocity of the system under consideration, as we will show later. The effect of the surface immobilization caused by the presence of a surfactant is also discussed in this article, at least for the special limiting case. Compared with the work done by Baygents and Saville,35 who considered the corresponding case of a very dilute suspension, the effect of neighboring liquid drops is examined theoretically in detail. The cases of both identical and distinct ion diffusion velocities are considered here. A pseudospectral method based on Chebyshev polynomial40 is adopted to solve the resulting general electrokinetic equations. This method has proven to be a powerful and efficient method in this field.24,25 Key parameters such as the magnitude of the surface potential, the double-layer thickness, and the viscosity of the internal liquid are examined to analyze their effect on the overall diffusiophoretic behavior of the system.
Lou and Lee
∇2φ ) -
2 Fe zjenj )ε ε j)1
∑
(1)
where ∇2 is the Laplace operator, φ is the electric potential, Fe is the space charge density, ε is the dielectric constant of the dispersion medium, e is the elementary charge, and nj is the number concentration of ionic species j. The conservation of ionic species j yields
[(
) ]
njezj ∂nj ) ∇ Dj ∇nj + ∇ φ - njV ∂t kBT
(2)
where ∇ is the gradient operator, Dj is the diffusivity of ionic species j, kB is the Boltzmann constant, T is the temperature, and V is the velocity of the dispersion medium. Because the Reynolds number is very small, the flow field is described by
∇V ) 0
(3)
µ∇ v - ∇ p - Fe ∇ φ ) 0
(4)
2
where p is the pressure and µ is the viscosity of fluid. A modified Boltzmann distribution is presented here to account for the polarization effect of the double layer as the droplet is set in motion.
(
nj ) nj0 exp -
)
zje (φ + δφ + gj) kBT e
(5)
That is, the electric potential is decomposed into φe, δφ, and gj, representing the equilibrium electric potential in the corresponding static problem, the induced electric potential arising from the movement of the liquid phase, and the equivalent perturbed potential arising from the double-layer polarization, respectively.24,25 Note that δφ and gj are regarded as perturbed quantities arising from the applied concentration gradient. Because the liquid drop remains stagnant when the applied concentration gradient is absent, the velocity vanishes, and the perturbed terms δφ and gj in eq 5 vanish. The governing equations and their associated boundary conditions can be grouped in the equilibrium system and the perturbed system,41 respectively. Note that diffusiophoresis can be characterized by an electrophoresis driven by the local gradient of the electric potential coming from the bulk concentration gradient. When the external electric field is present, but the external gradient of the
Theory Consider the diffusiophoretic problem illustrated in Figure 1 where a dispersion of electrolyte-free drops of radius a moves with a velocity U under an applied uniform concentration gradient ∇n0 in the z direction. The dispersion medium contains z1/z2 electrolyte, with z1 and z2 representing the valences of cations and anions, respectively. The electroneutrality constraint in the bulk liquid phase requires that n20 ) n10/R where n10 and n20 are the bulk concentrations of cations and anions, respectively, and R ) -z2/z1. Kuwabara’s unit cell model23 is adopted, where a representative liquid drop is surrounded by a concentric liquid shell of a radius b. The spherical coordinates (r, θ, φ) are used with its origin located at the center of the liquid drop. The physical properties of the electrolyte solution and those of the droplet fluid are assumed to remain constant, and the shape of the drop remains spherical, which will be explained later in this section. Moreover, creeping flow is assumed for the system under consideration. The electric potential of the system under consideration, φ, can be described by the Poisson equation
Figure 1. Schematic representation of the system under consideration.
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concentration is absent, the movement is called electrophoresis. When there is an external concentration gradient, but there is not an external electric field, the movement is called diffusiophoresis. This implies that the equations governing a diffusiophoresis problem can be deduced directly from those of the corresponding electrophoresis problem.15,17 It can be shown that the governing equations of the present problem in dimensionless forms are as follows:24,25 (A) Outside the Liquid Drop: Equilibrium System. The governing equations for the equilibrium electric potential φe can be derived by eq 1 and eq 5
∇*2φ*e ) -
(κa)2 [exp(-φrφ*e ) - exp(Rφrφ*e )] (6) (1 + R)φr
where φ*e ) φe/ζ. The inverse debye length κ and the scaled surface potential φr are defined, respectively, by
[∑
]
2
κ)
2
(7)
ζ kBT ⁄ z1e
(8)
φr )
where ζ is the surface potential on the droplet surface. We note that the conventional standard electrokinetic model is adopted in this study in that no surface conductivity is considered here. The slipping plane coincides with the droplet surface. The Stern layer conductivity, however, is a potentially important factor for the thorough understanding of the electrokinetic phenomena, as pointed out by Delgado et al.42 The boundary conditions for φ*e are
φ*e ) 1 at r* ) 1 ∂φ*e
b ) 0 at r ) * a ∂r
φr(∇*δφ* + ∇*gj*)∇*gj* ) 0 (14) where V* ) V/[ε(kBT/z1e)2/µa], and Pej ) ε(kBT/z1e)2/µDj is the corresponding Pe´clect number of ion j, taking into account the convection. We assume that the liquid drop is impermeable to ions and that there is an ionic concentration gradient across the outer virtual surface of the unit cell. Therefore
∂g*1
{
*
*
(10)
∂r
E*4ψ* ) -
(κa)2 1+R
{[
(13)
where δφ* ) δφ/ζ and g*j ) gj/ζ. The equation (∇*n*0) ) ∇n0/ (n10/a) refers to the dimensionless applied concentration gradient. The equation β ) (D1 - D2)/(D1 + D2) is a dimensionless parameter (an experimentally measurable property) where D1 and D2 are the diffusion coefficients of cations and anions, respectively. For example, in an aqueous solution, β is 0, -0.2,
(15)
(16)
n*1
∂g*1 *
∂r
[
+ n*2
∂g*1 n*1 * ∂θ
∂g*2 *
∂r
+ n*2
]
(Rn*2)
∂φ* ∂θ
∂g*2
] }
* * ∂φ (Rn ) sin θ (17) 2 ∂θ* ∂r*
where ψ* is the scaled stream function, n*j ) nj/n10, and E*4 is the operator of E*2E*2, which is defined as
E*2 )
1 ∂ ∂2 sin θ ∂ + *2 *2 ∂θ sin θ ∂θ ∂r r
(
)
(18)
Because the bulk liquid cannot penetrate the droplet surface and both the velocity and the shear stress are continuous across the droplet surface, the following conditions are assumed
∂ψ* ∂r*
exp[Rφr(φ*e + δφ* + g*2)]} +
(12)
) 0 at r* ) 1
By taking the curl on eq 4 and introducing the stream function in the spherical coordinates, we get rid of the continuity equation and obtain
|
and the corresponding boundary conditions are
*
1 * * (∇ n0) φr 1 (δφ* + g*2) ) (∇*n*0) at r* ) b Rφr a
(κa) {exp[-φr(φ*e + δφ* + g*1)] (1 + R)φr
∂δφ* ) 0 at r* ) 1 * ∂r 1 b δφ* ) - β(∇*n*0) at r* ) φr a
)
(δφ* + g*1) ) -
2
(κa)2 [exp(-φrφ*e ) - exp(Rφrφ*e )] (11) (1 + R)φr
∂g*2
∂r
(9)
Equation 9 means that the surface potential of a liquid drop at equilibrium remains constant and eq 10 implies that the unit cell as a whole is electrically neutral; thus, there is no electric current between adjacent cells. Perturbed System. By substituting eq 5 into eq 1, we obtain the governing equation of the induced electric potential from eqs 1 and 6 in the dimensionless form
∇*2δφ* ) -
∇*2gj* - φr∇*φ*e ∇*gj* - φr2Pejv*(∇*φ*e + ∇*δφ* + ∇*gj*) -
1⁄2
nj0(ezj) ⁄ εkBT
j)1
and 0.64 for KCl, NaCl, and HCl, respectively.43 The first condition implies that it is dielectric inside the drop and the other condition implies that the net flux for cations and anions is zero across the outer virtual cell. The conservation equation of ions, eq 2, is converted into a dimensionless form by our substituting eqs 5 and 6
|
) r*)1+
∂ψ* ∂r*
|
|
(19) r*)1-
ψ* r*)1+ ) ψ* r*)1- ) 0
(20)
* * µiπrθ r)1- ) µ0πrθ r)1+
(21)
|
|
π*rθ
where is the scaled shear stress tensor on the droplet surface and µi and µ0 are the viscosity of the liquid drop and the bulk solution, respectively. Equations 19–21 are the same as Rybczynski-Hadamard theory,36 which describes the motion of liquid drops in pure solution in a creeping flow region. According to Levich’s theory,37 a tangential force term must be added in eq 21 if the effect of the surfactant in the solution cannot be neglected. That is
|
)-
* µiπrθ
r)1-
|
1 * ∇ γ* + µ0πrθ Ca
(22) r)1+
where γ0 is the characteristic surface tension on the drop surface, and γ* is the dimensionless form of the surface tension. The value Ca ) ε(kBT/z1e)2/aγ0 is a capillary number indicating the relative importance of electrostatic and capillary stress.35 We note that the surfactant concentration considered here is suf-
12458 J. Phys. Chem. C, Vol. 112, No. 32, 2008
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ficiently low and does not affect the double layer, so the additivity of both kinds of diffusiophoresis may be assumed. At the outer virtual surface of the unit cell
1 ψ* ) r*2U* sin2 θ at r* ) b ⁄ a 2
(23)
E2ψ* ) 0 at r* ) b ⁄ a
(24)
Equation 23 states that the fluid is flowing toward the stationary liquid drop with a relative scaled velocity of U*, and eq 24 states that the virtual surface is curl-less, as proposed by the Kuwabara unit cell model.23 (B) Inside the Liquid Drop: Because there is no electrolyte inside the drop, nj is zero when 0 < r* < 1. The droplet fluid is supposed to be dielectric. Therefore, the governing equations for the electric potential and the velocity field inside the liquid drop are, respectively,
∇*2φ*e ) 0
(25)
E ψ )0
(26)
*4
*
The associated boundary conditions are
∂φ*e ∂r*
) 0 at r ) 0
ψ* ) 0 at r ) 0
Figure 2. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of φr at various values of σ when φ ) 0.1, β ) 0, κa ) 1, and R ) 1. The dashed line represents the hard spherical particle case.
(27) (28)
*
∂ψ ) 0 at r ) 0 ∂r*
(29)
which are obtained from the spherical symmetry of the present problem. The details of the above derivations can be found elsewhere,24 among our previous publications. The approach of Prieve and Roman15 is adopted when it was assumed that the concentration of the solute was only slightly nonuniform over the length of scale a, that is, a|∇n0| , n0. Under this condition, the perturbation terms higher than first order can be omitted because they are small. The problem can be simplified to a linear system, allowing a decomposition into two virtual subproblems.15,24 In the first problem, the liquid drop moves with a velocity in the absence of the applied concentration gradient, whereas in the second problem the liquid drop is somehow held stationary when the concentration gradient is applied. If the corresponding forces acting on the surface of spherical particles for the two problems are F1 and F2, respectively, then F1 ) f1′(∇*n*0) and F2 ) f2′U*, where f1′ and f2′ are proportional constants.24,25 Therefore, the diffusiophoretic mobility U*m can be written as
f1′ U * Um ) * )f2′ Ez *
(30)
Given the values of ∇*n*0 and U*, F1 and F2 are calculated first by solving the entire set of electrokinetic equations; f1′ and f2′ are then determined directly by their definitions, and U*m is calculated by eq 30. The governing equations, eqs 6, 11, 14, 17, 25, 26, and their associated boundary conditions are then solved by a pseudospectral method based on the Chebyshev polynomial, which is found to be a powerful and accurate algorithm for problems of the present type.24,25 The shape of the drops is assumed to remain spherical during diffusiophoresis, which is justified by the trivially small magnitude (around 10-10) of the dimensionless Weber numbers, We ) FU2a/σ,44 where σ is the surface tension of the droplet interface. The Weber number is the measurement
Figure 3. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of φr at various values of σ when φ ) 0.1, β ) -0.2, κa ) 1, and R ) 1. The dashed line represents the hard spherical particle case.
of the relative significance of the inertia force and the induced electric force compared with the surface tension, which is the shape-keeping force for the droplet to remain spherical. Results and Discussion Effect of Ionic Diffusion Coefficients (β ) 0 or β * 0). The crucial factors in determining the diffusiophoretic velocity of a colloid suspension are the ionic diffusion coefficients D1 and D2, which determine the magnitude of the Peclet numbers Pe1 and Pe2 in the dimensionless electrokinetic equations directly. Figures 2 and 3 depict the calculated normalized diffusiophoretic velocities (U*/U0) as a function of the surface potential φr at various values of the viscosity ratio σ for β ) 0 (Pe1 ) Pe2 ) 0.26, KCl solution) and β ) -0.2 (Pe1 ) 0.39 and Pe2 ) 0.26, NaCl solution), respectively, where U0 ) (ε/ µa)(kBT/z1e)2∇n0 is a reference diffusiophoretic velocity representing the corresponding velocity of an isolated sphere in
Diffusiophoresis of Concentrated Suspensions an unbounded electrolyte solution with β ) 0.9 Note that β ) 0 corresponds to identical ionic diffusion coefficients, whereas β * 0 for a distinct case. As shown in Figure 2, the φr dependence of (U*/U0) is an even function regardless of the σ considered when the ionic diffusion coefficients are identical; that is, β ) 0. Similarly, as in the case of rigid spheres, we observed symmetry behavior here again, and the reasoning behind it is essentially the same. Briefly speaking, the diffusiophoresis of the charged drops in this case comes solely from chemiphoresis,15,17 which is independent of any electric property of the charged drop itself because the diffusion velocities are identical for cations and anions. The term “chemiphoresis” implies that the motion of charged drops is due to the presence of the solute gradient that leads to an osmotic pressure gradient around the charged drops. No electricity effect, such as a local induced electric field, develops, which is a straight deduction from the fact that both cations and anions diffuse with the same mobility. Details are available elsewhere.24 This symmetric property of (U*/U0) with respect to φr, however, disappears if β * 0, as shown in Figure 3 somewhere between φr ) 0 and -0.2. The symmetry of the profile breaks down here for exactly the same reason as reported before for the corresponding case of rigid hard spheres25 because the environment external to the particle is essentially the same, whether the particle is rigid or not. In short, it is an induced electric field as a result of different diffusion mobilities of cations and anions. Depending on the charged condition on the droplet surface, this induced electric field may reinforce or compete against the bulk concentration gradient. If the induced electric field has a greater influence than the bulk concentration gradient, then the liquid drops may even move in the opposite direction, as shown in Figure 3 as φr > 0. In summary, the sign and the magnitude of β are the key factors in determining the behavior of the diffusiophoretic velocity as well as the surface potential on the droplet surface. Effect of Viscosity Ratio σ. When one characterizes the internal flow of a liquid drop in motion, the viscosity of the liquid drop is the most important factor. As a matter of fact, a rigid particle is considered to be a drop with infinitely high viscosity, whereas a bubble is considered to be a drop with zero viscosity. As the liquid drop is set in motion as a result of an electrolyte concentration gradient, the fluid inside is driven to flow as well. This internal flow lowers the shear rate at the liquid-liquid interface between the drop and the surrounding electrolyte solution, hence reducing the hydrodynamic drag force accordingly. Conceptually speaking, the lower the viscosity, the lower the hydrodynamic drag force and hence the higher the diffusiophoretic velocity. This is exactly what we found in our calculations, as shown in Figures 2 and 3. The Figures show that the diffusiophoretic velocity of liquid drops is always greater than that of rigid particles. Moreover, as the viscosity of the droplet fluid increases, the diffusiophoretic behavior of the drop approaches that of a rigid particle eventually. The behavior of a drop will approach that of a bubble as σ reduces to zero, by definition. For the purpose of comparison, corresponding results of rigid spherical particles24,25 under the same conditions are presented in Figures 2 and 3 as well. As one can clearly see from these Figures, the diffusiophoretic velocity of liquid drops is very close to that of hard spheres because σ ) 1000. The magnitude of the diffusiophoretic velocity of an inviscid liquid drop, however, is about 3 times that of a rigid particle. Perhaps more importantly is the observation of the liquid drop with σ = 1, which is typical in most oil drops suspended in an aqueous electrolyte solution. The diffusiophoretic velocity is
J. Phys. Chem. C, Vol. 112, No. 32, 2008 12459
Figure 4. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of κa when φ ) 0.1, β ) 0, φr ) (3, and R ) 1.
Figure 5. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of κa when φ ) 0.1, β ) -0.2, φr ) -3, and R ) 1.
about twice as high as that of a hard sphere, indicating a huge estimation error if the effect of the internal flow is not considered. Figures 4–6 depict the effect of the viscosity in a more detailed way, with σ as the abscissa, where the diffusiophoretic velocity at several specific κa values is shown. All of these Figures indicate that the viscosity effect is most significant when σ is around unity. The steepest slope in the profiles is shown, illustrating the necessity of considering the internal flow effect. Moreover, the profiles indicate that the thinner the double layer (higher κa), the higher the diffusiophoretic velocity. This is because a thinner double layer indicates a higher concentration gradient of ions, hence a higher driving force for the diffusiophoretic motion in general. Therefore, it is anticipated that the diffusiophoretic velocity of the charged drops should increase with the increase in κa. However, we see that the diffusiophoretic velocity becomes somewhat irregular between κa ) 3 and 5 in these Figures. The reason behind this phenomenon will be discussed in the following paragraph.
12460 J. Phys. Chem. C, Vol. 112, No. 32, 2008
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Figure 6. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of κa when φ ) 0.1, β ) -0.2, φr ) +3, and R ) 1.
Figure 8. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of κa at various values of φ when σ ) 0.8, β ) -0.2, φr ) -3, and R ) 1. (φ ) 0.06, 0.1, 0.2, 0.3, 0.5).
Figure 7. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of κa at various values of φ when σ ) 0.8, β ) 0, φr ) (3, and R ) 1. (φ ) 0.06, 0.1, 0.2, 0.3, 0.5).
Figure 9. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of κa at various values of φ when σ ) 0.8, β ) -0.2, φr ) +3, and R ) 1. (φ ) 0.06, 0.1, 0.2, 0.3, 0.5).
Effect of Double-Layer Thickness Ka. The variations of (U*/ U0) as a function of κa, where κ is the reciprocal of the doublelayer thickness at various volume fractions of drops φ, are presented in Figures 7–9. These Figures reveal that the scaled diffusiophoretic velocity of the charged drops increases with the increase in κa. We note that in the limit as κa f 0 the magnitude of the diffusiophoretic velocity always decreases to zero. This is because κa f 0 indicates an infinitely thick double layer (hence perfect overlapping), and a concentration gradient does not develop in such a system. As a result, an electric force is not generated upon the charged drop, and it remains stationary. This kind of behavior is observed in electrophoresis as well.45,46 As κa increases, the effect of the double-layer overlapping from neighboring drops decreases, and the velocity increases as a result. In particular, as shown in Figure 9, we find the appearance of both a local maximum and a minimum over some ranges of κa when φ is small. Similar behaviors can also be observed in the study of concentrated suspensions of rigid particles.24,25 The explanation there is also applicable here for the concentrated
suspensions of liquid drops. The polarization of the double layer strongly influences the direction of the droplet movement. A microscopic electric field arising from the polarization effect normally tends to retard the diffusiophoretic velocity. According to our analysis in rigid particles,24,25 the polarization effect must be considered when φr and κa are large. Judging from the Figures shown here, we can see that the polarization effect plays an important role in the motion of the droplet dispersion as well. Joint Effect of Polarization and Droplet Volume Fraction O. The effect of the volume fraction φ has several aspects here. In one way, the increase in the volume fraction has a tendency to slow down the droplet velocity with two mechanisms: the double-layer overlapping and the hydrodynamic hindrance effect. Both of these mechanisms increase with increasing volume fraction, with the former weakening the driving force and the latter increasing the hydrodynamic drag force of the droplet motion. However, as the volume fraction gets so high that the backyard of one liquid drop is virtually the front region of the next, the polarization effect associated with each individual liquid drop will offset one another to a certain degree;
Diffusiophoresis of Concentrated Suspensions
Figure 10. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of Pt when φ ) 0.1, κa ) 1, β ) 0, φr ) (3, and R ) 1.
hence the diffusiophoretic velocity will be higher than expected to some degree. With the interactions of these three competing factors, namely, the double-layer overlapping, the polarization effect, and the droplet volume fraction, the overall behaviors of Figures 7–9 get very complicated. Among these Figures, we can observe the occurrence of a local maximum shift to the right side along the κa axis with the increase in φ. This shiftto-the-right behavior is a key feature of the competition among these three factors, which agrees very well with the corresponding case of rigid particles.24,25 Although it is relatively difficult to pinpoint any specific factor as the major cause, the actual behavior is the joint performance of all three factors, following the reasoning mentioned above. Effect of Surfactant on Droplet Motion. To specify the effect of surfactant impurities on the droplet surface, we consider the effect of the surfactant on the diffusiophoretic velocity of liquid drops. It implies that we use eq 22 as the boundary condition instead of eq 21. For convenience, we use a variable Pt, where Pt ) -∇γ*/Ca, to describe the effect of the surfactant. For a 0.1-µm-sized liquid drop, if the characteristic surface tension γ0 is 50 mN/m, then Pt is typically on the order of 10-2 in the system considered. With the increase in the surfactant concentration, the surfactant concentration gradient along the droplet surface also increases accordingly. It leads to a growth of the surface tension gradient, that is, the value of Pt. When the liquid drop moves in a surfactant solution, it encounters more drag than it does in a pure solution. Besides, if the liquid drop moves quickly enough, then there will be more adsorbed surfactants accumulated on the rear of the drop. In Figures 10–12, we observe that the diffusiophoretic velocity decreases with increasing Pt for less-viscous droplets. For instance, for a droplet with a viscosity ratio of 0.01 (regardless of whether β ) 0 or β ) -0.2), as Pt increases to 0.1 the corresponding velocity will decrease 35% as a result. If the effect of the surface tension gradient had a greater impact on the motion of the liquid drops than that of the bulk concentration gradient plus that of the induced electric field, then the liquid drops would migrate to the opposite direction, as shown in Figure 12. However, this effect is less significant as the viscosity ratio gets large mainly because of the retardation of the surface motion in this case. We note that in the absence of the surfactant the surface tension of the liquid drop is a constant so that ∇γ* ) 0 in eq 22, and
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Figure 11. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of Pt when φ ) 0.1, κa ) 1, β ) -0.2, φr ) -3, and R ) 1.
Figure 12. Variation of the scaled diffusiophoretic mobility (U*/U0) as a function of σ at various values of Pt when φ ) 0.1, κa ) 1, β ) -0.2, φr ) +3, and R ) 1.
the velocity of the liquid drops reduces to that predicted in our previous results. Conclusions The diffusiophoretic behavior of a concentrated dispersion of liquid drops with an arbitrary surface potential and a doublelayer thickness is investigated theoretically here. The effect of the volume fraction of liquid drops and the viscosity ratio is examined in particular. In summary, we conclude the following: (i) The lower the viscosity of the droplet fluid, the higher the magnitude of the diffusiophoretic velocity. An invicid liquid drop has a magnitude that is about three times higher than that of the corresponding rigid ones, whereas a liquid drop with a viscosity similar to that of the electrolyte solution has a magnitude that is about two times higher. (ii) If the surface potential of charged liquid drops is high and the double-layer thickness is finite, then the effect of the double-layer polarization cannot be neglected. The velocity of liquid drops exhibits a local maximum as a result. (iii) In general, the magnitude of the
12462 J. Phys. Chem. C, Vol. 112, No. 32, 2008 diffusiophoretic velocity declines with the volume fraction of liquid drops as a result of the hindrance effect and the electrostatic interaction from neighboring drops. (iv) The effect of the double-layer overlapping gets even more significant in a very concentrated dispersion. It may even offset the effect of the double-layer polarization to at least a certain degree. (v) As liquid drops move in the solution, the surface tension gradient induced by the surfactant concentration difference on the droplet surface cannot be neglected if the viscosity of the droplet fluid is small. The higher the surface tension gradient, the slower the liquid drops move. References and Notes (1) Smith, R. E.; Prieve, D. C. Chem. Eng. Sci. 1982, 37, 1213. (2) Korotkova, A. A.; Deryagin, B. V. Colloid J. 1991, 53, 719. (3) Lin, M. J.; Prieve, D. C. J. Colloid Interface Sci. 1983, 95, 327. (4) Ebel, J. P.; Anderson, J. L.; Prieve, D. C. Langmuir 1988, 4, 396. (5) Prodi, F.; Santachiara, G.; Cornetti, C. Aerosol Sci. 2002, 33, 181. (6) Mun˜oz-Cobo, J. L.; Pen˜a, J.; Herranz, L. E.; Pe´rez-Navarro, A. Nucl. Eng. Des. 2005, 235, 1225. (7) Prodi, F.; Santachiara, G.; Travaini, S.; Vedernikov, A.; Dubois, F.; Minetti, C.; Legros, J. C. Atmos. Res. 2006, 82, 183. (8) Deryagin, B. V.; Dukhin, S. S.; Korotkova, A. A. Kolloidn. Zh. 1961, 23, 53. (9) Dukhin, S. S., and Deryagin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (10) Deryagin, B. V.; Dukhin, S. S.; Korotkova, A. A. Colloid J. 1978, 40, 531. (11) Korotkova, A. A.; Churaev, N. V.; Deryagin, B. V. Colloid J. 1990, 52, 176. (12) Anderson, J. L. PhysicoChem. Hydrodyn. 1980, 1, 51. (13) Lechnick, W. J.; Joseph, J. A. J. Colloid Interface Sci. 1984, 102, 71. (14) Lechnick, W. J.; Joseph, J. A. J. Colloid Interface Sci. 1985, 104, 456. (15) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287. (16) Pawar, Y.; Solomentsev, Y. E.; Anderson, J. L. J. Colloid Interface Sci. 1993, 155, 488. (17) Dukhin, S. S. AdV. Colloid Interface Sci. 1995, 61, 17. (18) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (19) Keh, H. J.; Hsu, J. H. J. Colloid Interface Sci. 2000, 221, 210. (20) Wei, Y. K.; Keh, H. J. Langmuir 2001, 17, 1437.
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