Diffusiophoresis of concentrated suspensions of spherical particles with
identical ionic diffusion velocities
James Lou, Yan-Ying He, Eric Lee
∗Department of Chemical Engineering, National Taiwan University, Taiwan 10617, Taipei Received 15 November 2005; accepted 20 January 2006
Available online 9 March 2006
Abstract
Diffusiophoresis of concentrated suspensions of spherical particles subject to a small electrolyte gradient is analyzed theoretically at arbitrary levels of zeta potential and double-layer thickness. The Kuwabara unit cell model is adopted to describe the system under consideration. The effect of double-layer polarization is taken into account. It is found that the diffusiophoretic mobility exhibits a local maximum as well as a local minimum with varying zeta potential or double-layer thickness, similar to the corresponding dilute dispersion. The direction of the particle movement may even change back and forth. The previous low-zeta-potential approach is found to significantly overestimate the diffusiophoretic mobility as the zeta potential goes high. The deviation may be several fold sometimes. The effect of the volume fraction ratio of colloids is also examined. The higher the ratio, the lower the mobility.
©2006 Elsevier Inc. All rights reserved.
Keywords: Diffusiophoresis; Polarization effect; Kuwabara unit cell model; Arbitrary zeta potential
1. Introduction
The electrokinetic behavior of colloidal particles dispersed in a continuous medium is of great practical interest. When an external electric field is applied to a uniform electrolyte solution containing charged colloidal particles, the resulting mi-gration of colloidal particles is called “electrophoresis”[1,2]. This well-known phenomenon has been treated extensively in the past literature[3–9]. However, the corresponding “diffusio-phoresis” has received very limited attention so far. If the con-centration of electrolyte solute is somehow originally nonuni-form in the solution, an induced electric force will arise, due to this macroscopic concentration gradient of electrolyte solute. Hence the colloidal particles will migrate nonetheless, even though the applied electric field is absent.
Deryagin et al.[10,11]first noticed this electrokinetic phe-nomenon and proposed the diffusiophoresis mechanism to ex-plain it. They found that diffusiophoresis was the underlying mechanism for industrial processes such as ionic deposition
* Corresponding author.
E-mail address:[email protected](E. Lee).
[10–12]. When a solid shape containing a salt was immersed in a latex solution, the growth rate of the latex film on the solid shape would be comparable to that from the electrodepo-sition process, where hundreds of volts are required. In other words, the new process was able to achieve about the same growth rate of latex films on a substrate without the applica-tion of any electric field[11,12]. A corresponding theoretical analysis in electrolyte solution was carried out by Dukhin and Deryagin[10,13]and experimentally verified by Korotkova and co-workers [10]. They proposed that the diffusiophoretic ve-locity of a charged particle subject to a constant concentration gradient∇n∞in an unbounded solution of a symmetrically bi-nary electrolyte is (1) U∗=εζ μ kBT ze ∇n∞ n0 β+ ζ−1ln(cosh ζ ),
where ζ = zeζ/4kBT, ze is the charge carried by a single ion, kB is the Boltzmann constant, T is the temperature, ζ is the zeta potential of the particle surface, ε is the fluid permittivity, μ is the fluid viscosity, and n0 is the macroscopic electrolyte concentration measured without the particle and concentration gradient. Note that β= (D1− D2)/(D1+ D2), where D1and D2are the diffusion coefficients for cations and anions, respec-0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
tively, is a dimensionless index indicating the relative diffusion velocities of cations and anions. β is an experimentally mea-surable property of the specific electrolyte solution under con-sideration. For example, β= 0 for KCl, β = −0.2 for NaCl, and β= 0.64 for HCl. Note also that β = 0 implies identical diffusion velocities of anions and cations. Based on the Gouy– Chapman model[1,2], Anderson[14]was able to predict the diffusiophoretic mobility of a charged spherical particle when the double-layer thickness and the zeta potential were small. He found that the diffusiophoretic mobility in an electrolyte solution was determined essentially by three factors: the zeta potential of the particle surface, the relative diffusion velocities of ions (β), and fluid properties such as viscosity, tempera-ture, etc. Prieve and co-workers[15–18] further loosened the restriction of low zeta potential in their theoretical analyses and observed strong evidence supporting their proposed diffu-siophoretic mechanism with experimental data. Concentration gradients of electrolytes were established and measured in their experimental setup, making use of a porous membrane [18]. Lechnick and Joseph [19,20]further extended Deryagin’s re-sults to ternary-component systems and constructed ternary dif-fusion coefficients both theoretically and experimentally. With the aid of the Maxwell–Stefan equation, they concluded that the Stokes–Einstein theory was not suitable to predict the bi-nary diffusion coefficient for a charged particle in electrolyte solution. Denisov and Zharkikh [21] solved the correspond-ing problem with arbitrary double-layer thickness. Afterwards Prieve and Roman [22] calculated diffusiophoretic mobility with arbitrary double-layer thickness, using the same numer-ical method as employed by O’Brien and White [6]. In their study, they compared the difference between cases where β= 0 (KCl) and β= 0 (NaCl). They also concluded that the diffu-siophoretic motion of colloidal particles cannot be totally pre-dicted by thermodynamic forces in electrolyte solution [23]. Dukhin [24] provided a thorough physical analysis later on. He figured that the external concentration had also induced a concentration and electrical dipole moment. This nonequilib-rium ionic distribution might play an important role in diffu-siophoresis. Baygents and Saville[25]studied numerically the diffusiophoresis of a droplet or a small bubble suspended in an electrolyte solution with the finite element method. Misra et al.[26]explored the effect of the soft layer on charged col-loids in diffusiophoresis. Following a similar treatment, Kos-mulski and Matijevic[27]proposed another mechanism named “solvophoresis.” They found that different solvents might have similar results even though the solutes were the same.
From this literature review, we noticed that essentially all the studies on diffusiophoresis, analytical or numerical, were confined to very dilute dispersion systems. When the concen-tration of colloidal particles is high, however, the existence of neighboring particles can no longer be ignored. This boundary effect will be felt by all the colloids in the solution. Hence the appropriate determination of associated boundary conditions on the virtual surface, representing the interface between neighbor-ing particles, is essential for a successful theoretical approach. As a result, theoretical studies on diffusiophoresis have been focusing on systems with boundaries, virtual or physical, in
re-cent years. The existence of a physical boundary is in itself an interesting classic problem. For example, Keh and co-workers
[28,29]considered the diffusiophoresis for a colloidal sphere in a nonelectrolyte or electrolyte solutions near a planar boundary. They showed that the retardation effect produced by a planar boundary might decelerate the motion of the particle. In addi-tion, Wei and Keh [30] and Keh and Wei [31]predicted the diffusiophoretic behavior in a suspension using both the Hap-pel model[32]and the Kuwabara [33]unit-cell model. They showed that the Kuwabara unit-cell model was more appropri-ate than the Happel model for the system under consideration. In recent years, Keh and Ma[34,35]also obtained diffusioos-motic flow near a plane boundary. However, their results were restricted to low surface potential, with no account of the con-vection of ions.
What we present here is a study of the diffusiophoretic be-havior of spherical particles dispersed in an electrolyte solu-tion, with arbitrary surface potentials and arbitrary double-layer thickness. In fact, diffusiophoresis exists not only in electrolyte solutions[36], but also in nonelectrolyte solutions[37]. How-ever, the electrokinetic phenomenon of diffusiophoresis in elec-trolyte solution is much more interesting and complicated. In our analysis, we focus on the situation where the mobility of the two ions is identical in the electrolyte solution (β = 0). Compared with the results in the earlier literature, our study of the diffusiophoresis of concentrated colloidal dispersion takes into account the arbitrary effects of zeta potential, double-layer thickness, and volume fractions of the particle. To solve the re-sulting general electrokinetic equations, which are highly non-linear, and the powerful pseudo-spectral method[8,38–40]is employed in this study.
2. Theory
We consider the diffusiophoretic behavior of concentrated spherical particles of radius a in a solution that contains z1:z2 electrolytes, z1and z2being respectively the valences of cations and anions. The Kuwabara[33]unit-cell model is adopted as the basis for subsequent analysis. Referring toFig. 1, a uniform concentration gradient∇n0is applied to the system in the z di-rection. The radius of each particle is a. Each individual particle is considered as surrounded by a concentric spherical shell of electrolyte solution phase of radius b, and moves upward with a constant velocity U due to the concentration gradient. As a re-sult, it is most convenient to use spherical coordinates (r, θ, φ) in our system. The electroneutrality constraint in the bulk liq-uid phase requires that n20= n10/α, n10 and n20 be the bulk concentrations of cations and anions, and α= −z2/z1.
It is assumed that the physical properties of the liquid phase take their macroscopic values. Also, the shape of the colloidal particle remains spherical when it is migrating in the fluid, which is generally true for low-Reynolds-number situations. It is further assumed that the concentration of solute is only slightly nonuniform over the length scale a; that is, a|∇n0| n0. Under these conditions, the diffusiophoretic motion is gov-erned by the well-known electrokinetic equations, which we shall elaborate as follows.
Fig. 1. Schematic representation of the system under consideration. First we assume that the electrical potential φ of our system under consideration can be described by the Poisson equation,
(2) ∇2φ= −ρ ε = − 2 j=1 zjenj ε ,
where ρ and ε are respectively the space charge density and the permittivity of the solution, e is the elementary charge, and nj and zj are respectively the number concentration and the va-lence of ionic species j . Moreover, the flow field is governed by the Navier–Stokes equation in the creeping flow region, to-gether with an incompressibility constraint,
(3) ∇ · v = 0,
(4) μ∇2v− ∇p − ρ∇φ = 0,
where p and μ are respectively the pressure and the viscosity of the fluid. The ionic concentration nj is governed by the con-servation of ion species,
(5) kT Dj ∇2n j+ zje kT ∇nj· ∇φ + nj∇2φ− ∇nj· v = 0, where Dj is the diffusion coefficient of ion species j , λjis the diffusion coefficient for ion j , and zjeis the charge carried by ion j .
We nondimensionalize the above quantities further to facili-tate subsequent treatments. Let φe∗= φe/ζ, where φemeans the equilibrium electric potential in the solution, with the absence of the applied electric field, r∗= r/a, and n∗j= nj/n10, where ζ represents the zeta potential on the shear plane. The conven-tional standard electrokinetic model is adopted in this study, in that no surface conductivity is considered here. The sur-face conductivity, however, is a potentially important factor for the thorough understanding of the electrokinetic phenomena, as pointed out by Dukhin, Delgado et al.[41,42]. Superscripts on all the quantities indicate corresponding dimensionless forms. If the surface potential is uniform on the colloidal surface, with
the introduction of the convection term in Eq.(5), the ion dis-tribution in the electrolyte solution can be described in a form similar to the Boltzmann distribution,
(6) nj= nj0exp −zje kBT (φe+ δφ + gj) ,
where nj0 is the bulk concentration of the species j , δφ the induced electric potential due to the difference of ion mobilities, and gj a perturbed potential adopted to describe the double-layer polarization. Note that to analyze separately the effects of double-layer polarization and induced electric field, we have decomposed φ into φe, δφ, and gj; that is, φ= φe+ δφ + gj. Note also that Eq.(6)is just a convenient analog, the generality of polarization effect due to the fact that inclusion of convection term in the ionic flux equation, gj, is intact[40]. Therefore, the equilibrium potential can be determined as
(7) ∇2φ∗ e= − (κa)2 (1+ α)φr exp−φrφ∗e− expαφrφ∗e ,
where the inverse Debye length κ and the scaled zeta potential φrare defined respectively by
(8) κ= 2 j=1 nj0(ezj)2/εkBT 1/2 , (9) φr= ζ z1e/kT .
The boundary conditions for φe∗are:
(10) φe∗= 1 at r∗= 1, (11) ∂φe∗ ∂r∗ = 0 at r ∗=b a.
Equation(10)means that the surface potential of the particle at equilibrium remains constant and Eq.(11)implies that the unit cell as a whole is electrically neutral; thus there is no electric current between adjacent cells.
Substituting Eq. (6) into Eq. (2), we obtain the governing equation of induced electric potential from Eqs.(2) and (7)in the dimensionless form
∇∗2δφ∗= − (κa) 2 (1+ α)φr exp−φrφ∗e+ δφ∗+ g∗1 − expαφr φe∗+ δφ∗+ g2∗ (12) + (κa)2 (1+ α)φr exp−φrφe∗− expαφrφe∗ , and the corresponding boundary conditions are:
(13) ∂δφ∗ ∂r∗ = 0 at r ∗= 1, (14) δφ∗= 0 at r∗=b a,
where δφ∗= δφ/ζ . The first condition represents that it is di-electric inside the colloidal particle and the other one implies that the net flux for cations and anions is zero across the outer virtual cell. These are also the boundary conditions adopted by Keh and Wei[31]. To simplify the system and focus on issues
of major interest, we consider the situation where β= 0 for the time being in this paper.
The conservation equation of ions, Eq.(5), is converted into dimensionless form by substituting Eqs.(2) and (6)into it, ∇∗2g∗j− φr∇∗φe∗· ∇∗g∗j − φ2 rPejv∗· ∇∗φ∗e+ ∇∗δφ∗+ ∇∗gj∗ (15) − φr∇∗δφ∗+ ∇∗gj∗· ∇∗g∗j= 0,
where gj∗= gj/ζ and Pej is the corresponding Peclet number of ion j , taking into account convection. We assume that the colloidal particle is impermeable to ions, and there is an ionic concentration gradient across the outer virtual surface of the unit cell. Therefore,
(16) ∂g1∗ ∂r∗ = ∂g2∗ ∂r∗ = 0 at r ∗= 1, (17) ⎧ ⎪ ⎨ ⎪ ⎩ (δφ∗+ g∗1)= −1 φr (∇∗n∗0) (δφ∗+ g∗2)= 1 αφr (∇∗n∗0) ⎫ ⎪ ⎬ ⎪ ⎭ at r ∗=b a.
Taking curl on Eq.(4)and introducing the stream function in the spherical coordinates, we get rid of the continuity equation and obtain E∗4ψ∗= −(κa) 2 1+ α n∗1∂g ∗ 1 ∂r∗ + n ∗ 2 ∂g∗2 ∂r∗ αn∗2∂φ ∗ ∂θ (18) − n∗1∂g ∗ 1 ∂θ∗+ n ∗ 2 ∂g∗2 ∂θ∗ αn∗2∂φ ∗ ∂r∗ sin θ,
where ψ∗is the scaled stream function and E∗4is the operator of E∗2E∗2, which is defined as (19) E∗2= ∂ 2 ∂r∗2+ sin θ r∗2 ∂ ∂θ 1 sin θ ∂ ∂θ . Corresponding boundary conditions are
(20) ψ∗= 0, ∂ψ ∗ ∂r∗ = 0 at r ∗= 1, (21) ψ∗=1 2r ∗2U∗sin2θ, at r∗= b/a, (22) E2ψ∗= 0 at r∗= b/a.
Equation (20) suggests no slip on the particle surface, and Eq.(21)states that the fluid is flowing toward the stationary col-loid with a relative scaled velocity of U∗. Equation(22)states that the virtual surface is curlless, as proposed by the Kuwabara unit-cell model[33].
Having done the mathematical manipulations mentioned above, we end up with governing equations identical to those in electrophoresis, except for the boundary conditions in Eqs.(14) and (17)at the outer virtual surface. As pointed out by Dukhin and Deryagin, diffusiophoresis and electrophoresis have a com-mon base. They are two cases of a more general phenomenon, namely particle movement under the influence of an external electrochemical potential gradient. When an external electric field is present, while an external gradient of concentration is
absent, the movement is called electrophoresis. When there is an external concentration gradient, while an external elec-tric field is absent, the movement is called diffusiophoresis. Moreover, both phenomena are described by the identical elec-trokinetic equation[13]. Note that this concentration gradient in diffusiophoresis can be huge near a surface reaction area at times, as pointed out by Prieve[16]and Smith and Prieve[17], which actually drives the colloids.
The details of the above derivations can be found elsewhere, such as Chu and co-workers[8], among our other previous pub-lications. We now follow the same approach as that of Prieve and Roman[22], who assumed the applied concentration gra-dient to be very weak. Under this condition, the problem can be simplified to a linear system, allowing decomposition into two virtual subproblems[6,22]. In the first problem the spheri-cal particle moves with a velocity in the absence of the applied concentration gradient, where as in the second problem the spherical particle is somehow held stationary when the concen-tration gradient is applied. If the corresponding forces acting on the surfaces of spherical particles for the two problems are F1and F2, respectively, then F1= f1(∇∗n∗0)and F2= f2U∗, where f1and f2are proportional constants[22]. Therefore, the diffusiophoretic mobility Um∗ can be written as
(23) Um∗=U ∗ E∗z = f1 f2.
Note that both F1 and F2 are computed given the values of ∇∗n∗
0 and U∗; thus the constants f1 and f2 are determined straightforwardly, and Um∗ can be calculated via Eq.(23).
3. Results and discussion
A pseudo-spectral method based on the Chebyshev polyno-mial[38,39]is adopted for the solution of the governing equa-tions, subject to the associated boundary conditions. Details of the pseudo-spectral method employed in analyzing electroki-netic phenomena can be found elsewhere, such as Lee and co-workers[8,40]. It proves to be a very powerful and suitable method for the fields of interest.
Fig. 2shows the calculated normalized diffusiophoretic mo-bilities as a function of surface potential φrwhen β= 0, α = 1, and Pe1= Pe2= 0.26 (KCl). The characteristic value of diffu-siophoretic mobility is chosen to be
(24) U0= ε μa kT z1e 2 ∇∗n∗0.
Note that we use the same characteristic value as Keh and Wei in 2002[31]. Compared with the results of Wei and Keh[30]
and Keh and Wei[31]for κa= 1 and 10 (dashed line inFig. 2), which is the limiting case of ours at low zeta potential and very thin double layer, we reproduce their results exactly when φr is low enough, indicating the accuracy of our results. As φr increases, however, deviation between our results and theirs in-creases, indicating that the low zeta potential approach runs into problems for|φr| higher than 1 approximately. This observation is consistent with our general conclusion in the previous study of electrophoresis[8,40].
Fig. 2. Variation of scaled diffusiophoretic mobility (U∗/U0) as a function of φrat various values of κa when ϕ= 0.1, β = 0, and α = 1. Dashed line,
low-surface-potential case.
We also find that the mobilities for all κa are even func-tions of φr, which is exactly the same observation as for Prieve and Roman[22], Keh and Wei[31], or Wei and Keh[43]. Ac-cording to their analysis, diffusiophoresis of the particle comes solely from chemiphoresis since the diffusion velocities are identical for cations and anions (β= 0). The term “chemiphore-sis” implies that the motion of a charged particle is due to the nonuniform adsorption of counterions within the electri-cal double layer [31,43]. It reflects the governing equations and corresponding boundary conditions in two ways: the con-tribution from the disturbed electric potential, δφ∗, is zero in Eq.(12), and the Peclet numbers are identical in Eq.(15)(here we set Pe1= Pe2= 0.26, that is, a KCl solution). Keh’s study focused on low zeta potential, whereas Prieve’s focused on di-lute colloidal suspensions. In the present paper, we extend it to concentrated colloidal dispersions and conclude that it is always an even function of φrat arbitrary κa, whether for di-lute or concentrated dispersions. Most important of all, we find that the diffusiophoretic mobility does not increase all the way with increasing φr. It actually reaches a local maximum first, and then decreases. In other words, when φrexceeds a certain threshold value, the mobility of the colloid no longer increases monotonously with it. This is contrary to the reports made be-fore by Keh and Wei [31], where they studied the limiting case of low zeta potential and thin double layer. Under these assumptions, they claimed that mobility increased monoton-ically with φr (β= 0). We compared with their results and found that the low-zeta-potential approach, which linearized the Poisson–Boltzmann equation, is applicable only for|φr| < 1. Once above this range, the deviation becomes more and more significant. For example, for|φr| = 3 and κa = 10, we find that their predictions extrapolated beyond|φr| = 2 in their paper can be as high as 40% overestimated, compared with the actual mo-bility based on our general approach, which allows arbitrarily
Fig. 3. Variation of scaled diffusiophoretic mobility (U∗/U0) as a function of κaat various values of φrwhen ϕ= 0.125, β = 0, and α = 1.
high values of zeta potential. At extremely cases as|φr| 5, the deviation may even be severalfold.
However, the diffusiophoretic mobility becomes somewhat irregular between κa= 3 and 5 inFig. 2. To explain what hap-pens there, we use κa as abscissa in Fig. 3for convenience. In Fig. 3, we see similar behavior over a certain range of κa. Briefly speaking, higher κa results in greater mobility of the particle because higher κa means higher ion concentration nj0, according to Eq.(8). In other words, as κa increases, the amount of electrolytes increases accordingly in the vicinity of the colloid surface; hence the electric force exerted upon it in-creases simultaneously. However, Malkin, Korotkova, and their collaborators [44,45]showed that the rate of diffusiophoresis of polystyrene latex particles decreases and then changes di-rection with the increase of electrolyte concentration. This was the first time that the inversion of diffusiophoresis direction was predicted. They also suggested that concentration polar-ization of the double layer strongly influences the direction of particle movement[44]. In our study, we also find the appear-ance of both a local maximum and minimum over some range of κa. This phenomenon, analyzed in depth by Dukhin [24], may arise from the polarization of the double layer surround-ing the particle, which induces a microscopic electroosmosis slip opposing the normal diffusiophoretic motion. Although the gradient of potential increases due to the reduction of double-layer thickness (larger κa), which implies a stronger driving force for particle motion, the opposing electric force becomes stronger also. Comparing the variation of electric force FE∗ for various φr, as shown inFig. 4, we find that the variations of FE∗ and mobility with κa have similar trends[8]. According to this analysis, the ultimate behavior of the diffusiophoretic mobility is mainly due to the dominant effect of FE∗. Lee et al.[8]also reported this result in their study of electrophoresis. However, the concentration polarization of the double layer here produces a coupled mass and charge flow near the surface of the particle. This makes the motion of the particle much more complicated
Fig. 4. Variation of scaled electric force FE∗on the particle surface as a function of φrat various values of κa when ϕ= 0.125, β = 0, and α = 1.
at high zeta potential or medium double-layer thickness. There-fore, even taking no consideration of the effect of discrepant dif-fusion speeds for cations and anions, it is still possible that the movement of the colloids may change direction more than once, as we actually observe from our calculation results. The cause of this phenomenon is the polarization effect, as we have elabo-rated above. Moreover, note that the mobilities for each κa are all symmetric functions of φrinFig. 3. If the diffusion veloci-ties are different for cations and anions, i.e., β= 0, an induced electroosmotic flow will appear near the particle surface. The resulting velocity direction of the colloid is much more compli-cated, as it is determined by the competing chemiphoretic and electrophoretic components under the circumstances[24].
We also plot equilibrium potential φe∗distribution at various surface potentials φrinFigs. 5 and 6. We find qualitatively sim-ilar contours between Figs. 5 and 6. The major difference is the quantitative value of φ∗e. On the other hand, a comparison between cases of thick (κa= 0.01) and thin (κa = 7.3) double-layer for contour plots of φ∗e is also shown inFig. 5. InFig. 5, it is obvious that the distribution for φe∗is a strong function of double-layer thickness. The electric double-layer represents the range of surface potential that can affect the ion distribution. When the thickness of double layer becomes greater than the radius of virtual surface, indicated by κa smaller than 1, the double-layer overlapping becomes significant. Under the same value of zeta potential, we find a steeper gradient of φe∗ near the particle surface in Fig. 5b, where κa is large, compared with Fig. 5a, where κa is small. Therefore, the correspond-ing mobility in Fig. 5and the electrical body force in Fig. 4
of κa= 7.3 are larger than those at κa = 0.01. This result also implies that κa has a greater influence on the distribution of φe∗than φrdoes. The corresponding comparison for the scaled counterion distribution n∗2is shown inFig. 7. When κa= 0.01, a lot of counterions are attracted into the cell, by the particle surface potential, resulting in a large area of double layer. At small κa, we observe that the distribution of n∗2forms
concen-(a)
(b)
Fig. 5. Contours of scaled equilibrium potential φe∗at two levels of κa when
ϕ= 0.1, β = 0, φr= 3.0, and α = 1: (a) κa = 0.01, (b) κa = 7.3.
tric shells around the spherical particle. The larger the value of n∗2, the closer the contour is to the particle surface. With the increase of κa, the contour of n∗2 distorts slightly toward the side of low ion concentration. If we compare the value of n∗2 at (0.7,−0.75) and (0.7, 0.75) in Fig. 7b, we find a small deviation of ionic concentration within the double layer. It is 0.984072 at (0.7,−0.75) but 0.983046 at (0.7, 0.75). As we have explained earlier, this is because the polarization effect reaches its peak at medium κa, since the double-layer overlap-ping eventually disappears. The different value of n∗2represents the possibility that the profile of double layer may be a little distorted toward the opposite direction from the particle
move-(a)
(b)
Fig. 6. Contours of scaled equilibrium potential φe∗at two levels of κa when ϕ= 0.1, β = 0, φr= 5.0, and α = 1: (a) κa = 0.01, (b) κa = 7.3.
ment. This nonuniform distribution of ions induces an electric force retarding particle movement[8]. Furthermore, if we set φr= 3 on the particle surface, more anions accumulate near the particle surface due to the electrostatic force. Consequently, the value of n∗2 is always greater than zero within the double layer. For completeness, corresponding streamlines around the particle are also shown inFig. 8. It is very interesting to note that when κa= 0.01, the mobility of colloidal particles and the velocity of the fluid are lower than for κa= 7.3. There-fore, the change of the value of the stream function is smaller inFig. 8a than inFig. 8b, indicating that the particle moves in
(a)
(b)
Fig. 7. Contours of n∗2at two levels of κa when ϕ= 0.1, β = 0, φr= 3.0, and
α= 1: (a) κa = 0.01, (b) κa = 7.3.
the same direction of∇∗n∗0 relative to the fluid on the virtual surface.
Finally, the influence of the particle concentration, measured by the volume fraction ϕ= (a/b)3of the particle in the cell, on the diffusiophoretic mobility in general is presented inFigs. 9 and 10.Fig. 9 shows the variation of scaled diffusiophoretic mobility as a function of volume fraction ϕ at various φr, for a representative case of κa= 1.0. Clearly the mobility decreases accordingly when the dispersion gets more and more trated. This is fully expected because the higher the concen-tration of particles, the more significant the interaction among
(a)
(b)
Fig. 8. Contours of stream function at two levels of κa when ϕ= 0.1, β = 0, φr= 3.0, and α = 1: (a) κa = 0.01, (b) κa = 7.3.
particles in the neighborhood, and thus the more important the steric hindrance for the fluid flow during diffusiophoresis. Here the steric hindrance refers to the resistance of fluid motion ow-ing to the existence of neighborow-ing particles. Surely it is difficult to swim fast in a crowded swimming pool. In addition, we also find that the steric hindrance delays the dominance of the po-larization effect on the diffusiophoretic mobility, as shown in
Figs. 9 and 10, where the occurrence of local maximum shifts to the right side along the φr or κa axis. The shift may be so profound that the local maximum, caused by the polariza-tion effect explained earlier, is pushed beyond the scope of the
Fig. 9. Variation of scaled diffusiophoretic mobility (U∗/U0) as a function of φrat various values of ϕ when ka= 1.0, β = 0, and α = 1.
Fig. 10. Variation of scaled diffusiophoretic mobility (U∗/U0) as a function of κaat various values of ϕ when φr= 4.0, β = 0, and α = 1.
figure. Though the specific case presented is for φr= 4.0 in
Fig. 10, this delay or shift-to-the-right behavior is typical in our other calculations. This is again due to the crowd of the par-ticles when the volume fraction is increased. When the double layer around each particle is thick (small κa), the overlapping is very profound at concentrated dispersions (large volume frac-tion). Therefore, the polarization effects are somewhat offset by one another, making the system shift toward the uniform phase.
4. Conclusions
We investigated here diffusiophoresis phenomena in con-centrated colloidal dispersions at arbitrary zeta potentials and double-layer thicknesses. When the diffusion coefficients of
cations and anions in the electrolyte solution are identical, each other, the diffusiophoretic mobility is found to exhibit both a lo-cal maximum and a lolo-cal minimum with increasing φrand κa, due to the impact of polarization effect on the electric force ex-erted on the particle. This is similar to the corresponding case of dilute colloidal suspensions at high zeta potential reported in Refs. [22,24]. The previous simplified model for low zeta potential and very thin double layer is acceptable only when |φr| < 1, and it overestimated the actual magnitude of mobil-ity[31]. The deviation gets more and more severe as φror κa increases. Moreover, the particle velocity may not even move in the same direction as that of the concentration gradient. In ad-dition, we compared the equilibrium electrical potentials and ionic concentration difference at various double layer thick-ness. We showed that the ion distribution around the particle, which determines the diffusiophoretic mobility, is affected by factors such as the double-layer overlapping, ion concentration gradient, and polarization effect. Finally, we find that the influ-ence of steric hindrance on potential distribution as well as fluid flow must be taken into account as the colloidal dispersions be-come more and more concentrated. Overall, we conclude that the polarization effect by far is the most important factor in the determination of the colloid mobility.
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