Diffusiophoresis of a charged, rigid sphere in a Carreau fluid

Journal of Colloid and Interface Science 465 (2016) 54–57

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Short Communication

Diffusiophoresis of a charged, rigid sphere in a Carreau fluid Shiojenn Tseng a, Chun-Yuan Su b, Jyh-Ping Hsu b,⇑ a b

Department of Mathematics, Tamkang University, New Taipei 25137, Taiwan Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history: Received 22 October 2015 Revised 18 November 2015 Accepted 20 November 2015 Available online 22 November 2015 Keywords: Diffusiophoresis Charged Rigid sphere Carreau fluids

a b s t r a c t Since non-Newtonian fluid behavior are not uncommon in practice, especially in modern applications of colloid and interface science, assessment of how serious is the deviation of the existing results for Newtonian fluids due to fluid nature is highly desirable and necessary. Here, we extend previous analyses for the diffusiophoresis of a particle in a Newtonian fluid to that in a non-Newtonian fluid choosing Carreau fluids as an example. Results gathered reveal that due to the shear-thinning property of the fluid considered, the difference between the particle mobility in a Carreau fluid and that in the corresponding Newtonian fluid can be on the order of 100%. In addition, this difference has a local minimum as the thickness of double layer varies. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Due to its potential application serving as an auxiliary mechanism in applied electric field based sensing technique [1], diffusiophoresis, the moving of particles driven by an applied concentration gradient, has drawn the attention of researchers in various fields recently. Although the theoretical efforts made on analyzing this mechanism are ample in the literature, most of them ⇑ Corresponding author. E-mail address: [email protected] (J.-P. Hsu). http://dx.doi.org/10.1016/j.jcis.2015.11.049 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.

focused on the case where the liquid phase is Newtonian (see e.g., [2–10]). In practice, non-Newtonian fluids are ubiquitous in various areas of colloidal science [11–15]. Non-Newtonian fluids have also been proposed to model the electrokinetic flow in micro- or nanoscaled devices, and to interpret relevant experimentally observed phenomena. Das and Chakraborty [16] and Chakraborty [17], for example, adopted a power-law model to simulate applied electric field or pressure gradient driven blood flow in a capillary and/or microchannel. Zhao and coworkers [18–21] conducted a series studies on deriving analytical solutions for the electroosmotic flow

S. Tseng et al. / Journal of Colloid and Interface Science 465 (2016) 54–57

of power-law fluids in simple geometries. Huang [22] examined the diffusioosmotic flow of a power-law fluid near a large, charged flat plate. Assuming thin double layers, an analytical solution is derived for the flow velocity of both dilatant and pseudo plastic fluids. Huang and Yao [23] modeled the diffusioosmotic flow of a viscoelastic Phan-Thien-Tanner fluid in a planar slit microchannel for an arbitrary level of surface potential. They concluded that both the strength of the flow velocity and that of volume flow rate increase with increasing Deborah number, regardless of their directions. Considering a finite electrical double layer thickness, Chang et al. [24] analyzed the diffusioosmotic flow of an electrolytic power-law fluid in a parallel plate microchannel. They found that, relative to a Newtonian fluid, the rheology of dilatant (pseudoplastic) fluid enlarges (reduces) the parametric regime in which flow rate directs towards downstream. This behavior is opposite to that for the case when double layer is thin. Although the diffusioosmotic flow in a non-Newtonian fluid has been studied, particle diffusiophoresis in that fluid has not in the literature. In recent experimental studies of particle electrophoresis in nonNewtonian fluids, Lu et al. [25,26] demonstrated that a particle phoretic motion in non-Newtoian fluids can be different significantly from that in Newtonian fluids. Considering the diffusiophoresis of an isolated particle in a symmetric salt solution subject to an applied uniform salt gradient, Deryagin et al. [27] derived the expression below for the diffusiophoretic velocity U:


 ef kB T rn0 D2  D1 ln cosh
f U¼ þ
f g Ze n0 D2 þ D1

ð1Þ

e; f; g, kB, T, Z, e, n0, and D1 (D2), are the fluid permittivity, the particle surface potential, the fluid viscosity, Boltzmann constant, the absolute temperature, the valence of ionic species, the elementary charge, the bulk salt concentration, and the cation (anion) diffusivity, respectively. r and
f ¼ Zef=4kB T are the gradient operator and the scaled surface potential, respectively. Eq. (1) suggests that the smaller the g the larger the U. Although this qualitative trend is obvious, a quantitative assessment on the influence of fluid nature on the diffusiophoretic behavior of a particle is both desirable and necessary from a practical point of view. This is done in the present study by extending Eq. (1), which is valid for low surface potentials and thin double layers, to the case of arbitrary surface potential and double layer thickness using Carreau fluid as an example. Among proposed non-Newtonian fluid models, Carreau fluid is a concise one capable of describing power-law fluids and recovering Newtonian fluid as the limiting case [28–31]. 2. Theory Let us consider the diffusiophoresis of an isolated, rigid, nonconducting, spherical particle of radius a and surface Xp subject to an applied salt gradient rn0 . The cylindrical coordinates r, h, z are adopted with the origin at the center of the particle, and rn0 is in the z direction. Suppose that the fluid phase is an incompressible Carreau fluid with its viscosity g expressed as [32,33] n1 g ¼ ½1 þ ðac_ Þ 2 g0

ð2Þ

g0 is the viscosity at zero shear rate, a the relaxation time constant, c_ the shear rate, and n the power-law index. Eq. (2) implies that if n ? 1 and/or a ? 0, the present Carreau fluid reduces to a Newtonian fluid; if a is sufficiently large, it simulates a powerlaw fluid.

55

Let /, v, and p be the electric potential, the fluid velocity, and the pressure, respectively. Then assuming that the system is at a pseudo steady state yields

D/ ¼ 

N X q zj enj ¼ e e j¼1

ð3Þ


 zj enj r/ Jj ¼ nj v  Dj rnj þ kB T

ð4Þ

r  Jj ¼ 0

ð5Þ

rv ¼0  rp þ r  s  qr/ ¼ 0

s ¼ gðc_ Þc_

ð6Þ ð7Þ ð8Þ

D and r are the Laplace operator and the gradient operator, respectively; s; c_ , q, e, e, kB, and T are the fluid shear stress tensor, the rate of deformation tensor, the space charge density of mobile ions, the fluid permittivity, the elementary charge, Boltzmann constant, and the absolute temperature, respectively. Jj, zj, Dj, and nj are the ionic flux, the valance, the diffusivity, and the number concentration of ion species j, respectively. We assume that the particle surface is non-slip, and maintained at a constant charge density r. The electric, flow, and concentration fields at a point far away from the particle are all uninfluenced by its presence. Therefore, the following conditions apply:

n  er/ ¼ r on Xp

v ¼ 0 on Xp

n  Jj ¼ 0 on Xp

ð9Þ ð10Þ ð11Þ

v ¼ Up ez far away from particle

ð12Þ

/ ¼ 0 far away from particle

ð13Þ

p ¼ 0 far away from particle

ð14Þ

C ¼ C high as z ! 1

ð15Þ

C ¼ C low z ! 1

ð16Þ

Here, n is the unit outer normal and ez is the unit vector in the z direction. In our case, the forces acting on the particle include the electrical force FE and the hydrodynamic force FD. The z components of these forces, FE and FD, respectively, can be evaluated by [34,35]

ZZ

FE ¼

Xp

ZZ FD ¼

Xp

ðrE  nÞ  ez dXp

ð17Þ

ðrD  nÞ  ez dXp

ð18Þ

rE and rD are the Maxwell stress tensor and the hydrodynamic stress tensor, respectively. Similar to the treatment of O’Brien and White [36], we partition the present problem into two sub-problems. In the first sub-problem the particle moves at a constant velocity in the absence of rn0 , and rn0 is applied but the particle is remained fixed in the second sub-problem. If we let Fi = FEi + FDi be the total force acting on the particle in the z direction in sub-problem i (i = 1, 2), then F 1 ¼ v1 U and F 2 ¼ v2 rn0 , where v1 and v2 are proportional constants. Because F1 + F2 = 0 at steady state, we obtain U¼

v2 rn v1 0

ð19Þ

3. Solution procedure The solution procedure can be summarized as following. (i) Solve Eqs. (3)–(8) subject to boundary conditions Eqs. (9)–(16). (ii) FE and FD are evaluated by Eqs. (17) and (18). (iii) Calculate v1 and v2 . (iv) Evaluate U by Eq. (19).

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S. Tseng et al. / Journal of Colloid and Interface Science 465 (2016) 54–57

Typically, using a total number of 170,000 mesh elements is sufficient for the flow field, and 300,000 for the electric and concentration fields. To examine the diffusiophoretic behavior of a particle under various conditions, numerical simulation is conducted through varying the values of the key parameters, n and a in Eq. (2). For illustration, we consider a n-heptane dispersion at 1 atm and 20 °C with g0 ¼ 4:21  104 Pa s, e ¼ 1:77  1011 F/m, and q ¼ 682 kg/m3 [37,38]. In addition, a = 10 nm. For convenience, we define the scaled mobility U  ¼ UU with U ref ¼ ern0nðk0eBgT=z1 eÞ being ref

0

a reference velocity. In addition, the percentage difference PD is defined as

PD ¼

U Carreau  U Newtonian  100% U Newtonian

ð20Þ

PD measures the difference between the particle mobility in a Carreau fluid and that in the corresponding Newtonian fluid. 4. Results and discussion

Fig. 1. Variation of the scaled particle mobility U⁄ (solid curves) and percentage difference PD (dashed curves) with scaled double layer thickness ja for various values of n at a = 5 (a), and that for various values of a at n = 0.8 (b).

The set of nonlinear, coupled equations, Eqs. (3)–(8), is solved numerically by COMSOL MultiPhysics (version 4.3a, www.comsol.com) operated in a high performance cluster. To ensure that the results obtained are reliable, mesh independence is checked.

As seen in Fig. 1(a), U⁄(Carreau fluid) is larger than U⁄(Newtonian fluid), and the percentage difference between the two, PD, increases with decreasing n. This is because the smaller the n the more serious the deviation of the Carreau fluid from the corresponding Newtonian fluid, or smaller the fluid viscosity. Note that PD shows a local minimum as ja varies. If ja exceeds ca. unity, PD increases monotonically with increasing ja, and can be on the order of 100%, implying the fluid nature can be crucial. Fig. 1 also reveals that U⁄ has a local maximum as ja varies. This is also observed in the case of a Newtonian fluid [39,40], and can be explained by that if ja is small (double layer is thick), the mobility of a particle is dominated by type I double layer polarization (DLP), the uneven distribution of ionic species inside double layer [41], driving it towards the high-concentration side (U⁄ > 0). As ja increases, so is the significance of type I DLP, and therefore, U⁄ increases with increasing ja. However, if ja is sufficiently large, because double layer is thin enough, so that type II DLP, the uneven distribution of ionic species immediately outside double layer [41], becomes significant and the effect of type I DLP is lessened. Note that if ja exceeds ca. unity, PD increases monotonically with ja, implying

Fig. 2. Contours of the fluid viscosity (Pa s) for n = 0.4 (a), 0.6 (b), and 0.8 (c), at a = 5 and ja = 3.25.

S. Tseng et al. / Journal of Colloid and Interface Science 465 (2016) 54–57

that the higher the salt concentration the more significant the shear-thinning effect of the Carreau fluid. The behaviors of the results presented in Fig. 1(b) are similar to those in Fig. 1(a), and can be explained by the same reasoning. Fig. 2 illustrates the simulated fluid viscosity near a particle for various values of parameter n. According to Eq. (2), the smaller the value of n the less viscous the fluid phase, as is seen clearly in Fig. 2. Because the closer the fluid to the particle surface the faster its velocity and, therefore, the smaller its viscosity. In summary, we modeled for the first time the diffusiophoresis of a rigid particle in a non-Newtonian fluid, taking Carreau fluids as an example. Due to the shear-thinning nature of this type of fluid, the diffusiophoretic velocity of the particle is faster than that in the corresponding Newtonian fluid, and the difference between the two can be on the order of 100%. In addition, this difference has a local minimum as the thickness of double layer varies. Since non-Newtonian fluid behavior are not uncommon in practice, especially in modern applications of colloid and interface science, further assessment of how serious is the deviation of the existing results for Newtonian fluids due to fluid nature is highly desirable and necessary. Acknowledgment This work is supported by the Ministry of Science and Technology, Republic of China. References [1] M. Wanunu, W. Morrison, Y. Rabin, A.Y. Grosberg, A. Meller, Nat. Nanotechnol. 5 (2010) 160–165. [2] S.S. Dukhin, B.V. Deryagin, Surface and Colloid Science, vol. 7, Wiley, New York, 1974. [3] J.L. Anderson, M.E. Lowell, D.C. Prieve, J. Fluid Mech. 117 (1982) 107–121. [4] J.L. Anderson, D.C. Prieve, Sep. Purif. Meth. 13 (1984) 67–103. [5] Y. Pawar, Y.E. Solomentsev, J.L. Anderson, J. Colloid Interface Sci. 155 (1993) 488–498. [6] D.C. Prieve, R. Roman, J. Chem. Soc. Faraday Trans. 2 (83) (1987) 1287–1306. [7] J.P. Hsu, J. Lou, Y.Y. He, E. Lee, J. Phys. Chem. B 111 (2007) 2533–2539.

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