Importance of Temperature Effect on the Electrophoretic Behavior of

ARTICLE pubs.acs.org/Langmuir

Importance of Temperature Effect on the Electrophoretic Behavior of Charge-Regulated Particles Jyh-Ping Hsu,* Yi-Hsuan Tai, and Li-Hsien Yeh Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617

Shiojenn Tseng Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137 ABSTRACT: The Joule heating effect is inevitable in electrophoresis operations. To assess its influence on the performance of electrophoresis, we consider the case of a charge-regulated particle in a solution containing multiple ionic species at temperatures ranging from 298 to 308 K. Using an aqueous SiO2 dispersion as an example, we show that an increase in the temperature leads to a decrease in both the dielectric constant and the viscosity of the liquid phase, and an increase in both the diffusivity of ions and the particle surface potential. For a particle having a constant surface potential, its electrophoretic mobility is most influenced by the variation in the liquid viscosity as the temperature varies, but for a charged-regulated particle both the liquid viscosity and the surface potential can play an important role. Depending upon the level of pH, the degree of increase in the mobility can be on the order of 40% for a 5 K increase in the temperature. The presence of double-layer polarization, which is significant when the surface potential is sufficiently high, has the effect of inhibiting that increase in the mobility. This implies that the influence of the temperature on the mobility of the particle is most significant when the pH is close to the point of zero charge.

’ INTRODUCTION Electrophoresis, the movement of charged particles in a liquid phase containing ionic species driven by an applied electric field,1 has been applied widely both in fundamental studies1 and in industrial operations;2 both experimental and theoretical results are ample in the literature. In practice, electrophoresis operations are almost always accompanied by the Joule heating effect,3
5 which can play a significant role because the physical properties of both the dispersed phase and the dispersion medium are usually temperature dependent. Evenhuis and Haddad6 pointed out that the heat dissipated during electrophoresis depends upon the strength of the applied electrical field, and the length and the radius of an electrophoresis device.6 Musheev et al.7 showed that if a capillary is not properly cooled, then its temperature could increase 15
25 °C when the strength of the applied electrical field exceeds 400 V/cm, and even if it is cooled its temperature could still increase ca. 5 °C. Chein et al.8 pointed out that imposing an electrical field yields a temperature difference between the entrance and the center of a channel, which is capable of influencing the performance of the operation.9 Several attempts have been made to assess the influence of temperature on the electrokinetic phenomena occurring in a microor nanoscaled device. In a study of the electrophoresis conducted in microstructured fibers, Rogers et al.10 found that the slope of the current against the electrical field strength curve alters when the latter exceeds ca. 80 kV/m. Seyrek et al.11 examined the r 2011 American Chemical Society

electrophoresis of carboxyl-terminated dendrimers, and observed that the influence of electrical field on its mobility varies with ionic strength; that is, the Joule heating effect depends upon the thickness of the double layer. Wang and Tsao12 analyzed theoretically the temperature effect on capillary electrophoresis. They concluded that the electrophoretic velocity of an entity is proportional to ionic current even if the Joule heating effect is significant. Considering band transport electrophoresis, Tang et al.13 simulated the performance of capillary electrophoresis in the presence of the Joule heating effect. They found that both the shape and the position of the peak of sample concentration curve are influenced by a thermal effect. Xuan and Li14 investigated the influence of the Joule heating effect on the electric current and the separation efficiency. It was concluded that if the applied electrical field is sufficiently strong, then the increase of ionic current with increasing electrical field strength becomes nonlinear. In addition, Joule heating effect is capable of raising the sample velocity; that is, the operation time can be shortened by increasing the temperature. Evenhuis et al.15 investigated experimentally the heating effect on the mobility of inorganic ions. They found that the mobilities of the considered inorganic ions all increase with increasing heat dissipation; however, the rate of increase Received: August 18, 2011 Revised: October 20, 2011 Published: November 29, 2011 1013

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’ THEORY

Figure 1. Electrophoresis of a rigid spherical particle of radius a along the axis of a cylindrical computation domain of radius b subject to an applied uniform electric field E in the z direction. r, θ, z are the cylindrical coordinates adopted with the origin on the axis of the computation domain. ΩP, ΩW, and ΩE are the surface of the particle, the side surface of the computation domain, and the end surface of the computation domain, respectively.

depends upon both the type of inorganic ions and the background materials. Previous analyses on electrophoresis usually are based on the assumption that the surface of a particle is maintained either at a constant electrical potential16 or constant charge density.17 Although this makes relevant analysis simpler, considering a more general and realistic model, charge-regulated surface,18,19 is highly desirable. In fact, both theoretical and experimental results reveal that assuming constant surface potential might yield appreciable deviation.20,21 Note that because the charged conditions of a charge-regulated surface involve dissociation/association reactions, the temperature can also play a role. Up to now, a general electrophoresis analysis which takes account of the temperature-dependent physical properties of the liquid phase and the dispersed particles has not been reported. In this study, the Joule heating effect on the electrophoretic behavior of a particle is modeled by considering the electrophoresis of a charge-regulated sphere at a temperature ranging from 298 to 308 K.11
15 The temperature dependence of the surface potential of the particle, the diffusivity of ions, and the pH and the viscosity of the liquid phase is taken into account for the case where the effect of double-layer polarization22 might be important. Another key issue in practice is the possible presence of multiple ionic species in the liquid phase. For instance, if the solution pH is adjusted by introducing electrolytes such as HCl and NaOH, as is usually done, then, in addition to the background electrolytes, the presence of those electrolytes might be significant, especially when the pH deviates appreciably from 7.23 This issue is almost always overlooked in relevant analyses, for simplicity, and is considered in this study.

As illustrated in Figure 1, we consider the electrophoresis of a rigid sphere of radius a subject to an applied uniform electric field E. For convenience, a cylindrical computation domain of radius b is defined, where E is along its axis. In subsequent analysis, we assume that λ = a/b = 0.1 so that the presence of the boundary of the computation domain can be neglected.23 The present problem is two-dimensional, and therefore, we choose to work on the cylindrical coordinates (r, z) with the origin at the center of the particle. Let ΩP, ΩE, and ΩW be the surface of the particle, the surface of the two ends of the computation domain, and its lateral surface, respectively. We assume the following: (i) The system under consideration is at a pseudosteady state, which has been verified to be valid in a similar electrokinetic problem.24 (ii) The liquid phase is an incompressible Newtonian fluid containing N kinds of ionic species. (iii) The flow field is in the creeping flow regime. (iv) The surface of the particle is nonslip, nonconducting, and impenetrable to ionic species. (v) Both the flow and the electrical fields at a point far away from the particle are uninfluenced by its presence. (vi) The linear size of the particle is much smaller than that of the electrophoresis device so that the boundary effect is negligible, which is usually satisfied if the latter exceeds ca. 10-fold the former. (vii) The dispersion is dilute so that the possible presence of the particle
particle interactions is unimportant. Suppose that the surface of the particle has functional groups AH undergoing the dissociation/association reactions below: AH S A
þ Hþ

ð1Þ

AH þ Hþ S AHþ 2

ð2Þ

A typical example for the type of surface considered includes silica dioxide.25 Let Ka and Kb be the equilibrium constants of the reactions expressed in eqs 1 and 2, respectively, with Ka = NA
[H+]/NAH and Kb = NAH2+/NAH[H+], where NA
, NAH2+, and NAH are the surface densities of A
, AH+2 , and AH, respectively, and [H+] is the concentration of H+. If we let Ntotal be the total number of AH on the particle surface, then Ntotal = NA
+ NAH + NAH2+. Similar to the treatments of O’Brien and White26 and Ohshima,27,28 where E is assumed to be much weaker than the electrical field established by the particle, each dependent variable is partitioned into an equilibrium component and a perturbed component. The former is the value of that variable when E is not applied, and the latter denotes that coming from the application of E. Then the equations governing the present problem and the associated boundary conditions can be summarized as following:
 zj ϕ e k2 ζa N zj nj0 ð3Þ exp
∇2 ϕe ¼
N z2 nj0 z1 ζa j ¼ 1 z1 n10 j

∑

∑

2 j ¼ 1 z1 n10

∇2 δϕ ¼

N

∑

k2 z2j nj0

N

z2 nj0

∑ 2j ðδϕ þ gj Þexp j ¼ 1 z1 n10





zj ϕe z1 ζa

 ð4Þ

2 j ¼ 1 z1 n10

∇2 gj ¼

1014

zj 1 1 ∇ϕ ∇gj þ ðδv
UP Þ 3 ∇ϕe , j ¼ 1, 2, :::, N Dj z1 ζa e 3

ð5Þ


∇δp þ η∇2 δv þ ∇2 ϕe ∇δϕ þ ∇2 δϕ∇ϕe ¼ 0

ð6Þ

∇ 3 δv ¼ 0

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zj ϕe nj ¼ nj0 exp
z1 ζ a n 3 ∇ϕe ¼


σS ε

"

# zj ðδϕ þ gj Þ , j ¼ 1, 2, :::, N 1
ζa z1

ð8Þ

Table 1. Percentage Difference between • at T, •(T), and that at T = 298 K, •(298), {100%  [•(T)
•(298)]/•(298)} |ϕS|

1  !2 ϕe þ C B Ka
Kb ½H 0 exp
C ζa FNtotal B C B ¼ ! C on ΩP B 2     C B ε @ ϕ ϕ A Kb ½Hþ 0 exp
e þ ½Hþ 0 exp
e þ Ka ζa ζa 0

ð9Þ n 3 ∇δϕ ¼ 0 on ΩP

ð10Þ

n 3 ∇g j ¼ 0 on ΩP

ð11Þ

n 3 ∇ϕe ¼ 0 on ΩW

ð12Þ

n 3 ∇δϕ ¼ 0 on ΩW

ð13Þ

n 3 ∇gj ¼ 0 on ΩW

ð14Þ

n 3 ∇ϕe ¼ 0 on ΩE

ð15Þ

n 3 ∇δϕ ¼
Ez on ΩE

ð16Þ

gj ¼
δϕ on ΩE

ð17Þ

δv ¼ UP ez on ΩP

ð18Þ

δv ¼ 0 on ΩW

ð19Þ

n 3 ∇δv ¼ 0 on ΩE

ð20Þ

Here, Up is the particle velocity; Ez is the z component of E; the subscript e and the prefix δ denote the equilibrium and the perturbed properties, 2 respectively; ε is the permittivity of the liquid phase; k = [∑N j = 1nj0(ezj) / εkBT]1/2 is the reciprocal of Debye length; ζa = kBT/z1e is the thermal potential with e, kB, and T being the elementary charge, Boltzmann constant, and the absolute temperature, respectively, and subscript 1 denotes a reference ionic species (H+ in this study); σS is surface charge density; zj, Dj, nj, and nj0 are the valence, the diffusivity, the number concentration, and the bulk number concentration of ionic species j, respectively, j = 1, 2, ..., N; η is the fluid viscosity; F is Faraday constant; ϕ is the electrical potential; p is the pressure; v is the fluid velocity; gj is a hypothetical potential function simulating a polarized double layer; [H+]0 is the bulk molar concentration of H+. 3 and 32 are the gradient and the Laplace operator, respectively, ez is the unit vector in the z direction; n is the unit normal vector directed into the liquid phase on ΩP and ΩW, and in the z direction on ΩE. Because the particle is stagnant at equilibrium, ve = 0, therefore v = δv. In addition, since no pressure gradient is applied, δp = 0 on ΩE. Equation 9 describes the equilibrium charge density on the particle surface, which can be derived through Gauss' law. Equations 10 and 11 imply that the particle surface is nonconductive and ion impermeable, respectively. Equations 12
17, 19, and 20 state that both the electrical and the hydrodynamic properties of the liquid phase far away from the particle is uninfluenced by its presence. Equation 18 implies that the particle surface is nonslip. In the present problem, ε, η, Dj, Ka, and Kb can all be influenced by the temperature. For simplicity, the temperature dependence of Kb is neglected because we are interested only in pH ranging from 3 to 9.5, where the charged nature of the particle surface is dominated by Ka. The relative permittivity of water, εr, ε = εrε0 with ε0 being the dielectric

T (K)

ε

η

Dj

Ka 6¼ Ka(T)

Ka = Ka(T)

303


2.27


13.3

1.59

1.79

11.7

308


2.26


9.15

1.67

1.77

11.4

constant of a vacuum, are 78.358, 76.581, and 74.846 for T = 298, 303, and 308 K, respectively.29 The viscosity of water, η, takes the values of 0.901, 0.797, and 0.720 mPa s for T = 298, 303, and 308 K, respectively.30 The diffusivity of each ionic species is estimated by the Nernst
Haskell equation31 " # RT 1=jzj j Dj ¼ 2 ð21Þ F 1=λ0j where R is gas constant and λ0j is the limiting conductance of ionic species j. The diffusivities of H+, Na+, Cl
, and OH
are 9.31  10
9, 1.33  10
9, 2.03  10
9, and 5.30  10
9 m2/s, respectively, at T = 298 K, 9.47  10
9, 1.35  10
9, 2.06  10
9, and 5.39  10
9 m2/s, respectively, at T = 303 K, and 9.62  10
9, 1.37  10
9, 2.10  10
9, and 5.48  10
9 m2/s, respectively, at T = 308 K. The dependence of Ka on the temperature can be expressed as ln Ka ¼


Δr G kB T

ð22Þ

where ΔrG is the standard Gibbs energy change of the reaction expressed in eq 1. For the dissociation of H+ on silica surface, ΔrG = 13.4 kcal/mol,32 and the corresponding Ka at T = 298, 303, and 308 K are 10
7, 1.4  10
7, and 2  10
7 M, respectively. The percentage difference between the values of ε, η, Dj, and ϕs at T with the corresponding values at T = 298 K are summarized in Table 1, where the percentage difference in Dj represents the averaged difference of the four kinds of ionic species considered. The equilibrium dissociation constant of water, Kw, is also affected by temperature: the values of pKw are 13.997, 13.830, and 13.680 at T = 298, 303, and 308°K, respectively.33 However, as will be discussed later, it affects only slightly the electrophoretic behavior of a particle. To estimate the electrophoretic mobility of the particle μE defined below, a trial-and-error procedure needs be applied: μE ¼ UP =Ez

ð23Þ

For a given Ez, this procedure is begun by assuming an arbitrary value for UP. The electrical force, Fe, and the hydrodynamic force, Fd, acting on the particle are then evaluated. Because the total force acting on the particle in the z direction should vanish at steady state, the condition Fe + Fd = 0 is checked. If it is satisfied, then the assumed value of UP is corrected; otherwise, another UP needs be assumed. In our case, Fe and Fd can be calculated by integrating the Maxwell stress tensor σE and the shear stress tensor σH over ΩP34   Z ∂ϕe ∂δϕ ∂ϕe ∂δϕ
nz dΩP Fe ¼ ð24Þ ΩP ∂n ∂z ∂t ∂t Fd ¼

Z ΩP

ðσ H 3 nÞ 3 e dΩP

ð25Þ

where n and nz are the magnitude and the z component of n, respectively, and t is the magnitude of the unit tangential vector t. 1015

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Figure 2. Variation of the average surface potential ϕs as a function of temperature at various levels of pH for the case of a = 20 nm, pKa = 7, pKb = 2, CNaCl = 10
3 M, and Ntotal = 5  10
7 mol/m2. Solid curves: Ka = Ka(T). Dashed curves: Ka 6¼ Ka(T).

’ RESULTS AND DISCUSSION The present problem is solved numerically by FlexPDE,35 a finite-element method based commercial software, which was verified previously by solving the electrophoresis of a rigid particle of constant surface potential solved analytically by Shugai and Carnie,36 and by fitting the experimental data of Sonnefeld et al.,37,38 where the electrophoresis of SiO2 particles of radius 20 nm was conducted. Suppose that the background electrolytes in the liquid phase are NaCl, and the solution pH is adjusted by NaOH and HCl. Then, four kinds of ionic species (i.e., N = 4) need be considered, namely, Na+, Cl
, H+, and OH
. Let [Na+]0, [Cl
]0, [H+]0, and [OH
]0 be the bulk molar concentrations of these ionic species, respectively, and CNaCl be the background molar concentration of NaCl. If we let Kw be the dissociation constant of water, then the following relations apply: ½Hþ 0 ¼ 10-pH , pH e pKw =2

ð26Þ

½Naþ 0 ¼ CNaCl , pH e pK w =2

ð27Þ

½Cl
0 ¼ CNaCl þ 10-pH
10
ðpK w -pHÞ , pH e pK w =2

ð28Þ ½OH
0 ¼ 10
ðpK w -pHÞ , pH e pKw =2

ð29Þ

½Hþ 0 ¼ 10-pH , pH g pKW =2

ð30Þ

½Naþ 0 ¼ CNaCl þ 10
ðpKw
pHÞ
10
pH , pH g pKw =2

ð31Þ ½Cl
0 ¼ CNaCl , pH g pKw =2

ð32Þ

½OH
0 ¼ 10
ðpKw
pHÞ , pH g pKw =2

ð33Þ

Figure 3. Variation of mobility as a function of pH at various levels of temperature for the case of Figure 2: (a) Ka 6¼ Ka(T), (b) Ka = Ka(T).

Note that for pH ranging from 3 to 9.5, the influence of pH on the ionic concentration is inappreciable; that is, the temperature dependence of pKw can be neglected. For illustration, we consider an aqueous SiO2 dispersion with pKa = 7, pKb = 2, and Ntotal = 5  10
7 mol/m2 in subsequent discussion.37,38 In addition, CNaCl is assumed the value of 10
3 M. Therefore, as pH varies from 3 to 9.5, the Debye length varies from 8.46 to 9.71 nm, which is comparable to the particle radius, ca. 20 nm, implying that the effect of double-layer polarization (DLP) can be significant.23 This effect comes from the convective flow of the unbalanced counterions inside the double layer.39 The effect of DLP usually retards the particle motion26,39 and plays an important role in electrophoresis when the particle size is compared to the Debye length.24,39
44 Influence of Temperature on Surface Potential. Let us examine first the influence of the temperature T on the surface potential of a particle because it relates directly to the electric driving force acting on the particle, and, therefore, its mobility. Figure R R2 illustrates the variation of the averaged surface potential, ϕs = ΩPϕe dΩP/(4πa2), as a function of T at various levels of pH. This figure shows that for a fixed T, |ϕs| increases with increasing pH. This is expected because eq 1 is the dominating reaction, and therefore, the higher the pH the more the amount of H+ dissociated from the particle surface and the higher the density of negative surface charge. For a fixed level of pH, 1016

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Table 2. Percentage Difference between the Mobility at T, μE(T), and that at T = 298 K, μE(298), {100%  [μE(T)
μE(298)]/ μE(298)} under Various Conditionsa pH = 3
10

a

pH = 3
7

pH = 7
10

T (K)

Ka 6¼ Ka(T)

Ka = Ka(T)

Ka 6¼ Ka(T)

Ka = Ka(T)

Ka 6¼ Ka(T)

Ka = Ka(T)

303

13.1

28.8

11.2

43.9

15.1

13.7

308

8.88

22.3

8.75

32.4

8.97

12.1

The results shown are the averaged values in each pH range.

if Ka = Ka(T), then |ϕs| is seen to increase with increasing temperature. This is because as Ka increases the reaction expressed in eq 1 shifts rightward, and therefore, the particle surface is more negatively charged. Table 1 reveals that, for the case where Ka = Ka(T), the increase in |ϕs| can exceed 10% for a 5 K increase in the temperature. However, if Ka 6¼ Ka(T), then that increase, which arises from the shrink of double layer measured by ka, 2 1/2 , is only ca. 2%. where k = [∑N j = 1nj0(ezj) /εkBT] The variation in the mobility μE as a function of pH at various levels of T is presented in Figure 3; both the results for the case where Ka 6¼ Ka(T) and those for the case where Ka = Ka(T) are shown. As can be seen in this figure, |μE| increases with increasing T in both cases. This is consistent with the experimental observation of Rodriguez-Santiago.45 As will be discussed later, the presence of a local maximum in |μE| near pH = 9 arises from the effect of DLP.23 Table 2 indicates that if Ka 6¼ Ka(T), then the percentage deviation in μE arising from neglecting the influence of temperature for 3 < pH < 10 is on the order of 10% for a 5 K increase in T. However, if Ka = Ka(T), then that deviation becomes much larger, ca. 25%. As will be discussed later, this comes from the variation in the surface potential of the particle. Influence of Dielectric Constant and Viscosity. According to Henry,46 μE can be expressed as μE ¼

εϕs f ðkaÞ η

ð34Þ

This expression is based on negligible effect of DLP. Nevertheless, it can be used to assess approximately the influence of the key parameters on the mobility of a particle. As can be seen in Table 1, a 5 K increase in the temperature yields ca. 2% decrease in ε, ca. 10% decrease in η, and ca. 2% increase in Dj. Note that a decrease in ε yields an increase in ka, and f(ka) increases accordingly. Therefore, for a particle having a constant surface potential,16 where Ka 6¼ Ka(T), its mobility is most influenced by the viscosity of the liquid phase as the temperature varies. On the other hand, for a charged-regulated particle, where Ka = Ka(T), both η and ϕs can play an important role. Influence of pH. Figure 3 summarizes the variation of the mobility μE as a function of pH at various levels of T. This figure reveals that, for 3 < pH < 7, the rate of increase in |μE| with increasing pH for the case where Ka = Ka(T) is faster than that for the case where Ka 6¼ Ka(T). This arises mainly from the behavior of ϕs; the rate of increase in |ϕs| with increasing pH is faster in the former case. However, if pH exceeds 7, then the influence of T on the behavior of μE becomes inappreciable. Table 2 indicates that if Ka = Ka(T), then for 3 < pH < 7, the increases in μE are 43.9% and 32.4% as T increases from 298 to 303 K, and from 303 to 308 K, respectively. These values are much larger than the corresponding results for the case where Ka 6¼ Ka(T), 11.2% and 8.75%, respectively. Again, this arises mainly from the influence in the surface potential of the particle. If pH exceeds 7, then the

Figure 4. Contours of the perturbed ion distribution δn along the z direction for the case of Figure 2 at pH = 8: solid curves, Ka = Ka(T); dashed curves, Ka 6¼ Ka(T). Dotted lines and breakings denote the outer boundary of double layer and the particle surface, respectively.

increase in μE for the case where Ka = Ka(T) reduces to 13.7% as T increases from 298 to 303 K, and 12% as T increases from 303 to 308 K. These values are close to those for the case where Ka 6¼ Ka(T), 15.1% and 8.97%, respectively. This interesting phenomenon can be explained by the effect of DLP. Influence of Temperature on DLP. Figure 4 shows the variation of the difference in the perturbed ionic distribution, δn, defined below as a function of the position along the z axis δn ¼ ðnanion
nanion, e Þ
ðncation
ncation, e Þ

ð35Þ

where nanion and ncation are the number concentrations of anions and cations, respectively, and nanion,e and ncation,e are the corresponding equilibrium concentrations. After E is applied, more cations (anions) accumulate near the top (bottom) region of the particle, yielding a local electrical field, the direction of which is opposite to E. The unbalanced distribution of cations and anions, or DLP, comes mainly from the relaxation effect; the movement of ions is unable to catch up with that of particle. The effect of DLP is important when the surface potential is sufficiently high, and for SiO2 particles, which occurs when pH exceeds ca. 7.23 The presence of DLP has the effect of reducing the electrical driving force acting on a particle, and, therefore, its mobility. Note that the magnitude of the mobility decreases with increasing pH if it exceeds ca. 10. This is because in this case the higher the pH the higher the concentration of OH
and the thinner the double layer, and, therefore, the less significant the effect of DLP23 and the smaller the magnitude of the mobility.19 Figure 4 reveals that the effect of DLP for the case where Ka = Ka(T) is 6 Ka(T). At pH = more appreciable than that for the case where Ka ¼ 8 and T = 308 K, the difference in δn between the two cases 1017

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Langmuir ranges from 0.5% to 10%, depending upon the position. That difference comes mainly from the increase of |ϕs| with increasing temperature. That is, the higher the temperature the more significant the effect of DLP, which explains the results shown in Table 2, where for 7 < pH < 10, the increase in |μE| is 13.7% as T increases from 298 to 303 K, but reduces to 12.1% as T increases from 303 to 308 K. As suggested by Table 2 the influence of the temperature becomes more important as pH approaches the point of zero charge (PZC), pH = 2.5 in the present case, where an increase of 5 K in the temperature can yield more than 30% increase in the mobility. If pH is much higher than PZC, then due to the enhanced effect of DLP, that increase reduces to ca. 13%. This provides valuable information for the design of an electrophoresis device and/or for the interpretation of experimental data.

’ CONCLUSIONS The influence of the Joule heating effect on the electrophoretic behavior of a particle is investigated by considering a chargeregulated particle in a solution containing multiple ionic species. Adopting an aqueous SiO2 dispersion as an example, we conclude the following: (i) An increase in the temperature leads to a decrease in the dielectric constant and the viscosity of the liquid phase, and an increase in the diffusivity of ions and the level of the surface potential of the particle. The net effect is that the mobility of the particle increases with increasing temperature. (ii) For a particle having constant surface potential, its electrophoretic mobility is most influenced by the variation in the liquid viscosity as the temperature varies. However, for a charge-regulated particle both the liquid viscosity and the surface potential can play an important role. (iii) Neglecting the effect of temperature will underestimate the surface potential of a particle, which correlates closely to its mobility, to an order of 10%. Depending upon the types of surface reactions, that deviation might become even larger. (iv) Depending upon the level of pH, an increase of 5 K in the temperature can yield ca. 40% increase in the mobility. (v) The rate of increase in the mobility with increasing pH for the case where the effect of temperature is considered is faster than that for the case where that effect is neglected, in general. This arises from the surface potential in the former case being higher than that in the latter case. (vi) That increase in the rate of mobility is inhibited if the pH is away from the point of zero charge. This is because the effect of double-layer polarization becomes important when the surface potential exceeds a certain level. For instance, if pH is below 7, then the increase in the mobility exceeds 30% for an increase of 5 K. However, if pH exceeds 7, then it reduces to only ca. 13%. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Phone: 886-2-23637448. Fax: 886-2-23623040.

’ ACKNOWLEDGMENT This work is supported by the National Science Council of the Republic of China. ’ GLOSSARY a = radius of particle (m) b = radius of cylindrical computation domain (m)

ARTICLE

Dj = diffusivity of ionic species j (m2/s) e = elementary charge (Coul) ez = unit vector in z direction E = strength of applied electric field (V/m) E = applied electric field (V/m) Ez = strength of applied electric field in z direction (V/m) F = Faraday constant (Coul/mol) Fe = electrical force acting on particle in z direction (N) Fd = hydrodynamic force acting on particle in z direction (N) gj = hypothetical perturbed potential associated with ionic species j (V) kB = Boltzmann constant (J/K) Ka = equilibrium constant for the dissociation of AH (M) Kb = equilibrium constant for association of AH (M
1) Kw = dissociation constant of H2O (M2) n = magnitude of n nz = z component of n nj = number concentration of ionic speciesj (no./m3) nj0 = bulk number concentration of ionic species j (no./m3) n = unit normal vector Ntotal = density of functional groups on particle surface (mol/m2) p = pressure (Pa) r = radial coordinate (m) t = magnitude of t tz = z component of t t = unit tangential vector on particle surface T = absolute temperature (K) UP = velocity of particle (m/s) v = fluid velocity (m/s) z = axial coordinate (m) zj = valence of ionic species j Greek Letters

δϕ = perturbed potential arising from E (V) ε = permittivity of liquid phase (Coul2/N/m2) ζa = kBT/ez1 (V) η = viscosity of liquid phase (kg/m/s) 2 1/2 ) reciprocal Debye length (m
1) k = ([∑N j = 1nj0(ezj) /εkBT] λ = a/b μE = electrophoretic mobility (m2/V/s) σH = hydrodynamic stress tensor (N/m2) σsurface = charge density on particle surface (Coul/m2) ϕe = equilibrium potential (V) ΩE = end surface (area) of cylindrical computation domain (m2) ΩP = surface (area) of particle (m2) ΩW = lateral surface (area) of cylindrical computation domain (m2) 3 = gradient operator (1/m) 32 = Laplace operator (1/m2) Subscripts

e = equilibrium property j = index of ionic species Prefix

δ = perturbed property

’ REFERENCES (1) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001. (2) Ennis, J.; Zhang, H.; Stevens, G.; Perera, J.; Scales, P.; Carnie, S. J. Membr. Sci. 1996, 119, 47. (3) Knox, J. H.; McCormack, K. A. Chromatographia 1994, 38, 207. (4) Fonslow, B. R.; Bowser, M. T. Anal. Chem. 2005, 77, 5706. 1018

dx.doi.org/10.1021/la203245n |Langmuir 2012, 28, 1013–1019

Langmuir

ARTICLE

(5) Fu, L. M.; Wang, J. H.; Luo, W. B.; Lin, C. H. Microfluid. Nanofluid. 2009, 6, 499. (6) Evenhuis, C. J.; Haddad, P. R. Electrophoresis 2009, 30, 897. (7) Musheev, M. U.; Filiptsev, Y.; Krylov, S. N. Anal. Chem. 2010, 82, 8692. (8) Chein, R. Y.; Yang, Y. C.; Lin, Y. S. Electrophoresis 2006, 27, 640. (9) Petersen, N. J.; Nikolajsen, R. P. H.; Mogensen, K. B.; Kutter, J. P. Electrophoresis 2004, 25, 253. (10) Rogers, B.; Gibson, G. T. T.; Oleschuk, R. D. Electrophoresis 2011, 32, 223. (11) Seyrek, E.; Dubin, P. L.; Newkome, G. R. J. Phys. Chem. B 2004, 108, 10168. (12) Wang, C. K.; Tsao, H. K. J. Phys. Chem. B 2004, 108, 17685. (13) Tang, G. Y.; Yang, C.; Gong, H. Q.; Chai, J. C.; Lam, Y. C. Anal. Chim. Acta 2006, 561, 138. (14) Xuan, X. C.; Li, D. Q. J. Chromatogr., A 2005, 1064, 227. (15) Evenhuis, C. J.; Hruska, V.; Guijt, R. M.; Macka, M.; Gas, B.; Marriott, P. J.; Haddad, P. R. Electrophoresis 2007, 28, 3759. (16) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (17) Qian, S.; Wang, A.; Afonien, J. K. J. Colloid Interface Sci. 2006, 303, 579. (18) Tang, Y. P.; Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2002, 242, 121. (19) Hsu, J. P.; Chen, C. Y.; Lee, D. J.; Tseng, S.; Su, A. J. Colloid Interface Sci. 2008, 325, 516. (20) Venditti, R.; Xuan, X. C.; Li, D. Q. Microfluid. Nanofluid. 2006, 2, 493. (21) Cetin, B.; Li, D. Q. Electrophoresis 2008, 29, 994. (22) Hsu, J. P.; Chen, Z. S. Langmuir 2007, 23, 6198. (23) Hsu, J. P.; Tai, Y. H. Langmuir 2010, 26, 16857. (24) Qian, S.; Wang, A.; Afonien, J. K. J. Colloid Interface Sci. 2006, 303, 579. (25) Binner, J.; Zhang, Y. J. Mater. Sci. Lett. 2001, 20, 123. (26) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (27) Ohshima, H. J. Colloid Interface Sci. 1995, 62, 189. (28) Ohshima, H. Electrophoresis 2006, 27, 526. (29) Owen, B. B.; Milner, C. E.; Miller, R. C.; Cogan, H. L. J. Phys. Chem. 1961, 65, 2065. (30) Cho, C. H.; Urquidi, J.; Robinson, G. W. J. Chem. Phys. 1999, 111, 10171. (31) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (32) Sefcik, J.; Goddard, W. A. Geochim. Cosmochim. Acta 2001, 65, 4435. (33) Lide, D. R. CRC Handbook of Chemistry and Physics, 70th ed.; CRC Press: Boca Raton, FL, 1990. (34) Hsu, J. P.; Yeh, L. H.; Ku, M. H. J. Colloid Interface Sci. 2007, 305, 324. (35) FlexPDE, version 2.22; PDE Solutions: Spokane Valley, WA, 2000. (36) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (37) Sonnefeld, J.; L€obbus, M.; Vogelsberger, W. Colloid Surf., A 2001, 195, 215. (38) Hsu, J. P.; Tai, Y. H.; Yeh, L. H.; Tseng, S. J. Phys. Chem. B 2011, 115, 3972. (39) Yeh, L. H.; Hsu, J. P. Soft Matter 2011, 7, 396. (40) Zhao, H.; Bau, H. H. Langmuir 2008, 24, 6050. (41) Ai, Y.; Liu, J.; Zhang, B. K.; Qian, S. Anal. Chem. 2010, 82, 8217. (42) Ai, Y.; Qian, S. Phys. Chem. Chem. Phys. 2011, 13, 4060. (43) Tsai, P.; Lee, E. Soft Matter 2011, 7, 5789. (44) Zhang, M.; Ai, Y.; Kim, D. S.; Jeong, J. H.; Joo, S. W.; Qian, S. Colloids Surf., B 2011, 88, 165. (45) Rodriguez-Santiago, V.; Fedkin, M. V.; Lvov, S. N. Rev. Sci. Instrum. 2008, 79, 093302. (46) Henry, D. C. Proc. R. Soc. London 1931, 133, 106.

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