Colloid vibration potential in a suspension of soft particles

Colloids and Surfaces A: Physicochem. Eng. Aspects 306 (2007) 83–87

Colloid vibration potential in a suspension of soft particles Hiroyuki Ohshima ∗ Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Received 29 September 2006; received in revised form 2 December 2006; accepted 13 December 2006 Available online 19 December 2006

Abstract We develop a theory of the vibration potential produced in a dilute suspension of soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution when a sound wave (i.e., an oscillating pressure gradient field) is applied to the suspension. The total vibration potential (TVP) consists of two components. One is the colloid vibration potential (CVP) and the other is the ion vibration potential (IVP). The obtained CVP expression covers two extreme cases, that is, the CVP in a suspension of hard particles (in the absence of the polyelectrolyte layer) and that in a suspension of spherical polyelectrolytes (in the absence of the particle core). A simple analytic expression for CVP, which is proportional to the dynamic electrophoretic mobility of a soft particle, is given. It is shown that the CVP of a suspension of a soft particle depends on the density of the volume charge distributed in the polyelectrolyte layer, on the frictional coefficient characterizing the frictional forces exerted by the polymer segments on the liquid flow in the polyelectrolyte layer, on the particle size, on the thickness of the polyelectrolyte layer, and on the frequency of the applied oscillating pressure gradient field. © 2007 Elsevier B.V. All rights reserved. Keywords: Colloid vibration potential; Soft particle; Dynamic electrophoretic mobility; Sound wave; Electroacoustic phenomena

1. Introduction Soft particles (i.e., hard particles covered with an ionpenetrable surface layer of polyelectrolytes) serve as a model for biological cells [1–6] and humic substances [7]. The purpose of the present article is to derive an expression for the colloid vibration potential (CVP) or current (CVI) in a dilute suspension of spherical soft particles in an electrolyte solution. When a sound wave is propagated in an electrolyte solution, the motion of cations and that of anions may differ from each other because of their different masses so that periodic excesses of either cations or anions should be produced at a given point in the solution, generating vibration potentials. This potential is called ion vibration potential (IVP) [8–13]. A similar electroacoustic phenomenon occurs in a suspension of colloidal particles. Since colloidal particles are much larger and carry a much greater charge than electrolyte ions, the potential difference in the suspension is caused by the asymmetry of the electrical double layer around each particle rather than the relative motion of

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cations and anions [6,13–24]. It has been shown that the CVP of a suspension of colloidal particles is proportional to the dynamic electrophoretic mobility of the particles. Approximate expressions for the dynamic electrophoretic mobility of hard particles are given in Refs. [16,25–28]. It must also be noted that in a colloidal suspension in an electrolyte solution, IVP and CVP are both generated simultaneously. Recently we have developed a general acoustic theory for a suspension of spherical rigid particles, which accounts for both CVP and IVP [23,24]. In the present paper we apply this theory to the suspension of spherical soft particles on with the help of an approximate expression for the dynamic electrophoretic mobility of soft particles [3]. 2. Colloid vibration potential, ion vibration potential, and total vibration potential Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(−iωt) in a symmetrical electrolyte solution of viscosity η and relative permittivity εr in an applied oscillating pressure gradient field p exp(−iωt) due to a sound wave propagating in the suspension, where ω is the angular frequency (2π times frequency) and t is time. We treat the case in which ω is low such that the dispersion of εr can be neglected.

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H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 306 (2007) 83–87

with K∗ = K∞ − iωεr εo   1 1 + K∞ = z2 e2 n λ+ λ− φ=

Fig. 1. A spherical soft particle in an applied pressure gradient field; a is the radius of the particle core and d is the thickness of the polyelectrolyte layer covering the particle core. b = a + d.

We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d (Fig. 1). Let the valence and bulk concentration (number density) of the electrolyte be z and n, respectively. We also denote the mass, valence, and drag coefficient of cations by m+ , V+ , and λ+ , respectively, and those for anions by m− ,V− , and λ− . The drag coefficients λ+ and λ− , are related to the corresponding limiting conductance of cations, Λ0+ , and that of anions, Λ0− , by NA e 2 z λ± = Λ0±

(1)

where e is the elementary electric charge and NA is Avogadro’s number. We adopt the model of Debye–Bueche [29,30] that the polymer segments are regarded as resistance centers distributed in the polyelectrolyte layer, exerting a frictional force on the liquid flowing in the polyelectrolyte layer, where the frictional coefficient is ν. We also assume that fixed charge groups of valence Z are distributed at a uniform density of N in the polyelectrolyte layer. We have recently proposed a general acoustic theory for a dilute suspension of particles, which accounts for both CVP and IVP [23,24]. Experimentally, the total vibration potential between two points in the suspension, which is given by the sum of IVP and CVP, is observed. That is, TVP = IVP + CVP where IVP and CVP are given by [23]   zen m + − ρ o V+ m − − ρ o V− IVP =

P − ρo K ∗ λ+ λ− CVP =

φ(ρp − ρo ) μ(ω) P ρo K ∗

(2)

(3)

(4)

(5) (6)

(4π/3)a3 Np V

(7)

Here p has been replaced with the pressure difference between the two points is P, K∞ and K* are, respectively, the usual conductivity and the complex conductivity of the electrolyte solution in the absence of the particles, εo is the permittivity of a vacuum, φ the particle volume fraction, V the suspension volume, ρp and ρo , are, respectively, the mass density of the particle and that of the electrolyte solution, and μ(ω) is the dynamic electrophoretic mobility of the particles. Eq. (4) is an Onsager relation between CVP and μ(ω), which takes a similar form for an Onsager relation between sedimentation potential and static electrophoretic mobility [31]. Eq. (2) is an expression for CVP applicable for dilute spherical hard particles. For a dilute suspension of spherical soft particles, Eq. (4) must be replaced by CVP =

[φc (ρc − ρo ) + φs (ρs − ρo )] μ(ω) P ρo K ∗

(8)

with φc =

Vc N p V

(9)

Vc =

4 3 πa 3

(10)

φs =

Vs Np V

(11)

where φc is the volume fraction of the particle core, Vc the volume of the particle core, φs the volume fraction of the polyelectrolyte segments, Vc the total volume of the polyelectrolyte segments coating one particle, ρc the mass density of the particle core, and ρs is the mass density of the polyelectrolyte segment. Eq. (8) can be derived from analogy between CVP and sedimentation potential [31]. For a spherical soft particle, the following expression for μ(ω) for the dynamic electrophoiretic mobility has been derived [3]:    2εr εo 1 − iγb a3 μ(ω) = 1+ 3 3η 1 − iγb − (γ 2 b2 /3) − Γ 2b   ψo /κm + ψDON /β × 1/κm + 1/β   1 − iγb − (γ 2 b2 /3)(1 − a3 /b3 ) ZeN + (12) 1 − iγb − (γ 2 b2 /3) − Γ ηβ2 with ψDON =

⎡ ZN kT ln ⎣ + ze 2zn



ZN 2zn

2

1/2 ⎤ ⎦ +1

(13)

H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 306 (2007) 83–87

⎛ ⎡ 1/2 ⎤   kT ⎝ ⎣ ZN ZN 2 ⎦ ψo = +1 ln + ze 2zn 2zn ⎡ +

2zn ⎣ 1− ZN





κm = κ 1 +

ZN 2zn



ZN 2zn

2

1/2 ⎤⎞ ⎦⎠ +1

CVI =

2 1/4 (15)



Γ = =

fc =

(16) (17)

(18)

(19)

γ 2 [Vc (ρc − ρo ) + Vs (ρs − ρo )] 6πbρo 2(γb)2 {fc (ρc − ρo ) + fs (ρs − ρo )} 9ρo

(20)

 a 3

(21)

b Vs fs = 4πb3 /3

(26)

where μ(ω) is given by Eq. (12). (14)

1/2 2z2 e2 n κ= εr εo kT  1/2 ν λ= η   iωρo ωρo γ= = (1 + i) η 2η   γ 2 β =λ 1− λ

[φc (ρc − ρo ) + φs (ρs − ρo )] μ(ω) P ρo

85

3. Results and discussion We have derived a general expression for the colloid vibration potential, the ion vibration potential, and the total vibration potential (TVP) for a dilute suspension of spherical soft particles (Eqs. (2), (3), and (8)) together with the corresponding currents, that is, CVI, IVI, and TVI (Eqs. (24)–(26)). Some results of the calculation of the magnitude and phase of the dynamic mobility via Eqs. (24)–(26) are given in Figs. 2–7, in which we have used the following values: ρo = 1 × 103 kg m−3 , εr = 78.5 (water at 25 ◦ C), and ρc = ρs = 1.1 × 103 kg m−3 in an aqueous KCl solution at 25 ◦ C (Λ0+ = 73.5 × 10−4 m2 −1 mol−1 and m+ = 39.1 × 10−3 kg mol−1 for K+ , and Λo− = 76.3 × 10−4 m2 −1 mol−1 and m− = 35.5 × 10−3 kg mol−1 for Cl− ). For the ionic volumes for K+ and Cl− ions, we have used the values of their partial molar volumes reported by Zana and Yeager [12], that is,V+ = 3.7 × 10−6 m3 mol−1 (for K+ ) and V− = 22.8 × 10−6 m3 mol−1 (for Cl− ). The values of φs and fs have approximately been set equal to zero. It is to be noted that the phase of CVI agrees with that of dynamic electrophoretic mobility μ(ω) (Eq. (12)) and the phase of IVI is zero (in the present approximation). Figs. 2 and 3 show the dependence of the magnitude (Fig. 2) and phase (Fig. 3) of each of CVI, IVI, and TVI on the frequency ω of the pressure gradient field due to the applied sound wave for the case where a = 1 ␮m for an aqueous KCl solution

(22)

Here k is Boltzmann’ s constant, T the absolute temperature, κ the Debye–H¨uckel parameter, ψDON the Donnan potential in the polyelectrolyte layer, ψo the potential at the boundary between the polyelectrolyte layer and the surrounding solution, which we call the surface potential of the soft particle, and κm is the Debye–H¨uckel parameter in the polyelectrolyte layer. Eq. (12) is a good approximation when the following conditions are satisfied: |β|b  1,

κb  1,

κd = κ(b − a)  1,

|β|d = |β|(b − a)  1, |β|  |γ|,

κ  |γ|

(23)

which hold for most practical cases [3]. The total electric current TVI = K* TVP generated in the suspension can also be observed [13], which is expressed as the sum of the ion vibration current (IVI) and the colloid vibration current (CVI), viz., TVI = IVI + CVI with zen IVI = ρo



m + − ρ o V+ m − − ρ o V− − λ+ λ−

(24) 

P

(25)

Fig. 2. Magnitudes of CVI, IVI, and TVI divided by P as a function of the frequency ω/2π of the applied pressure gradient field for a suspension of soft particles in a KCl solution of concentration n = 0.01 M. Calculated via Eqs. (24)–(26) as combined with Eq. (12) for a = 1 ␮m, d = 10 nm, N = 0. 05 M, Z = 1, 1/λ = 1 nm, ρo = 1 × 103 kg m−3 , ρc = ρs = 1.1 × 103 kg m−3 , φs = 0, fs = 0, η = 0.89 mPa s, T = 298 K. −4 2 −1 εr = 78.5, Λ+ m  mol−1 , V+ = 3.7 × 10−6 m3 mol−1 , 0 = 73.5 × 10 0 −3 −1 m+ = 39.1 × 10 kg mol , Λ− = 76.5 × 10−4 m2 −1 mol−1 , V− = 22.8 × 10−6 m3 mol−1 , and m− = 35.5 × 10−3 kg mol−1 .

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Fig. 3. Phases of CVI, IVI, and TVI as a function of the frequency ω/2π of the applied pressure gradient field for a suspension of spherical soft particles in a KCl solution. Calculated via Eqs. (24)–(26) as combined with Eq. (12). Numerical values used are the same as in Fig. 2.

Fig. 5. Phase of CVI, IVI, and TVI divided by P as a function of the radius a of the particle field for a suspension of spherical soft particles in a KCl solution. Calculated via Eqs. (24)–(26) as combined with Eq. (12) for ω/2π = 1 MHz and a = 1 ␮m. The other numerical values used are the same as in Fig. 2.

of concentration n = 0.01 M. It is seen that the ω-dependence is negligibly small ω/2π < 104 Hz and becomes appreciable for ω/2π > 104 Hz. That is, CVI is essentially equal to its static value at ω = 0 for ω/2π < 104 Hz and drops sharply to zero for ω/2π > 104 Hz, while the phase of CVI is zero for ω/2π < 104 Hz and increases sharply for the frequency range ω/2π > 104 Hz. The magnitude of IVI, on the other hand, is constant independent of ω, while the phase of IVI is always zero, since in the present approximation IVI is a real quantity. Figs. 4 and 5 show how the mobility magnitude and the phase of CVI depend on the particle core radius a at d = 10 nm and ω/2π = 1 MHz. Even in the static case (shown as doted line in Fig. 4), the mobility increases with a. This is due to the factor

(1 + a3 /2b3 ) in Eq. (12), which, for the case of dynamic electrophoresis, represents the distortion of the applied filed due to the presence of the particle core. It can also be shown that the CVI magnitude is almost equal to the corresponding static value in the region a < 0.1 ␮m. For larger a, the particle vibrates slower so that the CVI magnitude decreases, resulting in a decrease in the CVI magnitude. Thus, the magnitude of CVI exhibits a maximum and then decreases sharply with a. Figs. 6 and 7 show the dependence of the magnitude and phase of CVI, IVI, and TVI on the electrolyte concentration n at a = 1 ␮m, d = 10 nm, and ω/2π = 1 MHz. Fig. 6 shows that the magnitude of IVI becomes much greater than that of CVI, except for the range of very low electrolyte concentrations and

Fig. 4. Magnitudes of CVI, IVI, and TVI divided by P as a function of the radius a of the particle field for a suspension of spherical soft particles in a KCl solution. Calculated via Eqs. (24)–(26) as combined with Eq. (12) for ω/2π = 1 MHz and n = 0.01 M. The other numerical values used are the same as in Fig. 2. Dotted lines correspond to the results for ω = 0.

Fig. 6. Magnitudes of CVI, IVI, and TVI divided by P as a function of the electrolyte concentration n for a suspension of spherical soft particles in a KCl solution. Calculated via Eqs. (24)–(26) as combined with Eq. (12) for a = 1 ␮m and ω/2π = 1 MHz. The other numerical values used are the same as in Fig. 2.

H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 306 (2007) 83–87

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where A1 and A2 are, respectively, the magnitudes of IVI and CVI. References [1] [2] [3] [4] [5] [6] [7]

Fig. 7. Phases of CVI, IVI, and TVI divided by P as a function of the electrolyte concentration n for a suspension of spherical soft particles in a KCl solution. Calculated via Eqs. (24)–(26) as combined with Eq. (12) for a = 1 ␮m and ω/2π = 1 MHz. The other numerical values used are the same as in Fig. 2.

that the CVI tends to a nonzero limiting value at very high electrolyte concentrations, as in the case of other electrokinetics of soft particles [1–7]. This is a characteristic of the electrokinetic behavior of soft particles, which comes from the second term of the right-hand side of Eq. (12). The limiting CVI value is obtained from Eq. (26) (as combined with Eq. (12)) with the result that [φc (ρc − ρo ) + φs (ρs − ρo )] CVI = ρo   1 − iγb − (γ 2 b2 /3)(1 − a3 /b3 ) ZeN ×

P (27) 1 − iγb − (γ 2 b2 /3) − Γ ηβ2 Fig. 7 shows that, in contrast to Fig. 6, the phase of CVI becomes much greater than that of IVI. It can also be shown that the phase of CVI (φCVI ) is related to the phase of TVI by   sin φTVI φCVI = arctanh (28) cos φTVI − A1 /A2

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