Langmuir 2006, 22, 1911-1918
1911
Electrophoresis of Carreau-Drop Dispersions with a Charge-Regulated Surface Eric Lee, Chia-Ping Chiang, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed September 28, 2005. In Final Form: NoVember 23, 2005
The electrophoresis of a liquid-liquid dispersion, where the dispersed phase comprises drops of a shearing-thinning Carreau fluid with a charge-regulated surface and the dispersion medium is an aqueous electrolyte solution, is analyzed theoretically under the conditions of low surface potential, uniform weak applied electric field, and arbitrary double layer thickness. This is the first attempt for the description of the electrophoretic behavior of a dispersion containing non-Newtonian drops with a charge-regulated surface. We show that, in general, the more significant the shearthinning nature of the drop fluid, the lower the concentration of drops, the lower the pH of bulk solution, or the higher the concentration of dissociable functional groups on drop surface, the larger the mobility. Also, the mobility of a drop may exhibit a local maximum as the thickness of the electric double layer surrounding it varies.
1. Introduction Electrophoresis has various fundamental and practical applications in colloid science, including separation of charged entities, measurement of the stability of a dispersion, and estimation of the charged conditions on a surface, to name a few. Since the original theoretical analysis of Smoluchowski,1 for the case of a rigid, isolated particle in an infinite electrolyte solution, extensive studies have been conducted both theoretically and experimentally, most of them focused on rigid entities. The electrophoresis of nonrigid entities was initiated by Craxford et al.2 through considering the electrophoresis of a mercury drop. Due to its specific physical nature, which is ideal for model construction, mercury drops were adopted frequently by many workers in the analyses of the electrophoretic behaviors of nonrigid entities.3-6 One of the important colloidal systems is liquid-liquid dispersion, or emulsion, which plays a key role in modern technology. The preparation of nanosized particles, for example, can be conducted in such a system. Often, polyelectrolyte is introduced into an emulsion to raise its stability through electrostatic and/or steric interactions between neighboring drops. For the ionic polyelectrolyte, the dissociation of the functional groups of its molecules attached to the surface of a drop yields a charge-regulated surface, in general. This type of charged condition is somewhere between a constant surface potential model and a constant surface charge model, two simple yet widely assumed conditions in relevant analyses. Although the electrophoresis of rigid particles has been analyzed extensively, available theoretical results for the electrophoresis of emulsions are very limited, especially when the dispersed drops are of non-Newtonian nature. The only reported result is that of Lee et al.,7 where the electrophoresis of an aqueous * To whom correspondence should be addressed. Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail:
[email protected]. (1) Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (2) Craxford, S. R.; Gatty, O.; McCay, H. A. C. Philos. Mag. 1937, 23, 1079. (3) Levine, S.; O’Brien, R. W. J. Colloid Interface Sci. 1973, 43, 616. (4) Ohshima, H. J. Colloid Interface Sci. 1999, 218, 535. (5) Baygents, J. C.; Saville, D. A. J. Chem. Soc., Faraday Trans. 1997, 87, 1883. (6) Lee, E.; Hu, J. K.; Hsu, J. P. J. Colloid Interface Sci. 2003, 257, 250.
dispersion of Carreau drops under the conditions of low and constant surface potential and weak applied electric field is modeled. They show that, in general, the shear-thinning nature of drop fluid yields a larger mobility, and this effect is pronounced as the thickness of the double layer declines. The main difference between the electrophoresis of a rigid particle and that of a drop is that usually the electric, concentration, and flow fields inside a rigid particle need not be solved, which is not the case for a drop. Apparently, the solution procedure of the latter is more complicated than that of the former even if a numerical approach is adopted. Another possible difficulty is that the drop fluid can be non-Newtonian, which is not uncommon in practice. In this study, the analysis of Lee et al.7 is extended to the case where the surface of a drop is of charge-regulated nature, a more general condition which arises from the dissociation of functional groups. A pseudo-spectral method8 based on Chebyshev polynomials is adopted for the resolution of the governing electrokinetic equations and the associated boundary conditions. The influences of the thickness of a double layer surrounding a drop, the nature of the drop fluid and that of dissociable functional groups, and the concentration of drops on the electrophoretic behavior of a dispersion are discussed.
2. Theory We consider the electrophoresis of monodispersed spherical drops of radius a in a z1:z2 aqueous electrolyte solution, z1 and z2 being the valences of cations and anions. If we let n10 and n20 be respectively the bulk concentrations of cations and anions, then electroneutrality in the bulk phase requires that -n20z2 ) n10z1. Referring to Figure 1, the system under consideration is simulated by Kuwabara’s unit cell model,9 where the dispersion is described by a cell comprising a representative drop and a concentric liquid shell of radius b. The ratio H ) a/b is a measure for the concentration of drops. A uniform electric field E is (7) Lee, E.; Chang, C. J.; Hsu, J. P. J. Colloid Interface Sci. 2005, 282, 486. (8) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. Spectral Method in Fluid Dynamics; Springer-Verlag: Berlin, 1987. (9) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527.
10.1021/la052639s CCC: $33.50 © 2006 American Chemical Society Published on Web 01/05/2006
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eq 2 to take the electric force into account, and τ is the stress tensor. For an incompressible generalized Newtonian fluid, we have10
τ ) - η(γ˘ )γ˘
(5)
γ˘ ) ∇vC + (∇vC)T
(6)
where η, γ˘ , and γ˘ are respectively the apparent viscosity, the rate of strain tensor, and its magnitude and the superscript T denotes matrix transpose. Here, we assume that the drop liquid is a Carreau fluid11,12 with
η(γ˘ ) ) η0[1 + (λγ˘ )2](n-1)/2
(7)
where η0 is the zero shear rate viscosity, λ is the relaxation time constant, and n is the power-law index. Note that, for both n ) 1 and λ ) 0, eq 7 leads to a Newtonian fluid. The spatial variation in the electrical potential φ can be described by the Gauss’ law, which leads to
∇2φ ) -
Figure 1. (a) Problem considered where Carreau drops of radius a are dispersed in a Newtonian liquid. An electric field E is applied in the z direction, and U is the electrophoretic velocity of drops. (b) Kuwabara’s unit cell model9 in which a cell comprises a representative drop and a concentric liquid shell of Newtonian fluid with radius b.
applied in the z direction, and U is the electrophoretic velocity of the drop. The spherical coordinate (r,θ,φ) is adopted with its origin located at the center of the drop. We assume that the dispersion medium is an incompressible, Newtonian fluid with constant physical properties, and the drop fluid is an incompressible Carreau fluid, which is electrolyte-free, is nonconductive, and has constant physical properties. The system remains at a quasi-steady-state. For the present case, the flow field can be described by the equation of motion in the creeping flow region and the equation of continuity. We have
Fe
(8)
where is the permittivity of the dispersion medium. For convenience, φ is partitioned into the equilibrium electrical potential or the potential in the absence of the applied electric field, φ1, and a perturbed electrical potential outside the drop, φ2, which arises from the applied electric field, that is, φ ) φ1 + φ2. If the surface potential is low and the applied electric field is weak, it can be shown that eq 8 reduces to13,14
∇2φ1 )
1
2
∑zjenj0 j)1
( ) zjeφ1 kT
(9)
and
∇2φ2 ) 0
(10)
where k and T are respectively the Boltzmann constant and the absolute temperature and nj0 is the bulk concentration of ionic species j. In terms of scaled symbols, these equations become
∇‚v ) 0, a < r < b
(1)
∇*2φ/1 ) (κa)2φ/1
(11)
µ∇2v - ∇p - Fe∇φ ) 0, a < r < b
(2)
∇*2φ/2 ) 0
(12)
∇‚vC ) 0, 0 < r < a
(3)
-∇pC - ∇‚τ ) 0, 0 < r < a
(4)
In these expressions, ∇ is the del operator, v is the velocity of 2 the dispersion medium, ∇2 is the Laplace operator, and Fe ) ∑j)1 zjenj is the space charge density, e, zj, and nj are respectively the elementary charge, the valence, and the number density of ionic species j. p and pC are respectively the pressure of the dispersion medium and the dispersed phase, µ is the viscosity of the dispersion medium, and vC is the velocity of the dispersed phase. Fe∇φ is a body force term, which is included in
where ∇* ) a∇, φ/1 ) φ1/(kT/e), and φ/2 ) φ2/(kT/e), and the 2 reciprocal Debye length κ is defined by κ ) (∑j)1 (zje)2nj0/ kT)1/2. In terms of the scaled stream functions of the dispersion medium and the dispersed phase, ψ* and ψC*, the governing equations (10) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 2002. (11) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymer Liquids; Wiley: New York, 1987; Vol. 1. (12) Carreau, P. J. Ph.D. Thesis, University of Wisconsin, Madison, 1968. (13) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475. (14) Herny, D. C. Proc. R. Soc. London Ser. A 1931, 133, 514.
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for the flow fields can be rewritten as7
D*4ψ* ) - (κa) η*D*4ψ/C + sin θ
[(
(
∂φ/1 2
)
∂φ/2
∂r* ∂θ
σ)-
sin θ, 1 < r* < 1/H (13) /
∂η* / ∂2η* ∂η* ∂γ˘ r θ γ˘ r θ + r* 2 γ˘ /r θ + r* + ∂r* ∂r* ∂r* ∂r*
) (
/ / ∂2η* / ∂η*∂ γ˘ rr ∂η* ∂γ˘ θθ ∂2η* / γ˘ + γ˘ + + ∂r*∂θ θθ ∂θ ∂r* ∂r*∂θ rr ∂r* ∂θ
) (
2 / ∂3ψ/C ∂2ψ/C 1 2 cot θ ∂ ψC + + r*2 sin θ ∂r*∂θ r*2sinθ ∂r*∂θ2 r*3 sin θ ∂θ2
) (
∂ψ/C
∂ ψ/C *2 3
3
ψ/C 3
∂ 2cotθ 1 ∂η* 1 - *4 - *2 3 ∂θ ∂θ r* sinθ r sin θ ∂r ∂θ r sin θ ∂θ
)]
2 / ∂ψ/C cot θ ∂ ψC 1 + r*4 sin3 θ ∂θ r*4 sin θ ∂θ2
) 0, 0 < r* < 1 (14)
[
(
ψ/C
)]
(15)
) and UE ) Also, r* ) r/a, ψ* ) (kT/e)2/µa, which is a reference velocity similar to the electrophoretic velocity of an isolated particle predicted by Smoluchowski’s theory when an electric field (ζa/a) is applied / / with ζa ) kT/e. γ˘ * ) aγ˘ /UE, η* ) η/µ, and γ˘ /rr, γ˘ rθ , and γ˘ θθ are respectively the rr, rθ, and θθ components of the rate of strain tensor. Suppose that the dissociation reaction ψC/a2UE,
Ka
AH S A- + H+
(16) [A-][H+]
occurs on the surface of a drop, where Ka ) S/[HA] is the dissociation constant. Here, a term in square brackets denotes concentration, and the subscript S represents the property on drop surface. Suppose that [H+] follows the Boltzmann distribution
( )|
[H+]S ) [H+]B exp -
eφ1 kT
r)a
(17)
where [H+]B is the concentration of H+ in the bulk liquid phase. If we let NS be the surface concentration of the functional groups, then -
NS ) [AH] + [A ]
eNS 1 + [H+]B/Ka
-
(e2NS/kT)([H+]B/Ka) φ1, r ) a (1 + [H+]B/Ka)2 (21)
Assuming that the permittivity of drop fluid is much smaller that of the dispersion medium, then it can be shown that σ ) -(dφ1/ dr) and eq 21 becomes
([H+]B/Ka) (1 + [H+]B/Ka) dφ1 kT φ1 ) , r ) a (22) e (e2N /kT) dr (1 + [H+] /K ) S
B
a
If we let A ) (e2NSa/kT) and B ) [H+]B/Ka, then this expression becomes
NS
( )
[H+]B eφ1 1+ exp | Ka kT r)a
(23)
The boundary condition for φ/1 on cell surface is assumed as
dφ/1 ) 0, r* ) 1/H dr*
(24)
This implies that the unit cell as a whole is electrically neutral. The following boundary conditions are assumed for the scaled perturbed potential φ/2
∂φ/2 ) 0, r* ) 1 ∂r*
(25)
∂φ/2 ) -E/z cos θ, r* ) 1/H ∂r*
(26)
where E/z ) Ez/(kT/ea), Ez being the z component of the applied electric field. The first expression is based on the fact that drops are nonconductive and are impenetrable to ionic species. The second expression, proposed by Leivne et al.,15 arises from the nature of Kuwabara’s unit cell model;9 that is, since cell boundary corresponds to system boundary, the electric field over there is that contributed by the applied electric field only. We assume that both the velocity and the shear stress are continuous across the drop-dispersion medium interface, that is
νθ,C|r)a- ) νθ|r)a+
(27)
(18)
τrθ,C|r)a- ) τrθ|r)a+
(28)
(19)
where Vθ and νθ,C are respectively the θ component of the fluid velocity of the dispersion medium and the dispersed phase, and τrθ and τrθ,C are respectively the rθ component of the shear stress tensor of the dispersion medium and that of the dispersed phase. For convenience, we assume that a drop is stationary while the fluid in the electrolyte solution moves with the relative velocity
Equations 17 and 18 yield
[A-] )
(20)
/
sin θ ∂ 1 ∂ ∂ + 2 2 ∂r r ∂θ sin θ ∂θ
ψ/UEa2,
r)a
1 + B dφ1 B / φ ) 1, r* ) 1 A dr* 1 + B 1
where D*4 ) D*2D*2 with
D*2 ) a2D2 ) a2
( )
[H ]B eφ1 1+ exp Ka kT
If the surface potential is low, this expression can be approximated by13
σ=-
3 / / 1 ∂2η* / 1 ∂η* ∂γ˘ r θ ∂η* 1 ∂ ψC γ ˘ + + r* ∂θ2 rθ r* ∂θ ∂θ ∂r* sin θ ∂r*3
eNS +
Since the charge density on drop surface, σ, is - e[A-], we have
(15) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1996, 47, 520.
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U in the z direction. Since the drop liquid and the dispersion medium are immiscible, there is no net flow across dropdispersion medium interface. Therefore
Vr,C ) Vr ) 0, r ) a
(29)
where Vr and Vr,C are respectively the r component of the fluid velocity of the dispersion medium and that of the dispersed phase. On cell surface, we assume
Vr ) -U cos θ and (∇ × v)φ ) 0, r ) b
(30)
where the subscript φ denotes the φ component of ∇ × v. It is also required that both Vr,C and Vθ,C are finite at the center of a drop (r ) 0). It can be shown that in terms of the scaled stream function, the boundary conditions for the flow field become
∂ψ/C ∂ψ* ) | | ∂r* r*)1- ∂r* r*)1+
[
(31)
η0 2 ∂ 1 ∂2 [1 + (λ*γ˘ *)2](n-1)/2 µ r* ∂r*2 r*2 ∂r* 1 ∂2 cos θ ∂ / + ψC|r*)1- ) r*3 ∂θ2 r*3 sin θ ∂θ 1 ∂2 1 ∂2 2 ∂ cos θ ∂ - 3 2+ 3 - 2 ψ*|r*)1+ (32) 2 r* ∂r* r* ∂r* r* ∂θ r* sin θ ∂θ
]
[
ψ* ) ψ/C ) 0 and
]
∂ψ/C
∂ψ* ) ) 0, r* ) 1 ∂r* ∂r*
(33)
1 ψ ) U*r*2 sin2 θ and 2 cos θ ∂ ∂2 1 ∂2 ψ* ) 0, r* ) 1/H (34) + ∂r*2 r*2 ∂θ r*2 ∂θ2
(
)
ψ/C ) 0 and
∂ψ/C ) 0, r* ) 0 ∂r*
(35)
/ ∂φ/1 ∂φ/2 ∂ψ/ ∂ψC ) ) ψ* ) ψ/C ) ) ) 0, θ ) 0 or θ ) π ∂θ ∂θ ∂θ ∂θ (36)
where U* ) U/UE. Equation 36 arises from the symmetric nature of the present problem. Electrophoretic Mobility. The electrophoretic mobility of a drop can be evaluated by that the sum of the z component of the electrical force, FEz, and that of the hydrodynamic force, FDz, vanishes at steady state, FEz + FDz ) 0. FEz can be evaluated by6,16
FEz )
∫σ(- ∇φ)S‚δz dA ) ∫[(∇φ)S‚δr][(∇φ)S‚δz] dA
(37)
where δr, and δz are respectively the unit vector in the r direction and that in the z direction. It can be shown that6,16 / kT 2 dφ1 | FEz ) - 2π e dr* r*)1
( )
∫0π
∂φ2* 2 sin θ|r*)1 dθ ∂θ
(38)
(16) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240.
Figure 2. Variation of scaled mobility U/m as a function of κa for the case when H ) 0.5, A ) 10, and B ) 10. Curve 1, rigid particles, 2, Newtonian drops, 3, Carreau drops with n ) 0.8 and λ* ) 1.5.
FDz can be evaluated by6,16,17
( )∫ () ( ) [∫
FDz ) π
2
∂(E*2ψ/r*2) |r*)1 sin θ dθ + 0 ∂r* / 2 π ∂φ2 / kT (κa)2 0 sin 2 θ dθ ) π φ| e ∂θ 1 r*)1 2 2 2 π ∂(E* ψ/r* ) kT π r*)1 sin θ dθ + 0 e ∂r* / π ∂φ2 / sin2 θ dθ (39) (κa)2 0 φ ∂θ 1 r*)1 kT e
π
∫
|
∫
( )|
]
The electrophoretic mobility of a drop is defined as Um ) U/Ez, or in terms of scaled symbols as U/m ) U*/E/z . For a given applied electric field, φ1 and φ2 are determined by solving eqs 9 and 10, and the results obtained are used to evaluate FEz through eq 38. Since FEz + FDz ) 0, eq 39 can be used to solve the stream function ψ, which is used to evaluate the velocity of a drop, and the electrophoretic mobility can be calculated. The governing equations and the associated boundary conditions are solved numerically by a two-dimensional pseudo-spectral method12,18 coupled with a Newton-Raphson iteration scheme. For the electrokinetic problems of the present type, the numerical approach adopted has the merits such as a fast rate of convergence, the convergent properties independent of the boundary conditions considered, and the mini-max property typically associated with the Chebyshev polynomial is maintained.8
Results and Discussion In this section, the influences of the key parameters of the system under consideration on its electrophoretic behavior are investigated through numerical simulation. For illustration, we assume that (η0/µ) ) 1, E/z ) 1, and R ) 1 in subsequent discussions. Figure 2 illustrates the variation of the scaled mobility U/m as a function of κa; the corresponding results for the case of rigid particles and Newtonian drops are also presented for comparison. This figure shows that, for a fixed value of κa, the magnitudes of the mobility of various entities follow the order Carreau drops > Newtonian drops > rigid particles. It is known that the drag on a rigid sphere in a Newtonian fluid, FD ) 6πµaUE, is greater than that on a drop, which is between 4πµaUE and 6πµaUE,17 and therefore, the U/m in the former is smaller than (17) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff, 1983. (18) Chu, J. W.; Lin, W. H.; Lee, E.; Hsu, J. P. Langmuir 2001, 17, 6289.
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Figure 3. Variation of scaled viscosity η* at θ ) π/2 as a function of scaled radius r* for the case when H ) 0.5, A ) 10, B ) 10, and κa ) 1.995. Curve 1: Carreau drops with n ) 0.8 and λ* ) 1.5, 2, Newtonian drops.
Figure 5. Variation of scaled viscosity η* at θ ) π/2 as a function of scaled radius r* for various values of A for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, B ) 10, and κa ) 1.995.
Figure 4. Variation of scaled mobility U/m as a function of κa at various values of A for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, and B ) 10.
Figure 6. Variation of scaled surface potential φ/1|r*)1 as a function of κa at various values of A for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, and B ) 10.
that in the latter. Note that a no-slip boundary condition is assumed on the surface of a rigid particle and in contrast, the velocity is assumed to be continuous on the drop surface. Since the velocity gradient in the former is greater than that in the latter, so is the drag. Furthermore, because a rigid particle can be viewed as a fluid with an infinitely large viscosity, the drag on its surface is larger than that on a drop surface. For Carreau and Newtonian drops, because the shear-thinning nature of the former leads to a smaller viscosity, as illustrated in Figure 3, the corresponding drag becomes smaller, and the mobility is larger. Influence of A. Figure 4 shows the variation of the scaled mobility U/m as a function of κa at various values of parameter A. This figure indicates that for a fixed value of κa, U/m increases with the increase in A. This is because the definition of A ) e2NSa/kT implies that it is proportional to the number of dissociable functional group on liquid surface NS ()[HA]+[A-]). Since the equilibrium constant Ka is fixed when T is fixed, the larger the value of A the higher the concentration of [A-], the larger the amount of fixed charge carried by a drop, the greater the electric force acting on it, and the larger its mobility. This can also be explained from the flow field point of view. Figure 5, for example, shows that, for a fixed value of r*, the larger the value of A the smaller the viscosity of drop, which leads to a smaller drag and a larger mobility. The variations of the scaled surface potential φ/1|r*)1 as a function of κa at various values of A are presented in Figure 6. According to this figure, |(φ/1|r*)1)| declines with the increase in
κa, and the smaller the value of A, the faster the rate of decrease. According to eq 23, since (dφ/1/dr*)r*)1 increases with the increase in κa, for fixed values of A and B, |φ/1|r*)1| declines with the increase in κa; that is, φ/1|r*)1 increases from a negative value to zero as κa increases. Both Figures 2 and 4 reveal that U/m has a local maximum as κa varies. This is because, if κa is small, the double layer surrounding a drop is thick, the overlapping between neighboring double layers is serious, the movement of a drop is retarded, and its mobility becomes small. As κa increases, the degree of influence of double-layer overlapping declines and the mobility of a drop increases. However, because |φ/1|r*)1| decreases with the increase of κa, and as κa f ∞, |φ/1|r*)1| f 0, and U/m f 0. The result that U/m has a local maximum as κa varies can also be explained by the behavior of (dφ/1/dr*)r*)1. If κa is small, (dφ/1/dr*)r*)1 is small too, and since the amount of surface charge is proportional to (dφ/1/dr*)r*)1, the electric force driving force acting on a drop is small, so is its mobility. As κa increases, (dφ/1/dr*)r*)1 increases too, and the mobility increases accordingly. However, as κa increases, the thickness of the double layer surrounding a drop declines, and since the available space for the dissociated counterions decreases, the degree of dissociation of the functional groups on drop surface decreases, and the mobility declines accordingly. Influence of B. The influence of parameter (B ) [H+]0/Ka) on the mobility of a drop is illustrated in Figure 7, which reveals that the larger the value of B the smaller the mobility is. This is because the value of B is proportional to the bulk concentration
1916 Langmuir, Vol. 22, No. 4, 2006
Figure 7. Variation of scaled mobility U/m as a function of κa at various values of B for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, and A ) 10.
Lee et al.
Figure 9. Variation of scaled surface potential U/m|r*)1 as a function of κa at various values of B for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, and A ) 10.
Figure 8. Variation of scaled viscosity η* at θ ) π/2 as a function of scaled radius r* at various values of B for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, A ) 10, and κa ) 1.995.
of H+, [H+]0, and the higher the [H+]0 or the lower the pH, the higher the H+ on drop surface, [H+]S. This leads to a higher degree of association between H+ and A-, the amount of surface charge declines, and the mobility decreases accordingly. The behavior of the mobility of a drop as B varies can also be explained by the behavior of the viscosity of the drop shown in Figure 8, where the scaled viscosity of drop η* is plotted against the scaled radial distance r* at various values of B for the case when θ ) π/2. As can be seen from this figure, for a fixed value of r*, the larger the value of B the larger the value of η*, which leads to a smaller mobility. The behavior of φ/1|r*)1 shown in Figure 9 also provides justification that the larger the value of B the smaller the mobility of a drop, because the larger the value of B the smaller the value of |φ/1|r*)1|. It is worth noting that, although φ/1|r*)1 at different values of A approaches the same value as κa f 0, as seen in Figure 6, this is not the case for φ/1|r*)1 as B varies. Figure 9 indicates that the smaller the value of B the larger the value of |φ/1|r*)1| as κa f 0. On the other hand, regardless of the values of A and B, φ/1|r*)1f0 as κa f ∞, as suggested by Figures 6 and 9. Influence of n and λ*. The influence of the nature of a Carreau fluid, characterized by parameters n and λ* on the mobility of a particle is illustrated in Figure 10; the corresponding results for the case of Newtonian drops (n ) 1.0 or λ* ) 0.0) are also presented for comparison. This figure reveals that, in general, the more significant the shear-thinning nature of a Carreau fluid, that is, a smaller value of n or a larger value of λ*, the larger the mobility, which is expected since shear-thinning leads to a
Figure 10. Variation of scaled mobility U/m as a function of κa for various values of n at λ* ) 1.5, (a), and for various values of λ* at n ) 0.8, (b), for the case when H ) 0.5, A ) 10, and B ) 10.
decrease in the drag acting on a drop. Note that if κa is small, the double layer surrounding a drop is thick and the overlapping between adjacent double layers is significant, which limits the movement of the drop, and therefore, U/m is small. In this case, the influence of n and λ* on U/m becomes inappreciable. On the other hand, if κa is large, since the absolute value of the electrical potential gradient on the surface of a drop is large, so is the electric force acting on it, its mobility becomes large, and the effect of shear-thinning is pronounced. Figure 10 indicates that the influence of the nature of a Carreau fluid on the mobility of a drop seems to be not very important. This is because for the range of the parameters examined the variation of the viscosity of the drop fluid is inappreciable, as can be seen in Figure 11.
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Figure 13. Variation of scaled viscosity η* at θ ) π/2 as a function of scaled radius r* at various combinations of H and κa for the case when n ) 0.8, λ* ) 0.5, A ) 10, and B ) 10. Curves 1 and 4, H ) 0.56; 2 and 5, H ) 0.5; 3 and 6, H ) 0.45. Curves 1-3, κa ) 1.995; curves 4-6, κa ) 10.
Figure 11. Variation of scaled viscosity η* at θ ) π/2 as a function of scaled radius r* for various values of n at λ* ) 1.5, (a), and for various values of λ* at n ) 0.8, (b), for the case when H ) 0.5, A ) 10, B ) 10, and κa ) 1.995.
Figure 12. Variation of scaled mobility U/m as a function of κa at various values of H for the case when n ) 0.8, λ* ) 0.5, A ) 10, and B ) 10.
Influence of Drop Concentration. The influence of the concentration of drops, measured by parameter H ()a/b) on the electrophoretic behavior of a drop is illustrated in Figure 12. This figure shows that, for a fixed value of κa, the larger the value of H, the smaller the value of U/m. Also, for the range of drop concentration considered, the influence of H on U/m is not significant for the case when κa is either very large or very small. These are expected because the higher the concentration of drops the more important the interaction of neighboring double layers and the smaller the mobility. If κa is very small, regardless of the concentration of drops, the overlapping between neighboring double layers is always serious, and if κa is very large, it is
Figure 14. Contours of scaled stream function, (a), and scale viscosity, (b), for the case when H ) 0.5, n ) 0.8, λ* ) 1.5, A ) 10, B ) 10, and κa ) 1.
always not important except for a very high drop concentration. The behavior of U/m seen in Figure 12 can also be explained by the variation of the scaled viscosity of a drop illustrated in Figure 13. According to this figure, for a fixed value of r*, the smaller the value of H and/or the larger the value of κa, the smaller the viscosity of a drop, which leads to a larger mobility.
1918 Langmuir, Vol. 22, No. 4, 2006
Figure 14 shows typical contours for the scaled stream function and the scaled viscosity. In our case, the dissociation of the functional groups on the surface of a drop leads to a negatively charged surface. Since the applied electric field is in the z direction, the drop moves toward the -z direction, and therefore, the liquid on the right-hand side of the drop flows counterclockwise and that on its left-hand side flows clockwise. Note that since the fluid outside a drop is Newtonian the scaled apparently viscosity η* is unity. For the fluid inside the drop, due to the shearthinning nature of Carreau fluid, η* has the lowest value near drop surface. Note that the conditions that both the velocity and the shear stress are continuous on drop surface lead to a positiondependent apparent viscosity on that surface, that is, η*|r*)1- ) η*(θ)|r*)1-. Equation 9 is appropriate for the case when the surface potential of a drop ζp does not exceed 25.6 mV. Here, we assume that the applied electric field is weak relative to the electric field established by ζp. Since the strength of the latter is on the order of κζp, the magnitude of the strength of the applied electric field E must be smaller than this value. In our study, because κ ranges from 105 to 107 m-1, E must be lower than 25.6 kV/m, which is satisfied for conditions of practical significance.
Lee et al.
Conclusions The classic result of Smoluchowski for the electrophoresis of an isolated, rigid particle is extended to the case of a dispersion of non-Newtonian drops with a charge-regulated surface under the conditions of low surface potential, uniform weak applied electric field, and arbitrary double layer thickness. The results of numerical simulation can be summarized as follows: (a) For a fixed thickness of double layer, the magnitudes of the mobility of various entities follow the order Carreau drops > Newtonian drops > rigid particles. (b) The higher the concentration of dissociable functional groups on the surface of a drop, the larger its mobility. (c) The mobility of a drop exhibits a local maximum as the thickness of double layer surrounding it varies. (d) The lower the pH, the smaller the mobility. (e) In general, the more significant the shear-thinning nature of the drop fluid, the larger the mobility. (f) In general, the higher the concentration of drops, the smaller the mobility. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA052639S
