Langmuir 2002, 18, 10475-10485
10475
Osmosis through a Fibrous Medium Caused by Transverse Electrolyte Concentration Gradients Huan J. Keh* and Yeu K. Wei Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC Received April 29, 2002. In Final Form: September 30, 2002
The steady diffusioosmotic flow of an electrolyte solution in the fibrous porous medium constructed by a homogeneous array of parallel charged circular cylinders is analytically studied. The imposed electrolyte concentration gradient is constant and normal to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model which allows for the overlap of the double layers of adjacent cylinders is employed. The electrokinetic equations that govern the ionic concentration distributions, the electrostatic potential profile, and the fluid flow field in the electrolyte solution surrounding the charged cylinder in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the cylinder as the small perturbation parameter. Analytical expressions for the diffusioosmotic velocity of the electrolyte solution as a function of the porosity of the ordered array of cylinders correct to the second order of their surface charge density or zeta potential are obtained from a balance between the electrostatic and the hydrodynamic forces exerted on each cylinder. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic velocity of an isolated circular cylinder caused by the transversely imposed electrolyte concentration gradient.
1. Introduction The electrokinetic phenomena in porous media with charged surfaces are of much fundamental and practical interest in various fields of science and engineering. In general, driving forces for the flow of electrolyte solutions in a small pore with a charged wall include dynamic pressure differences between the two ends of the pore (a streaming potential is developed as a result of no net electric current) and tangential electric fields that interact with the electric double layer adjacent to the pore wall (electroosmosis). Problems of electrokinetic flow caused by these well-known driving forces were studied extensively in the past.1-4 Another driving force for the electrokinetic flow in a micropore, which has commanded less attention, involves tangential concentration gradients of an ionic solute that interacts with the charged pore wall. This solute-wall interaction is electrostatic in nature, and its range is the Debye screening length κ-1. The fluid motion associated with this mechanism, known as “diffusioosmosis” (and in contrast to the diffusiophoresis of colloidal particles in prescribed solute concentration gradients), has been discussed mainly for solutions near a plane wall.4-7 Electrolyte solutions with a concentration gradient of order * To whom correspondence should be addressed. FAX: +8862-23623040. E-mail:
[email protected]. (1) Smoluchowski, M. In Handbuch der Electrizitat und des Magnetismus (Graetz); Barth: Leipzig, Germany, 1921; Vol. II, p 336. (2) Burgreen, D.; Nakache, F. R. J. Phys. Chem. 1964, 68, 1084. (3) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017. (4) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (5) Prieve. D. C.; Anderson, J. L.; Ebel, J.; Lowell, M. E. J. Fluid Mech. 1984, 148, 247. (6) Aderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61. (7) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142.
1 M/cm along rigid surfaces with a zeta potential of order kT/e (∼25 mV; e is the charge of a proton, k is Boltzmann’s constant, and T is the absolute temperature) can flow by diffusioosmosis at rates of several micrometers per second. The analytical solution for the diffusioosmotic velocity of electrolyte solutions parallel to a charged plane wall5 can be applied to the corresponding flow in capillary tubes and slits when the thickness of the double layer adjacent to the capillary wall is small compared with the capillary radius. However, in some practical applications involving dilute electrolyte solutions in very fine pores, this condition is no longer satisfied, and the dependence of the fluid flow on the electrokinetic radius κR, where R is the radius of a capillary tube or the half thickness of a capillary slit, must be taken into account. Recently, the diffusioosmosis of electrolyte solutions in a capillary tube or slit with an arbitrary value of κR was analyzed for the case of small surface potential or surface charge density at the capillary wall.8 Closed-form formulas for the fluid velocity profile and average fluid velocity on the cross section of the capillary tube and slit were obtained. The capillary model of porous media is not a realistic model for either granular or fibrous systems, for it does not allow for the convergence and divergence of flow channels. For diffusioosmotic flow within beds of particles or microporous membranes, it is usually necessary to account for the effects of pore geometry, tortuosity, etc. To avoid the difficulty of the complex geometry appearing in beds of particles, unit cell models were often employed to predict these effects on the relative motions between a granular bed and the bulk fluid. These models involve the concept that a bed of identical particles can be divided into a number of identical cells, one particle occupying each cell at its center. The boundary value problem for (8) Keh, H. J.; Wu, J. H. Langmuir 2001, 17, 4216.
10.1021/la020400v CCC: $22.00 © 2002 American Chemical Society Published on Web 11/19/2002
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multiple particles is thus reduced to the consideration of the behavior of a single particle and its bounding envelope. The most acceptable of these models with various boundary conditions at the outer (virtual) surface of a cell are the “free-surface” model of Happel9 and “zero-vorticity” model of Kuwabara.10 In the past three decades, the unit cell models were used by many researchers to predict various electrokinetic properties such as the mean electrophoretic mobility,11-19 effective electric conductivity,14,17-20 mean sedimentation rate,21-23 and sedimentation potential21-24 of a suspension of charged spherical particles as well as the electroosmotic mobility of electrolyte solutions in a fibrous porous medium.25-28 Recently, using the cell models, the present authors derived analytical expressions for the mean diffusiophoretic velocity of a swarm of dielectric spheres,29,30 which can also apply for the diffusioosmotic velocity within a fixed bed of charged spheres. In this work, the Happel and Kuwabara cell models are used to obtain analytical expressions for the diffusioosmotic velocity of a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient within a homogeneous array of parallel charged circular cylinders. No assumption is made about the thickness of the electric double layers relative to the radius of the cylinders, and the overlap of adjacent double layers is allowed. In the next section, we present the fundamental electrokinetic equations and boundary conditions that govern the electrolyte ion distributions, the electrostatic potential profile, and the fluid flow field for the system of a dielectric cylinder in a unit cell. These basic equations are linearized assuming that the ion concentrations, the electric potential, and the fluid pressure have only a slight deviation from equilibrium due to the application of the electrolyte gradient. In section 3, the problem of diffusioosmosis around the charged cylinder in the cell containing the electrolyte solution is solved. Using the Debye-Huckel approximation, we first obtain the solution of the equilibrium electric potential distribution. The linearized electrokinetic equations are then transformed into a set of differential equations by a regular perturbation method with the surface charge density of the cylinder as the small perturbation parameter. The perturbed ion concentration (or electrochemical potential energy), elec(9) Happel, J. AIChE J. 1958, 4, 197. (10) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (11) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (12) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Yu. B. Colloid J. 1981, 43, 434. (13) Zharkikh, N. I.; Shilov, V. N. Colloid J. 1982, 43, 865. (14) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (15) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166. (16) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (17) Johnson, T. J.; Davis, E. J. J. Colloid Interface Sci. 1999, 215, 397. (18) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 236, 180. (19) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2001, 243, 351. (20) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 443. (21) Levine, S.; Neale, G. H.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 424. (22) Keh, H. J.; Ding, J. M. J. Colloid Interface Sci. 2000, 227, 540. (23) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Colloids Surf. A 2001, 195, 157. (24) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295. (25) Zharkikh, N. I.; Borkovskaya, Yu. B. Colloid J. 1982, 43, 520. (26) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (27) Ohshima, H. J. Colloid Interface Sci. 1999, 210, 397. (28) Lee, E.; Lee, Y.; Yen, F.; Hsu, J. J. Colloid Interface Sci. 2000, 223, 223. (29) Wei, Y. K.; Keh, H. J. Langmuir 2001, 17, 1437. (30) Wei, Y. K.; Keh, H. J. J. Colloid Interface Sci. 2002, 248, 76.
Keh and Wei
Figure 1. Geometrical sketch for the diffusioosmosis of an electrolyte solution around a circular cylinder in a coaxial cylindrical cell.
tric potential, fluid velocity, and dynamic pressure profiles are determined by solving this set of differential equations subject to the appropriate boundary conditions. Analytical expressions for the diffusioosmotic velocity of the electrolyte solution result from satisfying the requirement that the net force acting on the cylinder vanishes. Finally, a closed-form expression in the limiting case of maximum porosity and typical numerical results of the diffusioosmotic velocity in the fiber matrix are presented in section 4. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. 2. Basic Electrokinetic Equations We consider the diffusioosmotic flow of a fluid solution of a symmetrically charged binary electrolyte through a uniform array of parallel, identical, charged, circular cylinders at the steady state. The applied electrolyte concentration gradient ∇n∞ is a constant equal to |∇n∞|ex, and the bulk diffusioosmotic velocity of the solution is -Uex, where ex is a unit vector in the positive x direction which is perpendicular to the axes of the cylinders. The electrolyte ions can diffuse freely in the fibrous porous medium, so there exists no regular osmotic flow of the solvent. As shown in Figure 1, we employ a unit cell model in which each dielectric cylinder of radius a is surrounded by a coaxial circular cylindrical shell of the fluid solution having an outer radius of b such that the fluid/cell volume ratio is equal to the porosity 1 - φ of the fiber matrix; viz., φ ) (a/b)2. The cell as a whole is electrically neutral. The origin 0 of the polar coordinate system (F,φ) is taken at the axis of the cylinder, and the polar axis φ ) 0 points toward the positive x direction. Obviously, the twodimensional problem for each cell is symmetric about the x-axis. The differential equations describing diffusioosmosis of electrolyte solutions are the same as those for electroosmosis. The difference is in the boundary conditions prescribed at the outer (virtual) surface of the cell. Conservation of both ionic species requires that
∇‚J( ) 0
(1)
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Langmuir, Vol. 18, No. 26, 2002 10477
where J((F,φ) are the number flux distributions and the subscripts + and - refer to the cation and anion, respectively. If the solution is dilute, the fluxes can be given by
(
)
Zen( ∇ψ J( ) n(u - D( ∇n( ( kT
η∇ u ) ∇p + Ze(n+ - n-)∇ψ
(3)
∇‚u ) 0
(4)
where η is the viscosity of the fluid and p(F,φ) is the dynamic pressure distribution. The local potential ψ and the space charge density are related by Poisson’s equation
∇2ψ ) -
4π Ze(n+ - n-)
F ) b:
(5)
In this equation, ) 4π0r, where r is the relative permittivity of the electrolyte solution and 0 is the permittivity of a vacuum. In eqs 2, 3, and 5, the properties D(, η, and are assumed to be constant. The boundary conditions at the surface of the dielectric cylinder are
(7a)
∂ψ ∂ψ∞ ) ∂F ∂F
(7b)
uF ) -U cos φ
(7c)
[ ()
τFφ ) η F
u)0
(6a)
eF‚J( ) 0
(6b) (6c)
where eF is the unit normal outward from the surface and σ is the uniform surface charge density. In eq 6a, we have assumed that the “shear plane” coincides with the cylinder surface. Equations 6b and 6c state that no ions can penetrate into the cylinder and the Gauss condition holds at its surface, respectively. Because the bulk concentration of the electrolyte, n∞, is not uniform, it is required that the total fluxes of cations and anions are balanced in order to have no current arising from the diffusive fluxes of the electrolyte ions in an electrically neutral solution, and an electric field E∞ ) -∇ψ∞ occurs spontaneously due to the difference in ionic mobilities. At the virtual surface of the cell, the local ionic concentration gradient and electric field are compatible to the gradient ∇n∞ and induced field E∞, respectively.
]
∂ uφ 1 ∂uF + )0 ∂F F F ∂φ
(7d)
(for the Happel model)
(∇ × u)z )
1 ∂ 1 ∂uF (Fu ) )0 F ∂F φ F ∂φ
(7e)
(for the Kuwabara model), where U is the diffusioosmotic velocity to be determined and uF and uφ are the F and φ components, respectively, of u. The overlap of the adjacent double layers is allowed in eqs 7a and 7b. The Happel cell model assumes that the radial velocity relative to the bulk flow and the shear stress of the fluid on the outer boundary of the cell are zero, while the Kuwabara cell model assumes that this radial velocity and the vorticity of the fluid are zero there. Note that the Happel model has an advantage over the Kuwabara model in that the former does not require an exchange of mechanical energy between the cell and the environment.31 To calculate the diffusioosmotic velocity of the fluid, we shall assume that the prescribed electrolyte gradient is not high (R ) a|∇n∞|/n∞(0) , 1) and hence that the electric double layer surrounding the cylinder is only slightly distorted from equilibrium by the application of the gradient. Therefore, the concentration distribution of each ionic species, the electric potential distribution, and the dynamic pressure distribution have small deviations from equilibrium, and one can write
F ) a:
4π eF‚∇ψ ) - σ
∂n( ∂n∞ ) ∂F ∂F
(2)
where Z is the valence of the symmetric electrolyte which is positive, n((F,φ) and D( are respectively the concentration (number density) distributions and diffusion coefficients of the ionic species, u(F,φ) is the fluid velocity field relative to the cylinder, and ψ(F,φ) is the electrostatic potential distribution. The first term on the right-hand side of eq 2 is the convection of the ionic species by the fluid, and the second term denotes the diffusion and electrically induced migration of the species. We assume that the Reynolds number of the fluid motion is very small, so the inertial effect on the fluid momentum balance can be neglected. The fluid flow is governed by the Stokes equations modified with the electrohydrodynamic effect, 2
Thus, the boundary conditions there are11,30
p ) p(eq) + δp
(8a)
n( ) n(eq) ( + δn(
(8b)
ψ ) ψ(eq) + δψ
(8c)
(eq) where p(eq)(F), n(eq) (F) are the equilibrium ( (F), and ψ distributions of dynamic pressure, ionic concentrations, and electric potential, respectively, and δp(F,φ), δn((F,φ), and δψ(F,φ) are the small perturbations to the equilibrium state (in which no bulk concentration gradient or electric field is imposed). The equilibrium concentrations of the ions are related to the equilibrium potential by the Boltzmann distribution. Substituting eq 8 into eqs 1, 3, and 5, canceling their equilibrium components, and neglecting the products of the small quantities u, δn(, and δψ, one obtains
∇2δµ( ) (
(
)
Ze kT ∇ψ(eq)‚∇δµ( ∇ψ(eq)‚u kT D(
(9)
(31) Happel, J.; Brenner, H. In Low Reynolds Number Hydrodynamics; Nijhoff: Dordrecht, 1983.
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Langmuir, Vol. 18, No. 26, 2002
η∇2u ) ∇δp ∇2δψ )
Keh and Wei
2 (eq) (∇ ψ ∇δψ + ∇2δψ ∇ψ(eq)) 4π
[ (
)
(10)
4πZen∞(0) Zeψ(eq) exp (δµ- + Zeδψ) kT kT Zeψ(eq) (δµ+ - Zeδψ) (11) exp kT
(
)
]
equation, the boundary condition 6c, and the requirement of no electric current passing through the virtual surface F ) b at equilibrium. It is easy to show that
ψ(eq)(F) ) ψeq1σ j + O(σ j 3)
where σ j ) 4πZeσ/κkT, which is the nondimensional surface charge density of the cylinder, and
Here δµ((F,φ) is defined as a linear combination of δn( and δψ based on the concept of the electrochemical potential energy
δn( δµ( ) kT (eq) ( Zeδψ n(
(12)
The boundary conditions for δµ( and δψ resulting from eqs 6b,c and 7a,b and their equilibrium state are
F ) a: ∂δµ( )0 ∂F
(13a)
∂δψ )0 ∂F
(13b)
∂δµ( R ) kT(1 - β) cos φ ∂F a
(14a)
kT R ∂δψ )β cos φ ∂F Ze a
(14b)
F ) b:
(17)
ψeq1(F) )
kT I1(κb)K0(κF) + I0(κF)K1(κb) Ze I1(κb)K1(κa) - I1(κa)K1(κb)
(18)
In eq 18, In and Kn are the modified Bessel functions of order n of the first and second kinds, respectively, and κ is the reciprocal Debye length, defined by κ ) [8πZ2e2n∞(0)/kT]1/2, in which n∞(0) can be experimentally taken as the mean bulk concentration of the electrolyte in the vicinity of the cylinder (or in the cell). Expression 17 for ψ(eq) as a power series in the surface charge density of the cylinder up to O(σ j ) is the equilibrium solution to the linearized Poisson-Boltzmann equation that is valid for small values of the electric potential (the Debye-Huckel approximation). That is, the surface charge density or surface potential of the cylinder must be small enough for the potential field ψ(eq)(F) to remain small. Note that the contribution from the effects of O(σ j 2) to ψ(eq) disappears only for the case of symmetric electrolytes. Using eqs 17 and 18, one obtains a relationship between the surface potential and the surface charge density of the dielectric cylinder in a unit cell at equilibrium
σ j)W
Zeζ kT
(19)
where
where
β)
D+ - DD+ + D-
(15)
Expression 14b for the induced electric field is derived from the requirement that the ionic fluxes J+ and Jdefined by eq 2 are equal in the bulk solution.4,5,32 The fluid velocity u is a small perturbed quantity, and the boundary conditions for u have been given by eqs 6a, 7c, and either 7d or 7e. The boundary conditions of the ionic concentrations and the electric potential at the virtual surface F ) b may be taken as the distributions giving rise to the imposed gradient ∇n∞ in the cell when the cylinder does not exist.13,30 In this case, eqs 14a and 14b are replaced by
F ) b: R δµ( ) kT(1 - β) F cos φ a δψ ) -
kT R β F cos φ Ze a
(16a) (16b)
3. Solution for the Diffusioosmotic Velocity Before solving for the problem of diffusioosmosis around a charged cylinder in a unit cell filled with the solution of a symmetric electrolyte with a constant bulk concentration gradient ∇n∞, we need to determine the equilibrium electrostatic potential in the fluid phase first. The equilibrium potential ψ(eq) satisfies the Poisson-Boltzmann (32) Levich, V. G. In Physicochemical Hydrodynamics; Prentice Hall: Eaglewood Cliffs, NJ, 1962.
W)
I1(κb)K1(κa) - I1(κa)K1(κb) I1(κb)K0(κa) + I0(κa)K1(κb)
(20)
and ζ ) ψ(eq)(a), which is the equilibrium surface potential (known as the zeta potential) of the cylinder. Namely, the solution for ψ(eq)(F) given by eq 17 with the substitution of eq 19 is also valid for a dielectric cylinder with a constant surface potential in the cell. To solve the small quantities δµ(, u, δp, and δψ in terms of the diffusioosmotic velocity U when the parameter σ j is small, these variables can be written as perturbation expansions in powers of σ j
j + µ2(σ j 2 + ‚‚‚, δµ( ) µ0( + µ1(σ
(21a)
j + u2σ j 2 + ‚‚‚ u ) u1σ
(21b)
j + p2σ j 2 + ‚‚‚ δp ) p1σ
(21c)
δψ ) ψ0 + ψ1σ j + ψ2σ j 2 + ‚‚‚
(21d)
j + U2σ j 2 + ‚‚‚ U ) U1σ
(21e)
where the functions µi(, ui, pi, ψi, and Ui are independent of σ j . The zeroth-order terms of u, δp, and U disappear because the electrolyte solution around an uncharged cylinder will not move by imposing an electrolyte concentration gradient if only the electrostatic interaction is considered. Substituting the expansions given by eq 21 and ψ(eq) given by eq 17 into the governing eqs 4 and 9-11 and
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boundary conditions 6a, 7c, 7d or 7e, 13, and 14 or 16, and equating like powers of σ j on both sides of the respective equations, we obtain a group of linear differential equations and boundary conditions for each set of the functions µi(, ui, pi, and ψi with i equal to 0, 1, and 2. These perturbation equations can be analytically solved, and the results for the F and φ components of u, δp (to the j ) can be written order of σ j 2), δµ(, and δψ (to the order of σ as
{[
uF )
uφ )
U1F0F(F) -
{[
δp )
U1F0φ(F) -
] [
kT βRF1F(F) σ j + U2F0F(F) + ηa2 kT RF2F(F) σ j 2 cos φ (22a) 2 ηa
] [
]}
kT βRF1φ(F) σ j + U2F0φ(F) + ηa2 kT RF2φ(F) σ j 2 sin φ (22b) ηa2
{[
]}
kT η U1Fp0(F) - 2βRFp1(F) a ηa
] [
where
G1(F) ) G2(F) ) -
Fh ) a
j ] cos φ δµ( ) kT(1 - β)R[Fµ0(F) - Fµ1(F)σ
(23)
kT R[-βFψ0(F) + Fψ1(F)σ j ] cos φ δψ ) Ze
(24)
Here FiF(F), Fiφ(F), Fpi(F) (with i equal to 0, 1, and 2), Fµ0(F), Fµ1(F), Fψ0(F), and Fψ1(F) are dimensionless functions of F given by eqs A1 and A6-A8 in the Appendix. Note that j do not contain the solutions for δµ( and δψ to the order of σ the influence of the fluid motion. The total force exerted on the charged cylinder in a unit cell filled with an electrolyte solution can be expressed as the sum of the electric force and the hydrodynamic drag force. Since the net charge within a cell is zero, the electric force acting on the charged cylinder can be represented by the integral of the electrostatic force density over the fluid volume in the cell. Because the net electric force acting on the cylinder at the equilibrium state is zero, the leading order of the electric force per unit length of the cylinder is given by
∫0 ∫a
Fe ) 4π
2π
b
2
(eq)
(∇ ψ
2
∇δψ + ∇ δψ ∇ψ
(eq)
) F dF dφ. (25)
Substituting eqs 17 and 24 into eq 25 and performing the integration, we obtain
{ [
κ2a3 kT ψ (a)Fψ0(a) R -β 2 4Ze eq1 a b F 2 κ2a2b ψeq1(b)Fψ0(b) + 4π a G (F) dF σ j+ 4Ze a 1 κ2a3 κ2a2b ψeq1(a)Fψ1(a) ψeq1(b)Fψ1(b) + 4Ze 4Ze b F 2 4π a G (F) dF σ j 2 + O(σ j 3) ex (26) a 2
Fe )
∫
[
∫
()
()
]
]
}
dψeq1 Ze κ2a4 ψ (F)Fµ0(F) F (F) + 16πZeF µ1 kT eq1 dF (27b)
[
]
∫02π {-δpeF + η[∇u + (∇u)T]‚eF}F)a dφ
(28)
Substitution of eq 22 into the above equation results in
{[
kT βRC12 + a2
4πηU1C02 - 4π
] [
κ2akT βRψeq1(a)Fψ0(a) σ j + 4πηU2C02 + 4Ze
κ2akT kT βRψeq1(F)Fψ0(F) σ j + U2Fp0(F) + 2 RFp2(F) + 4πηZe ηa
]}
(27a)
In eqs 26 and 27, the function ψeq1(F) has been given by eq 18, and Fµi(F) and Fψi(F), with i equal to 0 and 1, are defined by eqs A6-A8. The hydrodynamic drag force acting on the cylinder per unit length is given by the integral of the hydrodynamic stress over the cylinder surface
Fh )
κ2akT Rψeq1(F)Fψ1(F) σ j 2 cos φ (22c) 4πηZe
dψeq1 κ2a4 Fµ0(F) 16πZeF dF
2
4π
]
}
kT κ akT Rψeq1(a)Fψ1(a) σ RC22 j 2 + O(σ j 3) ex 4Ze a2 (29)
For the case of a dielectric cylinder with a constant (equilibrium) surface potential in the cell, the parameter σ j in eqs 21-24, 26, and 29 can be replaced by Zeζ/kT in terms of eq 19. At the steady state, the total force acting on the cylinder is zero. Applying this constraint to the summation of eqs 26 and 29, one obtains
βR kT 2Θ1 4πηa Ze W
(30a)
R kT 2 Θ2 32πηa Ze W2
(30b)
U1 ) U2 )
( )
( )
where W is defined by eq 20, and Θi with i equal to 1 and 2 are functions of κa and φ ()(a/b)2) given by
{
(-8)i-2(κa)2Wi Ze 1 ψeq1(b)Fψ(i-1)(b) + 2 1/2 kT (1 + φ ) φ ω 2 16πZe b F2 F F 2F 1 1 2 ln + φ 1 + 2 ln a a κ2a3 a a2 a2 F2 Gi(F) dF (31a) a2
Θi )
∫
[
(
)
)]
(
}
for the Happel model, and
16πZe κ2a3
∫ab
[
{
Ze 3 ψ (b)Fψ(i-1)(b) + kT 2φ1/2ω′ eq1 F2 F 1 1 - 2 1 - 2ln - φ 1 a 2 a F2 F4 2 2 + 4 Gi(F) dF (31b) a a
Θi ) (-8)i-2(κa)2Wi
(
)
)]
(
}
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Langmuir, Vol. 18, No. 26, 2002
Keh and Wei
for the Kuwabara model. In the above equations
ω ) (1 - φ2 + ln φ + φ2 ln φ)-1
(
ω′ ) 1 -
1 4 2 φ + φ2 + ln φ 3 3 3
)
(32a)
-1
(32b)
From eqs 21e and 30, the diffusioosmotic velocity of the fluid can be expressed as
U)
1 R kT 2 Θσ β Θ1σ j+ j 2 + O(σ j 3) 4πηaW Ze 8W 2
( )[
]
(33)
Since the solutions for δµ( and δψ given by eqs 23 and 24 are not influenced by the fluid flow, the effect of the polarization (or relaxation) of the diffuse ions in the electric double layer surrounding the cylinder is not included in eq 33 up to the order σ j 2. Substitution of eq 19 into eq 33 yields an expression for the diffusioosmotic velocity as an expansion in powers of ζ
U)
1 R kT βΘ1ζ + Θ2ζ2 + O(ζ3) 4πηa Ze 8
[
]
(34)
Note that the diffusioosmotic velocity through the array of dielectric cylinders with a constant (equilibrium) surface potential ζ is the same as that for the cylinders with a constant surface charge density σ ) Wκζ/4π, and it can be expressed by either eq 33 or eq 34. 4. Results and Discussion It is understood that the diffusioosmosis of an electrolyte solution in a porous medium results from a linear combination of two effects: (i) chemiosmosis due to the nonuniform adsorption of counterions in the electric double layer over the charged surface, which is analogous to the diffusioosmosis of a nonionic solution;6,7 (ii) electroosmosis due to the macroscopic electric field generated by the concentration gradient of the electrolyte and the difference in mobilities of the cation and anion of the electrolyte, given by eq 14b or 16b. The terms in eqs 33 and 34 proportional to β (involving the function Θ1) represent the contribution from electroosmosis, while the remainder terms (containing the function Θ2) are the chemiosmotic component. Using the boundary condition 14b for the electric potential at the virtual surface of a unit cell, one can find that the function Θ1 given by eq 31b for the Kuwabara model is identical to that obtained by Kozak and Davis26 for the electroosmotic velocity of the fluid in an ordered array of charged cylinders. In the limit of the largest possible porosity of the fiber matrix (or φ f 0), eq 31 becomes
Θ1(φ)0) ) 1 + B(0,3,1,0) - 4B(0,5,1,0)
(35a)
Θ2(φ ) 0) ) 2 - 4B(0,3,1,0) + 2B(0,3,2,0) + K1(κa) - 2B(1,4,1,0)] - (κa)2[B(1,-1,2,0) [B 4 K0(κa) (1,2,1,0) B(1,1,2,0) + 2B(1,1,2,1)] (35b) where B(m,n,i,j) are functions of κa defined by
B(m,n,i,j) )
∫
∞ -n t (ln 1
t)j
[
]
Km(κat) K0(κa)
i
dt
(36)
The substitution of eq 35 into eq 33 or 34 also gives the diffusiophoretic velocity of an isolated circular cylindrical particle caused by the transversely imposed electrolyte
Figure 2. Plots of the function Θ2(φ)0) as calculated from the exact solution given by eq 35b (solid curve) and the curve-fit equation given by eq 38 (dashed curve). Table 1. Values of Θ1 and Θ2 as Functions of Ka Calculated from Eq 35 for the Limiting Case of φ ) 0 κa
Θ1(φ ) 0)
Θ2(φ ) 0)
κa
Θ1(φ ) 0)
Θ2(φ ) 0)
0 0.1 0.5 1 2 3 5
0.500 0.502 0.525 0.557 0.614 0.659 0.723
0.000 -0.043 -0.088 -0.086 -0.026 0.050 0.188
10 20 30 50 100 ∞
0.812 0.885 0.917 0.947 0.972 1.000
0.417 0.630 0.730 0.825 0.907 1.000
concentration gradient. Note that Θ1(φ)0) given by eq 35a is the same as that derived by Henry33 for the electrophoretic mobility of a dielectric circular cylinder in the direction normal to its axis. The values of Θ1 and Θ2 as functions of κa calculated from eq 35 for the limiting case of φ ) 0 are given in Table 1. It can be seen that Θ1(φ ) 0) is a monotonic increasing function of κa from Θ1 ) 1/2 at κa ) 0 to Θ1 ) 1 as κa f ∞. On the other hand, Θ2(φ)0) first decreases with κa from Θ2 ) 0 at κa ) 0 to a minimum value of about -0.0915 near κa ) 0.7 and then increases monotonically with κa until Θ2 ) 1 as κa f ∞. Ohshima27 suggested an approximate formula for Θ1(φ)0) as
Θ1(φ)0) )
{ [
]}
1 2.55 1+ 1+ 2 κa(1 + e-κa)
-2
(37)
which is applicable for all κa with relative errors less than 1%. On the other hand, a curve-fit equation for Θ2(φ)0) can be written as
Θ2(φ)0) ) 1 - 1.15[1 + 0.066(κa)1.15 0.02(κa)0.2]-1 + 0.15[1 + 15(κa)1.7]-0.8 (38) Figure 2 gives plots of the function Θ2(φ)0) as calculated from eqs 35b and 38. It can be seen that, for the most part of the entire range of κa, eq 38 can provide fair approximation for the values of Θ2(φ)0). In Figures 3 and 4, the numerical results for the function Θ1 calculated from eq 31 are plotted versus the parameters κa and φ for the unit cell model with various boundary conditions at the virtual surface of the cell. The calculations are presented up to φ ) 0.9, which corresponds to the maximum attainable volume fraction (or minimum po(33) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106.
Osmosis by Transverse Electrolyte Gradients
Figure 3. Plots of the function Θ1 as calculated from eq 31 versus κa with φ as a parameter: (a) when boundary condition 14 is used; (b) when boundary condition 16 is used. The solid and dashed curves represent the calculations for the Happel and Kuwabara models, respectively.
rosity) for a swarm of identical parallel cylinders (triangularly ordered).34 It can be seen that Θ1 is always positive and decreases monotonically with the decrease of κa (or with the increase of the double-layer overlap) for a specified value of φ. When κa ) 0, Θ1 ) 1/2 as φ ) 0, and Θ1 ) 0 for all finite values of φ. The effect of the porosity ()1 φ) of the fiber matrix on Θ1 can be significant even when the porosity is fairly high. When the boundary condition 14 for the electrochemical potential at the virtual surface of the cell is used, Θ1 is a monotonic decreasing function of φ for any given finite value of κa (and equals unity as κa f ∞, regardless of the value of φ) for the case of the Kuwabara model. However, for the case of the Happel model, Θ1 is not a monotonic function of φ and has a maximum (whose value can be greater than unity) for a given finite value of κa. The location of this maximum shifts to greater φ as κa increases. On the other hand, when the boundary condition 16 for the electrochemical potential at the outer surface of the cell is used, both the Happel and the Kuwabara models predict that Θ1 decreases monotonically with an increase in φ for an arbitrary fixed value of κa. For any combination of κa and (34) Berryman, J. G. Phys. Rev. A 1983, 27, 1053.
Langmuir, Vol. 18, No. 26, 2002 10481
Figure 4. Plots of the function Θ1 as calculated from eq 31 versus φ with κa as a parameter (a) when boundary condition 14 is used and (b) when boundary condition 16 is used. The solid and dashed curves represent the calculations for the Happel and Kuwabara models, respectively.
φ, the Kuwabara model results in a lower value of Θ1 (or a stronger porosity dependence for the electroosmotic velocity) than the Happel model does, which occurs because the zero-vorticity boundary condition yields a larger energy dissipation in the cell than that due to the drag on the cylinder alone,31 and the value of Θ1 calculated using the boundary condition 16 is always smaller than that calculated using the boundary condition 14. The results for the function Θ2 calculated from eq 31 are plotted versus the parameters κa and φ in Figures 5 and 6 for various cases of the cell model. It can be seen that Θ2 in general is not a monotonic function of φ for a specified value of κa. When the boundary condition 14 is used, Θ2 is not a monotonic function of κa for a given value of φ, and a local maximum and a minimum of this function would appear at some values of κa. The value of Θ2 is negative in the vicinity of each minimum and is positive otherwise. For relatively concentrated suspensions, the values of Θ2 in the vicinity of each local maximum may be greater than unity. For a combination of κa and φ not
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Langmuir, Vol. 18, No. 26, 2002
Figure 5. Plots of the function Θ2 as calculated from eq 31 versus κa with φ as a parameter (a) when boundary condition 14 is used and (b) when boundary condition 16 is used. The solid and dashed curves represent the calculations for the Happel and Kuwabara models, respectively.
close to these minima, the Kuwabara model predicts a smaller value of Θ2 than the Happel model does. On the other hand, when the boundary condition 16 is used, the value of Θ2, in general, is positive and increases with an increase in κa for a constant finite value of φ greater than about 0.1. For a fixed value of κa which is large, Θ2 decreases monotonically with an increase in φ. For a combination of κa and φ, the Kuwabara model leads to a smaller magnitude of Θ2 than the Happel model does. Figure 7 illustrates the dependence of the diffusioosmotic velocity within an array of identical charged cylinders on their dimensionless surface (zeta) potential at various values of κa and φ calculated for the Kuwabara model with boundary condition 16. The Kuwabara model with boundary condition 16 is chosen because the diffusiophoretic/diffusioosmotic mobility predicted by this case of the unit cell model agrees well with the ensembleaveraged result obtained by using the concept of statistical mechanics for dilute suspensions of dielectric spherical particles with thin but polarized double layers.29 The consequence that the boundary condition 14 is not as
Keh and Wei
Figure 6. Plots of the function Θ2 as calculated from eq 31 versus φ with κa as a parameter (a) when boundary condition 14 is used and (b) when boundary condition 16 is used. The solid and dashed curves represent the calculations for the Happel and Kuwabara models, respectively.
accurate as the boundary condition 16 is probably due to the fact that the angular component of the electrochemical/ electrostatic potential gradients at the virtual surface of the cell is not specified in eq 14. A possible reason for the outcome that the Kuwabara model is more appropriate than the Happel model for this suspension might be the fact that the zero-vorticity boundary condition is consistent with the irrotational-flow environment generated by a diffusiophoretic particle with a thin polarized double layer. However, note that except for the case with κa f ∞, the flow caused by two or more identical diffusiophoretic spheres is not irrotational.35 The magnitude of the diffusioosmotic velocity illustrated in Figure 7 is normalized by a characteristic value given by
U* )
R kT 4πηa Ze
( )
2
(39)
The case that the anion and cation diffusivities are equal
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Langmuir, Vol. 18, No. 26, 2002 10483
diffusion coefficients (β ) -0.2 is chosen). In this case, both the chemiosmotic and the electroosmotic effects contribute to the fluid flow and the net diffusioosmotic velocity is neither an even nor an odd function of the surface potential. It can be seen that U/U* is not necessarily a monotonic function of Zeζ/kT given constant values of κa and φ. Some of the curves in Figure 7b show that the fluid flow might reverse its direction more than once as the surface potential of the cylinders varies from negative to positive values. The reversals occurring at the values of Zeζ/kT other than zero result from the competition between the contributions from chemiosmosis and electroosmosis. In the limit κa ) 0, the diffusioosmotic velocity vanishes for any finite value of φ but reduces to the result U/U* ) βZeζ/2kT when the porosity of the fiber matrix approaches unity (with φ ) 0). Note that the situations associated with Figures 7a and 7b (taking Z ) 1) are close to the diffusioosmosis of the aqueous solutions of KCl and NaCl, respectively. In another cell-model analysis for the diffusioosmotic mobility of electrolyte solutions in an ordered array of identical charged cylinders, the assumption of thin but polarized double layers was made.36 It would be of interest to know the extent to which this assumption is valid. A comparison of the results of the reduced diffusioosmotic mobility obtained in this previous analysis with our present calculations for the Kuwabara model with boundary condition 16 is shown in Figure 8. For the case of Zeζ/kT ) -1 chosen in this figure, the polarization effect of diffuse ions in the double layers is negligible. It can be seen that the results of U/U* predicted by the thin-doublelayer analysis can be in significant errors when the value of κa is less than about 20. 5. Concluding Remarks
(β ) 0) is drawn in Figure 7a. Only the results at positive surface potentials are shown because the diffusioosmotic velocity, which is due to the chemiosmotic effect entirely, is an even function of the surface potential ζ as indicated by eq 34. Since our analysis is based on the assumption of small surface potential, the magnitudes of Zeζ/kT considered are less than 2. As expected, in this range of Zeζ/kT, the magnitude of the reduced diffusioosmotic velocity U/U* increases monotonically with an increase in Zeζ/kT for given values of κa and φ. There is no chemiosmotic motion for the special cases of Zeζ/kT ) 0 or κa ) 0. Figure 7b is plotted for the reduced diffusioosmotic velocity U/U* of the fluid in the fiber matrix for a case that the cation and anion of the electrolyte have different
In this paper, the steady-state diffusioosmosis of a solution of a symmetric electrolyte with a transversely imposed concentration gradient in an ordered array of identical charged cylinders with an arbitrary value of κa is analyzed using unit cell models with various boundary conditions at F ) b. Solving the linearized continuity equations of electrolyte ions, Poisson-Boltzmann equation, and modified Stokes equations applicable to the system of a cylinder in a unit cell by a regular perturbation method, we have obtained the ion concentration (or electrochemical potential energy) distributions, the electric potential profile, and the fluid flow field. The requirement that the total force exerted on the cylinder is zero leads to eqs 31-34 for diffusioosmotic velocity of the electrolyte solution as functions of the porosity of the array of cylinders correct to the order σ2 or ζ2. Equations 33 and 34 with Θ1 and Θ2 given by eq 35 for the limiting case of φ ) 0 can also be used to express the diffusiophoretic velocity of a charged circular cylinder in the direction normal to its axis. We note that the cell models with various boundary conditions at the virtual surface of the cell lead to somewhat different results of the diffusioosmotic velocity. Although these results provide meaningful information for the porosity effects on the diffusioosmosis through fibrous media, we still need to assess which boundary conditions are most likely applicable. As mentioned in the previous section, an earlier analysis of diffusiophoresis in suspensions of charged colloid spheres with thin but polarized electric double layers29 using unit cell models with the same boundary conditions at the virtual surface as the present work indicates that the tendency of the
(35) Tu, H. J.; Keh, H. J. J. Colloid Interface Sci. 2000, 231, 265.
(36) Keh, H. J.; Wei, Y. K. J. Colloid Interface Sci. 2002, 252, 354.
Figure 7. Plots of the reduced diffusioosmotic velocity of an electrolyte solution in an ordered array of identical cylinders versus the dimensionless zeta potential of the cylinders at fixed values of κa calculated for the Kuwabara model with boundary condition 16: (a) β ) 0; (b) β ) -0.2. The solid and dashed curves represent the cases of φ ) 0 and φ ) 0.2, respectively.
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Langmuir, Vol. 18, No. 26, 2002
Figure 8. Plots of the reduced diffusioosmotic velocity of an electrolyte solution in an ordered array of identical cylinders versus φ at Zeζ/kT ) -1 and several values of κa calculated for the Kuwabara model with boundary condition 16: (a) β ) 0; (b) β ) -0.2. The solid and dashed curves represent our present results and those obtained by using the thin-layer polarization model,36 respectively.
dependence of the normalized diffusiophoretic velocity on the particle volume fraction for the case of boundary conditions given by eq 14 is not correct, in comparison with the ensemble-averaged result obtained by using the concept of statistical mechanics. Therefore, it is recommended that the result calculated from the Kuwabara model described by eqs 7c and 7e with boundary condition 16 should be chosen for the system with thin double layers (large values of κa) because the zero-vorticity boundary condition 7e is consistent with the irrotational flow field outside the thin double layers. For systems with moderate to thick double layers, however, the result obtained from the Happel model specified by eqs 7c and 7d with boundary condition 16 should be chosen, since the free-surface boundary condition 7d has an advantage of not requiring an exchange of mechanical energy between the cell and the environment. Generally speaking, the predictions from the Happel and Kuwabara models are in numerical
Keh and Wei
agreement to within 20% and result in the same behavior qualitatively. In section 3, we set the total force (sum of the electric and hydrodynamic forces) on the charged cylinder equal to zero. Actually, this would give the velocity of the cylindrical fiber in a unit cell, and the fluid velocity is the negative of it. To hold a charged fiber fixed in an electrolyte gradient requires that a force be applied to the fiber, and the fluid will flow by it. This applied force would be equal in magnitude to the sum of the electric and hydrodynamic forces exerted on the fiber translating transversely with the obtained diffusiophoretic/diffusioosmotic velocity in the electrolyte solution with a uniform concentration having the prescribed value at the center of the fiber. For the case of a charged dielectric sphere in a cell filled with an electrolyte solution, the analytical relation between the applied force and the translational velocity of the particle was recently derived.22 Following the same procedure, one may calculate the applied force required to hold the charged cylinder fixed in a cell with an electrolyte gradient. Equations 31-34 are obtained on the basis of the Debye-Huckel approximation for the equilibrium potential distribution around the dielectric cylinder with a low zeta potential in a unit cell. The reduced formula for Θ1 of a single dielectric sphere with low zeta potential in an unbounded electrolyte solution was compared with the general solution and shown to give a good approximation for the case of reasonably high zeta potentials (with errors less than 4% for |ζ|e/kT e 2).37 Also, comparing with the numerical solution for the diffusiophoretic mobility of an isolated charged sphere in KCl and NaCl aqueous solutions obtained by Prieve and Roman38 valid for an arbitrary value of zeta potential, we find that the corresponding result for a dilute suspension of charged spheres with low zeta potential is also quite accurate for the entire range of |ζ|e/kT e 2.39 Therefore, our results in eqs 31-34 might be used tentatively for the situation of reasonably high electric potentials. To see whether our approximate solution can be extended to the higher values of electric potential, a numerical solution of the electrokinetic equations with no assumption on the magnitude of electric potential would be needed to compare it with the approximate solution. For a general case of diffusioosmosis/electroosmosis of a fluid solution in a fibrous system constructed by an array of parallel cylinders, the imposed electrolyte gradient/ electric field can be taken as a combination of its transversal and longitudinal components with respect to the orientation of the cylinders. Hence, the general problem can be divided into two, if it is linearized as done in Section 2, and they might be separately solved. The overall diffusioosmotic/electroosmotic velocity of the bulk fluid can be obtained by the vectorial addition of the twocomponent results. Since the problem of the transverse diffusioosmotic flow in a homogeneous array of parallel cylinders has been solved in the present paper, in a future work one only needs to treat the diffusioosmosis in the array generated by a longitudinal electrolyte gradient. This longitudinal problem is now under investigation using the unit cell model and will be presented in a subsequent article. (37) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (38) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday. Trans. 2 1987, 83, 1287. (39) Keh, H. J.; Wei, Y. K. Langmuir 2000, 16, 5289.
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Langmuir, Vol. 18, No. 26, 2002 10485
Acknowledgment. This research was supported by the National Science Council of the Republic of China under Grant NSC90-2214-E-002-019. Appendix: Definitions of Some Functions in Section 3 For conciseness, the definitions of some functions in section 3 are listed here. In eq 22
F a2 F2 FiF(F) ) Ci1 + Ci2 ln + Ci3 + Ci4 + a F a 2 2 F F F F (δi1 + δi2) ln F a G (F) dF - a ln FGi(F) dF a i a 1a2 F F4 1F2 F G (F) dF + G (F) dF , (A1a) 4a a i 4F a a i
()
∫
[
(
() ∫() ( )∫ ( )
()
)∫
Fiφ(F) ) -FiF(F) - F Fpi(F) ) -2Ci2
]
dFiF dF
(A1b)
a F F F + 8Ci4 - 2(δi1 + δi2) a Gi(F) dF + F a a a F F2 G (F) dF (A1c) F a a i
[
∫
∫
]
()
for i ) 0, 1, and 2. In the above equations, the functions Gi(F) are defined by eq 27, δij is the Kronecker delta which equals unity if i ) j but vanishes otherwise
Ci1 ) [(-1 + φ2)Ai + 2φ ln φBi]ω
(A2c)
Ci4 ) [-φ2Ai - (1 + ln φ)φBi]ω
(A2d)
for the Happel model, and
1 Ci1 ) [4(-1 + φ)A′i - (1 + 2φ ln φ - φ2)B′i]ω′ 6 (A3a) 1 Ci2 ) [4A′i + (1 - 2φ + φ2)B′i]ω′ 3
(A3b)
1 Ci3 ) [2(2 - φ)A′i + (1 - φ + φ ln φ)B′i]ω′ (A3c) 6 1 Ci4 ) [-2φA′i + (φ + φ ln φ - φ2)B′i]ω′ 6
(A3d)
for the Kuwabara model, where
b
a
4
b ln Gi(F) dF (A4a) F 2
i
i
a
b
i
i1
i2
b
i
a
2
2
i
a
b
a
i
(A5b) and ω and ω′ are defined by eq 32. In eqs 23 and 24
Fµ0(F) ) Fψ0(F) )
{
1F a + χa F
(
)
(A6)
2 ZeF a Aµ1(a,b) + (1 - χ)Bµ1(a,b) + 2kTaχ χF2 1-χ 1 1 Aµ1(a,b) + 2 Bµ1(a,b) + 2 Bµ1(a,F) + χ bφ F
Fµ1(F) )
[
]
Aµ1(F,b)
}
(A7)
Fψ1(F) ) g1I(κa)Aψ1(a,b) + g1K(κa)Bψ1(a,b) [g2K(κb)I1(κF) + g2K(κb)g1I(κa) - g2I(κb)g1K(κa) g2I(κb)K1(κF)] - I1(κF)Aψ1(b,F) + K1(κF)Bψ1(b,F) (A8)
Aψ1(x,y) )
( )
a2 dψeq1 dF F2 dF
(A9a)
dψeq1 dF dF
(A9b)
Aµ1(x,y) )
∫xy
Bµ1(x,y) )
∫xy (F2 - a2)
1-
Ze ψeq1(F)Fµ0(F)] dF ∫xy κ2FK1(κF)[Fµ1(F) + kT (A10a)
Bψ1(x,y) )
Ze ψeq1(F)Fµ0(F)] dF ∫xy κ2FI1(κF)[Fµ1(F) + kT (A10b)
g1I(x) ) I0(x) + I2(x), g1K(x) ) K0(x) + K2(x) (A11a-b) g2I(x) ) g1I(x), g2K(x) ) g1K(x), χ ) 1 - φ (A12a-c) (if eq 14 is used)
2
[( ) ∫ (aF) G (F) dF - (ab) ∫ G (F) dF] 2
2
(A2b)
Ci3 ) [Ai + (1 - ln φ)φBi]ω
1 a Bi ) (δi1 + δi2) 4 b
b
i
4
where
Ci2 ) [2(1 + φ )Ai + 4φBi]ω
∫ab (aF)
2
2
b
a
(A2a)
2
Ai ) δi0 + (δi1 + δi2)
[∫ (aF) ln bFG (F) dF + 1a 1b F G (F) dF - ( ) ∫ G (F) dF] (A5a) 4(b) ∫ (a) 4a F b B′ ) (δ + δ )[- ∫ ( ) G (F) dF + ( ) ∫ G (F) dF] a a
A′i ) δi0 + (δi1 + δi2)
b
a
i
(A4b)
g2I(x) ) -I1(x), g2K(x) ) K1(x), χ ) 1 + φ (A12d-f) (if eq 16 is used). LA020400V
