Simulation of Adsorption, Desorption, and Exchange Kinetics of

system of equations may be used to describe the kinetic- diffusion-controlled adsorption and desorption processes for multicomponent mixtures on a pla...
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Langmuir 1998, 14, 2864-2875

Simulation of Adsorption, Desorption, and Exchange Kinetics of Mixtures on Planar Surfaces. 1. Kinetic-Diffusion-Controlled Adsorption and Desorption for One-Component Mixtures Nadezhda L. Filippova* Russian Branch RTD Corporation, Moscow 111538, Russia Received February 20, 1996. In Final Form: September 10, 1997 The quantitative theory for the kinetic-diffusion-controlled adsorption and desorption onto planar substrates has been developed. A power series representation of the adsorption and desorption valid for short times and an asymptotic representation valid for long times are given for systems obeying any adsorption isotherms. For the diffusion-controlled adsorption and desorption the analytical solutions over a wide range of time are found for one-component mixtures satisfying arbitrary adsorption isotherms by using the consistent time scale (CTS) approach. It is shown that the relaxation function F(t) may be applied to describe the adsorption and desorption processes over a wide range of times. The equation in the form of F(t) ) log[Γ0/Γ(t) - 1] ) n log(t/trel) describes the kinetic-diffusion-controlled adsorption and desorption processes for arbitrary adsorption isotherms. The analytical analysis is shown to be asymptotical. The correlation n equals 0.5 for long times for different adsorption isotherms (linear, Langmuir, and nonlinear). Simple formulas are derived to calculate the parameters n, trel, and D* (effective diffusion coefficient in the adsorbed layer) and also the times of establishment of the equilibrium states for the adsorption and desorption processes obeying arbitrary adsorption isotherms.

Introduction Adsorption and desorption processes on solid-liquid interfaces have been intensively studied because of their applicability to various industrial processes. Therefore, it is of importance to understand how the equilibrium state and nonequilibrium processes of surfactants are affected by different parameters. Diffusion-controlled adsorption kinetics on planar interfaces has been treated theoretically and experimentally by a number investigators.1-11 The suggestion that diffusion of molecules of adsorbates to the surface was the cause of the alteration of adsorption with time was first made by Milner and described by Langmuir and Schaefer.1-3 The diffusioncontrolled adsorption kinetics of individual substances has been analyzed by Ward and Tordai4 using an integral equation. Hansen5,6 found analytical solutions in the form of three terms for the short-time approximation and in the form of the first term for the long-time approximation for the systems obeying the Langmuir isotherms. A critical analysis of numerical solutions has been conducted by Borwankar and Wasan7 for the individual component and by Miller and co-workers8 for multicomponent mixtures. They have analyzed the different numerical schemes and shown that numerical solutions of the nonlinear equations describing the adsorption process may be obtained only * Address for correspondence: 37 Veshnajkovskaja Street, Suite 235, Moscow 111538, Russia. (1) Milner, S. R. Philos. Mag. 1907, 13, 96. (2) Lottermoser, A.; Baumguret, B. Trans. Faraday Soc. 1935, 31, 200. (3) Langmuir, I.; Schaefer, V. J. J. Am. Chem. Soc. 1937, 59, 2400. (4) Ward, A. F. G.; Tordai, L. J. Chem. Phys. 1949, 14, 453. (5) Hansen, R. S. J. Phys. Chem. 1960, 64, 637. (6) Hansen, R. S. J. Colloid Sci. 1961, 16, 549. (7) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (8) Miller, R.; Lunkenheimer, K.; Kretzcmar, G. Colloid Polym. Sci. 1979, 257, 1118. (9) Hua, X. Y.; Rozen, M. J. J. Colloid Interface Sci. 1988, 124, 652. (10) Hua, X. Y.; Rozen, M. J. J. Colloid Interface Sci. 1991, 141, 180. (11) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1988, 43, 1323.

by making sets of simplifying assumptions. Therefore, it is reasonable to derive the analytical solutions which allow us to calculate the adsorption and surface concentration over a wide range of time. Rosen and Hua9,10 have used a phenomenological equation to describe the adsorption as a function of time and two adjustable parameters

F(t) ) log[Γ0/Γ(t) - 1] ) n log(t/trel)

(1)

where Γ(t) is the amount of surfactant (or polymer) adsorbed at t, Γ0 is the equilibrium adsorption (t f ∞), and n (dimensionless) and trel (units of time) are empirical constants. It is reasonable to analyze the theoretical foundations of the phenomenological equation in terms of the kinetic-diffusion theory. The effect of adsorption kinetics on the adsorption and desorption processes may be significant.7,11 As far as is known by the author, desorption kinetics are not considered theoretically in the literature. Therefore, in this paper attention is therefore focused on the theoretical analysis of the kinetic-diffusion desorption over a wide range. We examine the analytical approaches and the simplifying assumptions of the earlier investigations. Despite several attempts, there is no treatment of the problem that achieves a general analytical approach for solving the nonlinear equations describing the adsorption and desorption processes for individual components and multicomponent mixtures, without making sets of simplifying assumptions. It is shown that the adsorption (or desorption) process on a planar surface is governed (a) by the adsorption (or desorption) kinetics at the interface for short times, (b) by the adsorption (or desorption) kinetics at the interface as well as simultaneously diffusion for intermediate times, and (c) by diffusion for long times. The purpose of this paper is (a) to develop a theory of the kinetic-diffusion adsorption and desorption processes over a wide range of time for individual components and multicomponent mixtures and (b) to derive simple formulas to calculate the parameters

S0743-7463(96)00151-5 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/17/1998

Kinetic-Diffusion-Controlled Adsorption and Desorption

n and trel in the relaxation equation (eq 1) for the adsorption and desorption processes on a planar surface.

Langmuir, Vol. 14, No. 10, 1998 2865

Adsorption and Desorption Kinetics on a Planar Surface. Adsorption and desorption kinetics on a planar surface is described by13

Adsorption and Desorption Processes on Planar Surfaces The adsorption and desorption processes take place in a thin layer on a planar interface. In the general case these processes are governed by the kinetics of adsorption and desorption of surfactant molecules at the interface and diffusion of these molecules in the adsorbed layer. Therefore, the adsorption and desorption processes on a planar surface may be described in the framework of a one-dimensional model. Below we consider formulation of the problem for adsorption and desorption kinetics on a planar surface. In the framework of a one-dimensional model the mass-balance equation for the k-th component of a mixture is

∂ck(x,t) ∂jk(x,t) + ) 0, 1 e k e n ∂t ∂x

(2a)

where ck(x,t) is the concentration of the k-th component of a mixture at a coordinate x normal to the surface (x ) 0), jk(x,t) is the flux of the k-th component, and n is the number of components in a mixture. Mass Transport in the Bulk. According to the nonequilibrium thermodynamics, the flux of the k-th component is equal to12 n

∑ s)1

jk ) -

( )

Lks ∂µk° , 1 e k, s e n T ∂x ∂cs

n

jk ) -

∑Dks ∂x

(3a)

(3b)

s)1 n

Dks )

∑ j)1

()

(3c)

T ∂cs

(3d)

where Lmk are the Onzager coefficients, µk is the chemical potential of the k-th component, µ(n+1) is the chemical potential of the solvent, T is the absolute temperature, Dkk is the diffusion coefficient of the k-th mixture component, and Dkm is the mutual diffusion coefficient (k * m) of the k-th and m-th mixture components. In the general case the diffusion coefficients Dks depend on the surfactant concentration in the bulk;13 therefore, the flux of the k-th component is equal to n

∑ Dks(c1,c2,...,cn) s)1

∂cs(x,t) , 1 e k, s e n (4) ∂x

In this case the mass-balance equation for the k-th component of a mixture reduces to

∂ck(x,t) ∂t

n

)

∂

∑ ∂x

s)1

[

]

∂cj(x,t)

Dks(c1,c2,...,cn)

∂x

, 1 e k, s e n (5)

(12) Groot, S. P.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland Publishing Co.: Amsterdam, 1962. (13) Adamson, A. W. Physical Chemistry of Surfaces; Interscience: New York, 1986.

(6a)

Rk+ ) KkadsGk+[c1(0,t),c2(0,t),...,cn(0,t),Γ1,Γ2,...,Γn] (6b) Rk- ) KkdesGk-(Γ1,Γ2,...,Γn)

(6c)

where Γk is the adsorption of the k-th mixture component on a planar interface, ck(0,t) is the surface concentration of the k-th mixture component on an interface, Rk+ and Rk- are the forward and backward rates in the kinetic expression for the k-th mixture component, Kkad and Kkdes are the rate constants of the adsorption and desorption processes, respectively, for the k-th mixture component, and n is the number of adsorbed components of a mixture. In the general case functions of Gk+ and Gk-, taking into account the surfactant/interface, surfactant/surfactant, and surfactant/solvent interactions, may be found from the analysis of experimental kinetic data. Initial and Boundary Conditions. Initial conditions for the k-th component of a mixture are

ck(x>0,0) ) c0k, Γk(0) ) 0, ck(∞,t) ) c0k, 1 e k e n (adsorption process) (7a) ck(x>0,0) ) 0 Γk(0) ) Γ0k, ck(∞,t) ) 0, 1 e k e n (desorption process) (7b) The adsorbed amount per unit of surface, Γk(t), for the k-th component of a mixture, using the Gibbs convention,13 is given by eq 8

Γk(t) )

Lkj ∂µj°

µk° ) µk - µ(n+1)

jk(x,t) ) -

dΓk(t)/dt ) Rk+ - Rk-, 1 e k e n

∫0∞[c0k - ck(x,t)]dx,

1eken

(8)

where [c0k - ck(x,t)] is the excess concentration of the surfactant in the adsorbed layer (mass per volume unit) for the k-th component of a mixture. The boundary condition (x ) 0) may be found from the integral form of the mass-balance equation. After manipulations from eqs 2a, 5, and 8, the boundary condition (x ) 0) for the k-th component of a mixture is given by

dΓk(t) dt

) -jk(0,t) ) n

∂cs(0,t) , ∂x 1 e k, s e n (9)

∑Dks[c1(0,t),c2(0,t),...,cn(0,t)] s)1

The system of eqs 5, 6a-7b, and 9 is closed; therefore, this system of equations may be used to describe the kineticdiffusion-controlled adsorption and desorption processes for multicomponent mixtures on a planar surface. Let us consider the diffusion-controlled adsorption and desorption processes. The diffusion-controlled adsorption and desorption processes are realized in the limiting case when the rate of the adsorption and desorption is infinite, that is, Kkods f ∞, Kkdes f ∞ (1 e k e n), so that eq 6a reduces to

Kk(p)Gk+[c1(0,t),c2(0,t),...,cn(0,t),Γ1,Γ2,...,Γn] ) Gk-(Γ1,Γ2,...,Γn), Kk(p) ) Kkads/Kkdes (10a)

2866 Langmuir, Vol. 14, No. 10, 1998

Filippova

which may be written as

D* ) (Dads)2/D°

Γk(t) ≡ fk[c1(0,t),c2(0,t),...,cn(0,t)], 1 e k e n (10b) where Kk(p) is the equilibrium constant of adsorption for the k-th component of a mixture and fk is a function describing the adsorption isotherm for the k-th component of a mixture. The system of eqs 5, 7a, 7b, 9, and 10b is closed; therefore, this system of equations may be used to describe the diffusion-controlled adsorption and desorption processes for multicomponent mixtures on a planar surface. Kinetic-Diffusion-Controlled Adsorption for OneComponent Mixture. The adsorption model in the form of the system of eqs 5, 6a-7b, and 9 is complex. This nonlinear system of equations may be integrated only numerically by a computer. Therefore, it is reasonable to consider the simplified adsorption model for which the behavior of adsorption and desorption processes may be estimated in the analytical form. For the simplified adsorption model the diffusion coefficient in the bulk is assumed to be constant. In this case eqs 5 and 9 reduce to

∂2c(x,t) ∂c(x,t) ) D° ∂t ∂x2

(11)

∂c(0,t) dΓ(t) ) (Dads) dt ∂x

(12)

where D° is the diffusion coefficient of the surfactant in the bulk and Dads is the diffusion coefficient of the surfactant in the adsorbed layer. The simplified adsorption model in the form of eqs 11 and 12 has already been discussed by Ravera and co-workers;14 the authors interpreted the diffusion coefficient, Dads, in the framework of an interfacial potential barrier in the adsorbed layer. Inhomogeneity taking place within the adsorbed layer is assumed to be estimated by using the homogeneous model with the constant diffusion coefficient Dads, depending on the interaction between adsorbate molecules in the adsorbed layer. We also consider the simplified model for the adsorption kinetics. The simplified expression for the Henry, Langmuir, and Freundlich adsorption kinetics may be written as13

where D* is the effective diffusion coefficient. The system of eqs 12-14c may be applied to describe the kineticdiffusive-controlled adsorption on a planar surface. To describe the kinetic-diffusive controlled adsorption over a wide range of time, it is reasonable to use the system of eqs 13 and 14a for short times and the system of eqs 13 and 14b for long times. The diffusion-controlled adsorption is realized in the limiting case when Kads, Kdes f ∞, so that from eq 13, when dΓ/dt ) 0, one writes

KpG+[c(0,t),Γ] ) G-(Γ), Kp ) Kads/Kdes (15a) which can be rearranged as

Γ(c) ) f(c), c ≡ c(0,t)

where Γ°m is the maximum adsorption and m and s are the parameters of adsorption and desorption kinetics (for the Henry adsorption kinetics, m ) 1, s ) 0; for the Langmuir adsorption kinetics, m ) s ) 1; and for the Freundlich adsorption kinetics, 0 e m e 1, s ) 0). The system of eqs 7a, 11, and 12 may be reduced to the following integral Volterra equations for the adsorption Γ(t) and the surface concentration c(0,t):

Γ(t) ) c0(4D*t/π)1/2 -

c(0,t) ) c0 -

d dt

Γ(c) ) Kpc

(16a)

Γ(c) ) Γ°mKpc/(1 + Kpc)

(16b)

Γ(c) ) Kpcm

(16c)

The system of eqs 12, 14a-14c, and 15b-16b may be applied to describe the diffusive-controlled adsorption on a planar surface. Diffusion-Controlled Adsorption. Now we consider the adsorption kinetics on a planar surface when the adsorption process is controlled by the diffusion of molecules from the bulk for different adsorption isotherms. For arbitrary adsorption isotherms, which do not have the singularity at c ) 0, the solutions of eq 14a are given by

Γ(τ)/Γ0 ) a1τ1/2 - a2τ + a3τ3/2 - a4τ2 + a5τ5/2 - ... (17a) c(0,τ)/c0 ) b1τ1/2 - b2τ + b3τ3/2 - b4τ2 + b5τ5/2 - ... (17b) (Γ0/c0) t , t° ) t°D D D*

2

(17c)

where τ is the relative time and the ak and bk coefficients in solutions 17a and 17b are given by eqs A3-A20 in Appendix A. Now we consider the solution of eq 14a for the Freundlich adsorption isotherm

Γ(C)/Γ0 ) Cm

(18)

(14a)

where C ) c/c0, Γ0 ) Kpc0, and m is the parameter of the Freundlich adsorption isotherm (0 < m e 1). Since for the Freundlich adsorption isotherm (eq 18) the singularity at c ) 0 takes place, that is, dΓ(cf0)/dc f ∞, then the solution of eq 14a cannot be found in the form of eqs 17a and 17b and may be found in the following form:

(14b)

Γ(τ)/Γ0 ) 2τ1/2/π1/2 - a°2τ(1/2+1/2m)

∫0t(4D*/π)1/2c(0,t - F) dF1/2

∫0tΓ[c(x,t - F)] dF/(πD*F)1/2

(15b)

where Kp is the equilibrium constant and f(c) is an equation of the adsorption isotherm. Equations 13 and 15a for the Henry, Langmuir, and Freundlich adsorption kinetics reduce to

τ) dΓ(t)/dt ) Kads[c(0,t)]m[Γm° - Γ(t)]s - KdesΓ(t) (13)

(14c)

(14) Ravera, F.; Liggieri, L. J. Colloid Interface Sci. 1993, 156, 109.

(19a)

c(0,τ)/c0 ≈ 2τ1/2/(a0π1/2), a0 ) [dΓ(c)0)/dc]/[Γ(c)c0)/c0] (short time) (19b)

Kinetic-Diffusion-Controlled Adsorption and Desorption

Langmuir, Vol. 14, No. 10, 1998 2867

where a°2 ) (4/π)1/2mΓ(1 + 1/2m)/Γ(3/2 + 1/2m) and Γ(m) is the gamma function.15,16 Next we consider the relative amount of surfactant adsorbed for long times (t f ∞). For long times from eq 14b for the diffusive-controlled adsorption process obeying the arbitrary adsorption isotherm, the relative surface concentration c(0,τ)/c0 and the relative amount of surfactant adsorbed Γ(τ)/Γ0 are given by

c(0,τ)/c0 ≈ 1 - 1/(πτ)1/2 (long time)

(20a)

Γ(τ)/Γ0 ≈ 1 - a°/(πτ)1/2, a° ) [dΓ(c)c0)/dc]/[Γ(c)c0)/c0] (long time) (20b) From the previous analysis the important conclusion follows that for the diffusive-controlled adsorption on a planar surface obeying arbitrary adsorption isotherms the behavior of the surface concentration c(0,τ) and the adsorption Γ(τ) is different. For any adsorption isotherms the relative amount of surfactant adsorbed Γ(τ)/Γ0 and the relative surface concentration c(0,τ)/c0 are described by eqs 17a and 17b, respectively, for short and intermediate times and are described by eqs 20a and 20b for long times. Time-Dependence of Adsorption and Surface Concentration. As shown,9,10,17 the relaxation function Fad(t) is convenient to describe the behaviors of the adsorption process over a wide range of times. The relaxation function Fad(t) may be written in the following form9,10,17

Fads(t) ) -log[Γ0/Γ(t) - 1] ) n log(t/trel)

(21)

where trel is the relaxation time characterizing the rate of the adsorption processes on the interface and in the adsorption layer and n is the slope of the relaxation function Fad(t) versus log(t). In fact, eq 21 is in a form similar to that of the Fourier transform of a correlation function, often used in relaxation theory.17 Now we consider the behaviors of diffusive-controlled adsorption for short and long times. According to eqs 17a and 20b, for the diffusive-controlled adsorption obeying an arbitrary adsorption isotherm for short times, eqs 17a and 21 reduce to

Fads(τ) ) n0 log(4τ/π), Fads(t) ) 0.5 log(t/t°rel) ) 0.5 log(t) + s°D (short times) (22) where n0 ) 0.5, t°rel ) πt°D/4, and s°D ) -0.5 log(πt°D/4) is the shift of the straight line [Fads(t) versus log(t) for short times] for the diffusive-controlled adsorption. For the diffusive-controlled adsorption obeying arbitrary adsorption isotherms for long times, eqs 20b and 21 reduce to

Fads(t) ) -log[Γ0/Γ(t) - 1] ) 0.5 log(t/t∞rel) ) 0.5 log(t) + s∞D (long times) (23) where n∞ ) 0.5, t∞rel ) (a°)2t°D/π, and s∞D ) -0.5 log[(a°)2t°D/ (15) Arfken, G. Mathematical Methods for Physics; Academic Press: New York, 1970. (16) Abramowitz, M.; Stegun, I. Handbooks of Mathematical Functions with Formulas, Graphs and Mathematical Tables; Applied Mathematics Series; National Bureau of Standards: Washington, DC, 1964. (17) Atherton, N. M. Electron Spin Resonance Theory and Applications; Halsted Press: New York, 1973.

Figure 1. Change of the relative surface concentration for the diffusive-controlled adsorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 17b, 20a, and 24a.

π] is the shift of straight line [Fad(t) versus log(t) for long times] for the diffusive-controlled adsorption. Equations 22 and 23 may be used to find the relaxation time t°rel and t∞rel and the effective diffusion coefficient D*. From the previous analysis the important conclusion follows that the slope of the straight line of the relaxation function Fad(t) versus log(t) for short and long times for the diffusivecontrolled adsorption is to be 0.5. Therefore, it is reasonable to represent the experimental kinetic data in the form of the relaxation function versus log(t). In this case the values of the slopes n0 and n∞ may be used as the criteria to estimate that the adsorption process is governed by the diffusion when the slope of the straight line of the relaxation function versus log(t) equals 0.5 for the experimental kinetic data. Now we consider the new consistent time scale (CTS) approach to describe the diffusive-controlled adsorption process over a wide range of times in an analytical form for the arbitrary adsorption isotherm. As shown in Appendix B, the CTS approach may be useful for finding the solutions of eqs 14a and 14b in the analytical form for arbitrary adsorption isotherms. In particular, for the Langmuir adsorption isotherm the solutions of eqs 14a and 14b may be written in the following form

Γ(t)/Γ0 ) (1 + b)g(ξ(τ(t)))/[1 + bg(ξ(τ(t)))], c(0,t)/c0 ) g(ξ(τ(t))) (24a) ξ(τ) ) τ(1 + b)2/[1 + bg(τ)]2, τ ) t/{D°[Γ0/(Dadsc0)]2}, b ) Kpc0 (24b) Figures 1-3 show the time-dependent relative surface concentration c(0,τ)/c0, the relative adsorption Γ(τ)/Γ0, and the relaxation function Fad(τ) for the Langmuir adsorption isotherms in the form of eqs 16b, 24a, and 24b. From the preceding solutions and the data in Figures 1-3, it follows that eq B2 in Appendix B may be useful to describe the time-dependent diffusive-controlled adsorption over a wide range of time for any arbitrary adsorption isotherms. Kinetic-Diffusive-Controlled Adsorption. In the general case the adsorption process on a planar surface is governed by the adsorption kinetics on a planar surface and the diffusion of molecules of surfactant from the bulk simultaneously. The kinetic-diffusive-controlled adsorption on a planar surface is described by the system of eqs 6a, 14a, and 14b. First, we consider the kinetic-diffusive-controlled adsorption obeying arbitrary adsorption kinetics for short times (t f 0). From eqs 6a and 14a the amount of surfactant adsorbed is given by

2868 Langmuir, Vol. 14, No. 10, 1998

Filippova

Figure 2. Change of the relative adsorption for the diffusivecontrolled adsorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 17b, 20b, and 24a.

Γ(t) ≈ KadsG+(c)c0,Γ)0)t

(25)

where Kads is the rate constant of the adsorption and G+(c,Γ) is a function for the forward rate in the kinetic expression. Thus, the amount of surfactant adsorbed is proportional to t (time) for short times for the kineticdiffusive-controlled adsorption obeying arbitrary adsorption kinetics on a planar surface. For short and intermediate times (τ f 0) the solutions of eqs 13 and 14a may be found in the following form

Γ(τ)/Γ0 ) a°1τ - a°2τ3/2 + a°3τ2 - a°4τ5/2 + a°5τ3 - ... (26a) c(0,τ)/c0 ) 1 - b°1τ1/2 + b°2τ - b°3τ3/2 + b°4τ2 - b°5τ5/2 + ... (26b) Substituting eqs 26a and 26b in eqs 13 and 14a, establishing the coefficients in the power series by means of eqs 26a and 26b, we may find the coefficients a°k and b°k for arbitrary adsorption kinetics. In particular, for the Langmuir adsorption kinetics (eq 13, m ) s ) 1) the relative adsorption and the surface concentration are given by

Γ(τ)/Γ0 ) κτ(1 + b) - 4κ2(1 + b)2τ3/2/(3π1/2) + κ2(1 + b)2[κ(1 + b) - 1]τ2/2 8κ (1 + b)2{(1 + b)[κ(1 + b) - 2] 3b/2}τ5/2/(15π1/2) + ..., b ) Kpc0, κ ) t°D/t°kin, τ ) t/t°D (27a) 3

c(0,τ)/c0 ) 1 - 2κ(1 + b)τ1/2/(3π1/2) + κ2(1 + b)2τ 4κ2(1 + b)2[κ(1 + b) - 1]τ3/2/(3π1/2) + κ3(1 + b)2{(1 + b)[κ(1 + b) - 2] - 3b/2}τ2/2 + ..., t°kin ) 1/Kdes (27b) For the kinetic-diffusive-controlled adsorption process obeying arbitrary adsorption kinetics from eqs 13 and 14a for long times (τ f ∞), the relative adsorption and relative surface concentration are given by

Γ(τ)/Γ0 ) 1 - a°/(πτ)1/2, a° ) [dΓ(c0)/dc]/[Γ(c0)/c0] (28a) c(0,τ)/c0 ) 1 - 1/(πτ)1/2

(28b)

Figure 3. Change of the relaxation function for the diffusivecontrolled adsorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 17a, 21, 20b, and 24a.

From the previous analysis the important conclusion follows that for the kinetic-diffusive-controlled adsorption on a planar surface obeying arbitrary adsorption kinetics the behavior of the surface concentration c(0,τ) and the adsorption Γ(τ) is different. For any adsorption kinetics the relative amount of surfactant adsorbed Γ(τ)/Γ0 and the relative surface concentration c(0,τ)/c0 are described by eqs 26a and 26b, respectively, for short and intermediate times and are described by eqs 28a and 28b for long times. Time-Dependence of Adsorption and Surface Concentration. As shown,9,10,17 the relaxation function Fad(t) is convenient to describe the behaviors of the kineticdiffusive-controlled adsorption process over a wide range of times and to compare behaviors for the diffusivecontrolled adsorption and the kinetic-diffusive-controlled adsorption. From eqs 19a, 19b, 20b, 25, 28a, and 28b, it follows that for these models of the adsorption processes the time dependencies of adsorption are the same for long times (t f ∞) and are different for short times (t f 0). According to eqs 21 and 25 for the kinetic-diffusivecontrolled adsorption obeying arbitrary adsorption isotherms, the relaxation function Fad(t) for short times is given by

Fads(t) ) n0 log(t/t°rel) ) log(t) + s°kin

(29a)

t°rel ) KadsG+(c)c0,Γ)0)

(29b)

where n0 ) 1 and s°kin ) -log(t°rel) is the shift of the straight line [Fad(t) versus log(t) for short times] for the kineticdiffusion-controlled adsorption. For the Langmuir adsorption kinetics the relaxation time, t°rel, is to be Kadsc0 Γ°m. Figures 4 and 5 show the time-dependent relative adsorption and relaxation functions for the Henry adsorption kinetics in the form in eq 21. As shown in Figure 4, for short times the adsorption process is governed by the adsorption kinetics; the slope n0 of the straight line [Fad(t) versus log(t)] equals 1. For long times the adsorption process is governed by the diffusion; the slope n∞ of the straight line [Fad(t) versus log(t)] equals 0.5. Thus, the slope n may be used as the criterion to estimate the mechanisms of adsorption on a planar surface. Kinetic-Diffusion-Controlled Desorption for OneComponent Mixture. For the reversible process, the kinetic-diffusive-controlled desorption process is described by the system of eqs 7b and 11-13. The system of eqs 7b and 11-13 may be reduced to the following integral

Kinetic-Diffusion-Controlled Adsorption and Desorption

Langmuir, Vol. 14, No. 10, 1998 2869

and the relative amount of surfactant adsorbed Γ(τ)/Γ0 are given by

c(0,τ)/c0 ≈ 1 - 2τ1/2/(a°π1/2), a° ) [dΓ(c)c0)/dc]/[Γ(c)c0)/c0] (short time) (32a) Γ(τ)/Γ0 ≈ 1 - 2τ1/2/π1/2 (short time)

(32b)

Now we consider the relative amount of surfactant adsorbed for long times (t f ∞). For long times from eqs 30a and 30b for the diffusive-controlled desorption process obeying the arbitrary adsorption isotherm, the relative surface concentration c(0,τ)/c0 and the relative amount of surfactant adsorbed Γ(τ)/Γ0 are given by Figure 4. Change of the relative adsorption for the kineticdiffusive-controlled adsorption obeying the Henry adsorption kinetics at various parameters κ, according to eqs 26a and 28a.

c(0,τ)/c0 ≈ 1/(πτ)1/2 (long time)

(33a)

Γ(τ)/Γ0 ≈ a0/(πτ)1/2, a0 ) [dΓ(c)0)/dc]/[Γ(c)c0)/c0] (long time) (33b)

Figure 5. Change of the relaxation function for the kineticdiffusive-controlled adsorption obeying the Henry adsorption kinetics at various parameters κ, according to eqs 21, 26a, and 28a.

Volterra equations:

Γ(t) ) Γ0 -

∫0t(4D*/π)1/2c(0,t-F) dF1/2

c(0,t) ) Γ0/(πD*t)1/2 -

(30a)

∫0tΓ[c(x,t-F)] dF/(πD*F)1/2

d dt

(30b) To describe the kinetic-diffusive-controlled desorption process over a wide range of times, it is reasonable to use the system of eqs 13 and 30a for short times and the system of eqs 13 and 30b for long times. Diffusion-Controlled Desorption. Now we consider the desorption kinetics on a planar surface when the desorption process is controlled by the diffusion of surfactants from the bulk. For arbitrary adsorption isotherms, including the Freundlich adsorption isotherm, the solution of eq 30a may be found in the following form:

Γ(τ)/Γ0 ) 1 - a*1τ1/2 + a*2τ - a*3τ3/2 + a*4τ2 - a*5τ5/2 + ... (31a) c(0,τ)/c0 ) 1 - b*1τ1/2 + b*2τ - b*3τ3/2 + b*4τ2 - b*5τ5/2 + ... (31b) For arbitrary adsorption isotherms the coefficients a*k and b*k in eqs 31a and 31b are given by eqs C7-C18 in Appendix C. From the previous analysis it follows that, for the diffusive-controlled desorption process for short times (t f 0), the relative surface concentration c(0,τ)/c0

From the previous analysis the important conclusion follows that for the diffusive-controlled desorption on a planar surface obeying arbitrary adsorption isotherms the behavior of the surface concentration c(0,τ) and the adsorption Γ(τ) is different. (a) For any adsorption isotherms the relative amount of surfactant adsorbed Γ(τ)/ Γ0 is described by eq 32b for short times and the relative surface concentration c(0,τ)/c0 is described by eq 33a for long times. (b) For different adsorption isotherms the relative amount of surfactant adsorbed Γ(τ)/Γ0 is described by eq 33b for long times and the relative surface concentration, c(0,τ)/c0 is described by eq 32a for short times. As shown,9,10,17 the relaxation function Fdes(t) is convenient for describing the behavior of the desorption process over a wide range of times. The relaxation function Fdes(t) may be written in the following form9,10,17

Fdes(t) ) log[Γ0/Γ(t) - 1] ) n log(t/t*rel)

(34)

where t*rel is the relaxation time characterizing the rate of the desorption processes on the interface and in the adsorption layer and n is the slope of the relaxation function Fdes(t) versus log(t). The diffusive-controlled and kinetic-diffusive-controlled desorption over a wide range of time may be described by using the relaxation function Fdes(t). According to eq 32b, for the diffusive-controlled desorption obeying an arbitrary adsorption isotherm for short times, eq 34 reduces to

Fdes(t) ) n0 log(4τ/π) ) log(t/t*°rel) ) 0.5 log(t) + s*°D (35) where n0 ) 0.5, t*°rel ) πt°D/4, and s*°D ) -0.5 log(πt°D/4) is the shift of the straight line [Fdes(t) versus log(t) for short times] for the diffusive-controlled desorption. From eqs 33b and 34 for the diffusive-controlled desorption obeying an arbitrary adsorption isotherm for long times, we find

Fdes(t) ) log[Γ0/Γ(t) - 1] ) n log(t/t*∞rel) ) 0.5 log(t) + s*∞D (36) where n∞ ) 0.5 and s*∞D ) -0.5 log[(aq)2t°D/π] is the shift of straight line [Fdes(t) versus log(t) for long times] for the diffusive-controlled desorption. Equations 35 and 36 may be used to find the relaxation time t*°rel and t*∞rel and the

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0.5. Thus, the slope n may be used as the criterion to estimate the mechanisms of desorption on a planar surface. Kinetic-Diffusion-Controlled Desorption. The kineticdiffusion-controlled desorption on a planar surface is described by using eqs 13, 30a, and 30b. First, we consider the kinetic-diffusive-controlled desorption obeying arbitrary adsorption kinetics for short times (t f 0). From eqs 6a and 30a the amount of surfactant adsorbed is given by

Γ(t) ≈ Γ0 - KdesG-(Γ)Γ0)t

Figure 6. Change of the relative surface concentration for the diffusive-controlled desorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 24a and 31b.

(37)

where Kdes is the rate constant of the desorption and G-(Γ) is a function for the backward rate in the kinetic expression. Thus, the amount of surfactant adsorbed is proportional to t (time) for short times for the kineticdiffusive-controlled desorption obeying arbitrary adsorption kinetics on a planar surface. Now we consider the kinetic-diffusion-controlled desorption for arbitrary adsorption kinetics. For short and intermediate times the solutions of eqs 13 and 30a may be found in the following form

Γ(τ)/Γ0 ) 1 - a*1τ + a*2τ3/2 - a*3τ2 + a*4τ5/2 - a*5τ3 + ... (38a) c(0,τ)/c0 ) b*1τ1/2 - b*2τ + b*3τ3/2 - b*4τ2 + b*5τ5/2 + b*6τ3 - ... (38b)

Figure 7. Change of the relative adsorption for the diffusivecontrolled desorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 24a and 31b.

Substituting eqs 38a and 38b in eqs 13 and 30a and establishing the coefficients in the power series by means of eqs 38a and 38b, we may find the coefficients a*k and b*k for arbitrary adsorption kinetics. In particular, for the Langmuir adsorption kinetics (eq 13, m ) s ) 1) the relative adsorption and the surface concentration are given by

Γ(τ)/Γ0 ) 1 - κτ + 4κ2τ3/2/(3π1/2) - κ2(κ - 1)τ3/2 + 8κ3(κ - 2 - b/2)τ5/2/(15π1/2) (39a) c(0,τ)/c0 ) 2κτ1/2/π1/2 - κ2τ + 4κ2(κ - 1)τ3/2/(3π1/2) κ3(κ - 2 - b/2)τ2/2 (39b) τ ) t/t°D, t°D ) L2a/D*, t°kin ) 1/Kdes, κ ) t°D/t°kin (39c) Figure 8. Change of the relaxation function for the diffusivecontrolled desorption obeying the Langmuir adsorption isotherm at various parameters b, according to eqs 24a and 31a.

diffusion coefficient D* from the experimental kinetic data. Figures 6-8 show the time-dependent relative surface concentration c(τ)/c0, the relative adsorption Γ(τ)/Γ0, and the relaxation function Fdes(τ) for the Langmuir desorption isotherms in the form of eqs 24a and 24b for the CTS approach and eqs 31a-36. From the preceding solutions and data in Figures 6-8, it follows that eqs 24a and 24b may be used to describe the time-dependent desorption over a wide range of times for the diffusion-controlled desorption obeying arbitrary adsorption isotherms. As shown in Figures 6-8, for short and long times the desorption process is governed by the diffusion; the slopes n0 and n∞ of the straight line [Fdes(t) versus log(t)] equal

From the previous analysis the important conclusion follows that for the kinetic-diffusive-controlled desorption on a planar surface obeying arbitrary adsorption kinetics the behavior of the surface concentration c(0,τ) and the adsorption Γ(τ) is different. (a) For arbitrary adsorption kinetics the relative amount of surfactant adsorbed Γ(τ)/ Γ0 is described by eq 37 for short times and the relative surface concentration c(0,τ)/c0 is described by eq 33a for long times. (b) For different types of adsorption kinetics on a planar surface the relative amount of surfactant adsorbed Γ(τ)/Γ0 is described by eq 33b for long times, and the relative surface concentration c(0,τ)/c0 is described by eq 38b for short times. Next we consider the relaxation function Fdes(t) for the kinetic-diffusive-controlled desorption. According to eqs 33b, 34, and 37 for arbitrary adsorption kinetics, the relaxation function Fdes(t) for short and long times is given by

Kinetic-Diffusion-Controlled Adsorption and Desorption

Fdes(t) ) n*0 log(t/t*°rel) ) log(t) + s*°kin (short time) (40a) des

t*°rel ) 1/K

F

des

, n*0 ) 1, s*°kin ) -log(t*°rel) (short time) (40b)

(t) ) n*∞ log(t/t*

∞

rel)

) log(t) + s*

∞ D

(long time) (41a)

t*∞rel ) (a0)2t°D/π, n*∞ ) 0.5, s*∞D ) -0.5 log(t*°rel) (long time) (41b) where s*°kin and s*∞D are the shift of the straight line [Fdes(t) versus log(t) for short and long times, respectively] for the kinetic-diffusive-controlled desorption. For the kinetic-diffusive-controlled desorption obeying the Langmuir adsorption kinetics, the relaxation time is t*°kin ) 1/Kdes. From the previous analysis it follows that the kinetic-diffusive-controlled desorption obeying arbitrary adsorption kinetics is governed by the adsorption kinetics for short times; therefore, the slope n*0 of the straight line [Fdes(t) versus log(t)] equals 1.0. The kinetic-diffusivecontrolled desorption obeying arbitrary adsorption kinetics is governed by the diffusion for long times; therefore, the slope n*∞ of the straight line [Fdes(t) versus log(t)] equals 0.5. Therefore, it is reasonable to represent the experimental kinetic data in the form of the relaxation function Fdes(t) versus log(t). In this case the value of the slopes, n*0 and n*∞ may be used as the criterion to estimate the mechanisms of desorption on a planar surface. Results and Discussion The above theory for the diffusive-controlled and kineticdiffusive-controlled adsorption and desorption was applied in order to estimate the time dependence of adsorption and desorption over a wide range of times. In the literature to estimate the time-dependence of adsorption the systems of eqs 13, 14a, 14b, 30a, and 30b were integrated numerically by using a computer.7,8 However, it should be noted that to analyze by a computer the behavior of the adsorption and desorption over a wide range of time (eight and more the order of magnitudes for adsorption and twelve and more the order of magnitudes for desorption is a complex problem.7 As just mentioned the choice of mesh size, ∆τ, ∆C, and ∆Γ is of significant importance in the application of finite difference methods for the numerical integration of eqs 13, 14a, 14b, 30a, and 30b by using a computer. Therefore, the correct numerical integration of nonlinear equations requires a special mathematical analysis. In our opinion the analytical analysis of the system of eqs 13, 14a, 14b, 30a, and 30b is more reasonable, as the analytical expressions for the kinetic-diffusive-controlled adsorption and desorption allow us to estimate (a) the time dependence of adsorption and desorption for any order of magnitudes of times, (b) the constants of rate of adsorption and desorption, and the effective diffusion coefficient by using experimental kinetic data, and (c) the effect of the different mechanisms of adsorption and desorption processes on a planar surface. We apply the theory kinetic-convective-diffusive-controlled adsorption kinetics on a planar surface to describe the behavior of the amount of surfactant adsorbed over a wide range of times, and the values characterizing the adsorption process of surfactants are represented in Table 1. From the previous theoretical analysis and experimental data,9-11,13,14 it follows that the kinetics of formation of

Langmuir, Vol. 14, No. 10, 1998 2871 Table 1. Characteristics of the Adsorption Process for Surfactants on Interfaces c0 Γ°m Γ(c)c0)/c0 Kp (×10-7 mol/cm3) (×10-10 mol/cm2) (×10-4 cm) (×107 cm3/mol) 1-10

1-10

Kads

1-10

1-10

(×108 cm3/mol‚s)

D* (×10-6 cm2/s)

t°kin (s)

t°D (s)

1-8

1-5

3-300

1-500

adsorbed layers can be separated into three stages (regimes): (I) the kinetics of adsorption and desorption of surfactant molecules on the bare surface, (II) the kinetics of adsorption and desorption of surfactant molecules onto the interface and simultaneous diffusion of molecules within the adsorbed layer, and (III) the diffusion of surfactant molecules in the adsorbed layer. The first process (I) occurs for short times or at low surfactant concentrations in the adsorption layers. The relaxation time (or the time scale) characteristic of this process is relatively small (on the order of 10 or 100 seconds, as shown in Table 1). The second process (II) is governed simultaneously by the kinetics of adsorption and desorption and the diffusion of surfactant molecules in the adsorbed layer. The third process (III) takes place at long times or high surfactant concentrations in the adsorption layer when interactions between surfactant molecules are essential. The relaxation times of all three adsorption and desorption regimes for kinetic-diffusive-controlled adsorption and desorption kinetics depend on (a) the molecular weights of molecules, (b) the structure of the adsorbed layer, (c) the surface coverage, and (d) the interaction between molecules of adsorbate, solvent, and adsorbent. As shown above, the relaxation functions Fad(t) and Fdes(t) allow us to separate the regimes for the adsorption and desorption processes over a wide range of time. According to eqs 22 and 23 for the diffusivecontrolled adsorption obeying arbitrary adsorption isotherms, the time dependence of the adsorption over a wide range of time may be described by the three linear equations in the form of eq 42a as

F

{

0.5 log(t) + s°D, 0 e t e t1cr mid (t) ) nmid log(t) + s , t1cr e t e t2cr 0.5 log(t) + s∞D, t g t2cr

ads

}

(42a)

log(t1cr) ) (smid - s°D)/(0.5 - nmid)

(42b)

log(t2cr) ) (smid - s∞D)/(0.5 - nmid)

(42c)

The values of s°D and s∞D are found from eqs 22 and 23, respectively. The values of nmid and smid are given as

smid ) -0.5 log(tmid), nmid ≈ dF(tmid)/d log(t), Γ(tmid)/Γ0 ≈ 0.5 (42d) As shown in Figures 3 and 9, the slope nmid depends on the adsorption isotherms. Thus, from the preceding analysis it follows that for the diffusive-controlled adsorption the following relations take place:

n0 ) n∞ ) 0.5, nmid > 0.5

(43)

As shown in Figure 3, the slope smid increases when the parameter b for the Langmuir adsorption isotherm increases. The shifts s°D and s∞D of the straight lines [Fad(t) versus log(t) for short (t f 0) and long (t f ∞) times, respectively] for the diffusion-controlled adsorption char-

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Filippova

Figure 9. Change of the relaxation function for the diffusivecontrolled adsorption obeying the Langmuir adsorption isotherm at b ) 100, according to eqs 17a, 20b, 21, and 26a.

acterize the rate of the adsorption process. The difference of shifts ∆sads between these lines at short and large times, respectively, characterizes qualitatively the rate of the adsorption process. From eqs 22 and 23 for the diffusivecontrolled adsorption obeying arbitrary adsorption isotherms (eq 44) and the Langmuir adsorption isotherm (eq 45a), the difference of shifts forms respectively

∆sads ) s°D - s∞D ) log(2a°/π)

(44)

∆sads ) log{2/[π(1 + b)]} ) log[2(1 - θ)/π] Γ(c0)/Γ°m ) θ ) b/(1 + b)

{

}

(46a)

(46b)

log(t2cr) ) (smid - s∞D)/(0.5 - nmid)

(46c)

The values of s°kin and s∞D are found from eqs 23 and 29a, respectively. For the Henry adsorption kinetics, the time dependence of the adsorption over a wide range of time may be described by the two linear equations, as shown in Figure 5. From the preceding analysis it follows that for the kinetic-diffusive-controlled adsorption the following relations take place:

(47)

As follows from eqs 43 and 47, the value of the slope n0 may be used as a criterion to separate the kinetic-diffusivecontrolled and diffusive-controlled adsorptions for short times. In fact, for short times the adsorption process is controlled by the adsorption kinetics when n0 ) 1 and it is controlled by the diffusion when n0 ) 0.5. Equation 23 for the shift s∞D may be applied to find the relaxation time t°D, and the effective diffusion coefficient, D* is found from eqs 14c and 17c by using the experimental data for the adsorption kinetics in the form of the relaxation function

(48a)

From eq 48a and the data in Figure 2, it follows that the time tcads does not depend on the adsorption isotherms (or the parameter b for the Langmuir adsorption isotherm). By analogy, from eq 42a for an arbitrary adsorption isotherm, the time tΓads, when the adsorption Γ(tΓads), reaches the quasi-equilibrium state (1 - )Γ0 is given as

tΓads ) (a°)2t°D/(π2) + πt°D/4

(45b)

log(t1cr) ) (smid - s°kin)(1 - nmid)

n0 ) 1, n∞ ) 0.5, nmid > 0.5

tcads ) t°D/(π)2

(45a)

where θ is the surface coverage of a planar surface. According to eqs 23 and 29a, for the kinetic-diffusivecontrolled adsorption obeying arbitrary adsorption isotherms the time dependence of the adsorption over a wide range of time may be described by the three linear equations in the form of eq 46a as

0 e t e t1cr log(t) + s°kin, mid ads t1cr e t e t2cr n log(t) + s , F (t) ) mid ∞ 0.5 log(t) + s D, t g t2cr

Fad(t) versus log(t) for long times. Equations 17c, 27a, 27b, and 29b for the shift s°kin may be applied to find the rate constant of the adsorption Kad by using the experimental data for the adsorption kinetics in the form of the relaxation function Fad(t) versus log(t) for short times. Now we consider how the surface concentration c(0,t) and adsorption Γ(t) may approach the equilibrium states c0 and Γ0. Strictly speaking, these equilibrium states may be reached with infinitely long times, that is, with asymptotic t f ∞. However, if we reduce the ranges of the equilibrium surface concentration and the amount of adsorption values by some relative value  ( f 0), that is, if we take the following ranges of quasi-equilibrium surface concentration and adsorption (1 - )c0 and (1 )Γ0, then the quasi-equilibrium states become realizable at finite times. From eq 28a for an arbitrary adsorption isotherm, the time tcads when the surface concentration c(0,tcads) reaches the quasi-equilibrium state (1 - )c0 is equal to

(48b)

For the strong convex adsorption isotherms (b f ∞) the equilibrium state, Γ0 is reached at the time πt°D/4. From eq 48b for the Langmuir adsorption isotherm we write

tΓads ) t°D/{π[(1 + b)]2} + πt°D/4 ) (1 - θ)2t°D/(π2) + πt°D/4 (48c) From eq 48c and Figure 3, it follows that the time tΓads depends on the parameter b for the Langmuir adsorption isotherms. Now we consider desorption kinetics. According to eqs 35 and 36 for the diffusive-controlled desorption obeying arbitrary adsorption isotherms, the time dependence of the desorption over a wide range of time may be described by the three linear equations in the form of eq 49a as

F

{

0.5 log(t) + s*°D, 0 e t e t*1cr mid (t) ) nmid log(t) + s* , t*1cr e t e t*2cr 0.5 log(t) + s*∞D, t g t*2cr

des

}

(49a)

log(t*1cr) ) (s*mid - s*°D)/(0.5 - nmid)

(49b)

log(t*2cr) ) (s*mid - s*∞D)/(0.5 - nmid)

(49c)

The values of s*°D and s*∞D are found from eqs 35 and 36, respectively. The values of nmid and s*mid are given as

s*mid ≈ Fdes(tmid) - 0.5 log(tmid), nmid ≈ dFdes(tmid)/d log(t) (50) where tmid ) (t*cr1 + t*cr2)/2. From the preceding analysis and Figures 8 and 10 it follows that for the diffusioncontrolled desorption obeying arbitrary adsorption isotherms the following relations take place:

Kinetic-Diffusion-Controlled Adsorption and Desorption

Langmuir, Vol. 14, No. 10, 1998 2873

and adsorption Γ(t) may approach the equilibrium states c ) Γ ) 0. Strictly speaking, these equilibrium states may be reached with infinitely long times, that is, with asymptotic t f ∞. However, if we take the following ranges of quasi-equilibrium surface concentration c0 and adsorption Γ0, then the quasi-equilibrium states become realizable at finite times. From eq 33a for an arbitrary desorption isotherm, the time tcdes, when the surface concentration c(0,tcdes) reaches the quasi-equilibrium state c0, is equal to

tcdes ) t°D/(π)2

Figure 10. Change of the relaxation function for the diffusivecontrolled desorption obeying the Langmuir adsorption isotherm at b ) 100, according to eqs 24a and 31a.

n0 ) n∞ ) 0.5, nmid < 0.5

(51)

As shown in Figure 8, the slope s*mid increases when the parameter b for the Langmuir adsorption isotherm increases. The shifts s*°D and s*∞D, of the straight lines [Fdes(t) versus log(t) for short (t f 0) and long (t f ∞) times, respectively] for the diffusion-controlled adsorption characterize the rate of the desorption process. The difference of shifts ∆sdes between these lines at short and large times, respectively, characterizes qualitatively the rate of the desorption process. From eqs 35 and 36 for the diffusive-controlled adsorption obeying arbitrary adsorption isotherms (eq 52a) and the Langmuir adsorption isotherm (eq 52b) the difference of shifts forms respectively

∆sdes ) s*∞D - s*°D ) log(π/2a*)

(52a)

∆sdes ) log{π/[2(1 + b)]} ) log[π(1 - θ)/2]

(52b)

Γ(c0)/Γ°m ) θ ) b/(1 + b)

(52c)

According to eqs 40a and 41a, for the kinetic-diffusivecontrolled desorption obeying arbitrary adsorption isotherms the time dependence of the adsorption over a wide range of time may be described by the three linear equations in the form of eq 53a as

F

{

0 e t e t*1cr log(t) + s*°kin, mid (t) ) nmid log(t) + s* , t*1cr e t e t*2cr 0.5 log(t) + s*∞D, t g t*2cr

des

}

(53a)

log(t*1cr) + (s*mid - s*°kin)/(1 - nmid)

(53b)

log(t*2cr) ) (s*mid - s*∞D)/(0.5 - nmid)

(53c)

The values of s*°kin and s*∞D are found from eqs 40b and 41b, respectively. According to eq 52b and the numerical data in Figure 10, the rate of the desorption process decreases when the parameter b for the Langmuir adsorption isotherm increases. From the preceding analysis it follows that for the kinetic-diffusive-controlled desorption the following relations take place:

n0 ) 1, n∞ ) 0.5, nmid < 0.5

(54)

As follows from eqs 51 and 54, the value of the slope n0 may be used as a criterion to separate the kinetic-diffusivecontrolled and diffusive-controlled desorptions for short times. Below consider how the surface concentration c(0,t)

(55)

From eq 55 and the data in Figure 6, it follows that the time tcdes does not depend on the parameter b for the Langmuir adsorption isotherm. By analogy, from eq 33b for an arbitrary desorption isotherm, the time tΓdes, when the desorption Γ(tΓdes) reaches the quasi-equilibrium state Γ0, is given as

tΓads ) (a0)2t°D/(π2)

(56a)

From eq 56a for the Langmuir adsorption isotherm we write

tΓdes ) (1 + b)2t°D/(π2) ) t°D/[π2(1 - θ)2]

(56b)

From eq 56b and the data in Figure 7, it follows that the time tΓdes depends on the parameter b for the Langmuir adsorption isotherm. To compare the rates of establishment of the equilibrium states for adsorption and desorption processes, it is reasonable to use the following criteria Λc and ΛΓ, which from eqs 48a, 48b, 55, 56a, and 56b may be written for arbitrary isotherms (eqs 57a and 57b) and for Langmuir isotherms (eqs 57a and 57c) as

Λc ) tcdes/tcads ) 1

(57a)

ΛΓ ) tΓdes/tΓads ) (a*/a°)2

(57b)

ΛΓ ) (1 + b)4 ) (1 - θ)-4

(57c)

The criteria Λc and ΛΓ characterize the ratio of times of establishment of the equilibrium states for desorption and adsorption processes onto a planar surface for the surface concentration c(0,t) and adsorption/desorption Γ(t), respectively. From eq 57a it follows that the times of establishment of the equilibrium states of the surface concentration c(0,t) for the adsorption and desorption processes are the same, as Λc ) 1. The ratio ΛΓ of times of establishment of the equilibrium states for the adsorption/desorption Γ(t) depends on the surface coverage θ. For the linear adsorption/desorption isotherms this ratio is equal to 1. For nonlinear convex adsorption/desorption isotherms the value of ΛΓ can be much more than 1 (ΛΓ . 1) when θ f 1 (or b . 1). If b ) 10, then ΛΓ ) 104, and if b ) 100, then ΛΓ ) 108. From the preceding analysis it follows that for the strong convex adsorption isotherms (b . 1) the equilibrium state for the desorption process cannot be reached for real finite times. In these cases the desorption is an irreversible process. Conclusion We have developed a theory for the kinetic-diffusivecontrolled and diffusive-controlled adsorption and desorption of surfactants onto a planar surface obeying arbitrary adsorption isotherms. The consistent time scale (CTS) approach based on the use of the analytical solutions in

2874 Langmuir, Vol. 14, No. 10, 1998

Filippova

the form of power series representations is suggested to describe in the analytical form the surface concentration c(0,t) and the adsorption/desorption Γ(t). The simple relaxation equations (eqs 42a, 46a, 49a, and 53a) may be useful to predict the adsorption and desorption processes obeying arbitrary adsorption/desorption isotherms over a wide range times. It is shown that for nonlinear adsorption/desorption isotherms the rate of the desorption process can be much more than the rate of the adsorption process. The simple formulas (eqs 48a, 48c, 55, and 56b) are derived to calculate the times of establishment of the equilibrium states for the adsorption and desorption processes. It is shown that for strong convex adsorption isotherms the desorption processes are irreversible because the times of establishment of the equilibrium states in these cases are very large (1 year and more).

For arbitrary adsorption isotherms, which do not have the singularity at c ) 0, in the vicinity of c ) 0 the equilibrium adsorption Γ(c) and the surface concentration c(0,τ) are related by an equation for the adsorption isotherm, Γ(c) ) f(c); therefore, we write

U(C) ) f(C), C) f-1(U), U ) Γ(c)/Γ(c)c0), C ) c(0,τ)/c0, 0 e C, U e 1 (A1) ∞

∑ k)1

dkUk, dk )

U ) Γ(c)/Γ(c)c0), b ) Kpc0, C ) c/c0, 0 e U, C e 1 (A13) the coefficients of dk in eq A2 equal to

d1 ) 1/(1 + b), d2 ) b/(1 + b)2, d3 ) b2(1 + b)3, dk+1 ) bk/(1 + b)(k+1) (A14) therefore, the coefficients of ak and bk in eqs 17a and 17b are given by

a1 ) 2/π1/2, a2 ) 1/(1 + b),

Appendix A

C)

Γ(c) ) Γ°mKpc/(1 + Kpc), U ) f(C) ) (1 + b)C/(1 + bC), C ) f-1(U) ) U/(1 + b - bU) (A12)

()

(k) -1 1 d f (U)0) , k! ) 1‚2‚3‚...‚k k! dUk (A2)

From the previous analysis it follows that the coefficients in eqs 17a and 17b are related by eqs A3-A5

b1 ) d1a1, b2 ) d1a2 + d2a21, b3 ) d1a3 + 2d2a1a2 + d3a31 (A3) b4 ) d1a4 + d2(2a1a3 + a22) + 3d3a21a2 + d4a41 (A4) b5 ) d1a5 + 2d2(a1a4 + a2a3) + 3d3(a21 + a1a22) + d4(4a31a2 + 6a31a3) + d5a51 (A5)

a3 ) 4(1 - 4b/π)[3π1/2(1 + b)2] (A15) a4 ) (π + 6b2 - 4b - 3bπ)/[2π(1 + b)3] (A16) a5 ) 8[1 - 32b3/π2 + 24b2/π + 128b2/(3π2) 2b - 32b/(3π)]/[15π1/2(1 + b)4] (A17) b1 ) 2/[π1/2(1 + b)], b2 ) (1 - 4π)/(1 + b)2 (A18) b3 ) 4(π + 6b2 - 4b - 3bπ)/[3π3/2(1 + b)3]

(A19)

b4 ) [1/2 - 16b3/π2 + 15b2/π + 64b2/(3π2) 5b/2 - 22b/(3π)]/(1 + b)4 (A20) Appendix B Below we consider the consistent time scale (CTS) approach, which may be useful to find the solutions of eqs 14a and 14b in the analytical form for arbitrary adsorption isotherms. The CTS approach is based on the preceding analysis and the following assumptions:18 (A) The surface concentration c(0,t) is described by using the function g(ξ(τ)) in the following form

g(x) ) 1 - exp(x) erfc(x1/2),

∫0y exp(-z2) dz

erfc(y) ) 1 - (2/π1/2

(B1)

From eqs A1-A5 the coefficients a°k and b°k in eqs 17a and 17b are given by

(B) The consistent time scales ξ(τ), according to the preceding relations and the CTS approach, may be written as

a1 ) 2/π1/2, a2 ) d1, a3 ) 4(d21 - 4d2/π)/(3π1/2) (A6)

ξ(τ) ) τ[Γ0/Γ(g(τ))]2, τ ) t/{D°[Γ0/(Dadsc0)]2} (B2)

a4 ) d31/2 - d1d3(3/2 + 2/π) + 3d3/π

(A7)

a5 ) 8{d41 - 2d21d2[1 + 16/(3π)] + 128d22/(3π2) + 24d1d3/π - 32d4/π2}/(15π1/2) (A8) b1 ) 2d1/π1/2, b2 ) d21 - 4d2/π b3 ) 4[d31 - d1d2(3 + 4/π) + 6d3/π]/(3π1/2)

(A9) (A10)

b4 ) d41/2 - d21d2[5/2 + 22/(3π)] + 64d22/(3π2) + 15d1d3/π - 16d4/π2 (A11) In particular, for the Langmuir isotherms

The relation of eq B2 made consistent the time scales ξ(τ) with the local time scales τ, as for any adsorption isotherms the local time scales depend on [Γ(c)/c]2. By using eqs 17a-17c and the CTS approach, for any arbitrary adsorption isotherms Γ(c), the solution of eqs 14a and 14b may be written as

Γ(t) ) Γ(g(ξ(t))), c(0,t)/c0 ) g(ξ(t))

(B3)

ξ(t) ) τ[Γ0/Γ(g(t))]2, τ ) t/{D°[Γ0/(Dadsc0)]2} (B4) For the Langmuir adsorption isotherm eqs B3 and B4 reduce to eqs 24a and 24b. Equations B3 abd B4 may be (18) Filippov, L. K.; Filippova, I. V. Russ. J. Phys. Chem. 1986, 60, 1842.

Kinetic-Diffusion-Controlled Adsorption and Desorption

useful to describe the time-dependent diffusive-controlled adsorption over a wide range of time for any arbitrary adsorption isotherms.

For arbitrary adsorption isotherms, including the Freundlich adsorption isotherms, the relative equilibrium adsorption Γ(c)/Γo and the relative surface concentration c(0,τ)/co are related by an equation of the adsorption isotherm, Γ(c) ) f(c); therefore, we write

U(C) ) f(C), C ) f-1(U), U ) Γ(c)/Γ(c)c0), C ) c(0,τ)/c0, 0 e C, U e 1 (C1) ∞

∑ (-1)(k+1)d°k(1 - U)k,

k)1

a*1 ) 2/π1/2, a*2 ) d°1, a*3 ) 4[(d°1)2 + 4d°2/π]/(3π1/2) (C7) a*4 ) (d°1)3/2 + d°1d°2(3/2 + 2/π) + 3d°3/π (C8)

Appendix C

1-C)

Langmuir, Vol. 14, No. 10, 1998 2875

a*5 ) 8{(d°1)4 + 2(d°1)2d°2[1 + 16/(3π)] + 128(d°2)2/(3π2) + 24d°1d°3/π + 32d°4/π2}/(15π1/2) (C9) b*1 ) 2d°1/π1/2, b*2 ) (d°1)2 + 4d°2/π

b*3 ) 4[(d°1)3 + d°1d°3(3 + 4/π) + 6d°3/π]/(3π1/2) (C11) b*4 ) (d°1)4/2 + (d°1)2d°2[5/2 + 22/(3π)] +

()

(k) -1 1 d f (U)1) , k! ) 1‚2‚3‚...‚k (C2) d°k ) k! dUk

From the previous analysis it follows that the coefficients in eqs 31a and 31b are related by eqs C3-C6:

d°k ) (1/k!)d(k)[c(Γ)Γ0)/c0]/d(Γ/Γ0)k, k! ) 1‚2‚3‚...‚k (C3)

64(d°2)2/(3π2) + 15d°1d°3/π + 16d°4/π2 (C12) In particular, for the Langmuir isotherms the coefficients d°k are equal to

d°1 ) (1 + b), d°2 ) b(1 + b), d°3 ) b2(1 + b), d(k+1) ) bk(1 + b) (C13) therefore, eqs 31a and 31b are given by

a*1 ) 2/π1/2, a*2 ) (1 + b),

b*1 ) d°1a*1, b*2 ) d°1a*2 + d°2(a*1)2,

a*3 ) 4(1 + b)(1 + b + 4b/π)/(3π1/2) (C14)

b*3 ) d°1a*3 + 2d°2a*1a*2 + d°3(a*1)3 (C4) b*4 ) d°1a*4 + d°2[2a*1a*3 + (a*2) ] + 3d°3(a*1)2a*2 + d°4(a*1)4 (C5) b*5 ) d°1a*5 + 2d°2(a*1a*4 + a*2a*3) + 2

a*4 ) 3(1 + b)[(1 + b)(1 + b + 4b/π)/3 + b(1 + b) + 2b/π]/2, a*5 ) 16b*4/(15π1/2) (C15)

2

2

(C10)

b*1 ) 2(1 + b)/π1/2, b*2 ) (1 + b)(1 + b + 4b/π) (C16) b*3 ) 4(1 + b)[(1 + b)(1 + b + 4b/π)/3 +

3

3d°3[(a*1) a*3 + a*1(a*2) ] + d°4[4(a*1) a*2 + 2

5

6(a*1) a*3] + d°5(a*1) (C6) For arbitrary desorption isotherms the coefficients a*k and b*k in eqs 31a and 31b are given by

b(1 + b) + 2b/π]/(π1/2) (C17) b*4 ) (1 + b)2[(1 + b)(1 + 6b + 44b/π) + 128b2/(π2)]/2 + 15b2(1 + b)2/π + 16b3(1 + b)/π2 (C18) LA9601519