Adsorption and Desorption Kinetics of Molecules and Colloidal Particles Z. A D A M C Z Y K 1 AND J. P E T L I C K I
Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Cracow, Poland Received January 29, 1986; accepted July 30, 1986 Kinetic aspects of adsorption determined by both bulk diffusion and surface activation barrier are analyzed theoretically for a spherical interface in contact with a finite volume of molecular or colloid solution. Rigorous transport equations for the bulk and surface phases are formulated together with general nonlinear boundary conditions which can be used for describing adsorption governed by various isotherms including those of Henry, Langmuir, Frumkin, etc. Explicit analytical results are derived by applying the Laplace transformation in the case of the ideal Henry isotherm for (i) a spherical adsorbing interface in contact with a quiescent solution of a finite or infinite extent and (ii) for a sphere surrounded by a stagnant layer of a finite thickness in contact with an infinitesizedreservoir of a wellstirred solution. Limiting shorttime and longtime results as well as results derived in the special case of a plane geometry of the adsorbing surface are discussed. Using our general expressions the range of validity of equations previously known in the literature which were derived for the diffusioncontrolled or surface barriercontrolled adsorption regime is estimated. In the general case of the nonlinear boundary conditions (Langmuir isotherm) an implicit numerical scheme is applied based on an "exact" solution of the set of nonlinear ordinary equations arising from the discretization of the surface kinetic equations. © 1987 Academic Press, Inc.
INTRODUCTION
often preceded by an adsorption step. This situation is caused largely by experimental difficulties in measuring adsorption kinetics, which is usually a very fast process when small molecules or ions are concerned. However, for larger organic molecules, e.g., surfactants, polymers, and colloidal particles, the nonstationary (transient) adsorption conditions m a y last for long periods o f time, approaching hours and m o r e for a diffusioncontrolled process (for estimations o f adsorption relaxation times see Table I). Therefore, experimental investigations of adsorption kinetics can be readily performed using colloidal suspensions. Such experiments have been performed in either an indirect (15) or a direct way (67) u n d e r steadystate conditions when particle adsorption kinetics have been determined by fluid convection effects. Early attempts at a theoretical description o f adsorption kinetics were limited to the planar geometry o f the adsorbing surface and
Adsorption p h e n o m e n a have usually been studied according to their equilibrium aspects, i.e., the correlation between bulk concentration o f a given substance (solute) and its surface concentration, often erroneously called surface coverage or surface excess, at an interface u n d e r fixed temperature and transport conditions. By varying the bulk concentration o f solute a functional dependence between these two variables, usually called the adsorption isotherm, can be easily determined in an experimental way. In contrast to the extensive experimental and theoretical knowledge o f adsorption statics (thermodynamics), kinetics o f adsorption, and especially ofdesorption, has been relatively little studied despite its great practical significance in heterogeneous catalytic chemical and electrochemical reactions
' To whom correspondence should be addressed. 20 00219797/87 $3.00 Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of ColloM and Interface Science, Vol. 118, No. 1, July 1987
21
ADSORPTION KINETICS TABLE I Estimationsof DiffusionControlledAdsorptionLengthScaleL, and Adsorption RelaxationTime z~ (InfiniteVolumeAdsorption) Molecular solutions Bulk concentration Solute size and diffusion coefficient
106 M (6 × 1014cra 3)
103 M (6 X 1017 cra 3)
2 × 10 8 c m (2 A)
Nm = 1015 c m 2
0.13 s
2.1 × 10 5 cm2/s
1.7 × 10 9 m o l e c m 2 Nm = 1012 c m 2
1.7 × 10 3 c m 1.3 × 10 7 s
1.7 × 10 t2 m o l e c m 2
1.7 × 10 6 c m
Colloidal solutions 2.5 × 1012 c m 3
Nm = 2 × 109 c m 2 Arm = 2 × 107 c m 2
0.13s 1.7 × 10 3 c m
2.5 × 109 cm3 (0.001% vol)
(1% vol) 2 × 10 5 e m 2.1 × 10 8 cm2/s
1.3 × 105 s 1.7 c m
30 s 8 × 10 4 c m 3 × 10 3 s 8 × 10 6 e m
3 × 10 7 S 0.8 cm 3× 103 s 8 × 103 cm
Note. T = 2 9 3 K , rl = 10 2 g ] c m s ( a q u e o u s s o l u t i o n s ) , L a = Nmle~, "ra = L~IDoo, D o = kT]67r*la.
usually concerned the diffusioncontrolled regime when no microscopic motion of the suspending medium took place. Ward and Tordai (8), Sutherland (9), Hansen (10), and Bakker et al. (11) were the first to describe theoretically adsorption kinetics onto a plane interface (fluidair) from a quiescent fluid of an infinite extent. General integral dependence between surface concentration and the volume subsurface concentration of a solute has been formulated for an arbitrary adsorption isotherm and explicit analytical formulas have been derived in the case of the ideal isotherm (9, 11). McCoy (12) used a series expansion technique combined with the Laplace transformation to obtain solutions describing diffusioncontrolled adsorption kinetics onto a plane interface for the Langmuir and Freundlich isotherms, whereas Miller (13) applied a finitedifference numerical method to perform similar adsorption kinetics calculations for various nonlinear adsorption isotherms. In (14) Miller et al. performed numerical calculations of nonlinear, diffusioncontrolled adsorption kinetics of surfactant mixtures at the liquidair interface.
These results obtained for simple geometries have been generalized by Mysels et al. (1517) to an arbitrary geometry of the adsorbing surface. However, explicit analytical expressions have been given only for diffusioncontrolled adsorption onto a plane surface or on a sphere surrounded by a stagnant layer of a finite thickness in contact with a wellstirred reservoir of solution having an infinite volume. The theoretical results cited above have been derived assuming that a quasiequilibrium is being established between the solute surface concentration N and its subsurface concentration Co according to an appropriate isotherm equation, i.e., Co = F(N) (purely diffusioncontrolled adsorption regime). This model, leading to particularly simple forms of boundary conditions for the bulk transport equation, is expected to hold well when both adsorption and desorption processes are fast (large adsorption and desorption constants due to the lack of surface activation barriers) or when a system is close to equilibrium (longtime limit of adsorption). For short times and for slow adsorptiondesorption transitions (barrier controlled adsorption) a more general Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
22
ADAMCZYK AND PETLICKI
form of the boundary conditions at the interface is needed. This aspect of mixed adsorption kinetics, controlled by both bulk solute diffusion and surface barrier, has been discussed at length by Baret (18, 19) who formulated general kinetic equations connecting solute surface concentration with its instantaneous subsurface concentration via implicit integral relations for various nonlinear adsorption isotherms (localized and nonlocalized adsorption processes). Krotov (20) formulated general transport equations for multicomponent mixtures with general boundary conditions valid for any adsorption process, deriving explicit results for the special case when the bulk transport (diffusion) of solutes was very rapid so the overall adsorption kinetics was controlled by surface barriers. Miller and Kretschmar (21) numerically treated the mixed adsoprtion kinetics of surfactants onto a planar free interface using the Langmuir isotherm, whereas Pierson and Whitaker (22) applied the general boundary conditions to describe the adsorption kinetics of surfactants onto a growing spherical interface (drop). They solved the bulk transport equation using an explicit numerical scheme and derived limiting analytical expressions for the surface bartiercontrolled case. The concept of a reversible adsorption of colloidal particles was first introduced by Ruckenstein et al. (23, 24) who developed a theory of particle adsorption kinetics at solid interfaces (within energy minima) from both stagnant suspensions and suspensions undergoing macroscopic flow resulting in a forced convection of particles toward interfaces (25). In their papers equations describing particle adsorption rate as a function of the specific energy minimum depth, energy barrier height, and the intensity of fluid flow have been given. These expressions have been derived for the ideal isotherm, i.e., for low surface coverages of particles when the geometrical blocking effects and other nonlinear effects were absent. In (26) a more general theoretical approach has been developed to describe both a reversible and irreversible adsorption of colloidalJournal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
sized particles at solid interfaces (collectors) from suspensions undergoing welldefined flows. Kinetic equations have been derived for the ideal isotherm under quasistationary conditions, describing the number of particles accumulated within an energy minimum near a collector surface as a function of time, barrier height, flow intensity, collector geometry, etc. These equations can be used for predicting reversible adsorption or desorption of particles as well as for describing irreversible capture of particles (deposition) in an energy minimum of an infinite depth (which corresponds to a zero desorption rate and socalled perfect sink boundary conditions). From what was discussed above, it seems that there are no concrete theoretical results concerning transient adsorption and desorption from a finite volume of solution in the case of mixed kinetics (diffusion and barrier control). Therefore the aim of our paper is to derive such results for the general case of a spherically shaped interface (either a solid or a free one). From results obtained for the sphere many simpler subcases can be derived, e.g., adsorption onto planer interfaces from a finite or infinite volume. From our solutions we also determine the range of validity of previous results derived for diffusion and barriercontrolled adsorption onto planer interfaces. ADSORPTION KINETICS ONTO SPHERICAL INTERFACES
Transport Equation Let us consider a spherically shaped homogeneous interface (adsorber) of radius R, (see Fig. 1) which may represent either a liquid drop, a gas bubble, or a solid particle (collector) in contact on the exterior with a liquid or gaseous solution of one adsorbing substance (solute). The size of the solute particles is arbitrary, i.e., ranging from molecular to colloidal and larger; however, the size of the adsorber is assumed to be much larger than that of solute particles. If the solution is not concentrated the diffusional motion of solvent molecules may be neglected and the continuity
ADSORPTION KINETICS b
c I
23 T=O,
c=%
I
(i) OO~x =0 (ii)c=Cb
o "o
o rn 9=(r Ra)/L
×
<
H
A=L/Ra
~L
"
H={R1RaVL
FIG. 1. (a) Kineticsof adsorption onto a sphericalinterfacefrom a finitevolumeschematicdiagram. Case (i), a perfectlyreflectingouter boundary, thus j± = 0 at r = R~ (Jl is the normal component of the solute flux vector).Case (ii), contact with a wellstirredsolution,i.e., c = CDat r = RI (whereCbis the uniform bulk solute concentration for z = 0). (b) Kinetics of adsorption onto a planar interface schematic diagram of concentration changes as a function of the time z.
(mass balance) equation of the solute is given by the wellknown expression Oc   q  V ' j b = Qb, dt
[1]
where c is the volume concentration o f the solute, Jb is the bulk flux vector being a sum of a " r a n d o m " diffusional contribution due to a chemical potential gradient and a "systematic" migration contribution due to all specific hydrodynamic and external forces, and Qb is the source term describing bulk chemical reactions of the solute. By solving Eq. [1] with appropriate boundary conditions one can describe quantitatively the temporal evolution (which for convenience can be called adsorption or desorption) of the adsorbersolution system, starting from an arbitrary initial concentration of solute and moving toward an equilibrium concentration distribution. In order to do so, one should know all specific forces acting between solute particles and adsorbers and among the particle themselves as a function of position, surface coverage, solute particle configurations, and many other factors. This makes such ab initio calculations of adsorption kinetics prohibitive even when using powerful numerical methods. Results
have been obtained only for a simple onedimensional specific potential which was independent of surface coverage (linear model (26, 27)). In view of the complexity of such an "exact" approach one is forced to accept a simplified model of the adsorption process usually based on the somewhat artificial concept of an adsorption layer where a large condensation of solute particles takes place. Within this layer the solute concentration is expressed in terms of surface concentration (surface coverage) N, which can be a positiondependent quantity. Thus, the mass balance equation of solute has the following form in the adsorption layer, ON G+v~.js=G,
[2]
where Js is the surface flux vector (if solute particles remain mobile) due to surface diffusion and migration and Qs is the surface source term, being a sum of the chemical reaction contribution Qr and the source of particles exchanged with the external world Qe which must be equal to the normal component of the bulk particle flux vector J0 at the edge of the adsorption layer. We postulate, as in many adsorption theories (1820) that the net flux of solute jo is Journal of Colloid and Interface Science, VoL 118, No. 1, July 1987
24
ADAMCZYK AND PETLICK/
the sum of adsorption and desorption fluxes Ja and Jd according to the relation Qe =J0 =ja ja = k~cof(N)  koNg(N),
[3]
whereja is proportional to the adsorption constant/ca, the subsurface concentration of the solute Co, and a given function of surface concentration f(N), whereas ju is proportional to the desorption constant ka, surface concentration N, and eventually a correction function g(N) accounting for the lateral interactions among particles. Under equilibrium conditions, when j0 = 0 from Eq. [3] one has ( N ) = Uls(U), K~co = N g~~~
[41
where K~ = k~]kd is the equilibrium adsorption constant and Is(N) = g(N)/f(N). Equation [4] represents an isotherm equation written in a general form, from which one can derive various wellknown subcases, e.g., the Langmuir isotherm i f g = 1 and f = 1  (N/Nm) (where Nm is the maximum surface concentration), the linear Henry isotherm ifg = 1 and f = Nm, the Frumkin isotherm widely used in electrochemistry i f f = (1  (N/Nm))exp biN and g = exp  b~N (where bl is a parameter accounting for particle lateral interaction within the adsorption layer). In the general form Eq. [3] represents the link between surface and bulk continuity equations, constituting the boundary condition for Eq. [1] from a mathematical point of view. In due course we shall consider adsorption from stagnant solutions when convection and external force effects are absent. Equation [ 1] may then be written in the following simpler form for a dilute suspension (when there are no bulk chemical reactions), Oc ~ + ~7. (DUe) = O, [5] where D is the diffusion coefficient of solute particles; it can be position dependent but does not depend on solution concentration. We also confine ourselves to the special case of boundary and initial conditions possessing a radial symmetry. As a consequence of this limitation Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
the transport equation [5] becomes onedimensional with the following boundary condition at the adsorber surface, D Oc_ . _ Jo  Qe = k a c o f ( N )  kdNg(N)
[61
at r = R + a, where a is the particle radius. In the absence of surface reactions the surface continuity equation reduces to (for a constantarea interface) ON Ot
Qs = kaCof(N) kdNg(N).
[7]
Due to the fact that the functionsfand g may depend on N, Eq. [7] is nonlinear and so is the entire boundary value problem formulated by Eqs. [5][7]. As far as the boundary conditions far from the adsorber are concerned we shall consider two separate cases: (i) No flux boundary condition, i.e., •
Oc
0
Jb = Or =
[81
at r = RI (see Fig. 1). This type of boundary condition describes adsorption from a spherical, finitevolume reservoir of radius R~ whose walls are perfectly reflecting (nonadsorbing). (ii) Contact with a wellstirred solution, when at the distance r = R~ a uniform timeindependent solute concentration is being maintained during adsorption, i.e., C=Cb
at
r=R1.
[9]
In this case R~ can be interpreted as the thickness of the stagnant (unmixed) layer surrounding the adsorber (natural convection thickness). In order to describe adsorption kinetics quantitatively one also formulates initial conditions describing the spatial distribution of solute within suspension and the value of N at t = to. We introduce the uniform initial conditions C=Cb for
Ra
[10]
ADSORPTIONKINETICS N=N~
at
t=to
[111
(one usually assumes to = 0 for convenience). The transport equation and the boundary and initial conditions can be less cumbersome for a mathematical analysis if one introduces the following dimensionless variables (similarly done by Miller (13)): 2
rRa a L
[121
r = Ra(A2 + l + ~a) ~ Ra(A2 + 1); A =L/Ra
[13]
D~ r = tLT
[141
~=c/cb
[151
O=N/Nm.
[161
Consequently,
ka
= k L2cb ,~aD~Nm
L2 Tca= kd Do° ,
[17]
[181
where L is the characteristic length scale (to be discussed later); 2 is the dimensionless distance from the adsorber surface; r is the dimensionless adsorption time (D~ is the diffusion coefficient of solute particles in the bulk of the solution); Nm is the maximum surface coverage in molecules per unit of adsorber surface which is usually larger than its geometric surface; ? and 0 are the dimensionless solute concentration and surface concentration (coverage), respectively; and ~ and kd are the dimensionless adsorption and desorption constants. Considering the above definitions Eqs. [5] and [7] together with the boundary and initial conditions are transformed to
0r0c (jx+l 1)2~[ D ( Z ) ( A 2 + 120Y] ) ~~] [19] 00 Or ka?of(O)fCdOg(O)
[20]
25
0"~  ~ / ~ [ k a c 0 f ( 0 )
~_=0;
(i) or

kd0g(0)]; g= 1 at
at
2 =0
[211
2=H;
(ii)[221
O<2
[23]
Ox
~= 1r _ r 0 ; ) 0 = OiJ

where ~ = L J L , H = (Ra  R a ) / L . L. = Nm/cb is the characteristic "adsorption" length which is a natural length scale of the problem under consideration because if we choose L = La the boundary condition [21 ] simplifies to
0___~=kaYof(O) fcdOg(O) at 02
2 = 0.
[24]
In Table I values of the adsorption length La and corresponding values of the characteristic adsorption time ra = LZdD~ are calculated for molecular and colloidal solutions (in a liquid phase). It can be seen from Table I that for a molecular solution the adsorption time (calculated for difffusioncontrolled adsorption) is usually small for larger concentrations of the solution. In most cases, adsorption of molecules can be treated as an instantaneous process except for very dilute solutions and large equilibrium surface coverages. On the other hand, for colloidal suspensions the adsorption time can be as large as 106 s due to smaller values of the diffusion coefficients and much lower particle concentrations in the bulk in comparison with molecular concentrations. Equation [19] can be further simplified if one assumes that the particle diffusion coefficient is position independent, which is a good approximation for molecules and small colloidal particles because of very large adsorption distances La in comparison with particle dimensions. Considering this and introducing the new variable u = b(A2+ 1 ) 4 ~= u/(A2+ 1),
[25]
Eqs. [19][23] become OU
OZu
Or
022
[26]
Journalof Colloidand InterfaceScience, Vol. 118, No. 1, July 1987
26
ADAMCZYK AND PETLICKI alytical solutions to these equations derived for the linear isotherms (Henry) which are expected to be valid for low surface coverages. These analytical solutions, although limited in their applications, can provide one with a [281 deeper insight into the physical nature of adsorption kinetics than cumbersome numerical solutions. Moreover, the analytical solutions can serve as a good test of the accuracy and stability of the numerical scheme applied. [29]
O0 = tCaUof(O) k,#g(O) Oz
[271
du ,9.£ [Bkaf(O) + A ] u o  ~ O g ( O ) at
Ou Au 02 A 2 + l '
(i) or
(ii) at
~?=0
u=A2+l;
.~=H
u = A 2 + 117 "=7°; O= Oi J
LINEAR ISOTHERM ADSORPTION
O<.~
[301
From Eq. [25] we see that the dimensionless subsurface concentration Co is equal to Uo. In the case when A = L/R~ ~ 1, i.e., when the characteristic adsorption length is much smaller than the adsorber radius, the above set of equations reduces to the form OR
In the linear model, i.e., when riO) = g(O) = 1 adsorption kinetics is governed by the set of equations Ou 02u [351 07"
02U
at
Y= 0
u
1;
[32]
(ii) at
Y= H
u = l ]7" OiI_ =7°;
 
0.~ 2

kzd0
[33a]
~=(13Tq+A)uo~TcdO=jo Ou Au 0.~ A Y + I at
(i)or

Ou
00 O~ = k~uof(O)  k~Og(O)
Ou ~~= #[[c~uof(O)l&g(O)]
00
[31]
= 0~ 2
Ou OY 0;
1. General Solutions for a Finite Volume Adsorption
[33b]
0<)~
[34]
0
u=E These equations describe kinetics of adsorption or desorption onto a planar interface. However, even in this limiting case, the governing transport equations constitute nonlinear parabolictype boundary value problems which can only be solved in an exact way by means of numerical methods. Before we discuss the numerical technique applied for solving Eqs. [25][30] or [31][34] and the results obtained in the case of nonlinear isotherms, we shall present in the next section some anJournal of Colloidand InterfaceScience, Vol. 118, No. 1, July 1987
(i) or
[361
at
Y = 0 [37]
u=AY+I
Y= H;(ii)
u=A'~+ ll~'=ro; 0 = Oi J
O
[38] [39]
Since the above transport equations and boundary conditions are linear they can be readily solved by means of the Laplace transformation. If one defines u exp  sr
~7= so 0=
dr = L u ,
g = ~/(A2 + 1) 0 exp  sr
dr = CO
[401 [41 ]
and so on, Eqs. [35][38] are transformed to d2~ d22 = s ~  (A2+ 1)
[421
27
ADSORPTION KINETICS
Oi
~
~
[43]
u0.
ft = (A.~ + 1)/s S 112
+K.
The boundary conditions for Eq. [42] are
TS 2 "} a3s 3/2 ..~ a2 Ts + a is 1/2 + aoT s1/2(.,~  H ) A T s h S1/2(___X H).] C J R S 1/2 S 1'
x rch
 ~ = ( i l l +A)~7o flkag at d~ d2
A fi A2 + l
at
2=H;
2= 0 case(i)
[441 [45]
[
[49] where T = th sl/2H, S = sh s 1/2 (2  H), C = ch sl/2(2  H )
or
A2+ 1 zi=~ s
at
2=H;
case(ii)
R=AH+
[46]
ka
2
go(t)exp  Ed(r 
al = EaA 2 ~H , _
ao =
t)dt
U=
+ ClCh S1/2(2H)
s + C2sh sl/2(2 H),
d,
A2+ 1 s
[47]
A)?+ 1
A 2
[501
K = fl(Edoi ka). In ease (ii), ~7is given by
= 0iexp  Edz + kaexp  Ear £" ~o(t)exp( Eat)dt.
Therefore, in order to evaluate 0 explicitly from Eq. [47] one should know the subsurface solute concentration ?0 as a function of the time r. It is interesting to note that in the case when ~ = 0 the value of the surface coverage always decays exponentially according to the relation 0 = 0iexp  Ear although the solute subsurface concentration ~0 and its concentration within the adsorption volume are complicated functions of the time r. The general solution of the ordinary differential Eq. [42] is
A R'
a3=fl~+A
A a2 = Ed (BEa + A)~,
Considering the properties of the Laplace transformation (convolution theorem) it can be easily deduced by inverting Eq. [43] that the surface coverage 0 is given by the following general relationship: 0 = Oiexp  Edr +
I,
K CTs3/2+a2s+alCTsl/2+ao
X
s h S 1/207 
H)
S
[511
where, CT=cths~/2H=C/S, al = ka,
a2=flEa+A,
ao = EdA.
Explicit solutions for u, u0, ?, and 0 can be obtained from the above subsidiary equation by a direct application of the inversion theorem of the Laplace transformation which states that (28) 1 r x+i~~ exp(sr)ds = £  l g ( s ) . u =2~ax_i~
[521
After some algebraic manipulations (see Appendix A) one obtains the following results: [481 case (i)
where C1 and C2 are arbitrary constants of integration to be determined from boundary conditions [44][46]. Considering these boundary conditions one obtains for z7 in case (i),
, .. ~ aZn[cos a . ( 2  H )
C = Coo t/t 2a  /
n=lPn[
X
A
%R
sin a n ( Y  H ) ] ; ~ / e x p  a 2 r [53] an  a2an + ao I JournalofColloidandInterfaceScience,Vol. 118,No. 1, July 1987 4
a3a 2  a l
  7 cos an//
2
28
ADAMCZYK
?.=l+
K
R(a3 + azH + 1/2alH 2 + 1/6aoH3) '
where a . are the real positive roots of the nonlinear trigonometric equation given in Appendix A. The expression for 50 is given in Appendix A and using it one obtains from Eq. [47] the following result for 0: 
ka
AND
PETLICKI
equal to the initial one, i.e., unity (consequently 0~ = (~:a]~:d)5oo= Ka?~), and that both these equilibrium values are being approached exponentially for longer periods of time as in the previous case. Introducing the reduced surface concentration 0 = (0  Ooo)/(Oi  0o~), often used for interpreting results of experimental measurements, Eq. [56] can be written in the form oo
0 = 0iexp  ka7+ ~ 5o~(1  exp  L r )
0 = exp  kdr +/~kakd ~ Q,(exp  a2r n=l
oo
+ k~K ~ Q,(exp  an2r  exp  kdr),
A Q" =
1
a3o~al
R a 4 
a2o~ z. +
[57]
Since Q, does not depend on Oi, Eq. [57] has a universal character describing both adsorption and desorption runs (i.e., when 0r < 0 or 0; > 0) in exactly the same form.
where
o~
 e x p  Ear).
[54]
n=l
) ao
( L  a2)P,
Although expressions [53] and [54] are rather cumbersome for direct use one can easily deduce that the approach to the uniform equilibrium concentration ?~ and surface coverage O~ = (kJkd)c~ is exponential for a finite value of H (stagnant layer thickness). It should be also noted that the final concentration co i s different from the initial one (equal to one in dimensionless units). In case (ii) we obtain analogously (see Appendix A) 5 A£+ 1 1 sin an(Y H) l+Kn=~,/~, s ~ n a  ~ expan2~" [55]
2. Limiting Solutions for Adsorption onto a Sphere In this section we shall consider some limRing cases of the general solutions derived above. These solutions are much less cumbersome than the general ones and may be used easily for comparing experimental measurements of adsorption (or desorption) kinetics. First we consider the shorttime limit when the system is close to its initial state. In terms of the transforming variable s the shorttime limit means that ~ H >> 1 which can be readily inverted giving the inequality z ~ H 2. Considering this, the subsidiary equations for ~ in cases (i) and (ii) become respectively 1 S
K S 1/2
0 = 0iexp  kdr + ~ ( 1  exp  k~z)
(s2 +a3s3/2 +a2s+alsl/2 +ao) (A.~+ 1) [58]
oO
+ K Z Qn(eXp a2zexp~:d~'),
[56]
n=l
1 5=+ S
where an are the real positive roots of the equation given in Appendix I. I t can be seen from these equations that in case (ii) the final uniform concentration 5oo is Journalof Colloidand InterfaceScience, Vol. 118, No. 1, July 1987
e x p   s 1/22~
K (s3/2+a2s+alsl/2+ao)
exp  s 1/2)? (A£+ 1)
[59] Moreover if the following conditions are fulfilled,
29
ADSORPTION KINETICS
a3~.$1/2;
(a2c~ 1)
4flkakd 3/2.
0= 1  k d z +   f  ~ r
aE~s; al~sa/Z; a0~s2;
The solute flux to the adsorption layer j0 is given by
(al'r3/2,~ 1)
(a0~2,~1)
[60]
then both Eqs. [58] and [59] reduce to the same form: 1
+ 1 , a / ~ 2 ~ '1"2
[61]
This equation can be easily inverted giving the expression for ? as
g=1+~[2 V~/7rexp/t~r)'£2\ .£ _
,62,
where erfc z =
exp  ~2d(.
The subsurface solute concentration ?0 is given by
2K 1/2
1+ ~
r
.
[63]
From this dependence we see that for times short enough to fulfill inequalities [60] the subsurface solute concentration always changes as a square root of the time r. If the initial surface coverage Oi is equal to zero (so K = ka/~) then ~ decreases linearly with r uE for short times. Substituting Eq. [63] into expression [47] and performing the necessary integration gives 1
0 = 0iexp  kctr+ ~ (1  exp  kdr)
+
( OC) : [~(f~aSO__~dO)~_l~[T£a__f£dOi J0=~0
K
C=~s (Ax+ 1)s3/~expsl/2~.
~0=
[66]
(a2r ,~ 1)
~:,K e x p  kay" f ~ ]/~3/~ d0
~2exp ~2d~" [64]
If ~:d~"~ 1, Eq. [64] further reduces to
0 = Oi+ ( f q  Oi[¢d)r+ Tc,K4.~ z 3/2 [65] 3V~r
kd(kaOiled)r].
[67]
It is interesting to note that the surface concentration always increases linearly for short times, even for small surface barriers, and the solute flux is finite and constant equal to ka (time independent) in this limit. This is an important conclusion which differs markedly from what was found in previous theories (917) of diffusioncontrolled adsorption where the subsurface concentration ?0 was always zero when r * 0, and the solute flux Jo was infinite in this limit. This divergence between our work and previous results stems from the assumptions of a local equilibrium prevailing at the edge of the adsorption layer so that c0 is always connected with 0 via an isotherm equation. When 0 = 0 for r = 0 (as was assumed in previous theories) it follows from all isotherms that ?0 = 0 as well, and therefore an infinite concentration gradient is bound to appear at the adsorber surface. It seems that the assumption of local equilibrium is rather unjustified in the limit of short times. This point shall be discussed later in some detail. Presently we shall consider the longtime limit when the system is close to the final equilibrium conditions which in terms of the Laplace parameter s is equivalent to the postulate that ~ H ~ 1 (or after transformation r >> H2). In this limit from Eq. [49] we obtain for (in case (i)) ( =  q1
s
K
1
(AH+ 1)s(as+b)(1 + 1/2sH 2)
=~o~ K 1 s b(AH + 1) (as + b)'
[68]
JournalofColloidandInterfaceScience,Vol. 118, No. 1, July 1987
30
ADAMCZYK AND PETLICKI
where

0 = 0iexp  k~r + ~dd[1  exp  ~:dr]
Fo~= 1 + K/(AH + 1)b a = H + la3HZ +~a2H 1 3 + ~la ~ H 4 +ifgaoH5 1 b = a3 + azH+ ½al H 2 + laoH3.
[qKH [_ 1 + a(r2  rl) [kd  rl (exp rlr  exp  ~:dr)
In case (ii) the expression for 8 is ~:d 1"2(exp  r2r 1
1 = s
g(~n) (A£+ 1)(asZ+bs+c) '
[69]
[73]
1
b = 1 + a2H+ ~a~H 1 2 + ga0H3 1 c= a~ + aoH.
3. Adsorption from an Infinite Volume
1 5 = ½H2+~a2H 3 + ~la ~ H 4 +~6aoH
The inversion of Eq. [68] gives
c=coo
b(AH+l)exp
• .
[70]
This dependence indicates that the final uniform concentration F~ is attained in an exponential way with the rate constant equal b to. a The value o f 0 in the longtime limit is given by the expression 0 = 0iexp  kdr t ~ ka/C
×
exp  kar)]
In the following two paragraphs we discuss in more detail important limiting cases o f adsorption onto a sphere from an infinite volume and then we discuss adsorption onto a planar surface.
where a

 7
b
i_,/exp   r  exp \ a
1 s
A~+ 1

b(AH+ 1)/~Td a )
)
exp  s i/2Y
K
+ s3/Z + a'zs + a'ls~/2 + a'o A(Y + 1)
[74]
where
a'z= 3ka + A,
( I  exp  kd 7") /
If the adsorption volume is very large in comparison with the adsorber then the value of the parameter R is m u c h larger than unity and Eqs. [49] and [51 ] both reduce to the same limiting form
a'l = ka,
and
a'o= [caA.
The inversion o f this equation which requires some algebraic manipulations is performed in Appendix B. The final result is 3
[71]
F = 1 + K ~ Cnrnexp(rnY+r2~) n=l
In case (ii) the solutions for F and 0 are (if (b/a) z  4e/a > O) F=I
K(YH) (AY+ 1)a(r2  rl)
0 = 0iexp  ~:dr + ~ ( 1  exp  kdZ)
× (exp  rlr  exp  r2~), [72] 
3
r Cn n
F
2
+ kaK • ~ / e x p ( r n r ) e r f c ( r , n=l rn t KdL
where rl=~[b+
~(~)Z4c];
Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
Vz)
+(1 + r , I ) e x p   k d ~ ] , where
rz=l[b~(b)24c]
.r
I=
2
exp
[76]
ADSORPTION
where C1 = 1/(rl  r2)(r3  rl), C2 = 1/(rl  r2) × (r2  r3), C3 = 1/(r2  r3)(r3  rl), and z=   r , (n = 1, 2, 3) are the real roots of the characteristic cubic equation z 3 + a'2z 2 + a'~z + ab = 0.
and 1 3=~ S
K s */2(CTs + 3kas 1/2 + CTk~)
[77]
If only one of these roots is real (say r3) and two others are complex conjugates the solutions for 6 and 0 can be found in Appendix B. As done previously, Eq. [76] can be simplified by introducing the reduced surface concentration 0 showing that in the case of adsorption from an infinite volume, adsorption and desorption runs are identical in terms of the 0 variable, provided that the values of ~:a and ~:d remain the same.
× sh s 1/2()7_ H ) ; (ii). S
0o
Pn
n=l
In this section we discuss in more detail the important subcase when the adsorption surface is planar, which can be treated as the limit of adsorption onto the spherical collector when its radius tends to infinity in comparison with the characteristic adsorption length L; thus one can assume A ~ 1. All results presented below can be, in principle, derived from previous solutions for the spherical adsorber. However, this can sometimes be a very tedious procedure and therefore it seems that for us the limiting solutions for the planar surface can be more efficiently derived via the subsidiary equations (in the Laplace transformation space). Moreover, planar adsorption kinetics has been treated so extensively in the literature that presenting explicit expressions seems to be very useful for a comparison of our results with those previously obtained. Assuming that A ~ 0 the subsidiary equations for the sphere [49] and [51] are simplified to
[79]
It should be noted that Eq. [78] can also be used for describing adsorption or desorption kinetics onto two parallel planar surfaces (channel) located at a distance 2 H from each other. By inverting Eqs. [78] and [79] (see Appendix A) one obtains for ? and 0 the following solutions:
4. Adsorption onto a Planar Surface
×
COS O ~ n ( X  
H)
e x p  a~r
[80]
COS Ofn 
0 = 0iexp  kd/"+ _
oe
~ d C'oo( I  
exp  ~:dr)
1
+ kaK ~x(expa2zexp~:dz),
[81]
n=l l~n
where ~ = ( H + Oi~)/(H + Kal3) is the final uniform solute concentration within the adsorption volume and c~. are the real positive roots of the trigonometric equation given in Appendix A. In case (ii) we have J = 1 + K ~ (ka  a~) sin a , ( ) ?  H ) n=l Pn sin a . H ×expa~r
[82]
1
0 = 0iexp  kdr b ~ ( 1  exp  ~:dr) o~ 1
 kaK Z g(exp  aan'c exp  ~:dZ). [831
1
~_fi=
n=l In
S
K ch I s1/2(Tsnt_~SaS1/2_t._ T~d)
31
KINETICS
S 1/2(• 
H ) ; (i)
C [781
These solutions, although less complicated than those for the spherical adsorber are rather cumbersome for a direct use. We discuss therefore some important limiting cases which Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
32
ADAMCZYK AND PETLICKI
can be derived directly from Eqs. [80][83] or via simplified forms of the subsidiary Eqs. [78] and [79], which is usually a simpler way. When adsorption proceeds from an infinite volume or for very short times when the inequality $1/2H >> 1 (r 4. H 2) is fulfilled, Eqs. [78] and [79] both reduce to the same simpler form 1 ~=~ S
~= erfc~~rl[r+ eXp(r12 + r2r)erfc(~r~rl/r+ rl )
[871 Co= exp(r2r)erfc(rl ~r) O=
ka
2
Oi+~[exprlr
[88] 2ka ,t
erfc rll/Tr 11 + f~rrl Yr. [89]
K SI/2(S~[~TCaS1/2]fCd)
I
exp sl/2x
For short times, i.e., when/32~:2r @ 1 the subsurface concentration & and surface coverage 0 are given by
K
=~
s s x/2(sI/2 + rl)(s 1/2+ r2) exp 
s 1/2E, [84]
where rl and r2 are the roots of the quadratic equation z2 + 5~az + ~a = o. By inverting Eq. [84] one obtains the following solution for ~ and 0 (when the roots are real): Y= 1 + ~
K
[ 2 [exp(rl x + r l r)erfc

k:a
0 = Oiexp  kar + ~ (1  exp  kdr)
+ ~ kaK { ~ 1
(r22+ ~:d)[exp(r2r)
× erfc(r2frr)exp[qr]].
[86]
It is easy to show that for short times Eq. [86] reduces to the limiting form derived previously for the spherical adsorber given by Eq. [64]. If one of the roots, say r2, is equal to zero which is the case when the desorption constant ka is zero then rl =/3k~ and one has the following solutions for g and 0: Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
[901
0 = 0; + k,r + (4k, rl/3 VT~)r3/2.
[91 ]
Equation [91 ] indicates that for short times the surface concentration increases linearly with time r (this result can also be derived directly from the more general Eq. [64]) which contrasts previous results (914) suggesting a linear dependence of 0 on the square root of time r, This is so because in these works a pure bulk diffusioncontrolled adsorption was postulated (quasiequilibrium condition at the subsurface for all times). On the other hand, when k2r >> 1, which is the case for long times or even for short ones when the adsorption constant ka is large (for instance, in the "perfect sink" model), Eq. [89] reduces to the wellknown simple formula
O Oi = ~~ V~T.
[exp(r2l~')erfc(r,1/7) 1
 exp  kdr]
?0 = 1  (2/]/~)/3kafrr +/~2ka2r
[921
This equation describes the maximum diffusioncontroUed adsorption rate possible under any conditions. When the roots rl and rE are complex conjugates then the solutions for ? a n d 0 are more complicated as can be seen in Appendix B. It is also useful to derive limiting solutions for the case where the thickness of the solute layer in contact with the adsorbing interface is very small in comparison with the adsorption length scale L, so that the value of the dimensionless H parameter is much smaller than unity (thin film adsorption limit). For
ADSORPTION KINETICS the planar adsorption in case (i) (and also for the sphere when AH ~ 1) one has the following simple form of the subsidiary equation (which can be derived from Eq. [49] assuming H ~ 1), K
,
33
In case (ii) (contact with a wellstirred solution at the distance ~ = H) the solutions for and 0 are
K(H ,2)
fed
?=l+l+flkaHeXp
[93]
s
l+flkaH r
[96]
1
0 = 0iexp  ~:dr + ~ (1  exp  ~:dr) + (Oi /ca) 

where H + fl0i
X( e x p  k +=flk, a l H r  e x p  ~:dr) • [971
H+~Ka" This equation can also be easily derived from the longtime limiting expression for the sphere, i.e., Eq. [68] assuming H ~ 1. The inversion of Eq. [93] gives c = Coo (fl~:,+ ~:dH) exp  ~ + ~:a r. [941
If flkaH ~ 1 Eq. [97] simplifies further to
0  O~ = (Oi 0~)exp  kdr (Oi 0~)flkakdH e x p  koz. [98] 

Thus, for 0 we have the simple expression 0 = exp  ~:dr(1 /3~:a~:dH)
It should be remembered that Eq. [94] is valid after a short transition time on the order of H 2. As can be seen from Eq. [94] after this transition time the concentration distribution through the solution remains uniform at any time during the adsorption run, changing exponentially with the rate constant equal to
~L/H + ~. Using the above expression for g (equal to 50) one can derive the following expression for 0,
OiKa _{flka+k " O=O~4flK~~HHexp /~  d)*,
[95]
where
which has exactly the same form for both adsorption and desorption runs. Equations [78] and [79] can be used for determining the range of validity of previous models (917) describing the diffusioncontrolled adsorption kinetics onto a plane surface when the quasiequilibrium condition is postulated at the adsorbing surface, i.e., when the subsurface solute concentration 50is connected with the surface coverage 0 via an isotherm equation at any time. When one assumes that flka >~ S1/2 (which can be inverted giving B2k2 >> r) and kd > s(kdr >> 1) then Eqs. [78] and [79] reduce to
0o~= (kdkd)5oo. As can be seen from this equation 0 approaches the final 0oo value in an exponential way with the rate constant being the same as that for concentration changes, i.e.,/3kdH + kd, thus the smaller the thickness of the solute layer in contact with the adsorber the faster the adsorption (or desorption) kinetics at fixed values of ~:a and ~:d. Equations [94] and [95] can also be derived from Eqs. [72] and [83] assuming H ~ 1 and taking the first term of the series expansion only.
[99]
1
3=qS
(
K/flKa
S1/2 C S 1 / 2 + _ ~
1 s) Chsl/2(yH) a
[1001 and
1
= +
K/flKa
sh 
s 1/2(y_ H).
s s,/2(Ssl/2+_~KC ) [1011 For an infinite volume adsorption when H~s Journalof Colloidand InterfaceScience, Vol.
118, No. 1, July 1987
34
ADAMCZYK
>> 1 (so that T and CT  ) 1) Eqs. [100] and [101] both reduce to the following form:
1
(OdKa I)
5=  ~ S
$1/2(S1/2÷
\
exp  s 1/20?.
[ 102]
1
O=Ka+2(OiKa) ~ / H \ n=l I/3HKao/2n + 1 +  ~
)
~ga]
 2( O i  1 \ga
×
Thus
1...~_1
× e x p  o/~r, [106]
If we put 0i = 0 and/3 = 1, Eqs. [100][102] become identical to those previously derived in (14, 15). Therefore, it seems that results presented in those papers should be valid for values of the dimensionless adsorption time large enough so that the inequalities kZr >> 1 and ~:dr >> 1 are met simultaneously (which is always the case when r tends to infinity). The inversion of Eq. [ 100] gives the following expression for ? (see Appendix A):
c=c°~
AND PETLICKI
where o/n is to be determined from solutions of ctg o/nH =/3Kao/,. Similarly, the inversion of Eq. [102] gives 5=1 + (K0~/ 1)exp(r12+r2r)
X erfc(2X~r + rl~r )
[107]
O=K.+(OiKa)expr~r effcrl~r
[1081
1 (/3gKao/2 ÷ l + ~_KaKa)
COS O/n0~
H)
e x p  o/n2r, [103]
or
COS o / n n
= exprgr erfc r, frr,
where J~o = (/30i + H)/(/3Ka + H) and 0/, are the consecutive roots of
tga,H = ~Kao/n. The solution for 0 is o =Ka?o
where rl = 1//3Ka. Equations [103][106] when assuming Oi = 0, and /3 = 1 become identical to those derived in (1517). It is interesting to note that the short time limit of Eq. [102], i.e., ifrlV~z @ 1, is O = 0 i 
1 = Kacoo  2(Oi  g a ) ~ , / nffil

I/3Hgaa2 + 1 + X expo/2r.
H \
)
[104]
In case (ii) (from the inversion of Eq. [101]) one has
5=1+20i1
1
× sin o/n(~H) exp  a~r. sin a . H
2
(OiKa)7~r,1/7+ (OiKa)r~r VTr
[ 105]
Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
[109] or
~=1 ~ 2
VT.
[1101
SinceOi//3Ka>~ 1 for all desorption processes, it can be easily deduced from Eq. [ 109] that the surface overage 0 decreases from the initial value 0i as the square root of the time r, i.e., the desorption kinetics is nonlinear from the very beginning.
35
ADSORPTION KINETICS NUMERICAL SCHEME
The onedimensional partial differential equation [26] with the nonlinear boundary and initial conditions [271[30] was solved in the general case by the finitedifference implicit CrankNicolson scheme used previously (27) for solving similar transport equations with linear boundary conditions. The resulting set of linear firstorder equations (originating from the discretization of the bulk transport equations) was solved by the direct Gauss elimination method starting from the outermost point of the grid and proceeding toward the adsorber surface. In this way one was left with only one linear equation with two unknown values of solute concentration at two first points of the grid. This equation was then combined with the discrete analog of Eq. [28] describing the flux and with Eq. [27] describing the surface coverage changes. Finally, after rearranging these equations one was able to formulate two nonlinear equations with two unknowns: the subsurface concentration c0 and surface coverage 0 at time r + dr. This set of nonlinear equations was solved by the generalized secant method. Thus in our scheme, the new values of 0 and c0 were obtained in an "exact" way without using the tedious and not always converging iterative scheme applied by Miller (13) for solving the nonlinear difffusioncontrolled adsorption kinetics. The mesh size in the spatial domain was increased gradually when proceeding toward the bulk of the solution by using special transforming functions (27). Also the time step was increased monotonically with the increase in time. These transformations were applied to accommodate for the physical properties of the problem where the largest concentration and flux changes occurred at short times within the region close to the adsorber surface. Usually we used 200400 grid points in the domain and performed a few hundred time steps to complete a numerical run. For testing the accuracy and stability of numerical results we used the analytical expressions derived in
the previous section for the linear adsorption model. An excellent agreement (usually better than 0.1%) between analytical and numerical results was found in all runs suggesting that the nonlinear adsorption regime numerical calculations are also of good accuracy. Further details of the numerical scheme applied and the solving procedure are to be given elsewhere. ILLUSTRATIVE EXAMPLES
In this section we present some characteristic results derived from the analytical equations formulated in preceding sections in the case of the linear isotherm and results obtained from numerical solutions of the transport equation [26] for the nonlinear Langmuir isotherm. In the case of adsorption onto a spherical interface from a finite volume the dimensionless surface concentration 0 is, in general, a function of dimensionless time r, the initial surface concentration 0i, and four other dimensionless parameters, i.e.,
O= f(r, Oi, k~,kd,A,H).
[1111
This makes a systematic analysis of adsorption and desorption kinetics prohibitive since a presentation of results obtained would consume too much space. Therefore, in this paper, we limit ourselves to the special case of the planar geometry of the adsorber for which the analysis is much simpler and our results can be compared with those previously obtained in the literature for a diffusioncontrolled adsorption regime. For the planar adsorption 0 is independent of the A parameter (which is equal to zero in this limit) and is given by the dependence
O= f(r, Oi,ka,kd,H).
[1121
In the linear model in case (ii) (contact with a wellstirred solution) 0~ can be eliminated from Eq. [112] by introducing the reduced surface coverage 0 (as defined previously), i.e.,
0 = 0  0 ~ =f(r, fq, ka,H). 0r 0~
[1131
Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
36
ADAMCZYK AND PETLICKI
For adsorption from an infinite volume, which we shall discuss first, ~ is independent of H, thus ~ = f(7", ]~a' ~:d)
[114]
whereas for the nonlinear model (Langmuir) also depends on 6i. In Figs. 2a, 2b, and 2c results obtained in the case of linear adsorption from an infinite volume onto a planar interface are presented in the form of the dependence of 0 (Fig. 2a), the subsurface concentration 50 (Fig. 2b), and the solute flux J0 to the interface (Fig. 2c) on dimensionless time r = t D J L 2. The results shown in these figures were derived for 0; = 0 and a fixed value of the constant Ka  ~:J~:a equal to one. As mentioned earlier the analytical results and the numerical ones agreed to within 0.1%, which is well below the line thickness. Thus only the numerical results were plotted in all figures discussed in this section. In Figs. 2a2c the analytical solutions derived previously for the purely diffusioncontrolled adsorption, given by Eqs. [107], are also shown for comparison. In this case adsorption kinetics is independent of the value of adsorption or desorption constant provided that its ratio is fixed equal to Ka. As our numerical calculations show this is the case when both ~:aand ~:aassume large values (as expected from previous discussion). For ~:~and ~:dequal to 10 the difference between the difffusioncontrolled and mixed adsorption kinetics becomes practically negligible since the deviation between t9 values is on the order of a few percent for short times and decreases when r increases. The difference between mixed and diffusioncontrolled kinetics is larger, as far as the subsurface concentration and solute flux are concerned, because in the diffusioncontrolled model the value of c0 is equal to zero at the beginning of any adsorption run and Jo is infinitely large in this limit. When the dimensionless constants ~:~ and ~:a become smaller than unity the deviation between the "exact" and the diffusioncontrolled results becomes very large, especially at short times when the 0 values are overestimated by many Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
times in the diffusioncontrolled model. This discrepancy stems from the assumption of an equilibrium between 50 and 0 at any time during the adsorption process whereas, in reality, the subsurface concentration decreases within a finite time interval. From the initial value of unity it goes through a minimum and then increases again tending to the final value of 1 (see Fig. 2b). Since this concentration is always larger than assumed in the quasiequilibrium model (diffusioncontrolled) the solute flux toward the interface is smaller and as a consequence 0 (being an integral of j0 over time r) is also smaller. In Figs. 2a and 2b the shorttime asymptotic solutions derived from Eqs. [63] and [66] are also plotted for comparison. As can be seen, for short times, 50 and 0 always approach the asymptotic values, i.e., 0 increases linearly with time (parabolically with the square root of r) whereas c0 decreases from the initial value of 1 and V~z.This is so because for short times the overall adsorption kinetics is always controlled by the solute transfer rate through the adsorption layer or reorientation of solute molecules in this layer determined by activation energy (the diffusion process is infinitely fast when r ~ 0). When ka become very small (large activation barrier) diffusion plays practically no role, not even for very long times (50 remains close to unity for the entire adsorption run) and one proceeds to the barriercontrolled adsorption kinetics (curve 3 in Fig. 2a) discussed at length in relation to colloid particle adsorption and desorption kinetics (2327). For a dilute colloidal suspension the shortrange specific interaction energy between a particle and an interface can be calculated explicitly from the DLVO theory as a sum of van der Waals dispersion and the electrical doublelayer contribution. Knowing ¢ as a function of particleinterface separation 97the kinetic adsorption and desorption constants ~:aand ~:dcan be calculated in the linear model from the expressions (2327). l
~b
_
1
a
i
i
i
I
i
0
o.// / 0,y 0.4
/,/
0 b
~
//
i
i
I
i
i
1
2
3
/..
5
,
r ~
,
,
,
o,a \ 2
~

0.6 0'4/ 0,2 t ,/
L///
0 C [n'j.
0
I
I
I
I
t
1
2
3
/,
5
'
'
~
'
'
4
5
1
2
"~
3
2
4
6 0
1
2
3
FIG. 2. Infinite volume adsorption kinetics, planar interface, linear model. (a) The dependence of the dimensionless surface concentration 0 = N/N=, where Nm is the maximum surface concentration, on the square root of dimensionless time r = tD~/L 2. 1. ka = kd = 10. 2. k~ = kd = 1.3. ~ = ~:d = 0.1. The broken line denotes the diffusioncontrolled adsorption regime, i.e., ka '~ ov and kd ~ oo, ( .  . ) represents the shorttime limiting solution calculated from Eq. [65], i.e., 0 = ka (~f~r)2. (b) The dependence of the dimensionless subsurface concentration fro = Co/Cbon ~/~r;curves 13 are the same parameters as those for Fig. 2a. The broken line denotes the diffusioncontrolled results and ( . ) shows the shorttime results calculated from Eq. [63], i.e., if0 = 1  (2/V~)ka~r. (c) The dimensionless solute flux at the interface Jo = j±LJDmcb VS ~fTT.
Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
37
38
ADAMCZYK
ka=k./fexp(~T)dX=kaI2},
[116]
where D(x) is the positiondependent diffusion coefficient. The integral Ia should be calculated within the region where the energy barrier occurs (positive values of ~b),whereas 1Mis to be calculated in the region of energy minimum (either primary or secondary) where the interaction energy is negative. It is interesting to note that the equilibrium adsorption constant Ks is equal to 1M. It is worthwhile noting that IMCb is equal to the surface concentration of particles under the equilibrium conditions when their concentration within the adsorption layer assumes the Boltzmann distribution, i.e., c = cbexp(~/kT). If the energy barrier height is larger than a few kT units and the minimum depth is also large then one can express the integrals IB and IM by the following approximate formulas (25), _.
. / 3"B ~1/2
I~ 1=LJtXB)I2~} exp(k~) [ 3'M ~l/2ex [tM~
where
3'B = /~x2)~=xB
and
[d24~ where ~B is the energy barrier height, ~bm is the minimum depth, and D(xB) denotes the value of the diffusion coefficient at the distance x~ where the barrier is the highest. When q~M 1 the desorption rate contact ka assumes negligible values. The above expressions are valid for an unlocalized adsorption (no tangential specific forces and no lateral interactions among particles) under no flow within the adsorption layer. Their range of validity was discussed extensively in (2427). By using expressions [115] and [116] and applying equations deJournal of Colloid and Interface Science, Vol. l 18, No. 1, July 1987
AND
PETLICKI
rived in previous sections one is able to describe quantitatively the reversible unlocalized adsorption of colloidal particles without introducing any adjustable parameters. Now we present the adsorption kinetics results obtained in the Langmuir model by numerically solving Eqs. [26][30]. In Figs. 3a3c the dependence of 0, c0, and J0 against V~r is plotted for various values of ks and kd (their ratio, i.e., Ka, being fixed and equal to 1). The linear model results are also shown for comparison (broken curves in Figs. 3a3c). As can be seen there is no fundamental difference between the linear and Langmuir models. On the contrary, for short times (r < 1) when 0 remains smaller than about 0.3 these two models give results which almost coincide. This may suggest the conclusion that all the shorttime limiting analytical solutions discussed previously can be applied for predicting adsorption kinetics in the Langmuir model, in particular Eq. [65] indicating that 0 increases linearly with time when k2~ ~ 1 and kar ~ 1. Therefore, the linear model is especially well suited for approximating the Langmuir model for small k~ and ka, i.e., for bartiercontrolled adsorption kinetics. For larger and kd and longer times the deviation between these two models becomes significant and a general trend is such that the final values of 0 are approached in the Langmuir model much faster than in the linear one. This point is confirmed by the results shown in Fig. 4 where the dependence of 0 on ~ is plotted for various values of Ka (Oi = 0 and ka = 1). As is noted these two models give very similar results for all Ks when r < 1. The larger Ka and r the larger the deviation between the Langmuir and Henry adsorption kinetics.
Adsorptionfrom a Finite Volume In this section we consider some illustrative examples of adsorption kinetics from a finite volume in both the linear and the Langmuir models. First we shall note that the equilibrium solute concentration and surface coverage value are given in case (i), reflectingwall
ADSORPTION KINETICS
b_ Co
3
39 r
,
i
i
i 2
3
,
0,8 0,6
a
e 0.4
Y/ /
~
i ii
0.4
II1
///
0.2 ,
/
0,2 i
1
2
3
4
e tnj~ \
~
0
1
i
i
/.
.~
,
\

\
\
4 5
1
2
3
4
~
FIG. 3. N o n l i n e a r a d s o r p t i o n k i n e t i c s o n t o a p l a n a r i n t e r f a c e f r o m a n i n f i n i t e v o l u m e , L a n g m u i r m o d e l . (a) T h e d e p e n d e n c e o f 0 o n ~ f o r Ka = 1 a n d 0~o = 0 . 5 . 1. ka = kd = 10. 2. ka = kd = 1 . 3 . ka = kd = 0.1.
The broken line denotes the results derived from the linear model. (b) The dependence of c0 on ~ . (c) The dependence ofj0 on V~z.
boundary condition at 5 = H, by the expressions
H + Oi/5 Coo 


H+K~
Ooo=K~oo in the linear model and Ka(B  ~0~  H ) + H
2KaH
(~/
×
4(H + f30i)KaH ) 1f [Ka(t3_[30_H)+H]21
[118]
Ka 0oo Ka?oo +~ ?oo
[119]
in the Langmuir one.
In case (ii), when in contact with a wellstirred solution, ffoois always equal to 1; thus, 0oo = Ka in the linear model and 0oo = KJ(K~ + 1) in the Langmuir model. Results obtained in case (i) for various values of the adsorption volume thickness H are shown in Fig. 5 as the dependence of 0 on V~z.One can see in Fig. 5 some characteristic features already observed for the infinitevolume adsorption, most noticeably the linear dependence of O on r (parabolic on ~ ) for short times both in the Henry and Langmuir models. For H ~ 1 results obtained by using these two models coincide even if z >> 1 due to the fact that 0 is m u c h smaller than unity in this case. It should also be noted that the adsorption rates in the shorttime limit are inJournal of Colloid andlmerface Science, Vol. 118, No. 1, July 1987
40
A D A M C Z Y K AND PETLICKI
for adsorption times larger than about H 2, i.e., being on the order of 0.04 for H = 0.2 and 1 for H = 1. As can be seen in Fig. 5 this is really the case even for H values comparable with unity. In Fig. 6 adsorption kinetics from a finite volume is presented in case (ii), contact with a wellstirred solution. This model approximates well adsorption kinetics under natural or forced convection conditions provided that the characteristic length scale H (diffusion boundary layer thickness) can somehow be estimated. After a transient time on the order o f H 2 a linear solute concentration distribution is established through the solution (as Eq. [96] indicates) and the adsorption flux j0 is constant (the desorption flux increases linearly with 0). Under such circumstances the thinlayer limiting expression [57] can describe the adsorption kinetics well as seen in Fig. 6. We also see in this figure that for short times the dependence of Oon r is linear in both the Henry and the Langmuir models. The adsorption rate for longer times is larger in case (ii) than in the previous one (i) for the same values of H, ka, and kd due to a larger concentration gradient. We also see in Fig. 6 that the adsorption rate is fairy independent of H in the shorttime limit. Observing the approach of 0 to the
O
O.8
0.6
/
////'
z
2
/11I 0,4~S~"~
3
0,2
0
i
i
i
i
1
2
3
z,
~
FIG. 4. Planar adsorption kinetics from an infinite volume, Langmuir model. The dependence o f 0 on ~ for 0i = 0 and ka = 1. 1. ~d = 0.05, Ka = 20, 0~o = 0.95.2. kd =0.5, Ka=2,0o~=O.666.3. kd = 1, Ka = l, 0~o = 0.5.4. kd = 2, Ka = 0.5, 000 = 0.333. The broken line denotes the linear model results.
dependent of H as expected from the approximate limiting equation [65]. In Fig. 5 results derived from the thinlayer adsorption limiting formula
OiK~
are also plotted for comparison. As previously discussed this formula is expected to hold well
J
8
0,8
0,~
."///
..7/~.0.2
0
/;//
4 I
i
i
l
I
~
2
3
4
5
nl~
FIG. 5. Finite volume adsorption kinetics in case (i). The dependence of 0 on ~ for ~ = 1, ~:d = l, and 0i = 0. 1. H = ~ , 2. H  5, 3. H = 1, 4. H = 0.2. Continuous linelinear model, broken lineLangmuir model. (. . ) The thinlayer limiting results derived from Eq. [95]. ( . . . . . ) The shorttime limiting results derived from Eq. [65]. Journal of Colloid and InteoeaceScience, Vol. 118, No. 1, July 1987
ADSORPTION KINETICS
41
O
0.8 0.~ 0,~ 0,2
! 0
,
1
0
t
i
i
2
3
4
5
6
FIG. 6. Finite volume adsorption kinetics in case (ii), 0 vs ~ relationship for ks = 1 and Oi = O. 1. H = oo, 2. H = 5, 3. H = 1, 4, H = 0.2. Continuous linelinear model, broken lineLangmuir model. (. . ) The thinlayerapproximate results from Eq. [97]. ( . . . . . ) The results derivedfrom Eq. [65].
final equilibrium values one can conclude, as has been done by Mysels and Frisch (17) for difffusioncontrolled adsorption, that for ka and /~0 on the order of unity a gentle stirring of the solution, fixing the value of H, can considerably shorten the approach to adsorption equilibrium. On the other hand, when k~ and kd are much smaller than unity (barriercontrolled adsorption) the effect of stirring is minor. CONCLUDING REMARKS From our general solutions of "mixed" adsorption kinetics we conclude that in the shorttime limit when the inequalities r ~ H z, fC2ar 1, and kar ~ 1 are met simultaneously adsorption or desorption processes are always barrier limited and the surface concentration changes linearly with time according to the equation =
=
4fl[qK
o = ( k ~  kdOi)~ + ~
3/2
+" •
".
The solute flux at t h e adsorption layer J0 is finite when r ~ 0 and is given by the equation =
2flfcaK In r +" • ..
Jo = fl(~:a IQOi) + ~
These limiting forms apply to both a finiteand infinitevolume adsorption in cases (i) and (ii). When Oi = 0 they can also be used for describing adsorption kinetics for short times in the nonlinear Langmuir model. When the inequalities ~:2r ~> 1 and ~:ar >> 1 are met, (e.g., for large ka and kd or for long times) the adsorption or desorption kinetics is diffusion controlled. The condition Ka = kdka "> 1 was proved to be insufficient evidence for this conclusion to be true. Any desorption into a very dilute solution or into a pure solvent is governed by exponential decay, i.e., 0 = 0 i e x p  ~:~z,
in both the linear and the Langmuir models (finite or infinite volume). A similar exponential character shows an adsorption or desorption process in the case where the adsorption volume (unmixed) is much smaller than that needed for completing the maximum surface coverage, i.e., 0 = 1 (thinlayer adsorption limit). Thus, if H ~ 1 one has the following solutions for ? and 0, valid after a short transition time, on the order of H 2. Case (i), reflecting outer boundary, Journal of Colloid and InterfaceScience, Vol. 118, No. 1, July 1987
42
ADAMCZYK AND PETLICKI
HWl30i fl(OiKa) {13~:a ) C= H+/3Ka H+IfKa exp ~~ + kd.z 0
" H+30~ OiKa //~ka+~" =1% H + flKa H   ~ K a H exp ~~ ~)'r.
Case (ii), contact with a wellstirred solution at the distance Y = H, K ~:d 5 = 1 + 1 +/3k:an ('~' H)exp 1 + 3 k a n r
0=00.
~
Oi 0~ = exp
1 + ~:aH "r"
If the surface concentration 0 during any adsorption or desorption run remains smaller than about 0.25 both the linear and the Langmuir models give very similar results. Therefore, the linear model is especially well suited for describing adsorption phenomena in real systems when the adsorption is barrier controlled (when 0 is changing very slowly with time).
Since the functions Fl(S) and F2(s) do not appear in tables of Laplace transformations we derive the solution for u, (c), by directly applying the inversion theorem (28) which states that
_
1 (x+i~
u  2~axi~ ~(s)exps~'ds = ~lff(s)
where Z  I means the inversion of the Laplace transformation. The complex integral in Eq. [A5] can be calculated by Cauchy's theorem as a sum of residues at the poles of the integrand times 2~ri (28). After evaluating these residues according to the method described in (28) one arrives at the following expression for (, in case (i), oD 2[COSan(.~. H ) C~SS~nnH
C C°° "F K Z
A a3a 2  al +1~ ( ' n  a2"~n + ao)
APPENDIX A
sin a , ( Y  H)] exp  a2nr
Inversion of Subsidiary Equations for Adsorption from a Finite Volume
For the spherical adsorber we have the following subsidiary equations for ~ in case (i), A•+ 1
if= 5(A.~+ 1)     + [
1'2
X
A
×[chs/X+~s1/2shs1/__,
2y)
Sl~Otn'~
j
Pn
[106]
,
oLn are the positive real roots of the nonlinear trigonometric equation
where
[A7]
a,(a3a2n al) tg a , H = (a 4 _ aza2n + ao)
KFl(S)
S
[A5]
[AI]
and
where K
$1/2
FI(s) = Ss 2 + a3Cs3/2 + a2Ss + alCs 1/2+ aoS
[A21 and R = A H + 1, S = sh  s1/2H, C = ch  s 1/2H. In case (ii) the expression for ff is AY+ 1
=  
¢ (AH + 1)(~aoH3+ ½axHZ + a2H + a3) [A81 Pn = (~4
1
a2a2~ + a0)
[A4°tsn+ (A3a3  A4a2
+ A 2 ) a 6 + (Ao  A2a2  A 3 a l + A4ao)ot 4
+ A Ia3a 3 + (Azao  Aoa2)a2~ A lal a. + Aoao] KF2(s)sh  s 1/22,
s
[A3]
where 1
F2(S) = cs3/2 + a2Ss + alCsl/2 + aoS. Journalof ColloidandInterfaceScience,Vol.118,No. 1,Ju]y 1987
[A4]
[A91 Ao =  l (aoH + al);
AI = a2 + ½al H
A2 = ½a2H+~a3;
A3 =  ( ½ a a H + 2)
A4 = ½H.
[Am]
ADSORPTION KINETICS Consequently Co is given by o~ a2 / c ° = c ~ + K ~n[l~=lP~\ ~
43
0 = 0iexp  ~:dr + ~ ( 1  exp  kdz)
A a30z2  al R (a~a2a~+ao)4 2 × e x p  a2~".
)
ou
~:.K ~ Qn(eXpa2rexpfc,
Knowing ? one can calculate 0 from Eq. [471 and the result is
[A19]
where an = 1/Pn(~Zd or2).
[A20]
Adsorption f r o m a Finite Volume onto a Plane Interface
ka
0 = 0iexp  L z + ~~c~(1  exp  kd~') + Kk. ~ Q.(exp  a2z  exp  kay'),
ff),
n=l
[A11]
[A12]
In the case of planar adsorption the value ofA ~ 0 and Eqs. [A1] and [A4] reduce to
n=l
~
where Q.
1
1
u = s + Ks 1/2(Ss + ~kaCs ~/2+ kaS)
= a2[1
A(aaa 2  al)
× ch s 1/2(y_ H )
~ .~
1 L
[A13] In case (ii) after calculating the residues of the function F2(s) one obtains the following expressions for ? and c0, • ~ ~
1 sin a~(~? H)
~= 1 +/~ 2, n=len
"'77. sin C~ntl
e x p  a~r [A 14]
1
= s + K s l/2(Ts + {J]£aS1/2_]_ k~T) ×
n=l
H)
ch s I/2H
'
[A211
where T S I C and
u = s  Ks ~/2(Cs+ ~ L S s 1/2+ Ck~) × sh s 1/2(~2_H).
oo 1
~'o= 1  K ~ ~~ e x p  a]r,
c h s 1/20~ 
[A22]
[A151
n
where an are the consecutive roots of the trigonometric equations
Using the same inversion procedure as was used previously one obtains the following expression for ~, oo
• ~ a2ot2 + a0 ctg a~rl = ~ an(al  ~ )
[A16]
c = coo + K ~ (ko  a2n)cos a n ( x  H ) n=~ en COS otnH × e x p  a2~",
and Pn = [A3 a6 + (A2a2 +A3al  A l a l ) a 4 + (Alal + Aoa2 + A2ao)a 2 + Aoao]/a2(al  a 2) [A171 Ao =  ½ ( a o H + a l ) ;
A1 = ½alH + a2
A2
A3 = ½~.
½azH+ 3"
[A181
Applying the same integration procedure as was used previously 0 was found to be
[A23]
where K ~ = 1 ~ flka+/CdH
H + 0i/3 H+Ka[3
[A24]
Ka=ka/k~
H 4 + ~ (fl2k2aH_ 2 ~ d H + B~a)tX2 Pn=~". 1

+ ~(~k. + ~:dH)ka.
[A25]
Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
44
ADAMCZYK AND PETLICKI
T h e consecutive roots an are to be calculated from the equation
tgan H = ~2k~a~ n .
[A26]
an  ~d
By applying the inverse Laplace transformation to Eq. [A31] one obtains the following expression for ~, _
_
2 K oo
C=Co~ ~
T h e expression for 0 is
= flHK.ofl + l +  ~
0 = Oiexp  ~:dr + K.C~(1  exp  kdr)
cos a n n
[127]
n=l In
In case (ii) the solutions for ? and 0 are
5=lK~
cos a , ( 2  H)
×
1 + k a K ~ b(exp  a2r  exp  Ear).
(~:a  a 2) sin an(2 H) exp  a2nZ n=l Pn sin a . H
1
expan2r,
[A33]
where the roots o/n can be calculated f r o m the equation tg a , H = flKactn. [A34] Considering Eq. [47] one obtains the following expression for 0:
[A28] 0 = Ka?0 = Kac~
0 = 0iexp  ~:dr + Ka (1  exp  ~:dr) x
 ~:.K ~ e p n=l
 ~ 
o~
~
exp
Pn
[A29]
'
where an are the consecutive roots given by the equation:
ctg O~n  
/~TaOZ n ( k d   O~2n)
[A30]
exp  an2r
2(OiK~) ~ / [A35] +H " n=l( ~Hgaa2n+ l ~a) Proceeding in a way similar to that used previously, in case (ii) we obtained the solutions for 5 and 0 in the f o r m 2K ~
1
and P, is the p o l y n o m i n a l o f an identical to that found previously.
DiffusionControlled Adsorption W h e n the adsorption and desorption constants are large so that the inequalities ¢~k, }> S 112 (/~2~72T >~ 1) and ka >> s (kdr >> 1) are m e t simultaneously then Eqs. [121 ] and [ t 2 2 ] describing kinetics o f planar adsorption reduce to 1
K
5=_~ _ _
1
S [~kasl/2(S1/2C..[_~gaS)
ch sm£
X sin a . ( x H ) exp  aar sin o~nH
[t36]
oo
0 = K a  2(0i  Ka) 3"
exp  a2r
[1371 where an are the roots o f the equation ctg anH = BKaan.
[A381
[A311 APPENDIX B
and l
t~= ~
K
1
sh  st/2,~"
S ~l~asl/2(sl/2s_~_~KaC )
Inversion of Subsidiary Equations for Adsorption from an Infinite Volume 1. Adsorption onto the spherical collector.
[132]
JournalofColloidandInterfaceScience,Vol. 118, No, 1, July 1987
In this case the subsidiary equation is
45
ADSORPTION KINETICS
a = e ( a 2 + l)
zl = 2
A2+ 1
K
s
[s3/2+(lq+A)s+Tqsm+TcdA]
X e x p  sl/22.
z2=2
[B1]
First we find the denominator's zeros by substituting s ~/2 = z. In this way we obtain the following polynominal of the third order, [B2]
z3+a~z2+a'lz+a'o=O,
~p cos ~ o ~_1
z3=2
[1 ~\ ~p C o s ~ o + g )
5pcos ~og
where
a'~=ka;
a'2=3fca+A.
[BI0I
1 ~3/2
where a'o=kaA;
,
[B31
We then apply the standard Cardan method for solving Eq. [B2]. Let us first define the quantity Q = (p/3) 3 + (q/2) 2, [B4] where
Considering the above solutions for Zl, z2, and z3 and defining for the sake of convenience r~ = Zl; r2 =  z z ; and r3 = z3, Eq. [B1] can be written as (when the roots are unequal) A2+ 1
l~t2 ± p . P~~"   ~ 2 T a l ,
S
3
(a'l) 3 1 , , + q = 2 ~   ~ a 2 a l a ~ .
If Q > 0 there are two complex conjugate roots given by
.=l(S
i(PIP2)~
2
(P1 + / ° 2 ) 
= (a


2
+ i(P,

[BI1]
C1 = 1/(rl  r2)(r3  rl) C2 = 1/(rl  r2)(r2  r3)
[B6]
C3 = 1/(r2  r3)(r3  rl).
=  ( a + bi) Z 2 
+r.)
with
v5
(P1 + / ° 2 )
Z 1 
C
+ K e x p (  s l / 2 Y ) ~ . 1~ ~
[851
[B12]
When two of these roots are equal say r, = r2 then the expression for ~7has the form
v5
Pz)  ~
[B71 A)?+ 1 ~7=  + K exp
bi),
S
where Pl=~/q/2+V~, p2=~Cc/q/2V~
1/2[Cis1/2+C2
[B8I
7
C3
]
[813]
+(.,,.+.,)j
where
and one real root Z3 = P I + P
s
2.
[B9]
If Q is equal to zero the roots r~ and r2 are equal and real. If Q < 0 there are three various real roots that can be calculated from the equations
CI =  1 / ( r ,  0)2;
C2 = (r3 2rO/(rl  0)2;
C3 = 1/(r,  r3)2.
[B14]
The inversion of Eq. [B 11 ] can be easily performed using the table of Laplace transformations: Journal of Colloidand InterfaceScience, Vol. 118,No, 1, July 1987
46
ADAMCZYK AND PETLICKI 3
~=
u
= 1 + K ~ Cnrn(exprn£ + r2nr)
(AYe l)
n= 1
Y ×erfc(~~r+r~fr ) .
[B15]
On the other hand, the inversion ofEq. [BI 3] gives r3
c= l + K { ~ [ e x p ( r l x
where
a '= 2bZ;
b' = 2b(r3  a).
Thus, C1 and C2 are complex conjugates and Cs is real. For calculating the inverse of Eq. [BI 1], when C1 and C2 are complex conjugates we consider that
(s ~/~+ r~) ~ (s m + rE) exp
s m2
+ rZr) = (C1 + C2) ~
Y + rl frz)  exp(r3Y + r 23~') × erfc(~~z
FIs l/2(sC11/2rl + rO
exp  s 1/22C2r2
] v  _ .1/2,7.
s 1 / 2 ( 7 ~ r2)]e^v
~
[B20]
and take advantage of the following theorem (28): 2
2
×exp(r12+rlr)effc(~r + r,1/rr)  2rL~ ~ (r3  rl)
iris
e x p  ~r " [B161
When the roots r~ and r2 are complex conjugates and r3 is real, the inversion is not obtainable from the table of transformations. Therefore, in order to invert Eq. [B11] we proceed as follows. First, we calculate the values of the constants C~, C2, and C3, C1
1
2bi(r3  a  bi)
= 1/(a'+b'i)
c3 =
[B21 ]
2 £ c= l + K{C3r3exp(r32 + r3r)erfc(~r+ r3frr) + exp(aJ?+ a2r)[RelCOS(b:~+ 2ab~')Ii
=l/(a'b'i)
+ ImlCOS(b~ + 2abr)I2] 1,
[B22]
J
where
(rl  r2)(r2  r3)
2bi(a  h i  r3)
~rF( X)d~,
wheref(s ~/2) is an arbitrary function and F(~,) = .£lf(sl/2) is the inverse of this function. In this way we obtain the following solution for in the case of two complex conjugate roots,
[B17]
1
1
~ 1 fo°° exp  X2
+ R isin(b:? + 2abr)12  lm i sin(b:? + 2abr)ll
(rl  r2)(r3  rl) 1
C2=
1/2) 
st/2
[B18]
Rel = rlCl + r2C2 = r3/[b 2 + (r3  a) 2]
[B23]
Iml = (b 2 + a 2 _ ar3)/b[b 2 + ( r  a) 2]
[B24]
and 11 and 12 are the definite integrals given by the expressions
1
2 r, Ii = ~nnJt~+ZaO/2cos(ZbVrrOexp(~2)d~
(r2 /'3)(/'3  r,) 1 (a  bi  r3)(a + b i  r3)
2 I"~;'+2a°/247 =expb2r~J
°
cos(ZbV~r0
1
[b2  (a  r3)2] '
[BI9]
Journal of Colloid aad Interface Science, Vol. 118, No. 1, July 1987
× exp(~2)d~
[B25]
ADSORPTION KINETICS
47
t " oo
1
h =  ~ J(~+2,,)/2~ sin(2b~/rT0exp  ~2d~
0 = 0iexp  kdr + ~ ( 1  exp  ~dr)
= exp(bZr)erf(b~) 2
"
r ('£c+2ar)/2xl7
f ~ J0
~
~:.r~ [ 1 [ ~
[exp(r ~r)erfc(rl ~r)
sin(2b~r0exp(~ 2)d~" [B261
By deriving Eq. [B22] we took advantage of the fact that C1 + C2 + Ca = 0. The expressions for c0 in the case of three different real roots, two equal real roots, and two complex conjugates and one real root are
(1 +rlI)expkar]
1
(r2+kd)
× [exp(r~r)erfc(ra~r  (1 + r3I)exp  Tcdr]} J
~  ~ 4Kk"2h 3/2I3exp  kdz ~ Kk, r~ VTr(r3 rl)kd (r 2 t kd)
3
?o = 1 + K Z C~rnexp(r2nr)erfc(r,~r)
[B27]
x
{(
l
exp(r~r)erfc(rll/7)
n=l
[1 +r,(~d (r~ +/ka))]exp kd'} . [B321
r~[exp(rZlr)erfc(rl~r)
Co= 1 +K{
r 2
 exp(r ~r)erfc(r3 l~r) + ~
exp(r ~r)
When there are two complex conjugate roots, one obtains the solution for 0 from Eq. [47], 1
2,
0 = 0iexp  kdr + ~ (1  exp  ~:dr)
X erfc(rl Vrr) + (r3  r2~
C3r3~Ta 2 +K ~ [exp(r3r)erfc(r3V~r)  exp(kdr)
Co= 1 + K{ C3raexp(rEV~r)erfc(raVrz) + expa2r[Rel Ilcos(2abr) + Re11Esin(2abr)
£r
 Irnl Ilsin(2abr) + Iml IECOS(2abr)]}. [B29]
X (1 + r3I)l + Kk, exp(kar)
The expression for 0 can be obtained knowing c0 from the general relationship expressed by Eq. [47]. For three real roots one has the solution for 0
× [Iml(/zcos 2abt  I~sin 2abt)
i
0 = 0iexp  ~:dr + ~ (1  exp  ~:dZ)
exp(a 2 + ka)t
+ IiRel(cos 2abt + hsin 2abt)]dt,
[B331
where the functions I~ a n d / 2 are now given by 2 11 = ~ £./7 cos(2b~r()exp(~ 2)d(
Cnn + Kk, ~ (r2~ ~ca)[exp(rZ"r)erfc(r"f~r) 3
r

 ( 1 +r,I)exp~z~z],
[B30]
where 2
r,/V~
I=~Jo
2 r /2 = ~_ _J , 4 7 sin(2b~r~)exp(~2)d~ /3
exp~2d~"
.m
(2exp((2)d("
[B34]
[B311
For three real roots (two of them equal) one has the following expression for 0:
Defining p = a 2 + ~:a and q = 2ab after some algebra one obtains the following expression for 0: Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987
48
ADAMCZYK AND PETLICKI I
0 = Oi exp  fear + ~ (1  exp  Ear) f
KC3r3f~a
(r~ + ~)
× [exp(r~r)erfc(r3 ~r)  exp(kdr)(l + r3I)]
K~a
6 = 1 + ~ [ egx p ( r [l x +

2 rlr)
× erfc(~~z + rl ~r)  e x p ( r 2 Y + r~r)
q (p2 + q2) {Imlexp(aar)[(P cos qr + q sin qr)I2  (p sin qr  q cos qr)I1 1
 I m l q exp(kdr)(1 + aI)}
0 = O~exp  kar + ~ ( 1  exp  k:dr)
+ Rel (expa2r[(p cos qr + q sin qr)I1

+ (p sin qr  q cos qr)I2
Ropexp(kdr)(l
+aI)}.
1
2
{ ~
 exp  ~:ar]
(r~ + ~:d) [exp(r~r)
[B351
2. Adsorption onto planar interfaces. Equations describing adsorption kinetics onto a planar interface can be derived from those equations formulated in the previous section for the spherical collector, when one assumes that A ~ 0. Therefore, a0 = 0 in Eq. [B2] which reduces to z 2 + flfqz + kd = 0.
1
+k a K ~
[exp(r,r)erfc(rl ~ )
1
× erfc(r2 V~r) e x p  ~:dr]}. If the roots are complex the solution is K z ? = 1 + ~ e x p ( a £ + a r)[I2cos(b£+ 2abr)
[B36]  I l s i n ( b £ + 2abr)].
The roots o f this quadratic equation are either both real when A = (fl~:~)2 _ 4kd > 0, i.e.,
.lflk~+ z, =  r , = J~
21
z2 =  r z
2 V(flk~)2  4ka
[ 2
[B41]
V(/~/~a)2 4kd]
]
[B42]
F r o m Eq. [B35] considering Eq. [B39] one has for 0 i
0 = 0iexp  ~:dr + ~ (1  exp  ~:dz) [B37]
or a complex conjugate when A < 0, i.e.,
]
+ K ( p z k q 2 ) b {exp(aZr)[(p cos qr + q sin qr)12  (p sin qr  q cos qr)Ii]
Zl =  r , = [~ + iV4lcd  (ilk,) 2 =  ( a + bi)
qexp(fzdr)(1 + a b I ) }
[B43]
with p = a 2 + ~:d, q = 2 a b .
z2 =r2 = [@iV4/~d  (flka) 2]
= (a
bi).
[B38]
Since r3 = 0 we also have
C1 = 1/rl(r2  rl); Rel = 0;
Cz
=
1/r2(r2
l m l = 1/b.

rl) [B39]
Thus, from Eq. [B15] we obtain the following solutions for ? and 0 in the case of two real roots: Journal of Colloid and Interface Science, VUl. 118, No. 1, July 1987
ACKNOWLEDGMENTS The authors express their thanks to Professor A. Pomianowski for stimulating interest in our work and for valuable suggestions and to P. Warszyfiski for useful discussions. The work was supported by Grants MR. 1.17. and 03.10. REFERENCES 1. Matijevir, E., J. Colloid Interface Sci. 58, 314 (1977). 2. Sasaki, H., Matijevir, E., and Barouch, E., J. Colloid Interface Sci. 76, 319 (1980).
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Journal of Colloid and Interface Science, Vol. 118, No. 1, July 1987