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A general stochastic model of adsorption–desorption: Transient behavior
Original Research Paper
213
n
Chemometrics and Intelligent L&oratory Systems, 6 (1989) 273-280 Elsevier Science Publishers B.V., Amsterdam -
Printed in The Netherlands
A General Stochastic Model of Adsorption-Desorption: Transient Behavior B.L. GRANOVSKY
Department of Mathematics,
Technion-Israel
Institute of Technology, Haifa (Israe)
T. ROLSKI
Institute of Mathematics,
Wroclaw University, Wroclaw (Poland)
W.A. WOYCZYNSKI
*
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland
OH 44106 (U.S.A.)
J.A. MANN
Department of Chemical Engineering, Case Western Reserve University, Cleveland
OH 44106 (U.S.A.)
(Received 1 July 1988; accepted 28 June 1989)
ABSTRACT
Granovsky, B.L., Rolski, T., Woyczynski, W.A. and Mann, J.A., 1989. A general stochastic model of adsorption-desorption: transient behavior. Chemometrb and Intelligent Luborato~ Systems, 6: 273-280. A mathematical model for adsorption-desorption phenomena is provided. The model permits interaction of adsorbed particles and, in particular cases, displays a nonconcave kinetic behavior (i.e. the expected number of adsorbed particles is a nonconcave function of time) that was previously discovered experimentally for adsorption of short chain carboxylic acids at the water-air interface and for adsorption in the presence of chemical reactions.
INTRODUCTION
This paper is a continuation of our study of stochastic models of adsorption and desorption originally motivated by the underpotential deposition phenomena (cf. refs. l-3). The model considered in what follows is fairly general and permits account to be taken of the interaction of particles. In particular cases, such as when adsorbed particles have a graph structure (i.e. a possible flexi0169-7439/89/%03.50
0 1989 Elsevier Science Publishers B.V.
ble structure of interactions with no rigid geometric restrictions imposed on them, cf. Section 5), taking into account neighbor interactions, a more detailed analysis is possible, and a concave-convex-concave behavior of the expected number of adsorbed particles as a function of time is displayed in certain situations. A motivation for our model was provided by some experimental results that showed that r(t), the number of adsorbed molecules per unit area
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Chemometrics and Intelligent Laboratory Systems
0.0 0.0
0.2
0.4
0.6
0.8
I
1.0
t* Fig. 1. Kinetics of adsorption of the octanoic acid on the water air interface. The dependence of coverage r/I& on time was computed from experimental data obtained by Hansen and Wallace [4] for different values of I&, determined by the volume concentration of the octanoic acid. For purposes of comparison, a dimensionless parameter was used for scaling, t* = t/t= where t, was the time necessary to reach 0.95 of the equilibrium coverage. res is the coverage that obtains at equilibrium for a given concentration of octanoic acid.
(called the excess), can be a nonconcave function of time. In particular, Hansen and Wallace [4] demonstrated that the dependence on time of the spreading pressure of adsorbed, short chain, carboxylic acids at the water-air interface is qualitatively different from the behavior characteristic of the classical, Langmuirian, adsorption without lateral interaction in the interface. (See Adamson [5] for background about the definition of the excess and isotherms relating I’ to the surface tension and volume concentration.) Fig. 1 shows the effect of lateral interactions on the kinetics of adsorption of the octanoic acid and is adapted from Hansen and Wallace [4] using the isotherm data given in Lucassen and Hansen [6]. For large values of the surface coverage parameter @eq= Gq/Knax (I& corresponding to a given concentration of surfactant, I,, corresponding to the maximum excess of the isotherm of I vs. concentration) the curves show a nonconcave kinetic behavior expected from our theory (Fig. 4). Such a behavior is possible in octanoic acid since the strength of the intermolecular forces between, straight-chain, carboxylic molecules increases with chain length. For shorter chains of, for example, pentanoic acid the equilibrium isotherm is that of an ideal monolayer following, for all values of t&,
214
the Langmuir isotherm within experimental error; the molecules interact only weakly. However, octanoic acid deviates from ideal behavior; the seven CH2 groups provide lateral, attractive forces that can be observed. At relatively ‘low equilibrium = 0.6 the adsorption dynamics coverage, res/rmax are concave over the entire time scale. When rlX&ax = 0.99, intervals of convex and concave behavior are observed experimentally. The original data in ref. 4 related the surface tension to time. In order to compare them to our theory, which relates the excess I to time, it was necessary to make a conversion which used the assumption that the surface tension depends only on the excess even when the system is not in equilibrium. Since our purpose is only to compare shapes of the graphs of kinetic data, this local equilibrium assumption is satisfactory. The experimentally determined parameters for the Frumkin isotherm were taken directly from Lucassen and Hansen [6] (eq. (16), Table (l)), and includes a term that takes approximate account of interacting surface species. The fit was satisfactory and can be considered as providing an interpolation formula relating spreading pressure, rr = y,, - y (y. surface tension of water, y surface tension of solution) to excess. Then, Hansen and Wallace’s [4] interpolation formula (4) (Table II), which was fitted to ?r vs. time data for various short-chain carboxylic acids and octanoic acid in particular, was used in an obvious way to transform their 7~ vs. t data for octanoic acid to coverage vs. r data. The latter are displayed on Fig. 1. An issue that Hansen [7] and other groups discussed at the time was whether adsorption was kinetic or diffusion controlled. This concern is handled in our theory through postulating a Markov process mechanism for adsorption/desorption and two rate constants X and /J respectively. We allow particles to adsorb on empty ‘sites’ with a fixed rate X and then desorb at a rate p which depends on ‘near neighbor’ configurations. This amounts to assuming a ‘barrier’ to desorption. Indeed, our theory gives the linear dependence on time at short time as required by Hansen’s theory and various experimental results. While we can introduce a neighborhood dependence for both h and CL,we have chosen to keep A
Original Research Paper
275
constant at this stage of development. In the future, we plan to reanalyze quantitatively the kinetic data of Hansen and other groups. Moreover, neighbor effects can be computed quantitatively by molecular dynamic (Brownian dynamic algorithm) simulation and this will be done in the future. We remark that it is important to realize that for our theory to apply a cluster of surfactant molecules in the surface may have a graph of neighbors and next nearest neighbors and perhaps even more distant interacting neighbors. But these molecules need not be on a surface net (two-dimensional lattice) as, for example, is the case of CO adsorbed on a clean Ni, single crystal surface. In our language adsorbed particles may have a graph structure but not necessarily a lattice structure; our theory describes both situations. We also remark that similar kinetic effect was reported by Forster [8] wherein the influence of diffusion and chemical reaction on the adsorption kinetics in binary surfactant systems was studied. The adsorption of atoms and molecules onto the surface structure of solids does involve well-defined nets and strong lateral interactions are possible. The work done in this paper provides qualitative comparison of experimental and theoretical information. Quantitative comparison with experimental data and data generated by simulation will be done later.
2 GENERAL DESCRIPTION SORPTION MODEL
OF THE ADSORPTION-DE-
We assume that during deposition, the particles are adsorbed on a finite set S of N sites x. We denote by X= {OJ}’ the state space for our adsorption-desorption process i.e. the family of all 2N possible configurations r) = { n(x), x E S } of occupied and empty sites where n(x) is a 0 (empty) - 1 (occupied) valued function on S. If n E X is a particular configuration, it will be convenient to denote
H
for a configuration created from TI by just flipping the state of site x, and to denote ]?I] =#{xESn(x)=l} the number of occupied sites in configuration n. Now, we assume that the adsorption-desorption dynamics are described by a Markov process cp(t) = cp(t, w), t 3 0, u E at, (defined on a certain probability space (G!, 9, Prob)) with values cp(t, w) = n E X and transition function determined by the conditions: n-+nx
atrate
X
if
at rate
~(n,
n(x)=0
and n +nx
x)
if
n(x)
=l
wherep: XxS+R+,andp(n, x)=Oif ]n] =O. In other words, particles are allowed to adsorb on empty sites with a fixed rate h (e.g. think about deposition on a solid surface from an infinite solution), and then desorb at rate p which depends on state of the ‘neighborhood’ of the particle about to desorb. Denote by p,(t)
= Prob{cp(t)
= T)
the probability that at time t 2 0 the configuration of the process is 1. Then, from the forward Kohnogorov’s equation (cf. e.g. ref. 9, Chapter 2.7, (6)) we readily find that the collection of probabilities {p,(t) : v E X} satisfies the following system of 2N differential equations: &>
=
c
(x: +
Tj(x)=l) (
:,f3,=o)P(9x’ x
c (x:$(x)=0)
-(
+
x
X&(4
(
~~~)=l)16h x. x
+l,(t)
h,(t)
+w
(2.1)
for all n E X. Example 2.1. Assume that the desorption rate p(q, x) = p for all n E X, x E S i.e. the sites are
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Chemometrics and Intelligent Laboratory Systems
276
independent of each other. In this particular case, (2.1) gives equations
Proof. Summing up eqs. (2.1) over the set {q: 1~1 =k} we find that
z-%$)=h
P;(t)=
(x:
c
J+&)+CL
q(x)=l}
c
p?j,(t)
{x: q(x)=O)
l+Wl~l)P,(t)
-(UN-
c
(2.2)
= (7r(t))lV’(l
-T(t)pg’
ApJt)
{x: q(x)=l}
+c
for all n E X. A simple, probabilistic argument helps to guess that the solution is of the form p,(t)
c
‘$‘=k
c
Ph,
-
(2.3)
c APT(t) I$l=k (v(x)=01
where -
7r(t) = l--(1
_ e-(h+r)t)
x)p,,(t)
ltll =k {x: ~l(x)-O)
c
(2.4)
CL(W)P?Jt)
c
Iq’=k
{x: q(x)=l}
eq. (3.1) follows directly from the obvious fact that
Now,
which recovers the elementary formulas obtained in ref. 1.
c 3 COVERAGE DYNAMICS
In this section we study the integer valued process n(t) = 1‘p(t) I which represents the number of occupied sites at time t. Note that, in general, the process (n(t), t z 0) is not a Markov process, an obvious exception being the case of independent sites studied in Example 2.1. Denote by P-l(t)
tzo
= 0,
and Pk(t)
=
c
k=O,...,
p,(t),
N
Ivl=k
probabilities that at time t there are exactly k sites occupied, and by b(n) =
(~:&=llP(‘IV
x,
what could be called the global desorption rate for a given configuration n. 3.1.
The probabilities
0, , . . , N furfill the following equations P;(t)
=X(N-k+ +
system
pk(t), k = of differential
l)Pk--l(t) c
= A(N-
k + I)Pk-i(f)
and from the identities ((17, x)EXXS: = {(n,,
b(v)P&)
x)EXXS:
((1, x)EXXS:
1111=k,
1~1 =k-1,
= ((17x7 x)EXXS:
+)=O}
q(x)=O}
171 =k,
0(x)=1}
The first identity can be checked as follows: let (7,x) be such that Iq(=k+l and 17(x)=1. Take { = nx. Then, clearly, (S,, x) = (0, x) and I { ( = k, p(x) = 0. Conversely, let 9 be such that Iq I = k, v(x) = 0. Then Iq,I = k + 1, and again ({, x) = (n,, x) for 5 = nx. A proof of the second identity can be obtained in an analogous way. Q.E.D. Let us denote by = ;
kP,(t)
k=l
the expected value of the number of adsorbed particles at time t. The above Theorem 3.1 permits us to obtain the following. COROLLARY
‘q’=k+l
Iql =k+l,q(x)=l}
and
I(t) THEOREM
A&)
‘q’=k-1 s(x)=0
3.1. I(t)
satisfies
the differential
equation -UN--
k)pk(t)
-
c
b(dP,(t)>
Ivl=k
k=O,...,
N
I”(t)=AN-U(t)-E[b(cp(t))],
(3-I)
tzo (3.2)
211
Original Research Paper
which, for the initial condition r(O) = 0, has the following solution r(t)=N(l-e-“)
-/de-h(t-S)E
b(cp(s))
ds (3.3)
Proof. Multiplying the kth equation in (3.1) by k and summing over k = 1,. . . , N we get I”(t)=A
k=l -k(N-k)Pk(t))
+
kcl
-k
k
(
c Ivl=k
c Iql=k+l
+h&)
bbh+(t))
=XN-hr(t)
-
$ k=l
c Iql=k
%h+(t)
Q.E.D COROLLARY
obvious solution
r(t) = x-(l
-
e-(A+P)t)
In particular, Pk(d
=
c
&@>
=
( ;)ddktl
-a(t))k
Ivl=k
where a is as in formula (2.4). Theorem 3.1 relates the mean coverage function r(t) to the mean global desorption function /3(t) = Eb(cp(t)). For the independent sites model considered above, the explicit computation shows that I’(t) is concave, while, in general, on the basis of Corollary 3.2 we can only claim that r(t) is concave near the origin (with, obviously, r’(0) = XN). This can be interpreted as a statement that at the beginning of the adsorption process the qualitative and quantitative behaviors of all systems are similar. In the next section we will give sufficient condition which assures the global concavity of the mean coverage function r(t), while in Section 6 we will show examples of the mean coverage functions which are not always concave.
{(k(N-k+l))p&t)
f
W
3.2. If r(0) = 0 then
r”(o) < 0
(3.4)
Proof. Denote P(t) = E&q(t)) >, 0, t >, 0. Since r(0) = 0, we have p,,(O) = 0, n E X, and, consequently, /3(O) = 0. Thus p’(0) z 0 and, by (3.2) we obtain that r”(0) < 0 since I”(0) < 0. Example 3.1. In the case of independent (Example 2.1 continued) we have that
sites
4 CONCAVITY
OF THE MEAN
COVERAGE
FUNCTION
In this section we study some additional properties of the function r(t) = E 1q(t) I under an assumption that our interacting particle system is attractive (cf. eg. ref. 11, p. 134). This assumption, in our case, means that
W=Wl so that eqs. (3.1) take the form P;(t)
=X(N-h(N-
k+
l)Pk-l(f)
k)P,(t)
+
(k
+
1h‘P,+&)
- pkPk(t)
In this case 1cp(t) 1 is a Markov process, and Eb(cp(r))
= CIE I dt>
I = @(t)
Thus, from eq. (3.2) we obtain that
r’(t) =AN-m(t) -g(t) which under initial condition
r(0) = 0 has the
whenever TJ~5 and n(x) = l(x) = 1, which describes the situation when a particle is less likely to desorb if it is surrounded by more adsorbed particles. Physically, it is a very common situation (cf. refs. l-3 and the references quoted therein). Under the above assumption r(t) is nondecreasing whenever p(O) = 0 (mod P) where 0 E X denotes the empty configuration, i.e. when there are no adsorbed particles in the initial state (cf. ref. 3, Theorem 3.2.6 (1)). It turns out that, assuming additionally monotonicity of the global de-
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Chemometrics
sorption prove
rate b(7)
THEOREM b(v)Gb(S)
and Intelligent
Laboratory
(cf. Section
Systems
3), we can also
4.1. If the system is attractive for
17G
218
and (4.1)
then r(t), t z 0, is concave. Proof. Since cp(t) is attractive and the initial state is empty, it follows from (4.1) that the function E[b(cp(t))], t 30 is nondecreasing. Thus, Corollary 3.1 implies that I’(t) is decreasing which gives the concavity of I. Q.E.D.
Note that attractive systems fulfilling condition (4.1) are close in their qualitative behavior to the independent sites model.
Example 5.1. (Mutually sticky particles model.) Assume that, if v(x) = 1 then
ON A SET OF SITES WITH A
In this section we look at the adsorption process cp(t) in a special situation where the set of sites S has some topological structure, i.e. each site x E S has the set of its closest neighbors S(x) c S defined. Such a structure is mathematically best described by a graph on the set of vertices S. Physically, it could correspond, for instance, to absorption on the surface of a crystalline substrate, or could correspond to certain bond structure between adsorbed molecules. Such models have been studied in refs. 2 and 3 and from simulations reported therein, we infer that r(t) need not be always concave. This was especially evident in ref. 2 (Caption to Fig. 2), where the rates p were assumed to be of the form
Here Iv(q, x) 1 = #{x E S(x): TJ(X)= l} and K and d are positive constants. Since this model is rather difficult to study analytically and the explicit formulas are not readily available for it, we propose here the following simplification.
147, x) I =O
otherwise
0
which means that adsorbed particles desorb at a fixed rate I_Cif they have no neighboring sites occupied and do not desorb at all if at least one of the neighboring sites is occupied. Denote by a,,( 9) the number f occupied sites with no neighboring sites occupied i.e. n&l)=
c
{x:
I( Iv(ll. x)1=0)
q(x)=l)
where 1, is the indicator function of set A. Note that WV) = CLAN. Since in the considered case the stationary state of the process is, obviously, the fully occupied configuration, we see that with probability 1 n,(cp(t))
5 ADSORPTION PROCESS GRAPH STRUCTURE
CL if
4% 4 =
+O
as
and hence Eb(cp(t) Example S = {1,2,3} Fig. 2.
I + cc --, 0 as I + cc.
5.2. Consider the concrete example of with the graph structure described in
Then X = {(O,O,O), (O,O,I), (OLO), (I,O,O), (LOJ), (LLO), (LIJ)} and
~(9, i) =
p
if n(j)=O,
0
otherwise
(OJJ),
j#i
by (3.1) we have PW
= CLPIW - 3APoW
P;(t)
= 3hPoW
- (2X + CL)PlW
P;(t)
= 2XPlW
- hP&)
P5W
= AP&)
with initial conditions ~~(0) = 1, ~~(0) = 0, k = 1, 2, 3. The matrix of the system of the first three of 3
/\
2
Fig. 2. The graph structure of intermolecular sidered in Example 5.2.
interactions
con-
Original Research Paper
219
n
the above equations is
thus, 1 0
-3x
[
3x
-(21+I”)
0
0
2A
-A
for ~1z 2X, its eigenvalues are different and equal to (Yg= -A [-(p+5A)f~~2+lOXp+P]
al,2 = f
GO
3
and its eigenvectors are
rL
A
3x+cu,
3x+(Y,
1 2x
1 2x (Y1+ x
(Y2+ h
where (Ye+ X # 0. Thus, whenever, p # 2X, the general solution of the above system is as follows CL a1’ + c -e 23x+oL2
PO(t) = c&L? 3x+ff, pi(t)
= c,eal’ + c2eaZt
p2(t)
2x = cOeao’+ c1 -eal’ (Y1+ x
9’
a21 + c2-e 2x LX2+ x
where the constants c,,, cl, c2 are to be determined by the initial conditions. Then Eb(cp(l))
=j.kP1(t)
= --$$-(ealf-
Fig. 3. The concave dependence of excess function r on time for a simple system considered in Example 5.2 for X = 50 and p = 50.
Fig. 3 (A = 50, p = 50) or follows a concave-convex-concave pattern of Fig. 4 (A = 50, /J= 15 000) with two inflection points, and these two possibilities exhaust all the possible qualitative patterns of the behavior of this system. This follows from the fact that any function of the form F(t) = ale”” + a2ea2’ + ajea3’ has no more then two zeros. Indeed, f(t) = e’+‘g(t), where the derivative of the function g(t) has no more than one zero. Since I”‘(0) K 0, by Corollary 3.2, and I’(t) is bounded, one change of sign of r”(t) is not possible. The concave-convex-concave behavior displayed on Fig. 3, indicating the deceleration period followed by an acceleration and then another de-
eaZf)
Thus, by (3.3)
1.O
l?(t) = 3(1 -e-“)
- $emAf 2
2 L
x
feAs(eW
_
eW)
ds
J0
=
311 -e-X’)
0.5 - -?9L a1 a1 -
x
[
a2
a2
f
(X+a*)(X+a2)e-hr
1 +h+(Yle
alf alf - _e 1 x + lx2
(1.0e
1
0.0
(5.1)
The function l?(t) in (5.1) is either concave as in
1.0
2.0
t x 10
3.0
4.0
3
Fig. 4. The concave-convex-concave dependence of excess function r on time for a simple system considered in Example 5.2for X=5Oand p=15000.
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Chemometrics
and Intelligent Laboratory
Systems
celeration period in the approach to system’s equilibrium does indeed occur if the condition A + (Ye > 0 is satisfied. In fact, in the considered case, p > 2X, and we have (A + a,)(X + az) = 2X2 - PA < 0. Thus, from (5.1) we derive that I+(t) =-
3e-” p-22x
&(*:(A
2h3-
[
-af(X +
a2)e(a1+X)r
+ al)e U2+*)*
)I
Now, take t = i > 0, satisfying e(OL1-a2)i = &LX:, which is possible due to the fact that 1 a2 1 > 1 a1 1, a1 - a2 > 0. Hence y(q
=
2
[2~3 - p(yfe(X+al)i]
Therefore, if CL--, 00, then (pi + 0, a2 - -I_L and the relationship (Y~(Y* = 6A2 implies that ?G 0. Consequently, r”(i) > 0 for sufficiently large values of pL,p> 2X. Observe, that in the considered example I”‘(0) = -h(p + A) < 0, for all fixed values of X and p, in accordance with Corollary 3.2, while II”(E(p)) > 0, as j.4--, co, F(p) + 0.
ACKNOWLEDGEMENT
Research partially supported by an SK0 Grant from ONR during the visit of B.L.G. and T.R. at Case Western Reserve University.
280
REFERENCES
B. Ycart, W.A. Woyczynski, J. Szulga, J.A. Mann, Jr. and D.A. Scherson, Birth and death dynamics in adsorption: towards the theory of underpotential deposition, CWRU Preprint # 86-53. B. Ycart, W.A. Woyczynski, J. SzuIga, S. Reazor and J.A. Mann, Jr., Monte-Carlo simulation of some interacting particle systems in the plane: towards the theory of underpotential deposition, Chemometrics and Intelligent Laboratory Systems, 3 (1988) 141-149. B. Ycart, W.A. Woyczynski, J. Szulga, S. Reazor and J.A. Mann, Jr., An interacting particle model of adsorption, Applicationes Mathematicae, in press. R.S. Hansen and T.C. Wallace, The kinetics of adsorption of organic acids at the water-air interface, Journal of Physical Chemistry, 63 (1959) 1085-1091. A.W. Adamson, Physical Chemistry of Surfaces, Wiley-Interscience, New York, 4th ed., 1982. J. Lucassen and R.S. Hansen, Damping of waves on monolayer-covered surface II. Influence of bulk-to-surface diffusional interchange on ripple characteristics, Journal of Colloid and Interface Science, 23 (1967) 319-328. R.S. Hansen, Diffusion and the kinetics of adsorption of aliphatic acids and alcohols at the water-air interface, Journal of Colloid Science, 16 (1961) 549-560. R.T. Forster, The influence of diffusion and chemical reaction on adsorption kinetics in binary surfactant systems, Journal of Colloid and Interface Science, 96 (1983) 386-410. 9 P. Whittle, Systems in Stochastic Equilibrium, Wiley, New York, 1986. 10 B. Ycart, Interacting particle models of deposition, Universite de Pau, Laboratoire de Mathematiques Appliques, Preprint # 87/08. 11 T. Liggett, Interacting Particle Systems, Springer Verlag, Berlin, New York, 1985.
213
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Chemometrics and Intelligent L&oratory Systems, 6 (1989) 273-280 Elsevier Science Publishers B.V., Amsterdam -
Printed in The Netherlands
A General Stochastic Model of Adsorption-Desorption: Transient Behavior B.L. GRANOVSKY
Department of Mathematics,
Technion-Israel
Institute of Technology, Haifa (Israe)
T. ROLSKI
Institute of Mathematics,
Wroclaw University, Wroclaw (Poland)
W.A. WOYCZYNSKI
*
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland
OH 44106 (U.S.A.)
J.A. MANN
Department of Chemical Engineering, Case Western Reserve University, Cleveland
OH 44106 (U.S.A.)
(Received 1 July 1988; accepted 28 June 1989)
ABSTRACT
Granovsky, B.L., Rolski, T., Woyczynski, W.A. and Mann, J.A., 1989. A general stochastic model of adsorption-desorption: transient behavior. Chemometrb and Intelligent Luborato~ Systems, 6: 273-280. A mathematical model for adsorption-desorption phenomena is provided. The model permits interaction of adsorbed particles and, in particular cases, displays a nonconcave kinetic behavior (i.e. the expected number of adsorbed particles is a nonconcave function of time) that was previously discovered experimentally for adsorption of short chain carboxylic acids at the water-air interface and for adsorption in the presence of chemical reactions.
INTRODUCTION
This paper is a continuation of our study of stochastic models of adsorption and desorption originally motivated by the underpotential deposition phenomena (cf. refs. l-3). The model considered in what follows is fairly general and permits account to be taken of the interaction of particles. In particular cases, such as when adsorbed particles have a graph structure (i.e. a possible flexi0169-7439/89/%03.50
0 1989 Elsevier Science Publishers B.V.
ble structure of interactions with no rigid geometric restrictions imposed on them, cf. Section 5), taking into account neighbor interactions, a more detailed analysis is possible, and a concave-convex-concave behavior of the expected number of adsorbed particles as a function of time is displayed in certain situations. A motivation for our model was provided by some experimental results that showed that r(t), the number of adsorbed molecules per unit area
n
Chemometrics and Intelligent Laboratory Systems
0.0 0.0
0.2
0.4
0.6
0.8
I
1.0
t* Fig. 1. Kinetics of adsorption of the octanoic acid on the water air interface. The dependence of coverage r/I& on time was computed from experimental data obtained by Hansen and Wallace [4] for different values of I&, determined by the volume concentration of the octanoic acid. For purposes of comparison, a dimensionless parameter was used for scaling, t* = t/t= where t, was the time necessary to reach 0.95 of the equilibrium coverage. res is the coverage that obtains at equilibrium for a given concentration of octanoic acid.
(called the excess), can be a nonconcave function of time. In particular, Hansen and Wallace [4] demonstrated that the dependence on time of the spreading pressure of adsorbed, short chain, carboxylic acids at the water-air interface is qualitatively different from the behavior characteristic of the classical, Langmuirian, adsorption without lateral interaction in the interface. (See Adamson [5] for background about the definition of the excess and isotherms relating I’ to the surface tension and volume concentration.) Fig. 1 shows the effect of lateral interactions on the kinetics of adsorption of the octanoic acid and is adapted from Hansen and Wallace [4] using the isotherm data given in Lucassen and Hansen [6]. For large values of the surface coverage parameter @eq= Gq/Knax (I& corresponding to a given concentration of surfactant, I,, corresponding to the maximum excess of the isotherm of I vs. concentration) the curves show a nonconcave kinetic behavior expected from our theory (Fig. 4). Such a behavior is possible in octanoic acid since the strength of the intermolecular forces between, straight-chain, carboxylic molecules increases with chain length. For shorter chains of, for example, pentanoic acid the equilibrium isotherm is that of an ideal monolayer following, for all values of t&,
214
the Langmuir isotherm within experimental error; the molecules interact only weakly. However, octanoic acid deviates from ideal behavior; the seven CH2 groups provide lateral, attractive forces that can be observed. At relatively ‘low equilibrium = 0.6 the adsorption dynamics coverage, res/rmax are concave over the entire time scale. When rlX&ax = 0.99, intervals of convex and concave behavior are observed experimentally. The original data in ref. 4 related the surface tension to time. In order to compare them to our theory, which relates the excess I to time, it was necessary to make a conversion which used the assumption that the surface tension depends only on the excess even when the system is not in equilibrium. Since our purpose is only to compare shapes of the graphs of kinetic data, this local equilibrium assumption is satisfactory. The experimentally determined parameters for the Frumkin isotherm were taken directly from Lucassen and Hansen [6] (eq. (16), Table (l)), and includes a term that takes approximate account of interacting surface species. The fit was satisfactory and can be considered as providing an interpolation formula relating spreading pressure, rr = y,, - y (y. surface tension of water, y surface tension of solution) to excess. Then, Hansen and Wallace’s [4] interpolation formula (4) (Table II), which was fitted to ?r vs. time data for various short-chain carboxylic acids and octanoic acid in particular, was used in an obvious way to transform their 7~ vs. t data for octanoic acid to coverage vs. r data. The latter are displayed on Fig. 1. An issue that Hansen [7] and other groups discussed at the time was whether adsorption was kinetic or diffusion controlled. This concern is handled in our theory through postulating a Markov process mechanism for adsorption/desorption and two rate constants X and /J respectively. We allow particles to adsorb on empty ‘sites’ with a fixed rate X and then desorb at a rate p which depends on ‘near neighbor’ configurations. This amounts to assuming a ‘barrier’ to desorption. Indeed, our theory gives the linear dependence on time at short time as required by Hansen’s theory and various experimental results. While we can introduce a neighborhood dependence for both h and CL,we have chosen to keep A
Original Research Paper
275
constant at this stage of development. In the future, we plan to reanalyze quantitatively the kinetic data of Hansen and other groups. Moreover, neighbor effects can be computed quantitatively by molecular dynamic (Brownian dynamic algorithm) simulation and this will be done in the future. We remark that it is important to realize that for our theory to apply a cluster of surfactant molecules in the surface may have a graph of neighbors and next nearest neighbors and perhaps even more distant interacting neighbors. But these molecules need not be on a surface net (two-dimensional lattice) as, for example, is the case of CO adsorbed on a clean Ni, single crystal surface. In our language adsorbed particles may have a graph structure but not necessarily a lattice structure; our theory describes both situations. We also remark that similar kinetic effect was reported by Forster [8] wherein the influence of diffusion and chemical reaction on the adsorption kinetics in binary surfactant systems was studied. The adsorption of atoms and molecules onto the surface structure of solids does involve well-defined nets and strong lateral interactions are possible. The work done in this paper provides qualitative comparison of experimental and theoretical information. Quantitative comparison with experimental data and data generated by simulation will be done later.
2 GENERAL DESCRIPTION SORPTION MODEL
OF THE ADSORPTION-DE-
We assume that during deposition, the particles are adsorbed on a finite set S of N sites x. We denote by X= {OJ}’ the state space for our adsorption-desorption process i.e. the family of all 2N possible configurations r) = { n(x), x E S } of occupied and empty sites where n(x) is a 0 (empty) - 1 (occupied) valued function on S. If n E X is a particular configuration, it will be convenient to denote
H
for a configuration created from TI by just flipping the state of site x, and to denote ]?I] =#{xESn(x)=l} the number of occupied sites in configuration n. Now, we assume that the adsorption-desorption dynamics are described by a Markov process cp(t) = cp(t, w), t 3 0, u E at, (defined on a certain probability space (G!, 9, Prob)) with values cp(t, w) = n E X and transition function determined by the conditions: n-+nx
atrate
X
if
at rate
~(n,
n(x)=0
and n +nx
x)
if
n(x)
=l
wherep: XxS+R+,andp(n, x)=Oif ]n] =O. In other words, particles are allowed to adsorb on empty sites with a fixed rate h (e.g. think about deposition on a solid surface from an infinite solution), and then desorb at rate p which depends on state of the ‘neighborhood’ of the particle about to desorb. Denote by p,(t)
= Prob{cp(t)
= T)
the probability that at time t 2 0 the configuration of the process is 1. Then, from the forward Kohnogorov’s equation (cf. e.g. ref. 9, Chapter 2.7, (6)) we readily find that the collection of probabilities {p,(t) : v E X} satisfies the following system of 2N differential equations: &>
=
c
(x: +
Tj(x)=l) (
:,f3,=o)P(9x’ x
c (x:$(x)=0)
-(
+
x
X&(4
(
~~~)=l)16h x. x
+l,(t)
h,(t)
+w
(2.1)
for all n E X. Example 2.1. Assume that the desorption rate p(q, x) = p for all n E X, x E S i.e. the sites are
n
Chemometrics and Intelligent Laboratory Systems
276
independent of each other. In this particular case, (2.1) gives equations
Proof. Summing up eqs. (2.1) over the set {q: 1~1 =k} we find that
z-%$)=h
P;(t)=
(x:
c
J+&)+CL
q(x)=l}
c
p?j,(t)
{x: q(x)=O)
l+Wl~l)P,(t)
-(UN-
c
(2.2)
= (7r(t))lV’(l
-T(t)pg’
ApJt)
{x: q(x)=l}
+c
for all n E X. A simple, probabilistic argument helps to guess that the solution is of the form p,(t)
c
‘$‘=k
c
Ph,
-
(2.3)
c APT(t) I$l=k (v(x)=01
where -
7r(t) = l--(1
_ e-(h+r)t)
x)p,,(t)
ltll =k {x: ~l(x)-O)
c
(2.4)
CL(W)P?Jt)
c
Iq’=k
{x: q(x)=l}
eq. (3.1) follows directly from the obvious fact that
Now,
which recovers the elementary formulas obtained in ref. 1.
c 3 COVERAGE DYNAMICS
In this section we study the integer valued process n(t) = 1‘p(t) I which represents the number of occupied sites at time t. Note that, in general, the process (n(t), t z 0) is not a Markov process, an obvious exception being the case of independent sites studied in Example 2.1. Denote by P-l(t)
tzo
= 0,
and Pk(t)
=
c
k=O,...,
p,(t),
N
Ivl=k
probabilities that at time t there are exactly k sites occupied, and by b(n) =
(~:&=llP(‘IV
x,
what could be called the global desorption rate for a given configuration n. 3.1.
The probabilities
0, , . . , N furfill the following equations P;(t)
=X(N-k+ +
system
pk(t), k = of differential
l)Pk--l(t) c
= A(N-
k + I)Pk-i(f)
and from the identities ((17, x)EXXS: = {(n,,
b(v)P&)
x)EXXS:
((1, x)EXXS:
1111=k,
1~1 =k-1,
= ((17x7 x)EXXS:
+)=O}
q(x)=O}
171 =k,
0(x)=1}
The first identity can be checked as follows: let (7,x) be such that Iq(=k+l and 17(x)=1. Take { = nx. Then, clearly, (S,, x) = (0, x) and I { ( = k, p(x) = 0. Conversely, let 9 be such that Iq I = k, v(x) = 0. Then Iq,I = k + 1, and again ({, x) = (n,, x) for 5 = nx. A proof of the second identity can be obtained in an analogous way. Q.E.D. Let us denote by = ;
kP,(t)
k=l
the expected value of the number of adsorbed particles at time t. The above Theorem 3.1 permits us to obtain the following. COROLLARY
‘q’=k+l
Iql =k+l,q(x)=l}
and
I(t) THEOREM
A&)
‘q’=k-1 s(x)=0
3.1. I(t)
satisfies
the differential
equation -UN--
k)pk(t)
-
c
b(dP,(t)>
Ivl=k
k=O,...,
N
I”(t)=AN-U(t)-E[b(cp(t))],
(3-I)
tzo (3.2)
211
Original Research Paper
which, for the initial condition r(O) = 0, has the following solution r(t)=N(l-e-“)
-/de-h(t-S)E
b(cp(s))
ds (3.3)
Proof. Multiplying the kth equation in (3.1) by k and summing over k = 1,. . . , N we get I”(t)=A
k=l -k(N-k)Pk(t))
+
kcl
-k
k
(
c Ivl=k
c Iql=k+l
+h&)
bbh+(t))
=XN-hr(t)
-
$ k=l
c Iql=k
%h+(t)
Q.E.D COROLLARY
obvious solution
r(t) = x-(l
-
e-(A+P)t)
In particular, Pk(d
=
c
&@>
=
( ;)ddktl
-a(t))k
Ivl=k
where a is as in formula (2.4). Theorem 3.1 relates the mean coverage function r(t) to the mean global desorption function /3(t) = Eb(cp(t)). For the independent sites model considered above, the explicit computation shows that I’(t) is concave, while, in general, on the basis of Corollary 3.2 we can only claim that r(t) is concave near the origin (with, obviously, r’(0) = XN). This can be interpreted as a statement that at the beginning of the adsorption process the qualitative and quantitative behaviors of all systems are similar. In the next section we will give sufficient condition which assures the global concavity of the mean coverage function r(t), while in Section 6 we will show examples of the mean coverage functions which are not always concave.
{(k(N-k+l))p&t)
f
W
3.2. If r(0) = 0 then
r”(o) < 0
(3.4)
Proof. Denote P(t) = E&q(t)) >, 0, t >, 0. Since r(0) = 0, we have p,,(O) = 0, n E X, and, consequently, /3(O) = 0. Thus p’(0) z 0 and, by (3.2) we obtain that r”(0) < 0 since I”(0) < 0. Example 3.1. In the case of independent (Example 2.1 continued) we have that
sites
4 CONCAVITY
OF THE MEAN
COVERAGE
FUNCTION
In this section we study some additional properties of the function r(t) = E 1q(t) I under an assumption that our interacting particle system is attractive (cf. eg. ref. 11, p. 134). This assumption, in our case, means that
W=Wl so that eqs. (3.1) take the form P;(t)
=X(N-h(N-
k+
l)Pk-l(f)
k)P,(t)
+
(k
+
1h‘P,+&)
- pkPk(t)
In this case 1cp(t) 1 is a Markov process, and Eb(cp(r))
= CIE I dt>
I = @(t)
Thus, from eq. (3.2) we obtain that
r’(t) =AN-m(t) -g(t) which under initial condition
r(0) = 0 has the
whenever TJ~5 and n(x) = l(x) = 1, which describes the situation when a particle is less likely to desorb if it is surrounded by more adsorbed particles. Physically, it is a very common situation (cf. refs. l-3 and the references quoted therein). Under the above assumption r(t) is nondecreasing whenever p(O) = 0 (mod P) where 0 E X denotes the empty configuration, i.e. when there are no adsorbed particles in the initial state (cf. ref. 3, Theorem 3.2.6 (1)). It turns out that, assuming additionally monotonicity of the global de-
n
Chemometrics
sorption prove
rate b(7)
THEOREM b(v)Gb(S)
and Intelligent
Laboratory
(cf. Section
Systems
3), we can also
4.1. If the system is attractive for
17G
218
and (4.1)
then r(t), t z 0, is concave. Proof. Since cp(t) is attractive and the initial state is empty, it follows from (4.1) that the function E[b(cp(t))], t 30 is nondecreasing. Thus, Corollary 3.1 implies that I’(t) is decreasing which gives the concavity of I. Q.E.D.
Note that attractive systems fulfilling condition (4.1) are close in their qualitative behavior to the independent sites model.
Example 5.1. (Mutually sticky particles model.) Assume that, if v(x) = 1 then
ON A SET OF SITES WITH A
In this section we look at the adsorption process cp(t) in a special situation where the set of sites S has some topological structure, i.e. each site x E S has the set of its closest neighbors S(x) c S defined. Such a structure is mathematically best described by a graph on the set of vertices S. Physically, it could correspond, for instance, to absorption on the surface of a crystalline substrate, or could correspond to certain bond structure between adsorbed molecules. Such models have been studied in refs. 2 and 3 and from simulations reported therein, we infer that r(t) need not be always concave. This was especially evident in ref. 2 (Caption to Fig. 2), where the rates p were assumed to be of the form
Here Iv(q, x) 1 = #{x E S(x): TJ(X)= l} and K and d are positive constants. Since this model is rather difficult to study analytically and the explicit formulas are not readily available for it, we propose here the following simplification.
147, x) I =O
otherwise
0
which means that adsorbed particles desorb at a fixed rate I_Cif they have no neighboring sites occupied and do not desorb at all if at least one of the neighboring sites is occupied. Denote by a,,( 9) the number f occupied sites with no neighboring sites occupied i.e. n&l)=
c
{x:
I( Iv(ll. x)1=0)
q(x)=l)
where 1, is the indicator function of set A. Note that WV) = CLAN. Since in the considered case the stationary state of the process is, obviously, the fully occupied configuration, we see that with probability 1 n,(cp(t))
5 ADSORPTION PROCESS GRAPH STRUCTURE
CL if
4% 4 =
+O
as
and hence Eb(cp(t) Example S = {1,2,3} Fig. 2.
I + cc --, 0 as I + cc.
5.2. Consider the concrete example of with the graph structure described in
Then X = {(O,O,O), (O,O,I), (OLO), (I,O,O), (LOJ), (LLO), (LIJ)} and
~(9, i) =
p
if n(j)=O,
0
otherwise
(OJJ),
j#i
by (3.1) we have PW
= CLPIW - 3APoW
P;(t)
= 3hPoW
- (2X + CL)PlW
P;(t)
= 2XPlW
- hP&)
P5W
= AP&)
with initial conditions ~~(0) = 1, ~~(0) = 0, k = 1, 2, 3. The matrix of the system of the first three of 3
/\
2
Fig. 2. The graph structure of intermolecular sidered in Example 5.2.
interactions
con-
Original Research Paper
219
n
the above equations is
thus, 1 0
-3x
[
3x
-(21+I”)
0
0
2A
-A
for ~1z 2X, its eigenvalues are different and equal to (Yg= -A [-(p+5A)f~~2+lOXp+P]
al,2 = f
GO
3
and its eigenvectors are
rL
A
3x+cu,
3x+(Y,
1 2x
1 2x (Y1+ x
(Y2+ h
where (Ye+ X # 0. Thus, whenever, p # 2X, the general solution of the above system is as follows CL a1’ + c -e 23x+oL2
PO(t) = c&L? 3x+ff, pi(t)
= c,eal’ + c2eaZt
p2(t)
2x = cOeao’+ c1 -eal’ (Y1+ x
9’
a21 + c2-e 2x LX2+ x
where the constants c,,, cl, c2 are to be determined by the initial conditions. Then Eb(cp(l))
=j.kP1(t)
= --$$-(ealf-
Fig. 3. The concave dependence of excess function r on time for a simple system considered in Example 5.2 for X = 50 and p = 50.
Fig. 3 (A = 50, p = 50) or follows a concave-convex-concave pattern of Fig. 4 (A = 50, /J= 15 000) with two inflection points, and these two possibilities exhaust all the possible qualitative patterns of the behavior of this system. This follows from the fact that any function of the form F(t) = ale”” + a2ea2’ + ajea3’ has no more then two zeros. Indeed, f(t) = e’+‘g(t), where the derivative of the function g(t) has no more than one zero. Since I”‘(0) K 0, by Corollary 3.2, and I’(t) is bounded, one change of sign of r”(t) is not possible. The concave-convex-concave behavior displayed on Fig. 3, indicating the deceleration period followed by an acceleration and then another de-
eaZf)
Thus, by (3.3)
1.O
l?(t) = 3(1 -e-“)
- $emAf 2
2 L
x
feAs(eW
_
eW)
ds
J0
=
311 -e-X’)
0.5 - -?9L a1 a1 -
x
[
a2
a2
f
(X+a*)(X+a2)e-hr
1 +h+(Yle
alf alf - _e 1 x + lx2
(1.0e
1
0.0
(5.1)
The function l?(t) in (5.1) is either concave as in
1.0
2.0
t x 10
3.0
4.0
3
Fig. 4. The concave-convex-concave dependence of excess function r on time for a simple system considered in Example 5.2for X=5Oand p=15000.
n
Chemometrics
and Intelligent Laboratory
Systems
celeration period in the approach to system’s equilibrium does indeed occur if the condition A + (Ye > 0 is satisfied. In fact, in the considered case, p > 2X, and we have (A + a,)(X + az) = 2X2 - PA < 0. Thus, from (5.1) we derive that I+(t) =-
3e-” p-22x
&(*:(A
2h3-
[
-af(X +
a2)e(a1+X)r
+ al)e U2+*)*
)I
Now, take t = i > 0, satisfying e(OL1-a2)i = &LX:, which is possible due to the fact that 1 a2 1 > 1 a1 1, a1 - a2 > 0. Hence y(q
=
2
[2~3 - p(yfe(X+al)i]
Therefore, if CL--, 00, then (pi + 0, a2 - -I_L and the relationship (Y~(Y* = 6A2 implies that ?G 0. Consequently, r”(i) > 0 for sufficiently large values of pL,p> 2X. Observe, that in the considered example I”‘(0) = -h(p + A) < 0, for all fixed values of X and p, in accordance with Corollary 3.2, while II”(E(p)) > 0, as j.4--, co, F(p) + 0.
ACKNOWLEDGEMENT
Research partially supported by an SK0 Grant from ONR during the visit of B.L.G. and T.R. at Case Western Reserve University.
280
REFERENCES
B. Ycart, W.A. Woyczynski, J. Szulga, J.A. Mann, Jr. and D.A. Scherson, Birth and death dynamics in adsorption: towards the theory of underpotential deposition, CWRU Preprint # 86-53. B. Ycart, W.A. Woyczynski, J. SzuIga, S. Reazor and J.A. Mann, Jr., Monte-Carlo simulation of some interacting particle systems in the plane: towards the theory of underpotential deposition, Chemometrics and Intelligent Laboratory Systems, 3 (1988) 141-149. B. Ycart, W.A. Woyczynski, J. Szulga, S. Reazor and J.A. Mann, Jr., An interacting particle model of adsorption, Applicationes Mathematicae, in press. R.S. Hansen and T.C. Wallace, The kinetics of adsorption of organic acids at the water-air interface, Journal of Physical Chemistry, 63 (1959) 1085-1091. A.W. Adamson, Physical Chemistry of Surfaces, Wiley-Interscience, New York, 4th ed., 1982. J. Lucassen and R.S. Hansen, Damping of waves on monolayer-covered surface II. Influence of bulk-to-surface diffusional interchange on ripple characteristics, Journal of Colloid and Interface Science, 23 (1967) 319-328. R.S. Hansen, Diffusion and the kinetics of adsorption of aliphatic acids and alcohols at the water-air interface, Journal of Colloid Science, 16 (1961) 549-560. R.T. Forster, The influence of diffusion and chemical reaction on adsorption kinetics in binary surfactant systems, Journal of Colloid and Interface Science, 96 (1983) 386-410. 9 P. Whittle, Systems in Stochastic Equilibrium, Wiley, New York, 1986. 10 B. Ycart, Interacting particle models of deposition, Universite de Pau, Laboratoire de Mathematiques Appliques, Preprint # 87/08. 11 T. Liggett, Interacting Particle Systems, Springer Verlag, Berlin, New York, 1985.