Competition effects in surface diffusion controlled reactions: Theory and
Brownian dynamics simulations
D. V. Khakhar and U. S. Agarwal
Citation: J. Chem. Phys. 99, 9237 (1993); doi: 10.1063/1.465540
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Competition effects in surface diffusion controlled reactions: Theory
and Brownian dynamics simulations
D. V. Khakha,.a) and U. S. Agarwal b)
Department of Chemical Engineering, Indian Institute of Technology-Bombay, Powai,
Bombay 400076, India
(Received 6 April 1993; accepted 10 August 1993)
Surface diffusion controlled reactions on a heterogeneous catalyst surface comprising randomly
placed circular reactive sites are considered. The diffusing species adsorbs onto the surface
following Langmuir-Hinshelwood kinetics and reacts instantaneously on contact with a reactive
site. Approximate theories are formulated to describe the process for high concentrations of the
reactive sites, when competition between the sites is significant, following three different
approaches: (i) modification of the single sink theory; (ii) using a cell model; and (iii) using an
effective medium theory. The predictions of the theories are compared with the results of
multi particle Brownian dynamics simulations for the overall reaction rate, the bulk
concentration of the reactive species, and the ensemble averaged concentration profile around a
reactive site. The effective medium theory is found to give the best results among the theories
considered, and the predictions are in good agreement with the computational results.
I. INTRODUCTION
In rapid heterogeneous reactions on solid catalytic surfaces, such as the oxidation of CO on platinum,I,2 the overall reaction rate may be limited by the rate of surface diffusion of the adsorbed species on the catalyst surface to a
suitable configuration (e.g., diffusion to another molecule
of the adsorbed species, or diffusion to a reactive site). 3-5
Other examples of diffusion controlled aggregation processes in two dimensions (2D) are precipitation in 2D,6
thin film nucleation and growth, 7 coagulation between
drops floating on the surface of a turbulent liquid (e.g., oil
spills), etc. In addition, the reaction of diffusing molecules
in lipid bilayer membranes is effectively a 2D process because the diffusing molecules are of a size comparable to
the membrane thickness. 8,9
The trapping of diffusing species by spherical sinks in
3D space is a classical problem, and considerable work has
been reported in the literature (Refs. 10 and 11 for reviews
of theoretical analyses). However, relatively few theoretical studies of diffusion controlled processes in 2D, which
are known to have qualitatively different behavior as compared to 3D systems, 12 have been carried out. Freeman and
Do1l 3 compared the results of the theory (based on Smoluchowski's approach13) for the reaction of an adsorbed diffusing species with a single circular reactive sink to those of
a Langevin dynamics simulation. They obtained reasonable
agreement between the reaction rate constant obtained
from simulation and the theoretical prediction for physically practical values of the diffusion constant. However,
significant deviation between theory and computations was
reported for the rate constant and the concentration profile
for large values of the diffusion constant. Freeman and
Do1l 3 also carried out multiple absorber simulations. In
a)Author to whom all correspondence should be addressed.
b)Current address: Chemical Engineering Division, National Chemical
Laboratory, Pune 411008, India.
this case, the rate constants obtained for low values of the
diffusion constant were in excellent agreement with the
theory.
When the reactive sites cover a significant fraction of
the surface, interaction between the sites becomes important and the effective rate constant is found to increase. 4.14
Cukier4 considered such competition effects in a rigorous
effective medium theory for reaction of adsorbed species at
randomly placed circular line sinks. Kuan et al. 5 presented
an approximate treatment for reaction at the perimeter of
randomly placed circular disks based on the assumption of
no flux at the symmetry boundary between neighboring
sinks using a cell model. The validity of these theoretical
approaches for reaction rate constants of diffusion controlled reaction on surfaces has not been investigated in
detail. Fichthorn et al. 1 carried out Monte Carlo simulations of adsorption, surface diffusion, and reaction on catalytic surfaces, and they found that adsorbate islands were
formed on the surface. At low desorption rates, the growth
of these islands resulted in oscillatory reaction kinetics. In
the absence of reactant desorption, self-poisoning of the
surfaces was found to occur. Reaction rate constants for
the process, however, were not calculated.
Thus, while diffusion limitations regulate the rates of
many important aggregation processes in 2D, no exact theoretical treatments are available when competition effects
are significant. Here we consider the steady state diffusion
controlled reaction of adsorbed species with circular static
reactive sites on a heterogeneous catalytic surface. Approximate theoretical analyses and multiparticle Brownian dynamics simulations are presented for finding the effective
rate of the diffusion controlled reaction. The validity of the
approximate analyses is examined by comparison with the
results of the simulations.
The mathematical formulation of the problem and the
approximate theories using three different approaches are
given in Sec. II. The computational procedure used in the
Brownian dynamics simulations is given in Sec. III and the
J. Chern. Phys. 99 (11), 1 December 1993 0021-9606/93/99(11 )/9237/11/$6.00 © 1993 American Institute of Physics
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9237
D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
9238
results and discussion in Sec. IV. The conclusions are summarized in Sec. V.
so that the dimensionless parameters of the system are
(ka+kd) and kaNmax. The boundary conditions in dimensionless form are
N=O
II. THEORY
We consider the reaction of molecules A that adsorb on
the surface from the gas phase and diffuse on the surface
until they desorb or react at reactive sites
Agas+=±Aads+reactive site ..... product gas.
(1)
where k~ and kd are the adsorbtion and desorption rate
constants, and C and Cmax are the local and the maximum
number concentrations of the adsorbed species on the surface. DAis the surface diffusivity of the adsorbed species.
Assuming that the reaction at the reactive site perimeter is
instantaneous, the boundary conditions for the problem are
for r=ra , a= 1,2, ... ,ns ,
(2)
where ra is a position vector to the perimeter of the ath
reactive site and ns is the total number of reactive sites. In
addition to the above, far field boundary conditions for the
problem must also be specified (e.g., no macroscopic number density gradients of the adsorbed species on the surface). The main objective of the analysis is to calculate the
effective rate of reaction due to surface diffusion given by4
(3)
where A ~ is the total surface area, the integral is over the
perimeter of site a, and Da is the outward pointing unit
vector perpendicular to the perimeter of site a. Furthermore, direct adsorption of species A onto the reaction site
results in instantaneous reaction (i.e., reaction by the
Eley-Rideal mechanism).3 The reaction rate for such direct adsorption is
(4)
where 4J is the fraction of the total area covered by the sites.
In the presence of overlap between sites, the area fraction is
given by (Appendix A)
4J-;:::, l-exp( -4Jo),
a=I,2, ... ,n s
(7)
and the expressions for the dimensionless rates of reaction
by diffusion and direct adsorption are
(8)
and
The reactive sites (circular disks of radius a) are randomly
distributed on the surface, and overlap of the sites is allowed. The governing equation for the problem, assuming
that adsorption and desorption of the molecules follow
Langmuir-Hinshelwood kinetics and that a steady state
exists, is4
C=O
for S=Sa,
(9)
respectively, where R=R'a4/ DA and AT = A~/a2. The
number concentration of the sites (Ns = n/AT) is an additional dimensionless parameter of the system.
A. Theory for low reactive site concentration
When the dimensionless number concentration of the
reactive sites is sufficiently small (Ns <1), the sites are
located far apart and competition effects are negligible.
Hence the problem reduces to diffusion of the adsorbed
species to a single isolated reactive site on the surface, and
has been solved by Freeman and Doll. 3 We briefly review
the results of this single absorber (SA) theory below and
carry out modifications to improve the accuracy of the
theory at moderate concentrations.
The governing equation and boundary conditions for
this case are
a (5 aN)
g1 as
as +kaNmax- (ka+kd)N=O,
(10)
N=O,
(11 )
N=N B ,
5=1,
5 ..... 00,
(12)
where N B is the uniform average dimensionless concentration of the adsorbed species far from the reactive site. The
dimensionless concentration profile around a sink corresponding to the above equations is obtained as 3
(13)
where ,1= (ka+kd) 1/2. Using Eq. (8), the reaction rate
due to surface diffusion is
RD=kIftPB,
(14)
where the effective rate constant is given by
(15)
The far field concentration is obtained from Eq. (10) in the
limit 5..... 00 as
(5)
(16)
where 4Jo = 1T"a2n/A~ is the area fraction covered without
overlap.
Using the following dimensionless variables: N = Ca 2 ;
S=r/a; ka = k~2/DA; and kd = k;p2/DA, the governing
equation becomes
Here Ne is the eqUilibrium surface concentration of the
adsorbed species in the absence of reactive sites.
In addition to modifying the spatial variation of the
number density of the adsorbed species (N), increase in
the number density of sites (Ns ) has the following two
effects: (i) reduction in the total perimeter available for
reaction due to overlap of sites; and (ii) reduction in the
(6)
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9239
D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
far field concentration (N B) due to the reduced area available for adsorption and consumption by reaction of the
adsorbed species; both are in tum due to the presence of
reactive sites in the far field. We incorporate these two
effects into the SA theory, neglecting changes in the concentration profile due to overlap of reactive sites and competition effects.
The rate of reaction, with the above assumptions, becomes
(25)
The latter condition specifies that no diffusional flux of
particles from the symmetry element occurs.
The concentration profile for this case is obtained as
Ko(A-g) +KI (A-a2)10(A-g)111 (A- a2) ]
N=Ne [ 1
1)
1
l '
Ko(A-al) +KI (/l.a2 10(/l.al)111 (/l.a2)
(26)
(17)
where Ip is the fraction of the total site perimeter unavailable for reaction due to overlap. When only pairwise overlap is considered, we obtain Ip=1T'Ns' Taking into account
the consumption of the adsorbed species (A) in the far
field, the balance equation for A becomes
(18)
where we have included the factor (1- rp) to account for
the reduction in the adsorption rate due to the presence of
sites in the far field. Using Eq. (17), the bulk concentration
is given by
The corresponding reaction rate and bulk concentration
are
RD=21rNsC 1- Ip)A-Ne
KI (A-al) -KI (A-a2)11 (A-al)III (A-a2)
X Ko(A-al) +KI (A- a2)10(A-o l )1II (A- a 2)
(27)
and
(28)
where we have used a balance on A [Eq. (18)] to obtain the
bulk concentration.
(19)
N B ka+ k d+ k Jfls(1- Ip) .
The bulk concentration predicted by the modified SA theory thus decreases with an increase in the number density
of reactive sites in contrast to the SA theory [Eq. (16)].
The reaction rate [Eq. (17)] is lower compared to the SA
theory [Eq. (14)] because of this and because only a fraction of the total perimeter is available for reaction.
B. Equivalent annulus approximation
Kuan et al. 5 considered regular arrays of circular reactive sites and assigned a symmetry element around each
site defined as the contour across which the flux of diffusing particles vanishes. For randomly placed sinks, an
equivalent annular symmetry element was defined and the
inner and outer radii in dimensionless form were taken to
be5
(20)
C. Effective medium theory
The basic objective of the effective medium analysis is
to replace the heterogeneous surface containing discrete
reactive sites by an equivalent homogeneous surface with a
spatially uniform rate of consumption of reactants. Such an
effective medium representation for the system can be obtained by averaging the governing equation (6) over all
configuration of sites, taking into account the interactions
between sites. Since the governing equation is linear, the
effect of every reactive site perimeter (which essentially
acts as a line sink) is additive. An equivalent formulation
of the problem is then
(29)
where aa is the strength of the site a considering diffusion
from outside the site alone. The operator 3a is defined as
and
(30)
(21)
respectively, where Ae= lINs is the area of the symmetry
element and P is the active perimeter of the sites. (All
lengths are made dimensionless using a and areas are made
dimensionless using a 2.) For overlapping sinks, the equivalent radii are
al =rplrpo( 1- Ip),
a2=a l l
#.
where Sa is a vector to the perimeter of site a. In addition
to the above, the boundary conditions (7) are required to
obtain aa' Taking a configurational average (see Appendix
A) of Eq. (29) gives
V g(N)+A- 2 (Ne -(N» =
(22)
(23)
f
ds' W(g-g')(N)(s')
= W(N),
(31)
n
(~as=18aaa)
A
The method of solution followed here is the same as in the
previous section except that boundary conditions used are
where the form of the average
is assumed to
be as given above 15- 17 and the caret denotes a convolution.
As shown below, the operator Wcan be approximated as
(24)
(32)
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9240
D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
so that the rate of reaction, the bulk concentration, and the
averaged concentration profile around a site can easily be
obtained if W(s,s') is known.
We evaluate the function W(s,s') using the multiple
scattering approach developed independently by Muthuku~
mar l4 and Cukier and Freed 18 for static spherical traps
randomly distributed in a volume. The unique feature of
the approach is that the Green's function used in the anal~
ysis includes the unknown function W so that higher order
interactions between sites are taken in account in a relatively simple way. Cukier4 extended this approach to the
case of surface diffusion to circular reactive sites, however,
the reaction rate obtained was based on the diffusional flux
to the perimeter from outside the reactive site as well as
from within the site. 19 Hence the problem considered is
different from that defined here, and as might be expected,
the results obtained in Ref. 4 do not reduce to the SA
theory in the limit of low concentrations. Cukier4 also as~
sumed that the reactive sites do not overlap. In the follow~
ing analysis, we follow Cukier's approach4 for the case of
disk~like reactive sites taking into consideration overlap
between sites.
When the reactive sites are represented as circular line
sinks, the diffusional flux to the perimeter from inside is
due to the particles which adsorb inside the site. For the
problem under consideration, such particles would instantly react by the Eley-Rideal mechanism, so that N =0
everywhere inside the reactive site. To account for this, two
alternative methods can be used: (i) to assume a distribu~
tion of sinks inside the reactive site such that N =0 every~
where inside instead of only at the perimeter [Eq. (7)]; (ii)
to specify that no adsorption occurs within reactive sites,
so that at steady state, we have N =0 inside all sites. The
.latter is straightforward to implement and we use this ap~
proach in the analysis. The governing equation in this case
becomes
a=1,2, ... ,ns'
(37)
where baf=I dSS(s-sa)/(S)=/(Sa). Taking the pe~
rimeter inverse of the propogator for site a, we obtain the
strength of the line sink as
aa(Sa) = KabaGCWN +tf;) -
L
fJ=Fa
KabaG5{P[3'
where by definition KabaG5aaa=aa (Appendix B). Here
~[3=1=a denotes the sum over (3 excluding a. Taking a con~
volution with respect to Sa' we obtain
(39)
where Ta=5aKaba.
Substituting for Saaa on the right-hand side of Eq.
(39) an infinite number of times and using Eq. (36), we
obtain the concentration in implicit form as
(40)
XG(WN+tf;)
=Q( WN+tf;).
(41)
Similarly, substituting for N on the right-hand side of Eq.
( 41), we get the complete solution
(42)
Finally, to obtain W, take a configurational average of the
above equation
(43)
where we have used (N)=G(tf;). Keeping only the first
term in the above expansion, we obtain
(33)
where H= 1 inside sites and H=O outside. H(s) is defined
to take into account overlap between sites (Appendix A).
Defining the operator
2"=( -V~+A2+ W),
(34)
Eq. (33) becomes
2"N=WN+tf;-
~
a=1
5aa a ,
(35)
(44)
This approximation is referred to as the coherent potential
approximation" and essentially corresponds to taking a
configurational average keeping the position of one reactive
site fixed. The approximation is found to give good predic~
tions for 3D systems," but has not been tested for 2D
systems. The errors are expected to be larger for the latter. 20 Using Eq. (C6) and simplifying, we get
GWG(tf;)=
where tf;=A 2N e(1-H). The formal solution to Eq. (35) is
N(S)=G(WN+tf;)-
~
a=1
G5aa a ,
(36)
where G(S,S') is the Green's function (effective medium
propogator) of operator 2" (Appendix B). As mentioned
earlier, the effective medium propogator (G) contains the
averaged contribution of all the sites (W).
Using the boundary conditions (7), we get
(38)
~
a=1
G(i'a)G(tf;)-
~
a=1
G(TaGHa)(tf;),
(45)
where (tf;)=A 2N e(1-</J). Using (N)=G(tf;), the above
equation reduces to
W(N)
= (t) (N) -
(1-</J) (q),
(46)
where (T)=~:s=I(Ta) and (q)=A2N~:s=l(faGHa).
Taking Fourier transforms (see Appendix B for a definition) and expanding Wand (T) in a Taylor series about
k = 0, the above equation becomes
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D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
[W(O) +kW'(O) +
~ W"(O) ] (N)
J2-d2 _,
]_
_
= [ (T)(0)+2 dP (T) k=O (N)-(1-ifJ)(ij),
(47)
where the tilde denotes a Fourier transform. Equating
terms with equal powers of k, we obtain
(48)
(49)
W'(O)=O,
2
_
d
-,
W"(O) =2tJD=df? (T) k=O'
(50)
where we assume that there are no macroscopic concentration gradients, so that (N) =NB for k::::::O. Using the
results given in Appendix B, we finally obtain
(51)
tJD=rrNs ( 1+t5D)
[II (,U)~l (fL)
1
] (52)
I o(fL)Ko(ll) ,
where 1l 2= (A. 2+kr)/(1 +tJD).
The governing equation for the effective medium thus
becomes
(1 +tJD)V~(N) +A2[ (l-ifJ)Ne - (N)] =kj(N).
(53)
In the absence of macroscopic concentration gradients, we
have (N) =NB, and bulk concentration is given by
(54)
N B=A2NeC l-ifJ)/(A 2+kf ).
The rate of reaction is then
(55)
In the limit Ns-O, we have kf::::::O, oD::::::O, Il::::::A., ifJ::::::O,
and the above results reduce to the SA theory.
To obtain the configurationally averaged concentration
field around a site, we assume that Eq. (53) is valid. Using
the boundary conditions given in Sec. II A [Eq. (12)] the
concentration profile is
(56)
(N) =N B[ l-Ko(IlS)!Ko(Il)].
The above concentration profile is identical to the dilute
case [Eq. (13)] except that A. is replaced by Il and N B is
given by Eq. (54). We note that the rate of reaction computed from the above equation based on the average flux to
the sink is the same as that given in Eq. (55), so that the
equations are self-consistent. In addition, this also verifies
that the calculated reaction rate is due only to the diffusive
flux to the perimeter from outside the reactive site as required by the model.
9241
procedure used is similar to that of Freeman and Dolb for
multiple absorber simulations. The dimensionless variables
used in the computations are the same as given earlier.
In the simulation, we first generate the configuration of
reactive sites by fixing the positions of ns circular reaction
sites of unit radius at random positions (allowing overlap)
on a square cell ofside L' (dimensionless). At dimensionless time T=ta 21 DA=O, no particles of the diffusing species
are randomly distributed on the same surface. Here no is
only an initialization value and is not critical. The diffusion
of A is simulated as a random walk with discrete time steps
(.601'). The displacements ASx and ASy along the coordinate
axes during time step .601', for each particle, are picked from
a set of random numbers with a Gaussian distribution with
zero mean and variance 2AT, and the positions of the particles are updated. To approximate an extended system,
periodic boundary conditions are applied, i.e., if a particle
leaves the cell through one wall, it reenters at an identical
position through the opposite wall. In addition, during
each time step, each of the n ( 1') adsorbed species is removed with a probability (ka+kd)Ar [i.e., a fraction (k a
+kd)Ar of the n particles are removed, which results in a
local desorption rate of (ka+kd)N per unit surface area].
At the end of each time step, the position of every diffusing
particle relative to each reactive site center is calculated. If
separation of any particle is less than the reactive site radius, then the particle is removed and counted as reacted.
For each time step, an average of (kfimax)L,2 AT particles
are adsorbed on the surface at random positions. Those
landing directly on any reaction site are considered to have
reacted by the Eley-Rideal mechanism and are removed.
The number of adsorbed particles A [n ( T)] is updated at
the end of each of the adsorption, desorption, diffusion,
and reaction steps. The algorithm is repeated for long times
until n (T) becomes nearly constant with T, indicating that
a steady state is reached. Following this, the algorithm is
continued for several time steps, and time averages of the
rates of adsorbtion, desorption, and reaction at steady state
are calculated. For a given set of parameters, such simulations are carried out for typically 30 configurations of the
reactive sites, and the steady state averages obtained for the
reaction rate per unit area (R D) and the bulk concentration of species A (N B=nl L,2). We also calculate the average concentration profile of A around a reactive site by
generating a frequency distribution of the distance of every
particle from each site and averaging for different configurations of the sites and over time in the steady state regime.
The computational parameters of the simulation are
L' and AT. We have taken L' large enough and .601' small
enough that the results are not influenced significantly by
further increase and decrease, respectively, of these parameters. Typical values used in the computations are L' =20
and AT= 10- 3•
IV. RESULTS AND DISCUSSION
III. BROWNIAN DYNAMICS SIMULATIONS
We briefly outline the procedure used for the Brownian
dynamics simulation of the system defined in Sec. II. The
The simulation results for R D and N B are first considered for three sets of values of the parameters kfimax and
(ka+kd), keeping the equilibrium concentration constant
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D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
9242
050
0.10
0.40
008
0.30
0.06
cr:"
cc'"
0.04
0.20
~ Simulation
~EMT
~ Simulation
~EMT
G€EE8 Eqv. Annulus Th.
0.10
[333OEl
Modified SA Th.
koNmo • ;;;; 1.6
k.+k, = 40
000 +r-,...,..,.Tr1rrrrn""'"n,,n"", In'""n,,-,-,,...,n,',I"n,'''',',',,,,,,,I
0.00
0.05
0 10
0.15
0.20
Ns
(a)
G€EE8 Eqv. Annulus Th.
0.02
o.00
[333OEl
+r",,,rrrrn-,-,-nrrTr1rrrrn-,-,-,,,,,nornrr1
0.00
(a)
0.40
Modified SA Th.
k.Nm •• = 0.16
k.+k, = 04
0.05
010
0.20
0.15
Ns
0.30
00BE!El Modified SA Th.
_
Simulotior.
~EMT
GeEJElE) Eqv.
0.30
0.20
Annulus Th.
keNmo. = 0.16
ko+k. =
0.4
",020
~
0.10
00BE!El Modified SA Th.
_ _ Simulation
0.10
~EMT
GSeeO Eqv. Annulus Th.
kaN m•• = 1.6
ko+k, = 4.0
o 00
.:j...,r-n-,-,-nornrrrnrr-rr-rrrrn-,-,-rn-rr-rrrr-n..,..,
0.00
(b)
0.05
0.10
0.15
0.20
N.
0.00 +r-",-,-,-rrrrn-,-,-nrrTT"rrrrn-'-'-TTT-rr-rrrrn"
0.20
0.15
0.10
0.00
0.05
(b)
Ns
FIG. 1. Variation of (a) the dimensionless reaction rate (R D ) and (b)
the dimensionless bulk concentration of the diffusing species (N B), with
the number density of reactive sites (N,) for k.Nmax= 1.6, and (ka+kd)
=4.0 (Ne=0.4).
FIG. 2. Variation of (a) the dimensionless reaction rate (R D ) and (b)
the dimensionless bulk concentration of the diffusing species (N B), with
the number density of reactive sites (N,) for k.Nmax=0.16, and (ka+kd)
=0.4 (Ne=0.4).
N e=kfimax/(ka+kd=0.4). In Figs. 1-3, we plot the theoretical and the simulation results for the variation of the
overall reaction rate (R D ) and the bulk concentration of
the diffusing species (N B) with number density of the reactive sites (Ns ). The overall reaction rate (R D) first increases with Ns due to increasing total sink perimeter [Figs.
1 (a)-3(a)]. With further increase in N s ' RD decreases because the site-free fraction of the total surface available for
adsorption decreases. This suggests an optimum loading of
catalytic sites, when direct adsorption and reaction within
the sink is not possible, either by the Eley-Rideal mechanism, or by diffusion to the inner perimeter of the sink.
(An example of such a case is the reaction of one of the
adsorbed species at a perimeter of islands of adsorbed molecules of the second species. s ) The optimal number density
of sites shifts to higher values with an increase in the adsorption rate (kfimax) for a fixed value of N e . With an
increase in N s ' the bulk concentration decreases [Figs.
l(b)-3(b)] because the adsorption rate is not rapid
enough (due to reduction in available surface area for ad-
sorbtion) to make up for the increasing consumption by
reaction. The decrease in N B with Ns is faster at smaller
adsorption rates (kfimax).
A comparison of the predictions of the theories shows
that all three give the correct trends for both R D and N B
(Figs. 1-3). This is in contrast to the SA theory, in which
the bulk concentration is independent of the reactive site
concentration (N B=Ne =0.4). The deviation of N B from
Ne is significant even at the lowest reactive site concentration considered (Ns =0.02), which corresponds to about
6.5% of the total area covered by reactive sites [Figs. 1 (a)3(a)]. As is apparent from Eq. (54), the deviation from Ne
is larger for smaller (ka+kd)' The SA theory also predicts
a monotonic and linear increase of R D with Ns which is
qualitatively different from the observed behavior. For all
values of the parameters considered, the effective medium
theory (EMT) and the equivalent annulus theory give
good estimates of the overall reaction rate [Figs. 1 (b)3(b)]. For large values of the rate of adsorption (kfimax
=4.0), the EMT and the equivalent annulus theory both
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D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
0.014
9243
0.12
0.10
0012
0.08
0.010
0
a::
0.008
-
0.04
Simulation
~
~EMT
0006
0004
0.00
cea:B Eqv. Annulus Th.
[E33J Modified SA Th.
k.Hmo. = 0.016
k.+k. = 0.04
TI, 1'1""'" i i i '
0,05
010
J
cea:B Eqv. Annulus Th.
C£EEEJ Modified SA Th.
k,N mo • = 0.16
N, = 0.10
0.02
iii", iii
i'
iii,
0 15
0.00 +rT'T1-r-rr-rTTT"rrTTTO-rrrrr.-rrrrTTO-rrrrr-rnrn
~.oo
1.50
0.50
100
000
I 'I
O.~O
N.
(a)
Simulation
~EMT
~o
(a)
0,15
+
kd
0.40
0.30
G8I38EJ Modified SA Th.
_ _ Simulotion
0.10
G8I38EJ Modified SA Th.
~EMT
~Simulotiorl
GeeeO Eqv. Annulus Th.
~EMT
kaNmo. = 0016
ko+ka
=
0.04
.:;
GeeeO Eqv. Annulus Th.
0.20
kaNmo, = 0.16
Ns = 0.1
005
010
0.05
(b)
0.10
0.15
0.20
N.
o 00
-h-rrrTTTTr1;-rrrrrTTl...-rrTTTTTTr1rrrrrrTTTTTl
000
(b)
FIG. 3. Variation of (a) the dimensionless reaction rate (R D ) and (b)
the dimensionless bulk concentration of the diffusing species (N B), with
the number density of reactive sites (N,) for kaNmax=0.016, and (k.
+kd) =0.04 (N.=0.4).
give good estimates of N B for all the site concentrations
considered (Fig. 1). However, at small adsorption rates
(kfimax=O.04), the results for N B for both show large
deviations from the simulation results. The deviations are
because of the very long range influence of each reactive
site in this limit [see Eq. (56)], which results in very strong
competition effects.
In Fig. 4, we have plotted R D and N B results for a fixed
adsorption rate and site number density (kfi max = 0.16,
Ns=O.l) to examine the effect of varying (ka+kd)' The
results of EMT and equivalent annulus theory are again in
good agreement with the simulations at large (ka+kd)'
while considerable deviation in N B is seen at small values
of (ka+kd)'
In Fig. 5, we plot the theoretical and simulation results
for the variation of local concentration with S about a
reactive site for the indicated values of the parameters. The
original SA theory would obviously deviate considerably at
large S and is not included in the figure. The modified SA
theory predicts a larger zone of influence, indicated by a
0.50
1.00
ko
+
1.50
2.00
kd
FIG. 4. Variation of (a) the dimensionless reaction rate (R D ) and (b)
the dimensionless bulk concentration of the diffusing species (N B), with
(k.+kd) for kaNmax=0.16, and N,=O.l.
leveling out of the concentration profiles only at large S, as
compared to the simulation results. This is because the
influence of the neighboring reactive sites is ignored in the
modified SA theory, which is essentially based on Smoluchowski's "isolated sink" approach. 13 The equivalent annulus theory considers the zone of influence to be of a size
comparable to average intersink separation. However, this
is not correct because the zone of influence, as obtained
from the simulation, exceeds this (Fig. 5). In this regard,
only the EMT gives a correct estimation of the influence of
the reaction at the site boundary. Similar results are obtained for higher reactive site concentrations, and the
agreement between the EMT predictions for (N) and the
simulation results is good in all cases where the N B values
for the two are close to each other.
Finally, in Fig. 6, we show the variation of the dimensionless effective diffusivity with reactive site concentration
for the same parameter values used in Figs. 1-3. The predictions based on Cukier's theory4 are also shown. While
both theories match at low concentrations, there is a significant difference between the two at moderate concentra-
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9244
D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
0.15
/
0.10
0.05
o.00
Equivalent Annulus Th.
Modified SA Th.
EMT
Simulation
kaN mo • = 0.16. ko+kd
N. = 0.1
0.4
-t-n-rriTTT"tTT"rrnrrrrTTTTTTrrnrTTTTTTTTTrrncrrrTTTTTl
0.0
2.0
4.0
6.0
8.0
10.0
FIG. 5. Average steady state concentration (N) of diffusing species at a
radial distance S from the reactive site center as computed from different
theories and simulation.
tions. In addition, the EMT predictions show a nearly linear, monotonic increase in fJD with Ns in contrast to
Cukier's theory, which shows a maximum at moderate
concentrations and at high rates of adsorption.
tional area covered by the reactive sites (cpzO.065), and
these effects are accentuated at low adsorption rates. The
bulk surface concentration of the adsorbed species is found
to decrease monotonically with an increase in reactive site
concentration. The rate of reaction, however, increases
with reactive site concentration due at first to the larger
total perimeter available for reaction, and then decreases
due to the lower area available for adsorption. There is
thus an optimal reactive site concentration which increases
with increasing adsorption rates for a fixed equilibrium
concentration (Ne ).
While the modified SA theory, obtained by making ad
hoc corrections to the single sink theory, gives the correct
trends for the bulk concentration and the reaction rate, the
equivalent annulus theory and the EMT also give good
quantitative predictions over a wide range of parameters
and at high reactive site concentrations. Only the EMT,
however, gives the correct behavior of the ensemble averaged surface concentration profile around a reactive site.
The results presented in this work seem to indicate that the
approximations involved in the approach of Muthukumar 14 and Cukier and Freed 18 may not be very severe, and
such an effective medium theory gives good predictions at
high reactive site concentrations even for 2D systems.
APPENDIX A: AREA FRACTION COVERED BY
OVERLAPPING REACTIVE SITES
V. CONCLUSIONS
In this work, we have analyzed, theoretically and computationally, surface diffusion controlled reactions on a
heterogeneous surface containing randomly distributed circular disk-like reactive sites. The reactants adsorb onto the
surface following Langmuir-Hinshelwood kinetics. The
theories and Brownian dynamics simulations show that
competition effects become significant even at a low frac-
We consider ns circular disks of unit radius randomly
distributed over an area AT' In the absence of overlap, the
area fraction covered by the disks is
(Al)
To obtain the area fraction covered when the disks are
allowed to overlap, we define the function
1 inside any trap,
H(s)= {
o outside traps.
1.00
(A2)
This function can be written in terms of Ha(S) defined as
1,
Ha(S) = {0,
0.80
IS-Sal";1,
IS-Sal> 1,
(A3)
where Sa is a vector to the perimeter of disk a. It is
straightforward to verify by considering regions with different orders of overlap that
0.60
Q
<0
0.40
0.20
X(l-~ L
6=1=a,{3,y
0.00
0.00
0.05
0.10
0.15
0.20
Ns
FIG. 6. Variation of the dimensionless increase in effective diffusivity
(8D) with the number density of reactive sites (N.) for the indicated
values of k.Nmax and N e=O.4. The dashed lines are the predictions of
Cukier's theory (Ref. 4) and the solid lines are the predictions of the
EMT.
H6
("·»)]}
(A4)
satisfies the definition given in Eq. (A2). Here ~{3=1=a denotes the sum over f3 excluding a. For example, if S is
within a disk (say a= l), but not in a region of overlap
between any two disks, then all the terms in the curly
brackets vanish for a=1, and H(s)=H 1(s)=1. (All the
terms in the first sum vanish if a=¥= 1.) Similarly, if S is in
the region of intersection between any two disks (suppose
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9245
D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
a = 1 and a = 2), then all the terms inside the square
brackets vanish for a and /3= 1 or 2 and we get
00
L
I {f.1S')Km{f.1S)eim (9'-9),
21T(1 +5D) m=-oo m
(B4)
H(~) =H\ (~) [I -tH2(~)] +H2(~) [I -!H\ (~)] = 1.
(A5)
The above arguments are easily extended for regions of
intersection of three and more disks.
The average area fraction covered by the disks is obtained by taking an average of H(~) over all possible configurations of disks as
(A6)
where 1m and Km are modified Bessel functions of order m
of the first and second kinds, respectively, and f.12= (A,2
+kf )/(1+5D). The inverse Fourier transform is defined
as
1
f{~)=2ii2
{21T
kdk
(A7)
where fa is the position of the center of disk a. On simplifying, we get
Jo
_
d()kf{k)e-
ik
'
s,
(B5)
where ()k is the angle made by the wave number vector (k)
with the reference axis.
2. Perimeter inverse
X(I-~ {3=!=a
L ~T fdt/JH,k"})'
Ka«(}a,(J~)
Assuming that the coordinate system is located at the
center of site a, we have
(B6)
_ 1rns ( _~ 1T(ns-l) { _~ 1T(ns-2)
r/J- AT I 2
AT
I 3 AT
X[I_~1T{~~3)
{OO
Jo
L'"
Ameim(9a-9~),
(B7)
m=-oo
( ... )]})
(AS)
= 1- (l-r/JoIns)ns.
(A9)
where ()a is the angle made by the unit vector from the site
center to its perimeter and Am=Im{f.1)Km{f.1)/[21T{1
+SD)] using Eq. (B4). By definition, KabJ;Sa
= S{()a - ()~), which reduces to
(AlO)
(BS)
For ns> 1, the above equation becomes
1 2 1 3
I
4
r/J~r/J0-2 r/Jo+ 3.2 r/Jo- 4.3.2 r/Jo+'"
(All)
= l-exp{ -r/Jo).
An identical relation was been given by Richards \0 for
overlapping spheres in 3D, but without a derivation.
Assuming
00
L
Ka«()a'()~) =
Bp exp[ip«()~-()a)]
(B9)
P=-OO
and using the identity ~;;;=_ooeim(8-8')=21TS{()_()'), we
finally obtain
APPENDIX B: EVALUATION OF THE FORMAL
EFFECTIVE MEDIUM THEORY RESULTS
Most of the results given below are similar to those
derived in Ref. 4. Hence only the important definitions and
final results are given, wherever possible.
(BlO)
3. Configurationally averaged scattering operator
1. Effective medium propogator
The Fourier transform of the Green's function corresponding to the operator .!f is given by
(BI)
where the Fourier transform is defined as
(B2)
and () is the angle made by tEe position vector
reference axis. Expanding W{k) as
(~)
with the
(B3)
and taking inverse Fourier transforms, we obtain the
Green's function for the operator .!f as
«T»
The scattering operator (T) is defined as
T(~,~') =
I
a=\
(BI I)
SaKaba
= a~l
f dS Jds~
a
S{S-Sa)
xKa{Sa'S~)S{s' -S~)'
(BI2)
where Sa is a position vector to the perimeter of site a and
the integrals are line integrals over the perimeter of site a.
For randomly distributed reactive sites, the scattering operator averaged over all configurations is
J. Chern. Phys., Vol. 99, No. 11, 1 December 1993
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D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
9246
(T)(g,g') =
a~l ~T f
dfa
f f dR~
(B19)
dR a
XI5(g-Ra-fa)Ka(Ra,R~)
XI5(g' -R~-fa),
where
(B20)
(BB)
fa is a vector to the center of site a,
and Ra= (ga
is a unit vector from the center of the site to its
perimeter. On simplification, we obtain
-fa)
4. Configurationally averaged source term
«q»
The configurationally averaged source term is defined
as
xl5(g' -R~),
(BI4)
where ()a is the angle made by Ra with the reference axis.
Taking Fourier transforms with respect to p= (g-g'), we
obtain
(B2I)
a=l
2
A
A
A
where ga=A Ne(KabaGHa)' For a coordinate system centered on the reactive site a, we get
(B22)
Xexp[ik· (Ra-R~) ]Ka«()a'()~)'
(BI5)
baGHa = 2rr[Ko(/-L)I 1(/-L)//-L( 1+I5D)],
Using Eq. (B9) and the identity
I
In(k)rexp[in(()a-()k)],
n=-oo
ga=A 2Ne(2rr) 2 [Ko(/-L)I 1 (/-L) BoI/-L( 1+I5D)].
(BI6)
where I n are Bessel functions of order n, of the first kind,
and ()k is the angle made by k with the reference axis, we
finally obtain
00
(i)(k)
= (2rr)2Ns I
(B23)
and from the definition of K a , we finally obtain
00
exp[ik cOS«()a-()k)] =
[see Appendix A for a definition of Ha and Eq. (B4) for
G(g,g')]. Using Eq. (B22), we get
B".J~(k).
(B24)
Averaging over all configurations of the reactive sites, we
get
ns
(q)=
I <8a)ga=N~a'
a=l
(B25)
(BI7)
m=-oo
The coefficients Bm are given in Eq. (BlO). From the
above equation, we obtain
APPENDIX C: EVALUATION OF (Aa H )
(BI8)
Using the definition of H(g), we get
-~ (Aa p¥=a
I Hp #P
I Hr{I-~ 8¥=P,
I r Hf>[I--4 E¥=P,
I r,f> Hk")]})'
I
(Cl)
Using the results of Appendix A for n s >1, we get
(C2)
On simplifying, we get
J. Chern. Phys., Vol. 99, No. 11, 1 December 1993
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D. V. Khakhar and U. S. Agarwal: Surface diffusion controlled reactions
9247
(Aam = (Aalla)cfJ/cfJo + (Aa) (cfJo-cfJf/2+cfJt/6) + (Aalla) ( -cfJoI2+cfJf/6-cfJt/24)
+~ (Aa L
fJ,i=a
H{3
L Hlfa[l-~
#a,{3
L
E=/=a,{3, y,/j
Hk")])
(C3)
Continuing this way, we obtain
(Aall) = (Aalla)cfJ/cfJo + (Aa) (cfJo-cfJf/2!
+cfJt/3!-cfJV4!+"') + (Aalla> (-cfJoI2!
+2cfJf/3!-3cfJt/4!+"')'
(C4)
Using Eq. (AlO)
(Aall> = (Aalla) (1-2cfJoI2!+3cfJf/3!-4cfJt/4!+"')
+ (Aa)cfJ.
(C5)
On simplifying, we obtain the final result
(Aall) = (Aalla) (l-cfJ) + (Aa)cfJ,
which is exact in the limit
(C6)
ns-+ 00.
K. Fichthorn, E. Gulari, and R. Z. Ziff, Chern. Eng. Sci. 44, 1403
(1989).
2F. Bagnoli, B. Sente, M. Dumont, and R. Oagonnier, J. Chern. Phys.
94, 777 (1991).
3D. L. Freeman and J. D. Doll, J. Chern. Phys. 78, 6002 (1983); 79,
2343 (1983).
I
4R. I. Cukier, J. Chern. Phys. 79, 2430 (1983).
sO. Y. Kuan, H. T. Davis, and R. Aris, Chern. Eng. Sci. 38, 719 (1983);
38, 1569 (1983).
6L. A. Girifalco and D. R. Behrendt, Phys. Rev. 124, 420 (1961).
7M. J. Stowell, Philos. Mag. 21, 125 (1970).
8K. R. Naqvi, Chern. Phys. Lett. 28, 280 (1974).
9C. S. Owen, J. Chern. Phys. 62, 3204 (1975).
lOp. M. Richards, J. Chern. Phys. 85, 3520 (1986).
II M. Fixrnan, J. Chern. Phys. 81, 3666 (1984).
12 0. C. Torney and H. M. McConnel, Proc. R. Soc. London, Ser. A 387,
147 (1983).
13M. V. Srnoluchowski, Z. Phys. Chern. 92, 129 (1917).
14M. Muthukurnar, J. Chern. Phys. 76, 2667 (1982).
ISM. Tokuyarna and R. I. Cukier, J. Chern. Phys. 76, 6202 (1982).
16C. W. J. Beenakker and J. Ross, J. Chern. Phys. 84, 3863 (1986).
17Previous studies (Refs. 15 and 16) have shown that such a form may
not be valid due to nonlocal effects. The approximation is found to
break down when the variation in the averaged concentration profile is
significant over the length scale of a site (Ref. 16). However, since we
do not consider macroscopic gradients, the approximation gives reasonable results.
18R. I. Cukier and K. F. Freed, J. Chern. Phys. 78, 2573 (1983).
19 A similar effect is described in Ref. 16 for the analysis presented by
Muthukurnar (Ref. 14).
20G. Zurnofen and A. Blurnen, Chern. Phys. Lett. 88, 63 (1982).
J. Chern. Phys., Vol. 99, No. 11, 1 December 1993
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