kinetic lattice gas model of two species in · systems of co on ru(001), co on ru(llo), and co on...
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KINETIC LATTICE GAS MODEL OF TWO SPECIES IN
ONE DIMENSION
BY Stefan M. Patchedjiev
SUBMITTED IN PAKI'LAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
AT
DALHOUSIE UNNERSITY HALIFAX, NOVA SCOTIA
SEPTEMBER 1997
@ Copyright by Stefan M. Patchedjiev, 1997
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Contents
List of Figure8
Abstract
Abbreviations and Symbols used
Aeknowledgements
viii
1 Introduction 1
2 One Dimensional Lattice Gas Model 5
. . . . . . . . . . . . . . . . . . . . . 2.1 Thermodynamics of Adsorption 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Timescales 7
3.3 Hamiltonian . . . . O . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Master equation for two species . . . . . . . . . . . . . . . . . . . . . 11
2.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Closure approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Atomic and Nondissociative Molecular Adsorption 20
3.1 Transfer Matrix Method . . . . O . . . . . . . . . . . . . . . . . . . 20
3.2 One Mobile and One Immobile Atomic Species . . . . . . . . . . . . . 22
3.3 Cornpetitive Adsorption of Two Species . . . . . . . . . . . . . . . . . 23
3.4 Sticking Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Temperature Programrned Desorption
5 Diffusion
6 Conclusion
A Higher Order Correlator Equations
Bibliography
List of Figures
. . . . . . . . 1.1 The surface potentid of physisorption. Arbitrary units.
. . . . . . . 2.1 Open linear and circular chains as one dimensional lat tice
3.1 Chemical activity of the a species as a function of the two equilibrium
wverages a, b. T = 200 K Attractive interactions Vas = -600 K Vmb =
-2OOK& = -400K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Chemical potential of the b as a function of the two equilibrium cov-
erages a, b. T = 200K Attractive interactions V.. = -600KVab =
-200K VM = -400K. . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Time evolution curves for one mobile and one immobile species. The
immobile b has coverages .l, .2, -3, .4 Repulsive interactions V., = V,b =
Vas = 1000K. No diffusion. . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Time evolution curves for a for correlatoa < 66 >= -01, -09, .3. Repul-
sive V,, = Vas = Vu = 1000K. No diffusion. . . . . . . . . . . . . . . 27
3.5 Time evohtion curves for a and correlators. Repulsive Va) = 1000K,
attractive Vaa = VM = -1000K. NO diffusion. . . . . . . . . . . . . . 28
3.6 Time evolution curves for a and correlators. Repulsive Kb = 4000K,
attractive Va, = Vu = -4000 K. No diffusion. . . . . . . . . . . . . . 28
3.7 Time evolution m e s for a. and correlators. Attractive Va( = - 1000K, repulsive Va, = Vu = 1000K. No diffusion. . . . . . . . . . . . . . . 29
3.8 Time evolution curves for a,b and melators. Attractive Va, = -lOOOK,
repdsive VM = Vab = 1000K. Binding energys Vo, = 1200 KV* =
1300K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Time evolution c w e s for a,b and correlators. Attractive Vau = -1000K
Vu = -100 K, repulsive Vau = 10ûûK. Different partial pressures. . . 3.10 Time evolution curves for the partial mverages. Repdsive nearest
neighbor interactions Va= = Ka = Vu = lûû0K. Sticking coefncient
SOb = .8,.6,.4,.2,.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
a) Plot of the coverages and b) TPD spectra for two species, Kb =
OKV,. = 100Wss = 1000 K. No diffusion. . . . . . . . . . . . . . . . a) Plot of the coverages and b) TPD spectra for bb = (1.3,1.35,1.4,1.45) x
lOûûK, K b = OWaa = Vu = 1000K. NO diffusion. . . . . . . . . . . a) Plot of the coverages and b) TPD spectra for one of the two species
Ka = Kb = VW = (1.5,1,.5,0,-.5,-1,-1.5) x 1000K. No diffusion.
a) Plot of the coverages and b) TPD spectra for one of the two species
Ka = -lOOOKxb = OKVM = -1200K. No diffusion. . . . . . . . . . a) Plot of the coverages and b) TPD spectra for one of the two species
V, = -lOOOKVas = 1000KVM = -12ûOK. No diffusion. . . . . . . . a) TPD spectra for the correlaton for Ka = Va = lOOK va& =
-1000K b) TPD spectra for the melators for Ka = VM = 10OK
Kb = -1300K No difision . . . . . . . . . . . . . . . . . . . . . . . a) Plot of the coverages and b) TPD spectra for one of the two species
. . . . . . V,, = -lOOOKV,b = - l O O O U M = -1200K. No diffusion.
a) TPD spectra for the correlators for Ka = Kb = Vu = lOOOK
JOb = JOo = O b) TPD spectra for the correlatoa for Va. = Kb = VM =
IOOOK J~ = J~ = 10-~, IO-(, I O - ~ S - ~ . . . . . . . . . . . . . . . . .
vii
Abstract
A kinetic lattice gas mode1 of adsorption for two species is developed. The evolution
equations for up to three site correlators are derived and solved for the equilibrium
coverages. The time evolution for isosteric and isothermal process and temperature
programed desorption spectra are calculateci for different latteral interactions. Terms
describing surface diffusion are added to mode1 noneguiiibnum effects.
Abbreviations and Symbols used
ni Occupation number of site i.
n Vector occupation number
n, Occupation site number of a: adparticles.
n;b Occupation site number of b adparticles.
E, Energy of a single particle adsorbed on the surface
V,, Vd Binding energies of a and b adparticles on the surface.
Va., VM , Vas Nearest neighbor interaction energies
kB Boltzmann anstant
ti Planck's constant
T Temperature
t Time
va, y Vibrational frequencies of a and 6 adparticles
q,, Vibrational partition function
a, Area per site
Ath T h e d wavelength
&nta Intemal partition function of adparticles of type a
0 Total coverage on surface
8., es Partid coverages on surface of a and L adparticles
P., Pb Partial pressures gases a and b
Z Grand partition function
Cr,, Y Pgb Chernical potentials of o and b gases
ch, c l a a , c l a b ? CM, C2aoai C2b66, C2ba6, C2.u bteraction coefficients
N, Total number of sites on the surface
< a >, < b >, < au >, < ab >, < bb > Correlation functions
Sh, SM Sticking coeflicients
JO Hopping rate
Q Activation energy
Acknowledgements
1 would Iike to thank Professor H.J.Kreuza for providing me with this project. I
appreciate his guidance during the course of work, the excellent work conditions he created for me, and his support in every way, in and out of work.
I a m also happy to have worked with al1 the members of our research group.
Dr. S. H. Payne, R. Pawlitzek, were always ready to help m e when 1 asked them.
Chapter 1
Introduction
As physical devices become srnaller and srnaller, the need for understanding the nature
of the interactions of solid surfaces with other solids, fluids and gases becomes more
and more important. For instance knowledge of adsorption and desorption rates are
necessary if the chernical vapor deposition technique is used to grow well-defined films
and s t ~ c t u r e s of metals or semiconductors, [l].
Two types of adsorption can be distinguished. One speaks of physical adsorption
or physisorption when the electronic configuration of the adsorbing particle remains
more or less intact and produces an induced dipole that interacts with its image
in the metal. The binding energy then is typically less than 0.5 eV. If substantial
rearrangement of electronic orbitals of the adparticle occurs, Le. if chernical bonds are
established between the adsorbeci particle and the surface one speaks of chemisorption.
Because of the bond character of chemisorption it is usually restricted to less than
a monolayer (except for metal films), with adsorption energy typicdly a few eV. In t heory, the extremes of p hysisorpt ion and chemisorption are easily dehed, but in
real systems the tmth is somewhere in between, (21.
When a neutral particle approaches a solid, quantum-mechanidy it can be ex-
cited and the resulting rapidly ftuctuating dipole and higher multipole moments inter-
act with their image in the metal. Asymptotically far from the surface this interaction
leads to attraction between the individual gas atom and the solid. It can be modeled
by an attractive van der Waal+type potential:
V ( Z ) = -C/z3 (1.1)
Closer to the surface the electron overlap of the adparticle orbitals and those of
the solid produces strong tepulsion. It forces the partidea badt into the gaa phase
and h a the value of
V ( z ) = B/za (1-2)
The sum of Eq. (1.1) and (1.2) is just the remit of the Le~ard-Jones potential
for physisorption summed over a l l tw*body forces betwecn the adparticle and the
al l the atoms from the surfice of the soiid, [4]. The resulting potential is shown on
Fig. 1.1. For chemisorption both the attractive and repulsive contributions are of
01s:anct from suriaet
Figure 1.1: The surface potential of physisorption. Arbitrary units.
This potential function varies from v i t i o n to position on the d a c e in depth
and also in shape, producing a Nf- corrugation. The potential minimum to the vacuum level is defined as the binding energy (-P&, with positive).
For a pafect crystal (Le. one without defects) this pokntiai minimum will change
periodicaUy dong any direction on the crystal, going betweai pe& and valleys. Thuo
the adsorbed particle may not be free to move l a t e d y on the sudace. To do so it
has to jump over potential barriers that separate the potentid minima of the surface
potential. If the temperature is sficiently low the adparticle can only vibrate around
its equilibrium position, at the bottom of the potential well at a particular site of the
lattice structure. Increasing the temperature makes the hops from one potential
minimum to another more probable. In the extreme case, that is when the potential
bazrier pardel to the surface is much l e s than keT, the comgation is negligible and
the surface becornes flat. Then the adsorbate c m be treated as a two-dimensional
ideal gas.
The kinetic theories that describe the surface processes of adsorption, desorption
and diffusion c m be grouped into three categories:
(i) Theories that desaibe the system at a macroscopic level by a set of equations
for the macroscopic variables, in particular rate equations for the partial coverages.
In this approach the framework of nonequilibrium thermodynamics is used.
(ii) If it cannot be guaranted that the adsorbate remains in local equilibrium
throughout the adsorption-desorption process then a set of macroscupic variables is
not sdlicient and an approach based on nonequilibrium statistical mechaniw, involv-
ing t ime-dependent distribution h c t ions must be invoked. The kinetic lat tice gas
model, considered in this thesis, is an example of such theory. It was originally set
up in close analogy to the kinetic model for magnetic systems [5] and is based on a
phenomenological Hamiltonian and on postulated transition probabilities. The ob- servables are derived from a master equation that treats adsorption, desorption and
d a c e diffusion as Markovian processes. A hierarchy of equations of motion is de-
rived for the n-site correlation functions. This system is truncated to yield finite sets
of coupled equations and solved using different approximations. When the correlation
functions are considered site independent a few mrrelation functions are found to be
sufncient to describe the equilibrium and non-equilibrium properties of the adsorbate.
(iii) The proper theory of the time evolution of adsorption and desorption must
start from a microscopie Hamiltonian of the coupled gas-solid system. The transition
probabilities must be calculateci expücitly involving ody microscopie parameters.
Up to the present day the quasi-equilibriurn theory has been appiied successfuily to
systems of CO on Ru(001), CO on Ru(llO), and CO on Ru(ll1). The kinetic lattice
model of one species on a two dimension4 surface is discussed at length in [3]. It
gives essentidy exact results both for the quasi-equilibrium and nonequilibrium time
evolutions. In order to extend that model to include two different interacting species it
is useful to know the behavior of the system in one dimension. The one dimensional
problem is exactly soluble. It allows the development of different approximation
schemes that later can be generalimd for twdimensional systems.
Chapter 2
One Dimensional Lattice Gas
2.1 Thermodynarnics of Adsorption
In ordinary threedimensional thermodynamics, in the energy representation, the in-
temal energy U is expressed as a function of the entropy S, the volume V and the
number of particles Ni for each of the the species i present in the system. The fundamental t hermodynamic relation is writ ten as
dU = TdS - P U + C pidN;
or, after integrating when holding all intensive variables T, P, pi constant
U = T S - P V + & N ~ (2.2)
Consider now an adsorbatesubstrate system which contains Nmb particles of the
substrate, Na& adparticles of type a and Na& of type b in the adsorbate, both in
equiiibrium with the gas phase. The fundamental themodynamic relation (2.2) must
then be extended to read,
On the other hand for the pure substrate substance it has the form
duo = TdSo - P<No + h d N d
Subtract ing (2.4) from (2.3), and defining excess quanti t ies
We get for the differential energy for the two species adsorbate,
In the special case when the substrate molecules are inert, U, becornes just the
energy of Nado and Nad adsorbed molecules in the potential field of the inert adsor-
bent and the energy of the adsorbent subtracts out except for the interaction energy
between the adsorbent and the the adsorbate. As for S. aad K'., they are additive
extensive quantities describing the adsorbate. Nd is proportional to the surface area,
so that @ is proportional to the spreading pressure.
One usually takes lVd to be the total number of lattice sites N, in the surface
and Eq(2.6) becornes
Equation (2.7) has exactly the same fom as that for the three dimensional ther-
modynamics except for the extra term -OdN,, [2].
If the gas ph- is in equilibrium with the adsorbed gas particles then the chemical
potentials must be equal
Pa = Ppa, P b = P g b
Where ~1,. and pgb are the chemical potentials of the gases*
2.2 Time scales
The equilibrium thermodynamic properties of a large system are controlled by the
minimum of its free energy, whereas the kinetics involve the dynamics of energy
transfer. The relevant tirne scales for adsorption, desorption and diffusion govem the
tirne evolution of a gas-solid system. When a gas particle approaches the surface of
a solid it either bounces back elastically or, if it gets rid of enough energy within the
attractive region of the surface potential, it is trapped. Kowever, even if it descends
al l the way to the bottom of the surface potentid well, it will eventually evaporate
again; thus absolute trapping does not exist, there dways exists a possibility for the
particle to evaporate. For times to required for a particle to traverse the attractive
potential well, the particle will remain close to the top of the well within an energy
of kBT. In this time there is a fair chance that the particle acquires enough energy
from the heat bath of the solid to escape again. If this escape, which can be identified
with inelastic scattering, has not happened within a few round trips, the particle
will begin its descent to the bottom of the potential well. Li a quantum picture
this descent corresponds tu a cascade of transitions between the bound states of the
surface potential, each downward transition accompanied by emission of ~honons into
the solid and each upward transition with the absorption of phonons. This adsorption
process, characterized by a time scale ta is more likely at low temperatmes. After it
has happened, the particle will try again to dimb out of the potential well through a
sequence of phonon absorption and emission processes. It will eventually succeed in
doing so after a desorption time td. If ta is much shorter than t d , then the adsorption
and desorption are statistically independent, and the processes of sticking, energy
accommodation (i.e. thermalization) and desorption can be separated in different
terms. This is most likely the case if the thermal energy kBT is much less than the
depth of the surface potentid well.
2.3 Hamiltonian
To set up the kinetic lattice gas model for one dimension, one assumes that the surface
of the solid can be divided into one-dimensional ceils labeled i , in which the adparticles
can be adsorbai, [6]. W e assume here that all adsorption sites are quivalent and only
one particle can be adsorbed per site. The totai number of adsorption sites is N.. 2
Figure 2.1: Open linear and circular chahs as one dimensional lattice
Each site can be either occupied or empty; a microsapic occupation number for
each of the two species is introduced nh=O or 1, where the label m denotes the type
of particle adsorbed. In the following derivat ions na, c m be either species a or species
b. The condition n,.na = O specifies that two particles can not be adsorbed on the
same site at the same time.
W e define a vector of occupation nurnbers n as,
The model Hamiltonian describing the lateral interactions between the particles
on the surface then reads:
where the summations in the fist two terms are over all lattice sites, and < i , j > in the last three sums implies summation over the nearest neighbor pairs of sites. E.
is the Helmholtz fiee energy of an isolated particle on the surface, i.e.
Vh is the positive binding energy of the a-type adparticles. Moreover <h, is the
single particle partition function accounting for the vibrations of the adsorbed particle
in the surface potential and the qht, is the intemal partition function of the a-type
adsorbed particle. When the vibrational modes are considered independent we can
factorize
where
qza = ezp(kuztx / 2 k ~ T) (2.13)
e z p ( f i ~ z a / b T ) - 1 is the vibrational partition function of a harmonic oscillator accounting for motion
perpendicular to the surface. Likewise, q,, is the partition function for the motion
pardel to the surface, which for a l odzed adsorbate is usuaily taken to be the
product of terms like (2.13) but with different fiequencies v, and v,. If the adsorbate
is not localized, i.e, if the comigation of the surface potential parallel to the surface
is negligible, the one-dimensional ideal gas formula is used
where a, is the adsorption site area and Ath is the thermal wavelength if the
adpart ide.
The vibrational frequencies of atoms on metallic surfaces are close to the vibra-
tional frequencies of the lat tice of the solid and of order 1012 - 1013s-1. For instance
[7] gives for Ag/Mo(llO) system u, = u, = 3.4 x 1012s-'.
A h , Ka in (2.10) is the lateral interaction between two a particles adsorbed on
nearest neighbor sites. Vbb and Vh are the bb and a b interaction energies.
The coupling of the adsorbed particles to the gas phase is achieved via the chernical
potential. Using (2.8) one gets,
Where Pa is the partial pressure of the o gas and Zint, is the intemal partition
function of the free gas particle accounting for its vibrational and rational degrees of
freedom.
here the vibrational part is givea by terms of the type (2.13) for each vibrational
degree of freedom with a frequency v , b = kBTjGb/h in terms of the vibrational
temperature of the free molecule. For the rotational part the high temperature a p
proximation is taken T/aTfiOt where a is 1 for heteronudear diatomic molecules and
2 for homonucleat molecules. When the adparticle is adsorbed on the sudace it not
only descends into the surface potential well, but also changes the character of its
motion, e.g. the rotation is changed to a hindered rotation. That is why for the
adsorbed molecule qint is chosen as a product of factors like (2.16), where Zivb is of
hannonic type (2.13) and the zot is the rotation hindered oscillator given by
(2. l?)
From the canonical ensemble partition function it is relatively easy, (21, to derive
the formulae for the equiübrium partial coverages of two noninteracting immobile
gases on the surface,
2.4 Master equation for two species
In the following der i~ t ions of the equations of motion d the relevant processes like
adsorption, desorption and diffusion are assumed Markovian, Le. they do not depend
on the past history of the system. There is no hysteresis. Then a function P(n., nb; t)
can be introduced, which gives the probability that a given microscopie configuration
n is realized at time t. The time evolution of the system is controlled by master
equat ion,
t , = C W(n,, na; ni, ni) Po(n:, n$) - W(n:, n:; na, ns)Po(na, na) (2-20) lit n'
where W(& nt; na, ns) is the transition probability that a microstate n changes
into n' per unit time. The master equation simply states that the rate of change of
the population of a certain microstate is the sum of the probabilities of all the other
microsates t r a n d o d n g to that level minus the probability for leaving this microstate.
In equilibrium, i.e. when the RHS of (2.20) is O, this equation yields the conditions
of detailed balance,
is the equilibrium probability for the state n to be realized. 2 is the grand partition
function of the system and pa and pb are the chernical potentials of the species a and
b In principle the transition probability W(nong; na, nb) must be calculated from
the Hamiltonian that includes in addition to (2.10) coupling terms to the gas phase
and the solid that mediate mass and energy exchange. Here an approach initiated by
Glauber, [8] is used in which one sets up a kinetic lattice gas mode1 and chooses an
appropriate form for W(n2n;; na, ns), subject to detialed balance.
If the residence time in a given state is much longer than the time needed for
a transition to another state, as discussed in section 2.2, the transition probability
can be written as a s u m of independent transitional probabilities for adsorption,
desorption and diffusion on different adsorption sites.
Various choices for the adsorption term have b e n proposed, [9]. In the case of
"Langmuir kinetics" adsorption at site n; is impossible when the site is not empty
but otherwise it is independent of the local environment of that site. The adsorption
t em then reads
The s u m nins over a l l ceus i . The Kronecker deltas specify that the adsorption
event can occur ody on the desired place that wiU lead to microstate or.
The desorption term (with only nearest neighbor interactions) is
i.e. desorption from a site c m occur only if that site is already occupied by a
particle, with the condition that dl remaining particles don't move from their sites.
Substituting (2.24) and (2.25) into the detailed balance equations (2.21) gives the
values of the interaction terms, namely
and
where, for instance, CZkb accounts for the interaction of an adparticle of type a
with two nearest neighbors of type b. Not surprisingly the terms C2,k and C2hb are quai to C:ab due to the fact that, when the particle in the middle desorbs, two
bonds of ab types are broken. The exponential dependence implies that for strong
attractive interactions Ci = -1 and C2 =: 1, whereas for repulsive interactions the
large exponential term survives.
The diffusion term in (2.23) is written as,
A particle can jump only to a neighboring site if it is unoccupied. The summations
are over al1 adsorption sites.
Average occupation numbers of a site can be defined to make the connection with
the macroscopic observables
- nia(t) = x d ' ( n a , nb; t ) (2.36)
P - nib(t) = xnibf'(na,nb;t) (2.37) n
and the sum r u s over d microscopie configurations n vvith each n, = O and 1 ,nib =
O and 1.
The observable partial coverages of adsorbed species a and b are then given by
Their time evolution is obtained by substituting (2.38) into (2.20)
Exchanging n and n' in the fist term of the sum, we get
In terms of partial coverages the equation reads
de, 1 -=- C ( & - nia)w(no, ni; na, nb)P(na, nb, t ) dt Km,&,
2.5 Equations of Motion
For a homogeneous substrate multi-site correlations functions are not site specific and
one can define averages as,
where, for instance,
Not all of the above correlators are independent, e-g.,
Those correlators are also sub ject to a hierarchy of equations of motion. Because
of the nearest neighbor interactions the equation for the n-site correlator involves
tems up to (n + 2)-site correlators. For the fist five correlation functions these
equations are (for adsorption and desorption, i.e omitting the difision terms),
Equations for the t hree-site correlators are given in Appendix A. Comparing qua-
tion (2.54) with the phenomenological rate equations [IO] gives the d u e s for Wh and
wbb,
Wh and Wo6 are just the fluxes of ga9 particles, per unit cell, hitting the surface.
a, is the area of one adsorption site and So. is the stickïng coefficient of the a-species
at Mnishing coverage, defined as the ratio of the rate of adsorption on the surface
to the rate of collision of particles from the gag phase at the surface. From equation
(2.54) one can see that the sticking coefficient as a function of coverage, 9 = 8. +es, is So(B) = Soa(l - B ) , i.e. sticking is limited by site exclusion only ("Langmuir kinetics" ).
2.6 Closure approximations
In order to solve the system of equations of motion one must truncate the hierarchy.
The simplest scheme is the Kirkwood closure approximation in which all the higher
correlators are expressed as a product of twebody correlation functions. In the case
of two species the Kirkwood approximation is done in the following way
<au > < a b > < aab > = < a >
(2.6 1)
< a M > = < a b > < b o > < o b >
etc. < b > ( I - < a > - < b ) '
(2.62)
Thus only onesite correlators appear in the denominator. The five truncated
equations of motion axe (adsorption and desoption only)
In one dimension the above system of equations gives the exact equilibrium solu-
tion, but away from equilibrium it is oniy an approximation (in the twdimensional
case the twesite closure is equivalent to the quasi-chernical approximation to the
equilibrium corselators).
Solutions for the equilibrium values of the two-site correlators can easily be ob-
tained, as a function of the partial equilibrium coverages of o and b,
IR the next higher approximation one keeps all three-site correlators and uses
two site overlap. This results in fifteen equations given in Appendix A. All higher
correlation functions that appear on the right side have to be factorized in terms of
three-site correlaton in the numerator and twesite correlators in the denominator,
for instance,
< aaa >< uaa > < aaoa > = < au >
In one dimension the above factorization and its generalization is cailed maximum
overlap factorization, [Il]. It is unique for each correlator. Any other factorization
using less than (n - 1) correlators to factorize n-th correlator functions is not unique,
for example for mrrelators containhg an empty site one wodd get
Either choice is exact in equilibrium. However, away from it, equations (2.54)-
(2.58), or the ones in Appendix A, yield diRerent time evolutions. The best results
are obtained wit h maximum overlap factorization.
Chapter 3
Atornic and Nondissociative
Molecular Adsorption
3.1 Tkansfer Matrix Method
Calculation of the equilibriurn coverages on the surface for an isothermal and isosteric
process can be done by equating the kf t side of the time evolution equations (2.54)-
(2.58) and the ones for the three-site correlators fiom Appendix A to zero. The
chemicai potentials of the two gas phases are then given as a function of the two
equilibrium coverages. In Fig. (3.1) and (3.2), the chernical potentials for species
attracting each other on the surface are plotted. These results were compared and
found to be exactly the same as the results obtained using the traosfer matrix method.
In the t r a d e r matrix method the same Hamiltonian (2.10) is used to describe
the system [12]. The grand partition function is written as
The occupancy of each site is given by one of the vectors
and then the grand partition function takes the form,
over all states. Equation (3.4) has the form of a matrix product, where C is a 3 x 3
symmetric mat rix.
To evaluate (3.4) we first look at the case with N, = 3 and get,
In general, we have,
So that the grand partition function reads,
where Xi, Xz, Xa are the eigenvalua of the symmetric transfer matrix,
where the matrix Boltzmann factors are
Because the transfer matrix is positive and symmetric, Al > O and Xi > A2>X3.
For the limit N. + m causes the terms in the brackets in (3.7) to &sh. The gand
partition tunction is then given by the largest eigendue of the transfer matrix raised
to the power of N,. Differentiating the grand partit ion h c t ion with respect to the chernid potentials
gives the equilibrium coverages on the surface for the two species.
In one dimension this solution is exact and the same as the one obtained by solving
the master equation.
3.2 One Mobile and One Immobile Atomic Species
In the following numericd examples we have chosen the vibrational frequencies to be
10'~Hz, the adsorption "arean 3 kl, the sticking coefficients equal to one and the
masses of the gas particles equal to the mass of the Ni atom.
The easiest case to consider is when one of the species is fiozen on the surface,
i.e. nb(t) = const. When the partial coverage of that component is very s m d the
resulting equilibrium coverage of the other gas is just the equilibrium coverage for
one species in one dimension. It is given exactly by
wit h
A repulsive interaction between a and b particles will reduce, at a given tempera-
ture and Ob the coverage 8, as seen in Fig. 3.3 for equal repulsion between all species.
The precoverage varies from 0.1 to 0.4 and it leads to equilibrium coverages for a
from 0.33 to 0.201. The value for the < bb > currelator is b2 (random distribution).
The way the preadsorbed particles are distributed on the surface also affects the
final equilibrium coverage of the other species. If the repulsive preadsorbed partidea
are grouped together, more free space is left for adsorption. The same interaction
parameters as the ones in Fig. 3.3 lead to diflerent final coverage of a, depending
on the correlation function < bb >. Correlators much above and much less than the
random distribution on the s u d a c e may lead to as much as a 30% change of B., Fig.
3.4.
3.3 Cornpetitive Adsorption of Two Species
Depending on the difference of the interactions the final coverages of the a and the b
species on the suriace will vary. In all of the following figures at time zero there is a
clean surface and then both species start to adsorb. If the temperature is low enough
after some adsorption tirne all the surface will be covered with adparticles and the
total coverage 9 will be close to 1.
In the noninteracting case for 6. = Ob = 112, the equilibrium probability of hding
two adparticies next to one another wiU be the same as finding two b's, or ab, i.e. the
correlators are
< au >=< a >2=< b6 >=< B >2=< ab >=< a >< b >= 114.
Strong attraction between the a 's and between the b's, combinecl with strong
repulsion between the a's and the b's will lead to clustering of the adsorbed particles
of the same type. The probability of o b s e ~ n g long stnnga of a's or b's increases,
when the repulsion between a and b increases. Fig. 3.5 shows the time evolution
of the coverage of one of the gases and the lowest order mrrelators for interactions
Ka = Vu = -1000K and & = 1000K. In the extreme case, the probability for
hding the n-th correlator of a particles does not depend on the iength of the correlator
and is equal to a half, for B. = 4 = 112. Fig 3.6 shows this case. Strong binding
energies of the a and b, Vb = Va = 2000K leads to total coverage B = 1 and the
dues of all higher correlators with particles of the same kind approaches 0.5.
Strong repulsion between the a's and b's and strong attraction between the two
different species leads to ordered stmcture on the surface - every other site is taken
by the species of the same type. The < a6 > correlator is close to the equilibrium
coverage of the gases and the ptobability of h d i n g two particles of the same type
beside one another is close to zero, Fig. 3.7.
The clifference of the binding energies to the s&e Vk,Vos, is the domïnating
factor that determines the final cuverages. Higher binding energy leads to higher
equilibrium coverage at a given temperature. When the adsorption starts on a clean
surface both species start to adsorb at almost equal rates due to the equal impact
rates. Later, when the las bonded species starts to desorb the sites occupied by it
are taken over by the other species until the equilibrium is reached, Fig. 3.8. The
correlation function < aa > closely foliows the coverage m e s of o with the time
because of the strong attractive potential Va, = -1000K.
Fig.3.9 shows the time evolution for d i k e n t partial pressures of the gas phase
above the adsorbent. At the beginning almost immediately all of the adsorbing sites
are taken by the gas particles with the higher partial pressure. The system then
evolves to its equilibrium coverages, and due to the repulsive a-a interaction the
correlator < au > is almat 0. The strong a-b attraction leads to the same values of
the a and the correlator < ab >, meaning that every b particle has an a particle as a
neighbor .
3.4 St icking Coefficient
The sticking coefficient or sticking probability So is the ratio of the rate of adsorption
on the surface to the rate of collision of particles from the gas phase at the surface. The stidcing coefficient in generai depends on both the temperature and the coverage.
In our mode1 we have assumed simple site exclusion or the "Langmuirw kinetics.
The behavior of the system, when different sticking d c i e n t s for the two types
of adparticles are assumed, is plotted in Fig. 3.10. AU other parameters for the adsorbing species are chosen the same. The strongly stidùng gas paxticles dways
stick to the suffice when they hit it and at the beginning they have the bigger partial
coverage. The larger the difference in the sticking coefficients, the longer it takes for
the system to thermalize. However, the final equilibrium coverage is independent of
the difference in the sticking.
Figure 3.1: Chernical activity of the o species as a function of the two equilibrium coverages a, b. T = 200K Attractive interactions Va, = - 6 0 0 m b = -200 K Vu =
Y
Figure 3.2: Chernical potentid of the b as a fundion of the two equilibrium coverages a, b. T = 200K Attractive interactions V,, = - 6 0 0 ~ a b = -200 K VM = -400 K.
Figure 3.3: Time evolution curves for one mobile and one immobile species. The immobile b ha9 coverage, .i, -2, -3, .4 Repuleive interactions V, = Vd = Vu = 1000K. No diffusion.
T= m ~ V a F b m = l O M K Vaa:V&lMOQc a~== 01. -08. 3) a, 41b.09 M O &=O0
3, e . 3
tim [s] Figure 3.4: Time evolution m e s for a for correlators < lib >= .01, .09, -3. Repulsive
= VJ = Vu = 1ûûûK. No diffwion.
Figure 3.7: Time evolution m e s for o and correlators. Attractive V., = -1000K, repulsive V, = VM = 1000K. No diKusion.
tim [sl Figure 3.8: Time evolution curves for o,b and correlators. Attractive V.. = -1000K, repulsive VM = Kb = 1ûûûK. Binding energys VQ = 1200Wu = 1300K
Figure 3.9: Time evolution curves for a,b and the correlatonr. Attractive V, = -1000K VA = -100K, repulsive V.. = 10K. Different partiai pressures.
Figure 3-10: Time evolution cuves for the partial averages. Repulsive nearest neigh- bor interactions K,, = Kb = Vu = 10K- StiCking coefficient So, = .8, -6, .4, -2, .1
Chapter 4
Temperat ure Programmed
Desorption
The easiest way to study desorption processes is to measure the desorption rates as
the temperature of the substate is increased. In all of the following temperature
programmed desorption (TPD) spectra the heating of the surface is assumed linear
with the time with heating coefficient 1 Ks-l.
Fig. 4.1 shows the typical behavior of a system with repulsive interactions between
the a ' s and between the b's but no interactions between a and b. The initial two
and three-site correlators have the values for randomly distributed particles on the
surface, i.e. < au >=< o >2, < abb >=< a > . < b >' and so on. This assumption
is remonable because the adsorption at low temperature l d s to random sticking of
the adparticles on the surface.
On each of the desorption curves for the a's and the b's three peaks can be distin-
guished. They can be interpreted as "staged" desorption, with the low-temperature
peaks reflecting desorption of particles from an environment of two occupied neigh-
boring sites and the middle peak as desorption from the ends of chains of atoms. This
picture is confirmed by looking at the time evolution of the wrrelators. The < aaa > correlator has a maximum at the same temperature as the first desorption peak of
< a >, and the < au > correlator has a maximum at the second.
Higher binding energy shifts the TPD peaks to higher temperatures, Fig. 4.2. On
the same gaph four different desorption spectra are plotted, for four different binding
energies of the b species. As there is no interaction between the a's and the b's the
TPD spectra of a is unafkcted by changes in VOb. The threepeak structure of b is
more difficult to observe a9 the binding energy increases due to the relative decrease
of the mutual interaction VM to the VM. In Fig.4.3, the partial coverages and the desorption rates for interacting species
with different interactions are shown. The seven desorption spectra represent changes
of the nearest-neighbor interaction of the particles on the d a c e from repulsive to
attractive. A decrease of the repulsive potential shifts the low-temperature peak to
higher temperatures, and does not have any effect on the position of the single particle
desorption third peak. For noninteracting particles only this peak is obsefved, the
curve in the middle. Stronger attractive potentids shift the single desorption peak
to higher temperature.
The interactions between the different species have a sipnificant effect on their
desorption spectra Fig. 4.4 and 4.5 show the evolution of the coverages on the
surface and the TPD cuves for species that attract particles of the same kind and
repell the other species, with different K6. In the first case, Fig. 4.4, the a - b
interaction is zero. First to desorb then are the particles of type a with one or two
nearest neighbors b, having a desorption maximum that mincides with the maximum
for the correlator < au >. This suggests that the highest desorption energy for a is
when it is surrounded by a's. The same is true for the b partides.
The repulsion Kb = lOOOK leads to three well distinguishable peaks in the TPD of a, Fig. 4.5. The h t c m be explained by desorption fmm the sites bab, because
the correlator < ab > has a maximum at the same temperature. The second peak is
due to desorption of a 's that have b73 as a single nearest neighbor. The last unshifted
peak is the desorption peak of particles o with two neighbors of the same type. Much
more b's desorb at lower temperatures due to the repulsion of the a's.
When there is a repulsive interaction between the particles of the same species and
attractions between the ones of different kind even more interesting stmctures can
be obsewed. Fig. 4.6 a) shows the TPD spectra for initially randomly distributed
particles on the surface, with two different desorption energies. The repulsive p*
tentid for the a's and the b's have the same value, V.. = V& = 1000K, and the
attractive abpotentid is Vas = -1000K. As the temperature increases first to leave
the surface are particles of type a with two nearest-neighbors of the same type. Thia
gives the first macimum in the desorption rate for a. A maximum in the TPD c w e
of the < au > correlator shows that the next to desorb are a atoms with only one
nearest-neighbor of the same type. This process is slowed down by the more strongly
bound to the surface b particles, and the next peak of the desorption of a is shified to
higher temperature, representing desorption of a from the an environment < baab >. The big desorption peak of a without nearest-neighbor interactions is foliowed by
another peak, due to desorption of the last remaining on the surface a particles in
the structure < bab >. The three peak stmcture of the TPD spectra is also seen for
more strongly bound b particles, as the attractive nearest-neighbot interactions do
not change t heir desorption energy.
Fig. 4.6 b) shows the TPD spectra for desorption of a and b with stronger ab at-
traction, Ks = - 13OOK. The biggest difference is the shifting of the last temperature
peak of a to higher temperature, due to the stronger attraction. The single particle
peak of a is unaffectecl and its position does not depend on the a-b interaction.
When desorption starts with equilibrium coverages at low temperature the interac-
tion between the different types of particles have a significant eff't on the desorption
spectra. These ini t i d equili b n ~ m coverages and equili brium correlators were calcu-
lated using the time evolution equations, (2.54)-(2.58) and the higher order ones from
Appendix A. Fig. 4.7 shows the desorption from equilibrium. Because of the attrac-
tive a b interactions all the b particles have a as neighbors, and the initial < ua > correlator is close to zero. Then when o starts to desorb its desorption rate is almost
the same as the one for < ab >.
Figure 4.1: a) Plot of the coverages and b) TPD spectra for two species, Kb = OKV,. = 100KVb = 1000K. No diffusion.
-. rigure 4.2: a) Plot of the coverages and b) TPD spectra for Va = 1.3,1.35,1.4,1.45) x 1000K, V.) = O KV,, = VM = 1000 K. No diffusion.
Figure 4.3: a) Plot of the coverages and b) TPD spectra for one of the two species K. = K b = VW = (1.5,1,.5,0, -.5, -1, -1.5) x 10ûûK. No diKusion.
Figure 4.4: a) Plot of the coverages and b) TPD spectra for one of the two species I/.. = -1000KV.a = OKVu = -1200K. No difision.
Figure 4.5: a) Plot of the coverages and b) TPD specta for one of the two species V., = -lOOOKV.b = 1000KVM = -1200K. No diffusion.
Figure 4.6: a) TPD spectra for the wrrelators for V.. = VU = lOOK Kb = -1000K b) TPD spectra for the correlators for V.. = VM = lûûK b6 = -1300K No diffusion
Figure 4.7: a) Plot of the coverages and b) TPD specta for one of the two species K. = -lOOOKKb = -1000Kv& = -1200K. No diffusion.
Chapter 5
Diffusion
When taking into account the diffusion term on the surface from (2.35) one gets for
the first t hree correlators,
that have to be added to time evolution equations (2.54) -(2.58). The %site diffu-
sion t e rms are given in Appendix A. AU results for the desorption rate are obtained
for the three site correlators. For one species the three site approximation is almost
exact [II] and that justifies its use for two species. The difision constants Jo. and
Job can be chosen in the form of a thermdy act i~ted hopping process from one
adsorption site to a neighboring site. The rate constant of the process is then
where Q is the activation energy and uo the attempt
(5.4)
Frequency for a jump. In our
case as in [I l] the jumping coefficient is considerd constant and independent of the
temperature, i.e. we have Q = 0.
Diffusion on the surface will affect, the TPD spectra. Fig. 5.1 a) and b) shows the
TPD spectra for immobile and mobile particles with initial coverage 0 = 0.98. The a
and 6 species have the same interaction energies Va. = V.6 = Vu = lOOK but different
binding energies. The hopping rates, Jo. = JW = 10-3s11 were chosen larger than
the desorption rate constant, WObC., in the midrange of desorption temperatures.
Variation of JO between the immobile and mobile limit is approximately four orders
of magnitude.
The main feature on the TPD curves that changes as the hopping rate is increased
is the disappearing of the middle desorption peak. This desorption peak is due to
desorption from the ends of atorn chahs, Le. desorption from sites that have only one
occupied neighboring site. With hopping possible some atoms will jump away from
t heir repelliog nearest neighbors, thus decreaoing the probability of chain desorption
and increasing the single particle one. This effect explains the bigger third peak of
b in 5.1. Our mode1 takes into account only nearest neighbor interactions and does
not take into effect the final state, i. e. the environment of the empty site where the
particle will jump into. Thus the processes of the type < awa >+< aaw > has
the same probability as the process < crwa >+< oaou >. Consequently when the
coverages are close to a monolayer at the beginning of the desorption process even if
there are atoms without neighbors, for instance in the structure < a a w w >, they
will diffuse to < aaaooa >, giving the increase in the low temperature < aaa > peak
in Fig. 5.1.
Figure 5.1: a) TPD spectra for the correiators for V., = K6 = VM = lOOOK Job = Jo. = O b) TPD spectra for the correlators for V.. = Kc = Vu = IOOOK Joa = Jo. = 10-~, 104, I O - ~ S - ~
Chapter 6
Conclusion
In this work the kinetic lattice gas model for adsorption, desorption and difision on
surf'acea was extended to describe systems containhg two different species of adsorb-
ing particles. The kinetic lat tice gas mode1 developed in this t hesis describes surface
phenornena in a mast physical way making the connection between microscopie distri-
butions and macroscopic obsenrables without introducing too many ad hoc concepts
and parameters.
The assumption of statistical independence of ail processes on the d a c e d o w s
the separation of the transitional probabilities for adsorption, desorption and diffu-
sion. The Langmuir adsorption kinetics, in which the sticking coefficient decreases
linearly with the coverage was then studied.
In Chapter 2, time evolution equations describing two interacting gases on the
one dimensional surface were deriveci. The fidl set of up to three-site correlator
functions was given in Appendix A. These coupled equations were truncated using
the maximum overlap method and solved for the equilibrium coverages.
Chapter 3 summarizes the Transfer Matrix Method for two species. The partid
equilibrium coverages obtained using that method and the kinetic gas model ones
were found to be the same.
Time evolution and TPD curves were calculated for different surfiace interaction
potentials and hopping rates. The results were presented in Chapters 4 and 5.
The kinetic lattice gas mode1 for two species in one dimension cari be easily ex-
tended to explain the processes of adsorption, desorption and diffusion on a t w e
dimensional surface. Adding in the master equation terms that describe association
and dissociation processes will d o w the description of reactions on the surface. The lowest order time evolution equations have already been derived and the work for
solving them continues.
Appendix A
Higher Order Correlator Equations
In addition to the equations of motion (2.54)-(2.58) we have the fobwing for the
three site correlation functions.
d < uaa > dt
= Wo42 <mu > + < (100 >)
3 -~W&&'CJ~(~ < aaa > +Ci.,(2 < aaa > + < aaaa >)
1 +Clab < aaab > +C2aoo < aaaa > +Ckab < aaab > +-C2aao < m a >)
2 +2Jk(< aoaa > - (1+ Cl.,) < oaaa > +CI.. < aaoaa > +Clas < baoaa >)
d < aab > dt
= Wb(< mb > + < aob >) + Wa(< uao >)
-WhCQ(2 < aab > +Clau(< aaab > +2 < aa6 >) + Clab(< btzab > + < aab >)
d < aoa > dt
= W o o ( 2 ( < o o 4 > - < u o < 1 > ) - W ~ < a a >
+2&(< a m > +(1+ Cl..) < auo > +Cl., < aawa >
Bibliograp hy
[1] Finlay MacRitchie, Chemistq of Interfaces, Academic Press Co., Inc., San Diego,
California, 1990.
[2] A. Clark, The Theory of Adsorption and Catalysis, Academic Press Co., Inc.,
New York and London, 1970.
[3] A. Wierzbicki, H. J. Kreuzer S u d Science 1991, 257, 417
[4] H. J. Kreuzer, 2. W. Gortel, Physisorption Kinetics Springer-Verlag, Berlin Hei-
delberg, 1986
[5] K. Kawasaki, Phase Transitions and Critical Phenornena, Voi.2, Acadenic Press,
New York, 1972
[6] H. J. Kreuzer, Langmuir, 1992, 8 , No.3 (1992) 774-781
[7] K. J. Kreuzer, Surf. Science 231 (1990) 223-226
[8] R. J. Glauber, J. Math. Phys. 4 (1963) 294
[9] K. J. Kreuzer, Zhang Jun Appl. Phys. A 51 (1990) 183
[IO] H. J. Kreuzer,S. H. Payne Dynamics of Ga-Solid Collisions, Eds. C. T. Rentter
and M. N. Ashfold (Roy. Soc. of Chemistry, Cambridge, UKJ991)
[Il] S. H. Payne, A. Wierzbicki, H. J. Kreuzer Surf. Science 291 (1993) 242-260
[12] 2. Jun PhD Thesis, Dalhousie University, 1992
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