KINETIC LATTICE GAS MODEL OF TWO SPECIES IN

ONE DIMENSION

BY Stefan M. Patchedjiev

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MASTER OF SCIENCE

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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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