A theoretical study of the surface diffusion of - The Fritz Haber ...

A theoretical study of the surface diffusion of large molecules. I. n-alkane- 

type chains on W( 100) 

D. Cohen 

Department of Physics, Nuclear Research Center-Negev, Beer-Sheva, P.O. Box 9001, Israel 

and Department of Nuclear Engineering, Ben-Gurion University. Beer-Sheva, Israel 

Y. Zeiri 

Department of Physics, Nuclear Research Center-Negev, Beer-Sheva, P.O. Box 9001, Israel 

(Received 29 May 1991; accepted 10 March 1992) 

In the present study the surface diffusion of model n-alkane-type chains adsorbed on a W( 100) 

surface were simulated. The simulations were performed using molecular dynamics 

calculations where the thermal motion of the surface atoms was introduced via the generalized 

Langevin method. The potential function among the chain atoms used in these calculations 

described the nearest-neighbor interaction by a Morse potential while next-nearest-neighbors 

and next-next-nearest-neighbors interaction was described by a Lennard-Jones 12-6 and a 

repulsive exponential function, respectively. The length of the chains, N, considered were 

N = 3,6, 10, and 20. For each value of N the chain diffusion at three or four surface 

temperatures was examined. For all values of N it was found that the diffusion coefficient 

could be described by an Arrhenius expression. It was found, in good agreement with the 

experimental results, that the activation energy for the diffusional motion scales with the chain 

length while the preexponential factors were practically independent of N. In addition, various 

static (e.g., average mean square end-to-end distance and average mean square radius of 

gyration) and dynamic (e.g., autocorrelation functions) properties of the simulated systems 

were computed. The results of these simulations were used to obtain a qualitative 

understanding of the mechanism by which such chains diffuse on a solid surface. 

1. INTRODUCTION 

The surface diffusion of adsorbed species plays an im- 

portant role in many of the processes related to surface 

chemistry and physics. Surface diffusion constitutes a direct 

probe of the adsorbate-surface and adsorbate-adsorbate in- 

teraction potentials.’ In addition, the mobility of adsorbed 

particles on solid surfaces may be the rate limiting step in 

many technologically important processes. Thus, one would 

like to have a good understanding of the detailed mechanism 

of surface diffusion. 

It is well established that one may describe the surface 

diffusion process as a random walk of the adparticle between 

the various adsorption sites on the surface. On a single crys- 

tal surface one may assign, for a given adsorbate, adsorption 

sites whose arrangement and symmetry is related to that of 

the surface atoms. In such a situation the surface diffusion 

process will be described as random hops of the adparticle 

from one adsorption site to another. The rate of diffusion, for 

such situations, is related to the energy barrier between two 

neighboring sites and to the frequency of attempted jumps. It 

has to be realized that the energy barrier for surface migra- 

tion is determined by both the adsorbate-surface interaction 

potential as well as interaction among the adsorbed parti- 

cles. In most situations this activation energy associated 

with the adparticle’s motion along the surface is much larger 

than the thermal energy (i.e., kT, where k is the Boltzmann 

constant and T is the system temperature). This is true not 

only for strongly bonded adsorbates but also for many sys- 

tems in which the adparticles are physisorbed to the surface. 

A detailed description of the surface diffusion process, from 

both theoretical and experimental view points can be found 

in a number of review articles.* 

It has to be realized that the description of surface diffu- 

sion in terms of random walk-type process is limited to sys- 

tems in which the size of the adsorbate is small compared to 

the size of the surface unit cell. Only in such a situation can 

well defined adsorption sites and transition states be as- 

signed. Once the size of the adsorbed species is larger than 

the surface lattice constants, adsorption sites and transition 

states may not be defined uniquely. Moreover, the large 

number of degrees of freedom in such a situation may result 

in a large number of adsorption configurations, each one 

corresponds to a local minima in the potential energy surface 

describing the adsorbate-surface interaction. The activation 

energies between these local minima may be of the order of 

the thermal energy in the system, kT. As a result, the diffu- 

sional motion cannot be considered any more as random 

hops from one adsorption site to another. For such systems it 

is more realistic to describe the surface diffusion as a quasi- 

continuous motion along the surface. The nature of the sur- 

face diffusion dynamics in such systems is the subject of the 

present work. 

Recently, George and co-workers3 reported on an ex- 

perimental study of the surface diffusion of n-alkanes on a 

Ru (00 1) surface. The surface diffusion coefficients for pro- 

pane, n-butane, n-pentane, and n-hexane were measured us- 

ing the laser induced thermal desorption (LITD) tech- 

J. Chem. Phys. 97 (2), 15 July 1992 0021-9606/92/141531-11$06.00 @ 1992 American Institute of Physics 1531 

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1532 

nique.4 It was found that for all the above adsorbates the 

diffusion coefficient could be described by an Arrhenius be- 

havior. Moreover, the surface diffusion activation energies 

obtained increased linearly with carbon chain length, while 

the preexponential factors remained nearly constant. These 

measurements also indicated that the surface diffusion coef- 

ficients were coverage independent. In addition, the surface 

corrugation ratio (i.e., the ratio between the energy barriers 

for diffusion and desorption) was observed to have a con- 

stant value for all adsorbates considered.3 Similar studies of 

the surface diffusion of a number of other organic molecules 

have been reported by the same group.5 

In the present work we shall describe molecular dynam- 

ics simulations of the surface diffusion of chain like mole- 

cules. The length of the chains, N, in the calculations was 

varied in the range N = 3 up to N = 20. Since the interaction 

potentials among the chain atoms as well as between the 

chain and the surface are unknown for realistic surface-ad- 

sorbate systems, we employed model potential functions to 

describe the system. Thus, the emphasis in the present study 

is on the qualitative understanding of the surface diffusion 

mechanism of chain like adsorbates rather than a quantita- 

tive description of a realistic system. In order to simplify the 

calculations the chain was assumed to consist of carbonlike 

atoms while the hydrogen atoms were not included in the 

calculation. This approximation greatly reduces the amount 

of computer time needed for the simulations and is not very 

likely to alter significantly the qualitative behavior of the 

adsorbed chain on the surface. This approximation will be 

tested in a future study in which the hydrogen atoms will be 

included. In the calculations described below a single ad- 

sorbed chain was considered, namely, the low coverage limit 

was assumed. In the next section the theoretical model used 

in the calculations will be described, while Sec. III is devoted 

to the description and discussion of the results. In the last 

section a short summary will be given. 

II. THEORETICAL MODEL 

D. Cohen and Y. Zeiri: Surface diffusion of large molecules. I 

The aim of the present work was to study the mecha- 

nism of surface diffusion for n-alkane-type chains adsorbed 

on a W ( 100) surface at the low coverage limit where adsor- 

bate-adsorbate interactions may be ignored. The simula- 

tions were carried out using the molecular dynamics ap- 

proach where the generalized Langevin method (GLE) ,6*7 

was used to describe the thermal motion of the surface 

atoms. The W ( 100) surface was represented by a slab of 169 

atoms arranged as a single layer of 13 X 13 atoms. This slab 

constituted the “primary zone” with which the adsorbed 

chain was strongly interacting. The rest of the crystal was 

represented by 22 1 fictitious particles which surrounded the 

slab and which were attached to primary zone atoms whose 

bonds to atoms outside this zone were truncated. According 

to the standard GLE approach, the slab atoms motions were 

described by Newtonian equations of motion 

ff= ~~‘CdRS,R,) ~‘V,,UVLR,) 

JR* - dR2 

+ fds, 

(1) 

J. Chem. Phys., Vol. 97, No. 2,15 July 1992 

where R, S, and X are vectors containing the positions of the 

slab atoms, the fictitious particles, and the chain atoms, re- 

spectively. In Eq. ( 1) R, represents a vector which contains 

the positions of a number of solid atoms which were fixed at 

their lattice point. These atoms were added in order to obtain 

realistic interaction potentials for both slab and adsorbate 

atoms near the primary zone edges. These fixed solid atoms 

were added as two additional layers around the primary 

zone and below it. Notice that the coordinates of all the par- 

ticles in the system are mass weighted coordinates. The first 

two terms on the right-hand side of Eq. ( 1) represent the 

forces due to slab atom-slab atom and slab atom-adsorbate 

atom interactions, respectively, while the last term repre- 

sents the force due to the coupling of the slab atoms to the 

fictitious particles. Similarly, the motion of the chain atoms 

was described by 

g = _ ~f’,OGWo) WrxW 

ax - ax ’ 

while the motion of the fictitious particles is described by 

i$ = - a?$ +zifR, -ps, +J;(t). (3) 

Equation (3) describes the motion of the ith fictitious parti- 

cle which is coupled to the ith slab atom where the first term 

represents a self-force constant and the second term is a har- 

monic coupling to the slab atom. The last two terms in Eq. 

(3) correspond to a friction-type dissipation and a random 

force, respectively. These two terms are related by the sec- 

ond fluctuation-dissipation theorem 

(f(WVN) = 2kT’,&f), (4) 

where k is the Boltzmann constant, Tis the surface tempera- 

ture, p is the friction constant, and S represents a delta func- 

tion. All the coefficients in Bq. (3) may be obtained from the 

surface phonon spectra and if one uses the Debye model 

these constants are given in terms of the Debye frequency, 

w,, . A more detailed description of the GLE approach can 

be found in Refs. 6 and 7. It should be noted that similar (as 

well as less accurate) approximations to describe the solid 

surface were used in the study of various surface related 

processes. 7(a)*8 It is believed that this approximate treat- 

ment of the surface atoms motion does not influence the 

qualitative behavior of the adsorbate during the calculation 

of the trajectories. Moreover, a more accurate description of 

the surface motion (i.e., the use of more than a single layer 

for the primary zone) would result in a large increase of the 

time required for the computations which did not seem to be 

justified due to the approximate description of the adsorbate 

(discussed above) and the model potentials used to describe 

the system (see discussion below). 

In Eqs. ( 1) and (2) three different potential functions 

were defined namely, V,, , V,, and V,,. These potentials 

represent the interactions in and between the two subsys- 

tems, namely, the surface and the adsorbate. As discussed 

above, the interaction potential between n-alkane adsorbates 

and a metal surface are not available in the literature. More- 

over, the internal (e.g., chain atom-chain atom) potential of 

such adsorbates probably undergoes marked changes as 

compared to a gas phase molecule. This modification of the 

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(2)

internal chain potential function due to the presence of the 

solid surface is also unknown. Thus, the various potential 

functions were constructed using the available data on such 

systems and served as model potentials (see discussion be- 

low). Although such model potential functions are not ex- 

pected to yield quantitative information about a given sys- 

tem it is believed that they will help to obtain a reliable 

qualitative description of the surface diffusion of chainlike 

molecules. 

The potential among the surface atoms, V,, , was given 

as a sum of pairwise Lennard-Jones (L-J) ( 12-6) potentials 

where the sum contained nearest and next-nearest-neighbor 

interactions. The values of the two parameters for the L-J 

function, ERR = 24.63 kcal/mol and a,, = 2.56 A, were 

taken from the work of Halicioglu and Pound.’ This form of 

the potential function describing the interaction among the 

slab atoms has been used successfully in previous work.’ 

The potential describing the interaction among the 

chain atoms, V,,, was assumed to be given also by a sum of 

pairw&e interactions. The interaction potential between 

nearest-neighbor chain atoms was described by a Morse 

function where the parameters correspond to those of a C-C 

bond. The actual values used in the calculation were 

D, = 109.6 kcal/mol, w, = 1854.71 cm-‘, and R, = 1.526 

A. The interaction between next-nearest-neighbor atoms 

was described by an L-J ( 12-6) potential whose parameters 

were E = 1.0 kcal/mol and (T = 2.259 A. These values were 

chosen in such a way that the equilibrium distance between 

next-nearest-neighbors was 2.536 A and the corresponding 

angle between the two C-C bonds was 112.4 deg. The inter- 

action among third order neighbors in the chain was as- 

sumed to be described by a repulsive exponential function of 

the form A -exp( - a-X,), where X,, is the distance between 

next-next-nearest-neighbors iand jand the values of the two 

parameters were A = 100 kcal/mol and a = 2.0 A - ‘. This 

repulsive term was introduced in order to reduce the folding 

of the chain on itself. For more distant neighbors in the chain 

the interaction was assumed to be zero. Thus the potential 

V,, was calculated by 

v,Y,Y =I v.(xl -xi*l) + C vLJCxi -xX,*2) 

n’n “‘“‘” 

+ 2 Kx,(Xi-Xx,*3), (5) 

It.“‘“‘” 

where the three potential functions on the right-hand side of 

Eq. (5)) VM, Vu, and Vexp represent Morse, L-J, and the 

repulsive exponential potentials, respectively. The internal 

chain potential described above does not take into account 

the rotational energy barriers which occur in such n-alkane- 

type molecules when next-nearest-neighbor C-C bonds are 

rotated. The elimination of these rotational barriers served 

mainly to simplify V,, and reduce the computer time need- 

ed for the simulations. In addition, this approximation may 

be justified by the fact that the energetics involved in the 

rotational motion of one C-C bond relative to the other in 

the adsorbed phase is probably quite different than the corre- 

sponding value of an isolated gas phase molecule. Since the 

D. Cohen and Y. Zeiri: Surface diffusion of large molecules. I 1533 

J. Chem. Phys., Vol. 97, No. 2, 15 July 1992 

variation of these rotational barriers in the adsorbed phase is 

unknown we decided to ignore it in the present calculations. 

Finally, we did not find in the literature data related to 

the interaction of n-alkanes with a W( 100) surface except 

the work of Yates and Madey on the adsorption of methane 

on W( 100).lo In this study it was found that below 110 K 

methane adsorbs to the W surface molecularly with a bind- 

ing energy of 6.9 kcal/mol. However, no information about 

the adsorption site of molecular methane or its diffusion bar- 

rier were reported. In addition, data related to the binding 

energy of methane on other metal surfaces such as rho- 

dium’ ’ and nickel” exhibit lower values of binding energies 

(e.g., 5 and 2.9 kcal/mol, respectively). Inspection of the 

desorption energies, Edes, obtained for propane, butane, pen- 

tane, and hexane adsorbed on the Ru (00 1) surface3 shows 

that the values of Edes do not vary linearly with chain length. 

The lack of experimental information related to the details of 

the n-alkane/W( 100) interaction forced us to use a model 

potential function to describe this interaction. The form of 

V, chosen to be used in the present study was a sum of 

individual chain atom-surface interaction where each one of 

the chain atoms was assumed to interact with the nearest 

four W atoms via L-J potentials. Thus, the adsorbate-sur- 

face interaction potential was given by 

VRX= i i VuCXi --Rj), (6) 

i=l j=1 

where the index i runs over the chain atoms and the index j 

runs over the four nearest surface atoms for each ith chain 

atom. The values of the L-J parameters used in the calcula- 

tions were ERX = 0.722 kcal/mol and a,, = 2.67 16 A. For a 

single chain atom over the three symmetry sites of the 

W( 100) surface (when the surface atoms are fixed at their 

lattice points) the binding energies and equilibrium dis- 

tances as obtained using this potential were Eb = 0.91, 

1.735, and 2.89 kcal/mol and R, = 2.95, 2.5, and 2.0 A for 

the on-top, bridge and fourfold sites, respectively. 

Each trajectory started by placing the n-alkane-type 

chain along the surface where all chain atoms were posi- 

tioned 2 A above the surface. The distances between the 

chain atoms were chosen to correspond to the equilibrium 

distances, namely, 2.536 A. The initial velocities of the chain 

atoms as well as those of the slab atoms were chosen from a 

Boltzmann distribution at the surface temperature. Each 

trajectory started by an equilibration process, namely, the 

equations of motion were integrated for 3000-4000 integra- 

tion steps (i.e., 3-4 ps) . This equilibration process was ter- 

minated when the average kinetic energy of the particles in 

the system was < 4 deg off the required surface temperature 

(during a period of 1 ps). The positions and velocities after 

this initialization procedure were used as initial conditions 

for the trajectory. A trajectory was terminated if it exceeded 

40 000 integration steps (i.e., 40 ps) or if the adsorbate 

moved out of the slab boundaries. In some cases the ad- 

sorbed molecule desorbed from the surface, such trajectories 

were restarted. For each Nvalue and given surface tempera- 

ture ten trajectories were calculated. 


1534 D. Cohen and Y. Zeiri: Surface diffusion of large molecules. I 

III. RESULTS AND DISCUSSION 

The present calculations were devoted to the study of 

the surface diffusion of single n-alkane-type molecules ad- 

sorbed on a W ( 100) surface, namely, the low coverage limit 

was considered. The calculations were performed for four 

chain lengths, N = 3,6,10, and 20. For each chain 34 sur- 

face temperatures were considered in the range 300-1000 K. 

These relatively high temperatures were chosen in order to 

be able to obtain reliable diffusion coefficients during the 

limited time that could be simulated by molecular dynamics. 

For each case, given N value and surface temperature, we 

calculated ten trajectories. The results of these trajectories 

were used to obtain various averaged quantities and time 

correlation functions of the adsorbed chain. The main goal 

was to obtain both static and dynamic behavior of chain like 

molecules adsorbed on a solid surface. The remainder of this 

section will be subdivided into four parts which are devoted 

to the discussion of the diffusion coefficient, the static prop- 

erties of the adsorbate, chain relaxation properties, and to a 

qualitative description of the surface diffusion mechanism of 

adsorbed chains. 

A. Diffusion coefficient 

The diffusion coefficients of the various chains consid- 

ered were calculated according to the Einstein formula using 

the mean quadratic displacement of the chain’s center-of- 

massI 

D= t~,u/4)([x,,(~o -l) -&M(~o)]2)N,~f9 

where ( ) Nt represents an average over the time steps during 

the trajectory and X,, is the position vector of the center-of- 

mass of the chain. The value of the diffusion constant, D, was 

obtained as the average of the values obtained in the ten 

trajectories. The natural logarithms of the resulting values of 

D were plotted as a function of l/Tas shown in Fig. 1. It is 

clear from the linear relationship between In(D) and l/T 

that the diffusion coefficients can be described by an Arrhen- 

ius form 

-5.5 , I 

-13.5 

0 N=lO 

x N-20 

t 

-15.5 ’ I 

0 0.5 1 1.5 2 2.5 3 

1000/T [Kj 

FIG. 1. Arrhenius plot showing the temperature dependence of the surface 

diffusion coefficients for the four chainlike molecules examined. 

(7) 

D=D,exp -- , 


where Do and Edi are the preexponential factor and activa- 

tion energy for diffusion, respectively. The values of Do and 

Ediff for the four chains studied are shown in Table I. The 

two additional quantities shown in Table I are the activation 

energy for desorption, Ed=, and the surface corrugation ra- 

tio defined by fi = E,,/E,,. The magnitude of Edes was 

estimated by the following procedure: the adsorbatesurface 

interaction potential was monitored during the evolution of 

the trajectory. The value of this potential which corresponds 

to the largest binding energy of the adsorbate to the surface 

was designated as Eda. The activation energies for desorp- 

tion quoted in Table I correspond to the average of the Ed,.. 

values obtained in the 30-40 different trajectories calculated 

for any given chain length (these averages correspond to the 

results of ten trajectories at any given T considered). There 

are a few points which should be emphasized, first, in all 

cases Eq. (8) describes accurately the dependence of D on 

the surface temperature. However, the variation of D as a 

function of N cannot be described by a simple scaling de- 

pendence. Inspection of Table I shows that the values of both 

EdiR and Edes increase with increasing chain length, how- 

ever, the ratio between these two activation energies remains 

almost constant (see values of R in Table I). Moreover, the 

values of the preexponential factors for the different chains 

seem to be independent of the value of N. Inspection of the 

magnitude of the activation energies for diffusion indicate 

that this quantity is much more sensitive to the magnitude of 

N. To obtain a more quantitative relation between Edif (and 

Edes ) and Na log-log plot of these quantities is shown in Fig. 

2. The linear relationship between ln( activation energy) and 

ln( N) indicates that one may write a scaling law of the form 

EAc = CNv, (9) 

where EAc = EdiF or Ed=. The value of the scaling exponent 

obtained from the slopes of the best fits to the calculated 

results (the lines) shown in Fig. 2 are Y = 0.565 & 0.074 

(with intercept C = 1.014 f 0.11) and Y = 0.7 + 0.06 

(with intercept C = 3.353 + 0.28) for activation energies of 

diffusion and desorption, respectively. These results are in 

good qualitative agreement with the experimental data of 

Ref. 3. The experimental study was related to variation of 

chain length from N = 3 to 6. In this range of N values 

TABLE I. Preexponential coefficients and activation energies for diffusion 

as obtained from the Arrhenius plot (Fig. 1) together with the values of the 

activation energy for desorption and the corrugation ratio, R = EdlR/Ede.. 

Units of D,, in cm*/s and of the activation energies in kcal/mol. 

N DO E dll7 E @!n n 

3 (9.45 f 1.38) x lo-’ 1.161 f 0.2 6.93 + 0.11 0.255 

6 (6.69 f 2.25) x 10 - 4 2.955 f 0.5 12.05 f 0.09 0.245 

10 (2.39f0.116)~10-’ 4.077kO.06 18.24kO.14 0.224 

20 (1.49 f 0.24) x 10 - 3 5.111 *0.70 25.81 + 0.21 0.198 

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(8)

0 

0 0.5 1 1.5 2 2.5 3 

In(N) 

FIG. 2. A log-log plot of the surface diffusion activation energy and activa- 

tion energy for desorption as a function of the chain length. 

George and co-workers found that the magnitude of De was 

practically constant while the magnitude of Edie increased 

linearly with N. In addition, the experimental values ob- 

tained for R showed that the corrugation ratio remained 

constant for all n-alkanes studied. Inspection of the values 

for Do, Edif, and fi for the N values considered here (in 

Table I) shows that the magnitude of D,, is practically inde- 

pendent of the chain length. In addition, both activation en- 

ergies can be accurately fitted by a linear function of Nif only 

the first three values are considered (i.e., those of N = 3, 6, 

and IO). However, Eq. (9) results in a much better fit than a 

linear one if the activation energies for N = 20 are added. 

These results suggest that for relatively short adsorbates a 

linear relationship exists between the two activation energies 

and the chain length, once longer chains are considered de- 

viations of linearity are observed. A comparison of the R 

values for N = 3 and 6 (in Table I) shows that these two 

corrugation ratios are N independent as observed in the ex- 

perimental studies. The scaling laws obtained for the activa- 

tion energies for diffusion and desorption suggest that the 


surface corrugation ratio in the present calculations is relat- 

ed to Nby 0 = 0.303~N -o.136. 

B. Static properties of adsorbed chain 

In order to understand the static properties of the ad- 

sorbed chains on the solid surface we have calculated the 

mean quadratic distance between the end atoms in the chain 

(end-to-end distance) 

(R2) = ((Xl --XN12)N, (10) 

and the mean quadratic radius of gyration 

(l/N) 2 (X, - Xc, )I) , 

i= 1 N, 

(11) 

where Xi is the position vector of chain atom i and Xc, is 

given by 

X CM = (l/N) 5 X,. 

i= 1 

TABLE II. The values obtained for (R ‘),(S’), and the ratio between them together with the timeconstant for 

the initial decay of C( V,, , t) for the various N values and surface temperatures. 

N T(K) (R*) (A*) (S2) (A? (R ‘)/W Ty (ps) x 102 

3 300 6.67 f 0.13 1.26 + 0.015 5.24 &- 0.04 

5.2 f 0.4 

450 6.39 + 0.05 1.23 f 0.005 5.14 f 0.015 

7.7 + 0.2 

600 6.31 k 0.03 1.22 f 0.003 5.11 f 0.01 

7.5 f 0.2 

800 6.33 + 0.065 

1.23 f 0.008 5.098 f 0.02 

11.0*0.05 

6 300 31.11 + 1.27 

4.13 f 0.09 7.42 * 0.17 

4.3 * 0.4 

450 30.23 f 0.6 4.12 * 0.04 7.18 f 0.09 

4.9 * 0.3 

@xl 26.73 + 0.4 3.91 + 0.03 6.62 f 0.06 

6.3 k 0.3 

800 27.26 + 0.4 3.94 + 0.03 6.72 f 0.065 10.6 * 0.2 

IO 300 85.62 * 5.6 10.17 * 0.4 8.12 + 0.37 

450 67.96 f 4.5 9.03 f 0.3 6.9 + 0.35 7.4 f 0.2 

600 73.02 k 1.3 

9.42+0.11 7.34 + 0.08 

8.1 f 0.1 

800 58.83 + 2.4 

8.35 f 0.19 6.53 * 0.17 

9.5 + 0.2 

20 600 276.06 + 10.1 

31.79 f 0.1 8.32 f 0.23 

5.7 f 0.2 

800 211.3 f 16.4 

26.81 f 1.2 7.18 f 0.34 

9.6 * 0.2 

1000 158.25 f 10.6 

22.18 f 0.77 6.29 f 0.3 

12.4 f 0.2 

(12) 

In Table II we present the results for (R ‘) and (S’) and the 

ratio between these two quantities as obtained for the differ- 

ent chain lengths and surface temperatures. Note that these 

results correspond to an additional average over the (R “) 

and (S’) values obtained in the ten trajectories calculated 

for each Nand Tpair. Inspection of the results shows that for 

a given value of N> 3 both (R “) and (S’) decrease for in- 

creasing temperatures. For N = 3 both quantities seems to 

be independent of the temperature in the range considered. 

The constant (R ‘) and (S ‘) values obtained for N = 3 are 

related to the relatively small configuration space available 

for such a short chain. In other words, for N = 3 the vari- 

ation in the magnitude of the end-to-end distance, R,, , and 

in the radius of gyration, S,, are mainly due to the bending 

mode of the adsorbate. Since the variation of the magnitude 

of bending oscillations for increasing values of T is small 

both R,,, and S, are practically constant. In the case of long- 

er adsorbates, the decreasing values of (R “) and (S’) for 




(a) 

(b) 

FIG. 3. Adsorption configurations for two chains at T= 600 K. (a) N= 6 and (R ‘) = 5.56 A’; (b) N = 6 and (R ‘) = 2.06 A’; (c) N = 20 and 

(R ‘) = 473.95 A*; (d) N = 20 and (R ‘) = 96.3 A’. These configurations correspond to the largest [(a) and (c)] and smallest [ (b) and (d) ] (R ') values 

obtained in two typical trajectories. In order to distinguish between the adsorbate and the surface the distance of each atom (in the adsorbate) from the 

surface was multiplied by a factor of 1.5. The original distances of these atoms from the surface were in the range 2.045 A. 

increase of T is related to the occurrence of an additional 

bending mode associated with the bending of a segment of 

the chain (which may contain a few atoms) with respect to 

the remainder of the chain. The transition of a stretched 

chain into a bent configuration is associated with a passage 

over an energy barrier (since some of the chain atoms may be 

located at an energetically unfavorable site over the surface). 

It is clear that an increase in the surface temperature will 

correspond to a larger probability to reach such bent config- 

urations and hence to a decrease in the (R ‘) and (S “) values 

obtained. The variation of R,,, and S, during all the trajec- 

tories were monitored. The configurations of two adsorbates 

(N = 6 and 20) which correspond to the smallest and largest 

values of R,,, during two typical trajectories at T = 600 K 

are shown in Fig. 3. It is clear that large values of R,,, are 

related to configurations in which the adsorbate is basically 

stretched along the surface while it is highly bent in the case 

of small R,,, values. 

It has been shown that for unperturbed conditions the 

value of (R 2)/(S2) for long chains should be 6.13 As is 

evident from Table II in the case of N = 3 the corresponding 

value at all temperatures is < 6 while for longer chains this 

J. Chem. Phys., Vol. 97, No. 2.15 July 1992 

(d) 

value is > 6 for all T values. These deviations of (R ‘)/ (S “> 

from the free chain value are due to the constraints on the 

adsorbate configurations introduced by the presence of the 

solid surface. 

In order to obtain a better visualization of the variation 

of the end-to-end distance and the radius of gyration we have 

plotted in Fig. 4 the variation of R a,, as a function of time as 

obtained in two typical trajectories for N = 3 and 20 at sur- 

face temperature of 600 K. Figure 5 shows similarly the vari- 

ation of S: during the same two trajectories. Inspection of 

these plots clearly shows that for N = 3 both R ,‘,, and Si 

perform rapid oscillations which correspond to the bending 

mode of the triatomic adsorbate (note the relatively small 

magnitude of these oscillations). 

It is clear that these oscillations are random in nature 

although there are some indications for the trapping of the 

adsorbate in some contracted configurations. This trapping 

process manifests itself in a sudden decrease of the average 

values of R zt, and S : which can be observed for example in 

Figs. 4(a) and 5 (a) at 17 < t < 20 ps. Similar, however much 

more pronounced, trapping occurs in the case of N = 20 

[Figs. 4(b) and 5 (b) 1. Here various trapped configurations 


9 

6 

*- 

s 

7 

8 

-g! 6 

(4 

5 

4 

3' 

I 

0 5 10 15 20 25 30 35 40 

600 

500 

400 

n- 

2% 

9'300 

"2 

200 

Time [ps] 

100 

i,: 

0 

0 5 10 15 20 25 30 35 40 

(b) ime b4 

FIG. 4. Variation of R & as a function of time in two typical trajectories at 

T= 60 K for (a) N= 3 and (b) N= 20. 

can be observed (corresponding to very small R & and ,!i’i 

values) and they may exist for relatively long duration (i.e., 

5- 10 ps ) . The large variation in the values of R zt, and S : for 

N = 20 manifests the large variety of bent configurations in 

which the adsorbate can be trapped. It should be noted that 

for increasing values of the surface temperature the probabil- 

ity of the adsorbed molecule to be trapped in a bent configu- 

ration is markedly increased (this behavior is clearly exhibit- 

ed in the decreasing values of (R “) and (S2) as T is 

increased, see Table II). It has to be emphasized that the 

variation of R z,, and S: shown in Figs. 4 and 5 are typical of 

the behavior of the adsorbate on the surface as observed in 

the present study. The large variations in the magnitude of 

R z,, and S: obtained for large values of N and T demon- 

strate the various adsorption configurations the adsorbate 

may reach during its motion on the surface. These oscilla- 

tions are not due to the thermalization process of the adsor- 

bate on the surface which has been discussed above. We have 

verified this by initiating a trajectory at low temperature 

(i.e., 200 K) and increased it gradually during the trajec- 

tory. The variation in the magnitude of R z,, and Si in such a 

trajectory was minor at the beginning and increased appre- 

ciably with increase of T. 

Next the dependence of (R “) and (S2) on the chain 

length, N, was examined. In particular, the data of (R “) and 

(S ‘) from Table II were fitted to a power law of the form 


1.5 

1.4 

r 1.3 

-&, 

(4 

1.1 

1 

0.9 ’ 

I 

0 5 10 15 20 

Time [PSI 

25 30 35 40 

50 

45 

40 

35 

s 

*c% 30 

25 

al 

15 

10 

0 5 10 15 20 25 30 35 40 

08 Time [ps] 

FIG. 5. Variation 0f.S: as a function oftime in the same trajectories as those 

.ofFig.4for(a) N=3and(b) N=20. 

(A “) = C xN”‘, (13) 

where A = R or S. The results for T = 600 and 800 K are 

shown in Fig. 6. It is clear that in all cases a scaling of both 

(R ‘) and (S2) with chain length exists. Moreover, for each 

quantity the value of y seems to be independent of the system 

temperature. The exponents obtained for (R “) and (S2) 

were 0.953 f 0.05 and 0.833 f 0.02, respectively. It should 

be noted that these values of Y were obtained also for the 

lower temperatures considered. Thus, for (R ) practically a 

linear dependence on N is obtained while for (S) the observed 

dependence on chain length deviates from linearity. 

For large polymer molecules (N-, c/3 ) a number of analytical 

and numerical calculations have shown that 

(R “) a (S’) a N”‘, where for a two dimensional system 

Y = 3/4 while for three-dimensional systems Y = 0.588.14 

The deviations of the Y values obtained in the present work 

from the expected ones for very long polymer chains are 

related to both the small values of N considered and to the 

existence of a relatively strong interaction between the 

chains and the substrate. Indeed, it was verified that when 

the adsorbed chain is “stretched” along the surface the 

strongest adsorbate-surface interactions are obtained while 

bent configurations correspond to a weaker chain-surface 

bonding. It is expected in such a case that the existence of the 

solid surface will tend to “stretch” the adsorbate and to yield 




a- 

6 

-G 4 

z 

2- 

+ Q=Reta T=600 

0 C?=Rete T=800 

0' 

0 1 2 3 4 

FIG. 6. The variation ofln( (R ‘)) and In( (S’)) as a function of In(N) for 

T=Wand8OOK. 

an (R ‘) value which is roughly proportional to N. For much 

larger N values, the differences in the free energies of the 

system for “stretched” and “bent” configurations decrease 

due to the rapid increase in the number of adsorption states. 

Hence, the contribution of these bent configurations to (R ‘) 

will increase and the theoretical limit of the Y valueI is ex- 

pected. 

C. Chain relaxation 

Among the many dynamical properties which can be 

studied the chain relaxation is one which attracted a certain 

amount of attention.‘3,‘4 The relaxation process is charac- 

terized by the decay of the time-dependent autocorrelation 

functions of (R ‘) and (S*). These functions measure the 

extent to which a configuration at time to influences the con- 

figurations at subsequent times to + t. The autocorrelation 

function for a scalar property A is defined by 

C(A,t) = 

WJ) 

(A(t,M(to + t,> - (A(to))* 

(A(to)*) - (A(to>)* ’ 

(14) 

where A denotes R * or S *. The average ( ) should be evalu- 

ated as ensemble averages but these ensemble averages may 

be replaced by averages over different to values provided that 

the interval between successive times is sufficiently large for 

the configurations to be uncorrelated. It is clear from Eq. 

(14) that C(A,t = 0) = 1 and the correlation function 

should be zero for large values of t. Typical behavior of 

C(R *,t) and C( S *,t) are shown in Fig. 7 for T = 600 K and 

N = 3. In addition the behavior of these correlation func- 

tions for different surface temperature values and for N = 3 

are shown in Fig. 8 (note that the time scale in Fig. 8 is 

different than the one in Fig. 7). Qualitatively very similar 

results were obtained for longer adsorbed chains. Inspection 

of these results shows that in all cases there exists a very 

rapid initial decrease in the value of C( A,t) during the first 

0.25 ps which is followed by a much slower relaxation up to 

t = 4 ps. The rate of this slow relaxation exhibits a linear 

dependence on the time, t. Moreover, Fig. 8 shows that this 

slow relaxation rate is practically independent of the surface 

0.8 

0.6 

a 

"a 0.4 

si 

(a) 

0.2 

0.8 

0.8 

0 

"&I 0.4 

3 

(b) 

0.2 

0 

0 2 

2 4 6 8 10 

Time [ps] 

4 6 8 10 

Time [ps] 

FIG. 7. Thevariationof C(A,t) asafunctionoftimefor T= 600and N = 3, 

where (a) A = R * and (b) A = S*. 

temperature. The variation of C(A,t) in the range between 

t = 1 and 4 ps were examined by a best fit of a linear function 

to this region for all N and T values. It was found that for 

both A = R * and A = S * the relaxation time constants, r,., , 

obtained were in the range of 2-6 ps. No systematic depend- 

ence on the surface temperature (for a given N value) was 

observed. Moreover, we could not find any correlation be- 

tween 7A and the chain length. It should be noted that the 

relatively large values of the relaxation time constant for 

both R * and S* are similar in their magnitude to periods 

during which the adsorbate is trapped in the various bent 

configurations (see discussion above 1. 

An additional quantity of interest in the study of relaxa- 

tion processes in such systems13*‘4 is related to the time de- 

pendence of the velocity autocorrelation function of the 

chain’s center-of-mass. This quantity is defined by 

CW,,J) = 


w,,(to)~v,,(to +t,> 

w,,cto)*v,,cto)) - 

A 

(15) 

Typical results of C(V,, , t) for different surface tempera- 

tures and for N = 6 are shown in Fig. 9. Analysis of the 

variation of the center-of-mass autocorrelation as a function 

of time shows that for all Tvalues one obtains a fast exponen- 

tial decrease of C( V cM ,t> followed by a rapid decaying oscil- 

lation. It should be noted that the oscillations at t values > 1 


(a) 

.02 

0 

(b) ’ 

- T=300 

‘.-‘.‘..’ T=450 

- T=800 

I I 

0 1 2 3 4 5 

Time [ps] 

- T=3OU 

““.’ ” T=450 

----- T=600 

- T=EW 

1 2 3 4 5 

Time [PCS] 

FIG. 8. Same as in Fig. 7 for different surface temperatures (note the difference 

in time scale-s compared to Fig. 7) (a) A = R * and (b) A = S*. 

ps are mainly due to noise. Similar shapes of C( V,, ,t) were 

obtained for other values of N. At the lower surface tempera- 

ture values (T = 300 and 450 K) a negative minimum in 

cw cM ,t) is observed. These negative values of the autocor- 

relation function are associated with the existence of long 

lived configurations of the adsorbate on the surface. Similar 

behavior was observed for isolated polymer chains in solu- 

tion ‘3(a)P’5 where the negative regions of the velocity corre- 

lation function were associated with the cage effect. In the 

present study the local minima in the adsorbate-surface po- 

tential function are analogous to the cage formed by the sol- 

vent particles in the study of polymer dynamics in solutions. 

The fast initial decay of C( V,, ,t) was fitted to an exponen- 

tial function of the form 

a v,, , 0 aexp( - t/r”), (16) 

where r” is the time constant associated with the decay rate 

of the center-of-mass velocity correlation function. The val- 

ues obtained for the various T and N values are presented in 

the last column of Table II. These values clearly show that 

no obvious scaling law of ry as a function of N can be estab- 

lished. However, there exists a clear dependence of the relax- 

ation time constant on the surface temperature. It is clear 

that the magnitude of ry increases with increasing values of 

T (see Table II). This variation of ry as a function of T is 

related to the trapping of the adsorbate in highly bent config- 


I 

2 

” 

0.4 - T=800 

z 

0 0.2 : 

IL 

0 

i L, 

? 

if .: I‘.;;, py. 

z is 

; , s(,:.: 

%\‘,V . . ‘i 

>1 

“- ‘,Y 

. .,;> ..--y 

-0.2 

0.4 

Ii 

0 1 2 3 4 5 

Time [ps] 

FIG. 9. Center-of-mass velocity autocorrelation as a function of time for 

N = 6 at various surface temperatures. 

urations which are associated with relatively high barriers 

which the molecule has to overcome in order to get into 

them. Thus, the increase in Tresults in a larger probability of 

the adsorbate to reach these bent configurations. However, 

once these configurations are reached, the adsorbate will be 

trapped in them for a relatively long duration. Nevertheless, 

no simple functional dependence of this time constant on T 

was detectable for the various values of N. 

D. Surface diffusion mechanism 

In this subsection we shall try to establish a general de- 

scription of the detailed mechanism by which n-alkane-type 

chains diffuse on solid surfaces. The analysis described be- 

low is based on the results described above as well as on a 

detailed study of a large number of individual trajectories. 

This analysis consisted of three different and complemen- 

tary ways of representation of the adsorbate configurations 

on the surface and the variation of these configurations dur- 

ing the trajectory. First we have used drawings of the adsor- 

bate configuration on the surface (similar to those of Fig. 3) 

for a number of consecutive t values at different stages of the 

trajectory. In addition the variation of the three Cartesian 

components of the position vector of all the atoms in the 

chain were plotted. Finally, the variation of the normal and 

parallel components of R & [i.e., R zt, (n) and R zt, (p), re- 

spectively] as a function of time were examined. Typical 

results describing the variation of R z,, (n) and R z,, (p) as a 

function oft for N = 3 and 6 at T = 600 K are shown in Fig. 

10. Inspection of Fig. 10 shows that both quantities oscillate 

around an average value and there is a strong correlation 

between these oscillations. In the case of N = 3 [Fig. lO( a) ] 

any decrease/increase in the value of R & (n) is compensat- 

ed by an increase/decrease in the magnitude of R $, (p). 

Moreover, after a cycle of decrease and increase in the mag- 

nitude of R t (p) in most cases the ratio between the two 

components of R zt, (p) along the surface (i.e., along the X 

and Y directions) were markedly changed. In the case of 

N = 6 [Fig. lO( b) ] most of the oscillations in R 2, (n ) do 

not correspond to a marked change in R & (p) (except, for 

example, the oscillation at t = 3 ps). Similar behavior was 



16 

Q=Ret&) 

0 10 20 30 40 

(a) Time [ps] 

0 10 20 30 40 

lb] Time [ps] 

FIG. 10. Variation ofthe normal (solid line) and parallel (light dotted line) 

components of R * as a function of time for T = 600 K. (a) shows the results 

for N = 3 [note that the curve of R :,, (p) was shifted to higher values by 4 

A2 to obtain a better resolution of the two curves] while (b) shows the 

results for N = 6 [here the values of R z,, (n) were multiplied by 5 while 

those of R :,. (p) shifted to larger values by 10 A2 to obtain better resolution 

between the two curves]. 

observed for other N values and different surface tempera- 

tures. For all chains considered, an increase in the value of T 

resulted in an increase in the frequency of oscillations for 

bothR:,,(n) andRzt,(p). 

Analysis of individual configuration sequences during 

the oscillations described above, together with a detailed ex- 

amination of the variation in the positions of individual 

chain atoms during the trajectory suggest that two types of 

basic movements are involved in the motion of chainlike 

molecules on a solid surface. These two types of movements 

may be described by rotational motion of relatively small 

fragments of the chain as is shown schematically in Fig. Il. 

The first type [Fig. 11 (a) ] consists of the elevation of the 

chain edge above the surface followed by a rotation of this 

elevated segment around a bond to the surface. The move- 

ment ends by lowering the segment back to the surface above 

a new position. We shall denote this type of movement as 

“edge rotation” or ER. The second type of basic movement 

observed, consists of the elevation of a segment located in the 

middle of the chain above the surface which results in the 

D. Cohen and Y. Zeiri: Surface diffusion of large molecules. I 

(b] 

(a) 


Edge rotation 

central segment rotation 

- - 

/////////// /////////// 

FIG. 11. Schematic description of the two basic types of motion of different 

chain segments. (a) Edge rotation and (b) central segment rotation. 

formation of an inverted U shape. Next, this elevated seg- 

ment rotates around the axis connecting its two edges, hence, 

the center of the segment is lowered to the surface while one 

or both edges are elevated, a schematic description of such 

movement is shown in Fig. 11 (b). In the following, this type 

of movement will be denoted as “central segment rotation” 

or CSR. It is clear that the CSR is expected to occur only in 

the case of longer chains (i.e., when Nis 2 5) while the ER 

may take place for any N value. For long chains, both types 

of movements may occur as well as various combinations of 

them. These two different types of segment movements sug- 

gest that the activation energy for the diffusional motion of 

the molecule is determined by the energy required to elevate 

the segment from the surface. Moreover, the configuration 

which corresponds to an elevated segment (in both ER and 

CSR) may be identified with the transition state for the dif- 

fusional motion. It is clear that the energy required for the 

elevation process is determined by the segment-surface in- 

teraction as well as the interaction among the chain atoms. 

Thus, the surface diffusion of the n-alkane-type chains is a 

result of a sequence of these two basic types of movements 

and their combinations. According to this mechanism of 

surface diffusion (which involves the two basic movements 

described above) it is expected that the activation energy for 

diffusion will not scale linearly with N for chains longer than 

a few atoms. However, for short chains, when the ER-type 

movement dominates the diffusional motion a linear rela- 


tionship between EdiR and Nmay exist. These results are in a 

good qualitative agreement with the experimental observa- 

ti0ns.j The experimental results for n-alkane adsorbates 

with Nin the range 3-6 indicated that the adsorbed molecule 

performs a concerted motion along the surface during its 

surface diffusion. This suggestion is confirmed in the present 

study, namely, the motion ofshort chains along the surface is 

dominated by the ER-type movement in which the rotating 

segment includes most of the atoms in the chain. However, 

for longer adsorbates, both ER and CSR-types of move- 

ments contribute to the motion of the adsorbate along the 

surface and the motion cannot be viewed as a concerted one 

anymore. 

IV. SUMMARY 

The present study was devoted to the investigation of 

the nature of surface diffusion of n-alkane-type chains on a 

solid surface. The motion of a single adsorbate (low cover- 

age limit) on a W( 100) surface was simulated using the 

GLE approach. The calculations were performed for various 

chain lengths and a number of surface temperatures. It was 

found that for all values of N the variation of the diffusion 

coefficient as a function of surface temperature could be de- 

scribed by an Arrhenius form. Moreover, it was found that 

the preexponential factor is practically independent of the 

chain length while the activation energy for diffusion was 

found to be approximately proportional to the square root of 

N. These results are in good agreement with the experimen- 

tal findings3 except for the variation of EdiR with IV which 

were found experimentally to exhibit a linear relationship. 

This difference is attributed to the relatively short chains 

which were used in the experimental study. 

In addition a number of static properties of the adsorbed 

chain were examined. It was found that the adsorbates may 

be trapped in local minima of the potential energy surface 

describing the chain-surface interaction. These trapped con- 

figurations may exist for relatively long periods (for 2-10 

ps), and hence significantly influence the mean square end- 

to-end distance and the mean square radius of gyration of the 

adsorbed molecules. Moreover, this trapped configuration 

may serve as a cage for the adsorbate which is manifested in 

the existence of a negative minima in the time autocorrela- 

tion function of the center-of-mass velocity. 

The results obtained in this work allowed us to identify 

two different types of basic movements which dominate the 

motion of the adsorbate on the surface. These correspond to 

two different rotational movements of segments located at 

the edge and in the middle of the chain. The mechanism of 


surface diffusion of n-alkane-type chains is described by a 

sequence of these two basic movements and their combina- 

tions. 

Future work will be devoted to the study of the influence 

of the chain rigidity on its surface mobility.16 

ACKNOWLEDGMENT 

This research was supported by The Fund for Basic Re- 

search Administered by The Academy of Sciences and Hu- 

manities. 

‘See, for example, (a) G. Ehrlich and K. Stolt, Annu. Rev. Phys. Chem. 

30,603 (1980); (b) D. A. King, J. Vat. Sci. Technol. 17,241 (1980). 

*See, for example, (a) A. G. Naumovets and Y. S. Vedula, Surf. Sci. Rep. 4, 

365 (1985); (b) R. Gomer, Rep. Prog. Phys. 53,917 (1990). 

3J. L. Brand, M. V. Arena, A. A. Deckert, and S. M. George, J. Chem. 

Phys. 92,5136 (1990). 

‘See, for example, (a) S. M. George, A. M. DeSantolo, and R. B. Hall, 

Surf. Sci. 159, L425 ( 1985); (b) C. H. Mac, J. L. Brand, A. A. Deckert, 

and S. M. George, J. Chem. Phys. 85, 1676 ( 1986); (c) C. H. Mac, B. G. 

Koehler, J. L. Brand, and S. M. George, ibid. 87,234O ( 1987); (d) C. H. 

Mac, J. L. Brand, B. G. Koehler, and S. M. George, Surf. Sci. 191, 108 

(1987). 

’ (a) C. H. Mak, B. G. Koehler, and S. M. George, J. Vat. Sci. Technol. A 

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(1986); (b) Y. Zeiri and R. R. Lucchese, ibid. (in press); (c) Surf. Sci. 

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‘See, for example, (a) J. C. Tully, G. H. Gilmer, and M. Shugard, J. Chem. 

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Sci. 198, L335 (1988). 

9T. Halicioglu and G. M. Pound, Phys. Status Solid A 30,619 (1975). 

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(1989). 

I3 See, for example, (a) M. Bishop, M. H. Kalos, and H. L. Frisch, J. Chem. 

Phys.70,1299(1979);(b)D.C.Rapaport,ibid.71,3299(1979);(c)W. 

Bruns and R. Bansal. ibid. 74.2064 ( 1981): ,, Id) .-, J. Luque, J. Santamaria, 

and J. J. Freire, ibid.‘91, 584’( 1989). 

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I6 D. Cohen and Y. Zeiri (in preparation). 

J.Chem. Phys.,Vol. 97,No.2,15 July 1992

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