We have studied  										  the lattice gas model subject to  										  nearest neighbour repulsive  										  interactions with the Monte-Carlo  										  method leading to a clear  										  understanding of order-disorder  										  transition and its effect on the  										  adsorption and intercalation  										  processes. The lattice division into  										  two (three) sublattices in the case of  										  the square (triangular) lattice  										  enables us to underline the appearance  										  and the growth of the ordered phase. A  										  comparison between the mean field  										  approximation and Monte-Carlo results  										  is also presented.We have studied  										   the lattice gas model subject to  										  nearest neighbour  repulsive  										  interactions with the Monte-Carlo  										   method leading to a clear  										  understanding of order-disorder   										  transition and its effect on the  										  adsorption and  intercalation  										  processes. The lattice division into  						 				  two (three) sublattices in the case of  										  the square  (triangular) lattice  										  enables us to underline the appearance   										  and the growth of the ordered phase. A  										   comparison between the mean field  										  approximation and  Monte-Carlo results  										  is also presented

Boughaleb, Y.

Ennamiri, H.

Hekkouri, A.

English

We have studied  										   the lattice gas model subject to  										  nearest neighbour  repulsive  										  interactions with the Monte-Carlo  										   method leading to a clear  										  understanding of order-disorder   										  transition and its effect on the  										  adsorption and  intercalation  										  processes. The lattice division into  						 				  two (three) sublattices in the case of  										  the square  (triangular) lattice  										  enables us to underline the appearance   										  and the growth of the ordered phase. A  										   comparison between the mean field  										  approximation and  Monte-Carlo results  										  is also presented.We have studied  										  the lattice gas model subject to  										  nearest neighbour repulsive  										  interactions with the Monte-Carlo  										  method leading to a clear  										  understanding of order-disorder  										  transition and its effect on the  										  adsorption and intercalation  										  processes. The lattice division into  										  two (three) sublattices in the case of  										  the square (triangular) lattice  										  enables us to underline the appearance  										  and the growth of the ordered phase. A  										  comparison between the mean field  										  approximation and Monte-Carlo results  										  is also presented

Revues Scientifiques Marocaines

M.J. CONDENSED MATTER                        VOLUME 2, NUMBER 1                         1 NOVEMBER 1999Surface diffusion in the framework of lattice gas model: mean field treatment andMonte-Carlo methodH. Ennamiri, A. Hekkouri and Y. Boughaleb*.Laboratoire de Physique de la Matière condensée.Université Hassan II, Faculté des Sciences Ben M'Sik, CasablancaMorocco.We have studied the lattice gas model subject to nearest neighbour repulsive interactions with the Monte-Carlo method leading to a clear understanding of order-disorder transition and its effect on the adsorption andintercalation processes. The lattice division into two (three)  sublattices in the case of the square (triangular)lattice enables us to underline the appearance and the growth of the ordered phase. A comparison between themean field approximation and Monte-Carlo results is also presented.PACS. 66.30D- Theory of diffusion and Ionic conduction in solids.PACS. 64.60   - General studies of phase transition.I- INTRODUCTIONRecently there has been a vivid interest in themodelling of chemisorbed or physisorbedmonolayers at surfaces in terms of lattice gasmodels [1,2]. By these models, order-disorderphenomena in the monolayers can be interpreted interms of powerful tools of the modern theoriesabout critical phenomena [3-5]. While in previouswork most emphasis has been on static properties ofthe model, such as the phase diagram, adsorptionisotherms, order parameters, ordering energies, etc.,relatively little attention has been paid to thedynamics of such systems. Clearly, a betterunderstanding of surface diffusion at arbitrarycoverage may be an important ingredient oftheoretical analysis of complicated processes atsurfaces such as catalysis, corrosion, etc….In the present paper, we study surface diffusion inthe framework of lattice gas model, where theadatoms can occupy only preferred sites of a latticedetermined by the substrate surface. We study theeffect of lateral interactions on surface diffusion bythe Monte-Carlo simulation for a broad reducedinteraction.The outline of our paper is as follows. In section 2,we briefly define the quantities of interest forgeneral lattice gas models, we summarize therelevant static properties of our square (triangular)lattice gas model and we also discuss the mean fieldapproach results. In section 3  we introduce theMonte Carlo method and we present our simulationresults for different choices of interactionparameter, while section 4 contains ourconclusions.II- GENERAL THEORYA- Lattice  gas modelIn the lattice gas, a local occupation variable ni isassociated with each lattice site, where ni=1 if thelattice site i is occupied by a particle, while ni=0otherwise.Assuming pair interaction between particles, thehamiltonian H of the lattice gas takes its usual formas an expansion in terms of the {ni}.−−=iijiijij nnnH µε21              (1)where εij are the pairwise interaction parametersand µ is the chemical potential [6].The configuration {n} represents the set ofoccupation numbers for each site that we note for alattice with N sites.{n}={n1,n2,...,nN }The diffusion process of the particles is assured bythe change of the occupation numbers of the latticesites. In the lattice gas formulation this can be seenas the change of the lattice configurations. Thedynamic of the lattice is then governed by thephenomenological master equation [6],  whichdescribes the evolution in time of the probabilityP({n},t) of  the configuration {n}.{ }( ) { } { }( ){ }{ }( ){ } { }( ) { }( )tnPnntnPnnttnPn,',,',','ωω∂∂ −=   (2)The rate of the configurations exchange isexpressed by the transition frequency ω({n},{n'})which encloses all physical information of theconsidered system. The transition frequency issupposed to satisfy the detailed balance conditionthat insures the equivalence of transitions betweenthe system equilibrium states:* Corresponding author                                                      2     15               1999 The Moroccan Statistical Physical Society16                                           H. ENNAMIRI, A. HEKKOURI AND Y.BOUGHALEB                                      2{ } { }( ) { }( ) { } { }( ) { }( )','', nPnnnPnn eqeq ωω =     (3)The choice of an analytic expression for thetransition frequency is determinant for theinvestigation of the expected behaviour [8]. In ourcase, we restrict the jumps of the mobile particlesbetween nearest-neighboring sites only, hence, thetransition frequency has the same form as in ref.[7].The average concentration over all possibleconfigurations is defined by :{ }{ } ),()( tnPnnipnii ==          (4)Its time evolution is given by the following masterequation: (5){ } { } )1()()1()()( jiajiijij nnnwnnnwtip −−−=∂∂→→where wi→j ({n}) is the jump probability operator.This equation introduces a current operator Ji,j({n})along the bond i → j, considering the symmetricproperties of the lattice, eq. (5) takes the form :+−−=∂∂aijji aiJiJtip )()(21)(,,with  j = i +a{ } { } { } )1(),,()1(),,()(, ijjiji nnajnwnnainwnJ −−−−=The jump probabilities wi→j({n}) = w({n},i,a) arefunction of the energy difference, seen by thejumping particle between i and j sites. For anactivated process, the jump probability is taken as := { } )( nw ji→where ωο is a constant prefactor, ε=/kBT is thereduced energy that we call γ, kB is the Boltzmannconstant and ui is the vector connecting the initialand final sites.B-The break down of the lattice occupation symmetry: Mean field treatmentInvestigation of the time evolution of theconcentration is done in the limit of the mean fieldapproach; the time evolution of p(i) is given by thefollowing equation,{ }−−=∂∂ijji nJtip )()( ,        (6)For systems with repulsive nearest neighbourinteraction, the use of the lattice gas model allowsthe reproduction of the order-disorder transitioncharacterized by the break down of the occupationsymmetry of the lattice. Hence, we decide todistinguish between various sublattices. Then,square (triangular) lattice is supposed to be dividedinto two (three) interpenetrating sublattices A,B(A,B and C), in the way that any site of them say Awould have two (three) nearest neighbours in eachof the other sublattice B ( B and C) [9,10].The average concentration evolution in time of eachsublattice satisfies the equation (5).Then, we obtain two (three) kinetic coupledequations:{ }−=−=∂∂ijaji nJtip )()( λαα        (7){ } ))(1)(())(1)(()( ipjpwjpipwnJ ijjiji αλαλλαλαλα −−−= →→with the upper indexes α=A,B or (A, B, C) andλ=≠=α=are introduced to identify the sublattices.   The numerical resolution of those equations (7)leads to the results shown in figures(1,2) for thesquare lattice and figures (3,4) for the triangularlattice.0,0 0,2 0,4 0,6 0,8 1,00,00,20,40,60,81,0γ== =2pBpApFig.1: Occupation of  the square sublatices A and B.{ }( ) )exp(,,''≠+−=aiuiaiBo nTkainw εω2                      SURFACE DIFFUSION IN THE  FRAMEWORK OF LATTICE GAS MODEL…                   170,0 0,2 0,4 0,6 0,8 1,00,00,20,40,60,81,0γ===2ηpFig.2 :Evolution of the order parameter η = pA-pB as afunction of the mean concentrationWe notice that the square lattice gas model withrepulsive interaction presents an ordered phaseFig.1 and Fig.2, indeed we observe that onesublattice ‘A’ is occupied where the second ‘B’ isempty.Fig. 3 shows, for different reduced energies, thevariation of the concentrations of the threesublattices with respect to the  averageconcentration of the system.0,0 0,2 0,4 0,6 0,8 1,00,00,20,40,60,81,0p'(pB, pC)p'pAγ===1.4γ===1.5γ===1.6pApA, p'pFig. 3: The break down of the occupation symmetry ofthe triangular latticeWe notice that the lattice is disordered up to aminimum concentration which corresponds to thebreak down of occupation symmetry of the lattice.Then, we assist to the filling of one sublattice, thelong range order corresponds to the commensuratephase of concentration p =1/3. The increase ofaverage concentration leads the system to itsdisordered phase. The symmetry around p = 0.5 isa natural consequence of lattice gas model withinteraction limited between pair of particles.Onerealizes the equivalence of the occupation of twosublattices. Then, the dynamic of the triangularlattice can be characterized by only one orderparameter:η==  | pA - p' |where p'=(pB+pC)/2 is the mean concentration of theidentical sublattices.The representation of η as a function of averageconcentration p  is shown in Fig. 4. We remarkthat the order parameter η=presents maximums atthe concentrations 1/3 et 2/3 relative  to theformation of the ordered phases.0,0 0,2 0,4 0,6 0,8 1,00,00,10,20,30,40,50,60,70,8eqγ===1.4γ===1.5γ===1.6η===ηpFig.4: Evolution of the order parameter for differentreduced energies. Arrows indicate different values of thereduced energies.III- MONTE-CARLO SIMULATIONSThe study of the dynamic of atoms on randommedia has led to the development of varioussophisticated numerical methods as, Monte-Carloprocedures [11]. As an example, we treat in theframework of Kinetic lattice gas model the randomjump to neighbouring empty sites of the chosenlattice.  Particular attention is paid to the influenceof order-disorder transitions in the diffusion of theordered region. As an explicit example, the squareand triangular lattice with repulsion betweennearest neighbours.A- AlgorithmThe simulations have been performed on a semi -infinite lattice with periodic boundary conditionsalong the lateral side. For a given situation we startfrom a source where we choose randomly aparticle; then if this particle jumps randomly to oneof the first neighbour vacant sites, its site energyand that of the first neighbours particles of its initialposition will change. So for every possible jump tothe first neighbour sites we calculate the energychanges. The jump then occurs in one of the vacantsites with a probability proportional to theBoltzman factor with the appropriate energydifference and temperature.18                                            H. ENNAMIRI, A. HEKKOURI AND Y.BOUGHALEB                                     2The choice of the particle that jumps and the jumpdirection are identified by a random number0≤R≤1.The dynamic of the system is controlled by thetransition frequency w which encloses all thephysical information of the considered system [11]and satisfies the detailed balance condition.This transition frequency is proportional to ∆−TkHBexp , where ∆H is either equal 0 or ε(N-N') and it represents the energy difference betweeninitial and final state.TkN Boji eww/ε−=→       (8)Where ε is the interaction energy (ε < 0 withrepulsive interaction). wo is the Arrhenius factorchosen as the time of the simulation, following theMetropolis algorithm we write:( )),1(min / TkH Bo ew∆−=      (9)N(N’) is the number of the nearest neighbourparticles of the initial (and final) site.Fig 5.a Schematic illustration of the intercaled atoms(full circle) in a triangular gird of the interlayeroctahedral holes.Fig 5.b Represents a schematic view of the surface of thesubstrate, whose periodic potential provides a squarelattice of preferred adsorption sites.In this case the research has focused on the case ofdiffusing particles with repulsive nearest neighbourinteraction [12,13], the other interactions areignored. The key to the exact enumerationprocedure is that the probability of a random walkerbeing at any site (i) at some time t (one can speak oftime, t, or the t-th step as convenient) is determinedby the probabilities of being at the nearestneighbours of the site (i) at time (t-1).The simulation is a lattice gas simulation. Theparticles jump randomly on a square (triangular)lattice to the four (six) first neighbouring sitesunder the condition that if a given  site is alreadyoccupied no other particle can jump it.Experimentally diffusion (intercalation) ischaracterised by a concentration profile p(r,t),where r is the position vector r (i , j).The average concentration is a function of time andspace. As it follows we will consider only thesituations obtained at fixed successive times. Bothsquare and triangular lattices will be investigated.If p(r,t) is bigger than the random number R, we puta particle in that site. In this way we constitute onedetailed diffusion picture on which we look for thediffusion front [14-18] defined  geometrically as thelimit of the particles (solid circles) Fig.6 which areconnected via first neighbours to the source.Fig.6: The diffusion front as a geometric place of the lastinfinite cluster connected to the source.The dynamic of that object is developed in [15].Indeed, very generally, the difficulty with theinteracting case comes from the long rangecorrelation and the appearance of phase transitions.In the simple case considered here two correlationlengths are present : a thermal one associated with atemperature Tc where a phase transition appears,and a percolation one associated with aconcentration pc where the infinite connectedcluster appears.B-Square latticeIf one wishes to deal with a square lattice with thenearest neighbour interaction, one observes anordered structure at p  = 0.5, and it is convenient todivide the lattice into 2 sublattices as indicated inFig.7, and thus also introduces the sublatticesdensities pA and pB.In terms of the pA and pB we express the long-rangeorder parameter η== (pA-pB)/2.2                    SURFACE DIFFUSION IN THE  FRAMEWORK OF LATTICE GAS MODEL…                  19The  curve  exhibits  an  ordered   region  aroundp  = 0.5 where one of the two sublattices isoccupied.Fig.7 : Subdivision of the square lattice into twosublattices A and B.Indeed, in Fig. 8 we confront the results of themean field approach (continued line) and theMonte-Carlo calculation (dotted line), we remarkthat the methods are in good agreement.Fig.8: Representation of the sublattices concentrations pAand pB as a function of average concentration (pA+pB)/2,(… Monte Carlo results,        mean field result)C-Triangular latticesubdivision of the  triangular lattice into A, B and Csublattices,  such as that any site in one  of  themsay A has  three  nearest neighbours in each of theother tow sublattices B and C.Fig.9: Schematic view  of  the triangular latticesubdivisionAnalysis of the low temperature distribution ofparticles reported in the following figures givesclear evidence for some ordering of the particles onthe 2D triangular intercalation lattice due to theparticle-particle interaction.General schemes of Monte-Carlo simulation ofsurface diffusion are shown below. In fact, weobserve that a long range order can develop. Fig.10shows the intercaled particle distributions fordifferent concentrations ( p = 1/3, 1/2, 2/3) . Infigure 11,  we   report   the   mean  concentrationp  = (pA+ pB+ pC)/3 as a function of positionparameter x/L.In figure 12, we remark that for p = 1/3 thetendency for local ordering along an a 3 x a 3superlattice appears clearly. For p  =1/2 one cansee the tendency for forming alternatively full andempty parallel rows of sites. For p  =2/3 oneobserves the construction of the hexagonalsuperlattice complementary to the triangular a 3 xa 3  one, obtained for p  =1/3, where we caneasily calculate that at this concentration theparticles were randomly distributed on theoctahedral sites plane with a long range order, soeach particle would have only one particle in thefirst-neighbour. 20                                            H. ENNAMIRI, A. HEKKOURI AND Y.BOUGHALEB                                     2Fig.10a: Schematic view of the simulation, the front isindicated by the bold pointsFig.10b: Representation of the ordered region atp  = 1/3. (one sublattice occupied, two empty)Fig.10c: Schematic view of the situation at p  = 0.5, weremark that system is disordered.Fig.10d: At p  = 2/3 we remark that the system presentan ordered phase (two of the tree sublattices are occupiedand one is empty)Fig.11: The concentration profile dependency  on  thecoverage concentration p = f(x/L).Fig.12:Evolution of the order parameter as a function ofthe coverage      p = (pA+pB+pC)/3We remark that the lattice is disordered until aminimum concentration then, there appears orderedclusters,    the    growth    of    the    ordered    phase2                    SURFACE DIFFUSION IN THE  FRAMEWORK OF LATTICE GAS MODEL…                 21corresponds to the filling of one sublattice A, at theconcentration 0.33. The increase of the averageconcentration leads to the break down of the latticeorder, at the half-full lattice ( p  = 0.5). Whereas,the order at p = 2/3 corresponds to the filling oftwo sublattices A and B. The result is in agreementwith the mean field approach [13,20].IV-CONCLUSIONWith this work, the Monte-Carlo method isconfirmed to be a very powerful tool for studyingintercalation in adsorbed and lamellar compounds.The exact enumeration procedure may easily becontinued up to time and memory limitations of thecomputer. The computation time increases linearlywith the number of sites in the medium times thenumber of steps to which diffusion is performed.A comparison between the Monte-Carlo and themean field results is also presented. A qualitativeagreement is found between both results.The generalisation to other choices of interactionsis the object of our  perspectives._____________________[1] W. Dieterich, J. Stat. Phys. 39, 583 (1985).[2] Y.Chabre and P. Deniard, in chemical physics NatoAsi series B172 395-402 (1987).[3] A.Danani, R. Ferrando, E. Scalas and M. Torri,International journal of modern Physics B, Vol 11, 19,2217 (1997).[4] H.K. Janssen, in dynamical Critical Phenomena andRelated Topics, Vol. 104, ed. C.P. Enz, Lecture Notes inPhysics (Springer, Heidelberg, 1979).[5] G.Betz and H. Tribusch, Progress in solid StateChemistry 16, 195 (1985).[6] A. Natori &H. Ohtsubo, Surface Science 171-213(1986);181-289 (1987).[7] R. Nassif, Y. Boughaleb, A. Hekkouri, J. F. Gouyetand M. Kolb, Eur. Phys. J.B. 1,453 (1998).[8] J. F. Gouyet,  Phys. Rev. E 51, 1695 (1995).[9] J. F. Gouyet, Europhys. Lett. 21, 335 (1993).[10] H. Ennamiri, R. Nassif, and Y.Boughaleb Physica A3326 1-13 (1997).[11] K. Binder and D.W. Heerman," Monte CarloSimulation Methods in statistical Physics" Springer-Verlag (1988).[12] Burley D.M., Proc. Phys. Soc. (London) 85, 1163(1965).[13] H. Ennamiri, R. Nassif, Y. Boughaleb and J. F.Gouyet. J. phys: Condens. Matter , 9, 2433 (1997).[14] F.Family, Physica A 168, 561 (1990).[15] Y. Boughaleb and J.F. Gouyet, Phys. Rev. B,Vol. 407, 4760 (1989).[16] B.Sapoval, M. Rosso and J.F. Gouyet, J. Phys. Lett.46, 149 (1985).[17] T.Vicsek, in fractal Growth Phenomena (WorldScientific Publishing, Singapore, 1989).[18] J.F. Gouyet, in Physique et Structure Fractales(Masson, Paris, 1992).[19] A. Memsouk, Y. Boughaleb H. Ennamiri and R.Nassif. to be published in M. J. condens. Matter. (1999).[20] H. Ennamiri thèse de 3ème cycle’ Effets destransitions ordre désordre sur la dynamique des systèmesinhomogènes’ (1997).

Surface diffusion in the framework of lattice gas model: mean field treatment and Monte-Carlo method

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Surface diffusion in the framework of lattice gas model: mean field treatment and Monte-Carlo method

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Revues Scientifiques Marocaines

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