Cooperative sequential adsorption with nearest-neighbor exclusion and next-nearest neighbor interaction
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Cooperative sequential adsorption with nearest-neighbor exclusionand next-nearest neighbor interaction
C. J. Neugebauer ∗ Department of Chemistry, University of Cambridge, Cambridge, United Kingdom
S. N. Taraskin
St. Catharine’s College and Department of Chemistry,University of Cambridge, Cambridge, United Kingdom (Dated: November 1, 2018)A model for cooperative sequential adsorption that incorporates nearest-neighbor exclusion andnext-nearest neighbor interaction is presented. It is analyzed for the case of one-dimensional dimerand two-dimensional monomer adsorption. Analytic solutions found for certain values of the in-teraction strength are used to investigate jamming coverage and temporal approach to jamming inthe one-dimensional case. In two dimensions, the series expansion of the coverage θ ( t ) is presentedand employed to provide estimates for the jamming coverage as a function of interaction strength.These estimates are supported by Monte Carlo simulation results. PACS numbers: 05.70.Ln, 68.43.-h, 68.43.De
Surface adsorption phenomena are important in agreat number of physical, chemical and biological sys-tems. Equally large is the number of phenomena them-selves that occur when particles/molecules are adsorbedto a surface. The surface-adsorbate interactions can bebroadly classified into two categories, physisorption andchemisorption [1]. While the former is associated withelectrostatic interactions, van der Waals forces [2, 3],the latter is commonly associated with the formation ofchemical bonds. Due to the strength of the chemicalbond, chemisorption is often irreversible for temperaturesof interest. An well-known example is the adsorption ofwater on Fe(001) [4]. A simple but effective model forsuch irreversible adsorption is the random or cooperativesequential adsorption (CSA) [5]: particles are adsorbedrandomly in a sequential manner without diffusing or des-orbing.For irreversible adsorption, a central quantity of inter-est is the coverage, θ , of the final adsorbed monolayerattains, the jamming coverage, θ J . Detailed knowledgeof this jamming coverage might, for example, becomeimportant for chemical sensing devices, such as micro-cantilevers [6], in order to distinguish between differentspecies of adsorbates. The jamming coverages have beenestimated within several approaches [5]. However, impor-tant and significant effects due to interactions betweenadsorbates on the jamming coverage have not been con-sidered in detail and are the focus of this work.One of the simplest models for interaction between ad-sorbates is the nearest-neighbor exclusion (NE) mecha-nism that causes an adsorbed particle to block its nearest-neighbor binding sites from adsorption [4]. It has beenshown [7] that for the NE process on a two-dimensionalsurface with a square-lattice arrangement of binding sites ∗ Electronic address: [email protected] the jamming coverage approaches a non-trivial value of θ J = 0 . θ = 0 .
5. This is a consequence of the stochastic natureof the adsorption process which results in the exponen-tially small probability of an ideally covered surface.Here, we introduce, in addition to the NE, a short-range adsorbate-adsorbate interaction which provides amore accurate and realistic description of adsorption pro-cess. This interaction affects the rate of adsorption of anext-nearest neighbor (NNN) binding site and influencesthe jamming coverage. Such interactions might be causedby a variety of mechanisms. If the adsorbates have an ef-fective charge, then resulting electrostatic forces wouldlead to a repulsive interaction: for example, hydrogen onPd(100) acquires a dipole moment due to charge trans-fer from the surface [1]. In this case, assuming that theadsorption rates are of Arrhenius type and that each oc-cupied NNN of an available binding site contributes anequal amount to the binding energy ε int , the adsorptionrate including the interaction with n occupied NNN couldbe modeled by r n = r exp ( − n ε int /T ) [8] (where r isa typical rate of adsorption and T is the temperature).Attractive interaction might arise if an adsorbate (e.g.water on Pd(100) [9, 10]) induces a local change in thesurface structure [11] that increases the rate of adsorp-tion at NNN sites. Another mechanism that would leadto an effective attractive interaction could involve pre-cursor layer diffusion [12]: a gas particle might becomephysisorbed even if it collides with an already adsorbedparticle [13]. In that case, it might either desorb or dif-fuse to the next available site surrounding the adsorbedcluster for chemisorption.Below, we use a linear approximation in n for the ad-sorption rate, r n = 1 − nǫ ( r = 1 by rescaling the time).Such a linear dependence on n arises naturally for the at-tractive interaction of adsorption via the precursor state,as each occupied NNN should contribute equally to theadsorption of the surrounded site [14]. It is clearly thefirst order approximation of the Arrhenius-type rate pre-sented above, which is valid for high temperatures orsmall interaction energies ε int and also a possible as-sumption for the mechanism of morphology changes inthe surface. It should be mentioned, that the effects ofadsorption rates that depend on NNN occupation, espe-cially on the island structure, have been previously con-sidered [15], albeit with a different choice of rates.The effects of the NNN interaction on adsorption canbe investigated by means of rate equations [5] whichdescribe the evolution of the marginal probability den-sity P ( G ; t ) of finding a configuration G of lattice sitesempty at time t , irrespective of the state (occupied orvacant) of the remaining sites of the lattice. For the two-dimensional square lattice, the rate equations are [16] ∂ t P ( G ; t ) = − X i ∈ G z X n =0 r n P ( { G ∪ D i } n ; t ) . (1)The only way that in an irreversible adsorption processthe probability of finding a set G of sites can change isby adsorption at one of its binding sites i ∈ G . Dueto NE, i must have z empty nearest neighbors. There-fore if i lies on the boundary of G , this can only hap-pen if G is a subset of the larger set of empty sites, i.e. G ∪ D i . The subscript n in { G ∪ D i } n refers to the ad-ditional interaction with the environment of site i sur-rounded by n occupied NNN. Using the fact that themarginal probability densities obey the following rela-tion, P ( G ∪ { σ j = 1 } ; t ) + P ( G ∪ { σ j = 0 } ; t ) = P ( G ; t )( σ i denotes occupation of site i ) we can recast the RHSof Eq. (1) completely in terms of probability densities ofconfigurations of empty sites. For example, consideringonly the contribution to the rate equation due to adsorp-tion at the dotted site, Eq. (1) reads ∂ t P (cid:0) ◦◦ ◦◦·◦ ; t (cid:1) = . . . − (1 − ǫ ) P (cid:0) ◦ ◦◦ ◦ ◦· ◦◦ ◦ ; t (cid:1) − ǫP (cid:0) ◦◦◦ ◦◦· ◦◦◦ ◦ ; t (cid:1) . (2)Formally, we write such a rate equation as ∂ t P ( G ; t ) = −L P ( G ; t ) where L is the operator that generates theconfigurations G ′ of empty sites that can produce G bya single adsorption event.First, we analyze the situation in 1d. Monomer ad-sorption with NE and NNN interaction is equivalent todimer adsorption with with nearest-neighbour interac-tion in 1d [17, 18], which we will consider here. Thismodel has been solved for general cooperative rates [19].However, the temporal approach to jamming, which ismainly of our interest, is not readily available from sucha solution. Therefore, we present here a different form ofthe solution that is suitable for our purposes.For this process, the following rate equations for find-ing a stretch of m vacant sites can be written: ∂ t P (1; t ) = − − ǫ ) P (2; t ) − ǫP (3; t ) (3) ∂ t P ( m ; t ) = − [ m − − ǫ ] P ( m ; t ) − P ( m + 1; t ) − ǫ P ( m + 2; t ) for m ≥ . (4) With the initial condition of an empty lattice, Eq. (4) issolved exactly by P ( m ; t ) = exp[ − ( m − − ǫ ) t − − e − t ) − ǫ (1 − e − t )] . (5)for m ≥
2. The solution for the case m = 1 (see Eq. (3))is given by P (1; t ) = 1 − e − − ǫ ((1 − ǫ ) I ǫ ( t ) + 2 ǫI ǫ ( t )) (6)where I ǫm ( t ) = e ǫ Z t P ( m ; t ′ ) d t ′ . (7)The probability P (1; t ) is particularly important for eval-uation of the critical coverage, θ J = 1 − lim t →∞ P (1; t ).The integrals I ǫm ( t ) can be evaluated analytically onlyfor some special cases. Namely, by considering negativevalues for ǫ = − α ( α > I ǫm ( t ) in termsof a sum of lower incomplete gamma functions γ [20], I − αm ( t ) = 12 K αm ∞ X k =0 α k/ (cid:18) m + 2 α − k (cid:19) × (cid:20) γ (cid:18) k + 12 , w (cid:19)(cid:21) w ( α, w ( α,t ) (8)where K αm = e /α α / − m − α , w ( α, t ) = √ α ( e − t − /α ).The infinite series only converges in the interval α ∈ [0 , α = n/ n = 0 , , , . . . . Using the identi-ties γ ( a + 1 , x ) = aγ ( a, x ) − x a e − x , γ (1 , x ) = 1 − e − x and γ (1 / , x ) = √ π erf( √ x ) [20], we find, for example, P (1; t )for ǫ = − / P (1; t ) = 1 + 2 √ e (cid:20)r π w ) − (2 + √ w ) e − w (cid:21) w (1 / , w (1 / ,t ) (9)and for ǫ = − P (1; t ) = 1 + (cid:20)r π w ) − (2 + 3 w + 2 w ) e − w (cid:21) w (1 , w (1 ,t ) (10)where erf( x ) denotes the error function. It follows fromEqs. (9) and (10) that the time-dependent coverage, θ ( t ) = 1 − P (1; t ), asymptotically approaches the crit-ical value θ J ≃ . θ ( t ) = √ πe (cid:16) erf( −√ − erf( − / √ (cid:17) + 2 − e − − t + O (( w (1 / , t ) − w ∞ ) ) (11)and θ J ≃ . θ ( t ) = e − − − / e − − t +O (( w (1 , t ) − w ∞ (1)) ) (12)for ǫ = − / ǫ = −
1, respectively. These expressionshave been obtained by expanding Eqs. (9) and (10) about w ∞ ( α ) ≡ lim t →∞ w ( α, t ) = − / √ α such that w − w ∞ = √ αe − t . The time-dependent coverage for any ǫ = − n/ n , all terms of order smaller orequal n will drop out of the expansion so that the leadingtime-dependent term in the approach to jamming when t → ∞ is exp( − t/τ ) with τ = 1 / ( n +1). Extrapolating toany value for ǫ , the characteristic time τ to jamming forthe one-dimensional process is then given by τ = 1 / (1 − ǫ ) which can easily be verified numerically. This form ofthe temporal approach to jamming also follows from thetime dependence of the sticking probability P (2; t ) givenby Eq. (5).The 2d case of monomer adsorption with NE and NNNinteraction cannot be handled analytically and the meth-ods of series expansion (SE) and Monte Carlo simulations(MC) were employed. The SE is an Taylor expansion ofany probability density P ( G ; t ) in t using the rate equa-tions (1), P ( G ; t ) = P n =0 ( − t ) n n ! L n P ( G, t ) [5, 16]. Forthe coverage, the expansion of P ( ◦ ; t ) is of interest. Wehave implemented the algorithm introduced in [21] withthe extension of allowing for polynomial values in ǫ in or-der to compute the coefficients of the series numerically.The series coefficients up to order 13 were computed (seeTab. I) using this algorithm. With these coefficients andthe first terms in the expansion of P (cid:0) ◦◦◦ ◦◦ ; t ) ≡ P ( G ; t ),we can also calculate the expansion of the sticking prob-ability, S ( θ ) = P ( G , θ ), as a function of the coverage θ ( t ). The first few terms are S ( θ ) = 1 − θ + (6 − ǫ ) θ + 83 (1 + 3 ǫ − ǫ ) θ −
23 (1 − ǫ − ǫ + 84 ǫ ) θ + O( θ ) (13)The interaction hardly affects the sticking probability forsmall coverages (see inset in Fig. 1) and the effect can beseen only close to the jamming limit. This is what onewould expect, given the short-range nature of the interac-tion. The value of the sticking probability for attractive(repulsive) interaction are expectedly greater (smaller)than for the adsorption without NNN interaction.To obtain estimates for θ J from the series the stan-dard Pad´e approximants [7, 22] [ n, m ] (where n and m are the orders of the polynomials in the numerator anddenominator of the Pad´e approximant) were used. First,the transformation of variables, y = (1 − exp( − (1 − bǫ ) t )) / (1 − bǫ ) (similar to that used in Ref. [23]), wascarried out with b being an adjustable parameter. Itis clear from the preceeding discussion that this mim-ics the approach to jamming in 1d where b take thevalue b = 2. It has been found before [7] that usingthe knowledge of the temporal behavior of the coveragein 1d for the transformation of variables can consider-ably improve estimates in 2d. However, instead of choos- TABLE I: The series coefficients c nm for the Taylor expansion P ( ◦ ; t ) = P Nn =0 P n − m =0 ( − t ) n ǫ m c nm /n ! up to order N = 13. n m c nm n m c nm n m c nm ing a value for b , we use it to make the estimates of θ J independent of the choice of Pad´e approximant. In or-der to do so, the free parameter b has been found byminimizing the cost function C ( ǫ, b ) = (cid:0) θ J ( ǫ, [6 , , b ) − θ J ( ǫ, [6 , , b ) (cid:1) + (cid:0) θ J ( ǫ, [6 , , b ) − θ J ( ǫ, [7 , , b ) (cid:1) + (cid:0) θ J ( ǫ, [6 , , b ) − θ J ( ǫ, [7 , , b ) (cid:1) with respect to b where θ J ( ǫ, [ n, m ] , b ) is the jamming coverages obtained for thehighest-order Pad´e approximants available.The results of this analysis are presented in Fig. 1 andcompared with the values of jamming coverage calcu-lated numerically by MC simulations. In the MC sim-ulations, an event-driven algorithm [5, 16, 24] was used.Within this algorithm, all the susceptible binding siteswere grouped depending on the number of occupied NNNsites. A binding site for the next adsorption event is thendrawn randomly out of a group according to the rates r n and the waiting times are distributed exponentially. Theresults for the jamming coverage calculated numerically -6 -5 -4 -3 -2 -1 0 ε θ J θ S FIG. 1: (color online). Comparison of jamming coverage θ J for ǫ = − . , . . . , .
24 from the series expansion up to order13 using Pad´e approximant [6 , ,
7] and [7 ,
6] for the opti-mization described in the text (+) and from MC simulationsfor a 200 ×
200 - lattice and 100 iterations (red ). Errorsin simulation data points are of order 0 . S as function of coverage θ up to order N = 9for three different ǫ = 0 .
1( ), ǫ = 0 .
0( ), and ǫ = − . ·· ·· ). -1-100-10000-1e+06 ε θ J FIG. 2: (color online). Jamming coverage θ J for ǫ = − . , . . . , − . × from MC simulations for a 200 × .
001 and thus smaller than the symbol size. for a wide range of interactions are presented in Figs. 1and 2.In Fig. 1, one can clearly see that the series expansionand the MC simulations agree for ǫ ∈ [ − . , . ǫ = 0 and then flattens as ǫ becomes more negative,i.e. the interaction becomes more attractive. In Fig. 2,we can see that only for very large negative ǫ , ǫ . − ,the jamming coverage comes within a few percent of theideal coverage of θ J = 0 . ǫ takes nega-tive half- and integer values. This allowed us to computethe jamming coverage and to extract the temporal ap-proach to jamming. For the two-dimensional process, wehave computed the series expansion for the coverage asa function of time and have found the jamming coveragefor various strengths of interaction. Monte Carlo simu-lations convincingly support the series expansion resultsand provide estimates for the jamming coverage that areunaccessible to the series expansion, thus demonstratinga slow convergence to the ideal coverage with increasingattractive interaction.CJN would like to acknowledge the UK EPSRC andthe Cambridge European Trust for financial support. [1] A. Groß, Theoretical Surface Science (Springer-VerlagBerlin Heidelberg, 2003).[2] K. Autumn, Y. A. Liang, S. T. Hsieh, W. Zesch, W. P.Chan, T. W. Kenny, R. Fearing, and R. J. Full, Nature , 681 (2000).[3] G. Ehrlich, Br. J. Appl. Phys. , 349 (1964).[4] D. J. Dwyer and G. W. Simmons, Surf. Sci. , 617(1977). [5] J. W. Evans, Rev. Mod. Phys. , 1281 (1993).[6] Z. Hu, T. Thundat, and R. Warmack, J. Appl. Phys. ,427 (2001).[7] R. Dickman, J.-S. Wang, and I. Jensen, J. Chem. Phys. , 8252 (1991).[8] H. Pak and J. W. Evans, Surface Science , 550 (1987).[9] E. Kampshoff, N. Waelchli, A. Menck, and K. Kern, Surf.Sci. , 55 (1996). [10] W. Dong, V. Ledentu, P. Sautet, A. Eichler, andJ. Hafner, Surf. Sci. , 123 (1998).[11] C. J. Barnes, Phase Transitions and Adsorbate Restruc-turing at Metal Surfaces (Elsevier, 1994), vol. 7 of