adsorption–desorption processes of extended objects on a square lattice

Physica A 301 (2001) 174–180www.elsevier.com/locate/physa

Adsorption–desorption processes of extendedobjects on a square lattice

Lj. Budinski-Petkovi*c∗, U. Kozmidis-Luburi*cFaculty of Engineering, Trg D. Obradovi�ca 6, 21000 Novi Sad, Yugoslavia

Received 12 December 2000; received in revised form 29 June 2001

Abstract

Kinetics of adsorption–desorption process is studied by Monte Carlo simulation on a squarelattice. The depositing objects are squares covering nine lattice sites. At each Monte Carlo stepadsorption is attempted with probability Pa and desorption with probability Pdes = 1 − Pa. Theequilibrium coverage and the relaxation time are obtained for various ratios of the adsorptionprobability to the desorption probability. The dependence of the relaxation time � on (Pa=Pdes)is described by a power law � ˙ (Pa=Pdes)2. The equilibrium coverage decreases exponentiallywith Pdes=Pa for low desorption probabilities. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Adsorption; Desorption; Coverage

1. Introduction

The adsorption of large particles such as colloids, proteins or latexes on substratesis often a highly irreversible process [1]. For strong enough interactions between theparticles and the surface, energy barriers for desorption are much higher than thecorresponding barriers for adsorption. If the di@usion of the adsorbed particles is veryslow and if the particles do not adsorb on top of the previously adsorbed ones, such aprocess can be suitably described by the random sequential adsorption model (RSA) [2].

RSA, or irreversible deposition is a process in which the objects of a speciAedshape are randomly and sequentially deposited on a substrate. Once an object is placedit a@ects the geometry of all later placements, so the dominant e@ect in RSA is theblocking of the available substrate area and the “jamming” coverage �(∞) is less than

∗ Corresponding author. Fax: +381-21-350-770.E-mail address: [email protected] (Lj. Budinski-Petkovi*c).

0378-4371/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(01)00354 -5

Lj. Budinski-Petkovi�c, U. Kozmidis-Luburi�c / Physica A 301 (2001) 174–180 175

in close packing. The temporal approach to the jammed state is described by the timeevolution of the coverage �(t), which is the fraction of the substrate area occupied bythe adsorbed particles.

Depending on the system of interest, the substrate can be continuum or discrete andRSA models can di@er in substrate dimensionality. In one dimension most problemshave been solved analytically [3,4]. Theoretical studies of RSA also include seriesexpansions [5–7] and Monte Carlo simulations [8–13]. For lattice RSA models, theapproach to the jamming coverage is exponential

�(t) = �(∞) − Ae−t=� ; (1)

where A and � are parameters that depend on the shape and orientational freedom ofdepositing objects.

The RSA model, however, describes only the limiting cases of real adsorption pro-cesses. A more realistic treatment should take into account possible di@usion or des-orption of the adsorbed particles [14–17]. Such generalization is appropriate for manyphysical, chemical and biological systems [18–21].

Allowing desorption makes the process reversible and the system ultimately reachesan equilibrium state. Adsorption–desorption processes in one dimension were studiedanalytically [14] and an exact solution for the equilibrium properties was obtained.The density of the particles in the steady state was found to be the function only ofthe ratio of the adsorption rate to the desorption rate. The Monte Carlo simulationof the adsorption–desorption process in one dimension gave an exponential approachto the steady state: �eq − �(t) ˙ e−t=�. On the other hand, an adsorption–desorptionprocess of rods on a line with vanishingly small desorption was studied in Ref. [15]and a slow logarithmic approach to a close packed state was found.

Generalized RSA of hard spherical particles onto a Kat uniform surface has beenanalyzed in Ref. [17] by distribution function approach. Unfortunately, the validity ofthe coverage expansions are limited to low-to-intermediate coverages. A descriptionof the available surface function � by its Arst four terms in � is valid up to a coverageof 35% for the RSA and 30% for the equilibrium case [22,23]. This allows consideringvalues of the desorption=adsorption rate ratio (kdes=ka) that is of the order of 0.5 orlarger. For smaller values of kdes=ka results can be obtained by numerical simulations.

2. De�nition of the model and the simulation method

The Monte Carlo simulations of adsorption–desorption processes are performed ona square lattice of size 96 × 96. The adsorbing objects are squares of size 3 × 3,covering nine lattice sites. Periodic boundary conditions are used in both directions, sothat Anite-size e@ects are negligible.

At each Monte Carlo step, a lattice site is selected at random. Then the attemptedprocess is selected: adsorption is attempted with probability Pa and desorption withprobability Pdes = 1−Pa. In the case of adsorption we try to place the object with the

176 Lj. Budinski-Petkovi�c, U. Kozmidis-Luburi�c / Physica A 301 (2001) 174–180

center at the selected site, i.e., we search whether all relevant sites are unoccupied. Ifthe selected process is desorption and if there is a center of a previously adsorbed objectat the selected site, the object is removed from the surface. It should be emphasizedthat it is only the ratio of the adsorption to the desorption probability that matters.

The time � is counted by the number of attempts to select a lattice site and scaledby the total number of lattice sites L2. It is also convenient to rescale the time to theadsorption process, so the time is measured by the adsorption attempt per lattice site.Let us deAne the time variable

t=Pa� (2)

that corresponds to the Axed deposition attempt rate that is typically a directly experi-mentally controlled quantity.

The data are averaged over 100 independent runs for each adsorption probability.The number of attempts is always taken to be large enough to bring the system to theequilibrium state.

3. Kinetics of the adsorption−desorption processes

The simulations are performed for various values of the adsorption=desorption proba-bility ratio, i.e., for various adsorption probabilities given in the Arst column of Table 1.The dependence of the coverage on time is shown in Fig. 1 for the processes with ad-sorption probabilities Pa = 0:3; 0:6 and 0:9. We can see that the system indeed reachesa steady state in which the rate of adsorption is exactly balanced by desorption.

Table 1Values of the relaxation time � and the equilibrium coverage �eq forvarious adsorption probabilities

Pa � �eq

0.98 690 0.83820.96 185 0.79860.94 74.5 0.77280.92 39.8 0.75330.90 19.3 0.73710.80 4.43 0.67870.70 2.27 0.63620.60 1.20 0.59770.50 0.549 0.56030.40 0.312 0.52030.30 0.153 0.47280.20 0.0968 0.41160.10 0.0490 0.31240.08 0.0391 0.28290.06 0.0322 0.24600.04 0.0231 0.19630.02 0.0135 0.1248


Fig. 1. Evolution of the coverage for Pa = 0:3; 0:6 and 0:9. The system always reaches an equilibrium stateand the relaxation time, as well as the equilibrium coverage, depends on the adsorption=desorption rate ratioPa=Pdes.

Except in very early times, the plots of ln(�eq − �(t)) are straight lines for alladsorption probabilities. This suggests that the approach to the equilibrium state isexponential

�eq − �(t) ˙ e−t=� : (3)

The values of the relaxation time � are determined from the slopes of these lines andthey are given in Table 1 together with the equilibrium coverages. For large valuesof the adsorption=desorption probability ratio the approach to the equilibrium state isvery slow and � is large. When Pa=Pdes decreases, the relaxation time also decreasesand for small enough adsorption probabilities the steady-state coverage is reached inless than one adsorption attempt per lattice site. The plot of

√� vs. Pa=Pdes shown

in Fig. 2 indicates that a power law � ˙ (Pa=Pdes)2 holds. For the one-dimensionalmodel and for Pa=Pdes�1 analytic results were obtained [24]. The plot of ln(1=�) vs.ln(Pa=Pdes) is approximately a straight line suggesting a power-law dependence in thesmall desorption limit.

The equilibrium coverage also depends on the ratio Pa=Pdes. For large adsorptionprobabilities (and small desorption probabilities) the equilibrium coverage has greatervalues than the jamming coverage in the pure RSA process. Small desorption proba-bility enables a slow rearrangement of the adsorbed particles on the surface and someof the blocked sites become available for the adsorption. When Pa=Pdes decreases, theequilibrium coverage decreases and from Table 1 we can see that the equilibrium cover-age is equal to the jamming coverage in the process without desorption (�(∞) = 0:6804for Pa=Pdes � 4). For smaller values of this ratio adsorption is slow comparing to des-orption and the equilibrium coverage is less than the jamming coverage. The values


Fig. 2. Dependence of√� on Pa=Pdes. The data lie on a straight line, suggesting that a power law

� ˙ (Pa=Pdes)2 is valid.

Fig. 3. Equilibrium coverage vs. Pdes=Pa for Pdes ¡Pa and vs. Pa=Pdes for Pdes ¿Pa. The dotted linerepresents the value of the jamming coverage in the RSA case.

of equilibrium coverage are shown in Fig. 3 vs. Pdes=Pa for Pdes ¡Pa and vs. Pa=Pdesfor Pdes ¿Pa. The dotted line shows the value of the jamming coverage in the casewithout desorption. The results for Pdes=Pa ¡ 0:5 are particularly interesting, since theresults obtained by the distribution function approach are not valid for small desorption


Fig. 4. Equilibrium coverage vs. Pdes=Pa for Pdes�Pa. The dotted line represents the exponential At of theform: �eq = 0:751 + 0:194 exp(−(Pdes=Pa)=0:0267).

probabilities. The errors for both the relaxation time and the equilibrium coverage areestimated to the last given digits.

For Pdes�Pa the equilibrium coverage decreases exponentially with Pdes=Pa, asshown in Fig. 4. The dotted line represents the exponential At of the form

�eq = �0 + Ae−(Pdes=Pa)=� (4)

with �0 = 0:751, A= 0:194 and �= 0:0267. Thus, the presence of desorption, even ifslight, signiAcantly changes the coverage of the surface.

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adsorption–desorption processes of extended objects on a square lattice

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