Kinetics of irreversible deposition of mixtures N. Švrakić, Malte Henkel To cite this version: N. Švrakić, Malte Henkel. Kinetics of irreversible deposition of mixtures. Journal de Physique I, EDP Sciences, 1991, 1 (6), pp.791-795. <10.1051/jp1:1991170>. <jpa-00246371> HAL Id: jpa-00246371 https://hal.archives-ouvertes.fr/jpa-00246371 Submitted on 1 Jan 1991 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
PhysicsAbswacJs 68,10J 05.70L N-M- vrakib and Malte Henkel(*>**) Monte Carlo results are reported for the kinetics of random sequential deposition of Abstract. of line segments of two different lengths on the square lattice. It is shown that the rate of mixtures deposition is independent on the type of mixture, but that kinetics is governed by a novel late-stage not observed in single-species adsorption. On the basis of our data, a simple expression mechanism, Irreversible deposition, or the Random Sequential Adsorption (RSA) process is defined as are not allowed to overlap, and must stay permanently fixed, once deposited. Initially, objects substrate h empty, and the process ends when the jamniing limit is reached, I-e-, when it is the to place further objects. Due to the blocking of the area by the already deposited impossible the jamming density is less than the close packing. The emergence of this jammed state objects, theory. This model finds applications in the experiments on the adsorption of proteins mean-field latexes [2], and colloids [3,4], on solid substrates. In the earlier theoretical work [5] it was also [I], "the car parking problem" termed kinetics of the process is characterized by the time evolution of the coverage, 0(t), I-e-, the The RSA, which include Monte Carlo approaches [9-16], series expansions ii I], rate equations of etc., both continuous and discrete models were analyzed. Exact results are also available [8,10,12], discrete models) approach to the jamming limit. Thus, for discrete models, one writes for long Permanent address: D6pt. de Physique Th60rique, Universit6 de Gen~ve, 24 quai Ernest Ansermet, (* Gen~ve 4, Switzerland. CH-1211 1 Phys. I 1 (1991) 791-795 JUIN 1991, PAGE 791 Classification Short Communication Kinetics of irreversible deposition of mixtures HilchstJeistungsrechenzentrum des Forschungszentrums, Postfach 1913, D-5170 Jolich I, Germany (Received 29 March 1991, accepted 5Apfil 1991) describing the late stage approach to the jamming limit is obtained. follows: objects of a specified shape are sequentially and randomly deposited on a substrate. The is governed by the infinite-time memory effects, and its formation can not be described within the fraction of the substrate area covered by the deposited objects. In previous theoretical studies mostly for one-dimensional models. In such studies it was established that the latestage [6,7,13,14,17], deposition kinetics follows either a power law (for continuous models) or exponential (for times 6(t) 0(cc) A exp(-t la), I) (** Supported in part by the Swiss National Science Foundation.
shape, size, etc., of the adsorbing object. In the continuous models onc has oil 1l(c~c) -~ [7] and Swendsen [17]. Pomeau little is known about kinetics of mirture deposition, I-e-, adsorption of two (or Comparatively different kinds of objects. Available studies include various approximation schemes [19, more) but no systematic theoretical picture has emerged. Avery recent exact study of mixture de- 20], position [21] in d lattice. Let us first briefly Summarize relevant results for single length deposition. square the recent study [22] of Single-species deposition (lines of single length I) on the square In -~ ill), - Do, of the previously Fig. I. A typicaljammiug configuration for the mixture ofliue segments ufleugth ii 5 and 12 792 JOURNAL DE PHYSIQUE I N 6 where 1l(c~c) is the jamming coverage, while A and a are parameters which may depend on the ~/ ~, t~ d is the spatial dimensionality, a result first conjectured by Feder [18], and later proved by where I indicates that the time evolution of the coverage may be modified in a way. In this communication we report our results of a Monte Carlo study of mixture nonuniversal kinetics, and attempt to develop a more systematic approach to this problem. Specif- deposition ically, we study kinetics of deposition of lines segments of two different lengths (li and 12) on the lattice, the exponential form (I) was confirmed, and it was found that b( cxj) and A depend on the length (specifically, A while the rate a is independent of this length, I-e-, segments of any length approach the jamming limit with the same rate. The value of this rate is numerically found to be a ct 0.5. The jamming density1li(c~c) decieases with increasing I, I-e-, Shorter Segments more efficiently cover the suiface than the longer ones, as expected on heuristic grounds. As I this density approaches a definitive limit lli_n~ (c~c) m 0.583. the present simulation we randomly select a site on a square lattice of size L x L, and attempt In deposit the segment of randomly chosen length (li or 12) along the lattice axis. (For definite- to ness, we will always take fi < 12.) If the segment does not intersect with any ones, it is permanently deposited at this site. If it intersects with other lines, the attempt adsorbed abandoned, the segment h disposed, and the new site and new segmcni is selected. In either is the time is increased by one unit. The jamming limit is reached when no segment can be case, without intersection, on any site of the lattice. deposited, The typical jamming configuration is shown in figure I. In this figure, segments of length fi 5 20 on a 128 x 128 lattice.
the species separately(!). We believe that thin result has its origin in the specific kinetic process of mixture deposition which we discuss below. We have confirmed this relation by considering of -9 llj, j, > llj, j~, I The approach of the coverage 81,,ia (t) to the jamming limit di~,i~ Fig. is plotted using a semi log co with time t for segment lengths li 1, 12 scale 1, 2, 4, 8, 16, 32, 64, from bottom to top. order to analyze kinetics of deposition, we have followed the time evolution of the coverage In simulation. Guided by the single species results [22], and by the d I results of Bartelt during and Privman [21], we assume the following time-dependence lengths 12.) The lowest (separated) curve corresponds to the (I,I mixture, I.e., the single segment case. Other curves are for the (1,2), (1,4),...(1,64) mixtures, starting from the bottom. A Species N 6 KINEUCS OF IRREVERSIBLE DEPOSI~ON OF MIX'IIJItES 793 and 12 20 are deposited on the 128 x 128 square lattice with periodic boundary conditions. The density llj,,j~ (c~c) is found to obey the following interesting relationship llj,,j~ > llj, and jamming > llj~, I.e., the mirture of segments of two lengths covers the area more cficiently than either llj,,j~ mixtures of fi 1, 2, 4, 8 with 12 2, 4~ 8, 16, 32, 64, on a 1024 x 1024 lattice. The densities are obtained by averaging over 25 independent jamming configurations for each (Ii, 12 mixture. Furthermore, we found that jamming density obeys a more general law, I-e-, (2) '2 whenever 1[ > 12. We will discuss this relation later. o 2 4 6 8 lo t of,,12(~) " bl,,12( ~) Al,,12(~) ~~Pl~~/~l,,tj ). (3) The plot of In[bi,,i~ (c~c) bi,,i~ (t )] versus time is shown in figure 2, for the ii l~ mixtures of li I, with 12 1, 2, 4, 8, 64. (That is, we fix the (shorter) segment length ii, and mix it with variable at figure 2 reveals the following: (I) all curves become linear for longer times, I.e., the latetime glance kinetics follows the exponential law (3), with Al,,i~ (t) independent of time; (it) the lines are
$ ~f -2 O.25 794 JOURNAL DE PHYSIQUE I N 6 o '~~
thank M.C.TP Bartelt and V Pfivman for informing us of their results prior to publication, We to M. Konstantinov16 for the explanation of the experimental aspects of RSA. One of us fl4h) and thanks the HLRZ for its warm hospitality. 78 (1980) 144. Sci. and LINIGIER E.G., Phys, Rev A 33 (1986) 715. G,Y. TOMIt M., BI KUP B., KUNJA It I. and MATIJEVIt E., COT%ids Siif 29 (1987) 185. N., M. and R. PAmL, preprint (1991) KONSTANTINOVIt PJ., J Am. Chem Soc. 61(1939)1518. FLORY B.,1 Cheil~. Pliys. 44 (1966) 3888. WIDOM Y., J Phys. A 13 (1980) L193. POMEAU J. and NORD R-S-, I Siai. Phys. 38 (1985) 681. EVANS M., Phys. Rev A 36 (1987) 2384. NAKAMURA P., TALBOT J., RABEONY H-M- and REISS H., I Phj,s. Diem. 92 (1988j 4826. SCHAAF R-D- and ZIFF R-M-, J Chem. Phys. 91 (1989) 2599. VIGIL P and TALBOT J., Phys. Rev Lett. 62 (1989) 175. SCHAAF M-C- and PRIVMAN V, J Chew Phys. 93 (1990) 6820. BABTELT P, PRIVMAN v and WANG J-S-, f Phys. A 23 (1990) L1187; P/ij<s. Rev B 43 (1991) 3366. NIELABA J.D., f Phys. A 23 (1990) 2827. SHERWOOD R-M- and VIGIL R-D-, I Phys. A 23 (1990) 5103. ZIFF R., Phys. Rev A 24 (1981) 504. SWENDSEN f 7heor BioL 87, (1980) 237. FEDERJ., SCHAAF R, Pliys. Rev A 40, (1989) 422. and M-C- and PRIVMAN v, preprint (1991). BARTELT N 6 KINEUCS OF IRREVERSIBLE DEPOSI~ON OF MIXTURES 795 Our calculations took about 60 CPU hours on one processor IBM 3090 without a vector feature at HLR2L Acknowledgements. References NAKAMURA M., J Pliys. A 19, (1987) 2345. [22] MANNA S-S- and (vrakit N-M-, f Phys. A (1991) in print.