Russian Physis Journal, Vol. 48, No. 8, 5 CRITICAL EXPONENTS TAKING INTO ACCOUNT DYNAMIC SCALING FOR ADSORPTION ON SMALL-SIZE ONE-DIMENSIONAL CLUSTERS A. N. Taskin, V. N. Udodov, and A. I. Potekaev UDC 54.8 Adsorption on small-sie one-dimensional lusters is investigated using the Monte Carlo method. The effet of temperature and system sie variations on adsorption is studied. Critial exponents of the orrelation length and dynami ritial exponent are alulated taking into aount the hypothesis of dynami saling. The results obtained demonstrate that non-equilibrium adsorption in nanosystems an our in a muh different fashion than in marosystems. INTRODUCTION From the physial viewpoint, of great interest are those phase transitions that our under adsorption onto small lusters [, ]. An interation may be, for instane, suh that at low temperatures the attahment of adatoms to the rystal surfae might be advantageous, resulting in a ontinuous one-dimensional hain []. As the temperature is inreased, part of the adatoms evaporates, that is to say a phase transition takes plae. The thermodynami hypothesis of statistial saling used in this work was formulated for the Ising model in [], for the gas liquid transition in [4], for ferromagneti systems in [5], and later was extended onto non-equilibrium systems [6]. Here we disuss an investigation of the kinetis of non-equilibrium adsorption on small lusters taking into aount the hypothesis of dynami saling, and also present the alulated ritial exponents.. MODEL One of the models allowing for the desription of adsorption is the Ising model that allows one to obtain ritial exponents [7], with the lattie-gas version of this model used in this ase. We onsider here ergodi systems and perform alulations using the Monte Carlo tehnique (Metropolis s algorithm [8 ]), taking into aount the nearest neighbor interations and the external-field effets. The external field harateristi, H, is a ertain parameter denoting the interations of adatoms with the substrate. The interations of adatoms with eah other are taken to be muh stronger than with the substrate. Attention is foused on one-dimensional systems of the length L = N 6, whih are made by the sites to whih atoms an adhere. The energy of a system with free ends has the following form []: N N = i = i i i+ i= i= ([ ] ) ( ), () E E n H n J n n where n i are the filling numbers for the adatoms in the lattie gas sites [7, 9 ] (n i = or ), i is the site number, J is the interation energy of the nearest adatoms, and N is number of sites in one-dimensional lattie gas. As long as the model is taken to be parametri, it should be borne in mind that the presene of a substrate is inluded into J. N. F. Katanov Khakassiya State University, e-mail: udodov@khsu.ru; V. D. Kunetsov Siberian Phisial Tehnial Institute at Tomsk State University. Translated from Ivestiya Vysshikh Uhebnykh Zavedenii, Fiika, No. 8, pp. 8 87, August, 5. Original artile submitted February, 5; revision submitted July 8, 5. 64-8887/5/488-87 5 Springer Siene+Business Media, In. 87
ν.7.5.. -..5 <T> Fig.. Dependene of ν on <T> for different N : 5 (urve ), (urve ), and 5 (urve ). (H =., J =.) ν ν.8 a.8 b.6.6.4..4. 5 5 N 5 5 N Fig.. Dependene of ν on N for different <T>:.9 (urve ),.7 (urve ), and.5 (urve ). (H = J = (a) and H =., J = (b).). CRITICAL EXPONENTS y AND ν Using the Metropolis algorithm, we find the dependenes of relaxation time and orrelation length on the number of adatoms in the system and temperature. The variation in the orrelation length, ξ, with variations in temperature is harateried by its ritial exponent ν [ 4]: ξ = l = A Θ, () max ν where <l max > is the averaged maximum length of a ontinuous luster made of sites filled by adatoms, A is a ertain kt proportionality oeffiient, Θ= is the redued temperature (further in the text denoted as T ), and k is the Boltmann J onstant. Equation () takes into aount that the phase transition temperature (as the points of singularity of thermodynami funtions) is equal to ero for a finite system. For suh a system, А depends on the number of sites, N, in it. The dependenes of ν on temperature and number of sites are shown in Figs. and, respetively. Sine the onstant A is 874
unknown, so in order to alulate the ritial exponent of the orrelation length use was made of two lose values of temperature, for whih the orrelation length values had been alulated (. and.5,. and.,.5 and.,.5 and.5, et.), that is <T> is the simple average of these values. From Fig. it an be readily understood that as the temperature is inreased, v tends to inrease too. This implies that for eah inrease in the sie of the hain ξ dereases with inrease in temperature (note that А is a onstant or varies but slightly), and the longer the hain, the faster this happens. It is evident from Fig. that v inreases with N at any value of Т. This implies that as the sie of the adatom hain is inreased, the dependene of orrelation length on temperature is getting stronger, at elevated temperatures this happens at a muh faster rate. Under weak fields (Fig., b), this effet is more onspiuous (Fig. b versus Fig. a). When the phase transition point is approahed, the orrelation length ξ, aording to Eq. (), tends to infinity in an infinite system. For small systems, the value of ξ annot beome larger than the sie of the system. The phase transition point for a finite system is, therefore, smeared over to the ritial region [8 ], where ξ is equal to the sie of the system, in this ase ξ = N. The ritial exponent Y (Figs. and 4) harateries the variation in relaxation time with temperature [6] τ= onst. () From Fig. a and b, it an be readily seen that for simultaneous weakening of the external field Н and interation of the nearest neighboring adatoms J, the respetive values of Y are observed in the region of lower temperatures. One might expet that for simultaneous strengthening of the external field Н and interation J, the relaxation time τ would derease. With the interation of adatoms dominating (J > Н), the effet of the sie of the system is smaller (Fig. b versus Fig. a), i.e., the differene in Y for different N beomes onsiderably smaller. Note also, that the relaxation times in the systems of different sie would get loser in values, whih suggests that interation of adatoms exerts a dominating influene on relaxation time. Figure 4 implies that with inrease in temperature Т there is also an inrease in Y, that is to say that we observe a shorter relaxation time τ for any sie of the system N. T Y. PROCEDURE FOR CALCULATING THE DYNAMIC CRITICAL EXPONENT Z Coeffiient determines the relaxational behavior near the phase transition point [6] τ ~ ξ, (4) where ξ is the orrelation length and is the dynamial ritial exponent. For many models, in a thermodynami limit (in partiular, the Landau Khalatnikov theory) = [6]. For one-dimensional finite-sie models, the relaxation time of the ordering parameter in the ritial region for is determined as ξ = L = N (5) τ =. (6) ~ L N Let us now look at the relaxation time as a funtion of the sie of the system, with time τ alulated in dimensionless units (Monte Carlo steps) [9 ]. First, alulate the ritial dynamial oeffiient to a ero approximation []. Further, we will inlude into onsideration the orretions following from the hypothesis of dynami saling (similitude) [] 875
Y.8 a Y.8 b.6.4.6.4...6.8 <T>...5.7 <T> Y.8.6.4....5.7 <T> Fig.. Dependenes of Y on <T> for different N: 5 (urve ), (urve ), and 6 (urve ). (H = J = (a), H = J =.5 (b), and H =., J = ().) ξ τ= N f, T N, N (7) where f is an unknown funtion of two arguments. Assuming that Eq.(5) would be valid in the ritial region, we may from Eq. () obtain the width of this region T as follows: ~ N ν T. (8) Substituting Eq. (8) and ondition (5) into Eq. (7), we arrive at where (, ) ( ) Z τ= N f T N = N f α, (9) 876
Y.5 4.5 5 5 N Fig. 4. Dependenes of Y on N for different <T>:.5 (urve ),.9 (urve ), average over the entire set of data (urve ), and.6 (urve 4). (H =., J =.) N ν α=. () Expanding funtion f into a series with respet to the small parameter α (that depends on temperature, number of sites and ritial exponent ), we obtain where = and с, f, f, are the fitting parameters, where (...) τ= N f + f α+ f α +, () N f = () τ is the oeffiient of proportionality to a ero approximation. From the physial standpoint, the series, Eq. (), would onverge. Taking f = f = = and to a ero approximation, from Eq. () we might find to a first approximation by varying parameter f and seleting those values for whih the r.m.s. deviation of would be minimal. Further on, is found to a seond approximation by hanging parameter f, et. As a first approximation, on average., i.e., inreases ompared to. From the expansion, Eq. (), one an readily find the dependene of f on f (f = f 4 = = ) τ f f N = α f +. () α For the onvergene of the series, Eq. (), it is neessary to find suh values of с and f, for whih the terms in the series would derease in their absolute value. A system of inequalities was built 877
a > b < f f Fig. 5. Diagrams of dependene of с on f at N = 8, =., ν = (a) and.5 (b). The system has no solution (), either of the inequalities is not fulfilled (, ), the region where the system has a solution (). М = 6. f f >αf >αf,, (4) aording to whih the diagrams of dependene of с on f were onstruted (Fig. 5). Four regions are observed in the figure: the system has no solution (), either first or seond inequality is not fulfilled (, ), and the system has a solution (). Thus, it is for region that the terms of the expansion series derease in their absolute value. For instane, if we take N = 8, =., ν =, =.5 and f = 5 6, then we obtain a dereasing skew-symmetri series: τ N Z (.78.7 +.85 ). Thus, the values of the fitting parameters have been found for whih the first terms in the series are monotonially dereasing in their absolute value. An interesting regular feature has been also disovered: there exists suh a value of the fitting parameter с (in Fig. 5 it is a horiontal line above с = ), for whih the variation limits of f hange sharply. Sine the region of the diagram where, taking into aount saling, <, is muh smaller in area than the region in whih >, there is reason to expet that in the majority of ases would be larger than. This implies that exponent inreases (very likely, by more than a fator of 5) ompared to the ero approximation. Thus, the estimates of the ritial exponents of orrelation length,ν, dynamial exponent, Y, and dynamial ritial exponent,, for one-dimensional adsorption in small-sie systems have been performed. In partiular, an average value of the orrelation length exponent has been found <ν>.5, whih agrees with that from the Landau theory of phase transitions [4]. It has been established that, on average, ritial exponent Y inreases with temperature. The value of dynamial ritial exponent has been found as a ero approximation. Taking into aount the obtained values of these exponents, the values of have been derived in aordane with the hypothesis of dynami saling. It has been demonstrated that an inrease by a fator of 5 6 ompared to its value to a ero approximation, reahing a value of and over, whih is by far larger than for marosystems (typially = ). This suggests that the dependene of relaxation time on the sie of a system for the ase of adsorption on small-sie lusters is stronger than it is for marosystems. Adsorption on nanolusters, would, therefore follow a different route than it does in marosystems. SUMMARY The estimates of ritial exponents for the ase of adsorption in small-sie systems have revealed that the dependene of a number of harateristis on the system sie is muh stronger than it is in marosystems. This implies that non-equilibrium adsorption in nanosystems would our in a different fashion than it does in marosystems, provided the hypothesis of dynamial saling is valid. 878
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