Temperature and Composition Dependence of Kinetics of Phase Separation in Solid Binary Mixtures
We present results for the kinetics of phase separation in solid binary mixtures () from Monte Carlo simulations of the Ising model in two dimensions. The simulation results are understood via appropriate application of the finite-size scaling theory. At moderately high temperatures, for symmetric compositions of and particles the average size of the domains exhibit power-law growth with the exponent having the Lifshitz-Slyozov value of from very early time. However, our analysis shows that for low enough temperatures, the growth exponent at early time is smaller than the Lifshitz-Slyozov value. For composition dependence, we find that at moderate temperature, even for extreme off-critical composition, curvature dependent correction to the growth law is weak which is counterintuitive in this case that gives rise to droplet morphology. This is however consistent with the recent understanding on curvature dependence of surface tension. Results from rather general studies on the finite-size effects with the variations of temperature and composition have also been presented.
Understanding of the thermodynamic and dynamic properties of multi-component mixtures in the miscible as well as immiscible phases are of crucial technological importance (1); (2); (3); (5); (4); (6). Nevertheless, it appears that some basic understandings for mixtures involving even two components () are rather incomplete, particularly certain aspects in nonequilibrium dynamics. In this work we study the kinetics of phase separation in a solid binary mixture and address questions primarily related to coarsening dynamics involving nanoscopic length scales.
When a homogeneously mixed binary mixture is quenched inside the miscibility gap it becomes unstable to fluctuations and starts evolving towards its new equilibrium state where and phases coexist. During this evolution, - and -rich domains form and grow with time until the system entirely phase separates (3); (5); (4); (6). Such domain coarsening is a scaling phenomena where patterns at two different times are self-similar. This is reflected in the properties of various morphology characterizing functions, e.g., the two point equal time order-parameter correlation function, , obeys the scaling relation
where is a scaling function, independent of , the average domain size. The function provides information on the type of pattern formation during the evolution and is expected to be different for different situations (3) dictated, e.g., by the presence or absence of order-parameter conservation. Typically has power-law dependence on time as (3); (5); (4); (6)
where and are respectively the growth exponent and amplitude. The value of depends on the order-parameter conservation, system and order-parameter dimensions and other factors. Here, of course, we are dealing with phase separation where the order-parameter is conserved and for this particular problem, is a scalar quantity.
For conserved order parameter, depends upon the transport mechanism which is different in solids and fluids (3); (5); (4); (6). The growth amplitude, in each of these cases, depends on temperature, composition and material under consideration. In case of solid mixtures the growth occurs via diffusive mechanism. In that case one can relate the rate of increase of with the chemical potential () gradient and write (3); (7)
where is the interfacial tension. Solving Eq. (3) one obtains . This is referred to as the Lifshitz-Sloyozov (LS) law (8). The LS behavior is the only growth law expected for phase-separating solid mixtures when there is prominent bulk diffusion. As we will discuss later, depending upon the value of one may expect some correction to the LS law, of course. Also, dominant interface diffusion at very low temperature can give rise to different value of the exponent. In fluids, hydrodynamics being important, the growth should be much faster (3); (4); (6); (9); (10); (11); (12). In this case, in addition to the diffusive growth, there exist two more regimes, at late times (9); (10); (11); (12); (13): first is the viscous hydrodynamic regime where domains grow linearly with time and the next is inertial hydrodynamic regime where one expects . For the sake of brevity we do not provide further details on fluid phase separation here.
where one can identify the spin at lattice site with an -particle (-particle). In Eq. (4) stands for summation over the nearest neighbors (nn). An alternative method of studying the kinetics of phase separation in solid mixtures is via the coarse-grained dynamical equations, viz., the Cahn-Hilliard (CH) equation (17)
where is a coarse-grained space- and time-dependent order parameter. The CH equation, when supplemented with thermal noise, should be equivalent to MC simulations of Kawasaki Ising model. Kinetics of phase separation in such simple but useful models have been studied mostly after quenching the systems to moderately low temperature along the critical composition line (7); (18); (19); (20); (21). Beyond that, it is also of significant importance to understand the effects of temperature () as well as composition (22) on the growth law. In this paper we will deal with these issues using the Kawasaki Ising model.
The LS law () for diffusive growth is expected to be valid for the situation when one has dominant diffusion in the bulk (). In that case, variation of temperature (quench depth) is expected to bring changes in the growth amplitude only. On other hand, at very low temperature, diffusion along interfaces () may play the dominant role (23); (24); (25); (26). It is pointed out that with the lowering of temperature a change in the exponent (23); (24); (25); (26) from to a smaller value should happen. This can be justified via the following arguments (26). In this case, it may be useful to write the CH equation with concentration dependent mobility. A simple form for such mobility (26) is
and is the equilibrium domain magnetization. At moderate temperatures when and are comparable, one has nonzero mobility in the bulk. On the other hand, for very low , it is argued that and in the bulk everywhere. This gives rise to negligible bulk mobility. However, since at the interfaces, one has nonzero mobility of particles there. Such dominant interface mobility may be responsible for a smaller exponent () at low temperature. From a dimensional analysis of the mobility dependent CH equation, Gemmert et al. (26) worked out the crossover from to , as a function of . Further, it is pointed out that this interface diffusion mechanism will provide only at early time and in the large length scale limit the LS value will be recovered, even at low enough temperature.
In case of composition variation, as one moves away from the critical value, the domain pattern changes from interconnected structures to droplets of the minority species in the background of the majority sea. For fluid phase separation (27); (9); (28); (11); (29); (30), this difference in morphology brings in striking and important change in the mechanism and thus the exponent of the growth law. In solid binary mixtures, however, one does not expect the mechanism to change due to the variation of composition. Here, an interesting and important objective could be to search for the correction to the scaling law (7); (20); (21), if any, due to the finite, say nanoscopic, radius of curvature of well defined droplets. Possibility of such correction at early time was discussed even for composition (7). However, recently (20); (21) it has been demonstrated that for symmetric composition, such finite time correction is negligible due to the flat boundary in an “all time” elongated interconnected domain structure. Away from critical composition, a possible reason for the curvature dependence of the growth exponent could be due to the correction in the surface tension in droplet geometry as (31); (32); (33); (34); (35); (36); (37)
where and have dimensions of length. Another contribution can possibly come from the curvature dependence of the kinetic prefactor (7) (not shown) in Eq. (3). Here, we note that in Ising-like symmetric models, the Tolman length (31) () is absent (32); (35); (36); (37).
Athough the current interest in kinetics of phase separation lies in much more complicated systems (38); (39); (40); (41); (42); (43); (44); (45); (46); (47); (48) with realistic interactions and boundary conditions, many of the basic facts, as discussed above, are still incompletely understood even for simple Ising model. Further, a major challenge in computer simulations of nonequilibrium systems is the finite-size effects (22); (20); (21) that pose difficulty in extracting the growth exponent in the asymptotic limit. Thus understanding of finite-size effects is also of crucial importance.
In this work, we have undertaken a detailed study of kinetics of phase separation in solid binary mixtures via Monte Carlo (MC) simulations of Kawasaki exchange Ising model in two spatial dimensions (). Our studies have particular focus in the small domain size limit. We present results on the effects of variation of temperature as well as that of composition both on the morphology and growth dynamics. In addition, we provide important quantitative information on the finite-size effects. To obtain information on the growth law, the corresponding correction at early time and the finite-size effects, we relied on finite-size scaling analysis (15); (49); (50), among other techniques.
A. Method of Simulation
As already stated, we performed MC simulations (15); (16) of Ising model in , keeping the order parameter conserved. This conservation in the dynamics was implemented via the Kawasaki exchange mechanism (15); (14) where a trial move consisted of interchanging positions between a randomly chosen pair of nn spins. Standard Metropolis algorithm (15) was used to accept or reject a move. Such exchange trials for pairs ( being the linear dimension of a square lattice, in units of lattice constant , used for the simulations) of spins form one Monte Carlo step (MCS). In all our simulations, periodic boundary conditions were applied in both - and - directions.
By now it is clear that the central quantity of interest in this study is the average domain size . This can be calculated via the scaling properties of various morphological functions. In this work we have calculated by exploiting three such functions (3): (a) From the decay of the correlation function (51)
more precisely from the distance where crosses zero for the first time (note that for such conserved dynamics exhibits damped oscillation around zero before decaying to the later); (b) From the first moment of the structure factor ( being the wave vector) which is the Fourier transform of ; (c) Finally, from the first moment,
of the normalized domain-size distribution function , where is the separation between two interfaces in or directions. Results from all three different methods are found to be consistent with each other for all temperatures and compositions, differing only by constant numerical factors. So, for the presentation purpose, unless otherwise mentioned, we will use results from only one of them, viz., method (c).
Note that the above mentioned methods of calculating not always give the accurate measurment, particularly at high temperatures, due to the presence of noisy clusters of the size of equilibrium correlation length . Hence we have calculated from noise free morphology by implementing a majority spin rule (20); (21); (52). In this method one replaces a spin at a particular lattice site with the sign of the majority of the spins involving the spins sitting at that site and its nn. Depending upon the temperature, one may need to improve the quality of noise filtering by including distant neighbors or increasing the number of such iterations.
B. Method of analysis
In this work, we have extensively used the finite-size scaling method (15); (49); (50) for the interpretation of simulation results. So, it will be useful to provide a discussion on this technique to improve readability of the paper. Following Ref. (21), below we briefly outline the basic working concept.
In critical phenomena (53), the singularity of a quantity , in the thermodynamic limit, is quantified in terms of the reduced temperature ( being the critical temperature) as
where is a critical exponent. For example, the correlation length diverges as
A central objective in critical phenomena is to obtain accurate information about the exponents. In finite systems, however, such divergences are restricted, e.g., cannot be larger than the system size . In that case, to account for the size effect, one introduces a scaling function , to write (49); (50)
where is a dimensionless scaling variable that provides information on the deficiency of system size with respect to the thermodynamic limit. Note that in the limit ( or ) there is no size effect and one should recover Eq. (11). Hence should approach in that limit. Another important point is that should be independent of system size (see the definition of ). So, as the working priciple, one acquires data for different values of and uses the exponents involved as adjustable parameters to obtain data collapse in vs plot. The values that accomplishes this purpose describes the behaviors of the quantities correctly in the thermodynamic limit. In addition to adjusting the value of the critical exponents, sometimes (54) it may also be crucial to vary the background contribution (non critical part), coming from small length fluctuations, to obtain such data collapse. Here we note that the background parts usually have rather weak temperature dependence.
Similar scaling could be constructed in the nonequilibrium problems also, such as the present one, by identifying with and with . We point out that the domain growth in an infinite system should be quantified as (20); (21)
where and are temperature dependent quantities. The length could be interpreted as the average cluster size when the system becomes unstable to fluctuations at time since the quench. Hence the measurement of time starts from there, i.e. . Note that the scaling part in Eq. (14) is only , i.e, . This initial length is similar to the background in critical phenomena. Analogous to the weak temperature dependence of the backgrowund contribution in equilibrum critical phenomena, here we expect to be practically independent of time. Of course, if is very large, subtraction of length does not bring in major difference. However, in computer simulations, where one deals with finite systems, if is not subtracted correctly one can end up with wrong conclusion.
being the scaling variable. In Eq. (16), is the average length of domains in equilibrium that scales with . Both in Eqs. (15) and (16), is subtracted to work only with the scaling parts. The limiting forms of , which can be arrived at by using Eqs. (14), (15) and (16), are the following
Learning the full form of would also be an interesting task, though challenging. Our objective in this analysis would be to interpret the simulation results from the behavior of the scaling function which will be obtained from optimum data collapse by varying primarily . Here we mention that we have varied also to obtain optimum data collapse. In all the cases it appears that the value that serves the purpose best is .
In addition to the finite-size scaling, we have employed other methods of analysis as well which we will discuss subsequently while presenting the results.
In this section we present results for the kinetics of phase separation in Ising model with critical composition for various depths of temperature quench.
A. Temperature Dependence
In Fig. 1, we show the snapshots at MCS, from the evolutions at four different temperatures, after quenching a homogeneously mixed Ising system to the mentioned temperatures below . Two observations are in order. First, the growth is faster for higher temperature. Second, the thermal noise, as expected, increases as one goes closer to . As already mentioned, the later poses difficulty in accurate estimation of domain length . To overcome this problem, as previously discussed, we calculated from pure domain morphology after removing the noise. Quantitative justification of this is provided below.
In Fig. 2, we show the plots of vs , on a double log scale, for two different temperatures. For both the temperatures we have included results from noisy as well as noise-free environments. For the lower temperature, of course, both the curves almost overlap with each other. This is due to very low level of noise. Notice that the equilibrium values, , of (where the plots flatten) are different for the two temperatures for the calculation with noise. Here note that, since the results are presented for the same system size, must be same for both the temperatures. Indeed, this is the case when the calculation is done from configurations without noise. This justifies the method, of course. However, at very early time, when the domains are very small, the noise removal exercise brings some undesirable feature. In this regime, along with noise, many domains also get removed. But it could be seen that during very early time, for both the temperatures, calculated without removing the noise, match with each other, which is expected. In view of that, in our analysis, we will combine the data with noise at early time (say, upto MCS for the temperatures in Fig. 2) and the noise-free data from the rest of the period.
In addition to the methodological justification, as already mentioned, Fig. 2 also provides the quantitative information about the fact that the equilibration occurs faster for higher temperature. This implies that, if the exponent is same, the amplitude of growth is larger for higher temperature. We will investigate it in detail in the following.
In Fig. 3, we have shown the instantaneous exponent , calculated via
The results for both the temperatures are consistent with this form showing convergence to , for . From the temperature dependent slopes of these numerical results we can make a guess about the background length , and use it as a “good” trial value for the finite-size scaling analysis. From the exercise in Fig. 3, we obtain and for and , respectively. Here we note that the linear dependence of on should not be mistaken with curvature dependent corection (20); (21). This form merely appears due to nonzero value of .
In Fig. 4 (a) we show the exercise from the finite-size scaling analysis of data from different system sizes sizes at . (The original data on a direct plot are shown in the inset of this figure). This figure shows the plot of as a function a function with . Here was introduced to bring the infinity (corresponding to very early time) of the abscissa to a finite number. In this exercise we have fixed to and varied to obtain optimum data collapse. The corresponding value of is in strong agreement with that obtained from Fig. 3 using Eq. (20). Here, the flat part corresponds to the region unaffected by finite size of the systems. This flat behavior from very early time confirms about the absence of any significant length dependent correction to the LS value of the exponent. The deviation from this flat behavior provides us information about the onset of finite-size effects. At all temperatures we observe that this is rather weak. We will make quantitative statement about the size effect in the next subsection. Also note that the behavior of at small is linear, consistent with Eq. (17).
In Fig. 4 (b) we present a plot of amplitude , extracted from the value of the ordinate for the flat parts of the finite-size scaling, vs . One observes that monotonically increases with . From the quality of data, however, it is difficult to figure out if there is any specific critical behavior of this quantity.
Uptil now, all the results correspond to reasonably high temperatures. Next we present results for low enough temperature to see if there is any signature of , due to dominant interface diffusion at early time.
From the nature of data at early time, for , in Fig. 2, it appears that the interface diffusion mechanism (23); (24); (25); (26) is perhaps giving rise to a smaller value of . In fact we have struggled to obtain even a reasonable finite-size collapse of data at early time, for this temperature, by fixing . This is demonstrated in Fig. 5. A good collapse of data is obtained only for MCS. The nonscaling behavior of the data in the range (see inset) and the trend of the early time data coming from the lower side indicate that the exponent is lower at the beginning. This could, possibly be identified with the “expectation” of for lower temperature.
Further, our simulation results are also in nice agreement with the argument (26) that at lower temperature there should be crossover from to . This, however, means that bulk mobility, which is negligible at early time becomes significant at late time. By examining Eqs. (6) and (7), this is possible if the bulk order parameter decreases or increases with time. This is, we feel, counterintuitive and advocate for further studies to provide justification for that.
B. Composition Dependence
In this subsection, we present results for morphology and growth from the MC simulations of Ising model with various different compositions of and particles. Here all the results are obtained at .
Fig. 6 shows the snapshots during the evolution of a Ising system ( and ) starting from a random initial configuration. Requirement of energy minimization and very off-critical composition restricts the domain geometry of the minority species to droplets. Since the hydrodynamics is unimportant in solid, growth of droplets, after their nucleation, takes place via simple concentration diffusion mechanism. As already stated, our objective here is to find out correction to the LS growth law, if any, at early time, due to finite radius of curvature of the droplets.
In Fig. 7, we show scaling plots of the correlation function, for composition. Again, the length scale used here was obtained from the first moment of domain size distribution function. It is seen that one obtains good quality data collapse with the increase of time.
In Fig. 8, we show a comparative picture of snapshots from four different compositions at the same time ( MCS). It is apparent that as one moves closer to the symmetric or critical composition, the growth is faster. This is is due to the fact that with the decrease of overall concentration of particles, they need to travel longer to get deposited on a domain.
In Fig. 9, we show the plots for correlation functions from four different compositions for the late time snapshots shown in Fig. 8. No scaling behavior is seen because of the method of calculation of . Note that we have used only the -domains to calculate . But, from the definition of , it is clear that the information of both -rich and -rich regions are incorporated in it. Thus, we expect data collapse if in all the cases is obtained from the decay of . This is demonstrated in the inset of this figure. However, there is mismatch in the amplitude of damped oscillation. This discrepancy is due to the fact that integration of with respect to is proportional to the sum of the order parameter which is different for different compositions. Apart from that, reasonable collapse of data indicates that the basic structure is same in all the cases.
In Fig. 10, we show as a function of time for three different compositions, as indicated. Interestingly it appears that the equilibration time is a non-monotonic function of composition. Another observation is that the amplitude of growth decreases with increasing asymmetry of composition. This, combined with the fact that for extreme off-criticality smaller amount of particles need to assemble to reach equilibrium, can possibly explain the non-monotonic behavior of . In Fig. 11, we show the instantaneous exponents obtained for the plots in Fig. 10.
In all the cases it is seen that in the limit , there is a tendency of the data to converge to the LS value .
In Fig. 12, we show the plots of vs from various different system sizes, for the composition . Corresponding finite-size scaling analysis is demonstrated in Fig. 13. Best data collapse, that is presented here, was obtained for and MCS. This value of is consistent with the data in Fig. 11, for this composition.
Here again we have fixed to the LS value . Reasonable flat nature of the data from the begining confirms that the correction is negligible and if at all present, is buried in the statistical fluctuations of the simulation results.
Again, from the ordinate of this flat region, we extract the amplitude of growth. This is plotted in Fig. 14 as a function of composition.
Further, we have extracted information on the finite-size effects from the deviation of data from flat behavior in the finite-size scaling plots. In all the cases, both as function of temperature and composition, finite-size effects become visible at . Here we note that, for asymmetric composition, if this is quantified as a fraction of the system size, this pre-factor will, of course, be much smaller than . But we feel that it is more reasonable to quantify it with respect to the equilibrium length of the minority component, as it is done here.
Coming back to the point of curvature dependent correction, we provide a different analysis below. In our finite-size scaling analysis one may ask about the ambiguity in the estimation of , however small it may be. To avoid such criticism, we search for a method where can be gotten rid of in a different mathematical way. Writing
Since appeared microscopic in our previous analysis, here we have replaced by . By setting , one has
In Fig. 15, we plot vs for three different compositions, including the critical one. In all the cases, numerical results look consistent with linear behavior from very early time (see the inset), confirming about only weak correction to scaling. The solid lines there are guides to the eye for which we have used slopes by taking from Fig. 14.
Despite all these findings, we do not discard the fact that a correction is present. Our analysis might have failed to capture this aspect because of statistical fluctuation in the data combined with the fact that the corrections are of higher order in with very small value of prefactor. This argument stems from the curvature dependent interfacial tension for symmetric models, like the Ising one, with quadratic leading correction (36); (37) [see Eq. (8) and subsequent discussion about nonexistence of ]. From equlibrium studies of a symmetric binary fluid that belongs to the Ising universality class of critical phenomena, it is observed that in Eq. (8) is very small at temperatures comparable to the present study. For temperatures significantly close to , however, curvature dependent correction to the growth law may be significant. (We have seen such signature even for composition.) This statement can be justified from the finding that diverges as the equilibrium correlation length at criticality (36). But, closer to , computational study of kinetics of phase separation, particularly for off-critical composition, is limited by additional difficulties because of mixing of two diverging lengths, viz., the equilibrium correlation length and domain size . So we leave it as a future task when we acquire better computational resources.
We close this section with the following remarks. Irrespective of composition, at moderate temperatures, the correction to scaling appears very small and this contradicts and corrects our previous understandings about this phenomena in kinetics of phase separation. Further, the finite-size effects in all the cases appear rather weak.
We have presented detailed results for the kinetics of phase separation in solid binary mixtures via Kawasaki exchange Monte Carlo simulation of Ising model in two dimensions. Results are understood via sophisticated finite-size scaling analysis and other methods. Important aspects related to the variation of quench depth and mixture composition are discussed. A particular focus was on the early time dynamics.
In case of critical composition, for temperatures above , it is observed that the growth law is consistent with Lifshitz-Slyozov (LS) law () almost all along, with amplitude increasing as one moves closer to the critical point. For significantly low temperature, however, at early time, lower value of exponent was noticed which crosses over to the LS value only at much later time. In this case though the early time behavior is consistent with the prediction for interface diffusion mechanism, we feel that further studies are needed for appropriate understanding of growth over entire time range.
For the composition dependent case, our primary objective was to search for any significant correction to the scaling (7). The finding appears to be against it at least for the temperature considered. Our observation, however, is consistent with recent studies in the equilibrium context involving curvature dependent interfacial tension (36); (37). Nevertheless, in line of these equilibrium studies, we expect noticeable corrections for temperatures close to the critical value. Because of technical reasons, we, however, leave it as a future problem.
In addition to the growth law, we have presented results for the pattern formation also. These patterns were characterized via calculation of correlation functions. Lastly, our observation about the weak finite-size effects appears to be generic at all temperatures and compositions.
The authors are grateful to the Department of Science and Technology, India, for financial support via grant number SR/S2/RJN-13/2009. SM also acknowledges Council of Scientific and Industrial Research, India, for financial support in the form of research fellowship.
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