NON MEAN-FIELD BEHAVIOUR OF CRITICAL WETTING TRANSITION FOR SHORT RANGE FORCES.

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1 NON MEAN-FIELD EHAVIOUR OF CRITICAL WETTING TRANSITION FOR SHORT RANGE FORCES. Paweł ryk Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland* We revisit the long-standing problem of critical wetting transition for shortrange forces in three dimensions (d = 3). Using Monte Carlo simulation in conjunction with an anisotropic finite size scaling approach, as well as approaches that do not rely on finite size scaling, we show that the critical wetting transition shows clear deviation from meanfield behaviour. Our estimates for the ective critical exponent for J/kT = 0.35 are in accord with predictions of Parry et al. [Phys. Rev. Lett. 00, 3605 (2008)]. We also show, that the anisotropic finite-size scaling approach in d = 3 requires further modification in order to reach full consistency of simulational results. Introduction Wetting transition is a surface-induced transition in which the contact angle of a liquid deposited on a surface drops to zero from a non-zero value upon increasing temperature or chemical potential. This transition has been the subject of many theoretical and experimental studies. The wetting transition can be either first order or continous (second-order). In the case of the first-order wetting transition there is an additional prewetting (thin-thick film) transition, which occurs off-coexistence and the prewetting line joins the binodal exactly at the wetting point. The critical (second-order) wetting transition is not accompanyed by prewetting. The nature of the wetting transition and its universality class depends on (among others) the dimensionality of the system and the range of interparticle interactions. While in most cases the nature of the wetting transition has been well established and documented, it turns out that the case of 3-dimensional critical wetting transition for short-ranged for ces (i.e. decaying exponentially or faster) is still an unresolved issue. One of the important early discoveries was that the critical wetting transition for shortrange forces is strongly nonuniversal. Using renormalization-group calulations based on an ective interfacial Hamiltonian rezin et al. [] predicted that the critical exponent for this transition depends on a dimensionaless parameter k T 2 () 4 where T is the temperature, ξ is the correlation length of the bulk wetting phase and Σ is the interfacial stiffness. Capillary-wave-like fluctuations give rise to a diverging transverse correlation length ξ and the relvant exponent is belived to behave ( ) for 0 / 2 2 ( 2 ) for / 2 2 (2) for

2 Unfortunately, subsequent Monte Carlo calulations carried out of the Ising [2 5] and related [6] models revealed that while the general features regarding the wetting transition for the Ising model do agree with the theoretical predictions [7], the critical wetting transition is only very weakly nonuniversal. This disagreement has been the subject of ever-lasting orts in order to reconcile the theory and simulations. Recently a new attempt has been made towards an explanation of this discrepancy. Using the nonlocal ective interfacial Hamiltonian Parry et al. [8] argued that the spectrum of the interfacial fluctuations has a lower cutoff due to apperance of a new length scale NL l ln. This gives rise to an ective critical exponent and an ective wetting parameter,, of the form ln( l) 2 3 (3) l where, and l denotes film thickness. The ective wetting parameter is lower than the asymptotic critical value 0.8 for certain film thicknesses and attains a minimum value of 0.3 when the film thickness is approximately 5 bulk correlation lengths. The goal of the present study is to verify by simulation these recent predictions. The model We consider simple-cubic Ising L L D systems with two free surface L L layers, and periodic boundary conditions in two remaining directions. The local order parameter of the corresponding phase transition is a pseudospin variable si = ± at the lattice site i. The Hamiltonian for the system is H J bulk s s i j J s surf s s i k H bulk k ksurf si H s H s (4) D k ksurfd In the above Js is the surface exchange constant acting within the two free surface layers, J is the bulk exchange constant acting in the remaining layers. H is th bulk field, and surface fields acting on the first and last layer are H and HD, respectively. We have considered two different types of systems, namely the symmetric systems with H = HD, and the antisymmetric systems with H =-HD. Throughout this study we restricted ourselves to the case Js = J. The systems were simulated using highly icient multispin coding algorithm [9]. One of the simulational challenges is the critical slowing down that near the critical wetting point. This hampers the statistics of the accumulated data. In order to overcome this drawback we applied hyper-paralell tempering technique [0] and simulated many systems at the same time, and allowed for frequent swaps between the systems. The exchange of spins between the systems n and m was accepted with probabilisty where P nm min[,exp( E) ( H) m ( HD) md ( H) m] (5) 4-04

3 , E En Em (6) ktn ktm ( n) ( n) ( H ) H H, m m m (7) k T k T n m ( n) ( n) ( H D) H D H D, md md md (8) k T k T n m k T k T ( n) ( n) ( H) H H, m m m (9) n m During the course of simulation several quantities were accumulated. Of particular interest are: the magnetization in the surface layers the total magnetization and the mixed surface layer susceptibility m (0) 2 (2L ) s k ksurf 2 m ( L D) () i s i m 2 L D( mm m m ) (2) H Tracking down the critical wetting transition point from computer simulation proves to be a challenge. The original idea of inder, Landau and coworkers [2, 3] was to consider the symmetric system, H = HD. First part of the task is to consider the case where H = 0 and locate the critical wetting transition by varying H at a constant temperature. The maximum of χ as a function of H indicates the location of the critical surface field Hw at which the critical wetting transition occurs. The second part of the task is to determine the ective critical exponent from ff / 2 H e for τ = 0 (3) where H H w / J. Later Parry, Evans and inder [5] proposed an alternative mean of determination of by analyzing the amplitude ratios of χ at H = 0 in the vicinity of Hw but in view of rather poor statistics this method was not investigated. Results - symmetric surface fields 4-05

4 First simulations were carried out for symmetric field at J/kT = 0.35, in order to compare with [4, 5]. We assumed that D = 60 and L =26, 252, 35, 378 and 504. The statistical ort was spin flips per site. Fig.: χ for different system sizes, evaluated at J/kT = 0.35 for the symmetric surface fields. Here D=60. The big black circle with error bars denotes the simulational result of inder et al.[4] Figure shows the results of the simulations. Its clear that the statistics is much improved, when compared to the earlier orts. The data for L = 26 can be compared with that from Ref. 4. We note that the position of the maximum of Hw/J = ± agrees very well with earlier study (Hw /J = ± 0.004). However, there is a considerable shift for larger systems. This indicates that Hw(L = ) is lower. As the carefull reader noticed, there is a very disturbing feature of the results presented in Fig.. Namely the maxima of χ get smaller and smaller as the linear system size increases. This means that in the thermodynamic limit this peak disappears and this transition does not exist! In order to understand what happens we recall that the simulated system is not semi-infinite but a slit-like pore. In such system the only phase transition that remains stable is capillary condensation (and in general layering and (first order) prewetting, but we do not study such cases here). It s clear that the large statistical ort and hyperparalell tempering technique must give the correct result i.e. that the transition studied by simulation is not a stable transition in the thermodynamic limit. However, we argue that in this particular case one can still trust the positions of the susceptibility vs. system size even if their magnitude decreases with increasing system size. This situation can be interpreted in terms of finie size scaling at the first order transition, as formulated by inder and Landau in Ref.. One can approximate the probablity distribution of the magnetization of an Ising system by two Gaussians for the two phases that would coexist at the capillary condensation transition. At zero bulk field, the phase with the magnetization oppositely oriented to the surface fields is not the stable one. It still gives a 4-06

5 signal in a finite system, which ultimately in the thermodynamic limit - will be exponentially suppressed. However, for the surface susceptibility, the stable phase (magnetization parallel to the surface field) gives only a small background contribution, and hence one can still detect the developing singular behaviour of the surface layer susceptibility even though the magnitude of the signal decreases with increasing system size. If the sampling is insuffiencient to reach full equilibrium, one may get the wrong amplitude of the signal, as far as equlibrium is concerned, but still it will be possible to detect where the anomaly is located. This method yields the critical surface fields Hw(L= )/J = for J/kT = Results - Anisotropic finite size scaling Recently a new anisotropic finite size scaling (AFSS) theory which should be suitable for studying wetting transitions in general was proposed [2, 3]. Within this approach the thermodynamic limit D should be taken so, that the generalized aspect / * / ratio C D / L (or, alternatively C D/ L ) is constant. The system has antisymmetric surface fields, and in this way we avoid the ects connected to the capillary condensation. The scaling ansatz for the order parameter probability distribution is given by / ~ / D, P( C, L/, m ), m 0, (4) P L where P ~ is a scaling function, whereas β is the order parameter critical exponent. For d = 3, β = 0 while the exponent for the transverse correlation length 0. Consequently we keep fixed the generalized aspect ratio of the form C* = D/ ln(l). 4-07

6 Fig.2: Average absolute magnetization m and cumulants U4, vs. surface field H/J. Parts (a) and (b) show the results evaluated at J/kT = 0.35 and for the generalized aspect ratio C* = System sizes D are given in the Figure. Figure 2 shows the plots of the average absolute magnetization and the cumulant vs. H. In opposition to the case of d = 2, where both m and U4 exhibit well-defined intersection points, here the cumulants hardly intersect and the intersections of m have not converged to a unique location either. However, the nonexistence of intersection points is not as serious a problem, as it looks at first sight. Finite size can cause a shift as well as a rounding of a transition. oth should scale in the same way, should a straightforward application of finite size scaling work. However, even then it is possible that the amplitude prefactor for the shift is much larger than the rounding. In such case one would find no intersections for the cumulant. Fig.3: Estimation of the ective critical exponent using the AFSS approach. Plots show the maximum slope of the average absolute magnetization ( m / ( H / J)) the maximum slope of the cumulant ( U4 / ( H / J)) max vs. linear system size L. The maximum slope of the order parameter / 4-08 )) max / max and ( m / ( H / J L while ( U4 / ( H / J)) max L. This enables a direct estimation of the surface critical exponent. Figure 3 shows the plots of the maximum slopes vs. L. We note clear MF deviations from the mean-field value. Figure 4a shows a log-log plot of maximum susceptibility 2 2 ' L D( m m ) / kt vs. L. We note that the wetting transition in the thermodynamic limit occurs AT H = which is quite close to the value estimated in the previous section using the system with symmetric walls. The straight line extrapolation would not work if the mean-field value was used. Figures 4b and c show consistency checks. While the cumulants (cf. Fig. 2

7 4b) collapse nicely onto one master curve, the plot of m vs. ( H H w) / L does not collapse (cf. Fig. 4c). It seems that there are additional finite size ects, which are peculiar to d = 3. The analysis of diverging length scales for the wetting transition show that in three dimensions there is one more divergence, l / ln [4], where leq is the equilibrium film thickness. We speculate, that the collapse of the cumulants is connected with the fact, that these additional finite-size ect cancel out, since U4 is a ratio of moments of magnetization. Unfortunately, at present it is not clear how to incorporate this ect into the AFSS framework. This lack of full consistency prompts to look for other methods of determination 2 of the critical exponent. It has been shown [2, 3] that D '. In the region not affected by finite-size ects the plot of D ' vs. H H w for different system sizes should exhibit the same slope equal to 2. Figure 5a demonstrates, that the slope of such a quantity has the slope slightly less than 4, which is consistent with previous estimates. Finally Fig. 5b shows the estimation of the ective critical exponent using Eq. 3. The mean-field behaviour would imply a slope of -0.5 and such was the conclusion of the early reports. Instead we observe clear deviations from the mean-field value, which is again consistent with our estimates obtained from different methods. Putting together our results we obtain =.76 ± This leads to the value of the ective wetting parameter ω = 0.44 ± eq 4-09

8 Fig.4: (a) Estimation of the critical surface field for the wetting transition, Hw, for J/kT=0.35. The plot shows log-log plot of the position of the maximum susceptibility χ vs / L. The intercept with (b) Scaling plot of U4 vs ( H (c) Scaling plot of m vs / L H w) = 0 yields Hw. / L at H w ( H H w) / L at H w Fig.5: (a) Plots of the mixed surface susceptibility vs (H - Hw)/J. (b) Plot of χ s vs bulk field H calculated for H = Hw. The results were obtained using the symmetric system, H = HD, and for the system size given in the figure. The above results lead to a conclusion, that the early reports about the mean-field behaviour of critical wetting in d = 3 can be traced back to the inaccurate estimatation of the critical surface field. However, since those calculations took place more than 25 years ago, it was simply impossible to arrive at the correct conclusion at that time. Only now, with the lateral system sizes as large as 500 we are able to confirm the theoretical conjecture [8] about very slow crossover to the asymptotic regime. In order to reach the full nonuniversal behaviour of critical wetting one needs to consider system sizes exceeding tens of thousands of lattice spacing. In conclusion, we have carried out accurate Monte Carlo simulations of critical wetting transition in d = 3 and demonstrated clear deviations from mean-field behaviour. We estimate that the ective critical exponent =.7 ± 0. for J/kT = Our results clearly support the non-local Hamiltonian model [8]. We have also found that the under standing of finite size ects on critical wetting in d = 3 is still incomplete: analytical guidance to find the proper extension of the AFSS approach to deal with the weak logarithmic divergence of the perpendicular correlation length remains a future challenge, to Reach a full understanding of the simulation results. ACKNOWLEDGMENTS 4-0

9 I am grateful to prof. K. inder for his continued interest in this project. Support from 7 th Framework Programme under the contract PIRSES-GA is acknowledged. REFERENCES [] rezin E, Halperin.I., Leibler S, Critical Wetting in Three Dimensions, Phys. Rev. Lett. (983) [2] inder K, Landau D.P, Kroll D.M, Critical Wetting with Short-Range Forces: Is Mean-Field Theory Valid? Phys. Rev. Lett. (986) [3] inder K, Landau D.P, Wetting and layering in the nearest-neighbor simple-cubic Ising lattice: A Monte Carlo investigation. Phys. Rev. (988) [4] inder K, Landau D.P, Wansleben S., Wetting transitions near the bulk critical point: Monte Carlo simulations for the Ising model. Phys. Rev. (989) [5] Parry A.O, Evans R, inder K, Critical amplitude ratios for critical wetting in three dimensions: Observation of nonclassical behavior in the Ising model. Phys. Rev. (99) [6] Gomper G, Kroll D.M, Lipowsky R, Nonclassical wetting behavior in the solid-onsolid limit of the three-dimensional Ising model. Phys. Rev. (990) [7] Nakanishi H, Fisher M.E, Multicriticality of Wetting, Prewetting, and Surface Transitions. Phys. Rev. Lett. (982) [8] Parry A.O, Rascon C, ernardino N.R, Romero-Enrique J.M, 3D Short-Range Wetting and Nonlocality. Phys. Rev. Lett. (2008) [9] Wansleben S, Ultrafast vectorized multispin coding algorithm for the Monte Carlo simulation of the 3D Ising model. Comp. Phys. Comm. (987) [0] Yan Q, de Pablo J.J, Hyper-parallel tempering Monte Carlo: Application to the Lennard-Jones fluid and the restricted primitive model. J. Chem. Phys. (999) [] inder K, Landau D.P, Finite-size scaling at first-order phase transitions. Phys. Rev. (984) [2] Albano E.V, inder K, Finite-Size Scaling Approach for Critical Wetting: Rationalization in Terms of a ulk Transition with an Order Parameter Exponent Equal to Zero. Phys. Rev. Lett. (202) [3] Albano E.V, inder K, Wetting transition in the two-dimensional lume-capel model: A Monte Carlo study. Phys. Rev. E (202) [4] Dietrich S, Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, London, Academic, Vol. 2, p

PHYSICAL REVIEW LETTERS

PHYSICAL REVIEW LETTERS VOLUME 76 4 MARCH 1996 NUMBER 10 Finite-Size Scaling and Universality above the Upper Critical Dimensionality Erik Luijten* and Henk W. J. Blöte Faculty of Applied Physics, Delft