Non mean-field behaviour of critical wetting transition for short-range forces
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Non mean-field behaviour of critical wetting transition for short-range forces.
Pawe l Bryk
Department for the Modeling of Physico–Chemical Processes,Maria Curie–Sk lodowska University, 20–031 Lublin, Poland. ∗ Kurt Binder
Institu f¨ur Physik, Johannes Gutenberg-Universit¨at Mainz, Staudinger Weg 7, D-55099 Mainz, Germany (Dated: November 2, 2018)Critical wetting transition for short-range forces in three dimensions ( d = 3) is reinvestigatedby means of Monte Carlo simulation. Using anisotropic finite size scaling approach, as well asapproaches that do not rely on finite size scaling, we show that the critical wetting transitionshows clear deviation from mean-field behaviour. We estimate that the effective critical exponent ν eff k = 1 . ± .
08 for
J/kT = 0 .
35 and ν eff k = 1 . ± .
07 for
J/kT = 0 .
25. These values are inaccord with predictions of Parry et al. [Phys. Rev. Lett. , 136105 (2008)]. We also point outthat the anisotropic finite size scaling approach in d = 3 requires additional modification in orderto reach full consistency of simulational results. PACS numbers: 68.03.Cd, 68.08.-p, 68.03.-g, 68.35.Md, 68.47.Mn
Understanding interfacial properties of fluids is impor-tant for many applications including adsorption in porousmaterials[1], nanofluidic devices[2] and design of super-hydrophobic surfaces[3]. The common problem perti-nent to these research areas is the prediction and con-trol of the wetting properties of surfaces. The intro-duction of patterns on the nanoscale leads to substantialchanges in wettability and creates a host of new effects[4–6]. While the wetting phenomena at planar surfaces iswell understood[7], there is one notable exception. Crit-ical wetting has been a long standing and stubbornlydifficult problem to understand.Renormalization group (RG) calulations based on alocal interfacial Hamiltonian [8, 9] predict that the crit-ical wetting transition for short range forces is stronglynonuniversal. When temperature T approaches the wet-ting temperature T w , the critical exponent character-izing the divergence of the paralell correlation length, ξ k ∼ ( T w − T ) − ν k , depends on a nonuniversal dimension-aless wetting parameter, ω = kT π Σ ξ , where k is Boltz-mann’s constant, Σ is the interfacial stiffness (or surfacetension for simple liquids), and ξ is the correlation lengthin the phase that wets the wall. For the case of the Isingmodel in three dimensions ( d = 3) one has 1 / < ω < ν k ( ω ) = ( √ − √ ω ) − . (1)Surprisingly, subsequent simulation studies [10–12]showed only minor deviations from mean-field value, ν MF k = 1. In order to reconcile theory and simulationParry et al. [13] proposed a new non-local (NL) interfaceHamiltionian which removed various intrinsic inconsis-tencies of previous approaches. Reanalysis of the NLmodel showed the apperance of another diverging length ξ NL = √ lξ ∝ p ln ξ k , which cuts some of the interfa-cial fluctuations for small film thicknesses l . This in turn leads to a reduction in the effective value of the wettingparameter ω eff , and to an effective exponent ν eff k . -0.91 -0.9 -0.89 -0.88H /J0.30.40.50.60.7 < | m | > D=12D=14D=16D=18 -0.91 -0.9 -0.89 -0.88H /J0.30.40.50.60.7 U D=12D=14D=16D=18 -0.62 -0.6 -0.58 -0.56 -0.54 H /J < | m | > D=24D=28D=32D=36 -0.62 -0.6 -0.58 -0.56 -0.54 H /J U D=24D=28D=32D=36J/kT=0.35 J/kT=0.35 (a) (b)
J/kT=0.25 J/kT=0.25 (c) (d)
FIG. 1: (color online) Average absolute magnetization h| m |i ,and cumulants U , vs. surface field H /J . Parts (a) and(b) show the results evaluated at J/kT = 0 .
35 and for thegeneralized aspect ratio C ∗ = 2 . J/kT = 0 .
25 and for C ∗ =5 . D are given in the Figure. Very recently a new anisotropic finite size scaling(AFSS) theory which should be suitable for studying wet-ting transitions in general was proposed [14, 15]. Theseauthors have suggested that the previous [10–12] esti-mates for the location of the critical wetting transitionneed a revision, however the critical exponent has notbeen determined. In this Communication we reconsiderthis approach and show that the critical wetting transi-tion for short-range forces in d = 3 shows clear deviationsfrom mean-field theory. We also give evidence that theAFSS theory is problematic in d = 3, which was notanticipated before [14, 15]We consider simple-cubic Ising L × L × D systems withtwo free surface layers L × L , and periodic boundary con-ditions in two remaining directions. The local order pa-rameter of the corresponding phase transition is a pseu-dospin variable s i = ± i . The Hamiltonianfor the system is H = − J X bulk s i s j − H X bulk s i − H X k ∈ surf s k − H D X k ∈ surfD s k , (2)where J is the bulk exchange constant, H is the bulkfield. Surface fields H and H D act only on the firstand last layer, respectively. In order to avoid effectsconnected to capillary condensation we select ”antisym-metric” walls, i.e. H = − H D <
0. During thecourse of simulation several quantities were accumu-lated, including the average absolute value h| m |i of themagnetization m = ( L D ) − P i s i , susceptibility χ ′ = L D ( h m i − h| m |i ) /kT , and the fourth order cumulant U = 1 −h m i / h m i . When the system is in the partialwetting regime, the interface is bound to the wall k = 1or k = D with equal probability. Consequently, h| m |i isnonzero in the thermodynamic limit. On the other hand,for the wet state the interface is unbound from either ofthe walls and wanders around the middle of the system.Consequently, h| m |i is zero for D → ∞ . The systemswere simulated using highly efficient multispin coding al-gorithm [16]. In order to overcome critical slowing downnear the critical wetting point we applied hyperparalleltempering technique [17] and simulated many systems atthe same time, and allowed for frequent swaps betweenthem. Statistical effort was at least 5 × spin flips persite.Within AFSS approach the thermodynamic limit D →∞ must be taken in a special way, keeping the gen-eralized aspect ratio C = D ν k /ν ⊥ /L (or, alternatively C ∗ = D/L ν ⊥ /ν k ) constant[14, 15]. The scaling ansatz forthe order parameter probability distribution is given by P D,L ( m ) = ξ β/ν k k ˜ P ( C, L/ξ k , mξ β/ν k k ) , m → , ξ k → ∞ , (3)where ˜ P is a scaling function, whereas β is the orderparameter critical exponent. For d = 3 β = 0 while theexponent for the transverse correlation length ν ⊥ = 0.Consequently we keep fixed the generalized aspect ratioof the form C ∗ = D/ ln( L ).Following earlier papers [10–12, 14] we keep the tem-perature constant (which keeps fixed the bulk correlationlength) and vary the surface field H . The calculationswere carried out for two temperatures, J/kT = 0 .
35 with C ∗ = 2 . J/kT = 0 .
25 with C ∗ = 5 .
100 200 400L305070 ( ∂ < | m | > / ∂ ( H / J )) m ax
100 200 400L2030405060 ( ∂ U / ∂ ( H / J )) m ax
100 200 400 L ( ∂ < | m | > / ∂ ( H / J )) m ax
100 200 400 L ( ∂ U / ∂ ( H / J )) m ax J/kT=0.35 slope=0.535
J/kT=0.35 ν ||eff =1.87 (a) (b) J/kT=0.25 slope=0.604 ν ||eff =1.66 J/kT=0.25 (c) (d)slope=0.507 ν ||eff =1.97 slope=0.55 ν ||eff =1.82 FIG. 2: Estimation of the effective critical exponent ν eff k usingAFSS approach. Plots show the maximum slope of the av-erage absolute magnetization ( ∂ h| m |i /∂ ( H /J )) max and themaximum slope of the cumulant ( ∂U /∂ ( H /J )) max vs. lin-ear system size L . Parts (a) and and (b) denote the resultsobtained for J/kT = 0 .
35 while parts (c) and (d) are for
J/kT = 0 . and for several lateral system sizes.Figure 1 shows the plots of the average absolute mag-netization and the cumulant vs. H calculated for thetwo temperatures. Unlike the case of d = 2, where both h| m |i and U exhibit rather well-defined unique intersec-tion points, here the cumulants hardly intersect and theintersections of h| m |i have not converged to a unique lo-cation either. The nonexistence of intersection points isnot as serious a problem, as it looks at first sight. Finitesize can cause a shift as well as a rounding of a transition.Both should scale in the same way, should a straight-forward application of finite size scaling work. Howevereven then it is possible that the amplitude prefactor forthe shift is much larger than the rounding. In such caseone would find no intersections for the cumulant.The large statistical effort together with hyperparalleltempering technique yielded accurate, smooth data al-lowing for an estimation of the exponent ν eff k . It has beenestablished [14, 15] that ( ∂ h| m |i /∂ ( H /J )) max ∝ L /ν eff k ,giving a very convenient way of determination of the crit-ical exponent ν eff k . Likewise, a similar relation holds forcumulants, ( ∂U /∂ ( H /J )) max ∝ L /ν eff k . Figure 2 showsthe plots of the maximum slopes of h| m |i and U vs. L .We find that for both temperatures ν eff k is clearly differentfrom the mean-field value ν MF k = 1, and attains valuesslightly below 2.Fig. 3(a) shows log-log plot of the position of maxi- -H )/J)L ν || eff U D=24D=28D=32D=36 -H )/J)L ν || eff < | m | > D=24D=28D=32D=36 -1/ ν || eff (- H ) χ ’ = m ax H =-0.616(a) J/kT=0.25(b) (c) FIG. 3: (color online) (a) Estimation of the critical surfacefield for the wetting transition, H w , for J/kT = 0 .
25. Theplot shows log-log plot of the position of the maximum sus-ceptibility χ ′ vs L − /ν eff k . The intercept with L − /ν eff k = 0yields H w . (b) Scaling plot of U vs ( H − H w ) L /ν eff k ob-tained using H w = − .
616 and ν eff k = 1 .
97. (c) Scaling plotof h| m |i vs ( H − H w ) L /ν eff k obtained using H w = − . ν eff k = 1 . mum susceptibility χ ′ vs L − ν eff k for J/kT = 0 .
25. Wefind that the value of the surface field for the wettingtransition H w = − . ± . ν MF k = 1was used instead. Further consistency checks are dis-played in Fig. 3(b) and (c). We find a good scaling of thecumulants [cf. Fig. 3(b)] with the estimated value of theeffective exponent. Quite surprisingly, the plot of h| m |i vs. ( H − H w ) L /ν eff k do not collapse [cf. Fig. 3(c)]. Sim-ilar results were found for J/kT = 0 .
35 [18]. It seems,that there exists additional finite-size effect that shouldbe applied to the ordinate variable h| m |i ! These effectsdo not exist in d = 2. The comparison of the diverginglength scales indicates that for d = 3 there is still onemore divergence, l eq /ξ ⊥ ∝ √ ln τ [19], where l eq is theequilibrium film thickness, and τ is the distance from thetransition. In contrast, for d = 2 this ratio is constant.It is tempting to speculate, that the fact that the cumu-lants do collapse is connected with the fact, that theseadditional finite-size effect cancel out, since U is a ratioof moments of magnetization. Unfortunately, at presentwe do not see a straightforward way of incorporating thiseffect into AFSS framework.In view of the above it is natural to seek another evi- dence of the non mean-field behaviour of critical wettingin d = 3, which would not resort to AFSS approach. Ithas been demonstrated [10, 11] that the ”surface layersusceptibility” χ s = Dχ ‘ ∝ ξ k . It follows, that whenplotting χ s vs. H − H w for several system sizes, theregions unaffected by finite size effects should exhibit thesame slope equal to 2 ν eff k . This provides additional esti-mation of the paralell correlation length exponent, inde-pendent of finite size scaling. Figs. 4a and b demonstratethat for both temperatures the slope of χ s in the regionfree of finite size effects is a bit less than 4, which isconsistent with previous estimates for ν eff k .As a final check, in Figs. 4(c)-(d) we show log-log plotsof the surface susceptibility vs. non-zero bulk field H evaluated at the critical surface fields H w , for the twotemperatures in question. The calculations presentedhere were carried out using the ”symmetric” boundaryconditions, i.e. H = H D in order to follow exactly thecomputational procedure presented in the first simula-tional studies on critical wetting [10–12]. In such a sys-tem wetting films develop independently on both walls.During the simulations we monitor ”surface layer sus-ceptibility”, χ s = ∂m /∂H = L D ( h m m i − h m ih m i ).Since χ s ∼ H − / ν eff k for H > H = H w , the slopegives information about the universality class of the wet-ting transition[11]. The mean-field behavior would implya slope of -0.5 and such was the conclusion of the earlyreports. However, when the calculations are performedfor the new estimations of the critical surface fields H w ,we observe clear deviations from mean-field exponents,again consistent with values obtained by different meth-ods. The final values of ν eff k are obtained by averag-ing the exponents resulting from four different meth-ods. Putting together all the results we estimate that ν eff k = 1 . ± .
08 for
J/kT = 0 .
35 and ν eff k = 1 . ± . J/kT = 0 .
25. Once the exponents are determined, weare able to calculate the effective wetting parameter. Weobtain ω eff = 0 . ± .
03 and 0 . ± .
02 for
J/kT = 0 . d = 3 canbe traced back to the inaccurate estimation of the criticalsurface field H w . This is not to say that those simula-tions were wrong. Simply, using the computing resourcesavaiable almost thirty years ago it was not possible to ar-rive at the correct conclusions. It is now clear that sizeslike L = 50 [10, 11] were far too small. The fact that evenfor the lateral system size L = 504 we reach roughly onlyhalf of the full nonuniversal value of ν k ≈ . et al. [13]about very slow crossover to the asymptotic regime. Weestimate that the system sizes required to see in simula-tions the full non-universal behaviour of critical wettingmust be of the order of tens of thousands lattice spacings.In conclusion, we have carried out accurate MonteCarlo simulations of critical wetting transition in d =3. We have found clear deviations from mean-field be-haviour. We estimate that the effective critical exponent ν eff k = 1 . ± . J/kT = 0 .
35 and ν eff k = 1 . ± . J/kT = 0 .
25. Our results clearly support the non-local Hamiltonian model [13] and together with Ref.20(where related effects due to ξ NL for complete wettingwere studied) provide a strong evidence towards the va-lidity of this approach. We have also found that theunderstanding of finite size effects on critical wetting in d = 3 is still incomplete: analytical guidance to find theproper extension of the AFSS approach to cope with theweak logarithmic divergence of the perpendicular corre-lation length remains a future challenge, to reach a fullunderstanding of the simulation results. Thus, a long-standing puzzle may finally be close to its resolution. χ χ (H -H )/J D χ ’ D=12D=14D=16D=18 (H -H )/J D χ ’ D=24D=28D=32D=36J/kT=0.35 J/kT=0.25 (a) (b)slope=-3.40
J/kT=0.25 (c) (d)
J/kT=0.35 slope=-3.53 ν ||eff =1.7 ν ||eff =1.765slope=-0.27 ν ||eff =1.85slope=-0.278 ν ||eff =1.80 L=504, D=80H =H L=504, D=120H =H FIG. 4: (color online) (a) and (b) Plots of the ”mixed surfacesusceptibility” vs ( H − H w ) J . (c) and (d) Plots of χ s vs bulkfield H calculated for H = H w . The results were obtainedusing symmetric system, H = H D and for system sizes givenin the figure. P.B is grateful to W. R˙zysko and N.R. Bernandino formany discussions. K.B. thanks E.V. Albano and D.P.Landau for discussions. ∗ [email protected][1] L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, and M.Sliwinska-Bartkowiak, Rep. Prog. Phys. , 1573 (1999).[2] L. Bocquet and E. Charlaix, Chem. Soc. Rev. , 1073(2010).[3] X.-M. Li, D. Reinhoudt, and M. Crego-Calama, Chem.Soc. Rev. , 1350 (2007).[4] O. Gang, K.J. Alvine, M. Fukuto, P.S. Pershan, C.T.Black, and B.M. Ocko, Phys. Rev. Lett. , 217801 -0.2 0 0.2 0.4((H -H )/J)L ν || eff U D=12D=14D=16D=18 -0.2 0 0.2 0.4((H -H )/J)L ν || eff < | m | > D=12D=14D=16D=18 -1/ ν || eff (- H ) χ ’ = m ax H =-0.909(a) J/kT=0.35(b) (c) FIG. 5: (color online) Supplemental material: (a) Estimationof the critical surface field for the wetting transition, H w ,for J/kT = 0 .
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