Dynamical transitions and sliding friction of the phase-field-crystal model with pinning
J.A.P. Ramos, E. Granato, S.C. Ying, C.V. Achim, K.R. Elder, T. Ala-Nissila
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Dynamical transitions and sliding friction of the phase-field-crystal model withpinning
J.A.P. Ramos , , E. Granato , , S.C. Ying , C.V. Achim , , K.R. Elder , and T. Ala-Nissila , Departamento de Ciˆencias Exatas, Universidade Estadual do Sudoeste da Bahia, 45000-000 Vit´oria da Conquista, BA,Brasil Laborat´orio Associado de Sensores e Materiais, Instituto Nacionalde Pesquisas Espaciais,12245-970 S˜ao Jos´e dos Campos, SP, Brazil Department of Physics, P.O. Box 1843, Brown University, Providence, RI 02912-1843, USA Department of Applied Physics and COMP Center of Excellence, P.O. Box 1100,Helsinki University of Technology, FI-02015 TKK, Espoo, Finland Institut f¨ur Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universit¨at D¨usseldorf,Universit¨atsstraße 1, D-40225 D¨usseldorf, Germany and Department of Physics, Oakland University, Rochester, Michigan 48309-4487, USA
We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal(PFC) model with pinning including both thermal fluctuations and inertial effects. The modelprovides a continuous description of adsorbed layers on a substrate under the action of an externaldriving force at finite temperatures, allowing for both elastic and plastic deformations. We derivegeneral stochastic dynamical equations for the particle and momentum densities including boththermal fluctuations and inertial effects. The resulting coupled equations for the PFC model arestudied numerically. At sufficiently low temperatures we find that the velocity response of aninitially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitionsfor increasing and decreasing applied forces at different critical values. The main features of thenonlinear response in the PFC model are similar to the results obtained previously with moleculardynamics simulations of particle models for adsorbed layers.
PACS numbers: 64.60.Cn, 68.43.De,64.70.Rh, 05.40.-a
I. INTRODUCTION
In recent years, considerable attention has been givento the study of driven adsorbed layers in relation tosliding friction phenomena between surfaces at the mi-croscopic level . The nonlinear response of the ad-sorbed layer is a central problem for understanding ex-periments on sliding friction between two surfaces witha lubricant or between adsorbed layers and an oscillat-ing substrate . Various elastic and particle models havebeen used to study the driven dynamical transitions andthe sliding friction of adsorbed monolayers . A fun-damental issue in modeling such systems is the originof the hysteresis and the dynamical melting and freezingtransitions associated with the different static and kineticfrictional forces when increasing the driving force fromzero and decreasing from a large value, respectively. Indriven lattice systems, hysteresis can occur in the under-damped regime where inertial effects are present or in thepresence of topological defects such as dislocations andthermal fluctuations . Topological defects can be auto-matically included in a full microscopic model involvinginteracting atoms in the presence of a substrate potentialusing realistic interaction potentials. However, numeri-cal observation of the full complexities of the phenomenais severely limited by the small system sizes that can bestudied, even when simple Lennard-Jones potentials areemployed .Recently, a phase-field-crystal (PFC) model wasintroduced that allows for both elastic and plasticdeformations within a continuous description of the par- ticle density while still retaining information on atomiclength scales. By extending the PFC model to take intoaccount the effect of an external pinning potential ,a two-dimensional version of the model has been usedto describe commensurate-incommensurate transitions inthe presence of thermal fluctuations and the driven re-sponse of pinned lattice systems without thermal fluctu-ations and inertial effects. . However, in order to studyfully the sliding friction behavior, both inertial effects and thermal fluctuations need to be taken into account.In this work, we study sliding friction of an adsorbedlayer via the nonlinear response of the PFC model to anexternal force in the presence of a pinning potential. Tothis end, we first derive the general stochastic dynamicalequations for the particle and momentum density fieldstaking full account of the thermal fluctuations and iner-tial effects. The resulting coupled equations are studiednumerically. At low temperatures, we find that the veloc-ity response of an initially commensurate layer shows hys-teresis with dynamical melting and freezing transitionsfor increasing and decreasing applied forces at differentcritical values. The main features of the nonlinear re-sponse are similar to the results obtained previously withmolecular dynamics simulations of particle models. How-ever, the dynamical melting and freezing mechanisms aresignificantly different. In the PFC model, nucleation oc-curs via stripes rather than closed domains found in theparticles model. II. DYNAMICS OF THE PFC MODEL WITHINERTIAL EFFECTS
In the generalized PFC model, the system is repre-sented by a coarse grained effective Hamiltonian that isa functional of the number density field. To take intoaccount inertial effects in the dynamics, we need to con-sider the contribution of the kinetic energy to the to-tal energy of the system in addition to the configura-tional energy. Thus we consider the momentum den-sity, ~g ( ~x ) = ρ ( ~x ) ~v ( x ), a dynamical variable in the coarsegrained Hamiltonian in addition to the particle densityfield ρ ( ~x ). The total effective Hamiltonian in the pres-ence of an external force ~f can be written as H t = H kin + H int − Z d~xρ ( ~x ) ~x · ~f , (1)where H kin is the kinetic energy contribution given by H kin = Z d~x ~g ( ~x )2 ρ ( ~x ) , (2)and H int ( ρ ) is the configurational contribution to the ef-fective Hamiltonian in the original PFC model. The lastterm is due to the presence of the the external force ~f .In the absence of energy dissipation, the time depen-dence of ρ and ~g are determined by the Poisson brack-ets { H t , ρ } and { H t , ~g } . At finite temperatures, addi-tional dissipative noise terms are present in the dynam-ical equations. The noise satisfies the fluctuation dis-sipation relations which allows the system to reachthermal equilibrium in the absence of external perturba-tions. We include the dissipative noise term directly inthe dynamical equations for ~g , which is a nonconservedfield.For the particle density ρ ( ~x, t ) we have ∂ρ∂t = − X j Z d~x ′ { ρ ( ~x ) , g j ( ~x ′ ) } δH t δg j ( ~x ′ ) , (3)and for the momentum density ~g ( x, t ) ∂g i ∂t = − Z d~x ′ { g i ( ~x ) , ρ ( ~x ′ ) } δH t δρ ( ~x ′ ) − X j Z d~x ′ { g i ( ~x ) , g j ( ~x ′ ) } δH t δg j ( ~x ′ ) − η δH t δg i + ν i ( ~x, t ) , (4)where η is a dissipative coefficient and the noise ~ν ( ~x, t )has variance h ν i ( ~x, t ) ν j ( ~x ′ , t ′ ) i = 2 k B T ηδ ( ~x − ~x ′ ) δ ( t − t ′ ) δ i,j . (5)The Poisson brackets for the mass and momentum den-sities are given by { ρ ( ~x ) , g i ( ~x ′ ) } = ∇ i ( ρ ( ~x ) δ ( ~x − ~x ′ )); (6) { g i ( ~x ) , ρ ( ~x ′ ) } = ρ ( ~x ) ∇ i δ ( ~x − ~x ′ ); { g i ( ~x ) , g j ( ~x ′ ) } = ∇ j ( g i ( ~x ) δ ( ~x − ~x ′ )) − ∇ ′ i ( g j ( ~x ) δ ( ~x − ~x ′ )) . Substituting Eqs. (6) into Eqs. (3) and (4) gives ∂ρ∂t = −∇ · ~g ; (7) ∂g i ∂t = − ρ ∇ i δH int δρ − η g i ρ + ρf i + ν i ( ~x, t ) (8) − X j ∇ j g i g j ρ , for a spatially uniform external force ( f i ). We can rede-fine the coefficient η → ρη to remove the denominatorfrom the term g i /ρ in Eq. (8). With this change thevariance of the noise ~ν ( x, t ) becomes h ν i ( ~x, t ) ν j ( ~x ′ , t ′ ) i = 2 k B T ηρ ( ~x ) δ ( ~x − ~x ′ ) δ ( t − t ′ ) δ i,j (9)To leading order in the momentum density ~g , we dropthe quadratic term in Eq. (8) giving ∂ρ∂t = −∇ · ~g ; (10) ∂g i ∂t = − ρ ∇ i δH int δρ + ρf i − ηg i + ν i ( ~x, t ) . (11)Similar dynamical equations for PFC models with inter-nal dissipation were obtained in Ref. 20. If the effectiveHamiltonian H int is known, then these coupled stochas-tic dynamical equations should provide a full descriptionof the particle and momentum densities in presence offluctuations represented by the noise ν i ( x, t ) with corre-lations proportional to the temperature T and inertialeffects determined by the damping parameter η . In theoverdamped limit, ∂g/∂t = 0, with T = 0 and f = 0, theequation for the time evolution of ρ obtained by inserting ~g from Eq. (11) in Eq. (10) reduces to the deterministicequation for the density which has been obtained fromclassical density functional theory of liquids . III. PFC MODEL WITH PINNING ANDTHERMAL FLUCTUATIONS
In the presence of an external pinning potential ,a specific form of the configurational energy contribu-tion H int to the total effective Hamiltonian in Eq.(1)has been proposed which is an extension of the stan-dard PFC model free energy functional used in manyapplications . In dimensionless form, this effectiveinteraction Hamiltonian H int = H pfc can be written as H pfc = Z d~x { ψ [ r + (1 + ∇ ) ] ψ + ψ V ψ } , (12)where ψ ( ~x ) is a continuous field, V ( ~x ) represents the ex-ternal pinning potential and r is a parameter. The phasefield ψ ( ~x ) can be regarded as a measure of deviations ofthe particle number density ρ ( ~x ) from a uniform referencevalue ρ , such that ψ ( ~x ) = ( ρ ( ~x ) − ρ ) /ρ . It is a con-served field and its average value, ¯ ψ , represents anotherparameter in the model. The intrinsic wave vector of themodel ~k i has no preferred directions and its magnitudeis set to unity in the present work.In the absence of a pinning potential, the Hamiltonianof Eq. (12) is minimized by a configuration of the field ψ ( ~x ) forming a hexagonal pattern of peaks with recipro-cal lattice vectors of magnitude | ~k h | ≈
1, in an appro-priate range of values for the parameters r and ¯ ψ in themodel. This periodic structures of peaks in ψ can beregarded as a simple model of an atomic layer.To study the nonlinear dynamical behavior we takethe form of H int given by Eq.(12) together with the ki-netic energy and the external force terms for the totaleffective Hamiltonian H t in the dynamical equations Eq.(11) and in Eq. (9). We make the additional simplify-ing approximation that ρ ( ~x ) ≈ ρ in Eq. (9) and in thecoefficient of the first term in Eq. (11). This approxima-tion ensures that the effective diffusion coefficient in themodel is positive definite for any temperature and driv-ing force. Another motivation for this approximation isto show that the dynamical equations used in the previ-ous works follows from the more general Eqs. (10)and (11). Setting ρ = 1, we obtain, ∂ψ∂t = −∇ · ~g ; (13) ∂g i ∂t = −∇ i δH pfc δψ + ψf i − ηg i + ν i ( ~x, t ); h ν i ( ~x, t ) ν j ( ~x ′ , t ′ ) i = 2 k B T ηδ ( ~x − ~x ′ ) δ ( t − t ′ ) δ i,j . Here we have redefined ~g → ~g + ρ ~f /η to remove a uni-form term on right hand side of the equation for ~g .The above coupled equations can be combined in asingle equation by applying the operator ∇· to both sidesof the second equation and using the first one to eliminate ∇ · ~g , giving ∂ ψ∂t + η ∂ψ∂t = ∇ δH pfc δψ − ~f · ∇ ψ + ξ ( ~x, t ); (14) h ξ ( ~x, t ) ξ ( ~x ′ , t ′ ) i = 2 k B T η ∇ δ ( ~x − ~x ′ ) δ ( t − t ′ ) . When the driving force ~f and the external pinning po-tential are set to zero, the dynamical equation above isidentical to the one used in Ref. 22 to study propagat-ing density modes in the PFC model. It can also beobtained by introducing inertial effects in the dynamicalequation of the PFC model through a memory functionof exponential form . In the limit of large η when ∂g i /∂t in Eqs. (13) or, equivalently, ∂ ψ/∂t in Eqs. (14),can be neglected, these equations reduce to the familiaroverdamped dynamical equations used in the previousworks without inertial effects at zero temperature. IV. NUMERICAL RESULTS AND DISCUSSION
In this section, we present our numerical results forthe velocity response of the PFC model in presence ofan external pinning potential under a uniform appliedforce. For the numerical calculations, the phase field ψ ( ~x )and momentum density field ~g ( ~x ) are defined on a spacesquare grid with dx = dy = π/ L × L with L = 64 to 128 whereused. The Laplacians and gradients were evaluated usingfinite differences.We consider a pinning potential V ( ~x ) representing asubstrate with square symmetry V ( ~x ) = − V [cos( k x ) + cos( k y )] , (15)where k defines the period of the pinning potential forboth the x and y directions. The lattice misfit betweenthe phase-field crystal and the pinning potential can bedefined as δ m = (1 − k ). We choose the parametersof the PFC model as r = − .
25, ¯ ψ = − .
25, a latticemismatch δ = − . V = 0 . c (2 × , where every second site of the lattice of thepinning potential corresponds to a peak in the phase field ψ ( ~r ). This commensurate configuration is stable belowa commensurate melting temperature T c ≈ . S ( ~k ) is calculated from the the positions ~R j of the peaks in the phase field as S ( ~k ) = h N P X j,j ′ =1 N P e − i~k · ( ~R j − ~R j ′ ) i . (16)Here, h ... i denotes a time average which is equivalent toa thermal average at equilibrium. For increasing tem-peratures thermal fluctuations disorder the layer and thescaled structure factor S ( ~k c ) /N p evaluated at the pri-mary reciprocal lattice vector ~k c of the commensuratephase decreases from a value of unity at T = 0 rapidlythrough T c to zero at higher temperatures. The transi-tion is broadened due to finite size effects as shown inFig. 1(a). Another signature of this commensurate melt-ing transition is that the specific heat develops a broadpeak near T c corresponding to an increase in the fluctu-ations in ψ at the transition as shown in Fig. 1(b). Fi-nally, the mobility µ defined as lim f → ( v/f ) where v isthe drift velocity, also changes qualitatively through thetransition. In the commensurate phase, a finite thresholdfor the sliding of the overlayer exists and hence the mo-bility is vanishingly small. As the system melts above T c the mobility rapidly rises and reaches a plateau at highertemperatures. This behavior is shown in Fig. 1c. For thestudy of the nonlinear response and sliding friction of thesystem, we focus on an initial state which corresponds toa well pinned c (2 ×
2) phase initially, corresponding to T = 0 . ≪ T c , and a damping parameter η = 0 . x direction is increased from zero to a maximum valueabove a critical depinning force and then decreased backto zero. For each value of the force, the coupled equa-tions (13) are solved using an Euler algorithm with timestep dt = 0 . − . time steps were used to allow the sys-tem to reach a velocity steady state and an additionalperiod with the same number of time steps were used tocalculate the average velocity and other time averagedquantities.To study the velocity response of the PFC model, weneed to determine the velocity of the peaks in the phasefield ψ ( ~x ). This is done by determining the time depen-dence of the peak positions ( ~R i ( t )) in ψ . The steady statedrift velocity ~v for the system is obtained from the peakvelocities ~v i ( d ~R i /dt ) as ~v = h N P N P X i =1 ~v i ( t ) i , (17)where N P is the number of peaks and h ... i denotes timeaverage. The steady state structure factor S ( ~k ) which isa measure of translational order, is also calculated fromthe the peak positions ~R j , as in Eq.(16).The most notable qualitative feature of the velocityresponse to the driving force ~f (as shown in Fig. 2) isthat it shows hysteresis behavior with two different crit-ical force thresholds f a ≈ .
075 for increasing forces and f b ≈ .
045 for decreasing forces. These two threshold val-ues correspond to the static frictional force and kineticfrictional force respectively. As the force is increased be-yond f a , the velocity jumps abruptly from zero to a finitevalue whereas when the force is decreased below f b thevelocity of the sliding layer drops abruptly and becomepinned by the external potential to form an immobilecommensurate state again. Below, we present the micro-scopic details of the configurations for the system in theneighborhood of the two thresholds. This allows us tocharacterize the change in velocity response at f a as adynamical force induced melting transition of the initialcommensurate state, and the second transition at f b asa dynamical force induced freezing of the sliding phase.To study these transitions we first examine the behav-ior of the steady state structure factor, as shown in Fig.3. We focus on the dependence of S ( Q ) on the drivingforce, where ~Q is the dominant reciprocal lattice vectorof the layer. ~Q = ~k c corresponds to the primary recip-rocal lattice vector for the c (2 ×
2) phase and ~Q = ~k h to the reciprocal lattice vector of the hexagonal phase inabsence of the driving force. Consistent with the veloc-ity response behavior in Fig. 2, on increasing the forcebeyond f a , S ( k c ) drops abruptly to zero. This is theonset of the dynamical force induced melting transition (c)L=128 µ T (a) L=128 L=96 S ( k c ) / N P T (b)L=128 TC FIG. 1: Temperature dependence of the (a) scaled structure-factor peak S ( k c ) /N p ; (b) specific heat C , and mobility µ forthe model without an external driving force. increasing f x decreasing f x cb a V f x FIG. 2: Velocity response as a function of applied force. Ar-rows correspond to critical values f a , f b and f c . of the initial c (2 x
2) commensurate state. The behaviorof S ( k c ) is analogous to the temperature induced disor-dering transition shown in Fig. 1(a). On decreasing theforce, the value of S ( k c ) stays vanishingly small until theforce drops below the threshold f b , at which point S ( k c )rapidly increases to a value corresponding to the com-mensurate pinned state. The other interesting feature isthat for f > f c ≈ .
12, the structure factor shows clear
FIG. 3: Scaled structure-factor peak S ( Q ) /N p as a function ofapplied force. Here ~Q stands for the primary reciprocal latticevector for either the c (2 ×
2) phase ( ~k c ) or the hexagonal phase( ~k h ). Filled and open symbols correspond to increasing anddecreasing forces, respectively. p k fx p p p p FIG. 4: Fraction of density peaks p k with coordination num-ber k (4,5,6 and 7 nearest neighbors) for increasing appliedforce. ~Q = ~k h corresponding to a hexagonal phase, which growsas the driving force increases. This implies that at adriving force larger than this third threshold f c , thereis another dynamic continuous transition from a disor-dered phase into an incommensurate hexagonal phase.This state emerges as the average effect of the externalpinning potential becomes less and less important at highsliding velocities and the steady state then approximatelycorresponds to the phase-field crystal in the absence ofthe pinning potential which has hexagonal symmetry inthe equilibrium state. However, since the scaled struc-ture factor for the hexagonal phase is still much less thanunity, this incommensurate hexagonal phase is not fullyordered even at at the largest force values ( f < . p k fx p p p p FIG. 5: Fraction of density peaks p k with coordination num-ber k for decreasing applied force.FIG. 6: Snapshot of the density field in the sliding state at f x = 0 . p4p5p6p7 FIG. 7: Configuration of the density peak locations withcorresponding coordination numbers for the density plot inFig. 6.
To better understand the dynamic melting transitionat f a and the freezing transition at f b , we inspect thesteady state configurations near the two thresholds aswell as the region between the two thresholds in realspace, which yield more direct and detailed informationthan the structure factor at the peak values. Treatingthe peaks in the phase field as ”particles”, the coordina-tion number of each particle in the commensurate c (2 × p , p , p , p . The results are shownin Figs. 4 and Fig. 5 for increasing and decreasing ap-plied forces respectively. Fig. 4 shows that when theforce f is increased beyond f a , besides the rapid drop in p consistent with the structure factor data, the fractionof other coordination numbers p , p , p also increasessignificantly untill f reaches f c , beyond which p and p start to decrease and we have a continuous transition toan incommensurate hexagonal phase. Thus the natureof steady state above f a is a strongly disordered stateanalogous to the high temperature phase in the absenceof driving force above the commensurate melting tem-perature. On decreasing the force below f a , the data inFig. 5 shows that the system remains in a melted statewith large disorder until the threshold f b is reached, be-low which we have only 4-fold coordination number andthe system returns to a pinned c (2 ×
2) phase. Takentogether, the qualitative behavior of the structure factorand the coordination number strongly suggest that thetransitions at f a and f b can be regarded as a force in-duced dynamical melting and freezing transition respec-tively.Finally, we look at a snapshot of the phase field ψ ( x )obtained in the steady state for a driving force just abovethe dynamical freezing threshold f b . This is shown inFig. 6. It consists of stripes of commensurate c (2 × f b .The time sequence of the freezing is shown in Fig. 8. Inreturning to the pinned state, the domain wall regionsgradually shrink and eventually disappear.The main features of the dynamical melting and freez-ing and hysteresis effects in PFC model described aboveare similar to those found earlier by molecular dynamicssimulations of particle models with interacting Lennard-Jones potentials and the uniaxial Frenkel-Kontorova FIG. 8: Snapshots of the density field for increasing timesafter starting from the moving state of Fig. 6 and reachingthe pinned commensurate state, at a force f x = 0 .
045 justbelow f b . Times are in units of 10 time steps. model under a driving force starting from a commensu-rate state. The similarity of results from these very differ-ent models demonstrates the universality of the hysteresisloop for models with inertia effects and the macroscopicconsequence of stick and slip motion. When comparedwith the Lennard-Jones particle model one notable dif-ference is the mechanism of the dynamical freezing at f b .In the present PFC model, it involves parallel liquid-likedomain walls similar to the uniaxial Frenkel-Kontorovamodel, although in the absence of the driving force thereis no easy direction. In the Lennard-Jones model, nu-cleation and growth occur via closed pinned domains .The origin of this intriguing difference requires furtherinvestigation of both atomistic and PFC models. Onepossibility is that the nucleation of stripes is related tothe absence of a fixed constraint on the number of peaksin the PFC model. In this case, the mechanism of the dy-namical freezing found in the present PFC model shouldbe compared to the results of particle models with a con-stant chemical potential rather than with a fixed particlenumber. Unfortunately, such results are currently un-available. V. CONCLUSIONS
In this paper, we have derived general stochastic dy-namical equations for the particle and momentum den-sity fields including both thermal fluctuations and iner-tial effects. The new equations are applied to the studyof the nonlinear response to an external driving forcefor a PFC model with a pinning potential. The modeldescribes a driven adsorbed layer as a continuous den-sity field, allowing for elastic and plastic deformations.The numerical results showed that at low temperatures,the velocity response of an initially commensurate layershows hysteresis with dynamical melting and freezingtransitions for increasing and decreasing applied forcesat different critical values. The inclusion of both ther-mal fluctuations and inertial effects are crucial for a cor-rect description of these dynamical transitions. The mainfeatures of the nonlinear response, in particular the hys-teresis loop separating the static friction and sliding fric-tion thresholds are similar to the results obtained previ-ously with particle models. However, the details of thedynamical melting and freezing mechanisms are signifi-cantly different. In the PFC model considered here, theycorrespond to nucleation of stripes rather than closed do-mains found in particle models. It is possible to describemore realistic sliding adsorbed systems if the parametersof the model are adjusted to match experimental systems, similar to the recent works for the colloidal systems andFe . Acknowledgments
J.A.P.R. acknowledges the support from Secretaria daAdministra¸c˜ao do Estado da Bahia. E.G. was supportedby Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜aoPaulo - FAPESP (Grant No. 07/08492-9). S.C.Y. alsoacknowledges FAPESP (Grant No. 09/01942-4) for sup-porting a visit to Instituto Nacional de Pesquisas Espa-ciais. This work has been supported in part also by theAcademy of Finland through its COMP Center of Ex-cellence grant and by joint funding under EU STREP016447 MagDot and NSF DMR Award No. 0502737. K.R. E. acknowledges support from NSF under Grant No.DMR-0906676. E.G. thanks Sami Majaniemi for manyhelpful discussions. B.N.J. Persson, Sliding Friction: Physical Principles andApplications (Springer, Heidelberg, 1998). articles in Physics of Sliding Friction , edited by B.N. J.Persson and E. Tosatti (Kluwer, Dordrecht,1996). B. N. J. Persson, Phys. Rev. Lett. , 1212 (1993); J.Chem. Phys. , 3449 (1995). H. Yoshizawa, P. McGuiggan, and J. Israelachvili, Science , 1305 (1993) 14), 109 (1992); Klafter, D. Gourdon,and J. Israelachvili, Nature , 525 (2004). E. Granato and S. C. Ying, Phys. Rev. B. , 5154 (1999). E. Granato and S.C. Ying, Phys. Rev. Lett. 85, 5368(2000); Phys. Rev. B , 125403 (2004). J. Krim, D.H. Solina, and R. Chiarello, Phys. Rev. Lett. , 181 (1991) A. Carlin, L. Bruschi, M. Ferrari, and G. Mistura, Phys.Rev. B , 045420 (2003). D. S. Fisher, Phys. Rev. B , 1396 (1985). K.R. Elder, M. Katakowski, M. Haataja, and M. Grant,Phys. Rev. Lett. , 245701 (2002). K.R. Elder, and M. Grant, Phys. Rev. E , 051605 (2004). K.R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M.Grant, Phys. Rev. B , 064107 (2007). C.V. Achim, M. Karttunen, K.R. Elder, E. Granato, T.Ala-Nissila, and S.C. Ying, Phys. Rev. E , 021104(2006); J. Phys.: Conf. Ser. , 072001 (2008). J.A.P. Ramos, E. Granato, C.V. Achim, S.C. Ying, K.R.Elder, T. Ala-Nissila, Phys. Rev. E , 031109 (2008). C.V. Achim, J.A.P. Ramos, M. Karttunen, K.R. Elder, E.Granato, T. Ala-Nissila, and S.C. Ying, Phys. Rev. E ,011606 (2009). P.M. Chaikin and T.C. Lubensky,
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