Melting points of Lennard-Jones particles and trimers ... - dirac

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Melting points of Lennard-Jones particles and trimers ... - dirac

Melting points of Lennard-Jones particles and trimers computed by interfaces pinningUlf R. Pedersen ∗Institute of Theoretical Physics, Vienna University of Technology,Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria andFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria(Dated: November 22, 2012)Solid-liquid Gibbs free energy differences are computed from two-phase simulations with an externalspring-like field biasing configurations with an interface. This novel approach, proposed in[SHORT PAPER], is investigated in details for the well-studied Lennard-Jones model and appliedto isosceles Lennard-Jones trimers. The Gibbs free energy difference is computed from the averageforce of the external field needed to pin the interface. Melting points are subsequently approachediteratively using Newtons method for finding roots. The Lennard-Jones melting line is determinedto a high precession and reproduces result achieved with other methods. The temperature-densitycoexistence region is outlined by isomorphs. Statistic and systematic errors of the interface pinningmethod are investigated. Melting temperatures at ambient pressure of isosceles Lennard-Jonestrimers with angles ranging from 70 to 100 degrees are determined. Two crystal structures are considered:a near base centered cubic structure and a near face centered cubic structure with preferredangles of 77 and 96 degrees, respectively. Liquid dynamics are slowed when the angle is increased.A trimer angle of 83 yield the largest distance between isochrones and the melting temperature,suggesting that this is the optimal glass-forming angle. It is conjectured that better glass-formersmay be found at angles larger than the ones considered in this study and that near 65 degree trimersare prone to crystallization.Understanding highly viscous liquid dynamics near theglass-transition comprise a paramount unsolved problemof condense matter physics [? ]. In the last decades, theuse of computer simulations have played an increasinglyimportant role in investing underlying assumptions andpredictions of theories. Well-studied simulation models,such as the Kob-Andersen binary Lennard-Jones (LJ)mixture [1], the Wahnström LJ mixture [2] or the Lewis-Wahnström isosceles LJ trimers [3, 4], are characterizedby being computational cheap and not prone to crystallization.With increasing computer power, however,simulations nowadays reach timescales where these modelscrystallize [5–7]. Thus, there is a need for modelsthat inherit the simplicity of the well-studied models,while being better glass-formers. In this paper we investigatecrystalline stability of isosceles LJ trimers similar tothe model suggested by Lewis & Wahnström [3, 4]. Thestrategy is to design a good glass-former by changing thetrimer angle. For this, the crystal stability is identified bycomputing the melting temperature for various angles (atambient pressure considering two crystal structures). Forthe originally suggested angle of 75 ◦ , trimers crystallizeinto a structure where LJ particles occupies a near basecentered (BC) cubic lattice (consistent with the findingof References [7, 8]). At angles near 90 ◦ , however, nearclose-packed face centered (FC) cubic structures are morestable. At 83 ◦ destabilization of the crystal is optimal.Moreover, liquid dynamics are slowed when the angle isincreased.In general, it is non-trivial to locate first order transitionssince they rely on a rare event of nucleation [9].In computational studies they are in particular easy tobypass since they typically depend on homogeneous nucleationin a small periodic systems. Thus, constructingphase diagrams constitute an essential problem in computationalstudies of condense matter – atomic, molecular[10], nano-materials, macroscopic particles or biologicalsystems. We utilize a novel method briefly introducedin Ref. [11]. In short, the strategy is to compute the averagethermodynamic force needed to pin the interfaceof a two-phase system. This force, given by the Gibbsfree energy difference between the phases, is computedby applying an external spring-like field to the system.We will refer to this method as “interface pinning”. Themotivation is to improve the popular “direct approaches”[8, 12–17]. The strategy in these methods are to setupa two phase system – for instance a liquid and a crystal– and then simulate a trajectory at a given state point.Crystallization or melting determines whether the crystalor the liquid is the thermodynamical stable phase,respectively. The present method converts these out-ofequilibriumapproaches into a well-defined equilibriumcomputation. The method reproduces the coexistenceline of the Lennard-Jones (LJ) model found with othermethods, and computed Gibbs free energies are consistentwith thermodynamic integration.The aim of this paper is two-fold: i) to give a detaileddescription and investigation of the interface pinningmethod and ii) to compute stability of LJ trimercrystals as a function of trimer angle. The remainderof the paper is organized as follows: First we introducethe interface pinning method in general terms. Then westudy the method by applying it to molecular dynamicscomputations of the well-studied Lennard-Jones model.The last part of the paper use interface pinning to studystability of LJ trimer crystals, and discuss glass-forming


2ability.THE INTERFACE PINNING METHODThe following provide the theoretical background ofthe interface pinning method for computing solid-liquidchemical potential differences and discuss the location offreezing points.Consider a two phase system in a periodic orthorhombicbox as exemplified on Figure 1. Choose box lengthsX ≤ Y < Z so that the box is elongated in the z-direction. Sample configurations of the Np z T -ensembleof the unstrained crystal: i.e., the constant temperatureand pressure ensemble where the box lengths X and Yare fixed at values where the crystal is unstrained while Zare allowed to fluctuate for barostating. In the elongatedgeometry the two interface planes will be orthogonal tothe long axis in order to minimize the interface area andthereby the interface Gibbs free energy G i . Note that thesystem is only barostated in the z-directions since interfaceparticles will add additional stress to the p x and p ycomponents [15]. (We note that a orthorhombic box isnot a requirement. The angle between the box vectors⃗X and ⃗ Y may differ from 90 ◦ , but should then be keptconstant at an angle not straining the crystal).We will assume that the relative interface positionsdoes not contribute to the Gibbs free energy and thatbulk of pure phases are represented, at least in the centeroff the slaps (we will discuss and verify assumption later).Then, particles may be labeled either s = [solid], l =[liquid] or i = [interface] and the total number of particlesmay be written asN = N s + N l + N i (1)Let µ l be the chemical potential of the liquid and µ s thechemical potential of the solid so the total Gibbs freeenergy of the two-phase system isG = N s µ s + N l µ l + G i . (2)(Figure 2). By combining the two above equations andreplacing G i and N i by a nonessential constant:G = N s ∆µ + [constant], (3)where ∆µ ≡ (µ s −µ l ). Throughout the paper we let “∆”denote “[solid] − [liquid]” and let “[constant]” refer to aconstant with a nonessential value.External field pinning interfacesTo sample configurations in the two-phase region, weapply a spring-like bias field: For a given configurationR = {r 1 , r 2 , . . . , r N } define a global crystallinity orderFIG. 1. Two-phase configuration of LJ trimers (θ = 75 ◦ ) in aperiodic orthorhombic box at a state point where the liquidis thermodynamically stable and the crystal is metastable.An external bias field have, however, been applied stabilizingtwo-phase configurations. The average force exerted by theexternal field on the system is related to the Gibbs free energydifference between the phases. Trimes are colored accordingto cubatic orientational order (see Ref. [7]) making crystallineparticles red.parameter Q(R) having on average Q l when the system isin the liquid phase and Q s when the system is in the solidphase (at a given pressure p and temperature T ). LetQ have an linear dependency with the amount particlesin various phases when integrating out other degrees-offreedom:Q = N sN Q s + N lN Q l + N iN Q i = N s∆Q + [constant], (4)Nwhere NiN Q i is a constant contributions from interfaceparticles. Let U(R) be the energy of the unperturbedsystem, andU ′ (R) = U(R) + κ 2 (Q(R) − a)2 , (5)the energy of the system plus the external field where ais the spring anchor point and κ is the spring constant.The Gibbs free energy of the biased system along the Qcoordinate isG ′ (Q) = G(Q) + κ 2 (Q − a)2 + k B T ln(Z ′ /Z) (6)where Z ′ and Z are the partition functions of the biasedand the unbiased systems respectively. Insertion of equations3 and elimination of N s with equation 4 show that


3Gibbs free energy G2000computed20 40G iwith bias fieldnobiasfieldα∆Q = N∆µDistribution1T = 0.8p = 1.5κ = 2κ = 4κ = 20κ = 10a = 27liquidtwo phasescrystalCrystallinity order parameter Q022 24 26QFIG. 2. Sketch of Gibbs free energy G(Q) (solid; black) alonga crystallinity order parameter Q at a state point where theliquid is thermodynamic stable and the crystal is metastable.The double arrows indicates the interface contribution G i(red) and the bulk contribution N∆µ (blue), respectively.The dashed green curve indicates the Gibbs free energy G ′of a system with an spring-like external field applied. Theinsert show G(Q = |ρ k |) for the LJ model in the two-phaseregion (N = 5120; T = 0.8; p = 1.5).FIG. 3. P ′ (Q) distribution of the Lennard-Jones model atT = 0.8 and p = 1.5 of four bias-fields: a = 27 and κ ={2, 4, 10, 20} respectively. Fluctuations of crystallinity orderparameterQ are Gauss distributed (dashed lines; Eq. 7).arrives from isolating ∆µ from the average, 〈Q〉 ′ , of thedistribution given in Eq. 7.Q fluctuations are limited to a Gauss distribution:√{κP ′ (Q) =2πk B T exp −κ2k B T[Q − a + N∆µ ] } 2κ∆Q(7)(in the two phase regime). Figure 3 show this distributionfor the Lennard-Jones model.Computing ∆µ from average force exerted onexternal fieldThe chemical potential difference between the phases∆µ can be computed from the average force F fieldQ =−κ(〈Q〉 ′ −a) along the Q coordinate exerted by the externalfield on the system: When the system is equilibratedthe interface is stationary except for thermal fluctuations.where F systemThus F fieldQapplying the chain rule ∂G= −F systemQ∂Q∂Q ∂N sQ= ∂G∂N s= − ∂G∂Qand by= ∆µ then∆µ = − κ∆QN (〈Q〉′ − a) (8)where ∂Q∂N s= ∆Q/N is obtained from Equation 4. Alternatively,a statistical mechanical deviation of this relationComputing coexistence state pointsusing Newtons root finding methodA coexistence state point may be computed to highprecision using the following iterative procedure: First,imagine that we have computed ∆µ at some roughguesstimate of a coexistence state point (p, T ). The coexistencespressure p m at the temperature T is definedas ∆µ(p m , T = T m ) ≡ 0 and∫ pm0 = ∆µ(p, T ) + dp ′ ∆v(p ′ , T ). (9)pAssuming that ∆v is constants (a good approximationwhen p ≃ p m ) the above simplifies top m ≃ p − ∆µ∆v(10)where ∆µ and the specific volume change ∆v are evaluatedat (p, T ). To improve the estimate, a new computationis now preformed at the pressure estimated usingthe above equation. This is repeated until ∆µ is zerowithin the statistical error. Alternatively, a melting temperaturemay be estimated by iteratively evaluatingT m ≃ T + ∆µ∆s(11)


4where the specific entropy change ∆s = (∆u + p∆v −∆µ)/T and ∆u is the specific internal energy change.Recipe for finding a coexistence state pointThe following recipe may be used to find a coexistencepoint (for a given system size). First, make a roughguesstimate of a coexistence state point (p, T ). Then:i: Construct a crystal configuration in an elongated orthorhombicbox;ii: Determine the box lengths X and Y of the unstrainedcrystal in an NpT simulation where X, Y and Zare allowed to fluctuate independently;iii: Compute Q s and v s in an Np z T simulation of theunstrained crystal;iv: Construct a liquid configuration in an elongated orthorhombicbox having box lengths X and Y ;v: Compute Q l and v l in an Np z T simulation of the liquid;vi: Construct a two-phase configuration;vii: Compute 〈Q〉 in an Np z T simulation of the twophasesystem with an interface pinning κ 2(Q − a)2field applied;viii: Calculate ∆µ using Equation 8;ix: If ∆µ is non-zero (within the statistical error), repeatabove steps at the coexistence pressure estimatedby Equation 10.TRANSLATIONAL ORDER PARAMETERTo utilize the method we must choose an order parameterQ(R). Unlike liquids, crystals have long-rangedtranslational order. Thus, the collective density field maybe used:Q(R) = |ρ k | where ρ k = N − 1 2N∑exp(−ik · r j ) (12)where k = (2πn x /X, 2πn y /Y, 0) and the indices (n x , n y )should be chosen so k correspond to a Bragg peak (tomaximize the contrast between liquid and crystal). n zis set to zero since Z fluctuate in the Np z T -ensemble.The 1/ √ N ensure scale invariance for the average liquidvalue, Q l ∝ 1. The intensity of a (single) crystal will,however, scale as Q s ∝ √ N. Since the phase of thecrystal is independent of the phase of the liquid, Q willscale linearly with the amount of particles in the phases(Eq. 4).jIn order to preform molecular dynamics simulations,forces exerted on particles from the external field have tobe evaluated: The force acting on particle j isF ′ j = F j − κ(|ρ k | − a)∇ j |ρ k | (13)where F j is the force without external field, and∇ j |ρ k | = −k R[ρ k] sin(k · r j ) + I[ρ k ] cos(k · r j )|ρ k | √ N(14)where R[ρ k ] = ∑ Nj cos(k · r j)/ √ N and I[ρ k ] =− ∑ Nisin(k · r j )/ √ N are the real and imaginary partsof ρ k respectively. Note that, although the force on particlej depends on all particles in the system, using theabove equations allow for a N-scaling algorithm: i) Loopover all particles to compute R[ρ k ], I[ρ k ] and |ρ k |; ii)Compute particle forces using equation 13.When preforming Monte Carlo simulation, the energychange δU ′ when attempting to move a particle or changingZ box length have to be evaluated:where δ|ρ k | 2|ρ attemptk|−|ρ currentk|. These may be computed by eval-− ρ currentkif the current value ofis stored. Moving particle j yielduating δρ k = ρ attemptkρ k = ρ currentkδU ′ = δU + κ 2 [δ|ρ k| 2 − 2aδ|ρ k |] (15)= |ρ attemptk| 2 − |ρ currentk| 2 and δ|ρ k | =δρ k = [exp(−ik · r attemptj ) − exp(−ik · r currentj )]/ √ N.(16)We note that δρ k only involves information about particlej allowing for efficient (N-scaling) computations. Whenscaling the Z box length δρ k = 0 and δU ′ = δU.Choosing field parametersIn this section we discuss considerations for choosingthe field parameters κ and a: Consider the nature of Qfluctuations for a given external-field. As sketched onFigure 2 (dashed; green), the Gibbs free energy with theexternal field isG ′ (Q) = κ 2 (Q − a + α/κ)2 + [constant] (17)where α = ∆µ ∂Ns∂Q= N∆µ/(Q s−Q l ) is the slope of G(Q)in the two-phase region (solid; black). Thus, the probabilitydistribution P ′ (Q) is Gaussian with mean value〈Q〉 = a − α/κ and variance σQ 2 = k BT/κ. Combiningthis with Equation 8 we find that the statistical error of∆µ is [relative error] ∝ 1/ √ N s , where N s is the numberof independent configurations in a given run. To understandthe scaling of N s with κ, we need to consider thedynamics of Q(t) fluctuations.


5There are two processes contributing to fluctuations inthe Q(t) time-series: i) fast phonon vibrations in the kdirection and other rearrangements of the pure phases,and ii) fluctuations of the interface position (i.e., conversionfrom one phase to the other). For small valuesof κ, process ii will dominate. In this regime, the interfacefluctuate diffusively, and the number of independentconfiguration for a given simulation time will scale asN s ∝ 1/σQ 2 ∝ κ. For large values of κ, process i willdominate. Q fluctuation will not reflect interface fluctuations,but a under damped sound wave along the kvector. The conclusion is that there is a optimal choiceof κ. An analysis of Q(t) fluctuations is needed to illuminatethis further. In practice (discussed later), we find,that the method is quite forgiving towards the choiceof κ. Orders of magnitude different κ’s yield the samestatistical error. A pragmatic first choice is to chooseκ so Q fluctuations varies about one width of a crystalunit-cell, κ ≃ k B T Nz 2 /∆Q 2 where N z is the number ofunit-cells in the z-direction. The anchor point a shouldbe chosen to maximize the distance between the interfaces:a ≃ Q l + 1 N∆µ2∆Q +κ∆Q. ∆µ is unknown but canoften be estimated. E.i., near coexistence, ∆µ ≃ 0 anda = Q l + 1 2∆Q is a good choice. In practice, the methodis forgiving towards the choice of a (discussed later).PACKING OF LENNARD-JONES PARTICLESTo verify the method, we construct configurations offace centered cubic (FCC) super-cells of Lennard-Jonesparticles. Interactions are truncated and shifted: U =Ni>j u ij where u ij = 4(r −12ij − r −6ij ) − 4(r−12 c − rc−6 ) forr ij < r c and zero otherwise (in Lennard-Jones units).Two truncations are considered: r c = 2.5 and r c = 6.Molecular dynamics simulations are preform using theLAMMPS software package. The ρ k -field was implementedinto the package. The Parrinello-Rahman barostatis used [21] with a time constant of 8 Lennard-Jonestime units.Table I and II and list coexistence points computedwith the interface pinning method using r c = 6 andr c = 2.5 respectively (initial super-cell of 8×8×20 FCCunit cells; N = 5120; κ = 4; a = 26; n x = 16; n y = 0).The melting line agree with the Clausius-Clapeyron relation,m dpdT m= ∆s∆v(two last rows in Tables I and II)demonstrating agreement with thermodynamic integration.The left-hand side of the equations is computedby central differences and is associated with large errorcompared to the right-hand side.p3020100deviationsfrom fit0.8 1.2 1.6 2 2.4CP0.1p m+p tail(r c=6)p m+p tail(r c=2.5)cubic fit[Mastny 2007][Sousa 2012]0.8 1.2 1.6 2 2.4TFIG. 4. Coexistence line of the Lennard-Jones model in thepT -plane. Filled symbols are coexistence points computedwith the interface pinning method (Tables I and II) correctedfor truncation of long-range contributions to the pressure(Equation 18). The solid line is a cubic fit (r c = 6). +’sand ×’s are coexistence points reported in References [18]and [19]. The insert show deviations from the fit. The asterixis the gas-liquid critical point (T CP = 1.31; p CP = 0.15) [20].Melting line corrected for truncation of pairinteractionsTo compare with coexistence lines reported in literature[18, 19] (computed with other methods) the meltingpressure is corrected for truncation of long-range interactionsusing an approximate pressure correction p tail . Forthe long-range truncation r c = 6 the correction is smalland melting lines agrees to a high precision (Figure 4).For the shorter truncations r c = 2.5 the approximatecorrection is surprisingly good.Consider the correction in a simulations of bulk phases.It is convenient to assume that the radial distribution isconstant at distance larger than the truncation. Thenthe correction is analytic and only depend on ρ and r c[22]. Since the density of solid and liquid are different,the correction is different for the two phase. Using theaverage of the pressure corrections of the pure phases,thenp tail = 16 3 π ( 12 v−2 s0+ 1 2 v−2 l) ( 23 r−9c)− rc−3 . (18)Third rows in Tables I and II list the corrected meltingpressures (p m + p tail ). Deviations between the corrected


6TABLE I. LJ solid-liquid coexistence line of r c = 6, N = 5120.T m 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6p m -0.970 0.132 1.337 2.629 4.012 6.930 10.145 13.549 17.104 20.857 24.850 28.916 33.041p m + p tail -1.030 0.068 1.270 2.560 3.940 6.853 10.063 13.463 17.014 20.763 24.753 28.815 32.937v s 1.0553 1.0400 1.0242 1.0086 0.9933 0.9661 0.9412 0.9194 0.9002 0.8827 0.8663 0.8518 0.8388v l 1.2358 1.1804 1.1425 1.1120 1.0864 1.0470 1.0127 0.9852 0.9616 0.9403 0.9208 0.9038 0.8885u s -6.314 -6.125 -5.929 -5.722 -5.506 -5.037 -4.526 -3.972 -3.385 -2.768 -2.120 -1.451 -0.764u l -5.024 -5.008 -4.902 -4.755 -4.570 -4.106 -3.603 -3.021 -2.409 -1.762 -1.085 -0.379 0.325∆h -1.115 -1.135 -1.185 -1.239 -1.310 -1.492 -1.649 -1.842 -2.025 -2.206 -2.389 -2.575 -2.729∆s -1.858 -1.622 -1.480 -1.377 -1.311 -1.245 -1.177 -1.153 -1.125 -1.103 -1.085 -1.073 -1.050dp mdT m- 11.5 a 12.5 13.4 14.3 15.3 16.5 17.4 18.3 19.4 20.1 20.5 -∆s∆v10.3 11.6 12.5 13.3 14.1 15.4 16.5 17.5 18.3 19.1 20.0 20.6 21.1a :dp mdT mcomputed by central difference of values in the first two rows.TABLE II. LJ solid-liquid coexistence line of r c = 2.5, N = 5120.T m 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6p m -0.212 0.928 2.185 3.514 4.939 7.921 11.181 14.632 18.180 22.007 26.029 30.050 34.314p m + p tail -1.046 0.049 1.264 2.555 3.943 6.859 10.056 13.448 16.943 20.717 24.688 28.661 32.878v s 1.0614 1.0452 1.0277 1.0110 0.9951 0.9672 0.9421 0.9202 0.9014 0.8835 0.8671 0.8530 0.8394v l 1.2194 1.1714 1.1360 1.1080 1.0830 1.0446 1.0117 0.9838 0.9612 0.9399 0.9211 0.9043 0.8888u s -5.358 -5.156 -4.953 -4.742 -4.513 -4.020 -3.483 -2.907 -2.301 -1.663 -0.997 -0.315 0.394u l -4.294 -4.218 -4.075 -3.888 -3.683 -3.183 -2.627 -2.041 -1.400 -0.727 -0.009 0.696 1.446∆h -1.031 -1.055 -1.114 -1.195 -1.264 -1.450 -1.634 -1.797 -1.989 -2.178 -2.393 -2.552 -2.747∆s -1.718 -1.507 -1.392 -1.327 -1.263 -1.207 -1.168 -1.123 -1.107 -1.091 -1.087 -1.063 -1.055dp mdT m- 12.0 a 12.9 13.8 14.7 15.6 16.8 17.5 18.4 19.6 20.1 20.7 -∆s∆v10.9 11.9 12.9 13.7 14.4 15.6 16.8 17.6 18.5 19.3 20.1 20.7 21.4a :dp mdT mcomputed by central difference of values in the first two rows.melting pressures when truncating at r c = 2.5 or r c = 6are compatible to statistical noise (Tables I and II).Computed melting points are shown on Figure 4 asfilled symbols. In the same figure, +’s and × are coexistencepoints reported in references [18] and [19], respectively.The agreement is good. Differences are highlightedin the insert by showing deviations from a cubicfit to the melting line computed in this study. Deviationsfrom the points in Reference [19] are within statistical error,while the [18] melting pressure is systematically tolow by about ∆p ≃ 0.1. Systematic errors in computedmelting lines are common [18] and are typically related toapproximate tail corrections (like Eq. 18), finite size effectsor other method specific systematic errors. In latersections we will discuss possible sources of systematic andstatistical errors related to using the interface pinningmethod.Isomorph prediction of the ρT coexistence regionIn the following we will confirm a theoretical predictionof the melting region in the ρT -plane, Figure 5:A large class of systems have curves in the phase diagram,referred to as “isomorphs”, where structure, dynamicsand some thermodynamic properties are constant:T = Γρ γ(ρ) (19)where the Γ is a shift constant and the slope γ(ρ) is afunction of ρ. The latter is a slope in the sense that whenρ is change by 1% then T change by γ% (the slope in alog-log plot, Figure 5). This class of systems are coined“strongly correlating” since virial W (the potential partof pressure) and energy U fluctuations are strongly correlated,i.e. if δW = W − 〈W 〉 and δU = U − 〈U〉 thenR = 〈δW δU〉/ √ 〈(δW ) 2 〉〈(δU) 2 〉 is close to unity.For systems with inverse power-low pair interactions,u ij ∝ r −n , isomorph invariance is trivial with a constantslope γ = n/3. For systems with Lennard-Jones interactionsthe scaling is approximate and reflects an effectiveinverse power-low of repulsive interactions where the effectiveexponent is state point dependent [23]:ρ γ(ρ) = ρ 4 (γ ∗ /2 − 1) − ρ 2 (γ ∗ /2 − 2) (20)where γ ∗ is a constant that may be determined fromvirial-energy fluctuations at a single state point by using


7log 10(T)0.60.40.20-0.2γ = 5.24γ = 5.181 1.2ρr c= 6r c= 2.5[Sousa 2012]-0.1 -0.05 0 0.05 0.1log 10(ρ)6γ(ρ)5LiquidCoexistenceSolidG/kT + [constant]1.20.80.40.026 28 30 32Qp = 2.194T = 0.8κ = 0.5(8,8,20)0.0-0.1(G-G lin)/kTone planea = 26a = 27a = 28a = 29a = 30a = 31allG lin24 26 28 30 32QFIG. 5. Coexistence region of the Lennard-Jones model in theρT -plane on a log-10 scale. Filled symbols are computed withthe interface pinning method (Tables I and II). ×’s are fromReference [19]. The agreement is good. At high temperaturesand densities, the coexistence region is outlined by isomorphs(see text for details). The shape of the isomorphs (insert) isdetermined at the state points indicated by open circles.FIG. 6. Growing crystal planes is barrierless (∆G ≪ k BT ).Circles show G(Q) in units of k BT computed with umbrellasampling [22] of six fields (crosses; shifted for clarity). Prefactorsof Equation XXX is computed using the sing MBARalgorithm [25]. The inset show deviations from linear dependency.〈δW δU〉that γ(ρ) =〈(δU) 2 〉. The solid line on Figure 5 is a liquidisomorph where γ ∗ =? is determined from γ(0.94) = 5.24computed at T = 1.5 [24]. The dashed line is a crystalisomorph where γ ∗ =? is determined from γ(1.06) = 5.18computed at T = 0.8 [24]. At high temperatures and densitiesthe coexistence region is outline by thees isomorphs(Γ’s are determined by fitting to T > 2). Deviations fromthe prediction are, however, significant at low ρT . Thisis expected, and reflects that attractive interactions cannotbe treated in a mean-field way [? ]. Consistently,the virial-energy correlation coefficient R deviate significantlyfrom unity in this part of phase space (see Ref. [?]).The Gibbs free energy of interface positionsWe have assumed that the Gibbs free energy in thetwo-phase region is independent of the interface positions.There are, however, two effects that may spoilthis assumption: i) If the distance between the the interfacesis too small, particles in one or both phases willnot reflect bulk properties. ii) Particles added to a completedinterface contribute with a slightly different ∆µthan ones that are added to a partially build interface.This will introduce a rippling of G(Q) with a period of∆Q/N z (see e.g. Ref. [27]) and a shape reflecting thestructure of a unit cell.To investigated the Gibbs free energy dependencyof interface positions, we preformed simulations over arange of a’s with overlapping P ′ (Q) distributions, andconstructed G(Q) using Equation 6 (i.e. umbrella sampling[22]; constants we determined using the MBAR algorithm[25]). See Figure 6. The assumption is fulfilledwithin the statistical noise.Error dependency of κAs discussed in the first part of the paper, an analysisof Q(t) fluctuations is needed to investigate the role ofκ on the statistical error. Thus, we computed the Q(t)time autocorrelation function:C(t) = 〈δQ(t 0 )δQ(t 0 + t)〉/〈(δQ) 2 〉 (21)where δQ(t) = Q(t) − 〈Q〉. C(t) is shown on figure 7for four κ’s. There is a separation of time-scales betweenfast phonon fluctuations and slower interface fluctuations.The phonon vibrations have the characteristicsof a under-damped harmonic oscillator, while, interfacefluctuations follow exponential relaxation. For the investigateκ’s the characteristic time is constant, related to


8Q(t) autocorrelation function10κ = 2κ = 4Rel. err.κ = 10κ = 202%1%0%1 10κT = 0.8p = 1.50.1 1 10 100t10 3 ∆µ0-1-2-3-4-5-6640 LJ particles-70.35 0.4Interface position: (’-Q l)/(Q c-Q l)FIG. 7. Q time autocorrelation function for four spring constantsκ (see first row of Table ?? other simulations parameters).Decorrelation occurs on two timescales: i) a fast timescale related to sound waves and ii) interface movements.Dashed lines are A exp(−t/τ) fits to the slow interface process.The inset show the relative statistical error on the∆µ = 0.0811 estimate, computed by dividing runs into independentblocks [26] of length ∆t = 100.the time to rearrange the interface. For smaller values ofκ this timescale scales as 1/κ (data not shown).The relative statistical error on the ∆µ, estimated fromcorrelated data, is computed by dividing runs into independentblocks [26], see inset on Figure 7. For the investigateκ’s, spanning two orders of magnitude, the erroris independent of κ. We note that for small values of κwe expect that the error to scale as 1/ √ κ, as discussedin the introduction.G(Q) in general have (small) ripples with a period correspondingto the width of a unit-cell. Choosing a lowκ will average over several unit-cell, minimizing this systematicerror when using Equation 8. Thus there is acompetition between low “statistical error” and “systematicerror”. We have not been able to investigate thisaspect in more details, since the theorized systematicrippling error is hidden in the statistical error (Figure6). We emphasize that rippling may be accounted for byinvestigating G(Q) using Equation 6 and does not constitutea fundamental problem for the method. They are,on the other hand, interesting since they give informationsabout barriers to grow an interface.FIG. 8. Computed ∆µ in small system of N = 640 particles asa function of interface position defined as (〈Q〉 ′ −Q l )/(Q s−Q l )(4×4×10 FCC unit cells, T = 0.8, p = 2.194, κ = 10, 7.5 ≤a ≤ 9.0).Systematic errors for small systemsFigure 8 investigate the performance of the methodgoing down to systems of 640 particles. At this smallsystems size we find systematic errors within the statisticalnoise.TODO TODO TODOComputing surface tension from two-phasesimulationAn advanced of the interface pinning method is thatproperties of the interface may be computed. To illustratethis, we compute the surface tension γ of theinterface[28–30]. This can be done by integrating thedifference between the tangential p T (z) and the normalpart p N (z) of the pressure tensor, i.e. the virial expressionof the surface tension:γ = 1 2∫ Z0dz[p N (z) − p T (z)]. (22)The factor 1 2accounts for the integral representing twointerfaces. Since forces are non-local the integrandp N (z) − p T (z) is not uniquely defined (the integral isunique). Using the integrand suggested by Krikwood and


9Buff the surface tension may be computed as [28, 30]:〈γ = 1 ∑ rij 2 − 〉3z2 ijF ij (23)8A r iji≠jAt coexistence at T = 0.8 we find 0.41 ± 0.05 (r c = 2.5)in agreement with the surface tension computed by othermethods [31].Comparison to other methodsSeveral methods have been developed [15, 22] to locatecoexistence states in the phase diagrams: In the directapproaches a simulation is preformed of a two phasessystem with an explicit interface [8, 12, 13, 15–17]. Thefavorable phase will grow (in a proper ensemble) allowingto locate coexistent by changing the intensive variables,pressure, temperature or chemical potential. An alternativeis to apply a indirect method, i.e. computing theGibbs free energy of the pure phases individually. Thechemical potential of a bulk phase can be computed byparticle insertion [32, 33] or by thermodynamic integrationto a state with know chemical potential, e.g. idealgas [34], harmonic solid [35] or Einstein solid [36]. Anotheralternative is to introduce an implicit interface bytransferring particles between two bulk simulations [37].For details and more methods the reader is encouragesto explore Refs. [15, 22] and references within.The interface pinning method inherits the conceptualsimplicity ease to implement of the moving interfacemethod.PACKING OF TRIMERSNext, we apply the interface pinning method toLennard-Jones trimers. Consider molecules of three LJparticles placed in the corners of a isosceles triangle withtwo sides of length σ spanning an angle θ. Bonds andangles are constrained using harmonic interactions.In 1993 Lewis & Wahnström [3, 4] suggested an angleof 75 ◦ as a model of a glass former, specifically orthoterphenyl(OTP). Nowadays computers, however, reachtimescales where the supercooled liquid of this modelcrystallize into a structure where LJ particles occupies anear body centered (BC) cubic lattice [7, 8]. The optimalangle in the base centered cubic is 2 sin −1 (1/ √ 3) ≃ 70.5 ◦[7], however, since the typical LJ distance is larger than σ(r min /σ = 2 1 6 ≃ 1.12), the optimal angle larger. From anapproximate analytic expression of the p = 0, T = 0 stability,it was argued that the Lewis-Wahnström angle issurprisingly close to the optimal angle of 76.4 ◦ . This wasthe direct motivation to address the question: “What isthe angle dependency of crystal stability relative to theliquid?”. We will limit ourself to 70 ◦ < θ < 100 ◦ andFIG. 9. Close packed structures of isosceles LJ trimers. LJparticles (on triangle corners) occupies sites akin to the facecentered cubic (upper) and body centered cubic (lower) structure(Table III). Gray boxes outline orthorhombic unit cellscontaining four and two trimers respectively. These structuresmay accommodate a range of trimer angles.Coexistence temperature, T m[K]4003903803703601 K per 1 oBCFC35070 o 75 o 80 o 85 o 90 o 95 o 100 oTrimer angle, θFIG. 10. Solid-liquid coexistence temperature at ambientpressure as a function of trimer angle.


10Rotational correlation function C rot(t)10.80.60.40.270 o 75 o 83 o 90 oTrimersT = 300 K00.1 1 10t [ns]FIG. 11. Rotational autocorrelation function of the trimerlong-bond of trimers with angles of θ = {70 ◦ , 75 ◦ , 83 ◦ , 90 ◦ } atT = 300 at ambient pressure. An increase of the angle slowsdynamics.adapt LJ parameters suggested by Lewis & Wahnström:σ = 4.83 nm, ε = (600 K) × k B and m = 76.768 u.First we need to construct candidates for close packedstructures. We will use a philosophy of simplicity andintuition and leave it for future investigations for a moresystematic search. When the trimer angle is near thosebetween close packed spheres, 60 ◦ and 90 ◦ , LJ particlesmay occupy sites akin to hexagonal close packed or facecentered (FC) cubic. Triangles may decorate these latticesin numerous ways. Structures made up of lines of“herring bones” [7] may accommodate a range of anglesbond lengths and LJ-pair distances while keeping an orthorhombicunit cell. Figure 9 show two such structuresand their orthorhombic unit cells. Super-cells was equilibratedat p = 0, and varies T and θ. Average moleculecoordinates in orthorhombic unit-cells are listed in TableIII.Figure 10 show the coexistence temperature for closedpacked structures with respect to the liquid (at p = 0)computed by interface pinning. The optimal angle for theBC structure is ∼ 78 ◦ in good agreement with the findingsof Ref. [7]. Similar, the optimal angles for the FCstructure is 95 ◦ rather than 90 ◦ . As for the BC structure,this widening of the angle is due intermolecular neighbordistances being larger that intramolecular distances.Widening the angle slow dynamics of the liquid suggestingthat a wide angle is preferred to make a goodglass former. This is exemplified in Figure 11 showingthe rotational correlation function (RCF)C rot (t) = 〈u i (t ′ ) · u i (t ′ + t)〉 (24)τ rot[ns]10 210 110 010 -190 o83 o75 o70 o10 -2280 320 360 400 440 480T [K]FIG. 12. Rotational correlation time as a function of temperatureof trimers with angles θ = {70 ◦ , 75 ◦ , 83 ◦ , 90 ◦ }. Theinsert shows isochrones in the temperature-angle plane.where u i is a unit vector pointing along the longest bondof molecule i and 〈. . .〉 is average over molecules and t ′ .(This is a first order Legendre polynomial RCF and correspondto what is measured in dielectric spectroscopy [38]if cross-correlations between molecules can be ignored).Figure 12 show the temperature dependency of the characteristicrotational time τ rot defined asC rot (τ rot ) ≡ 1/e. (25)The slowing with persist when temperature is lowered.How significant is the slowing down related to thewidening of the angle relative to the lowering of themelting temperature? To answer this, we investigateisochrones in the temperature-angle plane shown in theinsert of Figure 13. In the low-temperature region of thephase diagram, a angle increase of 1 ◦ correspond to atemperature increase of about 1 K. Assuming that thispersist to lower temperature, the glass-transition temperatureT g have the same angle dependency. For comparison,the change in melting temperature from 83 ◦ to 93 ◦is 2.5 K per 1 ◦ (Figure 10). Thus, the change in meltingtemperature change more rapidly than isochrones (although,interestingly, changes are compatible in size forlarge angle changes).


11Isochrone slope [K per degree]2.01.81.61.41.21.0T [K]40036032028070 75 80 85Trimer angle0.1 1Isochrone [ns]0.1 ns0.2 ns0.5 ns1 ns2 nsFIG. 13. The insert show rotational isochrones in thetemperature-angle plane. Isochrones have a linear θ dependencywith the slope shown in the main panel. At low temperature,the slope approaches 1 K per 1 ◦ .To related these empirical findings to the glass-formingability as a function of angle, we will in the followingargue that the distance between the melting line andisochrones is an indicator of glass-forming ability. Recallclassical nucleation theory (CNT) where the predictionfor the nucleation rate per unit volume is [9, 10]√∆µ 24D s N 2 {}3k CNT =c6πk B T Nc v l λ 2 exp −16πγ3 v s3k B T (∆µ) 2 (26)where D s is the self-diffusion constant, λ is a diffusiondistance andNc = − 32 3 πv sγ 3 (∆µ) −3 (27)is the size of the critical nucleus (in number of particles).In the super-cooled regime, D s change superexponentially[? ]. A convenient way to account forthis is to consider the rate along a isochrone T τ (θ) definedas a curve where the self-diffusion D s (T τ ) ≡ 1/τis constant (the glass-transition may be defined as suchan isochrone). Along a T τ (θ) curve good glass-formersare characterized by having low nucleation rates. NearT m (where ∆µ = 0 and the rate is zero) the dominantterm in Eq. 26 is exp{−(∆µ) −2 }. Using that∆µ(T ) ≃ [T m − T ]∆s(T m ) the nucleation rate per unitvolume along the τ-isochrone iswhere K τ (θ) ≃ K τ is assumed near constant alongisochrones. Thus, the angle that minimize T m (θ) − T τ (θ)may be used as a prediction for good glass-former. Notethat it is not crucial that CNT nor the expansion isquantitatively correct, but only qualitatively correct inthe sense that ∆µ changes dominates rate changes alongisochrones. Equation 28 is convenient due to its simplicity,but approximations may be crude at temperatureswhere crystallization occurs. For a detailed investigationof this, crystallization rates could be computed usingtechniques for sampling rare events [? ].Using Equation 28, the empirical results suggest thatthe optimal glass-former angle is at about θ = 83 ◦ (determinedgraphically on Figure 10 as the θ with the minimumdistance between the melting line and the dashedred line). T τ (θ) have an significant angle dependencysuggesting that better glass-formers may be found at angleslarger than investigated in this study. Indeed extrapolationof the isochrone on Fig. 10 suggest thatθ = 100 ◦ may be an good angle for a glass former (investigationsof crystal structures that are stable at largeangles would enlighten this hypothesis). Trimers withangles near 65 ◦ (a little larger than the close-packing angle60 ◦ ) are expected to be prone to crystallization. Thisangle is optimal for the crystal, but not the liquid, suggestingthat they may be more prone than the atomic LJmodel. Slower liquid kinetics of trimers relative to theatomic liquid, however, may play an important role [8].SUMMARYIn summary, we have investigated the applicability ofthe interface pinning method for computing Gibbs freeenergy.TODO TOOO TODOACKNOWLEDGMENTSThis work was financially supported by the AustrianScience Fund FWF within the SFB ViCoM (F41).The author is grateful for valuable comments and suggestionsfrom Christoph Dellago, Gerhard Kahl, FelixHummel, Thomas B. Schrøder, Jeppe C. Dyre.∗ ulf.pedersen@tuwien.ac.at[1] Walter Kob and Hans C. Andersen, “Scaling behaviorin the β-relaxation regime of a supercooledlennard-jones mixture,” Phys. Rev. Lett. 73, 1376–1379 (Sep 1994), http://link.aps.org/doi/10.1103/PhysRevLett.73.1376.[2] G. Wahnström, Phys. Rev. A 44, 3752 (1991).k τ (θ) = K τ (θ) exp { −[T m (θ) − T τ (θ)] −2} (28)


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