The solid-liquid interfacial free energy out of equilibrium

Bingqing Cheng,1 Gareth A. Tribello,2 and Michele Ceriotti1, ∗

1Laboratory of Computational Science and Modeling, Institute of Materials,Ecole Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland

2Atomistic Simulation Centre, School of Mathematics and Physics,Queen’s University Belfast, Belfast, BT7 1NN

(Dated: August 20, 2018)

The properties of the interface between solid and melt are key to solidification and melting, as theinterfacial free energy introduces a kinetic barrier to phase transitions. This makes solidificationhappen below the melting temperature, in out-of-equilibrium conditions at which the interfacial freeenergy is ill-defined. Here we draw a connection between the atomistic description of a diffuse solid-liquid interface and its thermodynamic characterization. This framework resolves the ambiguities indefining the solid-liquid interfacial free energy above and below the melting temperature. In addition,we introduce a simulation protocol that allows solid-liquid interfaces to be reversibly created anddestroyed at conditions relevant for experiments. We directly evaluate the value of the interfacialfree energy away from the melting point for a simple but realistic atomic potential, and find a morecomplex temperature dependence than the constant positive slope that has been generally assumedbased on phenomenological considerations and that has been used to interpret experiments. Thismethodology could be easily extended to the study of other phase transitions, from condensation toprecipitation. Our analysis can help reconcile the textbook picture of classical nucleation theorywith the growing body of atomistic studies and mesoscale models of solidification.

Solidification underlies many natural phenomenasuch as the freezing of water in clouds and the forma-tion of igneous rock. It is also crucial for various criticaltechnologies including commercial casting, solderingand additive manufacturing [1–3]. Computational andexperimental studies of these phenomena are compli-cated by the fact that homogeneous nucleation of solidsoften occurs at temperatures well below the meltingpoint Tm [4]. The free energy change associated withthe formation of a solid nucleus containing ns particlesis usually written as

G(ns) = µslns + γslA(ns). (1)

In this expression the first bulk term stems from thedifference in chemical potential between the solid andthe liquid µsl = µs − µl, which is negative below Tm.The second term describes the penalty associated withthe interface between the two phases, and introduces akinetic barrier to nucleation. This surface term is theproduct of interfacial free energy γsl, and the exten-sive surface area A(ns). However, due to the diffusenature of the interface, there is a degree of ambiguityin the location and area of the dividing surface be-tween phases. In classical nucleation theory (CNT),an infinitesimally-thin dividing surface divides the solidnucleus from the surrounding liquid. These two phasesare usually taken to have their bulk densities so thesurface area of the solid nucleus can be calculated us-ing A(ns) = σn

2/3s , where σ is a constant that depends

on the shape, e.g. σ = (36π)1/3v2/3s for a spherical

nucleus with bulk solid molar volume vs. Under theseassumptions, the rate of nucleation can be estimatedby calculating the free-energy barrier to nucleationusing G? = 4

27σ3µ−2

sl γ3sl.

The experimental determination of nucleation ratesis challenging due to the non-equilibrium conditions,and is often affected by considerable uncertainties [4, 5].These difficulties have triggered the development of nu-

merous computational methods for evaluating the freeenergy change during nucleation. Inducing the forma-tion of a solid nucleus from the melt in an atomistic sim-ulation [6–8] allows one to directly evaluate G(ns) atany given temperature, without explicitly partitioningthe free energy as in Eq. (1). However, this approachis restricted by the system size, and usually only worksat unphysical undercoolings, where the critical nucleusconsists of a few hundred atoms. Extrapolating resultsforG? obtained from these simulations to the mesoscalecritical nuclei formed at experimental conditions in-evitably requires invoking models such as CNT, whichinvolves quantities that are ill-defined for nanosizednuclei [9, 10]. In addition, because the nuclei observedin these simulations are tiny, defining their shape andsurface area is problematic, which makes it almostimpossible to infer the anisotropy of γsl. For these rea-sons, an alternative strategy in which γsl is evaluatedfor an idealized planar interface, which resembles thesurface of a mesoscopic nucleus, is often adopted. Inthe planar limit the value of the surface area A(ns)is unequivocal. In addition, different crystallographicorientations can be treated separately, so γsl and itsanisotropy can be defined more rigorously. Techniquesof this type include the capillary fluctuation method,which relies on an analysis of the height profile for theatomically-rough solid-liquid interfaces [11, 12] and thecleaving method, which computes the reversible workrequired to cleave the bulk solid and liquid and to jointhe pieces to generate an interface [13–15]. These twomethods only allow one to examine how the systembehaves at the melting temperature, however.

To investigate the solid-liquid interface at conditionsthat mirror the ones in experiments, a method thatworks for planar interfaces and that operates underrealistic levels of undercooling is required. In thispaper, we derive just such a method by extendingan approach that uses a history-dependent bias to

arX

iv:1

511.

0866

8v1

[co

nd-m

at.s

tat-

mec

h] 2

7 N

ov 2

015

2

reversibly create and dissolve a solid-liquid interfaceat equilibrium [16, 17] so that it also works at T 6= Tm.We then use this method to compute the temperaturedependence of γsl for a realistic interatomic potential.

In order to relate atomic-scale descriptions withmesoscopic models and experiments, it is necessaryto introduce a consistent framework for quantifyingthe number of solid atoms ns. At the atomistic level,several families of structural fingerprints φ(i) can beused to determine whether the environment aroundatom i is solid-like or liquid-like [18]. In our simu-lations, for example, we used a modified version ofthe cubic harmonic order parameter κ(i) introducedin Ref. [16], which, as described in the SupplementalMaterial at [URL will be inserted by publisher] [19],we further refined so as to better discriminate betweenthe liquid, the solid, and the various different crys-tal orientations by applying a switching function Sand defining φ(i) = S(κ(i)). Instead of defining ns

by introducing an ad-hoc threshold on φ, which isquite common in simulations of this kind, we adoptedan alternative approach, that is closely related to theclassical definition of a Gibbs dividing surface basedon a thermodynamic variable such as the volume V .For each microstate, we construct a global collectivevariable (CV) Φ =

∑i φ(i) by summing over all the

atoms. Taking φs and φl to be the reference values ofthe order parameter averaged over atoms in the bulksolid and the bulk liquid at the same thermodynamicconditions, one can then define the number of atomsof solid as

ns(Φ) = (Φ−Nφl) / (φs − φl) . (2)

This is equivalent to choosing a reference state thatsatisfies Φ = φsns +φlnl, subject to the conservation ofthe total number of atoms in the system ns + nl = N .In other words, zero surface excess for the extensivequantity Φ has been assumed. Within this approach,the choice of the order parameter implicitly determinesthe position of the solid-liquid dividing surface whileautomatically averaging over roughness and thermalfluctuations, which is essential for determining themacroscopic value of γsl [10].

We simulated the solid-liquid interface for a simplebut realistic Lennard-Jones system [15, 16, 20]. Toaccelerate the sampling so as to obtain reversible for-mation of a solid-liquid interface in a viable amount ofsimulation time, we performed well-tempered metady-namics simulations with adaptive Gaussians [21, 22].using Φ =

∑i S(κ(i)) (see above) as the CV. A sim-

ulation supercell with dimensions commensurate tothe equilibrated lattice at the temperature T , wasemployed. This cell was elongated along the x axiswhich was chosen to be parallel to the crystallographicorientation of the solid-liquid interface. Simulationswere run at constant temperature and pressure, withonly the x direction left free to fluctuate. Under theseconditions, the planar interfaces, whenever present, arealways perpendicular to the chosen lattice directionand always have a surface area A = 2∆y∆z equal

0 1000 2000 3000 4000 5000ns(Φ)

0

50

100

150

200

250

300

350

Free

ener

gypr

ofile

(ϵ)

γ100Φ (Tm)A

γ100Φ (0.64)A

Gs- Gl(0.64)

FIG. 1. The free energy profile averaged from 36 inde-pendent metadynamics simulation runs, and shown as afunction of ns(Φ), for an interface perpendicular to 〈100〉and a simulation box composed of 16 × 9 × 9 fcc unitcells. Representative atomic configurations for differentvalues of ns(Φ) are shown, with atoms colored accordingto the local order parameter φ. The two sets of curvescorrespond to the free energy profiles at the melting tem-perature (γsl(Tm) = 0.371(2)) and at a moderate degreeof over-heating (γsl(0.64) = 0.365(3)). All quantities areexpressed in Lennard-Jones units.

to twice the supercell cross section. The samplinghas to be restricted to the relevant regions of phasespace: the melt, the coexistence state with correctly-oriented solid-liquid interfaces, and the defect-free fcccrystal. Choosing an appropriate CV is critical forachieving this end and, in addition, several judiciouslytuned restraints are required to prevent the formationof grain boundaries and twin defects. The presenceof these defects was already observed in biased sim-ulations at Tm [17] and they become very likely atundercooled conditions. Fast implementation of thiscomplex simulation setup was made possible by theflexibility of the PLUMED code [23] in combinationwith LAMMPS [24]. See the Supplemental Materialfor sample input files and simulation details [19].

Figure 1 shows the free energy G as a function ofns(Φ) at two different temperatures. These curveswere reconstructed from the metadynamics simula-tions using on-the-fly re-weighting [25, 26]. At T = Tm,G(ns(Φ)) displays a broad horizontal plateau that cor-responds to the progressive movement of the solid-liquid interface. As discussed in Ref. [16], γsl(Tm)Acan be unambiguously taken as the difference betweenthe free energy of the plateau and the free energy atthe bottom of the solid and liquid minima. The biaspotential allows the interface to form even when µsl

is non-zero, at temperatures away from Tm. Figure 1thus also reports the free energy profile computedfor T > Tm. The free energies of the minima cor-responding to the bulk solid and liquid phases nowtake different values. The slope of the line that joinsthese minima is equal to µsl(T ). The region that wasa plateau at T = Tm has become an oblique line, with

3

-2000 0 2000 4000 6000ns(V)

0

1000

2000

3000

4000

5000

n s(Φ

)

0

50

100

150

Free

ener

gysu

rface

(ϵ)

FIG. 2. The free energy as a function of ns(Φ) andns(V ) for a simulation at Tm = 0.6185 computed using asupercell 16× 9× 9 times the fcc unit cell. The dashed lineindicates the ns(Φ) = ns(V ) ideal line, whereas the full lineindicates the mean value of ns(V ) for a given value of ns(Φ).The horizontal offset between the two curves correspondsto the surface excess for the atom count associated withthe Φ-based dividing surface, ∆ns (Φ, V ), which for thisparticular calculation is equal to 259± 2 atoms.

a slope that is also equal to µsl as moving the dividingsurface now has an energetic cost that is associatedwith forcing atoms to undergo the phase transition [27].The vertical red arrow in Figure 1 indicates the freeenergy difference between the simulated system witha solid-liquid interface and a reference state composedof ns(Φ) atoms of bulk solid and N − ns(Φ) atoms ofbulk liquid. This quantity is equal to γsl(T )A.

At this point it is important to note that at T 6= Tm

a change in the definition of the order parameter φaffects how the free energy of a given state is parti-tioned between the bulk and surface terms. We willdemonstrate later that the value we obtain for γsl(T )thus depends on φ. As discussed in the SupplementalMaterial [19], this does not change the free energybarrier for the formation of a critical nucleus, pro-vided that a consistent framework is used to definethe nucleus’ size and surface. It can, however, lead toconfusion when comparing simulations, phenomenolog-ical models and interpretations of experiments as thesemay have been analysed using different definitions. Inparticular, CNT-based models generally assume thatthe nucleus has the bulk density. This is equivalent totaking a reference state based on the molar volume, orto using a zero-excess-volume dividing surface. A V -based value of γsl is therefore desirable as one can thenestablish a direct link between simulations and CNTmodels. In addition, simulations run with differentorder parameters can be compared straightforwardlyif the volume-based interfacial free energy γV is alsocomputed.

In principle it is possible to compute γsl using adifferent local order parameter θ by on-the-fly reweigh-ing the biased trajectory. In practice, however, anyresults obtained are only reliable when the sampling ofΘ =

∑i θ(i) is as thorough as the sampling of the CV

used in the metadynamics simulations. In addition,in order to extract γsl, the free energy profile as afunction of ns(Θ) must exhibit a clear plateau regionthat corresponds to solid-liquid coexistence. Unfor-

tunately, the molar volume does not fulfill these twocriteria. To illustrate this problem, we performed 2Dmetadynamics simulations in which Φ =

∑i S(κ(i))

(see above) and the total volume V were used as CVs.In Figure 2 the resulting free energy surface is dis-played as a function of the solid atom counts obtainedwhen these two extensive quantities are inserted intoEq. (2). The fluctuations in the molar volume for thetwo bulk phases are so large that they overlap with theplateau region. The molar volume, by itself, is thusnot an effective fingerprint for distinguishing the solidand the liquid phases at the atomic scale.

Even though one cannot immediately obtain γVslfrom the free energy profile as a function of ns(V ), itcan still be determined indirectly. To do so, one simplycomputes the change in atom count that occurs whenV rather than Φ is used to define the reference state:

∆ns (Φ, V ) = 〈ns(V )δ [ns(Φ)− ns(Φ′)]〉 − ns(Φ). (3)

In the above, the Dirac δ ensures that only states witha certain value of ns(Φ) are selected, and 〈·〉 repre-sents a NPT ensemble average, which can be obtainedby appropriately re-weighting the biased simulation.∆ns(Ψ, V ) can be determined to a very high statisticalaccuracy as long as one of the two order parameterscan identify the coexistence region. Graphically, thisquantity corresponds to the horizontal offset betweenthe solid black line and the dashed diagonal in Fig-ure 2, which is constant over the entire plateau re-gion. This implies that, when a solid-liquid interfaceis present, a change in the definition of the Gibbs di-viding surface leads to a constant shift in ns. Theinterfacial free-energy based on the volume, γVsl (T ),can thus be inferred from the quantity γΦ

sl (T ) usingγVsl = γΦ

sl − µsl∆ns (Φ, V ) /A.

Figure 3 shows the temperature dependence for µsl

and γsl computed for the 〈100〉 and 〈111〉 crystal latticeorientations. µsl is a bulk property so its value shouldnot depend on the orientation of the surface as is ob-served in the top panel. The magnitude and anisotropyof the interface free energy at T = Tm match the valuesreported in the literature Ref. [15, 16, 20]. Its temper-ature dependence, however, depends strongly on thechoice of reference state. The lower panel shows thatγΦ

sl has a near-constant negative slope for both orienta-tions which indicates that the excess entropy associatedwith the Φ-based dividing surface is positive. The im-portance of using a consistent reference state whenstudying the solid-liquid interface in out-of-equilibriumconditions is highlighted by the dramatically differentbehavior observed when the volume is used to definethe reference state. γV100 increases with temperatureas ∆ns(Φ, V ) is large and negative. When the molarvolume is used to define the reference state a largerfraction of the interface region is seen as an orderedliquid rather than a disordered solid. The interface forthe 〈100〉 crystal orientation thus has a negative excesssurface entropy. For the 〈111〉 orientation, meanwhile,∆ns(Φ, V ) is smaller and the slope of γV111 remainsnegative.

4

γ100Φ γ100

V

γ111Φ γ111

V

0.58 0.60 0.62 0.64 0.66Temperature

0.34

0.36

0.38

0.4

Inte

rfaci

alfre

een

ergy

(ϵ/σ

2)

(100)

(111)

-0.1

-0.05

0

0.05

0.1

μs-μ

l(ϵ)

FIG. 3. µsl = µs − µl (upper panel) and γsl (lowerpanel) for two different crystal orientations, as a functionof temperature. γΦ

sl and γVsl were computed using a Gibbs

dividing surface that has zero surface excess for Φ andV , respectively (see Eq. (2)). These two different ways ofpartitioning the free energy of the system between bulkand surface terms ensure that there are significant differ-ences between γΦ

sl and γVsl when T 6= Tm. Each data point

represents an average over 24 independent metadynamicsruns and statistical uncertainties are indicated by the errorbars. The uncertainties in γΦ

sl and µsl∆ns (Φ, V ) were as-sumed to be independent during the error estimation. Thedata points were obtained by restricting the sampling toprevent complete melting, as discussed in the SupplementalMaterial [19]. The simulation boxes comprised about 1200atoms, which implies a small finite-size effect that shiftsTm to 0.62. At Tm the interfacial free energies, for thetwo orientations we considered are γ100(0.62) = 0.373(2)and γ111(0.62) = 0.352(1). All quantities are expressed inLennard-Jones units.

It is important to remember that the total free en-ergy of the nucleus is the only quantity that affectsits thermodynamic stability. However, the dependenceof γsl on the dividing surface will have consequenceson predictions and data analysis if the same referencestate is not used consistently. Usually, CNT models arebuilt assuming that the nucleus’ density is constantand equal to its bulk value. To be consistent withthis assumption, one should thus use γVsl when makingpredictions using this theory. In the SupplementalMaterial at [URL will be inserted by publisher] [19]we show that, when using a reference state with a non-zero surface volume excess, additional correction termsmust be included on top of the usual CNT expressions.In addition to changing the value of γsl, one must use adensity that differs from the bulk value when inferringthe surface area of the cluster. This density dependson the nucleus size in a manner that is reminiscentof the well-known Tolman correction, which has beenshown to be effective when extrapolating the value of

G? obtained from simulations of nanoscopic nuclei tothe mesoscale [10].

The observation of a positive slope for γVsl (T ) is con-sistent with a simple phenomenological model for ahard-sphere system [28], and with the effective γsl val-ues that were determined by fitting the values of G?(T )obtained from simulations run at deeply undercooledconditions using a CNT expression [8]. Our resultspaint a somewhat more nuanced picture, however. Thebehavior of γV100 deviates significantly from linearityeven at small undercoolings. In addition, the temper-ature dependence of the solid-liquid interface energycomputed relative to a zero-excess-volume referencedepends strongly on orientation, with the 〈111〉 direc-tion showing a small negative slope. As a consequence,the anisotropy in γVsl between the two orientations weconsidered decreases down to T = 0.58, suggestingthat the shape of the critical nucleus may depend onthe degree of undercooling. Fully determining the sizeand shape of the critical nucleus would require one todetermine γsl and ∂γsl/∂T for other high-symmetryorientations and for small mis-orientations. A directcomparison between planar-interface calculations and3D nucleation simulations at intermediate undercool-ings would also allow one to test the validity of thevarious assumptions within CNT. For example, suchsimulations would allow one to determine whether ornot the Laplace pressure caused by the curvature ofthe interface has a significant effect on the chemicalpotentials of the two phases. These calculations, aswell as the investigation of more realistic inter-atomicpotentials, will be the subject of future work, whichwill provide more accurate parameters for mesoscalephase-field models of nucleation and growth [29, 30].

Using a thermodynamic definition for the size ofthe solid region allows one to treat atomic-scale finger-prints and macroscopic observables on the same footing,thereby providing a practical procedure for convert-ing the value of γsl obtained with a computationally-effective order parameter into the value consistent withthe assumptions of mesoscopic theories of nucleation.The combination of this framework and acceleratedmolecular dynamics makes it possible to generate andstabilize planar interfaces at T 6= Tm. Consequently,simulations can be performed with conditions thatmore closely resemble those in experiments. Such sim-ulations could be used to shed light on the changesin morphology of the critical nucleus, to monitor cap-illary fluctuations at T 6= Tm [11], and to give atom-istic insight on the evolution of interfaces away fromequilibrium[31]. In addition, the methodology and thethermodynamic considerations we make here, whilstexamining solidification, would apply with minor mod-ifications to the study of nucleation in other contexts,be it melting [32], precipitation [33], condensation,order-disorder transitions, or other situations in whicha phase transition is hindered by a surface energyterm[34].

We would like to thank Michel Rappaz for insightfuldiscussion. MC acknowledges the Competence Centrefor Materials Science and Technology (CCMX) for

5

funding.

∗ michele.ceriotti@epfl.ch[1] J. A. Dantzig and M. Rappaz, Solidification (EPFL

press, 2009).[2] D. Li, X.-Q. Chen, P. Fu, X. Ma, H. Liu, Y. Chen,

Y. Cao, Y. Luan, and Y. Li, Nature Comm. 5, 5572(2014).

[3] S. V. Murphy and A. Atala, Nature biotechnology 32,773 (2014).

[4] D. T. Wu, L. Granasy, and F. Spaepen, MRS bulletin29, 945 (2004).

[5] J. Hoyt, M. Asta, and A. Karma, Materials Scienceand Engineering: R: Reports 41, 121 (2003).

[6] P. R. ten Wolde, M. J. Ruiz-Montero, and D. Frenkel,The Journal of chemical physics 104, 9932 (1996).

[7] P. R. ten Wolde and D. Frenkel, Physical ChemistryChemical Physics 1, 2191 (1999).

[8] F. Trudu, D. Donadio, and M. Parrinello, Physicalreview letters 97, 105701 (2006).

[9] W. Lechner, C. Dellago, and P. G. Bolhuis, Physicalreview letters 106, 085701 (2011).

[10] S. Prestipino, A. Laio, and E. Tosatti, Physical reviewletters 108, 225701 (2012).

[11] J. Hoyt, M. Asta, and A. Karma, Physical ReviewLetters 86, 5530 (2001).

[12] C. Becker, D. Olmsted, M. Asta, J. Hoyt, and S. Foiles,Physical Review B 79, 054109 (2009).

[13] J. Q. Broughton and G. H. Gilmer, The Journal ofchemical physics 84, 5759 (1986).

[14] R. L. Davidchack and B. B. Laird, Physical reviewletters 85, 4751 (2000).

[15] R. L. Davidchack and B. B. Laird, The Journal ofchemical physics 118, 7651 (2003).

[16] S. Angioletti-Uberti, M. Ceriotti, P. D. Lee, and M. W.Finnis, Phys. Rev. B 81, 125416 (2010).

[17] S. Angioletti-Uberti, J. Phys. Cond. Matt. 23, 435008(2011).

[18] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti,Phys. Rev. B 28, 784 (1983).

[19] “Supplemental material,” .[20] R. Benjamin and J. Horbach, The Journal of chemical

physics 141, 044715 (2014).[21] A. Barducci, G. Bussi, and M. Parrinello, Physical

review letters 100, 020603 (2008).[22] D. Branduardi, G. Bussi, and M. Parrinello, Journal

of Chemical Theory and Computation 8, 2247 (2012).[23] G. A. Tribello, M. Bonomi, D. Branduardi, C. Camil-

loni, and G. Bussi, Computer Physics Communica-tions 185, 604 (2014).

[24] S. Plimpton, J. Comp. Phys. 117, 1 (1995).[25] G. M. Torrie and J. P. Valleau, J. Comp. Phys. 23,

187 (1977).[26] P. Tiwary and M. Parrinello, J. Phys. Chem.. B

(2014).[27] U. R. Pedersen, F. Hummel, G. Kresse, G. Kahl, and

C. Dellago, Physical Review B 88, 094101 (2013).[28] F. Spaepen, Solid State Phys. 47, 1 (1994).[29] W. Curtin, Physical review letters 59, 1228 (1987).[30] L.-Q. Chen, Annual review of materials research 32,

113 (2002).[31] M. E. Gurtin and P. W. Voorhees, Acta Materialia

44, 235 (1996).[32] A. Samanta, M. E. Tuckerman, T.-Q. Yu, and W. E,

Science 346, 729 (2014).[33] M. Salvalaglio, T. Vetter, F. Giberti, M. Mazzotti,

and M. Parrinello, J. Chem. Am. Soc. 134, 17221(2012).

[34] R. Z. Khaliullin, H. Eshet, T. D. Kuhne, J. Behler,and M. Parrinello, Nature materials 10, 693 (2011).