Physica B 406 (2011) 4163–4169

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physb

General construction of mean-field potential and its application to themultiphase equations of state of tin

Lin Zhang n, Ying-Hua Li, Yu-Ying Yu, Xue-Mei Li, Yun Ma, Cheng-Gang Gu, Cheng-Da Dai, Ling-Cang Cai

National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics P.O. Box 919-102, 621900 Mianyang, Sichuan, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 8 August 2010

Received in revised form

8 January 2011

Accepted 10 January 2011Available online 18 July 2011

Keywords:

Equation of state

Phase equilibrium

Mean-field approach

26/$ - see front matter & 2011 Published by

016/j.physb.2011.01.018

esponding author.

ail address: zhanglinbox@263.net (L. Zhang).

a b s t r a c t

The mean-field potential (MFP) approach is an efficient way to evaluate the free energy contribution of

ion motions for both solid and liquid states. In this paper the MFP is generally constructed with a

volume-dependent term and a shape function. The former is derived in accordance with quasi-

harmonic approximation. The latter is given semi-empirically. Application to multiphase equations of

state for b-, g- and liquid-tin has been examined. The theoretical phase diagram and thermodynamic

properties of isotherm, thermal expansion, heat capacity, Hugoniot states as well as phase transitions

are all in excellent agreement with experiments.

& 2011 Published by Elsevier B.V.

1. Introduction

Multiphase equations of state (EOS) are of considerableimportance for high-pressure and high-temperature propertiesof condensed matter and engineering applications. However, it isimpossible to develop completely rigorous EOS; the computationof more accurate equations of state is always valuable. A generalmethod for EOS construction is to calculate the free energy as afunction of specific volume and temperature [1,2], for which aproper account of the ion motions and electron thermal excita-tions is essential. This paper mainly focuses on the calculation ofthe contribution of ion motions for a wide range of material statesfrom solid to liquid.

For solid, one simple way to examine the free energy of ionmotions is the quasi-harmonic approximation, which can beaccomplished either with classical models [3,4] or in ab-initio[5–7], but it is usually restricted to a lower temperature range asthe an-harmonic contribution of ion vibrations is very small.Molecular dynamics (MD) is another popular approach to studythis issue [8], it is applicable for a wide range of temperatures andpressures, as well as solid and liquid states. Unfortunately it stillremains hard to perform large-scale ab-initio MD simulations[9,10] due to the computer resources. It is now possible to carryout classical molecular dynamics simulations in a scale of 108

atoms [11], but it continues to be a great challenge to construct

Elsevier B.V.

the intermolecular interactive potential applicable for widespreadtemperatures and pressures as well as phase changes.

Another useful approach suitable for solid and liquid is the mean-field approximation, including the particle-in-a-cell model (PIC)[12–14] and the mean-field potential approach (MFP) [15–18]. Forsolid, mean-field approximation has some advantages over the quasi-harmonic approximation under high temperatures due to betteraccounting for the an-harmonic effects. The main difference betweenPIC and MFP is the treatment of the configuration integral. The PICdeals with each atom as moving in its Wigner–Seitz cell experiencingthe potential due to all the other atoms fixed at their equilibriumpositions, and the integration over the coordinates of each atom isrestricted to one particular Wigner–Seitz cell. PIC calculations aretime consuming and inconvenient. On the contrary, the MFP, which isbased on the free volume theory [19,20], pictures each atom asvibrating in the mean-field potential due to the surrounding homo-geneous atom cloud occupying the space outside the nearest atom,which is averaged from all other atoms. By proper construction of thepotential, Wang et al. [15–17] and Song and Liu [18] demonstratedMFP may accurately interpret the high-pressure and high-tempera-ture properties of both high- and low-symmetry materials, includingliquids. In addition, it is convenient to perform the calculations.

The central issue of MFP approach is to construct the mean-field potential. In the pioneer works, Wang proposed the specialformula [15]

gðr,VÞ ¼ 12 ECðRþrÞþECðR�rÞ�2ECðRÞ½ � ð1Þ

then revised it as [16]

gðr,VÞ ¼1

2ECðRþrÞþECðR�rÞ�2ECðRÞ½ �

L. Zhang et al. / Physica B 406 (2011) 4163–41694164

þl2

r

RECðRþrÞþECðR�rÞ½ � ð2Þ

where Ec is the static 0-K energy, R the lattice constant withrespect to specific volume V and r represents the distance that thecentral atom deviates from center. Corresponding to differentexpressions of the Gruneisen parameter due to Slater, Dugdaleand MacDonald as well as free volume theory, l takes the value�1, 0 and 1, respectively. In this paper, we show the mean-fieldpotential can be more generally constructed with better physicalinterpretations and suitable for development of high-accuracymultiphase EOS, including solid and liquid states. The applicationto b-, g- and liquid-tin is presented, examinations in isotherm,thermal expansion, heat capability, Hugoniot states as well asphase diagram illustrate our calculations high-accurately repro-duce the available experimental data of the thermodynamicproperties of tin.

2. Brief overview of the mean-field potential approach

For a given specific volume V and temperature T, the specificHelmholtz free energy F(V,T) can be written as

FðV ,TÞ ¼ ECðVÞþFionðV ,TÞþFelðV ,TÞ ð3Þ

where EC is the static 0-K energy, Fion the free energy of ionmotions and Fel the free energy due to the thermal excitation ofelectrons.

The static 0-K energy EC can be calculated with either first-principle techniques [21,22] or classical models [1,23,24]. Withrespect to Fel in Eq. (3), theoretically it can be written as Fel¼

Eel�TSel, in which Eel and Sel take the forms [15,16]

EelðV ,TÞ ¼

Znðe,VÞf ede�

ZeF nðe,VÞede ð4Þ

SelðV ,TÞ ¼ �kB

Znðe,VÞ½f ln f þð1�f Þlnð1�f Þ�de ð5Þ

where kB is Boltzmann’s constant, n(e,V) the electronic density ofstates (DOS), f the Fermi distribution and eF the Fermi energy. Atlower temperatures (kBT5eF, usually up to 104–105 K), Fel reducesto [25]

Fel ¼�p2

6ðkBTÞ2nðe0

F ,VÞ ð6Þ

nðe0F ,VÞ is the electronic DOS at 0-K Fermi energy level.Under the mean-field potential approximation, the contribu-

tion of ion motions to the total specific free energy can beformulated as [15,16]

Fion ¼�NkBT3

2ln

mkBT

2p_2þ lnvf

� �ð7Þ

vf ¼ 4pZ

r2 exp �gðr,VÞ

kBT

� �dr ð8Þ

Eq. (8) represents the configuration integral. N is the atomnumber per unit mass, m the mass of atom, _ the reduced Planck’sconstant. If the MFP g(r,V) is provided, the ion motion free energymight be calculated at any given V and T, then other thermo-dynamic functions and the thermal properties could be deduced.

3. General construction of mean-field potential

It is comprehensive if MFP g(r,V) is expanded to second orderin powers of r, it represents a potential of a harmonic oscillation.On the other hand, when the center ion moves close to thesurrounding atom cloud, the MFP will tend to infinity, so we write

the MFP in the general form

gðr,VÞ ¼ g0ðVÞhðr,VÞ ð9Þ

Fundamentally, h(r,V) is required to satisfy the constraints

hð0,VÞ ¼ 0

hðR,VÞ ¼1

h0ð0,VÞ ¼ 0

h00ð0,VÞ40 ð10Þ

where R represents the radius of the spheric inner edge of thesurrounding atom cloud, which is only volume-dependent. Weassume

R¼V

N

� �1=3

ð11Þ

h0(r,V) and h00 (r,V) are the first- and second-order derivatives ofh(r,V) with respect only to the variable r.

The volume-dependent function g0(V) in Eq. (9) may bederived by the following process. Expanding Eq. (9) in theTaylor-power to the second order with respect to r, we get

gðr,VÞ ¼1

2g0ðVÞh

00ð0,VÞr2þOðr3Þ ð12Þ

The first term in the right hand of Eq. (12) represents aharmonic oscillation with the frequency n(V)

nðVÞ ¼ g0ðVÞh00ð0,VÞ

m

� �1=2

ð13Þ

where m is the mass of atom. Consequently the Gruneisenparameter can be estimated by [16,25]

gionðVÞ ¼�@ lnnðVÞ@ lnV

ð14Þ

On the other hand, the Gruneisen parameter due to Slater (seeRef. [26]), Dugdale and MacDonald [26] as well as free volumetheory [20] may be expressed in a general form

gionðVÞ ¼a2�

2

3

� ��

V

2

d2½PCðVÞVa�=dV2

d½PCðVÞVa�=dVð15Þ

with a taking the values 0, 2/3 and 4/3. PC(V) stands for the coldpressure curve which can be obtained either by ab-initio calcula-tions or by parameterized models.

Let:

f ðVÞ ¼ PCðVÞVa ð16Þ

From Eqs. (13) to (15), we obtain

g0ðVÞ ¼mn2

0

h00ð0,VÞ

f 0ðVÞ

f 0ðV0Þ

V

V0

� �ð4=3Þ�að17Þ

where n0 is the vibration frequency of the central ion at zero coldpressure, V0 is the corresponding specific volume. The prime onf(V) represents its derivative. For solids n0 in Eq. (17) can beevaluated from zero-point vibrational energy (e0) through3Nð1=2Þ2p_n0 ¼ e0. For liquid, there is no simple way to evaluaten0 due to the active diffusion of the ions, so we propose to dealwith it as a fitting parameter.

It is hard to identify the expression of the function h(r,V) inEq. (9) in accordance with physical principles; the problem issimilar to that of the expression of intermolecular interactivepotential in molecular-dynamics studies, so in mathematics, anyfunction, which satisfies the constraint Eq. (10) can be used forh(r,V). In this paper, the following combination of power functionsis tried.

hðr,VÞ ¼1

ðRþrÞnþ

1

ðR�rÞn�

2

Rnð18Þ

Fig. 1. Schematic of the MFP expressed by Eq. (19), (a) overall view of the shape;

(b) the enlarged bottom part of (a).

L. Zhang et al. / Physica B 406 (2011) 4163–4169 4165

where n is a model parameter. Subsequently the mean-fieldpotential is derived as

gðr,VÞ ¼mn2

0R2

2nðnþ1Þ

f 0ðVÞ

f 0ðV0Þ

V

V0

� �ð4=3Þ�a1þ

r

R

� ��n

þ 1�r

R

� ��n

�2

� �

ð19Þ

A schematic of the MFP expressed by Eq. (19) is presented inFig. 1(a), and its bottom part is enlarged in Fig. 1(b). The figuresimply: for liquid, ions have higher kinetic energy, the MFP Eq. (19)is much close to the hard-sphere potential [27]. On the contrary,for solid the MFP reduces to that of a modified Einstein model,which takes an-harmonic effects into account, similar to theworks of Dorogokupets and Oganov [28] and Zharkov andKalinnin [29]. So we expect the MEP (Eq. (19)) may describe bothsolid and liquid well.

4. Application in EOS calculations of b-, c- and liquid-tin

Tin lies in a special location in the periodic table. It belongs togroup IV together with C, Si, Ge and Pb. Above Sn, the elements C,Si and Ge tend to form a diamond structure with strong covalentbonds, while below Sn the stable phase for Pb is metallic and fcc.This makes the structural, electronic and thermodynamic proper-ties of Sn very sensitive to temperature and pressure. At lowtemperatures and pressures, tin exhibits a diamond structuretypical of covalently-bonded materials, above 286 K at atmospheric

pressure it turns into metallic b structure. At room temperatureand about 9.2 GPa [30], 9.4 GPa [31], b-Sn transforms into body-centered tetragonal g phase. The triple point of b-, g- and liquid-Snlocates at about (580 K, 3 GPa) [31], (583.15 K, 3.3 GPa) [32]. Theelasticity, phase equilibrium and thermodynamic properties ofsolid- and liquid-tin have been extensively and accurately measuredin experiments [30–45], this makes tin a good candidate to examinethe capability of the generalized MFP. Furthermore, the multiphaseequations of state of tin have absorbed many authors’ interests[46–48].

4.1. Details of calculations

Due to the characteristic of lower melting temperatures, theelectronic contributions to total free energies for b-, g- and liquid-tinmight be evaluated by Eq. (6). In principle, the volume-dependentDOS nðe0

F ,VÞ might be computed with quantum electronic structurecalculations such as the Hartree–Fock method [49]; however, in thispaper we evaluate it in a more simple way by rewriting Eq. (6) basedon free electron theory [50] as

Fel ¼�GðNe,VrÞV

Vr

� �2=3

T2 ð20Þ

where Vr is the specific volume of a reference point, Ne the totalnumber of free electrons and G(Ne,Vr) a constant related to Vr and Ne.If we assume approximately that Ne remains unchanged before andafter phase transformation and select the same reference point forb-, g- and liquid-tin, then G(Ne,Vr) has the same value for the threephases. In our calculations the value of Vr is set to 0.1373 cm/g (thevalue at ambient conditions), correspondingly G(Ne,Vr) is evaluatedto be 7.497 mJ/kg K2 from the low-temperature experimental dataof heat capacity [50].

The 0-K isotherm takes an essential role in the generalized MFPapproach, which might be calculated in ab-initio or with classicalmodels. Considering extensive data from experiments and theoryare available in literatures for tin [34,37,38,41,42,51–55]; here wechoose the classical model of Vinet’s universal formula [23,24] toexpress the cold pressure, in the meanwhile, this may save time-consuming ab-initio calculations

PCðVÞ ¼ 3B01�x

x2exp Zð1�xÞ

�

x¼ ðV=V0Þ1=3

Z¼ 3ðB00�1Þ=2 ð21Þ

where B0, B00 and V0 are the bulk modulus, its pressure derivativeand volume at the state of zero temperature and pressure,respectively.

The expression of Eq. (19) with a¼4/3 is used for the MFP.Noticing the parameter n in Eq. (19), it is hard to be determined aslacking clear physical meaning. However it will be demonstratedlatter that the EOS is quite insensitive to it. In our calculations thevalue 0.5 is used.

Corresponding to the cold pressure treatment, the cold energyEC(V) is calculated by integration of Eq. (21)

ECðVÞ ¼ E0�

Z V

V0

PCðVÞdV ð22Þ

where E0 is the cohesive energy at the point V0, which is set thevalue zero for b-tin, so the values for g- and liquid-tin areaccordingly adjusted. The other parameters used in the calcula-tions alone with some comparing data from literatures are givenin Table 1.

Table 1Parameters used in calculations alone with comparing data from literatures. All values with no reference label are from this work.

b g Liquid

B0 (GPa) 55.0 (this work) 50.0 (this work) 46 (this work)

55.0 (expt. Ultrasonic) [34] 49.72 (expt. X-ray) [42]

55.4 (expt. Ultrasonic) [37] 54.8 (theory, LDA) [51]

54.6 (expt. Ultrasonic) [38] 52.0 (theory, LDA) [52]

56.6 (expt. X-ray) [41] 60.8 (theory, LDA) [55]

55.3 (theory, LDA) [51]

54.4 (theory, LDA) [52]

60.5 (theory, LDA) [53]

61.0 (theory, LDA) [54]

63.7 (theory, LDA) [55]

B00 5.7 5.5 5.8

V0 ðA=atomÞ 26.588 (this work) 26.489 (this work) 26.903 (this work)

26.420 [51] 27.090 [51]

27.280 [52] 26.500 [52]

25.296 [53]

25.180 [54]

n0ð�1013 s�1Þ 1.426 1.234 1.0

E0 (mRy/atom) 0.0 1.780 3.453

Fig. 2. Comparison of the isotherm at room temperature between calculated line

and experimental data for b-tin.Fig. 3. Comparison of the isotherm at room temperature between calculated line

and experimental data for g-tin.

L. Zhang et al. / Physica B 406 (2011) 4163–41694166

4.2. Thermodynamic properties

A series of checks has been performed to test the generalizedMFP and its precision in the EOS calculations. The first one, alsobasic, is the property of compressibility. The calculated isother-mal curves of b- and g-tin at room temperature are plotted inFigs. 2 and 3, respectively, showing good agreement with experi-mental results [39,41,42]. Room-temperature isotherm is mainlycontributed from the cold energy, the heat-related properties maybe better examined through the thermal expansion, heat capacity,entropy, etc. In Table 2, the calculated values of specific volume,linear thermal expansion coefficient, entropy and heat capacityare compared with experimental data, also with the simulationresults by Ravelo and Baskes [56] via the modified embeddedatom potential MD method. One can notice all quantities of ourcalculations are accurately consistent with experiments. Fig. 4shows the density of liquid-tin as a function of temperature underconstant ambient pressure condition. Again, the theoretic curve isin good agreement with the recent experimental data measuredby Alchagirov and Chocheava [43].

4.3. Phase diagram

Phase diagram, we think, gives the most sensitive examination.In Fig. 5, the calculated diagram is plotted together with experi-mental measurements [30–32], showing good agreement.Although no experimental data of melting curve in high-pressurerange is available for comparison, however, the shock-inducedmelting pressure (at about 43.5 GPa) predicted from our EOScoincides with the most recent measurements (constrainedbetween 35 and 45 GPa) performed by Hu et al. [45]. Bernardand Maillet [57] had ever studied the melting curve of tin viaextensive first-principles and classical molecular dynamics simula-tions. At ambient pressure they predict b-tin melts at 450 K, lowerthan experimental value by about 11%. On the contrary, theircalculated melting temperatures of g-tin obviously tend to behigher than true values (the filled squares in Fig. 5). Similar trend isalso observed in Davis and Foiles’ MD simulations [58] (the opencycles in Fig. 5). The comparisons illustrate that the precision ofEOS predicted with molecular-dynamics simulations and ab-initiocalculations still needs to be improved for high-quality engineering

Table 2Calculated thermodynamics properties compared with experimental data and the molecular dynamics

simulation results carried out by Ravelo and Baskes [56].

Quantity Experiment Theory

Vb (cm3/g, ambient condition) 0.1373 [59] 0.1373 (this work)

0.1440 [56]

Vb (cm3/g, ambient pressure,453 K) 0.1390 [60] 0.1388 (this work)

0.1473 [56]

ab (e�5 K�1, ambient condition) 2.12 [61] 2.13 (this work)

2.35 [60] 1.77 [56]

aliquid (e�5 K�1, melting point at ambient pressure) 2.92 [62] 2.91 (this work)

1.87 [56]

Sb (J/g K, ambient condition) 0.4338 [63] 0.4332 (this work)

0.4701 [56]

CbV (J/g K, ambient condition) 0.2087 [62] 0.2137 (this work)

0.2236 [56]

0.2228 [63]

0.2271 [64]

Fig. 4. The density of the liquid phase as a function of temperature confined to the

constraint of constant ambient pressure. The solid line is calculated from this

equation of state, the dashed line is experimental result from recent measure-

ments by Alchagirov and Chocheava [43].

Fig. 5. Calculated phase diagram of tin compared with experimental data and the

molecular-dynamics simulation results from literatures.

L. Zhang et al. / Physica B 406 (2011) 4163–4169 4167

simulations, phenomena method, which may accurately reproducethe experimental data is still in need. The volume changesaccompanying phase transitions are also examined. At roomtemperature, our calculated relative volume collapse accompany-ing b-g transition is 1.3%, which is consistent with experimentalvalue 1.1–1.3% [30]. At ambient pressure condition, experimentsreveal b-tin melts with about 2.3% volume expansion (see Ref.[57]), our calculated value is 2.1%, they are in close agreement.

4.4. Hugoniot curve

Shock-wave technique is an important experimental methodto study the thermodynamic properties under extreme high-temperature and high-pressure conditions. The last examinationis to compare the Hugoniot curve, the Hugoniot pressure (PH)versus volume (VH), which can be determined experimentally bymeasuring the shock-wave velocity (D) and the particle velocity(up), then VH/Vini¼(D�up)/D and PH¼riniDup, where rini is theinitial density, and Vini is the reciprocal of rini. Theoretically, theHugoniot curve can be derived by solving the Rankine–Hugoniotconservation equation [65] (1/2)PH(Vini�V)¼EH�Eini, where Eini

and EH are the internal energies at the initial and Hugoniot state,respectively. Illustrated in Fig. 6 are the theoretic calculatedHugoniot curve and experimental data [66], again good agree-ment is exhibited.

4.5. Effects of model parameter

As pointed out above, the parameter n in Eq. (19) is hard to bedetermined; however, the EOS is insensitive to it. If substituteEq. (17) into Eq. (12), it can be found the second-order Taylor-power expansion of the generalized MFP has no relation with thespecial expression of h(r,V), so the parameter n only appears inthe terms after the fourth power, subsequently it can be con-cluded the effects of the parameter n on thermodynamics mightonly be observed in the range of very high temperatures. Ourcalculations with different value of n confirmed this point. As asensitive examination, the comparison of phase diagrams ispresented in Fig. 7, which are calculated with n¼0.5 and 2.0(other parameters the same). One can notice the differencemainly appears in the high-temperature part of melting curve,but the difference is very small.

Fig. 7. The effect of the parameter n in Eq. (19) on phase diagram. The solid lines

are calculated with n¼0.5, the dashed lines with n¼2.0. This figure implies that

the multiphase equations of state are insensitive to the parameter.

Fig. 6. The comparison between the calculated and experimental Hugoniot

curves. Open squares are experimental data, lines are calculated from the multi-

phase EOS of this paper, the dashed line parts correspond to metastable states.

L. Zhang et al. / Physica B 406 (2011) 4163–41694168

5. Summary

The mean-field potential (MFP) approach is an efficient way tocalculate the motional free energy of atom ions. The main issue ofthe approach is to construct MFP. In this paper, a generalexpression for MFP is proposed (Eqs. (9) and (17)), which isconsisted of a volume-dependent term and a shape function. Thevolume-dependent term is derived in accordance with quasi-harmonic approximation. The shape function is given semi-empirically. For liquid, ions have higher kinetic energy, the MFPproposed here is much close to the hard-sphere potential [27]. Onthe contrary, for solid the MFP reduces to that of a modifiedEinstein model, which takes an-harmonic effects into account.Application to the multiphase equations of state of tin is exam-ined, results show the calculated isotherm, thermal expansioncoefficients, heat capacity, density (as a function of temperature),the Hugoniot curve, volume changes accompanying phase transi-tions as well as phase diagram are all in good agreement withexperiments. We believe this generalized MFP approach exten-sively applicable for a wide range of temperature and pressurecovering solid and liquid states.

Acknowledgment

We acknowledge the support by the National Key Laboratoryof Shock Wave and Detonation Physics fund under Grant no.9140C6701010901, the NSAF fund under Grant no. 10776029/A06and the science and technology development fund of CAEP underGrant no.2009A0101004.

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