ISSN 0018�151X, High Temperature, 2013, Vol. 51, No. 5, pp. 621–625. © Pleiades Publishing, Ltd., 2013.Original Russian Text © V.G. Baidakov, 2013, published in Teplofizika Vysokikh Temperatur, 2013, Vol. 51, No. 5, pp. 692–696.

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INTRODUCTION

Transfer coefficients (coefficients of self�diffusionD, shear viscosity η, thermal conductivity λ) relate thefluxes of the mass, momentum, and energy in irrevers�ible processes to the gradients of density, velocity, andtemperature. The transfer coefficients for weakly non�ideal systems are determined within the kinetic theoryby the Chapman–Enskog solution [1]. Such anapproach makes it possible to obtain the explicit formof the dependences D, η, λ on the thermodynamicparameter of the state. In systems with an arbitrarydensity and any type of interparticle interaction, thetransfer coefficients can be expressed in terms of inte�grals over time of the autocorrelation functions of thecorresponding fluxes. Such expressions are known asGreen–Kubo formulas [2]. Being strict relations ofthe linear thermodynamics of irreversible processes,the Green–Kubo formulas do not make it possible toobtain an explicit form of the dependences D, η, λ onthe temperature and density but can be used for thesolution of this problem in molecular�dynamic exper�iments [3].

Of particular interest in the behavior of the macro�scopic system are the states in which the system has areduced thermodynamic stability. Such states areimplemented near the points of phase transitions,when one of the phases turns out to be metastable.Keeping stability with respect to the infinitesimallysmall (continuous) changes of the parameters, themetastable phase is unstable with respect to finitechanges. The recovery reaction of the metastablephases to the infinitesimally small long�wavelengthperturbations is broken on the spinodal, the boundaryof the thermodynamic stability, which is determinedby conditions [4]

(1)( ) 0, ( ) 0,T p pp T s T c∂ ∂ = ∂ ∂ = =v

where p is pressure, is volume, T is temperature, s isentropy, and cp is the isobaric heat capacity. If the spi�nodal exists, the region of the metastable states is lim�ited on the thermodynamic surface F(p, T, ) by twospatial curves: the line of the phase equilibrium (bin�odal) and the line of the boundary of the thermody�namic stability (spinodal).

Conditions (1) are a consequence of the thermody�namic theory of stability. In determining the boundaryof stability, the thermodynamics makes it possible tojudge about the character of the variation of someother thermodynamic quantities (coefficients of sta�bility) when approaching the spinodal. The problemabout the behavior of the transfer coefficients near thespinodal is more complicated. According to the sec�ond law of thermodynamics, the transfer coefficientsshould be positive in any stable state (stable and meta�stable). However, the thermodynamics of the irrevers�ible processes has no information about the behaviorof D, η, λ when approaching the boundary of stability.

The van der Waals equation of state qualitativelycorrectly reproduces the picture of the liquid–vaporphase transition with the metastable states. The analogof the van der Waals equation is absent for the transfercoefficients in the nonequilibrium statistical mechan�ics, though attempts to describe the transfer coeffi�cients in the van der Waals fluid were undertaken [5,6].

As to its nature, the spinodal cannot be reached inthe quasi�statistical experiment. It is also not reach�able in computer experiments (molecular dynamicsand Monte Carlo methods). In fluctuating systems,this is due to process nucleation, which serves as thenatural limiter of the penetration depth in the regionof the metastable states.

Usually the boundary of the thermodynamic stabil�ity is determined asymptotically from the data

v

v

THERMOPHYSICAL PROPERTIESOF MATERIALS

Transfer Coefficients Near the Boundaryof Thermodynamic Stability

V. G. BaidakovInstitute of Thermophysics, Ural Branch, Russian Academy of Sciences, Ekaterinburg, Russia

e�mail: baidakov@itp.uran.ruReceived September 6, 2012

Abstract—It has been established according to the data of the molecular�dynamic calculations of the transfercoefficients in the stable and metastable states of the Lennard�Jones fluid that in the variables p, T the familiesof the lines of constant value of the self�diffusion coefficient D, excess thermal conductivity Δλ and excessshear viscosity Δη have an envelope, which coincides with the spinodal. Thus,

when approaching the spinodal.

DOI: 10.1134/S0018151X13050015

( ) ,TD p∂ ∂ → ∞

( ) ,Tp∂Δλ ∂ → ∞ ∂Δη ∂ → ∞( )Tp

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BAIDAKOV

obtained for the stable and weakly metastable regionsof the phase diagram. Though the behavior of the sys�tem near and at the very boundary of stability differsfrom that before it, the asymptotic way of the determi�nation of the spinodal in almost all cases turns out tobe satisfactory if it is in agreement with the propertiesof the system in the region of the stable state [4].

In this work, the asymptotic way is used for deter�mination of the transfer coefficient behavior on thespinodal of the superheated liquid and supersaturatedvapor. The analysis is based on the results of molecu�lar�dynamic calculations of the coefficients of self�dif�fusion, shear viscosity, and thermal conductivity of theLennard�Jones fluid.

RESULTS OF THE MOLECULAR�DYNAMIC CALCULATIONS OF TRANSFER

COEFFICIENTS

The coefficients of self�diffusion, shear viscosity,and thermal conductivity of the Lennard�Jones fluidas a function of the temperature, density, and pressureare calculated according to the Green–Kubo formulaby the molecular dynamics method. The results of thecalculations of D, η are given in [7–9]. The data on thecoefficient of thermal conductivity are used for thefirst time and will be published later.

The studied system contained 4000 Lennard�Jonesparticles. The cut�off radius of the potential was

Here σ, ε are the parameters of the poten�6.78 .cr = σ

tial of the interaction, which, along with the Boltz�mann constant kB, are parameters of the reduction ofthe thermodynamic and kinetic values to the dimen�sionless form. Below, the dimensionless quantities aredenoted by the symbol “*”. The calculations were per�formed in the temperature range from T* = =

0.4 to 2.0 and densities from to 1.1. Themaximal supersaturation of the liquid and vaporphases were limited by spontaneous nucleation.

The thermodynamic properties of the Lennard�Jones fluid within the specified model were calculatedearlier in [7–9], and data about the spinodal and binodalwere also presented there. The parameters of the critical

point are and In [10–12], as in the earlier [13–15], the transfer

coefficients in the Lennard�Jones system were calcu�lated and analyzed mainly as functions of the temper�ature and density. Here their temperature and baricdependences are discussed.

Figures 1 and 2 show the pressure as a function of

the coefficient of self�diffusion D* = andthe excess thermal conductivity Δλ* =Δλm1/2ε–1/2 ×

= at the temperatures above and belowthe temperature of the critical point. An analogous form

has the dependence of p* on Δη* = =

Here are the thermal conductivity andviscosity of the rarefied gas at and m is the massof the particle.

It follows from Figs. 1 and 2 that the character ofthe dependence of the pressure on the transfer coeffi�cients is similar to the dependence of the pressure onthe volume (density) in the van der Waals fluid. It ispossible to assume the presence of the points on iso�therms, where the derivatives have the form

(2)

and points where also the second derivatives are zero

Does the line determined by the conditions (2) corre�spond to the spinodal, where according to (1)

The determination of the points of the minima ofisotherms in the coordinates p, is con�nected with the large error. Therefore we will use thegeometric method, which was used for the first timefor interpreting the spinodal in [16, 17].

THE SPINODAL AS THE ENVELOPE OF THE FAMILY OF LINES OF THE CONSTANT VALUE

OF THE TRANSFER COEFFICIENTS

It is assumed for the homogeneous system that theequation exists

(3)

k T εB3 0ρ = ρσ =*

* 1.330,cT =* 0.311,cρ = * 0.137.cp =

1 2 1 2 1Dm − −

ε σ

2 1k −

σ B*Tλ − λ*

2 1 2( )m −

Δησ ε

*.Tη − η* *,Tλ*Tη

0ρ =*

( ) 0, ( ) 0, ( ) 0,T T Tp D p p∂ ∂ = ∂ ∂Δλ = ∂ ∂Δη =

2 2( ) 0,Tp D∂ ∂ =2 2( ) 0,Tp∂ ∂Δλ =

2 2( ) 0.Tp∂ ∂Δη =

( ) 0?Tp∂ ∂ =v

{ }, , ,D Δλ Δηv

( , , ) 0,F p T ϕ =

–0.61.0–1.5 0.50–0.5–1.0

–0.4

–0.2

0

0.2

0.4

0.6

logD*

p*

12345

6789

С

Fig. 1. Pressure as a function of the logarithm of the self�diffusion coefficient over isotherms: (1) T* = 0.85, (2) 1.0,(3) 1.15, (4) 1.2, (5) 1.25, (6) 1.3, (7) 1.35, (8) 1.5, (9) 2.0;dash�dotted line—binodal, C—critical point.

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TRANSFER COEFFICIENTS NEAR THE BOUNDARY 623

linking two thermodynamic quantities (p, T) and thetransfer coefficient in the region ofstable, metastable, and labile states. In the three�dimensional p,T,ϕ�space a certain surface corre�sponds to Eq. (3), which is considered to be smooth.

Let us choose the transfer coefficient ϕ as a param�eter. Then Eq. (3) sets the one�parameter family ofcurves on the plane p, T. The condition

(4)

along with Eq. (3) determines the envelope of thisfamily of curves. Additional conditions [18]

(5)

form the sufficient sign that the envelope Eqs. (3), (4)is a simple regular curve, and each of the curves of thefamily touches the envelope at a single point.

If we set the pressure as a function of the tempera�ture and kinetic coefficient in the explicit form

(6)then for the function F we have F = Con�dition (4) takes the form

(7)

The inequalities (5) are written as

(8)

which excludes the possibility of the existence of criti�cal points on the curves ϕ = const.

The envelope has the return point if at this point,along with Eqs. (3), (4), the conditions are fulfilled

(9)

At the return point of the envelope according toEqs. (6), (7), (9),

(10)

The conditions (7), (8), (10) do not contradict theresults of the molecular�dynamic calculations of thetransfer coefficients [7–9].

According to the geometric interpretation, thespinodal in variables p, T is the envelope of the familyof isochors continued into the region of the metastablestates [16, 17]. The property of the envelope isexpressed most vividly by the condition of the contact

Here the derivative in the left�hand side refers tothe envelope (the index “s” indicates that it belongs tothe spinodal), the derivative in the right�hand siderefers to the curve of the family of isochors. The prop�erty of the spinodal as an envelope is widely used dur�ing the approximation of the boundary of the thermo�

{ }, ,Dϕ ≡ Δλ Δη

0FFϕ

∂≡ =∂ϕ

( , )0, 0, 0

( , )p T

F FF F F

p Tϕ

ϕϕ

∂+ ≠ ≠ ≠

∂

( , ),p f T= ϕ

( , ).p f T− ϕ

0.T

p⎛ ⎞∂ =⎜ ⎟∂ϕ⎝ ⎠

2 2

21 0, 0, 0,

T

p p pT Tϕ

⎛ ⎞∂ ∂ ∂+ ≠ ≠ ≠⎜ ⎟∂ ∂ϕ∂∂ϕ⎝ ⎠

0, 0.F Fϕϕ ϕϕϕ= ≠

2 3

2 30, 0, 0.

T T T

p p p⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂= = ≠⎜ ⎟ ⎜ ⎟⎜ ⎟∂ϕ ∂ϕ ∂ϕ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )∂=

∂.

s

dp pdT T v

dynamic stability according to the experimental data[4, 19, 20].

Figure 3 shows the spinodal of the overheated(stretched) Lennard�Jones fluid in variables p, T. Thespinodal is built according to the results of the molec�ular�dynamic calculations of p, ρ, T—the propertiesof the Lennard�Jones fluid [10]. The lines of the con�stant excess viscosity are built there according to thedata of [8]. These lines for the liquid phase are close tothe straight lines in both stable and metastable regions.Extrapolating the lines Δη* = const to the spinodal, weobtain an envelope that coincides with the spinodal ofthe stretched liquid within the accuracy of such a pro�cedure. The spinodal of the stretched liquid is envel�oped only by the Δη* = const isolines, for which

0.165. Isolines with formanother envelope, which coincides with the spinodalof the supersaturated vapor. Both branches of the spin�odal close at the critical point, forming the returnpoint of the envelope on the plane p, T. The branchesof the spinodal are convex toward each other. The signof the curvature of the gas branch of the spinodal coin�cides with the sign of the curvature of the binodal.

The fact that the isolines of the transfer coefficientsare straight is an exception rather than the rule. There�fore, their extrapolation should be performed takinginto account the curvatures. Since the thermodynamicand kinetic properties in the molecular�dynamicexperiments can be calculated under the deep pene�tration into the metastable region, taking into accountthe curvature of the isolines ϕ = const does not lead toa large error. The region of the extrapolation of the

*KΔη > Δη* �

*KΔη < Δη*

–0.6–1.5 0.50–0.5–1.0

–0.3

0

0.3

0.6

logΔλ*

С

p*

Fig. 2. Pressure as a function of the logarithm of the excessthermal conductivity over isotherms (notations as in Fig. 1).

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HIGH TEMPERATURE Vol. 51 No. 5 2013

BAIDAKOV

lines of the constant excess viscosity in Fig. 3 is sepa�rated by dashes from the region in which the viscositywas directly calculated in the computer experiment.

Figure 4 illustrates the behavior of the isolines D* =const of the gas phase in the projection onto the planep, T. The spinodal of the supersaturated vapor is builtaccording to the data of [10]. At the critical point

It should be noted that the error of themolecular�dynamic calculations of the transfer coeffi�cients in the gas phase and critical region is muchhigher than in the liquid.

Thus, the isotherms of three kinetic coefficients onthe plane p, ϕ contain the points at which the deriva�tives are zero. These points coincide with thepoints of the branches of the spinodal of the over�heated liquid and supersaturated vapor. It follows fromthe relation for the derivatives

and the finite value of the derivative that, atthe point where the derivative is also zero, i.e., the spinodal on the plane T, ϕ is thegeometric place of the extrema of isobars.

CONCLUSIONS

The dynamic behavior of the metastable system isdetermined by the transfer coefficients. The transfercoefficients limit the velocity of the growth of thenucleus of a new phase and the kinetics of the decay ofthe system during the transfer via the spinodal. Themolecular�dynamic calculations indicate that thecoefficients of self�diffusion, shear viscosity, and ther�mal conductivity in the one�component system havethe final value on the spinodal. Nevertheless, the spin�odal turns out to be the separated line not only in thespace of thermodynamic variables but also on the sur�face F(p, T, ϕ).

At least for simple systems, data of computerexperiments indicate that the pressure (temperature)as a function of the transfer coefficient has a point ofthe extremum on isotherms (isobars). These points invariables p, T form lines that coincide with the gas andliquid branches of the spinodal. Thus, the spinodal invariables p, T is not only the envelope of the family ofisochors and adiabats [4] continued into the region ofthe metastable phase states, but also an envelope of thelines of constant values of the coefficients of self�diffu�sion, excess thermal conductivity, and shear viscosity.

This property of the spinodal is one more possibil�ity of its approximation (as an envelope) according tothe results of the measurement of the transfer coeffi�cients.

* 0.572.cD �

( )Tp∂ ∂ϕ

( ) 1Tp

pTp T ϕ

⎛ ⎞ ∂ϕ⎛ ⎞ ∂∂ = −⎜ ⎟⎜ ⎟∂ϕ ∂ ∂⎝ ⎠⎝ ⎠

( )p T ϕ∂ ∂

( ) 0,Tp∂ ∂ϕ = ( ) ,pT∂ ∂ϕ

–2

1.60.4 1.41.21.00.80.6

–1

0

11 2 3 4 5 6

7

89

С

p*

T*

Fig. 3. Lines of the constant excess viscosity: logΔη* = 0.8(1), 0.4 (2), 0.2 (3), 0.1 (4), 0 (5), –0.1 (6), –0.4 (7), –0.6(8), –0.78 (9); dashed line—spinodal of the overheatedliquids, dash�dotted line—binodal, C—critical point.Dashes in lines logΔη* = const separate the regions wherethe actual data of the molecular�dynamic calculations areterminated and their extrapolation begins.

00.6

0.15

0.10

0.05

0.9 1.2 1.5

С

p*

T*

12 3 4

5

6

7

8

Fig. 4. Lines of the constant value of the self�diffusioncoefficient: (1) – D* = –0.2, (2) 0, (3) 0.1, (4) 0.2, (5) 0.3,(6) 0.4, (7) 0.6, (8) 0.8; dashed line—spinodal supersatu�rated vapor, dash�dotted line—binodal, C—critical point.

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TRANSFER COEFFICIENTS NEAR THE BOUNDARY 625

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research (project no. 12�08�00467) andthe Program of the Presidium of the Russian Academyof Sciences no. 18 (project no. 12�P2�1049).

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Translated by L. Mosina