Physics Letters A 366 (2007) 1–6

www.elsevier.com/locate/pla

A theoretical explanation for possible temperature discontinuitiesacross a macroscopically-sharp plane solidification front

Suman Chakraborty ∗,1, Franz Durst

Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, Cauerstr. 4, 91058 Erlangen, Germany

Received 8 November 2006; received in revised form 3 January 2007; accepted 19 January 2007

Available online 9 February 2007

Communicated by R. Wu

Abstract

This work aims to theoretically investigate the possible situations of temperature discontinuities prevailing across propagating plane frontsolidification interfaces. A generalization of interfacial conditions is proposed, which can accommodate finite macroscopic jumps in the interfacialtemperature. This is achieved by implicitly accounting for two disparate and unresolved time-scales, which are respectively associated withthe interfacial heat conduction and the molecular rearrangements that are necessary for phase transformation. Through the present theoreticalestimates, possibilities of extending continuum-based formulations towards accommodating deterministic rates of plane front propagation in caseof directional solidification of pure materials can be explored.© 2007 Elsevier B.V. All rights reserved.

1. Introduction

Under certain conditions, plane solidification fronts of purematerials, propagating at high speeds, may be associated withsharp discontinuities in temperature across the interface. Al-though rather non-intuitive, such macroscopic temperaturejumps at the interfaces have been observed in a variety of phys-ical situations, such as in oceanography [1], geophysics [2], andadiabatic phase changes in thermo-elastic solids [3]. The classi-cal theories behind macro-scale solidification modelling [4–6],however, have not shown much success in accounting for suchmacroscopic temperature jump conditions in a sharp interfacelimit.

Typically, interface-tracking macroscopic models, charac-terizing solidification of pure substances, have followed theclassical approach of employing separate energy conservationequations for each phase. These equations are connected by thewell-known Stefan condition [7], assuming an infinitesimally

* Corresponding author.E-mail address: suman@mech.iitkgp.ernet.in (S. Chakraborty).

1 On leave from Department of Mechanical Engineering, Indian Institute ofTechnology, Kharagpur-721302, India.

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.01.058

thin interface between phases, with a continuity of temperatureacross the interface. Such models, in essence, necessitate thedetermination of interfacial locations as a function of time, inorder to predict the interface speed implicitly. An explicit quan-tification of interfacial speeds from the fundamental transportequations, however, turns out to be an impossible propositionfrom such models, since a continuity in interfacial temperaturesrenders the velocity of the interface indeterminate. This, how-ever, is contradictory to experimental observations supportingdeterministic speeds of interfacial propagation under specifiedconditions, in which the classical Stefan model of solidificationmay turn out to be degenerate. In a recent work, Danescu [8]proposed a generalized Stefan model that allows for jumps inthe temperature field across a nonmaterial evolving interface.In this work, the author postulated a general form of the interfa-cial dissipation inequality characterizing the temperature jump,based on an additional superficial field that needs to be pre-scribed by a constitutive function.

The present work aims to theoretically establish the unique-ness of the above-mentioned temperature jump condition fromfundamental physical considerations and postulates a newproposition to account for the temperature discontinuitiesacross the faces of an advancing solidification front. In order

2 S. Chakraborty, F. Durst / Physics Letters A 366 (2007) 1–6

to search for a solution from a macroscopic viewpoint, contin-uum considerations of mass, momentum and energy balanceare first invoked, with an understanding that under certain con-ditions, the time-scales for molecular rearrangements that arenecessary for solidification may turn out to be considerable, incomparison to the time-scales for interfacial conduction. Underthese circumstances, latent heat release on account of solid-ification phase change may not be able to compete with therate of heat conduction away from the interface, leading to anet unbalanced instantaneous interfacial heat flux. It is shownthat a correct accounting of this interfacial flux can lead to awell-posed problem with deterministic interfacial speeds, re-solving the above-mentioned apparent anomaly with regard tothe uniqueness of the interfacial temperature jump conditions.A systematic development of this fundamental proposition isoutlined in the foregoing discussions.

2. Problem formulation

The generic problem of plane front solidification of a puresubstance, characterized by a sharp interface, is considered, inwhich the interface is described by its direction normal, n, asrepresented schematically in Fig. 1. The fundamental equationsof energy conservation, consistent with the pertinent thermo-dynamic constraints, as appropriate for this generic physicalproblem, are briefly outlined here through Eqs. (1)–(13), for thesake of completeness. For detailed derivations of these equa-tions, one may refer to the work of Fried and Shen [9]. Thenotation ‘−’ is used to designate the ‘solid’ side of the in-terface and ‘+’ to designate its ‘liquid’ side. For simplicity,the densities of the solid and liquid are assumed to be equal,i.e., ρ+ = ρ− = ρ (say). In particular, the coordinate axes arechosen such that n = −i, V = V i (velocity of interface propa-gation), and x = xi, where x is the direction of freezing frontadvancement and i represents a unit vector along the same.Further, a conduction dominated transport is assumed, for sim-plicity. The jump and interfacial average values of temperature,respectively, are denoted by the following: �T �= T +−T −, and〈〈T 〉〉 = 1

2 (T + + T −). The quantities T + and T − in principle,can be solved from the thermal energy conservation equationsin the solid and liquid phases, as

(1)

{ρCl

∂T∂t

= λl∇2T for the liquid phase,

ρCs∂T∂t

= λs∇2T for the solid phase

Fig. 1. Schematic diagram of the model problem.

where C is the specific heat capacity, ρ is the density, λ isthe thermal conductivity, and subscripts l and s refer to liquidand solid phases, respectively. In the liquid phase, the far-fieldboundary condition implies that

(2)T → T∞ as (x − V t) → ∞.

For a solution of the temperature profile in the liquid phase,a Galilean transformation can be employed as z = x − V t , toyield

(3)ρClVdT

dz+ λl

d2T

dz2= 0

with the following boundary conditions:

T = T + as z → 0+ and T = T∞ as z → ∞.

The above can be solved, together with the solid phase solution,to obtain

(4)T (z, t) ={

T − for z < 0,

T∞ + (T + − T∞) exp(−ρClV z

λl) for z > 0.

The above equation can be substituted in an expression for in-terfacial energy balance (see Appendix A, Eq. (A.2), whichsummarizes the detailed derivations presented in Ref. [9]), toobtain

(5)T − = L + ClT∞Cs

where L is the latent heat of freezing. Next, a combined ex-pression of the first and second laws of thermodynamics (seeAppendix A, Eq. (A.7)) can be invoked to yield the followingthermodynamic constraint:

(6)E + JH � 0

where J is a scaled temperature jump, quantified as follows:

J = �T �〈〈T 〉〉 , s is the specific entropy and q is the heat flux vector.

In expression (6), E = �a� may be interpreted as an interfa-cial energy release term, where a is the specific Helmholtz freeenergy. The parameter H = (m〈〈T s〉〉− 〈〈q . n〉〉) can be concep-tualized as an interfacial heating term. From the literature [9], itis well known that the parameters E and H are functions of thequantities m and J , respectively. Although, in reality, compli-cated dependences may exist between these parameters, simplelinear relationships [9] may be used to establish the centraltheme of this work, in principle, in the absence of any cou-pling between the individual dissipative effects. Accordingly,one may write

(7a)E = αm,

(7b)H = γ J

where α and γ are constitutive parameters. To satisfy inequality(6) in a definitive manner, one must have

(8)α,γ � 0.

Utilizing an expression for L from Eq. (5), and employingthe temperature distribution as described by Eq. (4), one cannow obtain the following simplified forms of the parameters E

S. Chakraborty, F. Durst / Physics Letters A 366 (2007) 1–6 3

and H :

(9)E = αρV = Cl

(T + − T∞

) − ClT+ ln

T +

Tl

+ CsT− ln

T −

Ts

,

H = γ J

(10)

= −1

2ρV

(Cl

(T + − T∞

) − ClT+ ln

T +

Tl

− CsT− ln

T −

Ts

).

For deriving the above two expressions, Fourier’s law of heatconduction is assumed to remain valid in each of the con-stituent phases. Further, for the derivation of the above, fol-lowing supplementary thermodynamic relationships for incom-pressible pure substances are utilized:

(11)s(T ) ={

Cs ln TTs

for the solid phase,

Cl ln TTl

for the liquid phase,

(12)a(T ) ={

CsT (1 − ln TTs

) for the solid phase,

ClT (1 − ln TTl

) + L for the liquid phase,

here Ts and Tl , respectively, are the reference temperatures forsolid and liquid state property calculations. It can also be notedhere that the above definitions are consistent with the followingdefinition of internal energy:

(13)e ={

CsT for the solid phase,

ClT for the liquid phase.

Eqs. (9) and (10) essentially pose two coupled expressionswith two unknowns V (with V > 0) and T +. However, thesolvability of V from the above system of equations, under allcircumstances, is by no means trivial. It is imperative to note inthis context that with α,γ � 0 (to satisfy the second law of ther-modynamics), and J � 0 (since T + � T −, on account of latentheat release across the interface for the liquid to become solid),the inequalities imposed by the second law of thermodynam-ics (Eq. (6), for example) are trivially satisfied [9] by Eqs. (9)and (10), irrespective of the value of V . Thus, the second lawof thermodynamics gives no additional information to specifypermissible values of interfacial speed, in this case. As an alter-native way of assessing the solvability conditions for interfacialspeed, Eqs. (9) and (10) can together be manipulated to yield

α

γρ2(T + + T −)

V 2 − 2

(ρCs

γ

(T + + T −)

lnT −

Ts

)V

(14)− 4(T + − T −) = 0.

With T − � T +, the sole feasible solution for V can be obtainedfrom Eq. (14) as

V = Cs

ραT − ln

T −

Ts

(15)+ Cs

ρα

√(T − ln

T −Ts

)2

+ 4αγ (T − − T +)

C2s (T − + T +)

.

Eq. (15) can be substituted into Eq. (9) and squared to yield

C2l

(T +

(1 − ln

T +

Tl

)− T∞

)2

+ 4αγ (T + − T −)

T + + T −

(16)− C2s

(T − ln

T −

Ts

)2

= 0.

From Eq. (16), the following important observations can bemade:

(a) The uniqueness of the value of V depends on the unique-ness of the solution of Eq. (16).

(b) With α > 0 and γ → ∞, there is temperature conti-nuity across the interface, but there is a discontinuity in thespecific free energy. In that case, Tm > T + = T − = ClT∞+L

Cs.

Under these circumstances, V = Cl

ρα

(T −(1 − ln T −

Tl) − T∞ +

CsT−

Clln T −

Ts

).

(c) When α = 0 and γ < ∞, the temperature is discontinu-ous but the specific free energy density is continuous across the

interface, with V = 2γ (T +−T −)

ρCs(T ++T −)T − ln T −Ts

.

(d) When α = 0 and γ → ∞, the specific free energy den-sity and temperature are both continuous across the interface,T + = T − = Tm (i.e., the so-called equilibrium conditions, asencountered under slow growth rate conditions), but V is left asindeterminate.

(e) A generalized interfacial condition will not necessitateboth the conditions α = 0 and γ → ∞. It needs to be notedhere that the parameter α has a unit of velocity per unit mass.In a physical sense, a higher value of α would imply a reducedinterfacial mobility. The parameter γ can be interpreted as aninterfacial heat flux, which acts as a pivotal entity for the presenttheoretical development.

With an understanding that the conditions α = 0 and γ → ∞leave the interfacial velocity indeterminate (Stefan condition),it may be instructive to investigate further the interfacial condi-tions that may essentially relax the above-mentioned situation,by accommodating a possible temperature jump. Since the pa-rameter γ essentially controls this behaviour, it is imperativeto investigate specifically the situations in which γ < ∞. Oneneeds to appreciate at this point that the Stefan condition essen-tially specifies an energy balance between the heat conductedaway from the solidification front and an instantaneous releaseof heat as a consequence of phase transformation at interfaciallocations. This mathematical statement has one inherent as-sumption that the time-scale for molecular rearrangements nec-essary for solidification is vanishingly small in comparison withthat associated with interfacial heat conduction. However, forfast rates of front propagation, the characteristic time-scales forconduction of heat away from the interface are reduced from therespective values at slower growth rates, and the above two dis-parate time-scales come closer to each other and offer with aninteresting interplay. A major implication of this occurrence canbe appreciated from the fact that the latent heat release dependson the time-scale for molecular rearrangements of liquid phaseto be transformed into solid phase. In this respect, the solidifi-cation zone (essentially constituted of a few atomic layers) may

4 S. Chakraborty, F. Durst / Physics Letters A 366 (2007) 1–6

be conceptualised as narrow, but of finite thickness, in compari-son with the length scale for interfacial conduction heat transfer.However, in a microscopic/mesoscopic/macroscopic sense, thiszone cannot be resolved, and is eventually manifested in theform of a sharp freezing front. The solidification zone is charac-terized with finite-rate molecular level changes associated withthe formation and growth of solid nuclei. The solidificationzone, therefore, can be thought as being associated with a rateof solid production, depending on local temperatures and small-scale morphologies present within the same. This considerationcan be utilized to estimate an interfacial heat flux that eventu-ally leads to a volumetric source term in the solidification zone,on the basis of a dynamic equilibrium between microscopicmelting and freezing. The individual liquid-to-solid and solid-to-liquid transformations can be described by Arrhenius-typekinetic relationships, with molecular ‘freezing’ and ‘melting’activation energies of QF and QF + L, respectively, where L

is the enthalpy of fusion per molecule. In a non-dimensionalform, the pertinent heat source term in the solidification zonecan be described as [10]

(17)Q = tsA exp

(−QF

kT

){1 − exp

[−L

k

(1

T− 1

Tm

)]}where ts is the time-scale for solidification in the interfacialzone and k is the Boltzmann constant. The coefficient A (withunits of 1/s) appearing in Eq. (17) accounts for interfacial mor-phology over unresolved scales, atomic site availability, atomicjump frequencies and atomic surface densities in interfacial lo-cations. Since the above features are not resolved in the lengthscales under concern here, the parameter A is treated as amaterial-dependent constant. By noting that Eq. (17) leads toa prediction of Q that is of the order of unity when Q is a max-imum (since, Q is normalized between 0 and 1), it follows that

(18)ts ∼ kTm exp(1/ε)

εAL

where ε = kTm

QF. Expression (18) is essentially derived by ex-

panding Eq. (17) in orders of ε and neglecting non-linear terms.This is in consistency with the fact ε may be typically of theorder of 10−3 or less, as apparent from thermophysical datareported in the literature [10]. Physically, expression (18) im-plies that an increase in activation energy of phase change in-creases the characteristic solidification time-scale, whereas anincrease in latent heat of fusion decreases the same, which isconsistent with physical intuition. A comparison of the solidifi-cation time-scale, ts , with the conduction time-scale, tc, can bemade by noting that the interfacial conduction length scale isLc ∼ λ

ρC√

L/CTmVref, where Vref is the characteristic interfacial

speed. From this estimation, it follows that

(19)tc ∼ L2cρC

λ= λTm

ρLV 2ref

.

The parameter ε is related to these time-scales as ε ∼ ts/tc.The solidification zone length scale can accordingly be esti-mated as � ∼ εLc . Estimates of the order of magnitude of theabove-mentioned scales can be obtained, based on typical prop-erty data, corresponding to the solidification of a pure material.

For estimates made in this Letter, upper limits of the parame-ter ε (∼ 10−3) and the interfacial speed, Vref (∼ 102 m/s) areconsidered, so that the possibility of an interfacial temperature-jump phenomenon can be clearly manifested. It needs to benoted at this points that such extreme conditions are not rou-tinely realized in usual solidification experiments, but can occurunder certain non-equilibrium phase-change situations [3]. Fornumerical estimates, following property data are considered:λ = 90 W/m K, ρ = 8900 kg/m3, L = 2.9 × 105 J/kg, Tm =1726 K. These result in the following estimates from the presentmodel: Lc ∼ 10−7 m, tc ∼ 10−8 s, ts ∼ 10−11 s, � ∼ 10−10 m.A physical feel of the length scales, derived as above, can be ob-tained by recalling that atomic-scale dimensions are typically ofthe order of 10−15 m, typical lattice spacings in crystals are ofthe order of 10−10 m, and grain boundary length scales of purematerials are around 10−8 m. All these length scales, however,are impossible to resolve during the macro- or meso-scale simu-lations of solidification processes. The solidification time-scaleturns out to be two orders of magnitude above the time-scale as-sociated with crystal lattice vibrations (of the order of 10−13 s),but shorter than interfacial heat conduction time-scales.

The above exercise, in effect, leads to a scaling estimation ofthe interfacial heat flux as (noting that Q, being a normalizedheat flux, is of the order of unity):

(20)γ ∼ ρL�

ts.

With �ts

< ∞, for the solidification time-scale (ts ) to be of fi-nite duration (noting that ts can be infinitesimally small, onlyunder interfacial equilibrium conditions), γ needs not be infi-nitely large in all cases, explaining the possibility of a non-zeromagnitude of the scaled temperature jump (J ) for a finite valueof the parameter H .

3. Discussions and conclusions

For a quantitative assessment of the present proposition, thenormalized interfacial temperature jump values are ascertained,as a function of the interfacial speed, which in turn is a func-tion of prevailing undercooling conditions that are implicitlycontrolled by the far-field boundary temperature, T∞. The out-come of this analysis is depicted in Fig. 2, under the conditionsgiven by α = 0, and γ ∼ 1010 W/m2. It can be seen from Fig. 2that lower values of T∞ essentially lead to faster front prop-agation speeds, characterised by more prominent temperaturejumps. However, as the interfacial heat flux tends to infinity,local equilibrium conditions are practically regained, and devia-tions from classical theory turn out to be virtually infinitesimal.In such cases, the Stefan model is degenerate, and the interfacialspeed becomes indeterminate. On the other hand, for extremelyrapid rate of solidification front propagation, local equilibriumconditions at solid-liquid interface may need to be relaxed.

In conclusions, simple theoretical estimates of interfacialheat flux have been successfully derived, allowing interfacialequilibrium requirements to be relaxed under rapid growth rateconditions. The physics of the above postulate have been at-tributed to a reduction in the time-scale of interfacial con-

S. Chakraborty, F. Durst / Physics Letters A 366 (2007) 1–6 5

Fig. 2. Typical variation of interfacial speed and normalized temperature jumpas a function of far-field temperature.

duction at high rates of solidification, in comparison with thetime-scales associated with molecular rearrangements that arenecessary for solidification phase transformation. The paradigmof the present proposition, which accounts for the presence ofthese dual time-scales, differs from other existing approaches inthe sense that this theory does not attempt to resolve explicitlythe microscopic length scales to quantify a macroscopic jump,but rather expresses those in terms of macroscopically mea-surable parameters. A penalty associated with this upscalingis that the present theory cannot directly describe the complexmicrostructures and the associated morphology in typical solid-ification systems, as the entire burden is implicitly embeddedin the kinetic parameter A, which is of the order of kTm exp(1/ε)

εts L.

However, at the same time, it needs to be kept in mind that it isnot essential to resolve the complex topographical parametersin a physically-consistent estimation or accounting of planarfront solidification speeds, in accordance with the continuumconservation requirements. Future efforts, therefore, can be di-rected towards generalizing the present approach further, to takeinto account some more complicated constitutional relation-ships, and to perform molecular dynamics simulations to yieldmore accurate and precise estimates of the interfacial tempera-ture jump conditions.

Acknowledgements

The first author (Dr. Suman Chakraborty) gratefully ac-knowledges the fellowship provided by the Alexander von Hum-boldt Foundation, Germany, for executing this research.

Appendix A. A general description of interfacialconstraints in terms of average values and jumpconditions, for the Stefan problem [9]

The conservation of mass, as applied at the interface, can bestated in terms of an interfacial average as

(A.1)m = ⟨⟨ρ(V − v . n)

⟩⟩

where V is the interfacial velocity, v is the velocity of a fluidparticle located at the interface, and m is the interfacial massflux. A combined statement of interfacial momentum and en-ergy balance can similarly be written as:

(A.2)m�e�+ �v�〈〈τ n〉〉 = �q . n�where e is the specific internal energy density, τ is the Cauchystress tensor, q is the interfacial heat flux, and �χ� representsa jump in the quantity χ . The second law of thermodynamicscan be invoked at this stage to write an interfacial inequality onentropy as

(A.3)m�s� ��

qT

. n

�where s is the specific entropy density. Consistent with theabove expressions, the jump and interfacial average values oftemperature can be described as �T � = T + − T −, and 〈〈T 〉〉 =12 (T + + T −), respectively. A scaled temperature jump, J , can

be accordingly quantified as follows: J = �T �〈〈T 〉〉 . Based on this

definition, the jump quantities can be described as

(A.4)

�1

T

�= −

⟨⟨1

T

⟩⟩J,

(A.5)�s� =⟨⟨

1

T

⟩⟩(�T s�− J 〈〈T s〉〉).Using Eq. (A.4), Eq. (A.2) can be rewritten as

(A.6)

m

�e − n . τ n

ρ+ 1

2(V − v . n)2

�+ P�v� . P〈〈τ n〉〉 = �q . n�

where P = 1 − n ⊗ n.Using Eq. (A.6) in inequality (A.3), and utilizing the defi-

nition of Helmholtz free energy density, a = e − T s, one canwrite

m

�e − n . τ n

ρ+ 1

2(V − v . n)2

�+ P�v� . P〈〈τ n〉〉

(A.7)+ J(m〈〈T s〉〉 − 〈〈q . n〉〉) � 0.

Here, E = m�e − n.τ n

ρ+ 1

2 (V − v . n)2� may be interpreted asan interfacial energy release term, F = P〈〈τ n〉〉 as an interfa-cial friction term, and H = (m〈〈T s〉〉−〈〈q . n〉〉) as an interfacialheating term. From the literature [9], it is well known that theterms E, F and H are functions of the quantities m, P�v�, andJ , respectively. In the absence of any coupling between theindividual dissipative effects, a simple linear functional rela-tionship in this regard can be assumed, and one may write

(A.8a)E = αm,

(A.8b)F = βP�v�,(A.8c)H = γ J,

where α, β , and γ are constitutive parameters. To satisfy in-equality (A.7) in a definitive manner, one must have

(A.9)α,β, γ � 0.

6 S. Chakraborty, F. Durst / Physics Letters A 366 (2007) 1–6

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