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https://doi.org/10.1016/j.amc.2013.12.003

The effect of surface tension and kinetic undercooling on a

radially-symmetric melting problem

Julian M. Back, Scott W. McCue, Mike H.-N. Hsieh, Timothy J. Moroney

School of Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia.

Abstract

The addition of surface tension to the classical Stefan problem for melting a spherecauses the solution to blow up at a finite time before complete melting takes place.This singular behaviour is characterised by the speed of the solid-melt interface andthe flux of heat at the interface both becoming unbounded in the blow-up limit. In thispaper, we use numerical simulation for a particular energy-conserving one-phase versionof the problem to show that kinetic undercooling regularises this blow-up, so that themodel with both surface tension and kinetic undercooling has solutions that are regularright up to complete melting. By examining the regime in which the dimensionlesskinetic undercooling parameter is small, our results demonstrate how physically realisticsolutions to this Stefan problem are consistent with observations of abrupt melting ofnanoscaled particles.

Keywords: Stefan problem, surface tension, kinetic undercooling, nanoparticlemelting2010 MSC: 80A22, 35R37

1. Introduction

Solidification processes are modelled by moving boundary problems, called Stefanproblems, which in their most simple form involve solving the heat equation in boththe solid and liquid domains subject to a so-called Stefan condition on the solid-meltinterface. This condition is an energy balance that describes the manner in which latentheat is released at the interface. For classical well-posed solidification problems, we havethe additional condition that the temperature u∗ on the solid-melt interface is fixed tobe the freezing temperature. Problems of this sort have been dealt with extensively inthe literature, the results of which are reported in books such as Gupta [39] and others[6, 18, 22, 46]. There is the analytic ‘Neumann’ solution in one Cartesian coordinate[6, 18, 22, 39, 46], but otherwise exact practical useful solutions are extremely rare. Ofparticular interest to the present study, we note that the classical radially-symmetricStefan problem has no known exact solution, but turns out to have a rather interestingasymptotic structure in the limit of complete freezing (we shall refer to this limit asthe extinction limit) [43, 59, 78, 79]. Further formal results for this radially-symmetric

Preprint submitted to Applied Mathematics and Computation December 3, 2013

problem have been generated using a variety of numerical and analytical techniques[7, 21, 23, 32, 45, 64, 71]. Asymptotic studies of classical Stefan problems in more thanone spatial variable exist, but are less common [49, 57, 58, 86]. From a more rigorousperspective, much attention has been devoted to proving existence and uniqueness inone dimension (see [5, 36, 72], for example) and higher dimensions [35, 40].

An interesting and very well studied class of Stefan problems, which in their mostbasic form are ill-posed, arises from modelling solidification of pure substances froman undercooled melt. One-dimensional problems of this sort have solutions in whichthe speed of the solid-melt interface becomes infinite at some finite time [42, 50, 52];in higher dimensions, the solutions can exhibit more complicated forms of finite-timeblow-up, for example via the birth of cusps or corners [44, 84, 85]. In order to provide aphysical regularisation for such ill-posed problems, we may apply the Gibbs–Thomsoncondition

u∗ = u∗m(1− σ∗K∗)− ε∗v∗n, (1)

on the interface. Here the freezing temperature (the right-hand side of (1)) is not as-sumed to be simply equal to the constant bulk freezing temperature, u∗m, but insteadis corrected by two regularising terms. The first, and most studied, involves surfacetension σ∗, which acts to penalise regions of the interface with high mean curvature K∗.This term can be derived using thermodynamic arguments by considering a system inequilibrium [55]. To take into account the departure from equilibrium due to the mov-ing solid-melt interface, a kinetic correction term is required. This kinetic correction isthe second regularising term in (1) and involves a parameter ε∗, referred to here as thekinetic undercooling parameter, multiplied by the interface’s normal velocity v∗n. Therelationship between the melting temperature and the solidification rate represents anadditional driving force generated by undercooled liquid near the interface [1]. Condi-tion (1) has also been considered as a linearised version of a more complicated kineticrelationship [22, 30, 33]. In the past decade there have been numerous numerical studiesof crystal formation using (1) with either ε∗ > 0 or ε∗ = 0, with intricate descriptions ofpattern and finger formation [13, 37, 48, 82]. Some details of existence and uniquenessfor this class of Stefan problem are provided in [14, 28, 56].

The stability of Stefan problems with the Gibbs–Thomson condition have beenexamined by several authors by tracking small perturbations of solutions for problemsin one spatial dimension. Results for the surface-tension only case (ε∗ = 0, σ∗ > 0 in(1)) include the melting and freezing of a planar solid [8, 90], propagation of a planarfront into a supercooled liquid [54, 73] and the growth of a spherical crystals [9, 65].Less attention has been paid to the kinetic undercooling only case (ε∗ > 0, σ∗ = 0) butthere exists results for planar solidification [16, 17]. Lastly, for Stefan problems withthe full Gibbs–Thomson condition (ε∗ > 0 and σ∗ > 0), we have stability results forplanar problems [25, 83] and spherical problems [68, 74].

The above studies on solidification also apply to melting problems since mathemati-cally both melting and freezing problems are equivalent, the only difference arising fromswitching the sign of the temperature throughout. For example, the asymptotic results

2

Liquid u(r, t)

Solid

r = 1

r = s(t)

Figure 1: A schematic of a melting particle with the dimensionless variables used in Section 2.1. Theouter region (shaded) is the liquid melt layer which surrounds the inner phase: the solid core. Thesetwo phases are separated by the moving front r = s(t), which propagates inwards during the meltingprocess. It is the temperature u(r, t) in the outer liquid phase that we are interested in here.

for the classical well posed problem of freezing a spherical ball of liquid also describe themelting of a spherical particle. Further, the results for the ill-posed crystal formationproblem also apply for the ill-posed problem of melting a superheated solid (for whichthere are fewer examples in nature). In the present paper we shall use the language ofmelting, not freezing, for reasons that should become apparent, with the understandingthat results hold for both cases. Furthermore, we shall continue to use the term “kineticundercooling parameter” for ε∗, even though the “undercooling” arises from freezingproblems, not melting problems.

It has been observed that adding surface tension (via the Gibbs–Thomson condi-tion with σ∗ > 0 and ε∗ = 0) to the classical well-posed problem of melting a sphericalparticle has the unexpected result of introducing a singularity, with the resulting prob-lem exhibiting finite-time blow up [38, 60, 63]. The full two-phase problem takes intoaccount temperature variations in both the two phases, as depicted in the schematicin Figure 1. The problem develops an infinite temperature gradient in the inner solidphase [60, 89] shortly after this inner phase becomes locally superheated (here we usethe term “locally superheated” in the sense that the temperature in the solid phaseis everywhere greater than the melting temperature; this feature has also been notedin [34]). Similar observations have been made for a special class of initial conditionsfor which the temperature profile of the outer phase is held at the spatially-dependentmelting temperature u∗ = u∗m(1− σ∗/r∗) for all time, resulting in a one-phase problemwhich focuses on the inner phase [3, 63]. This type of blow up, characterised by aninfinite temperature gradient in the inner phase and an unbounded moving front speed,appears to be of the same nature as the finite-time blow-up of the one-dimensionalStefan problem for the freezing of a supercooled liquid [3, 24, 42, 50].

In the present paper we are concerned with extending a particular energy-conserving

3

one-phase version of the radially-symmetric melting problem with surface tension (i.e.,with σ∗ > 0 and ε∗ = 0), treated by Wu et al. [88], which also exhibits an unboundedphase boundary speed at some finite time before the particle has completely melted.Our goal is to generalise the work of Wu et al. [88] to include the full Gibbs–Thomsoncondition (1), with both nonzero surface tension σ∗ > 0 and kinetic undercooling ε∗ >0, and study the effect that kinetic undercooling has on the finite-time blow-up justmentioned. Section 2.1 provides the governing equations in question, and discusses themanner in which the relevant single-phase limit is formulated [29, 50, 77, 88]. Themodel is solved using a front-fixing numerical scheme and the method of lines, detailedin Section 2.2. In Section 3 we summarise the key findings of Wu et al. [88] (which is forε∗ = 0), while in Section 4 we provide numerical results that show the addition of kineticundercooling regularises the problem, so that for ε∗ > 0 the solution exists up untilcomplete melting. For small kinetic undercooling, the transition through the blow-upregime is smooth but rapid, so that in the limit that the kinetic undercooling vanishes,the system approaches what could be referred to as ‘abrupt melting’ [51, 53, 62, 69, 75].These results are consistent with experimental observations of melting nanoparticles,as we discuss in Section 5.

2. Mathematical model and numerical scheme

2.1. One-phase melting problem

A solid ball of radius a∗ is held at its bulk melting temperature u∗m. Now, supposethe outer boundary is raised to a higher temperature u∗a such that the ball beginsto melt. The solid-melt interface r∗ = s∗(t∗), which is initially at s∗(0) = a∗, thenevolves towards the centre of the ball. For the one-phase limit we are interested in, thedimensionless equations governing the melting of the radially symmetric particle are

in s(t) < r < 1,∂u

∂t=∂2u

∂r2+

2

r

∂u

∂r, (2)

on r = s(t),∂u

∂r= −(u+ β)

ds

dt, (3)

on r = s(t), u = σ

(1− 1

s

)− εds

dt, (4)

on r = 1, u = 1. (5)

Here, the temperature in the outer liquid phase has been scaled u(r, t) = (u∗(r∗, t∗) −u∗m(1 − σ∗/a∗))/∆u∗, where ∆u∗ = u∗a − u∗m(1 − σ∗/a∗). Lengths and time have beenscaled by r = r∗/a∗ and t = kut

∗/ρcua∗2, while the position of the moving boundary

has been scaled like s(t) = s∗(t∗)/a∗. There are three dimensionless parameters in thisproblem: these are the Stefan number β = L/cu∆u∗, which represents the ratio oflatent heat to sensible heat; the surface tension coefficient σ = σ∗u∗m/a

∗∆u∗; and thekinetic undercooling coefficient ε = ε∗ku/ρcua

∗∆u∗. Here the thermodynamic constantsare the density ρ, the thermal conductivity of the liquid phase ku, the specific heat ofthe liquid phase cu and the latent heat L.

4

The Dirichlet boundary condition (5) together with the initial condition s(0) = 1 waschosen for simplicity and also to allow a direct comparison with previous mathematicalstudies and known results (such as [61, 88]). An alternative approach is to treat theproblem on the domain s(t) < r < rout, where rout > 1. An initial condition forthe temperature, such as u(r, 0) = 1, is then required on 1 < r < rout. Solving thisalternative problem requires only minor changes to the numerical scheme presented inSection 2.2. An interesting exercise is to study the regime rout � 1, as a means tomodel the melting sphere in a large body of liquid, and make comparisons with theresults for (2)–(5). We comment on these comparisons later in the paper.

We now discuss the derivation of (2)–(5). It is important to first recall that theclassical two-phase Stefan problem for a melting particle with σ = 0 and ε = 0 can bereduced to a one-phase problem in the usual way by assuming that the temperatureof the inner solid core is everywhere equal to the melting temperature (which is u = 0in this dimensionless model), such that we need to study the outer liquid region only.This straight-forward simplification is no longer valid when either σ > 0 or ε > 0, asthe nonconstant temperature data on the boundary via (4) will result in temperaturefluctuations throughout the material. We can, however, consider an asymptotic approx-imation to the full two-phase problem with σ > 0 and ε > 0 with the understandingthat ratio of thermal conductivities κ is small, so that the liquid phase conducts heatfar more easily than the solid phase. The resulting moving boundary problem is (2)-(5).While the details are presented elsewhere [29, 50, 77, 88], we note the presence of aninterior layer in the inner phase in which the temperature changes rapidly, resultingin an extra term −u ds/dt in the Stefan condition (3). Further, we note that if weallow the solid ball to be initially at temperature less than the bulk temperature, or ifwe take into account the differences between the specific heats in the liquid and solidphases, then the only change in the boundary condition (3) is a redefinition of the pa-rameter β [88]. An assumption of different specific heats can also lead to a generalisedGibbs–Thomson law [34], which would affect (4).

One important advantage of the one-phase formulation leading to (2)–(5) is thatenergy is conserved in the system [29, 50]. On the other hand, the ratio of thermalconductivities κ is not small in practice, and in fact is roughly 2.1 for gold, 3.0 forsilver, and 2.2 for lead. Thus the shortcoming of using this one-phase model is thatit holds for a parameter regime that does not correspond to melting nanoscaled metalspheres. We return to this point shortly.

We mention here that a different one-phase approximation that leads to a movingboundary problem in the outer liquid phase, which also conserves energy, is presentedby Myers et al. [67] for planar geometry with ε > 0 (note that in a planar geometrysurface tension has no effect, as the interface has zero curvature). In this case the ratioof thermal conductivities κ is large, and the inner phase is assumed to conduct heat sowell that the temperature in the solid rapidly settles to the melting temperature. Thisleads to a heat conduction problem for the solid phase (that can be solved in terms of sand its derivatives) and a modified one-phase Stefan problem for the liquid phase. Thebehaviour of this alternative one-phase problem in a spherical geometry, including the

5

effects of surface tension, is not yet known. A further complementary approach is toessentially ignore the inner phase by keeping the Gibbs–Thomson rule (4) but applyingthe Stefan condition (3) without the extra term −u ds/dt. Results for this case, whichdoes not conserve energy, are provided for σ > 0 and ε = 0 in Wu et al. [87] and Herraizet al. [41]. Apart from not conserving energy, a disadvantage of this simplified modelis that it does not capture the type of blow-up behaviour that is observed for σ > 0and ε = 0 in the full two-phase problem.

It is worth emphasising that our ultimate interest is in studying the manner inwhich kinetic effects regularise the finite-time blow-up that occurs in the full two-phaseproblem with surface tension [60]. The present work can be thought of as a steppingstone in that direction. It is in this context that we choose to deal with the one-phaseproblem (2)–(5), even though it is derived under the physically unrealistic assumptionthat the ratio of thermal conductivities κ is small. An argument in favour of thisapproach is that the one-phase problem (2)–(5) with ε = 0 is known to exhibit thesame qualitative features of the full two-phase problem (studied in [60]), which suggeststhat the mathematical structure of (2)–(5) with ε > 0 is likely to mimic the full two-phase problem to a significant extent. In contrast, the alternative energy-conservingone-phase reduction outlined by Myers et al. [67], which is based on the more physicallysensible assumption that the ratio of thermal conductivities κ is large, has not beenapplied to the spherically symmetric melting problem with surface tension. Thus it isnot clear whether this approach leads to the kind of finite-time blow-up that the fulltwo-phase problem displays, and so for this reason we do not pursue the large thermalconductivity limit in this study.

We close this subsection by noting that one-phase models with surface tension thatgive rise to moving boundary problems in the inner phase have also been studied [3, 63].For these formulations, the inner solid phase appears to exhibit the same type of finite-time blow-up as in the full two-phase problem. While such models deserve furtherattention, we shall not be treating them here.

2.2. Front-fixing numerical scheme

We solve the problem (2)-(5) using a front-fixing transformation combined withfinite difference spatial discretisation and the method of lines. The Landau-type trans-formation

ξ =r − s(t)1− s(t)

with φ(ξ, t) = u(r, t)

6

is first applied, which has the effect of fixing the expanding domain s(t) ≤ r ≤ 1 to0 ≤ ξ ≤ 1. The equations (2)-(5) in terms of the new spatial variable ξ become

in 0 < ξ < 1,∂φ

∂t=

1

(1− s)2∂2φ

∂ξ2+

1

1− s

(2

s+ ξ(1− s)+ (1− ξ)ds

dt

)∂φ

∂ξ, (6)

on ξ = 0,∂φ

∂ξ= −(1− s)(φ+ β)

ds

dt, (7)

on ξ = 0, φ = σ

(1− 1

s

)− εds

dt, (8)

on ξ = 1, φ = 1. (9)

A uniform mesh is then introduced, and the spatial derivatives in (6) are replaced withsecond order finite difference approximations. For the moving boundary equation (7)we use one-sided differences to discretise the spatial derivative. At ξ = 0 we imposethe melting temperature condition (8) and at ξ = 1 we impose the outer boundarycondition (9).

The resulting semidiscrete system of coupled nonlinear ordinary differential equa-tions in time is solved using MATLAB’s ode15i solver. With this variable order,variable step size, fully implicit approach, we are solving for the location of the movingboundary simultaneously with the field equations, and allowing the time-step sizes toautomatically adjust over the course of the simulation. This is particularly crucial forlate times, when we are concerned with accurately tracking the moving front in themoments before blow-up occurs. The time-step size is observed to (correctly) decreasedramatically in response to the large rates of change encountered during this part ofthe simulation.

One benefit of solving the system of coupled nonlinear ordinary differential equationswith MATLAB’s ode15i is the opportunity to specify options such as the relative orabsolute error tolerances. With the appropriate options, this method presents no signif-icant barrier to performance. In particular, it is essential to specify the sparsity patternof the Jacobian matrix, so that it may be formed efficiently using shifted evaluationsand forward difference quotients [76]. With this option in place, accurate simulationswith tens of thousands of mesh nodes can be performed with just one or two minutesof runtime on a standard desktop machine. Recall, we are solving for the transformedfunction φ on the fixed domain 0 ≤ ξ ≤ 1, however the original temperature u is de-fined on the physical domain s(t) ≤ r ≤ 1, which is expanding in time. Thus, as timeprogresses, the mesh becomes more coarse in the physical domain. With this in mind,10000 mesh points were used in all of our numerical experiments, and the solutionwas observed to have converged within visual accuracy for both the temperature andinterface profiles.

An interesting question is how to commence the simulations, given there is no spatialdomain at time t = 0. We chose to start from time t0 = 1× 10−6 using an asymptoticsolution valid in the limit t → 0+. For the classical case ε = σ = 0 we used the‘Neumann’ solution for this purpose [6, 18, 22, 39, 46]. For σ > 0, ε = 0 we used the

7

small time solution derived by Wu et al. [88]. Finally, for ε > 0, σ ≥ 0 we used the smalltime solution presented in McCue et al. [61], noting that effects of kinetic undercoolingwill dominate surface tension in the small time limit.

3. Summary of results with zero kinetic undercooling

It is instructive to provide a brief outline of the main results of Wu et al. [88],who study the one-phase problem (2)-(5) for the case ε = 0. As mentioned in theintroduction, solutions of this problem evolve until a singularity forms in the speed ofthe moving front; in particular, we have ds/dt→ −∞, s(t)→ s+c as t→ t−c , where sc isthe critical radius and tc is the critical time at which finite-time blow-up occurs. In thiscase, the temperature gradient at the interface ∂u/∂r(s(t), t) is defined for 0 < t ≤ tc,so from (3) it must be that u(sc, tc)+β = 0. This gives the temperature at the interfaceat the blow-up time to be

u(sc, tc) = −β (for ε = 0), (10)

while combining this result with (4) gives the critical radius

sc =σ

σ + β(for ε = 0). (11)

Thus we can see that as the surface tension σ decreases, the critical radius sc decreases,and the onset of blow-up is delayed. Furthermore, the large Stefan number regimeβ � 1 also corresponds to a small value of sc. Indeed, the formal asymptotic limitβ → ∞ considered by Wu et al. [88] predicts that sc = 0 to leading order, whichsuggests that we cannot study the finite-time blow-up in this limit.

Representative numerical solutions for this regime are presented in Figure 2, whichshows temperature profiles for the case β = 1, σ = 0.15, ε = 0. The final temperatureprofile at t = tc is called the blow-up profile uc(r). Figure 3 (top) shows the dependenceof the moving front position s(t) on t for a variety of surface tension values. We havechecked that the numerical values for uc(sc) and sc agree with the analytical results(10) and (11) to six decimal places.

It is worth emphasising that the remarkable feature here is that the problem withσ = 0 and ε = 0 is well-posed, with the solution remaining regular and evolving untilthe interface reaches the centre at some finite time. It is the addition of surface tensionto the model that has caused the singular behaviour. This is unusual because surfacetension is often added to regularise models which lack the physics required to suppressunphysical singularities [13, 37, 48, 82].

A discussion on the blow-up behaviour seen in Figure 3 is given in Wu et al. [88].In summary, as seen in Figure 2, we have that for β = O(1) and σ � 1,

u ∼ σ

(1− 1

r

)for r − s� 1 as s→ s+c (for ε = 0), (12)

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

r

u(r, t)

uc(r)

Figure 2: Numerically calculated temperature profiles for β = 1, σ = 0.15 and ε = 0, for timest = 0.0092, 0.0303, 0.0683, 0.1150, 0.1471 and the critical time tc = 0.1700. The Gibbs–Thomsonequation (4) with ε = 0 is also plotted (dashed). For times near blow-up, the temperature profileappears to ‘hug’ the Gibbs–Thomson equation, leading to the approximation (12).

and substituting (12) into (3) gives

tc − t ∼1

2(s− sc)2 as s→ s+c (for ε = 0). (13)

Equation (13) gives the asymptotic scaling for the interface location in the limit s→ s+c .This prediction is plotted against the numerical results in Figure 3 (bottom). This figuremakes it clear that as s→ s+c , the speed of the boundary becomes unbounded, and thesolution cannot be continued past the critical time t = tc.

4. Results for nonzero kinetic undercooling

4.1. Extinction behaviour

We now extend the results in Wu et al. [88], summarised above in Section 3, byincluding both surface tension σ > 0 and kinetic undercooling ε > 0 in (3). Equa-tions (2)-(5) with σ > 0 and ε > 0 are solved using the numerical scheme describedin Section 2.2. Temperature profiles for the representative case σ = 0.15, ε = 0.1 andβ = 1 are shown in Figure 4. Here the solution remains regular until the moving in-terfaces reaches the centre of the sphere at the extinction time t = te. At this timethe entire ball has completely melted, so there is no longer an inner solid phase. Thefinal temperature profile, which we refer to as the extinction profile ue(r), is thereforedefined for all 0 < r < 1. Figure 5 (top) shows the position of the moving interfacefor the same parameter values as in Figure 3 (top), except that now ε = 0.1 insteadof ε = 0. We see that, regardless of the surface tension σ, the interface for ε = 0.1continues inwards until it reaches the centre. These numerical solutions suggest thatthe addition of kinetic undercooling has regularised the singularity.

9

0.00 0.05 0.10 0.15 0.180

0.2

0.4

0.6

0.8

1

t

s(t)

σ = 0.15

σ = 0.3σ = 0.5

σ = 1

0.1 0.2 0.3 0.4 0.5 0.6−0.20

−0.15

−0.10

−0.05

0.00

s

dt

ds

σ = 0.15

σ = 0.3

σ = 0.5

σ = 1

Figure 3: (Top) The position of the moving front, calculated numerically for σ = 0.15, 0.3, 0.5 and1, and ε = 0. For all cases, the numerical solution ceases to exist at a finite value of s at which theinterface speed ds/dt becomes very large. These results support the prediction (11), which providesthe exact value of the interface position when the solution blows-up. (Bottom) For times near blow upwhen tc− t� 1, the asymptotic approximation (13) (dashed) is in good agreement with the numericalsolution (solid). For the numerical solutions, finite-time blow-up occurs at the following critical times:when σ = 0.15, tc = 0.193 and sc = 0.130; when σ = 0.3, tc = 0.124 and sc = 0.231; when σ = 0.5,tc = 0.089 and sc = 0.333; and when σ = 1, tc = 0.047 and sc = 0.500. In all cases, as s→ s+c we havethat ds/dt→ −∞.

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

r

u(r, t)

ue(r)

Figure 4: Temperature profiles for σ = 0.15, ε = 0.1 and β = 1, at t = 0.019, 0.066, 0.150, 0.229, 0.274,and te = 0.286. The melting process continues past the ε = 0 critical radius and the front moves tothe centre. The final temperature profile at t = te is referred to as the extinction profile ue(x).

In Figure 5 (bottom) we have plotted dt/ds versus s, in an analogous way to Figure 3(bottom). While for ε = 0 we observed that dt/ds → 0− as s → s+c (with sc > 0), wesee for ε > 0 that dt/ds→ 0− as s→ 0+. So while the addition of kinetic undercoolinghas acted to suppress the blow-up that occurred at s = sc, we see that ultimately thespeed of the interface still becomes unbounded. The difference is that for ε = 0 wehave ds/dt → −∞ as s → s+c , while for ε > 0 we have ds/dt → −∞ as s → 0+. As aconsequence of this behaviour, the Stefan condition (3) implies that

ue(0) = −β (for ε > 0), (14)

which is similar to (10), and is consistent with our numerical simulations (see in Figure4, for example). Further, substituting (14) into (4) shows that ds/dt scales like 1/s ast→ t−e , and so, including one correction term, we have

s(t) ∼√

2σ

ε

√te − t−

2(β + σ)

3ε(te − t) as t→ t−e (for ε > 0), (15)

which is compared to the position of the moving front calculated numerically in Figure5 (bottom). The approximation (15) gives reasonably good agreement to the numericalsolution, and of course improves as s decreases.

We close this subsection by mentioning the alternative problem described in Section2.1 with (5) replaced by u(rout, t) = 1 and rout � 1. For the case σ > 0 and ε = 0,this problem also exhibits finite-time blow-up at the critical radius (11), with both (12)and (13) still appropriate. When σ > 0 and ε > 0, our numerical results (not includedhere) show the behaviour is qualitatively similar to that outlined above for (2)-(5), withthe addition of kinetic undercooling acting to suppress the blow-up at s = sc. Further,regardless of how large rout is, the temperature on the interface at extinction is stillgiven by (14), while the interface still behaves as (15) in the extinction limit.

11

0.00 0.05 0.10 0.15 0.20 0.25 0.300

0.2

0.4

0.6

0.8

1

t

s(t) σ = 0.15

σ = 0.3

σ = 0.5σ = 1

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10−0.04

−0.03

−0.02

−0.01

0

s

dt

ds

σ = 0.15 σ = 0.3

σ = 0.5

σ = 1

Figure 5: The position of the moving front for when σ = 0.15, 0.3, 0.5 and 1 with ε = 0.1 and β = 1(top). The addition of nonzero kinetic undercooling has suppressed the singular behaviour, allowingthe front to reach the centre at te = 0.286 such that the entire particle is in the liquid phase. (Bottom)The asymptotic approximation (15) (dashed) and the numerically calculated position of the movingfront as t→ te. In the extinction limit, the speed of the front becomes unbounded.

12

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0−1

−0.5

0

0.5

1

r

u(r, t)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

−0.5

−0.75

−1

r

u(r, t)ue(r) uc(r)

Figure 6: Temperature profiles for σ = 0.15, ε = 0.001 and β = 1 at times t = 0.006, 0.032, 0.081,0.127 and te = 0.172 (top). The extinction profile ue(r) for ε > 0 appears to be a perturbation awayfrom the ε = 0 blow-up profile uc(r) from Figure 2 (dashed) when r = O(1). When r � 1 (bottom),the temperature deviates away from the blow-up profile as the moving front continues to the centre.

4.2. Small kinetic undercooling, ε� σ

The numerical simulations presented in Section 4.1 demonstrate that the inclusionof kinetic undercooling ε > 0 in the model prevents finite-time blow-up from occuringat a critical radius sc > 0, instead delaying the blow-up until complete melting at theextinction time. In this context it is of interest to consider the limit ε → 0. For thispurpose we show temperature profiles in Figure 6 which are computed for a very smallvalue of the kinetic undercooling parameter, ε = 0.001. For the temporal period inwhich s > sc (which for this figure is sc = 0.13), the solution profiles for ε� 1 appearto be a small perturbation of those for ε = 0. On the other hand, for s < sc, thetemperature profiles for ε � 1 appear to be made up of two parts: an almost flatcomponent in which u ≈ −β for s < r < sc, and a component which closely mimics

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0.00 0.05 0.10 0.150

0.2

0.4

0.6

0.8

1

t

s(t)

σ = 1σ = 0.5

σ = 0.3σ = 0.15

Figure 7: The position of the moving front for σ = 0.15, 0.3, 0.5 and 1, when ε = 0.001 and β = 1(solid). For the ε = 0 cases (dashed), the speed of the front becomes unbounded at some critical time,corresponding to a critical radius, and the solution cannot be continued past this critical time. Theposition of the moving front for the ε > 0 case follows the ε = 0 case very closely up until just beforethe critical time. Then, the speed of the front rapidly increases as the front moves past the criticalradius towards the centre of the sphere.

the ε = 0 solution for r > sc. This limiting behaviour is illustrated further in Figure 6(bottom), which shows a close-up of the extinction profile for ε = 0.001. Also included,as a dashed line, is the blow-up profile for ε = 0, which is valid for r > sc.

In Figure 7 we show the dependence of the interface position on time for differentsurface tension parameters, keeping the kinetic undercooling fixed to be the very smallvalue ε = 0.001. Again, we see that for t < tc, the trajectory of the interface is a smallperturbation of the ε = 0 solution. Then, for a very short period after t = tc, the speedof the moving front increases very rapidly, the interface moves to the centre so thatcomplete melting is achieved, and ds/dt → −∞ as s → 0+. The limiting behaviour isaccording to (15).

4.3. Small surface tension, σ � ε

The problem (2)–(5) with kinetic undercooling (ε > 0) but no surface tension (σ = 0)has been studied by McCue et al. [61] in the context of modelling solvent penetrationinto a glassy polymer, with applications to drug delivery devices. In that case u rep-resents the concentration of the solvent, and the interface r = s(t) separates the innerglassy core of the polymer ball from the outer rubbery region. Similar models in onedimension have been analysed [2, 15, 31, 47, 66], while in the context of melting and/orfreezing, Stefan problems with kinetic undercooling in one dimension have been treatedby a number of authors [11, 12, 23, 29, 30, 50] The kinetic undercooling condition onthe interface has also been applied in the Hele-Shaw context [10, 20, 70]; here the ap-propriate model is derived from our Stefan problem in the limit of large Stefan numberβ →∞. Identical govering equations approximate the initial streamer phase of electric

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0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

r

ue(r)

σ = 0.001

σ = 0.1

σ = 0.05

σ = 0.01

Figure 8: Extinction profiles for ε = 0.1 and β = 1, and for σ = 0 (dashed), 0.001, 0.01, 0.05 and 0.1(solid). For small σ the extinction profile is a perturbation away from the ε = 0 case, with a boundarylayer in space rapidly reducing the temperature on the moving front to ue(0) = −β.

breakdown in simple gasses, during which time a weakly ionized region propagates intosome non-ionized region due to a strong applied electric field [26, 27, 81].

As discussed in McCue et al. [61], the problem (2)–(5) with ε > 0 and σ = 0 does notexhibit finite-time blow-up, and solutions are regular for t ≤ te. That is, the interfacespeed ds/dt remains finite up to and including the extinction time t = te. Indeed, thetemperature profiles are sufficiently well behaved that ∂u/∂r(0, te) = 0, which meansthe no-flux condition is satisfied at the centre of the ball at extinction so that thesolution (to the subsequent linear problem) can be continued past t = te. Note that forthe case ε > 0 and σ = 0, we have ue(0) = −ε ds/dt (c.f. (14), which is for σ > 0), whichimplies that ue(0) > 0. The dashed curve in Figure 8, denoting the extinction profilefor a representative value of ε > 0 with σ = 0, clearly demonstrates this property.

With this summary of known results for ε > 0 and σ = 0, it is clear that the fullproblem (2)-(5) is singular in the limit σ → 0, since for ε > 0 and σ > 0 we haveds/dt → −∞ as t → t−e and ue(0) = −β < 0. Indeed, for σ � ε, a boundary layerdevelops near extinction, as shown in Figure 8. Here the top solid curve shows theextinction profile for a very small value of σ. This curve appears very close to theextinction profile for σ = 0, except near r = 0, where it undergoes a rapid drop tothe value ue(0) = −β. The present numerical scheme used to solve this problem is notdesigned for this type of extreme boundary layer behaviour, and we are unable to fullyresolve the temperature for small r with such small surface tension values. Increasing σresults in a smooth transition to the regime in which neither surface tension nor kineticundercooling is dominant, as shown, for instance, in Figure 4.

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5. Discussion

The Gibbs–Thomson law (1) with ε∗ = 0 (or, equivalently, the dimensionless version(4) with ε = 0) has been used by physicists as a model for the observed size-dependentmelting temperature that occurs for melting nanoscaled particles [4, 80]. For example,the bulk melting temperature of gold is 1337K, whereas the melting temperature of aparticle of gold with a radius of 20nm is approximately 1000K [4].

From a mathematical perspective, the use of (4) with ε = 0 has the unexpectedeffect of introducing a singularity in the solution at a finite radius, which occurs ata finite time before complete melting can take place. This blow-up is unexpected inthe sense that surface tension acts to penalise regions of high curvature, and normallyhas the effect of smoothing solutions that may otherwise be singular. From a physicalperspective, while the blow-up for σ > 0 cannot occur in reality (as it is characterisedby an solid-melt interface whose speed becomes unbounded in the blow-up limit), itmay be interpreted as modelling the onset of abrupt melting [51, 53, 62, 69, 75], knownto occur in experiments of melting nanoscaled particles like tin or lead.

In this broader context, the meaning of our results is two-fold. First, the inclusionof kinetic undercooling acts to regularise the unphysical singularity that is caused byintroducing surface tension into the continuum model. This type of regularisation isperhaps reminiscent of that described by King and Evans [50], although their workin one Cartesian coordinate (for which surface tension does not apply) would bettercorrespond to the inner solid phase that we ignore. More generally, kinetic undercoolingis known to penalise high interface velocities, which explains why its inclusion suppressesthis sort of blow-up.

Second, physically we expect kinetic undercooling to be extremely small, and so themost meaningful regime is likely to be ε� 1. Thus, for the time period before blow-up,the solution with kinetic undercooling will behave much like the solution without it,with the dimensionless melting temperature approximately equal to σ(1 − 1/s). Onthe other hand, the inclusion of a small amount of kinetic undercooling in the model(4) allows the solution to continue past the blow-up time right through to extinction(complete melting). This appealing feature of the model is also consistent with thenotion of abrupt melting, with the interface speed naturally increasing in a smooth butrapid fashion as it approaches the critical radius r = sc. No matter how small kineticundercooling is, by taking its effects into account we see that as s→ 0 the leading ordermagnitude of the terms σ(1 − 1/s) and ε ds/dt in (4) balances, so the dimensionlessmelting temperature does not decrease below the finite value −β.

Of course, the validity of any continuum model must be questioned as the particleradius becomes extremely small. So while the inclusion of kinetic effects in the Gibbs–Thomson law provides sensible mathematical solutions that do not blow-up at a finiteparticle radius, have temperature profiles bounded below at the fixed value −β, andcontinue through until the particle is completely melted, we do not propose that phys-ically the results hold literally until extinction. Detailed arguments about how smallparticles can be before Stefan-type models cease to be valid are outlined recently by

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Font and Myers [34], for example. These authors cut off their results for gold particlesof radius 1nm.

The model treated in this paper is a one-phase Stefan problem derived under theassumption that the heat conduction occurs much more easily in the liquid phase thanthe solid phase (that is, the ratio of thermal conductivities κ is small). This assumptionwill not hold for melting nanoscaled metal particles, which have the property thatκ > 1. However, we note that our one-phase problem with ε = 0 exhibits blow-upbehaviour that is qualitatively similar to the full two-phase problem [60], and so theinsight provided by including kinetic effects in our one-phase problem is likely to beinstructive for the full two-phase model, even for κ > 1. The extent to which thisproves true can only be tested by treating the two-phase problem directly. This workis ongoing.

We close by noting that a further worthwhile extension to our study would be totrack the solid-melt interface for a genuine two- or three-dimensional inward meltingproblem. One-phase Stefan problems of this sort without surface tension or kinetic un-dercooling have been studied using formal asymptotics [57, 58] and numerically [13]. Ofparticular interest, we mention that the Hele-Shaw model with surface tension and ki-netic undercooling, which is the one-phase Stefan problem for melting a two-dimensionalparticle in the limit β →∞, has been shown recently to provide a rich bifurcation struc-ture [19]. The ultimate shape of the moving boundary is shown to be dependent on therelative strength of surface tension versus kinetic undercooling (and in some cases theprecise form of the initial condition). The question of whether these results carry overfor finite Stefan number β remains unclear.

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