Surface melting and crystal shape

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Surface melting and crystal shape

P. Nozières

To cite this version:

P. Nozières. Surface melting and crystal shape. Journal de Physique, 1989, 50 (18), pp.2541-2550.

�10.1051/jphys:0198900500180254100�. �jpa-00211080�

2541

Surface melting ^and crystal shape

P. Nozières

Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, ^France (Reçu ^{le 19} juin 1989, accepté ^{le 26} juin 1989)

Résumé.

Pour expliquer ^les expériences ^{récentes de} Heyraud et ^{al. sur} des cristallites de plomb, quelques spéculations ^{sont avancées} concernant l’effet de la fusion de surface sur la formation des facettes. Partant d’une facette non mouillée par le liquide, l’apparition de marches sur une surface vicinale peut induire le mouillage 2014 donc la fusion de surface. Le raccord de la facette aux parties

arrondies est alors anguleux. ^La taille de la facette et l’angle de raccordement dépendent ^{de la} température : ^diverses possibilités ^sont envisagées. Au-dessus de la fusion, ^{une « lentille »} liquide peut apparaître ^entre deux facettes sèches. Si on admet que la fusion de la facette n’est pas nucléée au contact de cette lentille, ^on _peut calculer le domaine de stabilité d’une telle structure, autorisant la surchauffe du cristal observée expérimentalement.

Abstract. 2014 In order to explain ^recent experiments ^of Heyraud et ^{al. on lead} crystallites, ^some speculations ^are put ^forward concerning the effect of surface melting ^on facet formation. Starting

from a nonwetted facet, the appearance of steps ^{on a} vicinal surface can induce wetting ^{2014 hence}

surface melting. ^The matching ^{of the facet to round} parts ^{is then} angular. Facet size and matching angle ^both depend ^on temperature : ^various possibilites ^are envisaged. ^Above melting, ^a liquid

« lens » may appear between ^two dry ^facets. Assuming that facet melting ^{is not nucleated at the}

contact with such a lens, ^one may calculate the stability ^{of such a} structure, thereby accounting ^for

the observed superheating ^{of the} crystal.

J. Phys. ^{France 50} (1989) ^{2541-2550 15 SEPTEMBRE} 1989,

Classification

Physics ^Abstracts

61.50J

68.20 - 68.45

Very recently, Heyraud ^{et al.} [1] ^{were able to} observe the equilibrium shape ^{of small lead} crystallites very close ^to the melting triple point. ^{These beaùtiful in situ} experiments provide ^a

harvest of unexpected ^results.

(i) ^The (111) facet is found to persist up ^to the triple point Tm. ^{While at low} temperatures the matching of the facet to the round parts ^is tangential, it becomes angular ^{some 20 K below} Tm. Moreover, the relative width of the facet increases as melting ^is approached.

(ii) ^{Close to} Tm, ^Rheed measurements provide unambiguous ^evidence of surface melting (the liquid layer is thick and it effectively ^{screens the beam from the} underlying substrate). ^It

is found that surface melting ^{occurs on the round} parts, ^{but not on} (111) facets. Put another way, the liquid ^{wets the round} part, ^{but not} the facet. Similar conclusions were reached for the (101) ^facet through shadowing techniques [2].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180254100

(iii) ^Non equilibrium octahedral crystals, made up of eight (111) ^facets separated by ^small

round parts, display ^clear superheating ^a very unusual feature in crystals. They may persist

several K above Tm.

The purpose of this brief note is to speculate ^{on a} qualitative explanation ^{of these} findings.

For simplicity, ^{we consider} only cylindrical crystals, characterized by their cross-section. The

analysis ^{thus does not} apply ^{to real} crystals, which have two curvatures. We believe that the main ideas are nonetheless the same : they ^{are more} evident in such a simple geometry.

1. A _survey of surface melting.

Consider a planar solid-gas interface, ^{and assume that a thickness f of} liquid is molten. If

f is large, the energy Ex) of this molten layer may be written ^as

S ⁼ 7sv " ’Y sL - Y LV is the gain in surface energy upon melting : ^{it is} positive ^{if the} liquid

wets the solid. The bulk energy is H = P L 111 L - 1£ S 1, ^{where 1£ is the free} enthalpy per unit

mass of each phase and p L the specific ^{mass of the} liquid. ^{H vanishes at the} melting temperature Tm, ^{and it is} positive ^below Tm (the ^{solid is more} stable than the liquid). ^{The last}

term is, ^{within a constant} absorbed in S, ^the change in Van der Waals attraction due to

melting : ^{the constant A is} positive if p s > p L, the most usual situation.

Assume that H and S are positive. ^{If we} ignore the Van der Waals term, E(l) ^is depicted by

the full curve of figure ^{1 : a} liquid layer of vanishing thickness appears ^at the surface. In such ^a case, the concept ^{of two distinct} interfaces is meaningless. ^{A detailed} description ^{of the}

interface profile ^is needed, ^{based on a} Landau-Ginzburg formulation. Such a calculation ^was

performed by Lipowski ^and Speth [3]. Basically, ^{it acts to round} off the cusp of

E (Q ) ^{at f} = 0, ^{over a} range qu d, the thickness of each interface. When H goes ^{to zero,} the minimum f * of E (f ) is controlled by ^the exponential tail of the Landau Ginzburg profile ^{- as}

a result f ^{* -} log ^H.

If we restore the van der Waals term, with A > 0, E(l) displays ^{a minimum at} f* = (2A/H)1/3. ^{If H is} large, f * ^{is of order d, and a detailed} description ^{of the} profile ^is again required. ^{Close to} melting, however, ^H is small and f ^* is large : ^a relatively ^thick liquid layer develops ^{- the} only ^{feature we} need for further discussion.

Fig. ^{1. - The extra} energy E(l)ofa liquid layer ^of which f (i) ^{Full curve :} neglecting ^{Van der Waals}

interaction and interface broadening. (ii) ^{Dotted curve : with a} finite interface width (iii) ^Dot-dash

curve : with Van der Waals interaction (A > ⁰ ).

2. Crystal shape ^{as an} equilibrium ^of steps.

Whether surface molten or not, ^a crystalline interface is characterized by ^{a net} surface energy

y (corresponding ^to the minimum of E (f )) ^{which is} anisotropic. ^The resulting equilibrium

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shape is then obtained through the Wulff construction. In the vicinity ^{of a} facet orientation

(taken ^{as the} origin ^of angles), it is convenient to describe the shape ^{as a} distribution of crystal

steps (height a), ^{with a} density n (x ) along ^{the x} crystal planes. ^The equilibrium shape ^then corresponds ^{to the} equilibrium ^of steps, subject ^{on the one hand to a} supercooling ^{force, on}

the other hand to their mutual interaction [4]. ^Let E (n ) be the surface energy per unit ^area

along ^the crystal plane (E ⁼ y /cos 0 ). ^{In the vicinity} of the facet orientation, ^we may expand

E

where yo is the facet surface energy, the energy of ^a single step, 0 /d2 ^the leading step interaction (whether statistical or elastic), ^{etc. Such an} expansion ^{makes sense as} long ^{as the} steps ^{are well} separated, ^{i.e. if} ne « 1, where e ^{is the} step width, ^{of order} ya2lB. ^When

n) > 1, E (n ) returns to the usual non singular angular dependence.

If the exchange with the vapour is easy, the supercooling ^{force F is fixed} by the vapour pressure : its work is the gain of energy upon freezing. ^Let 5pv ^{be the} departure from nominal

equilibrium ^at temperature ^{T : then} [4] :

If the crystal ^is isolated, ^{F becomes a} Lagrange multiplier ^that ensures mass conservation : it

can still be viewed as deriving ^{from a} fictitious 6pv.

In order to characterize step interactions, ^{we note that a} step ^{has a} local energy

It is thus subject ^{to a force}

which expresses the repulsion ^{of its} neighbours. Equilibrium is achieved if

It is easily shown that (1) is identical to the Wulff condition

where R is the radius of curvature of the interface, ^while

is the so-called ^« surface stiffness ^».

Plotting ^{E as a} function of n is of course equivalent ^to plotting ^{y as a} function of 0. The advantage ^{of such a} representation (known ^{as the «} {3-plot ») is that it allows ^a simple analysis ^{of the} equilibrium shape. ^The steps may be viewed ^as « particles ^{», with a}

« compressibility » E". A given orientation is locally stable if Ex 0. If regions ^{of the} E (n ) plot ^have negative curvature, ^a « phase separation » ^{ensues, obtained} by ^{the usual}

double tangent construction (Fig. 2a) : ⁰ jumps ^from 61 ^to 02, the interface displaying ^an

angular point. ^{Such a} phase separation may also ^occur with the cusp ^{at n = 0 =} 0, ^{as shown in}

Fig. ^2.

Formation of angular points ^{at the} interface ; (a) ^{between two} rough regions ; (b) ^{between a}

facet and around part.

figure ^{2b : the flat facet at 0 =} 0 then goes into ^a round part ^{with a} finite orientation

0, : ^{1 the} matching ^{at facet} edge ^is angular (Fig. 3), ^{while it is} tangential in the usual case in which Ex 0.

The equilibrium facet width 2 L ^* is such that the energy be stationary ^{if a terrace is added}

or suppressed ^at the facet. For tangential matching, ^{one more terrace means a} gain ^{in bulk}

energy 2 L * F, ^at the expense of ^{two more} steps with energy 2 0, ^hence

The facet size reflects the step energy. If the matching ^is angular, a ^is replaced by ^the slope ^of

the double tangent /3 (which is the real energy of the ^new step, taking ^{account of} step

interactions) : the facet is thus smaller than it would be for tangential matching (fi ^{3 ).

Fig. ^3.

^A flat facet with angular matching ^{to its} surroundings.

The result (3), ^{obtained on} physical grounds, follows also formally ^from (2). The cusp in

y (0 ) implies ^a function contribution to y" (6) equal ^to & (0 ) 2 a la. ^{The Wulff} equation

may thus be written ^as

Integrating ^{across the} singularity immediately yields the width Ax ⁼ 2 L ^* of the facet

hence (3). ^{If the} matching ^is angular, ^{the real} y (6) follows the double tangent up ^to

0, : /3 ^is replaced by e -

In conclusion, ^the equilibrium ^{width 2 L *} of the facet and its matching angle 6c ^{to the round} parts provide direct information of E(n) - ^{hence on} y (0).

3. Steps ^{vs. surface} melting.

We compare two extreme situations : (i) ^{no surface} melting ^{at all, with a} surface energy

YSy(6), ^and (ii) ^a sizeable surface liquid layer (which implies T very ^{close to} Tm), ^{with a net}

surface energy YSLV(o). ^{If the} liquid ^{does not wet the facet, we have}

2545

Fig. ^4.

Matching ^{of a non} wetting facet with a wetting ^round part.

We now argue that the step energy is much larger ^{at a} solid vapour interface than it would be for a solid-liquid interface : 13 sv > {3 SLV. As a result S (0 ) ^increases rapidly ^{with 0,} and it may well become positive ^{for a} reasonably ^small 6m: ^{the curves} Ysv(6) ^and YSLV(O) ^{cross. Put}

another way, starting ^{from a non} wetting ^{facet, the} larger step energy {3 sv may induce wetting

for a vicinal surface 8 > 6 m.

Before exploring its consequences, ^we discuss briefly ^{the reasons for our} statement. A solid vapour interface is abrupt, ^{and the} step ^{is an atomic} entity, ^{with a width - a}

^hence /3 - ^{ya. In} contrast, ^{a solid} liquid interface is progressive : ^{the solid} periodicity ^« penetrates ^» into the liquid, ^{over a distance d -} may be ^a few atomic spacings. ^A step ^is simply ^{a shift of}

this transition region by ^{an amount a : its} width e ^{in the} x-plane ^will accordingly ^be large

and the step energy 8 - ’}’ a 2/ g will be small. The argument ^{is valid as} long ^as the interface is free to extend into the liquid, i.e. if the liquid ^{width f} is such that f » d - a condition which is met close enough ^to Tm (1).

If this situation holds, ^the E (n ) ^{curve behaves as} depicted ⁱⁿ figure 4. The transition at n =,n. is actually ^{rounded off, since at that} point ^the width of the liquid layer is very small

(Sm ⁼ 0) : ^the (SV) ^and (SLV) configurations ^are hardly distinguishable, (SL) steps ^{can no} longer expand ^{into the} liquid ^{and a} microscopic ^treatment of the transition region is needed.

However, ^{such a} rounding ^{does not affect an} obvious consequence : the global ^E (n ) ^{curve has}

a downward convexity, ^{and a double} tangent may be drawn from the n ⁼ 0 cusp. The

matching of ^{a non} wetting facet ^{with a} wetting ^round part ^is necessarily angular. If the facet wets (So > 0), ^{the whole} Esv ^{curve is above} EsLv : ^the matching will then be usually tangential (although they may exist double hump EsLv ^curves yielding ^{a finite} 6e).

In practice, experiment provides only ^relative crystal shapes, since F is usually ^unknown.

Using ^{the facet} edge ^{as a} reference, ^we may infer from the round parts (n > ne) :

in which we have set

(1) ^One may ^even envisage ^a situation in which f3SLv ⁼ 0, ^{i.e. the} solid-liquid interface is above its

roughening transition, while the solid vapour interface is below. A proper description ^{of such a situation} implies ^{a detailed treatment} of thermal fluctuations, which control both the roughening transition and the wetting properties

^see for instance [5].

-

Similarly the facet size L * gives ^{access to the} slope E’ (n ) ^{at n =} nc :

The shape ^of ESLv (n) for n:::- n, ^{is thus} determined, ^{within an} unknown additive constant.

The only ^other piece ^of information is the matching angle Oc, which would provide ^{access to} So ^{is we could} extrapolate ESLv below nc.

A naive approximation ^{is to assume an} isotropic ySL (which implies ^that steps ^{at the solid} liquid ^{interface are} completely ^washed out). Since the bulk and Van den Waals energies ^are independent of e, y sLV is ^{a constant} and

The round parts ^are circular with radius R (a ^{test of such a} simplification !), ^{and moreover}

The facet thus gives ^{access to} So ^{- but of course it} only ^{senses the} total ySLV (0 ).

In practice, ^the liquid solid interface is certainly anisotropic : ^a specific model is needed in order to extract So ^{out of} 6 c.

4. Température dependence ^{of facets.}

The facts to be explained ^{are the} growth ^{of L* and} 6 c ^as temperature approaches Tm, ^{in a} range of roughly 20 K below melting. The situation is here much less clear.

As a first try, ^one may ^assume that the shape ^of ysv (0 ) ^and YSLV( 6) ^{does not} change ^much

as temperature increases, ^the step energies 8 sv and 16 SLV being essentially ^{constant in such a}

narrow range of temperature. ^The only relevant evolution is that of So, which becomes

negative ^{at some} temperature Tc Tm (roughly Tc - ^{300 °C} compared ^to Tm ⁼ 327 °C), ^and

which keeps going ^{down as one} approaches Tm. ^Since dy ^{= - (T} dT, where a is the surface entropy, ^{such a behaviour} implies asv > (T SL v. Such a simple model does in fact account for all experimental facts : the more negative So, ^the larger ^the slope of the double tangent (hence L *) ^{and the} larger ^the matching angle Oc.

Unfortunately, ^the assumption ^{of a constant} {3SLV ^is questionable. ^{As the} temperature ^is lowered, ^the thickness f ^* of the liquid layer ^{narrows, and} {3SLV ^{should increase,} ultimately approaching ^the step energy Ssv of ^a straight solid vapour interface. We may thus try ^the opposite ^{view :} So ^is essentially ^{constant, but the} slope /3sLv grows rapidly ^when

T is small enough ^{for f * to be} comparable ^{to d} (the width of the liquid ^solid interface). ^{If this}

view holds, ^the conclusions are reversed : the slope ^{of the double} tangent ^{as well as} 6 c should increase when T is lowered. Since the whole behaviour of {3 SL v ( 6) ^is changed ^the

effect on crystal shape ^{is not} obvious : it seems likely ^{that a} larger a implies ^a larger ^relative

facet width.

Experiment pleads for the first choice - and indeed it is ^not so clear that a SLV ^should

increase as temperature is lowered. For a vicinal surface with step density ^n, the surface energy acquires ^a contribution n{3 ^which depends ^{on f} through {3 : ^{this extra f} dependence

must be included, together with bulk and Van der Waals energies, in the minimisation that

determines f *. Since {3 ^{decreases with} increasing f, the minimum will be shifted to higher

2547

£ *. If the step energy turns out to be larger than the Van der Waals term, ^it becomes the

leading ^{factor in} fixing f *, ^which basically ^{will be} d, just ^where /3 begins growing. ^{In this} picture, ^the step energy a SLV cannot grow much, ^since as soon as it does, ^it stops the decrease of f*. In the end the first picture may ^not be that stupid !

It is clear that these simple considerations are far too. naive. A proper ^treatment should account for thermal fluctuations of the two coupled interfaces : actually the condition asv > OSLV looks very queer ! Here, ^{we want} only ^to point ^the issue raised by experiment ^and

to suggest possible trends - the problem ^is definitely ^{not settled.}

5. Superheating of faceted crystals.

Consider ^a round part joining ^two facets that make an angle ^{2 a, fixed} by ^the geometry. ^{If the} facets do not grow, their width is arbitrary, ^{and the} global shape ^{is not the} equilibrium shape.

The round parts may be very small

typically ^a radius R - 50 A - corresponding ^{to a} large

effective supercooling of the solid vapour transition. We ^{want to} study ^the shape of this round part ^{as the} temperature goes beyond Tm.

We assume that the liquid ^{does not wet the facets,} whose surface energy is ysv(0) ^{= yo. On}

the other hand, ^{it does wet} the round parts. ^{In order to make the} algebra ^as simple ^as possible, ^{we make the} simplifying assumption ^{of an} isotropic ^solid liquid interface : y sL and _{y Lv} are taken as constant, with the condition

In practice, yLV is much larger than ysL (yLV ^{= 0.46} JIm2, ysL roughly ^{10 times} smaller).

The various phases ^{in contact are} characterized by ^{the same chemical} potential

g and temperature ^{T. For two} phases ^{1 and 2 we} may define the relevant supercooling ^{as the}

difference of pressure p(g, T) :

(The phase ^{1 is more} stable if _’P12 > 0.) We thus define _’PSL and _’PLV. The former global

« supercooling » force F is simply

Note that _{cp SL} becomes negative ^above melting.

Suppose ^{now for a moment that a finite} liquid ^meniscus develops ^{at the corner, as shown in} figure 5. Within our isotropic assumption, ^the liquid-vapour ^and solid-liquid interfaces are

circles, ^{with radii} respectively ^{R and R’, counted} positive ^{if the} convexity ^is outward, negative

if it is inward. These radii are related to the corresponding supercooling :

The ratio R/R’ is thus fixed by thermodynamics :

Three cases are then possible

(i) x > ^{1 : the SL interface} should extend beyond ^{the LV} interface, ^{which is} impossible : ^it

is consequently pressed upon upon it, forming ^{a thin} liquid film whose thickness is controlled

by other forces. This is the standard wetting situation, ^{in which}

(ii) ^{0 x 1 : then a} finite liquid meniscus appears, with 1/R’ 1/R, ^both positive.

Presumably, ^{the two surfaces} join ^on the facet (Fig. 5), ^{and the} matching angles 0 ^and

8 ’ must be such as to respect ^force equilibrium ^{at the} junction ^{A. Such a} stationary ^{state must}

also be stable, ^{in the sense that a small} displacement ^{of A must induce a} restoring ^force opposite ^{to the} displacement. ^Note that in this regime ^{we are} still below melting : ^{there is no}

desire to melt the facet.

Fig. ^5.

Possible formation of a liquid lens above Tm ^{between two} dry ^facets.

(iii) x - ^{0 : we enter the} superheating region. There may exist ^a range where ^a stationary,

stable lens of liquid exists between the two facets, ^with opposite convexities. Then the liquid

will not proceed further inside the solid. However, ^another question ^{remains : can this}

« liquid lens » nucleate melting along ^the facet ? Returning ^to the notations of section 1, ^H and S are now negative : E(l) ^{has a} potential barrier, ^{which must be overcome in order to}

melt the facet. This will not happen spontaneously ^{- but the} liquid may creep from the ^corner A along the facet. Whether this occurs or not is unclear ^{- an} unambiguous ^{answer would} require ^{a detailed treatment of the corner} region, ^{which is} extremely complicated. The very fact that superheating is observed indicates that the corners A do not nucleate facet melting :

we will take it as granted, although admittedly ^{this is a} very questionable point. Superheating

is then controlled only by ^the stability ^{of the} liquid ^lens.

The algebra ^{is now} straightforward. Let h be the distance HA in figure 5, ^{related to}

R and R’ by ^the geometrical relationships

The two angles 0 ^{and 0’} obey the dimensionless equation

The other equation ^{is the} balance of forces acting ^{on A} parallel ^{to the facet} (the only ^direction

in which A can move) :

(5) ^and (6) determine 0 and 0’ as a function of x.

2549

(i) ^{For x} > 1, ^{we are forced to 0 = 8’ -} 6 c’ given by

We recover the angular matching of section 3 ((7) ^is equivalent ^to the double tangent construction used previously).

(ii) ^{If x} 1, 9 and 0’ should be different : we can then write (6) ^as

In this form, it is clear that (8) ^{has no} solution if Oc ⁼ 0 : it is the angular matching ^{to the} facet

that makes possible ^the appearance o f a finite ^{meniscus at the corner. As x} decreases, ^{6 will be} smaller, while 0’ grows. In the usual situation yLv > ysL, ^we may expand (8) ^{as :}

(iii) ^The stability ^{of this} stationary ^state requires ^that

the derivatives being ^{taken at constant} supercooling, ^{i.e. at R and R’ constant.} (10) ^{reduces to}

Since 9 > a, the second term is destabilizing, while the first _one, due to the LV interface, ^is stabilizing. ^The stationary lens becomes unstable when

In the usual case yLV > ysL, (11) ^{means 0 = a +} -u /2, ^{i.e. a} LS interface which is a half circle. Beyond ^that threshold, ^the liquid invades the bulk. Such a conclusion is hardly surprising. (11) is in fact ^a refinement valid if _yLV and ysL ^are comparable.

We have thus shown that a liquid ^{meniscus can} persist ^above melting, up ^{to a} threshold

given by (11), ^{based on} geometrical arguments. ^{It remains to} translate this threshold into

physical ^variables

namely up ^to which temperature ^{can an} interface of radius R be stable ? It is convenient to use the triple point ^{as the} origin ; instead of (J..L, T) ^{we take as} independent

variables the liquid pressure correction 5 PL and 5 T. The Gibbs Duhem equation yields

Since _{cp Lv} and _(PSL are related to R and R’, ^{we can} obtain the temperature ^{8 T at which a}

given geometry (R, R’ ) ^{occurs :}

in which £ is the melting ^latent heat). ^{A finite} liquid ^meniscus appears when R ⁼ R’ ⁼ Rc,

hence at

For lead the last term is dominant : 8 Tl ^is negative. ^{For the} equilibrium crystals ^studied previously, ^{R was - 1 IL,} yielding 6 Tl - 0.04 K : the effect is not observable. For the non

equilibrium octahedral crystals, ^R is - 50 Â : 8 Tl ^{is much} larger.

In order to obtain the stability ^limit 8 T2, ^{we need to} calculate the radii R and R’ at threshold (the ^{two terms of} (13) ^{have the same} sign). Qualitatively, ^{R -} 50 A implies 8 T2 ’" ³⁰ K, ^much larger than the observed superheating - ^{3 K. The reason for this} discrepancy ^{is not} clear. It may well happen ^{that the} stability is limited by ^facet melting ^{at the}

contact A with the round parts, ^{which for some unknown reason} would have to overcome a

small potential barrier. The issue remains open.

Conclusion.

This note raises more questions ^{than it answers. The} only ^firm result is the statement that a

dry facet matches with a wet round part ^{at a} finite angle Oc. ^The temperature dependence ^of (J c and of facet width remains controversial. Experiments ^{seem to be} explained by ^{a model in}

which the facet would « surface melt » when the temperature ^{is lowered some 20° below} Tm. This is indeed a very queer assumption. (Usually, melting ^occurs upon heating !). ^Before exploring ^{more « realistic »} models, ^an experimental ^test of that feature would be desirable.

We have also shown that angular matching allows the appearance of ^a finite liquid ^{lens at}

the corner between two dry facets. In principle, ^{such a lens can} persist ^{in a finite} region ^above Tm, i f ^{the contact} with the facets does not initiate melting of the latter (which ^{much overcome}

a potential barrier). ^Here again, ^{such an} assumption looks queer

but we must face the

experimental ^{fact :} superheating is observed. The estimated superheating 8 T2 ^{is too} large :

the mechanism that limits metastability of the solid remains mysterious.

Acknowledgments.

The author wants to express his gratitude ^to J. J. Metois and J. C. Heyraud, who drew his attention to this problem. ^{Needless to} say, they ^{bear no} responsibility ^{in the} speculations put forward here.

References

[1] HEYRAUD J. C., METOIS J. J., ^{BERMOND J.} M., ^J. Cryst. ^Growth (in press).

[2] ^{PLUIS B., DENIER VAN DER GON A. W.,} FRENKEN J. W. N., ^{VAN DER VEEN} J. F., Phys. ^{Rev. Lett.}

59 (1987) ^2678.

[3] ^{LIPOWSKI R., SPETH W.,} Phys. ^{Rev. B 28} (1983) ^3983.

[4] ^{UWAHA M., NOZIÈRES P.,} Proceedings of the 1985 Oji ^{Seminar on} Morphology ^and Growth Unit of

Crystals, ^I. Sunagawa ^Ed. (Terra Scientific Publishing, Tokyo).

[5] ^{LIPOWSKI R.,} Phys. Rev. B 32 (1985).