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Adiabatic nucleation
425
Journal of Crystal Growth 74 (1986) 425—438 North-Holland, Amsterdam
ADIABATIC NUCLEATION Erich MEYER Instiluto de flsica da flnirersidade Federal do Rio de Janeiro, Bloco A, 4 Andar, C/dade Uniuersitdria, 21945
-
Rio de Janeiro, Brasil
Received 1 March 1985: manuscript received in final form 10 October 1985
A simple adiabatic nucleation model, based on an extended Mollier (enthalpy—entropy) diagram, predicts a definite stability limit for the liquid—solid phase transition and is in good agreement with the lowest known experimental supercooling temperatures of low viscosity liquids. In the case of liquid gallium this model also correctly predicts that, at maximum supercooling, the metastable phase Gafl will nucleate rather than the stable phase Ga,, or the metastable phase Gay.
1. Introduction Supercooling and supersaturation phenomena were well known already at the end of the 19th century. Ostwald [1], the creator of the term “metastahility”, postulated the existence of three ranges for every structural phase: stability, metastability and labile (unstable) states, basing himself on a large number of experimental observations. This means that, besides the coexistence curve, a stability limit was also postulated. In the case of the liquid—solid phase transition, the following interpretation was given: Above the equilibrium melting temperature the liquid is stable. Below, down to a certain temperature (stability limit), the liquid crystallizes only in the presence of a crystalline seed (heterogeneous nucleation). Below the stability limit, spontaneous crystallization (homogeneous nucleation) will occur after a certain time. With the later appearance of isothermal nucleation theories of Volmer and Weher [2], Becker and Doring [3], Frenkel [4] and Turnbull and Fisher [5] (see also refs. [6—14]),the interpretation changed. Only two qualitative ranges were postulated: stability and metastability, where in the latter range the nucleation frequency changed quantitatively, well below the melting temperature (in a relatively narrow temperature interval) from “almost never” to an “astronomic” number. This could again cx-
plain qualitatively the same experimental facts. Recently, Rasmussen [15,16] modified the classical nucleation theory, introducing a size dependent surface tension and obtained a stability limit. This limit, however, until now could not be evaluated numerically. The relation (T, T,,) —
K
=
1(1~ L~)was predicted for critical point systems, where 7~,T5 and 1~are the temperatures of the critical point, of homogeneous nucleation and of phase equilibrium, respectively. It could be shown further that K1 changed experimentally from 1.13 to 1.3 depending on the system. For solute systems the relation ~17~ K2LXT~~was shown experimentally, where ~T5 and LXTm are the variations of the homogeneous nucleation temperature and of the melting temperature with concentration, respectively. K2 could not be predicted theoretically and changed from system to system. According to the zeroth law of thermodynamics it is reasonable to admit that nucleation is an isothermal phenomenon, if this event is supposed to occur with the help of an equilibrium distribution of heterophase fluctuations. However, the existence of such heterophase fluctuations never has been proven experimentally. Further, one has to consider the experimental evidence that the overall crystallization process of a supercooled melt need not necessarily be (quasi)isothermal, but, in fact, can be very fast and adiabatic, producing eventually a real temperature “jump”, as shown by
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
=
426
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Adiabatii nucleation
Turnbull and Cech [17] (“...a sudden brightening ‘blick’ accompanies solidification ), Nelson [18]. Glicksman and Schaefer [191and Meyer and Rinderer [20]. Consequently the question arises whether homogeneous nucleation could not he a spontaneous adiabatic phenomenon, occurring at an until now unknown adiabatic stabtlity limit, in really pure liquids. The idea that nucleation could he an adiabaticisenthalpic phenomenon had first been suggested by Krestovnikov, Rozin and Vigdorovtch [21]. hut interfacial problems were neglected in this work. Later, this hypothesis was reconsidered by Cheynet, Eustathopoulos. Hicter and Desre [22]. These authors maintained heterophase fluctuations and came to the conclusion that nucleation should he (in the limiting case) an isentropic phenomenon. In both works, an additional hypothesis was necessary: that the nucleus should appear with a (higher) temperature, identical to the melting temperature. However. supercoolings considerably
greater. than can he expected by these two models. have been realized in low latent heat materials, e.g. phosphorus. In these cases the melt can he supercooled so much that the appearing crystal ts far below the melting temperature (Hildebrand and Rotariu [23] and Glicksman and Schaefer [19]). This effect was called hypercooling. The stability limit can he encountered, if one considers not only the usual representation of the first order phase transition in terms of the Gibbs potential of the two phases as a function of the temperature at constant pressure (see fig. 1), hut also its Legendre transformation, the enthalpy of the two phases as a function of the entropy at constant pressure, as shown in fig. 2 (Kestin [24] and Weinreich [25]). This representation is known (for the liquid—vapour transition and limited to the stable range) as the Mollier diagram. The stability limit of the liquid can then he indicated by the intersection of the enthalpy curve of the liquid phase at the external pressure with the
g g1
A
B
X
I
Fig. I - This figure shows quaIitativels~the Gibbs potentials g~ and g~ of the liquid and solid phase of a pure substance. I he equilibrium isothermal phase transition occurs at the melting temperature T. 0 - at point B. Nonei.~uilihriuniphase transitions of an isoihcrniically enclosed s~siemat the temperature T~nlav occur slowI~,directly from point A to point C. or rapidls. being of the type A B--C.
E. Mes’er
/ Adiabatic
enthalpy curve of the solid phase at a pressure corresponding to the external pressure increased by the capillary pressure of the nucleus. Below this stability limit, the system can evolve with increasing entropy at constant enthalpy and constant external pressure of the system, in agreement with the first and second law of thermodynamics. In order to prepare the discussion of the nucleation phenomenon, different types (limiting cases) of crystallization will be analysed first, with the help of the traditional (Gibbs potential) and the extended Mollier representation. Fig. 1 shows qualitatively the Gibbs potentials of the liquid. g1 and solid phase, g5 of a pure substance, at constant pressure p as a function of the temperature T. The equilibrium isothermal phase transition occurs at the melting temperature TM (point B), where the Gibbs potentials of the two phases are equal. A non-equilibrium phase transition of an
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isothermically enclosed system may occur slowly, quasi-isothermically at the temperature I~,directly from point A (g1(7~))to point C (g5(l~))in the case of a high viscosity, slowly crystallizing liquid. Or, such a phase transition can occur rapidly. in the case of a low viscosity, fast crystallizing liquid, where the whole system is heated up, eventually to the melting point B, “due to the liberated latent heat”, being afterwards cooled down again to point C. The Gibbs potential is used in fig. 1, because we know from thermodynamic that at constant pressure a system in contact with a heat bath at a temperature T5, will have a minimum Gibbs potential in the stable state. Fig. 1 is not useful for showing a nonequilibrium phase transition of an adiabatically enclosed system. Fig. 2 shows qualitatively the enthalpy of the liquid, h1 and of the solid, h5 as a function of the entropy s, at constant pressure. The melting tern-
h
~hL
6 L
h1
S Fig. 2. This figure shows qualitatively the enthalpies h1 and h~of the liquid and solid phase as a function of the enirop~.~h1 and are the latent heat and latent entropy. The points a. /3 and y represent the system in the solid and liquid single phase and in the solid—liquid phase equilibrium at the melting point, respectively. Nonequilibrium phase transitions of an isothermically enclosed. modestly supercooled system may occur slowly, directly from point ~ to point c. or rapidly, being of the type ~—y—~.
421
L. ‘Iei cc
..-l il/abutii ,iui/eati,oi
perature T\
1 is characterized by the common tangent of the two potentials and it is easy to see that the two tangential points a and /3 have the same Gibbs potential: =
h,
—
T~1 .v,
=
=
h1~
—
The latent heat ih1 and latent entropy is1 of the isothermal equilibrium phase transition are also indicated. The points a and /3 represent the system in the solid and liquid phase. respectively, at the melting point. The equilibrium isothermal phase transition can be represented by a point y “travelling” between a and /3’ where the “lever rule” would be valid, indicating that the (mass) quantity of the system in the solid state divided by the quantity in the liquid state would he equal to (h11 — Ii )/( h — h,) or (s6 — .v~)/( S., .v,). A non-equilibrium transition of an isothermically enclosed system can be represented by an arrow from point 6 (modestly supercooled liquid) to point c (stable solid), where =
((~(h/~s),,) = T~.
in the case of a slowly crystallizing liquid. Or. in the case of a rapidly crystallizitig liquid, such a phase transition can be shown, by the arrows from point 6 to point y (corresponding to internal entropy production) and from point y to point (corresponding to entropy/enthalpy removal by heat conduction). It is interesting to note, that in these transitions the total amount of the entropy of the system decreases. This is evidently compensated or overcompensated by the increase of the entropy of the surroundings. to where the latent heat was conducted. Fig. 3 shows the same potentials as fig. 2. The arrows 6 — y and K — A represent nonequilibrium transitions of a modestly and a strongly supercooled adiabatically enclosed system. respectivel~. These transitions occur irreversibly, with increasing entropy, constant enthalpy and at constant external pressure of the system. Details of this type of transition and justifications will he given below. These examples illustrate the well known fact, that phase transitions are not necessarily isothermal and suggest again the question: What about
the first step of a structural first order phase transition, nucleation? Is nucleation necessarily an isothermal phenomenon. as most authors helieve? Of course the group of atotlis or tTiOleculeS which is going to form the nucleus does not “feel”, whether the system is enclosed isotherniicallv (Sr adiabatically. Nucleation will not depend on these external conditions of the system. A simple adiabatic nucleation model is presented in order to respond to the questions above. Further discussions will then he given at the end of this article.
2. Basic thermodynamics If the only work considered is the mechanical work —pit’, then the first law of thermodynamics can be written in the following was’:
(I) where ui is the change of the ititernal energy. iq the beat quantit~given to the system. p the external pressure and it’ the change of vol uiiie of t lie closed systetil. It’ the system is isolated. i.e. adtahatically enclosed and of constant volume.
it’
=
0.
(2)
=
0.
(3)
then it follows from the first and second laws of thermodynamics (e.g. Guggenheim 1261 and Mandel [27]) that in
=
i.s
0,
(4)
I).
(5)
If the systetii is adiahatieallv enclosed and the external pressure is maintained constatit. eq. ( leads to itt +
J)
it-
=
i(
it + pr)
=
ih
=
0.
(6)
where Ii is the enthalpy of the system. (‘onsequently the conditions iq ip
=
=
0.
(7)
0.
lead with the firsl and second laws of thermodv-
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/ Adiabatic
narnics [26,27] to =0. is ~ 0.
(10)
From the experimental point of view eqs. (7)—(l0) are much more important than eqs. (2)—(5), because (with the possible exception of gases) it is much easier to maintain the external pressure of a system constant, than its volume. Eqs. (7)—(10) are valid for a single phase system, as well as for a multiphase system. One does not have to worry about capillary pressures occurring inside the enclosed phases nor about the corresponding interfacial tensions. This is because eqs. (7)—(10) have been deduced using the globally (for the whole system) valid energy conservation principle. For a single phase and for a reversible change, the variables occurring in eqs. (7)—(10) are related by dh
=
Tds
+
r’ dp.
(11)
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nucleation
Integrating this equation with p = const., individually for the liquid and solid phase of. e.g., one gram-atom substance each, relations would be ohtamed, as shown qualitatively in figs. 2 and 3. However, analytically this can be done easily only in the simple case, where the specific heat at constant pressure is constant, a condition which is approximately valid for solids and liquids, not too far from the melting point. That is why the liquid—solid phase transition is especially convenient for this study. Further, the specific heats of the liquid and solid phases are usually also very close to each other at the melting point, so that both can be substituted by =
(ci, +
~)/2
=
const.,
(12)
where c~,and c
1~, are the specific heats at constant pressure of the liquid and solid phase at the melting point and where c,, is the common mean and constant value. This approximation will simplify
h
S Fig. 3. The same potentials are shown as in fig. 2. The arrow S—y represents a nonequilibrium phase transition of an adiabatically enclosed system, from the modestly supercooled liquid phase to the stable solid—liquid phase equilibrium. The arrow —X shows an analogous transition from the highly supercooled liquid to the stable single phase solid.
430
L.
i/iii’s
all further considerations, hut of course also reduce the significance of the results in cases, where eq. (12) is a had approximation. From eqs. (11) and (12) fol lows:
(ah/a.v),,=T. (~ ~h/a.~2),, =
(~“h/a.v5),, n Ii
=
=
(13) (‘,,
‘( ~(/i/~).v
)
(~)/i/~)s),,.
(14)
1. 2 (15)
1 ili,i1~ ilis
si isi’/eiiti,oi
2 an.l 3 [24): Ii
~‘e.,T\l~exp[(.s’
s’,,)~.,]
--~
l~÷Ii,,. (17)
c 1,/~1
{ exp[(
~
~,:
--
I
f
(18)
Ii..
these equations confirm the qualitative he— haviour of the curves shown in figs. 2 and 3. Up to here it was tint absolutely necessary to define the size of the phases. hut from now on it is helpful to consider all extensive functtotis (e.g. h,. Ii
.x.
c1,.
=
1( ) where h is an integration constant, which takes different values for the liquid and solid phase. The function satisfying eqs. (13) -(15) is evidently of expotiential form and can he written for the solid and liquid phase, considering the boundary conditions at the points a and /3 in figs.
etc.) as atomic functions, if not otherwise specified. This has the advantage that it is also always possible to consider the values per gram-atom. when all extetisive functions are ti~ultiplied by, Avogadro’s numher. The enthalpy of an atom of the homogeneous phase of a small solid sphere (possible nucleus) in the liquid wouild then he h,, =c,,T\1~exp[(s
~~s)/~j
‘--
i} +h,+ih,.. (19)
H
~hL
ASL
S I— ig. 4. ‘t he qualitative behaviour of the enthalp~per atom ( oi’ per grain—atom) of ihc I iqin ii /i1 . lid /i and f the sisl iii homogeneous phase of a nucleus of radius r. /i~, is shown as a function of the entrop’, per atom (or per grani—atoill). In addition the following is indicated: The latent enthalpv ~1h i and entropy ~ ~i of melting, the latent enthalp’, ~/i , and entropy i , of melting of the homogeneous phase of a nucleus of radius r and the stability limit for formation of such a nucleus. ~s,
E. Meyer
/ Adiabatic
where ihr is the enthalpy due to the capillary pressure 2a~/r, (20) ip where again a, is the interfacial tension, which is a function of the radius r. With this one obtains =
ihr
2arl~’s/1’,
=
(21)
where v 5 is the volume per atom of the solid, which is supposed to be incompressible, The enthalpies h1, h5 and h5, are qualitatively shown as a function of the entropy in fig. 4, where ihLf and iSLf are the latent heat and entropy of melting per atom of the homogeneous phase of a small sphere of radius r in the liquid. Both values, ihLr and iSL, are independent of the temperature (in this model). Considering points of equal temperature on all three curves in fig. 4, one finds (note that the three curves are in fact identical ones, which have just been displaced along the ~ and h axes)
(22)
ihLr=ihL~ihr,
which is a know relation (Defay et al. [91), and iSL,.
=
(23)
is1.
In fig. 4, ihLr and iSLr are indicated for 13ra special temperature whereEach the atom pointsof a, have a common tangent. theand liquid and of the homogeneous phase of a solid sphere of radius r has there the same Gibbs potential g,,,.
=
h,,,
—
TM,s,,,
=
g1~,= h~,— TM,s~,,
(24)
where the temperature is indicated by TM,. and is usually interpreted as the melting temperature of a solid sphere of radius r in the liquid (see below). Evidently it is true that TM,/TM
=
ih1,/ih1,
(25)
and eqs. (21), (22) and (25) lead then to 2a,vS/rihL, (26) I which is a known relation [8,11]. The radius r in eq. (26) is called the critical radius. This is because an atom of the homogeneous phase of a sphere of a smaller or larger radius at the same temperature TM,/TM
=
—
would have a higher or lower Gibbs potential than
nucleation
431
has an atom of the liquid phase, both situations being shown out ofthat equilibrium (consider fig. 4). has been the equilibrium at TMr is a Itlabile one and that smaller or larger spheres have the tendency to disappear or to grow, respectively, in a supposed strictly isothermal process [11]. Consequently, TM, can be defined as the labile equilibrium melting temperature for supposed isothermal melting or solidification processes. However, such solidification processes are only expected for the limiting case of an extremely slowly growing material of very high liquid viscosity. Common materials can grow directly, with increasing atomic entropy and increasing temperature of the solid sphere (and eventually of the immediate environment), without any restriction. In these cases TM, no longer corresponds to an equilibrium, either stable or labile, but corresponds to a non-equilibrium situation and loses the real sense of a melting temperature. So far we considered in eqs. (17)—(19) individually one atom of the solid and liquid phase and of the homogeneous solid phase of a small sphere, where the capillary pressure was included. But we did not study the interface itself. If now a system is investigated, which consists of a solid sphere in the liquid, eq. (11) has to be replaced [91by dH=TdS+ V~dp 1+ V5dp5+~dA, (27) where H and S are the total enthalpy and entropy of the system, V 1 and 1<, are the total volumes of
the liquid and solid phase, p and p~ are the pressure of the liquid and solid phase and A is the interfacial area. Evidently, with eq. (27), a, can be defined as =
(8H/8A ) ~
I
(28)
and can be understood as a corrective enthalpy term per unit of Gibbs’ surface of tension, up to which phases considered be homogeneous both [9,28]. This are model is the tomathematical idealization of the real situation, where we have an inhomogeneous interfacial layer of a certain (unknown) thickness d. One gram atom of molecules of this interfacial layer will have a corresponding approximate interface area of n*v 1/d, where n’ is Avogadro’s number and v~is the mean volume per
432
,
E.3’lei’ci
atom in the interfacia! layer. The extra enthalpy contribution of this interfacial layer, in relation to the low enthalpy (solid) phase, is evidently due to the fact that the molecules of the layer are forced itito an intermediate enthalpic (atid entropic) position by the interaction with the molecules of the two neighbouring homogeneous phases. Consequently this extra enthalpy contribution of a gram-atomic (pure) interface may he written as n*aihj ,, where a is an unknowti value. With these arguments one obtains approximately =
1795
aih1,.
‘
which leads, if ad/t’~ can be considered as approximately independent of the radius, to =
i/i
1,./i h1,.
(30)
where a,.. is the interfacial tension of the flat interface. With eqs. (25). (26). (29) and (30). one oh tai ns (31) = (I + 2adt’5/t’~r) and substituting
8
=
37) —
(at’5/t’~)d.
one gets Tolman’s [281 equation
a,/a~=(l+26/r)
(33) I,
In Tolman’s theory. 6 was defined as the distance from the surface of tension to the equimolecular surface and was supposed to he constant, when the approximation. eq. (33) was deduced. Evidently 8 = const. leads together with eq. (32) to ad/i’, = const.. if the solid is considered to be incompressible, and the latter approximation was used in the present work for deducing eq. (30). Substituting eq. (32) in eq. (29) one obtains (34) a, = 8i 1z~,/t’~ (35) a,,.
=6ih1/i’5.
To the knowledge of the author, there exist no calculations of 6 for the solid—liquid interface. All one knows from Tolman’s [281 and Otio and Kondo’s [29] estimates is that 26 should be approximately equivalent to the intermolecular distance and that 8 should be. at the largest, of the
.‘ts/ia/,atis
,iui’/eatiosi
order of magnitude of the thickness d of the interface layer. respectivel~.
3. A simple adiabatic nucleation model Fig. 4 shows, that there are two principal ranges of the liquid phase. h~~ h1~and h1 < h4, where the liquid is stable and metastable respectively. The questioti remains then, whether there is also a stahility limit, where the ~iquid hecomes unstable. For this we may first consider a small solid sphere at the temperature T\15.. Let us suppose that the system is a pure liquid at point /3,. Admit then that a favourable statistical (temperature) fluctuation brings so many atoms or molecules, as cotitained in a sphere of radius i’. to point y, (see fig. 5). Then this group of atoms cati immediately (sirnultaneouisly) evolve to point 3,~_~y, is an ordinary statistical fluctuation a,. Stepwhen / which, of the order of a rms (root mean square) fluctuation is highly probable to occur. Step y,—-a, is an ordering process (creation of a crystal nucleus) with increasing entropy. This seems to he paradoxal. as au increase of entropy is always connected with an increase of disorder. However, as the newly created crystal nucleus appears at a higher temperature (and at a higher pressure), the total disorder (configurational and thermic) will he higher. so that there is no cotitradiction. The tiew crystal at point a,. will cotitinue to grow, absorbing atoms (or tilolecutles) of higher atomic enthalpy, corresponding to point /3,. the system increasing its entropy, until reaching point c (if the system is adiabatically enclosed) where the system is in the (single) solid phase (see arrow a, c iti fig. 5). While a fluctuation process of the type /3,. ‘y,. is more probable the less atoms participate. the process y,.--a,. is only possible for a minimum of molecules, which are tiecessary to define the structure of the solid phase. This number is the number one needs to define the Wigner—Seitz primitive cell [301.13 molecules in the case of a fcc or hcp structure or 15 molecules in the case of a bce structure. Compared with Tolman’s parameter 26. which we suppose now to he identical with the intermolecular distance, such a Wigner—Seitz
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433
A diahazic nucleanon
h
h sr L
S
Fig. 5. This figure is basically identical to fig. 4. In addition arrow /3,— -y,. shows a favourable (temperature) fluctuation process for the simplified case described by eq. (37). Arrow’ y,—a, represents the spontaneous adiabatic nucleation event at constant atomic enthalpy and increasing atomic entropy and arrow a,.—, represents qualitatively the growth of the nucleus and the adiabatic solidification of the system.
primitive cell determining group of molecules (only approximately spherical) has a radius of
only in isothermal and not in adiabatic processes, as discussed above. For adiabatic processes the only reference point is the limit p., p.~(see fig. 5). This point is the real stability limit, because only below this limit, processes of the type Yr’~, are possible with increasing entropy. Consequently one may assume, that when the supercooled liquid is at a temperature i—, from where the metastability limit temperature 7~can be attained by mean (rms) temperature fluctuations of radius r, iTrn~sr, that then the probability is high for the whole nucleation process to occur. This is an approximation, the real nucleation event may need a larger fluctuation (in order to extract a heat corresponding to the latent heat ihLr), occurring at 1/ at a reduced probability. The temperatures discussed above are related =
r
36.
~36)
This would give together with eq. (25), (30) and (33) (37) T~/TM TM~/TM 0.6, =
=
where T~ has the meaning of the temperature of maximum supercooling for a nucleation process occurring with a nucleus of a radius 36, in this simple model. The probability of the nucleation event depends of course still on the probability of fluctuations of the type /3,—’y,, as described above, being higher for smaller latent heats i h L,’ Eq. (37) is based on the labile equilibrium melting temperature for isothermal melting of a Wigner—Seitz primitive cell determining sphere, TM~. But this temperature is really of importance
by =
7;,-;
+
(38)
434
L. %leis’r
,,‘
A diahatic nucleation
where the last term is given by
i7;,,,5,/7;’
=K(iT/T,.
)2)t/2
one obtains [32] =
(k/ç~ )“
(39)
T’,1~,
3x
T51 (Landau and Lifshitz [31]), where k is Boltzmann’s constant and is the specific heat per number of fluctuating atoms (of the liquid phase, at constant pressure and at the temperature 7/) and consequently depends on the radius r of the fluctuation. Eq. (38) leads then together with eq. (39) to 7;
/T51
=
(7;;,/T5, )Fl
1 21 —
t
J
(k/ç~)
-
=
h13 — ih1. .~, —
i~i.
c~ T51 (exp [ ( S
—
s~+ is~ )/c1,
J
—
x(1
+
26/r)
(43)
(c”
‘
—
1)
(44)
“.
(e’ —1) (47)
1331
and together with eqs. (13). (18). (22). (30) and (33) one obtains =
JI
i c’,,QN~~ /I
(42)
I}
+h13—ih1+ih,.
I :1
(41)
(40)
in eq. (19). It follows =
—
k
where T~,> is the maximum supercooling temperature of liquids which are supercooled in relation to solid phases having a Wigner—Seitz number iV.,~..The value c’) in eq. (47) is the value at the temperature 1//~. However, as the specific heats of supercooled liquids are tiot expected to vary tsmuch with the temperature in the range of — O.STM—TM and as eq. (47) is not very sensitive to modest variations of c),. the value at the meltitig temperature can he used as well. This value again does not change much from element to element
The temperature of the liquid phase at the stability limit, 7/ can easily be calculated if one substitutes =
1’
(the gram-atomic values are typically slightly higher than Dulong and Petit’s value) so that c) finally can be substituted by 3k. With these approximations one obtaitis
T51
—
3A
—
~[1 5
—(3QN~..)
2]
I
(e’
—
I) (48)
and with the further approximation. lows the simple result [34]
14, fol-
‘V~ =
.v=iht/cf,T\l=ih~/c~TM,
T14/Tsj=x(1.67—0.26VQ)
where iht and c,~are the latent heat and specific heat per gram-atom. Substituting then eq. (44) in eq. (40) one obtains
where Q is the number of atoms per molecule of the solid phase and
7;
—
-
[( 1—
)
k
j
1,2]
I (
~ 1+~~ ri I
(e’—l)
.
.
(45) Supposing further that the nucleus amid the fluctuation leading to the nucleation contains so many molecules N~, as are necessary to define the Wigner—Seitz primitive cell, it follows that =
~
(46)
Q is the number of atoms per molecule of the solid phase. Together with eqs. (36) and (45),
where
(e’
—
I)
‘.
(49)
.v=ih~/c’j,Ty1 =iht/c’~T~1. both variables being dimensionless. Eq. (49) isshown in fig.6forQ=l,2,4and the limiting value ~. which in fact represents the stability limit 7;,~4/T51.The full squares are experimentally observed maximum supercoolings. where nucleation occurred. The open ones show attained supercoolings of liquid Ga in relation to the solid phases Ga,, and Gay, where instead of these phases Ga15 nucleated. Near the experimental values of P (Hildebrand and Rotariu [23]). Sn (Takagi [35]). Ga (Bosio. Defrain and Epelboin [36], see also ref. [37]), Hg (Zhdanov and Vertsner [38] and Perepezko and Paik [39,40]), Pb (Stowell
E. Meyer
/
435
Adiabatic nucleation
[101) and Bi (Perepezko and Paik [39.40]), the corresponding curves are fully drawn. TI;
4. Discussion The good general agreement of the lowest known experimental supercooling temperatures with eq. (49) (see fig. 6) confirms the present adiabatic nucleation model, especially if one takes into account that in the future possibly lower experimental values will be attained for Sn and Bi. Unfortunately, we have no way of knowing whether a nucleation event was due to homogeneous or heterogeneous nucleation. The present nucleation model shows, that nuclei are very small and that a single impurity atom or molecule eventually could be responsible for heterogeneous nucleation. Of course these arguments are not only applicable to ‘Sn and Bi, but also to all other elements. The lower value of Hg (Zhdanov and Vertsner [38]) shows that the evolution could go in the future to slightly lower values than predicted by eq. (49). However this lower value for Hg could only be shown for extremely small samples (< 50 A). The most interesting case is Ga, where our model predicts correctly. that the metastable phase Ga~should nucleate first, if one takes into account that Ga,, consists of Ga2 molecules (Laves [41]). This agreement contrasts with the predictions of the empirical Ostwald [1] step rule, followingwhich the (least stable) ‘y-phase should nucleate first, as well as with the general conclusions of Singh and Holz [13], T//TM const. (where T is the calculated lowest supercooling temperature). which would predict that the a-phase should nucleate first. Unfortunately Ga is the only known example for high liquid supercooling in relation to 3 solid phases, so that definite conclusions can hardly be arrived at. It is possible to calculate the “gram-atomic” interfacial tension of the flat interface, using eqs. (32) and (35) = a,,v~n*/d= a iht = iht/2, (50) where a could be substituted approximately by 1/2 in the case of a flat interface, indicating that the atoms of the interface have at the average a
p
-.
p’4
TM
of o i
0.5
Fig. 6. The reduced maximum supercooling temperature Ti’~/TMis shown as a function of the dimensionless parameter ~h1 /c,,T,,1, eq. (49). where ~ and c1, are the (atomic or gram-atomic) latent heat of melting and the mean specific heat of the solid and liquid phase at the melting temperature TM. Q is the number of atoms per molecule of the solid phase. The full squares represent experimental maximum supercoolings. where the indicated solid phases nucleated and the open squares show attained supercoolings of liquid Ga in relation to the phases Ga,, and Gay where, instead of these phases. Ga15 nucleated. Near the experimental values, the corresponding curves are fully drawn.
mean enthalpy in relation to the solid and liquid phase. Unfortunately eq. (50) is useless for determining a,.~.because v and specially d are unknown, hut it is interesting to compare this equation with Turnbull’s [6] conclusion (51) for most materials and (52)
for H,O, Bi, Sb and Ge, where is the gramatomic interfacial tension, which was considered to be curvature independent and where the assumption was made that the interfacial layer is a monoatomic layer. However, this latter assumption was later modified and larger interfacial thicknesses were admitted (Spaepen and Turnbull 142] and Thompson and Spaepen [43]). The comparison of
436
L..iiem’cr
4diabaiii siucleatiosi
,‘
eqs. (50)-(52) is interesting, because Turnhull’s [61 values of the interfacial tetision have always heeti considered to he at east of the correct order of
magnitude (e.g., Nash amid Glicksman [44]). The present treatment of the nucleation problem may give the impression that the interfacial tension was used only indirectly for calculating the capillary pressure. etc.. but that the interfacial layer itself was neglected. This is riot so. because a temperature fluctuation, reaching Yr in fig. 5 may he considered to he surrounded by a sort of intermediate layer with mean atomic properties (mean
enthalpy. entropy and temperature) corresponditig to a point halfway between /3, and -y, (near p.,) in fig. 5. This intermediate layer can evolve (simultaneously with the transition y,—a,) at constant enthalpy and increasing entropy to a point halfway
on the tangent /3,—a,. creating the interfacial layer. We did not include this intermediate layer in the fluctuation term. eq. (39), for simplicity and also, because the thickness of this layer is unknown. Because of this. ç~ in eq. (46) is underestimated and iT.msr and 7/ are slightly overestimated. However, considering all other approximations, this fact is of little importamice. Eq. (49) can be approximated for large molecules (large Q) to T
14/T51= 7;514/T51=0.6x(e’— I)
(53 I, -
during the freezing process, so that the amorphous structuire freezes-in before there is a chatice for homogeneous nucleatioti. A similar relation (eqs. (55) and (56)) has been indicated for glasses by Zanotto [511. i.e. glasses which do tiot show homogeneous (voluime) nuclea— tuon duiring slow cooling (Si02. 1320,. Na ~ 2SiO2. K 2O—2SiO2, PhO--Si02, (‘aO --Al 2O~ 2SiO2. (‘aO—MgO- 2SiO2. Na20—Al20,--ôSiO, follow the relation > T14/T~.
and glasses which show (supposedly) homogeneous (volume) nucleation during slow cooling [14] (Li 2°~ 2S1O2, BaO—2SiO2. Na20 -2CaO--3SiO2, Na 70--SiO2 ) follow the relation <
T14~/T~.
7;
However. ~ is a function of the quenching speed and the inequality above can eventually he inverted in a quenching process. which explains. why the compounds of the second group can also he obtained as pure (non-crystalline) glasses.
The glass transition temperatures of amorphous alloys are mostly not well known. It is possible
that for these materials, during slow freezing. the ‘
following relation is valid (57) while during fast cooling one has
(54)
in all 27 cases where the necessary thermodynamic data are available, except polypropylene where and polyoxymethylene where is still disputed (0.41 ~ 7;/T51 ~ 0.58). and where the reviewers recommend a low value. = 0.42. compared with 7/4/T51 = 0.51 [501. (The number Q of a polymer evidently does not correspond to the total number of atoms of the macromolecule, but to the number of atoms of the monomer as indicated by the chemical formula.) Consequently, eq. (54) shows that pure amorphous
7;
polymers do not crystallize (by homogeneous
7;
=
of polymers (Gaur
>
nucleation), because
(56)
CaO ‘ Si02 being a limiting case (7;/T51 T1~/T~1). belonging to the second group.
and conferring this equation with revised values of glass transition temperatures et al. [45—49]). one sees
(55)
is attained before
7/4.
7;/T~1> T14/T~1
(58)
with Ty1 = (T1 + 7;)/2. where T1 and 7; are the liquidus and solidus temperatures, respectively. It is empirically well known that 7; imicreases in fast freezing processes. This would explaiti. why metallic glasses usually can only he produced by
fast quenching. For pure metals the relation Tg/T’,i:~’~l/4
(59)
has heen proposed [43]. If this relation is correct. then there is little hope of producing aniorphous pure metals by quenching from the melt. in agreemerit with experimental observatiotis. This hy-
E. Meyer
/ Adiabatic nucleation
pothesis is not in contradiction with the fact, that pure metals could be quenched from the vapour phase directly to thin amorphous films at low temperature (Buckel and Hilsch [52]), because in this way the temperature, where the liquid would crystallize, could be “bypassed” in the vapour phase.
437
model on glasses and for useful discussions. Valuable discussions with Professors Michel Bienfait, Hans J. Scheel and J. Leonardo M.D. de Souza are gratefully acknowledged. as well as the financial support of the institutions CNPq, FINEP and CEPG.
References
5. Conclusions An alternative and simpler explanation could be given for the nucleation of the solid in the liquid phase and a definite stability limit for supercooled liquids is predicted by the present nucleation model. The lowest known experimental supercooling temperatures of pure low viscosity liquids are in good agreement with the theoretical predictions, and experimentally determined glass transition temperatures of high viscosity liquids are shown to be higher than the stability limit. This explains why the latter do not crystallize (by homogeneous nucleation). These facts suggest that nucleation could be an adiabatic phenomenon, instead of isothermal, and could be preceded by common statistical (temperature) fluctuations, instead of heterophase fluctuations. These ideas should be tested in the future in the context of the vapour—liquid and vapour—solid transition, as well as in the case of nucleation in dilute solutions. Heterogeneous nucleation was not treated in this work. It is evident that in typically heterogeneous nucleation events, as nucleation from the vapor phase on a substrate or precipitation from solid solutions, the heterogeneous nucleus may contain less molecules than are necessary to determine the Wigner—Seitz primitive cell. This is, because some molecules of the substrate or the primary solid solution may contribute to determine the symmetry of the new phase by epitaxy, in analogy with isothermal heterogeneous nucleation models.
Acknowledgements
The author thanks Professor Edgar D. Zanotto for the communication, prior to publication, of the conclusions of the test of the present nucleation
[1] W. Ostwald, Z. Physik, Chem. 22 (1897) 289. 121 M. Volmer and A. Weber. Z. Physik. (‘hem.
119 (1926)
277.
131 R. Becker and W. Doring. Ann. Physik 24 (1935) 719. 141 J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1946).
151 D. Turnbull and J.C. Fisher. J. Chem. Phys. 17 (1949) 71. [6] D. Turnbull, J. AppI. Phys. 21 (1950) 1022. 171 D. Turnbull, J. Chem. Phys. 20(1952)411. [8] J.W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, Oxford. 1965). [91 R. Delay, I. Prigogine and A. Bellemans, Surface Tension and Adsorption (transl. D.H. Everett)(Longmans, London,
1966).
1101 M. Stowell, Phil. Mag. 22 (1970) 1. 1111 D.P. Woodruff, The Solid—Liquid Interface (Cambridge University Press, London, 1973). [12] V.P. Skripov. in: Crystal Growth and Materials. Eds. F. Kaldis and H.J. Scheel (North-Holland, Amsterdam, 1976) p. 328. [131 H.B. Singh and A. Holz. Solid State Comm. 45 (1983) 985. [141 P.F. James, J. Non-Crystalline Solids, in press. [151 D.H. Rasmussen, J. Crystal Growth 56 (1982) 45. [161 Dl-!. Rasmussen, J. Crystal Growth 56 (1982) 56. [17] D. Turnbull and RE. Cech, J. Appl. Phys. 21(1950) 804. 118] L.S. Nelson, Nature 207 (1965) 741: 210 (1966) 410. [19] M.E. Glicksman and R.J. Schaefer. J. Crystal Growth I
(1967) 297. [20] E. Meyer and L. Rinderer, J. Crystal Growth 28 (1975) 199. 1211 A.N. Krestovnikov, K.M. Rozin and V.N. Vigdorovich, in: Surface Phenomena in Metallurgical Processes, Ed. Al. Belyaev (Consultants Bureau, New York, 1965) p. 45. [221 M.C. Cheynet, N. Eustathopoulos, P. Hicter and P. Desre, J. Chim. Physique 77 (1980) 919. [231 J.H. Hildebrand and G.J. Rotariu, J. Am. Chem. Soc. 73
(1951) 2524. [24] J. Kestin, A Course in Thermodynamics (Blaisdell, Waltham, MA, 1966). [25] G. Weinreich, Fundamental Thermodynamics (AddisonWesley, Reading, MA 1968) p. 167. 1261 E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, (1959). [27] F. Mandel, Statistical Physics (Wiley, New York. 1977). [281R.C. Tolman, J. Chem, Phys. 17 (1949) 333.
431
I’.. ‘s/es’s.,’’ .1 s/iisl,aii,’ inn /,‘siin’fri
12’)] S. Ono arid S. Kortdo, U,: Eue~’cIopediaot Physics. Vol. 10. Ed. S. Fl’Ogge (Springer, Berlin. 1960) p. 153. 3))] (.‘. Kiitel, Introduction to Solid State Physics (Wiles, Nes York. 1976). [Ii] I.,D. LandaLi and I-NI. Lif~hit,. Statistical Physics (Per— gaillon. London. 195S).
[41] I’. Eases, L. Krist. 14 (1933) 256, [42] F. Spaepen and 1). l’urnhull. ri: Proc. 2nd Intern. (ssi,). on Rapidly Quenched Metals, F.ds. N.J. (lrant and B.C. (iiessen (MIT Press. Cambridge. MA. 1971,) p. 21)5. [43] (V. l’honipssin and F. Spaepen Ac Ia Met. 31 (913) 2021.
132] F. Meyer. Helv. Phys Ac Li 55)1 912( I 61, [33] R. Hultgren. PD. Desai. I).]’. hawkins. NI. (ileiser. K. K. Kelles and 1)1,). Waginan. Selected Values ol the I hernio—
I~4IG.E.
[34] [35] [36] [37]
dynamic Properties of the Elements I Am. Soc. ( or Metal Metals Park, OF!, 1973). II. Meyer. in: Ahsiracts 7th Intern. (‘on). Oil (‘rvst:il Growth 1(10—7). Stuttgari. 1913. p. 2.1)1. NI. Takagi, .J. Phys. Soc. Japan 9 11954) 359. I.. Bosisi. A. Defrain and I. Epelhoni .1. I’hvsique 27 (l966( 61. 1.. Bosio, R. (‘ortes, A. Defrain and ( i . Folcher. .1. (‘hii’i’i. PhssiqLie 70 (1973( 357.
[31] (1.5. Zhdanos and V. liVe rtsner. 51,5 let Phvs.—Sol id State 1(1966)117. [39] .I.H. l’erepei.kss and J.S. Paik. in: Rapidly Solidified Amorphous and (‘rs’stalline Alloys, Eds. B.h1. Rear, B.( Giessen and NI. (‘ohen ( NortIi— Holland. Amsterdam, 1 912) p. 49. [40].1. H. Perepeiko and IS. Pa i k . .J . Non—Crystal line Solids 61,62(1914)113.
Nash and MI’.. (ilicksnian, Phil. Mag. 24)1971)577. [45[ U. Gaur. I I.—(’. Shu. ‘\. Metha and It. Wsinderlicli. .1. Phvs. hen~.Ref. I.)at~i10(1911) 19. [46] t . Gaiir and B. Wunderlich,. .1. Ohs’s. (‘liens. Re(. Data I)) (1911) II’). lUll. 11)51: II (1912) 313. [47] t H (jaur, RB. Wunderhicli and B. Wuiiderlich. .1. I’hys. ‘lien~.Ref. Data 12 (1983) 29. [48] 1. . G:iur. 5.1’. I,au, 13.13. Wunderlichi ,ind It. Wunderlichi. .1. F’hys.Cherri. Ret’. Data 12 (1913) 65. [49] t . Gaur. SI’. l.:iu and It. Wunderlich i. Phvs. (‘hem. Re)’. Data 12 (1913) 91. [50] F. Meyer. in: .An,iis do I’rimeirs, Simposis l,atino— Americans, she F ,sie:i dos Sistenias “s nis)rfs)s, Lii. I . .5 i~da ((‘eniro l.:ilinss—Anieric:insi de l”isi,’a, Rio sIc _Eaneirs,. 1915) Vii. 2. p. 465. [SI] 1:1), Zanotto. .1. Non—Crystalline Solids. ssihnuitted. [52] W. Buckel and R. Hilsch. 7. Physik 131 (1954)1(19.
Journal of Crystal Growth 74 (1986) 425—438 North-Holland, Amsterdam
ADIABATIC NUCLEATION Erich MEYER Instiluto de flsica da flnirersidade Federal do Rio de Janeiro, Bloco A, 4 Andar, C/dade Uniuersitdria, 21945
-
Rio de Janeiro, Brasil
Received 1 March 1985: manuscript received in final form 10 October 1985
A simple adiabatic nucleation model, based on an extended Mollier (enthalpy—entropy) diagram, predicts a definite stability limit for the liquid—solid phase transition and is in good agreement with the lowest known experimental supercooling temperatures of low viscosity liquids. In the case of liquid gallium this model also correctly predicts that, at maximum supercooling, the metastable phase Gafl will nucleate rather than the stable phase Ga,, or the metastable phase Gay.
1. Introduction Supercooling and supersaturation phenomena were well known already at the end of the 19th century. Ostwald [1], the creator of the term “metastahility”, postulated the existence of three ranges for every structural phase: stability, metastability and labile (unstable) states, basing himself on a large number of experimental observations. This means that, besides the coexistence curve, a stability limit was also postulated. In the case of the liquid—solid phase transition, the following interpretation was given: Above the equilibrium melting temperature the liquid is stable. Below, down to a certain temperature (stability limit), the liquid crystallizes only in the presence of a crystalline seed (heterogeneous nucleation). Below the stability limit, spontaneous crystallization (homogeneous nucleation) will occur after a certain time. With the later appearance of isothermal nucleation theories of Volmer and Weher [2], Becker and Doring [3], Frenkel [4] and Turnbull and Fisher [5] (see also refs. [6—14]),the interpretation changed. Only two qualitative ranges were postulated: stability and metastability, where in the latter range the nucleation frequency changed quantitatively, well below the melting temperature (in a relatively narrow temperature interval) from “almost never” to an “astronomic” number. This could again cx-
plain qualitatively the same experimental facts. Recently, Rasmussen [15,16] modified the classical nucleation theory, introducing a size dependent surface tension and obtained a stability limit. This limit, however, until now could not be evaluated numerically. The relation (T, T,,) —
K
=
1(1~ L~)was predicted for critical point systems, where 7~,T5 and 1~are the temperatures of the critical point, of homogeneous nucleation and of phase equilibrium, respectively. It could be shown further that K1 changed experimentally from 1.13 to 1.3 depending on the system. For solute systems the relation ~17~ K2LXT~~was shown experimentally, where ~T5 and LXTm are the variations of the homogeneous nucleation temperature and of the melting temperature with concentration, respectively. K2 could not be predicted theoretically and changed from system to system. According to the zeroth law of thermodynamics it is reasonable to admit that nucleation is an isothermal phenomenon, if this event is supposed to occur with the help of an equilibrium distribution of heterophase fluctuations. However, the existence of such heterophase fluctuations never has been proven experimentally. Further, one has to consider the experimental evidence that the overall crystallization process of a supercooled melt need not necessarily be (quasi)isothermal, but, in fact, can be very fast and adiabatic, producing eventually a real temperature “jump”, as shown by
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
=
426
L.Meier
Adiabatii nucleation
Turnbull and Cech [17] (“...a sudden brightening ‘blick’ accompanies solidification ), Nelson [18]. Glicksman and Schaefer [191and Meyer and Rinderer [20]. Consequently the question arises whether homogeneous nucleation could not he a spontaneous adiabatic phenomenon, occurring at an until now unknown adiabatic stabtlity limit, in really pure liquids. The idea that nucleation could he an adiabaticisenthalpic phenomenon had first been suggested by Krestovnikov, Rozin and Vigdorovtch [21]. hut interfacial problems were neglected in this work. Later, this hypothesis was reconsidered by Cheynet, Eustathopoulos. Hicter and Desre [22]. These authors maintained heterophase fluctuations and came to the conclusion that nucleation should he (in the limiting case) an isentropic phenomenon. In both works, an additional hypothesis was necessary: that the nucleus should appear with a (higher) temperature, identical to the melting temperature. However. supercoolings considerably
greater. than can he expected by these two models. have been realized in low latent heat materials, e.g. phosphorus. In these cases the melt can he supercooled so much that the appearing crystal ts far below the melting temperature (Hildebrand and Rotariu [23] and Glicksman and Schaefer [19]). This effect was called hypercooling. The stability limit can he encountered, if one considers not only the usual representation of the first order phase transition in terms of the Gibbs potential of the two phases as a function of the temperature at constant pressure (see fig. 1), hut also its Legendre transformation, the enthalpy of the two phases as a function of the entropy at constant pressure, as shown in fig. 2 (Kestin [24] and Weinreich [25]). This representation is known (for the liquid—vapour transition and limited to the stable range) as the Mollier diagram. The stability limit of the liquid can then he indicated by the intersection of the enthalpy curve of the liquid phase at the external pressure with the
g g1
A
B
X
I
Fig. I - This figure shows quaIitativels~the Gibbs potentials g~ and g~ of the liquid and solid phase of a pure substance. I he equilibrium isothermal phase transition occurs at the melting temperature T. 0 - at point B. Nonei.~uilihriuniphase transitions of an isoihcrniically enclosed s~siemat the temperature T~nlav occur slowI~,directly from point A to point C. or rapidls. being of the type A B--C.
E. Mes’er
/ Adiabatic
enthalpy curve of the solid phase at a pressure corresponding to the external pressure increased by the capillary pressure of the nucleus. Below this stability limit, the system can evolve with increasing entropy at constant enthalpy and constant external pressure of the system, in agreement with the first and second law of thermodynamics. In order to prepare the discussion of the nucleation phenomenon, different types (limiting cases) of crystallization will be analysed first, with the help of the traditional (Gibbs potential) and the extended Mollier representation. Fig. 1 shows qualitatively the Gibbs potentials of the liquid. g1 and solid phase, g5 of a pure substance, at constant pressure p as a function of the temperature T. The equilibrium isothermal phase transition occurs at the melting temperature TM (point B), where the Gibbs potentials of the two phases are equal. A non-equilibrium phase transition of an
427
nucleation
isothermically enclosed system may occur slowly, quasi-isothermically at the temperature I~,directly from point A (g1(7~))to point C (g5(l~))in the case of a high viscosity, slowly crystallizing liquid. Or, such a phase transition can occur rapidly. in the case of a low viscosity, fast crystallizing liquid, where the whole system is heated up, eventually to the melting point B, “due to the liberated latent heat”, being afterwards cooled down again to point C. The Gibbs potential is used in fig. 1, because we know from thermodynamic that at constant pressure a system in contact with a heat bath at a temperature T5, will have a minimum Gibbs potential in the stable state. Fig. 1 is not useful for showing a nonequilibrium phase transition of an adiabatically enclosed system. Fig. 2 shows qualitatively the enthalpy of the liquid, h1 and of the solid, h5 as a function of the entropy s, at constant pressure. The melting tern-
h
~hL
6 L
h1
S Fig. 2. This figure shows qualitatively the enthalpies h1 and h~of the liquid and solid phase as a function of the enirop~.~h1 and are the latent heat and latent entropy. The points a. /3 and y represent the system in the solid and liquid single phase and in the solid—liquid phase equilibrium at the melting point, respectively. Nonequilibrium phase transitions of an isothermically enclosed. modestly supercooled system may occur slowly, directly from point ~ to point c. or rapidly, being of the type ~—y—~.
421
L. ‘Iei cc
..-l il/abutii ,iui/eati,oi
perature T\
1 is characterized by the common tangent of the two potentials and it is easy to see that the two tangential points a and /3 have the same Gibbs potential: =
h,
—
T~1 .v,
=
=
h1~
—
The latent heat ih1 and latent entropy is1 of the isothermal equilibrium phase transition are also indicated. The points a and /3 represent the system in the solid and liquid phase. respectively, at the melting point. The equilibrium isothermal phase transition can be represented by a point y “travelling” between a and /3’ where the “lever rule” would be valid, indicating that the (mass) quantity of the system in the solid state divided by the quantity in the liquid state would he equal to (h11 — Ii )/( h — h,) or (s6 — .v~)/( S., .v,). A non-equilibrium transition of an isothermically enclosed system can be represented by an arrow from point 6 (modestly supercooled liquid) to point c (stable solid), where =
((~(h/~s),,) = T~.
in the case of a slowly crystallizing liquid. Or. in the case of a rapidly crystallizitig liquid, such a phase transition can be shown, by the arrows from point 6 to point y (corresponding to internal entropy production) and from point y to point (corresponding to entropy/enthalpy removal by heat conduction). It is interesting to note, that in these transitions the total amount of the entropy of the system decreases. This is evidently compensated or overcompensated by the increase of the entropy of the surroundings. to where the latent heat was conducted. Fig. 3 shows the same potentials as fig. 2. The arrows 6 — y and K — A represent nonequilibrium transitions of a modestly and a strongly supercooled adiabatically enclosed system. respectivel~. These transitions occur irreversibly, with increasing entropy, constant enthalpy and at constant external pressure of the system. Details of this type of transition and justifications will he given below. These examples illustrate the well known fact, that phase transitions are not necessarily isothermal and suggest again the question: What about
the first step of a structural first order phase transition, nucleation? Is nucleation necessarily an isothermal phenomenon. as most authors helieve? Of course the group of atotlis or tTiOleculeS which is going to form the nucleus does not “feel”, whether the system is enclosed isotherniicallv (Sr adiabatically. Nucleation will not depend on these external conditions of the system. A simple adiabatic nucleation model is presented in order to respond to the questions above. Further discussions will then he given at the end of this article.
2. Basic thermodynamics If the only work considered is the mechanical work —pit’, then the first law of thermodynamics can be written in the following was’:
(I) where ui is the change of the ititernal energy. iq the beat quantit~given to the system. p the external pressure and it’ the change of vol uiiie of t lie closed systetil. It’ the system is isolated. i.e. adtahatically enclosed and of constant volume.
it’
=
0.
(2)
=
0.
(3)
then it follows from the first and second laws of thermodynamics (e.g. Guggenheim 1261 and Mandel [27]) that in
=
i.s
0,
(4)
I).
(5)
If the systetii is adiahatieallv enclosed and the external pressure is maintained constatit. eq. ( leads to itt +
J)
it-
=
i(
it + pr)
=
ih
=
0.
(6)
where Ii is the enthalpy of the system. (‘onsequently the conditions iq ip
=
=
0.
(7)
0.
lead with the firsl and second laws of thermodv-
E. Meter
/ Adiabatic
narnics [26,27] to =0. is ~ 0.
(10)
From the experimental point of view eqs. (7)—(l0) are much more important than eqs. (2)—(5), because (with the possible exception of gases) it is much easier to maintain the external pressure of a system constant, than its volume. Eqs. (7)—(10) are valid for a single phase system, as well as for a multiphase system. One does not have to worry about capillary pressures occurring inside the enclosed phases nor about the corresponding interfacial tensions. This is because eqs. (7)—(10) have been deduced using the globally (for the whole system) valid energy conservation principle. For a single phase and for a reversible change, the variables occurring in eqs. (7)—(10) are related by dh
=
Tds
+
r’ dp.
(11)
429
nucleation
Integrating this equation with p = const., individually for the liquid and solid phase of. e.g., one gram-atom substance each, relations would be ohtamed, as shown qualitatively in figs. 2 and 3. However, analytically this can be done easily only in the simple case, where the specific heat at constant pressure is constant, a condition which is approximately valid for solids and liquids, not too far from the melting point. That is why the liquid—solid phase transition is especially convenient for this study. Further, the specific heats of the liquid and solid phases are usually also very close to each other at the melting point, so that both can be substituted by =
(ci, +
~)/2
=
const.,
(12)
where c~,and c
1~, are the specific heats at constant pressure of the liquid and solid phase at the melting point and where c,, is the common mean and constant value. This approximation will simplify
h
S Fig. 3. The same potentials are shown as in fig. 2. The arrow S—y represents a nonequilibrium phase transition of an adiabatically enclosed system, from the modestly supercooled liquid phase to the stable solid—liquid phase equilibrium. The arrow —X shows an analogous transition from the highly supercooled liquid to the stable single phase solid.
430
L.
i/iii’s
all further considerations, hut of course also reduce the significance of the results in cases, where eq. (12) is a had approximation. From eqs. (11) and (12) fol lows:
(ah/a.v),,=T. (~ ~h/a.~2),, =
(~“h/a.v5),, n Ii
=
=
(13) (‘,,
‘( ~(/i/~).v
)
(~)/i/~)s),,.
(14)
1. 2 (15)
1 ili,i1~ ilis
si isi’/eiiti,oi
2 an.l 3 [24): Ii
~‘e.,T\l~exp[(.s’
s’,,)~.,]
--~
l~÷Ii,,. (17)
c 1,/~1
{ exp[(
~
~,:
--
I
f
(18)
Ii..
these equations confirm the qualitative he— haviour of the curves shown in figs. 2 and 3. Up to here it was tint absolutely necessary to define the size of the phases. hut from now on it is helpful to consider all extensive functtotis (e.g. h,. Ii
.x.
c1,.
=
1( ) where h is an integration constant, which takes different values for the liquid and solid phase. The function satisfying eqs. (13) -(15) is evidently of expotiential form and can he written for the solid and liquid phase, considering the boundary conditions at the points a and /3 in figs.
etc.) as atomic functions, if not otherwise specified. This has the advantage that it is also always possible to consider the values per gram-atom. when all extetisive functions are ti~ultiplied by, Avogadro’s numher. The enthalpy of an atom of the homogeneous phase of a small solid sphere (possible nucleus) in the liquid wouild then he h,, =c,,T\1~exp[(s
~~s)/~j
‘--
i} +h,+ih,.. (19)
H
~hL
ASL
S I— ig. 4. ‘t he qualitative behaviour of the enthalp~per atom ( oi’ per grain—atom) of ihc I iqin ii /i1 . lid /i and f the sisl iii homogeneous phase of a nucleus of radius r. /i~, is shown as a function of the entrop’, per atom (or per grani—atoill). In addition the following is indicated: The latent enthalpv ~1h i and entropy ~ ~i of melting, the latent enthalp’, ~/i , and entropy i , of melting of the homogeneous phase of a nucleus of radius r and the stability limit for formation of such a nucleus. ~s,
E. Meyer
/ Adiabatic
where ihr is the enthalpy due to the capillary pressure 2a~/r, (20) ip where again a, is the interfacial tension, which is a function of the radius r. With this one obtains =
ihr
2arl~’s/1’,
=
(21)
where v 5 is the volume per atom of the solid, which is supposed to be incompressible, The enthalpies h1, h5 and h5, are qualitatively shown as a function of the entropy in fig. 4, where ihLf and iSLf are the latent heat and entropy of melting per atom of the homogeneous phase of a small sphere of radius r in the liquid. Both values, ihLr and iSL, are independent of the temperature (in this model). Considering points of equal temperature on all three curves in fig. 4, one finds (note that the three curves are in fact identical ones, which have just been displaced along the ~ and h axes)
(22)
ihLr=ihL~ihr,
which is a know relation (Defay et al. [91), and iSL,.
=
(23)
is1.
In fig. 4, ihLr and iSLr are indicated for 13ra special temperature whereEach the atom pointsof a, have a common tangent. theand liquid and of the homogeneous phase of a solid sphere of radius r has there the same Gibbs potential g,,,.
=
h,,,
—
TM,s,,,
=
g1~,= h~,— TM,s~,,
(24)
where the temperature is indicated by TM,. and is usually interpreted as the melting temperature of a solid sphere of radius r in the liquid (see below). Evidently it is true that TM,/TM
=
ih1,/ih1,
(25)
and eqs. (21), (22) and (25) lead then to 2a,vS/rihL, (26) I which is a known relation [8,11]. The radius r in eq. (26) is called the critical radius. This is because an atom of the homogeneous phase of a sphere of a smaller or larger radius at the same temperature TM,/TM
=
—
would have a higher or lower Gibbs potential than
nucleation
431
has an atom of the liquid phase, both situations being shown out ofthat equilibrium (consider fig. 4). has been the equilibrium at TMr is a Itlabile one and that smaller or larger spheres have the tendency to disappear or to grow, respectively, in a supposed strictly isothermal process [11]. Consequently, TM, can be defined as the labile equilibrium melting temperature for supposed isothermal melting or solidification processes. However, such solidification processes are only expected for the limiting case of an extremely slowly growing material of very high liquid viscosity. Common materials can grow directly, with increasing atomic entropy and increasing temperature of the solid sphere (and eventually of the immediate environment), without any restriction. In these cases TM, no longer corresponds to an equilibrium, either stable or labile, but corresponds to a non-equilibrium situation and loses the real sense of a melting temperature. So far we considered in eqs. (17)—(19) individually one atom of the solid and liquid phase and of the homogeneous solid phase of a small sphere, where the capillary pressure was included. But we did not study the interface itself. If now a system is investigated, which consists of a solid sphere in the liquid, eq. (11) has to be replaced [91by dH=TdS+ V~dp 1+ V5dp5+~dA, (27) where H and S are the total enthalpy and entropy of the system, V 1 and 1<, are the total volumes of
the liquid and solid phase, p and p~ are the pressure of the liquid and solid phase and A is the interfacial area. Evidently, with eq. (27), a, can be defined as =
(8H/8A ) ~
I
(28)
and can be understood as a corrective enthalpy term per unit of Gibbs’ surface of tension, up to which phases considered be homogeneous both [9,28]. This are model is the tomathematical idealization of the real situation, where we have an inhomogeneous interfacial layer of a certain (unknown) thickness d. One gram atom of molecules of this interfacial layer will have a corresponding approximate interface area of n*v 1/d, where n’ is Avogadro’s number and v~is the mean volume per
432
,
E.3’lei’ci
atom in the interfacia! layer. The extra enthalpy contribution of this interfacial layer, in relation to the low enthalpy (solid) phase, is evidently due to the fact that the molecules of the layer are forced itito an intermediate enthalpic (atid entropic) position by the interaction with the molecules of the two neighbouring homogeneous phases. Consequently this extra enthalpy contribution of a gram-atomic (pure) interface may he written as n*aihj ,, where a is an unknowti value. With these arguments one obtains approximately =
1795
aih1,.
‘
which leads, if ad/t’~ can be considered as approximately independent of the radius, to =
i/i
1,./i h1,.
(30)
where a,.. is the interfacial tension of the flat interface. With eqs. (25). (26). (29) and (30). one oh tai ns (31) = (I + 2adt’5/t’~r) and substituting
8
=
37) —
(at’5/t’~)d.
one gets Tolman’s [281 equation
a,/a~=(l+26/r)
(33) I,
In Tolman’s theory. 6 was defined as the distance from the surface of tension to the equimolecular surface and was supposed to he constant, when the approximation. eq. (33) was deduced. Evidently 8 = const. leads together with eq. (32) to ad/i’, = const.. if the solid is considered to be incompressible, and the latter approximation was used in the present work for deducing eq. (30). Substituting eq. (32) in eq. (29) one obtains (34) a, = 8i 1z~,/t’~ (35) a,,.
=6ih1/i’5.
To the knowledge of the author, there exist no calculations of 6 for the solid—liquid interface. All one knows from Tolman’s [281 and Otio and Kondo’s [29] estimates is that 26 should be approximately equivalent to the intermolecular distance and that 8 should be. at the largest, of the
.‘ts/ia/,atis
,iui’/eatiosi
order of magnitude of the thickness d of the interface layer. respectivel~.
3. A simple adiabatic nucleation model Fig. 4 shows, that there are two principal ranges of the liquid phase. h~~ h1~and h1 < h4, where the liquid is stable and metastable respectively. The questioti remains then, whether there is also a stahility limit, where the ~iquid hecomes unstable. For this we may first consider a small solid sphere at the temperature T\15.. Let us suppose that the system is a pure liquid at point /3,. Admit then that a favourable statistical (temperature) fluctuation brings so many atoms or molecules, as cotitained in a sphere of radius i’. to point y, (see fig. 5). Then this group of atoms cati immediately (sirnultaneouisly) evolve to point 3,~_~y, is an ordinary statistical fluctuation a,. Stepwhen / which, of the order of a rms (root mean square) fluctuation is highly probable to occur. Step y,—-a, is an ordering process (creation of a crystal nucleus) with increasing entropy. This seems to he paradoxal. as au increase of entropy is always connected with an increase of disorder. However, as the newly created crystal nucleus appears at a higher temperature (and at a higher pressure), the total disorder (configurational and thermic) will he higher. so that there is no cotitradiction. The tiew crystal at point a,. will cotitinue to grow, absorbing atoms (or tilolecutles) of higher atomic enthalpy, corresponding to point /3,. the system increasing its entropy, until reaching point c (if the system is adiabatically enclosed) where the system is in the (single) solid phase (see arrow a, c iti fig. 5). While a fluctuation process of the type /3,. ‘y,. is more probable the less atoms participate. the process y,.--a,. is only possible for a minimum of molecules, which are tiecessary to define the structure of the solid phase. This number is the number one needs to define the Wigner—Seitz primitive cell [301.13 molecules in the case of a fcc or hcp structure or 15 molecules in the case of a bce structure. Compared with Tolman’s parameter 26. which we suppose now to he identical with the intermolecular distance, such a Wigner—Seitz
E. Meter
/
433
A diahazic nucleanon
h
h sr L
S
Fig. 5. This figure is basically identical to fig. 4. In addition arrow /3,— -y,. shows a favourable (temperature) fluctuation process for the simplified case described by eq. (37). Arrow’ y,—a, represents the spontaneous adiabatic nucleation event at constant atomic enthalpy and increasing atomic entropy and arrow a,.—, represents qualitatively the growth of the nucleus and the adiabatic solidification of the system.
primitive cell determining group of molecules (only approximately spherical) has a radius of
only in isothermal and not in adiabatic processes, as discussed above. For adiabatic processes the only reference point is the limit p., p.~(see fig. 5). This point is the real stability limit, because only below this limit, processes of the type Yr’~, are possible with increasing entropy. Consequently one may assume, that when the supercooled liquid is at a temperature i—, from where the metastability limit temperature 7~can be attained by mean (rms) temperature fluctuations of radius r, iTrn~sr, that then the probability is high for the whole nucleation process to occur. This is an approximation, the real nucleation event may need a larger fluctuation (in order to extract a heat corresponding to the latent heat ihLr), occurring at 1/ at a reduced probability. The temperatures discussed above are related =
r
36.
~36)
This would give together with eq. (25), (30) and (33) (37) T~/TM TM~/TM 0.6, =
=
where T~ has the meaning of the temperature of maximum supercooling for a nucleation process occurring with a nucleus of a radius 36, in this simple model. The probability of the nucleation event depends of course still on the probability of fluctuations of the type /3,—’y,, as described above, being higher for smaller latent heats i h L,’ Eq. (37) is based on the labile equilibrium melting temperature for isothermal melting of a Wigner—Seitz primitive cell determining sphere, TM~. But this temperature is really of importance
by =
7;,-;
+
(38)
434
L. %leis’r
,,‘
A diahatic nucleation
where the last term is given by
i7;,,,5,/7;’
=K(iT/T,.
)2)t/2
one obtains [32] =
(k/ç~ )“
(39)
T’,1~,
3x
T51 (Landau and Lifshitz [31]), where k is Boltzmann’s constant and is the specific heat per number of fluctuating atoms (of the liquid phase, at constant pressure and at the temperature 7/) and consequently depends on the radius r of the fluctuation. Eq. (38) leads then together with eq. (39) to 7;
/T51
=
(7;;,/T5, )Fl
1 21 —
t
J
(k/ç~)
-
=
h13 — ih1. .~, —
i~i.
c~ T51 (exp [ ( S
—
s~+ is~ )/c1,
J
—
x(1
+
26/r)
(43)
(c”
‘
—
1)
(44)
“.
(e’ —1) (47)
1331
and together with eqs. (13). (18). (22). (30) and (33) one obtains =
JI
i c’,,QN~~ /I
(42)
I}
+h13—ih1+ih,.
I :1
(41)
(40)
in eq. (19). It follows =
—
k
where T~,> is the maximum supercooling temperature of liquids which are supercooled in relation to solid phases having a Wigner—Seitz number iV.,~..The value c’) in eq. (47) is the value at the temperature 1//~. However, as the specific heats of supercooled liquids are tiot expected to vary tsmuch with the temperature in the range of — O.STM—TM and as eq. (47) is not very sensitive to modest variations of c),. the value at the meltitig temperature can he used as well. This value again does not change much from element to element
The temperature of the liquid phase at the stability limit, 7/ can easily be calculated if one substitutes =
1’
(the gram-atomic values are typically slightly higher than Dulong and Petit’s value) so that c) finally can be substituted by 3k. With these approximations one obtaitis
T51
—
3A
—
~[1 5
—(3QN~..)
2]
I
(e’
—
I) (48)
and with the further approximation. lows the simple result [34]
14, fol-
‘V~ =
.v=iht/cf,T\l=ih~/c~TM,
T14/Tsj=x(1.67—0.26VQ)
where iht and c,~are the latent heat and specific heat per gram-atom. Substituting then eq. (44) in eq. (40) one obtains
where Q is the number of atoms per molecule of the solid phase and
7;
—
-
[( 1—
)
k
j
1,2]
I (
~ 1+~~ ri I
(e’—l)
.
.
(45) Supposing further that the nucleus amid the fluctuation leading to the nucleation contains so many molecules N~, as are necessary to define the Wigner—Seitz primitive cell, it follows that =
~
(46)
Q is the number of atoms per molecule of the solid phase. Together with eqs. (36) and (45),
where
(e’
—
I)
‘.
(49)
.v=ih~/c’j,Ty1 =iht/c’~T~1. both variables being dimensionless. Eq. (49) isshown in fig.6forQ=l,2,4and the limiting value ~. which in fact represents the stability limit 7;,~4/T51.The full squares are experimentally observed maximum supercoolings. where nucleation occurred. The open ones show attained supercoolings of liquid Ga in relation to the solid phases Ga,, and Gay, where instead of these phases Ga15 nucleated. Near the experimental values of P (Hildebrand and Rotariu [23]). Sn (Takagi [35]). Ga (Bosio. Defrain and Epelboin [36], see also ref. [37]), Hg (Zhdanov and Vertsner [38] and Perepezko and Paik [39,40]), Pb (Stowell
E. Meyer
/
435
Adiabatic nucleation
[101) and Bi (Perepezko and Paik [39.40]), the corresponding curves are fully drawn. TI;
4. Discussion The good general agreement of the lowest known experimental supercooling temperatures with eq. (49) (see fig. 6) confirms the present adiabatic nucleation model, especially if one takes into account that in the future possibly lower experimental values will be attained for Sn and Bi. Unfortunately, we have no way of knowing whether a nucleation event was due to homogeneous or heterogeneous nucleation. The present nucleation model shows, that nuclei are very small and that a single impurity atom or molecule eventually could be responsible for heterogeneous nucleation. Of course these arguments are not only applicable to ‘Sn and Bi, but also to all other elements. The lower value of Hg (Zhdanov and Vertsner [38]) shows that the evolution could go in the future to slightly lower values than predicted by eq. (49). However this lower value for Hg could only be shown for extremely small samples (< 50 A). The most interesting case is Ga, where our model predicts correctly. that the metastable phase Ga~should nucleate first, if one takes into account that Ga,, consists of Ga2 molecules (Laves [41]). This agreement contrasts with the predictions of the empirical Ostwald [1] step rule, followingwhich the (least stable) ‘y-phase should nucleate first, as well as with the general conclusions of Singh and Holz [13], T//TM const. (where T is the calculated lowest supercooling temperature). which would predict that the a-phase should nucleate first. Unfortunately Ga is the only known example for high liquid supercooling in relation to 3 solid phases, so that definite conclusions can hardly be arrived at. It is possible to calculate the “gram-atomic” interfacial tension of the flat interface, using eqs. (32) and (35) = a,,v~n*/d= a iht = iht/2, (50) where a could be substituted approximately by 1/2 in the case of a flat interface, indicating that the atoms of the interface have at the average a
p
-.
p’4
TM
of o i
0.5
Fig. 6. The reduced maximum supercooling temperature Ti’~/TMis shown as a function of the dimensionless parameter ~h1 /c,,T,,1, eq. (49). where ~ and c1, are the (atomic or gram-atomic) latent heat of melting and the mean specific heat of the solid and liquid phase at the melting temperature TM. Q is the number of atoms per molecule of the solid phase. The full squares represent experimental maximum supercoolings. where the indicated solid phases nucleated and the open squares show attained supercoolings of liquid Ga in relation to the phases Ga,, and Gay where, instead of these phases. Ga15 nucleated. Near the experimental values, the corresponding curves are fully drawn.
mean enthalpy in relation to the solid and liquid phase. Unfortunately eq. (50) is useless for determining a,.~.because v and specially d are unknown, hut it is interesting to compare this equation with Turnbull’s [6] conclusion (51) for most materials and (52)
for H,O, Bi, Sb and Ge, where is the gramatomic interfacial tension, which was considered to be curvature independent and where the assumption was made that the interfacial layer is a monoatomic layer. However, this latter assumption was later modified and larger interfacial thicknesses were admitted (Spaepen and Turnbull 142] and Thompson and Spaepen [43]). The comparison of
436
L..iiem’cr
4diabaiii siucleatiosi
,‘
eqs. (50)-(52) is interesting, because Turnhull’s [61 values of the interfacial tetision have always heeti considered to he at east of the correct order of
magnitude (e.g., Nash amid Glicksman [44]). The present treatment of the nucleation problem may give the impression that the interfacial tension was used only indirectly for calculating the capillary pressure. etc.. but that the interfacial layer itself was neglected. This is riot so. because a temperature fluctuation, reaching Yr in fig. 5 may he considered to he surrounded by a sort of intermediate layer with mean atomic properties (mean
enthalpy. entropy and temperature) corresponditig to a point halfway between /3, and -y, (near p.,) in fig. 5. This intermediate layer can evolve (simultaneously with the transition y,—a,) at constant enthalpy and increasing entropy to a point halfway
on the tangent /3,—a,. creating the interfacial layer. We did not include this intermediate layer in the fluctuation term. eq. (39), for simplicity and also, because the thickness of this layer is unknown. Because of this. ç~ in eq. (46) is underestimated and iT.msr and 7/ are slightly overestimated. However, considering all other approximations, this fact is of little importamice. Eq. (49) can be approximated for large molecules (large Q) to T
14/T51= 7;514/T51=0.6x(e’— I)
(53 I, -
during the freezing process, so that the amorphous structuire freezes-in before there is a chatice for homogeneous nucleatioti. A similar relation (eqs. (55) and (56)) has been indicated for glasses by Zanotto [511. i.e. glasses which do tiot show homogeneous (voluime) nuclea— tuon duiring slow cooling (Si02. 1320,. Na ~ 2SiO2. K 2O—2SiO2, PhO--Si02, (‘aO --Al 2O~ 2SiO2. (‘aO—MgO- 2SiO2. Na20—Al20,--ôSiO, follow the relation > T14/T~.
and glasses which show (supposedly) homogeneous (volume) nucleation during slow cooling [14] (Li 2°~ 2S1O2, BaO—2SiO2. Na20 -2CaO--3SiO2, Na 70--SiO2 ) follow the relation <
T14~/T~.
7;
However. ~ is a function of the quenching speed and the inequality above can eventually he inverted in a quenching process. which explains. why the compounds of the second group can also he obtained as pure (non-crystalline) glasses.
The glass transition temperatures of amorphous alloys are mostly not well known. It is possible
that for these materials, during slow freezing. the ‘
following relation is valid (57) while during fast cooling one has
(54)
in all 27 cases where the necessary thermodynamic data are available, except polypropylene where and polyoxymethylene where is still disputed (0.41 ~ 7;/T51 ~ 0.58). and where the reviewers recommend a low value. = 0.42. compared with 7/4/T51 = 0.51 [501. (The number Q of a polymer evidently does not correspond to the total number of atoms of the macromolecule, but to the number of atoms of the monomer as indicated by the chemical formula.) Consequently, eq. (54) shows that pure amorphous
7;
polymers do not crystallize (by homogeneous
7;
=
of polymers (Gaur
>
nucleation), because
(56)
CaO ‘ Si02 being a limiting case (7;/T51 T1~/T~1). belonging to the second group.
and conferring this equation with revised values of glass transition temperatures et al. [45—49]). one sees
(55)
is attained before
7/4.
7;/T~1> T14/T~1
(58)
with Ty1 = (T1 + 7;)/2. where T1 and 7; are the liquidus and solidus temperatures, respectively. It is empirically well known that 7; imicreases in fast freezing processes. This would explaiti. why metallic glasses usually can only he produced by
fast quenching. For pure metals the relation Tg/T’,i:~’~l/4
(59)
has heen proposed [43]. If this relation is correct. then there is little hope of producing aniorphous pure metals by quenching from the melt. in agreemerit with experimental observatiotis. This hy-
E. Meyer
/ Adiabatic nucleation
pothesis is not in contradiction with the fact, that pure metals could be quenched from the vapour phase directly to thin amorphous films at low temperature (Buckel and Hilsch [52]), because in this way the temperature, where the liquid would crystallize, could be “bypassed” in the vapour phase.
437
model on glasses and for useful discussions. Valuable discussions with Professors Michel Bienfait, Hans J. Scheel and J. Leonardo M.D. de Souza are gratefully acknowledged. as well as the financial support of the institutions CNPq, FINEP and CEPG.
References
5. Conclusions An alternative and simpler explanation could be given for the nucleation of the solid in the liquid phase and a definite stability limit for supercooled liquids is predicted by the present nucleation model. The lowest known experimental supercooling temperatures of pure low viscosity liquids are in good agreement with the theoretical predictions, and experimentally determined glass transition temperatures of high viscosity liquids are shown to be higher than the stability limit. This explains why the latter do not crystallize (by homogeneous nucleation). These facts suggest that nucleation could be an adiabatic phenomenon, instead of isothermal, and could be preceded by common statistical (temperature) fluctuations, instead of heterophase fluctuations. These ideas should be tested in the future in the context of the vapour—liquid and vapour—solid transition, as well as in the case of nucleation in dilute solutions. Heterogeneous nucleation was not treated in this work. It is evident that in typically heterogeneous nucleation events, as nucleation from the vapor phase on a substrate or precipitation from solid solutions, the heterogeneous nucleus may contain less molecules than are necessary to determine the Wigner—Seitz primitive cell. This is, because some molecules of the substrate or the primary solid solution may contribute to determine the symmetry of the new phase by epitaxy, in analogy with isothermal heterogeneous nucleation models.
Acknowledgements
The author thanks Professor Edgar D. Zanotto for the communication, prior to publication, of the conclusions of the test of the present nucleation
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dynamic Properties of the Elements I Am. Soc. ( or Metal Metals Park, OF!, 1973). II. Meyer. in: Ahsiracts 7th Intern. (‘on). Oil (‘rvst:il Growth 1(10—7). Stuttgari. 1913. p. 2.1)1. NI. Takagi, .J. Phys. Soc. Japan 9 11954) 359. I.. Bosisi. A. Defrain and I. Epelhoni .1. I’hvsique 27 (l966( 61. 1.. Bosio, R. (‘ortes, A. Defrain and ( i . Folcher. .1. (‘hii’i’i. PhssiqLie 70 (1973( 357.
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