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Activation energy for nucleation of the supercooled hard-sphere fluid
Volume87. number
ACTlVATION
19 hkuch 1982
CHELIICAL PHWCS LCITLRS
2
ENERGY FOR NUCLEATION
OF THE SUPERCOOLED HARD-SPHERE FLUID *
R. TENNE * Barrel/e. Germa
Research Cerrtre, I 22
7 Carouge-Gerrcw
Rccclvcd 16 February 1979; m iin;ll form 28 Jsnuvy
The free cncrgy change iollowmg the (concepkd) IS calculxcd
A muumal xflv3tlon
Swrt:erland
1982
process
activxlon energy dccrcxcs with mcrcxc mstabdrtg ws found
tn the supercooled hxd-sphcrc’ tluld
m the dqrcc of undcrcoohng ;rnd dti3ppcxs
Thus
at the dcnslty 31 which structural
2. The free energy change following the spherical embrion formation
1. Introduction Gordon
ofa germ formatIon
free cncrgy must bc dchvcrcd IO the nucleus if growth of the fcrm bc Lontmucd
et nl. [l]
concluded
that no second-order
tranwmn of the kmd found for glass-iormmgfluidscan
2. I.
The compmal process
exist in a hard-sphere (HS) supercooled fluid, smce the configurational entropy (the hfference between the entroptes of the supercooled fluid and the crystal at the same temperature) for the HS system does not vamsh at any (posttive) temperature. However, a glass transition, ensuing from purely kmettc factors, was found at Hughcooling rates (lOI1 K/s). Later. Gordon [‘_I demonstrated that a stabtlity limtt at g = 0 65 may not be excluded for a HS flutd. The present work addresses Itself to this problem in an mdlrect way through a dtfferent approach. We examine the free energy change following the formation of a spherical germ, havmg the lattice symmetry of the eventual crystal but deprived of any internal degrees of freedom, m the supercooled HS fluid. We fid that for (reduced) densities C0.65, an acttvatton energy is indispensable for continued growth of the germ.
We examine the process of bringing M (supercooled) fluid particles, fixed in their posttions at Infinite separation from one another, to a final configuration m which these M particles form a sphertcal germ (cluster) having the symmetry of the crystal to which thrs fluid crystallizes eventually at a given temperature T. and denstty p = IV/Y. WCcall I$,(r, q) the work (function) to create this germ m the supercooled HS flutd, and W(T, q) the work function of the single HS fluid parttcle. Then. the free energy change following the process of bringing these AI particles from mfmite separatton to the fmal (germ) configuratlon ISgiven by 6 y,,(T.
l?) = ‘$(T,
9) - AWT.
11),
(1)
smce m the imtial configuration the fiscd particles do not Interact wnh one another, then their total work functton IS the sum of the single-parttcle work functtons MW(T, q). Here, q= pug. with uo = : rru3 the volume occupted by a smgle HS particle, a being the HS diameter. 2.2. Calculation of IV(T. V)
* Worksupported
by BxcUeCorpontcTcchniwl
Developmrnt
* Present address: Dcpxtmcnt of Phnics Research, The Wcizmxm lnsututcof Saence. Rchovot 76100, Israel.
0 009-7-614/82/0000-0000/S
02.75 0 1982 North-Holland
To calculate of the flutd:
W(T,
v) we
use the equation of state
177
19 March 1982
CHEMICAL PHYSICS LII-I’CRS
Volume 87. number 1
-3.3. Cahrlatio~r 01 IV,,(T, q) whcrc
(kT)-I . with X:the Boltrmann constant,p sLcm’spressure and / 1sa function of the rc-
/I =
is the s) duced
dcnslry
abh
p. Then.
the Gbbs-Duhem
reimon
= b9hjabia~
(3)
can bc mtc’grakd to ylcld the chcm~cal potential F my be reprcsenrcd in a formal way ihrough
which
&.4X 71)= LGO(r) + In rl + /MI)
.
(4
Here. &O(r) = In X3/,0 IS the idcal-gas standard chemical potential, X bcng the dc Broghe thermal wavelength, and /3lt’(n) is the required work function (multiplied by 0) From compurcr e\pernnents [3]. the freering and mcltmg densities of the HS flurd and crystal are rcspcct1vcly l& = 0.197 .
7$-j’ = 0.3457
.
thcsc densmes. rhc chemcal potcnr~~l of the fluid equals that of the sohd. Thus. WCmay USCrhe Carnalian-Starlmg equation of state for the HS tlutd. +r1+$
-nJ)/(l
-11)s)
(6)
to obtain the chenucal potential of the system m the Iranslt1on density using cq. (3).
s-
97); + 3
P(p-pO)=Insfj+vL
(1
u,\r = (M/I&)‘& .
(10)
Also, we consider the simple case of an infimtely dilute solutron \nth respect to the germs. In this simple case. eqs. (8) and (9) are sttll appropriate for the descrtption of the supercooled HS fluid To calculate 11’,,,(7,r)), we shall employ the e\pression from the scaled particle theory for HS nurtures [a], which. in the case of a bmary mtirutely dilute solution, reads
(9
At
PugP’/V=(l
To calculate IV,,I(T, n),),we shall rely on the followmg slmphfications We first assume that we have a perfectly spherical germ (thus Al must be large enough), having the structure of the equihbrium fee (or hcp) crystal, at the density qs to be specified below. In this case, the diameter of the embrion a,,f may be found from a simple geometrical argument.
(l$$
-L#
-
(7)
For the supercooled HS tluid (q > 0.497) a good cquatlon of state was found by Gordon et al. [I](3uopsc/~= 3.00/a,, + 7.90 + 0_28a,, (% = 0.6537/q - I).
(8)
Llsmg agam eq (3) and eqs. (7) and (8). we may write the work function for the smgle HS particle in the supercooled fluid as
=-In(l
-t7)+6-/=[1
f8uoppy3.
(II)
Here, Y = ia,rr/a and z =Gxl -II) Strictly speaking eq. (I I) IS not applicable for the estimation of the work necessary to create spherlcal clusters In the high-dennty fluid. However. for large embrions the major contribution to IV,+,, comes from the last ferm on the rhs of eq. (I 1) which is the correct PVcontrlbutlon at any fluid density. For large densities this term accounts for most of the work to create the embnon even for small Mvalues. For densities which esceed the freezing density of the HS fluid slightly (as In curve 1of fig. 1) It can again be expected that IVbfISnot m great error for any value 0fMThisargument shows that eq. (11) avalid for our application. To find 77sat the pertaining temperature and pressure, we make use of the HS crystal equation of state [31: fluopsltl = 3/a, + 1.56 + 0 560, (crs = 0.7305/rl - 1) ,
8 - 917; + 3(7$$
+(:!+3:)Y]
(12)
IMa) = $ (1 - $3 +
I 78
’ d,j a&# s i II) a7i ‘i0
- h&V&
.
wtuch may be solved for qs IO be used in eq. (I 0) for determinatton of a,,,. Finally, it is easily verified that the entropy change in the process islust
CHEMICAL PHYSICS LETTERS
Volume 87. number 7
sSfif fk
= -fls Ii{,, + /3psV*r ,
19 March 1982
(13)
where 6 y;rf is the molar volume change upon clustering. --1.10”
3. Results and discussion In figs. 1 and 2 we summarize our results. We see that p6 Ir:j~ is almost a linear function of III for large enough M. Thus IS because the major contribution
to WIl, in
-
-5000
-
this case, comes from the last term on the rhs of eq.
(I I). At low densities, and small number of particles, flk:,f deviates markedly from linearity and goes through a maGmum. Another significant result is that the formaCon of a metastable germ in the supercooled HS fluid IS econonuc (/%I It’,,, <0) for large values of AI only.
As we decrease AI. /JSW,,f goes through
posuive
values for 71< 0 65. For higher densltles,
formatlon
of a two-particle
cluster is already
cxogenic.
Thus means that for densltles 9 < 0 65, the formatlon of a metastable rivauon
germ IS possrble only if a mimmal
energy had been delivered
- -3.19’
4w.4 - -z.,o’ 05vr
zero to the
ac-
to the germ.
-lOcoo
‘. .
’ .-
1
In rcahty, the
non-sphcncal amounts
-200
-
cordmg
calculate
II’,,
for
clusters in a simple way, our procedure
to the followmg
to our procedure
process for small Al. wc
is smaller than the real value.
S 1t$1 IS more negallvc
Thus is confirmed -
for smallrlf.cspecial-
Smce we cannot
calculstc fllr’for the single splicrrcal partvles, then We deform them (conceptually) such that they can rccommodate mto a spherical cluster ha\mg rhe appropriate density vs. Thus. the work functron I($, calculated acand therefore
-3w
shape of the cluster would devm
from spherical symmetry
ly at low dcnatles.
-so0
-5”IG”
rig. 1. TIICfm cncrgj (entropy) change follownt:the processoia germ iormaIlon m IIIC supcrcoolcd hxd-sphcrc 11u1d (3) qf = 0.65 (qs = 0.7358). (4) qi= 0 653 (~5 = 0 7396) ni IS lhc hard-sphcrcsupcrLoolcddcnslt}. q, ISlhc cryst.d fcrrn dcnslty
markedly
-
-2.10’.
than the real one.
by fhe extrapolation
of 6 IL:,, from
Iugh values of M to smaller values of Al which tends to
-
-1500
Fig 1. The ircc energy (entropy) chxngc followmg rhc proccssof a germ formstlon LII the supercooled hard-sphereiluld. (I) rlf=O 55 (q=O 6119). (2) vi=0 60 (qs=O.6729). qf IS the hud-sphere supercooleddcnsny. 9s is the cr, SKII germ densIt>.
grve more realistic vahrcs of S lt’,rf for small 111.Note, that for very small values of Al. 61t’,,f must go through a mallmum (the ac1Ivatlon energy for the cluster formatIon) which is Just what we have m fig. I. Some words of caution are in order: (I) Any a priori corrclaiions between the fLcd smglc partrcles as well as between the diffcrent fL\ed germs arc escluded. (3) We disregard any contrrbutions due to the kinetic degrees of freedom. however.
(3) We mcorporated
some rncvltable
approxima179
Volume87, numba
1
CHEMICALPHYSICSLETTERS
ttons in our procedure, particularly in the calculation of I’{,l. Taking into account these limitations, one may however arrive at the following consequences: (1) The formatton of a (metastable) germ in a quasi-equdlbrium manner is favourable, for large enough germs, at any density of the supercooled HS fluid. (2) It appears that for slightly undercooled fluid, a constderable activation free energy must be dehvered to the embnon if crystallization should occur This activation free energy decreases with the degree of undercooling and at q = 0.65 the process of cluster formatlon is exogenlc from the very beginnmg (3) The latter phenomena may be connected with the finding [2] that at q = 0 65 the first peak in the structure factor of the supercooled HS fluid diverges which amounts to a structural msrabdlty of
180
19 Much 1982
this fluid, so that 9 = 0.65 is the maxmal accessible density of the supercooled HS fluid. (4) Although crystallization of the supercooled HS fluid ts favoursd thermodynamically, a very high coolmg rate may prevent the molecules from crystallizmg due to a threshold energy barrier for nucleation. However, for densities higher than ~0.65 this barrier disappears, rendering crystsllrzation immediately.
References [ I] J M.Gordon, J.H. Gibbs
snd P.D. l%mrng. J. Chcm
Phys. 65 (1976) 277 1.
[Z?]JAI Gordon, Phys Rev.Al8 (1978) 1272 13) B.J. Alder.WG Hoover and DA Young. J
Chcm. Pbys. 49 (1968) 3688. [-II J.L Lebowtz. II. Hclbnd and 11. Pneslgaard. J Chcm Phys. 13 (1965) 774
ACTlVATION
19 hkuch 1982
CHELIICAL PHWCS LCITLRS
2
ENERGY FOR NUCLEATION
OF THE SUPERCOOLED HARD-SPHERE FLUID *
R. TENNE * Barrel/e. Germa
Research Cerrtre, I 22
7 Carouge-Gerrcw
Rccclvcd 16 February 1979; m iin;ll form 28 Jsnuvy
The free cncrgy change iollowmg the (concepkd) IS calculxcd
A muumal xflv3tlon
Swrt:erland
1982
process
activxlon energy dccrcxcs with mcrcxc mstabdrtg ws found
tn the supercooled hxd-sphcrc’ tluld
m the dqrcc of undcrcoohng ;rnd dti3ppcxs
Thus
at the dcnslty 31 which structural
2. The free energy change following the spherical embrion formation
1. Introduction Gordon
ofa germ formatIon
free cncrgy must bc dchvcrcd IO the nucleus if growth of the fcrm bc Lontmucd
et nl. [l]
concluded
that no second-order
tranwmn of the kmd found for glass-iormmgfluidscan
2. I.
The compmal process
exist in a hard-sphere (HS) supercooled fluid, smce the configurational entropy (the hfference between the entroptes of the supercooled fluid and the crystal at the same temperature) for the HS system does not vamsh at any (posttive) temperature. However, a glass transition, ensuing from purely kmettc factors, was found at Hughcooling rates (lOI1 K/s). Later. Gordon [‘_I demonstrated that a stabtlity limtt at g = 0 65 may not be excluded for a HS flutd. The present work addresses Itself to this problem in an mdlrect way through a dtfferent approach. We examine the free energy change following the formation of a spherical germ, havmg the lattice symmetry of the eventual crystal but deprived of any internal degrees of freedom, m the supercooled HS fluid. We fid that for (reduced) densities C0.65, an acttvatton energy is indispensable for continued growth of the germ.
We examine the process of bringing M (supercooled) fluid particles, fixed in their posttions at Infinite separation from one another, to a final configuration m which these M particles form a sphertcal germ (cluster) having the symmetry of the crystal to which thrs fluid crystallizes eventually at a given temperature T. and denstty p = IV/Y. WCcall I$,(r, q) the work (function) to create this germ m the supercooled HS flutd, and W(T, q) the work function of the single HS fluid parttcle. Then. the free energy change following the process of bringing these AI particles from mfmite separatton to the fmal (germ) configuratlon ISgiven by 6 y,,(T.
l?) = ‘$(T,
9) - AWT.
11),
(1)
smce m the imtial configuration the fiscd particles do not Interact wnh one another, then their total work functton IS the sum of the single-parttcle work functtons MW(T, q). Here, q= pug. with uo = : rru3 the volume occupted by a smgle HS particle, a being the HS diameter. 2.2. Calculation of IV(T. V)
* Worksupported
by BxcUeCorpontcTcchniwl
Developmrnt
* Present address: Dcpxtmcnt of Phnics Research, The Wcizmxm lnsututcof Saence. Rchovot 76100, Israel.
0 009-7-614/82/0000-0000/S
02.75 0 1982 North-Holland
To calculate of the flutd:
W(T,
v) we
use the equation of state
177
19 March 1982
CHEMICAL PHYSICS LII-I’CRS
Volume 87. number 1
-3.3. Cahrlatio~r 01 IV,,(T, q) whcrc
(kT)-I . with X:the Boltrmann constant,p sLcm’spressure and / 1sa function of the rc-
/I =
is the s) duced
dcnslry
abh
p. Then.
the Gbbs-Duhem
reimon
= b9hjabia~
(3)
can bc mtc’grakd to ylcld the chcm~cal potential F my be reprcsenrcd in a formal way ihrough
which
&.4X 71)= LGO(r) + In rl + /MI)
.
(4
Here. &O(r) = In X3/,0 IS the idcal-gas standard chemical potential, X bcng the dc Broghe thermal wavelength, and /3lt’(n) is the required work function (multiplied by 0) From compurcr e\pernnents [3]. the freering and mcltmg densities of the HS flurd and crystal are rcspcct1vcly l& = 0.197 .
7$-j’ = 0.3457
.
thcsc densmes. rhc chemcal potcnr~~l of the fluid equals that of the sohd. Thus. WCmay USCrhe Carnalian-Starlmg equation of state for the HS tlutd. +r1+$
-nJ)/(l
-11)s)
(6)
to obtain the chenucal potential of the system m the Iranslt1on density using cq. (3).
s-
97); + 3
P(p-pO)=Insfj+vL
(1
u,\r = (M/I&)‘& .
(10)
Also, we consider the simple case of an infimtely dilute solutron \nth respect to the germs. In this simple case. eqs. (8) and (9) are sttll appropriate for the descrtption of the supercooled HS fluid To calculate 11’,,,(7,r)), we shall employ the e\pression from the scaled particle theory for HS nurtures [a], which. in the case of a bmary mtirutely dilute solution, reads
(9
At
PugP’/V=(l
To calculate IV,,I(T, n),),we shall rely on the followmg slmphfications We first assume that we have a perfectly spherical germ (thus Al must be large enough), having the structure of the equihbrium fee (or hcp) crystal, at the density qs to be specified below. In this case, the diameter of the embrion a,,f may be found from a simple geometrical argument.
(l$$
-L#
-
(7)
For the supercooled HS tluid (q > 0.497) a good cquatlon of state was found by Gordon et al. [I](3uopsc/~= 3.00/a,, + 7.90 + 0_28a,, (% = 0.6537/q - I).
(8)
Llsmg agam eq (3) and eqs. (7) and (8). we may write the work function for the smgle HS particle in the supercooled fluid as
=-In(l
-t7)+6-/=[1
f8uoppy3.
(II)
Here, Y = ia,rr/a and z =Gxl -II) Strictly speaking eq. (I I) IS not applicable for the estimation of the work necessary to create spherlcal clusters In the high-dennty fluid. However. for large embrions the major contribution to IV,+,, comes from the last ferm on the rhs of eq. (I 1) which is the correct PVcontrlbutlon at any fluid density. For large densities this term accounts for most of the work to create the embnon even for small Mvalues. For densities which esceed the freezing density of the HS fluid slightly (as In curve 1of fig. 1) It can again be expected that IVbfISnot m great error for any value 0fMThisargument shows that eq. (11) avalid for our application. To find 77sat the pertaining temperature and pressure, we make use of the HS crystal equation of state [31: fluopsltl = 3/a, + 1.56 + 0 560, (crs = 0.7305/rl - 1) ,
8 - 917; + 3(7$$
+(:!+3:)Y]
(12)
IMa) = $ (1 - $3 +
I 78
’ d,j a&# s i II) a7i ‘i0
- h&V&
.
wtuch may be solved for qs IO be used in eq. (I 0) for determinatton of a,,,. Finally, it is easily verified that the entropy change in the process islust
CHEMICAL PHYSICS LETTERS
Volume 87. number 7
sSfif fk
= -fls Ii{,, + /3psV*r ,
19 March 1982
(13)
where 6 y;rf is the molar volume change upon clustering. --1.10”
3. Results and discussion In figs. 1 and 2 we summarize our results. We see that p6 Ir:j~ is almost a linear function of III for large enough M. Thus IS because the major contribution
to WIl, in
-
-5000
-
this case, comes from the last term on the rhs of eq.
(I I). At low densities, and small number of particles, flk:,f deviates markedly from linearity and goes through a maGmum. Another significant result is that the formaCon of a metastable germ in the supercooled HS fluid IS econonuc (/%I It’,,, <0) for large values of AI only.
As we decrease AI. /JSW,,f goes through
posuive
values for 71< 0 65. For higher densltles,
formatlon
of a two-particle
cluster is already
cxogenic.
Thus means that for densltles 9 < 0 65, the formatlon of a metastable rivauon
germ IS possrble only if a mimmal
energy had been delivered
- -3.19’
4w.4 - -z.,o’ 05vr
zero to the
ac-
to the germ.
-lOcoo
‘. .
’ .-
1
In rcahty, the
non-sphcncal amounts
-200
-
cordmg
calculate
II’,,
for
clusters in a simple way, our procedure
to the followmg
to our procedure
process for small Al. wc
is smaller than the real value.
S 1t$1 IS more negallvc
Thus is confirmed -
for smallrlf.cspecial-
Smce we cannot
calculstc fllr’for the single splicrrcal partvles, then We deform them (conceptually) such that they can rccommodate mto a spherical cluster ha\mg rhe appropriate density vs. Thus. the work functron I($, calculated acand therefore
-3w
shape of the cluster would devm
from spherical symmetry
ly at low dcnatles.
-so0
-5”IG”
rig. 1. TIICfm cncrgj (entropy) change follownt:the processoia germ iormaIlon m IIIC supcrcoolcd hxd-sphcrc 11u1d (3) qf = 0.65 (qs = 0.7358). (4) qi= 0 653 (~5 = 0 7396) ni IS lhc hard-sphcrcsupcrLoolcddcnslt}. q, ISlhc cryst.d fcrrn dcnslty
markedly
-
-2.10’.
than the real one.
by fhe extrapolation
of 6 IL:,, from
Iugh values of M to smaller values of Al which tends to
-
-1500
Fig 1. The ircc energy (entropy) chxngc followmg rhc proccssof a germ formstlon LII the supercooled hard-sphereiluld. (I) rlf=O 55 (q=O 6119). (2) vi=0 60 (qs=O.6729). qf IS the hud-sphere supercooleddcnsny. 9s is the cr, SKII germ densIt>.
grve more realistic vahrcs of S lt’,rf for small 111.Note, that for very small values of Al. 61t’,,f must go through a mallmum (the ac1Ivatlon energy for the cluster formatIon) which is Just what we have m fig. I. Some words of caution are in order: (I) Any a priori corrclaiions between the fLcd smglc partrcles as well as between the diffcrent fL\ed germs arc escluded. (3) We disregard any contrrbutions due to the kinetic degrees of freedom. however.
(3) We mcorporated
some rncvltable
approxima179
Volume87, numba
1
CHEMICALPHYSICSLETTERS
ttons in our procedure, particularly in the calculation of I’{,l. Taking into account these limitations, one may however arrive at the following consequences: (1) The formatton of a (metastable) germ in a quasi-equdlbrium manner is favourable, for large enough germs, at any density of the supercooled HS fluid. (2) It appears that for slightly undercooled fluid, a constderable activation free energy must be dehvered to the embnon if crystallization should occur This activation free energy decreases with the degree of undercooling and at q = 0.65 the process of cluster formatlon is exogenlc from the very beginnmg (3) The latter phenomena may be connected with the finding [2] that at q = 0 65 the first peak in the structure factor of the supercooled HS fluid diverges which amounts to a structural msrabdlty of
180
19 Much 1982
this fluid, so that 9 = 0.65 is the maxmal accessible density of the supercooled HS fluid. (4) Although crystallization of the supercooled HS fluid ts favoursd thermodynamically, a very high coolmg rate may prevent the molecules from crystallizmg due to a threshold energy barrier for nucleation. However, for densities higher than ~0.65 this barrier disappears, rendering crystsllrzation immediately.
References [ I] J M.Gordon, J.H. Gibbs
snd P.D. l%mrng. J. Chcm
Phys. 65 (1976) 277 1.
[Z?]JAI Gordon, Phys Rev.Al8 (1978) 1272 13) B.J. Alder.WG Hoover and DA Young. J
Chcm. Pbys. 49 (1968) 3688. [-II J.L Lebowtz. II. Hclbnd and 11. Pneslgaard. J Chcm Phys. 13 (1965) 774