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Clustering phenomena in adhesive hard sphere fluids. Theory of freezing
Chemical Physics 34 (1978) 231-252 0 North-Holland Publishing Company
CLUSTERlNG PHENOMENA IN ADHESIVE HARD SPHERE FLUIDS. THEORY OF FREEZING
B. BARBOY * Labomtoire de Photopltysiqrre MoEculaire, UnivcrsitC de Paris&d, 9140.5 Orsoy, France
Received 31 May 1978
We develop P version of the physical cluster theory of the equntion of stnte bnsed on 3 definition of a “cluster” which depends both on particle positions zmd relative moments, namely we choose to call a group of i molecules a physical cluster if their total energy (kinetic energy plus interactions) does not exceed the translation energy of the cenrrc-of-mass of the whole group. This defmition in addition with the “microcrystal” model of a cluster allows us to calculate the properties of individual clusters, to consider the clustering phenomena and to wrivritedown an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken into consideration by using the adhesive hard sphere model and the Percus-Yevick approximation. Based on the relations obtained we develop the theory of freezing and perform a numerical investigation of the equilibrium behavior of the system at various temperatures and densities. In particular, the theory enables us to describe the regular phase diagram which includes (in coordinates pressure-temperature) three branches intersecting at the triple point. It is interesting that the coesistence curve of the fluid-solid as well as liquid-gas phase transition has an end point.
1. Introduction One of the most important aims of statistical mechanics is to predict the equilibrium behavior of substance. Mayer and his collaborators [I] developed a formally exact molecular theory of fluids which was based on the concept of the mathematical cluster. In contrast to this Frenkel [7_,3] and Band [4] regarded the imperfection of real gases as a result of molecule association into actual physicnZ clusters of various sizes, caused by intermolecular attractions. In this picture, they viewed the imperfect gas as a mixture of aggregates, each characterized by the number i of molecules involved, in dynamic association-dissociation (%hemical” equilibrium) with one another. In the original physical cluster theory of Frenkel and Band as well as in most applications, these molecular nggregates are treated as noninteracting; the system is considered to be an ideal mixture of clusters, each size corresponding to an independent species. The total pressure is calculated then by simply summing the ideal gas partial pressures. Based on these principles, and using Wergeland’s cluster defmition [S], ter Hasr [6,7] has derived the equation of state for an imperfect gas and shown that the isotherms in the Frenkel-Band approach to the P- u diagram consist of two analytically different parts: in the first part, for specific volume u larger than a definite critical value II,, the pressure P is a monotonically decreasing function of u; in the second part, for u
of California, Los Angeles, California, 90024, USA.
232
B. BarboyfCkstenig
phenomena
in adhesive hard sphere fluids
III the present article we develop a version of the physical cluster theory of the equation of state based on a definition of a cluster which depends both on particle positions and relative momenta [cf. eq. (SC), section 31. In section 4 we consider the clustering phenomena and write down an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken there into consideration by using the sticky hard sphere model [P] [eq. (36)] and the Percus-Yevick approximation [IO]. Section 5 is devoted to the investigation of the equilibrium behaviour of the system and to the freezing theory. The nucleation phenomena will not be delt with at all here.
2. Association-dissociation
equihbrium
We consider an assembly ofNmonoatomic particles obeying classical statistics and interacting via a short-range pair potential energy, I. The system is confined in a vessel of volume Vat temperature T and total pressure P. Let us denote by~i the number of cluster consisting of iatoms. Then the Hehnholtz free energy, F, of the system is given by F=C~iNi-PV, i where pi is the chemical potential of a cluster regarded as a distinct molecular species. The most probable distribution {N;} is derived by minimizing F with respect toNr keepingN, T and V fued
and under subsidiary conservation condition: N=CiNi. i
Keeping in mind the Gibbs-Duhem
relation we obtain in this way:
i=l.2,3,...,
I-$ =i.n9
(3)
where _uis the Lagrange multiplier. The meaning ofw is derived from eq. (I) by substituting (3) into (1): F=/.Uv--PV,
(4)
indicating that p is the macroscopic chemical potential. On the other hand, it follows from (3) that !.ris equal to the chemical potential, r_cl,of the monomers. Substitution ofp =,ul into (3) yieids the standard condition of “chemical” equilibrium among clusters, &=i!rl
9
W
which follows from the possibility of free exchange of atoms between aggregates. Ifall/li are given as functions of the temperature and cluster concentrations,pk =Nk/V(k= I, 2,3, _._),eqs. (2) and (3) permit, in principle, the calculation of the most probable distribution {pp} of molecules over clusters for any temperature and average number density p = N/V_
3. Definition of a physical cluster RI. Gene& considerations From the thermodynamic
point of view, the definition of a cluster is arbitrary and it is clear that any defmition
B. BarboyjChisteringphenomena in adhesivehard spherefluids
233
will lead to the same equation of state and macroscopic properties of the fhrid if all subsequent theoretical manipulations are exact. From the mechanical point of view, two atoms are bound as a pair (cluster of size 2) if their relative kinetic energy, Tre* 2 , is less than the negative of their potential energy of interaction, if2 = ~(‘1~): H2=T+J2<0.
(53
For larger aggregates Hill [II ,121 suggested a deftition which required each pair of atoms in a cluster to be bound in the sence of inequality (Sa). This leads to a formal theory which is analogous to Mayer’s (mathematical) cluster expansion [l]. However, this definition would exclude from a cluster an atom which is not bound in the sence of (5a) but which is nevertheless bound to the cluster because of attractive interaction with sofne of the atoms in the cluster. An alternative definition would consider a particle bound if its kinetic energy (relative to the centre-of-mass) does not exceed the negative of its interaction with other members of the cluster;
$--(~;iel)~ +iZiku(r&
(k= 1,2, _.., i)
Unfortunately, one encounters serious mathematical difficulties in calculation of the cluster partition function when one uses this (as well as Hill’s) definition. For this reason Stillinger [13] (see also refs. [8,14]) develops an alternative theory which is based on a definition of physical cluster depending only on particle positions_ Considering the pair energy potential n(r) cut-off by a defmite distance b he calls a group of i molecules a physical cluster if each of its molecules is in a distance less than b from at least one other member of the group and in a distance greater than b from each nonmember_ We emphasize that the range of moleculr interaction is infinite so that there is no natural cut-off distance. The definition arbitrariness aside, the theory encounters the fundamenta! difficulties in the investigation of the excluded volume problem. Thus we prefer to choose a different definition as a basis for the theory proposed hereafter. According to it, by analogy with @a), i molecules form a cluster if their total kinetic energy (with exception of translation energy of the centre-of-gravity), Tie’, does not exceed the negative of the total potential energy, Vi:
PC) Herepk is the momentum of particle k and rkt is the distance between particles k and 1. While Hill’s deftition is too strict for relatively weak molecule interaction, inequality (SC) can insert into the cluster as well a “hot” particle k whose high kinetic energy ~m-lp~ exceeds its interaction-ZII+ku(rkI), with the rest of the cluster but which stilt leaves (because of sufficiently ‘%old” particles) the total kinetic energy less than the total potential. 3.2. Properties of individual clusters In view of (5c) the cluster partition function has the form: Zi = ( Vi3/2/k3h3(ihi)
1 ___ _fdrl___ dri_
1
I&
... dpi_
I
ewPHi ,
(6)
Hi<0
where X= (@~~/27r~)r/~ is the de Broghe wavelength of the atoms in the system, and 0, h, and m have their usual meaning. or is introduced to account for the indistinguishability of the particles_ tf the particles can move freely inside the cluster (as, for instance, Wergeland [5] assumes), then or. = i!. We, however, accept here the “microcrystal” model [Xl, i.e. we treat the cluster as a very small crystallite and oi is the number of the rotational symmetry of the aggregate. The exact microcrystal partition function contains contributions from all stable configurations of
234
5. Barbay/Ciusrenig
phenomena in adh&ve hard sphere fluids
the cluster. But we shall ignore all configurations
except the single one corresponding to the miniium value of the potential energy of particle interactions. This consideration is suggested by Monte Carlo studies of Lee et al. [14] which show that only compact, roughly spherical clusters are important at low temperatures. In addition we shall use the harmonic approximation to evaluate the vibrational part of ZiPerforming the integration in (6) we obtain the following expressions for the cluster partition function:
zi>3
(74
= Ve-
G’b)
z, = If/?? .
(759
Here Ur! is the static energy of the cluster, J, is its principal momentum r(n) is the gamma-function of Euler (n - I)! z I’(n) =
4:emr t’*-1dt
of inertia, V~ is its cwth natural frequency,
(8)
,
0
and -y(fz,X) is the incomplete
y(n. x) = i e-I 0
gamma-function
tt*-1 dr .
(see appendix)
(9)
The structure of the cluster involving 3,4,5 and 6 molecules, and corresponding to the minimum of the interaction energy, is the equilateral triangle, the tetrahedron, the trigonal bipyramid (slightly contracted akng the symmetry axis) and the octahedron, respectively_ The larger aggregates can be designed with help of polyhedra with equai edges or a combination of them. These polyhedra are regular tetrahedra, octahedza, trigonal prisms, archimedian antiprisms, and tetragonal dodecahedra [16]. Among the several configurations obtained in this way for a given number of particles the construction corresponding to the lowest static energy is considered T*. The intermolecular interaction is described by the (12 : 6) Lennard-Jones potential:
withn=9 orR= 12. Figs. 1-4 and tables 1 and 2 show the results of our calculations for fourteen different molecular aggregates *. The data obtained are used to suggest interpolation formulae for calculation of the static energy Uz?, the average frequency w;, and product J1J2J3~lT2 charactetizing the rotational properties of the cluster. We fmd: 7 We have taken into consideration, of course, only such configurations which correspond to local minima of the interaction energy. To check whether a construction considered is the minimum energy configuration, 3inatural frequencies were calculated from the secular equation det (m~i - a* Uj/a#axij = 0 with xf(l C s Q 3, I < j c i) being cartesian coordinate. In local equilibrium six of 3i frequencies equai zero. ’ The potential energy surfaces for cIusters of some three to sixty atoms under (12 : 6) Lennard-Jones forces have been system&: tally explored using numerical optimization techniques by Hoare and Pal [17]. Unfortunately, this work came to our too late to be incorporated into our own investigations. In table 1 we compare semi of their results with our calculations. ’ The distance between the nearest neighbours d, = adO was calculated by minimizing U&Q) = Zl
attention
B_ Barboyjclustering phenomena in adhesive hard sphere fluids
235
w
lL.0
1
.a 12.0
osa
10.0 ’
0.4
0.6
0.8
1.0
o
a8
l
0 CD
o
2
0
93
0.6
lJf/ie, as a function of the number i of molecules involved in a cluster. (1) - (12 : 6) potential; (2) - (9 : 6) potential of Lennard-Jones.
Fig. 1. The &tic energy per molecule,
0.8
1.0
Fig. 2. The average frequency, c+ as a function of the cluster size for (1) - (12 : 6); (2) - (9 : 6) Lennard-Jones potential.
a
-6.0
Fig. 3. Dependence of the rotational properties of the cluster on its size. (1) - (12 : 6) potential; (2) - (9 : 6) potential.
2.o"‘o
A
Fig. 4. The standard chemical potential of a molecule in i cluster for three reduced temperatures: kT/.c = 0.4; 1.0; 10.0. (a) - (12 : 6) potential;(b) - (9 : 6) potential.
0 17
vui 0 e _ cj[a(O)
_ a(l) f-1/3 ,3i-6
+ or’2,i-2/3
1
I
0.4
I
,
0.6
1
I
-
8
0.8
1.0
01)
I
,1/3i-6
oi = do(~141~2 ( oFlv=)
zz
f&U _ &I
j -1 P + ~(2) i-213
(12)
I
and In Ii =$ In (J1J2J3/iym3$)
= -I(O) - I(1)i-1’3
,
where the coefficients &I, &), and I@) are listed in table 3. Inserting (I 1) - (13) into (7) yields the interpolation formula
(13)
for the cluster partition
function
(the standard
236
B. Barboy/clustering
phenomena in adhesive hard sphere fluids
Table I The cluster configurations and tie value of U!, Zi, oi, wi and the contraction Factor [IIFor (12 Number of molecules i
Contiguration
2 3 4 5 6 7=) 8 9 10 11 12 17 19 25 30 b) 40 54 66
dumb-bell equilateral triangle tetrahedron trigonal bipymmid octahedron 1:S.l 1:3:3:1 3:3:3 1:4:4rl 1:3:3:3:1 1:5:5:1 5:1:5:1:5 6:7:6 3:6:7:6:3
- U,’iis
- @lie
: 6) Lennard-Jones potential
l&P,
Oi
wi
Q
-4.9698 -4.9698 -2.3776 -4.3035 -1.6292 -1.0080 2.9359 0.7445 1.9473 2.5842 3.0730 4.6670 6.2218
2 6 12 6 24 10 12 2 8 6 5 10 6 6
12.0000 11.6650 11.3265 11.0660 11.6161 11.5359 10.5174 11.4599 11.4316 10.9392 11.0014 10.7960 12-4846 12.6101
1.0000 1.0000 1.0000 1.0006/0.9971 0.9955 0.9952 0.9957 0.9920 0.99 14 0.9903 0.9871 0.9832 0.9889 0.9881
[171 0.5000 1.0000 1.5000 1.8208 2.1187 2.3554 2.3712 2.5845 2.6763 2.8753 3.1026 3.4714 3.6339 3.9040 3.9857 4.2281 4.4573 4.5983
0.5000 1.0000 1.5000 1.8208 2.1187 2.3578 2.4773 2.6792 2.8420 2.9786 3.1639 3.6063 3.8242 4.0948 4.2724 4.4904 4.7341 4.6723
a) Coniigumtion marked m : n :p consists of m particles in the fmt, n in the second, and p in the third molecular layer. For instance, 1 I 5 : 1 is the pentagonal bipymmid: 1 : 3 : 3 : 1 is the tetrahedron capped by four tetrahedra (the fully stellated tetrahedron); and so an. @The last four values of Vi” are calculated with the help of eq. (11).
chemical potenti& of the cluster): /3$
= -ln(zi/v)
=
i(3 kl(Aw#-*)
+ (3 - Ai) ln(T*/w;)
- [CP -&-l/3
+ cP-q/T*)
- pi - 4 In i i- In 6 u. + I(O) -i-l(lklB
,
(14)
Table 2 The cluster ConfigUmtionSand the ValueSof UzP,f+ UjsWj and contraction factor 0 for (9 : 6) Lennard-Jones potential Number of molecules
Configuration
- lZ,?/jE
dumb-b& equilateraLtriangle tetrahedron trigonal bipyramid octahedron 1:5:1 2:2:4 3:3:3 1:4!4:L lr3:3:3:1 i:s:s:L 5:1:5:1:5 6:7:6 3:6:7:6:3
0.5000
Ln(ZFl-5)
9
wi
Q
2 6 12 6 24 10 2 2 8 6 5 10 6 6
10.3923 10.1022 9.8090 9.6075 10.1330 10.0753 8.7958 10.0454 10.0334 9.7435 9.9836 9.9319 11.0900 11.2315
1.0000 1.0000 1.0000 1.0001/0.9966 0.9934 0.9934 0.9874 0.9877 0.9866 0.9863 0.9825 0.9768 0.9824 0.9810
i
2 3 4 5 6 7 8 9 10 11 12 17 19 25
1.0000 1.5000 1.8271 2.1458 2.3815 2.4253 2.6520 2.7552 2.9463 3.1912 3.6095 3.7795 4.0766
-4.9698
-4.9698 -2.3811 -4.3162 -1.6403 2.3264 2.9101 0.7154 1.9229 2.5560 3.0340 4.6277 6.1845
E. BarboyjCIustering
Table 3 The values of coefficients a (k). &)
andI
237
in eqs. (ll)-(13)
(9 : 6) Lennard-Jones potential
(12
interval of i
intervalof i
a)
c)
b)
: 6) Lennard-Jones potential
a)
b)
c)
o(o) ,(I) &)
6.9648 126863 5.7215
6.9648 8.7645 0
9.6751
6.6362 11.9437 5.3075
6.6362 8.2356 0
8.6110
w(o) &) &)
16.6968 23.7086 19.8633
16.6968 23.7086 19.8633
32.6723
17.0535 20.1682 18.1444
17.0535 20.7682 18.1444
50.2825
13.3718 -11.7597 0.6854 0.6653
4.2371 1.9702 -
13.3718 -11.7597 0.6987 0.6549
4.2789 2.1252 -
I(O) I(‘) (iz)-‘n ($)-I” a) (i*)-‘”
phenomena in adhesive hard sphere fluids
< yin
< 1.
b) i-‘”
< (i*)-1”.
c)i*
1.
where
T* = kT]e , Ai=-(3.6988
-5_6416i-‘p
+ 0.4428 i-*/3)H(3
-i) ,
(17)
and
r(3i _ 4.5 + Aj; [o@)i - (y(t) i2j3 •r-LX(~)N~]/T*] lf?‘=j In
r(3i - 4.5 + Aj)
H(x) is the unit step function and A = h/2sdo(r7ze) Ii2 is de Boer parameter which expresses the influence of quantum effects on macroscopic properties of the system (in this work, we neglect it). This expression will be exploit in the next section to derive the equation of state of the system. We confine OUCselves to analysis of the relations obtained and did not perform numerical calculations of cluster distribution since our aim is a description of the macroscopic (thermodynamic) properties rather than microscopic ones.
4. Clustering phenomena and an equation of state 4.1. Dilute
systems
In a system of low density
(1%
9l-J=f+)~1, we can ignore cluster interaction
and write
(20)
238
B. BarboylCusteringplienomena in cidhesive hard sphere fluids
Then eqs. (3) and (20) permit the calculation of~the concentrations pi{; = 1,2,3, _.-); substituting these {p-} into (2) and using the relation flP= Ejipj yield the following equation of state in-the parametric repiesentationl Y
6uo pP =
Y2
[exp(llr*)
27-*(27rT+)1~2 + 2w2T*(T7-*)‘12
+&& i=3
I I (T*j3 exp(9.I - pUIp)(y/~?)~
6u0p=
[exp(l/T*)
27-*(27Tz-*)1~2 + W2T*(7rz-*)= + 2
i5 liWi6 -exp(cp. Q-*)3
+3
1
-/WP)(y/ I
mj
(21)
’
Y2
J’
- 1 - l/T*]
3)i
- 1 - l/T*]
(22)
2
where y = 12uoT*(2rT*)11~
epul/X3 _
(23)
By eliminating y from (21) and (22) we obtain the virial expansion fect gas: pp=p+c,
,+2
k
of the equation
of state of the dilute imper-
pk,
(24)
with B2 = -(24u,+~)(rT*)~~~
[exp(l/T*)
B3 = 4Bz - (3625v~/w~)nT*(2~T*)1/2 and so on. It follows from (21) that the concentration i41iWf Pi=
12uOT*(2rT*)1/2 w;
~u,,(T*)~ [PI
Comparing
- 1 - l/T*] exp(3/T*)
,
(25) ,
y(4-5’3’T*) JY(4.5)
of clusters containing
(26)
i atoms is:
i 3 expC+ - W,“) -
(27)
(27) with (25) and (26) one has
P~/P;
= -B2
(274
,
and p 3 fp31 =2B2-LB 2 3 _ 2
G’b)
The series on the rhs of eqs. (2 1) and (22) converge if the following inequality &.f < - (c@/T*
_
+ lim cpili>+ !.n[(h~~~~)~/l2q,T*(2nT*)~~~]
f-+02
is satisfied:
(28)
Here (see appendix) h
i-km
pi/i = 3 [ln(d”)/3T*)
- 0t(~)/3T* f l]H(l
- &‘)!?T*)
_
(29)
B. Barboy/Cbtstering
phenomena in adhesive hard sphere fluids
239
At a given density p the excess of the chemical potential, fl,u - ln pX3, is limited in its magnitude, and thus condition (28) holds if-the temperature is sufficiently high. Similarly, at a given temperature T the series in eqs. (21) and (22) are convergent when the density p is small enough (as follows from the fact that Iim,,0 /3~ = --)_ Therefore the equation of state given by (21) and (22) will hold as long asp is less than a certain critical value pf which depends on the temperature. It is possible to show (for example, in the same way that ter Haar proposed [6] that the analytical continuation of eqs. (21) and (22) for p >p, is: @la)
6uopP=fDf),
whereyfis given by eq. (23) at p =~~andfDf) is the sum of the series on the rhs of eq. (21) aty =I’~ Thus the breakdown of convergence of eqs. (21) and (22) can be associated with the onset of a constant pressure portion of the P-p isotheml (i.e. a phase transition). It follows from eq. (28) that at low temperatures the density pf satisfies inequality (19): pf CCexp(-JO’/T’)
(T*+l).
Q 1
This relation leads to the following equation In Pv =A(T)
(30) describing
the vapour pressure Pv:
+ In(T) - C/T _
(31)
Since the temperature is low enough we can ignore the second term in the rhs of (31) and the temperature dependence of the coefficient A _Thus the low density theory predicts that In Pv varies approximately linearly with l/T, in quaiitative agreement with experiment. 4.2. Clustering in a f&lid at moderate and high densities 4.21. Hard sphere approximation for repulsion between clmte~~ In dense fluids the interaction between clusters is not negligible and must be explicitly included in the theoretical description. For the moment we ignore the attraction between the cIusters and take into consideration only the volume ui which they occupy. Then we can describe the pressure of the system and the chemical potential of an i cluster by use of the known solution of the Percus-Yevick equation for the hard sphere mixture [18,19] : $$p=-
g0
1 -E3
+3 - hE2
+3
(32)
(1 - &)2
~~~=~~~=~~~+in~i-In(l-~~)+~nd~pP+~
3d1 _ g3 (& +digl)
(33)
where (34) and di is an equivalent diameter of an i cluster so that the volume Ui of the cluster is equal to $Kd:. Substituting pi from eq. (33) into (34) allows us to obtain E, in terms of 0.u:
Inserting eq. (35) into (32), (33), and (2) then yields expressions for the pressure P and the density p in terms of fip in the form of an infinite series (but not necessarily a power series of exp&) like (21) and (22)). This representatidn will give us the equation of state only if the series on the rhs of (35) converges. This occurs for:
240
B. Barboy/Clustering phenomena in
p(p - +7rP kn dT/fj< j-+m
adhesivehard @herefluids
-(oI(O,p*+ h (pi/i) + 3 ln(AcJ”)/T*) , j+m
(284
and again this inequality, like condition (28), holds for high enough temperatures or low densities. In other words, eqs. (32) and (33) also predict a phase transition in the system where p = pf(T) but here, unlike eq. (21) and (22), we are not limited by the case (19) of very low densities_ The character of this transition will be discussed below (section 5). 4.2-2. Sticky hard sphere model for descnption of the whole ckSte?- interaction Consider now the whole interaction between molecular aggregates, that is, take into account an attraction a~ well as a repulsion. It seems reasonable (at least For large aggregates) to describe this interaction by the use of the “sticky” (i.e. with a surface adhesion) hard sphere model which was proposed and considered for the first time by Baxter [9] _This model is characterized by a rectangular well potential in which the range of the well becomes zero and (simultaneoudy) its depth infinite, so that for two clusters consisting ofiand~molecules we can write their interaction potential, uiP in the form 5 exp [-@all]
= [(di+ dj)/4rJ6_ [r - ~(di + dj)J f H[r - ~(di f dj)] .
(36)
Here S_(x) is the asymmetrical Dirac delta function, i.e. for any function f(x) there holds the relation: f(b - 0) = j
dXf(X)&_(X
-b)
(b >Q)
,
(37)
di is the equivaleit cluster diameter characterizing the size of i clusters, and T is a dimensionless parameter which measures the temperature (and the strength of cluster attraction), so that as T goes from 0 to infmity as does 7; We assume hereafter that cluster collisions do not influence the inner structure and properties of the aggregates, i-e_$, di and r depend only on i and T. Then, in the Percus-Yevick approximation, the compressibility equation of state of such a system is written as [20,21]:
where Xi/ are given by (38) Note that eq. (32) is obtained from (32a) by taking ail Xii = 0. (We can see from (36) that in the absence of Bttraction, T is infinitely large and thus it follows from (38) that all hii equal zero). When considering the attraction between the clusters (r is finite) we encounter great difficulties. Although one can write down the expression for the derivatives $.+/&I~ [20] we are not able to perform the necessary integrations and obtain thereby the chemical potential pi as a function of temperature and all concentrations &(k = I, 2,3 ...)_Therefore we have to introduce an additional approximation. The simplest that we shall discuss is a model according to which the system consists mainly of individual molecules (monomers) but where there are also clusters (i > 1) in very small concentrations: UipilUlp1 < 1
(i 2 2) _
z Our T is six times as Baxter’s symbol T.
(39)
B. BarboyfClustering
phenotnenn
in adhesive hard sphere j7uids
241
Assume that in this case eq. (38) permits the following power expansion for parameters XV: (40) Insertion into (38) enables us to obtain the coefficients of the expansion in terms of the density pL(=p) and diameters di (and r, of course). Then we can integrate the expressions for a~ii/api and obtain the chemical potential also in the form of power expansion like (40) Using the first terms of these expansions yields the following distribution of clusters: Pj-/rOl = exp[F& - $33
(ia 2) ,
(41)
where function tii(~, pl) and the chemical potential p = pL1are given by expressions (44) and (44a) in ref. [20]. It can be shown [20] that in the case when: Idi - 3dl I/@. f dl) < [(r& - I)(1 - 2/T&] 112,
(42)
with
(43)
rC = 2 - 2112
the function J/i is continuous_ Otherwise $+ has two poles. In the neighbourhood following asymptotic form (X\y = A,): 2d_&$. - 3dl)(A,xil NPjlP1)
f di/dl - 1)2 InIX:? -X,1
=* 34(di
- dl)(h,
of these poles eq. (41) has the
,
(414
- A_)
where [20]
A* = x;’ { 1 - 7 f 6xi1 f [(I - ~~~ - 6x&
- 3~,)]~‘~)
,
(44)
and Xi1 = $(di
- dl)/(di
f dl) _
(45)
In this case (A,, (‘I = h ) the power expansion (40) is divergent. It follows from els. (40a) and (44) that at sufficientiy high temperature (T > 1) parameter Xi!) is not equal to h, for any value of the density and ratio di/dl (the ratio of the cluster equivalent diameter, di, to the hard core diameter, d,, of a molecule). Hence in this case eq. (41) describes adequately the cluster distribution_ Unfortunately, if r < 1 there exists a range of the density p1 where Xfy = X, for sufficiently large clusters (di/d, > 3,i.e. i> 27). Since expansion (40) includes all values of i 2 2 we conclude that in this case (T < 1) eq. (41) does not hold. This precludes our being able to derive the cluster distribution in the most interesting region of temperatures and densities. However, we shall see in the next section that it is still possibIe to describe the macroscopic behaviour of the system.
5. The theory of freezing 5. I. General considerations
In this section we focus our attention mainly on the thermodynamic (macroscopic) behaviour of matter. For this goal discuss a model that describes the system as an assembly of identical clusters, i-e_ the ensemble of molecular aggregates of different sizes is replaced by an equivalent system of clusters each of which consists of the same
242
B. BorboylCl~tstering
phenomena in adhesive hard sphere fluids
number of particles, i* (i* need not be an even integer although, of Course, i* > 1). It is worthwhile to note that this model is of interest to the chemistry of monodisperse systems of solid particles [22]. In this case we can omit the sum sign in (1) and rewrite it in the form: F/N= p - P/p = pi Ni /N - P/p = pi/i - P/p _
(la)
That is, relation (3) is satisfied automaticahy. Thus we define i* as the value of i which minimizes F (with the temperature and the density held fmed). In the low density system (pid; < 1) one has: ~PIPi=l + O@,L$) )
(46)
0~; = 0~: + In Pi f 0 @id: ) a
(47)
and thus: OF/N= i-l
f 6 (1) _
l&/i)
Since for any pair i>j> i-1 ln(&
(48)
1 (4%
> j-l In(pjj) ,
it follows from (48) that F(i=
l)
(50)
l),
i.e. Fis higher for a system of iclusters than for a system of l-clusters (individual molecules) and thus for any given temperature:
(51)
1iIlli*=1 P-+0
The same result is obtained in the case of very high temperatures for any ftied density p. Indeed, one gets from (14) and (A-1) (appendix): (PIN)F(T+
3- I;i<
0~; &rpid:
< 1) = - 1Si-1
In T* + 6 (1) ,
(52)
and therefore lim i*= ;p*-+m
1.
(53)
Of course, relations (51) and (53) are trivial. A more interesting case is represented by the system at moderate and high densities and at low temperatures_ In the model considered the pressure P of the system and the chemical potential pi are described by the following expressions [9,23]: lJ#P =
Vj+VF +s,? (I - Uj13
- Aiqi
18q.(2 +q.) - h?$ I
1
I L ,
CQb)
36(1 - TQ)~
3xiqiViqi - 8qi - 2) -- 3nli l -qi (54) Here
243
9. BarboyjClustering phenomena in adhesive hard sphere fluids
(5%
qf - viPj and Xi is given by (40a), where vlpl is replaced by vi_ Remember When pi + 1 one obtains from (32b), (54) and (40a)
that vi = n/M:
is the volume of an i cluster.
-Vi)3l II1+O (l -77i)l 3
uiPP= 113(1-rJ2/(1
(56)
and pal, _ #)
_ m Pi - 13plpi = [3(1 - ~,)~/2(1
Thus when vi 41
we get for any temperature
-
q-j21[1 + 60 - $1 _
(57)
and j < i < ~0
F(i)
(58)
that is, in the system at very high densities,
i* > 1
(59)
In the low temperature finite. Furthermore: &f = (T-)-r
q/c
limit (T* 6 1) at any fixed density qj < 1, the values of pi exp[/3& + (3 - 3i - Ai)ln T* + 6 (1) ~
- $)I
and vipP are
(6’3)
and thus inequalities
(58) and (59) are seen to hold. Therefore we can conclude that a considerable increase in the density or decrease in the temperature yields a transition of the system, which was originally in completely disordered state (i* = l), to the state in which most particles are located in lattice sites of very large clusters (i* > 1). This ordered state can be naturally identified with the solid. Thus this phase transition is the freezing of the system and, of course, the fluid density, pr, at which it happens depends on the temperature: pf = pr(T)_ To form a notion about the character of this dependence let us rewrite eq. (la) for the free energy of the system in the form:
fiF/N= bl(T*) -I-b2(T*, i)/i’p
- 5 III i/i
+ b3(T*, p, i)/i ,
(61)
where the explicit expressions for the coefficients b, can be easily obtained from (14), (32b) and (54). Note that in the most important cases, b2 and b3 are almost independent of i. So for qualitative analysis of the function F = F(i) we can ignore the changes in b, and b3 arising from increasing i. Then it follows from (61) that for 62 < 0 the function F = F(i) has a minimum at i = i,,,in, which is a single root of the equation b2i2~3-151ni+15+3b3=0.
(W
If b2 > 0 and 2b, + 25 t 15 ln(2b2/45) > 0 the free energy decreases monotonically when i increases from zero to infinity, and finally, in the case when b2 >0 and 2b3 f 25 + 15 ln(2b7/45) < 0, one fmds that F= F(i) has a minimum at a low value of i (imin < (22.5/b2)3/2) and a maximum at a higher i(i,,, > (22.5/b2)3/2). The latter case obtains for low values of p and fl= I/kT(If imin corresponding to the minimum of F= F(i) is less than unity it has no physical meaning and the free energy F obtains its minimum value at i = 1.) Increasing p orb yields an increasing Fmin (and i,,,i,, as well) so that at a certain p = pr one gets:
F(i*, T*, p%) = F(i + m, T*);
1-* = imin or i* = 1 _
(62)
Further increasing p (or lowering the temperature) causes F(&, or i = 1) > F,, and thus a system of high density (pressure) has to undergo the phase transition from fluid to solid f. In the above analysis the system is considered to be in a single phase. So it is clear that the density pf of the fluid which is in equilibrium with the solid is less *For footnote see next page
244
B. Barboy/C%steriig
&MT*, Pf) =
Pji(T*, Pf) i
j=i*
phenomena in adhesive hard-sphere fluids
F(i, T*,~3
= lim
N
i+m
.
(63)
We see that inequality (28) yields the same condition for the phase transition while (28a) gives too large a value for pt. It is even greater than pp defined by (62): T*, p) =N-IF,
N-lF(i*,
+Pp-l(l
Since at low temperature PFJN pf = I$ exp [-c@/T*]
- lim qi) _ f-t-
CW
= -ol’o,/T*, one gets from eqs. (63) and (51):
< 1
(3Oa)
(which coincidences with (30)). On the other hand at high temperatures we have: &~=aF(i=
l)[&V=-1Sln
PF,/N=3{ln[3Aw(0)/,(0)]
--u~P&~ ,
T* + 3(1 -rJ2/(1 -
1)
(64) $55)
and thus qpf=
1
.
(66)
Therefore the density pf is an increasing function of the temperature, being zero at zero temperature and large at T’%-1. It is known [9,20] that at sufficiently low temperature the system described by eqs. (32b) and (54) undergoes a phase transition between two disordered states with different densities. One can identify this transition with that of liquid-gas_ Since the liquid density at evaporation, pn, increases when the temperature is lowered the curve pa = pa(T) must be truncated by the curve of = pf(T) at a certain temperature T, which is known as the triple point. Hence the model discussed above enables us to describe two phase-transitions: liquid-gas (at temperatures below the critical one, T,) and fluid-solid, and thus to obtain the regular phase diagram which includes (in coordinates pressure-temperature) three separate branches intersecting at the triple point, Tt. 5.2. Some properties of the fluid-solid
transition
The case of high temperatures requires additional considerations. We describe the solid as a very big cluster (infinitely large, in the limit) and hence the free energy of the solid, Fs, is given by:
W-*1
P,=
lim
p
i-+03
F(i.
T*, P) = iv
3
ln
ArJ”) --+3do) T*
T*
(67)
We assume here that the clusters and solid are incompressible, i.e. the cluster diameter di is undistorted through collisions and increasing the pressure does not affect the solid volume. Hence the solid free energy F, is dependent on the temperature, but not on the pressure_ Moreover, it follows from our assumption that the solid density, ps, is constant and represents the limit density allowed. In fact, this assumption is reasonable only at very low pres- Fmtd Z+1 the system can during a long time be in the metastable state corresponding to F,,,h but not to F,. We have pertkmed numerical calculationsin the wide range ofthe temperatureand the cluster size (0.16 < T*_< 63 and 1 < i < 300). The results show that &(T*, rJ is positive_Therefore at very high density F falls monotonically with increasing i and thus the
* IfLV,,,
system can exist only as a solid. However, for solutions where aggregates of solute molecules are in surroundings of a solvent it is possible that 62 < O_If at a certain condition ~(i min) is significantly lower than F(i = 1) and F(i = -) and besides 100 < &,., < 1000 the system exists in the form of thermodynamically stable colloid solutions [22] (see a!so ref. (241.).
B. Barboy/CIustenttg
phenomena in adhesive hard sphere jluids
245
is alrvfoos* not quantitative~ valid at very high pressures (e.g. P= Pcs; see below). But we do not expect this neglect of the solid’s compressibility to affect substantially any of our conclusions. It follows from (67) that the chemical potential of the solid, /.+ = FJN, and its first derivative, ap,/aT, are continuous at any temperature while the second derivative, a2@fl, has a discontinuity at the temperature: srres - it
That is, at ‘&is temperature the heat capacity of the solid, c, = -M@
aQgap2 ,
(6%
has a jump AC = 9kN(eo(“))-2
(lim
a2flpslas2 - r_hm+o
a20~,idfi2) = -3kN
T-T,,-0
ISa condition for the second order phase-transition beEq. (69a) (and the fact of continuity of i&laT at T = T,,) _ tween the solid with the heat capacity C, = 3 kN and a state (which is also solid because i goes to infinity as well) with C, = O(!)_ TO resolve this contradiction we rewrite eq. (68) in a slightly different form: Es, = N(3k T,, + lhh ui”/i) = 0 According to the cluster definition (SC), the internal energy of the solid, Es = Oii~_,, Hi), is always negative Es = N(3kT f lint l&i)
=0 ,
(70)
and thus above the critical temperature T,, the solid is unstable. Eq. (70) has a clear physical meaning* _We have defined the solid state as a state where the atoms occupy the sites of an ideal lattice and oscillate near their equilibrium places. When the temperature is less than Tc, the average kinetic energy of these oscillations (3kT) does not exceed the average potential energy per atom due to its interaction with the other particles in the solid (-hrr~~_+, @/i). Hence these interactions are able to hold the atom in its position. But if T > T,, these forces are insufficient and the atom “wanders” off. Since it happens to any “average” atom such state cannot be associated with a solid and must be identified instead with the fluid state in spite of its high density (which is equal top,). We can also conclude that the values of pf obtained from eq. (63) for T > T,, must be bigger that ps and hence do not have a physical meaning. The above interpretation of T,, permits us to understand better the existance of a certain temperature above which the solid is unstable at any pressure. This is connected merely with the definition (5~) and (following - in the frame of the microcrystal model - from it) inequality (70) though, of course, the value of T,., itself, the density ps and so on, depend on the particular approximations adopted. The temperature T,, is the critical temperature for the solid-fluid phase transition and we emphasize that the pressure,Pc,, at this point is finite though very high*. This fact has no consequences in the present theory where ps = const. is the highest density allowed. But in a theory which does not ignore the compressibility of the solid, the finite value,Pcs, of the critical pressure means that the continuous change of the system from fluid to sclid is possible (like the known continuous change ofgas to Liquid). In thinking about how a completely disordered system is changed continuously into a state with some kind of symmetry, it should be born * Landau and Lifshitz [25] note, as an evidental fact, that for the body to be solid it must in any case have kT much less than the static energy attributed to a single molecule in the solid. This requirement coincides apparently with (70). ’ This result (Pcs < -) has been obtained on the base of the Percus-Yevick approximation (32b) and mi&t, but unlikely, be abolished in a more correct theory.
246
B. BarboylCllis~ering p~leno?ne~ in adhesive hard sphere fluids
in mind that the in&r structure of all real solids is far from a perfect lattice. Since deviations from ideality increase with the temperature, the symmetry of the solid becomes more and more local as the temperature increases. At the same time in fluids of very high density, there undoubtedly exist regioni possessing a sjmmetrical structure. Thus at high temperature and pressure the solid and fluid states differ from each other because of the differing character of molecule motions rather than structure. 5.3. Quantitative rest&s In order to obtain quantitative results we need to relate 7 and d;(or uj) to the temperature and molecular parameters do and E [defined by zlpair,. eq. (IO)]. It can be done via the second virial coefficient [23], i.e. we choose r and the ratio: Qo = u1/uo =(d&))3
,
(71)
so that the second virial coefficient given by** B2 =4IJ,(l
- 3/%)
(72)
(the expression can be easily obtained from eq. (32b) and the condition i = 1) is the same as that for the LennardJones potential (iO), that is, the following equality holds: aO(l - 3/2r) =3 [
(1 - ex~[~*)(-$
-$)])GdX
(73)
In the low temperature limit, r Q 1 and a0 = (6/~)~/(“-~)_ Thus r(T* Q 1) a (r*)-l/*
exp(-l/T*)
.
(744
This regularity follows also from combination (25) and (72) at T* < 1. It suggests the following form of dependence r = T(T): l/2 exp( l/T:) exp(l/T*)
- 1 - l/T; + c/Ti*r - 1 - l/T* f C/T*“’’
(74b)
where TE is the Boyle temperature at which&(Ti) = 0. The coefficients c and y > 1 in eq. (74b) can be calculated with the help of the following conditions: (i) &(T* > T,‘,) >ps, where pf satisfies eq. (63) and pf=ps, when T* = T,*,; (ii) at the critical point of the liquid-gas phase transition (where both first and second derivatives of P are zero) r(T,*) Z TV,since 7c is the upper limit point of the domain where no state of a homogenuous system can exist [9,23] (we can note a posteriori that since at the critical point i* = 1 then r(Tz) = rJ; (ii) at the triple point which is defined by: P&T;) = N-,+, P*) = M,+, P,)
(75a)
P,) = W,*, /JJ -
Wb)
and JV:,
T has to be bigger than the lower limit point, in, = 0.292 ~~~of the restricted range of the two-phase (liquid-gas) region of the compressibility equation of state [26] ; (iv) the isotherms do not cross each other [and again a posteriori we can note that condition (iii) holds as long as condition (iv) holds] _ BP The second vidal coefficient of the cluster assembly described in section 4.2.2 [eq. (32a)j is B sum of the expressions in the rhs of (25) and (72).
B. Barboy /Clusteringphenomena
in
adhesivehard sphereflaids
247
Table 4 Constants characterizing the interaction between molecules and between clusters Potential
c
Y
7.;
9:6 12~6
1.7347
1.2045
1.7193
1.2190
The volume per molecule in the solid, p;l, p;l
= 2-‘~2~o(6/7r)(s,/s6)(3/~-r-9
- lim qqei
lim wi
0,
i-(0) =CV
j+m = JO)
4.5554
1.1672
3.4183
1.2372
9.6751 8.6110
50.2825
32.6723
is obtained by (for the hexagonal close packed lattice [27])
= uoa,
,
(76)
where s~(IC=R or k = 6) is the ratio of the static energy of an atom arising from its interaction with the rest of the system to the energy of interaction with the nearest neighbours. If the distance between the nearest neighbours is &., and the molecule interaction is described by the inverse power potential u(d,) =Adn$, the constants sk has the following values for different k:+j= 1.2046, sg = 1.0411, and s12 = 1 .Ol 10. The above conditions do not enable us to evaluate c and y in (74b) but only the range of their values conforming to those conditions. But it turns out that these ranges are surprisingly small (a few fraction of percent) and hence practically exact values of c and y are obtained. In addition, the above conditions permit us to caIculate the average frequency of the solid, w(O) = limi_, Wi. Results of these calculations are listed in table 4. As soon as r is given, the ration a0 can be obtained by eq. (73). It depends on the temperature being (6/1~)~/“-e for T* + 0 and zero (a0 0~(T*)-1/4) as T* +a- but still a0 = l/3 when T* = 60. Suppose we denote Vi = iVlai
,
OW
and take for simplicity ai=a,
-(a,
- l)F1j3
_
(77b)
Using eqs. (73)-(77) we have made numerical calculations of properties of the system considered. The results are collected in tables 5 and 6 and shown in figs. 5-7 (the isotherms P = P@) and 1_1=,u(p) can be seen in figs. 2 and 3 of ref. [23]). We have included also. the results of some other theories, and numerical simulations and experimental data, for the purpose of comparison_ Agreement between data listed is satisfactory in spite of higbly idealized character of the adhesive hard sphere potential (36) and imperfection of the model considered in this section. In addition we have calculated the first coordination number, NC1 defined by [38] : 4 Iv,1 = 47r/I s g(r)r2&= 0
[2VrP/(l - U#)IXt
2
(78)
whereg(r) is the radial distribution function_ It characterizes the average number of molecules being in contact with a given one (the number of the nearest neighbours) in a fluid. We can see (fig. 7) that at gas-liquid and fluidsolid phase transitions NC1 changes from very small values (for gas) to sufficiently large (at freezing) which are close to the coordination number in the solid (NC1 = 12).
248
--
B. BarboyjCiustering@enomena
Table S The values of the critical constants
and the properties kT/e
the c&i&l
point of solid-fluid
(9 : 6) potential (12 : 6) potential
(9 :6) potential (12 : 6) potential (a) PY (e) [281 (b) pY (c) [281 (c) perturbation theory 1281 (d) numeriul simulations [29] (e) esperimental data [30] (fJ square-well potential (g) PY (e) 1311 (h) PY (c) [311 (i) HNC (c) [32] 0 HNC (v) [321 (k) YBG [33] (1) perturbation theory f34] (m) numerical simulations [35]
.I
of the system in the triple point
%P
VOPIE
0.8568 0.8083
9.0254 10.0784
E-lp-T*lnA3
PPIP
7
transition 3.2250 2.8703
the critical point of liquid-gas
in adhesive hard sphere fluids
12.7284 16.0443
1.1413 1.2989
-0.5349 -0.1461
0.5858 0.5858
-1.0986 -0.7934
0.2930 0.2557
transition 0.08810 0.06432 0.104 0.096 0.126 0.111 ?r 0.015 0.0866
1.4803 1.1866 1.34 1.32 1.41 1.34 * 0.02 1.26
0.1569 0.1429 0.25 0.21 0.24 0.25 t 0.01 0.234
1.27 f 0.01 1.205 f 0.003 1.30 + 0.02 1.16 f 0.02 1.56 2 0.02 1.39 1.28
0.24 ) 0.01 0.148 5 0.002 0.17 f 0.02 0.19 f 0.02 0.31 * 0.02
0.3794 0.3194 0.31 0.36 0.38 0.33 -c 0.03 0.293 0.313 + 0.025 0.37 f 0.01 0.42 i 0.03 0.3 2 5 0.04 0.48 + 0.02 0.393 0.306
the triple point (9 : 6) potential [12:6) potential (n) perturbation theory 1281 (0) perturbation theory [28J (p) numerical simulationsj36] (q) experiment?l data [37]
0.7602 0.5851 0.64 0.704 0.68 -C0.02 0.699
Data (a) -(d) and (a) -(p) are related to diameter to hard core diameter) and ue = in the Percus-Yevick approximation; (i) state); (k) - Yvon-Born-Green theory;
Table 6 Comparison
of calculation
0.004311 0.002435 0.00044 0.0015 0.00121
(12 : 6) potential; (g)-(m) to the square-well potential with d/u = 1.5 (ratio of the web $ros 2l”; (a) and (g) - the energy equation, (b) and (h) - the compressibility equation and (j) - hypemetted Ichain approximation (the compressibility and virkd equation of (n) using celt model for solid phase; (0) using machine calculation results for solid phase.
results and experimental pgCTt)
PC
properties PQ(T,)
PC
of simple liquids pt
Tt
WC
TB
K
<
PC
TC
experimental data present caIcuIations
0.0079
2.66
0.014
0.555
0.293
2.67
(9 : 6) potential (12 : 6) potential
0.03997 0.03203
3.843 4.023
0.04893 0.03786
0.5135 0.493 1
0.3794 0.3794
3.0774 2.8807
B.
Barboy/Clusteringphenomena in adhesive hard sphere fluids
249
Gas-vapour -----------p:
1.0-
Solid-vapour
.
rcs
b
kT Tc.
Liquid-solid Liquid 7
. \
r: _ i
\ I
10-3 Fig. 5. The phase diagram of the system with (12 : 6) potential (a) and (9 : 6) potential (IJ) (the pressure and temperature are given in logarithmical scale).
tic1
I
US*
r;
I
10-l
“OP
Fig. 6. The phase diagram in coordinates of dimensionless density uop and temperature T* (both in logarithmical scale) for (12 : 6) potential (a) and (9 I 6) potential (b).
b
6-
.%-
2-
-/;
0
:0
1 0.50
0
0.50
Fig. 7. The fmt coordination number at phase-transitions. plotted against the density (a) - (12 : 6) potential, (b) - (9 : 6) potential.
5.4. Concluding remarks One of the important results obtained is the conclusion that at any temperature .* r -1 3
(7%
i.e. in the model approximating the system by an assembly of identical clusters, the molecules do not form aggre-
gates when p
Acknowledgement The author would like to thank S. Baer for his interest and many very helpful discussions in every phase of this work. He is grateful to W.M. Gelbart for reading the manuscript and valuable remarks. He wishes also to express his thanks for the kind hospitality of the Laboratoire de Photophysique Moleculaire, Orsay.
Appendix: Selected properties of the incomplete gamma-function A full treatment of the incomplete gamma-function y(rz, x) is given in special literature (see, for instance, ref. [401). We shall only list here the properties which are important for our investigation. When lfiU:l is sufficiently small, power series expansion of exp(-r) in integrand of (9) gives: (_fl$l)3i-4.5+Ai
exp(%)=
m c @!J,P)k
(3i : 4.5 -I-A,)!
,@O
k!
3i-4.5
+A, (A-1)
X-f 3i - 4.5 + Ai -
For IS@ I % 1 and fmite values of i the following asymptotic expression holds [40]: ([email protected]+Ai
exp(‘i) = ’ -
(3j
f
5.5
f
m Ai)!
(kt4.5
-Af--3i)!
(4.5 - 4- _ 3j)!
@U%k
-
(A-2)
We also need the asymptotic behaviour Of47i when i increases indefinitely, i.e. when both arguments of the gamma-function become inftite. One has for i> 3:
Lx@- CIf+1,3) exp(rpi,l,3 j =exp(& + Texp~~)(-llui0)3’-4-5/(3i
- 4.5)!] [ J
dteAf(l
0
1
- 1 _ - tj/3U~)3i-4a5 I
(A-3)
Using the inequality eX(l -x2/n)<(1
+~/n)~
for estimating the integral on the rhs of eq. (A.3) we obtain in the limit case of i % 1:
(A-4)
B. Barboy/Chsrering
exp(yi& X [
phenomena in adhesive hard sphere fluids
= exp&) + [exp(llVio)(-pV,P)3’~.5/(3i
,,P~@) If=PP - cu@)/3T*] -
251
- 4.5)!]
3T*/c#J~) f 0 (l/i) I
.
(A.3
Thus in the case when T* < 3 do) Cl< exP(&
(A-6)
< eXP(cpicl/3 ) < 1 .
Therefore (A.7)
(A-8) In order to check the case T* > iot*) rewrite eq. (9) in the following form: x/(,2-1) y(n, x) = (rz - l)n el-n
J
(tel-r)n-*
dt .
(A-9)
0 From the properties of function t exp(1 - r) one can easily obtain: O~tG$
exp[l-x/(rz-l)]t
exp [-xl@
- l)] ; (A.10)
<$J
exp[l -xl(n
- l)],
--&e,p(-,/(I,-
i)]
Hence y(n, xj > (x’*/n) eex
(A.1 1)
and r(n, x) < (x”/rz) eAx +x” edx [l - e-xl(n-l)]
.
(A.12)
Together with Stiriing’s asymptotic formula for the factorial estimations (A.1 1) and (A-12) give: pi > (3i - 4S)ln[-f3U~/(3i
- 4.5)] + 3i - 4.5 + &Cl,”- $ln 2a(3i - 4.5) - 12(3is_ 4 5) =A ,
(A-13)
and cpi
-eXp[pclio/(3i-
5.5)])+
1) ,
(A-14)
where 0< 0 < 1, lpUl~l(3i- 5.5)] < i. It follows from (A-13) and (A-14) that in the case CY(~)/~T*< 1 lim ~Ji=3{ln[oL(0)/3T*] i-+0
•t-1 -&‘)/3T*}.
(A.lS)
252
B. BarboyfClustering phenomena
in adhesive hard
spherefluids
References [L 1 J-E- Mayer ad M-G- Mayer, Statistical mechanics (Wiley, New York, 1974) chs 14,1X I21 J. Frenkel, J. Chem. Phys. 7 (1939) 200,583. [31 J- Frenkel, Kinetic theory of liquids (Oxford Univ. Press, London, 1946) ch. 7. [4] W- Band, J. Chem. Phys. 7 (1939) 324,927. I51 H. Mkrgeland, Avhandl. Norske Vedenskaps Akad. Oslo 11 (1943). 161 D. ter Haar, Rot. Cambridge Phil. SIX. 49 (1953) 130. [7] D. ter Haar, Elements of statistical mechanics (Holt. Rinehart and Winston, New York, 1961) pp. 216-222. [8] H-P. Gillis, D.C. Marvin, and H. Reiss, J. Chem. Phys. 66 (1977) 214. [9] R-J. Baxter, J. Chem. Phys. 49 (1968) 2770. [lOI J-K. Percus and G.L. Yevick. Phys. Rev_ 110 (1958) l_ [ill T-L- Hill, J. Chem. Phys. 23 (19.55) 617_ [12] T-L Hill, Statistical mechanics (McGraw-Hili, New York, 1956) pp_ X2-164. 1131 F-H_ StiUinger, I. Chem. Phys. 38 (1963) 1486. [14] J-K_ Lee, J.A. Barker and F.F. Abraham, J. Chem. Phys. 58 (1973) 3166. 11.51 J-J. Burton, J. Chem. Phys. 52 (1970) 34.5. 1161 J.D. Bernal. Nature 185 (1960) 68. 1171 M-R. Hoare and P. Pal. Advan. Phys. 20 (1971) 161. [18] J.L. Lebowitz, Phys. Rev. 133 A (1964) 895. [19] J.L. Lebowitz and J.S. Rowlinson, J. Chem. Phys. 41 (1964) 133. [20] B. Barboy, Chem. Phys. ll(1975) 3.57. [21] J-W- Perram and E-R. Smith, C&m. Phys. Letters 35 (1975) 138. [22] F-1. Khgman and A-1. Rusanov. Kolloidn Zh_ 39 (1977) 44 in Russian [23] B_ Barboy, J. Chem. Phys. 61 (1974) 3194. [24] V.M. Barboy. Yu.M. Glazman and G-1. Fucks, Kolloidn. Zh. 32 (1970) 511, in Russian. 12.51 L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon Press, London, 1968) p. 191. [26] EL Barboy and S. Baer, Chem. Phys. Letters 36 (1975) 175. [27] E-A. Moelwyn-Hughes. Physical chemistry (Pergamon Press, Oxford, 1961) ch. 7. [28] D. Henderson, J.A. Barker and R-0. Watts, IBM J. Rcs. Develop. 14 (1970) 668. [29j L. Verlet, Phys. Rev. 159 (1967) 98. [301 J.S. Rowlinson. Liquids and liquid mixtures (Butterworths, London, 1959). [31] Y. Tago, J. Chem. Phys. 60 (1974) 1528. [32] D. Levesque, Physica 32 (1966) 1985 [33] D.A. Young and S-A. Rice, J. Chem. Phys. 47 (1967) 4228. [34] D. Henderson and J.A. Barker, in: Physical chemistry, an advanced treatise, Vol. Sa, ed. D. Henderson (Academic New York, 1971) ch. 6_ 1351 B.J. Alder, D.A. Young and M.A. Mark, J. Chem. Phys. 56 (1972) 301. [361 J.P. Hansen and L. Verlet, Phys. Rev. 184 (1969) 151. 1371 A.M. Clark, F_ Din, J. Robb, A. Michels, T. Wassenaar and T.N. Zwieteriug. Physica 17 (1951) 879. [38] C.J. Pings, in: Physics of simple liquids, ed. H-N-V. Temperly et al. (Amsterdam, 1968), ch. 10. 1391 H. Reiss, J. ColIoid Interface Sci. 53 (1975) 61. 1401 Higher transcendental functions, Vol. 2, (McGraw-Hill Book, New York, 1953) ch. 9.
Press,
CLUSTERlNG PHENOMENA IN ADHESIVE HARD SPHERE FLUIDS. THEORY OF FREEZING
B. BARBOY * Labomtoire de Photopltysiqrre MoEculaire, UnivcrsitC de Paris&d, 9140.5 Orsoy, France
Received 31 May 1978
We develop P version of the physical cluster theory of the equntion of stnte bnsed on 3 definition of a “cluster” which depends both on particle positions zmd relative moments, namely we choose to call a group of i molecules a physical cluster if their total energy (kinetic energy plus interactions) does not exceed the translation energy of the cenrrc-of-mass of the whole group. This defmition in addition with the “microcrystal” model of a cluster allows us to calculate the properties of individual clusters, to consider the clustering phenomena and to wrivritedown an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken into consideration by using the adhesive hard sphere model and the Percus-Yevick approximation. Based on the relations obtained we develop the theory of freezing and perform a numerical investigation of the equilibrium behavior of the system at various temperatures and densities. In particular, the theory enables us to describe the regular phase diagram which includes (in coordinates pressure-temperature) three branches intersecting at the triple point. It is interesting that the coesistence curve of the fluid-solid as well as liquid-gas phase transition has an end point.
1. Introduction One of the most important aims of statistical mechanics is to predict the equilibrium behavior of substance. Mayer and his collaborators [I] developed a formally exact molecular theory of fluids which was based on the concept of the mathematical cluster. In contrast to this Frenkel [7_,3] and Band [4] regarded the imperfection of real gases as a result of molecule association into actual physicnZ clusters of various sizes, caused by intermolecular attractions. In this picture, they viewed the imperfect gas as a mixture of aggregates, each characterized by the number i of molecules involved, in dynamic association-dissociation (%hemical” equilibrium) with one another. In the original physical cluster theory of Frenkel and Band as well as in most applications, these molecular nggregates are treated as noninteracting; the system is considered to be an ideal mixture of clusters, each size corresponding to an independent species. The total pressure is calculated then by simply summing the ideal gas partial pressures. Based on these principles, and using Wergeland’s cluster defmition [S], ter Hasr [6,7] has derived the equation of state for an imperfect gas and shown that the isotherms in the Frenkel-Band approach to the P- u diagram consist of two analytically different parts: in the first part, for specific volume u larger than a definite critical value II,, the pressure P is a monotonically decreasing function of u; in the second part, for u
of California, Los Angeles, California, 90024, USA.
232
B. BarboyfCkstenig
phenomena
in adhesive hard sphere fluids
III the present article we develop a version of the physical cluster theory of the equation of state based on a definition of a cluster which depends both on particle positions and relative momenta [cf. eq. (SC), section 31. In section 4 we consider the clustering phenomena and write down an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken there into consideration by using the sticky hard sphere model [P] [eq. (36)] and the Percus-Yevick approximation [IO]. Section 5 is devoted to the investigation of the equilibrium behaviour of the system and to the freezing theory. The nucleation phenomena will not be delt with at all here.
2. Association-dissociation
equihbrium
We consider an assembly ofNmonoatomic particles obeying classical statistics and interacting via a short-range pair potential energy, I. The system is confined in a vessel of volume Vat temperature T and total pressure P. Let us denote by~i the number of cluster consisting of iatoms. Then the Hehnholtz free energy, F, of the system is given by F=C~iNi-PV, i where pi is the chemical potential of a cluster regarded as a distinct molecular species. The most probable distribution {N;} is derived by minimizing F with respect toNr keepingN, T and V fued
and under subsidiary conservation condition: N=CiNi. i
Keeping in mind the Gibbs-Duhem
relation we obtain in this way:
i=l.2,3,...,
I-$ =i.n9
(3)
where _uis the Lagrange multiplier. The meaning ofw is derived from eq. (I) by substituting (3) into (1): F=/.Uv--PV,
(4)
indicating that p is the macroscopic chemical potential. On the other hand, it follows from (3) that !.ris equal to the chemical potential, r_cl,of the monomers. Substitution ofp =,ul into (3) yieids the standard condition of “chemical” equilibrium among clusters, &=i!rl
9
W
which follows from the possibility of free exchange of atoms between aggregates. Ifall/li are given as functions of the temperature and cluster concentrations,pk =Nk/V(k= I, 2,3, _._),eqs. (2) and (3) permit, in principle, the calculation of the most probable distribution {pp} of molecules over clusters for any temperature and average number density p = N/V_
3. Definition of a physical cluster RI. Gene& considerations From the thermodynamic
point of view, the definition of a cluster is arbitrary and it is clear that any defmition
B. BarboyjChisteringphenomena in adhesivehard spherefluids
233
will lead to the same equation of state and macroscopic properties of the fhrid if all subsequent theoretical manipulations are exact. From the mechanical point of view, two atoms are bound as a pair (cluster of size 2) if their relative kinetic energy, Tre* 2 , is less than the negative of their potential energy of interaction, if2 = ~(‘1~): H2=T+J2<0.
(53
For larger aggregates Hill [II ,121 suggested a deftition which required each pair of atoms in a cluster to be bound in the sence of inequality (Sa). This leads to a formal theory which is analogous to Mayer’s (mathematical) cluster expansion [l]. However, this definition would exclude from a cluster an atom which is not bound in the sence of (5a) but which is nevertheless bound to the cluster because of attractive interaction with sofne of the atoms in the cluster. An alternative definition would consider a particle bound if its kinetic energy (relative to the centre-of-mass) does not exceed the negative of its interaction with other members of the cluster;
$--(~;iel)~ +iZiku(r&
(k= 1,2, _.., i)
Unfortunately, one encounters serious mathematical difficulties in calculation of the cluster partition function when one uses this (as well as Hill’s) definition. For this reason Stillinger [13] (see also refs. [8,14]) develops an alternative theory which is based on a definition of physical cluster depending only on particle positions_ Considering the pair energy potential n(r) cut-off by a defmite distance b he calls a group of i molecules a physical cluster if each of its molecules is in a distance less than b from at least one other member of the group and in a distance greater than b from each nonmember_ We emphasize that the range of moleculr interaction is infinite so that there is no natural cut-off distance. The definition arbitrariness aside, the theory encounters the fundamenta! difficulties in the investigation of the excluded volume problem. Thus we prefer to choose a different definition as a basis for the theory proposed hereafter. According to it, by analogy with @a), i molecules form a cluster if their total kinetic energy (with exception of translation energy of the centre-of-gravity), Tie’, does not exceed the negative of the total potential energy, Vi:
PC) Herepk is the momentum of particle k and rkt is the distance between particles k and 1. While Hill’s deftition is too strict for relatively weak molecule interaction, inequality (SC) can insert into the cluster as well a “hot” particle k whose high kinetic energy ~m-lp~ exceeds its interaction-ZII+ku(rkI), with the rest of the cluster but which stilt leaves (because of sufficiently ‘%old” particles) the total kinetic energy less than the total potential. 3.2. Properties of individual clusters In view of (5c) the cluster partition function has the form: Zi = ( Vi3/2/k3h3(ihi)
1 ___ _fdrl___ dri_
1
I&
... dpi_
I
ewPHi ,
(6)
Hi<0
where X= (@~~/27r~)r/~ is the de Broghe wavelength of the atoms in the system, and 0, h, and m have their usual meaning. or is introduced to account for the indistinguishability of the particles_ tf the particles can move freely inside the cluster (as, for instance, Wergeland [5] assumes), then or. = i!. We, however, accept here the “microcrystal” model [Xl, i.e. we treat the cluster as a very small crystallite and oi is the number of the rotational symmetry of the aggregate. The exact microcrystal partition function contains contributions from all stable configurations of
234
5. Barbay/Ciusrenig
phenomena in adh&ve hard sphere fluids
the cluster. But we shall ignore all configurations
except the single one corresponding to the miniium value of the potential energy of particle interactions. This consideration is suggested by Monte Carlo studies of Lee et al. [14] which show that only compact, roughly spherical clusters are important at low temperatures. In addition we shall use the harmonic approximation to evaluate the vibrational part of ZiPerforming the integration in (6) we obtain the following expressions for the cluster partition function:
zi>3
(74
= Ve-
G’b)
z, = If/?? .
(759
Here Ur! is the static energy of the cluster, J, is its principal momentum r(n) is the gamma-function of Euler (n - I)! z I’(n) =
4:emr t’*-1dt
of inertia, V~ is its cwth natural frequency,
(8)
,
0
and -y(fz,X) is the incomplete
y(n. x) = i e-I 0
gamma-function
tt*-1 dr .
(see appendix)
(9)
The structure of the cluster involving 3,4,5 and 6 molecules, and corresponding to the minimum of the interaction energy, is the equilateral triangle, the tetrahedron, the trigonal bipyramid (slightly contracted akng the symmetry axis) and the octahedron, respectively_ The larger aggregates can be designed with help of polyhedra with equai edges or a combination of them. These polyhedra are regular tetrahedra, octahedza, trigonal prisms, archimedian antiprisms, and tetragonal dodecahedra [16]. Among the several configurations obtained in this way for a given number of particles the construction corresponding to the lowest static energy is considered T*. The intermolecular interaction is described by the (12 : 6) Lennard-Jones potential:
withn=9 orR= 12. Figs. 1-4 and tables 1 and 2 show the results of our calculations for fourteen different molecular aggregates *. The data obtained are used to suggest interpolation formulae for calculation of the static energy Uz?, the average frequency w;, and product J1J2J3~lT2 charactetizing the rotational properties of the cluster. We fmd: 7 We have taken into consideration, of course, only such configurations which correspond to local minima of the interaction energy. To check whether a construction considered is the minimum energy configuration, 3inatural frequencies were calculated from the secular equation det (m~i - a* Uj/a#axij = 0 with xf(l C s Q 3, I < j c i) being cartesian coordinate. In local equilibrium six of 3i frequencies equai zero. ’ The potential energy surfaces for cIusters of some three to sixty atoms under (12 : 6) Lennard-Jones forces have been system&: tally explored using numerical optimization techniques by Hoare and Pal [17]. Unfortunately, this work came to our too late to be incorporated into our own investigations. In table 1 we compare semi of their results with our calculations. ’ The distance between the nearest neighbours d, = adO was calculated by minimizing U&Q) = Zl
attention
B_ Barboyjclustering phenomena in adhesive hard sphere fluids
235
w
lL.0
1
.a 12.0
osa
10.0 ’
0.4
0.6
0.8
1.0
o
a8
l
0 CD
o
2
0
93
0.6
lJf/ie, as a function of the number i of molecules involved in a cluster. (1) - (12 : 6) potential; (2) - (9 : 6) potential of Lennard-Jones.
Fig. 1. The &tic energy per molecule,
0.8
1.0
Fig. 2. The average frequency, c+ as a function of the cluster size for (1) - (12 : 6); (2) - (9 : 6) Lennard-Jones potential.
a
-6.0
Fig. 3. Dependence of the rotational properties of the cluster on its size. (1) - (12 : 6) potential; (2) - (9 : 6) potential.
2.o"‘o
A
Fig. 4. The standard chemical potential of a molecule in i cluster for three reduced temperatures: kT/.c = 0.4; 1.0; 10.0. (a) - (12 : 6) potential;(b) - (9 : 6) potential.
0 17
vui 0 e _ cj[a(O)
_ a(l) f-1/3 ,3i-6
+ or’2,i-2/3
1
I
0.4
I
,
0.6
1
I
-
8
0.8
1.0
01)
I
,1/3i-6
oi = do(~141~2 ( oFlv=)
zz
f&U _ &I
j -1 P + ~(2) i-213
(12)
I
and In Ii =$ In (J1J2J3/iym3$)
= -I(O) - I(1)i-1’3
,
where the coefficients &I, &), and I@) are listed in table 3. Inserting (I 1) - (13) into (7) yields the interpolation formula
(13)
for the cluster partition
function
(the standard
236
B. Barboy/clustering
phenomena in adhesive hard sphere fluids
Table I The cluster configurations and tie value of U!, Zi, oi, wi and the contraction Factor [IIFor (12 Number of molecules i
Contiguration
2 3 4 5 6 7=) 8 9 10 11 12 17 19 25 30 b) 40 54 66
dumb-bell equilateral triangle tetrahedron trigonal bipymmid octahedron 1:S.l 1:3:3:1 3:3:3 1:4:4rl 1:3:3:3:1 1:5:5:1 5:1:5:1:5 6:7:6 3:6:7:6:3
- U,’iis
- @lie
: 6) Lennard-Jones potential
l&P,
Oi
wi
Q
-4.9698 -4.9698 -2.3776 -4.3035 -1.6292 -1.0080 2.9359 0.7445 1.9473 2.5842 3.0730 4.6670 6.2218
2 6 12 6 24 10 12 2 8 6 5 10 6 6
12.0000 11.6650 11.3265 11.0660 11.6161 11.5359 10.5174 11.4599 11.4316 10.9392 11.0014 10.7960 12-4846 12.6101
1.0000 1.0000 1.0000 1.0006/0.9971 0.9955 0.9952 0.9957 0.9920 0.99 14 0.9903 0.9871 0.9832 0.9889 0.9881
[171 0.5000 1.0000 1.5000 1.8208 2.1187 2.3554 2.3712 2.5845 2.6763 2.8753 3.1026 3.4714 3.6339 3.9040 3.9857 4.2281 4.4573 4.5983
0.5000 1.0000 1.5000 1.8208 2.1187 2.3578 2.4773 2.6792 2.8420 2.9786 3.1639 3.6063 3.8242 4.0948 4.2724 4.4904 4.7341 4.6723
a) Coniigumtion marked m : n :p consists of m particles in the fmt, n in the second, and p in the third molecular layer. For instance, 1 I 5 : 1 is the pentagonal bipymmid: 1 : 3 : 3 : 1 is the tetrahedron capped by four tetrahedra (the fully stellated tetrahedron); and so an. @The last four values of Vi” are calculated with the help of eq. (11).
chemical potenti& of the cluster): /3$
= -ln(zi/v)
=
i(3 kl(Aw#-*)
+ (3 - Ai) ln(T*/w;)
- [CP -&-l/3
+ cP-q/T*)
- pi - 4 In i i- In 6 u. + I(O) -i-l(lklB
,
(14)
Table 2 The cluster ConfigUmtionSand the ValueSof UzP,f+ UjsWj and contraction factor 0 for (9 : 6) Lennard-Jones potential Number of molecules
Configuration
- lZ,?/jE
dumb-b& equilateraLtriangle tetrahedron trigonal bipyramid octahedron 1:5:1 2:2:4 3:3:3 1:4!4:L lr3:3:3:1 i:s:s:L 5:1:5:1:5 6:7:6 3:6:7:6:3
0.5000
Ln(ZFl-5)
9
wi
Q
2 6 12 6 24 10 2 2 8 6 5 10 6 6
10.3923 10.1022 9.8090 9.6075 10.1330 10.0753 8.7958 10.0454 10.0334 9.7435 9.9836 9.9319 11.0900 11.2315
1.0000 1.0000 1.0000 1.0001/0.9966 0.9934 0.9934 0.9874 0.9877 0.9866 0.9863 0.9825 0.9768 0.9824 0.9810
i
2 3 4 5 6 7 8 9 10 11 12 17 19 25
1.0000 1.5000 1.8271 2.1458 2.3815 2.4253 2.6520 2.7552 2.9463 3.1912 3.6095 3.7795 4.0766
-4.9698
-4.9698 -2.3811 -4.3162 -1.6403 2.3264 2.9101 0.7154 1.9229 2.5560 3.0340 4.6277 6.1845
E. BarboyjCIustering
Table 3 The values of coefficients a (k). &)
andI
237
in eqs. (ll)-(13)
(9 : 6) Lennard-Jones potential
(12
interval of i
intervalof i
a)
c)
b)
: 6) Lennard-Jones potential
a)
b)
c)
o(o) ,(I) &)
6.9648 126863 5.7215
6.9648 8.7645 0
9.6751
6.6362 11.9437 5.3075
6.6362 8.2356 0
8.6110
w(o) &) &)
16.6968 23.7086 19.8633
16.6968 23.7086 19.8633
32.6723
17.0535 20.1682 18.1444
17.0535 20.7682 18.1444
50.2825
13.3718 -11.7597 0.6854 0.6653
4.2371 1.9702 -
13.3718 -11.7597 0.6987 0.6549
4.2789 2.1252 -
I(O) I(‘) (iz)-‘n ($)-I” a) (i*)-‘”
phenomena in adhesive hard sphere fluids
< yin
< 1.
b) i-‘”
< (i*)-1”.
c)i*
1.
where
T* = kT]e , Ai=-(3.6988
-5_6416i-‘p
+ 0.4428 i-*/3)H(3
-i) ,
(17)
and
r(3i _ 4.5 + Aj; [o@)i - (y(t) i2j3 •r-LX(~)N~]/T*] lf?‘=j In
r(3i - 4.5 + Aj)
H(x) is the unit step function and A = h/2sdo(r7ze) Ii2 is de Boer parameter which expresses the influence of quantum effects on macroscopic properties of the system (in this work, we neglect it). This expression will be exploit in the next section to derive the equation of state of the system. We confine OUCselves to analysis of the relations obtained and did not perform numerical calculations of cluster distribution since our aim is a description of the macroscopic (thermodynamic) properties rather than microscopic ones.
4. Clustering phenomena and an equation of state 4.1. Dilute
systems
In a system of low density
(1%
9l-J=f+)~1, we can ignore cluster interaction
and write
(20)
238
B. BarboylCusteringplienomena in cidhesive hard sphere fluids
Then eqs. (3) and (20) permit the calculation of~the concentrations pi{; = 1,2,3, _.-); substituting these {p-} into (2) and using the relation flP= Ejipj yield the following equation of state in-the parametric repiesentationl Y
6uo pP =
Y2
[exp(llr*)
27-*(27rT+)1~2 + 2w2T*(T7-*)‘12
+&& i=3
I I (T*j3 exp(9.I - pUIp)(y/~?)~
6u0p=
[exp(l/T*)
27-*(27Tz-*)1~2 + W2T*(7rz-*)= + 2
i5 liWi6 -exp(cp. Q-*)3
+3
1
-/WP)(y/ I
mj
(21)
’
Y2
J’
- 1 - l/T*]
3)i
- 1 - l/T*]
(22)
2
where y = 12uoT*(2rT*)11~
epul/X3 _
(23)
By eliminating y from (21) and (22) we obtain the virial expansion fect gas: pp=p+c,
,+2
k
of the equation
of state of the dilute imper-
pk,
(24)
with B2 = -(24u,+~)(rT*)~~~
[exp(l/T*)
B3 = 4Bz - (3625v~/w~)nT*(2~T*)1/2 and so on. It follows from (21) that the concentration i41iWf Pi=
12uOT*(2rT*)1/2 w;
~u,,(T*)~ [PI
Comparing
- 1 - l/T*] exp(3/T*)
,
(25) ,
y(4-5’3’T*) JY(4.5)
of clusters containing
(26)
i atoms is:
i 3 expC+ - W,“) -
(27)
(27) with (25) and (26) one has
P~/P;
= -B2
(274
,
and p 3 fp31 =2B2-LB 2 3 _ 2
G’b)
The series on the rhs of eqs. (2 1) and (22) converge if the following inequality &.f < - (c@/T*
_
+ lim cpili>+ !.n[(h~~~~)~/l2q,T*(2nT*)~~~]
f-+02
is satisfied:
(28)
Here (see appendix) h
i-km
pi/i = 3 [ln(d”)/3T*)
- 0t(~)/3T* f l]H(l
- &‘)!?T*)
_
(29)
B. Barboy/Cbtstering
phenomena in adhesive hard sphere fluids
239
At a given density p the excess of the chemical potential, fl,u - ln pX3, is limited in its magnitude, and thus condition (28) holds if-the temperature is sufficiently high. Similarly, at a given temperature T the series in eqs. (21) and (22) are convergent when the density p is small enough (as follows from the fact that Iim,,0 /3~ = --)_ Therefore the equation of state given by (21) and (22) will hold as long asp is less than a certain critical value pf which depends on the temperature. It is possible to show (for example, in the same way that ter Haar proposed [6] that the analytical continuation of eqs. (21) and (22) for p >p, is: @la)
6uopP=fDf),
whereyfis given by eq. (23) at p =~~andfDf) is the sum of the series on the rhs of eq. (21) aty =I’~ Thus the breakdown of convergence of eqs. (21) and (22) can be associated with the onset of a constant pressure portion of the P-p isotheml (i.e. a phase transition). It follows from eq. (28) that at low temperatures the density pf satisfies inequality (19): pf CCexp(-JO’/T’)
(T*+l).
Q 1
This relation leads to the following equation In Pv =A(T)
(30) describing
the vapour pressure Pv:
+ In(T) - C/T _
(31)
Since the temperature is low enough we can ignore the second term in the rhs of (31) and the temperature dependence of the coefficient A _Thus the low density theory predicts that In Pv varies approximately linearly with l/T, in quaiitative agreement with experiment. 4.2. Clustering in a f&lid at moderate and high densities 4.21. Hard sphere approximation for repulsion between clmte~~ In dense fluids the interaction between clusters is not negligible and must be explicitly included in the theoretical description. For the moment we ignore the attraction between the cIusters and take into consideration only the volume ui which they occupy. Then we can describe the pressure of the system and the chemical potential of an i cluster by use of the known solution of the Percus-Yevick equation for the hard sphere mixture [18,19] : $$p=-
g0
1 -E3
+3 - hE2
+3
(32)
(1 - &)2
~~~=~~~=~~~+in~i-In(l-~~)+~nd~pP+~
3d1 _ g3 (& +digl)
(33)
where (34) and di is an equivalent diameter of an i cluster so that the volume Ui of the cluster is equal to $Kd:. Substituting pi from eq. (33) into (34) allows us to obtain E, in terms of 0.u:
Inserting eq. (35) into (32), (33), and (2) then yields expressions for the pressure P and the density p in terms of fip in the form of an infinite series (but not necessarily a power series of exp&) like (21) and (22)). This representatidn will give us the equation of state only if the series on the rhs of (35) converges. This occurs for:
240
B. Barboy/Clustering phenomena in
p(p - +7rP kn dT/fj< j-+m
adhesivehard @herefluids
-(oI(O,p*+ h (pi/i) + 3 ln(AcJ”)/T*) , j+m
(284
and again this inequality, like condition (28), holds for high enough temperatures or low densities. In other words, eqs. (32) and (33) also predict a phase transition in the system where p = pf(T) but here, unlike eq. (21) and (22), we are not limited by the case (19) of very low densities_ The character of this transition will be discussed below (section 5). 4.2-2. Sticky hard sphere model for descnption of the whole ckSte?- interaction Consider now the whole interaction between molecular aggregates, that is, take into account an attraction a~ well as a repulsion. It seems reasonable (at least For large aggregates) to describe this interaction by the use of the “sticky” (i.e. with a surface adhesion) hard sphere model which was proposed and considered for the first time by Baxter [9] _This model is characterized by a rectangular well potential in which the range of the well becomes zero and (simultaneoudy) its depth infinite, so that for two clusters consisting ofiand~molecules we can write their interaction potential, uiP in the form 5 exp [-@all]
= [(di+ dj)/4rJ6_ [r - ~(di + dj)J f H[r - ~(di f dj)] .
(36)
Here S_(x) is the asymmetrical Dirac delta function, i.e. for any function f(x) there holds the relation: f(b - 0) = j
dXf(X)&_(X
-b)
(b >Q)
,
(37)
di is the equivaleit cluster diameter characterizing the size of i clusters, and T is a dimensionless parameter which measures the temperature (and the strength of cluster attraction), so that as T goes from 0 to infmity as does 7; We assume hereafter that cluster collisions do not influence the inner structure and properties of the aggregates, i-e_$, di and r depend only on i and T. Then, in the Percus-Yevick approximation, the compressibility equation of state of such a system is written as [20,21]:
where Xi/ are given by (38) Note that eq. (32) is obtained from (32a) by taking ail Xii = 0. (We can see from (36) that in the absence of Bttraction, T is infinitely large and thus it follows from (38) that all hii equal zero). When considering the attraction between the clusters (r is finite) we encounter great difficulties. Although one can write down the expression for the derivatives $.+/&I~ [20] we are not able to perform the necessary integrations and obtain thereby the chemical potential pi as a function of temperature and all concentrations &(k = I, 2,3 ...)_Therefore we have to introduce an additional approximation. The simplest that we shall discuss is a model according to which the system consists mainly of individual molecules (monomers) but where there are also clusters (i > 1) in very small concentrations: UipilUlp1 < 1
(i 2 2) _
z Our T is six times as Baxter’s symbol T.
(39)
B. BarboyfClustering
phenotnenn
in adhesive hard sphere j7uids
241
Assume that in this case eq. (38) permits the following power expansion for parameters XV: (40) Insertion into (38) enables us to obtain the coefficients of the expansion in terms of the density pL(=p) and diameters di (and r, of course). Then we can integrate the expressions for a~ii/api and obtain the chemical potential also in the form of power expansion like (40) Using the first terms of these expansions yields the following distribution of clusters: Pj-/rOl = exp[F& - $33
(ia 2) ,
(41)
where function tii(~, pl) and the chemical potential p = pL1are given by expressions (44) and (44a) in ref. [20]. It can be shown [20] that in the case when: Idi - 3dl I/@. f dl) < [(r& - I)(1 - 2/T&] 112,
(42)
with
(43)
rC = 2 - 2112
the function J/i is continuous_ Otherwise $+ has two poles. In the neighbourhood following asymptotic form (X\y = A,): 2d_&$. - 3dl)(A,xil NPjlP1)
f di/dl - 1)2 InIX:? -X,1
=* 34(di
- dl)(h,
of these poles eq. (41) has the
,
(414
- A_)
where [20]
A* = x;’ { 1 - 7 f 6xi1 f [(I - ~~~ - 6x&
- 3~,)]~‘~)
,
(44)
and Xi1 = $(di
- dl)/(di
f dl) _
(45)
In this case (A,, (‘I = h ) the power expansion (40) is divergent. It follows from els. (40a) and (44) that at sufficientiy high temperature (T > 1) parameter Xi!) is not equal to h, for any value of the density and ratio di/dl (the ratio of the cluster equivalent diameter, di, to the hard core diameter, d,, of a molecule). Hence in this case eq. (41) describes adequately the cluster distribution_ Unfortunately, if r < 1 there exists a range of the density p1 where Xfy = X, for sufficiently large clusters (di/d, > 3,i.e. i> 27). Since expansion (40) includes all values of i 2 2 we conclude that in this case (T < 1) eq. (41) does not hold. This precludes our being able to derive the cluster distribution in the most interesting region of temperatures and densities. However, we shall see in the next section that it is still possibIe to describe the macroscopic behaviour of the system.
5. The theory of freezing 5. I. General considerations
In this section we focus our attention mainly on the thermodynamic (macroscopic) behaviour of matter. For this goal discuss a model that describes the system as an assembly of identical clusters, i-e_ the ensemble of molecular aggregates of different sizes is replaced by an equivalent system of clusters each of which consists of the same
242
B. BorboylCl~tstering
phenomena in adhesive hard sphere fluids
number of particles, i* (i* need not be an even integer although, of Course, i* > 1). It is worthwhile to note that this model is of interest to the chemistry of monodisperse systems of solid particles [22]. In this case we can omit the sum sign in (1) and rewrite it in the form: F/N= p - P/p = pi Ni /N - P/p = pi/i - P/p _
(la)
That is, relation (3) is satisfied automaticahy. Thus we define i* as the value of i which minimizes F (with the temperature and the density held fmed). In the low density system (pid; < 1) one has: ~PIPi=l + O@,L$) )
(46)
0~; = 0~: + In Pi f 0 @id: ) a
(47)
and thus: OF/N= i-l
f 6 (1) _
l&/i)
Since for any pair i>j> i-1 ln(&
(48)
1 (4%
> j-l In(pjj) ,
it follows from (48) that F(i=
l)
(50)
l),
i.e. Fis higher for a system of iclusters than for a system of l-clusters (individual molecules) and thus for any given temperature:
(51)
1iIlli*=1 P-+0
The same result is obtained in the case of very high temperatures for any ftied density p. Indeed, one gets from (14) and (A-1) (appendix): (PIN)F(T+
3- I;i<
0~; &rpid:
< 1) = - 1Si-1
In T* + 6 (1) ,
(52)
and therefore lim i*= ;p*-+m
1.
(53)
Of course, relations (51) and (53) are trivial. A more interesting case is represented by the system at moderate and high densities and at low temperatures_ In the model considered the pressure P of the system and the chemical potential pi are described by the following expressions [9,23]: lJ#P =
Vj+VF +s,? (I - Uj13
- Aiqi
18q.(2 +q.) - h?$ I
1
I L ,
CQb)
36(1 - TQ)~
3xiqiViqi - 8qi - 2) -- 3nli l -qi (54) Here
243
9. BarboyjClustering phenomena in adhesive hard sphere fluids
(5%
qf - viPj and Xi is given by (40a), where vlpl is replaced by vi_ Remember When pi + 1 one obtains from (32b), (54) and (40a)
that vi = n/M:
is the volume of an i cluster.
-Vi)3l II1+O (l -77i)l 3
uiPP= 113(1-rJ2/(1
(56)
and pal, _ #)
_ m Pi - 13plpi = [3(1 - ~,)~/2(1
Thus when vi 41
we get for any temperature
-
q-j21[1 + 60 - $1 _
(57)
and j < i < ~0
F(i)
(58)
that is, in the system at very high densities,
i* > 1
(59)
In the low temperature finite. Furthermore: &f = (T-)-r
q/c
limit (T* 6 1) at any fixed density qj < 1, the values of pi exp[/3& + (3 - 3i - Ai)ln T* + 6 (1) ~
- $)I
and vipP are
(6’3)
and thus inequalities
(58) and (59) are seen to hold. Therefore we can conclude that a considerable increase in the density or decrease in the temperature yields a transition of the system, which was originally in completely disordered state (i* = l), to the state in which most particles are located in lattice sites of very large clusters (i* > 1). This ordered state can be naturally identified with the solid. Thus this phase transition is the freezing of the system and, of course, the fluid density, pr, at which it happens depends on the temperature: pf = pr(T)_ To form a notion about the character of this dependence let us rewrite eq. (la) for the free energy of the system in the form:
fiF/N= bl(T*) -I-b2(T*, i)/i’p
- 5 III i/i
+ b3(T*, p, i)/i ,
(61)
where the explicit expressions for the coefficients b, can be easily obtained from (14), (32b) and (54). Note that in the most important cases, b2 and b3 are almost independent of i. So for qualitative analysis of the function F = F(i) we can ignore the changes in b, and b3 arising from increasing i. Then it follows from (61) that for 62 < 0 the function F = F(i) has a minimum at i = i,,,in, which is a single root of the equation b2i2~3-151ni+15+3b3=0.
(W
If b2 > 0 and 2b, + 25 t 15 ln(2b2/45) > 0 the free energy decreases monotonically when i increases from zero to infinity, and finally, in the case when b2 >0 and 2b3 f 25 + 15 ln(2b7/45) < 0, one fmds that F= F(i) has a minimum at a low value of i (imin < (22.5/b2)3/2) and a maximum at a higher i(i,,, > (22.5/b2)3/2). The latter case obtains for low values of p and fl= I/kT(If imin corresponding to the minimum of F= F(i) is less than unity it has no physical meaning and the free energy F obtains its minimum value at i = 1.) Increasing p orb yields an increasing Fmin (and i,,,i,, as well) so that at a certain p = pr one gets:
F(i*, T*, p%) = F(i + m, T*);
1-* = imin or i* = 1 _
(62)
Further increasing p (or lowering the temperature) causes F(&, or i = 1) > F,, and thus a system of high density (pressure) has to undergo the phase transition from fluid to solid f. In the above analysis the system is considered to be in a single phase. So it is clear that the density pf of the fluid which is in equilibrium with the solid is less *For footnote see next page
244
B. Barboy/C%steriig
&MT*, Pf) =
Pji(T*, Pf) i
j=i*
phenomena in adhesive hard-sphere fluids
F(i, T*,~3
= lim
N
i+m
.
(63)
We see that inequality (28) yields the same condition for the phase transition while (28a) gives too large a value for pt. It is even greater than pp defined by (62): T*, p) =N-IF,
N-lF(i*,
+Pp-l(l
Since at low temperature PFJN pf = I$ exp [-c@/T*]
- lim qi) _ f-t-
CW
= -ol’o,/T*, one gets from eqs. (63) and (51):
< 1
(3Oa)
(which coincidences with (30)). On the other hand at high temperatures we have: &~=aF(i=
l)[&V=-1Sln
PF,/N=3{ln[3Aw(0)/,(0)]
--u~P&~ ,
T* + 3(1 -rJ2/(1 -
1)
(64) $55)
and thus qpf=
1
.
(66)
Therefore the density pf is an increasing function of the temperature, being zero at zero temperature and large at T’%-1. It is known [9,20] that at sufficiently low temperature the system described by eqs. (32b) and (54) undergoes a phase transition between two disordered states with different densities. One can identify this transition with that of liquid-gas_ Since the liquid density at evaporation, pn, increases when the temperature is lowered the curve pa = pa(T) must be truncated by the curve of = pf(T) at a certain temperature T, which is known as the triple point. Hence the model discussed above enables us to describe two phase-transitions: liquid-gas (at temperatures below the critical one, T,) and fluid-solid, and thus to obtain the regular phase diagram which includes (in coordinates pressure-temperature) three separate branches intersecting at the triple point, Tt. 5.2. Some properties of the fluid-solid
transition
The case of high temperatures requires additional considerations. We describe the solid as a very big cluster (infinitely large, in the limit) and hence the free energy of the solid, Fs, is given by:
W-*1
P,=
lim
p
i-+03
F(i.
T*, P) = iv
3
ln
ArJ”) --+3do) T*
T*
(67)
We assume here that the clusters and solid are incompressible, i.e. the cluster diameter di is undistorted through collisions and increasing the pressure does not affect the solid volume. Hence the solid free energy F, is dependent on the temperature, but not on the pressure_ Moreover, it follows from our assumption that the solid density, ps, is constant and represents the limit density allowed. In fact, this assumption is reasonable only at very low pres- Fmtd Z+1 the system can during a long time be in the metastable state corresponding to F,,,h but not to F,. We have pertkmed numerical calculationsin the wide range ofthe temperatureand the cluster size (0.16 < T*_< 63 and 1 < i < 300). The results show that &(T*, rJ is positive_Therefore at very high density F falls monotonically with increasing i and thus the
* IfLV,,,
system can exist only as a solid. However, for solutions where aggregates of solute molecules are in surroundings of a solvent it is possible that 62 < O_If at a certain condition ~(i min) is significantly lower than F(i = 1) and F(i = -) and besides 100 < &,., < 1000 the system exists in the form of thermodynamically stable colloid solutions [22] (see a!so ref. (241.).
B. Barboy/CIustenttg
phenomena in adhesive hard sphere jluids
245
is alrvfoos* not quantitative~ valid at very high pressures (e.g. P= Pcs; see below). But we do not expect this neglect of the solid’s compressibility to affect substantially any of our conclusions. It follows from (67) that the chemical potential of the solid, /.+ = FJN, and its first derivative, ap,/aT, are continuous at any temperature while the second derivative, a2@fl, has a discontinuity at the temperature: srres - it
That is, at ‘&is temperature the heat capacity of the solid, c, = -M@
aQgap2 ,
(6%
has a jump AC = 9kN(eo(“))-2
(lim
a2flpslas2 - r_hm+o
a20~,idfi2) = -3kN
T-T,,-0
ISa condition for the second order phase-transition beEq. (69a) (and the fact of continuity of i&laT at T = T,,) _ tween the solid with the heat capacity C, = 3 kN and a state (which is also solid because i goes to infinity as well) with C, = O(!)_ TO resolve this contradiction we rewrite eq. (68) in a slightly different form: Es, = N(3k T,, + lhh ui”/i) = 0 According to the cluster definition (SC), the internal energy of the solid, Es = Oii~_,, Hi), is always negative Es = N(3kT f lint l&i)
=0 ,
(70)
and thus above the critical temperature T,, the solid is unstable. Eq. (70) has a clear physical meaning* _We have defined the solid state as a state where the atoms occupy the sites of an ideal lattice and oscillate near their equilibrium places. When the temperature is less than Tc, the average kinetic energy of these oscillations (3kT) does not exceed the average potential energy per atom due to its interaction with the other particles in the solid (-hrr~~_+, @/i). Hence these interactions are able to hold the atom in its position. But if T > T,, these forces are insufficient and the atom “wanders” off. Since it happens to any “average” atom such state cannot be associated with a solid and must be identified instead with the fluid state in spite of its high density (which is equal top,). We can also conclude that the values of pf obtained from eq. (63) for T > T,, must be bigger that ps and hence do not have a physical meaning. The above interpretation of T,, permits us to understand better the existance of a certain temperature above which the solid is unstable at any pressure. This is connected merely with the definition (5~) and (following - in the frame of the microcrystal model - from it) inequality (70) though, of course, the value of T,., itself, the density ps and so on, depend on the particular approximations adopted. The temperature T,, is the critical temperature for the solid-fluid phase transition and we emphasize that the pressure,Pc,, at this point is finite though very high*. This fact has no consequences in the present theory where ps = const. is the highest density allowed. But in a theory which does not ignore the compressibility of the solid, the finite value,Pcs, of the critical pressure means that the continuous change of the system from fluid to sclid is possible (like the known continuous change ofgas to Liquid). In thinking about how a completely disordered system is changed continuously into a state with some kind of symmetry, it should be born * Landau and Lifshitz [25] note, as an evidental fact, that for the body to be solid it must in any case have kT much less than the static energy attributed to a single molecule in the solid. This requirement coincides apparently with (70). ’ This result (Pcs < -) has been obtained on the base of the Percus-Yevick approximation (32b) and mi&t, but unlikely, be abolished in a more correct theory.
246
B. BarboylCllis~ering p~leno?ne~ in adhesive hard sphere fluids
in mind that the in&r structure of all real solids is far from a perfect lattice. Since deviations from ideality increase with the temperature, the symmetry of the solid becomes more and more local as the temperature increases. At the same time in fluids of very high density, there undoubtedly exist regioni possessing a sjmmetrical structure. Thus at high temperature and pressure the solid and fluid states differ from each other because of the differing character of molecule motions rather than structure. 5.3. Quantitative rest&s In order to obtain quantitative results we need to relate 7 and d;(or uj) to the temperature and molecular parameters do and E [defined by zlpair,. eq. (IO)]. It can be done via the second virial coefficient [23], i.e. we choose r and the ratio: Qo = u1/uo =(d&))3
,
(71)
so that the second virial coefficient given by** B2 =4IJ,(l
- 3/%)
(72)
(the expression can be easily obtained from eq. (32b) and the condition i = 1) is the same as that for the LennardJones potential (iO), that is, the following equality holds: aO(l - 3/2r) =3 [
(1 - ex~[~*)(-$
-$)])GdX
(73)
In the low temperature limit, r Q 1 and a0 = (6/~)~/(“-~)_ Thus r(T* Q 1) a (r*)-l/*
exp(-l/T*)
.
(744
This regularity follows also from combination (25) and (72) at T* < 1. It suggests the following form of dependence r = T(T): l/2 exp( l/T:) exp(l/T*)
- 1 - l/T; + c/Ti*r - 1 - l/T* f C/T*“’’
(74b)
where TE is the Boyle temperature at which&(Ti) = 0. The coefficients c and y > 1 in eq. (74b) can be calculated with the help of the following conditions: (i) &(T* > T,‘,) >ps, where pf satisfies eq. (63) and pf=ps, when T* = T,*,; (ii) at the critical point of the liquid-gas phase transition (where both first and second derivatives of P are zero) r(T,*) Z TV,since 7c is the upper limit point of the domain where no state of a homogenuous system can exist [9,23] (we can note a posteriori that since at the critical point i* = 1 then r(Tz) = rJ; (ii) at the triple point which is defined by: P&T;) = N-,+, P*) = M,+, P,)
(75a)
P,) = W,*, /JJ -
Wb)
and JV:,
T has to be bigger than the lower limit point, in, = 0.292 ~~~of the restricted range of the two-phase (liquid-gas) region of the compressibility equation of state [26] ; (iv) the isotherms do not cross each other [and again a posteriori we can note that condition (iii) holds as long as condition (iv) holds] _ BP The second vidal coefficient of the cluster assembly described in section 4.2.2 [eq. (32a)j is B sum of the expressions in the rhs of (25) and (72).
B. Barboy /Clusteringphenomena
in
adhesivehard sphereflaids
247
Table 4 Constants characterizing the interaction between molecules and between clusters Potential
c
Y
7.;
9:6 12~6
1.7347
1.2045
1.7193
1.2190
The volume per molecule in the solid, p;l, p;l
= 2-‘~2~o(6/7r)(s,/s6)(3/~-r-9
- lim qqei
lim wi
0,
i-(0) =CV
j+m = JO)
4.5554
1.1672
3.4183
1.2372
9.6751 8.6110
50.2825
32.6723
is obtained by (for the hexagonal close packed lattice [27])
= uoa,
,
(76)
where s~(IC=R or k = 6) is the ratio of the static energy of an atom arising from its interaction with the rest of the system to the energy of interaction with the nearest neighbours. If the distance between the nearest neighbours is &., and the molecule interaction is described by the inverse power potential u(d,) =Adn$, the constants sk has the following values for different k:+j= 1.2046, sg = 1.0411, and s12 = 1 .Ol 10. The above conditions do not enable us to evaluate c and y in (74b) but only the range of their values conforming to those conditions. But it turns out that these ranges are surprisingly small (a few fraction of percent) and hence practically exact values of c and y are obtained. In addition, the above conditions permit us to caIculate the average frequency of the solid, w(O) = limi_, Wi. Results of these calculations are listed in table 4. As soon as r is given, the ration a0 can be obtained by eq. (73). It depends on the temperature being (6/1~)~/“-e for T* + 0 and zero (a0 0~(T*)-1/4) as T* +a- but still a0 = l/3 when T* = 60. Suppose we denote Vi = iVlai
,
OW
and take for simplicity ai=a,
-(a,
- l)F1j3
_
(77b)
Using eqs. (73)-(77) we have made numerical calculations of properties of the system considered. The results are collected in tables 5 and 6 and shown in figs. 5-7 (the isotherms P = P@) and 1_1=,u(p) can be seen in figs. 2 and 3 of ref. [23]). We have included also. the results of some other theories, and numerical simulations and experimental data, for the purpose of comparison_ Agreement between data listed is satisfactory in spite of higbly idealized character of the adhesive hard sphere potential (36) and imperfection of the model considered in this section. In addition we have calculated the first coordination number, NC1 defined by [38] : 4 Iv,1 = 47r/I s g(r)r2&= 0
[2VrP/(l - U#)IXt
2
(78)
whereg(r) is the radial distribution function_ It characterizes the average number of molecules being in contact with a given one (the number of the nearest neighbours) in a fluid. We can see (fig. 7) that at gas-liquid and fluidsolid phase transitions NC1 changes from very small values (for gas) to sufficiently large (at freezing) which are close to the coordination number in the solid (NC1 = 12).
248
--
B. BarboyjCiustering@enomena
Table S The values of the critical constants
and the properties kT/e
the c&i&l
point of solid-fluid
(9 : 6) potential (12 : 6) potential
(9 :6) potential (12 : 6) potential (a) PY (e) [281 (b) pY (c) [281 (c) perturbation theory 1281 (d) numeriul simulations [29] (e) esperimental data [30] (fJ square-well potential (g) PY (e) 1311 (h) PY (c) [311 (i) HNC (c) [32] 0 HNC (v) [321 (k) YBG [33] (1) perturbation theory f34] (m) numerical simulations [35]
.I
of the system in the triple point
%P
VOPIE
0.8568 0.8083
9.0254 10.0784
E-lp-T*lnA3
PPIP
7
transition 3.2250 2.8703
the critical point of liquid-gas
in adhesive hard sphere fluids
12.7284 16.0443
1.1413 1.2989
-0.5349 -0.1461
0.5858 0.5858
-1.0986 -0.7934
0.2930 0.2557
transition 0.08810 0.06432 0.104 0.096 0.126 0.111 ?r 0.015 0.0866
1.4803 1.1866 1.34 1.32 1.41 1.34 * 0.02 1.26
0.1569 0.1429 0.25 0.21 0.24 0.25 t 0.01 0.234
1.27 f 0.01 1.205 f 0.003 1.30 + 0.02 1.16 f 0.02 1.56 2 0.02 1.39 1.28
0.24 ) 0.01 0.148 5 0.002 0.17 f 0.02 0.19 f 0.02 0.31 * 0.02
0.3794 0.3194 0.31 0.36 0.38 0.33 -c 0.03 0.293 0.313 + 0.025 0.37 f 0.01 0.42 i 0.03 0.3 2 5 0.04 0.48 + 0.02 0.393 0.306
the triple point (9 : 6) potential [12:6) potential (n) perturbation theory 1281 (0) perturbation theory [28J (p) numerical simulationsj36] (q) experiment?l data [37]
0.7602 0.5851 0.64 0.704 0.68 -C0.02 0.699
Data (a) -(d) and (a) -(p) are related to diameter to hard core diameter) and ue = in the Percus-Yevick approximation; (i) state); (k) - Yvon-Born-Green theory;
Table 6 Comparison
of calculation
0.004311 0.002435 0.00044 0.0015 0.00121
(12 : 6) potential; (g)-(m) to the square-well potential with d/u = 1.5 (ratio of the web $ros 2l”; (a) and (g) - the energy equation, (b) and (h) - the compressibility equation and (j) - hypemetted Ichain approximation (the compressibility and virkd equation of (n) using celt model for solid phase; (0) using machine calculation results for solid phase.
results and experimental pgCTt)
PC
properties PQ(T,)
PC
of simple liquids pt
Tt
WC
TB
K
<
PC
TC
experimental data present caIcuIations
0.0079
2.66
0.014
0.555
0.293
2.67
(9 : 6) potential (12 : 6) potential
0.03997 0.03203
3.843 4.023
0.04893 0.03786
0.5135 0.493 1
0.3794 0.3794
3.0774 2.8807
B.
Barboy/Clusteringphenomena in adhesive hard sphere fluids
249
Gas-vapour -----------p:
1.0-
Solid-vapour
.
rcs
b
kT Tc.
Liquid-solid Liquid 7
. \
r: _ i
\ I
10-3 Fig. 5. The phase diagram of the system with (12 : 6) potential (a) and (9 : 6) potential (IJ) (the pressure and temperature are given in logarithmical scale).
tic1
I
US*
r;
I
10-l
“OP
Fig. 6. The phase diagram in coordinates of dimensionless density uop and temperature T* (both in logarithmical scale) for (12 : 6) potential (a) and (9 I 6) potential (b).
b
6-
.%-
2-
-/;
0
:0
1 0.50
0
0.50
Fig. 7. The fmt coordination number at phase-transitions. plotted against the density (a) - (12 : 6) potential, (b) - (9 : 6) potential.
5.4. Concluding remarks One of the important results obtained is the conclusion that at any temperature .* r -1 3
(7%
i.e. in the model approximating the system by an assembly of identical clusters, the molecules do not form aggre-
gates when p
Acknowledgement The author would like to thank S. Baer for his interest and many very helpful discussions in every phase of this work. He is grateful to W.M. Gelbart for reading the manuscript and valuable remarks. He wishes also to express his thanks for the kind hospitality of the Laboratoire de Photophysique Moleculaire, Orsay.
Appendix: Selected properties of the incomplete gamma-function A full treatment of the incomplete gamma-function y(rz, x) is given in special literature (see, for instance, ref. [401). We shall only list here the properties which are important for our investigation. When lfiU:l is sufficiently small, power series expansion of exp(-r) in integrand of (9) gives: (_fl$l)3i-4.5+Ai
exp(%)=
m c @!J,P)k
(3i : 4.5 -I-A,)!
,@O
k!
3i-4.5
+A, (A-1)
X-f 3i - 4.5 + Ai -
For IS@ I % 1 and fmite values of i the following asymptotic expression holds [40]: ([email protected]+Ai
exp(‘i) = ’ -
(3j
f
5.5
f
m Ai)!
(kt4.5
-Af--3i)!
(4.5 - 4- _ 3j)!
@U%k
-
(A-2)
We also need the asymptotic behaviour Of47i when i increases indefinitely, i.e. when both arguments of the gamma-function become inftite. One has for i> 3:
Lx@- CIf+1,3) exp(rpi,l,3 j =exp(& + Texp~~)(-llui0)3’-4-5/(3i
- 4.5)!] [ J
dteAf(l
0
1
- 1 _ - tj/3U~)3i-4a5 I
(A-3)
Using the inequality eX(l -x2/n)<(1
+~/n)~
for estimating the integral on the rhs of eq. (A.3) we obtain in the limit case of i % 1:
(A-4)
B. Barboy/Chsrering
exp(yi& X [
phenomena in adhesive hard sphere fluids
= exp&) + [exp(llVio)(-pV,P)3’~.5/(3i
,,P~@) If=PP - cu@)/3T*] -
251
- 4.5)!]
3T*/c#J~) f 0 (l/i) I
.
(A.3
Thus in the case when T* < 3 do) Cl< exP(&
(A-6)
< eXP(cpicl/3 ) < 1 .
Therefore (A.7)
(A-8) In order to check the case T* > iot*) rewrite eq. (9) in the following form: x/(,2-1) y(n, x) = (rz - l)n el-n
J
(tel-r)n-*
dt .
(A-9)
0 From the properties of function t exp(1 - r) one can easily obtain: O~tG$
exp[l-x/(rz-l)]t
exp [-xl@
- l)] ; (A.10)
<$J
exp[l -xl(n
- l)],
--&e,p(-,/(I,-
i)]
Hence y(n, xj > (x’*/n) eex
(A.1 1)
and r(n, x) < (x”/rz) eAx +x” edx [l - e-xl(n-l)]
.
(A.12)
Together with Stiriing’s asymptotic formula for the factorial estimations (A.1 1) and (A-12) give: pi > (3i - 4S)ln[-f3U~/(3i
- 4.5)] + 3i - 4.5 + &Cl,”- $ln 2a(3i - 4.5) - 12(3is_ 4 5) =A ,
(A-13)
and cpi
-eXp[pclio/(3i-
5.5)])+
1) ,
(A-14)
where 0< 0 < 1, lpUl~l(3i- 5.5)] < i. It follows from (A-13) and (A-14) that in the case CY(~)/~T*< 1 lim ~Ji=3{ln[oL(0)/3T*] i-+0
•t-1 -&‘)/3T*}.
(A.lS)
252
B. BarboyfClustering phenomena
in adhesive hard
spherefluids
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