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2, the percolation threshold p,, becomes z-dependent. The various critical exponents associated with the percolation problem are expected to be z-dependent as well, and we have no knowledge of their values. However, the general structure of the theory must remain unchanged, and it is only this which we use in the following. The p,. We know that as p+O, S,+O, while as p+l, S,-+Nkln(Nu). From (5.9), S, is a monotonic function of p and is sketched in Fig. 8. The slope of S,, (5.21) diverges as p+p, for 4, < 1. The behavior of the slope of S, and, therefore, the value of p, turns out to be critical in determining the order of the phase transition at T,, as discussed in Section VI. There is also the possibility that will equal or exceed unity. Such a situation may occur in more complex glasses (e.g., polymer glasses). In such systems, the moving units are molecular segments constrained by their connections to the rest of the molecule. Such constraints imposed by the complexity of the material increase t and otherwise decrease the growth rate of the clusters with increasingp, increasing the value of p. If this were the case, then (5.23) p,. We set D = D' and E = E' because these contributions arise from analytic terms in S, that are continuous through p,. We also set A' > A , because the presence of the infinite cluster makes a large additional contribution to S, and therefore abovep,. To complete the calculation of C,, we must know the temperature dependence of ij,, u,., and most important, p . This can only be done by solving the self-consistency condition p = h ( p ) , (6.13), in more detail. To reduce the number of free parameters, we use the viscosity data fitted by the parameters in Table I1 for the temperature dependence of 6,. For this reason, we limit our discussion to systems in which 17 has been measured over a wide temperature range. The remaining parameters are then uo, u,, and K to describef(u) andp,, a,A , A', D, and E in If we scale all volumes by u, taken equal to urn, that leaves only uc/uo and ~ = K U ; as unknowns in f(u). The latter is constrained, since we know
478
G . S . GREST AND M. H . COHEN
value of z appropriate to the free-volume theory is uncertain; a reasonable estimate is z = $ n, where n is the average number of nearest neighbors. At this point one is tempted to anticipate the results of the quantitative analysis and suppose that the liquid phase has p >p,,, the glass phase has p < p c z , and p =pcz at the glass transition temperature. If so, the transition would be second order because the infinite cluster is formed sharply."6-'2' Calculations based on the model show that it cannot be second order in most circumstances, but is first order, with a range of values of p aroundp, excluded. Elimination of the simplifications we have introduced wipes out the second-order phase transition, but the first-order phase transition persists in the circumstances we believe to hold experimentally, as we shall discuss after presenting the calculations. We note also that atomic mobility occurs within finite liquidlike clusters that exist below the transition. Thus the fluidity of the system would in principle persist below T,. In the usual percolation problem with z = 1, all pN liquidlike cells are in clusters if one counts all isolated liquidlike cells as clusters of size one. That is no longer true when z # 1. Only a fraction a,( p) of the pN liquidlike cells are now in the cluster [ a , ( p )= 11. The cluster distribution C,,(p), Y = 1,2,.. ., is normalized so that"'
where P , ( p ) is the percolation probability, the probability of being in the infinite cluster [ P,( p) = 0, p
which defines the critical exponent p,. For z = 1, PI-0.39 in three dimensions, and pc1-0.15 for the continuum percolation problem and pCI-0.18 for the site percolation problem on a face-centered cubic lattice. Numerical results are not available for z > 2. However, we do expect a larger value of p,, for z > 2 and a slower growth of the finite cluster, that is, a larger value
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
479
of p, for z > 1. Stauffer has shown that in mean field theory for z = 1, the cluster distribution function Cuz(p ) has a particularly simple form. Generalizing his results for all z, we find”’ -
C v , ( p ) = q o v ‘~exp(Ev‘~) &=4I(Pcz 0, =p,-%;I
(4.4a)
-P) =(y, + & ) - I ,
(4.4b) 7,
=2+6,-’
(4.4C)
where the qo and q 1 are positive quantities that may depend analytically on p and depend on the specific percolation problem. The terms T, and u, are the critical indices appropriate to the cluster distribution function Cuz(p ) . They are related to other critical indices, /?,, y,, and 6, through (4.4~). Equation 4.4a can be generalized beyond the mean field approximation toll9
where g is a universal function of its argument, depending only on dimension and z . In the absence of theoretical results for the exponent &, which we need in later sections, we turn to some experiments on magnetic alloys for useful information. There are several systems- for example, Ni-Cu, in which a magnetic atom m i ) must have a minimum number of llke neighbors before it can have a magnetic For Ni-Cu, n = 12 and z=8. The percolation threshold appears at pc8 = 0.44, and our estimates give 0.5 < & < 0.7 for &. These estimates were made by analyzing the p-dependence of the low-temperature magnetization above but near pc8. They are necessarily very crude because of the limited accuracy of the data, contributions to the magnetization from finite clusters (superparamagnetism plus weak ferromagnetic coupling of the finite clusters), and a probable dependence of the Ni moment on the number of Ni neighbors above the critical value of t = 8. A more refined description must await further investigation (e.g., by Monte Carlo techniques). The essential point is that there exists a welldefined p,, and a characteristic exponent p, whose values we d o not yet know. V. COMMUNAL ENTROPY AND THE FREE-ENERGY FUNCTIONAL Atoms can diffuse when a fluctuation in cellular volume of atomic size urn or greater occurs. Such large fluctuations can arise with significant probability only from redistribution of the free volume (free exchange of
480
G . S . GREST A N D M. H.COHEN
free volume) within a liquidlike cluster in the present picture; otherwise an activation free energy is required. The total free volume within a cluster of size v must therefore be greater than urnfor diffusion to occur. Y
2 ( u i -uc)>urn i- 1
for ui >uc
(5.1)
The average free volume within a liquidlike cluster is U,, given by (3.6). Thus for diffusive motion to take place within a given cluster, its size v must be at least urn/C,,
We call all clusters liquidlike. However, a cluster for which (5.2) holds is liquid, rather than liquidlike, in the sense that each atom or molecule within it moves in time through the entire cluster. That is, each molecule finds accessible the configuration space of every other molecule in the cluster. We now suppose that exchange of free volume between solidlike and liquidhke cells is so slow compared to exchange between liquidllke cells that we can ignore it in the computation of equilibrium properties. We return to this point later. As we shall see in Section X, the two time scales differ by much more than 2 orders of magnitude, In a liquid cluster, an atom or molecule is not confined to a particular cell or cage but can wander over the entire volume of the cluster. The communal entropy'". 12' of a single cluster of size v > v m is then given by kvln(v - l)V,, where U, is the average configurational volume of a liquidlike cell,
where s(u)= - af( - k In u,( u
aT
( 5 -4)
When the infinite cluster is present, the communal entropy is greatly enhanced, since many atoms have the possibility of extending their movement over the entire system. Therefore,% the communal entropy is that entropy associated with the accessibility of all the configurational volume within
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
48 1
the finite liquid clusters and within the infinite cluster when present. That is,% 03
v-Y,
where Np is the total number of liquidlike cells, N p a , ( p ) is the number of liquidlike cells in clusters, and k is the Boltzmann constant. The first term in (5.5) arises from finite clusters and the second from the infinite cluster. Note that an extra Nln N in (5.5) arises from treating the molecules as distinguishable. Although this term could be eliminated by treating the molecules as indistinguishable, it is very convenient to keep it and simply regard the Nln N correction as understood. Using (4.2), we can bring (5.5) into a more convenient form,
where
We have thus expressed S, in terms of p , V,, v,, a , ( p ) , C,,(p), and P , ( p ) . The first two quantities depend only on P ( u ) , and in Section VII we develop a method of determining P ( u ) . The a , ( p ) and P , ( p ) depend on CYz( p). Because CYz( p ) enters the free energy through S,, (5.6), it should be determined by minimization of the free energy. As we see below, the results are the same as those of a percolation problem somewhat different from the environmental percolation problem described in Section IV. The essential point is that clusters with v 2 v,,, are favored because they contribute to the communal entropy, while the formal structure of the problem remains that of a percolation problem. We now consider the effects of the communal entropy on the features of the percolation problem. Clusters of sizes less than v, tend to be suppressed; a , ( p ) and A , ( p ) move toward unity. The problem moves away
482
.
G . S . GREST AND M H . COHEN
from environmental percolation based on an uncorrelated distribution of cells toward the continuum percolation problem studied by Scher and Zaland Webman et al.I3’ These, together with the considerations in Section V on Ni-Cu alloys, give us very rough bounds on the values of p,, and p, appropriate to the present percolation problem,94
0.15 5 p c , 50.45
for 0.4 5 p, 50.7
Qualitatively speaking, as p increases toward p,,, the mean cluster size grows; that is, the scale of v’s important in (5.6), grows until at p,, an infinite cluster emerges. Above p,,, the percolation probability P,( p) increases. Thus S, shows a monotonic increase with p , with maximum and possibly divergent slope as p+p,, t O + . We can go no further toward the explicit evaluation of S, without introducing more information about C,,,( p). Since the results we obtained for the cluster distribution function in Section IV are valid only in the pure percolation problem, they do not apply here. The cluster distribution function can be obtained in a mean field approximation by minimizing the free energy with P( u), therefore p fixed. The C,,,(p) enters only through the communal entropy and a cluster surface free energy discussed below, which has not yet been explicitly introduced into the free energy. The procedure developed by Fisher13’ in his study of the droplet model of condensation can then be followed with only slight m~dification.”~ The entropy now contains an additional vln v contribution for “droplets” with v > v, which must be included. We consider the energy E,, of a cluster of v molecules. For all but the very small clusters, we can decompose E, into a bulk term proportional to the number of molecules in the cluster and a cluster surface energy contribution, l 3 I
E, =vEo
+W
(5.8a)
where E, is the average energy per molecule. A positive surface energy contribution W is given by
w-ws
(5.8b)
where s is the surface area of the cluster. Similarly, we expect the entropy of the cluster to have the forrnl3’
s,,=vSo( v ) + 0 s
(5.9a)
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
483
Here S,(u) is the average entropy per molecule So(v ) = So + ka, ( p ) In v
=so
for
for v >vm
v
(5.9b)
where So is independent of v and w is the surface entropy density. We have added the vln u contribution for v > vm, which follows from (5.5) and (5.6). This is where our result differs from the usual droplet model. The surface entropy density w is a measure of the number of different configurations of the cluster that have the same surface area. Fisher constructed the classical configurational partition f ~ n c t i o n , ' ~ ' (5.10) where the integrations are restricted to configurations in which u molecules form a drop. Assuming further that the pair interaction potential has an infinite hard core and an attractive square well, Fisher showed that (5.1 1) where the combinatorial g ( u , s ) is the number of configurations of v indistinguishable molecules that form a drop of surface area s. Although our problem differs from that of Fisher in that we have clusters within a condensed phase instead of drops in a gas, we can reinterpret what he does in terms of a Landau-Ginzburg-Wilson type of theory and carry (5.11) over into the present problem. The derivation of the Landau-Ginzburg-Wilson theory is discussed in the next section. The crux of the analysis is the factor g( u , s ) entering (5.1 1). It is convenient to define the additional quantityI3'
"ElJ
) 4AT) V
= exp( - - -
kT
(5.12)
where G u ( T )is related to the bulk entropy per particle So(u) in a larger cluster by the relation 1 So(u)=k lim - l n G v ( T ) (5.13) Y+cc
y
484
G . S . GREST A N D M. H . COHEN
Fisher points out further that the surface of a cluster cannot exceed some constant multiple of the number of the particles, nor can it be less than some minimum surface attained by some approximately spherical cluster; thus u p ' - I'd
(5.14)
Is 5 a,v
where a, and a, are constants. Since the terms in (5.12) are all positive, the standard maximum term argument leads one to the conclusion that for v large, G,(T) must have the form13' InG,(T)=lng[ v , S( v , T ) ]- B w S ( v , T )
+ ?In v + O( y o )
(5.15)
+
where S=S(v, T) is the most probable or mean surface area and is a positive constant. For the surface to be well defined, we must have S/v+O as v+m, in which case it is convenient to write simply for O
S=aOvo
(5.16)
where (I could be a function of T. From (5.13) to (5.16),we see that k In g( v, S) varies as So(v ) for large v, and the difference for smaller v is a measure of the surface entropy wS kln g ( v , S) - vSo( v ) = U S
(5.17)
as v+m. Combining these results one finds
vSo(v )
lnGv(T)m-
k
-
S( w - w T )
kT
-ilnv+lng,
(5.18)
This expression is different for v > v,,, and for v < v,,, because only clusters with v > v,,, contribute to the communal entropy So( v) and are therefore favored. This difference in the dependence of So(v) and the resulting discontinuity at v = v,,, is the only difference from the usual percolation theory. The desired cluster distribution function CV2( p) is porportional to q , ( T ) / V, obtained by substituting (5.18)into (5.12) and treating p as a function of temperature T. Thus C,,(p) is the same as that already given by (4.4)with ~ = ? - u ~ ( p ) f o r v > v , , ,Forv
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
485
We can now evaluate S, near p,. Inserting (4.5) into (5.6) and replacing the sum over v by a n integral leads for p near p, and /3 < 1 to: (we drop the subscript z for convenience): S,
= NkPU,(
P 1{ A,( P )In 6,+ BT I P -Pc
+ B I P -p, 1%
I
P -P,)ln[ NPU,( P ) B I P -P,I P ] } +Sf
(5.19)
for the communal entropy, where u ( x ) is the unit step function, S: stands for analytic contributions to S, and (5.20) the plus corresponding to p
L’m
where A, is a constant. Thus the slope of S, is constant as p+pc for /3 2 1.
N k en Vs
0
pc 2
P
’
Fig. 8. Sketch of the communal entropy S, as a function of the fraction of liquidlike cells p .
486
G . S . GREST A N D M. H. COHEN
VI. MEAN FIELD THEORY A. Statistical Mechanics
Now that we have an expression for the communal entropy, we can derive the probability distribution P(w ) . We start from the configurational free energy 9given by ,-PF=
1
D[Nle-BY"1
(6.1)
Here D [N ] is the volume element in the N-particle configuration space, and V[ N ] is the sum of the pairwise interaction potentials among the N molecules. We deal with the case of spherical molecules for simplicity; the results are general. We must convert (6.1) into an integral over a set of N independent dynamic variables that have a monatomic relationship to the cell volume variable u we have used thus far. Consider the configurations [ N - i ] of the N- 1 molecules other than the ith that contribute significantly to (6.1). These produce an interaction potential with the ith molecule,
where V(Ic -61) is the pairwise interaction, which for the densities we are dealing with has closed contours (defining a cell for the i th molecule) up to values of Yso large that they are unimportant in (6.1), a consequence of the steepness of the repulsive part of V. For fixed [ N - i ] , the volume T~ accessible to the center of the ith molecule is the volume inside the contour of constant Y[ { I N- i] corresponding to the largest value of Vof significant probability Y*. Fortunately, we do not need to know V* because of the steepness of the repulsive part V ; using the contour V[ I N - i] = 0 gives us adequate accuracy. We therefore define, formally,
<
where u= 1 for Y
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
487
where
There is a monotonic relationship between T ~ the , volume accessible to the center of the molecule within its cell, and ui,the volume of the cell. We can thus transform (6.4) into an integral over the ui, exponentiating the Jacobian and absorbing it into the free energy F [ u ] ,
The results embodied in (6.4) and (6.5) are obviously too formal to be directly useful, so we assume further that the probability P [ u ] = II,P(u,). We can therefore take for F [ u ] form we have developed thus far, (3.3) as modified in Section V to include cluster surface energies. The F(u) depends only on the probability distribution P(u), the cluster sue distribution C J p ) , and p. Thus (6.5) can be converted into a functional integral over P ( u ) , C,,(p), and p , and F is replaced by F [ P ,C, p ] , a LandauGinzburg-Wilson free-energy functional
The transformation from [ N ] to [ T I has not been shown to exist. One requirement for its existence is that T~ be finite for any configuration [ N - i ] of significant probability. Assign each moleculej P i a diameter equal to the separation at which V(I 6- I) goes through zero. Define a continuous percolation problem withp' the fraction of the space outside the molecules. As long as p' is below a percolation threshold p i , all the T~ will be finite. We can expect pi to be substantially larger than the value for the purely random case, 0.15, because large voids are suppressed by the large free energy they require and because the minimum linear dimension of a continuous void must exceed a molecular diameter. For hard spheres, the density of melting is two-thirds of the density of the ideally close-packed crystal. The density for dense random packing (the glass form) is 15% larger than that of the crystal. Thus the thermal expansion on melting is about 18'36, close to the random percolation threshold and presumably much less than p i . We expect therefore that the T~ are finite and that the transformation [ N 1 - 4 T ] exists.
5
488
G . S. GREST A N D M. H. COHEN
To find CV2( p), P( u), and p itself, we do a mean field calculation. We have in fact already found C,,(p) by such a mean field calculation in Section V. We require that 6 F / 6 X =0 subject to normalization conditions- for example, JornP(o)du=l where X is Cvz(p), P( u), and p in turn. In as much as CPz(p) has already been studied, we find P(u) and p in the present section. We choose F [ P ,C , p] to be of the form given by (3.3), insertf(u) from (3.4) and (3.8), and S, from (5.6). We note thatf(u) in (3.3) is regarded as independent of P ( u ) and yet that in (3.4) and (3.8)f(u) has in it an implicit dependence on P( u) through the presence of 4. The most convenient way to resolve this inconsistency is to treat f( u) as though it were a self-consistent field, itself the results of a first functional derivative of the total local free energy with respect to P ( u ) , and not differentiate it further in the variation of F [ P, C , p]. Otherwise, to avoid multiple counting, we would have to deal more explicitly with a complex nonlinear functional of P( u) with no direct physical interpretation. Moreover, the introduction of Cf into f(u) forces it into the role of a self-consistent field. The term P(u) enters both explicitly in F [ P ,C , p] [cf. (3.3)] and implicitly through the presence of v,,,, O,,and p in S, [cf. (5.2), (5.3), and (4. I)]. A convenient simplification of (5.3) is
v, = ~ , ( ~ , ) + u ; ( u , ) ~ f
(6.7)
We now write the communal entropy in the form
Its variation is, from (6.7), (3.6), (5.2), and (4.1), SS, = N k J m [ R , ( u -uc) - R 2 ] u( u - u,)6P( u ) du
(6.9a) (6.9b) (6.9~) (6.9d)
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
489
where (6.9~)is obtained from (6.9b) and (5.6). Inserting this into the variation of F [ P , C , p ]gives us the desired expression for P ( u ) ,
P(v)=pG;'exp(
-
T) v-vc
for u > v C
(6.10b)
where (6.1 1) (6.12) (6.13) (6.14) Our result for v > uc is essentially identical to that derived earlier by Cohen and Turnbul18" for the most probable distribution of free volume x, P(x)=y/v,exp( -yx/uf), where u, is the free volume averaged over all cells, u, =pG,, and y is a numerical factor between and 1 introduced to correct for overlap of the volume between neighboring cells. Comparing the exponent yx/u, = ( y / p ) x / G , with the exponent in (6.10b), (v- uc)/u,, we see that the two distributions are identical if x is taken as (v-uc) and y is taken a s p , which would be close to $ in the temperature region considered in Ref. 88. Equation 6.13 is a self-consistency condition for p , since its right-hand side contains p through the presence of R I in r of (6.12), and R , in Q of (6.14). We must therefore solve (6.13) before we can calculate the heat capacity or thermal expansion and characterize the glass transition. Indeed, (6.13) is precisely the mean field equation f o r p obtained by the variation of F directly, as shown below.
B. p and the Order of the Transition The self-consistency condition (6.13) is the key to the glass transition. To make its content and meaning clearer, we derive it by an alternative procedure. Instead of considering all possible variations of P(v) that leave it
490
G . S . GREST AND M. H . COHEN
normalized, we consider only those that leavep invariant. Variation of F [ PI subject to these two constraints leads again to (6.10) for P ( u ) but with p unspecified, that is, without (6.13). Insertion of P( u ) in the form (6.10) back into (3.3) gives
=4 (p )
+ Nfo
(6.15)
for the free energy as a function of p . In (6.15), - NkT( 1 -p)ln ug is the part of the free energy 4 ( p ) associated with the solidlike cells,
that associated with the liquidlike cells,
the entropy of mixing of solid- and liquidlike cells, and N k S the communal entropy. The values of p that make F [ P , C,p ] in (6.10) stationary are obtained by differentiating (6.15) with respect to p and setting the result equal to zero. Carrying out the differentiation and simplifying the result by using the stationarity of F [ P , C , p ] with respect to P ( u ) at constant p gives (6.10) as the condition of stationarity of the free energy with respect t o p . We now study its solution in each of the three cases /3 = 1, /3 < 1, and /3 > 1. Returning then to (6.13), we see that it has the form p = h ( p ) = [ l + Q u , T ] - ' , where ug is independent of p and r is a smoothly decreasing function of p . Both R , and 5 , are essentially independent of p as shown below, but R , is nonmonotonic, going to - 00 atp, because of the divergence to + 00 of a S / a p for /3< 1. Thus p , goes to - 00 atp,, Q to zero, and h ( p ) to unity. Therefore (6.13) does not possess a solution for p in the vicinity of p,, and p cannot increase continuously through p , with increasing temperature. The graphical solution of (6.13) is sketched in Fig. 9 for several temperatures. There is a bifurcation from one solution p I > p , at high temperatures to three solutions p I >p,, p z
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
49 1
pq
h(p)
,/
;
_...’ 0
~~
-’ PC
Fig. 9. Graphical solution of (6.13). Solid ( u ) , dotted (b), and dashed ( c ) curves correspond to h ( p ) (see text) for p< 1 at three , different and increasing temperatures. The circles give the values of p that satisfy (6.13). Case u has three solutions, and case c has one. Case 6 corresponds to the bifurcation point between u and c.
/
~~
P
Recall that these pi are stationary points for the free energy ‘ 3 ( p ) . We therefore pass from a single minimum in T ( p ) at high temperatures with p >pc, corresponding to the liquid, to three extrema in T(p ) at lower temperatures. Figure 10 sketches these features of ‘3(p). There are two minima a t p , > p , andp, < p , , separated by a maximum at p z at lower temperatures. At some temperature T,, T ( p l ) = 9 ( p , ) , and there is a first-order phase transition. Below q, ‘3(p , ) > ‘3( p , ) , and the minimum free-energy state has p = p 3
,
0
-
P
Fig. 10. Sketches of the free energy Gs(p) as a function of the liquidllke cell fraction p . Curves u , 6, and c correspond to those so labeled in Fig. 9. The positions of the solutions of Fig. 9 are indicated by dots. Curve u’ corresponds to S ( p , ) = S ( p , ) at the temperature Tp, where the first-order phase transition occur. Crosses emphasize the l n f i t e negative slope of S a t pc, the percolation threshold.
492
G . S . GREST AND M . H . COHEN
p -
~
-L
TP
._
T
Fig. 1 I . Sketch of the probability of liqludlike cells p versus T near T, for p< I .
The first-order transition is a direct result of the divergence of a S / a p as p+pc for /?<1. The latter is caused by the rigid and arbitrary manner in which the cells were divided into liquid- and solidllke cells. We discuss below the consequences of eliminating this unphysical feature of the model. For /?>1, a S / a p is finite a t pc, though nonanalytic, and therefore R , is always finite. The order of the transition now depends critically on the value of ( a S / a p ) I p c .When this quantity is large, the function h ( p ) is similar to that found for /?<1. That is, h ( p , ) approaches 1 and there are three solutions for p for some range of temperatures. In our search of the parameter space, as discussed below, this occurred most of the time. For smaller values of ( a S / a p ) l , coupled with a slow variation in a S / a p versus p , h ( p ) will vary more slowly as a function of p and can give rise to only one solution p for each T that satisfies (6.13). Nevertheless, S changes its functional dependence on p at p,, and there is a phase transition. Since p is a continuous function of T, the transition is second order, with corresponding to p =p,. Thus the transition for p 2 1 may be either first or second order, depending on the value of (aS/dp)l,,, and the magnitude of the variation in a S / a p versus p . We find, as discussed in Section VIII, that the former is most often the case. This situation with / ? 2 I may correspond to the case of the complex organic and polymeric glasses, where the infinite cluster has difficulty developing because of the additional constraints. Therefore /? increases and may become greater than or equal to 1. Even if /? remained less than unity in the complex glasses, the interplay between the configurational entropy (present in them but not in the simple glasses) and the communal entropy could lead to a variation in the total entropy, with p characterized by an exponent fi near pc, with f i = / ? + x > 1. Thus x describes the retardation in the growth of the internal configurational entropy imposed by the constraints on the free volume. If either /? o r b > 1, our thermodynamic results for some values of the parameters may be compared to those for polymers of Gibbs and Dimarzio,” who predict that the glass transition is a second-order phase transition in the limit of slow cooling. We make this comparison in Section VIII.
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
493
VII. FREE VOLUME AND VISCOSITY In the preceding section, we found that p is usually a discontinuous function of T. Here, we show that both the free volume o/ and the viscosity 77 are only weakly dependent on p and therefore do not reflect strongly the existence of the transition. The free volume i$ is given by the self-consistency condition, (6.13). The contribution to R , from aS/av, can be neglected because C,,(p), the number of clusters of size vm, is exponentially small compared to the total number of clusters present. We have, after combining (3.8), ( 6 . 9 ~ )and (6.12), a simple equation for v/,
Here A , ( p ) measures the number of clusters that are larger than some minimum value v, and a , ( p ) measures the number of liquidlike cells that belong to liquid clusters. As we found in Section VI, clusters of size V
494
G . S . GREST A N D M. H . COHEN
valid at all temperatures,
I"
( T - T o ) 2 k+ A 4u {
(7.2)
+
where kTo = kT, u&,. Equation 7.2 is noteworthy because U;vanishes only as T goes to zero, in contrast to the earlier view that the free volume vanishes at some finite temperature. This limiting behavior is independent of the exact form of R , or { and follows from (6.12), since R, is finite as T goes to zero. Because of the explicit free-volume dependence in 1 in (3.8), 6,has a characteristic dependence proportional to (T- To)at hgh temperatures. This introduction of 6 , into {, (3.7), as a phenomenological way to account for the effect of neighboring cells, does not affect the lowtemperature behavior of ?if, only the high-temperature region. The original work on the free-volume model by Cohen and Turnbull*'*" showed that the fluidity obeyed the Doolittle equation (2.8). We show that the percolation ideas developed in this paper give rise to the same equation for the fluidity. Let D, be the diffusion coefficient for a particle moving in a cluster of size u. The total diffusion coefficient D is then given by
From Ref. 88, we have that
where U is the average gas kinetic velocity, ?(u) is a correlation factor that may be associated with the magnitude of each displacement within the cluster, and a ( u ) is a mean net displacement, which depends on the immediate neighborhood of the moving particles and incorporates cooperative effects. The ?( u ) must go from 0 at u = 0 to 1 at u = 00. The basic assumption is that ?(u ) is a step function such that it is zero for u < u* and unity for u > u * ; that is, for u < u* the only motion that takes place is oscillation within a cage, oscillation having a vanishing correlation factor. Here u* = u, +urn - bii/, where b is a constant and u* is the minimum size of the cell necessary to accommodate two atoms. The correction b6, in u* is present because the second atom does not have to fit entirely into the cell but can use some of the free volume of the neighboring cells. The value of a ( u ) can be chosen to be proportional to u in the small u limit. Then since P,( u )
495
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
= p r exp[ - I '( u - q)], we have
D = Dopa,( p ) A , ( p ) e
(7.5)
-'m/'~
where we have added the contribution from the infinite cluster forp >pcand Do is a constant. We have included in D only the contributions from the liquid clusters. Equation 7.5 can be rewritten in the form
when om/u/ <
za P & l
v- 1
and one obtains the Doolittle equation. The latter is valid forp near and greater thanp,. Forp<
5.
l o g , , ~ = A+
2B
(7.7) T- T , + [ ( T - T , ) ~ + ~ " ~ s ~ T ] ~ / ~
where B=v,S,log,,e. The fit was excellent for all available data, and the parameters of best fit are shown in Table 11. The goodness-of-fit parameter x 2 =Z[(log,, qcal - log,, qexp)* ]/(N-4), where N is the number of data
496
G . S . GREST A N D M. H . COHEN
points, was typically between 0.001 and 0.007 for the glasses considered. The error in the fit is comparable to the experimental error. For all the glasses in Table 11, 7 was measured over at least 12 orders of magnitude. We find from the fits that To > T,, for each of these glasses. Included in the fit were two glasses, o-terphenyl and tri-a-naphthylbenzene for which ToL#O, whereas the other four showed Arrhenius behavior at low temperature, ToL+O. The fits for tri-a-naphthylbenzene and for o-terphenyl are shown in Figs. 12 and 13, respectively. The best fits to the Doolittle equation with ,51 a ( T - TO")are shown by the dashed curves in Figs. 12 and 13. We also tried fits where the preexponential A was temperature dependent, since q - T ' / ' D - ' . However this dependence is so weak compared to the exponential dependence on u,,, /C, that the fits are comparable to those with A constant. The generalization of these results to include the effects of the pressure P is straightforward. The free energyf( ti) then contains an additional term Po. The effect of the term on P( u ) , u >u,, is simply to change lo+lo+ P. Thus the free volume has the same form as in (7.2) with lo-$o P. The characteristic temperature To then has the form
+
To( P)'
Ua p To+ k
High-temperature viscosity measurements3* a t elevated pressure for 0.62KN03.0.38Ca(N03), were fitted to the usual three-parameter Doolittle equation with a To that had a linear dependence on pressure as in (7.8).
TABLE I1 Parameters Used in the Fit for the Viscosity 7 (7.7)
Tri-a-naphthylbenzene' o-Terpheny1 a-Phenyl-c-cresol' Sal0lC 8203
0.60KNO3.0.40Ca(NO,), *Ref. 53. bRefs. 54 and 55. 'Ref. 54. Ref. 60. =Ref.40.
-2.44 -2.65 -1.11
-0.52 0.63 - 1.79
345.3 253 .O 92.4 25.3 1825.4 362.O
10.6 6.4 2.9 0.94 72.6 6.4
401.8 278.7 252.8 264.6 609.5 365.4
497
LIQUIDS, GLASSES, A N D THE GLASS TRANSITION
~~, ~ ~ - 7 Q,~~ (3
14----
o TEHPHENYL.
~
12-
lo-
0
,8-
-
; 4-
s 6 :2
0-
,/. ,*++
I'
-2,.,.." L -L
Fig. 12. Viscosity versus temperature for tri-a-naphthylbenzene, based on data of Ref. 53. The solid curve is the best fit to (7.2) and (7.6), for the parameters given in Table 11. The dashed curve is the best fit (Ref. 53) with t 7 , aT- TOH,TOH-342°K.
i
/
0
m
2 ~
Fig. 13. Viscosity versus temperature for o-terphenyl based on data of Refs. 54 and 55. The solid curve is the best fit to (7.2) and (7.6). for the parameters given in Table 11. The dashed curve is the best fit with 0, a T TO",TOM=248.K.
The fit gave a value of u, = 1.1 A3. Unfortunately, we know of n o highpressure, low-temperature studies of 77 that can be used to check fully the temperature and pressure dependence of (7.2) and (7.6). Recently there have been some high-pressure studies ( ~ 5 1 0 P) ~for a number of inorganic glasses at room t e m p e r a t ~ r e . Further '~~ studies as a function of both T and P would be very useful for testing the present model. From Table I1 and (7.7), we obtain the value of 260.5 for um/u, in 0.60KN03 .0.40Ca(N03), . The two compositions (0.62 - 0.38 and 0.60 0.40) are close enough for the difference to be ignored. These values of um/uu and u, lead to a value of om of 237 A3. This corresponds to a radius rm of 3.8 which is to be compared to the ionic radii of 1.33 A for K + , 0.99 A for Ca2+,and approximately 3.0 A for NO;. The comparison shows clearly that the diffusive units are most probably not the individual ionic species but complexes instead. Angel1 and c o - ~ o r k e r shave ' ~ pointed out that T, = TOHand are nearly equal in many materials (e.g., see Table I). This can be understood in our
A,
498
G . S . GREST A N D M . H. COHEN
model as follows. From (7.6) and the Stokes-Einstein relation G, extrapolates to zero at T, in our theory. However, if 6 ,+O, then p+O from (6.13), so that the communal entropy and entropy of mixing also vanish. The only nonvanishing contribution to S is the entropy of the solidlike cells, which is essentially the same as the entropy of the crystal. This is true because no liquidlike cells remain when p goes to zero. Thus according to our theory, extrapolating the entropy to the crystal values using C, data from essentially the same range of temperature as that from which the extrapolation of 9 to infinity was made, must yield
T, N T,
(7.9)
because p=O both at T, and at T, and p is a single-valued function of T. VIII. EQUILIBRIUM THERMODYNAMIC PROPERTIES A. Specific Heat
The specific heat can be calculated in the vicinity of Tg from C, = T(aS/aT),where from (6.15), S=Nk
{
-(l-p)ln
( - ),(: lip)
-pln
-pR 6 + S I'
)
(8.1)
The heat capacity is the sum of contributions from the configuration and communal entropy,
and
as ap
cpc0-
= T-ap aT
where we assume that p is a smooth function of T away from rp so that we can use the chain rule. We know little about 6, = a S / a P except its scaling form near p,. From (5.21), we know that a Ip-p,I - a where a = 1 -p.
ep
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
499
With our earlier estimates of /3, we crudely estimate a to be positive and in the range 0.3-0.6. Note that the usual exponent a for the percolation problem is negative, since the entropy is related to the configuration probability of being in a cluster. Here the communal entropy is present and results in a different expression for a,dependent only on 0. For the critical contribution to which is dominant in the vicinity of p,, we take
c,,
P -P,
-a
E(P, -P) Pc
for p
ep
c,.
that is, the derivative of the free energy is rising more rapidly in the solid than in the liquidlike regimes. This gives
where we can take GJequal to its value at T,. The value of u,/w,is taken to be between 1.05 and 1.15. This leaves only the parameters in C, to be determined. However, these are constrained by the requirement that for three solutions of p in the vicinity of T, and '3(p ,) = '3(p j ) at Tp, which should satisfy the condition T, I T, < T,. Here T, is the temperature at which the system falls out of thermodynamic equilibrium for the experimental heat
500
G . S . CREST AND M . H . COHEN
capacity measurements. We also require that ACp = CpOiquid) - Cp(glass) > 0. This reduces the range of parameter space that is allowed. In practice, choosing values for p , , a, r7 and uc/uo effectively reduces the freedom in choosing the remaining parameters in 6'. We have not tried to span the complete parameter space but show here typical results for C , versus T. In the examples below, we choose a = 0.3. The results are not critically dependent on a because we have found similar results for both larger and smaller values of a. The important effect of the parameter a is on the size of the latent heat at T,. In general, the larger the value of a, the larger the latent heat. In our first paper,94we presented the equilibrium results for C' and 0 for the organic glass tri-a-naphthylbenzene. For comparison, we present here results for the ionic glass 0.60KN0,. 0.40Ca(N03),. We find similar results for o-terphenyl and B,O, glasses. In Fig. 14 we show the result for the probability distribution function P( u ) versus v for 0.60KN0, -0.40Ca(N03), for the parameters given in the figure legend. Note the bimodal distribution in P ( u ) , which is discontinuous at u=u,. In Fig. 15 we show a result for C, versus T near T,, for a set of parameters given in the figure legend. Since T, is unknown for this system, we have chosen parameters to give T, =335" K. The rise in C, in the equilibrium liquid phase is characteristic of several of the organic glasses. Since the transition is first order, there is a latent heat at T,,, denoted by a delta-function spike. There may also be other contributions to C, that we have not included, but they should be smooth near T,. I
I
20
2
Fig. 14. Plot of the logarithm of the probability distribution function P ( u ) for 0.60KN03.0.40Ca(N0,)2 at T = 340°K and p -0.20. The free-volume parameters from Table I1 and (7.2) are used for c,, and we have chosen ~ = = K u ; = 10S"Kan uc/uo = 1.1.
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
50 1
T(K)
Fig. 15. Equilibrium heat capacity Cp versus T near Tp. The transition is first order and the latent heat is represented by a splke at Tp in Cp.We have used the free-volume parameters from Table I1 for 0.60KN03.0.40Ca(N03),, To = 365.4"K, o,,,To=416.8"K, and ua/u,,, = 0.0038. The other parameters chosen are ;= I05"K,o,/u, = 1 . 1 , R ,=O,p, -0.15, a=0.3, A = 0.6, A'=2.0, D = 1.0, and E=0.5, which give -335°K.
This result cannot be compared directly to the experimentally measured C', since that is greatly affected by lunetic phenomena occurring around T,. Those measurements do not give an equilibrium result for C,, which is what we have calculated. As a second example, we have calculated the heat capacity for the metallic glass Auo,,,Geo,L36Sio,w4. Using the viscosity and heat capacity measured by Chen and Turnbull," we learn that CF/u,,, =(T-241.3)/540.6 and T, =295" K. Results for C' versus T a r e shown in Fig. 16 for parameters given in the legend. Here we have chosen parameters to give T, =280" K, but the results are similar for other values of T,. Note that the heat
11
2511
f,
300
T
(OK1
)O
Fig. 16. Equilibrium heat capacity Cp versus T near Tp using the free-volume parameters for the metallic glass Au,,,, Geo,,36Sio,w: To =241.3"K, u,J0 =590.6"K, and u, =O. The latent heat is represented by a spike at T p . The other parameters chosen that give =280"K are ;= I05"K, uc/oo = 1 . 1 , R ,= O , pc ~ 0 . 2a, ~ 0 . 3A, -0.7, A'=2.0, D=O.5, and E=0.30.
502
G . S. GREST A N D M. H . COHEN
capacity decreases with increasing temperature above Tp, as observed experimentally. As discussed above, it is difficult to compare our results directly with the measurements, which are not carried out at equilibrium. For p 2 1, the temperature dependence of p relates to the value of (i3S/i3p)lPr=- D > 0 in (8.5). For most values of D,the transiton is similar to that found for p< 1. That is, Q is very small nearp, and h(pc)S1. T h ~ s gives three solutions of the equation p = h ( p ) for some range of temperatures. The only differences are a reduction in the latent heat and the introduction of a critical end point for the liquid. The latter is the temperature below which the liquid state does not exist. There is a much smaller range of parameter space for whichp is a continuous function of T through Tp, and a second-order phase transition occurs at T= Tp when p =pc. This requires values of D,A, and A' in (8.5) so small that C, becomes a very smooth function of T, with no anamolous rise. The decrease in C, with decreasing T is then too gradual to avoid the entropy crisis. We therefore conclude that although a second-order transition is possible in our model, it is very improbable and does not correspond to the type previously proposed by Gibbs and DiMarzio.l7 In our model, internal configurational entropy is omitted, but entropy of mixing and communal entropy are included explicitly. In the entropy model of Gibbs and D i M a r ~ i o ,the ' ~ communal entropy is omitted, but entropy of mixing and internal configurational entropy are included explicitly. By increasing the exponent p above 1, we have attempted to subsume the internal configurational entropy missing from our model into the communal entropy. The loss of the entropy catastrophe thereby indicates that this is an inadequate way to deal with the internal configurational entropy. The problem of how to incorporate communal entropy into the entropy theory or internal configurational entropy into the free-volume theory remains unsolved.
B. Thermal Expansion Experimentally, the volume depends on the cooling and heating rates during the measurement and shows hysteresis. Here we calculate the average volume U as a function of T; it is given by (8.6) Inserting (6.10a) and (6. lob) into (8.6), we have
+
a=p( u, + 6,) (1 -p)u,
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
503
Thus with the exception of p , the parameters in (8.7) for 0 are smooth functions of temperature and show no anomaly at Tp. However, p is a discontinuous function of T at 5 and, as a result, this model predicts a jump discontinuity in 6at T. The volume will show a change of slope near T,, as the contributions from the infinite cluster dominate for p >p,, while the solidlike and finite-size clusters dominate f o r p
T( K )
parameters are the same as in Fig. 15.
504
.
. .
G S . GREST AND M H COHEN
C. Robustness of the Theory Now that we have described in some detail the analysis of the free-volume model, it is worthwhile to pause and consider the effect of the numerous simplifications introduced. The important question is, of course, how do they affect the physics of the equilibrium phase transitions as developed in the preceding sections? Fortunately, we do not expect that any of our assumptions will affect qualitatively the nature of the liquid-glass transition. To see this explicitly, let us review the most important simplifications introduced. These include the artificially sharp distinction between liquid and solidlike cells, which allowed us to approximate the local free energy by quadratic and linear regions. From this assumption, we were led immediately to the conclusion that only liquidlike cells exchange volume freely. However, there surely exist activated processes by which solid- and liquidlike cells can exchange volume, and these will be particularly important near the surface of liquidlike clusters. We have ignored all thermally activated motion, which should be important at low temperatures where the number and sizes of the liquidlike cells are reduced. There are also time-dependent effects, in which solidllke cells become liquidlike, and vice versa. We have assumed that the time scale on which this occurs is substantially longer than that for atomic motion with a liquid cluster. For T near and just above 5 , this should not be important, since the relaxation rates are long and the cage structure should be long-lived. However, for T >>q, this effect will be important. Finally, we have factored the joint probability distributions of cell volumes and treated the total free energy in a mean field theory. We assumed that f (u) and P( u ) depend only on the cellular volume u and neglected any interaction with neighboring atoms. The only local environment effect we included was in the dependence of {, on G,, (3.7), and this we treated as a mean field correction. However, we expect none of these effects to change the qualitative nature of the transition. This can most easily be seen by considering how one might begin to generalize the solution of the model within mean field theory, to eliminate these artificial discontinuities. First, one would introduce the probability u ( u ) that a cell of volume u will be liquidlike. If a cell’s local environment were such that it could participate in a free exchange of free volume, we would call that cell liquidlike. This requires only that in a localized region of configuration space (i.e., in cell u-space) the consequences of finite curvature in the free energy F on a scale given by kT can be neglected. That is, ( A X ) 2 6 2 F / S 2 Xis small in comparison to kT, where X is a generalized configuration coordinate and AX is its probable range of variation. In the local regions in which this condition is not satisfied, the cells are considered to be solidlike. One would expect that u ( u ) would be
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
505
a n S-shaped function such that u( u ) = 0 for u much less than a critical value u, and u ( u ) = 1 for u>u,. There would be a narrow range of u about u, in which the transition from 0 to 1 is made. Most important, the probabilityp survives elimination of the discontinuities, since one can define
The theory now proceeds as developed in Sections V and VI, essentially unchanged. For example, P ( u ) will have the same bimodal structure as shown in Fig. 14, but will now be continuous. Similar smoothing of all artificially introduced discontinuities will not affect the theory in any essential way. The loss of a sharp distinction between liquid- and solidlike cells could vitiate use of the percolation theory. The nonanalyticity in S will certainly be lost, leading to a communal entropy for which i3S/ap is always less than infinity. However, the first-order phase transition should be preserved, just as it was for most of the parameter space even when p> 1. The discontinuity i n p and I7 would be reduced, as would be the latent heat. One important effect of this smearing will be the appearance of a critical end point for the liquid, a temperature below which the liquid phase is n o longer even metastable. The second-order transition, which is only a small region of parameter space for p 2 1, is now wiped out completely by the restoration of analyticity. Our theory thus leads to a first-order phase transition or no transition at all. However, the entropy catastrophe can be resolved within our theory only if a transition occurs. We have now completed our discussion o f the equilibrium liquid-glass transition. However, as discussed above, the results for Cp and V obtained in this section are not seen experimentally. This leads us to consider nonequilibrium phenomena in the next section.
IX. STRUCTURAL RELAXATION AND T H E FREE-VOLUME MODEL The marked changes characteristic of a first-order transition shown in Figs. 15 and 17 are not seen experimentally. The observed anomalies in C, and V are the consequences of the falling out of complete thermodynamic equilibrium of the system as the time of measurement becomes comparable to the relaxation time of the system. The results just described cannot therefore be compared directly with experiment. First we must develop a relaxation theory of the kinetic effects observed around and below The changes of the volume I7 and enthalpy H consequent to stepwise or continuous changes in temperature have been investigated by a number of
G.
506
G . S. GREST AND M. H . COHEN
workers, following the early work of The essential features that relate the dependence of Tg on q to the temperature dependence of the structural relaxation time T( T) determined from cooling the liquid was first presented by Ritland.’34 Results for heating are more difficult to interpret, since the temperature dependences of U and H are affected by the detailed thermal history of the glass during cooling and annealing. To account for these effects quantitatively, it is necessary to develop a multiparameter approach, which has been done by N a r a y a n a ~ w a m y , ’Moynihan ~~ and coworker~,’~. 78 and Kovacs and c o - ~ o r k e r sThese . ~ ~ theories all involve the same basic assumptions but use different techniques and approximations. However, they are all built on the assumption that the volume U and enthalpy H of the nonequilibrium glass would always relax to its metastable equilibrium value for the liquid, in the limit of infinitely slow cooling. That is, none of these theories has allowed for the possibility of a n “ideal” glass phase in which the long time values of V and H differ from their extrapolated liquid value, as suggested by Ref. 71. The results of the recent work by Moynihan et al.I5,78 and Kovacs et al.79have shown that one can understand quantitively the experimental results for both Cp and V by introducing only a limited number of free parameters. Both theories stress the importance of including both a distribution of relaxation times T ; ( T for ) the structural modes and keeping track of the thermal history of the sample. However, no microscopic specification of the structural modes is made. One crucial result obtained by Moynihan et aL7’ is the dependence of In(q on temperature. They find
where Ah* is an activation enthalpy and R is the ideal gas constant. This result is in agreement with the experimental observation that In1 q I vanes linearly with l / T g for all the glass formers that have been systematically studied. Since T is large, a plot of lnlql over a limited temperature range would be approximately linear with respect to T, as shown in Fig. 14 for &02 and 0.60KN03.0.40Ca(N03),. The reader is referred to the recent reviews by Moynihan et al.78and Kovacs et al.79 for further details of their analysis. Here we wish to describe how the effect of these strongly temperaturedependent structural relaxation rates near can be incorporated into the free-volume Our analysis differs significantly from that described above because we have at our disposal an equilibrium theory of the heat capacity and volume to begin our discussion. By starting from a concrete
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
507
model of a glass, the free-volume model, we can identify the dominant structural mode and establish its relaxation mechanism. From our discussions above, we have that the free energy is a function of the form
A characterization of the free energy and therefore the structure of the system is obtained by a specification of P ( u ) , C , , ( p ) a n d p , which we can denote by the set { X , } , and which we refer to as the structural variables. The equilibrium state was obtained from the variational condition SF =o -
SX;
(9.3)
which determines the equilibrium variables of X I .found in Section VI. A study of relaxation near equilibrium can be made by examining the generalized equation of motion which we write in the form, X .=A.
'
.-
SF
"SXJ
(9.4)
where the set of parameters A,, describes all possible relaxation processes and a sum over repeated indices has been assumed. By considering the possible processes that contribute to the structural relaxation, we can separate out the rapidly relaxing quantities from those that are more slowly relaxing. The rapidly relaxing quantities can then be assumed to be in secular equilibrium during the variation of the slowly relaxing quantities. The variable P ( u ) should reach its equilibrium value very rapidly for fixedp, since both the vibrations of the atom or molecule in the solidlike cells and the free exchange of free volume associated with the oscillations in cages for liquidllke cells occur readily. Since the cluster size distribution C,,( p ) is fixed for fixedp, this leaves p as the one fundamental variable that determines structural relaxation. The value of p can change only by exchange of volume between solid and liquidlike cells, that is, by a reorganization of the cell structure. Therefore it can change only by self-diffusion, which is not localized within a cluster. In more detail, this exchange of volume between solidlike and liquidlike cells occurs at a cluster boundary and requires a cage rearrangement. This requires a diffusive step and becomes more difficult as T approaches T,, since the diffusion rates are decreasing exponentially. This same process that changes p allows C,,(p) to equilibrate with a new value of p . Thus p is the controlling variable. Everything else is in secular equilibrium with p .
508
G . S . CREST A N D M. H . COHEN
In the free-volume model, p becomes the dominant “structural mode” of the relaxation theories.” The rearrangement of the cage structure requires diffusion, which is slowly being frozen out. As T approaches T,,p can no longer follow its equilibrium value, but becomes frozen at a value p >p,. Since the variation in p no longer contributes to the heat capacity, C, decreases. The relevant contribution to C, arises from the communal entropy, which depends primarily on p and ceases to change. However, at temperatures above T,,the structure can equilibrate rapidly compared to the measurement time and both p and Cpapproach their equilibrium value. Since the heat capacity is measured using a differential scanning calorimeter, the dependence of C, on time and temperatures is complicated because C, is measured continuously while the sample is being heated or cooled at a constant rate. Therefore, the time for the heat capacity measurement is not well defined, and thermal history effects complicate the shape of the step at T,. To overcome this difficulty, one can approximate the real situation by changing the temperature in small discrete steps AT at time intervals A t . Stephens’* and Moynihan et al.78follow this procedure and calculate the change in the sample’s enthalpy H and its heat capacity from C’ = A H/AT. However, we already have an explicit expression for the equilibrium enthalpy and heat capacity. Since the result for C,, (8.2), depends partially on the rate of change of p with respect to T, we need only include the effect of p falling out of equilibrium (i.e., @/dT+O) for T < 7‘’ in that re~ult.~’ The exact dependence of the probability p on the detailed microscopic mechanisms that determine the interchange of volume between liquidlike and solidlike cells is unknown. In the development of the equilibrium results, we have assumed the local free energy4 of each cell depends only on its volume ui, f=f(u) as shown in Fig. 7. Consistent with this assumption, we assume a single average relaxation time T ( T ) for which the structural modes (i.e.,p) equilibrate. We can then allow in an average way for the fact that these modes may not completely relax before the next time increment A t . Although we do not expect results for C, that are quantitatively correct, we do expect the results to be qualitatively correct. The goal of the present analysis is to develop a clear, consistent picture of the freezing process, devoid of any unnecessary complications. We leave open at this point the more involved study that would be essential to obtain quantitative comparisons with experiment. We can then write an expression for p ( T ) , the probability of having a fraction of liquidllke cells at temperature T,9’
where p’( T) is the value of p at the end of the previous step and peq(T) is
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
509
the equilibrium probability for that temperature. For high temperatures hr/.r(T)is large andp(T)=peq(T).However as T approaches T',p(T) can no longer follow p (T) and will finally freeze out a value pfroz,which will e9 depend on the cooling rate and T ( T ) ,for all T < q.That is,9' (9.6) even though pq( T-tO) = 0. The program is then simply to start at a high temperature, where p ( T )= p,(T) and lower the temperature at a fixed q < O . The result forp(T) can then be used in (8.2) to (8.4) for C, to obtain results that can be compared directly with experiment. The only quantity that we must specify in addition to those in the equilibrium theory is the relaxation time T ( T ) .Since T ( T )is to describe the relaxation by diffusion of structural modes represented by the variation of p , it should have the same temperature dependence as the shear viscosity 77. That is, we suppose that the same microscopic movement processes underlie self-diffusion, viscosity, and structural relaxation. This supposition is consistent with existing theories and with a number of experimental results indicating that the activation enthalpy Ah* for volume or enthalpy relaxation is generally the same as the activation enthalpy for the viscosity v.'* We therefore assume that T ( T) can be expressed by the Doolittle equation, (9.7) in analogy with the result for 17. The only additional free parameter so introduced will be the preexponential factor T ~ since , the three parameters for u,,,/fi, can be taken from a best fit of (7.7) to the viscosity data. Using the values for p , , ( T ) obtained for the glass formers, tri-anaphthylbenzene, 0.60KN0,. 0.40Ca(N03), , and B,O,, we have calculated the dependence of C' on heating and cooling rates and history of the sample. Our results convince us that the effects are universal, as found experimentally for all glass formers. The value of 7o was chosen so that Ts is consistent with the experimentally observed values. We found95results that were insensitive to the choice of increments AT and Ar for fixed q, provided AT< 1°K. In the present analysis, we choose AT=0.05"K. The final results are dependent only on the product 4r0, and not q and T~ independently, since they enter only through this product in ( 9 . 3 , where Ar is replaced by AT/q. The results for tri-a-naphthylbenzene have been described elsewhere.95Here we show results for 0.60KN03.O . 4 ~ ( N o 3 )and , B203.
.
5 10
.
G S . GREST A N D M H . COHEN
In Fig. 18, we show results for C, versus T for 0.60KN03.0.40Ca(N03), for three values of 470 obtaining on cooling the system from high temperature. The parameters used to determine peq(T ) are given in the legend of Fig. 15. If we consider these curves to have 70 fixed, say T~ = lo-'' sec, curves a to c correspond to q= -0.1, - 10.0, and - 1000.O°K/min, respectively. Another way to think of these curves is to have q fixed, say q = - IO.O"K/min, then curves a to c correspond to T~ = lo-',, lo-", and l o - * sec, respectively. As expected, C, is a monotonic, decreasing function. For faster cooling rates, Cp departs from its equilibrium value at a higher temperature, since the system falls out of equilibrium sooner. The value of pfrozis larger for larger cooling rates. In most materials, C, cannot be measured on cooling, since crystallization occurs before the glass phase is reached. However, in our calculations, we have ignored the channel to crystallization and are able to study C, under cooling conditions. Most experimental measurements are made upon heating the sample from its glassy state. To consider this case, we start at low temperatures wherep'(T) equals its frozen-in value and 7 ( T ) is very large. Then we take AT>O and follow the same procedure described above. In Figs. 19 and 20, we show results for 0.60KN03 -0.40Ca(N03), obtained by heating. Figure 19 gives results for five values of qT0, in which the initial frozen-in state is obtained by cooling at the same respective rates. If we consider q = 10.O°K/min, then curves a to e correspond to T~ = lo-',, lo-", lo-'', and sec, respectively. Another way to think of these curves is to have T~ fixed, say r0 = 10 - lo sec, then curves a to e correspond to q =0.01, 0.1,1.0,10.0,and 100.O°K/min, respectively. As expected, the larger the to value of 470, the larger the transition temperature Tg. If we consider be the midpoint of the rise in C, (an arbitrary, but useful definition), we
I .o
320
360
3 40
T(K)
380
Fig. 18. Heat capacity Cp versus T for 0.60KN0,. O.4OCa(NO3), cooled through the transition for three values of 4 ~ For ~ . T~ = 10 - lo sec, these curves correspond to ( 0 )' 4 - 0.1, ( b ) - 10.0, and ( c ) - 1000.O°K/min. If, instead we consider q fixed, say equal to - lO.O"K/min, then (a) ( b ) lo-'', and (c) lO-"sec. T ~ -
1.0
320
340
360
380
T("K)
Fig. 19. Heat capacity C, versus T for 0.60KN03.0.40Ca(N03), heated through the transition region at various rates after cooling the glass at the same respective rate. For T~ = 10-Io sec, these curves correspond to ( a ) q = O . O I , ( b ) 0.1, (c) 1.0, (d) 10.0, and (e) 100.O°K/min. If instead we consider q fixed, say equal to IO°K/min, then (a) T~ = I0-l3 ( b ) lo-',, ( c ) lo-", (d) and (e) sec. The inset shows our results for (determined by the midpoint in the nse in C,) as a function of loglo(qTo). The slope of the h e is in excellent agreement with experiment.
10
I
1
Fig. 20. Heat capacity Cp versus T for 0.60KN0,. 0.4OCa(NO3),heated through the transition region at a fixed rate, after coolingatatdifferent differentrates. rates.For ForqT~ ,==1010- ' -I I ' cooling sec and and aa heating heating rate rate qq- - 10"K/min, 10"K/min, sec coolingatat thesecurves curvescorrespond correspondtoto( a( a) )cooling these - I . O " K / mi n , p r r o l= 0 . 2 0 3 : ( b ) - 10.O°K/min, pfrol= 0.207; and ( c ) - 1OO.O"K/min,prroz =0.211.
I
51 1
512
.
.
G S . GREST AND M . H COHEN
find that log,,(q~,) depends linearly on T,, as shown in the inset. This linear dependence of log,,(q7,) also occurs if one uses the temperature T,, for the maximum in C,, or T,, for the minimum in C, to describe the transition temperature. We find the slope of T, versus log,,(q.r,) equals 3.9"K, in remarkable agreement with the experimental result 3 3 ° K found by DeBolt et al.78 and shown in Fig. 4. This result is remarkable because we make no reference in our equilibrium calculation for peq(T)and Cp to this slope. Once p,,(T) is calculated, the only free parameter is T,, which shifts the position of T,, not the slope of T, versus log,,(qT,). The important conclusion to be drawn from this agreement is the correctness of the choice of T ( T ) ,(9.7), to have the same temperature dependence as the viscosity. This conclusion is further supported by the results for B203shown below (Fig. 21), for which the slope of T, versus log,,(qT,) also is remarkably close to its experimental value. From the published data for 0.60KN03.0.40Ca(N0,), at q = 10"K/min, we can estimate that ~ , - 1 0 - ' ~ sec. In Fig. 20 we plot results for heating at a constant rate 47, = 10-"/6O0K starting from different frozen-in states. These frozen-in states could be obtained by cooling at different rates or annealing at low temperatures. If we take T, = lo-'' sec and q = 10.O°K/min, the curves correspond to cooling as shown in the legend; the large dip in C, for case c results because pfrozis larger than it would have been for a cooling rate q= - lO.O"K/min, and p decreases to approach its equilibrium value when A t first becomes comparable to T ( T ) ,instead of increasing. In this region &/dT
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
513
for ti,(more free parameters) to 17 would eliminate it. However, this is a small point and should not detract from the more important results near T,. To check that these effects are universal, we have now calculated C' for a total of three glass formers. Results for tri-a-naphthylbenzene were shown in Ref. 95. We choose a third example, the network glass \O,, for two reasons. First, it should be very different from both the inorganic tri-anaphthylbenzene and the ionic 0.60KN0,. 0.40Ca(N03),, and second, data for 17 are available over a wide range, which we can use to fit the parameters in U,. The dependence of T, and C, on scanning rates q have also been studied in detail and are shown in Fig. 4. First we calculatedp,,(T) following the procedure outlined in Section VI. We chose the unknown parameters to give an equilibrium liquid-glass transition at a temperature T, below the observed T,. The parameters used in our calculation forp,(T) are i= 2x1O5"K, u c / u o = l . l , R,=0,pc=0.15, a=0.3, A=0.6, A'=3.0, D=l.O, and E=0.5, which give T, =480"K. Results for equilibrium C, are similar to those shown in Fig. 18 for 0.60KN03~0.40Ca(N03),.We then calculated C, following the procedure outlined above. Results for cooling at fixed q were similar to those shown in Fig. 21, which shows results for heating through the transition region at various rates after cooling the glass at the
(a)
(b)
;:ILL ,
7 , ,
w?
,
I
I
10
~ 520 1 ,
540
I
I
560
T(Kl
001 520
I
I
540
I
I
560
580
I
I
T(K)
Fig. 21. Heat capacity Cpversus T for q0,heated through the transition at various rates after cooling the glass at the same respective rate. For T~ = 10 - l o sec, these curves correspond to ( 0 ) q p O . 1 , ( 6 ) 1.0, ( c ) 10.0, and ( d ) IOO.O"K/min. If instead we consider q fixed say equal to lO.O"K/min, then (a)T~ = lo-", ( b ) lo-", ( c ) lo-", and ( d ) sec. The inset shows as a function of log,o(q70). our results for
514
.
G . S . GREST AND M. H COHEN
same respective rates for B,O,. As seen in the inset, T, was found to depend linearly on log,,( qTo) over the temperature range studied. We find that the slope of versus l ~ g , ~ ( q equals ~ , ) 13.1OK, also in good agreement with the experimental result 14.5"K shown in Fig. 4. From the results of these three glass formers, we believe the results for the nonequilibrium heat capacity can easily be explained in terms of the free-volume model. Although we have made several simplifying assumptions, particularly in regard to the existence of a single relaxation time, we believe that we can describe the essential features of the transition without introducing any additional free parameters beyond those used in the equilibrium calculation, apart from 7,. It is interesting to attempt a simple theory of the parameter T ~ Relaxa. tion of p occurs at the cluster surfaces via a diffusive jump across the interface. Using the free-volume model of self-diffusion (Section VII), we obtain the following for T ~ ,
where S / V is the surface-to-volume ratio of the liquid clusters. Since we are near the percolation threshold and the minimum-sized liquid cluster contains v,,, /C, molecules, S / V is well approximated by 5 ( v i /ti/) - ' I 3 .The result for T~ is lO-'Osec for 0.60KN03~0.40Ca(N03),,in excellent agreement with the value found from the lnq versus T, data.
X. TUNNELING MODES We saw in Section IX that the system's falling out of equilibrium at has a profound effect on p, the fraction of liquidlike clusters. Since the decrease of p requires a structural rearrangement of molecules in the dense liquid, it becomes more difficult as T approaches T,,where the value of p is frozen at pfroz>p,,. This freezing in of liquid clusters affects the lowtemperature properties of glasses by giving rise to the tunneling levels, described in Section 11. A number of liquid clusters of size v > v,,, is frozen in at T I because of the finite cooling rate used experimentally to avoid crystallization. Within those clusters, there remains the possibility that large (-atomic size) displacements can occur for T I by the coupling of translational motion and density fluctuations that lies at the heart of the free-volume model of diffusion. For temperatures near T,, the curvature in f(o) is negligible, and the local free energy is independent of the relevant configuration coordinates. However, when the glass is cooled down, T<
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
515
significant and gives rise to local energy minima separated by saddlepoint barriers on the order of kTg in size. The energy scale is set by k q because it is near that the system falls out of equilibrium, freezing in potential variations of at most this size. This last result is quite general and does not depend on the free-volume model. At high temperatures ( 2 G), the curvature in the free energy of the liquid clusters can be ignored, diffusion can occur, and, ultimately, extended motion can occur. At low T, the system freezes into one of its many total freeenergy minima. The most probable pattern is a freezing inward into the cluster from the interface with the solidlike or glassy regions. However, there is no significant relaxation between the clusters and the solidlike en~ironment.’~ Whatever excess volume Au, was in the cluster stays within the cluster. Within each cluster of size v, this excess volume Au, is .[(TI, -u,)+ V,]. The vG,comes from the free volume that is still freely redstributed a t high temperatures ( Iq).The ~ ( u-uo) , is the thermal expansion of each cell required to bring it into the liquidlike range. Some fraction of the v( u, - uo) will become available for redistribution as the cluster freezes into the minima. The freezing of a liquid into a glass requires atomic movement, which occurs by the concentration the free volume into ephemeral voids of the size This process can be regarded as a diffusion of the void away from the interface into the interior, where it becomes trapped as the freezing is completed. Thus within the originally liquid clusters there forms one void of volume somewhat greater than urnfor each vm = urn/V, atoms. The free-volume model leads us directly to an estimate of the number NT of these voids present at low T. There is a void of size urnfor each group of approximately vm atoms in a cluster. We find
G
where i = p a , ( p ) N is the total number of atoms in clusters, P,(p) is the probability of being on the infinite cluster, and C,,(p) is the cluster distribution. In general a,( p ) < 1, since we require each atom in a cluster to have at least t nearest neighbors that are also in the cluster. The quantities a , ( p ) , P , ( p ) , and C,,(p) have not yet been determined theoretically, but we can estimate useful upper bounds on vm and NT.
5 16
G . S. GREST AND M. H . COHEN
From Section IX, our best fits to the measured viscosity data for six glass That is, for each 20 to 40 atoms within formers give v, =urn/C, x 3 0 a t a cluster, we expect to find a void of the size mu,,,. Having chosen the percolation threshold value of p , =0.15, we found that pfroz-0.2 was a typical result from our relaxation studies of the specific heat near T,. Then if we consider
G.
we find
A realistic range would be lo-’ - 10 - 5 . A more precise estimate must await a study of the cluster distribution function C,,(p). It is important to note that NT depends only onp,,,, which in turn depends on the product 47(T).In our simulations of several glass formers we find that pfrozchanges by only a small amount for very different cooling rates q or during annealing at temperatures near T,. There should be little variation of the number of tunneling levels NT with the cooling rates used to produce the glass. To our knowledge, there has not yet been any systematic study of the dependence of NT on q. We assert that these voids are the tunneling centers. Any one of the neighboring atoms can move into the void, but in contrast to the crystal, the void itself and the configuration of the surrounding atoms will be irregular, so that such motion can be expected to be substantially easier for one particular atom. That atom then tunnels into the void by muftiparticle tunneling along a suitable one-dimensional path. Ignoring path curvature, we have for the effective Hamilt~nian,’’~
=P2
2P
+V(x)
(10.2)
provided the tunneling energy separations are much smaller than the mean phonon frequency w , ( x ) . Here p is the mass of the atom, V o ( x )is the equilibrium potential energy, Q0( x ) is the ground-state eigenfunction for all the
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
517
remaining degrees of freedom A, and the integration in the last term is over those degrees of freedom. The V ( x ) wd1 be of the general form shown in Fig. 22, assumed by Anderson et a1.lWfor the voids as large as those we consider here. It is possible but highly unlikely that more than two minima exist. There will be tunneling between the two minima, with a tunneling energy
A, =w oepA
(10.3)
E = ?(Az +A:)’/’
(10.4)
and energy
relative to the mean energy f w, above the lower minimum. Here A is the usual splitting between the lowest two energy levels. An estimate of A for the symmetric-well case can be obtained by using the triangular barrier approximation, which gives
*=!h![ 2p( v-
;wO)]l/z
(10.5)
where d is the separation in x of the two minima. Because the liquid clusters freeze in at T I Tg,the energy scale of A is set by Tg.Either of the two can be the lower minimum, so that A has a symmetric, roughly Gaussian distribution around zero with a root-mean-square value for which T, sets the scale. The 1 A 1 can be represented as distributed over a rectangular distribution of width I T,. Since w, is a typical vibrational frequency, it will be on the same scale; V is also governed by the same scale. Therefore values of V so close to f w, that according to (10.5) A is small enough for A: =Az are improbable. Consequently, we almost always have A, << 1 A 1, and the density of states for the tunneling levels is given by the distribution of
l*l* \
I
I
+d.
~-
Fig. 22. Potential energy surface producing a tunneling state characterized by a barrier height Y , asymmetry A , and generalized distance d .
518
G . S . CREST AND M . H . COHEN
At low T, the important values of I A I for the specific heat are -kT<
p(A’ (C)
Amin
h
A,
Fig. 23. Probability distribution P ( A ) versus A .
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
5 19
that we are invoking to explain the existence of tunneling levels. A freeenergy flat in localized regions of the configuration space on a scale of kTg simply cannot be regarded as flat any longer at temperatures much lower. Instead k q becomes the bound to the magnitudes of the variations, that is, maxima, minima, and saddle points (enhanced somewhat by thermal contraction) of importance for tunneling at low temperatures. In this view, tunneling is a continuation of diffusion. This view is independent of the particular set of configuration coordinates convenient for any particular material and thus portrays tunneling as a general phenomenon, universal in the same way that diffusion, viscosity, and the glass transition are universal. XI. CONCLUSIONS In this chapter we have tried to give a thorough description of the transition from a liquid to a glass within the context of a single model, the freevolume model. The major results of the theory are more general, however, and are not specific to one model. The model, in spite of its phenomenological nature, can provide a useful understanding of the complex phenomena occurring as a dense liquid solidifies without crystallizing. The free-volume model was originally derived to explain the temperature dependence of the viscosity. We have shown that it has a much broader application and can explain many of the outstanding experimental observations. This includes the existence of an entropy catastrophe at T, and the approximate equality of T, and T,, first observed by Angel1 and coworker~.'~ The relation between lnlql and measured by Moynihan et al.,78 also follows naturally and quantitatively from the notion that the liquidlike cell fractionp is the important variable that ceases to reach equilibrium when the relaxation rates become longer than the time scale for the measurement. We also find that a simple estimate of T,, within the model also produces remarkable agreement with estimates from the experimental observations. The connection between the low-temperature properties of the glass and its behavior at the transition region follows as a natural consequence of our nonequilibrium theory. The estimate of the number of tunneling levels is very close to the usual predictions. The remarkable agreement of these predictions with the experimental results, of course, cannot be taken too seriously, but it does support our belief that the model correctly describes many of the essential features of the transition. We should point out explicitly that our theory is not a theory of melting, though at first sight it has many similar features. Since this is an important point, let us make it very explicit. In our theory we derived a free energy F [ XI = F [ P ( u ) ; Cvz(p); p]. The equilibrium values for P( u ) , Cvz( p) and p
G,
520
G . S . GREST AND M. H . COHEN
are obtained from S F / S X = O . This free energy and, therefore, the equilibrium values of p ( ~ )C,,(p) , andp, depend parametrically on the interfacial tension a,, between the liquidlike and solidlike regions. If one were to develop a theory for melting, a similar argument could be made in which one introduces a free energy F,[ X ] of the crystalline state containing clusters of liquidlike cells. This F,[ XI depends on the interfacial tension a,, between crystal- and liquidlike regions. However, the structure of the solidlike regions is very close to that of the liquidlike regions; for example, for simple substances, the cage structure is based on dense random packing in both cases. Therefore, very little mismatch occurs across the interface. However, the mismatch is much greater at the crystal-liquid interface, so that a,, >as,. Following (5.8) to (5.18), one sees that as a increases, p and C,( p ) for v > v, are rapidly suppressed. Thus, although the same theory that yields a first-order phase transition between the glass and the liquid at yields also a first-order phase transition between the crystal and the liquid at T,, it follows from a,, >>a,, that T, >>q and that there are no premelting phenomena. The conceptual structure of our theory emphasizes the importance of entropy. There are in general three possible contributions to the entropy: communal, internal configurational, and mixing. The movement of the molecules or molecular segments within a cluster gives rise to the communal entropy and is fundamental to it; it also gives rise to the configurational entropy, though less directly. Although the entropy of mixing was included in our model, we did not include the internal configurational entropy, as is done in the complementary “entropy” theory. Remember that entropy is central to both theories. Both theories yield what is essentially a vacancy model for molecular motion, the free-volume model theory giving a more realistic description by including the fluctuations of the voids. The free-volume theory, however, admits in addition a description of the evolution of the communal entropy, which is the most fundamental aspect of the passage from the solid to the liquid state. The essential features of the glass transition do not require the freevolume concepts. On a more abstract level, the theory is based only on the notion that there exists a configuration space X in which F [ X ] is locally flat on the scale of k q or greater and that these configuration coordinates control movement and, therefore, the buildup of configurational and communal entropy. At low temperatures, far below q, one can no longer assume that F [ X ] is locally flat. The finite curvature gives rise to tunneling levels because the atoms or molecules can no longer move freely within the configuration space. In this general view, the localized tunneling motions are the low-temperature residuum of the free exchange of coordinate values and of the attendant classical diffusion at higher temperatures.
5
52 1
LIQUIDS, GLASSES, AND THE GLASS TRANSITION
Although the free-volume theory has been successful, there are, of course, places where the model fails or additional work is needed. The model does not produce a satisfactory dependence of volume u on temperature. This is due in large part to our approximate form forf(u): this local function neglects, except within mean field theory, the interaction of atoms with their neighbors. The form used for f(u) gives rise to a temperature dependence for u that is not linear, as found experimentally. We have also not included in any explicit manner the fact that in many glass formers the motile segment is a flexible molecule with internal configuration coordmates. This effect should be included for the polymer systems in which neither the internal configurational entropy nor the communal entropy dominates. The explicit elimination of the artificially introduced simplifications should be pursued, as discussed in Section VIII. We d o not expect it to make qualitative changes in the nature of transition, but it is essential if any quantitative agreement with experiment is to be expected. Moreover, a reduction in the number of sensitive free parameters in the theory is needed. There has been no direct verification of the conceptual structure of the theory. That is, a microscopic determination of the cluster distribution function has not been made, and the effects of percolation have not been seen. Assuming that the structure of the glass is well-defined liquidlike clusters in a denser solidlike background, one might expect to be able to see these clusters by either neutron or X-ray scattering. Since urnis probably between 100 and 400 A3 and vrn-30 at TST,, one would expect that a probe compatible with scattered wave vectors on the order of 0.1 could be used.
A-'
Acknowledgments Tlus work supported in part by the Materials Research Laboratory Program of the National Science Foundation at the University of Chicago by Grant No. DMR78-24698 and at Purdue University by Grant No. DMR77-23798.
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G . S . GREST A N D M . H. COHEN
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79.
523
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Advance in Chemical Physics, VolumeXLVIII Edited by I. Prigogine, Stuart A. Rice Copyright © 1981 by John Wiley & Sons, Inc.
AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that the author’s work is referred t o although his name is not mentioned in the text. Numbers in italics show the pages o n which the complete references are listed. Abkowitz, M., 459(71), 465(71), 523 Abraham, F. F., 344(72), 394,399( 11, 14), 419(14),424(14),425(14), 432(11, 14),435(11, 14), 442(14),450,472(112), 473(112),524 Abricossova, 1. I., 331(3), 333(3), 392 Adarns, D. J., 229(53), 246(53), 248(77), 250(53, 77), 252(53, 77), 253(53, 77), 258(5 3), 259(53), 26 l(5 3), 263(53), 269(53), 325 Adams, G., 467(84), 524 Adams, G. E., 341(62), 394 Adelman, S . A., 158(114), 181, 209(27), 262(88), 264(88), 304(139, 1401, 306 (147), 309(139), 310(139), 311(139), 314(140), 324, 326, 327, 328 Agrofonov, Yu.V., 244(70), 245(70), 325 Aguirre, 3 . L., 135(94), I80 Ahrnad, J., 362(97), 395 Akarna, H., 132(91), 180 Akcasu, A. Z., 106(64), 110(64), I79 Albertsen, P.,23(89), 68 Alder, B. J., 107(69), 179, 229(54), 246 (54), 250(54), 252(54), 259(54), 260 (54), 261(54), 266(54), 267(54), 268 (54), 269(54), 273(95), 278(106), 279 (95), 292(116), 293(116), 295(126, 127), 325, 326, 327, 407(82), 428(82), 452, 458(28, 29), 522 Alder, B. S., 399(6i, 422(6), 449(6), 450 Alexander, S. A., 442(103), 453 Alwattar, A. H., 153(102), I 8 0 Andersen, H., 425(80), 452 Andcrsen, H. C., 96(46), 179, 224(39,41), 225(41,44), 226(44), 237(64), 324,325,399(13),424(13),425 (13), 435(13),450
Anderson, E., 6(62, 63), 7(62, 63), 25(62, 63), 27(62, 63), 34(62, 63), 68 Anderson, E. W., 407(45), 451 Anderson, P. W.,470(100), 517(100), 524 Andronov, A., 342(65), 344(65), 394 Angell, C. A., 399(10, 23, 25j, 407(46, 48), 408(51), 411(23, 59), 415(67), 416(48), 417(68, 69), 418(69), 419(23), 420(59), 421(67, 721, 424(10), 427(81), 432(23), 435(10), 436(10), 437(10), 438 (23). 441(51), 443(106), 444(106), 445 (108), 450, 451, 452, 453, 457(14), 458 (34, 37, 38, 39), 460(14, 38, 39), 465 (14, 39), 467(39, 82), 468(14), 472(34), 496(38), 497(14), 519(14), 522, 524 Arai, K., 459(62), 523 Armstrong, R. L., 230(56), 231(56), 232 (56), 233(56), 234(56), 242(56), 306 (56), 325 Aroeste, H., 85(29), I 7 8 Atake, T., 427(81), 452 Baer, S., 188(6), 189(6, 7), 202(6), 214 (61,323 Baer, Y., 42(102), 69 Bagley , B. G., 446( 1 12), 453, 45 7( 1O ) , 522 Baker, A. D., 33(96), 35(96), 42(96), 69 Baker, C., 33(96), 35(96), 42(96), 69 Balescu, R.,91(40), 109(40), I79 Ball, M. A., 14(84), 15(84), 68 Banerjee, A., 7(78, 79), 23(78, 79), 61(78, 79). 63(78, 79), 68 Bararn, A., 447(117), 453 Baret, J. F., 377(136), 396 Barkatt, A., 421(72), 452 Barker, J. A., 224(40), 248(75), 253(75, 80), 324, 325, 423(77, 78), 435(78), 436(78), 438(95), 452
527
528
AUTHOR INDEX
Barlow, A., J., 458(47), 523 Barrer, R. M., 469(99), 524 Basak, S., 432(94), 452 Batchinski, A. J., 408(52), 451 Batschinski, A. J., 469(96), 524 Baym, G., 85(30), 178 Bearman, R. J., 209(27), 324 Bedeaux, D., 177(117), 181, 293(119), 294( 122), 327 Beers, J . C., 11, 295(127), 327 Benedek, G . B., 376(119), 377(135), 395, 396 Bennett, C. H., 436(90), 452 Ben-Reuven, A,, 85(31), 178 Berendsen, H. J. C., 254(86), 326 Bergmark, T., 42(102), 69 Berkowitz, M., 262(88), 264(88), 326 Bernal, 1. D., 422(75), 449(75), 452, 472(115), 524 Berndtssson, A., 42(103), 48(103), 69 Berne, B. J., 75(7), 96(47), 125(84), 161 (84), 178, 179, 180, 377(125), 396 Berry, G . C., 457(1 l), 522 Berry, R. S., 39(101), 69 Bestul, A. B., 459(51, 66), 460(51), 523 Beuche, F., 469(98), 524 Bhatnagar, P. L., 123(82), 180 Bird, M., 377(131), 396 Birks, J . D., 153(102), 180 Bixon, M., 421(74), 452 Black, J . L., 470(104, 1 0 9 , 518(104, 105), 524 Blaser, S., 458(40), 461(40), 496(40), 522 Blomberg, N., 160(115), 181 Blum, L., 230(56), 231(56), 232(56), 233(56), 234(56), 242(56), 304(144, 1451, 305(145), 306(56, 146), 314(144, 143,325, 327, 328, 338(38, 40, 42), 393, 394 Bohm, D., 123(82), 180 Boley, C., 106(65), 179 Bongiorno, V., 344(75), 394 Bose, R. A., 458(41), 522 Bose, S., 98(52), 100(52), 116(52), I79 Bossis,G., 253(84,85), 254(85), 325, 326 BGttcher, C. J. F., 271(90), 272(90), 326
Bouchiat, M. A., 346(77), 352(77), 362 (98), 376(122, 123). 377(122, 123, 132, 137), 394, 395,396 Boys, C. V., 334(24), 393 Bradley, R. S., 339(50), 394 Brawer, S. A., 399(24), 440(24), 441 (102),451,453 Brenner, H., 135(93),180 Brossel, J ., 376( 122), 377( 122), 395 Brot, C., 253(85), 254(85), 326 Brown, B . C., 423(77),452 Brown, W.F., Jr., 274(101), 276(101), 290( 1lo), 326, 327 Bruckner, R., 459(70), 462(70), 523 Brun, P., 440(98), 452 Brundle, C. R., 33(96), 35(96), 42(96), 69 Brush, S. G., 403(37), 451 Buckingham, A. D., 273(95), 279(95), 290(11 lb), 293(11 lb), 326, 327 Buff, F. P., 343(69), 394 Bunker, D. L., 157(109), 162(109), 180 Buscall, P., 334(33), 393 Buttner, H. J., 459(64), 523 Cade, P. E., 35( loo), 69 Cahn, J . W., 355(83), 356(83), 395 Canfield, F. B., 224(42), 324 Cape, J. N., 399(20), 402(36), 404(20), 407(20), 41 1(20), 419(20), 420(20), 422(20), 425(20), 426(20), 429(20), 436 (91), 437(91), 441(36, 91), 442(91), 443 (36,91), 445(20), 447(20), 449(20), 450 (20),451,452 Cargill, G . S., 430( 87), 432(87), 435 (87), 452 Cargill,G. S., II1,457(12),522 Carnahan, N. F., 448(120), 453 Carpenter, M. R., 459(52), 523 Casimir, H. B. G., 339(53), 394 Caufield, K., 352(79), 367(79), 394 Cederbaum, L. S.,5(16, 17, 18, 19, 20, 21, 22,23,24,25,26,27,28,29,30,31,32), 23(16), 28(16, 17, 18, 19.20, 21, 22, 23, 24, 25, 26, 27, 28, 29,30), 29( 19), 3 1(16, 24,25,26,27,28, 29, 30), 35(21, 23), 45(16,24,31,32), 46(24), 66, 6 7 Ceperley, D., 229(55), 245(55), 246(55), 253(55), 264(55), 268(55), 325 Chan, D.Y.C., 304(141, 142), 309(141), 3 12(14I), 32 1( 141) ,327
AUTHOR INDEX Chan, M., 303(135), 327 Chan, M. H. W., 301(132), 327 Chandler, D., 86(32), 104(32), 105(32), 125(84), 155(105), 161(84), 178, 180, 224(39, 41), 225(41, 44), 226(44), 237 (64), 274(104, 1 0 3 , 281(104, 105), 282 (104), 283( 104), 324, 325, 326, 425(80), 452 Chandrasekar, S., 82(25), 84(25), 178 Chang, S. S., 459(5 1, 66). 460(5 1). 523 Chanthanasinh, M., 440(99), 452 Chatain, D., 458(44), 523 Chaudhari, P., 457(13), 458(13), 462(13), 522 Cheeseman, P., 399(10, 23), 411(23), 419 (23), 424(10), 432(23), 435(10), 436 (lo), 437(10), 438(23), 450, 451 Cheeseman, P. A., 408(51), 441(51), 451 Chen, H. S., 404(39), 451, 458(22, 23), 460(20), 462(20), 501(20), 522 Chen, J . -H., 304(140), 314(140), 327 Chen, T. T., 6(48, 57, 58), 7(57, 58), 25 (57, 58), 32(48), 33(57), 34(57, 58), 35 (57), 67, 68 Chong, D. P., 3(11, 12), 66 Chu, B., 377(124), 396 Chuang, T. J., 156(108), 180 Chung, H. S., 468(92), 524 Churaev, N. V., 341(61), 394 Cichos, C., 333(5), 393 Clark, N. A., 377(141), 396 Clarke, J . H. R., 399(10, 12,15),408 (12), 411(12),412(12), 416(12),418 (12), 419(12), 420(12), 421(12), 424 (10). 425(12), 431(12), 432(12), 433 (12), 435(10, 12), 436(10, 12), 437(10, 12), 441(12), 442(12), 444(12), 450, 458(34), 472(34), 522 Clarke, R., 432(94), 452 Clausius, R., 290(109), 326 Clint, J. H., 333( 15), 393 Clunie, J. S., 333(15), 334(31), 393 Cohen, E. G. D., 107(72), 179 Cohen, M . H., 430(86), 445(109), 452, 453, 457(18, 19), 458(19), 468(87, 88, 94, 9 3 , 469(18, 87, 88), 470(87, 88, 94), 472(18), 473(18, 87, 88), 475(87,
529
88, 94), 477(94), 480(94), 481(94), 482(94, 130), 489(87, 88), 494(87, 88), 500(94), 506(95), 508(95), 509(95), 513(95), 515(95), 518(140), 522, 524, 525 Cole, R. H., 295( 123), 327 Collie, C. H., 303(137), 313(137), 327 Collings, A. F., 407(44), 451 Collins, F. C., 78(12, 14), 178 Colonamos, P., 303(136), 304(136), 312 (136), 327 Comly, J. B., 477(125), 479(125), 525 Condon, E. U., 457( 16). 522 Cowen, J . A., 377(143), 396 Croxton, C. A., 456(3), 521 Csanak, G., 3(9), 5(9), 66 Cukier, R. I., 98(53), 108(75, 76), 120 (53), 121(53), 123(53), 128(86), 130 (87), 132(86), 134(92), 173(53), 174 (53), 179, 180 Cukierman, M., 459(55), 460(55), 461 (55), 496(55), 497(55), 523 Cunimins, H. Z., 376(121), 377(121, 126, 127), 379(121), 395, 396 Dahler, J. S., 115(80), 116(81), 180 Dalgaard, E., 10(81), 20(88), 68 Damrnel, R., 157(110), 162(110), 180 Danielli, J. F., 334(30), 393 Dasannacharya, B. A., 412(60), 452 Davidson, E. R., 51(109), 69 Davies, D. B., 459(52), 523 Davies, H. A., 457(7), 458(7, 25), 521,
522 Davis, H. T., 344(75), 394 de Boer, J., 290(111a), 293(111a), 327, 338(44), 339(52), 394 DeBolt, M. A., 414(66), 415(66), 419 (66), 421(66), 452, 460(78), 462 (78), 463(78), 464(78), 506(78), 508 (78), 509(78), 512(78), 519(78), 523 Debye, P., 81(22), 178, 187(2), 262(2), 323, 338(47), 394 Dechow, F. J . , 377(136), 396 de Feijter, J. A., 333(18), 393 de Gennes, P. G., 357(86), 3 95 de Groot, S. R., 76(9), 178 DeLeener, M., 96(49), I79
530
AUTHOR INDEX
DeLeeuw, S., 201(23), 229(23), 246(23), 249(23, 78), 250(23), 251(23, 78), 252 (23), 261(23, 78), 263(23), 269(23), 324, 325 DeNeufville, J. P., 459(59, 63, 65, 69), 523 Derjaguin, B. V., 81(23), 178, 331(3), 333(3), 334(29), 338(36), 341(61), 392, 393, 394 Desai, R. C., 149(98), 180, 412(61), 452 Deutch, J . M.,80(18), 135(94), 178,180, 192(10), 198(10, 20), 200(10, 20), 209 (27), 232(10), 294(121), 306(147), 316 (20, 155), 317(155),323,324,327,328 de Vries, A. J., 331(4), 393 DeWitt, H. E., 403(38), 451 DeWitte, D. L., 459(61), 523 Dexter, A . R., 412(64), 452 Diercksen, G. H. F., 5(21), 28(21), 35 (211, 67 Dill, J. F., 414(66), 415(66), 419(66), 421(66), 452 DiMarzio, E. A., 457(17), 467(17), 492 (17), 502(17), 522 Dimitrov, D. S., 356(80),394 Dixon, M.,398(2), 408(49), 450 Doiron, T., 301(131), 327 Dom, B . E., 414(66), 415(66), 419(66), 421(66), 452 Domb, C., 216(34), 225(34), 324,477 (ll8),478(118), 524 Domcke, W., 5(16, 22, 23, 24,25, 26, 28, 29, 30, 31, 32), 23(16), 28(16, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32), 31 (16, 24, 25, 26, 28, 29, 30), 35(23), 45(16, 24, 31, 32), 46(24), 66, 67 Donnella, J., 445(108), 453, 467(82), 524 Donners, W. A. B., 340(59), 362(108), 367(111, 112), 368(111, 112), 369 (1111, 370(111), 371(111), 372(111), 373(108, 112), 374(59, 112), 375(112), 376(112), 394, 395 Doolen, G. D., 403(38), 451 Doolittle, A. K., 430(85), 452, 469(97), 470(97), 524 Dorfman, J. R., 96(48), 98(48), 105(56), 107(72), 108(78), 146(78), 147(78), 177(78), 179
Dorian, T., 301(131), 327 Dorrestein, R., 360(106), 395 Douglas, D. C., 407(45), 451 Dowell, F., 457(17), 467(17), 492(17), 502(17), 522 Drake, P. W., 414(66), 415(66), 419(66), 421(66), 452 Drummond, C. H., 111, 459(59), 523 Duderstadt, J. J., 106(64), 110(64), 179 Dunning, T. H., 6(49), 67 Duwez, P., 457(9), 458(9), 522 Duyvis, E. M.,333(10), 368(10), 393 Dynes, R. C., 518(138, 139), 525 Dzyaloshinskii, I. E., 339(55), 394 Easteal, A. J., 414(66), 415(66), 419(66), 421(66), 452, 460(78), 462(77, 78), 463(78), 464(78), 506(78), 508(78), 509(78), 512(78), 519(78), 523 Edmonds, A. R., 230(57), 325 Egelstaff, P. A., 421(73), 452, 456(2), 521 Einstein, A., 334(22), 342(22), 393 Eisenthal, K. B., 156(108), 180 Ellison, F. O . , 43(105), 69 Elterman, P. B., 414(66), 415(66), 419 (661, 421(66), 452 Emeis, C. A., 81(24), 178 Ernst, M. H., 96(48), 98(48), 105(56), I79 Essam, J . W . , 477(117), 478(117), 524 Evans, G. T., 131(88), 158(113), 180 Everett, D. H., 374(116), 395 Exerowa, D., 333(8), 393 Eyring, H., 83(27), 103(27), 153(102), 178, 180 Fehder, P. L., 81(24), 178 Felderhof, B. U., 80(18), 178, 251(79), 279(107), 294(120), 325, 326, 327, 357(88), 364(88), 365(102), 395 Feltz, A., 459(64), 523 Ferry, J. D., 468(86), 469(86), 524 Fetter, A. L., 2(4), 66 Fijnaut, H. M.,334(23), 357(89, 94), 360 (94), 361(94), 364(89), 365 (89), 377(23), 138, 139, 140), 380(138), 383(138), 387 (139, 140), 388(139), 390(89, 138), 391(138), 393,395,396
AUTHOR lNDEX Finney, J. L., 402(36), 422(76), 423(78), 425(76), 435(76, 78), 436(78), 438(95), 441(36, 76), 443(36), 449(76), 451, 452, 472(114), 524 Fisher, J. C., 401(28), 402(28), 451 Fisher, M. E., 482(131), 483(131), 484 (131), 525 Fixman, M., 77(11), 80(11), 158(113), I78, 180, 338(41), 394 Fleming, P. D., 399(17), 446(17), 447 (17), 448(17), 449(17), 451, 458(30), 472(30), 522 Flory, P. J., 468(85), 469(85), 470(85), 524 Fontana, E. H., 459(58), 523 Forman, R. E., 158(114), 181 Forrester, A. T., 376(118), 395 Forster, D., 76(10), 178 Fortune, P. J., 27(76), 68 Fowler, R. H.,399(22), 438(22), 451 Fox, T. G., 457(11), 468(85), 469(85), 470(85), 522, 524 Franck, J., 106(68), I79 Frankel, S . , 330(1), 334(25), 368(114), 392, 393, 395 Freasier, B., 316(155), 317(155), 328 Freasier, B. C., 155(106), 157(111), 162 ( l l l ) , 180 Freed, J. H., 153(102), 180 Freed, K., 516(136), 525 Freed, K. F., 3(6), 4(14, 15), 6(43, 44, 45, 67, 68, 69, 70, 71, 72), 7(72), 11 (72), 12(70, 721, 13(67), 14(44, 45, 701, 15(72), 16(72), 20(43, 44, 451, 23(91), 25(91), 27(70), 28(71), 29(71), 30(43), 31(69), 32(67, 68, 69, 70, 711, 33(68), 34(91), 35(67), 36(68, 91), 41(68), 43 (69), 44(69), 45(69), 46(69), 49(71), 51(71), 52(71), 53(71), 55(71), 60(72), 63(69), 65(69), 66(69), 66, 67, 68, 69 Frenkel, D., 400(27), 446(27), 451 Friedman, H.L., 158(114), 18I, 199(21), 253(82), 324, 325 Frisch, H. L., 78(12), 178, 292(114), 32 7 Frolich, H., 253(81), 273(95), 279(95), 325. 326 Frommhold, L., 295(126), 327 Fulcher, G. S., 410(57), 451, 459(74), 523
53 1
Fulton, R. L., 312(148), 328 Fumi, F. G., 438(97),452 Furtado, P. M., 149(98), 180 Gamba, A., 449(119),453 Cans, R., 342(66), 344(66), 394 Garno, J . P., 518(139), 525 Garrison, B. J., 158(114), 181 Garisto, F., 92(38), I 7 8 Gass, D. M., 107(69), 179, 407(82), 428 (82), 452 Gelbart, W. M., 295(126), 327 Gelius, U.,42(102), 69 Gershogorn, Z., 7(74), 27(74), 68 Giaque, W. F., 458(36), 460(36), 465 (36), 522 Gibbs, J . H., 399(17), 411(58), 420(58), 443(58), 444(58), 445(58), 446(17), 447(17), 448(17), 449(17), 451,458 (30), 467(84), 472(30), 522,524 Gibbs, J . W., 357(95),395,457(17), 467 (17), 492(17), 502(17),522 Giessen, W. C., 404(40), 451 Giglio, M., 377(135),396 Gillan, M. J., 408(49),451 Gilrnan, J. J., 458(27), 522 Glasstone, S., 83(27), 103(27), 178 Golden, S., 85(28), I 7 8 Goldstein, M., 419(71), 421(71), 432(71), 452,458(22,43), 522, 523 Goodman, J. F., 333(15), 334(31), 393 Gordon, J. M., 399(17), 446(17), 447(17), 448(17), 449(17),45I, 458(30), 472(30), 522 Goscinski, O., 5(36,42), 15(36), 20(36), 67 Goubau, W . M., 518(137),525 Goulon, J., 301(37),324 Graben, H. W., 292(117),327 Grabowski, E., 377(143), 396 Gray, C. G., 219(36), 299(36), 230 (56), 23 1(56), 232(56), 233(56), 234(56), 236(63), 238(36), 242 (56), 243(63), 244(71), 245(63,71), 262(63), 306(56), 324,325
532
AUTHOR INDEX
Gray, P., 91(39), I79 Gray, S. G., 339(18), 402(18), 403(18), 404(18), 422(18), 423(18), 441(18), 447(18), 451 Green, M. S . , 216(34), 225(34), 324 Greer, S. C . , 301(130), 327 Greet, R. J., 459(49, 50), 460(50), 523 Greffe, J. L., 301(37), 324 Grest, G. S . , 445(109), 453,468(94, 9 3 , ,470(94), 475(94), 477(94), 480(94), 481(94), 482(94), 500(94), 506(95), 508(95), 509(95), 513(95), 515(94, 95), 5 18(140), 524, 525 Griffing, K. M.,6(60, 61, 62), 7(60, 61, 62), 25(60, 61, 62), 27(60, 61, 62), 34(60, 61, 62), 68 Grigera, J . R., 304(145), 305(145), 314 (145), 327 Gross, E. P., 105(59), 123(82), 179, I80 Guang, S . G., 407(43), 408(43), 417(43), 45I Gubbins, K. E., 236(63), 243(63), 244(71), 245(63, 71), 262(63), 325 Guilmin, P.,477(123), 525 Gulden, G., 352(79), 367(79), 394 Gupta, P. K.,414(66), 415(66), 419(66), 421(66), 452 Halperin, B. I., 470(100, 103), 517(100), 518(103), 524 Hamaker, H. C., 339(51), 394 Hamer, N. D., 155(106), 180 Hammerius, Y.,377(129), 396 Hamrin, K., 42(102), 69 Hansen, J. P., 456(1), 521 Hansen, R. A., 377(136), 396 Hansen, R. S . , 362(97), 395 Happel, J., 135(93), I80 Hard, S., 366(105), 377(105, 129), 395, 396 Harris, F. E., 273(95), 278(106), 279(95), 326 Hartley, C. L., 301(129), 327 Hasha, D., 407(43), 408(43), 417(43), 451 Hasted, J. B., 303(137), 313(137), 327 Hawkins, G., 377(135), 396 Haworth, W . L., 459(69), 523 Haydon, D. A., 334(20), 393 Heden, P. F., 42(102), 69
Hedman, J., 42(102, 103), 48(103), 69 Helfand, E., 160(115), I81 Hellwarth, R. W . , 292(114), 327 Hemmer, P.C . , 202(26),324 Henderson, D., 153(102), 180, 224(40), 227(46), 228(46), 324, 338(38, 40, 42), 393, 394,423(77), 452 Henderson, R. L., 219(36), 229(36), 238 (39), 324 Herman, M. F., 4(15), 6(67, 68, 69, 70, 71, 72), 7(72), 11(72), 12(70, 72), 13 (67), 14(70), 15(72), 16(72), 27(70), 28(71), 29(71), 31(69), 32(67, 68, 69, 70, 71), 33(68), 35(67), 36(68), 41(68), 43(69), 44(69), 45(69), 46(69), 49(71), 51(71), 52(71), 53(71), 55(71), 60(72), 63(69), 65(69), 66(69), 66 Herring, F. H., 3(11), 66 Hertzfeld, K. F., 412(65), 445(65), 452 Hesse, G., 459(75), 523 Hesse, W., 407(47), 410(47), 451 Hesselink, F. T., 357(90, 91), 358(90), 359(90), 360(90), 361(90), 362(91), 363(91), 364(91), 366(103), 395 Hetherington, G., 459(57), 523 Hicks, T. J., 477(125), 479(125), 525 Hildebrand, J. H.,408(53,54,55,56), 410 ( 5 5 ) , 429(55), 430(55), 451 Hill, T. L., 471(110), 480(110), 524 Hills, G., 377(131), 396 Hiwatari, Y.,399(19), 402(33), 404(19), 422(19), 429(19), 441(19), 443(33), 446(92), 447(19,47), 451, 452 Hoare, M. H., 423(78), 435(78), 436(78), 452 Hoare, M. R., 432(88), 435(88), 438(95), 446(115), 452, 453 Hocken, R., 301(128, 130), 327 Hoegy, W . R., 96(48), 98(48), I79 Hoffman, G. W . , 156(108), I80 Hohlneicher, G., 5(17, 18), 28(17, 18), 66 Homsy, G. M.,333(6), 393 Hoover, W. G., 399(18), 400(26), 402 (18), 403(18), 404(18), 422(18), 423 (18, 26, 77), 441(18), 446(111), 447 (18, l l l ) , 4.51, 452, 453, 458(29), 522 Horman, J. A., 459(51), 460(51), 523 Horowitz, M. A., 301(130), 327
AUTHOR INDEX Howell, F. S., 458(41), 522 H h e , 1. S., 187(3), 188(4, 5), 189(4, 5), 190(4), 191(4), 192(4, 5), 195(4, 5 , 13, 14), 196(13, 16), 197(4), 199(22), 201 (24, 25), 204(4), 205(4), 206(4, 5), 209 (4, 5), 211(4), 212(13), 213(32), 217 (32), 226(13, 16), 232(4), 236(61), 238(13, 16, 61, 65), 245(61), 247(25), 249(4), 271(61), 274(102), 279(102), 282(25), 283(24), 288(102), 292(118), 294(122), 296(118), 301(102, 134), 316 (134), 317(4), 318(4), 321(22, 25), 323, 324, 325, 326, 327 Hsu, C. S . , 402(34, 35), 443(34, 35), 451, 471(111), 473(111), 524 Hubbard, J. B., 303(136, 138), 304(136, 138), 312(136), 327 Hudson, S., 93(42), 179,399(13),424 (13), 425(13), 435(13), 450 Huisman, H. F., 333(14), 393 Hutchinson, J. M., 462(79), 464(79), 465 (79), 506(79), 523 Hyde, B. G., 408(49), 451 Hynes, J. T., 79(15), 81(15, 21), 103(15), 108(77), 138(21), 143(97), 150(15, 101), 152(101), 154(103), 158(103), 160(115), I 79, 180, 181 Ingard, U., 362(117), 376(117, 120), 377 (117, 120), 395 Ingram, B. T., 334(31), 393 Isbister, D., 209(27), 316(155), 317(155), 324, 328 Isenberg, C., 334(26), 393 Israelachvili, J. N., 34 1(62),394 Ivanov, I. B., 356(80), 357(92, 93), 394, 395 Iwata, S., 47(106), 69 Jaccarino, V., 477(124), 479(124), 525 Jack, K. H., 459(57), 523 Jackle, J., 470(102), 524 Jacobs, D. T., 301(129), 327 Jacobson, B. S . , 157( log), 162(109), 180 Jacucci, G., 458(33), 472(33), 473(33), 522 Jain, R. K., 356(84), 395 Jansen, L., 290(111b, 112), 293(111b),327
533
Jansoone, V. M., 229(51), 246(5 l), 250(51), 252(51), 324 Jepsen, D. W., 199(21), 236(62), 262(62), 324, 325 Jhon, M., 115(80), 180 Johansson, G., 42(102), 69 Johnson, K. W., 399(18), 402(18), 403 (18), 404(18), 422(18), 423(18, 77), 441(18), 447(18), 451, 452 Johnson, R. I., 459(67), 523 Johnston, H. S., 157(112), 180 Joiner, B. A., 459(68), 523 Jolly, D. L., 155(106), 157(111), 162 ( l l l ) , 180 Jonas, J., 407(43), 408(43), 417(43), 451 Jones, H., 458(26), 522 Jones, M. N., 333(1 l ) , 393 Joosten, J. G . H., 357(94), 360(94), 361 (94), 363(99), 366(104), 377(138, 139, 140), 380(138), 383(138), 387(139, 140), 388(139), 389(139), 390(99, 138), 391(138), 395, 396 Jordan, K. D., 6(48, 61, 62, 64), 7(61, 62, 64), 25(61, 62,64), 27(61,62,64), 32 148), 34(61, 62, 64), 67, 6 8 Jdwnsen, P.,5(37), 6(37,46,47), 7(77), 23(89, go), 24(77), 32(46), 59(77), 61 (77), 67, 68, 69 Jortner, I., 482(130), 525 Jost, W., 153(102), 180 Kadanoff, L. P., 85(30), 178 Kallmann, H., 339(49), 394 Kapral, R., 80(19), 92(38), 93(42), 98(50, 51, 53), 106(66), 108(73, 77), 110(66), 111(66), 114(73), 116(51), 120(53), 121 (53), 123(53), 128(86), 130(87), 132 (86), 134(92), 144(51, 66), 149(98, 99), 150(100), 152(100), 154(103), 158(103), 171(50), 173(53), 174(53), 178, 179, 180 Kapral, R. E., 125(83), 180 Karplus, M., 10(80), 16(86), 68 Katyl, R . H., 362(117), 120), 377(117, 120), 395 Kauzmann, W.,443(105), 444(105), 453, 467(81), 524 Kawasaki, K., 91(41), 107(71), 111(41), 142(41, 95), 179, 180
534
AUTHOR INDEX
Keck, J. C., 87(34), 178 Keesom, W. H., 338(46), 394 Kelly, H. P., 2(5), 66 Kennedy, J. C., 459(57), 523 Kenney, J. W., 7(79), 23(79), 61(79), 63(79), 68 Kenny, J., 6(59, 61, 62), 7(59,61, 6 3 , 25(59, 61, 62), 27(59, 61, 62), 34(59, 61, 62), 68 Kerker, M., 334(21), 346(21), 393 Kerr, E. C., 295(124), 297(124), 327 Keyes, T., 108(74), 179 Kim, S. K., 293(119), 327 Kimball, G. E., 78(14), 178 Kirkpatrick, S., 477(116, 121), 478(116, 121), 524 Kirkwood, J. G., 197(17, la), 198(17), 200(17), 232(17), 250(17), 273(17, 93), 276(93), 290(93), 323, 326, 471(109), 524 Kitchener, J. A., 334(30), 393 Klare, H., 333(9), 393 Klein, M. J., 338(45), 394 Kolarov, T., 333(19), 393 Kolbeck, A. G., 459(61), 523 Kolgour, D. M., 449(118), 453 Konijnendijk, H. H. U., 174(116), 181 Koopmans, T., 15(85), 35(85), 68 Koppel, H., 5(26), 28(26), 31(26), 6 7 Kosugi, N., 47(106), 69 Kouvel, J. S., 477(125), 479(125), 525 Kovacs, A. J., 462(79), 464(79), 465 (79), 506(79), 523 Kraemer, W. P., 5(20, 22, 29), 28(20, 22, 29), 31(29), 67 Kramers, H. A., 82(26), 178 Kratky, K. W., 446(114), 453 Gistensen, W., 399(7), 411(7), 418(7), 419(7), 420(7), 432(7), 435(7), 450 Krook, M., 123(82), 180 Kubo, R., 75(5), 178 Kummer, R. B., 518(138), 525 Kunken, W., 292(114), 327 Kunzler, J. E., 458(36), 460(36), 465 (36), 522 Kuroda, H., 47(106), 69 Kusakov, M. M., 341(61), 394 Kuschick, J., 96(47), 179 Kutz, H. D., 85(31), 178
Laberge, N., 458(40), 461(40), 496(40), 522 Lacabanne, C., 458(44), 523 Ladd, A. J. C., 229(50), 246(50), 251(50), 324 Lado, F., 240(67), 309(67), 325 Laidler, K. J., 83(27), 103(27), 178 Lamb, J., 458(47), 523 Lamoreaux, R. H., 408(55,56), 410(55), 429(55), 430(55), 451 Landau, L. D., 316(154), 328 Landel, R. F., 468(86), 469(86), 524 Lane, J. W., 459(55), 460(55), 461(55), 496(55), 497(55), 523 Langbein, D., 340(57), 394 Langevin, D., 346(77), 352(77), 362(98), 367(77), 377(128, 137), 389(128), 394, 395, 396 Langhoff, S. R., 51(109), 69 Lantelme, F., 158(114), 181 Larsen, B., 236(61), 238(61), 245(61), 271(61), 325 Lasaga, A. C., 10(80), 16(86), 68 Lasocka, M.,462(80), 523 Lastovka, J. B., 376(119), 395 Laughlin, W. T., 459(54), 460(54), 461 (54), 496(54), 497(54), 523 Leadbetter, A . J . , 459(56),523 Leath, P. L., 477(120, 122). 478(120),524 Lebenhaft. J. R., 80(19), 178 Lebowitz, J. L., 105(57), 179, 188(6), 189 (6,7), 195(13), 196(13), 202(6), 212 (13), 214(6), 226(13), 238(13),323 LeFevre, E. J., 446( 113), 449( 113),453 Leontowicz, M., 342(65), 344(65), 394 Levelt Sengers, J. M. H., 301(131),327 Levesque, D., 229(52), 231(58,59), 233 (58,59), 234(59), 237(59), 238(52), 239(58,59), 240(58, 59), 241(58,59), 243(58,59), 246(52), 247(52), 248(52, 58,59), 249(52), 255(52,58,59), 256 ( 5 8 ) , 257(58), 258(52), 259(52), 260 (59), 261(58,59), 262(59), 263(52), 264(58), 265(58), 266(52,58), 267(89). 269(58,59), 270(59), 271(59), 304(143), 305(143), 306(143), 307(143), 308( 143), 309(143), 310(143), 31 1(143), 312(143), 313(143), 314(58), 321(143), 322(143), 325,326.327
AUTHOR INDEX Levich, V. G., 357(96), 395 Levine, S., 338(39), 391(142), 394, 396 Lewis, B. G., 457(7), 458(7), 521 Lifshitz, E. M., 316(154), 328, 339(54, 59,394 Light, J. C., 95(45), I79 Lindergerg, J., 2(3), 5(3), 66 Ling, A. C., 459(48), 460(48), 523 Lipatov, Y.,468(93), 524 Lippmann, F. J., 459(64), 523 Litovitz, T. A., 412(65), 418(70), 445 (65), 452, 468(90), 524 Liu, B., 6(71), 28(71), 29(71), 32(71), 34(98), 49(71), 51(71, 107, 108), 52 (71), 53(71), 55(71), 68, 69 Lo, S. M., 142(95), 180 Lofgren, H., 366(105), 377(105), 395 London, F., 338(48), 394 Loponen, M. T., 518(139), 525 Lorentz, H. A., 290(109), 326 Lovett, R., 312(149), 328, 343(69), 394 Low, G. G., 477(125), 479(125), 525 Lowdin, P. O., 24(93), 69 Luban, M., 447(117), 453 Lucas, J., 440(98, 99), 452 Lucassen, J., 356(85), 357(90,91), 358 (90), 359(90), 360(90), 361(90), 362 (91),363(91), 364(91), 365(85), 366 (85),395 Lucassen-Reynders,E. H., 356(85), 365 (85), 366(85),395 Lukman, B., 5(42), 6 7 Lumb, M. D., 153(102),180 Lundstrom, I., 377(130),396 Lyklema, J., 333(7), 368(113), 393,395 Lyon, J., 418(70),452 McCall, D. W., 407(45), 451 McClure, C. F., 108(78), 146(78), 147 (78), 177(78), I79 McCurdy, C. W., 6(55), 67 McCurdy, W., 31(94), 69 McDonald, I. R., 229(49), 237(49), 246 (49), 247(49), 248(77), 250(77), 252 (77), 253(77), 254(49), 324, 325, 398 (4), 404(4), 412(4), 450,456(1), 521 Macedo, P. B., 414(66), 415(66), 419 (66), 421(66), 452, 458(40, 41), 459 (60), 460(60, 78), 461(40, 76), 462
535
(78), 463(78), 464(78), 468(90), 496(40, 60), 506(78), 508(78), 509(78), 5 12(78), 519(78), 522, 523, 524 McKenzie, J. D., 419(71), 421(71), 432 (71), 452 McKinney, J . E., 458(43), 523 McKoy, V., 6(49,50, 51, 52, 53, 54,55, 56, 67), 13(50, 67), 15(50, 51, 521, 16 (53), 22(50, 51, 52), 23(50, 51, 52, 54, 56), 25(50, 51, 52, 56), 27(56), 31(56), 32(56, 67), 35(67), 36(56), 66(67), 67, 68, 69 McLachlan, A. D., 14(84), 15(84), 68 McQueen, N., 377(130), 396 McTague, J., 399(8), 400(27), 401(30), 402(30, 32), 41 1(8), 425(8), 435(8), 443(32), 446(27), 450, 451 McTague, J. P., 442(103), 453, 458(32), 471(32), 472(32), 473(32), 522 McWilliams, D., 3(11), 66 Magee, J. L., 80(16), I 7 8 Magill, J. H., 459(49,53), 460(53), 461 (53). 496(53), 497(53), 523 Maguire, J . F., 399(15), 450 Mahanty, J., 340(56,60), 341(56), 394 Malkina, A. D., 331(3), 333(3), 392 Mandel, M., 273(95), 279(95), 326, 399 (8), 401(30), 402(30, 32), 411(8), 425 (81, 435(8), 443(32), 450, 451 Mandelkern, L., 458(45), 523 Mandell, M. J., 458(32), 471(32), 472(32), 473(32), 522 Mandelstamm, L., 341(64), 342(64), 394 Mann, J. A., Jr., 334(32), 352(79), 362 (32), 367(79, 110), 377(136), 393, 394, 395, 396 Manne, R., 20(87), 68 Mansoori, G. A., 224(42), 324 March, N. H., 2(2), 9(2), 66 Marriott, J. R., 391(142), 396 Martin, R. L., 65(110), 69 Martina, E., 209(27), 315(150), 316 (151), 324, 328 Martinov, G. A., 244(70), 245(70), 325 Mason, E. A., 125(83), 180 Mason, J., 472(115), 524 Mason, S. G., 333(13), 393
536
AUTHOR INDEX
Matheson, A. J., 412(64), 452, 458(47), 459(52), 468(91), 523, 524 Matsuda, H., 399(19), 402(33), 404(19), 422(19),429(19), 441(19), 443(33), 447(19), 451 Maul, W., 459(64), 523 Mazenko, G. F., 106(60, 61, 62), 149 (98), 179, 180 Mazur, P., 76(9), 95(44), 117(117), 178, 179, 181, 273(95), 279(95), 293(119), 326, 327 Mehaffey, J. R., 98(53), 108(75, 76), 120 (53), 121(53), 123(53), 128(86), 130 (87), 132(86), 134(92), 149(98), 173 (53), 174(53), 179, 180 Messiah, A., 11(83), 68 Meunier, J., 376(122,123), 377(122, 123, 132), 395, 396 Meyer, H., 301(131), 327 Michels, A., 295(125), 327 Michels, J. P. J., 428(83), 452 Miller, A. A., 443(107), 453, 458(42), 462 (42), 522 Miller, W. H., 103(55), 179 Mills, R., 407(44), 451 Minns, C., 253(83), 325 Mistura, L., 301(37), 324 Mitchell, D. J., 304(141, 142), 309(141), 312(141), 321(141), 327 Mockler, R. C., 301(129), 327 Moeller, R. P., 414(66), 415(66), 419(66), 421(66), 452 Monchick, L., 80(16, 20), 125(83), 178, 180 Montgomery, J. A., 125(84), 161(84), 180 Montros, C. J., 414(66), 415(66), 419 (66), 421(66), 452 Mori, H., 89(36), 90(36), 91(36), 112(36), 172 Mossotti, 0. F., 290(109), 326 Mountain, R. D., 432(93), 443(93), 452 Moynihan, C. T., 414(66), 415(66), 419 (66), 421(66), 452, 457(15), 458(41), 460(78), 462(15, 77, 78), 463(15, 78), 464(78), 506(15, 78), 508(78), 509(78), 512(78), 519(78), 522, 523 Muller, V. M., 81(23), 178 Mundy, J. N., 408(50), 451
Munro, R . G., 497(133), 525 Murphy, T. J., 135(94), 180 Murrell, J. N., 157(110), 162(110), 180 Mysels, K. J., 330(1), 333(7, 11, 14), 334(25), 368(113, 114), 392, 393, 395 Nagel, S. R., 432(94), 452 Naghizadeh, J., 407(41), 412(41), 416 (41), 451, 468(89), 524 Nakajima, T., 3(13), 66 Napolitano, A., 459(60), 460(60), 496 (60), 523 Narang, H., 236(61), 238(61), 245(61), 271(61), 325 Narayanamurti, V., 518(138, 139), 525 Narayanaswarny, 0.S., 506(135), 525 Narten, A. H., 399(22), 438(22), 451 Nascirnento, M., 6(53), 16(53), 67 Nehrkorn, C., 5(39), 32(39), 67 Nemzak, T. L., 108(79), 179 Nesbet, R . K., 3(8), 23(92), 34(99), 40 (92), 45(99), 66, 69 Netzel, J., 333(9), 393 Neumann, M., 247(74), 325 Newton, I., 331(2), 392 Ng, K.C., 243(69), 325 Nielsen, A . E., 78(13), 178 Nienhuis, G., 192(10), 198(10, 20), 200 (10, 20), 209(27), 232(10), 316(20, 155), 3 17(155), 323, 324, 328 Nijboer, B. R. A., 412(63), 452 Nilsson, O., 377(129), 396 Nilsson, R.,42(103), 48(103), 69 Ninham, B. W., 304(141, 142), 309(141), 312(141), 321(141), 327, 340(56, 58, 60), 341(56), 394 Nitzan, A., 158(114), 181 Nordholrn, S., 155(106), 157(111), 162 ( l l l ) , 180 Nordling, C., 42(102, 103), 48(103), 69 Northrup, S. H., 79(15), 81(15, 21), 103 (15), 138(21), 143(97), 150(15, 101), 152(101), 160(115), 178, 180, 181 Noyes, R. M., 75(8), 77(8), 143(8,96), 155(107), 178, 180 Nyholm, R., 42(103), 48(103), 69 Oddershede, J., 5(37), 6(37), 67
AUTHOR INDEX Ogawa, T., 399(19), 402(33), 404(19), 422(19), 429(19), 441(19), 443(33), 447(19),451 Ogita, N., 399(19), 402(33), 404(19), 422(19), 429(19), 441(19), 443(33), 447(19), 451 Ohmichi, N., 3(13), 66 Ohno, M., 5(34), 67 Ohm, Y.,2(3), 5(3, 38, 39, 40, 41), 23 (38), 28(40, 41), 32(39, 95), 36(95), 47(38), 66, 67, 69 O’Keefe, M., 408(49), 451 Onsager, L., 198(19), 211(19), 262(19), 265(19), 272(19), 303(138), 304(138), 324, 327, 357(87), 395 Oppenheim, I., 85(31), 107(71), 108(74), 135(94), 178, 179, 180 Orcutt, R. H., 295(123), 327 O’Reilly, D. E., 407(42), 451 Ornstein, L. S., 344(71), 394 Ortoleva, P., 98(52), 100(52), 116(52), 179
Oseen, C. W., 135(93), 180 Oster, G., 197(18), 323 O’Sullivan, W. J., 301(129), 327 Ottewill, R. H., 334(33), 393 Outhwaite, C. W., 338(39), 394 Overbeek, J. T. G., 330(1), 334(27), 337 (34), 355(82), 392, 393, 395 Overdijk, D. J., 295(125), 327 Oxtoby, D. W., 295(126), 301(37), 324, 32 7
Pagitsas, M., 98(51), 106(67), 111(67), 116(51), 121(67), 144(51), 173(67), 179
Pai, D. M., 459(71), 465(71), 523 Pailthorpe, B. A., 304(141), 309(141), 312(141), 321(141), 327 Pankhurst, K. G . A., 334(30), 393 Parker, R., 10(82), 68 Parr, R. G., 10(82), 68 Parsegian, V. A., 340(58), 394 Patey, G. N., 213(30), 223(38), 229(47, 48, 52, 55), 230(38), 231(30, 38, 58, 59), 232(30), 233(20, 38, 58, 59), 234 (59), 237(38, 47, 59), 238(30, 38,52, 59), 239(30, 38,58,59), 240(30, 38.58, 59), 241(38,58,59), 243(30,58,59,
537
69), 245(47, 48, 52), 246(47, 48, 52, 55), 247(47, 48, 521, 248(30, 47, 48, 52, 58, 59), 249(30, 47, 52), 253(55), 254(47, 88), 255(30, 52, 58, 59), 256 (58), 257(58), 258(52), 259(52), 260 (59), 261(58, 59), 262(59), 263(52), 264(30,55,58), 265(58), 266(52,58, 267(89), 268(55), 269(58,59), 270 (59), 271(48, 59), 284(108), 285(108), 288(108), 289(108), 304(143), 305 (143), 306(143), 307(143), 308(143), 309(143), 310(143), 311(143), 312(143), 314(58), 321(143), 322(143), 324, 325, 326, 32 7
Pechukes, P., 103(54), 179 Pecora, R., 75(7), 178, 377(125), 396 Pedersen, J. B., 153(102), 180 Pgrez-Hernandez,W., 304(144), 314(144), 32 7
Perram, J. W., 201(23), 229(23), 246(23), 249(23, 78), 250(23), 251(23, 78), 252 (23), 261(23, 78), 263(23), 269(23), 324, 325
Perrier, J. P., 477(126), 479(126), 525 Peterson, E. M., 407(42), 451 Peyerimhoff, S., 5(17), 28(17), 66 Phillips, W. A., 470(101, 108), 524 Pickup, B. T., 5(36), 15(36), 20(36), 67 Pierrnarini, C. J., 497( 133), 525 Pike, E. R., 377(126, 127), 396 Pitaevskii, L. P., 339(55), 394 Plazek, D. J., 459(53), 460(53), 461(53), 496(53), 497(53), 523 Plurnmer, W. A., 459(58), 523 Pohl, R. O., 470(106), 524 Polk, D. E., 404(40), 451, 458(21), 522 Pollard, L. J., 458(38), 460(38), 496 (38), 522 Pollock, E. L., 229(54), 246(54), 250(54), 252(54), 259(54), 260(54), 261(54), 266(54), 267(54), 268(54), 269(54), 325
Pomeau, Y., 107(70), 111(70), 117(70), 169
Pople, J. A., 273(95), 279(95), 290(111b), 293(11 lb), 326, 327 Porzio, K. C., 334(32), 362(32), 393 Poulain, M., 440(98, 99), 452
538
AUTHOR INDEX
Prager, S., 292(114),327 Pratt, L., 155(105), 180 Pratt,L. R., 274(103, 104, 105), 281(104, 105), 282(104), 283(104),326 Princen, H. M., 333(13, 17), 393 Prins, A., 333(16), 393 Prins, C., 338(44,45), 394 Purvis, G. D., 5(38, 39,40,41), 23(38), 28(40, 41), 32(39,95), 36(95), 47(38), 67,68 Quentrec, B., 253(85), 254(85),326 Rabinowitch, E., 106(68), 179 Radoev, B., 333(19),393 Rahman, A., 248(92),326,399(8, 22), 401(30), 462(30, 32, 34, 35), 408 (SO), 41 1(8), 412(62,63), 425(8), 435 (8),438(22),443(32,34,35),450,451, 452,458(32), 471(32, l l l ) , 472(32), 473(32, 11l), 522,524 Rainford, B., 477(125), 479(125),525 Raman, C. V., 342(67), 394 Ramdas, L. A,, 342(67), 394 Ramshaw, J., 192(8,9), 193(11),323 Ramshaw, J. D., 213(28), 316(153), 324, 328 Rao,K. J., 443(106),444(106),453,458 (39), 460(39), 465(39), 467(39), 522 Rao, K . R., 412(60),452 Rasaiah, J., 224(42), 236(6 l), 238(6 l), 245(61), 271(61), 324,325 Rasaiah, J. C., 316(152), 328 Ravarhe, H. J., 399(9),'402(31), 432(93), 441(31), 443(31,93), 435(9),450,451, 452 Rayleigh, Lord, 344(76), 347(76), 394 Razouk, R. I., 333(14), 393 Redmon, L. T., 5(40), 28(40), 67 Ree, F. H.,446(111),447(111),453 Reich,G. R.,477(120, 122),478(120), 524 Renk, F., 333(6),393 Rescigno, T. N., 6(55), 67 Risibois, P., 96(49), 105(57,58), 107(70), 111(70), 117(70),179 Rice, S. A., 91(39), 179,407(41), 412(41), 416(41),451 Richmond, P., 374(116), 395
Riddiford, A. C., 334(30), 393 Rijnbout, J. B., 340(59), 367(111, 112), 368(111, 112, 115), 369(111), 370(111), 371(111), 372(111), 373(112), 374(59, 112), 375(112), 376(112),394,395 Ritland, H. N., 506(134), 525 Ritson, D. M., 303(137), 313(137),327 Robinson, K., 391(142), 396 Rockstad, H. K., 459(65),523 Rogers, S., 458(45), 523 Rose, J., 6(5 1,52), 15(5 1, 52), 22(5 1, 52), 23(51,52), 25(51,52),67 Rosenberg, B. J., 27(76), 68 Ross, J., 74(4), 93(42), 95(44,45), 125 (83), 178, 179, 180 Ross, M.,400(26), 423(26,27), 429(84), 451,452 Rowe,D. J., 3(10), 6(10), 13(10), 14(10), 15(10), 22(10), 66 Ruckenstein, E., 356(84), 395 Rushbrooke, G. S., 236(61,63), 238(61), 243(63), 245(61, 63), 262(63), 271(61), 292(115, 117),325,327 Sahlin, H. L., 403(37), 451 Saito, S., 459(62), 523 Sales, K. D., 35(100), 69 Sampanthar, S., 2(2), 9(2), 66 Samuel, A. H., 80(16), 178 Sanfeld, A , , 338(37), 393 Sangster, M. J . L., 398(2),450 S u e , E. J., 417(69), 418(69), 452 Sare, J. M.,417(69), 418(69), 452 Sarkisov, G. N., 244(70), 245(70), 325 Sarrach, D. J., 459(69), 523 Sasabe, H., 414(66), 415(66), 419(66), 421(66),452 Scaife, B. K . P., 273(95), 279(95),326 Schaefer, H. F., 2(1), 5 1(1), 66 Sche, S., 357(89), 364(89, 101), 365(89, 101), 390(89),395 Schell, M., 149(99), 180 Scheludko, A., 333(8, 12, 19), 355(12), 334(28),393 Schirmer, J., 5(24, 25, 27, 28, 29, 30), 28(24,25, 27.28, 29, 30), 31(24, 25, 27, 28, 29, 30),45(24), 46(24), 6 7 Schofield, P., 429(84), 452 Scholten, P. C.. 368(1!3),395
AUTHOR INDEX Schulze, H. J., 333(5),393 Scott, G. D., 449( 118), 453 Scott, J. D., 425(79), 452 Scriven, L. E., 344(75),394 Segal, G. A , , 7(75), 27(75), 68 Shante, V . K . S., 477(116), 478(116), 524 Shapiro, S. L., 156(108),180 Shavitt, I., 7(74), 27(74), 68 Shenoy, G. K., 408(50), 451 Shepard, R . , 7(78), 23(78), 61(78), 63(78),
68 Sherman, R . H., 295(124), 297(124), 327 Shibuya,T., 6(50, 51,52), 13(50), 15(50, 51,52), 22(50,51,52), 23(50, 51, 52), 25(50,51,52), 67 Shin, K. J., 98(53), 106(66), 110(66), 111(66), 120(53), 121(53), 123(53), 128(86), 132(86), 144(66), 150(loo), 152(100), 173(53), 174(53), 179, 180 Shinoda, K., 330(1), 334(25), 392,393 Shirley, D. A , , 65(110), 69 Shugard, M., 158(114), 181 Shuler, K. E., 95(45), 179 Sichina, W., 407(48), 416(48), 451,457 (14), 460(14), 465(14), 468(14), 497 (14), 519(14), 522 Siegbahn, K., 42( 102), 69 Siegbahn, P., 5 l(5 I), 69 Siegel, A . , 132(91), 180 Silbcrstein, L., 292( 113),327 Simons, 1.,6(46,47,48,56,57,58,59, 60,61,62,63,64,65), 7(56,57,58, 5 9 , 6 0 , 6 1 , 6 2 , 6 3 , 6 4 , 6 5 ,78, 79), 10 (81), 23(56,78,79), 25(56,57,58, 59, 60,61,62,63,64,65), 27(56,57,58, 59,60,61,62,63,64,65), 31(56), 32(46,48, 56), 33(57), 34(57,58,59, 60,61,62,63,64,65), 35(57,97), 36 ( 5 6 ) , 61(78,79), 63(78, 79), 67. 68, 69 Simmons, J. H., 461(76),523 Sinanoglu, O., 3(7), 66 Skinner, J. L., 86(33), 125(33, 85), 131 (90), 161(33,85), 178,180 Slattery, W. L., 403(38), 451 Smith, D., 458(39), 460(39), 465(39), 467(39), 522 Smith, E. R . , 201(23), 229(23), 246(23), 249(23, 78), 250(23), 251(23, 78),
539
252(23), 261(23,78), 263(23), 269(23), 324,325 Smith, 1. W., 377(135),396 Smith, W. D., 6(56,57,58), 7(56, 57, 58), 23(56), 25(56,57,58), 27(56,57,58), 31(56), 32(56), 33(57), 34(57, 58), 35 (57), 36(56), 67, 68 Smith, W. R., 244(72), 325, 338(40), 394 Snider, N. S., 74(2,4), 178 Solem, A . D., 290(112), 327 Sonntag, H., 333(9), 393 Soules, T. F., 441(100), 453 Spaepen, F., 457(8), 458(8), 521 Sparnaay, M. J., 338(35), 341(35), 393 Stace, A . J., 157(110), 162(110), 180 Stanley, H . E., 477(127),525 Starling, K. E., 448(120),453 Stauffer, D.,477(119), 478(119), 479 (119), 482(132), 484(119, 132),524, 525 Steinhauser, O., 247(74), 325 Stell, G., 187(3), 188(4,5,6), 189(4,5, 6, 7), 190(4), 191(4), 192(4, 5), 195(4, 5 , 13,14), 196(i3, 16), 197(4), 199(22), 201(24, 25), 202(6), 204(4), 205(4), 206(4, 5), 209(4, 5), 211(4), 212(13), 213(32), 214(6), 216(33, 34,35), 217 (32), 224(42), 225(34,43), 226(13, 16, 45), 227(46), 228(46), 232(4, 5), 234 (108), 236(61), 237(45), 238(13, 16,45, 61,65), 245(61), 247(25), 249(4, 5), 254(45), 271(61), 274(102), 279(102), 282(25), 283(24), 285(108), 288(102, 108), 289(108), 292(115, 117, 118), 296(118), 301(102, 133, 134), 315(150), 316(134, 151, 152), 317(4), 318(4), 321 (22,25),323,324,325,326,327,328 Stephens, R . B., 459(72), 470(107), 508 (72), 523, 524 Stillinger, F. H., 88(35), 178, 248(92), 273(91), 312(149), 326,328,343(69), 394,399(21), 402(21), 407(21), 41 1 (21), 432(21), 441(21),451,472(113), 524 Strauss, H. L., 292(116), 293(116), 295 (127), 327 Strauss, W., 458(38), 460(38), 496(38), 522
540
AUTHOR INDEX
Strcett, W. B., 402 (31), 432 (93), 441 (31), 443 (31,93), 451,452 Sullivan, D. E., 294 (121),327 Sutherland, W.,209 (27), 324 Swanstrom, P., 23 (go), 69 Swinney, H. L., 376 (121). 377 (121), 379 (121), 395 Tait, R. H., 518 (137),525 Takahata, Y., 3 (12), 66 Takayarna, S., 458 (24), 522 Tarnrnann. G., 407 (47). 410 (47), 451, 459 (75), 523 Taneniura, M., 402 (33), 443 (33), 451 Tate, J. R., 333 (IS), 393 Taylor, H. S., 3 (9), 5 (9), 66 Taylor, J . L., 334 (20), 393 Teixeira, J., 477 (127),525 Teller, E., 403 (37), 451 Tellinghuisen, J., 154 (104), 180 ten Seldarn, C. A . , 290 (1 1la), 293 (1 1 la), 295 (125), 327 Theurnann, W. K., 189 (7), 323 Thijsse, 9. J., 301 (131),327 Thompson, J. C., 459 (68), 523 Thornburg, 0 . D., 459 (67),523 Tissier, B., 477 (126). 479 (126),525 Titijevskaja, A . S., 331 (3), 333 (3), 392 Torrie, G. M.,154 (103), 158 (103), 180, 243 (69), 325, 338 (43), 394, 399 (16), 422 (16),436 (16), 446 (16), 447 (16), 450 Torruella, A. J., 230 (56), 231 (56), 232 (56), 233 (56), 234 (56), 242 (56), 306 (56), 325 Tosi, M. P., 438 (97), 452 Tournier, R., 477 (126), 479 (126),525 Trappeniers, N. J., 428 (83),452 Traykov, T. T., 357 (93). 395 Triezenberg, D. G., 343 (70), 394 Tsui, I:. S. M., 6 (43,44,45), 14 (44,45), 20 (43,44,45), 30 (43), 67 Tucker, J., 460 (78), 462 (77, 78), 463 (78), 464 (78), 506 (78), 508 (78), 509 (78),512(78),519 (78),523 Tucker, J. C., 399 (25),451, 458 (39), 460 (39), 465 (39), 467 (39), 522 Tully, J. C., 158 (114), 181 Turban, L., 477 (123), 525
Turg, P., 158(114), 181 Turnbull, D., 401(28), 402(28), 430(86), 446(112),451, 452,453,457(4,5, 8, 10, 13, 18, 19),458(4,5,8, 13, 19, 21), 459 (50,59), 460(20, S O ) , 462(13, 20), 468 (87, 88), 469(18, 87, 88), 470(87, 88), 472(18), 473(18, 87,88),475(87, 88), 489(87,88), 494(87,88),521, 522,523, 524 Turner, D. W., 33(96), 35(96), 42(96), 69 Twcer, H., 458(40), 461(40, 76). 496(40), 522 Uhlmann, D. R., 401(29),451,457(6),458 (6), 459(54,55,61), 460(54,55), 461(54, 55). 496(54,55), 497(54,55),521,523 Valleau, J. P., 229(47,48), 237(47), 243 (69), 245(47,48, 73), 246(47,48, 73). 247(47,48), 248(47,48), 249(47,73), 250(73), 253(83), 254(47, 481,271 (48), 324, 325, 338(43), 394, 398(1),450 van Beijeren, H., 108(78), 146(78), 147(78), 177(78),I79 van den Ternpel, M . , 357(90,91), 358(90), 359(90), 360(90, 107), 36 1(90), 362(9 I), 363(91), 364(91), 395 van de Riet, R. P., 360( 107), 395 van der Macsen, I:., 290(111a), 293(111a), 327 van Karnpen, N . G., 160(115), 181 Van Kranendonk, J., 230(56), 23 1(56), 232 (56), 233(56), 234(56), 242(56), 306(56), 325 van Leeuwen, J. M. J . , 96(48), 98(48), 174 (116),179,181 van Vleck, J . H., 274(100), 326 Varma,C. M.,470(100), 517(100),524 Vashishta, P., 408(50), 451 Veda, A , , 399(19), 402(33), 404(19), 422 (19),429(19), 441(19),443(33), 447(19), 451 Vericat, l:., 304(145), 305(145), 314(145), 327 Verlet, L., 213(31), 217(31), 224(41), 225 (31,41), 229(31), 237(31), 240(68), 246 (31), 247(31), 254(31),324, 325 Verwey, E. J., 337(34), 393 Vischer, P. B., 160(115),181
AUTHOR INDEX Vogel, H., 459(73), 523 von Himendahl, L., 438(96), 452 von Nicssen, W., 5(18, 20, 21, 22,23,24, 25, 26, 28, 29, 30), 28(18, 20, 21, 22, 23, 24, 25, 26, 28,29, 30), 31(24,25,26, 28, 29,30), 35(21,23),45(24),46(24),66,67 von Smoluchowski, M., 73( 1). 77(1), 78 ( l ) , 80(1), 178, 341(63),394 Vrij, A,, 333(18), 334(21, 23). 340(59), 342(68), 344(68), 346(21,68, 78), 347 (78), 349(78), 351(68), 352(78), 354 (78), 355(68,81,82). 356(81), 357(90, 91, 94), 358(90), 359(90), 360(90, 94), 361(90,94), 362(91, 108), 363(91), 364 (91). 367(78, 111, 112), 368(111, 112), 369(111), 370(111), 371(111), 372(111), 373(108, 112), 374(59, 112). 375(112), 376( 112), 377(23), 393, 394,395 Wagner, F., 367( 109), 395 Wahl, A. C., 35(100), 69 Wainwright, T. E., 107(69), 179, 399(6), 407(82), 422(6), 428(82), 449(6), 450, 452,458(28),522 Waisman, E., 227(46), 228(46), 324 Waite, T. R., 80( 17), I 7 8 Walecka, J. D., 2(4), 66 Walker, L. R., 477(124), 479(124),525 Ware, W. R., 108(79), I79 Watts, R. O., 238(66), 242(66), 248(75, 76), 253(75),325 Wayner, P. C., Jr., 333(6),393 Webb,W. W., 377(133, 134),396 Weber,M. J.,441(102),453 Weber, T. A,, 399(21), 402(21), 407(21), 4 11(21),432(2 l ) , 44 1(21),451.472 ( 113), 524 Webman, I., 482(130), 525 Weeks, J . D., 224(39,41), 225(41), 237 (64), 324.325,344(73), 394 Weeks, R. A., 425(80), 452 Weiler, R., 458(40), 461(40), 496(40),522 Weinberg, M., 108(73, 77). 114(73),179 Weiner, J. H., 158(114),181 Weis, J . J., 213(31), 217(31), 224(41), 225 (31,41), 226(45), 229(31,52), 231(58, 59),233(58,59), 234(59), 237(31,45,59), 238(45,52), 239(58,59), 240(58,59,68), 24 1(58,59),243(58,5 9),246(3lm,247
54 1
(31, 52), 248(52, 58, 59), 249(52), 254 (31,45), 255(52,58, 59), 256(58, 87), 257(58, 87), 258(52, 87j, 259(52), 260 (59), 265(58), 266(52,58), 267(89), 269 (58,59), 270(59), 271(59). 292(116), 293(116), 295(127), 304(143), 305(143), 306(143), 307(143). 308(143), 309(143), 310( 143). 311( 143), 312(143), 313(143), 314(58), 321(143), 322(143),324, 325, 326,327 Wcitz, A , , 458(42), 462(42), 522 Welch, B. J., 407(46),451 Wendin, D., 5(34), 6 7 Wendin, G., 5(33, 35), 29(35), 30(35), 6 7 Wendt,H. R., 399(11),432(11),435(11), 450,472(112),473(112),524 Wcntzel, G., 43(104), 64(104), 65(104), 69 Werne, L. O., 42(102), 69 Wertheim. M. S., 185(1), 194(12), 196(12, 15), 201(1), 207(12), 212(12), 213(15, 29), 231(12), 232(12), 233(12), 236(12, 60). 238(29), 241(12), 242(12), 274(29, 96,97,98,99), 276(96), 277(29), 278 (29, 96,98), 279(98), 283(29), 284(97, 99), 285(96,97), 286(29), 287( 12,29), 292(98), 323, 324, 325, 326, 344(74), 394 Wetmore, R. W., 7(75), 27(75), 6 8 Whittington, S . C., 245(73), 246(73), 249 (73), 250(73), 325 Widom, B., 74(3), 75(3), I 7 8 Wilcox, L. R., 301(128),327 Wilder, J., 462(77), 523 Wilder, J. A., 414(66), 415(66), 419(66), 421(66),452 Wilemski,G., 77(11), 80(11), 131(89), 178, I80 William, M . L., 468(86), 469(86), 524 Williard, J . E., 459(48), 460(48), 523 Willstatter, M., 339(49),394 Wittingham, S. G., 398(1), 450 Wolf, E., 376(121), 377(121), 379(121), 395 Wolpert, S. M., 458(42), 462(52), 522 Wolynes, P., 303(136), 304(136), 312(136), 32 7 Wolynes,P. G., 86(33), 125(33, 85), 131 (90), 161(33, 85),178, I80 Wong, J ., 399(25), 4 15(67), 42 1(67),451,452
542
AUTHOR INDEX
Wood, W. W., 443(104),453 Woodcock, L. V., 398(3), 399(5, 10, 20, 23), 402(36), 404(20), 407(5,20), 41 1 (5, 20, 23),418(16b),419(5, 16b, 20, 23), 420(20), 422(20), 424(10), 425(20), 426(20), 429(20), 432(23), 435(20), 436(10), 437(10), 438(23), 441(36), 443(36), 445(20), 447(20, 116), 448 (116), 449(16a, 20), 450(20),450,451, 453, 458(31, 34, 35), 472(34), 480(128), 522,525 Woods, L. A., 458(46), 523 Wright, A . C., 459(56), 523 Wu, E. S., 377(133, 134),396 Wunderlich, B., 458(42), 462(42), 522
91), 24(77), 25(91), 27(66, 70), 28(71), 29(71), 31(66,69,94), 32(66,76,68, 69,70, 71), 33(68), 34(66, 91), 35(67), 36(68,91), 41(68),43(69), 44(69), 45 (69), 46(69), 49(71), 51(71), 52(71), 53 (71), 55(71), 59(77), 61(77), 63(69, 77). 65(69), 66(66,69), 66, 67, 68, 69 Yip, S., 106(62,63), 107(63), 108(63), 117(63), 149(98), 179, 180, 412(61), 452 Young, C., 377(141), 396 Young, D. A., 458(29), 522 Young, E., 334(24), 393 Young, J. B., 334(24), 393 Young, W. H.,2(2), 9(2), 66 Yvon, J., 273(94), 290(94), 326
Xystris, N.. 116(81), 180
Yamamoto, T., 93(43), 94(43), 179 Yaris, R . , 3(9), 5(9), 66 Yeager, D. L.,4(14, 15), 6(53,54,55,66, 67,68,69, 70,71), 7(66, 77), 12(70), 13(66,67), 14(70), 16(53), 23(54, 66,
Zachariasen, W. H., 467(83), 524 Zeller, R. C., 470(106),524 Zernike, F., 344(71), 394 Zoliweg, J., 377(135), 396 Zwanzig, R., 75(6), 89(37), 92(6), 112 (37). 178, 343(70), 394
Advance in Chemical Physics, VolumeXLVIII Edited by I. Prigogine, Stuart A. Rice Copyright © 1981 by John Wiley & Sons, Inc.
SUBJECT INDEX Agrofonov, Martinov, and Sarkisov theory, 244-245 Amorphous phase, 457 Analytic direct correlation (ADC) approximation, 217 for polarizable molecules, 288 Andersen and Chandler approximation, 224 Approximate ground state functions, 22-23 Epstein-Nesbet perturbation theory and, 23,40 iterative improvement of, 16, 59 Rayleight-Schrodinger perturbation theory and, 23,40 reference state of, 23 multiconfigurational, 24,61 single determinant of, 23 Arrhenius equation, 407 Asymptotic expression for correlations, 198 Attenation, ultrasonic, 470 Batchinski’s equation, 410 Batchinski’s-Hildebrand equation, 430 Bhatnager, Gross, and Krook equation, 123-125 Binary collision operators, 97 Black spot formation, 373 Boltzman equation, 9 3 Born-Meyer-Huggins potential, 438 Borneppenheimer hamiltonian, 8 Boundary condition, 72 of complete absorption, 78 by Ewald, 249 periodic, 246 truly, 249-253 of radiation, 78 Cage effects, 106 Cavity potential, 121 Chargedipole systems, 308
Chemical reactions, type o f bimolecular, 72,94 dissociation, 82 fluorescence quenching, 75 isomerization, 82 recombination, 86 Chemical relaxation time, 74 Clausius-Mossotti approximation, twoparticle, 20 Clausius-Mossotti, continuum, 210 Cluster series expansion, 237-238 Collision events, uncorrelated, 107 Competition effects, 80 Computer simulation studies, see Crystal nucleation; Glass transition Configuration space equations, 127-139 Constant-polarizabity model, 273 Core parameter, 201-206 Correlation fluctuations, 344 Correlation functions, 377-378 for amplitude, 378 direct, 189 of intensity, 379 pair, 188-201,254 Coupling, weak, 92 Coupling mode, 91 Critical density, 295 Critical exponent of dielectric constant, 301 Critical temperature, 295 Crystallization tendency, for soft-sphere fluids, 404 Crystal nucleation, computer simulation, 402,441 body-centered-cubic nuclei, 443 characterization of, 441-443 face-centered-cubic nuclei, 443 factors involved, 403 structure factor, 443 Voronoi polyhedra, 443 see also Homogenous nucleation
543
544
SUBJECT INDEX
Crystal nucleation temperature, 445 Damping matrix, 89 Dense random packing, 472 Density fields, 85 Diagrammatic description of approximation, 228,229 Diagrammatic many body perturbation theory, 2 Dielectric constant, 185 critical exponent of, 301 critical properties of, 295-303 dielectric anomaly of, 297 high-frequency, 278 solutedependent, 304 see also Fluids Dielectric constant expressions: by Chandler, site-site for polar-polarizable molecules, 283 by H@yeStell,190 general, 204 for polar molecules, 282 by Kirkwood, 232 by Lebowitz, Stell, and Baer, 192 by de Leeuw, 250 by Nienhuis and Deutch, 192 by Ramskaw, 192 by site-site correlation functions, 20 1 by Wertheim, 196,212 see also Mean spherical approximation Diffusion coefficient, 74 autocorrelation function for, 91 relative, 137 Diffusion eigenfunction, 145 construction of, 175-176 Diffusion equation, 72 Diffusion tensor, 80, 130-136 Dipolar particles, polarizable, 273 Dipole-induceddipole, 276 Direct correlation function, 189 Dispersion relation: for bonding mode, 362 for squeezing mode, 362,383,384 Doolittle equation, 430,509 Double layer: electric, 336-338, 369-372 free energy of interaction in, 338 pressure in, 337
Droplet model, 482,483 Electrodynamic theory, 357 Electrolyte solutions, 303-3 14 Electrostatic repulsion forces, 336-338 Emulsion, 332 Energy transfer mechanism, 157 Entropy: communal, 481,485,488,505 and cluster distribution function, 482 definition of, 480 in free energy equation, 471 in free volume equation, 493 and heat capacity, 498,502 and order of phase transition, 490, 492 vanishing of, 498 configurational, 467,473,492,498 crisis of, 465,502 of crystal, 465,498 excess of, 468 of mixing, 467,490,498,502 residual, 458,469 of surface, 483,484 theory of, 456,467,468,502 vanishing excess of, 465 temperature, 467 Equations of motion calculations, 31-61 and configurational interactions, 49 derivation of equations, 8-16 electron affinities for: fluorine, 5 3 hydroxyl, 5 3 extended, 49 ionization intensities, 43, 64 ionization potentials for: boron hydride(BH), 5 2 hydrogen fluoride, 5 3 neon, 5 3 nitrogen, 31 orbital basis set dependence, 34 shake-up energies of nitrogen, 42 shifted vs. unshifted denominators, 38, 55 Simon’s method, 31-34 of third order, 36 Yeager’s method, 3 1-34 Equations of motion mcthod, 2-66
SUBJECT INDEX Equations of motion operators, 10-22 annihilation condition, 11 approximate ground state improvement, 15,59 conditions satisfied by, 11 deexcitation, 15, 20 transition matric elements calculation, 16 Equations of motion operators basis: completeness of, 15, 17 electron affinities and ionization potentials, 20 excitation energies, 21 linear independence of, 16-18 orthogonalization of, 25 unorthodox, use of, 56 many body, 18 Equilibrium, metastable, 506 Equipartition theorem, 342, 343 Excluded volume effects, 81 Expotential approximation, 226 Expressions for correlations, asymptotic, 198 Expressions for dielectric constant, see Dielectric constant, expressions Felderhof rederivation of WerhteimFrohlich expression, 279 Fermi’s golden rule, 4 3 Field point formulation, 164-167 Fluctuating-polarizability model: by Kirkwood and Yvon, 273 by van Vleck, 274 Fluctuation-dissipation theorem, 84 Fluids: dipolar, hard-sphere, 231 polar-nonpolarizable, 187-229 computer simulation studies of, 245254 quantitative results for, 254-273 polar-polarizable, 273-303 of Stockmayer, 231 generalized, 240 see also Dielectric constant Flux, one-way, 83 Fokker-Plank equation, 82-83 Forces: colloidal, 330, 331 of dispersion, 339-340 electrostatic, 332, 369
545
mean, 80, 198-199 random, 84 repulsive, 96 van der Waals-London, 332, 373-375 interpolation formula for, 374 Free energy, 476,487,504 cofigurational, 486 of liquidlike cells, 475,490,499 local, 473,474,475 of solidlike cells, 490,499 Free volume, 493 of amorphous phase, 470,471 distribution of, in liquids, 473 free exchange of, 479 of liquidlike cells, 475 of phase transition, 493 Free volume model, 456,468,475, 476 and fluidity, 470,494 in glass transition, 470 and heat capacity, 514 and internal configurational entropy, 502 limitation of, 476 robustness of, 504-505 with structural relaxation rates, 506 Free volume theory, see Free volume model Frenkel’s hole theory, 468 Friction coefficient, 83 relative, 158 Friction tensor, 133 Frohlich expression, 279 Fulcher equation, 410 Gaussian core model, 402 Geminate recombination: primary, 155 secondary, 155 Generalized mean field approximation, 219,238 Generalized mean spherical approximation, 23 7-2 38 by HGye, Lebowitz and Stell, 226 Glasses, 491 hard-sphere, 399 ideal, 457,467,506 inorganic, 5 1 3 ionic, 458,459 Lennard-Jonnes, 399,472
546
SUBJECT INDEX
metalic, 458 monoatomic, 457 network, 459,462,513 organic, 459,492,500 polymeric, 458,492 real systems, 399 repellent Gaussian, 399 soft-sphere, 399 viscosity, 457 Glass transition, 456-513 computer simulation: of io+c systems, multicomponent, 44 1 isothermal compressibility, 433 radial distribution functions, 436 radial fluctuation functions, 432434 relaxation process, 435 relaxation time, see Relaxation times in glass transition structural characterization of glass, 435-441 transport properties vs. pressure, volume and temperature, 429430 see also Pair potential models enthalpy and heat capacity of, 462 entropy of, 443445 fust order, 457,467,478,491,492, 500,505 and free volume model, 470 second order, 467,478,492,502, 5 05 simulation studies, 403,412414 compression rates, 436 densities, 417418,436437 and diffusion coefficients, 406407 and diffusivity-temperature relation, 410 viscosity in, 461 Glass transition temperature, 445 dependence on heating and cooling rate, 464 lower, 419 upper, 419 GMF approximation, 219, 238 GMS approximation, 237-238 Green’s function, see Many body Green’s function Ground state functions, see Approximate ground state functions
Hankel transform, 235 Hard-spheres with dipoles and quadrupoles, 269-271 Hartree-Fock theory, timedependent, 15 in configurational freezing, 462 dependence on heating and cooling rates, 462,503,509 and entropy of melting, 465 in isobarical cooling, 462 for metallic glasses, 501 Heat capacity, 456,463,498,500, 508 HNC approximation, see Hypernettedchain approximation Homogenous nucleation, 401. See also Crystal nucleation HQye and Stell theory of nonlinear effects, 316 Hydrostatic pressure term, disjoining pressure, 341 Hypernetted-chain approximation, 23824 3 linearized, 218,238 perturbation expansion of, 244-245 quadratic, 223,238 Initial conditions effect, 149-152 Integral equation solution: diffusion eigenfunction and, 175-177 theories for polar-nonpolarizable fluids, 232-245 Intensity fluctuation spectroscopy, 376 Interface: of liquid film, 360 viscoelastic properties of, 365 Ionization cross sections, 63 intensity borrowing, 44 Jepsen and Friedman equation, 199 Kinetic effects, 457,464,465, 472,501,505 Kinetic equations, 105-125 pair, 120-123 derivation of, 173-175 singlet, 115-120 derivation of, 167-172 Kinetic theory, renornialized, 106 Kirkwood equation, 198 Koopman’s theorem, 15,35
SUBJECT INDEX Langevin equation, 82-84 generalized, 106 L3 approximation, 237 Lattice gas, 296 LHNC approximation, 218, 238 Light scattering, 341 Linear (LIN) approximation, 237 Linearized hypernetted-chain approximation, 218, 238 Liouville equation, 89 pseudo, 96-97 Liouville operator, 4,62, 88 Liquid films, free, see Soap films Liquid-glass transition, see Glass transition Liquidlike cells, 48 1,493,508 free energy of, 475,476,490 free volume of, 475 exchange, 477,504 Liquidlike clusters, 478,480,493 Liquids: dense, 456 Lennard-Jones, 472,473 monotonic, 457 rubidiumlike, 471 supercooled, 459,462,471,476 viscosity, 459 Liquid surfaces, 34 1 Local density fields, 85 LOGA, 216,224,225 Long time tail, 80 Lowest-order-a-ordered approximation, 216,224 for soft-core potentials, 225 Macroscopic law, 76 Many body Green’s function: diagrammatic expansion of, 529 hierarchy of equations, 5 one electron, 5, 15 orbital basis set dependence, 35 shake energies, calculation of, 31 two electron, 5 Many body perturbation theory, diagrammatic, 2 Marangoni effect, 336 Markov approximation, 90 Maxwell stress tensor, 364 Mean field theory, 479,484,462492 free energy in, 504 Mean force, 80
547
potentials of, 80, 198-199 Mean reaction field method, 253-254 Mean spherical approximation, 216, 224, 232 Mean spherical approximation solution: of Blum for multipolar molecules, 233 for dipole fluids, 237 for dipolequadrupole fluids, 237 for quadrupole fluids, 237 Memory kernel, 89 dynamic, 112-113 static, 112-1 13 Metastable state, 456,457 Minimum image method, 246-249 Mode coupling, 91 Molecular dynamics, 398,458,471, 472 Molecules: rigid-polar , 188-201 surface active, 335 Monte Carlo method, 398 MSA, 216,224,232 Multiple expansion for electrostatic interactions, 230 Network model, random, 467 Nonequilibrium phenomenon, 456, 46 1 Nonhydrodynamic contributions, 149 Nonlinear effects theory, of Hoye and Stell, 316 Nonperiodic boundary conditions, 253254 Nucleation theory, 458. See also Glass transition Onsager continuum, 210 Onsager continuum Limit, 199 Onsager’s approximation, 198 two-particle, 209 Onsager’s regression hypothesis, 357 Onsager’s theory, 262 Operators: of binary collision, 97 EOM, see Equations of motion operators Liouville, 97 pseudo-, 97 Optimized random phase approximation, 224 ORPA approximation, 224
548
SUBJECT INDEX
Orstein-Zernike equation, 189 for site-site correlation function, 28 1 Oseen interaction, 135 Packing, dense random, 472 Pad6 approximant of rushbroke, Stell and H@ye,238 Pad&producing approximation, 238 Pair correlation function, 188-201, 254 Pair distribution function, 81 Pair kinetic equations, see Kinetic equations Pair potential, 229 Pair potential models: hard-sphere, 422423 Lennard-Jones, 422423 properties: thermodynamic, 423427 isobaric heat capacity, 426427 transport, 428430 see also Glass transition, computer simulation soft-sphere, 422423 Partitioning theory, 24-29 Born expansion, 25,36 denominators, w-dependent, unshifted, 25 primary operator space, numerical selection of, 27 primary space for ionization potentials, 26-29 Patey approximation: LHNC, 218,238 QHNC, 223,238 Percolation theory, 477,493 ability distribution function, 500 cluster distribution function, 478,479, 482,487,507 clusters: finite, 477 infinite, 477 critical exponents, 478,482 and mean field approximation, 479 per colation : continuum, 482 environmental, 477,481 probability, 478,482 threshold, 477, 479,499
probability distribution, 487 Percolation transition, 445 Perturbation expansion of HNC equation, see Hypernetted-chain approximation Phase transition, see Glass transition Photodissociation, 149 Plane-wave approximation, 43,63-69 density of states, 64 Poisson-Boltzman equation, 3 37 Polarizable molecules, 27 2 Pollock and Alder results for Stockmayer fluids, 267-268 PPA approximation, 238 Propagator, 5 Quadratic hypernetted-chain approximation, 223,238 Radical complex mechanism, 157 Rate coefficient, 73 correlation function expression for, 88-94 diffusive, 81 of one-way flux, 105 time dependent, 75 Rate constant, intrinsic, 78 Rate kernel, 75 analysis of, 139-149 dynamic process contributing to, 139143 kinetic theory expression for, 125-127 nonhydrodynamic states contribution to, 147-149 from pair theory, 138-139 projection onto diffusion modes, 143-146 Smoluchowski equation result for, 162164 t-space expression for, 79 structure of, 101-105 Rate law, 7 3 generalization of, 75 Reaction coordinate, 86 Recombination, geminate, 155 Regression hypothesis, 116 Relaxation time: chemical, 74 structural, 506 Relaxation times in glass transition, 404 determination of, 412 dielectric, 412
SUBJECT INDEX and displacement from equilibrium, 419-421 sample size effect on, 421 temperature dependence of, 416,419 viscosity: bulk, 412 shear, 412 Repulsive forces, 96 electrostatic, 336-338 Ring collision events, 107 Rotational invariants, 230 Self-consistent field method (SCF), 35 Self-consistent Ornstein-Zernike approximation (SCOZA), 238 by H@yeand Stell, 226 Single superchain (SSC) approximation, 238,288 Single kinetic equation, see Kinetic equations, 238, 288 Sink Smoluchowski equation, 128, 136139. See also Rate kernel Sink term, 81 Site-site correlation function, OrnsteinZernike equation for, 281 Smoluchowski equation, 80-81, 130. See also Rate kernel Soap films, 330,335-338 black spots on, 330 colors interference, 330 contact angles, 333 diffraction of light by, 347-350 diffusion in, 365-366 equilibrium film,341 equipartition of energy, 334 evaporation disturbing effect on, 373, 374 fluctuation dynamics, 356-367 instability regions in, 355 growing fluctuations, 375 scattering of light by, 334. See also Surface light scattering on substrate, 367 surface fluctuations in, see Surface fluctuations thickness profile of, 333 Soap molecule, monolayer, 335 Solidlike cells, 475,490,499,504 Solutions, ionic, 303-314 Spccics, microscopic specification of, 84-88
Specific heat, 295 Spherical cutoff method, 246 Spherical sample, 200 Stochastic trajectories, 84 Stockmayer fluids Pollock and Aldei results for, 267-268 Stokes law, 107 Sum-of-the-pairs theories, 3 Surface fluctuations, 352-354 modes of: bending, 350 squeezing, 350 uncoupled, 350 Surface light scattering, 344-346 angular dependence of, 346 by carbon dioxide, 342 instrumental broadening effects, 389 by water, 352 Surface pressure, 335 Timedependent Hartree-Fock theory, 15 Transition state theory, 72 Truly periodic boundary condition, 24925 3 Tunneling centers, 470 Two-particle approximation: by Clausius-Mossotti, 208 by Onsager, 207 Virial series vs. molecular dynamics observations, 446-449 Viscosity, 456,493,503,509 by Arrhenius, 459,468 by Doolittle, 469,476,494,495, 5 09 experimental results, 460,495,496 non-Arrhenius, 468 pressure dependence of, 496 by Vogel-Fulcher, 461,468 Volume, 456,464,502,505 configurational, 480 effects, excluded, 81 free, see Free volume Voronoi polyhedra, 443,472,473, 416 Wertheim continuum, 212 Wertheim-Frohlich expression, Felderhof rcdcrivation of, 279
549