Dept. BL0111780
Research Paper
| Materials Chemistry Research Dept. BL0111780 Research Paper | Frank H. Stillinger |
|
Daniela Kohen* and Frank H. Stillinger**
*Department of Chemistry, University of
California at Irvine, Irvine, CA 92717
**Bell Laboratories, Lucent Technologies Inc.,
Murray Hill, NJ 07974
and
Princeton Materials Institute, Princeton University Princeton, NJ 08544
May 11, 1998
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Diversity in Liquid Supercooling and Glass
Formation Phenomena Illustrated by a Simple Model
Daniela Kohen and Frank H. Stillinger
May 11, 1998
Bell Laboratories, Lucent Technologies Inc.
600 Mountain Avenue, Murray Hill, NJ 07974
The opportunity to map condensed-phase inherent
structures (potential energy minima) approximately onto the
vertices of a high-dimensional hypercube provides simple
conceptual and numerical modeling for first-order
melting/freezing transitions, as well as for liquid
supercooling and glass formation phenomena. That approach
is illustrated here by examination of three interaction
examples that were selected to demonstrate the diversity of
thermodynamic behavior possible within this hypercube
modeling technique. Two of the cases behave, respectively,
as ``strong'' and ``fragile'' glass formers, at least as
judged by their heat capacities. The third presents a novel
``degenerate glass,'' wherein full equilibration of the
supercooling liquid (_i._e. no kinetic arrest) leads to (a)
residual entropy in the limit of absolute zero temperature,
and (b) a linear temperature dependence of heat capacity in
- 2 -
the same limit. None of the three cases displays a
positive-temperature ideal (intrinsic) glass transition.
I. _I_n_t_r_o_d_u_c_t_i_o_n
Substances that readily supercool as liquids below
their equilibrium melting temperatures, and rigidify to form
glasses upon further cooling, constitute a chemically very
broad group.1,2 As a result, their physical properties, both
static and dynamic, present a wide range of behaviors.3 This
diversity continues to generate challenges to basic research
on glass-forming materials, while at the same time offering
many opportunities for technological application.4-6
Because so many different molecular structures and
interactions can be involved, it is difficult to construct a
purely theoretical explanation of supercooling and glass
formation that is both universally applicable and
quantitatively predictive. Nevertheless it is reasonable to
expect that some theoretical approaches and/or models might
attain some limited insights.3 The present work is offered
in this latter spirit; it is devoted to the further
development of a previously introduced ''hypercube''
__________
* Present address: Department of Chemistry, University of
California at Irvine, Irvine, CA 92717.
- 3 -
model7,8 to show that it is capable of imaging glass
diversity while raising some new conceptual issues.
On the phenomenological side, Angell has advocated a
particularly useful classification scheme for glass-forming
substances that arrays them between ``strong'' and
``fragile'' extremes.9 The initial distinguishing feature in
this scheme is the curvature of the Arrhenius plot of the
logarithm of shear viscosity versus inverse temperature:
the strong extreme shows none, the fragile extreme shows a
substantial amount. The heat capacity behavior could as
well have been used for the same classification, since
strong materials (such as SiO2) display virtually no change
in heat capacity upon cooling through the glass transition
temperature, while fragile materials (such as ortho-
terphenyl) present a sudden large drop in heat capacity as
the temperature declines through the narrow glass transition
region.10
Brief presentations of motivation and implementation
for the hypercube model have appeared previously,7,8 but the
following Section II revisits that background in a somewhat
novel form, for clarity and completeness. Section III
introduces three alternative choices for interactions that
operate in the model, chosen to illustrate strong, fragile,
and ``degenerate glass'' behavior, respectively. The last
of these is not usually considered in discussions of glass
properties, but its low temperature residual entropy and
- 4 -
_c_l_a_s_s_i_c_a_l linear heat capacity merit examination.
Section IV presents detailed numerical results for each of
the three cases. Section V offers some conclusions, and
raises some general issues that deserve further study in the
future.
II. _H_y_p_e_r_c_u_b_e__M_a_p_p_i_n_g
The present study focuses on the mechanically stable
configurations of the particles (atoms, ions, molecules) in
a dense glass-forming substance. These are local minima of
the potential energy function that comprises all
interactions in the many-particle system, and have been
called ``inherent structures''.11-14 These distinguished
configurations form a discrete set; any other configuration
can be resolved into an inherent structure and an intra-
basin vibrational distortion.15,16 An important technical
point to bear in mind is that the potential energy function
and its inherent structures depend upon whether constant
volume or constant pressure conditions apply.17
Suppose the glass-forming system of interest contains @
particle species, present respectively in numbers
N1, . . . ,N@. Any one inherent structure is substantially
equivalent to many others that differ only by permutation of
positions of identical particles. It has been established18
that under constant pressure or constant number density
- 5 -
conditions, the large-system-limit behavior of the total
number of inherent structures, Z, exhibits the following
asymptotic form:
9 lnZ 9~8AN +9 j=18R8@9 ln(Nj!) (2.1)
9
where
A > 0 , (2.2)
9 N =9 j=18R8@9 Nj .
9
The quantity A concerns the exponential rise rate, with
increasing system size, of the number of distinguishable
(_i._e. unrelated by permutation) inherent structures; it can
be expected to vary among substances, and to depend on
pressure or number density.
Temperature dependence of the occupation probabilities
for the various inherent structure basins constitutes a
basic feature of liquid supercooling and glass formation.
Constructing an isomorphism between the exp(AN)
distinguishable inherent structures and the vertices of a
D-dimensional hypercube offers a simplifying first step in
understanding that temperature dependence. The number of
hypercube vertices, 2D, must equal the number of
distinguishable inherent structures, so
- 6 -
D = (A/ln2)N . (2.3)
Thus the hypercube dimension scales linearly with N as does
the dimension of the original configuration space for the
N-particle system.
Euclidean coordinate locations for hypercube vertices
will be assigned by the following unit vectors:7,8
II = D-1/2(+_1, +_1, +_1,..., +_1) . (2.4)
Each vertex has exactly D first neighbors whose locations
differ by sign change of just one of the D entries in
expression (2.4). The distance between first neighbors is
2D-1/2. More generally, each vertex has
9 n!(D-n)!7D!9________ (2.5)
9
n-th neighbors (1 _< n _< D), all at distance
2(n/D)1/2 . (2.6)
The isomorphism envisaged is not unique. Ideally,
pairs of inherent structure configurations that are close in
the original configuration space should map onto pairs of
neighboring hypercube vertices. In particular, basins in
the original space that share a common boundary should, to
the extent possible, map onto nearest neighbor vertices.
Typically, basins will contact on the order of N other
basins, and on account of Eq. (2.3) the number of hypercube
9
- 7 -
vertex nearest neighbors is also proportional to N, as
required. It will be assumed in the following that the
isomorphism has been selected so as to preserve these
neighbor relationships in an optimal fashion. Among other
attributes this implies that the potential energies of
inherent structures which become nearest hypercube
neighbors, although both O(N) quantities, differ only by
O(1).
Without incurring any significant loss of generality or
error in the intensive quantities to be calculated below, it
will be convenient to suppose that D is an even integer.
Then it is possible to select a pair of unit vectors, IIx and
IIy, from the set (2.4) with the orthogonality property
IIx . IIy = 0 . (2.7)
This merely requires that D/2 of the components of IIx and IIy
agree, while the remaining D/2 components differ in sign.
The entire collection of 2D hypercube vertices can then be
projected into the x,y plane defined by IIx and IIy with
positions
x = II . IIx ,
y = II . IIy . (2.8)
These plane-projected positions all fall on or within the
square
- 8 -
|x + y| _< 1 ,
|x - y| _< 1 . (2.9)
The pattern of projected vertex locations involves a
regular array of (D/2 + 1)2 points, far less than the total
number of vertices when D (_i._e. N) is large. Consequently
most of the locations host a large number of vertices. This
characteristic is illustrated by Figure 1, which explicitly
displays the positions and multiplicities in the x,y plane
for the specific case D = 14. Only the four hypercube
vertices at +_IIx and +_IIy do not share positions (they are at
the four corners of the square (2.9)).
In the large system limit, with N and D increasing to
infinity, the set of projected positions becomes dense in
square (2.9). As a result of simple combinatorial
considerations7 it is straightforward to show that the
multiplicity at position x,y has the asymptotic form
exp[Dw(x,y)], where
w(x,y) = ln2 - (1/4)[(1+x+y)ln(1+x+y)
+ (1+x-y)ln(1+x-y) + (1-x+y)ln(1-x+y)
+ (1-x-y)ln(1-x-y)] . (2.10)
The selection of basis vectors IIx and IIy is not unique.
The number of possibilities rises rapidly with increasing D.
One can show that the number of distinct projection planes
generated this way is given by the expression
- 9 -
99 (D/2)!(D/2-1)!72D-2(D-1)!9______________ . (2.11)
9
This equals 7,028,736 for the modest D = 14 example
illustrated in Figure 1.
III. _H_y_p_e_r_c_u_b_e__I_n_t_e_r_a_c_t_i_o_n_s
Calorimetric measurements on a wide variety of glass
forming substances indicate that the vibrational heat
capacities of the crystal and of the amorphous glass states
are approximately equal.10,19-21 Vibrational degrees of
freedom with high frequencies will manifest considerable
quantum effects, especially at low temperature. But because
these effects appear to be nearly identical in crystalline
and amorphous phases, they can be dropped in calculation of
the influence of temperature on inherent-structure basin
occupation. This influence is the objective of the present
class of hypercube models, for which classical statistical
mechanics is now appropriate, and only the potential
energies of the inherent structures themselves are relevant.
The large number of available plane projections,
Eq. (2.11), offers a simplifying strategy. We choose that
basis pair IIx, IIy which comes closest statistically to
having identical potential energies for all inherent
structures that have a common projection location x,y. Then
assuming that this requirement has been met to a
- 10 -
satisfactory level of precision, the potential energy may be
expressed simply as
DU(x,y) , (3.1)
in other words as just a function of x and y. The factor D
has been included to account for the fact that N-body
potential energies are extensive quantities (_i._e.
proportional to N or equivalently to D), so that U(x,y) is
intensive.
The remaining two variables x and y should be regarded
as measures of the amount and kind of disorder that is
present in the many-particle system. Disorder in the
crystalline state alone takes many forms, including
vacancies, interstitials, orientational and conformational
defects, dislocations, stacking faults, and grain
boundaries. Liquid and amorphous solid states likewise must
display disorder diversity. Consequently it is reasonable
to suppose that at least two ``disorder parameters'' are
required to generate a unified description of crystalline
and amorphous phases. At this level of description the
crystalline solid on the one hand, and the liquid and glass
on the other hand, could be expected to appear at distinct
locations in the basic x,y square region (2.9), and indeed
in the following Section IV this will be seen to occur.
- 11 -
An exploration of several reasonable forms for the
potential energy function U(x,y), in conjunction with the
combinatorial entropy quantity w(x,y), Eq. (2.10), shows
that the hypercube model has the capacity to exhibit first-
order melting, and a substantial variety of liquid
supercooling behaviors. Two examples have been reported
earlier,7,8 both of which could be classified as
thermodynamically illustrating ``strong'' glass-former
behavior. Three new cases will now be examined. These
correspond respectively to the following three assignments,
in suitable reduced energy units:
U1(x,y) = x + y + 0.6(x-y+0.05)3 - (x-y+0.05)2
-0.11/(x-y+1.11); (3.2)
U2(x,y) = 3(x+y) + 1.2(x-y+0.05)3 - (x-y+0.05)2
+ 0.15(x+y)(x-y+1)4 - 0.11/(x-y+1.11); (3.3)
U3(x,y) = x + y + (x-y+0.1)3 - (x-y+0.1)2
- 0.11/(x-y+1.11) . (3.4)
The ground state (lowest potential energy) for each of these
cases occurs at the square vertex
x = -1 ,
y = 0 . (3.5)
Consequently the ground state is non-degenerate, and
corresponds to the structurally perfect crystal.
- 12 -
IV. _N_u_m_e_r_i_c_a_l__R_e_s_u_l_t_s
Let T* stand for reduced temperature. The
configurational free energy F(T*) for the hypercube model
arises from the following minimization with respect to x and
y over the square (2.9):
F(T*)/DT* =9 (x,y)8min [U(x,y)/T*-w(x,y)] . (4.1)
9
Locating the position of the minimum, or minima, as T9*
9 varies is a simple numerical task for each of the three
cases defined above, Eqs. (3.2), (3.3), and (3.4). In the
event that two (or more) local minima were to be found at
some T*, the one with the lower (or lowest) F of course
would correspond to the thermodynamically stable phase, the
other(s) to metastable phase(s).
Numerical analysis reveals that the three cases under
present consideration share several qualitative attributes.
Two local free energy minima exist at low T*, the more
stable one of which emerges from the vertex (3.5) as T*
increases above absolute zero. However above a melting
temperature Tm8*9 the other local minimum yields the lower free
energy, and so can be identified as the ``liquid''. The
metastable ''crystal'' minimum persists above Tm8*9 until it
vanishes at a higher finite instability temperature Tc8*9. For
T* > Tc8*9 only the liquid minimum exists, and its location
approaches x = y = 0 as T* approaches infinity. Table 1
9
- 13 -
collects the computed values of Tm8*9, Tc8*9, and several other
quantities to be discussed below for each of the three
interactions.
Figures 2, 4, and 6 show the paths traced by the free
energy minima as temperature varies, for each of the three
cases. The respective thermodynamic melting points are
graphically identified by pairs of small open circles,
between which the system discontinuously jumps upon passing
the melting/freezing first-order phase transition. The
crystal branch (cr) behaves similarly for all three cases,
emanating from the x = -1, y = 0 square vertex and moving
diagonally upward nearly along a side of the square as T*
rises from zero. And in all three cases the liquid (liq)
path is well separated in the square from that of the
crystal, indicating clear structural distinctiveness for the
two phases.
The liquid-phase paths for cases 1 (strong) and 2
(fragile), Figures 2 and 4 respectively, are qualitatively
similar though clearly differing in shape detail. Both
begin at vertex x = 0, y = -1 at T* = 0 as non-degenerate
(zero entropy) glasses. The potential energies of these
ideal glass states, Ug(T* = 0), along with the corresponding
ideal crystal potential energies, Ucr(T* = 0), appear in
Table 1.
- 14 -
Case 3 presents a glass-state anomaly. Its zero
temperature limit lies along a square edge, at position
x = -0.227017 ,
y = -0.772983 . (4.2)
Consequently this state is configurationally degenerate.
The extent of this degeneracy is measured by the value of w
at the square-side position (4.2), and is listed in Table 1
as the quantity wg(T*=0).
Figures 3, 5, and 7 show the curves for heat capacity
c = dU/dT* calculated for the three cases, with distinct
branches for the crystal and the liquid phases. As T*
approaches Tc8*9 from below, the crystal heat capacity for all
three cases diverges to infinity (one can show that these
are inverse-square-root singularities). The roughly
comparable heat capacities for crystal and liquid in case 1
merit the classification ``strong'', while the large
difference between them in case 2 (particularly below Tm8*9)
justifies the label ``fragile''. On this basis the
``degenerate'' case 3 (Figure 7) would also be classified as
strong.
All numerical results displayed in Figures 2-7 assume
that local equilibration in the x,y space for location of
free energy minima is operative. In the case of real
glass-forming substances this equilibration becomes
- 15 -
kinetically arrested at and below a glass transition
temperature Tg8*9,
0 < Tg8*9 < Tm8*9 . (4.3)
Kinetics of configurational transitions are an attribute of
the hypercube models that is logically independent of the
thermal features, and are therefore outside the scope of the
present paper. However the reader should keep in mind that
a ``realistic'' glass transition would terminate a liquid-
branch configurational heat capacity at Tg8*9, which thereupon
substantially vanishes at lower T*. Analogously, one must
realistically expect a superheated crystalline phase
kinetically to melt well before reaching its instability
temperature Tc8*9.
All three crystal heat capacity curves, and the liquid
curves for cases 1 and 2, vanish exponentially as T*
approaches zero. Once again the ``degenerate'' case 3 is
anomalous; its configurational heat capacity is linear in T*
in the low temperature limit. By using low-order expansions
for w(x,y) and U3(x,y) in the vicinity of position (4.2),
one can show
c3(T*) = 0.05134T* + 0(T*2) . (4.4)
Such linear dependence is reminiscent of that observed in
low-temperature amorphous solids, and associated with
quantum-mechanical two-level tunneling degrees of
- 16 -
freedom.22,23 However the present example is quite
different, arising as it does in a classical statistical
mechanical setting. In view of the fact that a positive-
temperature glass transition would preempt direct
calorimetric observation of a linear heat capacity of type
(4.4), it will be a substantial challenge to determine if
any real substances fall into our ``degenerate'' glass
category.
V. _C_o_n_c_l_u_s_i_o_n_s__a_n_d__D_i_s_c_u_s_s_i_o_n
The three interaction choices examined in this paper,
Eqs. (3.2)-(3.4), supplementing the two cases previously
studied,7,8 demonstrate that the hypercube model possesses
considerable diversity in the thermodynamic behavior it can
display. In particular it is now clear that insofar as heat
capacity is concerned, the model can span the full range
between ``strong'' and ``fragile'' extremes by relatively
simple alteration in the interaction function U(x,y). This
flexibility in behavior is sufficient in fact to have
produced a theoretically novel type of glass former, the
``degenerate'' type exemplified by U3(x,y), Eq. (3.4).
Some glass-forming substances exhibit crystal
polymorphism. Silicon dioxide (SiO2) is a well-known
example,24,25 with quartz, cristobalite, tridymite, coesite,
and stishovite predominating in distinct temperature-
- 17 -
pressure regimes. An adroit choice of interaction function
U(x,y) should allow the hypercube model also to possess two
or more low temperature ``crystal'' phases, each stable in
some temperature interval below the melting point at Tm8*9.
For example, their individual paths in the fundamental x,y
square could emanate from distinct vertices of the square as
T* increases from absolute zero.
The three examples studied in the present paper seem to
be more realistic in at least one important respect than the
examples that were presented in references 7 and 8. One can
see from entries in Table 1 that the ratio Tc8*9/Tm8*9 measuring
the maximum possible extent of crystal superheating is close
to 1.3 for each of the present cases. By contrast the
corresponding ratios in references 7 and 8 were
approximately 2.3 and 1.7, respectively. To stress a
previous point, flexibility in choice of U(x,y) beyond that
already exercised should permit Tc8*9/Tm8*9 to be reduced even
further toward unity, should experimental observation so
dictate.
One obvious, but benign, shortcoming of the hypercube
model examples as thus far implemented concerns their
behavior at the thermodynamic melting point Tm8*9. This is a
first-order phase transition that should permit, in
principle, coexistence of the two phases in arbitrary
relative amounts without changing the free energy
(interfacial terms are insignificant in the present context
- 18 -
that only concerns the large-system limit). This implies
that at Tm8*9 some continuous path must exist in the x,y plane
connecting the pure-phase locations (pairs of open circles
in Figures 2, 4, and 6) along which the free energy is
invariant. Hypercube model cases examined thus far do not
show this behavior, a shortcoming that can be patched up
``after the fact'' by redefining U(x,y) appropriately within
a domain that exists between (and is tangent to, at the Tm8*
9 points) the phase paths already traced out. The
correspondingly modified U(x,y) should be continuous within
the x,y square. This _a _p_o_s_t_e_r_i_o_r_i procedure is analogous to
the Maxwell double tangent construction26 that identifies
liquid-vapor coexistence regions for equations of state of
the van der Waals type.
Equation (2.3) above presented the formal relation
between the hypercube dimension D for a given number N of
particles, and the parameter A that measures the number of
distinct inherent structures. Recently, A has been
estimated27 for the fragile glass former prototype ortho-
terphenyl (OTP), using accurate calorimetric data for that
substance,10 with the result:
A(OTP) ~= 13.14 . (5.1)
Consequently Eq. (2.3) requires
D/N ~= 18.96 . (5.2)
- 19 -
To the extent that the ``fragile'' case (U2) is a reasonable
statistical model for OTP, this becomes its dimension
assignment.
Historically, one of the prominent concepts in glass
science concerns the second-order ``ideal glass transition''
at some positive temperature T08*9 less than the observed glass
transition temperature.28 This is the point hypothetically
at which cooling of an equilibrated, supercooled, liquid
would attain substantially vanishing configurational
entropy. The concept seems to be especially attractive for
fragile glass formers (such as OTP) because of near
coincidence between the calorimetric Kauzmann temperature
and the apparent divergence temperature of shear viscosity
and of mean structural relaxation time.3 However none of the
hypercube models previously investigated7,8 or examined in
this paper exhibit a second-order ideal glass transition.
Indeed general counterarguments exist against such a
possibility.29 Nevertheless it is legitimate to ask whether
the hypercube model is capable _i_n _p_r_i_n_c_i_p_l_e of producing a
second-order ideal glass transition in its supercooled
liquid phase, and if so what are the corresponding
requirements on the interaction function U(x,y). It is not
appropriate to pursue this point in great detail here, but
suffice it to say that such transitions can be produced if
U(x,y) has a bounded logarithmic singularity (of type zlnz)
located at the liquid-state square vertex.
- 20 -
Finally, kinetic properties of the hypercube model
deserve mention in passing. One approach involves
development of a Fokker-Planck equation in order-parameter
(x,y) space,7 and its extension to incorporate time-lag
phenomena.8 These formalisms remain largely unexplored at
present, and may be productive directions for future work.
- 21 -
_R_e_f_e_r_e_n_c_e_s
1. P. G. Debenedetti, _M_e_t_a_s_t_a_b_l_e _L_i_q_u_i_d_s (Princeton
University Press, Princeton, 1996), chap. 4.
2. C. A. Angell, Science _2_6_7, 1924 (1995).
3. J. Ja8..9ckle, Rep. Prog. Phys. _4_9, 171 (1986).
4. R. H. Doremus, _G_l_a_s_s _S_c_i_e_n_c_e, 2nd edition (Wiley-
Interscience, New York, 1994).
5. J. Fricke, Sci. American _2_5_8, No. 5, 92 (1988).
6. A. L. Greer, Science _2_6_7, 1947 (1995).
7. F. H. Stillinger, J. Phys. Chem. _8_8, 6494 (1984).
8. F. H. Stillinger, Physica D _1_0_7, 383 (1997).
9. C. A. Angell, J. Non-Cryst. Solids _1_3_1-_1_3_3, 13 (1991).
10. S. S. Chang and A. B. Bestul, J. Chem. Phys. _5_6, 503
(1972).
- 22 -
11. H. Tanaka and K. Nakanishi, J. Chem. Phys. _9_5, 3719
(1991).
12. A. Z. Patashinski and M. A. Ratner, J. Chem. Phys.
_1_0_6, 7249 (1997).
13. F. L. Somer, Jr., G. S. Canright, T. Kaplan, K. Chen,
and M. Mosteller, Phys. Rev. Letters _7_9, 3431 (1997).
14. D. S. Corti, P. G. Debenedetti, S. Sastry, and F. H.
Stillinger, Phys. Rev. E _5_5, 5522 (1997).
15. F. H. Stillinger and T. A. Weber, J. Chem. Phys. _8_0,
4434 (1984).
16. F. H. Stillinger, Science _2_6_7, 1935 (1995).
17. F. H. Stillinger, Phys. Rev. B_4_1, 2409 (1990).
18. F. H. Stillinger, Phys. Rev. E. (to be published).
19. S. S. Chang, J. A. Horman, and A. B. Bestual, J. Res.
Nat. Bur. Standards _7_1A, 293 (1967).
20. C. A. Angell and J. C. Tucker, J. Phys. Chem. _7_8, 278
(1974).
- 23 -
21. Reference 1, p. 244.
22. W. A. Phillips, J. Low Temp. Phys. _7, 351 (1972).
23. P. W. Anderson, B. I. Halperin, and C. M. Varma,
Philos. Mag. _2_5, 1 (1972).
24. R. B. Sosman, _T_h_e _P_h_a_s_e_s _o_f _S_i_l_i_c_a (Rutgers University
Press, New Brunswick, 1965).
25. N. R. Keskar and J. R. Chelikowsky, Phys. Rev. B_4_6, 1
(1992).
26. K. Huang, _S_t_a_t_i_s_t_i_c_a_l _M_e_c_h_a_n_i_c_s (John Wiley, New York,
1963), p. 45.
27. F. H. Stillinger, J. Phys. Chem. _1_0_2, 2807 (1998).
28. M. Goldstein, J. Chem. Phys. _6_7, 2246 (1977).
29. F. H. Stillinger, J. Chem. Phys. _8_8, 7818 (1988).
- 24 -
_T_a_b_l_e _1. Properties calculated for the hypercube model with
the three interaction choices.
Case 1a 2b 3c
Tm8*9 6.6862 7.6970 7.0148
Tc8*9 8.642 10.042 9.232
(Ww)m 0.34407 0.30103 0.34382
(Cliq/Ccr)m 0.70 1.83 0.69
Ug(T*=0) -1.4601 -5.1655 -1.2142
wg(T*=0) 0 0 0.2678
Ucr(T*=0) -3.4169 -5.9314 -3.5390
wcr(T*=0) 0 0 0
a "strong"; b "fragile"; c "degenerate".
- 25 -
_F_i_g_u_r_e _C_a_p_t_i_o_n_s
1. Pattern of plane-projected vertex positions for the
D = 14 hypercube. The integers shown at the 82 = 64
positions are the respective degeneracies, the numbers
of hypercube vertices that project to the same
position.
2. Paths traced out in the x,y square for the crystal
(cr) and liquid (liq) phases for interaction choice
U1. The arrows indicate the direction of increasing
temperature, and open circles locate Tm8*9, the
thermodynamic melting/freezing transition.
3. Heat capacity curves for interaction choice U1. The
arrow locates the equilibrium melting/freezing
transition.
4. Crystal and liquid paths in the x,y square for
interaction choice U2. The notation is the same as
that used in Figure 2.
5. Heat capacity curves for interaction choice U2. The
arrow locates the equilibrium melting/freezing
transition.
- 26 -
6. Crystal and liquid paths in the x,y square for
interaction choice U3. The notation is the same as
that used in Figures 2 and 4.
7. Heat capacity curves for interaction choice U3. The
arrow locates the equilibrium melting/freezing
transition.
=================== end research paper ===================
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