Physica A 205 (1994) 738-746
North-Holland
I
SSDI 0378-4371(93)E0490-6
Repulsive potentials, clumps and the metastable
glass phase
W. Klein a, Harvey Gould b, Raphael A. Ramos", I. Clejan a and
A n d r e w I. Mel'cuk b
aDepartment of Physics, Boston University, Boston, MA 02215, USA
bDepartrnent of Physics, Clark University, Worcester, MA 01610, USA
Received 15 September 1993
Revised manuscript received 5 October 1993
We present the results of simulations and theoretical investigations of a system of particles
with a weak, long-range repulsive interaction. At fixed density the fluid phase, which is stable
at high temperatures, becomes metastable and then unstable as the temperature is lowered.
For low temperatures the particles form approximately equal size clumps that interact weakly
with neighboring clumps. The global free energy minimum at low temperatures corresponds
to a crystalline lattice clump structure. However the free energy surface has many minima
associated with metastable amorphous phases. Quenches from the fluid phase to low
temperatures almost always will result in an amorphous clump structure with the number of
clumps dependent on quench history. The association of the amorphous phase with a large
number of metastable minima is similar to a mean-field spin glass.
T h e understanding of the processes of crystallization and glass f o r m a t i o n has
p r o v e n to be a formidable task [1,2]. A high d e g r e e of non-linearity and
evolution on several length and time scales m a k e s these processes difficult to
attack theoretically, experimentally, and by simulations. In addition, glass
formation and the crystallization processes often c o m p e t e with each other,
further complicating our ability to u n d e r s t a n d t h e m [3].
T o gain m o r e insight into these processes, we have b e g u n a study of systems
with a weak, long-range repulsive interaction, i.e., systems in the mean-field
limit. T h e advantage of studying mean-field systems is that several aspects o f
the p r o b l e m can be addressed theoretically. I n this p a p e r we r e p o r t o u r first
results which strongly indicate the presence of a metastable glass p h a s e and a
new phase of m a t t e r that we call the " c l u m p " phase. We will see that the
clumps play a significant role in the f o r m a t i o n of the glass phase in these
models.
We consider a pairwise, radially symmetric K a c potential [4] of the f o r m
0378-4371/94/$07.00 © 1994-Elsevier Science B.V. All rights reserved
W. Klein et al. / Repulsive potentials and the metastable glass phase
V(r) = ydr(rr) >1o,
739
(1)
where r = Irl, y is a parameter related to the strength and range of the
potential, and d is the spatial dimension. In the y ~ 0 limit, it is known [5] that
the system can be described exactly by a mean-field theory in which the single
particle distribution fu:~tion p~(x) satisfies the equation
p,(x) = z exp(-13 f ¢k(lx- x'D o,(x') ar') ,
(2)
where z is the activity, / 3 - ~ = kBT, and x = yr. For high temperatures (and z
fixed), the solution to (2) is a unique spatial constant [5]. For potentials whose
Fourier transform ~ ( q ) is negative for some value of q = q0 (the dimensionless
quantity q = k / y ) , the spatially constant solution of (2) becomes unstable for
the smallest value of/3p such that [5]
1 + flpqb(qo) = 0 ,
(3)
where ~(q0) is the minimum of q~(q). The number density p is the spatially
constant solution of (2). It has been shown [5] that at the critical value
/3p = (/3p) o found from the solution of (3), the structure function diverges at
q = q0, and hence (3) defines the location of the spinodal.
Unless otherwise noted, we will consider the step potential
YdS'(X) =
ya
0,
x~l
X>I.
(4)
It is easy to verify that there is a range of q for which ~s < 0 and that the first
minimum of as(q) occurs for q0 = 5.76. Hence a system with the step potential
(4) exhibits a spinodal in the mean-field limit y---~0.
Our main interest is the structure of the system for the step potential in (4)
for/3p > (/3P)0, that is, for temperatures below the spinodal at fixed density.
The interaction energy per particle for d = 3 when the system is in a uniform
state with density p is E u -- ~'rrp.14 (The factor of ½ accounts for the sharing of
the energy of interaction between two particles). Now consider a nonuniform
configuration where the particles are grouped into "clumps" with an equal
number of particles in each clump. Each particle interacts with all of the
particles in the same clump. If there is no interaction between particles in
different clumps, the energy of interaction per particle is E c = ½"y3N/Nc, where
N is the number of particles in the system, N c is the number of clumps, and
N / N c is the mean number of particles in a clump. To estimate No, we calculate
the number of clumps that just fit in a box of volume L 3 in a simple cubic (sc)
740
W. Klein et al. / Repulsive potentials and the metastable glass phase
arrangement. If we take the mean distance between the clumps, (L3/N ]1/3 to
be the interaction range y -1, we have N c = L3/T -3, and hence Esc = 1 p, which
implies that Esc < E u. The energy per particle for fcc, bcc, and random close
packing (rcp) arrangements of the clumps is given by Ef¢¢ = p/2V2~O.35p,
E b c c ~- 2p/(3/V~) ~ 0.38p, a n d E r c p ~ 0.41p respectively.
These qualitative considerations suggest that as T---~0, the arrangement of
particles into non-interacting clumps will have a lower energy and hence lower
free energy than the uniform distribution. This result appears to be true
whether the centers of mass of the clumps are arranged in a crystalline
configuration or are random. Our conclusion is that the clumps will determine
the structure of the system for large ]3p.
To test this hypothesis we have performed Monte Carlo simulations on a
system of N = 8000 and N = 1000 particles with the step potential (4) with
-1
T = 3. The density is fixed at p = 1.95 and periodic boundary conditions are
used. For this density, it is easy to show that the spinodal temperature T O given
by the solution of (3) equals 0.705 (in units where k B = 1). The simulations
show a pronounced increase in the structure factor at q ~ 5.3 as T approaches
T ~ 0 . 7 5 . The increase in the structure factor as the spinodal is approached
from above is associated with our visual observations of large fluctuations in
the density. In addition, we observe evidence for critical slowing down for
fluctuations whose wavelength is on the order of the interaction range.
However because of critical slowing down, our observations of the density
fluctuations as we approach the spinodal from above are limited by the
relatively short duration of the runs. These results are consistent with the
mean-field predictions which are applicable in the y---~ 0 and L, N---~ oo limits.
If we begin with a random configuration of particles and then quench to
T = 0.4, a temperature less than To, the system appears unstable since the
evolution of the clump phase begins without delay after the quench. After
equilibrium is established, quantities such as the mean potential energy do not
change. As can be seen from the typical configuration of particles shown in fig.
1, the particles form distinct clumps. Quenches from a random configuration to
T = 0.5 and T = 0.6 are qualitatively similar to the quenches to T = 0.4.
In fig. 2 we show the pair correlation function g(r) for a clump phase at
T = 0.4. The position and width of the first maximum implies that the particles
in the same clump are located close to one another. The position of the second
maximum of g(r) can be interpreted as the mean distance between particles in
neighboring clumps [6]. Some of the other results of the simulations for
N = 8000 are summarized in table I. Note in particular that the number of
clumps obtained after a quench to T < T O depends on the nature of the
quench. For example, an instantaneous quench from a random configuration
( T = oo) to T = 0.4 yields approximately 96 clumps, while a quench from
W. Klein et al. / Repulsive potentials and the metastable glass phase
741
Fig. 1. A typical configuration of N = 8000 particles with density p = 1.95 for the step potential (4)
with 3, -~ = 3. The temperature is quenched from T = 0.8 to T = 0.4. The points are the positions of
the centers of the particles. The particles form 87 clumps.
30
20
en
15
5
0
,
0
2
3
,
4
5
6
7
8
r
Fig. 2. The pair correlation function g(r) for the same configuration as shown in fig. 1. Note that
g(r) has a peak at r = 0 consistent with the presence of clumps and that the second peak occurs at
~ 3 ' 1. There is an insufficient number of clumps to determine the position of the third peak of
g(O.
1
I
742
W. Klein et al. / Repulsive potentials and the metastable glass phase
Table I
Summary of Monte Carlo simulations of quenches from the initial temperature T~= o~and T~ = 0.8
to T = 0.4 for N = 8000 particles in a cubical box with periodic boundary conditions. The density
p = 1.95 and the interaction range 3' -~ = 3. The quantities shown are n c, the number of clumps;
mcsp, the number of Monte Carlo steps per particle; me, the mean number of particles per clump;
de, the mean distance between the clumps as measured by the second peak in g(r); and El, the
mean energy per particle after the quench. The energy is measured in units of 3' 3. Note that the
number of clumps for T~= ~ differs from the number of clumps for T~= 0.8. We also show the
results of raising the temperature of a fcc system of clumps from T i = 0 to T = 0.4 at the same
density for N = 2600.
Run
Mcsp
Ti
nc
m~
dc
E~
1
2
1
2
fcc
200
200
1000
2000
500
oo
oo
0.8
0.8
0
94
97
89
87
26
84.6
82.0
89.6
91.7
81.2
3.725
3.625
3.875
3.875
0.926
0.915
0.959
0.972
0.845
T = 0.8, a t e m p e r a t u r e just a b o v e t h e s p i n o d a l , to T = 0.4 p r o d u c e s a p p r o x i m a t e l y 88 clumps. T h e d i f f e r e n c e a p p e a r s to be statistically significant. D u r i n g
o u r runs o f 2000 M o n t e C a r l o steps p e r p a r t i c l e ( m c s p ) , w e o b s e r v e d n o c h a n g e
in t h e n u m b e r o f c l u m p s after e q u i l i b r a t i o n , e v e n t h o u g h i n d i v i d u a l p a r t i c l e s
c h a n g e c l u m p s d u r i n g t h e runs. B e c a u s e o f t h e r e l a t i v e l y s h o r t d u r a t i o n o f t h e
runs for N = 8000, we also o b s e r v e d t h e n u m b e r o f c l u m p s a f t e r e q u i l i b r a t i o n
for t h e N = 1000 a n d f o u n d t h a t t h e n u m b e r o f c l u m p s d i d n o t c h a n g e o v e r
105 mcsp.
After the clumps form, their centers of mass are localized even though their
c o n s t i t u e n t p a r t i c l e s c o n t i n u e to diffuse. T h e t e m p e r a t u r e - d e p e n d e n c e o f t h e
single p a r t i c l e diffusion coefficient D as d e t e r m i n e d f r o m t h e m e a n s q u a r e d
d i s p l a c e m e n t is s h o w n in fig. 3. N o t e t h a t D d r o p s s h a r p l y at T = T 0, b u t
r e m a i n s n o n z e r o since p a r t i c l e s can c h a n g e c l u m p s if T is n o t t o o low. T h e
t e m p e r a t u r e - d e p e n d e n c e o f D for l o w e r t e m p e r a t u r e s n e a r T - - 0 . 2 will b e
r e p o r t e d in f u t u r e w o r k .
F r o m t a b l e I w e see t h a t the e n e r g y p e r p a r t i c l e in a fcc c o n f i g u r a t i o n at
T = 0.4 is l o w e r t h a n in t h e a m o r p h o u s c l u m p p h a s e at t h e s a m e t e m p e r a t u r e .
W e w e r e a b l e to e s t a b l i s h a crystalline c l u m p c o n f i g u r a t i o n for T < T o o n l y b y
s t a r t i n g t h e particles in such a c o n f i g u r a t i o n at T = 0 a n d t h e n raising t h e
t e m p e r a t u r e . W h e n we r a i s e d t h e t e m p e r a t u r e o f t h e fcc p h a s e to T = 0.8, w e
f o u n d t h a t the fcc c l u m p p h a s e d i d n o t c h a n g e significantly for at l e a s t
500 mcsp. In c o n t r a s t , a similar i n c r e a s e in the t e m p e r a t u r e o f an a m o r p h o u s
c l u m p p h a s e l e d to a significant a n d i m m e d i a t e c h a n g e in t h e n u m b e r a n d
n a t u r e of t h e clumps. W e i n t e r p r e t this c o n t r a s t i n g b e h a v i o r as e v i d e n c e o f
hysteresis effects a s s o c i a t e d with a f i r s t - o r d e r p h a s e t r a n s i t i o n . T h a t is, t h e r e is
a f i r s t - o r d e r p h a s e t r a n s i t i o n at a t e m p e r a t u r e h i g h e r t h a n t h e s p i n o d a l .
743
W. Klein et al. / Repulsive potentials and the metastable glass phase
In D
-3
0.07
-4
0.06
-5
0.05
-6
0.04
-7
0.03
-8
0.02
-9
0.01
-10
D
0
0.4
0.5
0.6
0.7
0.8
0.9
T
Fig. 3. The temperature-dependence of the single particle diffusion coefficient D for p = 1.95, the
step potential (4), and N = 1000. The results for D were obtained from the slope of the single
particle mean squared displacement for runs of 105 mcsp. The circles represent in In D and the
squares represent D.
The implications of the Monte Carlo results can be better understood in the
context of our theoretical investigations. It is straightforward to show that the
single particle distribution function that is a solution to (2) is an e x t r e m u m of
the mean-field free energy functional [7]
(x),
__1 f (Ix - ~ f p , ( x ) d x + t 3 - ' f p , ( x ) { l n [ p , ( x ) l - 1} d r .
(5)
We have found numerical solutions for pl(x) from (5) using a conjugate
gradient m e t h o d and used these solutions to m a p the free energy surface. For
d = 2 and T < To, we found that the free energy surface has m a n y minima, and
the absolute minimum corresponds to a clump phase with triangular symmetry.
There also exist m a n y minima that correspond to polycrystalline triangular
configurations. For d = 3 and T > To, we find a global free energy m i n i m u m
corresponding to the uniform phase and other free energy minima corresponding to a m o r p h o u s clumps. For T < T 0, there is a global m i n i m u m
corresponding to a crystalline clump phase and m a n y minima corresponding to
a m o r p h o u s clumps. Note that in the limit 3,---->0, the nucleation barriers
between the multiple free energy minima diverge as 7 -d, a result of nucleation
theory [8,9]. In d = 2 it appears that only crystalline clump structures can be
stabilized. (Note that long-range translational order in d = 2 is possible because
of the mean-field nature of the system.)
744
W. Klein et aL / Repulsive potentials and the metastable glass phase
The following picture emerges from the simulations and the numerical
solutions of (2). For potentials of the form given in (4), there is a well-defined
spinodal or limit to the metastable phase at/3p = (/30)o. Below the spinodal the
particles form clumps with a diameter d c - y -1 . I n d = 3 , the clump phase can
have an amorphous or crystalline structure. Quenches from above T O to below
T O will almost always result in an amorphous phase because of the large
number of amorphous free energy minima relative to the number of crystalline
minima. The spinodal is not sharp for 3, ¢ 0, but the width of the spinodal
region can be estimated using a Ginsburg criterion. The Ginsburg criterion for
the applicability of mean-field theory is that the fluctuations in a volume the
size of the correlation length must be small in comparison to the order
parameter pl(k). This assumption implies that ~dgT/[~dpl(k)]2~ 1, where ~ is
the correlation length and )¢T is the isothermal (k-dependent) susceptibility.
From the free energy function in (5), it can be shown [5] that ~ ~ R(AT) -~/2,
)Cry(AT) -~, and p~(k-1/~)--2AT, where A T = T - T O and R = y - I is the
range of the interaction. Hence for d = 3 the Ginsburg criterion can be written
as (AT)-3/z/(4R3)~ 1. If we assume that the Ginsburg criterion is satisfied if
(AT)-3/2/(4R3) ~ l / e , we find that A T - - 0 . 0 9 for R = 3, where the spread of
0.09 indicates the smearing due to the finite range of the interaction. This
reasoning indicates that the spinodal is at --0.705 +_0.09, and that the spinodal
becomes better defined as the interaction range, y - ~, increases. We expect that
the clumps will become more diffuse as we approach the spinodal from below
reflecting the approaching breakdown of the mean-field regime. Our observations of the clumps at T = 0.6 are consistent with this interpretation.
We conclude that for temperature quenches from above to below the
spinodal, the system forms a glass phase in three dimensions. That is, the
spinodal in our model can be associated with at least part of the glass
transition. The glassiness of the system is manifested not only by its apparent
quench history dependence, but also by the relatively confined movement of
the centers of mass of the clumps which show very little displacement during
the course of the simulations. The glass phase we have found, which we call a
"metastable glass", appears to be due to the large number of metastable
minima associated with random or amorphous phases. In this sense the
metastable glass phase has some characteristics in common with the mean-field
spin glass [10]. The relatively small center of mass motion of the clumps occurs
because the mass of the clumps diverges as 7 -a as 3'--* 0 while the temperature
of the system remains finite. The kinetic energy goes into the internal degrees
of freedom of the clumps.
We have begun a study of a system with the step potential (4) plus a hard
core of diameter or. As expected, there is little change in the clump behavior
for y o - ~ 1. For larger 7~r, the clumps are not visible. Since the hard core
W. Klein et al. / Repulsive potentials and the metastable glass phase
745
interaction yields a kinetic glass for sufficiently high density, we plan further
study of the combined hard core/step potential to determine the nature of the
crossover from metastable to non-ergodic glass behavior. To understand the
dependence of the clump phase on the nature of the interaction, we also have
begun to investigate a system with a Gaussian potential [11] of the form
d~b6(yx) = Td exp[-y21x21]. Because the Fourier transform of the Gaussian
potential is positive definite, there is no spinodal in the y---~0 limit as is found
for the step potential.
In summary, we have presented a system that forms clumps at low
temperatures despite the purely repulsive nature of the potential. The structure
of the clump phase can be either crystalline or amorphous. The amorphous
clump phase, which the system enters upon quenching, behaves as a metastable
glass. The glass phase is associated with a free energy surface that has a large
number of metastable minima. The glass transition is well-defined in the
mean-field limit and is associated with the limit of stability of the liquid phase.
In addition to exhibiting a previously unknown clump structure for systems
with repulsive potentials, our mean-field model should help us gain insight into
the kinetic as well as the thermodynamic aspects of glass formation. We
currently are investigating the kinetic properties of the glass phase near the
ergodic-to-nonergodic transition. There also are physical systems such as
polymers [12], lipid monolayers, and one-component plasmas that might
exhibit clumping phenomena, although we have not yet investigated these
systems.
Acknowledgements
This work was supported in part by a grant from the Division of Materials
Research of the NSF. Acknowledgement is made to the donors of the
Petroleum Research Foundation, administrated by the American Chemical
Society, for partial support of this research. We thank R. Mountain and L.
Colonna-Romano for useful conversations.
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