Tuning of tetrahedrality in a silicon potential yields a series of
monatomic (metal-like) glassformers of very high fragility.
Valeria Molinero*, Srikanth Sastry#, and C. Austen Angell*
*Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287, U.S.A.
#Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus, Bangalore, 560064, India.
We obtained monatomic glassformers in simulations by modifying the tetrahedral character in a
silicon potential to explore a triple point zone between potentials favoring crystals of
incompatible short and intermediate range order. The glassforming liquids are extraordinarily
fragile and their heat capacity behavior mimics that of metallic glassformers. Our results suggest
that Si and Ge liquids may be vitrified at a pressure close to the diamond-β−tin-liquid triple
point.
PACS numbers:
The recent development of metallic systems with bulk
glassforming character BMGs[1, 2] adds interest to any
study aimed at decreasing the tendency of atomic liquids to
crystallize. Many single component molecular liquids fail
to crystallize on cooling, forming glasses at lower
temperatures. Atomic liquids, however, are more
challenging to vitrify, and to date no monatomic glass was
obtained in the laboratory by cooling of the liquid.
Advances in understanding supercooled liquid behavior
of atomic liquids have come from computer simulation
methods, molecular dynamics (MD) in particular. Most
studies of atomic glassfomers have been made with model
binary mixtures, e. g. “binary mixed Lennard-Jones”
(BMLJ), in which the rapid crystallization of pure LJ is
avoided.[3] But the different size atoms, and the chemical
order imposed by the potentials, inevitably complicate
interpretations. To date, the only monatomic system which
permits study of the “low temperature” viscous regime of
liquids is the icosahedral model developed by
Dzugutov[4]. This is a spherically symmetric pair potential
which is antithetic to three dimensionally ordered
structures.
Because Dzugutov’s potential favors
quasicrystal formation, it is desirable to develop alternative
monatomic models resilient to crystallization on simulation
time scales up to microseconds. While the idea of
modifying the silicon potential to this end is now quite
old[5], it has not been explored except in the analog case
of water under perturbation towards a simple LennardJones liquid[6]. In ref [6] the liquid state anomalies, rather
than glassforming ability, were in focus.
The cases of silicon and its periodic table relative,
germanium, are of special interest because of the manner in
which they (i) undergo tetrahedral semiconductor-to-metal
transitions on melting (ii) decrease melting point on
increase in pressure, and, (iii) exhibit liquid-liquid (LL)
transitions on supercooling.[7, 8] Here we show how pure
atomic glassformers can be obtained by modifying the
Stillinger-Weber (SW) model for silicon[9] to explore a
frustration zone between potentials favoring diamond cubic
(DC) vs body-centered cubic (BCC) crystals. This is the
zero pressure analog of pressure destabilization of crystal
packing. The results are described by a novel temperaturepotential phase diagram with a triple point glassforming
domain.
In the SW model, tetrahedral coordination is favored by
adding, to a basic pairwise potential v2(r), a three-body
term v3(r,θ) which induces repulsion for angles that are not
tetrahedral, v=v2(r) + λ v3(r,θ), The repulsion parameter, λ,
and the pair potential parameters, were adjusted in [9] to
best reproduce the melting point and cohesive energy for
the laboratory substance.
In our study, we have kept the pair potential, defining an
invariant temperature scale, and varied λ to tune the
repulsive potential systematically on either side of the SW
value. Variations to smaller values provide the material for
the present report. The results were obtained from a series
of NPT MD simulations for 512 atoms (686 if starting
from perfect BCC crystals) at p=0, using procedures given
elsewhere.[8] Run lengths ranged from 1 to 130 ns (8.5 107
steps). The average displacement of the atoms was at least
4 atomic diameters during each of the equilibrium runs.
The first findings we discuss are those related to the
melting points, and the diffusive mobility at the melting
points, since it is the relationship between these, and the
increase in the driving force for crystallization with
supercooling, that determines the rate at which crystals
form during cooling at a given rate[1, 10], hence
the.glassforming ability GFA.
We confirm that[7, 8] the SW liquid (λ = 21) does not
crystallize at temperatures above the LL transition, at 650
K supercooling, and that crystallization to DC occurs from
the low temperature (low density) liquid. The crystal that
forms contains defects and is our starting point for the
melting line determination.[11] We repeat the melting
study with decreasing λ until the DC crystal becomes so
metastable that it “melts” exothermically. By this λ value,
an alternative BCC crystal form has become stable, as the
lattice energies (Supporting Material,[12]) suggest. We
repeat the procedure to obtain the melting line for BCC. It
has the opposite slope to that of the DC crystal.
We assemble these data into a new type of phase
diagram, Fig. 1, in which the potential parameter λ replaces
pressure on the horizontal axis of the familiar onecomponent T-p diagram. The diagram has the same general
V-shape as (a) the T-p phase diagrams for silicon and
germanium[13, 14] and (b) the phase diagrams for twocomponent glassformer systems like Au-Si, in which
metallic glass formation was first reported.[15] Whether or
not rapid quenching of liquid Si under high pressure can
produce glassy silicon is not yet known, but in the
analogous case of water under pressure ”easy” vitrification
has recently been reported[16]. The metastable extensions
of the melting lines (truncated in Fig. 1) continue until
there is an ambient temperature mechanical stability limit.
This is related to the much-studied phenomenon of
pressure-induced amorphization (most recent case,
silicon[17]).
Figure 1. Phase diagram of the modified Si potential. Melting
points for DC (down-pointing triangles) and BCC (squares)
crystals, in relation to the “tetrahedrality” parameter λ. The
glassforming domain corresponds to λ=17.5 to 20.25 (bold
line on λ axis). The minimum melting point (a triple point or
“pseudo-eutectic”) occurs at λ=18.75, very close to λ=18.6
(arrow) at which the lattice energies of BCC and DC are equal.
The variation of lattice energies of several crystalline
structures as a function of λ is shown in Ref. [12]. The uppointing triangles and circles locate the temperature of
dynamic arrest To and isoentropy Kauzmann temperature T K,
respectively. The errorbars for T o (±35 K) and TK (±12 K) are
of the size of their symbols. The minimum in TK occurs at the
λ value where the isothermal diffusivity maximum is found
(Figure 2). Using the observation of Swallen et al[18] that D =
10-20 m2s-1 for fragile liquids at T g, we estimate that Tg for each
of the present glassformers would lie 20-80 K above T K. The
onset temperatures of the liquid-liquid transition TLL are also
shown. The beginning of the glassforming regime at the
intersection of Tg and T LL suggests that the low density liquid
(LDL) is an intermediate step towards DC crystallization.
Uncertainty limits are explained in the supporting material
[12]. A β-tin phase, marginally stable in the range λ=18.218.7[12] is never seen.
Cooling of the high temperature liquid produces i) a
sharp transition to a metastable low density liquid phase
LDL[7, 8] which then crystallizes, for 20.5 < λ < 21.5l, ii)
a continuous transition to a glassy state for 17.5< λ
<20.25, iii) sharp crystallization to BCC for λ <17.5 On the
tetrahedral crystal side of the phase diagram we find a
correlation between glassformation and loss of density
maximum, as experimentally observed for water with
increasing concentration of electrolytes.[19]
In Fig. 2 we examine the diffusivity-temperature
relations in the glassforming domain, using Arrhenius plots
to emphasize the strongly super-Arrhenius character of the
diffusivity. The liquid with λ=17.5-18 has the highest
isothermal diffusivity. Nevertheless, the diffusivity
evaluated on the melting lines reaches a minimum of 0.95 x
10-5 cm2s-1 at the λ value of the triple point (inset of Fig. 2).
This diffusivity is the same as that of Ni in the marginal
glassformer NiP at the Ni-P eutectic temperature of 1171
K.[20]
Figure 2. Arrhenius plots of diffusivities for non-crystallizing
melts of different λ values. Diffusion coefficients D were
obtained from linear fits of the MSD, 〈r2(t)〉=6Dt. Solid curves
are best fits of Eq. (2) to the data. The strength parameter D of
Eq.(2) for these liquids is 3±1 in the glassforming range
λ=17.5-20.25[12]. Insert shows the value of the liquid
diffusivity at the melting points, D(Tm), as a function of λ
(circles correspond to DC melting and triangles to BCC
melting). Empirically, laboratory glassforming ability for
fragile liquids depends on the viscosity being greater than ~
0.1 Pa.s at the melting temperature (~1.5Tg, see any fragility
plot for viscosity). Viscosity and diffusivity are strongly
correlated in this diffusivity range,[18] and η=0.1 Pa.s
corresponds to D=0.73 x 10-6 cm2s-1, for T = 1000K and r = 0.1
nm.
Ultimately, the glassforming ability is decided by
nucleation and growth kinetics, the detailed study of which
exceeds the scope of this letter. In classical nucleation
theory the activation free energy to form a nucleus is
inversely proportional to the square of the crystallization
driving force[1], Gex=Gliquid-Gcrystal, that increases with
supercooling. This quantity has been held crucial for the
glassforming ability of metal alloys, where values as small
as 1.5 kJ/mol for T/Tm=0.8 typify the best glassforming
mixtures.[21] (cf 1.9 kJ/mol below). We computed the
excess thermodynamic properties shown in Figure 3 from i)
the melting temperatures Tm, ii) the melting enthalpy
ΔΗm evaluated as the difference between H of the liquid
and perfect crystal, at Tm and iii) the heat capacities Cp
(derived from the enthalpies) of the supercooled liquids and
perfect crystals. The excess entropies are computed as
liquid
"H m T C p
S (T) =
#$
Tm
Tm
ex
# C pcrystal '
dT ,
T'
ex
!
(1)
!
ex
ex
while the excess free energies are G (T)=H (T)- TS (T).
Figure 3c shows Gex(T/Tm) for the supercooled liquids with
potentials λ=16 to 20.25. The increase of Gex with
supercooling is minimal for the triple point potential. For
λ=18.5 Gex=1.9 kJ/mol at T/Tm=0.8, comparable to metallic
glassforming alloys that vitrify for 104K/s cooling rate,
such as Zr62Ni38.[21] These results indicate that the
potentials close to the triple point avoid crystallization
because they have i) the lowest diffusivity at Tm while also,
ii) a very low rate of increase in G ex with supercooling.
Figure 3. Excess thermodynamic functions of the liquid with
respect to the stable perfect crystal (red curves: BCC, black
curves: DC) as a function of temperature for systems beyond
the liquid-liquid transition region of the system. (a) Excess
heat capacities Cpliquid- Cpperfect crystal. Symbols indicate the lowest
temperature of equilibrated liquid data in the simulation. Heat
capacities were derived from the fitted enthalpies of each
phases. H data are provided in [12]. (b) Decrease of the excess
entropy, Sex=Sliquid-Scrystal, from its ΔS m value at the defect
crystal fusion point Tm, by integration of the excess heat
capacity shown in panel (a). Symbols mark the fusion
temperature and entropy for each lambda value. The fusion
entropy for the BCC form is always smaller than that for the
DC. (c) Excess free energies Gex in the supercooled regime
from T m to TK. The symbols at the end of each Gex curve are
only for labeling purposes.
Having a continuous series of one-component
(isothermally melting) systems, all of which are
glassforming, presents us with an optimum opportunity to
study the relation between thermodynamics and dynamics
of different glassforming systems at zero pressure - which
we now examine.
From the curvature of the Arrhenius plots of Fig. 2,
parameters for the VFT equation
D = Do exp("DTo /(T " To ))
(2)
may be obtained. The values of To of Eq.(2) are assessed in
the λ range 17.5-20.25 in which neither liquid-liquid nor
crystallization transitions interfere, and supercooling range
is limited only by computer time.
We obtained the Kauzmann temperatures by
extrapolation to zero of the excess entropies (Fig 3b) of the
liquids over DC or BCC crystals (Eq. 2), for each λ system.
The excess heat capacity (Cpliquid–Cpcrystal) data for λ < 20.25
(beyond the LL transition domain) are shown in Fig. 3(a),
constructed from fits to the enthalpy vs temperature data
(see [12]). The heat capacity of the glasforming liquids
shows the striking uprising form of the more fragile
metallic glassformers[21].
These To and TK values are displayed on the phase
diagram in Fig. 1, where the liquid-liquid transition onset
temperatures are also shown. The coincidence of To and TK,
despite the considerable λ-dependence of each, is
striking[22] (like that found earlier[23] for different
densities in the BMLJ system) and in agreement with the
prediction of theories like that of Adam and Gibbs[24]
which connect the two.
Is interesting to note that the glassforming domain starts
(from the DC side) where these ideal glass temperatures
intersect the expected continuation of the TLL (TLG at lower
λ), pointing to the central role of the LDL “intermediate
phase” in the crystallization pathway to DC. Consistent
indications of polyamorphous transformation in the path of
ice crystallization from high density amorphous water [25]
suggest that this scenario may be general to tetrahedral
liquids.
The question of liquid fragility remains to be addressed.
We find remarkably small D values, about 3±1 (m fragility
~ 200) in the λ=17.5-20.25 range (Fig. 2, inset). That single
component metallic liquids should be extremely fragile has
been the expectation based on observations of decreasing
fragility in optimized bulk metallic glassformers of
different component numbers[21]. Our values may be
compared with the value for tri-phenyl phosphite, which is
one of the most fragile liquids known (D = 3.7[26]]), and
one in which a liquid-liquid transition is also known to
occur[27]- slightly above the standard glass transition
temperature Tg.
How does all this relate to any real materials? Clearly
our weakening of the tetrahedral bonding tendency is what
is achieved in the periodic table group IV by making the
atomic core larger down the series. Thus our progression,
with decreasing λ, from DC to BCC is to be related to the
change from DC to β-tin structure down group 4 (though
we note that DC becomes the stable phase, even of tin,
below 12ºC – the medieval cathedral organ pipe “tin
disease”). Because we keep the pair potentials constant, the
cohesive energies in our system increase with decreasing λ,
whereas in group IV, they merely remain very high (while
the melting points decrease by a factor of ~3). What we
achieve by “potential tuning” at zero pressure can also be
achieved with individual group 4 elements Si and Ge, by
“pressure tuning”. Results of laboratory studies of rapid
cooling of liquid Si and Ge at pressures close to their DCβ-tin-liquid triple point, and MD simulations of Ge and Si
under pressure, will be reported separately [Bhat, Molinero,
Yarger, Sastry, Angell, unpublished work].
By exploiting a situation that abounds in Nature - the
existence of conditions where crystals of incompatible
short and intermediate range order coexist with a liquid
phase - we have found glassformers amongst simple atomic
systems. This strategy was highlighted before in the
framework of the two-parameter order model of
liquids.[28] Mishima and coworkers’ method[16] of
pressure-vitrifying water (a “good” glassfomer under the
hyperquenching conditions of computer simulations, albeit
a “bad” one in the lab at one atm.) illustrates the principle
used in this work for molecular systems, and highlights the
similarities of silicon and water.
The strategy presented here to enhance glassforming
ability of monatomic systems differs from the usual one for
making molecular glassformers that is based on the
frustration of the packing by decreasing the symmetry of
the molecule. The hindrance of the molecule’s rotation in
the viscous supercooled liquid (needed to form the crystal)
has no analog in atomic liquids. In the atomic case it is the
relative position with respect to the neighbors (with whom
they may have angle-dependent three body interactions)
and not the molecular shape that determines the ability of
the atoms to pack into a crystal.
In summary, by tuning a single parameter in a successful
interaction potential for an important experimental
substance, silicon, we have been able to generate a wide
range of liquid behavior incorporating most of the
interesting liquid and polyamorphic phenomena of
experimental glassformers - in simple monatomic systems.
Properties of current interest, such as infinite frequency
shear moduli in relation to inherent structure[29], Poisson
ratios[30], dynamic heterogeneities[31], and structural
order[32] in the glassformers will be reported in future
communications.
This research was sponsored by NSF under Chemical Sciences
grant no. 0404714.
____________
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