# Note: Shoving model and the glass transition in one-component plasma

Sergey Khrapak<EMAIL_ADDRESS>Joint Institute for High Temperatures,
Russian Academy of Sciences, 125412 Moscow, Russia

###### Abstract

A modified shoving model is applied to estimate the location of the glass
transition in a one-component plasma. The estimated value of the coupling
parameter $\Gamma\simeq 570$ at the glass transition is compared with other
predictions available in the literature.

A one-component plasma (OCP) model is an idealized system of point charges
immersed in a uniform neutralizing background of opposite charge Brush  _et
al._ (1966); Baus and Hansen (1980); Ichimaru (1982). This model is of
relevance in a wide interdisciplinary context, including laboratory and space
plasmas, planetary interiors, white dwarfs, liquid metals, and electrolytes.
There are strong relations to various soft matter systems such as charged
colloidal suspensions and complex (dusty) plasmas Fortov  _et al._ (2005);
Fortov and Morfill (2019); Ivlev  _et al._ (2012); Chaudhuri  _et al._ (2011);
Beckers  _et al._ (2023). OCP represents a very convenient system to test and
verify the applicability of different theoretical approaches used in condensed
matter research.

The mobile particles forming the OCP are interacting via a very soft and long-
ranged Coulomb repulsive interaction potential, $\phi(r)=e^{2}/r$, where $e$
is the electric charge and $r$ is the distance between a pair of particles.
The particle-particle correlations and thermodynamics of the OCP are
characterized by a single dimensionless coupling parameter $\Gamma=e^{2}/aT$,
where $a=(4\pi n/3)^{-1/3}$ is the Wigner-Seitz radius in three dimensions
(3D), and $T$ is the temperature in energy units ($\equiv k_{\rm B}T$). The
coupling parameter is equivalent to the inverse temperature in conventional
matter. Since the interaction potential is purely repulsive, the OCP does not
exhibit a gas-liquid phase transition, gas-liquid coexistence, critical and
gas-liquid-solid triple points. The one-dimensional phase diagram of the OCP
is very simple. There is a phase transition from the fluid phase to the body-
centred cubic (bcc) solid phase at sufficiently strong coupling (low
temperature), $\Gamma_{\rm fr}\simeq 174$ Dubin and O’Neil (1999); Khrapak and
Khrapak (2016), where the subscript “fr” refers to freezing. There is also a
gas-to-liquid dynamical crossover, which has been recently located at
$\Gamma/\Gamma_{\rm fr}\simeq 0.05$, that is at $\Gamma\sim 10$ Huang  _et
al._ (2023); Yu  _et al._ (2024).

Over the years there have been predictions that supercooled OCP fluid might
exhibit a glass transition at strong coupling. However, the location of this
transition differs greatly in these studies. Ichimaru and Tanaka used a
generalized viscoelastic theory to demonstrated the possibility of a glass
transition at $\Gamma_{\rm g}=900-1000$ Ichimaru and Tanaka (1986). Cardenas
and Tosi Cardenas and Tosi (2004) studied the supercooled-fluid region and the
transition to an amorphous glassy state in the OCP within the replica-
symmetry-breaking scenario developed by Franz and Parisi Franz and Parisi
(1997). They obtained the threshold value $\Gamma_{\rm g}\simeq 1500$ for the
glass transition. Glass transition properties for the Yukawa (screened
Coulomb) potential were investigated by Yazdi et al. using the traditional
mode coupling theory (MCT) with the structural information obtained from the
Ornstein-Zernike relation and the hypernetted-chain (HNC) approximation
closure Yazdi  _et al._ (2014). In the infinite screening length limit,
corresponding to the OCP, they obtained $\Gamma_{\rm g}\simeq 590$. Lucco
Castello and Tolias combined the traditional MCT with three closures of the
integral equation theory of liquids to estimate the glass transition line in
the Yukawa system Lucco Castello and Tolias (2021). With HNC closure they
located the glass transition point in the OCP at $\Gamma_{\rm g}\simeq 575$,
slightly lower than the result of Yazdi et al. With the isomorph-based
empirically-modified hypernetted chain (IEMHNC) approach Tolias and Castello
(2019) and the variational modified hypernetted chain (VMHNC) approximation
Rosenfeld (1986), the obtained coupling parameters are almost a factor of two
lower, $\Gamma_{\rm g}\simeq 290$ and $\Gamma_{\rm g}\simeq 280$,
respectively.

Considerable variation in the locations of the glass transition point reported
in different studies suggests checking these predictions with other methods
and tools. The purpose of this Note is to apply a simple model of the glass
transition based on elastic arguments – the shoving model Dyre  _et al._
(1996); Dyre and Olsen (2004). The model considers a “flow event”, which
requires a local volume increase. The activation energy for a flow event is
identified with the work done in shoving aside the surrounding liquid. This
work can be expressed using the infinite frequency elastic moduli (infinite
frequency shear modulus in the original formulation Dyre  _et al._ (1996)).
The shoving model is one of the elastic models discussed in connection to
glass-forming liquids Dyre (2006). In the simplest approximation these elastic
models lead to a Lindemann-like criterion for the glass transition Dyre
(2006), an analogy that will be elaborated further below. At the moment we
remind that the energy-landscape version of the shoving model predicts that
the following dimensionless combination is nearly constant at the glass
transition Dyre and Wang (2012):

$\frac{G_{\infty}}{nT}\frac{K_{\infty}+4G_{\infty}/3}{2K_{\infty}+11G_{\infty}/3}\simeq{\rm
const}.$ (1)

Here $K_{\infty}$ and $G_{\infty}$ are the infinite frequency bulk and shear
moduli and no distinction between the high-frequency plateau moduli and
idealized instantaneous affine moduli (see Ref. Dyre and Wang (2012) for
details) is made in case of the OCP. The value of the constant in Eq. (1) is
expected to be quasi-universal. For a wide range of metallic glasses the
values of the const are scattered in the vicinity of ${\rm const}\simeq 30$.
Actually, in Fig. 2 of Ref. Dyre and Wang (2012) the data points are scattered
around the value of $\simeq 0.03$, but the results are expressed in units of
GPa cm3/J = $10^{3}$.

The instantaneous bulk modulus of the OCP system is infinite: Due to very soft
and long-ranged character of the Coulomb interaction, the dispersion relation
has a plasmon dispersion instead of the conventional acoustic one. If we
formally substitute $K_{\infty}\rightarrow\infty$ in the Eq. (1) and adopt the
constant appropriate for metallic glasses, we obtain $\tilde{G}_{\infty}\simeq
60$ as a preliminary estimate of the glass transition point in the OCP. Here
$\tilde{G}_{\infty}=G_{\infty}/nT$ is the reduced shear modulus. Within the
quasi-localized charge approximation (QLCA) Golden and Kalman (2000), the
elastic moduli of the OCP system can be directly expressed via the excess
internal energy. At strong coupling, the OCP excess energy is not very
sensitive to whether OCP forms a fluid, solid, or glass. The ion sphere model
provides a reasonable estimate Baus and Hansen (1980); Ichimaru (1982),
resulting in Khrapak (2021, 2019)

$\tilde{G}_{\infty}\simeq 0.1936\frac{e^{2}n^{1/3}}{T}\simeq 0.12\Gamma.$ (2)

In terms of the coupling parameter this yields $\Gamma_{\rm g}\simeq 500$.
Still, this estimate is not very convincing, because it neglects the
contribution from the longitudinal collective mode.

Equation (1) should be modified to account for the non-acoustic character of
the longitudinal collective mode in the OCP system. We discuss this
modification below. Let us start with estimating the amplitude of the atomic
vibrations in the harmonic approximation. We have

$\left\langle\delta
r^{2}\right\rangle=\frac{3T}{m}\left\langle\frac{1}{\omega^{2}}\right\rangle,$
(3)

where $m$ is the atomic mass. Averaging can be performed over normal modes,

$\left\langle\frac{1}{\omega^{2}}\right\rangle=\frac{1}{3N}\sum_{\rm
k}\omega_{\bf k}^{-2}.$ (4)

Furthermore, the sum over frequencies can be converted to an integral over $k$
using the standard procedure

$\frac{1}{V}\sum_{\bf k}(...)\rightarrow\frac{1}{(2\pi)^{3}}\int(...)d{\bf
k},$ (5)

where $V$ is the volume. One longitudinal (compressional) and two transverse
(shear) modes are supported in the solid state. We get

$\left\langle\frac{1}{\omega^{2}}\right\rangle=\frac{1}{6\pi^{2}n}\int_{0}^{k_{\rm
max}}k^{2}dk\left(\frac{1}{\omega_{l}^{2}}+\frac{2}{\omega_{t}^{2}}\right),$
(6)

where $\omega_{l}$ and $\omega_{t}$ are the frequencies of the longitudinal
and transverse modes and the cutoff $k_{\rm max}=(6\pi^{2}n)^{1/3}$ ensures
that $\langle{\mathcal{X}}\rangle={\mathcal{X}}$ for a quantity
${\mathcal{X}}$, which is independent of $k$. Substituting the acoustic
dispersion relations $\omega_{l}=c_{l}k$ and $\omega_{t}=c_{t}k$ into Eq. (6)
and taking into account the relations between the sound velocities and elastic
moduli $mnc_{l}^{2}=M_{\infty}=K_{\infty}+\tfrac{4}{3}G_{\infty}$ and
$mnc_{t}^{2}=G_{\infty}$, we get

$\frac{\langle\delta
r^{2}\rangle}{\Delta^{2}}=\frac{3nT}{(6\pi^{2})^{2/3}G_{\infty}}\frac{2K_{\infty}+11G_{\infty}/3}{K_{\infty}+4G_{\infty}/3}.$
(7)

Here $\Delta=n^{-1/3}$ is the average interatomic separation and Eq. (7)
represents an expression for the Lindemann measure in the harmonic
approximation. According to the Lindemann melting rule, melting of a crystal
occurs when the average vibrational amplitude exceeds a universal fraction
($\sim 0.1$) of the inter-atomic distance. By virtue of Eq. (1), an analogue
of the Lindemann rule applies to the glass transition, as discussed previously
Dyre (2006). The only difference is in the relative vibrational amplitude.
Adopting ${\rm const}\simeq 30$ in Eq. (1) we arrive at

$\langle\delta r^{2}\rangle/\Delta^{2}\simeq 0.0066$ (8)

at the glass transition point.

It is now obvious how the shoving approximation should be modified in case of
the OCP. Performing averaging in Eq. (6) we have to substitute the plasmon
dispersion $\omega_{l}(k)\simeq\omega_{\rm p}$ instead of the acoustic one.
Here $\omega_{\rm p}=\sqrt{4\pi e^{2}n/m}$ is the plasma frequency. The
acoustic approximation for the transverse dispersion relation remains
adequate. Even more accurate results can be expected by substituting almost
exact QLCA long-wavelength dispersion relations for $\omega_{l}(k)$ and
$\omega_{t}(k)$ Khrapak  _et al._ (2016); Khrapak and Khrapak (2018) in Eq.
(6). There is no need to perform averaging here. This has been already done in
connection to the self-diffusion mechanism and the Stokes-Einstein relation
between the self-diffusion and viscosity coefficients in the strongly coupled
OCP Khrapak (2021). The result, which applies equally to strongly coupled
fluid and amorphous solid phases is $\langle\omega_{\rm
p}^{2}/\omega^{2}\rangle\simeq 9.76$ Khrapak (2021, 2024). Combining Eqs. (3)
and (8) we get the following condition for the glass transition in the OCP
model

$\Gamma_{\rm g}\simeq 570.$ (9)

This is our best estimate for the glass transition point within the shoving
model. It should be pointed out, however, that the shoving model is an
approximation and the value of the constant in Eq (1) based on metallic
glasses may not be optimal for the OCP system. Nevertheless, the present
estimate correlates much better with recent MCT predictions of Refs. Yazdi
_et al._ (2014); Lucco Castello and Tolias (2021), compared to earlier works
Ichimaru and Tanaka (1986); Cardenas and Tosi (2004).

To summarize, the energy-landscape version of the shoving model has been
modified to account for the non-acoustic character of the longitudinal
collective mode in the OCP system. This modified version of the shoving model
predicts the glass transition in OCP at $\Gamma_{\rm g}\simeq 570$. Similar
modifications can be straightforwardly implemented in other simple fluids for
which deviations from acoustic asymptotes are important. This includes for
instance 2D OCP as well as 2D and 3D Yukawa fluids and related systems with
soft interaction potentials Khrapak  _et al._ (2017). Generalization of the
shoving model to mixtures seems desirable in view of applications to complex
plasma experiments targeting the glassy state with polydisperse dust.

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