# Hidden scale invariance in the Gay-Berne model

Saeed Mehri  Glass and Time, IMFUFA, Department of Science and Environment,
Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark  Jeppe C. Dyre
<EMAIL_ADDRESS>Glass and Time, IMFUFA, Department of Science and Environment,
Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark  Trond S.
Ingebrigtsen<EMAIL_ADDRESS>Glass and Time, IMFUFA, Department of Science and
Environment, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark

###### Abstract

This paper presents a numerical study of the Gay-Berne liquid crystal model
with parameters corresponding to calamitic (rod-shaped) molecules. The focus
is on the isotropic and nematic phases at temperatures above unity. We there
find strong correlations between the virial and potential-energy thermal
fluctuations, reflecting the hidden-scale invariance symmetry. This implies
the existence of isomorphs, which are curves in the thermodynamic phase
diagram of approximately invariant physics. We study numerically one isomorph
in the isotropic phase and one in the nematic phase. In both cases, good
invariance of the dynamics is demonstrated via data for the reduced-unit time-
autocorrelation functions of the mean-square displacement, angular velocity,
force, torque, and first- and second-order Legendre polynomial orientational
order parameters. Deviations from isomorph invariance are only observed at
short times for the orientational time-autocorrelation functions, which
reflects the fact that the moment of inertia is assumed to be constant and
thus not isomorph invariant in reduced units. Structural isomorph invariance
is demonstrated from data for the radial distribution functions of the
particles and their orientations. For comparison, all quantities were also
simulated along an isochore of similar temperature variation in which case
invariance is not observed. We conclude that the thermodynamic phase diagram
of the calamitic Gay-Berne model is essentially one-dimensional in the studied
regions as predicted by isomorph theory, a fact that potentially allows for
simplifications of future theories and numerical studies.

##  I Introduction

Liquid crystals (LCs), rod-like polymers, and disk-formed particles all
involve molecules with a high degree of shape anisotropy [1]. LCs occur in
many contexts, ranging from the well-known display applications to biological
systems [2, 3, 4]. Depending on the density and the temperature, the
anisotropy of LCs may lead to different structural phases. For the
thermometric LCs of focus here, changing temperature and density may cause
transitions from the ordinary crystalline state to smectic, nematic, and
isotropic phases [2].

Pure fluids and mixtures consisting of aspherical particles have been subject
to many theoretical, experimental, and simulations studies [5, 6, 7, 8].
Theoretical studies are typically based on the Fokker-Planck equation [9, 10],
generalized Langevin equations [11, 12], Onsager theory [13], density-
functional theory [14], or generalized van der Waals descriptions [15].
Different numerical techniques like Monte-Carlo and molecular dynamics (MD)
have been applied for studying the phase behavior, thermodynamics, structure,
and dynamics of rigid anisotropic molecules forming LCs [16].

Depending on the type of interaction between the molecules, one can classify
LC models into two main groups [17]. The first group considers models of hard
particles with a non-spherical shape [18]. In such models there are no
attractive interactions, i.e., the potential is purely repulsive and short-
ranged. The main motivation for this approach is the success of the hard-
sphere model in explaining the properties of simple liquids [19, 20].
Extensive simulation studies have used this approach to investigate structure
and dynamics of LC fluids for different shape anisotropies (prolate
ellipsoids, spherocylinders, rods, disks, etc) [21, 22, 23]. In the other main
class of LC models, both short-range repulsive and long-range attractive
interactions are taken into account. Several potential models for fluids of
aspherical particles have been introduced for LC studies, e.g., the Kihara
potential [24], the site–site potential [25], the Gaussian overlap model [26],
and the Gay–Berne (GB) model [27]. By using site-site potentials one can
realistically mimic the structure of LC molecules and compare results to
experiments [28, 29, 30, 31, 32, 33], but unfortunately such models usually
require huge computational resources. This is why most simulations so far have
been conducted for relatively small system sizes. An exception is the GB model
based on the Lennard-Jones (LJ) pair interaction, which is computationally
cheap and still realistic. For this reason, the GB model has become popular
almost as the generic LC model. The GB model gives rise to a rich mesogenic
behavior [4]. Previous numerical studies of this model have focused on its
phase behavior [34, 35, 36, 37, 38, 39, 40], per-particle translational and
orientational dynamics [41], interfacial properties [42], elastic constants
[43], thermal conductivity [44, 45], and viscosity [46, 47]. Analytical
perturbation theories have also been applied in order to explore the phase
diagram of GB fluids [48, 49].

The GB model allows one to describe different shape anisotropy, spanning from
elongated ellipsoids to thin disks [27]. The GB potential depends on four
dimensionless parameters, which is often signaled by the notation
$GB(\kappa,\kappa^{\prime},\mu,\nu)$. The four parameters control the shape of
the molecules and the strength of the interaction between them. $GB(3,5,2,1)$
is the most studied case, leading to rod-shape molecules, and in this case the
phase diagram and orientational order parameter are known [50]. Moreover, the
velocity time-autocorrelation function [41], viscosity [47], elastic constants
[51], free energies and enthalpy [14], isotropic-nematic transition [36],
liquid-vapor coexistence curve [52], stress-tensor components [53], and self-
diffusion coefficient [54] have been studied for the $GB(3,5,2,1)$ model.

This paper presents a study of the Gay-Berne model with parameters
corresponding to calamitic, i.e., rod-shaped elongated, molecules at high
temperatures where the model is shown to obey the symmetry of hidden scale
invariance. According to this symmetry, the system is expected yo have so-
called isomorphs, which are curves in the thermodynamic phase diagram along
which structure and dynamics are almost invariant when given in properly
reduced units. A recent study of ours showed this for the more exotic discotic
Gay-Berne model $GB(0.345,0.2,1,2)$ in the isotropic phase [55]; the present
paper demonstrates the existence of isomorphs in both the isotropic and the
nematic phase of a more standard Gay-Berne model. The study given below nicely
confirms the recent work by Liszka and co-workers [56], although that paper
however focused more on density-scaling (a particular consequence of isomorph
theory) than on demonstrating isomorph invariance of structure and dynamics.

##  II Gay-Berne Potential

The GB potential between pairs of particles (“molecules”),
$GB(\kappa,\kappa^{\prime},\mu,\nu)$, is characterized by the following four
dimensionless parameters that are all defined below:
$\kappa\equiv\sigma_{e}/\sigma_{s}$ where $\sigma_{e}$ and $\sigma_{s}$ are
lengths, $\kappa^{\prime}\equiv\varepsilon_{ss}/\varepsilon_{ee}$ where
$\varepsilon_{ss}$ and $\varepsilon_{ee}$ are energies, and $\mu$ and $\nu$
are exponents.

The GB pair potential $v_{\rm GB}$, which is basically a direction-dependent
LJ pair potential, is defined as follows

$\displaystyle v_{\rm
GB}(\textbf{r}_{ij},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})$
$\displaystyle=4\varepsilon(\hat{\textbf{r}},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})\left[\left(\sigma_{s}/\rho_{ij}\right)^{12}-\left(\sigma_{s}/\rho_{ij}\right)^{6}\right],$
(1a) $\displaystyle\rho_{ij}$
$\displaystyle=r_{ij}-\sigma(\hat{\textbf{r}},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})+\sigma_{s}\,.$
(1b)

Here, $r_{ij}$ is the distance between molecules $i$ and $j$,
$\hat{\textbf{r}}\equiv\textbf{r}_{ij}/r_{ij}$ is the unit vector along the
vector from molecule $i$ to molecule $j$ denoted by $\textbf{r}_{ij}$, and
$\hat{\textbf{e}}_{i}$ and $\hat{\textbf{e}}_{j}$ are unit vectors along the
major axis of molecules $i$ and $j$. The GB molecule is roughly an ellipsoid
of two diameters $\sigma_{s}$ and $\sigma_{e}$, and one defines

$\displaystyle\sigma(\hat{\textbf{r}},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})$
$\displaystyle=\sigma_{s}\bigg{[}1-\dfrac{\chi}{2}\bigg{(}\dfrac{(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{r}}+\hat{\textbf{e}}_{j}\cdot\hat{\textbf{r}})^{2}}{1+\chi(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{e}}_{j})}+\dfrac{(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{r}}-\hat{\textbf{e}}_{j}\cdot\hat{\textbf{r}})^{2}}{1-\chi(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{e}}_{j})}\bigg{)}\bigg{]}^{-1/2}$
(2a) $\displaystyle\chi$ $\displaystyle=\dfrac{\kappa^{2}-1}{\kappa^{2}+1}\,.$
(2b)

Physically, $\chi$ is a shape anisotropy parameter and $\kappa$ quantifies the
molecular elongation. The case $\kappa=1$ ($\chi=0$) represents spherical
molecules, the case $\kappa\rightarrow\infty$ ($\chi\rightarrow 1$)
corresponds to very long rods, and the case $\kappa\rightarrow 0$
($\chi\rightarrow-1$) corresponds to very thin disks. The energy term is given
as follows

$\displaystyle\varepsilon(\hat{\textbf{r}},$
$\displaystyle\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})=\varepsilon_{0}\,\left(\varepsilon_{1}(\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})\right)^{\nu}\left(\varepsilon_{2}(\hat{\textbf{r}},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})\right)^{\mu}$
(3a) in which
$\displaystyle\varepsilon_{1}(\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})$
$\displaystyle=\big{(}1-\chi^{2}(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{e}}_{j})^{2}\big{)}^{-1/2}$
(3b)
$\displaystyle\varepsilon_{2}(\hat{\textbf{r}},\hat{\textbf{e}}_{i},\hat{\textbf{e}}_{j})$
$\displaystyle=1-\frac{\chi^{\prime}}{2}\biggl{(}\dfrac{(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{r}}+\hat{\textbf{e}}_{j}\cdot\hat{\textbf{r}})^{2}}{1+\chi^{\prime}(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{e}}_{j})}+\dfrac{(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{r}}-\hat{\textbf{e}}_{j}\cdot\hat{\textbf{r}})^{2}}{1-\chi^{\prime}(\hat{\textbf{e}}_{i}\cdot\hat{\textbf{e}}_{j})}\biggr{)}$
(3c) and the energy anisotropy parameter is given by
$\displaystyle\chi^{\prime}$ $\displaystyle=\frac{\kappa^{\prime
1/\mu}-1}{\kappa^{\prime 1/\mu}+1}\,.$ (3d)

The energies $\varepsilon_{ss}$ and $\varepsilon_{ee}$ are the well depths of
the potential in the side-side and end-end configurations, respectively.
Henceforth, unless isomorph-theory reduced units are used (see Sec. III),
$\sigma_{s}$ defines the length unit used and $\varepsilon_{0}$ the energy
unit. The density $\rho$ and the temperature $T$ are below always given in
these units.

The $GB(3,5,2,1)$ model was introduced in 1981 by Gay and Berne, inspired by
the Gaussian overlap model of Berne and Pechukas [26, 27]. For realistic LCs
the length-to-width ratio is at least $3$, leading to the choice of $\kappa=3$
by Gay and Berne [27]. To obtain the other parameters, the GB pair potential
was compared to the case of a pair of linear molecules consisting of four LJ
particles placed on a line such that the length-to-width ratio equals $3$.
This results in $\kappa^{\prime}=5,~{}\mu=2$, and $\nu=1$ [27].

As mentioned, the $GB(3,5,2,1)$ model shows a rich phase behavior with
isotropic, nematic, and smectic B phases [57, 35, 50]. Actually, the model has
also the following phases: smectic A [35], tilted smectic B [50], and rippled
smectic B phases [40]. In some cases, more involved versions of the GB
potential have been investigated by introducing, e.g., dipolar forces [58],
flexibility [59], more complex shapes [60], or biaxial molecules [61]. Other
sets of parameters have also been studied, and other properties have been
examined, e.g., the effect of the $\nu$ exponent on the orientational order
parameter [62], elastic constant for $GB(3,5,1,3)$ [63], diffusion coefficient
in the smectic A phase of $GB(4.4,20,1,1)$ [64], stability of the smectic
phase, radial distribution function, orientational order parameter [65], and
rotational viscosity coefficient [66]. Satoh $et~{}al.$ studied the effect of
an external magnetic field on GB fluids [67]. The isotropic-nematic region has
been explored for different values of $\kappa$ [68, 69]. Varying
$\kappa^{\prime}$ while keeping the other parameters fixed at
$\kappa=3,~{}\mu=2$ and $\nu=1$ has been investigated in detail, the liquid-
vapor region has been analyzed [70, 71], and so has the equation of state,
structure, and diffusion coefficient [72]. For discotic GB fluids the phase
diagram has been obtained for different $\kappa$ and $\kappa^{\prime}$
parameters [73, 74, 75, 55]. The most studied discotic model is
$GB(0.345,0.2,1,2)$ [76], and this incidentally led to an improvement of the
angle of view of liquid crystals displays [77].

In this work we study the $GB(3,5,2,1)$ model because, as already mentioned,
its phase diagram, structure, and dynamics are known. Fig. 1 presents
snapshots of the system at equilibrium in the isotropic, nematic, and smectic
phases. There is no positional or orientational ordering in the isotropic
phase. In the nematic phase there is no positional ordering, but some long-
range orientational ordering. In the smectic phase the molecules form parallel
layers with a robust orientational ordering within the layers.

Figure 1: Snapshots of the calamitic GB model $GB(3,5,2,1)$ at three state
points. (a) shows the isotropic liquid phase at the state point
$(\rho,T)=(0.27,1.2)$; (b) shows the nematic phase at $(\rho,T)=(3.3,1.2)$;
(c) shows the smectic phase at $(\rho,T)=(3.9,1.2)$.

##  III R-simple systems and isomorphs

Recalling that the virial $W$ quantifies the part of the pressure $p$ deriving
from molecular interactions via the defining identity $pV=Nk_{B}T+W$ (in which
$V$ is the volume and $N$ is the number of molecules) [78], liquids and solids
may be classified according to the correlation between the equilibrium
fluctuations of the virial and the potential energy $U$ [79]. The so-called
R-simple systems, which are those with strong such correlations, are
particularly simple because in this case the thermodynamic phase diagram is
basically one-dimensional instead of two-dimensional in regard to structure
and dynamics [80, 79, 81, 82].

Isomorph theory dealing with R-simple systems was developed over the last
decade [83, 84, 85, 86]. The $WU$ Pearson correlation coefficient is defined
by

$R(\rho,T)=\frac{\langle\Delta W\Delta U\rangle}{\sqrt{\langle(\Delta
W)^{2}\rangle\langle(\Delta U)^{2}\rangle}}\,.$ (4)

Here the angular brackets denote $NVT$ ensemble averages, $\Delta$ is the
deviation from the equilibrium mean value, and $\rho$ is density. Many
systems, including the LJ fluid, have strong $WU$ correlations in the liquid
and solid phases, whereas $R(\rho,T)$ usually decreases significantly below
the critical density [87]. A system is considered to be R-simple whenever
$R>0.9$ at the state points in question.

Figure 2: Scatter plot of $WU$ correlations for the $GB(3,5,2,1)$ model at the
state point $(\rho,T)=(0.33,1.2)$. The system is here strongly correlating
with $R=0.92$; the density-scaling exponent $\gamma$ is $8.22$.

The density-scaling exponent $\gamma$, which is characterized by $\Delta
U\cong\gamma\Delta W$, is found from linear-regression fits to a $WU$ scatter
plot as shown in Fig.2 using the defining equation

$\gamma=\frac{\langle\Delta W\Delta U\rangle}{\langle(\Delta
U)^{2}\rangle}\,.$ (5)

R-simple systems have curves in the phase diagram along which structure and
dynamics are approximately invariant, and these curves are termed isomorphs.
It is important to emphasize that isomorph invariance only applies when data
are presented in so-called reduced units. In the system of reduced units,
which in contrast to ordinary units are state-point dependent, the density
$\rho\equiv N/V$ defines the length unit $l_{0}$, the temperature defines the
energy unit $e_{0}$, and the density and thermal velocity define the time unit
$t_{0}$:

$l_{0}=\rho^{-1/3},~{}~{}e_{0}=k_{\rm
B}T,~{}~{}t_{0}=\rho^{-1/3}\sqrt{m/k_{\rm B}T}\,.$

Here $m$ is the molecule mass. Quantities given in the isomorph-theory reduced
units are marked with a tilde.

Strong virial potential-energy correlations arise whenever the hidden-scale-
invariance symmetry applies. This is the condition that the potential-energy
ordering of same-density configurations is maintained under a uniform scaling
of all coordinates [88], formally expressed as follows:

$U(\mathbf{R}_{a})<U(\mathbf{R}_{b})\Rightarrow
U(\lambda\mathbf{R}_{a})<U(\lambda\mathbf{R}_{b})$ (6)

where $\lambda$ is scaling factor. Consider two configurations with same
potential energy, i.e., $U(\mathbf{R}_{a})=U(\mathbf{R}_{b})$. After a uniform
scaling one has by Eq. (6)
$U(\lambda\mathbf{R}_{a})=U(\lambda\mathbf{R}_{b})$. By taking the derivative
of this with respect to $\lambda$ one easily derives
$W(\mathbf{R}_{a})=W(\mathbf{R}_{b})$ [88]; thus same potential energy implies
same virial, i.e., 100% correlation between $W$ and $U$. Equation (6) only
applies approximately for realistic systems, however, so in practice one
observes strong but not perfect virial potential-energy correlations.

It can be shown that Eq. (6) implies that the reduced-unit structure and
dynamics are invariant along the lines of constant excess entropy, the
isomorphs [88]. Recall that a system’s entropy $S$ can be expressed as that of
an ideal gas plus a term deriving from the intermolecular interactions,
$S=S_{\rm id}+S_{\rm ex}$. For an ideal gas one has $S_{\rm ex}=0$, for all
other systems $S_{ex}<0$ because these are less disordered than the ideal gas.

Along an isomorph one has

$dS_{\rm ex}=\left(\frac{\partial S_{\rm ex}}{\partial
T}\right)_{V}dT+\left(\frac{\partial S_{\rm ex}}{\partial
V}\right)_{T}dV=0\,.$ (7)

Using Maxwell’s volume-temperature relation for the configurational degrees of
freedom, $\left({\partial S_{\rm ex}}/{\partial
V}\right)_{T}=\left({\partial\left(W/V\right)}/{\partial T}\right)_{V}$, we
can rewrite the Eq. (7) as

$\left(\frac{\partial S_{\rm ex}}{\partial T}\right)_{V}T\,d\ln
T=\left(\frac{\partial W}{\partial T}\right)_{V}d\ln\rho\,.$ (8)

Using $dU=TdS_{\rm ex}-(W/V)dV$ leads to

$\left(\frac{\partial U}{\partial T}\right)_{V}d\ln T=\left(\frac{\partial
W}{\partial T}\right)_{V}d\ln\rho\,,$ (9)

which via the fluctuation relations $\left({\partial W}/{\partial
T}\right)_{V}=-\langle\Delta W\Delta U\rangle$ and $\left({\partial
U}/{\partial T}\right)_{V}=-\langle(\Delta U)^{2}\rangle$ for $\gamma$ leads
to the above Eq. (5),

$\gamma\equiv\left(\frac{\partial\ln T}{\partial\ln\rho}\right)_{S_{\rm
ex}}=\frac{\langle\Delta W\Delta U\rangle}{\langle(\Delta U)^{2}\rangle}\,.$
(10)

Equation (10) is completely general [84]. This equation is of particular
interest, however, when the system has isomorphs because the equation can then
be used for tracing out the isomorphs without knowing the equation of state,
which is done as follows. At a given state point $(\rho_{1},T_{1})$ one first
calculates $\gamma$ from the equilibrium fluctuations of the potential energy
and virial. Then, by scaling the system to a slightly different density
$\rho_{2}$ and numerically calculating $\left({\partial\ln
T}/{\partial\ln\rho}\right)_{S_{\rm ex}}$ from Eq. (10), one predicts the
temperature $T_{2}$ with the property that $(\rho_{2},T_{2})$ is on the same
isomorph as $(\rho_{1},T_{1})$. In the simulations of this paper we used
fourth-order Runge-Kutta integration to generate isomorphs [89] involving
density step sizes of approximately 1%.

##  IV Properties studied

We simulated a system of $1372$ particles. The pair potential was cut and
shifted at $r_{c}=4.0$ and the time step was $\Delta t=0.001$. Because of the
shape anisotropy, supplementing the standard $NVT$ Nose-Hoover algorithm for
the center-of-mass motion we used the IMP algorithm for the rotational motion
[90]. Different thermostats were applied for translational and rotational
motion (we eventually concluded that using a single thermostat did not result
in any noticeable differences, however). The molecular moment of inertia was
set to $I=1$. At each simulated state point, 20 million time steps were taken
to equilibrate the system before the production runs, each of which involved
67 million time steps. As a consistency check of our GB implementation, we
compared simulation results with those of the literature and found good
agreement in all cases. The quantities evaluated in these studies, which are
all defined below, were the following (where we also list the reference(s) to
which data were compared): the radial distribution function $g(r)$ [91, 50],
the radial distribution orientational correlation function $G_{2}(r)$ [50, 36,
34], the $S_{2}$ orientational order parameter [50, 36, 92], and various time-
autocorrelation functions [41, 93].

An order parameter is a physical quantity that distinguishes between two
phases. We proceed to define the second-rank orientational order parameter
$S_{2}$ that quantifies how much the molecular orientations vary throughout
the system [94]. For a uniaxial phase, $S_{2}$ is defined as the following sum
over all molecules

$S_{2}=\left\langle\frac{1}{N}\sum_{i}P_{2}(\hat{\mathbf{e}}_{i}\cdot\hat{\mathbf{e}}_{d})\right\rangle\,.$
(11)

Here $P_{2}$ is the second-order Legendre polynomial, $\hat{\mathbf{e}}_{d}$
is the director of the phase, and the angular brackets denote a time or
ensemble average. This quantity takes values between 0 and 1; for a perfectly
aligned system $S_{2}=1$ whereas $S_{2}=0$ implies an isotropic system.

In a simulation $\hat{\mathbf{e}}_{d}$ is unknown. Here the order parameter
can be evaluated by maximizing $S_{2}$ with respect to $\hat{\mathbf{e}}_{d}$,
which is done by rewriting Eq. (11) as follows [21, 94]

$S_{2}=\langle\hat{\mathbf{e}}_{d}\cdot\mathbf{Q}\cdot\hat{\mathbf{e}}_{d}\rangle\,\,.$
(12)

If $\otimes$ denotes a tensor product and $\mathbf{I}$ is the unity matrix,
$\mathbf{Q}$ is defined by

$\mathbf{Q}=\frac{1}{2N}\sum_{i}(3\hat{\mathbf{e}}_{i}\otimes\hat{\mathbf{e}}_{i}-\mathbf{I})\,.$
(13)

It can be shows that $S_{2}$ is the largest eigenvalue, $\lambda_{\rm max}$,
of the $\mathbf{Q}$ tensor.

Figure 3: Density-temperature phase diagram of the $GB(3,5,2,1)$ model with
(a) showing the virial potential-energy correlation coefficient $R$ (Eq. (4))
and (b) showing the second-order orientational order parameter $S_{2}$ (Eq.
(12)). Dark green triangles connected by dark green dashed lines delimit the
phase boundaries [95]. $I$ stands for the isotropic, $N$ for the nematic, and
$S$ for the smectic phase. The solid yellow curves are isomorphs not
investigated further here (results for these are presented in Ref. 96), the
two black curves are two isomorphs for which results are reported below, one
in the isotropic phase and one in the nematic phase. The light blue triangles
mark the isomorph that continues the I-N phase boundary numerical data of Ref.
95. The isomorphs were determined by numerical integration of Eq. (7) starting
from the following reference state points marked by the red filled circles:
$(\rho_{\rm ref},T_{\rm ref})=(0.25,1.2)$, $(\rho_{\rm ref},T_{\rm
ref})=(0.27,1.2)$, $(\rho_{\rm ref},T_{\rm ref})=(0.30,1.2)$, $(\rho_{\rm
ref},T_{\rm ref})=(0.32,1.2)$, $(\rho_{\rm ref},T_{\rm ref})=(0.33,1.1)$,
$(\rho_{\rm ref},T_{\rm ref})=(0.33,1.2)$, and $(\rho_{\rm ref},T_{\rm
ref})=(0.35,1.2)$.

Figure 3 shows a “heat-map” phase diagram of the $GB(3,5,2,1)$ model with
respect to the virial potential-energy correlation coefficient $R$ and the
orientational order parameter $S_{2}$. By definition, the regions with $R>0.9$
are R-simple; this is where one expects isomorph theory to apply. This is not
a sharp distinction, however, and many systems with $R$ between 0.8 and 0.9
have also been found to have good isomorphs. In Fig. 3 $I$ stands for the
isotropic, $N$ for the nematic, and $S$ for the smectic phase; the regions in-
between are those of coexisting phases. The dark green triangles marking the
phase boundaries were extracted from Ref. 95. Selected isomorphs are marked as
solid yellow lines, each of which starts from a “reference state point” marked
as a red full circle. The main paper presents results for two isomorphs
(black): one in the isotropic phase and one in the nematic phase. Results for
the remaining five isomorphs are reported in Ref. 96.

We conclude from Fig. 3(a) that there are strong correlations whenever the
temperature is above unity (roughly). The isomorph reference state points were
selected to obey $R>0.9$; from here on $R$ increases when density and
temperature are increased along each isomorph. The Appendix give details of
the two isomorphs studied by listing for each isomorph several state points
and the corresponding values of $R$ and $\gamma$. Note that invariance of the
physics along the isomorphs – the main focus of this paper – is manifested
already in Fig. 3(b) as regards the orientational ordering because $S_{2}$ is
clearly approximately isomorph invariant. In particular, the phase boundary
approximately follows an isomorph [97, 98]. Based on this, the light blue
triangles mark the expected isotropic-nematic phase boundary. This gives an
example of how isomorph theory may be used for estimating the phase boundary
by allowing one to go beyond the numerical phase-boundary data of Ref. 95
without having to perform extensive additional simulations.

Having established the phase diagram of the $GB(3,5,2,1)$ model, we next
define the quantities studied. We probed the system’s dynamics at the
different state points by calculating the mean-square displacement (MSD) as a
function of time, as well as four time-autocorrelation functions defined by

$\phi_{A}(t)=\langle\mathbf{A}(t_{0})\cdot\mathbf{A}(t_{0}+t)\rangle.$ (14)

Here $\mathbf{A}(t)$ is a vector or scalar molecular property and the angular
brackets denote an ensemble and particle average. We evaluate below Eq. (14)
from simulations for $\mathbf{A}$ equal to velocity, angular velocity, force,
and torque. We also study the first- and second-order orientational order
parameter time-correlation function defined by

$\phi_{l}(t)=\langle
P_{l}(\hat{\mathbf{e}}_{i}(t_{0})\cdot\hat{\mathbf{e}}_{i}(t_{0}+t))\rangle\,,$
(15)

in which $P_{l}$ is a Legendre polynomial ($l=1~{}\text{and}~{}2$). To
quantify the structure we measured the standard radial distribution function,
$g(r)$, as well as the radial-distribution orientational correlation function
defined by

$G_{2}(r)\equiv\langle
P_{2}(\hat{\mathbf{e}}_{i}\cdot\hat{\mathbf{e}}_{j})\rangle\,$ (16)

where the brackets imply an average over all pairs of molecules $i$ and $j$
that are the distance $r$ apart.

##  V Results

This section investigates to which degree the reduced-unit structure and
dynamics are invariant along two isomorphs. We present data for one isomorph
in the isotropic phase and one in the nematic phase (the black lines in Fig.
3). In realistic models isomorph invariance is only approximate, so in order
to put the findings into perspective we compare the results for each isomorph
with results for the isochore defined by the reference-state-point density
(red points in Fig. 3) with the same temperature variation as that of the
isomorph. For the isotropic-phase isomorph the reference state point is
$(\rho_{\rm ref},T)=(0.27,1.2)$; here we cover a density variation of about
$40\%$ and temperatures in the range $1.2<T<27$. For the nematic-phase
isomorph the reference state point is $(\rho_{\rm ref},T)=(0.33,1.2)$; here
the density varies by about $35\%$ and temperatures are in the range
$1.2<T<16$.

Figure 4:  Reduced mean-square displacement as a function of the reduced time
$\tilde{t}$ along the isochore and the isomorph in the isotropic phase (left)
and in the nematic phase (right). In both cases, the data collapse to a good
approximation along the isomorph but not along the isochore.

Fig. 4 provides data for the reduced-unit MSD along the isochore and the
isomorph in the isotropic and nematic phases, respectively. The level of
invariance in the center-of-mass dynamics is clearly higher along the isomorph
than along the isochore. At long times, the MSD is proportional to time and
the reduced diffusion coefficients may be extracted from these data.

Figure 5: Reduced diffusion coefficient as a function of temperature along the
isochores (upper panel) and the isomorphs (lower panel). The approximate
invariance in the latter case is clearly visible.

Figure 5 shows the reduced diffusion coefficient as a function of temperature
along the isomorphs and isochores – approximate isomorph invariance is again
clearly visible.

Figure 6: The upper figures show the reduced-unit time-autocorrelation
functions of the velocity $v$ and the angular velocity $\omega$ along the
isochore and the isomorph in the isotropic (left) and the nematic phases
(right). The lower figures show the analogous results for the reduced-unit
time-autocorrelation functions of the force $f$ and the torque $\tau$. The
color codes are the same as in Fig. 4. Good isomorph invariance is generally
observed except for significant short-time deviations for the two rotational
autocorrelation functions (see the text).

Next we show data for four time-autocorrelation functions. Figure 6 gives in
the two upper figures the velocity ($v$) and angular velocity ($\omega$) time-
autocorrelation functions while the two lower figures give the force and
torque time-autocorrelation functions. As in Fig. 4, all functions are given
in reduced units and as functions of the reduced time $\tilde{t}$. Overall, we
see in both the isotropic and the nematic phases good isomorph invariance and
a sizable variation along the corresponding isochore. The short-time angular
velocity and torque time-autocorrelation functions violate isomorph invariance
significantly, however. This is due to the fact that the moment of inertia in
the simulations was kept fixed, implying that this quantity is not constant in
reduced units. As a consequence, the short-time ballistic motion is not
isomorph invariant. At intermediate and long times, we do find good isomorph
invariance also for the rotational time-autocorrelation functions; here the
moment of inertia plays much less of a role for the dynamics, which for a
given molecule is dominated by interactions with the surrounding GB molecules.
A weaker, but still clearly visible violation of isomorph invariance occurs at
short times for the force autocorrelation function. In our understanding, this
reflects the fact that the density-scaling exponent $\gamma$ changes with
density along an isomorph, resulting in a changing effective inverse-power-law
interaction. The lowest densities have the largest $\gamma$ (see the
Appendix), leading to the highest average force squared coming from
collisions. The collapse of the isochore rotational autocorrelation functions
at short times is a consequence of the definition of reduced units, just as
the short-time reduced-unit MSD collapse is.

Figure 7: The left figures show the first- and second-order orientational
order parameter time-autocorrelation function in the isotropic phase, the
right figures show the same in the nematic phase. Good isomorph invariance is
observed in both phases.

Fig. 7 shows data for the first- and second-order orientational time-
autocorrelation function, again plotted as functions of the reduced time. In
the isotropic phase these functions go to zero at long times, confirming that
there are no preferred orientations. This is not the case, of course, in the
nematic phase. In both phases, however, we observe good isomorph invariance.
According to the phase diagram (Fig. 3), the isochore defined from the nematic
isomorph reference state point enters the isotropic phase at high
temperatures; this is reflected in the figures by the fact that both time-
autocorrelation functions converge to zero as the temperature increases.

Figure 8: Reduced-unit radial distribution functions (upper figures) and
radial-distribution orientational correlation function (Eq. (16), lower
figures) along the isochore and the isomorph in the isotropic (left) and
nematic phases (right). Good isomorph invariance is observed in both phases,
with some deviation at the first peak (see the text).

So far we have discussed different dynamic signals and seen good isomorph
invariance. The isomorph theory also predicts that the reduced-unit structure
should be invariant. This is tested in Fig. 8, in which the upper panels show
the center-of-mass radial distribution function along the isochores and
isomorphs. The data are close to invariant along the isomorph, though with
visible deviations around the first peak. This is often observed when isomorph
theory is tested over a large density range; it reflects a non-invariance
arising when the density-scaling exponent $\gamma$ varies along the isomorph
similar to the non-invariance of the short-time force time-autocorrelation
function discussed above. As in the latter case, a large $\gamma$ implies a
large effective inverse-power-law exponent, which decreases the probability of
near contacts and is “compensated” by a higher peak in order to arrive at the
same coordination number (defined by integration of the radial distribution
function over its first peak). This phenomenon was recently rationalized in
terms of isomorph invariance of the so-called bridge function of liquid-state
theory [99]. Confirming this interpretation, the highest first peaks along the
isomorphs correspond to the lowest temperatures where $\gamma$ is largest
(Appendix).

Data for the orientational structure quantified in terms of $G_{2}(r)$ are
given in the lower panels of Fig. 8. In the nematic phase, this function does
not converge to zero at long times as in the isotropic phase. Except for the
first peak in the isotropic phase, there is good isomorph invariance of the
structure. Note that the tail of $G_{2}(r)$ goes to zero as the temperature is
increased along the isochore in the nematic phase. This reflects a phase
transition into the isotropic phase. In contrast, the tail remains invariant
as we increase the temperature along the isomorph.

##  VI Conclusions

When given in reduced units the dynamic and structural properties of the
$GB(3,5,2,1)$ model are invariant to a good approximation along isomorphs in
both the isotropic and the nematic phases, with some deviations at short times
for orientational time-autocorrelation functions reflecting that the moment of
inertia was assumed to be constant and thus not isomorph invariant in reduced
units. In contrast, structure and dynamics are not invariant along isochores
with the same temperature variation. Overall, our findings confirm isomorph-
theory predictions and are consistent with the fact that the calamitic
$GB(3,5,2,1)$ model obeys hidden scale invariance at high temperatures in both
phases, i.e., has a virial potential-energy correlation coefficient above 0.9.
For future work, it would be interesting to investigate the smectic phase of
the model for which we based on Fig. 3 expect good isomorphs even at lower
temperatures than unity.

###### Acknowledgements.

This work was supported by the VILLUM Foundation’s Matter grant (16515).

## Appendix: Isomorph state points

This Appendix provides details of the two isomorphs studied, giving for each
of these at several state points: density, temperature, virial potential-
energy correlation coefficient $R$ (Eq. (4)), and density-scaling exponent
$\gamma$ (Eq. (7)).

$\rho$ |  $T$ |  $R$ |  $\gamma$
---|---|---|---
0.2700 |  1.2000 |  0.9077 |  8.4553
0.2727 |  1.3057 |  0.9158 |  8.5107
0.2754 |  1.4215 |  0.9231 |  8.5535
0.2782 |  1.5480 |  0.9285 |  8.5737
0.2810 |  1.6859 |  0.9330 |  8.5795
0.2838 |  1.8361 |  0.9367 |  8.5756
0.2866 |  1.9995 |  0.9396 |  8.5615
0.2895 |  2.1770 |  0.9416 |  8.5383
0.2924 |  2.3697 |  0.9433 |  8.5126
0.2953 |  2.5788 |  0.9445 |  8.4820
0.2982 |  2.8056 |  0.9457 |  8.4555
0.3012 |  3.0514 |  0.9461 |  8.4252
0.3042 |  3.3177 |  0.9467 |  8.3962
0.3073 |  3.6062 |  0.9469 |  8.3646
0.3104 |  3.9186 |  0.9468 |  8.3364
0.3135 |  4.2570 |  0.9469 |  8.3084
0.3166 |  4.6231 |  0.9465 |  8.2815
0.3198 |  5.0194 |  0.9463 |  8.2542
0.3230 |  5.4483 |  0.9460 |  8.2276
0.3262 |  5.9125 |  0.9456 |  8.2032
0.3295 |  6.4148 |  0.9450 |  8.1802
0.3327 |  6.9583 |  0.9446 |  8.1647
0.3361 |  7.5463 |  0.9442 |  8.1479
0.3394 |  8.1827 |  0.9436 |  8.1289
0.3428 |  8.8712 |  0.9428 |  8.1121
0.3463 |  9.6164 |  0.9423 |  8.0987
0.3497 |  10.4226 |  0.9416 |  8.0867
0.3532 |  11.2953 |  0.9410 |  8.0735
0.3567 |  12.2396 |  0.9403 |  8.0628
0.3603 |  13.2616 |  0.9396 |  8.0562
0.3639 |  14.3680 |  0.9390 |  8.0462
0.3676 |  15.5657 |  0.9383 |  8.0443
0.3712 |  16.8622 |  0.9376 |  8.0396
0.3749 |  18.2660 |  0.9370 |  8.0345
0.3787 |  19.7859 |  0.9362 |  8.0315
0.3825 |  21.4318 |  0.9356 |  8.0285
0.3863 |  23.2145 |  0.9350 |  8.0295
0.3902 |  25.1457 |  0.9342 |  8.0295
0.3941 |  27.2379 |  0.9335 |  8.0343

Table 1: Variation of density, temperature, correlation coefficient $R$, and
density-scaling exponent $\gamma$ for the state points on the isotropic-phase
isomorph with reference state point $(\rho_{\rm ref},T_{\rm ref})=(0.27,1.2)$.
At each step we increased density by 1% up to 40% overall.

$\rho$ |  $T$ |  $R$ |  $\gamma$
---|---|---|---
0.3300 |  1.2000 |  0.9169 |  8.2217
0.3333 |  1.3027 |  0.9230 |  8.2809
0.3366 |  1.4149 |  0.9274 |  8.3187
0.3400 |  1.5371 |  0.9315 |  8.3349
0.3434 |  1.6700 |  0.9342 |  8.3301
0.3468 |  1.8142 |  0.9359 |  8.3183
0.3503 |  1.9705 |  0.9375 |  8.2957
0.3538 |  2.1398 |  0.9385 |  8.2664
0.3573 |  2.3229 |  0.9389 |  8.2333
0.3609 |  2.5209 |  0.9393 |  8.2066
0.3645 |  2.7349 |  0.9394 |  8.1697
0.3682 |  2.9660 |  0.9392 |  8.1395
0.3719 |  3.2156 |  0.9393 |  8.1057
0.3756 |  3.4851 |  0.9387 |  8.0731
0.3793 |  3.7759 |  0.9384 |  8.0433
0.3831 |  4.0898 |  0.9375 |  8.0058
0.3870 |  4.4283 |  0.9367 |  7.9788
0.3908 |  4.7934 |  0.9362 |  7.9487
0.3947 |  5.1873 |  0.9354 |  7.9212
0.3987 |  5.6119 |  0.9345 |  7.8922
0.4027 |  6.0697 |  0.9337 |  7.8677
0.4067 |  6.5632 |  0.9321 |  7.8438
0.4108 |  7.0953 |  0.9315 |  7.8229
0.4149 |  7.6689 |  0.9306 |  7.7996
0.4190 |  8.2870 |  0.9296 |  7.7791
0.4232 |  8.9533 |  0.9286 |  7.7620
0.4274 |  9.6715 |  0.9277 |  7.7464
0.4317 |  10.4455 |  0.9264 |  7.7286
0.4360 |  11.2797 |  0.9258 |  7.7155
0.4404 |  12.1786 |  0.9244 |  7.6942
0.4448 |  13.1473 |  0.9232 |  7.6844
0.4492 |  14.1912 |  0.9223 |  7.6712
0.4537 |  15.3160 |  0.9209 |  7.6535
0.4583 |  16.5277 |  0.9197 |  7.6442

Table 2: Variation of density, temperature, correlation coefficient $R$, and
density-scaling exponent $\gamma$ for the state points on the nematic-phase
isomorph with reference state point $(\rho_{\rm ref},T_{\rm ref})=(0.33,1.2)$.

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