We study thermotropic nematic LCs doped with NPs of volume concentration
in the diluted regime (i.e.,
). We consider the two experimental observations addressed above, for which generic mechanism are not known: (i) non-monotonous
dependence [
17,
18], and (ii)
-driven crossover in
dependence [
19]. We first describe a possible mechanism yielding the non-monotonous
behavior. Afterward we develop a simple phenomenological model originating from the Lebwohl-Lasher-type lattice approach [
20,
21,
22] to explain crossover behaviour in nematic ordering on varying
.
2.1. Non-Monotonous Dependence
To identify a possible origin of non-monotonous
behavior, we first discuss the most probable behavior on decreasing temperature across the I-N phase transition for different concentrations of NPs. For this purpose, we consider simple model systems, which roughly mimic the experimentally studied systems in Reference [
17,
18,
19]. We assume that NPs are essentially spatially homogeneously distributed in the LC medium. Furthermore, the NP-LC coupling is sufficiently weak, so that NPs do not enforce topological defects in the LC medium. We set the NPs to be spherical and enforce a relatively weak homeotropic anchoring [
2]. In this case, LC molecules tend to be aligned along the surface normal of an infinitesimally small NP-LC interface area. Due to the weak anchoring condition, a local LC ordering reflects the compromise between local elastic forces, surface anchoring, and wetting tendencies.
In our treatment, we use the volume concentration, which is defined as
where
and
stands for the number of NPs, the volume of an average spherical NP, and sample volume, respectively. In addition to the NP radius
, there are several other lengths which play an important role in our study. These are the average separation length
between neighboring NPs, the nematic order parameter correlation length
, and the surface extrapolation length
Their values are estimated by [
2]
Here (see Methods and Reference [
2]),
is the representative nematic Frank elastic constant,
is the second derivative of the nematic phase condensation free energy expressed at the equilibrium, and
stands for the representative anchoring strength (see Methods). The correlation length estimates the length over which a locally-induced perturbation in the amplitude of nematic order persists. On the other hand,
estimates the typical length scale on which the nematic director varies near a surface imposing an elastic distortion.
We next describe expected LC structural behaviour on decreasing temperature from the isotropic phase. We assume that the temperature is gradually decreased (i.e., mixtures are not “quenched” into the nematic phase). Firstly, we consider diluted samples, in which
in the whole temperature regime. For such conditions, the most probable configuration for
is schematically sketched in
Figure 1. NPs act as seeds for paranematic (weakly ordered nematic) ”islands” in the isotropic “sea”, to which we henceforth refer to as
clusters. The effective radius and volume of approximately spherical
clusters equals to
and
, respectively. Due to the weak anchoring, the average preferential paranematic orientation of each cluster exists, to which we henceforth refer to as the
cluster director, that is aligned along some symmetry breaking direction. Namely, a paranematic ordering favours parallel alignment of LC molecules, which breaks the isotropic symmetry imposed by the homeotropic anchoring condition. On the other hand, the coupling between
cluster directors is relatively weak. Consequently, the orientational distribution function of
cluster directors is expected to be isotropic if
. When the phase transition temperature is reached, the nematic order nucleated at NP-LC interfaces gradually pervades all the LC volume. This growth is slow enough (due to sluggish kinetics of domain growth) so that previously randomly aligned
cluster directors realign along a common (in general domain) direction. In this concentration regime, the NP-imposed disorder strength is relatively weak. On increasing
, the “paranematic stiffness” increases and remains partially quenched in a weakly distorted metastable state on entering the nematic phase. We expect that the disorder strength increases with increasing
in the regime
. At
, the
clusters are expected to enter “percolated” regime [
23]. Note that at the percolation threshold [
24] the structures typically exhibit fractal-like patterns and fractals in general fingertip “edge of chaos”. In the regime
, the coupling between paranematic
clusters becomes relatively strong (with respect to thermal fluctuations) even in the isotropic (i.e., paranematic phase). If this interaction is strong enough, it can partially realign neighbor
clusters along a similar direction, and consequently, the disorder strength is decreased.
Such a configuration could be treated as a binary system consisting of LC molecules and
paranematic clusters. The corresponding effective free energy density could be approximately expressed as [
25]
, where
The terms
and
represent spatially averaged condensation contributions of the bulk-like LC component and
clusters, respectively. Here,
and
determine amplitudes of the respective nematic (paranematic) orderings;
,
,
,
,
,
,
are material dependent quantities; and
For example, for it follows The term models the clusters as effectively lyotropic LC molecules diluted in an isotropic fluid. The interaction term models spatially averaged coupling between LC molecules and clusters, where the coupling constant W is positive. This coupling favors mutually supporting the ordering of LC molecules and clusters. Therefore, in cases the LC component experiences an external field-like coupling term , where in “real” samples certainly exhibits a relatively strong spatial dependence. This introduces a certain degree of randomness in the system, in particular in the regime where percolation-like order of clusters is expected.
The fingerprint of this phenomenon could be non-monotonous
dependence observed in some LC mixtures, where examples are given in
Table 1. In it, we list
measured in mixtures of different LCs and NPs [
17,
18] in the diluted regime. The local minimum in
signals peaks in NP-imposed disorder strength. Namely, theoretical studies on the impact of random type disorder [
26] on nematic ordering reveal that the bistability region decreases on increasing the disorder strength. A rough insight into the latter behaviour is also evident from a relatively simple phenomenological model presented in the Methods (see Equation (24)), where the dimensionless field
estimates NPs-induced disorder strength. On increasing
, the nematic-paranematic coexistence temperature interval decreases in the regime
and vanishes at
.
2.2. Impact of NPs on Average Nematic Ordering
Polarized micro-Raman spectroscopy and optical birefringence measurements [
19] revealed bulk-like equilibrium ordering in the nematic phase within the experimental resolution in the diluted regime (
). However, the uniaxial order parameter displayed two different regimes on increasing
as shown in
Figure 2. In the first regime (
), where
,
monotonously decreased. However, in the second regime (
), it gradually saturated to a roughly constant value.
To reproduce the basic mechanism behind this observation, we derive a simple mean-field-type expression at temperature
for the nematic response in a randomly perturbed nematic phase. Our interest is in the structural behaviour which is expected to emerge in weakly disordered phases reached via a continuous symmetry breaking phase transition in which a spatially uniform orientational ordering is formed in the absence of disorder. We used the simplest possible modeling to reproduce the experimental observations reported in Reference [
19]. In this study, homogeneously dispersed NPs enforce a relatively weak disorder. In the modeling, we assumed that each NP enforces in general a different local orientation which we roughly mimic by a kind of preferentially ordered random field of constant amplitude. Reasons for such randomness are described in the previous section. We used a Lebwohl-Lasher-type lattice modelling which despite its simplicity well describes general properties of nematic ordering [
20,
21,
22].
We consider a
-dimensional lattice (we study cases
and
) of
nematic spins pointing along a local direction of rod-like LC molecules at the site
of a cubic lattice interacting by Lebwohl-Lasher interaction [
20]. To take into account the nematic head-to-tail invariance, we introduce the traceless nematic tensor order parameter [
2]
where
is the
-dimensional unit tensor and
marks the tensor product. We define the global nematic order parameter
of the system, where
stands for the spatial average. The
global nematic director points along the average nematic direction in the system and
stands for the global uniaxial nematic order parameter. Furthermore, we set that some randomly chosen sites are occupied by NPs, where their volume concentration is determined by
. In our modeling, we consider dilute regimes characterized by
. We set that a NP located at the
ith site enforces a local “easy direction” determined by the tensor
and orientations of unit vectors
are distributed according to some probability distribution
.
Taking into account Equations (5)–(6), we obtain
Therefore, for
and
it follows
For later convenience, we introduce also
The angles
and
are defined as
Note that and . Furthermore, () fingerprints that the nematic director field (NP enforced easy direction) is strictly aligned along . Isotropic distribution of yields . Similarly, for isotropic distribution of , it follows .
In the spirit of classical mean-field approaches [
21,
27], we replace the molecular field
(see Equation (17)) which is introduced in the Methods by an effective molecular field acting on each
nematic spin:
Here,
,
describes the interaction between nearby
nematic spins,
stands for the number of first neighbors,
and
represents the local “impurity” field strength (see Methods).
Next, we assume that a local nematic orientation is determined by the effective molecular field [
21,
27]. Hence, we request that the frames of
and
coincide (are parallel) [
21], i.e.,
. We determine the constant
from the condition
, where
and
for
and
, respectively. Therefore,
, and it follows
Finally, we take the spatial average of Equation (12), which yields the self-consistent integral equation for
for a given probability distribution
The integration is carried over all possible orientations
,
and
. In our approximate treatment, we assume a cylindrically symmetric distribution of
values about
. Consequently,
Note, that Equation (13) yield for . Furthermore, in the limit and the isotropic distribution P it follows .
In
Figure 3, we plot
as a function of
for different distributions
P and
. In the calculations, we impose a step-like distribution
and
where
is a constant. The
dependence monotonously decreases on increasing
, displaying a relatively steep crossover between the two regimes, characterized by significantly different characteristic values of S. Saturated value of
in the second (plateau) regime monotonously decreases with
and is finite for
. Note that the behaviour shown in
Figure 3 is relatively robust with respect to
P shape. For example, in
Figure 3c, we compare
behaviors for a step-like and Gaussian distribution. One sees that
well fingerprints the essential property of
We stress that in our simulations
is increased either by increasing
or
. However, our preliminary semi-microscopic lattice simulations using interactions at finite temperatures reveal that experimentally observed crossover behavior on varying
could be reproduced if the anchoring strength
is relatively weak. To estimate the anchoring strength in experiments, we use a simple mesoscopic model where the corresponding free energy contributions are given by Equation (20) (see Methods). Experimental measurements reveal negligible shifts in
on varying
. Our mesoscopic estimate (see Equation (26a)) reveals that elastic and NP-LC interface contributions tend to decrease and increase
, respectively. We set that the contradicting effects cancel each other, i.e.,
. It follows
where
,
estimates a typical linear length over which distortions in the nematic director field are realized, and we assume
(see Methods, Equation (25)). By setting
, we obtain (see Equation (2))
. For
,
J/m, and
nm [
19] it follows
J/m
2, confirming a relatively weak anchoring regime. Note that the interaction between NPs and LCs depends on material properties of both components and also on NP surface treatment [
28].