Critical and Glassy Dualism in Glass-Forming E7 Nematogenic Mixture and Fullerene C₆₀ Nanocolloids

1. Introduction

Nematogenic mixture E7 is a unique liquid crystalline (LC) material, originally designed to achieve a nematic phase over a very wide temperature range, from $~65°C$ to even $-50°C$. It is composed of rod-like LC materials with relatively large permanent dipole moment parallel to the long molecular axis [1,2]. These features led to enormous success in applications, including widespread use in displays, smart windows, soft robotics, or laser optics [2,3,4,5,6].

Liquid Crystalline (LC) E7 mixture is also a significant material for studies in glass transition studies [7,8,9,10], a cognitive mystery considered amongst grand challenges of 21^st -century Science [11,12,13,14,15,16,17,18,19,20,21,22]. The hallmark of this phenomenon is a set of universalistic features in the previtreous domain above the glass temperature $T>T_g$ [12,13,14,15,16]. One can recall: (i) non-Arrhenius changes of primary relaxation time ($\tau , {\tau }_{\alpha }$), viscosity (η), DC electric conductivity (σ) …, (ii) decoupling between translational and orientational dynamics, (iii) the non-Debye distribution of the primary relaxation time, (iv) the dynamic crossover, often linked to the ‘magic’ time scale $\tau \left(T_B\right)={10}^{-7\pm 1}s$, (v) the secondary relaxation emerging for $T<T_B$, and related to the time-scale $\tau \alpha {\tau }_{\beta }$, (vi) ‘~universality high frequency distribution parameter $n\left(T\to T_g\right)\to 1/2$, of the primary relaxation time,… (viii) common presentation of primary relaxation time via the normalized scale in the Angell plot, …

This list of ‘universalities’ is far incomplete, but it shows the dominance of dynamics-related features. There is still no ultimate model coherently addressing these checkpoint-validating properties [12,13,14,15,16,17,18,19,20,21,22]

The strong influence of nanoparticles on the properties of LC reference materials has remained the subject of numerous intensive studies in recent decades [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. It is significantly motivated by the expectation of obtaining composite materials with innovative features that enhance the applications mentioned above [50,51,52,53,54,55].

Surprisingly, studies focused on previtreous features of LC + nanoparticles systems are very limited. This includes E7-based nanocolloids, despite the fact that E7 nematogenic mixture is a particularly important for glass transition studies [23,24,25,29,30,31,36,37,41,48,49]. Probably, the only explicitly focused studies have only recently been reported, for E7 + BaTiO₃ nanoparticles (diameter $2r=50nm$, paraelectric) nanocolloids [56].

This report presents results of broadband dielectric spectroscopy (BDS) studies on E7 + C₆₀ fullerene ($d=0.7nm$) nanocolloids, focusing on previtreous and critical properties. They implement extended analysis when comparing with ref. [56], which led to the discovery of exceptional features significant both for the fundamental insight and applications.

2. Materials and Methods

E7 is the eutectic mixture composed of rod-like cyanobiphenyl and cyanoterphenol components, at a specific composition, namely (1) 4-cyano-4’-n-pentyl-biphenyl (5CB, 51%), (2) 4-cyano-4’-n-heptyl-biphenyl (7CB, 25%), (3) 4-cyano-4’-n-oxyoctyl-biphenyl (8OCB, 16%), and (4) 4-cyano-4’’-n-pentyl-p-terphenyl (5CT, 8%) [1,2]. E7 nematogenic mixture was purchased from Synthon at the highest available quality. Prior to measurements, it was degassed and purified in subsequent and repeated steps: (i) solidification by freezing, supported by liquid nitrogen, (ii) removal of air and vapors via a vacuum pump, (iii) heating up to ca. $90°C$ i.e., deeply into the isotropic liquid state. The tested samples exhibit the following mesomorphism: Solid Glass - ($220\pm 5K$) - Nematic - ($332.9K$) - Isotropic Liquid, in agreement with referenced results [1,2,3,4]. The transition to the glass state is ‘diffused’, as indicated in the description.

For each LC compound in E7 mixture, there is a notable permanent dipole moment ($\mu \approx 5 Debye$), approximately parallel to the long molecular axis. The length of the molecules in E7 mixture ranges from $~1.8nm$ (5CB) to $~2.2nm$ (8OCB) [1,2,3,4].

Fullerene C₆₀ nanoparticles (diameter $2r\approx 0.7nm$) were purchased from Sigma-Aldrich. From dielectric tests in dilute solution, the dielectric constant of such fullerenes was estimated as $\epsilon \approx 3.8$ [57]. Nanocolloids were prepared using relatively large volumes of E7,$~5{cm}^3$, in sample preparation to estimate reliably nanoparticles (NPs) concentration. Nanocolloidal samples were sonicated for ca 1 hour prior to filling the measurement module.

Studies focus on small concentrations of nanoparticles NPs, following findings of refs. that this can be a region of surprisingly strong influence on the system compared with the native E7. For the tested concentration range ($x<1\%$), no sedimentation occurs, allowing us to avoid a supplementary macromolecular compound that is often used to stabilize the colloid [50,51,52]. However, the presence of such supplements complicates the analysis of BDS spectra due to a very strong ‘parasitic’ effect from these macromolecules, both attached to nanoparticles and ‘free’.

Broadband dielectric spectroscopy (BDS) is essential for liquid-crystalline materials and their colloid-based systems, owing to their enormous sensitivity to the action of an electric field, which underlies numerous applications. BDS studies were carried out using a Novocontrol impedance analyzer, coupled with the extended Quattro Novocontrol temperature control unit. The facility is shown in Figure 1.

Tested samples were placed in a flat-parallel capacitor, with $d=0.3mm$ gap, and $2r=20mm$ diameter, made from Invar. The applied voltage for measuring the electric field was $U=1V$, which enabled a permanent 6-digit resolution during measurements and ensured a weak, ‘negligible ’ intensity of the electric field.

3. Results

Figure 2 shows representative spectra of the real and imaginary parts of dielectric permittivity in E7 +NPs (NanoParticles: C₆₀ fullerene) nanocolloid. They are selected from spectra of over 260 temperatures tested. Characteristic features and frequency domains are indicated. Notable is the horizontal static domain for the real part of dielectric permittivity ${\epsilon }^{\prime }\left(f\right)$, defining the dielectric constant. As the frequency increases above the static domain, the impact of the permanent dipole moment diminishes, and ${\epsilon }^{\prime }\left(f\right)$ significantly decreases. This is the High Frequency (HF) domain. On decreasing the frequency below the static domain ${\epsilon }^{\prime }\left(f\right)$ values strongly rise. It is the low-frequency (LF) domain. This exceptional rise in LF domain is most often heuristically commented as the influence of residual ionic ‘contaminations’. For the authors, it can also be due to local translational shifts of the basic molecules that constitute the system. Such an origin supports translational-orientational coupling/decoupling that links the LF and HF domains, as shown below for systems tested in the given report.

The real part of dielectric permittivity is associated with the relative change of electric capacitance, ${\epsilon }^{\prime }\left(f\right)=C\left(f\right)/C_0$, where $C_0$ is the reference capacitance of the capacitor with no sample. This is related to the upper part of Figure 2. The lower part of Figure 2 shows the coupled behaviour of the imaginary part of dielectric permittivity ${\epsilon }^{″}\left(f\right)=1/\omega R\left(f\right)C\left(f\right)$, where $\omega =2\pi f$. For ${\epsilon }^{″}\left(f\right)$ spectrum loss curves reflecting the leading relaxation process – particularly associated with the orientation of permanent dipole moments appear. The peak (maximum) of the loss curve is related to the relaxation time characterizing given process, namely: $\tau =1/{\omega }_{peak}$ and ${\omega }_{peak}=2\pi f_{peak}$ [58]. Loss curves ‘wings’ enable the insight into the distribution of relaxation times. Jonsher [59,60] introduced their following ‘universal’ empirical scaling characterization:

$${\epsilon }^{″}\left(f\right)\propto f^m \Rightarrow log{\epsilon }^{″}\left(f\right)\propto m\times logf \mathrm{for} f<f_{peak}$$

(1)

$${\epsilon }^{″}\left(f\right)\propto f^{-n} \Rightarrow log{\epsilon }^{″}\left(f\right)\propto -n\times logf \mathrm{for} f>f_{peak}$$

(2)

where $f_{peak}$ is the frequency related to the loss curve maximum, and $0<m,n<1$ are parameters characterizing the low- and high- frequency related distributions of relaxation time.

Eqs. (1) and (2) can be applied for the direct estimation of the distribution parameters via the following transformation of empirical data [15,61,62,63]: $dlog{\epsilon }^{″}\left(f\right)/dlogf=m$ - for $<f_{peak}$ and $dlog{\epsilon }^{″}\left(f\right)/dlogf=-n$ for $>f_{peak}$, also allowing to estimate the loss curve peak frequency: $dlog{\epsilon }^{″}\left(f=f_{peak}\right)/dlogf=0$.

Formally, Jonscher scaling [59,60] is a consequence of the behavior near the inflection point, where rapid functional growth transforms to the slow one.

The alternative path of analyzing loss curves is the direct portrayal via the Havriliak-Negami (HN) equation [58,64,65]:

$${\epsilon }^{\ast }\left(\omega \right)={\epsilon }_{\infty }+{\displaystyle \frac{\Delta \epsilon }{{\left[1+{\left(i\omega \tau \right)}^a\right]}^b}}+{\displaystyle \frac{{\sigma }_{DC}}{i{\epsilon }_0{\omega }^{\varphi }}}$$

(3)

where $\omega =2\pi f$, ${\epsilon }_{\infty }$ is the ‘infinite frequency’ terminal values related to the sum of atomic and electronic polarizabilities, ϕ is related to the distortion from the DC conductivity limit related $\varphi =1$. Parameters $a,b<1$ describe the distribution of relaxation times: $a,b=1$ are for the single relaxation time Debye model.

Distribution-related parameters in Jonscher’s and HN scalings are related, namely: $m=b$ and $n=ab$. The analysis employing HN Eq. (3) requires multi-parameter nonlinear fitting, and then it is inherently associated with a notable error. Nevertheless, it is the only tool when loss curves overlap, limiting the reliable application of Eqs. (1) and (2).

When decreasing the frequency below the loss curve manifestation in ${\epsilon }^{″}\left(f\right)$ spectrum, one enter the part of the spectrum enabling the estimation of the DC electric, ${\sigma }^{\prime }\left(f\right)={\omega \epsilon }^{″}\left(f\right)$ →${\sigma }_{DC}=\sigma =const$. It is related to the pattern ${\epsilon }^{″}\left(f\right)\propto f^{-\varphi }$, with $\varphi =1$ [15,58], as visible in Figure 2.

To obtain insight into temperature changes of properties influencing BDS spectra, over 260 temperatures were tested for the isotropic and nematic phases, down to the vicinity of the glass temperature. The dielectric constant is historically the first and remains an essential characterization of dielectric properties. Most often, it is determined by following ${\epsilon }^{\prime }\left(f,T\right)$ changes for frequencies $f=10kHz$ or $f=100kHz$. However, this cannot be the case for glass-forming liquids, in which a significant shift towards lower frequencies occurs as the glass temperature is approached. This is well illustrated by the shift of the primary relaxation time from $\tau ~{10}^{-8}s$ near I-N transition to $\tau \approx {10}^{2}s$ at the glass temperature $T_g$. Hence, in the given report, dielectric constant has been taken at frequencies in the middle of the static domain, i.e., at frequencies that naturally shift to lower frequencies on cooling towards $T_g$.

Figure 3 shows dielectric constant temperature changes for the isotropic and nematic phases of E7 and related nanocolloids. The focused insight into the isotropic liquid phase is presented in Figure 4.

The isotropic liquid–nematic (I-N) phase transition is a classic example of a weakly discontinuous phase transition, assisted by long-range critical-type pretransitional effects. The model for scaling dielectric constant changes was developed by Drozd-Rzoska et al [66,67], drawing on Critical Phenomena Physics. As one approaches a continuous phase transition, long-range pretransitional effects arise from pretransitional fluctuations with the local symmetry linked to the next, approaching phase. Their lifetime (${\tau }_{fl.}$) and size (correlation length, $\xi$) increase infinitely as they approach the critical singularity. In the given case, it can be associated with the extrapolated critical-like or pseudospinodal-type temperature $T^{\ast }$ [68,69,70]:

$${\tau }_{fl.}\left(T\right)={\tau }_0^{fl.}{\left(T-T^{\ast }\right)}^{-zv}$$

(4)

$$\xi \left(T\right)={\xi }_0{\left(T-T^{\ast }\right)}^{-v}$$

(5)

where $T>T_{I-N}$, is the extrapolated singular $T^{\ast }=T_{I-N}-{\Delta T}^{\ast }$, ${\Delta T}^{\ast }$ is the metric of I-N transition singularity, the correlation length exponent $\nu =1/2$ (in the given case), and $z=2$ is the dynamic exponent for the non-conserved order parameter.

Pretransitional, critical-like fluctuations are associated with the basic characterization of the next, approaching phase. For the isotropic liquid phase on approaching I-N transition, it refers to the local prenematic ordering of rod-like molecules. For rod molecules with the permanent dipole moment approximately parallel to the long molecular axis, as in E7, it leads to their antiparallel arrangement. Consequently, within prenematic fluctuations, the contribution from permanent dipole moments to dielectric constant is negligible, and thus $\epsilon \left(fluct.\right)\ll \epsilon \left(surrounding\right)$. The rise of the correlation length on cooling causes the volume occupied by prenematic fluctuations become even stronger, namely: $V_{fluct.}\left(T\right)={\xi }^3\left(T\right)\propto {\left(T-T^{\ast }\right)}^{-3/2}$. Following these, one can expected the dominance of the volume occupied by pretransitional fluctuations at some temperature above $T_{I-N}$ and consequently the decrease of $\epsilon \left(T{\to T}_{I-N}\right)$ finally takes place on cooling.

The model analysis developed in refs. [66,67] showed that it leads to the following relation for dielectric constant pretransitional scaling in the isotropic liquid phase:

$$\epsilon \left(T\right)={\epsilon }^{*}+a\left(T-T^{\ast }\right)+A{\left(T-T^{\ast }\right)}^{\phi }$$

(6)

where $T^{\ast }<T_{I-N}$ is for the extrapolated continuous phase transition temperature, $a,A=const$; the exponent $\phi =1-\alpha$, where the exponent $\alpha =1/2$ is the heat capacity (internal energy) critical exponent.

The portrayal via the above relation is visible in the inset of Figure 3. The effect is more pronounced for the derivative of dielectric constant, as shown in the central part of Fig. 3 [66,67]:

$${\displaystyle \frac{d\epsilon }{dT}}\propto {\left(T-T^{\ast }\right)}^{-1/2}$$

(7)

where the exponent $\phi -1=-\alpha =-1/2$.

Fig. 3 also shows the overlap of pretransitional effects expressed as the derivative of dielectric constant $d\epsilon /dT$ for tested nanocolloids. Hence, the addition of nanoparticles changes only the constant term ${\epsilon }^{\ast }$ in Eq. (6). Superior scaling of the pretransitional effect in the isotropic phase via Eqs. (6) and (7) explicitly validates the dominance of multimolecular prenematic characterization, with critical type – or, more precisely, pseudospinodal-like – characterization [67].

For the nematic phase, dielectric constant is most often used for testing the order parameter behaviour [70]: $S={\epsilon }_{\parallel }-{\epsilon }_{\perp }\propto {\left(T-T^{\ast \ast }\right)}^{\beta }$, where $\beta$ is the order parameter critical exponent and $T^{\ast \ast }>T_{I-N}$ is the extrapolated from the nematic phase singular temperature; ${\epsilon }_{\parallel }$ and ${\epsilon }_{\perp }$ are dielectric constants for the ‘parallel; and ‘perpendicular’ arrangement of rod-like molecules with respect to the measurement electric field in the capacitor. The distortions-sensitive analysis explicitly showed the mean field tricritical (TCP) behavior of the I-N transition, associated with $\beta =1/4$.

In bulk samples, the orientation of molecules is realized by the external strong magnetic field of the value $~1-2 Tesla$ [1,2,70]. For capacitors with a micrometric gap (‘thin layer’) the orientation by special preparation of plates, for instance covering by a polymeric layer, is used [1,2]. However, it introduces side effects that require supplementary model assumptions in the analysis [1,2].

A generally accepted heuristic assumption is that just a few degrees below $T_{I-N}$ both ${\epsilon }_{\parallel }\left(T\right)$ and ${\epsilon }_{\perp }\left(T\right)$ follow a ‘parallel pattern’ [1,2]. This leads to the basic way of determining the anisotropy of the dielectric constant $\Delta \epsilon \left(T\right)={\epsilon }_{\parallel }\left(T\right)-{\epsilon }_{\perp }\left(T\right) \to \Delta \epsilon ={\epsilon }_{\parallel }-{\epsilon }_{\perp }=const$ (‘parallel pattern’), one of the most important material characterizations of nematogenic LC materials [1,2]. However, there is a formal problem here: as we show below in the given report and earlier in ref. [56,71], the ‘parallel pattern’ assumption is incorrect. Namely the scaling of ${\epsilon }_{//}\left(T\right)$ is scaled by the following equation [68]:

$$\Delta \epsilon \left(T\right)={\epsilon }_{//}-{\epsilon }_{\perp }\approx {\epsilon }_{\Delta }^{\ast \ast }+B{\left(T^{\ast \ast }-T\right)}^{\beta }$$

(8)

However, as shown in ref. [70] one can also consider the anomalous pretransitional behavior of the ‘diameter’ in the nematic phase ’$\delta \left(T\right)$:

$$\delta \left(T\right)={\displaystyle \frac{1}{3}}{\epsilon }_{//}+{\displaystyle \frac{2}{3}}{\epsilon }_{\perp }\approx {\epsilon }_{\delta }^{\ast \ast }+D{\left(T^{\ast \ast }-T\right)}^{1-\alpha }+d\left(T^{\ast \ast }-T\right)$$

(9)

where $T<T_{I-N}$, $B,d,D=const$, $T^{\ast \ast }$ is the hypothetical continuous transition extrapolated from the nematic phase.

Linking Eqs. (8) and (9), one obtains relations describing changes of dielectric constant for the ‘perpendicular’ and ‘parallel’ contributions [70]:

$${\epsilon }_{\perp }\left(T\right)\approx a_1-a_2{\left(T^{\ast \ast }-T\right)}^{\beta }+a_3{\left(T^{\ast \ast }-T\right)}^{1-\alpha }+a_4\left(T^{\ast \ast }-T\right)$$

(10)

$${\epsilon }_{//}\left(T\right)=b_1+b_2{\left(T^{\ast \ast }-T\right)}^{\beta }+b_3{\left(T^{\ast \ast }-T\right)}^{1-\alpha }+b_4\left(T^{\ast \ast }-T\right)$$

(11)

where $a_i$,$b_i=const$ denote empirical parameters related to Eqs. (8) and (9).

Recalling the evidence indicating the near-tricritical nature of the I-N transition for exponents, one can assume: $\beta =1/4$ and $\alpha =1/2$.

Already in refs. [56], it was indicated that for selected concentrations of BaTiO₃ ($2r=50nm$) dispersed in nematogenic LC ‘matrix’ (concentration $x\le 1\%$) the endogenic, permanent orientation in the nematic phase appears. Consequently, the dielectric constant in non-oriented samples, i.e., without the impact of an external strong ‘oriented’ field, followed the pattern of ${\epsilon }_{\perp }\left(T\right)$ or ${\epsilon }_{//}\left(T\right)$, depending on the concentration of nanoparticles.

Figure 3 shows that for E7 + C₆₀ fullerene nanocolloids, the dielectric constant changes in non-oriented samples spontaneously follows the ${\epsilon }_{//}\left(T\right)$ pattern. This parameterization is expressed by Eq. (11). The validity of such scaling is shown in Figure 3.

The above pattern of $\epsilon \left(T\right)$ evolution undergoes a qualitative change from the pattern described by Eq. (11) at approximately 25 K above the glass temperature $T_g$, manifested via a significant decrease in detected values. Notably, for E7+BaTiO₃ nanocolloids, which initially follow the ${\epsilon }_{\perp }\left(T\right)$ pattern (Eq. (10)) dielectric constant increases following the explicit critical-like scaling [56].

In the isotropic liquid phase of nematogens, the primary loss curve maximum (${\epsilon }_{peak}^{″}\left(T\right)$) follows a pattern parallel to Eq. (6) for the ‘critical’ anomaly of dielectric constant $\epsilon \left(T\right)$ [63,70]. Both anomalies are governed by the volume occupied by prenematic fluctuations. For dielectric constant, they are detected by the decrease of dielectric constant due to the cancellation of the contribution from permanent dipole moments within prenematic fluctuations. The pretransitional anomaly of ${\epsilon }_{peak}^{″}\left(T\right)$ detects fluctuations via energy loss impact required for molecular reorientations. Figure 5 shows changes of ${\epsilon }_{peak}^{″}\left(T\right)$ in the nematic phase of tested nanocoloids: the pattern of changes also resembles the pattern for the dielectric constant presented in Fig. 3. Such scaling is explicitly shown in Figure 5. For ${\epsilon }_{peak}^{″}\left(T\right)$ changes in the immediate vicinity of $T_g$ are even stronger and more characteristic than for dielectric constant, as shown by the derivative-based plot in the inset in Fig. 5.

is related to the parallel of Eq. (11), with following parameters: $b_1=18.3$, $b_2=-14.8$, $b_3=\mathrm{5,2}$, $b_4=-0.15$ and $T^{\ast \ast }=334.2$.

The evidence presented in ref. [56], where nanocolloids E7 +BaTiO₃ ($2r=50nm$, paraelectric) nanoparticles and the results presented above for nanocolloids E7 +C₆₀ fullerenes ($2r=0.7nm$) can be considered a significant argument for pretransitional (previtreous) effect on $T{\to T}_g$ with the critical-like characterization. It is further supported by the critical-like scaling pattern for the parameter $n$ describing the high-frequency part of the distribution of the dominant (primary, alpha) relaxation time, shown in Figure 6:

$$n\left(T\right)\approx n_{ref.}+{\left(T-T_g\right)}^{\theta }$$

(12)

with $n_{ref.}=0.5\pm 0.05$, and $\theta =0.5\pm 0.1$Notably, a parallel relation was evidenced for the isotropic liquid phase of octyloxycyanobiphenyl (8OCB) and its nanocolloids with BaTiO₃ nanoparticles [72].

for the primary relaxation time in the nematic phase of E7 and related nanocolloids with C₆₀ fullerene nanoparticles. Note the changes at $T_B$, indicated by the dashed arrow in blue.

The map of relaxation time in E7 +C₆₀ fullerene nanocolloids is shown in Figure 7. The single dominant primary relaxation time in the isotropic liquid phase relaxation is continued in the nematic phase. However, in the LC mesophase additional relaxation processes emerge. Below $T_B~280K$ the third – and much faster - relaxation time can be detected. All processes follow explicitly non-Arrhenius (SA) patterns, which is validated by nonlinear changes in Figure 7, prepared using the Arrhenius scale: $ln\tau$ or ${log}_{10}\tau$ vs. $1/T$.

Such behavior is commonly scaled by the Vogel-Fulcher-Tammann (VFT) equation, which is the practical replacement for the general super-Arrhenius (SA) equation with the apparent, temperature-dependent activation energy $E_a\left(T\right)$ [13,14,15]:

$$\tau \left(T\right)={\tau }_{\infty }exp\left({\displaystyle \frac{E_a\left(T\right)}{RT}}\right) \Rightarrow \tau \left(T\right)={\tau }_{\infty }exp\left({\displaystyle \frac{E}{T-T_0}}\right)={\tau }_{\infty }exp\left({\displaystyle \frac{D_T{T}_0}{T-T_0}}\right)$$

(13)

where the left part is for the general SA relation and the right one is for the VFT counterpart, with $E_a=ER/t={RD}_T{T}_0/\left[\left(T-T_0\right)/T\right]$, $E=const$, $D_T$ is the fragility strength parameter, $T_0<T_g$ is the extrapolated VFT singular temperature; $R$ is fro the gas constant.

The VFT equation has become so popular that it is often considered a heuristic ‘universal’ pattern for general previtreous dynamics [12,13,14,15,16]. The enormous success of the VFT relation results from its ’functional flexibility’ and simple application to empirical data. Such a portrayal was also successfully applied for pure E7 and E7 in nano-sieves [7,8,9,58].

However, recent in-depth analyses explicitly evidenced that in glass-forming systems VFT equation can be considered only an effective portrayal tool [15]. The analysis of a set of model equations for scaling previtreous dynamics showed the explicit preference for only two scaling dependencies.

The first is the MYEGA equation, which can be obtained by introducing $E_a\left(T\right)=RKexp\left(C/T\right)$, $K,C=const$, to the basic SA equation [73,74]. It was successfully implemented for testing the previtreous dynamics in E7 + BaTiO₃ NPs nanocolloids [56].

The second equation, composed of the ‘Critical’ and ‘Activated’ terms, has been derived by Drozd-Rzoska [15,75]:

$$\tau \left(T\right)=C_{\Gamma }{\left({\displaystyle \frac{T-T_g^{\ast }}{T}}\right)}^{-\Gamma }{\left[\mathit{exp}\left({\displaystyle \frac{T-T_g^{\ast }}{T}}\right)\right]}^{\Gamma }=C_{\Gamma }{\left(t^{-1}\mathit{exp}t\right)}^{\Gamma }$$

(14)

where $t=\left(T-T_g^{\ast }\right)/T$ and $T_g^{\ast }<T_g$ is the extrapolated singular temperature.

The power exponent in Eq. (14) can be expressed via basic empirical metrics of the glass transition:

$\Gamma =m\ln 10\left(T_g/T_g^{\ast }\right)/\left(1/\left(\Delta T_g^{\ast }/T_g\right)-1\right)$, $\Delta T_g^{\ast }=T_g-T_g^{\ast }$ and $m$ is the fragility metric of a glass-forming system. The value of the exponent $\Gamma$ determines their relative share in the previtreous effect [75].

For systems composed of molecules with uniaxial symmetry, which is also the case of E7-based systems, the critical–like contribution dominates and a fair portrayal can be obtained even via a single critical-type term [15,75,76]:

$$\tau \left(T\right)=C{\left(T-T_g^{\ast }\right)}^{{\mathsf{\Gamma }}^{\prime }}$$

(15)

Eq. (14) results from the empirical ‘universalistic’ finding for the ‘steepness index’ $s\left(T\right)$ or alternatively apparent activation enthalpy $H_a\left(T\right)$ or the apparent fragility $m_P\left(T\right)$ previtreous behavior [15,75]:

$$s\left(T\right),m_P\left(T\right),H_a\left(T\right)={\displaystyle \frac{A}{T-T_g^{\ast }}} \Rightarrow {\left[s\left(T\right),m_P\left(T\right),H_a\left(T\right)\right]}^{-1}=A^{-1}T-A^{-1}{T}_g^{\ast }=aT-b$$

(16)

where the left-hand related magnitudes are associated with the following transformation of $\tau \left(T\right)$ empirical data [15]:

$$\tau \left(T\right) \Rightarrow s\left(T\right)={\displaystyle \frac{d\mathit{ln}\tau \left(T\right)}{d\left(1/T\right)}}={\displaystyle \frac{H_a\left(T\right)}{R}}=\left(T_{g}ln10\right)m_P\left(T\right)$$

(17)

with $m_P\left(T\right)=\left(T_g/{log}_{10}e\right)\left(d{log}_{10}\tau \left(T\right)/d\left(T_g/T\right)\right)$ is the apparent fragility and the fragility metric $m=m_P\left(T_g\right)$; $m_P\left(T\right)$ is the steepness index for the normalized Arrhenius plot ${log}_{10}\tau$ vs. $T_g/T$, known as the Angell plot.

Figure 8 shows the evolution of the apparent activation enthalpy (alternatively: apparent fragility or the steepness index) of E7+C₆₀ fullerene nanocolloids showing the ‘universalistic’ behavior’ defined by Eq. (16). Notable is the strong impact of the I-N transitions surrounding, which can be associated only with the impact of pretransitional fluctuations associated with its weakly discontinuous near-critical nature of this phase transition. We stress this issue because evidence for the influence of critical fluctuations on the primary relaxation time has been poorly documented to date.

). The linear behavior is consistent with Eq. (16). The plot relates to three relaxation times, as shown in Figure 7. The extrapolated singularity for the nematic phase is related to $T_g^{\ast \ast }\left(nem.\right)\approx 190K$ and for the isotropic liquid phase: $T_g^{\ast \ast }\left(iso.\right)\approx T_B^{\ast \ast }=280K$.

Notable is the appearance of the behavior defined by Eq. (16) also in the isotropic liquid phase, with the singularity at $T_{iso}^{\ast }\approx 280K$. For such functional dependence it can be associated only with the Mode Coupling Theory (MCT) singular behavior [14,15,77], predicted for so-called ergodic high temperature dynamic domain in glass forming systems expected for the time scale $\tau \left(T_B\right)\gg {10}^{-7\pm 1}s$ [14,15]. Hallmarks of the crossover at $T_B\approx 280K$ are also visible for other properties discussed in this section.

Figure 9 presents the behavior of the DC electric conductivity in the nematic and isotropic phases of tested nanocolloids. The inset in Fig. 9 shows the related apparent enthalpy, scaled via the counterpart of Eq. (16), namely:

$$\sigma \left(T\right)=C_{\Gamma }{\left(t^{-1}\mathit{exp}t\right)}^{-{\mathsf{\Gamma }}^{\prime }}\iff {\displaystyle \frac{d{ln\sigma }^{-1}}{d\left(1/T\right)}}=H_{\sigma }\left(T\right)={\displaystyle \frac{S}{T-T_g^{\ast }}}$$

(18)

One of the specific universal features of the previtreous dynamics in glass-forming molecular liquids, extending even up to $~T_g+$150 K, is the translational – orientational decoupling. It is expressed by the fractional Debye-Stokes-Einstein (DSE) law linking DC electric conductivity (${\sigma }_{DC},\sigma$) and the primary relaxation time [78]:

$$\sigma \left(T\right)\times {\left[\tau \left(T\right)\right]}^F=C=const \Rightarrow log\sigma \left(T\right)=C-F\times log\tau \left(T\right)$$

(19)

where $F$ is the fractional exponent, showing the degree of the translational - orientational decoupling.

$F=1$ is for the standard DSE law, when translational processes are in timing with orientational ones. The explicit fractional DSE behavior is for delayed or speed-up orientational processes in comparison to translational ones, related to $F<1$ and $F>1$, respectively. Generally, in glass-forming molecular liquids the standard pattern is evidenced by $F=1$ for $T>T_B$ (high temperature dynamic domain) and $F<1$ for $T_B>T>T_g$ (low temperature dynamic domain).

Figure 10 shows the F-DSE analysis for the tested nanocolloids, focused on presentation using the left side of Eq. (19). Notable is a non-standard nonlinear pattern in the broad surrounding of the I-N transition for $T>T_B$. However, as the amount of C₆₀ fullerene nanoparticles increases, the rising nonlinearity appears as the glass transition is approached. The direct insight into crucial for testing the pattern of translational-orientational coupling/decoupling exponent $F$enables the transformation of empirical data via the following equation:

$$F=-{\displaystyle \frac{log\sigma \left(T\right)}{log\tau \left(T\right)}}$$

(20)

Its implementation is shown in the inset in Figure 10, revealing two domains that can be explicitly linked to the high- and low-temperature dynamical domains discussed above.

There is also an explicit ‘anomaly’ indicated by the vertical red arrow for the time scale coupled to the temperature $T_B\approx 280K$, noted above. Such an ‘anomaly’ has to be linked to the impact of fullerene nanoparticles, since it is absent for ‘pure’ E7 [56]. For times scale $\tau <{\tau }_B=\tau \left(T_B\right)$ a systematic drop of the exponent $F$, until reaching $F\approx 1$ for some part of the low-temperature dynamic domain $\tau >{\tau }_B$. Near $T_g$ the decoupling appears again, but in this case associated with $F>1$.

4. Discussion

4.1. I-N Transition: Critical Behavior and Glassy Dynamics

4.1.1. I-N Transition: Critical Behavior

I-N transition is included in the canon of Critical Phenomena Physics, as the model case of a weakly discontinuous transition inherently associated with long-range pretransitional effects driven by multimolecular fluctuations [68,69]. It shows the symmetry of the neighboring phases, i.e., they are pre-nematic in the isotropic liquid and pre-isotropic in the nematic phase. This is particularly evident in heat capacity $C_P\left(T\right)$ studies or in the order parameter $S\left(T\right)$ tests, in the nematic phase. The most classic are pretransitional effects related to the Cotton-Mouton Effect, Rayleigh Light Scattering ($R_{LS}$), or Kerr Effect (KE), for which the exceptional ‘common pretransitional pattern inspired the Landau-de Gennes (LdG) model development - one of the most important phenomenological concepts in Liquid Crystals and Soft Matter Physics, namely [67,68]:

$$KE,R_{Ls},CME\propto {\displaystyle \frac{1}{T-T^{\ast }}}$$

(21)

To derive such behavior, De Gennes developed Landau’s expansion of the free energy with respect to the local order parameter to also $C_P\left(T\right)$ or $S\left(T\right)$ behavior [67,68].

KE and CME describe optical birefringence due to the strong electric or magnetic fields impacts, namely $\Delta n/E^2$ and $\Delta n/H^2$, respectively. The above methods directly detect pretransitional/critical-like fluctuation. They employ light, so the observation time-scale: $t_{obs.}=1/f_{ligth}\ll {\tau }_{fluct.}$ Later, the same pattern was noted for Nonlinear Dielectric Effect (NDE), detecting changes of dielectric constant under the strong electric filed ${\Delta \epsilon }^E/E^2$. However, for NDE: $t_{obs.}>{\tau }_{fluct.}$ [67]. Since it employs radio-frequency detection in the kHz-MHz range. The inclusion of NDE in research has led to finding a set of discrepancies between experimental results and the Landau - de Gennes model. This puzzle was only recently solved by the new model proposed by Drozd-Rzoska [67]. It also enabled, for the first time, a model-based derivation of Eq. (6) describing pretransitional changes of the dielectric constant.

In ref. [67], special attention was paid to the 'Contrast Factor’ $\left(CF\right)$ in studies of pretransitional effects. Namely, fluctuations-'heterogeneities' should differ significantly from their local environment with respect to the given research method. An example can be the absence of a pretransition anomaly for the dielectric constant in the isotropic phase of nematogens with a permanent dipole moment perpendicular to the long axis of the molecule, where the lack of $\epsilon \left(T\right)$ pretransitional anomaly is related to $CF \approx 0$ [79]. For LC molecules with the permanent dipole moment parallel to the long molecular axis $CF\gg 0$, which leads to a significant pretransitional effect, as shown in the Results section.

The anomaly is even more pronounced for ${\epsilon }_{peak}^{″}\left(T\right)$, since the energy loss required for the reorientation is essentially different withu fluctuations in comparison to their surrounding.

Notable, that NDE defined as ${\Delta \epsilon }^E/E^2=\left(\epsilon -\epsilon \left(E\right)\right)/E^2$ detects directly pretransitional fluctuations, i.e. ‘fluctuations’/ ‘heterogeneities in ‘different’ surrounding [42,67].

Dielectric constant changes $\epsilon \left(T\right)$ detects pretransitional changes ‘indirectly’, i.e., via the impact on the volume they occupy in the sample.

5CB is one of E7 mixture components. In ref. [63] it has been shown that the impact of prenematic fluctuations in the isotropic liquid phase tested via the range of $\epsilon \left(T\right)$ critical anomaly (Eq. (6)) can extend even up to $~T_{I-N}+100K$ in the isotropic liquid phase, for phase $T>T_{I-N}$. In the nematic phase, for $T<T_{I-N}$, the critical-like behavior extends down to $~T_{I-N}-80K$, as shown in Figure 3 by the critical-like portrayal via Eqs. (10) and (11). This was possible only due to the unique endogenic ordering of LC molecules induced by C₆₀ fullerene nanoparticles in the nematic phase.

4.1.2. I-N Transition: Glassy Dynamics

A quarter of a century ago, Letz et al. [80] and Theenhas et al. [81] modeled the hard ellipsoid fluid using a mean-field mode-coupling approach. They showed that this is a unique model ‘glassy’ system, exhibiting uniaxial fluctuations/heterogeneities and two basic relaxation processes. The first one relates to collective processes ('fluctuations') and the four-point (4) correlation function. The second one concerns single-particle processes and the two-point (2) correlation function. Both are described by the same functional dependence, characteristic of the MCT approach in the previtreous domain [80,81]:

$${\tau }_{\mathrm{4,2}}\propto {\left(T-T_{MCT}^{\mathrm{4,2}}\right)}^{-\varphi }$$

(22)

where $T_{MCT}^4$ and $T_{MCT}^2$ are extrapolated singular MCT temperatures, coupled to ${\tau }_4\left(T\right)$ and ${\tau }_2\left(T\right)$, respectively.

However, the collective relaxation time (4-point correlation function) singularity $T_{MCT}^4$ is located just below the solidification temperature $T_S$ and single element related $T_{MCT}^2$ ca. 30-40 K below $T_S$ . Such scenario behavior for dynamics fairly coincides with the behavior observed in the isotropic liquid phase of nematogens for NDE detected relaxation time and the primary relaxation time (alpha) from BDS tests, as can be noted from the given report and refs. [77,82,83,84]. Hence, $T_{MCT}^4$ correlates with $T^{\ast }$ and then ${\tau }_{\mathrm{4,2}}{\to \tau }_{fluct.}$. for two-point related relaxation ${\tau }_2\left(T\right)\to \left({\tau }_{\alpha },\tau \right)$ and $T_{MCT}^2\to T_B$ [77,82,83,84]. Indeed, such behavior was validated by Drozd-Rzoska et al. [42,63], in the isotropic phase of numerous LC materials, with a clear prevalence over the often used Vogel-Fulcher-Tammann (VFT) equation. Notably, that the Critical & Activated Eq. (14), is fairly approximated the MCT critical-like equation ${\tau }_2\left(T\right)\propto {\left(T-T_{MCT}^2\right)}^{-\varphi }$, well above the singular temperature [15,75].

The above behaviour for ${\tau }_2\left(T\right)$ (i.e., ${\tau }_{\alpha }\left(T\right)$) and ${\tau }_4\left(T\right)$ (i.e., ${\tau }_{fluct.}\left(T\right)$) was broadly confirmed in BDS and $NDE\left(f\right)$ spectroscopic studies in the isotropic phase of nematogens [61,62,63,67,70,75,76,77]. Notable can be also Transient Grating Optical Kerr Effect (TG OKE) studies, with the extreme time-scale insight that explicit yield evidence for ${\tau }_2\left(T\right)$ and ${\tau }_4\left(T\right)$ is possible using single scans for subsequent temperatures [82,83,84].

4.2. Glass Transition: Glassy Dynamics & the Critical Behavior

It is generally believed that the glass transition is a specific 'dynamic' phenomenon, where universalistic or singular features are restricted to dynamic properties, as indicated in the Introduction section. In glass-forming systems, such behavior is most often scaled by the VFT equation, but as evidenced in ref. [15] it has only an effective meaning. The optimal description for different types of glass-formers offers Eq. (14), and for systems with the dominant uniaxial molecular symmetry also the explicit critical-like portrayal [15,75,76]:

$${\tau }_2\left(T\right)\propto {\left(T-T_g^{\ast }\right)}^{-\phi }\mathrm{and} {\sigma }_{DC}\left(T\right)\propto {\left(T-T_g^{\ast }\right)}^{-{\phi }_1}$$

(23)

Among the dominant universalistic dynamic properties in the immediate previtreous area is particularly significant is the fractional translational-orientational decoupling, particularly for time scales $t>t_B~{10}^{-7\pm 1}$s (low temperature dynamic domain). Generally, it is linked to constant fractional exponent in the subsequent dynamic domain, and the near-smooth crossover: $F\approx 1\to F<1$ (see Eqs. (19 and (20). For E7+C₆₀ fullerene nanocolloids, a more complex picture appears, with increasing decoupling (exponent $F$) in the immediate pretransitional/previtreous region - as shown in the inset of Fig. 10

Two decades ago, Nielsen et al. [85] empirically pointed out that the parameter describing the high frequency distribution of low-molecular weight glass distribution of the primary relaxation time can be universal: $n\left(T{\to T}_g\right)\to ~1/2$. This report confirms this finding, and also shows explicit critical-type behavior (Eq. (12)) for $n\left(T\right)$ extended even up to $60K$ above $T_g$. It is worth mentioning that such a critical characteristic of $n\left(T\right)$ was also observed in the isotropic liquid phase of 8OCB, which is one of E7 components.

For reaching a breakthrough inspirations regarding the nature of the glass transition reliable scaling relations describing previtreous/pretransitional changes are particularly important. For such insight methods directly detect multimilecular heterogeneities are essential

This could TG OKE, time - frequency resolved NDE or time resolved EKE but they are essentially tested only in the isotropic phase of nematogens, eventually considered a model case for the glass transition. This results from the ratio between single molecules related and collective relaxation times, namely: ${\tau }_4/{\tau }_{42} ~{\tau }_{fluct.}/{\tau }_{\alpha }={10}^3-{10}^4$ [61,62,63,82,83,84]. For the I-N transition $\tau ={\tau }_{\alpha }={10}^{-9}-{10}^{-8}$ [61,62,63,82,83,84]] and then the detection by mentioned heterogeneities/fluctuations coupled methods is possible in a wide range of temperatures. For the glass transition ${\tau }_{\alpha }\left(T_g\right)\approx 100s$, and then ${\tau }_{fluct.}\left(T_g\right)~{10}^6$s – which makes experimental tests impossible in practice. Nevertheless, the ‘operational window’ for such tests can open slightly remote ${\tau }_{\alpha }\left(T_g\right)$, as shown in time-resolved NDE studies [86,87]. However, direct studies of multimolecular heterogeneities are still primarily tests for detecting their presence [88,89], without attempts, or even the possibility, to determine temperature scaling.

In such a context, the pretransitional changes shown in this work and ref. [56], for $\epsilon \left(T\right)$ and ${\epsilon }_{peak}^{″}\left(T\right)$ can be particularly significant as they reduce the problem related to the time scale of collective process. Notably, such evidence was also recently reported for standard glass-forming molecular liquids, particularly composed of molecules with dominant uniaxial symmetry [90]. Of particular importance here is the discovery of critical-like scaling of the relaxation time distribution parameter. To complete this picture, one can also recall recent evidence of critical-like scaling in the configurational entropy or the related contribution to the heat capacity for standard glass formers [91].

5. Conclusions

This work presents the results of studies on the nematogenic LC E7 mixture + C₆₀ fullerene nanoparticle nanocolloids, in bulk samples and using BDS scans to obtain insights beyond the standard patterns reported so far. This led to the discovery of several new properties that may have implications for both fundamental modeling and applications. Both of these issues are addressed by demonstrating that even small amounts of C₆₀ fullerene nanoparticles can induce a stable endogenous arrangement of rhodium-like molecules, a phenomenon previously achieved in bulk samples using strong external fields (magnetic or electric).

Another significant result of this work may have significant implications for glass transition physics. First, it is one of the few studies to date demonstrating the endogenous influence of nanoparticles on the previtreous properties of glass-forming systems. Second, it demonstrates the coexistence of coherent dual glassy and critical-like features in both the I-N transition and the glass transitions—potentially important experimental evidence in the still-unresolved debate on the nature of the glass transition. In this context, it is also worth recalling the authors' recent work on E7 + BaTiO₃ nanoparticle nanocolloids.

The authors also want to draw attention to new, previously unreported critical-like changes upon approaching the glass temperature, as well as to a number of similarities emerging for pretransition effects near the I-N transition and previtreous changes.

This report and ref. [56] indicate a significant and even decisive influence of nanoparticles on the properties of the supercooling E7 nematogenic LC matrix. The type and size of the nanoparticles are important here, as are the significant influences of multimolecular pre-transition fluctuations, whose size for the I-N transition changes from the microscale in the immediate vicinity of the phase transition to the nanoscale with distance.

The results of this work and ref. [56] also indicate the significant role of nanoparticle-induced 'frustration' on previtreous properties, which may have significant implications for further research into this great cognitive challenge. This is a research direction still underexplored. However, the influence of nanoparticles is particularly pronounced in some systems, such as LC materials, where strong interactions occur.