Crystallization Kinetics of an Equimolar Liquid Crystalline Mixture and Its Components

Abstract

This new equimolar mixture comprises the liquid crystalline compounds MHPOBC and partially fluorinated 3F2HPhF6. The phase sequence of the mixture was determined by differential scanning calorimetry, polarizing optical microscopy, X-ray diffraction, and broadband dielectric spectroscopy. The enantiotropic smectic A*, C*, and C_A* phases were observed for the mixture. Only partial crystallization of the mixture was observed during cooling at 2–40 K/min, and the remaining smectic C_A* phase underwent vitrification. In contrast, the crystallization of the pure components was complete or almost complete for the same range of cooling rates. The kinetics of the non-isothermal and isothermal crystallization of the mixture and its pure components were investigated by differential scanning calorimetry. The non-isothermal data were analyzed by the isoconversional method, while the isothermal data were analyzed using the Avrami model. As is typical, the nucleation-controlled crystallization kinetics were observed.

1. Introduction

Smectic liquid crystals (Sm LCs), possessing a lamellar order, can show switching in the electric field if particular conditions are fulfilled: the molecules are chiral, the tilted smectic phase is present (SmC* or its sub-phases), the component of the molecular dipole moment is non-zero in a direction perpendicular to the plane of tilt, and the helical twist of the molecular tilt along the layer normal is removed [1,2]. Smectogenic compounds with such properties can be used in LC displays [3]. To control the temperature range of the tilted smectic phase, which defines the operating conditions of the display, numerous mixtures have been investigated, e.g., as described in [4,5,6,7,8,9,10,11,12].

The sequence of chiral smectic phases upon cooling is as follows: SmA* → SmC_α* → SmC* → SmC_FI2* → SmC_FI1* → SmC_A*, although particular LC compounds do not always exhibit all the smectic phases among these [13,14,15]. In the SmA* phase, the average tilt angle of the molecules equals zero, while in SmC* and its variations, the average tilt angle is larger than zero. The SmC_α* phase is characterized by a helical order with a very short pitch, equal to a few smectic layer spacings. The SmC* phase has a synclinic order of the molecular tilt, and shows ferroelectric properties—bistable switching in the electric field—in the bookshelf surface-stabilized geometry. The SmC_FI2* and SmC_FI1* phases are characterized by a twist of the tilt angle with a period of four and three smectic layers, respectively. Finally, the SmC_A* phase has an anticlinic order of the molecular tilt in the neighboring layers, and shows antiferroelectric properties—tristable switching in the electric field—in the bookshelf surface-stabilized geometry [2,15,16].

For practical purposes, the LC should not crystallize in the temperature range within which the display or another LC-based device operates. The formulation of mixtures enables lowering of the melting point compared to the pure compounds [4,11,12]. It is also beneficial to use glass-forming compounds/mixtures, which can stay in the metastable LC state, below the melting point, for a long time. The crystallization process consists of two stages: nucleation followed by crystal growth. If the temperature range optimal for nucleation is lower than the temperature range for crystal growth, and these ranges do not overlap or overlap poorly, then the crystallization upon cooling (melt crystallization) is hindered. In such case, the formation of the crystal phase occurs upon heating (cold crystallization) if the sample is cooled down previously to the temperature range where the stable nuclei are formed [17,18,19]. LC glasses with helical order are considered as materials for optical filters [20,21]. Materials which show glass transition and cold crystallization are also proposed for application in energy storage [22,23]. The most convenient materials are those which undergo glass transition at moderate cooling rates of ~20 K/min or lower [19].

The 4-[(1-methylheptyloxy)carbonyl]phenyl 4′-octyloxy-4-biphenylcarboxylate compound, abbreviated as MHPOBC (Figure 1), was the first LC in which tristable switching in the electric field was detected, indicating the existence of the antiferroelectric smectic C_A* phase (SmC_A*) [16]. MHPOBC has been tested as a component of mixtures with other LCs [5,6,7,8] and as a nanoparticle matrix in composites [24,25,26,27,28]. The phase sequence upon cooling given in [29] is Iso (421 K) SmA* (395 K) SmC_α* (394.1 K) SmC* (392.4 K) SmC_γ* (391.6 K) SmC_A* (337.7 K) SmX_A* (304 K) Cr, where Iso is the isotropic liquid, Sm denotes various smectic phases, and Cr is the crystal phase. In the SmA* phase, the average tilt angle of the molecules is zero, and this phase has paraelectric properties. The next three phases, present in a narrow temperature range, are interpreted in various ways in the literature. Resonant X-ray diffraction study of (S)-MHPOBC films, with a small admixture (up to 5% molar percentage) of the selenophene PB237, leads to the identification of these phases as ferroelectric SmC*, four-layer SmC_FI2*, and three-layer SmC_FI1*, in order of decreasing temperature [14]. However, it is underlined in [14] that the PB237 dopant, necessary for the resonant X-ray experiment, might alter the structure of the smectic phases. The presence of optical impurities also changes the phase sequence. According to [13], in pure (S)-MHPOBC, these are SmC_α*, SmC_FI2*, and SmC_FI1*. Meanwhile, for (S)-MHPOBC with a small admixture of (R)-MHPOBC, the SmC_FI2* phase is replaced by SmC*, and SmC_α* is interpreted as a phase with a short helical pitch, both for optically pure (S)-MHPOBC, and (S)-MHPOBC with a (R)-MHPOBC admixture up to 12% [13]. The sequence SmC_α*, SmC*, SmC_FI1* is assumed in a recent paper [30]. The SmC_A* phase has a wide temperature range. After supercooling, it transitions to the monotropic SmX_A* phase by the appearance of a hexatic bond-orientational order within the layers [31]. The SmX_A* phase is either SmI_A* or SmF_A* (in [29] denoted as SmI_A*), which are difficult to distinguish, as the only difference in their structures is the direction of the molecular tilt in respect to the local hexagonal order within the layers. This usually cannot be inferred from powder X-ray diffraction (XRD) patterns [31]. We did not find any XRD study of the oriented MHPOBC sample in the literature, which would distinguish SmI_A* and SmF_A*; therefore, we prefer to use the SmX_A* notation for the hexatic phase.

The second investigated compound, (S)-4′-(1-methylheptyloxycarbonyl)biphenyl-4-yl 4-[2-(2,2,3,3,4,4,4-heptafluorobutoxy)etyl-1-oxy]-2-fluorobenzoate, is abbreviated as 3F2HPhF6 (Figure 1), and exhibits only two smectic phases, with a sequence Iso (377.2 K) SmC* (364.6 K) SmC_A* (307 K) Cr1 (299.5 K) Cr2 on cooling [32]. The SmC_A* phase of 3F2HPhF6 is orthoconic, meaning that the molecules’ tilt angle reaches almost 45° [32]. This property improves the quality of the dark state in an LC display [33,34]. The results for one LC mixture containing 3F2HPhF6 have been published [4]. The polymorphism of the MHPOBC/3F2HPhF6 system has not been investigated yet.

The paper aims to investigate the crystallization process of MHPOBC, 3F2HPhF6, and their equimolar mixture by differential scanning calorimetry (DSC), for various cooling rates and in isothermal conditions at different temperatures. This is part of the broader study of crystallization kinetics in antiferroelectric smectic liquid crystals, which has been focused on pure compounds up until now [35,36,37,38]. The sequence of the smectic phases in the mixture was also determined by DSC and polarizing optical microscopy (POM). X-ray diffraction (XRD) was used to measure the smectic layer spacing in the mixture, as a function of temperature in comparison to the values for the pure components. Broadband dielectric spectroscopy (BDS) facilitated the identification of the smectic phases based on the relaxation processes observed in the mixture. The main goal of the study was to test (1) the broadening of the stability range of the SmC_A* phase, (2) the slowing down of the melt crystallization, and (3) the glass-forming properties for moderate cooling rates (2–40 K/min) in the mixture, compared to its components.

2. Materials and Methods

The MHPOBC [13] and 3F2HPhF6 [4] compounds, both S-enantiomers, were synthesized at the Institute of Chemistry, Military University of Technology, in Warsaw. The components were mixed in the molar fractions 0.5002(7): 0.4998(6), respectively. The mixture was prepared by the dissolution of both components in acetone at 313 K, and the evaporation of the solvent. Then, the precipitate was heated to 433 K, to ensure proper mixing in the isotropic liquid phase. The obtained mixture is referred to as MIX2HF6. An earlier test with ethanol as a solvent was performed. However, the components showed lower solubility in ethanol, especially MHPOBC, and acetone proved to be a better solvent for the mixture preparation.

The DSC measurements were performed for the 2.180 mg, 2.260 mg, and 6.000 mg samples of MHPOBC, 3F2HPhF6, and MIX2HF6, respectively, in aluminum pans, for cooling/heating rates in the 2–40 K/min range, using a DSC 2500 TA Instruments (New Castle, DE, USA) calorimeter. Additionally, a modulated-temperature DSC (MTDSC) measurement for MIX2HF6 was performed during heating at 3 K/min and a modulation of 1 K per 60 s, after quenching the sample to 153 K. The DSC thermograms were analyzed using the TRIOS and OriginPro programs.

The samples for POM observations were prepared as a film between two glass slides without aligning layers. The textures were collected using the Leica (Wetzlar, Germany) DM2700 P polarizing microscope with the Linkam (Waterfield, UK) temperature stage. The phase transition temperatures were determined during cooling and heating at 5 K/min. For MIX2HF6, the additional POM observations for the 30 K/min rate were performed in a broader temperature range. The data analysis was performed using the TOApy program [39].

X-ray diffraction measurements with CuKα radiation ($\lambda$ = 1.540562 Å [40]) were performed in Bragg–Brentano geometry for flat samples, inserted into a holder with an area of 13 mm × 10 mm and a depth of 0.2 mm. The diffraction patterns were registered upon cooling from the isotropic liquid, using an X’Pert PRO PANalytical (Malvern, UK) diffractometer with a TTK-450 Anton Paar (Graz, Austria) temperature chamber. The fluorophlogopite mica Mg₃K[AlF₂O(SiO₃)₃] [41,42], which is the NIST Standard Reference Material 675 (SRM 675) for low 2θ angles [43], purchased from Merck (Darmstadt, Germany), was used for the calibration of the diffraction peak positions. The data analysis was performed using the FullProf package [44] and OriginPro.

BDS measurements were conducted for samples with a thickness of 80 μm, placed between two gold electrodes, with polytetrafluoroethylene spacers preventing shortcut. The dielectric spectra were collected using a Novocontrol (Frankfurt, Germany) impedance spectrometer for frequencies between 0.1 Hz and 10 MHz, upon slow cooling for all samples, and upon heating after slow and fast cooling for MIX2HF6. The data analysis was performed using OriginPro.

3. Results and Discussion

3.1. Phase Sequence

The DSC results for MHPOBC, 3F2HPhF6, and MIX2HF6 are presented in Figure 2, Figure 3, and Figure 4, respectively. The POM results are shown in Figures S1–S4 in the Supplementary Materials. The DSC curves of MHPOBC collected during cooling (Figure 2a,b) reveal the phase sequence Iso (418.2 K) SmA* (388.3 K) SmC* (386.6 K) SmC_A* (337.3 K) SmX_A* (305.3 K) Cr3, where the phase transition temperatures are extrapolated to zero cooling rate. For the cooling rates of 25–40 K/min, the melt crystallization is completed during subsequent heating (Figure 2c,d), and the onset temperature of the cold crystallization is 290–293 K. Upon further heating, another crystal phase or phases appear. For the 6–40 K/min heating rates, there is a Cr3 → Cr2 transition at 319–329 K, which is followed by a Cr2 → Cr1 transition at 336–338 K. For the 2–5 K/min heating rates, there is a direct Cr3 → Cr1 transition at 304–305 K. The phase transition temperatures of MHPOBC, extrapolated to zero heating rate, are Cr3 (317.1 K) Cr1 (351.6 K) SmC_A* (385.0 K) SmC* (389.3 K) SmA* (416.6 K) Iso. All the transition temperatures are the onset temperatures $T_o$ of anomalies in the DSC thermograms [45], except from the SmC* → SmA* transition, for which the peak temperature $T_p$ was used. Under the polarizing microscope, the MHPOBC sample (Figure S1) shows mainly a fan-shaped texture in SmA* and a broken fan-shaped texture in SmC* and SmC_A* [30,31]. In some parts, a homeotropic texture is visible [31]. The areas oriented homeotropically are usually dark, except for the 340–360 K region, where an increase in luminance is observed, with a peak at 349 K. It is explained by a helix inversion, which occurs in this temperature range [46]. SmC* or its sub-phases are not visible in the plot of luminance vs. temperature; the texture collected in 394 K is attributed to them based on visual inspection. In the SmX_A* phase, the texture is still a broken fan-shaped type, but looks closer to that of the SmA* phase. The decrease in luminance below 305 K indicates crystallization. During heating, the transition between the crystal phases Cr2 and Cr1 shows a broad valley in their luminance in the 317–342 K range. The Cr1 phase melts at 358 K.

The phase transitions of 3F2HPhF6 observed in the DSC thermograms during cooling (Figure 3a,b), extrapolated to zero cooling rate, are as follows: Iso (377.6 K) SmC* (365.1 K) SmC_A* (311.2 K) Cr1. At 2–5 K/min, the sample crystallizes in the phase denoted as Cr1. At 6–10 K/min, a mixture of two crystal phases, Cr1 and Cr2, is formed. At 15–40 K/min, the crystallization to the Cr2 phase is observed. Upon subsequent heating (Figure 3c,d), the cold crystallization of Cr2 from the remaining SmC_A* phase occurs at 30–40 K/min, with an onset temperature of 291–292 K. A Cr2 → Cr1 transition is observed at 320–321 K, upon heating at 8–40 K/min (for 6 K/min, this anomaly was not visible, due to the small amount of Cr2 formed during cooling). The phase transition temperatures extrapolated to zero heating rate are as follows: Cr1 (325.9 K) SmC_A* (369.2 K) SmC* (376.9 K) Iso. Under the polarizing microscope, 3F2HPhF6 shows broken fan-shaped textures in the SmC* and SmC_A* phases (Figure S2). The SmC* → SmC_A* transition has a weak impact on luminance during cooling and a significant impact during heating. A SmC_A* → Cr1 transition occurs gradually below 310 K with visible growing crystallites, and melting of Cr1 during heating is observed at 328 K.

The phase sequence of MIX2HF6, determined from DSC thermograms (Figure 4a,b) at the limit of zero cooling rate, is Iso (395.9 K) SmA* (386.9 K) SmC* (373.6 K) SmC_A* (286.7 K) Cr3. The crystallization is only partial, and a step-like anomaly with an inflection at $T_g$ = 241.5 K is observed, indicating the glass transition of the remaining SmC_A* phase. Glass softening upon heating occurs at 244.9 K, and cold crystallization follows (Figure 4c,d). A few endothermic anomalies related to the melting of the crystal phases are visible upon further heating. Melting of the Cr3 phase is observed for the 15–40 K/min heating rates at 286–288 K, and melting of the Cr2 phase is observed for the 2–25 K/min heating rates at 292–296 K. For all the heating rates, the Cr1 phase is observed, which melts at 298–307 K. Differences between the melting temperatures indicate that, actually, a mixture of Cr1, Cr2 and Cr3 with various mass ratios is formed, depending on the cooling/heating rate. Additionally, minor anomalies originating from the melting of the crystal phases denoted as Cr′ and Cr″ are visible for the 2, 3 K/min heating rates, with onset temperatures of 325–326 K and 328–329 K, respectively. The phase sequence in the limit of zero heating rate is as follows: Cr2 (295.2 K) Cr1 (308.0 K) [Cr′ (~325 K) Cr″ (~328 K)] SmC_A* (375.6 K) SmC* (386.2 K) SmA* (395.9 K) Iso. The Cr3 → Cr2 transition is not visible in the DSC results. This transition possibly occurs simultaneously with cold crystallization, or melt crystallization during slower cooling may already include the formation of Cr2.

The POM textures of MIX2HF6 are partially homeotropic and partially strongly defective. For the textures collected during cooling and heating at 5 K/min (Figure S3), the numerical analysis was performed not only for the whole textures, but also for the homeotropically oriented fragment (indicated for 273 K in Figure S3a), to better visualize the helix inversion and beginning of crystallization. The Cr3 → Cr1 (or Cr2 → Cr1) transition is not visible during heating. The crystallization and glass transition during fast cooling at 30 K/min (Figure S4) show a continuous increase and decrease in luminance, respectively. The glass softening and cold crystallization during heating do not show clearly in the luminance vs. temperature plot. The Cr3 → Cr1 and Cr1 → SmC_A* transitions are visible, which is in accordance with the DSC results for the 30 K/min rate. However, an additional transition occurs at 330 K, attributed to the melting of the Cr1″ phase, observed in the DSC thermograms only for the 2–3 K/min rates. The irregular presence of the Cr1′ and Cr1″ phases may be related to local deviations from the equimolar ratio of components in the solid state, resulting in small fractions of crystal phases with elevated melting temperatures. The texture in the SmC_A* phase after melting of the crystal phase is mainly strongly defected and fan-shaped, without homeotropic fragments. Therefore, helix inversion is not visible during heating. The polymorphism of the smectic phases is summarized in

Table 1. Polymorphism of the smectic phases of MIX2HF6 and its components. The presence and absence of a smectic phase are indicated by ● and -, respectively. The first row (normal font) contains the transition temperatures, in K, obtained by POM (5 K/min), the second row (bold) contains the transition temperatures, in K, obtained by DSC (extrapolated to 0 K/min), and the third row (italics) contains the enthalpy changes, in kJ/mol, determined by DSC.

Sample	SmXA*		SmCA*		SmC*		SmA*		Iso
cooling
MHPOBC	●	339 337.3 1.5	●	394 a 386.6 >0.1	●	394 a 388.3 >0.1	●	424 418.2 6.0	●
3F2HPhF6	-		●	366 365.1 >0.1	●		-	380 377.6 6.8	●
MIX2HF6	-		●	377 373.6 >0.1	●	388 386.9 1.1	●	394 395.9 4.6	●
heating
MHPOBC	-		●	394 a 385.0 >0.1	●	394 a 389.3 >0.1	●	424 416.6 5.9	●
3F2HPhF6	-		●	371 369.2 >0.1	●		-	380 376.9 6.8	●
MIX2HF6	-		●	377 375.6 >0.1	●	388 386.2 0.9	●	394 395.9 4.8	●

^a only the textures collected at 394 K during cooling and heating were attributed to SmC*.

. The effect of the sample contained is visible, as the phase transition temperatures determined by POM for a thin sample are shifted to higher temperatures compared to the DSC results. The identification of the smectic phases agrees with the XRD patterns (Section 3.2), and is confirmed by the dielectric spectra (Section 3.3), although for MHPOBC, the BDS method suggests the presence of two other smectic phases besides SmC* in a narrow temperature range between SmA* and SmC_A*.

3.2. Structure of Smectic Phases

The XRD patterns of the smectic phases (Figure 5a) are characterized by sharp diffraction peak or peaks located at low 2θ angles. The peak position is related to the smectic layer spacing by the Bragg equation [31]:

$$n\lambda =2d\sin \theta ,$$

(1)

where $n$ is the order of the peak ($n$ = 1, 2, 3…—the number of visible peaks of the order $n$ > 1 varies between the compounds and smectic phases), $\lambda$ = 1.540562 Å is the wavelength characteristic of CuKα₁ radiation [40], $d$ is the layer spacing, and $\theta$ is the peak position corrected for the systematic shift (in this study, results for the SRM 675 reference material were used for this correction). The XRD patterns at higher 2θ angles depend on the intra-layer order in the smectic phase [31]. For smectics A and C, the intra-layer positional order is short-range, described by the average intermolecular distance $w$ and correlation length $\xi$. It corresponds to the wide maximum in the XRD patterns located around 2θ ≈ 18–20°, defined by the Lorentz peak function [47]:

$$I\left(q\right)\propto {\displaystyle \frac{1}{1+{\xi }^2{\left(q-q_0\right)}^2}},$$

(2)

where $q=4\pi /\sin \theta$ is the scattering vector and $q_0=2\pi /w$ is the peak position. Upon transition to the hexatic phase, the diffuse maximum becomes narrower and has a sharper peak [31]. For MIX2HF6, sharpening of the peak at 2θ ≈ 19° is observed at 298 K (full-width at half-height equal to 1.0°, compared to 4.9° at 303 K). However, it is interpreted as the beginning of crystallization, because two other sharp peaks at 2θ ≈ 16° and 23° appear also at this temperature.

The smectic layer spacing in MIX2HF6 is between the results for the pure compounds (Figure 5b). The layer shrinkage at the SmA* → SmC* transition equals 3.4(3)%, and the overall layer shrinkage down to 298 K equals 11.2(2)%. For comparison, the overall layer shrinkage in the smectic phases of MHPOBC and 3F2HPhF6 equals 5.2(3)% and 0.5(2)%, respectively. The wide maximum at higher angles was fitted using Equation (2), with the linear background added. The correlation length in MIX2HF6 reaches almost 6 Å at 303 K, before crystallization at 298 K (Figure 5c). The maximal $\xi$ value obtained for 3F2HPhF6 was about 5 Å at 323 K [32], and for MHPOBC, it was ca. 7 Å at 348 K [25]. Thus, the values for the mixture are intermediate between these for the pure components. The distance $w$ equals 4.6–4.8 Å, as usually obtained for liquid crystals [32]. The $\xi /w$ ratio does not exceed 2. Therefore, the local positional order in MIX2HF6 includes only the next neighbors.

3.3. Dielectric Relaxation Processes

The dielectric spectra show the complex dielectric permittivity ${\epsilon }^{*}$ as a function of the frequency $f$ of the external electric field. The models used for fitting the experimental ${\epsilon }^{*}\left(f\right)$ dependences in this study are the Debye [48], Cole–Cole [49], and Havriliak–Negami [50] models, together with the contribution of ionic conductivity [48]:

$${\epsilon }^{*}\left(f\right)={\epsilon }^{\prime }\left(f\right)-i{\epsilon }^{″}\left(f\right)={\epsilon }_{\infty }+\sum _j{\displaystyle \frac{\Delta {\epsilon }_j}{{\left(1+{\left(2\pi if{\tau }_j\right)}^{1-a_j}\right)}^{b_j}}}-{\displaystyle \frac{iS}{{\left(2\pi f\right)}^{n_S}}},$$

(3)

where ${\epsilon }^{\prime }$ and ${\epsilon }^{″}$ are the real and complex components of ${\epsilon }^{*}$ (corresponding to dielectric dispersion and absorption, respectively), ${\epsilon }_{\infty }$ is the ${\epsilon }^{\prime }$ value in the limit of high frequencies, $\Delta \epsilon$ is the dielectric increment of each process, $\tau$ is the relaxation time of each process, and $S$ and $n_S$ are parameters describing the background in ${\epsilon }^{″}$ at low frequencies. If $n_S$ = 1, then $S=\sigma /{\epsilon }_0$, where $\sigma$ is the ionic conductivity and ${\epsilon }_0$ is the vacuum permittivity. The parameters $a$ and $b$ describe the shape of the step in ${\epsilon }^{\prime }$ and peak in ${\epsilon }^{″}$ originating from each relaxation process. For the Debye model, $a=0$ and $b=1$. For the Cole–Cole model, $a\in \left(0,1\right)$ and $b=1$. For the Havriliak–Negami model, $a\in \left(0,1\right)$ and $b\in \left(0,1\right)$.

The presence of some relaxation processes facilitates the identification of the smectic phases. The soft mode (SM) is related to collective fluctuations in the tilt angle value, and is observed in the SmA* and SmC* phases. Upon cooling, the $\tau$ and $\Delta \epsilon$ of the SM increase in SmA* and decrease in SmC*, showing the characteristic V-dependence on temperature [51]. The Goldstone mode (GM) is related to collective fluctuations in the tilt azimuth around the tilt cone, and is observed in the SmC* phase. It shows a much larger $\Delta \epsilon$ than the SM [51]. In the SmC_A* phase, there are two GM-like phasons, P_L and P_H, related to collective fluctuations in the tilt azimuth, which are, respectively, in-phase and anti-phase in the neighboring smectic layers. P_L has a higher $\tau$ than P_H, and it can overlap with the molecular s-process (rotations around the short molecular axis) [52,53].

The representative fits of Equation (3) to the experimental BDS spectra of MIX2HF6 are shown in Figure 6. The $\Delta \epsilon$, $\tau$, and $\sigma$ values, determined as a function of temperature, are presented in Figure 7, Figure 8 and Figure 9 for MHPOBC, 3F2HPhF6, and MIX2HF6, respectively. The SM with $\tau$ and $\Delta \epsilon$ increasing upon cooling confirms the SmA* phase in MHPOBC and MIX2HF6. The GM with high $\Delta \epsilon$ values confirms the SmC* phase and the pair of weaker phasons P_L and P_H confirm the SmC_A* phase in MIX2HF6 and both components. The GM is also observed in the SmC_FI1* phase [53], which can be present for MHPOBC. Thus, the strongest value of the GM’s $\Delta \epsilon$ at 391 K is attributed to SmC*, while the lowered $\Delta \epsilon$ values at 389–390 K can be attributed to SmC_FI1*. From the high-temperature side, the relatively strong SM may indicate the SmC_α* phase [53,54]. The final phase sequence in MHPOBC would be SmA*, SmC_α*, SmC*, SmC_FI1*, and SmC_A*. On the other hand, the DSC thermograms show only one phase between SmA* and SmC_A*. We refer to this phase as SmC*, because this phase is the most probable based on the BDS results. The SmC_A* → SmX_A* transition in the supercooled MHPOBC results in an increase and decrease in the $\Delta \epsilon$ of P_L and P_H, respectively (Figure 7a), and an increase in the slope of the activation plot of $\tau$ of both phasons (Figure 7b). For some compounds, an additional P_hex process arises at the low-frequency side of P_H in the SmX_A* phase [36,55]. However, this was not visible for MHPOBC. Compared to MIX2HF6 and 3F2HPhF6, which show comparable relaxation times, the GM process is much slower and the P_L process is much faster for MHPOBC. The P_L process, probably due to overlapping with the molecular s-process, shows a linear dependence in the activation plot. The activation energy $E_a$, determined by fitting the Arrhenius formula, is ca. 110 kJ/mol for all the samples. For the P_H process, the difference in the relaxation times between the mixture and its components is insignificant. The ionic conductivity is the highest for MIX2HF6, slightly smaller for 3F2HPhF6, and the lowest for MHPOBC.

The MIX2HF6 can be supercooled enough to investigate the next relaxation process on the high-frequency side of the P_H process (Figure 6). It can be interpreted as the molecular l-process (rotations around the long molecular axis) [56]. However, for the glass-forming materials, this process becomes the collective α-process upon approaching the glass transition temperature $T_g$ [57]. The α-relaxation time usually shows a nonlinear dependence in the activation plot, described by the Vogel–Fulcher–Tammann formula [48,57]:

$${\tau }_{\alpha }\left(T\right)={\tau }_{\infty }\exp \left({\displaystyle \frac{B}{T-T_V}}\right),$$

(4)

where ${\tau }_{\infty }$ is a pre-exponential constant, $T_V$ is the Vogel temperature, and $B$ is a parameter with a unit of temperature. For the Arrhenius dependence, $T_V=0$ and $B=E_a/R$. The α-relaxation in MIX2HF6 cannot be investigated in the vicinity of $T_g$ due to crystallization and overlap with the cr-process from the crystal phase. $T_g$ is defined as the temperature at which ${\tau }_{\alpha }$ = 100 s [57]. The α-relaxation time determined at higher temperatures, down to 259 K, shows a linear dependence (Figure 9). Only two points collected upon heating after fast cooling (Figure 9h) at 257 K and 255 K deviate from the Arrhenius formula, but at the same time, they do not follow the VFT equation. Thus, the α-relaxation time cannot be used for determination of the fragility index $m_f$, defined as follows [57]:

$$m_f={\left.{\displaystyle \frac{d{\log }_{10}{\tau }_{\alpha }}{d\left(T_g/T\right)}}\right|}_{T=T_g},$$

(5)

and describing the deviation of ${\tau }_{\alpha }\left(T\right)$ from the Arrhenius formula. For strong glassformers, ${\tau }_{\alpha }\left(T\right)$ follows the Arrhenius formula and $m_f$ = 16. For very fragile glassformers, for which ${\tau }_{\alpha }\left(T\right)$ dependence shows a strongly nonlinear dependence in the activation plot, $m_f$ may reach more than 200. The Arrhenius behavior of ${\tau }_{\alpha }$ could be interpreted as MIX2HF6 being a strong glassformer, but the deviations from the Arrhenius formula may appear at lower temperatures (as suggested by results from 257 K and 255 K), at which ${\tau }_{\alpha }$ was not investigated.

There is an alternative way for determining $m_f$, based on DSC thermograms [58]:

$$m_f={\displaystyle \frac{c{T}_g\Delta C_p}{\Delta H_m}},$$

(6)

where $\Delta C_p$ is the jump in the heat capacity at $T_g$, and $\Delta H_m$ is the enthalpy of melting. Based on the experimental data for more than 40 glassformers, the empirical $c$ parameter was usually ca. 56 [58]. The modulated-temperature DSC results were used to determine the $m_f$ parameter of MIX2HF6 (Figure 10). The MTDSC technique separates the non-reversing and reversing parts of the heat flow [59]. For MIX2HF6, these parts of the heat flow correspond to the cold crystallization and glass transition, overlapping in the standard DSC thermogram. The glass transition occurs at $T_g$ = 245.3 K, with $\Delta C_p$ = 0.16 kJ/(mol·K). As the melting enthalpy, the summed melting enthalpies of the Cr1 and Cr2 phases were included, giving $\Delta H_m$ = 9.2 kJ/mol. Using Equation (6) and $c$ = 56, one obtains $m_f$ = 233, which is an exceptionally high value [57]. However, for a few substances presented in [58], $c$ deviated strongly from 56, which can also be the case for MIX2HF6. Recently, we published the DSC and BDS results for the 3F7FPhH6 compound [37]. 3F7FPhH6 has a molecular structure similar to MIX2HF6 components, and it forms the SmC_A* glass. The fragility index obtained from the α-relaxation time is $m_f$ = 72. Using the $T_g$ = 233.9 K, $\Delta C_p$ = 0.18 kJ/(mol·K), and $\Delta H_m$ = 15.7 kJ/mol values determined from the DSC thermograms of 3F7FPhH6, one can conclude that the kinetic fragility agrees with the thermodynamic fragility for $c$ = 26. If the same $c$ parameter is applied for MIX2HF6, then $m_f$ = 108, a more reasonable value, within the 72–149 range obtained for similar compounds [36,37,38,56].

Returning to the BDS spectra of MIX2HF6, one can see that the P_H-relaxation time has a nonlinear dependence in the activation plot, and it is expected to reach 100 s at a temperature only slightly higher than $T_g$. Moreover, P_H appears at much lower frequencies than the cr-process from the crystal phase; therefore, upon cooling, it is less affected by crystallization than the α-process. The P_H-relaxation time fits the VFT formula well (Figure 9b). The approximate $T_g$ obtained from the fitting parameters is 249 K, close to the calorimetric 245.3 K. The fragility index obtained from the P_H-relaxation time is 100, also surprisingly close to the calorimetric 108. However, it does not mean that P_H can always be used to determine $m_f$. In publication [56], in which both P_H and α-relaxation times were analyzed by the VFT formula, the obtained $T_g$ values differed only by 2 K. At the same time, the $m_f$ values were less consistent (117 and 149, respectively).

The last relaxation process in MIX2HF6 is the cr-process from the Cr3 phase, which is absent in Cr2 and Cr1. Its relaxation time also follows the VFT equation. While the fitting parameters are similar for slow cooling and subsequent heating (Figure 9b,e), they are different for heating after fast cooling (Figure 9h)—one can see that in the latter, the cr-relaxation time is longer, and diverges at higher temperatures. The relaxation process in the crystal phase can be interpreted as conformational changes, corresponding to the conformationally disordered crystal phase (CONDIS phase), which can undergo glass transition [60,61,62]. Thus, the Cr3 phase is probably the CONDIS phase, while in Cr2 and Cr1, conformational disorder is lower or absent. If $T_g$ of Cr3 is defined as the temperature at which the cr-relaxation time reaches 100 s, then the VFT fits give $T_g$ = 188 K, $m_f$ = 52–54 for slow cooling and heating after slow cooling, and $T_g$ = 209 K, $m_f$ = 161 for heating after fast cooling. The differences in the cr-process in the slow- and fast-cooled samples may result from a larger fraction of the remaining SmC_A* phase during fast cooling, and the formation of both the Cr3 and Cr2 phases during slow cooling. Both can influence the microstructure of the Cr3 phase and, consequently, affect the cr-relaxation in Cr3 due to the confinement effect [63,64].

3.4. Non-Isothermal Crystallization

The melt crystallization of MIX2HF6 and its components were investigated by DSC for cooling rates in the 2–40 K/min range (Figure 2, Figure 3 and Figure 4). The relative crystallization degree $x$ was determined as an integral of the heat flow $\Phi$ between temperatures $T_{start}$ and $T_{end}$, indicating the temperature range of the exothermic anomaly related to crystallization [65]:

$$x\left(T\right)={\displaystyle \frac{{\int }_{T_{start}}^T\Phi \left(T\right)dT}{{\int }_{T_{start}}^{T_{end}}\Phi \left(T\right)dT}},$$

(7)

There are various models for the analysis of non-isothermal crystallization kinetics, including the Kissinger [66], Ozawa [67], Matusita [68], and Augis–Bennett [69] models, and the isoconversional method [70,71,72,73]. In the Kissinger, Ozawa, and Matusita models, the rate of temperature change appears under logarithm, and it is not certain whether it can be applied for cooling, where the rate is negative. In the Augis–Bennett model, the value under logarithm is positive both for cooling and heating. However, it is based on the peak and onset temperatures. Some of the results in this study include overlapped crystallization processes, in which the determination of these temperatures could be vague. Thus, the data were analyzed with the isoconversional method (Figure 11, Figure 12 and Figure 13), which enables the determination of the effective activation energy $E_{eff}$ as a function of $x$ and $T$ [65,70,71]. This method is based on the following formula [70,71]:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=f\left(x\right)A\exp \left(-{\displaystyle \frac{E_{eff}}{RT}}\right),$$

(8)

where $f\left(x\right)$ is the transition model and $A$ is a pre-exponential factor. The logarithm of the crystallization rate $dx/dt$ is plotted against the inverted temperature $T_x$ for a selected $x$, for all cooling rates. The slope of such an activation plot is equal to $-E_{eff}/R$. The obtained $E_{eff}$ can be plotted against temperature if the average temperature for each range of the linear fit is used [65,71].

The isoconversional analysis for MHPOBC reveals three crystallization regimes (Figure 11). The effective activation energies $E_{eff1}$ and $E_{eff2}$, which correspond mainly to a melt crystallization above the room temperature, are negative and equal to –(382–329) kJ/mol and –(171–84) kJ/mol, respectively. The negative values mean that the crystallization kinetics depend mainly on nucleation, i.e., the thermodynamic driving force controls it. Below the room temperature, the effective activation energy $E_{eff3}$ takes values between −121 kJ/mol and 37 kJ/mol. Positive $E_{eff3}$ values, obtained for the intermediate degree of crystallization, mean that at lower temperatures, the diffusion rate has a bigger impact on the crystallization kinetics than at above the room temperature, when the impact of the nucleation rate was the most significant.

The results for 3F2HPhF6 (Figure 12) show four regimes of crystallization. Above the room temperature, $E_{eff1}$ = −(66–11) kJ/mol, around 290 K, $E_{eff2}$ = −(170–134) kJ/mol, and below 285 K, $E_{eff3}$ changes between −81 kJ/mol and 22 kJ/mol. Thus, the crystallization kinetics above the room temperature is controlled by nucleation; the influence of the nucleation rate increases down to 290 K, and at lower temperatures, the diffusion rate has a gradually growing impact. By comparing these results with the phase sequence of 3F2HPhF6 for various cooling rates (Figure 3), one can conclude that $E_{eff1}$ and $E_{eff3}$ correspond to the SmC_A* → Cr1 and SmC_A* → Cr2 transitions, respectively. At the same time, $E_{eff2}$ describes the temperature range within which the mix of Cr1 and Cr2 is formed. For the final stages of crystallization at 30–40 K/min, one can observe the fourth regime, where the activation energy is negative at $E_{eff4}$ = −(80–65) kJ/mol (Figure 12b).

For MIX2HF6, the isoconversional analysis indicates two regimes, both described by the negative effective activation energies: $E_{eff1}$ = −(122–92) kJ/mol for the 2–10 K/min cooling rates, and $E_{eff2}$ = −(165–119) kJ/mol for the 15–40 K/min cooling rates (Figure 13). In the beginning stages of crystallization ($x$ = 0.1–0.2), the activation energy $E_{eff2}$ corresponds to the crystallization kinetics for all the applied cooling rates. $E_{eff1}$ and $E_{eff2}$ cover approximately the same temperature range, except the region below 265 K, at which only $E_{eff2}$ describes the crystallization kinetics. The negative $E_{eff1}$ and $E_{eff2}$ values for MIX2HF6 show that its melt crystallization is controlled mainly by nucleation, while the diffusion rate does not have a significant impact, unlike for the pure compounds, for which the $E_{eff}$ was also generally negative, but positive $E_{eff}$ values were obtained at certain temperatures.

3.5. Isothermal Crystallization

Isothermal crystallization was investigated by the DSC method. Each sample was heated to isotropic liquid and cooled at 20 K/min to the crystallization temperature $T_{cr}$. The ranges of $T_{cr}$ were selected based on the DSC results from Figure 2, Figure 3 and Figure 4. The isothermal DSC thermograms as a function of time are presented in Figure 14a–c. The crystallization degree vs. time was obtained using Equation (7), but the integration was performed over time, not over temperature. The $x\left(t\right)$ dependences were fitted by the Avrami model [73,74], commonly used to analyze isothermal crystallization [22,74,75,76,77]:

$$x\left(t\right)=1-\exp \left({\left({\displaystyle \frac{t-t_0}{{\tau }_{cr}}}\right)}^n\right),$$

(9)

where $t_0$ is the initialization time, ${\tau }_{cr}$ is the characteristic crystallization time, and $n$ depends on the type of nucleation (constant number of nuclei or constant nucleation rate) and dimensionality of the crystal growth (shape of crystallites). For a constant number of nuclei, $n$ = 1, 2, 3 indicates 1D, 2D, and 3D crystal growth. For a constant nucleation rate, the respective $n$ values are 2, 3, 4 [73,74,75]. Higher $n$ can be interpreted as the sheaf-like growth of crystallites [75]. The results for 3F2HPhF6 (Figure 14d) can be fitted with Equation (9). For MHPOBC, two overlapping crystallization processes are observed (Figure 14e), which requires the fitting of two Avrami models, where $A$ is the fraction of the crystal developed in the first stage of crystallization [78]:

$$x\left(t\right)=\left\{\begin{array}{c}A\left[1-\exp \left(-{\left({\displaystyle \frac{t-t_{01}}{{\tau }_1}}\right)}^{n_1}\right)\right] \mathrm{for} t_{01}\le t\le t_{02}\\ A\left[1-\exp \left(-{\left({\displaystyle \frac{t-t_{01}}{{\tau }_1}}\right)}^{n_1}\right)\right]+\left(1-A\right)\left[1-\exp \left(-{\left({\displaystyle \frac{t-t_{02}}{{\tau }_2}}\right)}^{n_2}\right)\right] \mathrm{for} t\ge t_{02}\end{array}\right.,$$

(10)

The crystallization of MIX2HF6 occurs in two stages, but the second stage starts after the completion of the first stage (Figure 14f). Therefore, it is also possible to fit their $x\left(t\right)$ dependences separately with Equation (9).

The isothermal crystallization of MHPOBC results in a mixture of Cr2 and Cr1 phases, with the fraction of the Cr2 phase increasing with increasing $T_{cr}$ (Figure 14g). The 3F2HPhF6 sample consists mostly of the Cr2 phase, which transforms into the Cr1 phase shortly before melting. Only for $T_{cr}$ = 308 K, the Cr2 → Cr1 transition is not visible at the slope of the melting anomaly; therefore, the isothermal crystallization at this temperature results in the Cr1 phase (Figure 14h). The structural differences between Cr2 and Cr1 are presumed to be small, as the Cr2 → Cr1 transition shows a small anomaly in the DSC thermograms. The melting temperature of MIX2HF6 is 303–308 K, and increases with increasing $T_{cr}$ (Figure 14i). It is comparable with the melting temperature of the Cr1 phase formed upon heating from the Cr2 or Cr3 phase (Figure 4d). After the isothermal crystallization of MIX2HF6, two closely overlapping melting anomalies are observed, which indicates two crystal phases with slightly different structures or fractions of the components.

The parameters of the Avrami model obtained for MIX2HF6 and its components are presented in Figure 15. The initialization time is shown in the time–temperature–transition (TTT) diagram [19] (Figure 15a). In the idealized TTT diagram, $t_0$ is expected to have a minimum at a certain $T_{cr}$. Above and below this temperature, the crystallization kinetics are controlled by the nucleation and diffusion rates. Therefore, $t_0$ decreases and increases with decreasing $T_{cr}$, respectively. For MHPOBC, the minimum of $t_{01}$ is observed at 315 K, and of $t_{02}$ at 312 K. For 3F2HPhF6, the shortest $t_0$ is obtained at 298 K. However, the $t_0$ value at 295 K is likely overestimated. For MIX2HF6, only the upper branch of the TTT diagram is observed in the investigated temperature range.

The activation plot of the characteristic crystallization time (Figure 15b) does not show a linear dependence for MHPOBC and 3F2HPhF6. The ${\tau }_1$ and ${\tau }_2$ of MHPOBC initially decrease between 323 K and 320 K ($E_{eff}$ < 0), then the activation plot is almost horizontal ($E_{eff}$ ≈ 0). For 3F2HPhF6, ${\tau }_{cr}$ shows an approximately decreasing trend with decreasing $T_{cr}$ ($E_{eff}$ < 0). The ${\tau }_1$ values for MIX2HF6 show a linear dependence in the activation plot, with $E_{eff}$ = −141(3) kJ/mol. Thus, the first stage of MIX2HF6 crystallization is controlled by the nucleation rate to a larger extent than the crystallization of the pure compounds, and by the second stage of MIX2HF6 crystallization, where ${\tau }_2$ shows a roughly linear dependence at $E_{eff}$ = −33(6) kJ/mol. Comparing the $t_0$ and ${\tau }_{cr}$ values for all the samples, one can say that 3F2HPhF6 shows the fastest crystallization. The first stage of MIX2HF6 crystallization occurs faster than that of MHPOBC in most cases, but the second stage of MIX2HF6 crystallization is much slower than the overall crystallization of MHPOBC. Additionally, the crystallization range of MIX2HF6 is shifted to lower temperatures compared to the pure compounds.

The last parameter is the Avrami exponent (Figure 15c). The lowest $n$ values of 1–1.5 are obtained for the first stage of MIX2HF6 crystallization, and are interpreted as mainly 1D crystal growth. For the first stage of MHPOBC crystallization, $n$ = 1.5–2.5, which means a mix of 1D and 2D crystal growth. The second stages of MHPOBC and MIX2HF6 crystallization, as well as 3F2HPhF6 crystallization, are characterized by $n$ = 2.5–4, which indicates a mix of 2D and 3D growth ($n$ = 2.5–3) or mainly 3D growth ($n$ = 3–4) of crystallites. The maximal $n$ ≈ 5 for the second stage of MIX2HF6 crystallization at 280 K indicates the contribution of sheaf-like crystal growth.

4. Summary and Conclusions

Two liquid crystalline compounds with the acronyms MHPOBC and 3F2HPhF6 were used to formulate an equimolar mixture, denoted as MIX2HF6. The phase sequences of the pure compounds and their mixture were investigated using the DSC, POM, XRD, and BDS methods. The kinetics of the melt crystallization in non-isothermal conditions (various cooling rates) and isothermal conditions (various crystallization temperatures) was investigated by DSC. The main conclusions regarding the behavior of the mixture and its components are as follows:

• According to the DSC and XRD results, the investigated MHPOBC sample exhibits the SmA*, SmC_A*, and hexatic SmX_A* phases. At least one smectic phase between SmA* and SmC_A* is visible in a narrow temperature range in the DSC thermograms. Based on the BDS spectra, the phase sequence is probably SmA* → SmC_α* → SmC* → SmCγ* → SmC_A* → SmX_A*.
• The equimolar mixture MIX2HF6 shows the SmA*, SmC*, and SmC_A* phases. The experimental results do not indicate the presence of SmC_α*, SmCγ*, and SmX_A*.
• While the pure components crystallize easily during cooling, MIX2HF6 shows only partial crystallization, and the remaining supercooled SmC_A* phase forms glass at $T_g$ = 241.5 K. The glass softening during heating occurs at a slightly higher temperature of ca. 245 K. The energy released during cold crystallization upon slow heating of a quickly cooled sample, as obtained by the modulated-temperature DSC method, equals 3.8 kJ/mol (6.1 J/g).
• The melt crystallization kinetics is usually controlled by the nucleation rate for the pure components (effective activation energy $E_{\mathit{eff}}$ < 0), although in certain conditions, the diffusion rate has a dominant contribution to the overall crystallization kinetics ($E_{\mathit{eff}}$ > 0). Meanwhile, the melt crystallization of MIX2HF6 is controlled by the nucleation rate in all the tested conditions. MIX2HF6 crystallizes at lower temperatures and a lower rate than the pure compounds.
• The melting temperature of MIX2HF6, $T_m$ = 308 K, is also lowered compared to 352 K for MHPOBC and 326 K for 3F2HPhF6. However, in some DSC thermograms of MIX2HF6, small fractions of the crystal phases with higher $T_m$ = 325 K and 328 K are observed. This is attributed to minor inhomogeneities in the mixture’s composition, which occur supposedly in the solid state. Nevertheless, the stability range of the antiferroelectric SmC_A* phase in MIX2HF6 is broader than in its components.

The next planned step is to mix the MHPOBC compound with higher 3FmHPhF6 homologs, and to investigate of the influence of the C_mH_2m chain length on the phase sequence and glass-forming properties of the obtained mixtures.