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Kinetics of Inverse Melting/Crystallization of Poly(dialkoxyphosphazenes) V. S. Papkov,† M. I. Buzin,† S. S. Bukalov,† M. N. Il’ina,† M. A. Shcherbina,*,‡,§ and S. N. Chvalun‡,∥

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†

Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Science, 28 Vavilova Str., Moscow 119991, Russian Federation ‡ Enikolopov Institute of Synthetic Polymer Materials, Russian Academy of Science, 70 Profsoyuznaya Str., Moscow 117393, Russian Federation § Moscow Institute of Physics and Technology, 4 Institutsky Line, Dolgoprudny, Moscow Region 141700, Russian Federation ∥ National Research Centre Kurchatov Institute, 1 Kurchatov Sq., Moscow 123098, Russian Federation S Supporting Information *

ABSTRACT: We have carried out a study of an unusual phase behavior of poly(dialkoxyphosphazenes) with pentoxy, butoxy, and propoxy substituents forming columnar mesophases which undergo a reversible transition into amorphous rubberlike state on cooling and reverse conversion into the original mesophases upon subsequent heating. On heating to around 200 °C they also yield isotropic melts. The DLI technique and polarized light microscopy were used for monitoring the overall rate of inverse 2D crystallization and linear growth rate of mesomorphic lamellae. The obtained kinetic data were treated in the framework of the Avrami approach and classical nucleation-controlled crystallization theory. Summarizing the results of our optical studies, we have concluded that the formation of 2D columnar mesophases in PDAP-C(3÷5) both in the usual way from the isotropic melt on cooling and via inverse freezing (crystallization) on heating the low-temperature reentrant amorphous phases is induced by heterogeneous athermal nuclei growing in two dimensions and yielding eventually a lamellar mesophase morphology.

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INTRODUCTION Great efforts have been devoted in about the last three decades to the phenomena of reversible inverse melting and inverse f reezing/crystallization.1−10 The former term pertains to a transformation in which a crystal amorphizes or melts on cooling and the latter to the reversal, i.e., melt crystallization, on heating.1 Both phenomena can be defined also as reentrant phase transitions in a physical system. A phase transition is said to be reentrant if it involves transformation of a system from one state into a macroscopically similar (or identical) state via at least two phase transitions through the variation of a single thermodynamic parameter (e.g., temperature).11 Reentrant isotropic phases have contradictory phase behavior, i.e., more symmetric isotropic phases are formed due to a polymorphous transition as the temperature decreases. This transition is called inverse melting, and the temperature below which the undercooled liquid or amorphous phase is thermodynamically © 2019 American Chemical Society

more stable than the crystalline state has been termed inverse melting temperature.9 The inverse melting is opposite to the usual case when a liquid freezes (crystallizes) exothermally on cooling and a crystal melts endothermally on heating. Considering in his short essay the physical background for occurrence of this, at first glance, counterintuitive phenomenon in polymer systems and metallic alloys, Greer5 has figuratively defined the situation as “too hot to melt”. It implies that inverse melting happens if the so-called “ordered” phase (formed high-temperature crystal) admits more entropy than the low-temperature “disordered” liquid state. This may occur e.g. if in the liquid phase some of the degrees of freedom Received: January 17, 2019 Revised: May 28, 2019 Published: June 4, 2019 3722

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Formation of such crystals, possessing higher entropy and being more disordered than a liquid, involves freezing of the center of mass location of the system constituents (whole macromolecules or repeating motifs). The loss of configurational entropy due to this freezing is compensated even more by other microscopic degrees of freedom that are coupled to the center of mass position, where localization of the center of mass increases the amount of such excitations. Specifically, a significant difference in vibrational entropy between liquid and crystal can occur when a difference in their specific volumes is present.12 The inverse melting and reentrant behavior of the singlephase systems described above, i.e., the metastable metallic alloys and some polymers, can be regarded as prototypes of transitions in widely differing systems where an “order parameter” (e.g., structural) does not reflect a real order in both low-temperature “disordered” and high-temperature “ordered” phases, since the absolute measure for order and disorder is the entropy of the system, which always increases with temperature.14 Such transitions of the first and second order in single-phase and multiphase systems, including phase separation of polymer blends and solutions and isotropic reentrant phases in liquid crystals, were surveyed in ref 12 and also described in later publications. In this paper we devote our attention to an unusual phase behavior of a number of poly(dialkoxyphosphazenes) [−N P(OCnH2n+1)2−]m (PDAP). PDAPs with pentoxy, butoxy, and possibly propoxy substituents form a mesomorphic state at ambient temperature, undergo a reversible transition into amorphous rubberlike state on cooling, and yield an isotropic melt on heating up to around 200 °C.15 Mesophases in these polymers are similar to those observed in some linear siloxane polymers, polyphosphazenes, and polysilanes16 and different from well-known liquid crystalline phases of polymers with mesogenic groups in the main or side chains. They are regarded as a special case of partially disordered crystalline modification in thermodynamic equilibrium: i.e., a “condis” crystal17 or 2D pseudohexagonal columnar phase.16 Thus, the observed phase transitions in PDAP match the above definitions for inverse melting (taking this term as a common one for melting on cooling and freezing on heating) and reentrant phase behavior with respect to the existence of lowtemperature amorphous phases and high-temperature isotropic melts separated by a 2D crystalline columnar mesophase. The most noticeable difference in the inverse melting of P4MP1 and PDAPs is that this phenomenon takes place under normal pressure in PDAP, leading to a formation of a soft rubber instead of an amorphous solid. This makes the above three PDAPs convenient objects for studies of the inverse melting kinetics that is of substantial interest for a better understanding of the origin of this phenomenon as well as for practical applications of smart elastomeric materials with specific lowtemperature properties. Here we present and discuss data on the kinetics of isothermal formation of a 2D pseudohexagonal columnar phase from overheated low-temperature reentrant amorphous phases (inverse crystallization) and undercooled isotropic melt of these polymers. The depolarized light intensity (DLI) technique and light microscopy were used for monitoring the overall rate of inverse 2D crystallization and linear mesomorphic domain growth rate. The obtained kinetic data were treated in the framework of the Avrami approach and classical nucleation-controlled crystallization theory.18 Their

of the structural units are frozen and melt in the crystalline phase. Inverse melting was discovered and begun to be studied practically simultaneously in polymer systems and in a number of binary metastable alloys of early transition metals (Ti, Nb, Zr, Ta) with later transition metals from groups V and VI. Note that we are dealing with inverse melting in binary systems within a single phase but not in a catatectic equilibrium (solid 1 ↔ solid 2 + liquid) when, on cooling, partial transformation of the initial amount of substance into a more disordered phase occurs.12 It is a first-order phase transition and its kinetics can, in principle, be treated using the Avrami approach, although corresponding detailed data have still not been reported. As the inverse melting in metastable alloys is a phase transformation far away from the equilibrium state, the nucleation of stable intermetallic compounds must be prevented. At low temperatures, formation of an amorphous or liquid phase is kinetically preferred due to its lower activation energy of formation, as the nucleation and growth of the liquid or amorphous phase requires only the diffusion of faster atoms, whereas the formation of equilibrium compounds is only possible if all components are mobile. The inverse melting starts at the grain boundaries of the crystalline phase and then propagates into the grains.7−10 A similar unusual phase behavior of polymers, which matches the concept of inverse melting, has been described by Rastogi et al. in the early 1990s.2,3 Using X-ray diffraction, the authors investigated phase transitions in the semicrystalline polymer poly(4-methyl-pentene-1) (P4MP1) induced by changes in pressure and temperature and found its amorphization under pressure, ordering on heating, and disordering on cooling, i.e., the attributes of inverse melting, and also the sign inversion of the melting point pressure coefficient, dTm/dp, with increasing pressure. Such behavior is consistent with the T−p looplike phase boundary deduced by Tammann13 on the basis of the Clausius−Clapeyron equation, calling it a universal phase diagram for crystalline and liquid/amorphous phases. Tammann supposed that a looplike phase boundary would be found for some materials in the T−p plane. Outside this loop is the liquid phase, and inside is the crystalline phase which melts at low pressures in a regular way, when a normal density relationship exists between the crystal and the amorphous phases, but at higher pressures, after a temperature maximum value is reached (dTm/dp = 0), the melting temperature decreases with increasing pressure. Then, the phase boundary curve passes through a maximum pressure point where ΔV = ΔS = ΔHf = 0 and dTm/dp = ∞ and, down to lower temperatures, the heat of fusion ΔHf becomes negative. Accordingly, on heating at a fixed high pressure, a liquid/ amorphous phase would reversibly freeze, absorbing heat, and on further heating the crystal would reversibly melt. The existence of inverse melting suggests the possibility of a reentrant appearance of the two widely separated “liquid”/ “amorphous” phases on the p−T phase diagram. In the inverse melting of binary metastable metallic alloys only configurational input in the thermodynamic driving force is meaningful. In contrast, reversible inverse melting ↔ inverse freezing (crystallization) in P4MP1 and the α phase of syndiotactic polystyrene is suggested to be predominantly affected by a vibrational entropy component which determines an excess of the overall entropy of crystal formed upon heating over that of the low-temperature isotropic reentrant phase. 3723

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presentation is preceded by a description of the main features of the investigated phenomenon, which were revealed by thermomechanical and mechanical measurements, DSC, X-ray diffraction, and optical microscopy.

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EXPERIMENTAL SECTION

I∞ − It I∞ − I0

RESULTS AND DISCUSSION

Before an examination of the kinetics of inverse melting in PDAP-C3, PDAP-C4, and PDAP-C5, it is expedient to position them among homologous series of PDAPs with side alkoxy substituents of different lengths in order to emphasize the particularity of their phase behavior in a wide temperature range. For this purpose, we reproduce here the partially modified generalized phase diagram for PDAP (Figure 1) based on the previous15 and newly obtained data on structural and other physical characteristics of all phases involved in the inverse melting phenomenon.

Materials. Poly(dialkoxyphosphazenes) (−[NP(OR)2−]n−) with alkoxy substituents OR = OC3H7, OC4H9, OC5H11 (designated as PDAP-C3, PDAP-C4, and PDAPC5, respectively), were prepared and kindly provided by Dr. Dzidra Tur. Their synthesis was described in detail earlier;19−22 the digest is provided in the Supporting Information. Thermomechanical Studies. The details of the studies are presented in ref 15. Integral and differential thermomechanical traces were recorded on a UIP-70M (USSR) thermomechanical analyzer, permitting the measurement of the strain with an accuracy of 0.001 mm. Polymer samples about 1 mm thick were placed in a steel cup (1 mm in depth, inner diameter of 6 mm). The diameter of a cylindrical indentor with the flat end connected to a quartz measuring probe was 2.52 mm. The applied compression stress varied from 0.02 to 0.1 MPa, depending on the compliance of polymer samples. Under a dead load the overall deformation (reversible and irreversible) was monitored. The time regime used in the alternating mode of measurements (24 s under load followed by 96 s recovery in the unloaded state) allowed separation of the reversible strain at different temperatures. To avoid possible artifacts, a small remaining stress was imposed on the sample in the unloaded state (about 0.0001 MPa). The heating/cooling rate was 5 °C/min. Differential Scanning Calorimetry. Calorimetric measurements were carried out on a Mettler-Toledo DSC-822e calorimeter at heating rates of 10 and 20 °C/min using samples of about 20 mg by weight. Optical Microscopy. We carried out our optical measurements using a POLAML-213M polarization microscope (LOMO, Russia) fitted with a Luminera Infinity 1 USB camera and equipped with an LTS-350 commercial hot stage, LINKAM-C194 System Controller, and LNP94/2 automatic cooling system (LINKAM, England). A maximum heating/cooling rate of 30 °C/min was used to attain predetermined experimental temperatures which differed from the inverse and normal melting temperatures by 6−250 °C. Thus, the initial nonisothermal heating lasted less than 1 min, in which crystallization could proceed by no more than 1%. This fact was confirmed by visual monitoring of the crystallization in the sample under a microscope in polarized light, which manifests itself as the appearance and longitudinal growth of birefringent mesomorphic lamellae. In that period plus the observed induction time (0.5−5 min) isothermal heating regime could be attained. In the course of phase transitions, photomicrographs of the observed developed optical texture in the sample were subsequently obtained. The PhotoM 1.21 program23 (A. Chernigovskii, Russia) was used for the calculation of the optical density of downloaded photomicrographs (intensity of depolarized light transmitted through the sample). Differential Light Intensimetry (DLI). The DLI technique is based on measurements of the intensity It of depolarized light transmitted through the system of birefringent crystals (polycrystalline polymer film) placed between crossed polarizers. Since the 1960s this technique has been widely used to monitor phase changes associated with birefringence. It provides a relatively simple and quick way of precise determination of such kinetic parameters as the induction time τ and half-time t1/2. Formally, the overall phase transformation rate of the polymer also may be obtained from the measured change in time of It in isothermal plots for different crystallization temperatures: α(t ) =

Article

Figure 1. Generalized phase diagram for PDAP-Cn with various numbers n of carbon atoms in side alkoxy groups.15

In principle, PDAPs belong to the kind of linear symmetrically substituted polymers with different organic side groups, which, in spite of the absence of classical mesogenic moieties in their structure, lend themselves to form thermotropic 2D columnar mesophases. As a rule, such mesophases are intermediate between crystalline phases and an isotropic melt. Their existence reflects a stepwise disorder of 3D crystals due to disturbance of the existing hierarchy of discrete intermolecular interactions which are partially released with increasing temperature. Eventually, on further heating all of the intermolecular interactions break and columnar phases melt. On cooling, the reverse consequence of transitions should be observed: i.e., isotropic melt → columnar mesophase → 3D crystal. The formation of columnar mesophases is observed only in PDAP with alkoxy substituents whose length varies in a shortrange from three to five carbon atoms. It should be noted that the driving force for their formation appears to be very small, as 0.2 mol % of defect monomer units (containing PO, P− OH, and P−Cl bonds) in PDAP macromolecules is already enough to prevent formation of a columnar mesophase. When they are cooled to low temperatures, instead of crystallization, mesomorphic PDAPs undergo a transition into an amorphous state exhibiting rubberlike properties. This inverse melting transition determines the boundary between a 2D columnar

(1)

where I∞ and I0 are the intensities of transmitted light after full transition and before its start, respectively.24,25,29 3724

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Figure 2. Optical micrograph of PDAP-5 film in polarized light between crossed polars (a) and scanning electron micrographs of the free surface of mesomorphic PDAP-C3 (b) and PDAP-C4 (c) films.

network are shown in Figure 3. For comparison, an extension curve of the mesomorphic PDAP-C4 network at 20 °C is given

mesophase (or crystalline state) and a low-temperature reentrant amorphous phase. The PDAP columnar mesophases are opaque semisolid substances capable of flowing under shear stress like plastic crystals. Polarizing optical micrograms of PDAP films cast from solution or crystallized from the melt reveal negative birefringent band like domains (Figure 2a). Their intensity is maximal at a 45° angle to the direction of a polarizer, and they are extinguished if they are aligned parallel or normal to that. By analogy to similar optical textures observed for mesomorphic polydiethylsiloxane,26 poly[bis(trifluoroethoxy)phosphazene], and poly[bis(phenylphenoxy)phosphazene]27−29 possessing a lamellar morphology, such bright bands were attributed to the lateral faces of mesomorphic lamellae properly oriented in space (the end faces of which make right or nearly right angles with the free surface of the sample).26,30,31 As the birefringence of oriented samples of mesomorphic PDAPs is positive, macromolecules in the lamellae should be arranged perpendicular to the end faces. Scanning electron micrographs of the free surface of mesomorphic PDAP-C3 and PDAP-C4 films are represented in Figure 2b,c, respectively. Visible supramolecular structures look like upper parts of single platelike domains emerging on the surface; this observation is consistent with the supposed lamellar morphology of the polymers. The thickness of such platelike domains is 1000−1500 Å, suggesting a possible folded PDAP macromolecular conformation. The considerably larger width of the bright bands manifesting theirselves in the microscope pictures (about 30000 Å) would be a result of corresponding aggregation of the above single mesomorphic lamellae. The 2D pseudohexagonal packing of macromolecules was suggested to be the PDAP mesomorphic lamellae by analogy with some linear polydiorganosiloxanes and polyphosphazenes. The amorphization process (inverse melting) is kinetically hindered due to the high viscosity of PDAP at low temperatures; it extends over a wide low-temperature range. This fact greatly complicates quantitative monitoring of the transition by thermomechanical methodsa substantial difference exists between the inverse melting and inverse crystallization temperatures corresponding to a sharp increase and decrease in PDAP-C4 sample compliance (rubberlike strains). It is necessary to repeat several cooling−heating cycles from +20 to −70 °C (and even below) or to carry out annealing of the material at such low temperatures to achieve a complete inverse melting transition. Thus, thermally treated weakly cross-linked PDAP-C4 and PDAP-C5 films below the inverse melting temperature manifest traditional rubberlike behavior with high reversible deformations on extension and mechanical loss hysteresis loops at subsequent contraction that are especially characteristic of filled and structured rubbers. Corresponding stress−strain curves for amorphous PDAP-C4

Figure 3. Stress−strain curves for the PDAP-C4 weakly cross-linked network in a mesomorphic state (1) and amorphized at −100 °C in extension−contraction cycles at −70 °C (2−4) and on extension at 20 °C (1): first cycle (2, 3); second cycle (3, 4). The extension/ contraction rate was 0.9 min−1.

as well. One can see that slightly cross-linked PDAP-C4 films in the mesomorphic state are capable of high plastic deformations on extension which are preserved to a large extent on unloading. However, on subsequent cooling below the inverse melting temperature, free samples contract and return gradually to their initial lengths. A jumpwise decrease of the mesomorphic reflection intensity, a substantial change in an initial optical texture, or an increase in reversible rubberlike deformation was not found.15 These facts imply that, even if inverse melting in PDAP-C3 takes place, it appears to be most hindered and to proceed only to a small degree. In contrast to the inverse melting, a reverse transition in PDAP-C4 was observed to take place in a fairly narrow temperature range (see Figure 4). We emphasize that the heat of transition ΔHim on heating is positive (endothermal), which is a characteristic feature of an inverse melting phenomenon. It is rather small (0.51 ± 0.03 J g−1), but the thermal event is reproducible and clearly discernible on DSC traces. Unfortunately, we failed to find a corresponding thermal effect of inverse crystallization on DSC traces for PDAP-C5 samples, likely due to a smaller magnitude or to its extension over a wider temperature range. At high temperatures, the existence of the mesomorphic 2D columnar phases is limited by their isotropization, which above 200 °C can be affected by a superposition of destructive processes.15 This complicates DSC measurements of the isotropization and mesophase formation heats (ΔHi and ΔHmf) and temperatures (Ti and Tmf). Therefore, we could 3725

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longitudinal growth of birefringent bright bands−mesomorphic lamellae (Figure 2). This process is common for all three studied materials. Birefringent domains appear after a certain induction period, and their number remains approximately unchanging over a sufficiently long time (Figure 5).

Figure 4. DSC traces (1) and DLI (2) and TMA (3) curves of PDAPC4 samples in a regime of alternating loading−unloading. The heating rates are 20, 10, and 5 °C/min, respectively.

determine only ΔHi = 1.8 and 1.4 J g−1, and Ti = 202 and 181 °C in the first and second DSC heating runs of PDAP-C3 samples, respectively; their mesophase transition heat and temperature on cooling were found to be ΔHmf = 0.84 J g−1 and Tmf = 149 °C. In PDAP-C4 ΔHi° = 3.4 J g−1 and Ti = 243 °C in the first heating run. No thermal events on DSC traces were observed on cooling PDAP-C4 samples from the melt and on heating PDAP-C5 above 260 °C, where according to the TMA data it melts.15 To understand better the specificity of the reentrant behavior of the aforementioned PDAP-C3, PDAP-C4, and PDAP-C5 samples, it would be more expedient to compare thermodynamic, kinetic, and morphologic characteristics of all their transitions. Unfortunately, this was impossible because of a partial degradation of PDAP-C4 and PDAP-C5 after heating to the isotropization temperature that suppressed formation of the mesophase. Thus, we investigated 2D crystallization of PDAP-C3 from the melt, suggesting that this might be a model for similar processes in PDAP-C4 and PDAP-C5 in the absence of degradation. PDAP-C4 and PDAP-C5 were in turn used for studies of the kinetics of low-temperature inverse 2D crystallization. A quantitative description of the kinetics of phase transitions in the isothermal conditions was carried out using the Avrami equation, which is useful for the determination of crystal nucleation mode (heterogeneous or homogeneous), of crystal growth habits, and of the overall crystallization rate:18 α(t ) = 1 − exp( −kt n)

Figure 5. Optical textures observed in polarized light for a supercooled melt of PDAP-C3 at 177 °C (a), PDAP-C4 at 6 °C (b), and PDAP-C5 at 60 °C (c) curves of PDAP-C4 samples in a regime of alternating loading−unloading. Heating rates are 20, 10, and 5 °C/min, respectively.

The conversion α of an amorphous polymer into the mesophase was calculated using eq 1 and is presented as a function of time in ordinary and Avrami coordinates (Figure 6). The values of Avrami exponents indicated in Figure 6 and equal or close to 2 could be used as a nominal confirmation of athermal heterogeneous nucleation of growing 2D columnar mesophase crystals. Lower values of n for PDAP-C4 might tentatively be related to such factors as a decrease of nucleation rate in thermal heterogeneous nucleation when the nuclei are exhausted,18 to variations of the linear growth rate of the mesomorphic crystals, and to the effect of transport process with increasing conversion. The induction time of the mesomorphic lamellae appearance is temperature-dependent (Figure 7), as is the rate of isothermal lamellar growth, which is constant over a rather wide time interval. The growth rate increases and induction time decreases with increasing supercooling and overheating PDAPs at their 2D crystallization from the melt and lowtemperature amorphous phase. Linear and polynomial extrapolations of the lamellar growth rate to zero magnitude lead to the quasi-equilibrium inverse melting temperatures for PDAP-C3, PDAP-C4, and PDAP-C5 being equal to 190, −3, and 42 °C, respectively. Summarizing the results of the optical studies, one can conclude that formation of a 2D columnar mesophase in PDAP-C3, PDAP-C4, and PDAP-C5 both in the usual way from the isotropic melt on cooling and via inverse freezing (crystallization) on heating the low-temperature reentrant amorphous phases is induced by heterogeneous athermal nuclei growing in two dimensions and yielding eventually a lamellar mesophase morphology. In this respect, a question arises as to the origin of the induction times observed, as athermal nucleation implies that all crystals start to grow at the same time. Note that induction times were observed earlier for athermal nucleation of the thermotropic columnar mesophase

(2)

where α(t) represents the fraction of crystallized (transformed into a mesophase) polymer, k is a temperature-dependent rate constant, and n describes the nucleation mode and is usually an integer between 1 and 4. It should be noted that, in practice, DSC is a convenient and widely applied technique for monitoring isothermal crystallization of polymers. However, thermal effects accompanying all the transitions in the PDAP samples are very low. Thus, we have made an attempt to measure the kinetics of the observed transitions by means of the differential light intensimetry technique. In polarized light, transformations of both optically isotropic high-melt and lowtemperature amorphous phases of the PDAPs into a 2D columnar mesophase proceed as the appearance and 3726

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Figure 6. Avrami plots of melt conversion of (a) PDAP-C3 at 175 °C (1), 177 °C (2), 180 °C (3), and 182 °C (4), (b) of the PDAP-C4 lowtemperature amorphous phase at 3 °C (5), 6 °C (6), and 10 °C (7), and (c) of the PDAP-C5 low-temperature amorphous phase at 60 °C (8), 63 °C (9), and 67 °C (10). Numbers above the lines indicate the values of Avrami exponent n.

Figure 7. Temperature dependences of induction time of the mesophase lamellae appearance for PDAP-C3 (a) and PDAP-C4 (b).

Figure 8. Linear growth rate of columnar mesophase lamellae in PDAP-C3 (a), PDAP-C4 (b), and PDAP-C5 (c) as functions of 1/TΔT and 1/T and the Avrami temperature-dependent parameter K of the overall rate of mesophase formation in PDAP-C3 (d) and PDAP-C5 (e) as functions of 1/TΔT and 1/T. The values of K were obtained from the linear plots (a) and (c), respectively.

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parable with those (1.85−5.55 erg2 cm−4) observed in the formation of 2D pseudohexagonal thermotropic mesophases in PDES26 and polyphosphazenes.27,28 As follows from eq 5, this fact correlates with a large difference between the heat of fusion of common homopolymers and the heat of isotropization of the mesophases. A nonlinear Arrhenius plot of ln v versus 1/T, given for comparison in Figure 8a, is in line with the conclusion that a nucleus activation barrier governs the kinetics of the 2D pseudohexagonal mesophase lamellae growth from PDAP-C3 melt. The Avrami plot of ln K yields a straight line by a leastsquares fit with the slope Sk = −43200 K2 (see Figure 8d). This linear dependence together with the aforementioned kinetic and optical data manifests a two-dimensional growth of PDAPC3 mesophase lamellae which is controlled by the secondary nucleation, as well as 2D heterogeneous primary nucleation of the first step of the overall melt transformation to the 2D pseudohexagonal mesophase. In principle, the parameter K in the Avrami equation can be expressed as18

formation in polydiethylsiloxane (PDES)26athermal nucleation suggests that, in the isotropic melt at the initial moment of transition, nuclei exist whose size should be equal to or larger than critical thermal nuclei dimensions. It was assumed that athermal nuclei in the PDES melt exceed the size of thermal nuclei insignificantly; thus, noticeable lateral growth of lamellae is initiated only when the longitudinal size of athermal nuclei achieves a length of the lamellar thickness. Stable aggregates of longer macromolecules or macromolecules extended under cooling induced stress may be thought to serve as such athermal nuclei. A similar explanation seems to be acceptable in the case of PDAP as well. According to the classical crystallization theory,18 the crystallite growth rate V is controlled by secondary nucleation and may be described by the equations i ΔG* yz i ΔE yz zz zz expjjjj− V = C expjjj− z k kT { k kT {

ΔG* =

aσσeb0Tm Δhf ρc ΔT

(3)

K = cBV n (4)

where c is a factor dependent on the geometry of macroscopic crystals, B stands for a number of heterogeneous nuclei per unit volume in the case of athermal nucleation, V is the linear growth rate, and n is the Avrami exponent. The ratio Sk/Sv may be close to a mean value of n being between 1.7 and 2.0. In fact, however, this ratio is 4.42, which is more than twice the estimated average value of n. This fact reveals a complicated temperature-dependent relationship between the birefringence of a single crystalline lamella and the overall birefringence of the whole sample with their random spatial orientation. Inverse freezing/crystallization of PDAP-C4 and PDAP-C5, i.e., formation of a 2D pseudohexagonal crystalline phase (columnar mesophase) from the low-temperature reentrant amorphous phase, proceeds on heating just like formation of the mesophase in PDAP-C3 on cooling from the isotropic meltthrough 2D growth of mesophase domains yielding eventually a lamellar mesophase morphology. The difference between the transitions is that the former is characterized by a higher entropy of the growing crystal in comparison to the preceding amorphous phase, whereas the latter is accompanied by a decrease of the system entropy. As the free enthalpy of crystallization (ΔG = ΔH − TΔS) should always decrease on transition, a correlation between magnitudes and signs of ΔH and ΔS determines the type of transition. In the case of the usual melt crystallization on cooling, a negative ΔG value results from a negative ΔH value. In contrast, a positive ΔS yalue yields a negative ΔG value of inverse crystallization (entropic origin of thermodynamic driving force). Obviously, classical crystallization theory can be applied to deriving the main kinetic relationships of inverse crystallization as well. Thus, the lamellae growth rate v is suggested to be controlled by secondary nucleation and may be described by equations similar to eqs 3 and 4 with appropriate corrections of the meanings: i.e., an increase of an overheating ΔT over the inverse melting temperature Tim, with rising crystallization temperature T. This entails a decrease in the free enthalpy of formation of critical nucleus ΔG* and, as a result, a parallel increase in both exponential factors in eq 3. This suggests a continuous increase of v with temperature. Such a situation is opposite to the usual crystallization, when at large supercoolings ΔT the increasing melt viscosity hinders shortdistance diffusion of the crystallizing element across the phase

where ΔE is the activation energy of molecular transport across the interface approximated by the activation energy of viscous flow, ΔG* is the free enthalpy of formation of critical nucleus (free enthalpy of nucleation), σ and σe are free surface energies per unit area parallel and perpendicular to the macromolecular axis, respectively, b0 is the thickness of the monomolecular layer (diameter of a macromolecule);, Δhf is the heat of fusion per gram of polymer, ρc is crystal density, ΔT = Tm − T is the supercooling, Tm is the melting point of crystals, and a is a factor equal to 4 or 4/3 depending on single or multiple surface nucleation. In accordance with eqs 3 and (4), the

(

quantity ln V +

ΔE kT

) must be a linear function of TΔ1 T with a

slope of S=−

aσσeb0Tm k bΔhf ρc

(6)

(5)

These relationships proved to be applicable for a successful description of the isothermal kinetics of transformation of the isotropic PDAP-C3 melt into a 2D pseudohexagonal mesomorphic phase as in the case of mesophases in PDES26 and polyphosphazenes.27,28 We carried out our measurements of the linear growth rate of mesophase lamellae in a temperature range of 175−185 °C in PDAP-C3 samples after their preliminary isotropization upon heating. In our calculations, the melting (isotropization) temperature Tm was taken to be equal to the extrapolated quasiequilibrium value of 190 °C, the heat of isotropization Δhf has a value of 1.8 J g−1 according to the DSC traces of the material, b0 = 1.38 nm,22 and the mesophase density was assumed to be 1.1 g/ cm3. The slope value Sv = −9760 K2 was determined from the linear plot of ln V vs 1 (see Figure 8), assuming ΔE/kbT to T ΔT be constant. This dependence is strictly linear, manifesting that the growth of a two-dimensional pseudohexagonal mesophase from PDAP-C3 melt is controlled by the secondary nucleation. The product of the free energies of side and end surfaces, σσe, was estimated to be 0.1 or 0.31 erg2 cm−4 depending of the value of parameter a (4 or 4/3) in eq 5. It is 1 or more orders of magnitude less than those σσe products usually observed for 3D crystallization of common homopolymers and is com3728

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boundary and, therefore, the linear growth rate v should exhibit a maximum value at a certain temperature. Applicability of eqs 3−5 to the description of isothermal kinetics of inverse crystallization of PDAP-C4 and PDAP-C5 into 2D pseudohexagonal mesophases is confirmed by the linear dependences of ln v and ln K versus 1/TΔT presented on Figure 8b,c. The corresponding slopes Sv were determined to be equal to 2210 and 15700 K2 for PDAP-C4 and PDAPC5, respectively, and the slope Sk for PDAP-C5 turned out to be equal to 72300 K2. Thus, the growth of 2D pseudohexagonal mesophases from PDAP-C4 and PDAP-C5 reentrant amorphous phases is controlled by the secondary nucleation and the latter implies a possible 2D heterogeneous primary nucleation as the first step of the overall inverse crystallization. Interestingly, the ratio Sk/Sv for PDAP-C5 is equal not to the expected Avrami exponent n = 2 but to n = 4.6: i.e., to a value practically coincident with that obtained for PDAP-C3 (Sk/Sv = 4.42). In this respect, it is appropriate to remember the peculiarities of DLI: a large variance of the evaluated values of n at different temperatures of isothermal inverse crystallization of PDAP-C4 made analysis of ln K vs 1/TΔT plots inappropriate. On the basis of the values of Sv determined for PDAP-C4 and PDAP-C5, we attempted to assess the products of the surface free energies σσe of the critical nuclei which induce the growth of mesomorphic lamellar crystals in the reentrant amorphous phases. The calculations were made using a modified form of eq 5, in which (i) Tm is the quasiequilibrium inverse melting temperature, (ii) the heat of inverse transition Δhit substitutes the heat of fusion Δhf, (iii) overheating is ΔT = T − Tm. For PDAP-C4 (ρc = 1.07 g cm−3, Tm = 270 K, Δhit = 0.51 J g−1) σσe was estimated to be equal to 9.34 × 10−3 erg2 cm−4 for a single and 2.8 × 10−2 erg2 cm−4 for multiple surface nucleation. Isotropization/melting of the PDAP-C4 columnar mesophase (formation of a high-temperature amorphous phase) manifests itself in DSC traces of the material as a clear-cut endothermal peak at 243 °C with an area of 3.4 J g−1 taken as the heat of melting. Unfortunately, partial degradation of this polymer in the course of its heating just above isotropization temperature prevents the usual melt crystallization on subsequent cooling. Thus, we could not monitor the development of a lamellar morphology and evaluate the value of σσe, characterizing the energetics of secondary nucleation during crystallization of PDAP-C4 from the melt. However, taking into account the aforementioned correlation between σσe and the heat of fusion of common polymers, one can suggest that this value should be comparable with that obtained for the PDAP-C3 melt crystallization (0.1 or 0.31 erg2 cm−4). We can also estimate entropies of both inverse and usual melting transitions estimating them as the ratios of the corresponding heats to the transition temperatures (−3 and 243 °C, respectively): ΔSic = 1.9 × 10−3 J g−1 K−1 (0.45 J mol−1 K−1) and ΔSm = 6.6 × 10−3 J g−1 K−1 (1.56 J mol−1 K−1). Their sum represents, in principle, the entropy difference of the low-temperature reentrant amorphous phase and of the isotropic melt without regard for an additional change in entropy of the polymer sample in the mesophase state in the temperature range of −3 to +243 °C. In principle, inverse crystallization of the low-temperature reentrant amorphous phase in PDAP-C5 could be expected to proceed in a similar way but perhaps with an even smaller heat

of the transition. As was noted above, we were not able to measure the heat of fusion of PDAP-C5 unambiguously using DSC, as it is rather small, though we can evaluate it from Figure 8c, as the slope of the ln V (1/TΔT) plot is inversely proportional to Δh (see eq 5). It was found to be 15700 K2. We can also make a reasonable assumption that the value of σσe, as it is determined mainly by the intermolecular interaction of alkyl side substituents, should be close to that obtained in studies of PDAP-C4 (∼10−3 erg2 cm−4). Thus, with b0 = 1.9 nm, an estimated heat of inverse 2D crystallization (mesophase formation) should actually be very small (0.11 J cm−3). Therefore, the calculated entropy of such inverse crystallization ΔSic should be very small as well in PDAP-C5 (3.5 × 10−4 J g−1 K−1 or 7.6 × 10−2 J mol−1 K−1). Linear extrapolation of PDAP-C3 and PDAP-C4 DSC data to a virtual melting of the 2D columnar mesophase in PDAPC5 (taking no regard of degradation) leads to the evaluations of Tm = 284 °C and ΔHm = 5 J g.−1 Consequently, the melting entropy was estimated to be ΔSm = 9 × 10−3 J g−1 K−1 (1.97 J mol−1 K−1). One can see that, in both PDAP-C4 and PDAPC5, the entropy of the high-temperature amorphous phases (melts) in both PDAPs is noticeably higher than that of the low-temperature reentrant amorphous phases. Thus, the major distinction of the low- and high-temperature amorphous phases lies in the different levels of molecular mobility with the main increase in the degrees of freedom arising due to a melting of columnar mesophases on heating and to an appearance of new vibration modes of side alkoxy groups. The latter assumption is indirectly confirmed by the larger increase in entropy of the melt of PDAP-C5 in comparison to that of PDAP-C4 due to the supposedly larger weight content of alkoxy groups. The difference between the melt and reentrant amorphous phase can be in principle a reason for the possible kinetic particularities of the formation of a thermotropic columnar mesophase. In the range of mesophase existence, its growth rate can nominally be expressed as functions of an undercooling against the melting (isotropization) temperature and of an overheating relative to the inverse melting/crystallization temperature (eqs 3−5). In this respect one can imagine that for a polymer with close melting and inverse melting temperatures there exists a point at which both kinetic ways of the mesophase growth will be equivalent. That should lead to secondary nuclei of two different sizes arising and, eventually, to a mixed terminal mesophase morphology. Finally, let us briefly formulate the influence of the length of alkoxy substituents on the thermodynamic parameters of both the usual and inverse transitions in the PDAP-C3, PDAP-C4, PDAP-C5 row. The melting (isotropization) temperature, enthalpy, and entropy of their usual melting at high temperatures increase in parallel with the length of side chains. In contrast, the enthalpy and entropy of inverse crystallization/ melting appear to be smaller in PDAP-C5 than in PDAP-C4, while the temperature of inverse melting/crystallization of the former is higher, as determined by the ratio of transition enthalpy and entropy. This means that with an elongation of butoxy substituents by one −CH2− group the entropy of inverse transition decreases disproportionately with the enthalpy.

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CONCLUSIONS We have carried out studies of an unusual phase behavior of poly(dialkoxyphosphazenes) with pentoxy, butoxy, and 3729

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propoxy side chains. Such compounds form columnar mesophases that undergo a reversible transition into an amorphous rubberlike state on cooling and the reverse conversion into the original mesophases upon subsequent heating. When they are heated to 200 °C, these compounds yield an isotropic melt. Historically, along with poly(4methylpentene-1), poly(dialkoxyphosphazenes) turn out to be the first examples of the occurrence of inverse melting in polymer systems. Even more, PDAPs are experimentally favorable, as inverse melting/crystallization takes place under normal pressure and forms a soft rubber instead of an amorphous solid in poly(4-methylpentene-1). In polarized light between crossed polarizers, transformations into 2D columnar mesophases of both optically isotropic high-temperature melt and low-temperature amorphous reentrant phases manifest theirselves as an appearance and longitudinal growth of birefringent mesomorphic lamellae. The DLI technique and light microscopy were used for monitoring the overall rate of inverse 2D crystallization and linear mesomorphic domain growth rate. The kinetic data obtained were treated in the framework of the Avrami approach and classical nucleation controlled crystallization theory. The formation of 2D columnar mesophases in PDAP-C(3−5) both in a usual way from the isotropic melt on cooling and via inverse freezing (crystallization) on heating the low-temperature reentrant amorphous phases is induced by heterogeneous athermal nuclei growing in two dimensions and yielding eventually a lamellar mesophase morphology. Entropies of inverse crystallization and high-temperature melting of the columnar mesomorphic phases in PDAP-C4 and PDAP-C5 were estimated. Their sum (about 9 × 10−3 J g−1 K−1) represents, in principle, the difference between the entropy of the low-temperature reentrant amorphous phase and the entropy of the isotropic melt. This quantity points to a noticeable distinction of the low- and high-temperature amorphous phases manifested primarily in the different levels of molecular mobility in them. A difference in the degrees of freedom arises due to melting of the columnar mesophases on heating and to an appearance of new vibration modes of side alkoxy groups. The latter assumption is in line with the larger increase in entropy of the melt of PDAP-C5 in comparison to PDAP-C4 due to supposedly the larger weight content of alkoxy groups in the former.

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AUTHOR INFORMATION

Corresponding Author

*E-mail for M.A.S.: [email protected]. ORCID

S. S. Bukalov: 0000-0003-2342-7093 M. A. Shcherbina: 0000-0002-5569-958X Funding

This work was supported by the Russian Scientific Foundation (Grant No. 14-13-01402). Notes

The authors declare no competing financial interest. 3730

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