Thermodynamics and kinetics of crystallization of flexible molecules

Abstract

Early structure models of crystals go back to the 17th century.1, 2 By studying the external regularity of the crystals and their optical properties, an arrangement of the fundamental particles as shown in Figure 1(a) was suggested without actual knowledge of their size and nature. It took about 200 years until the molecular details could be ascertained, as shown for NaCl in Figure 1(b).3 Note that this was before X-ray diffraction was available for precise crystal structure determination.8 At about the same time, thermodynamics reached the present-day precision.4, 9 It represents the macroscopic tool to describe phases. Thereafter, microscopic experimental crystal structures were amassed by X-ray diffraction. Parallel, large numbers of heat capacities (at constant pressure), Cp, were determined by adiabatic calorimetry, often covering the temperature region from close to absolute zero to beyond the equilibrium melting temperature, T. Tables of the integral thermodynamic functions enthalpy, H, free enthalpy, G, and entropy, S, were collected for many substances as a function of temperature.10, 11 On the basis of the macroscopic thermodynamics and the microscopic structure, the molecular motion within the crystals was assesses by approximation and in detail.12-14 (a–d) Development of the knowledge about crystal structure (a2 and b3), phase property and size (c4, 5, 6), and the 10 basic states of matter (d7). The next step involved the evaluation of the properties of smaller and disordered crystals. As the main change in property, experiments on small crystals showed a lower melting temperature, Tm. This change in Tm could be described by the Gibbs-Thomson equation based on their surface area and the specific surface free energies.5 The same treatment was useful later, when analyzing lamellar polymer crystals.6 Figure 1(c) summarizes the historical definitions of the thermodynamic properties of macrophases and microphases, with the former being, in at least one dimension, larger than 1000 nm (1 μm). The Gibbs-Thomson equation listed in Figure 1(c) applies to the melting point lowering, ΔT, of lamellar, microphase crystals (𝓁 < 1.0 μm). The earlier suggestion that microphases in the form of colloids where a “fourth state of matter” could be discarded after understanding the effects of surface free energy and surface charges. More recently, nanophases are of general interest for objects marginally larger than the limit of a few ångstroms, set by the homogeneity of the phase due to the atomic structure and the fluctuations of the thermodynamic properties because of a too small sampling volume.15 Experiments following the glass transition of unsupported polymer spheres of decreasing size16 suggested that as long as a small phase has unchanged bulk material within its center, there is no reason to apply a different name to a small microphase.17 As one, however, approaches a size so small that the opposing surfaces leave no unchanged bulk material, it was found that there is a size-range for an entirely new phase, a true “nanophase.”15, 17 An updated thermodynamic description of phases is given in Figure 1(d). It contains a list of the possible phase types when considering not only degrees of condensation and ordering, but also the differences in modes of molecular motion. Figure 1(d) expands the basic, classical states of matter to ten.7, 18 The phases known since antiquity, have been linked in the early 19th century to their newly proven atomic nature.19, 20 The expansion with intermediate phases (mesophases) was discovered over the last 150 years.21 On the left side of the figure, it is indicated that the mesophases become increasingly more “solid” when changing in order toward the crystal. Unfortunately, crystals may have a broad range of solidity, so that the term “solid” is not a scientific, operational definition22 for their state. Glasses, in turn, have an easily measured glass transition temperature, Tg, which can be identified as a solid/liquid transition.23 The mesophases often possess a Cp close to that of the melt. On quenching, they undergo a glass transition similar to liquids. This leads to the three mesophase glasses indicated (from top to bottom: liquid crystal glass, plastic crystal glass, and condis crystal glass). Recently, it was observed, that some polymeric crystals, like aliphatic nylons and polyoxides, may even approach liquid-like mobility on heating before melting or undergoing a disordering transition. This change in mobility causes a change in heat capacity as in a glass transition, that is, the crystal displays a glass transition.24 Crystals that remain “truly” solid up to Tm become a mobile liquid on fusion. Turning to the right side of Figure 1(d), one notes that only the gas connects to all condensed phases. The entropy change of a liquid to a gas without change in the molecular structure is expressed empirically by Trouton's rule.25 The disordering of a crystal to a liquid follows the empirical rule of Richards as long as the ordering species are spherical.26 Nonspherical species follow Walden's rule,27 and for conformational disordering, a similar empirical entropy increase was observed as for spheres.28 The possible transitions between the condensed phases are marked in Figure 1(d) and their overall entropy of fusion is expressed in terms of the three types of disorder by the boxed equation.29, 30 Today, this development of the knowledge about crystal structures, phase properties and sizes, and the states of matter are supported by a wide range of experiments. The macroscopic picture is supplied by the equilibrium and nonequilibrium thermodynamics, based on calorimetry, and is supported by direct experimental evidence about the microscopic structure as well as the molecular motion. Furthermore, the enormous increase in computational capability allows to simulate molecular structure and motion. The time scale of importance, the picosecond, however, is far removed from human experience. The present summary is to establish the needed developments to attain a base for the detection of flaws in the still incomplete description of the thermodynamics and kinetics of crystallization of flexible molecules and their phase structures. In the next three sections, the often neglected problems of nucleation of a new phase of increased order will be analyzed and the nanophase structure of macromolecules will be probed as to its influence across the interfaces. In the conclusions, a view towards the enormous job of supplying details about the resolution of the indicated problems is summarized. The development of the idea of primary and secondary nucleation as they were ultimately applied to the crystallization of semicrystalline macromolecules are described in this section.31 The classical concept of crystal nucleation was already suggested by Gibbs about 130 years ago.32 The description of small phases as a function of size was discussed in Figure 1(c) and led to the free enthalpy plots describing primary nucleation given in Figure 2(a). The curves are scaled to the free enthalpies of polyethylene crystals. The boxed numbers in the graph on the right of Figure 2(a) represent the free enthalpy in convenient units at an approximately 40 K supercooling. They indicate a saddle point (*), calculated by the given equations. The system must travel across it to become stable. Obviously, at T the size of the nucleus is infinity, that is, no nucleation is possible. The larger the supercooling, the lower is the barrier for a move into the region of negative ΔG. About 60 years ago, a mathematical expression for the nucleation rate was derived by Turnbull and Fisher33 written in Figure 2(b). This was fitted to experiments on homogeneous nucleation with sufficiently small polyethylene droplets in silicone oil, eliminating the effect of accidental, heterogeneous nuclei. The result is the graph in Figure 2(b), based on the two-dimensional plot of Figure 2(a).36 There is a region of about 30 K in polyethylene where primary, homogeneous nucleation is not observed, followed by increasingly fast nucleation which slows as the melt viscosity, η, increases and reaches zero when the glass transition at 250 K is approached.31 (a–d) Free enthalpy of primary nucleation of a tetragonal crystal of dimension i = a × a × 𝓁 and expressed by the equations (a31); rate of primary nucleation, calculated analogous to part a (b31, 33); and the basic surface effects leading to secondary nucleation (c,34 d35). To assess the further crystallization after homogeneous or heterogeneous nucleation, the Kossel model of a crystal was used.34 Figure 2(c) illustrates that on a cubic crystal there are five distinguishable locations of different surface free energy for crystallization or melting. Only position 3 has no change in surface free energy, that is, at position 3 crystallization (or melting) should occur at equal rates. Position 1 would require a secondary nucleation on the smooth crystal surface, and position 2, a tertiary nucleation of a new row on a step in the surface. Positions 4 and 5 would be stable and require a positive free enthalpy for removal from the crystal. This model was transferred 30 years later to the crystallization of polymers by Lauritzen and Hoffman35 by simplifying a polymer crystal as shown in Figure 2(d). It was the basis for the description of polymer crystallization for many years with numerous improvements and fine tunings,37 and is often still applied today. Figure 3(a) the for the secondary nucleation of analogous to homogeneous needed to be given for the observed crystallization in form of a but 𝓁 with a positive ΔG. Furthermore, it was that an of similar as for crystallization with a atomic step should the experiments on Early of the polymer as shown in Figure in a to the of the and from the of a molecular of one can that a with 𝓁 = nm which a crystal is only one of about In the in energy so that there is not only an but also an which of Figure temperature In present-day of secondary nucleation, the thermodynamic functions are as of temperature, the that of heat capacities are available for many for the heat capacity been measured by and was linked to its and by crystals of close to could be by crystallization in the condis phase of the condis phase to the phase by removal of the the equilibrium melting temperature was by to the from by and could be shown to the experimental equilibrium melting temperature = but the was leads to a 3 K a when small degrees of supercooling. Today, the contains and thermodynamic functions for 200 polymers and with the of the rate equation of Figure to secondary nucleation is the of the secondary polymer on a smooth crystal surface, as shown in Figure 2(d). as the on its molecular model of a surface a with new surface. This the or surface as the only to secondary It will be shown in the last of this that surfaces to to crystal Furthermore, and thermodynamic evidence of the of secondary nucleation will be in the next (a–d) and and experiments of possible secondary nucleation of all it was that for years the model for polymer crystallization was the to secondary at found The evidence for secondary nucleation was a large of on crystal as shown in Figure have an and could be fitted to nucleation rate as in Figure It was from the experimental of Figure that in the model would ultimately have to to a mathematical description of similar The direct experimental of the of secondary nucleation in polymers is about 40 years An of a of crystals of polyethylene is shown in Figure that should be for secondary nucleation can be on the surface. the was the was to melt some and to the surface with crystals. The area as in Figure is in Figure The on the surface should have secondary nucleation, but by following the crystal it is that not as crystal nucleation The crystals, however, are in indicated by their the of secondary surface of crystals of after heating to K for 3 to a small of crystals. indicate 1.0 from with from the crystallization of of was by experiments with the equilibrium phase by crystals were by up to a of about was The polymer was melting more than from the phase the of a solid This was to crystals from and from the at The after crystallization from can be and by were analyzed and the of is shown as 2 in Figure on crystallization from the melt are as 1 after the by K due to the increase of the melting temperature on of The melt was and so that the could only and were by calorimetry, was to the temperature and to on of the At the crystallization temperature, the curves up to far the by the phase as is this all curves approach the same limit at temperature, at there must be a for to the different the equilibrium melting or temperature of the given The that this is the equilibrium at the saddle point of secondary nucleation in Figure 3(a) is not Only one of many molecules could in this be by secondary On the basis of was molecular nucleation, similar to secondary nucleation was needed for of the molecular nucleation is as Figure It the step of nucleation by and the which sufficiently of the could at a different This nucleation of a could be followed by similar The in more than one form as were discovered earlier by On the molecules in different locations form a to the melting crystal. The are by after by This picture of the thermodynamics and kinetics of crystallization of flexible molecules is by years on the molecular of the of species their equilibrium melting temperature of molecular the of the of the nucleation and of and polyethylene crystals from the the description of the further developments must be In the in the of crystallization of flexible macromolecules a The possible of or at the lamellar surfaces of polymer crystals a It that were a could be found for by of the however, is based on with only by by all on it was from on to for polymer and interest It up only with a new of and new The is in the of more on the The new development with the of surface into the description of polymer crystals, or the secondary nucleation The of a was an early in the crystallization of small from the gas flexible were for the a model of was it no to the nonequilibrium of Figure suggestion of the a mesophase it is that the in Figure must intermediate order before the and a of polymers possess stable or for polyethylene (at and many mesophases are to be stable at the of following the molecular motion of of on a to a view of the crystallization in the to and follow possible in polymer crystals and to the motion involved in the of crystal but the time of to follow crystallization was still too could be To the to more was the many and the cubic an nucleation of nucleation and could be supported by the ultimately also the of molecular was in the as a new tool to and It is applied as calorimetry, Figure the limit of the of crystallization and melting of and experiments and were is a where the is about a temperature with a The were three of an of the and and broad The is to a of This limit of also with Figure where true is up to about It was that in the of glass or surfaces no in the crystallization of by homogeneous This found an by the of of long on by atomic The only K the bulk T. Figure also that the is not by the in the crystal with larger is by heating and as with Figure for As long as T is within the melting and crystallization are The and incomplete phase transitions due to time and a of the and 5 are indicated by the measured by the and the which to the and liquid heat The in Figure the of the of melting and crystallization with a of than the experimental limit of the heating rate to 30 K not change the of melting (at beyond the which was for in Figure 1 K 10 K The of in at larger than 10 K and reached a of 10 K at 30 K experiments a of the of crystallization by the enormous of possible of the in the to the no of crystal with the rate of after primary More on the kinetics of molecular nucleation and its influence on in the Turnbull and equation of Figure has not been as Today, with calorimetry, can be to as as K range of may a direct of molecular nucleation kinetics with changing molecular (a–d) of melting and crystallization of for a of melting in nucleation as step of crystallization after homogeneous or heterogeneous nucleation, in to an between crystal and the melt which must influence the This effect would in the Turnbull and equation of Figure and will be in the last On of the of semicrystalline it was found when the that the glass transition was to temperature and the increase in Cp at was often smaller than from the This in was the and is as the development of it possible to heat effects from the heat capacity in the temperature range between and Tm. from the of this it was found that many crystals a of of by the melting from Cp, a glass transition could be identified for a of polymers the of the phase, for the By a transition temperature, the of a that the not only of a but represents a phase between the crystal and the bulk phases. the influence on the heat capacity by melting and the it possible to the molecular motion within the crystals in this temperature It was found that polymer crystals an of conformational motion as the melting temperature is In many this molecular motion could also be and by solid state X-ray and molecular In of which is available as material, this increase in conformational mobility was discovered to at about In some this motion even reaches a glass transition of the crystal as in the of Figure 1(d). In a of polymer crystals, a transition to a mesophase as is long for for polyethylene at and To a semicrystalline material and to its it is to and three possible phase and more than one, The influence of phases on the overall thermodynamics and the kinetics of crystallization are analyzed in this The observed on is in Figure for It at the phase and is in the temperature range were melting is also observed, a close between the small of different surface melting was earlier by X-ray In this the at the surface over a wide temperature not the crystal and at a lower temperature than the By a larger of experiments on it was that only semicrystalline with and The melting is by the of to At lower temperature the is larger and it approaches zero at the of melting. The the the is the melting. It increases with and with The effect is in Figure on the of a and melting occur to the glass transition region and are at The crystallization in Figure on at 10 K a On with only of this because of to the by the of the is The broad melting is at a temperature than the main crystallization To be linked to the the melting must be a It can be by considering the molecular nucleation of Figure The of the are from the more of the so that melting can only occur at a The however, can as a molecular nucleus for the which within the of Crystals of macromolecules with or no of melting have also been Figure the for a without At this it is to crystals of and or of no (a–d) of the melting and crystallization of a of different polymers for melting of crystals of of different of polymers are or melting was observed for of Figure a of similar as in Figure similar melting was also observed for the crystal was determined by on from It of a lamellar structure as for of a to of within the The of the polymer was in this not to the homogeneous to like an their of than the limit of The polymeric was the that the crystallization and melting are close to that of a of similar The of of the different to influence the molecular nucleation and to be The has a influence on the crystallization of flexible in there is no of the as in there is a of the glass transition to temperature in all semicrystalline the intermediate phase was proven by on This intermediate phase was and on the top and of the This intermediate phase for of the and in solid state experiments it an intermediate mobility between that of the and This has also been by a between X-ray and The size of this intermediate phase was to be about nm for melt More is a of the but from the of the Figure for The Cp for the solid and the mobile could be calculated from of the The glass transitions can easily be from the K all polymer is and the indicated equation for the can be to represent of the and the of the all three phases must be when the by K no can be determined because glass and crystal have close to the same this for the and with it the of the crystallization not apply to by X-ray diffraction. is the of a phase the melting temperature, as for and in Figure experiments of this polymer by already that there was no glass transition the melting that is, the a the the melting the crystals as but it also the The of a is not far the melting temperature, but increases sufficiently when in the phase, as indicated by the experiments in Figure The melting rate is by the glass transition and the T is the of the but melting is at and the glass transition The melting kinetics could be followed and after some of the one of could melt for three units of Note that there is also no of melting in This an not by from the melt. The glass transition is sufficiently that nucleation leads to the of the its glass transition and further To the glass transition to be by the of a On one not only of the crystals, but also the On this is transferred to the the of it was possible to this by X-ray The of the that of a similar result was with The phase structure of semicrystalline polymers is a more system than for small On the basis of the updated phase description in Figure 1 of macroscopic and as well as molecular order and a arrangement of and nanophases was the classical model of Figure 2, one can the homogeneous nucleation of crystals of small as well as large the of the term may be to to this approach to the kinetics of polymer crystals the secondary nucleation of 2 and 3 4 and 5 supported the to a molecular (or nucleation to for direct of crystal without secondary nucleation, of of molecules the equilibrium temperature of their phase and the on of the limit of molecular nucleation a given set of experimental and the on melting and their They indicate that of sufficiently long molecules with a of the macromolecules are for and one can with the to heat capacities and to the thermodynamic functions in the temperature range between and Tm. to the picture that crystal in semicrystalline polymers is by an phase that has at least a glass transition from the phase, and in many leads to a with a glass transition a nanophase semicrystalline as and the may remain and on a structure which at their with a possible heat the of and that no one picture applies to all The details of must be before a description of the thermodynamics and kinetics of crystallization is an enormous which has by only The are available for this for evaluation of the thermodynamics, its and by even the of X-ray diffraction can the structure of all may assess the molecular and the The expression shown in Figure could still describe the experimental crystal as a function of temperature in Figure however, must a set of and the step of the crystallization can be by the of Figure The of may be from at fast the entropy and energy of the The of should into the by the because the of the in Figure are not to have the should be to represent all thermodynamic functions with their temperature In the this was supported by the of and the of and of of at and by for the of The a to or the form of this or to for