Thermodynamics crystallization

Thus far, we have discussed a number of key experimental observations regarding the effects of counit incorporation on the solid-state structure and the crystallization kinetics in statistical copolymers. In order to better quantify these experimental observations, various thermodynamic models have been proposed. Rory s model, as outlined in Section 11.2.1, correctly describes the equilibrium melting behavior of copolymers in the limit of complete comonomer exclusion however, it is often found to be inadequate at predicting experimentally accessible copolymer melting temperatures [11-14]. An alternative was proposed by Baur [91], where each polymer sequence is treated as a separate molecule with an average sequence length in the melt given by [91]  

Although Baur s copolymer model is built on the rather unrealistic premise that each polymer sequence of average length in a polymer chain may be treated as a disconnected single molecule, it has been found to match experimental data much better than Rory s 

Besides Baur s model, other relationships, both theoretical and empirical, have been proposed to describe the experimentally observed crystallization and melting behavior of statistical copolymers [94-96]. In one study, Pakula presented two models for determining copolymer crystallinity as a function of critical nucleus size [94]. In model one, crystallizable sequences longer than the critical nucleus are allowed to participate in the formation of stable crystallites however, chain folding is assumed to be absent, so any portion of the crystallizable sequence longer than the critical nucleus size must be rejected to the amorphous phase. In model two. 

Burfield proposed an empirical relation between crystallinity and mole fraction of crystallizable units for ethylene copolymers [95,96]. If the monomer sequence is truly random, then the probability of having m ethylene units in an uninterrupted sequence can be expressed as Xe- It was then further hypothesized that a correlation between the melting enthalpy of the copolymer, and Xe may be written in the form  

This relationship was applied to analyze the crystallization data of six different sets of ethylene statistical copolymers [96]. In all cases, a linear relationship between logA// and logX was observed. Furthermore, the values of m determined for ethylene-olefin copolymers were found to be in the range of 6-13, in good agreement with the critical sequence length proposed by Randall and Ruff based on NMR data and modeling considerations [97]. 

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Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

The development of the -modification is controlled by the relative crystallization thermodynamics and kinetics of the stable a-modification and of the smectic phase towards the metastable / -phase. For PP homopolymers, it is generally accepted that under isothermal conditions, the a-phase grows more rapidly at temperatures below 105 and above 140 °C than its counterpart, which in turn is more prone to develop in between these two temperatures in the presence of selective -promoters [52,70,122]. An elegant way to get fully nucleated /3-PP specimens would consist of pressing /3-PP pellets above their melting temperature (ideally more than 250 °C to erase any a-nuclei in the system), cool the melt quickly up to a crystallization temperature in between 100 and 130 °C, let the sample crystallize, and then quench it to room temperature [70]. However, such a processing method is too time-consuming to be of industrial relevance. 

Bock has studied a number of systems in which different polymorphs were obtained under thermodynamic and kinetic conditions. (2-pyridyl)(2-pyrimidyl)amine 3-II is dimorphic. Modification I is readily crystallized thermodynamically from any solvent (toluene was actually used) while modification II is obtained kinetically by fast evaporation of an ethereal solution or by resolidification of the melt (Bock et al. 1997). 

Crystallization kinetics generally is very sensitive to temperature flucmations and related factors, such as cooling rate or thermal history. As can be expected from nucleation theory and crystallization thermodynamics, presence of contaminants. 

Crystallinity 0.6(L0.95 Properties dependent on crystallinity, with amorphous phase providing flow, flexibility, and toughness. Kinetically stable folded chain crystals thermodynamically stable extended chain crystals can be formed by gel drawing... 

The procedure most commonly used to alter the crystallization thermodynamics of true racemate systems involves the formation of dissociable diastereomer species [44]. These are most often simple salts... 

Frenkel defects in general do not play an important role in molecular crystals. The asymmetric shape of the molecules and the steeply increasing repulsive potential between molecules at short intermolecular distances make the occurrence of interstitial molecules in molecular crystals thermodynamically improbable. The... 

DRIVING FORCE OF PROTEIN CRYSTALLIZATION THERMODYNAMICAL ASPECTS... 

Some crystals appear to become prone to attrition once they have been grown beyond a certain critical size in an agitated crystallizer. To some extent this can be attributed to increased damage from the agitator as higher rotational speeds are needed to keep them in suspension. Sometimes the critical size coincides with the onset of polycrystalline growth which tends to make the crystals friable. Polycrystalline growth, however, may not only render the crystals mechanically weak, but may even make the crystals thermodynamically... 

The emergence of a new phase due to local structural fluctuations in a lattice of the original solid phase Ag occurs most often at the boundaries and defect sites of the crystals. Thermodynamically stable nucleus of new phase has often a critical size close to unit cell volume, which differs from the normal one. This gives rise to mechanical stresses in the transformation zone, and even the destruction of the original crystal. Such effect leads to smaller quantity of a desired product. Thermal decomposition reaction, like all topochemical reactions, proceeds more rapidly... 

How solution chemistry determines co-crystal thermodynamic stability... 

The criteria for co-crystal thermodynamic stability were presented in Section 11.3. An important feature of co-crystals is that they coexist in equilibrium with solution. This occurs when their molar free energy or chemical potential is equal to the sum of the chemical potentials of each co-crystal component in solution. Thus, the individual component chemical potentials in a solution saturated with co-crystal can vary as long as their sum is constant. In terms of activities, it is the activity product that is constant. 

Eutectic points and stability regions for co-crystals of carbamazepine-4-aminobenzoic acid in ethanol are shown in Figure 11.13(a) and (b). This system has two co-crystals with different stoichiometries (1 1) and (2 1) thus, there are four eutectic points. Variations in [drug] u relative to [drug] at saturation in the absence of co-former provide a measure of co-crystal and drug solubility dependence on solution phase interactions. Co-crystal thermodynamic stability can be quantitatively predicted by combining eutectic point measurements with equations that describe solution behavior. " ... 

Aeu dependence on external conditions is critical to establish co-crystal thermodynamic stability. For instance, > 1 for a 1 1 co-crystal or A eu >0.5 for a 2 1 co-crystal indicates that the co-crystal is thermodynamically unstable with respect to the drug. A decrease of below the co-crystal stoichiometric ratio, for example, < 1 for a 1 1 co-crystal or <0.5 for a 2 1 co-crystal indicates a reversal in the thermodynamic stability, where the co-crystal is more stable... 

As described in previous sections, solution phase interactions play an important role in co-crystal solubility. The influence is greater than for single component crystals (or their hydrates) since each co-crystal component will modify solution behavior to different extents depending on their interactions with the environment. Kinetic studies are useful when informed by co-crystal thermodynamic solubilities and their solution phase dependence. Simply adding a cocrystal to a solution and measuring drug concentration as a function of time may fail to capture important properties of the co-crystal and lead to inaccurate assessment of its performance. 

Thermodynamics and kinetics are the two most fundamental theoretical aspects of polymer crystallization. Thermodynamics addresses why, or under which circumstances, polymer crystallization will begin, or in the opposite direction polymer crystals will start to melt. Kinetics addresses how fast polymer crystallization will be initiated, be developed, and be further improved. Both aspects decide crystal morphologies that eventually influence the properties of polymer materials. 

A few comparisons of molecules which differ stereochemically are available. Aqueous-phase data on P-hydroxysulfoxides have been quoted which show that whereas the two racemic diastereomeric compounds display virtually identical liquid- and liquid-crystal-phase boundaries, they differ substantially in the temperature range of their crystal solubility boundaries [39]. In particular, the higher melting isomer displays the higher Krafft eutectic temperature. The situation is one in which the l drophilicity of the two isomers appears to be virtually identical—but their ability to pack into a crystal differs substantially as a result of their differences in shape. Molecular shape is very important to crystal thermodynamics. 

In the section on structure and fundamental properties of SPS, Chapter 9 summarizes the polymorphic behavior of this polymer, the structure of the different forms, and the crystallization and melting behavior. Chapter 10 describes co-crystals and nanoporous crystalline phases of SPS regarding preparation, structure, properties, and new interesting applications, for example, molecular sensors. The section concludes with Chapter 11 on selected topics of crystallization thermodynamics and kinetics of SPS. 

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