Application of Continuous Thermodynamics to the Stability of Polymer Systems. III

Based on the stability theory of continuous thermodynamics for polymer solutions, necessary and sufficient conditions for multiple critical points are derived assuming the segment-molar excess Gibbs free energy to be independent of the distribution function. Equations for calculating double and triple critical points are given. Higher order critical points may be obtained in a successive way. For polymers possessing a Schulz-Flory molecular weight distribution, general conditions for an m-fold critical point are presented.     

APplication of Continuous Thermodynamics to the Stability of Polymer Systems. II

A determinant criterion for the critical state in solutions and mixtures of polydisperse polymers is established within the general framework of Gibbs theory. The treatment continues an earlier paper by considering more general Gibbs free energy relations: The function replacing the x-term in the classic Flory-Huggins equation is permitted to depend on a finite number of moments of the polymer distribution(s) so as to embrace most Gibbs free energy relations of practical use. The new criterion leads to a very large reduction of computer time and of needed storage capacity compared to the traditional Gibbs determinant criterion. Some relations known from the literature are shown to be special cases of the established new criterion.     

Enthalpy distributions and densities of states in linear polymers

Using polypropylene as an example, we applied a method we have recently developed to calculate the probability distribution of enthalpy from the temperature dependence of the heat capacity. The method involves the use of local temperature expansions of the heat capacity to calculate a set of moments of the enthalpy distribution. Using the maximum‐entropy method, one can then construct the enthalpy distribution for the system. The method is completely model free. The enthalpy distribution so obtained is the analogue of the Maxwell–Boltzmann distribution of kinetic energies for a gas, and like that function, tells one the distribution of enthalpies that an average unit in the polymer chain can have, a quantity that is crucial to understanding the chemical and physical properties of a polymer. Given the enthalpy distribution, one can then calculate the Gibbs free energy and the density of states that correspond to a particular value of enthalpy, thus giving one an expanded thermodynamics of the system in addition to the usual average quantities. We illustrate the fact that the Gibbs free‐energy distribution for this system scales as a simple function of temperature and that the density‐of ‐states function yields a simple empirical partition function for the system giving both the average thermodynamics and the distribution functions. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1513–1518, 2001     

Influence of Molar Mass Distribution on the Compatibility of Polymers

Abstract

Phase equilibria were calculated by means of a new method (direct minimization of the Gibbs energy of mixing) for polymer blends consisting of monodisperse polymer A and polydisperse polymer B. The results obtained for a Schulz-Flory distribution of B (molecular nonuniformity U = (M _w/M _n) ?1 = 1 and 100 components of model B) agree quantitatively with that of computations on the basis of continuous thermodynamics. The influence of U _B on the miscibility of A and B in 1:1 mixtures was studied for constant M _w of B, quantifying the incompatibility of the polymers by the length of the tie lines. The outcome of these calculations demonstrates that the typical effect of an augmentation of U _B (keeping M _w and the overall composition constant) consists in an enlargement of the mutual solubility of A and B. However, for an almost compatible pair of polymers (i.e., interaction parameters g are only slightly larger than the critical values for U _B = 0), this statement remains true only in the case of sufficiently small U _B. In order to gain some understanding of these findings, calculations were also performed for ternary systems (A and two species B). They demonstrate that it is the distance of the overall composition in the Gibbs phase triangle to the critical line (connecting the critical points for different U _B) which governs the changes in compatibility. Normally the critical point comes closer to the overall composition as U _B is raised, except for low g values where the critical point — after an initial approach — drifts apart as U _B becomes larger.     

Polydispersity Effects on the Demixing Behavior of Poly(Vinyl Methyl Ether)/Polystyrene Blends

Phase equilibrium calculations for solutions or mixtures of synthetic polymers become considerably more difficult when there is polydispersity of the polymers. To simplify the calculations, polydispersity is often neglected in the calculations or accounted for in a summary way only, and often only relatively simple free energy relations are applied. For example, Halary et al. published experimental demixing data on poly(vinyl methyl ether)/polystyrene blends. In evaluating the data the following assumptions were made: 1) the minimum of the demixing curve equals the critical point, 2) the X-parameter is independent of concentration and molecular weight, 3) the polydispersity may be roughly taken into account by using the formulas for monodisperse polymers and using the weight-average molecular weight. Continuous thermodynamics proves to be a suitable method to overcome the difficulties caused by polydispersity. Therefore, this method permits one to obtain detailed information on the phase equilibria in polymer solutions and in polymer blends in a relatively easy way. To show this, the data of Halary et al. are reanalyzed by means of continuous thermodynamics. In this way, more profound knowledge may be obtained from the experimental material, e.g., a more precise determination of the critical point and a more correct location of the spinodal.     

Hybridization thermodynamics of DNA bound to gold nanoparticles

Isothermal Titration Calorimetry (ITC) was used to study the thermodynamics of hybridization on DNA-functionalized colloidal gold nanoparticles. When compared to the thermodynamics of hybridization of DNA that is free in solution, the differences in the values of the Gibbs free energy of reaction, Δ_rG°, the enthalpy, Δ_rH°, and entropy, Δ_rS°, were small. The change in Δ_rG° between the free and bound states was always positive but with statistical significance outside the 95% confidence interval, implying the free DNA is slightly more stable than when in the bound state. Additionally, ITC was also able to reveal information about the binding stoichiometry of the hybridization reactions on the DNA-functionalized gold nanoparticles, and indicates that there is a significant fraction of the DNA on gold nanoparticle surface that is unavailable for DNA hybridization. Furthermore, the fraction of available DNA is dependent on the spacer group on the DNA that is used to span the gold surface from that to the probe DNA.     

The interaction between n-alkyl trimethylammonium bromides with poly(l-aspartate): a thermodynamics study

Specific conductivities of a homologous series of n-alkyl trimethylammonium bromides (C₈,C₁₀,C₁₂ and C₁₄TABs) in the presence of poly(l-aspartate) in glycine buffer at pH 3.2 and 25 ^∘C have been measured over a range of C _n TAB concentrations. From the conductivity changes, the number of surfactant molecules absorbed onto the polymer, the Gibbs free energies of adsorption and the equilibrium constants have been calculated. A statistical thermodynamics analysis was used to obtain the Gibbs free energies of adsorption. The results obtained using both methods are compared and analysed. Received: 15 December 1999 Revised form: 29 February 2000 Accepted: 3 March 2000     

Nucleation kinetics of a new nonlinear optical crystal

The induction period of various proportion of urea–thiourea zinc chloride crystal in water has been measured experimentally using the visual observation method. The induction period, which is inversely proportional to the nucleation rate, has been used to estimate the interfacial tension between the urea–thiourea zinc chloride and water; hence, the nucleation parameters like critical radius (r*), number of molecules in the radius (r*) and Gibbs free energy change for the formation of a critical nucleus (∆G*) have been calculated.     

Molecular-mechanics calculations of the geometry and racemization energies of an N-aryl-2(1H)-quinolone and N-aryl-4-pyridones

The geometries and energies for ground and possible transition states of the quinolone (1) and pyridones (2)–(4) were calculated by the molecular-mechanics method. The calculated energy differences between ground and the lowest transition state are in good correlation (r = 0.994) with the corresponding experimental racemization energies for interconversion of enantiomers (P) (M) in (1) – (4). However, the calculated potential energy differences do not correspond to measured differences in Gibbs energies. The fact was tentatively attributed to neglection of the entropy contribution to Gibbs energies.     

A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures

First-order phase transitions of binary mixtures at the given pressure (P) and temperature (T) are studied by taking into account the composition fluctuations. Isothermal-isobaric semigrand canonical ensemble is adopted to find the relations among the total number of molecules, the composition fluctuations and Gibbs free energy density. By combining two identical subsystems of mixtures successively, the free energy density is transformed until being stable and its linear segments represent phase transitions. A new method is developed to calculate the phase equilibriums of binary mixtures. The method handles multiple types and number of phase equilibriums at single time and its solutions are physically justified. One example is shown for calculating the phase diagram of binary Lennard-Jones mixture. It demonstrates that the fluctuations of the total number of molecules in mixtures are fundamental behind phase transitions and the van der Waals loops in Gibbs free energy are reasonable.     

Cloud-Point Curves and Coexistence Curves of Several Polydisperse Polystyrenes in Cyclohexane

Cloud-point curves, critical points, and coexistence curves with feed concentrations close to the critical concentration were measured in three systems involving cyclohexane + different polydisperse polystyrenes. The shape of the coexistence curves is analyzed by using a scaling expression. In two systems the critical exponent β possesses values somewhat larger than in actual binary systems (where β ≈ 1/3), whereas in the third system a somewhat smaller value is found. By using a three-parameter Gibbs free energy relation, cloud-point curves and coexistence curves are calculated from the critical point data and from the slope of the cloud-point curve at this point. To account for polydispersity, the method of continuous thermodynamics is applied. The cloud-point curves are well described, but the prediction of the coexistence curves is bad due to the mean-field character of the Gibbs free energy relation resulting in β = 1/2. Hence, the often used practice of fitting the parameters of a mean-field Gibbs free energy relation to the critical point and to some cloud points and then to calculate the coexistence data is to be considered with great care.     

A method to estimate the Gibbs free energy of non-equilibrium alloys by thermal analysis

A method has been developed to estimate the Gibbs free energy $ \left( {G_{\text{S}}^{\text{NE}} } \right) $ of the non-equilibrium solid alloys with multicomponents based on differential scanning calorimetry (DSC) analysis. In this method, the DSC curves of the non-equilibrium and equilibrium alloys during heating up to fully melting and those of the alloys during solidifying were measured. Then the thermal effects of the solid phase transformations from non-equilibrium to equilibrium states and the equilibrium solidification could be calculated. By evolving the traditional equal-G curve principle to equal-G point, the Gibbs free energy of the equilibrium solid alloy with multicomponents could be obtained on condition that the free energy of the liquid alloy was known. Considering the thermal effects of the solid phase transformations from non-equilibrium to equilibrium states, the Gibbs free energy value of the non-equilibrium alloys with a given composition could be achieved although the phase constitution of the equilibrium solid alloys and the Gibbs free energy of each phase were not known, and the calculation errors could be reduced by dividing the alloys into many infinitesimal virtual pure metals. The Gibbs free energy of the non-equilibrium Al?CSi?CMn alloys was calculated by using this method, confirming the validity of this method.     

Partial miscibility of liquid mixtures of protonated and deuterated high polymers

Certain low-angle neutron scattering experiments on bulk protonated polymers containing a fraction of the deuterated species (or vice versa) suggest that clustering of like isotopes is occurring. In order to see whether this is likely, a form for the Helmholtz free energy of mixing is proposed, using thermodynamics and the ideal solution (Flory-Huggins) entropy of mixing. From this free energy the upper consolute point T_c is found. T_c is calculated for several isotopic molecular solutions. The results suggest that while isotope segregation phenomena are impossible for small hydrocarbons, they may become important for large polymer systems.     

Reaction pathways of propene pyrolysis

The gas‐phase reaction pathways in preparing pyrolytic carbon with propene pyrolysis have been investigated in detail with a total number of 110 transition states and 50 intermediates. The structure of the species was determined with density functional theory at B3PW91/6‐311G(d,p) level. The transition states and their linked intermediates were confirmed with frequency and the intrinsic reaction coordinates analyses. The elementary reactions were explored in the pathways of both direct and the radical attacking decompositions. The energy barriers and the reaction energies were determined with accurate model chemistry method at G3(MP2) level after an examination of the nondynamic electronic correlations. The heat capacities and entropies were obtained with statistical thermodynamics. The Gibbs free energies at 298.15 K for all the reaction steps were reported. Those at any temperature can be developed with classical thermodynamics by using the fitted (as a function of temperature) heat capacities. It was found that the most favorable paths are mainly in the radical attacking chain reactions. The chain was proposed with 26 reaction steps including two steps of the initialization of the chain to produce H and CH₃ radicals. For a typical temperature (1200 K) adopted in the experiments, the highest energy barriers were found in the production of C₃ to be 203.4 and 193.7 kJ/mol. The highest energy barriers for the production of C₂ and C were found 174.1 and 181.4 kJ/mol, respectively. These results are comparable with the most recent experimental observation of the apparent activation energy 201.9 ± 0.6 or 137 ± 25 kJ/mol. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

芳酰胺-吖啶分子钳对中性分子的识别性能. II. 新型吖啶类分子钳的研究

采用差紫外谱法研究了新型芳酰胺-吖啶分子钳(1～7)对苯胺、苯二胺(邻, 间, 对)等中性分子的识别性能. 测定了结合常数(K_a)和自由能变化(ΔG°), 结果表明, 所有的分子钳受体与所考察的客体分子均形成1∶1型超分子配合物. 识别作用的主要推动力为多重氢键、van der Waals等的协同作用. 主客体间尺寸/形状匹配、几何互补等因素对识别性能均有重要的影响. 利用核磁氢谱与计算机模拟作为辅助手段对主要的实验结果与现象进行了解释.     

Morphology of PU/PMMA hybrid particles from miniemulsion polymerization: Thermodynamic considerations

The morphology of PU/PMMA hybrid particles prepared by miniemulsion polymerization was predicted through the consideration of their Gibbs free energy changes. Five morphological states of PU/PMMA hybrid particles were proposed and their Gibbs free energy changes were calculated. Before the formation of hybrid particles, the initial state included a monomer mixture of PU prepolymer, MMA, a chain extender, TMP, and an initiator, which was in droplets suspended in water containing SDS. Two assumptions were made. First, the densities of all states were the same. Secondly, secondary nucleation of particles was negligible. Thus the size of initial droplet and final particle was unchanged through miniemulsion polymerization. The interfacial tensions were measured by a pendant drop method and were used for calculation. The preferred morphology of PU/PMMA hybrid particle had the minimum value of ΔG_phase. Different NCO/OH ratios of PU and initiators of MMA were used to study the morphological change of PU/PMMA hybrid particles. When BD was used as the chain extender of PU, the hybrid particles showed the PU‐rich phase as the shell and PMMA‐rich as the core. When incorporating bisphenol A into PU polymer, the homogeneous structure of hybrid particle was preferred. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3359–3369, 2007     

Statistical thermodynamics in the framework of the lattice fluid model, 2. Binary mixture of polydisperse polymers of special distribution

In this paper, the Gibbs free energy, the equation of state and the chemical potentials of polydisperse multicomponent polymer mixtures are derived. For general binary mixtures of polydisperse polymers, we also give the Gibbs free energy, the equation of state and the chemical potentials and derive the stability criteria and spinodal. Furthermore, binary polydisperse polymer mixtures of special distribution, i.e., Flory distribution, uniform distribution and Schulz distribution, are discussed and the influence of polydispersity on the interaction energy parameter is considered. For these special-distribution systems, the spinodal curves are simulated and the influence of chain length and polydispersity on the spinodal curves is discussed. The results suggest that the spinodal temperature of the mixture with a given volume fraction of one component decreases with increasing polydispersity and the extent of the shift decreases with increasing degree of polymerization when η = M̄_w/M̄_n is given. In addition, the variations of the spinodal curves with polydispersity and chain length are shown and they are qualitatively compared with the experimental results.     

Multiple discrete energy levels and the bistable state of weak anchoring NLC cells

The weak anchoring nematic liquid crystal (NLC) cell is investigated with regard to energy. Because the Gibbs free energy of liquid crystal system used in theory does not include temperature and entropy, and because the equations and boundary conditions for δG=0 are also the mechanical equilibrium conditions of the continuum, the Gibbs free energy G is equivalent to the energy E of the liquid crystal continuum. There are multiple solutions which satisfy these equations and boundary conditions, each solution corresponding to a certain energy value. We call these discrete energy values and energy levels. Adopting a simple liquid crystal cell model, the energy levels are calculated in detail by means of analytical and numerical methods. The results show that there are three energy levels (or more in certain cases). The values and sequence of the energy levels are related to the external field and anchoring parameters. The relationships between the energy level structure and the bistable. Fréedericksz transition are disussed, together with their influence on the response time. The physical condition for the existence of more than three energy levels is also given.     

Reaction paths of BCl3?+?CH4?+?H2 in the chemical vapor deposition process

The reaction paths in the chemical vapor deposition preparation of boron carbides with BCl₃?CCH₄?CH₂ precursors were investigated theoretically in detail with a total number of 82 intermediates (IM) and 118 transition states (TS). The geometries of the species were optimized with B3PW91/6-311G(d,p) method and the TS as well as their linked IM were confirmed with the frequency and the intrinsic reaction coordinates analyses at the same theoretical level. The energy barriers and the reaction energies were determined with the accurate model chemistry method G3(MP2) after a diagnosis of the non-dynamic electronic correlations. The heat capacities and entropies were obtained with statistical thermodynamics. The Gibbs free energies at 298.15?K for all of the reaction steps were reported and the data at any temperature can be developed with the classical thermodynamics by using the fitted (as a function of temperature) heat capacities. All the possible elementary reactions, including both direct decomposition and the radical attacking dissociations for each reaction step were examined. It was found that there are nine reaction steps in the lowest reaction pathway to produce the final boron carbide and five steps to produce boron. The highest energy barrier in the lowest reaction pathway is 238.6?kJ?mol^?1 at 298.15?K and 346.0?kJ?mol^?1 at 1,200?K for producing BC, and is 294.7?kJ?mol^?1 at 298.15?K and 314.2?kJ?mol^?1 at 1,200?K for producing B.     

An Energy Interconversion Principle Applied in Reaction dynamics for the determination of equilibrium standard states

Chemical and other reaction theories involving thermodynamical equilibrium states utilize statistical mechanical equilibrium density distributions. Here, a definition of heat-work transformation termed thermo-mechanical coherence is first made, and it is conjectured that most molecular bonds have the above heat-work transformation property, which models a chemical bond as a “centrifugal heat engine”, where the internal energy state need not correspond to any of the standard equilibrium densities. Expressions are derived for the standard Gibbs free energy, enthalpy, and entropy where the bond coordinates need not conform to a non-degenerate Boltzmann state, since bond breakdown and formation are processes that have direction, whereas equilibrium distributions are derived when the Hamiltonian is of fixed form, which is not the case for chemical reactions using localized Hamiltonians. The empirically determined Gibbs free energy from a known molecular dynamics simulation of a dimer reaction , accords rather well with the theoretical estimate. A relation connecting the rate of reaction with the equilibrium constant and other kinetic parameters is derived and could place the commonly observed linear relationship between the logarithms of the rate constant and equilibrium constant on a firmer theoretical footing. These relationships could include analogues of the Hammett correlations used extensively in physical organic chemistry, as well as others which are temperature dependent. One prediction of the principles developed here is that the equilibrium standard reaction free energy is more dependent on the height of the intermolecular potential than its depth, so that the sign of the ΔG ^θ can change for varying barrier height with fixed well depth, which may appear counter-intuitive. All the above developments can be tested directly in simulations and therefore provides a fertile ground for further research with significant implications on how standard states are determined in relation to the direction of chemical reaction.This work treats the molecular bond using standard thermodynamics as if it were a system, and it is anticipated that with the advent of single-molecule science and experiment, that might be one direction in which molecular statistical thermodynamics would develop.