Twinkling fractal theory of the glass transition: Rate dependence and time–temperature superposition

The twinkling fractal theory (TFT) of the glass transition temperature T_g provides a new method of analyzing rate effects and time–temperature superposition in amorphous materials. The rate dependence of T_g was examined in the light of new experimental and theoretical evidence for the nature of the dynamic heterogeneity near T_g. As T_g is approached from above, dynamic solid fractal clusters begin to form and eventually percolate rigidity at T_g. The percolation cluster is a solid fractal and to the observer, appears to “twinkle” as solid and liquid clusters interchange in dynamic equilibrium with a vibrational density of states g(ω) ∼ ω. The solid-to-liquid twinkling frequencies ω_TF are controlled by the Boltzmann population of intermolecular oscillators in excited energy levels of their anharmonic potential energy functions U(x) such that ω_TF = ω exp −B(T*² − T²)/kT in which T* ≈ 1.2T_g. An oscillator changes from a solid to a liquid when a thermal fluctuation causes it to expand beyond its inflection point in the anharmonic potential. This leads to a continuous solid fraction P_s near T_g given by P_S ≈ 1−[(1 − p_c) T/T_g] where p_c ≈ 1/2 is the rigidity percolation threshold. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near T_g produces a continuous relaxation spectrum with many different length scales and times associated with the fractal clusters. The twinkling frequencies control the kinetics of T_g such that for a given observation time t when the rate γ > 1/t, only those parts of the twinkling spectrum with ω > γ can contribute to relaxation or percolation upto time t. The most important results in this article are as follows: The TFT describes the rate dependence of T_g, both for DSC thermal heating/cooling rates and DMA frequencies as the classic T_g − lnγ law as T_g(γ) = T_go + (k/2B) ln γ/γ_o in which the constant B = 0.3 cal/mol K². The constant B appears quite universal for the 17 thermoset polymers investigated in this study and 18 linear polymers investigated by others. Many other amorphous metal and ceramic glass materials exhibited the same rate law but required a new B value approximately half that for polymers. The same B = 0.3 value was also used to successfully describe the TTS shift factors using the twinkling fractal frequencies ω_TF = ωexp −B(T*² − T²)/kT, as ln a_T(TFT) = exp B(T_R² − T²)/kT, which gave comparable results with the classical WLF equation, log a_T = [−C₁(T − T_R)]/[C₂ + (T − T_R)]. The advantage of the TFT over the WLF is that C₁ and C₂ are not universal constants and must be determined for every material, whereas the TFT uses one known constant B which appears to be the same for all polymers. The TFT has also been found to describe the strong and fragile nature of the viscosity behavior of liquids and the rate and temperature dependence of the yield stress in polymers. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2578–2590, 2009     

Frequency spectra of disordered metals: especially liquid Rb

The phonon frequency spectrum g(ω) of a crystal, such as body centred cubic (bcc) Rb, is known to be characterized by the Van Hove singularities at ω?≠?0. However, for a liquid metal like Rb, g(ω) has a single, hydrodynamic-like singularity, namely a cusp ∝ ω ^(1/2), at ω?=?0. Here, we note first that computer simulation on liquid Rb near freezing has revealed a rather well-defined Debye frequency ω_D. Therefore, we propose here a zeroth-order model g ₀ (ω ) of g(ω) for Rb, which combines the Debye model with the ‘hydrodynamic’ ω ^(1/2) cusp. The corresponding velocity autocorrelation function 〈 v (t)·v (0)〉 has correctly a long-time tail ∝ t ^-(3/2). The terms from g ₀ (ω ) involving ω_D are then damped by weak exponential factors exp (-α _i t), and the resulting first-order approximation, g ₁ (ω ) say, to the frequency spectrum is found to have features in common with the molecular dynamics (MD) simulation form. Thus ω_D is fixed, as well as transport coefficients for the known thermodynamic state. The article concludes with a more qualitative discussion on supercooled liquids, and on metallic glasses such as Fe, for which MD simulations exist.     

Gelation Studies on a Radical Chain Cross‐Linking Copolymerization Process: Comparison of the Critical Exponents Obtained by Dynamic Light Scattering and Rheology

Summary: The sol–gel transition of a radical chain cross‐linking copolymerization system [N‐vinylcaprolactam/2‐hydroxylethyl methacrylate/allyl methacrylate] has been studied using in situ time‐resolved dynamic light scattering (DLS) and in situ rheology. A critical dynamic behavior was observed near the sol–gel transition, which was characterized by the presence of a power‐law spectra over three decades in the time–intensity correlation function g₂(t) − 1 ∼ t^−μ and over two decades in the oscillatory shear experiment G′(ω) ∼ G″(ω) ∼ ωⁿ. A comparison of the obtained critical exponents μ ≈ 0.62 and n ≈ 0.75 was made. The theory predicts a relationship between these exponents, but up to now no experimental comparison has been done. The experimental results favor the percolation model, with a fractal dimension d_f of the gel clusters of 1.67.

Double‐logarithmic plot of time–intensity correlation functions g₂(t) − 1 versus the delay time t.     

The physical properties of bisphenol-A-based epoxy resins during and after curing

A series of epoxy resins derived from diglycidyl ethers of bisphenol A with differing initial linear molecular chain lengths have been studied during and after curing with the diamines MDA (4,4′-methylene dianiline) and DDS (4,4′-diamino diphenyl sulfone). The properties that were measured during curing were the volume, the fictive temperature T_f, the gel fraction, the viscosity, and the equilibrium compliance. Graphs of T_f as a function the time of curing t_c obtained at four curing temperatures between 40 and 100°C have been reduced to a common curve. After curing, creep compliance curves J(t) were determined which characterize the viscoelastic response from the glassy compliance level to a rubbery equilibrium compliance level. The change in properties that occurs during the time-dependent spontaneous densification below the glass temperature T_g was monitored with repeated measurements of J(t). Time-scale shift factors as a function of volume contraction obtained during this physical aging below T_g were reduced to a common curve.     

Quasielastic light scattering study of chemically crosslinked gelatin solutions and gels

Quasielastic light scattering measurements are reported for experiments performed on mixtures of gelatin and glutaraldehyde (GA) in the aqueous phase, where the gelatin concentration was fixed at 5 (w/v) and the GA concentration was varied from 1×10⁻⁵ to 1×10⁻³ (w/v). The dynamic structure factor, S(q,t), was deduced from the measured intensity autocorrelation function, g ₂(τ), with appropriate allowance for heterodyning detection in the gel phase. The S(q,t) data could be fitted to S(q,t)=Aexp(−D _f q ² t)+Bexp(−t/τ_c)^β, both in the sol (50 and 60 ^∘C) and gel states (25 and 40 ^∘C). The fast-mode diffusion coefficient, D _f showed almost negligible dependence on the concentration of the crosslinker GA; however, the resultant mesh size, ξ, of the crosslinked network exhibited strong temperature dependence, ξ∼(0.5−χ)^1/5exp(−A/RT) implying shrinkage of the network as the gel phase was approached. The slow-mode relaxation was characterized by the stretched exponential factor exp(−t/τ_c)^β. β was found to be independent of GA concentration but strongly dependent on the temperature as β=β₀+β₁ T+β₂ T ². The slow-mode relaxation time, τ_c, exhibited a maximum GA concentration dependence in the gel phase and at a given temperature we found τ_c(c)=τ₀+τ₁ c+τ₂ c ². Our results agree with the predictions of the Zimm model in the gel case but differ significantly for the sol state. Received: 25 May 1999 /Accepted in revised form: 27 July 1999     

On the conjoint gradient correction to the Hartree—Fock kinetic and exchange energy density functionals

The kinetic and the exchange energy functionals are expressed in the form T[ρ] = C_TF∫ drρ^5/3(r)f_t(s) and K[ρ] = C_D∫ drρ^4/3(r)f_K(s), where C_TF = (3/10)(3π²)^2/3 and C_D = −(3/4)(3/π)^4/3 are the Thomas-Fermi and the Dirac coefficients, respectively, and s = |∇ρ(r)|/C_sρ^4/3(r), with C_s = 2(3π²)^1/3. These expressions are used to perform a comparison of f_T(s) and f_K(s) in terms of their generalized gradient expansion approximations. It is shown that f_κ(s) and is congruent to f_T(s) in the range characteristic of the interior regions of atoms and many solids and that the second-order gradient expansion of the kinetic energy provides a rather reasonable approximation to the generalized gradient expansion approximation of both the kinetic and the exchange energy functionals. © 1996 John Wiley & Sons, Inc.     

Kinetic model of the evaporation process of benzylbutyl phthalate from plasticized poly(vinyl chloride)

The kinetic model of the physical process of evaporation of plasticizer from plasticized PVC foils was developed from the results of isothermal thermogravimetric investigation of evaporation of benzyl-butyl phthalate in the temperature range 120-150 °C under nitrogen flow. The kinetic parameters were estimated by integral method of analysis. Mathematical modeling of the kinetic of plasticizers evaporation was performed on the basis of function c=f(T,t) and kinetic equation of evaporation −dc/dt=f(T,c₀,c(t)). The developed mathematical model was described by the general kinetic equation . The differential quotients δ(−dc/dt)/δT=f(T,c₀,c(t))=f(T,c₀,t) and δ(−dc/dt)/δc₀=f(T,c(t))=f(T,c₀,t) were performed, and mathematical definition of the changes of the evaporation rate constant with the change of temperature and the change of the initial plasticized concentration were discussed.     

Diffusion of nonionic azo dyes in nylon-6

Diffusion of 2,4-dinitroaniline and three nonionic azo dyes in Nylon-6 film was studied by analysis of the concentration-distance curves (profiles) of penetrants in the polymer. Actual diffusivities D_(c) of penetrants in polymer, diffusion coefficients as a function of the concentration C_f of penetrant in polymer, were calculated from the profile. It was found that D_(c) is almost constant or decreases gradually with decreasing C_f in the range of high-medium C_f but decreases appreciably with decreasing C_f at low C_f. The change in D_(c) with C_f was explained in terms of the dual-mode sorption-diffusion model. The penetrants diffuse in the polymer as two distinct species, i.e., a dissolved species and an adsorbed species. The former is the penetrant taken up by the polymer by a partition mechanism (dissolved species) and the latter is that taken up by Langmuir sorption (adsorbed species). The actual diffusivity D_P(c) of the dissolved species decreases with decreasing C_f. While the actual diffusivity D_L(c) of the adsorbed species normally increases gradually with decreasing C_f. D_P(c) is usually larger than D_L(c). © 1993 John Wiley & Sons, Inc.     

Thermochemical properties of free radicals from G3MP2B3 calculations

For a set of 32 selected free radicals, energy minimum structures, harmonic vibrational wave numbers ω_e, principal moments of inertia I_A, I_B, and I_C, heat capacities C°_p(T), entropies S°(T), thermal energy contents H°(T) ? H°(0), and standard enthalpies of formation Δ_fH°(T) were calculated at the G3MP2B3 level of theory in the temperature range 200–3000 K. In this article, thermodynamic functions at T = 298.15 K are presented and compared with recent experimental values. The mean absolute deviation between calculated and experimental Δ_fH°(298.15) values resulted in 3.91 kJ mol^?1, which is close to the average experimental uncertainty of ± 3.55 kJ mol^?1. The influence of hindered rotation on thermodynamic functions is studied for isopropyl and tert‐butyl radicals. © 2002 Wiley Periodicals, Inc. Int J Chem Kinet 34: 550–560, 2002     

Darstellung,Schwingungsspektren und Normalkoordinatenanalyse von Hexachlororhenat(V) sowie Kristallstruktur von [P(C6H5)4][ReCl6]

Preparation, Vibrational Spectra, and Normal Coordinate Analysis of Hexachlororhenate(V) and Crystal Structure of [P(C₆H₅)₄][ReCl₆] By oxidation of A₂[ReCl₆], A = [(n-C₄H₉)₄N]⁺, [P(C₆H₅)₄]⁺, with Cl₂ in dichloromethane/trifluoracetic acid A[ReCl₆] is formed. [P(C₆H₅)₄][ReCl₆] crystallizes with tetragonal symmetry, space group P4/n-C, a = 12.967(4), c = 7.6992(8) Å, Z = 2. The octahedral complexion [ReCl₆]^? is compressed (C_4v) with the bond lengths, axial Re? Cl1 = 2.28 and Re? Cl3 = 2.24 Å, equatorial Re? Cl2 = 2.31 Å. The infrared active antisymmetric Re? Cl stretching vibration is split into v′₃ = 346 an v″₃ = 326 cm^?1. The assignment of all IR and Raman modes is confirmed by a normal coordinate analysis. The different valence force constants, f_d(ReCl1) = 2.09, f_d(ReCl3) = 2.10, f_d(ReCl2) = 1.88 mdyn/ Å result from the distortion of the octahedron. On excitation with the Ar laser line 514.5 nm a resonance Raman spectrum is observed, showing 8 overtones of v′(A₁) = 382 cm^?1, from which the harmonic frequency ω₁ = 382.1 cm^?1, the anharmonicity constant X₁₁ = ?0.76 cm^?1, and the maximum bond dissociation energy of the [ReCl₆]^? ion to be 138 kcal/mol, are calculated. The vibrational fine structure of the intraconfigurational transitions in the near infrared has been resolved by measuring the absorption spectrum of [(n-C₄H₉)₄N][ReCl₆] at low temperature (10 K), resulting in the assignment of the following electronic origins: Γ₃(³T_1g) → Γ₄(³T_1g): 7 512, Γ₃(³T_1g) → Γ₁(³T_1g): 7 624 and Γ₃(³T_1g) → Γ₅(¹T_2g), Γ₃(¹E_g): 8 368 cm^?1.     

The thermodynamic functions of fullerene derivative (<Emphasis Type="Italic">t</Emphasis>-Bu)<Subscript>12</Subscript>C<Subscript>60</Subscript> over the temperature range from <Emphasis Type="Italic">T</Emphasis> → 0 to 420 K

The temperature dependence of heat capacity C _p ^o = f(T) of fullerene derivative (t-Bu)₁₂C₆₀ has been measured by a adiabatic vacuum calorimeter over the temperature range T = 6–350 K and by a differential scanning calorimeter over the temperature range T = 330–420 K for the first time. The low-temperature (T ≤ 50 K) dependence of the heat capacity was analyzed based on Debye’s the heat capacity theory of solids and its fractal variant. As a consequence, the conclusion about structure heterodynamicity is given. The experimental results have been used to calculate the standard thermodynamic functions C _p ^o (T), H ^o(T)−H ^o(0), S ^o(T) and G ^o(T) − H ^o(0) over the range from T → 0 to 420 K. The standard entropy of formation at 298.15 K of fullerene derivative under study was calculated. The temperature of decomposition onset of derivative was determined by differential scanning calorimetery and thermogravimetric analysis. The standard thermodynamic characteristics of (t-Bu)₁₂C₆₀ and C₆₀ fullerite were compared.     

The thermodynamic properties of 2-ethylhexyl acrylate over the temperature range from <Emphasis Type="Italic">T</Emphasis> → 0 to 350 K

The temperature dependence of the heat capacity C _p ^o= f(T) 2 of 2-ethylhexyl acrylate was studied in an adiabatic vacuum calorimeter over the temperature range 6–350 K. Measurement errors were mainly of 0.2%. Glass formation and vitreous state parameters were determined. An isothermic shell calorimeter with a static bomb was used to measure the energy of combustion of 2-ethylhexyl acrylate. The experimental data were used to calculate the standard thermodynamic functions C _p ^o(T), H ^o(T)-H ^o(0), S ^o(T)-S ^o(0), and G ^o(T)-H ^o(0) of the compound in the vitreous and liquid states over the temperature range from T → 0 to 350 K, the standard enthalpies of combustion Δ_c H ^o, and the thermodynamic characteristics of formation Δ_f H ^o, Δ_f S ^o, and Δ_f G ^o at 298.15 K and p = 0.1 MPa.     

Extraction of certain actinide and lanthanide elements from different acidic media by polyurethane foams loaded with di-(2-ethylhexyl)phosphoric acid (HDEHP)

The extraction of cerium(III) from weakly acidic chloride solutions by HDEHP-nitrobenzene-loaded polyurethane foams could be analyzed quantitatively in terms of the equation: log(9.056 D_c)=log K_c+2.14 log (C_d?6C_c)+3 pH+log f_c where D_c is the distribution ratio of cerium(III) between the foam and aqueous phases, C_d and C_c are the total HDEHP and Ce(III) concentrations on the foam, respectively, log f_c=[Ce³⁺]_(sq)/[ΣCe(III)]_(aq), and K_c is the equilibrium constant of the equation: Ce _(aq) ³⁺ +2.14(HX)_2.8(o) ? ? CeX₆·H_3(o)+3H _(aq) ⁺ . Values of K_c under the different extraction conditions tested are given.     

Thermodynamic properties of graphite-like nanostructures prepared by thermobaric treatment of fullerite C60

Temperature dependences of the heat capacities of disordered graphite-like nanostructures prepared by the thermobaric treatment of fullerite C₆₀ (p = 2 and 8 GPa, T = 1373 K) were measured in the temperature ranges from 7 to 360 K in an adiabatic vacuum calorimeter and from 330 to 650 K in a differential scanning calorimeter. At T < 50 K, the dependences obtained were analyzed using the Debye theory of the heat capacity of solids and its multifractal version. The fractal dimensions D were determined and some conclusions on the heterodynamic character of the structures studied were made. The thermodynamic functions C _p ^o T), H ^o(T) − H ^o(0), S ^o(T) − S ^o(0), and G ^o(T) − H ^o(0) were calculated in the temperature range from T → 0 to 610 (650) K. The thermodynamic properties of the graphite-like nanostructures studied and some carbon allotropes were compared. The standard entropies of formation Δ_f S ^o of the graphite nanostructures studied and diamond were calculated along with the standard entropies of the reactions of their synthesis from the face-centered cubic phase of fullerite C₆₀ and their interconversions at T = 298.15 K. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1940–1945, September, 2008.     

Thermochemical properties from G3MP2B3 calculations,Set‐2: Free radicals with special consideration of CH2CH?C•CH2, cyclo−•C5H5, •CH2OOH,HO?•CO,and HC(O)O•

After a set of 32 free radicals was presented (Int J Chem Kin 34, 550–560, 2002), an additional 60 free radicals (Set‐2) were studied and characterized by energy minimum structures, harmonic vibrational wave numbers ω_e, moments of inertia I_A, I_B, and I_C, heat capacities C^o_p(T), standard entropies S^o(T), thermal energy contents H^o(T) ? H^o(0), and standard enthalpies of formation Δ_fH^o(T) at the G3MP2B3 level of theory. Thermodynamic functions at T = 298.15 K are presented and compared with recent experimental values where these are available. The mean absolute deviation between calculated and experimental Δ_fH^o(298.15) values by the previous set of 32 radicals is 3.91 kJ mol^?1. For the sake of comparison, only 49 species out of the 60 radicals of Set‐2 are characterized by experimental enthalpies of formation, and the corresponding mean absolute deviation between calculated and experimental Δ_fH^o(298.15) values is 8.96 kJ mol^?1. This situation is cause for demand of more and also more accurate experimental values. In addition to the above properties, parent molecules of a large set of the respective radicals are calculated to obtain bond dissociation energies D^o(298.15). Radical stabilization owing to resonance is discussed using the complete sets of total atomic spin densities ρ as a support. In particular, a short review about recent developments of the first‐order Jahn–Teller radical c‐C₅H₅^? is presented. In addition, radicals with negative bond energies are described, such as ^?CH₂OOH where the reaction path to CH₂O + HO^? has been calculated, as well as radicals which have two different parent molecules, for example C?N? O^?. For the reaction HO^? + CO → H^? + CO₂, two reaction paths are characterized by a total of 14 stationary points where the intermediate radicals HO? ^?CO and HC(O)O^? are involved. © 2004 Wiley Periodicals, Inc. Int J Chem Kinet 36: 661–686, 2004     

Predictions of a recently proposed non-markovian model for polymer melts and concentrated solutions

Several predictions for a recently proposed mesoscopic model for polymer melts and concentrated solutions is presented. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by a fractal “waiting-time” distribution function, with a single adjustable parameter β, which describes the long-time behavior of the distribution: ∼ 1/t^1+β. We find that the model predicts D ∼ 1/N² and η₀ ∼ N^3.4 for β ≈︁ 1.4, where N is the degree of polymerization. The generalized model predicts that the relaxation spectrum has a plateau regime whose height is independent of N, but whose width is strongly N dependent, in agreement with experiment. The model also predicts that rings will diffuse somewhat more slowly than linear chains of the same molecular weight (about 80% as fast), with the same scaling dependence on N as linear chains, also in agreement with preliminary data.     

The impact of higher polarization basis sets on molecular ab initio results.: III. The ground state of Cl2 in comparison with other diatomics

The ground state potential curve of Cl₂ has been computed near R_c by means of the SCF, MC SCF, CI(SD), and the recently proposed CPF methods. The convergence of the total energy, of D_c and R_c is studied with the aid of computations for various basis sets which include up to three d, two f and one g set. Higher polarization functions have a larger effect than for F₂ for all methods, the g set still affects D_c by 0.15 eV and R_c by 0.02 a₀ on the CPF level. The most elaborate calculation, on the CPF [7,4,3,2,1] level, yields D_c and R_c with an accuracy of 0.08 eV and 0.02 a₀. The same accuracy is obtained for the MC-178 CAS SCF treatment employing a 2d1f polarization basis set. The present results allow us to order the polarization functions according to their relative importance as: d⁽¹⁾ > f⁽¹⁾ > d⁽²⁾ > f⁽²⁾ = d⁽³⁾ > g⁽¹⁾ for the SCF and d⁽¹⁾ > f⁽¹⁾ > d⁽²⁾ > g⁽¹⁾ > f⁽²⁾ = d⁽³⁾ for methods including external correlation. CI(SD) or CPF. A comparison of the results for N₂, F₂, P₂, and Cl₂ shows the higher polarization functions d⁽³⁾, f⁽²⁾, g⁽¹⁾ to contribute 0.1 (F₂) to 0.3 eV (F₂, Cl₂) to D_c and affect R_c by 0.005 a₀ (N₂) to 0.01 7 a₀ (P₂).We have focused our attention mainly on the impact of the atomic basis set incompleteness on the accuracy of results obtained from various methods of computation. Cl₂ turns out to be more demanding than the first-row counterpart F₂ on all levels of theory, which is mainly due to the slower angular convergence rate for E, D_c and R_c (for Cl₂), tables 2 and 3. The more pronounced influence of higher polarization functions is already noticed on the SCF and MC SCF level: f⁽¹⁾ increases D_c by 0.18 eV and reduces R_c by 0.052 a₀ on the MC-178 level for Cl₂, table 6, typical corresponding results for F₂ are only 0.04 eV and 0.013 a₀ [1]. CAS SCF calculations furthermore appear to require larger active spaces for Cl₂, as discussed in section 3.3.On the CI(SD) or CPF level — which aim to account for the entire external correlation — one even finds a pronounced influence of the first g set which contributes ≈ 0.15 eV to D_c and reduces R_c by ≈ 0.02 a₀ (on the CPF level, table 3), the corresponding effects for F₂ were only ≈ 0.04 eV and 0.01 a₀ [1]. The 2d1f polarization basis, which will remain the “standard” large basis for treatments of tri- and tetraatomic molecules, appears to underestimate D_c by still 0.5 eV and to overestimate R_c by ≈ 0.02 a₀ for P₂ and Cl₂, table 7, and probably all molecules in-between. This conclusion emerges from the cumulative effect of adding d⁽³⁾, f⁽²⁾ and g⁽¹⁾ which amounts already to 0.3 eV and 0.015 a₀, table 7.     

Folded chain crystals as micro-phases

Summary A thermodynamic treatment of homo-polymer systems out of linear chains with folded chain crystals is developed outgoing from appropriate models for single component systems. An expansion of thermodynamics to multi-micro-phase systems the structure of which is partially or totaly frozen is indispensable. General properties of melt crystallized homopolymers with folded chain crystals can be recognized indeed when the thermodynamic formalisms developed are applied.

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Zusammenfassung Das Schmelzen in polymeren Einteilchensystemen mit Faltungskristallen einheitlicher Dicke kann thermodynamisch als Umwandlung 1. Ordnung in einer Richtung behandelt werden, wenn die Faltungslänge bis zur Umwandlungstemperatur konstant bleibt (Faltungslänge als innerer Zusatzparameter). Eine wesentliche begriffliche Erweiterung ist für eine phänomenologische Beschreibung mit den Mitteln der Thermodynamik unumgänglich, wenn eine Faltungskristallit-Dickenverteilung existiert, weil dann prinzipiell nur noch partielle Koexistenz bestimmter Fraktionen metastabiler autonomer Mikrophasen mit der Schmelze möglich ist. Partielles Aufschmelzen und Rektistallisation können so dann auch in Betracht genommen werden. Die entwickelten Konzeptionen bewähren sich in der Anwendung auf bekannte Experimente.

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Notation g ^c (y);g ^m (Y) molar Gibbs-free energy of a chain of a lengthy within an extended chain crystal and the melt rsp - g _o ^c ;g _o ^m molar free enthalpy of the unit in the crystal lattice and the melt rsp - g(y,y, f) molar Gibbs-function of an ideally folded chain crystal with the fold heighty _f - gco(y, y _ef,y _f) molar free enthalpy of the crystal corey _co - g ₀ ^ex ((y_ef) excess free enthalpy of the longitudinal layers of folded chain crystals - g ^f(y_ef,g _o ^ex ) molar free enthalpy of the longitudinal layers of the folded chain crystals - g ^tot molar free enthalpy of a chain of the lengthy within a folded chain crystal with longitudinal layers - h _o ^1c ,h _o ^m molar enthalpy of the chain unit within the crystal lattice and the melt rsp - h =h _o ^m -h _o ^c molar heat of fusion of the unit - C _p=C _p ^m -C _p ^c difference of the molar specific heat of a unit within the melt and within the chain crystal - h ^D molar defect enthalpy of local defects within the crystal lattice - h ^D molar defect enthalpy of the unit - s _o ^c ,s _o ^m molar entropy of the chain unit within the crystal lattice and the melt rsp - s _c ^m conformational entropy of a chain in the melt - s _gk conformational entropy of a chain of lengthy within a super-lattice as indicated in figure 5, - s molar entropy of fusion of the melt - s _n ^c nematic configurational entropy - T absolute temperature - T _M melting temperature of extended chain crystals of infinite size - T _M(y) melting temperature of extended chain crystals containing only chains of the lengthy - T _M (y, y _f) melting temperatureof folded chain crystals of the thicknessy _f composed of chains of the lengthy - T _M(y _f) melting temperature of folded chain crystals of the thicknessy _fy - _eh excess free enthalpy of the chain ends occupying crystallographic places - _ef excess free enthalpy of a single fold loop - z coordination number of the lattice - 7 Euler's constant - R Boltzmann's constant - y number of chain units - y _f height of lamelliform folded chain crystals - f=(y/y _f - 1) number of fold loops of a chain of a lengthy when being built into a folded chain crystal of the thicknessy _f - y _co thickness of the crystal core of the simplified twophase model - y _et average thickness of the surface layers of folded chain crystals - N _c number of crystallized units of a chain of the lengthy - x _c molar number of crystallized units of a chain of the lengthy - x _nc molar number of noncrystallized units - excess free enthalpy parameter - (y _f) thickness distribution of the fold heightsy _f With 15 figures and 2 tables     

Thermodynamics of C70 fullerence in the 0–390 K temperature range

The temperature dependence of heat capacity of C₇₀ fullerene was studied by calorimetry in the range between 6 and 390 K. Phase transitions were established and their thermodynamic characteristics were determined. From the experimental data obtained, the thermodynamic functionsH ^o (T)-H ^o(0),S ^o(T),G ^o(T)-H ^o(0) for temperatures between 0 and 390 K were calculated. The results were used to calculate the standard values of Δ_f S ^o, Δ_f G ^o, and logK _f ^o for the formation of C₇₀ from graphite. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 647–650, April, 1998.