Tarasov Equation Research Articles

Prediction of the heat capacity of main-chain-type polymers below the glass transition temperature

This study predicts the absolute values of heat capacities from the molecular formula per monomer for main-chain-type polymers below the glass transition temperature. The frequencies of the skeletal and group-vibration modes are calculated using the Tarasov and Einstein equations, respectively, and the heat-capacity differences at constant pressure and constant volume are used to correct the predicted heat capacity. The contributes of skeletal vibrations to the heat capacity can be expressed by one- and three-dimensional Tarasov equations, and the contribution of group vibrations can be determined by summing the group-vibration heat capacities for functional groups and atoms constituting the monomer as obtained from the Einstein equation. The absolute value of the heat capacity is predicted from this combination of equations. The heat capacities of poly(4-methyl-1-pentene) are predicted within an error range of 8.0% from 90 to 180 K and ±2.0% from 180 to 300 K. The heat capacities of poly(vinyl benzoate) are within ±2.0% agreement from 190 to 350 K, while for poly(1,4-butylene adipate), the agreement is within ±2.0% from 80 to 200 K. This study predicted the absolute values of the heat capacities from the molecular formula per monomer for a main-chain-type polymer below the Tg. The calculations combined the Tarasov equation, the Einstein equation, and the (Cp − CV) correction term, accounting for the degrees of freedom of the monomer unit. The difference of predicted and experimental heat capacities of poly(4-methyl-1-pentene) was within 8.0% from 90 to 180 K and within ±2.0% agreement from 180 to 300 K. For poly(vinyl benzoate), the values were within ±2.0% from 190 to 350 K, and for poly(1,4-butylene adipate), they were within ±2.0% from 80 to 200 K. The predicted heat capacity was accurate, especially in the high-temperature region above 180 K.

Heat capacities of polymer solids composed of polyesters and poly(oxide)s, evaluated below the glass transition temperature

We evaluated the absolute values of the heat capacity of polymer solids composed of polyesters and poly(oxide)s below the glass transition temperature. The frequencies of the skeletal and group-vibration modes were calculated by the Tarasov and Einstein equations, respectively. Furthermore, the estimated heat capacity was corrected by the difference between the heat capacities measured at constant pressure and at constant volume. The heat capacity contributing to the skeletal vibration can be expressed by one- and three-dimensional Tarasov equations, and the contribution of the group vibration can be determined by substituting the absorption frequency obtained from infrared absorption measurements for the frequency value in the Einstein equation. In this combination of equations, the absolute value of the heat capacity was obtained with only three fitting parameters. We thus reproduced the measured heat capacities of eight polyester and five poly(oxide) polymer solids with a carbon and oxygen backbone. The reproduced and experimental heat capacities of all samples except polycaprolactone and poly(3,3-bis(chloromethyl)oxetane) agreed within ±2.5%, and the errors for polycaprolactone and poly(3,3-bis(chloromethyl)oxetane) were within ±4.0%. To better understand the heat capacities of polymer solids and considering that more than a dozen types of data on the absolute heat-capacity values of polymer compounds already exist, we evaluated the heat capacities of eight polyesters and five poly(oxide)s polymer solids with a carbon and oxygen backbone. Our calculations combined the Tarasov equation, the Einstein equation, and the (Cp − CV) correction term, accounting for the degrees of freedom of the monomer units. Using the combined Tarasov and Einstein equations, the heat capacities of the analyzed polymer solids were reproduced by only three fitting parameters. The reproduced and experimental heat capacities of all samples except polycaprolactone and poly(3,3-bis(chloromethyl)oxetane) agreed within ±2.5%, and the errors for polycaprolactone and poly(3,3-bis(chloromethyl)oxetane) were within ±4.0%.

Evaluation of the heat capacity of amorphous polymers composed of a carbon backbone below their glass transition temperature

Despite being a useful variable for the quantitative analysis of the thermodynamic properties of polymers, the absolute value of heat capacity remains poorly understood for amorphous polymers. This study evaluates the absolute values of the heat capacities of amorphous polymers by using the Tarasov and Einstein equations to calculate the frequencies of the skeletal and group vibration modes, respectively. Moreover, the difference between the heat capacity measured at constant pressure and that obtained at constant volume is added as a correction factor when estimating the heat capacity. The heat capacity that contributes to skeletal vibrations can be expressed using the one- and three-dimensional Tarasov equations, and the contribution of group vibrations can be determined by substituting the absorption frequency obtained from infrared absorption measurements into the frequency value of the Einstein equation. Only three fitting parameters were used to reproduce the absolute value of the heat capacity obtained from the combination of these equations. We used this approach to reproduce the heat capacities of 16 main chain-type amorphous polymers having a carbon backbone. The reproduced and experimental heat capacities of all the samples below their glass transition temperature agreed within less than ±2.5%. With the aim of gaining more insight into the heat capacity of amorphous polymers and considering that there are more than a dozen types of data on the absolute value of the heat capacity of polymer compounds, we evaluated the heat capacity of 16 main-chain-type amorphous polymers composed of a carbon backbone using a combination of the Tarasov equation, Einstein equation, and the (Cp − CV) correction term by taking into account the degree of freedom of monomer units. We found that the heat capacity of the analyzed amorphous polymers could be reproduced using only three fitting parameters from a combination of the Tarasov and Einstein equations below glass transition temperature.

Vibrational heat capacity of the linear 6,4-polyurethane

The calculated vibrational heat capacity of the linear 6,4-polyurethane (6,4-PU) in the solid state was estimated using its vibrational motions. The experimental low-temperature heat capacity of 6,4-PU collected from literature was related to approximate vibrational spectra based on the group and skeletal vibration contributions of 6,4-PU using the Advanced Thermal Analysis System (ATHAS) Data Bank. The group vibrational heat capacity was calculated based on ninety molecular vibrational motions derived from Raman and infrared spectroscopy. The skeletal vibrational heat capacity was described by a general Tarasov equation with Debye temperatures of Θ1 = 452 K, and Θ2 = Θ3 = 96 K and thirty skeletal vibrations. The liquid heat capacity of semicrystalline polyurethane (6,4-PU) was approximated from experimental data using a linear regression of the data above the melting region and the heat capacity of the fully amorphous material above the glass transition temperature. The equilibrium liquid heat capacity was expressed as Cp(liquid) = 1.0041T+233.24 in. J·K-1· mol-1. The solid and liquid heat capacities of linear polyurethane (6,4-PU) were applied as equilibrium baselines for advanced thermal analysis of the experimental, apparent heat capacity data.

Molecular interpretation of low-temperature heat capacity of aliphatic oligo-urethane

The low-temperature heat capacity of semi-crystalline aliphatic oligo-urethane obtained from the reaction between butane-1,4-diol and hexamethylene 1,6-diisocyanate was measured using a Quantum Design PPMS (Physical Property Measurement System) in the temperature range of (2.04–292.38)K. The experimental heat capacity data below the glass transition temperature of 280.2K (7.05°C) were interpreted in terms of molecular motion and were linked to the vibrational spectrum of oligo-urethane structure. The presented approach applies the classical Einstein, Debye and Tarasov treatments using the ATHAS Scheme. The low-temperature solid heat capacity was estimated by separately approximating the group and skeletal heat capacities from their vibrational spectra. The group vibrational heat capacity was calculated based on the chemical structure and molecular vibrational motions (Ngr=90) derived from infrared and Raman spectroscopy. The skeletal vibrational heat capacity contribution was estimated by a general Tarasov equation with thirty skeletal modes (Nsk=30). The solution of this equation gave the values of characteristic Debye temperatures as: Θ1=493.6K, Θ2=133.9K, and Θ3=51.6K. The result indicates the existence of planer (Θ2) interactions in the oligo-urethane molecules, in addition to linear (Θ1) and special (Θ3) interactions, which are attributed to a possible branched structure mixed with the linear form of the oligomer. The total vibrational heat capacity, being the sum of the group and skeletal heat capacities, was extended to higher temperatures and analysed further.The liquid heat capacity of semi-crystalline aliphatic oligo-urethane was approximated from experimental data by a linear regression and was compared with the estimated linear contributions of polymers that have the same constituent groups and were expressed as Cp(liquid)=0.406T+428.5 in J·K−1·mol−1. The solid and liquid heat capacities of oligo-urethane were applied as equilibrium baselines for advanced thermal analysis of the experimental, apparent heat capacity data.Using estimated parameters of transitions and solid and liquid heat capacities at equilibrium, the integral thermodynamic functions of enthalpy, entropy and free enthalpy as functions of temperature were calculated.

Vibrational heat capacity of Poly(N-isopropylacrylamide)

The heat capacity of poly(N-isopropylacrylamide) (PNIPA) has been measured from (1.8–460) K using a quantum design PPMS (Physical Property Measurement System) and differential scanning calorimeters. The low-temperature experimental heat capacity below the glass transition temperature 415 K (141.85 °C) was linked to the vibrational molecular motions of PNIPA to establish the baseline of the solid, vibrational heat capacity. The vibrational heat capacity of PNIPA was computed based on group and skeletal vibrations. The group vibrational heat capacities were calculated based on the chemical structure and molecular vibrational motions derived from infrared and Raman spectroscopy. The skeletal heat capacity was fitted to a general Tarasov equation using sixteen skeletal modes to obtain three Debye characteristic temperatures Θ1 = 650 K and Θ2 = Θ3 = 68.5 K. The fit agreed well with the experimental data (±0.2%) from T = (1.8–250) K, and the vibrational heat capacity, which was extended to higher temperatures, served as a baseline of the solid heat capacity for quantitative thermal analysis of the experimental, apparent heat capacity data of PNIPA. The liquid heat capacity of fully amorphous PNIPA was approximated by a linear regression and expressed as Cpliquid(exp) = 0.3379 ⋅ T + 126.69 in J K−1 mol−1. This was then compared to the estimated linear contributions of polymers that have the same constituent groups. Using estimated parameters of transitions and solid and liquid heat capacities at equilibrium, the integral thermodynamic functions of enthalpy, entropy and free enthalpy as functions of temperature were calculated.

Heat capacity of poly(3-hydroxybutyrate)

The heat capacity of poly(3-hydroxybutyrate) (P3HB) has been measured using a quantum design Physical Property Measurement System (PPMS), differential scanning calorimetry (DSC), and temperature-modulated differential scanning calorimetry (TMDSC) over the temperature range of (1.9 to 460)K. The results within the range of (1.9 to 250)K were obtained using the quantum design PPMS, and established the baseline of the solid heat capacity. This experimental low-temperature heat capacity was linked to the vibrational molecular motion of P3HB. The solid heat capacity of P3HB was computed based approximately on groups of vibration and skeletal vibration spectra. The skeletal vibration heat capacity contribution was estimated by a general Tarasov equation with three Debye characteristic temperatures Θ1=549.1K and Θ2=Θ3=71.8K, and ten skeletal modes, Nskeletal=10. The experimental and calculated solid heat capacities agree with an error of ±0.2% over the temperature range from (5 to 250)K. The vibrational, solid heat capacity was extended to higher temperatures to judge additional contributions to the experimental heat capacity from other large-amplitude motion or latent heat during the quantitative thermal analysis of semi-crystalline P3HB. The liquid heat capacity of semi-crystalline P3HB above its melting temperature and of fully amorphous P3HB above the glass transition temperature was approximated by a linear regression and expressed as Cpliquid(exp)=0.1791 T+94.722 in units of J·K−1·mol−1. The calculated solid and liquid heat capacities can serve as equilibrium baselines for the quantitative thermal analysis of semi-crystalline P3HB. Also, the integral thermodynamic functions of enthalpy, entropy and free enthalpy for the equilibrium condition were calculated using estimated parameters of transitions.

Heat capacity of poly(vinyl methyl ether)

AbstractThe heat capacity of poly(vinyl methyl ether) (PVME) has been measured using adiabatic calorimetry and temperature‐modulated differential scanning calorimetry (TMDSC). The heat capacity of the solid and liquid states of amorphous PVME is reported from 5 to 360 K. The amorphous PVME has a glass transition at 248 K (−25 °C). Below the glass transition, the low‐temperature, experimental heat capacity of solid PVME is linked to the vibrational molecular motion. It can be approximated by a group vibration spectrum and a skeletal vibration spectrum. The skeletal vibrations were described by a general Tarasov equation with three Debye temperatures Θ1 = 647 K, Θ2 = Θ3 = 70 K, and nine skeletal modes. The calculated and experimental heat capacities agree to better than ±1.8% in the temperature range from 5 to 200 K. The experimental heat capacity of the liquid rubbery state of PVME is represented by Cp(liquid) = 72.36 + 0.136 T in J K−1 mol−1 and compared to estimated results from contributions of the same constituent groups of other polymers using the Advanced Thermal AnalysiS (ATHAS) Data Bank. The calculated solid and liquid heat capacities serve as baselines for the quantitative thermal analysis of amorphous PVME with different thermal histories. Also, knowing Cp of the solid and liquid, the integral thermodynamic functions of enthalpy, entropy, and free enthalpy of glassy and amorphous PVME are calculated with help of estimated parameters for the crystal. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2141–2153, 2005

Heat capacity of poly(butylene terephthalate)

AbstractThe low‐temperature heat capacity of poly(butylene terephthalate) (PBT) was measured from 5 to 330 K. The experimental heat capacity of solid PBT, below the glass transition, was linked to its approximate group and skeletal vibrational spectrum. The 21 skeletal vibrations were estimated with a general Tarasov equation with the parameters Θ1 = 530 K and Θ2 = Θ3 = 55 K. The calculated and experimental heat capacities of solid PBT agreed within better than ±3% between 5 and 200 K. The newly calculated vibrational heat capacity of the solid from this study and the liquid heat capacity from the ATHAS Data Bank were applied as reference values for a quantitative thermal analysis of the apparent heat capacity of semicrystalline PBT between the glass and melting transitions as obtained by differential scanning calorimetry. From these results, the integral thermodynamic functions (enthalpy, entropy, and Gibbs function) of crystalline and amorphous PBT were calculated. Finally, the changes in the crystallinity with the temperature were analyzed. With the crystallinity, a baseline was constructed that separated the thermodynamic heat capacity from cold crystallization, reorganization, annealing, and melting effects contained in the apparent heat capacity. For semicrystalline PBT samples, the mobile‐amorphous and rigid‐amorphous fractions were estimated to complete the thermal analysis. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 4401–4411, 2004

Heat capacity of poly(lactic acid)

The heat capacity of poly(lactic acid) (PLA) is reported from T=(5 to 600) K as obtained by differential scanning calorimetry (d.s.c.) and adiabatic calorimetry. The heat capacity of solid PLA is linked to its group vibrational spectrum and the skeletal vibrations, the latter being described by a Tarasov equation with Θ 1=574 K, Θ 2= Θ 3=52 K, and nine skeletal vibrations. The calculated and experimental heat capacities agree to ±3% between T=(5 and 300) K. The experimental heat capacity of liquid PLA can be expressed by C p(liquid)=(120.17+0.076 T) J · K −1 · mol −1 and has been compared to the ATHAS Data Bank, using contributions of other polymers with the same constituent groups. The glass transition temperature of amorphous PLA occurs at T=332.5 K with a change in heat capacity of 43.8 J · K −1 · mol −1. Depending on thermal history, semi-crystalline PLA has a melting endotherm between T=(418 and 432) K with variable heats of fusion. For 100% crystalline PLA, the heat of fusion is estimated to be (6.55 ± 0.02) kJ · mol −1 at T=480 K. With these results, the enthalpy, entropy, and Gibbs function of crystalline and amorphous PLA were obtained. For semi-crystalline samples, one can check changes of crystallinity with temperature and judge the presence of rigid-amorphous fractions.

Conformational contribution to the heat capacity of the starch and water system

AbstractThe heat capacities of starch and starch—water have been measured with adiabatic calorimetry and standard differential scanning calorimetry and are reported from 8 to 490 K. The amorphous starch containing 11–26 wt % (53–76 mol %) water shows a partial glass transition decreasing from 372 to 270 K, respectively. Even the dry amorphous starch gradually increases in heat capacity above 270 K beyond that set by the vibrational density of states. This gradual increase in the heat capacity is identified as part of the glass transition of dry starch that is, however, not completed at the decomposition temperature. The heat capacities of the glassy, dry starch are linked to an approximate group vibrational spectrum with 44 degrees of freedom. The Tarasov equation is used to estimate the heat capacity contribution due to skeletal vibrations with the parameters Θ1 = 795.5 K, Θ2 = 159 K, and Θ3 = 58 K for 19 degrees of freedom. The calculated and experimental heat capacities agree better than ±3% between 8 and 250 K. Similarly, the vibrational heat capacity has been estimated for glassy water by being linked to an approximate group vibrational spectrum and the Tarasov equation (Θ1 = 1105.5 K and Θ3 = 72.4 K, with 6 degrees of freedom). Below the glass transition, the heat capacity of the solid starch—water system has been estimated from the appropriate sum of its components and also from a direct fitting to skeletal vibrations. Above the glass transition, the differences are interpreted as contributions of different conformational heat capacities from chains of the carbohydrates interacting with water. The conformational parts are estimated from the experimental heat capacities of dry starch and starch—water, decreased by the vibrational and external contributions to the heat capacity. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 3038–3054, 2001

Derivation of thermal equations of state for quantum systems using the quasi-Gaussian entropy theory

In this article, the quasi-Gaussian entropy theory is derived for pure quantum systems, along the same lines as previously done for semiclassical systems. The crucial element for the evaluation of the Helmholtz free energy and its temperature dependence is the moment generating function of the discrete probability distribution of the quantum mechanical energy. This complicated moment generating function is modeled via two distributions: the discrete distribution of the energy-level order index and the continuous distribution of the energy gap. For both distributions the corresponding physical–mathematical restrictions and possible systematic generation are discussed. The classical limit of the present derivation is mentioned in connection with the previous semiclassical derivation of the quasi-Gaussian entropy theory. Several simple statistical states are derived, and it is shown that among them are the familiar Einstein model and the one-, two-, and three-dimensional Debye models. The various statistical states are applied to copper, α-alumina, and graphite. One of these states, the beta-diverging negative binomial state, is able to provide an accurate description of the heat capacity of both isotropic crystals, like copper, and anisotropic ones, like graphite, comparable to the general Tarasov equation.

Heat capacity of poly(trimethylene terephthalate)

Thermal analysis of poly(trimethylene terephthalate) (PTT) has been carried out using standard differential scanning calorimetry and temperature-modulated differential scanning calorimetry. Heat capacities of the solid and liquid states of semicrystalline PTT are reported from 190 K to 570 K. The semicrystalline PTT has a glass transition temperature of about 331 K. Between 460 K and 480 K, PTT shows an exothermic ordering. The melting endotherm occurs between 480 K and 505 K with an onset temperature of 489.15 K (216 C). The heat of fusion of typical semicrystalline samples is 13.8 kJ/mol. For 100% crystalline PTT the heat of fusion is estimated to be 28--30 kJ/mol. The heat capacity of solid PTT is linked to an approximate group vibrational spectrum, and the Tarasov equation is used to estimate the skeletal vibrational heat capacity ({Theta}{sub 1} = 542 K and {Theta}{sub 3} = 42 K). A comparison of calculation and experimental heat capacities show agreement of better than {+-}2% between 190--300 K. The experimental heat capacity of liquid PTT can be expressed as a linear function of temperature: C{sub p} {sup L}(exp) = 211.6 + 0.434 T J/(K mol) and compares well with estimations from the ATHAS data bank using groupmore » contributions of other polymers with the same constituent groups ({+-} 0.5%). The change of heat capacity at T{sub g} of amorphous PTT has been estimated from the heat capacities of liquid and solid to be 86.4 J/(K mol). Knowing C{sub p} of the solid, liquid, and the transition parameters, the thermodynamic functions: enthalpy, entropy and Gibbs function were obtained.« less

Computation of Heat Capacities of Solids Using a General Tarasov Equation

The general Tarasov function is fitted to the skeletal heat capacities of materials with widely different crystal structures. Examples are chosen from flexible macromolecules (polyethylene, polypropylene, poly(ethylene terephthalate), selenium, rigid macromolecules (diamond and graphite), and a small molecule (fullerene, C60). A new optimization approach using the MathematicaTM software is developed. It results in one-, two-, and three-dimensional Debye temperatures, Θ1, Θ2 and Θ3 the fitting parameters of the Tarasov function. In addition to the Tarasov function, the evaluation of the heat capacities makes use of approximate group-vibrational spectra. The results support the earlier assumption that Θ2=Θ3 for simple, solid, linear macromolecules. In more complicated bonding situations, Θ1, Θ2 and Θ3 are used as averaging fitting parameters. This general approach provides an improvement in the quantitative thermal analyses of polymers and other substances included in the ATHAS Data Bank. Sufficient programming information is provided to enable anyone the computation with a copy of the popular MathematicaTM software. The programming file is also downloadable from the WWW.

Computation of heat capacities of solid state benzene,p-oligophenylenes and poly-p-phenylene

Heat capacities (Cp) of solid benzene, biphenyl,p-terphenyl,p-quaterphenyl, and poly-p-phenylene were analyzed using the ATHAS Scheme of computation. The calculated heat capacities based on approximate vibrational spectra of solid benzene and the series of oligomers containing additional phenylene groups were compared to experimental data newly measured and from the literature to identify possible additional large-amplitude motion. The skeletal heat capacity was fitted to the Tarasov equation to obtain the one- and three-dimensional vibration frequencies Θ1 and Θ3 using a new optimization approach. Their relationship to the number of phenylene groupsn is: Θ1=426.0−150.3/n; and Θ3=55.4+81.8/n. Except for benzene, the quantitative thermal analyses do not show significant contributions from large-amplitude motion below the melting temperatures.

The computation of heat capacities of solid high polymers from vibrational spectra

Based on experimental data collected over the last 15 years (ATHAS data bank) a system has been developed that permits the computation of the heat capacities of solid polymeric materials. It relies on separation of the vibrational spectrum into group and skeletal vibrations. The former are known from computations fitted to IR and Raman data, the latter can be fitted to low temperature heat capacities using the Tarasov equation. Knowing the chemical structure, the parameters of the Tarasov equation may be predicted by comparison with known heat capacities of related materials. Agreement between prediction, computation and experiment is usually better than ± 5 %.

Calculation of the heat capacity of linear macromolecules from theta-temperatures and group vibrations

An automated computer program is presented for the calculation of heat capacity of macromolecules from skeletal vibration frequencies described by the Tarasov equation and group vibration frequencies. The heat capacities calculated for crystalline polyethylene and polytetrafluoroethylene fit well with the experimental data from our ATHAS data bank. The program will be part of the planned ATHAS Computation Center.

Specific heat of Tl2Se between 3 and 640K

The specific heat capacity of Tl2Se has been measured by applying a heat pulse technique in the temperature range 3-300K and differential scanning calorimetry in the temperature range 190-640K. The specific heat of the material can be described by Tarasov's equation for a layer structure. Values of thermodynamic functions have been calculated.