Equation Of State For Polymers Research Articles

Experimental Determination of Coefficients for the Renner Model of the Thermodynamic Equation of State of the Poly(butylene succinate) and Wheat Bran Biocomposites.

This paper presents the assumptions of a thermodynamic equation of state for polymers according to the Renner model. The experiments involved extruding a biocomposite based on poly(butylene succinate) that was filled with ground wheat bran with its size not exceeding 200 μm. The biocomposite was produced in pellet form with three different contents by weight of wheat bran, i.e., 10%, 30% and 50%. All specimens were examined for their thermodynamic p-v-T characteristics. Taking advantage of the SimFit module of Cadmould 3D-F, experimental results were used to determine the coefficients of thermodynamic equation of state for the tested biocomposite according to the Renner model. The coefficients were then used to calculate transition temperature and to create diagrams illustrating the relationship between pressure, temperature and specific volume for the tested biocomposite. The obtained results can serve as a basis for assessing the suitability of the biocomposite for injection molding, selecting technological parameters of this process, as well as for analyzing shrinkage and defects of injection-molded parts.

Shock-Driven Decomposition of Polymers and Polymeric Foams.

Polymers and foams are pervasive in everyday life, as well as in specialized contexts such as space exploration, industry, and defense. They are frequently subject to shock loading in the latter cases, and will chemically decompose to small molecule gases and carbon (soot) under loads of sufficient strength. We review a body of work—most of it performed at Los Alamos National Laboratory—on polymers and foams under extreme conditions. To provide some context, we begin with a brief review of basic concepts in shockwave physics, including features particular to transitions (chemical reaction or phase transition) entailing an abrupt reduction in volume. We then discuss chemical formulations and synthesis, as well as experimental platforms used to interrogate polymers under shock loading. A high-level summary of equations of state for polymers and their decomposition products is provided, and their application illustrated. We then present results including temperatures and product compositions, thresholds for reaction, wave profiles, and some peculiarities of traditional modeling approaches. We close with some thoughts regarding future work.

Equation of state for polymers based on crystallization data

A simplified procedure with minimum input information is presented to predict an analytical equation of state for crystallizable polymer melts from enthalpy of fusion and crystalline density as scaling constants. The second virial coefficients are calculated from a two-parameter corresponding states correlation obtained from two constants which are heat of fusion, ΔHfus, and the molar density at the crystalline point, ρcrys. This new correlation has been applied to the Ihm-Song-Mason equation of state to predict the volumetric behavior of some polymer melts at compressed state with temperature range from 311.5K to 557.25K and pressures up to 200MPa. For 924 data points, the overall average absolute relative deviation (AARD) of pVT properties was found to be 0.46%. The isothermal compressibility coefficients and liquid bulk modulus have also been predicted using mTM EOS. From 840 data points examined, the AARD of both isothermal thermodynamic properties was equal to 9.71×10−2%. Also the enthalpy of fusion (ΔHfus) and entropy of fusion (ΔSfus) of three polymer melts are predicted using the mTM EOS. Our calculations on the volumetric and thermodynamic properties of studied polymers reproduce the literature data with reasonably good accuracy.

Analysis of equations of state for polymers

In the literature there are several studies comparing the accuracy of various models in describing the PvT behavior of polymers. However, most of these studies do not provide information about the quality of the estimated parameters or the sensitivity of the prediction of thermodynamic properties to the parameters of the equations. Furthermore, there are few studies exploring the prediction of thermal expansion and compression coefficients. Based on these observations, the objective of this study is to deepen the analysis of Tait, HH (Hartmann-Haque), MCM (modified cell model) and SHT (simplified hole theory) equations of state in predicting the PvT behavior of polymers, for both molten and solid states. The results showed that all equations of state provide an adequate description of the PvT behavior in the molten state, with low standard deviations in the estimation of parameters, adequate sensitivity of their parameters and plausible prediction of specific volume, thermal expansion and isothermal compression coefficients. In the solid state the Tait equation exhibited similar performance to the molten state, while HH showed satisfactory results for amorphous polymers and difficulty in adjusting the PvT curve for semicrystalline polymers.

Equation of state for polymers based on glass transition data

This paper addresses a method for predicting the participating constants in equation of state (EOS) for compressed polymeric fluids using two scaling constants, the surface tension γg and the molar density ρg, both at the glass transition point. The theoretical EOS undertaken is the one attributed to Tao and Mason. The second virial coefficients are calculated from a two-parameter corresponding states correlation, which is constructed with two constants as scaling parameters, i.e., the surface tension γg and the molar density ρg. This new correlation has been applied to the Tao–Mason (TM) EOS to predict the volumetric behavior of several polymer melts. The operating temperature range is from 291.25 to 603.4 K and pressures of up to 202.5 MPa. A collection of 516 data points has been examined for the aforementioned polymers. The average absolute deviation between the calculated densities and the experimental ones is of the order of 0.44%.

Generic van der Waals equation of state for polymers, modified free volume theory, and the self-diffusion coefficient of polymeric liquids

In this paper, a molecular theory of self-diffusion coefficient is developed for polymeric liquids (melts) on the basis of the integral equation theory for site–site pair correlation functions, the generic van der Waals equation of state, and the modified free volume theory of diffusion. The integral equations supply the pair correlation functions necessary for the generic van der Waals equation of state, which in turn makes it possible to calculate the self-diffusion coefficient on the basis of the modified free volume theory of diffusion. A random distribution is assumed for minimum free volumes for monomers along the chain in the melt. More specifically, a stretched exponential is taken for the distribution function. If the exponents of the distribution function for minimum free volumes for monomers are chosen suitably for linear polymer melts of N monomers, the N dependence of the self-diffusion coefficient is N − 1 for the small values of N , an exponent predicted by the Rouse theory, whereas in the range of 2.3 ≲ ln N ≲ 4.5 the N dependence smoothly crosses over to N − 2 , which is reminiscent of the exponent by the reptation theory. However, for ln N ≳ 4.5 the N dependence of the self-diffusion coefficient differs from N − 2 , but gives an N dependence, N − 2 − δ ( 0 < δ < 1 ) , consistent with experiment on polymer melts in the range. For polyethylene δ ≈ 0.48 for the parameters chosen for the stretched exponential. Because the stretched exponential function contains undetermined parameters, the N dependence of diffusion becomes semiempirical, but once the parameters are chosen such that the N dependence of D can be successfully given for a polymer melt, the temperature dependence of the self-diffusion coefficient can be well predicted in comparison with experiment. The theory is satisfactorily tested against experimental and simulation data on the temperature dependence of D for polyethylene and polystyrene melts.

Refrigeration plants using carbon dioxide as refrigerant: measuring and modelling the solubility and diffusion of carbon dioxide in polymers used as sealing materials

Because of increased environmental pressure, there is currently a movement away from more traditional refrigerants such as HCFC's toward refrigerants with lower global warming potential such as carbon dioxide (CO 2). However, the use of CO 2 as a refrigerant requires a refrigeration cycle with greater extremes of pressure, placing greater demands on the polymer materials used for seals and packing. In this work we have measured the solubility and diffusivity of gaseous CO 2 in two polymers used as sealing materials in CO 2 refrigeration plants. These are Hydrogenated Nitrile Butadiene Rubber (HNBR) and Ethylene Propylene Diene Monomer (EPDM) which are used in seals such as O-rings. The experiments were performed on a high-pressure microbalance. Solubility results were modelled using an equation of state for polymers (simplified PC-SAFT). The necessary polymer parameters were obtained using a previously published method. The measured results can be successfully correlated using simplified PC-SAFT.

Thermodynamic study of surfaces of liquid polybutadienes and their interfaces with poly(dimethylsiloxane)

AbstractSurface tension of liquid polybutadienes (PBD) as well as interfacial tension between them and poly(dimethylsiloxane) (PDMS) were measured in the temperature range from 25 to 150°C. The measured pressure–volume–temperature (PVT) data were used for the determination of reduction parameters of pressure, volume, and temperature in several equations of state for polymers. The reduction parameters were used for the estimates of surface tension and compared with experimental data. Interfacial tensions were used to determine Flory‐Huggins interaction parameters using the Roe and Helfand theories for the system PBD‐PDMS. The results obtained using both the theories were somewhat different; the difference being the least with the segment chosen as a part of molecule containing 4–5 nonhydrogen atoms.Modifications of PBD that were investigated increase the density. Maleic ester end groups increase both surface tension and interfacial tension and bring a positive contribution to the Flory‐Huggins interaction parameter (to its enthalpic component), whereas the pending maleic anhydride groups have shown the opposite effect; their negative contribution to Flory‐Huggins parameters concerns mainly its entropic component. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009

A novel method for static equation-of state-development: Equation of state of a cross-linked poly(dimethylsiloxane) (PDMS) network to 10 GPa

Pressure-volume-temperature (PVT) equation-of-state (EOS) information for polymers and polymeric composites is valuable for predicting their response to extreme conditions. An obstacle in determining equations of state for polymeric materials is the lack of a simple, static experimental method for acquiring PVT data for solid networks and liquids at pressures greater than several kilobars. Here, we report a novel approach in determining static EOS for polymers using high-pressure diamond-anvil cells coupled with optical microscopy and image analysis. Results are presented for a cross-linked poly(dimethylsiloxane) polymer, Sylgard 184. Static isothermal results were fitted to empirical and semiempirical equations of state, including the Tait, Birch-Murnaghan, and Vinet forms. Static PV data were also converted to pseudoshock velocity-pseudoparticle velocity (U(s)-u(p)) for comparison to dynamic Hugoniot data. A linear Rankine-Hugoniot fit U(s)=s(T)u(p)+c(T) gives c(T)=1.572 km/s and s(T)=1.703. s(T) is related to the pressure derivative of the bulk modulus B(0) (') by s(T)=(B(0) (')+1)/4 and B(0) (')=5.8. A comparison of the static and shock data is given, along with an estimate of the Grüneisen parameter, and a discussion of the free volume content in the polymer network, and limitations of this novel method.

An Equation of State for Polymers and Normal Fluids Using the Square-Well Potential of Variable Well Width

A perturbation theory equation of state for mixtures of freely jointed square-well fluids of variable well width that takes into consideration the influence of the attractive part of the intersegment potential on the chain properties is applied to describe thermodynamic properties of pure normal fluids, polymers, and their mixtures. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference sphere fluid and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of spheres to form chains. Pure polymer parameters are determined from low molecular weight pure substance parameters, from pure polymer volumetric data, and from mixture data. Good results are obtained when this equation of state is used in the calculation of vapor−liquid equilibrium data of highly asymmetric mixtures, with polymer properties determined from low molecular weight substance properties, molecular weight and tempe...

Application of the van der Waals equation of state to polymers: I. Correlation

Kontogeorgis G.M., Harismiadis V.I., Fredenslund A., and Tassios D.P., 1994. Application of the van der Waals equation of state to polymers. I. Correlation. Fluid Phase Equilibria, 96: 65-92. A reliable methodology for evaluating the energy and co-volume parameters of cubic equations of state for polymers is proposed. Only two low-pressure volumetric data points are used. The procedure is applied to the van der Waals equation of state and bubble-point pressure calculations are performed for a number of polymer solutions. The usual combining and mixing rules are used. It is shown that the van der Waals equation can correlate the equilibrium pressures of polymer solutions using one binary interaction parameter. The accuracy of the correlation is very good; it is comparable to or better than that achieved using the Flory-Huggins model. It is also shown that the necessary binary interaction parameters for the van der Waals equation of state can be estimated using a simple scheme based on the Berthelot combining rule. It is, thus, demonstrated that the van der Waals equation of state can be a simple and successful tool for the description of the vapor-liquid equilibria in polymer solutions.

A new isothermal equation of state for polymers

A new isothermal equation of state for polymers in the solid and the liquid is given by P= B(T,0) (n−m) {[ V(T,0) V(T,P)] ] n+1−[ V(T,0) V(T,P) ] m+1} where n = 6.14 and m = 1.16 are general constants for polymer systems. Comparison of the equation with experimental data is made for six polymers at different temperatures and pressures. The results predict that the equation of state describes the isothermal compression behaviour of polymers in the glass and the melt states, except at the transition temperature.

Semiempirical equation of state for polymers and simple liquids under high pressure

The semiempirical equation of state for polymers polyethylene (PE), poly( n-butyl methacrylate) (PNBMA) and polystyrene (PS) for pressures up to 2 kbar, five hydrocarbons from n-heptane to n-octadecane over 0 to 1.2–9.0 kbar, five organic liquids, including carbon tetrachloride (CCl 4), over 0 to 2.0–12.0 kbar, water up to 10 kbar and argon up to 3 kbar has been derived by using experimental data for specific volume V, compressibility β T and thermal expansion coefficients α P under high pressure published by Simha et al., Hellwege et al., Bridgman, Streett and others on the basis of the homogeneous function approach. The equations derived in this work under a constant temperature are: ln V∼-(P+P 0) 1-m 0 β T∼(P+P 0) -m 0 α P∼(P+P 0) 1-m 0 where P 0 is a constant with respect to pressure but depends on temperature, and m 0 and n 0 are constants determined from the experimental data. Values of m 0 range from 0.76 to 0.99, with an average value 0.87 for polymers and simple liquids including water and argon, while values of n 0 for α P for argon are around 0.50 over 90–150 K.

Semi-empirical equation of state for polymers and simple liquids in non-critical and critical regions

The semi-empirical equation of state for the polymers polystyrene (PS), polyisobutylene (PIB) and polydimethyl siloxane (PDMS) and the simple liquids benzene, n-heptane, carbon tetrachloride and argon at a negligible external pressure has been determined using the experimental data of thermal pressure coefficient γ v , molar volume V and thermal expansion coefficient α p . The equation of state obtained in this work is expressed by P = ( X β o V )c 1 exp(α −1 oI o)T − c o exp(a −1 oI o) , where X = (V c − V) V and I o = ∫ V V c X β o V dV , V c is the critical volume and α 0, β 0, a 0, c1 and c 0 are constants determined experimentally. We have obtained good agreement between the values of α p and γ v calculated and observed, within 1–2% over a wide temperature range. The semi-empirical equation of state in the critical region is expressed by V c − V∝( T c − T) 0.33, P − P c∝( V c − V) 5.0, β −1 T ∝( T c − T) 1.33 and α −1 p∝( T c − T) 0.67 for β 0 = 2.0, which agrees semi-quantitatively with the experimental critical indices.

Free Volume Concepts Connecting PVT Behavior and Gaseous Diffusion Through Polymers

Free volume concepts in polymers, originally suggested by Bueche, have been modified and extended and now generate a good equation of state for polymers above Tg. This paper is concerned with the extension of the concepts to describe gaseous diffusion. In order to do this, the notion of activated diffusion was introduced. A gas molecule in a matrix can jump under three conditions. (1) A hole opens next to it (from segmental oscillations of the polymer) large enough to allow the molecule to jump (passive diffusion). (2) A hole opens which is smaller than the gas molecule, but the kinetic energy of the gas molecule allows it to open the hole and jump by compressing neighboring segments (activation). Once compressed, the segments cannot relax on the segmental oscillation time scale, and thus can assist the gas molecule to jump into other small holes (activated diffusion). The segments at the old site relax, but the segments around the new site are now compressed. This continues until a hole as large or larger than the gas molecule opens. The jump into such a hole allows all segments to relax (deactivation). This approach was successful in correlating diffusivity with activation energy for a wide variety of penetrants in polyethylene and poly(vinyl acetate). It also reproduced the non‐Arrhenius change in diffusivity with temperature for diffusion in rubber using the same constants. This paper gives the initial formulation for diffusion, but much work still needs to be done.

Comparison between some empirical equations of state for polymers

AbstractSome commonly used empirical equations of state for polymers are considered: the Spencer‐Gilmore equation with two and three adjustable parameters, the Whitaker‐Griskey equation, and the Rehage‐Breuer equation. Also, a new equation is proposed: the Inverse Volume equation. These equations are evaluated with regard to fitting experimental P‐V‐T data and agreement with experimental data on isothermal compressibility and thermal expansion coefficient. The adjustable parameters for each equation are determined with the help of Rosenbrock's optimum‐seeking technique. Analysis of the residuals on specific volume for a variety of materials suggests that the Spencer‐Gilmore equation with three adjustable parameters, the Rehage‐Breuer and the Inverse Volume equations yield the smallest and most random residuals and thus the least systematic error. The same three equations mentioned above yield results in good agreement with experimental isothermal compressibility data. However, among all the equations considered in this study, the Inverse Volume equation yields the best agreement with experimental thermal expansion coefficient data. Furthermore, it is the only equation to correctly predict the rise in thermal expansion coefficient with increasing temperature.

Compressibility of elastomers with crystalline fillers and microvoid inhomogeneities related to various empirical equations of state for liquids and solids

AbstractModified classical compressibility techniques are described which measure both the adiabatic and isothermal bulk moduli of polymers. These differ from familiar methods principally in the utilization of sensitive electronic deformation sensors (differential transformers) to assess volume changes, the autographic recording of pressure and volume changes during an essentially quasi‐static pressurization cycle. The uniaxial and volumetric methods both characterize the compressibility behavior of polymers in the relatively low pressure region (0–100 atm.). Compressibility data for several well‐known liquids and solids are reported. The nonlinear nature of the PV behavior of liquids are shown to be adequately characterized by a simple equation of state which is applicable to rubbery polymerics as well. Equations of state for polymers and filled polymers containing dispersed void inhomogeneities are discussed. Relationships based upon the infinitesimal theory of the deformations of an externally pressurized hollow sphere are compared with more complex equations of state which consider finite deformations at the center of a hollow sphere. It is concluded that the experimental data for the compressibility of voided solids containing less than 0.5% of initial void volume are best represented by the simpler equations of state derived from infinitesimal theory.

A generalized equation of state for polymers

AbstractA simple, generalized equation of state has been developed that describes the pressure–volume–temperature behavior of both addition and condensation polymers. Use of the equation requires only that the polymer's glass temperature and density at 25°C. and one atmosphere be known. The equation applies to both amorphous and crystalline polymers for pressures up to 10,000 atm. It also appears to hold for copolymers. Polymer glass temperatures can also be estimated with the equation.