## The Cohesive Energy Density Of Polymer Liquids

DuPont Experimental Station Wilmington, DE, USA

### Introduction

One of the most important thermodynamic properties of a liquid is its cohesive energy density (CED). It is the energy per unit volume of the liquid required to move a molecule from the liquid state to a vapor state. It is a measure of the interactions between the molecules in the liquid state. The CED is defined as the ratio of the molar energy of vaporization (AU) divided by the molar liquid volume (V). Polymer molecules have a vanishingly small vapor pressure and we cannot compute an energy of vaporization. This has led to a number of approximate methods which are currently used to compute the CED of polymer liquids. Many of these are empirical and involve group contribution methods1 based on small molecule liquids. Other methods employ pressure-volume-temperature (PVT) data to approximate the CED based on its relationship to the internal pressure n = (dU/dV) | T)1,2 This later method is commonly used in equation of state calculations for polymer systems, but the validity of the relationship between the internal pressure and the CED is rarely discussed and is sometimes misunderstood or ignored. A limited experimental method used to estimate the CED of a polymer liquid is to study the "swelling" of the polymer by the liquid of known CED.3 This approach is not practical for a detailed determination of the properties of the CED for polymer liquids, however.

The solubility parameter (5) is defined as the square root of the CED. This quantity was indroduced by Hildebrand2 and used as a predictor of liquid-liquid solubility. The enthalpy of mixing is proportional to square of the difference (81-82) in the solubility parameters for the two liquids. Thus in combination with the entropy of mixing, the magnitude of this quantity determines the free energy of mixing. For polymers, where we do not have an experimental method for the determination of the CED of the liquid, we cannot expect to be able to make accurate predictions for the enthalpy of mixing. This point is elaborated further below.

In this paper we attempt to construct a scaling or corresponding states methology for the prediction of the CED using a combination of the surface tension and PVT

measurements for polymer liquids. It has long been understood that the surface tension is a direct manifestation of the cohesive forces that hold liquids together. Correlations between the CED and the surface tensions of small molecule liquids have been demonstrated.3 However, in the case of polymer liquids, accurate surface tension data have not been available until recently.4 The measurment of the thermodynamic properties of polymer liquids is difficult due to the fact that the temperature domain of the liquid state is frequently high enough for degradation reactions to occur which detract from a accurate measurement of many thermodynamic properties. PVT data for polymer liquids have also only recently become available due to the need for special dilatometers for the measurement of viscoelastic liquids5 The above mentioned stability problem necessitates a rapid measurement of the surface tension. This rules out conventional methods due to the long relaxation times associated with these polymer liquids. Sauer6 developed a new method which allows for the rapid determination of the surface tension of a highly viscous liquid using a very fine glass fiber.

### The Internal Pressure

It was Hildebrand that first made the observation that the internal pressure (n) was approximately equal to the CED for a number of hydrocarbon liquids.2 The internal pressure is defined by the following equation:

n= (dU/dV) I t=T a/p - P ~ T(a/p) (forP<1MPa) (1)

where a is the thermal expansivity, p is the isothermal compressibility, T is the temperature, and P is the pressure. Many authors have collected data at room temperature which supports this approximate observation.7 The ratio of the internal pressure to the CED was found to be close to 1 for many hydrocarbon liquids at room temperature. Hildebrand adopted the notation n (n = n /CED) for this ratio.2 For polar and hydrogen bonded liquids, the value of n deviated substantially from 1 and usually took on values less than 1. Weakly bonded liquids such as the perfluoro alkanes exhibited values of n greater than 1. There is no physical reason why the value of n should be equal to 1 except for the special case of a van der Waals fluid.3 The assumption that the internal pressure is equal to the CED is subject to substantial errors depending on the chemical structure of the molecules making up the liquid.

Many equation of state theories make the assumption that the n is equal to the CED. It is implicit in the theories of Flory, Orwoll, and Vrij,8 (FOV) and in the Sanchez Lacombe.9 Therefore, when one uses PVT data to define the parameters in these models one is using the internal pressure as a measure of the CED of that liquid. Any quantity computed in this manner will be subject to the error mentioned in the previous paragraph. Since n is related to the CED by a simple multiplicative constant, one might assume that this is not a major problem and in many cases this shortcoming simply disappears into an adjustable parameter. This is not the case when one is dealing with mixtures. If one computes the enthalpy of mixing with one of the above mentioned equations of state, one arrives at a term with the following form if we assume ideal random mixing:

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