The pressure-volume-temperature properties of polyethylene, poly(dimethyl siloxane), poly (ethylene glycol ) and poly(propylene glycol) as a function of molecular weight G. T. Dee, T. Ougizawa and D. J. Walsh Central Research and Development, E.L Du Pont de Nemours, Experimenta/ Station, PO Box 80356, Wilmington, DE 19880-0356, USA (Received 29 August 1991; accepted 28 September 1991 ) The pressure-volume-temperature (PVT) properties of a number of polymer liquid samples as a function of molecular weight have been investigated. The polymers studied included polyethylene (PE), poly(dimethyl siloxane) (PDMS), poly(ethylene glycol) (PEG) and poly(propylene glycol) (PPG). The properties of the end-groups and their effect on the density, compressibility, thermal pressure coefficient and expansivity as a function of their concentration are discussed. The specific volume exhibits a clear 1/M, dependence for those molecules with weakly interacting end-groups, i.e. PE and PDMS. The specific volume is found to be independent of M, for PEG and PPG, which can hydrogen-bond to each other. The data sets have been compared using equation-of-state theories. ( Keywords: P VT properties; polymer liquids; molecular weight; polyethylene; poly (dimethyl siloxane ); poly ( ethylene glycol); poly(propylene glycol); end-groups) INTRODUCTION fitting functions 1. The physical properties of the polymer are then represented by three or more reduction The change in density as a function of pressure, parameters. The limitation in the use of equations of state temperature and molecular weight is an important lies in their inability to describe the PVT data to within fundamental material property. Thermodynamic properties the experimental accuracy except for very restricted such as the expansivity, compressibility and pressure temperature and pressure domains. coefficient (dP/dT)v, can be calculated once this is A class of equations of state called cell models have known. Other equilibrium optical, electrical and material been particularly useful in describing the thermodynamic properties such as the refractive index, dielectric properties of polymer systems. The Flory, Orwoll and coefficient, glass transition temperature and surface Vrij z (FOV) theory can be derived in the context of the tension are also functions of the density. cell model formalism, developed by Lennard-Jones and The density is a function of molecular weight for Devonshire 3 to describe the properties of small molecular polymer liquids. A 20% change in the density is typical liquids. The N polymer molecules are divided into r mers, for changes in the degree of polymerization from 10 to with each mer occupying one cell. The connectivity of 1000. This change in the density with molecular weight the polymer is taken into account by assuming that each can be thought of as due to a change in the concentration mer has 3c degrees of freedom, where c is a constant less of end-groups with molecular weight. The end-group than one. The FOV theory does not give as accurate a segments are assumed to have different molar volumes description of PVT data as other theories such as the from the main-chain segments. The molar volume of the cell model 4 or the modified cell model 5. It is, however, end-group is determined by the size of the end-group and one of the most commonly used, and when making the fact that the end-groups have more degrees of freedom comparisons between different polymers will give than the main-chain mers. qualitatively the same picture. In this paper we describe the pressure-volume- Details of the derivation of the FOV equation of state temperature (P VT) properties, as a function of molecular and its application to mixtures are presented in detail weight, for four polymers, polyethylene (PE), poly- elsewhere 6. The equation of state has the following form : (dimethyl siloxane) (PDMS), poly(ethylene glycol) (PEG) and poly (propylene glycol) (PPG). We interpret p~,/~,= ~.,/3/(~,1/3 _ 1)- 1/7'P" (1) the results in terms of the concentration of end-groups Equation (1) is written in terms of reduced pressure, and their composition. temperature and volume variables /3, 7" and ~'. The reduction variables P*, T* and v* are defined as follows : THEORY 7F= Tck/e* = T/T* (2) Equation-of-state theories can be used to characterize P = Pv*/ckT* = P/P* (3) PVT data. In their simplest application they provide ~r ~-. V/V* : Dsp//)~p (4) 0032-3861/92/1634624)8 © 1992 Butterworth-Heinemann Ltd. 3462 POLYMER, 1992, Volume 33, Number 16 PVT properties as function of MW: G. T. Dee et al. where e* is the characteristic interaction energy per mer, the temperature. This is indeed what we observe in the v* is the hard-core mer volume and V*p is the hard-core case of PE and PDMS, as we will show below. In the specific volume. The parameters r, v*, e* and c are the case where the end-group is constrained by the presence microscopic parameters of the model. The relationships of a hydrogen bond to another polymer, we expect the between these microscopic parameters and the reduction number of degrees of freedom of the end-group to be parameters P*, T* and v*sp' which specify the comparable to those of the main-chain mers. Under these macroscopic physical properties of the system, are: circumstances we might expect that the physical size of the end-group dominates the dependence of the specific Nrv* = Mv*p (5) volume on molecular weight. For PEG and PPG with Nre* = MP*v*p (6) hydroxy end-groups we observe that the specific volume is independent of molecular weight. This implies that the Nrc = MV*v*p/k T* (7) enhanced mobility of the end-groups is the principal where M is the molecular weight of the molecule, N is factor determining the molecular-weight dependence of the number of molecules and r is the number of mers per such equilibrium properties as the specific volume. molecule. In the case of a polydisperse polymer sample, these equations apply as long as the same average is used EXPERIMENTAL for r and M. Let us assume that the effect of molecular weight on Materials some property F is simply a volume-fraction average of The linear hydrocarbons, obtained from Alfa Products, that property between the main-chain mers and the were specified as 99% pure, and were used as supplied. end-group mers. Then, within the context of the cell All of the polymer samples were obtained from model, we can write the following expression for the Polysciences. They were dried overnight in a vacuum dependence or F on r or the molecular weight: oven before use. The properties of the materials used are r(r) = r1¢1 + F2492 (8) shown in Table 1. Some of the PPG samples were treated where ~bi and Fi are the volume fraction and value of F Table 1 Properties of polymers for each component mer, respectively. If we denote by Vl and v2 the volumes of the main-chain and end-group Density mers, respectively, then 4) 1 and q52 take the following at 22°C form: Polymer M. a Mw/M" (g cm -3) ~b 1 = 1/(1 + 2v2/rvl) CllH24 156.3 0.740g C14Hao 198.4 0.760g ~b2 = 1/(1 + rvl/Zv2) (9) C16H34 226.4 0.770° C24H5o 338.7 0.933 where we have assumed that there are two end-groups C36H74 507.0 0.948 attached to each chain. In the limit where r is large, we C44H90 619.2 0.956 can expand these expressions. Using equation (5) we can Polyethylene 1000 1100b 1.07b 0.956 Polyethylene 2000 2100b 1.09b 0.974 write equation (8) in the form: High-density PE 28 000b 4.5 b 0.949 F(M.) = rl - (2V2/MnVlsp)(r 1 - rz) Poly (dimethyl siloxane) 340c 1.01/ 0.850o + (2Vz/MnVl~p)Z(F1 - F2) +... (10a) 770c 1.13/ 0.916o 2600c 1.5: 0.953 where V2 is the molar volume of the end-group mers, M. 5200c 1.86: 0.957 7830c 2.2: 0.960 is the number-average molecular weight of the polymer 68 000c 3.3: 0.966 chain and V~,p is the specific volume of the main-chain 187 000c 8.0t 0.966 mers. Keeping the lowest-order term one obtains the well Poly (propylene oxide) 400 1.3y 1.004 known relation : 1025~ 1.7 f 1.003 2000 1.3: 0.999 F(M.) = F= - e/M, (10b) 4000 1.7y 0.999 dimethyl ether 400 0.960 where F~ is the value of F(M.) at infinite molecular dimethyl ether 1000 0.972 weight and ~ is a constant. Using equations (10a,b) we dimethyl ether 2000 0.990 can estimate the value of M. for which we start to see Poly(ethylene oxide) 300 302e 1.18: 1.12M deviations from linearity in equation (10b). To do this 600 692e 1.06: 1.1240 we simply ask when the third term in equation (10a) 1540 1470e 1.16y 1.215 becomes of the order of a tenth of the second term, for 18 500 1.212 100000 1.207 example, i.e. when 2Vz/Mnvls p = 0.1. The theory implies monomethyl ether 350 1.0910 that we should see deviations from equation (10b) when monomethyl ether 750 0.929h the molar volume of the chain is of the order of 10 times dimethyl ether 600 1.0720 that of the end-groups. dimethyl ether 1000 0.944h If the end-groups have more degrees of freedom, then "Unmarked molecular weights are manufacturers' quotation we expect that their molar volume will have different bG.p.c, relative to polyethylene standards temperature and pressure dependences than those of the CMw by viscosity, then adjusted by Mw/M .