Measurement and Correlation of Solubility and Diffusion Coefficient of Ethylene in Molten Propylene-Co-Polymers

Summary
Solubility and diffusion coefficients of ethylene in molten poly(propylene-ethylene-1-butene)co-polymers (PPs) with different crystallinities were measured at 140 °C and 180 °C and up to 2.5 MPa. There was no significant difference in solubility of ethylene in PPs at each temperature. Experimental solubility was successfully correlated by the Sanchez-Lacombe equation of state. On the other hand, the diffusion coefficient of ethylene in PPs was not identical. The difference of diffusion coefficient was possibly caused by the difference of specific volume of the PPs and/or of glass transition temperature of the PPs. Diffusion coefficient was successfully correlated by the Kulkarni’s Free-Volume-Theory.

Introduction
Propylene-co-polymer (co-PP) is widely used in the society due to its unique mechanical and molding characteristics. In the industry, co-PP is usually manufactured by the gas phase polymerization process which is progressed as monomer molecules dissolve and diffuse into polymer and reach catalysts. Co-PP is known as crystalline polymer having both crystalline and amorphous regions in it and it has reported that monomer solution and diffusion only take place in the amorphous region [1]. Therefore crystallinity of co-PP strongly affects solubility and diffusivity of monomer in co-PP. Furthermore, figuring out the relationship between crystallinity and solubility and diffusivity of monomer is needed to optimize the polymerization process. However, it is difficult to obtain phase equilibrium data under a constant crystallinity since crystallinity has temperature and pressure dependences and is changeable by thermal histories. Hence there have been reported only a few studies that quantify the effect of crystallinity on solubility and diffusivity of monomer in co-PPs with a wide range of crystallinity. In this work, solubility and diffusion coefficient of ethylene in molten poly(propylene-ethylene-1-butene)co-polymers (PPs) with different crystallinities were particularly measured at 140 °C and 180 °C. The reason for the selection of temperature above the melting point of PPs was to examine the affinity between ethylene and PPs polymer chain without the effect of crystalline. In addition, solubility and diffusion coefficient were correlated by the Sanchez-Lacombe equation of state and the Kulkarni’s Free-Volume-Theory respectively.

Experiments
Ethylene was used as monomer and four poly(propylene-ethylene-1-butene)-co-polymers with different crystallinities were used as polymer samples. Characteristics of polymer samples are listed in Table 1. Melting point was determined by DSC measurement and set as using peak top temperature. Crystallinity was determined by density. Measurement of solubility and diffusion coefficient was operated by the gravimetric method using a Magnetic Suspension Balance (MSB) under 140 °C and 180 °C and up to 2.5 MPa with 0.5 MPa steps.

Table 1 The Characteristics of PPs

Models
Solubility of ethylene in PPs was correlated by the Sanchez-Lacombe equation of state (SL-EoS) [3, 4] shown in Equations 1 and 2.
(1)
(2)

Where P*, r*, and T* are characteristic parameters of the SL-EoS. The characteristic parameter of ethylene and PPs were referred by Sato’s experiment [5, 6] as assuming that the characteristic parameters of homo-polypropylene can be used to the ones of PPs used in this work. In order to apply the SL-EoS to binary systems, the mixing rule expressed in Equation 3 was used with binary interaction parameter k_ij as follows,
(3)

Where k_ij was a fitting parameter for correlation.
On the other hand, diffusion coefficient was correlated by the Kulkarni’s Free-Volume-Thoery (FVT) [7] expressed in Equations 4, 5 and 6.
(4)
(5)
(6)

Where A_d, B_d and g are fitting parameters of the FVT and determined for each ethylene/PP system as temperature independent parameters. The chemical potential of monomer in PPs, m₁^P, expressed in Equation 4 was determined by the SL-EoS. The thermal expansion coefficient, a, and the compressibility coefficient, b, were evaluated by the method which Kulkarni et al. [7] suggested with the data of PVT measurement. The standard free volume fraction of polymer, v_fs, expressed in Equation 6 was referred by the universal value suggested by Wiliams et al [8].

Results and Discussions
Figures 1 and 2 show the solubilities of ethylene in PPs at 140 °C and 180 °C, respectively. The solubilities increased with decreasing temperature and increasing pressure linearly. Solid lines in Figures 1 and 2 denote the correlation by the SL-EoS. Experimental solubility of ethylene in PPs was successfully correlated with the SL-EoS within the AAD defined by Equation 7 of 0.7%. Moreover, solubility of ethylene in four different PP samples showed almost similar values and the deviation of solubility in PP1 and PP4 defined by Equation 8 was 2.8% in 140 °C and 1.4% in 180 °C. Figure 3 shows the temperature dependence of k_ij, the binary interaction parameter in the SL-EoS in the series of correlation. The maximum deviation of k_ij of PPs at both temperatures was 9.6% in 140 °C, which changes solubility less than 3%. Hence, k_ij can be treated as a constant for all PPs at each temperature.
(7)
(8)

Figure 1 Solubility of ethylene in PPs at 140 °C

Figure 2 Solubility of ethylene in PPs at 180 °C

Figure 3 Temperature dependence of k_ij in SL-EoS

Figures 4 and 5 show the diffusion coefficients of ethylene in PPs at 140 °C and 180 °C, respectively. Diffusion coefficients of ethylene in PPs increased with increasing temperature and concentration of ethylene. Solid lines in Figures 4 and 5 are the results of correlation by the FVT. Experimental diffusion coefficients were successfully correlated with the FVT within the AAD defined in Equation 9 of 4.0% at both temperatures. Additionally, the deviation of diffusion coefficients of ethylene in PP1 and PP4 derived by Equation 10 was about 22% in both temperatures. So it can be noticed that diffusion coefficients of ethylene in PP4 is systematically larger than in PP1, while there wasn’t a significant difference in the solubility.
(9)
(10)

Figure 6 shows the relationship of diffusion coefficients at infinite dilution and free volume fraction of PPs. The diffusion coefficients at infinite dilution were calculated by extrapolating the FVT to zero concentration. Free volume fractions of PPs were derived by PVT measurement. It can be seen in Figure 6 that the diffusion coefficients at infinite dilution of PP1 and PP4 at the same value of free volume fraction were almost equal and ones of PP2 and PP3 were lower than of PP1 and PP4.

Figure 4 Diffusion coefficient of ethylene in PPs at 140 °C

Figure 5 Diffusion coefficients of ethylene in PPs at 180 °C

Figure 6 Diffusion coefficient vs free volume fraction of PPs at infinite dilusion

Conclusion
The solubilities and diffusion coefficients of ethylene in molten PPs were measured at 140 °C and 180 °C and up to 2.5 MPa. Both experimental solubilities and diffusion coefficients were correlated by the SL-EoS and the FVT, respectively. It can also be noted that the free volume fraction of PPs might be the key parameters for relating the solubility and diffusion of ethylene in polymers. Hence the PVT measurement of PPs would lead better understandings of the relationship between the diffusion coefficients and the specific volumes of PPs.

References
[1] A.S. Micheals and H. Bixler, J. Polym. Sci., 50, 413 (1961). [2] I.C. Sanchez and R.H. Lacombe, J. Phys. Chem., 80, 2353 (1976). [3] I.C. Sanchez and R.H. Lacombe, Macromolecules, 11, 1145 (1978). [4] Y. Sato, M. Yurugi, T. Yamabiki, S. Takishima and H. Masuoka, J. Appl. Polym. Sci., 79, 1134 (2001). [5] Y. Sato, A. Tsuboi, A. Sorakubo, S. Takishima, H. Masuoka, T. Ishikawa, Fluid Phase Equilibria, 170, 49 (2000). [6] S.S. Kulkarni and S.A. Stern, Journal of Polymer Science: Polymer Physics Edition, 21, 441 (1983). [7] M.L. Williams, R.F. Landel, and J.D. Ferry, J. Am. Chem. Soc., 77, 3701 (1955).

Symbols
P [MPa]: pressure, r [kg/m³]: density, T [K]: temperature, P^* [MPa], r^* [kg/m³], T^* [K]: characteristic parameters in the SL-EoS, r [-]: number of segment, f [-]: volume fraction, k_ij [-]: interaction parameter, D_mutual [m²/s]: mutual diffusion coefficient, D_self [m²/s]: self diffusion coefficient, x [-]: mole fraction, R [J/mol/K]: gas constant, m^P [J/mol]: chemical potential of polymer, A_d [m² mol/(s J)], B_d [-], g [-]: characteristic parameters, T_g [K]: glass transition temperature of polymer, a [K^-1]: thermal expansion coefficient, b [MPa^-1]: compressibility coefficient, v_f [-]: free volume fraction of polymer, v_fs [-]: free volume fraction of polymer at standard state, v_f,∞ [-]: free volume fraction of polymer at infinite dilution, Sol [g_gas/g_polymer]: solubility, D [m²/s]: diffusion coefficient

Index
1: monomer, 2: polymer, s: standard state