(218h) 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 Kulkarniâ??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 Kulkarniâ??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 Satoâ??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 Kulkarniâ??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 wasnâ??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 [m²mol/(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,â?? [-]:
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