Determining the Compatibility of Polymers Determine the liquid-liquid phase boundaries for the mixture of polymers polystyrene (PS) and polymethylmethacrylate (PMMA) over the temperature range from 25°C to 600°C. The polystyrene has a degree of polymerization (number of monomer units in the polymer

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Determining the Compatibility of Polymers

Determine the liquid-liquid phase boundaries for the mixture of polymers polystyrene (PS) and polymethylmethacrylate (PMMA) over the temperature range from 25°C to 600°C. The polystyrene has a degree of polymerization (number of monomer units in the polymer), [latex]N_{ PS }[/latex], of 1500, and the volume of a monomer unit, [latex]\underline{V}_{ PS , m}[/latex], is 107.8 [latex]cm ^{3} / mol[/latex]. The polymethylmethacrylate has a degree of polymerization, [latex]N_{ PMMA }[/latex], of 1700 and a monomer unit volume, [latex]\underline{V} PMMA , m[/latex], of [latex]89.7 cm ^{3} / mol[/latex]. The Flory parameter for the PS–PMMA mixture is given by

[latex]\chi=\frac{0.982 imes N_{ PS }}{T}[/latex]

where T is in K.

Accepted Answer

The equations for the activity coefficients of PS and PMMA are, from Eqs. 9.5-18,

[latex]\begin{aligned}\ln \gamma_{1} &=\ln \frac{\phi_{1}}{x_{1}}+\left(1-\frac{1}{m} ight) \phi_{2}+\chi \phi_{2}^{2} \\ln \gamma_{2} &=\ln \frac{\phi_{2}}{x_{2}}+(m-1) \phi_{1}+m \chi \phi_{1}^{2}\end{aligned}[/latex] (9.5-18)

[latex]\ln \gamma_{ PS }=\ln \frac{\phi_{ PS }}{x_{ PS }}+\left(1-\frac{1}{m} ight) \phi_{ PMMA }+\chi \phi_{ PMMA }^{2}[/latex]

and

[latex]\ln \gamma_{ PMMA }=\ln \frac{\phi_{ PMMA }}{x_{ PMMA }}+(m-1) \phi_{ PS }+\chi \phi_{ PS }^{2}[/latex]

where

[latex]m=\frac{\underline{V}_{ PMMA }}{\underline{V}_{ PS }}=\frac{N_{ PMMA } imes \underline{V} PMMA , m}{N_{ PS } imes \underline{V} PS , m}=\frac{1700 imes 89.7}{1500 imes 107.8}=0.943[/latex]

and

[latex]\chi=\frac{0.982 imes 1500}{T}=\frac{1473}{T}[/latex]

The liquid-liquid equilibrium equations to be solved are

[latex]x_{ PS }^{ I } \gamma_{ PS }^{ I }=x_{ PS }^{ II } \gamma_{ PS }^{ II } \quad ext { and } \quad x_{ PMMA }^{ I } \gamma_{ PMMA }^{ I }=x_{ PMMA }^{ II } \gamma_{ PMMA }^{ II }[/latex]

which, using the activity coefficient expressions above, can be written as[latex]^{10}[/latex]

[latex]\begin{aligned}\ln \left(\frac{\phi_{ PS }^{ I }\left(x_{ PS }^{ I } ight)}{\phi_{ PS }^{ II }\left(x_{ PS }^{ II } ight)} ight)+\left(1-\frac{1}{m} ight)\left(\phi_{ PMMA }^{ I } ight.&\left.\left(x_{ PMMA }^{ I } ight)-\phi_{ PMMA }^{ II }\left(x_{ PMMA }^{ II } ight) ight) \&+\chi\left[\left(\phi_{ PMMA }^{ I }\left(x_{ PMMA }^{ I } ight) ight)^{2}-\left(\phi_{ PMMA }^{ II }\left(x_{ PMMA }^{ II } ight) ight)^{2} ight]=0\end{aligned}[/latex]

and

[latex]\ln \left(\frac{\phi_{ PMMA }^{ I }\left(x_{ PMMA }^{ I } ight)}{\phi_{ PMMA }^{ II }\left(x_{ PMMA }^{ II } ight)} ight)-(m-1)\left(\phi_{ PS }^{ I }\left(x_{ PS }^{ I } ight)-\phi_{ PS }^{ II }\left(x_{ PS }^{ II } ight) ight)+\chi\left[\left(\phi_{ PS }^{ I }\left(x_{ PS }^{ I } ight) ight)^{2}-\left(\phi_{ PS }^{ II }\left(x_{ PS }^{ II } ight) ight)^{2} ight]=0[/latex]

These equations can be solved using MATHCAD or another equation-solving program. The results are given below.

T (°C)	[latex]x_{ PS }^{ I }[/latex]	[latex]x_{ PS }^{ II }[/latex]
25	[latex]7.71 imes 10^{-3}[/latex]	0.992
100	0.023	0.978
150	0.04	0.962
200	0.063	0.941
250	0.096	0.913
300	0.141	0.878
350	0.211	0.834
375	0.272	0.808
380	0.292	0.802
385	0.325	0.795
387	0.352	0.791
>390	Complete miscibility

The decomposition temperature has been reported to be 364°C for polystyrene and lower than 327°C for PMMA. Consequently, any reprocessing of these polymers would have to be done at temperatures considerably below 327°C; at such temperatures there are only small regions of composition in which the polymers are compatible (that is, do not phase-separate) in the melt.

For example, at 250°C the polymers will be mutually soluble only for PS concentrations less than 0.096 mole fraction or greater than 0.913 mole fraction. Thus the two polymers can be commingled for recycling only in limited proportions.

[latex]^{10}[/latex]Note that [latex]\phi_{ i }^{ J }\left(x_{ i }^{ J } ight)[/latex] is used to indicate that the volume fraction of species i in phase J is a function of its molefraction, [latex]x_{ i }^{ J }[/latex].

Question 11.2.6: Determining the Compatibility of Polymers Determine the liqu...

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