Page 116 - Numerical methods for chemical engineering

102 2 Nonlinear algebraic systems

The contact free energy of mixing is modeled as

G contact = RT n 1 φ 2 χ (2.171)
mix
χ isatemperature-dependentdimensionlessparameterthatisspecifictothesolvent/polymer
pair, where χk b T is a measure of the free energy penalty paid whenever a solvent molecule is
in contact with a polymer segment, k b being the Boltzmann constant. Usually χ = a + b/T ,
where the contributions are from nonideal entropic and enthalpic effects respectively. As χ
increases, the solvent and polymer segments dislike each other more, resulting eventually
in phase separation.
From this lattice theory, the free energy per mole of solvents and segments, g mix =
G mix /n 0 ,is
φ 2

g mix = RT φ 1 φ 2 χ − φ 1 ln φ 1 − ln φ 2 (2.172)
x
We see that as the chain length x increases, the polymer contribution to the entropy of mixing
decreases, due to the constraints that neighboring segments always must be next to each
other.
The chemical potentials, with respect to the pure reference states, at constant temperature
and pressure, are

∂
0
µ i − µ = G mix (2.173)
i
∂n i
T, P, n j =i
Using the free energy expression above, we have
µ 1 − µ 0 1 µ 2 − µ 0
1 2 2 2
= ln φ 1 + 1 − φ 2 + χφ 2 = ln φ 2 − (x − 1)φ 1 + xχφ 1
RT x RT
(2.174)
At χ = 0 we have only the positive ideal entropy of mixing and G mix < 0. As χ increases,
we eventually reach a point where phase separation into two coexisting phases, (I) and (II),
occurs. We wish to compute the properties of the coexisting phases. Let the volume fractions
(I) (II)
of solvent in each phase be φ and φ . From these values, we compute the volume fraction
1 1
of the system occupied by phase (I) from the lever rule
(II)
φ 1 − φ 1
(I)
= (I) (II) (2.175)
φ − φ
1 1
(I) (II)
To compute φ and φ , we equate the chemical potentials in each phase,
1 1
(I) 0 (II) 0
µ − µ i RT ) = µ i − µ /(RT ) i = 1, 2 (2.176)
i
i
This yields the following system of two nonlinear algebraic equations

(I) 1 (I) (I) 2 (II) 1 (II) (II) 2

ln φ 1 + 1 − 1 − φ 1 + χ 1 − φ 1 = ln φ 1 + 1 − 1 − φ 1 + χ 1 − φ 1
x x

(I) (I) (I) 2 (II) (II) (II) 2

ln 1 − φ − (x − 1)φ + xχ φ = ln 1 − φ − (x − 1)φ + xχ φ
1 1 1 1 1 1
(2.177)