Handbook of Solvents - George Wypych - ChemTech - Ventech!

5.1 Solubility parameters 247
Accepting that Eq. [5.1.12] represents a valid means of assignment of a constant δ 2 to
polymer, the rearrangement of this equation gives:
2
2
δ1 χ 2δ2
δ2 χS
− = δ1
RT V RT RT V
⎛ ⎞
⎜ ⎟ − − [5.1.14]
⎝ ⎠
1
1
Now it is assumed that χS is of the order
of magnitude suggested above and that,
in accordance with the Huggins equation, it
is not a function of δ2. Therefore χS/V1 is
2
only about 3% or less of δ2/RT for reasonable
values of δ2 of 10-20 (MJ/m 3 ) 1/2 .
Hence Eq. [5.1.14] gives δ2 from the slope
and intercept on plot against δ1 (see Figure
5.1.1).
This method was improved 21 by using
calculations that exclude strong deviations
of χ. When (χSRT/V1 ≈const), Eq. [5.1.14]
is close to linear (y =A+Bx), where
2 2
Figure 5.1.1. Dependence for equilibrium swelling of y= δ1 − χS(
RT/V1),A=-χS(RT/V1)-δ2,
crosslinked elastomer on the base of polyether urethane. B=2δ2,x=δ1. [Adapted, by permission, from V.Yu. Senichev in Syn- But δ2 enters into expression for a tanthesis
and properties of cross-linked polymers and compositions
on their basis. Russian Academy of Sciences gent of a slope angle and intercept which is
Publishing, Sverdlovsk,1990, p.16]
cut off on the ordinates axes. This can be
eliminated by introduction of a sequential
approximation of χS(RT/V1) and grouping of experimental points in areas characterized by
a definite interval of values χS(RT/V1). Inside each area χS(RT/V1) →const and Eq. [5.1.14]
becomes more precise.
The intervals of values χS(RT/V1) are reduced in the course of computations. For n experimental
points, the files X (x1,x2,.... xn)andY(y1,y2, .... yn) are gathered. Tangent of the
slope angle is defined by the method of least squares and the current value (at the given
stage) of a solubility parameter of a polymer is:
δ2 j
n
∑
=
⎛
⎜
⎝
i = 1
n
n
∑
x y −n
xy
i = 1
2
n
∑
i = 1
n ⎞
x⎟ −n
x
⎠
∑ ∑
i = 1
i = 1
2
[5.1.15]
where:
j a stage of computation
χS(RT/V1) is then calculated using the equation derived from Eqs. [5.1.13] and
[5.1.14]:
⎛ RT ⎞
2
⎜
χ ⎟ S yi δ2j 2xiδ2j
⎝ V ⎟
=− − + [5.1.16]
1 ⎠
j