Solubility Parameter
of
Poly dimethy1 Siloxane)
a
a Function
of
Temperature and Chain Length
INTRODUCTION
Gas chromatography is often used
to
study the thermodynamics of polymer solutions in the limit of infinite dilution of the lowmolecularweight component.'.' We recently reported some thermodynamic characteristics of solutions of five hydrocarbons in eight samples of poly( dimethyl siloxane) (PDMS) of different mean chain
length^ ^
In this note, smoothed values of the activity coefficients from the previous study are manipulated to yield the solubility parameter of PDMS. Effects of temperature and mean chain length on the solubility parameter are discussed.
THEORY
The infinitedilution limit of the FloryHuggins interaction parameter,
X ,
is related to the mass fractionbased activity coefficient
Qp
of the solute
(1)
at infinite dilution in the polymer (2) by' where
VYL
s the molar volume of the pure liquid solute,
A zn
is the numberaverage molar mass of the polymer, and
u1
and
u2
are the specific volumes of the solute and polymer, respectively. Provided that dispersion interactions only are operative between the solute and the polymer, the Flory Huggins interaction parameter may be expressed by an approximate relation5 where
R
is the molar gas constant,
T
is the thermodynamic temperature,
l
and
62
are the solubility parameters of the solute and the polymer, respectively, and is
a
dimensionless parameter related to the entropy of mixing. Rearranging eq. (2
,
one obtains6 Equation
3)
ontains both
6
and as unknowns, therefore, separate values for the two parameters cannot
be
obtained from eq.
(3)
without introducing an additional independent assumption? Early theoretical arguments5 suggest that
P
should be in the range of 0.1 to
0.2
and that it should be independent of
6 .
This means that the term
RTB/VyL
may amount to only a few percent of
6
for a plausible value
of
6 .
and that
RT@/VyL
hould be even less significant when compared with
2~46 .
Accordingly, if values of
X
are available for several different solutes at a given temperature, the polymer solubility parameter
6
is usually estimated from the slope of a straight line obtained by plotting the lefthand side of eq.
(3)
against
6 .
In principle, the square of the slopebased value of
6
may be compared with the intercept of the line to check the relative magnitudes of
6:
and the mean value of
RTj3/VyL
for the solutes employed. In order to estimate the values of at several different temperatures, the values of
Xa
and for the respective solutes at the respective temper atures have to be known. The solubility parameter
of
a
solute is defined by where
AU
is the cohesive energy of the solute. At temperatures well below the critical temperature of the solute, the volumetric behavior
of
the saturated vapor of the solute conforms
to
the virial Journal of Polymer Science: Part
B:
Polymer Physics, Vol. 28, 27152719 (1990) 1990 John Wiley Sons, Inc.
CCC
08876266/90/0130271505 04.00
2716
J.
POLYM SCI : PHYS
ED :
VOL
28
(1990)
equation of
state
truncated after the second term. The cohesive energy of the solute is then given by where
AHuI
s the enthalpy
of
vaporization
of
the solute,
Py
is the vapor pressure of the pure solute at saturation, and
Ell
is the second virial coefficient
of
the solute. The pure component parameters appearing in eq.
5)
may
be
obtained from suitable generalized correlations. The molar volume
V~L
an be calculated using the CampbellThodos equation.' The quantities
Bll
and
dBll/dT
may
be
derived from the HaydenO'Connell correlation? The vapor pressures
of
the solutes may
be
calculated from the Antoine equation, In
Py
=
A
B/
+
C)
(6)
For a number of compounds, the constants
A
B,
nd
C
of
eq.
(6)
have been compiled by Reid et al? If the temperature dependence
of
the vapor pressure
is
described by eq.
6),
he enthalpy of vaporization of the solute may
be
calculated from where
VyG
s the molar volume of the saturated vapor of the solute, Eqs.
(4)(8)
provide
a
tool to calculate the solubility parameter
of
the solute as
a
function of temperature. In a previous rep~rt,~ he temperature dependence of the infinite dilution Raoult law activity coefficient of the solute in the polymer was expressed by In
yl
=
cln
T+ d/T+
e
(9) The constants c,
d,
and
e
were obtained by regression analysis
of
the experimental data. In the limit of infinite dilution of the solute, the Raoult law and the mass fractionbased activity Coefficients are related by where
MI
is the molar mass
of
the solute. From eqs. (9) and
10).
the activity coefficient may
be
calculated for any temperature within the range
of
the experimental
data
(see below). The resultant value of In
fly
is then used in eq.
(
1
to obtain the interaction parameter at the respective temperature. The specific volume
of
the solute can
be
calculated from the molar volume (see above) and the molar mass. The temperature dependences of the specific volumes of the PDMS samples may be taken from the lowpressure results of Lichtenthaler et a1.I'
EXPERIMENTAL
The experimental procedures have been described elsewhere? The hydrocarbon solutes employed were C5 to CB nalkanes and cyclohexane. The samples of PDMS were commercial gas chro matographic stationary phases obtained from various suppliers. The numberaverage and mass average molar maas of the PDMS samples are listed in Table
I.
These values were determined
by
size exclusion chromatography. For all the solutepolymer pairs, the activity coefficients were derived from the specific retention volumes measured at temperatures within
30
to 95°C in about 5OC increments.
NOTES
2717
TABLE
I
NumberAverage and MassAverage Molar Mases of PDMS Samples Sample no.
2,410 3,480 3,480 4,820 15,100 20,700 208,000 218,000
18,000
8,810
16,600 28,800 28,700 95,300 580,000 480,000
RESULTS AND DISCUSSION
Employing the procedure described in the Theory section, the solubility parameters
of
the eight samples of PDMS were calculated
at
temperatures of
30, 40, 50.60, 70.80,
and
90°C.
The values of
62
derived from the
slopes
of the respective linear
fits
according to eq.
(3)
6,
=
slope/2)
are listed in Table
11,
and a typical plot is shown in Figure
1.
At the
95%
confidence level, the regression analysis yields Student's testbased uncertainty limits of up to
2%
and up to
4%
for the
slope
and the intercept, respectively. A comparison of the slope with the intercept (see Theory section) shows that, in all the samples of PDMS at all temperatures, the average values of
RTO/VYL
or the five solutes are positive and less than
5%
of the respective slopebased values of
6:.
However, the precision of the regression analysis is not sufficient to yield reliable absolute data on the average values of
RT@/VYL.
n all the samples of PDMS, the values of
A
decrease systematically as the temperature increases. The most significant source of uncertainty in the resultant solubility parameters is the lack of
a
univocal way to derive separate values for
a
and
B
from eq.
(3)
(see
Theory section). The dominant source
of
experimental error in the determination of
XW
s the uncertainty in the amount of polymer in the column packing?*  The uncertainty in the calculated solubility parameters
of
the solutes generally increases with increasing vapor pressures
of
the solutes. Together with an adverse effect of a broad distribution of chain length of the PDMS samples used in this study, the above sources of error effectively obscure the dependence
of
the resultant solubility parameters on the numberaverage molar mass
of
PDMS. It can be seen from Table
I1
that,
at
a constant temperature, the solubility parameters do not vary with the chain length,
at
least within the range of
A?zn
covered in this study. Therefore, the solubility parameters of the PDMS samples Nos.
18
in Table
I1
may
be
averaged
to
yield the mean values,
.
The temperature TABLE
I1
Solubility Parameters
of
PDMS Samples
[(
J/C~~)~'~] Temperature
( C)
Sample no.
30 40 50 60 70 80
90
14.9 14.9 14.9 14.9 14.9 14.9 15.0 14.9 14.6 14.7 14.7 14.7 14.7 14.7 14.7 14.7 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.1 14.2 14.2 14.2 14.2 14.2 14.2 14.2 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.6 13.6 13.6 13.6 13.6 13.6 13.7 13.6 13.3 13.4 13.4 13.3 13.4 13.4 13.4 13.3
2718
J.
POLYM.
SCI.:
PHYS.
ED.:
VOL.
28 (1990)
2bO

z
0
>
220
c
m
I
r
m
200
100
5
1
16
Fig.
1
Plot to determine solubility parameter of PDMS sample no.
8
t
6OoC.
npentane.
2
nhexane,
3
nheptane,
4
noctane, 5 cyclohexane. dependence of the average solubility parameter for PDMS within
30
o
95OC
may be fitted by an empirical linear function
2
=
15.7
0.026t
(11)
where
t
is the temperature ( C), and the units of
8
are (J/cm3)'/'. At 25OC. eq.
(11)
ields a value of
8
=
15.1
(J/cm3)'l2. This result is within the range of
14.9
o
15.6
J/cm3) ' quoted in the literat~re,'~ nd compares favorably with the values of
15.0
and
15.1
(J/cm3) ', derived in the same way from the interaction parameters measured by static vapor sorptionI6 and by gas chromatography,17 respectively.
I
am indebted to a referee for bringing
Ref
6
to my attention.
References
1.
J.
R.
Conder and C. L. Young, Physicochemical Measurement by
Gas
Chromatography,
2.
J.
S.
Aspler, Theory and applications
of
inverse
CC.
in Pyrolysis and
GC
in Polymer Analysis,
3.
M. Roth and
J.
NovHk, Macromolecules.
19.
364 1986). 4.
D
Patterson,
Y.
B.
Tewari,
H.
P.
Schreiber, and
J.
E.
Guillet, Macromolecules,
4,
356
Wiley, Chichester
(UK),
979,
.
190.
S.
A. Liebman and
E.
J.
Levy (Eds.), Marcel Dekker, New York,
1985,
p.
399523.
(
1971
.