1 Gel Permeation Chromatography Use in the Determination of Polymer-Polymer Downloaded via 185.50.251.65 on July 22, 2018 at 09:17:29 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
Interaction Parameters V E N K A T A R A M A N NARASIMHAN, C H A R L E S M . BURNS, and ROBERT Y. M. H U A N G Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 DOUGLAS R. L L O Y D Department of Chemical Engineering, The University of Texas at Austin, Austin, T X 78712 The use of gel permeation chromatography for the quantitative analysis of composition in mixed polymer systems is presented. Equations are developed for the determination of polymer-polymer interaction parameters for monodisperse and polydisperse polymers. Typical interaction parameters determined from the experimental data for the polystyrene-polybutadiene systems are presented. P O L Y M E R I N C O M P A T I B I L I T Y A N D S U B S E Q U E N T P H A S E S E P A R A T I O N have been the subjects of growing interest. The phenomenon of phase separation i n mixtures of two polymers i n a mutual solvent or two polymers i n the solid state is known as incompatibility and is of considerable practical impor tance. L i m i t e d miscibility plays a role i n the preparative and analytical fractionation of polymers; i n the preparation of plastic films, including paint and varnish coatings; and i n the determination of service properties of certain systems such as high impact styrene-butadiene products. Many investigations dealing w i t h polymer-polymer incompatibility in a common solvent have been conducted i n the past 35 years (1-31). Phase separation between the two incompatible polymers polystyrene (PS) and polybutadiene (PBD) is of considerable industrial importance and has been studied i n solution (15-18, 24-26, 31) and i n the solid state (32-36). I n solution, the binodal equilibrium curve on the triangular diagram has fre quently been approximated by cloud point isotherms determined by
0065-2393/84/0206-0003$06.00/0 © 1984 American Chemical Society
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
4
POLYMER BLENDS A N D COMPOSITES IN MULTIPHASE SYSTEMS
turbidimetric titration. Rigorous determination of the binodal curve, i n cluding tie lines and the critical point, requires lengthy equilibration and subsequent analysis of the conjugate phases. The method of analysis is se verely restricted by the necessity of maintaining an antioxidant in the solu tion to inhibit the cross-linking of the P B D . The antioxidant normally used, 2,6-di-teri-butyl-4-methylphenol, masks the U V analysis of a solution of the two polymers. T o overcome these problems of analysis, we used a gel permeation chromatograph equipped w i t h both differential refractive i n dex (RI) and U V absorbance detectors. The antioxidant was separated from the two polymers by the columns before the polymers entered the de tectors. The use of gel permeation chromatography (GPC) for the determi nation of composition i n the PS/PBD system is not new, although it has not been reported extensively (20, 37, 38). W e successfully used G P C w i t h se quential RI and U V detectors to give the compositional analysis of the con jugate phases i n the incompatible system of PS and P B D w i t h tetrahydrofuran (THF) as mutual solvent. Application of this method to equilibrated samples yields tie lines, binodal curves, and plait points. A detailed discus sion of the experimental procedure and of the determination of the tie lines, the plait point, and the binodal curves was given earlier (27, 28, 39). The object of the present work is to evaluate the polymer-polymer interaction parameter χ 3 from the quantitative analysis of mixtures of PS and P B D i n toluene by using G P C . This parameter is valuable as a means of characterizing the incompatibility of the two polymers. In the past, several methods have been used for the determination of polymer-polymer interaction parameters. These methods were mostly based on the study of ternary systems consisting of the two polymers i n question and a common solvent. Stockmayer and Stanley (40) calculated X23 from light scattering measurements; Sakurada et al. (41) calculated χ 23 by measuring the extent of swelling of polymers by a swelling agent; and Allen et al. (15) and Berek et al. (30) calculated χ 3 from the parameters of phase equilibrium by using polymer-solvent interaction parameters from two component systems. W e now discuss the theoretical basis for the determination of χ 23 from the phase equilibrium data. 2
2
Theoretical Discussions Scott (6) and Tompa (7) were the first to investigate different mathematical treatments of the Flory-Huggins theory to derive expressions that would help in studying the behavior of polymer-polymer-solvent systems. Scott (6) discussed mixtures of two polymers i n the presence of a solvent; that is, a three-component mixture. H e obtained equations that lead to Gibbs free energy of mixing. T o m p a (7) developed equations to express spinodals for such ternary systems.
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
1.
Gel Permeation Chromatography
NARASIMHAN ET AL.
5
Hsu and Prausnitz (42) described a numerical procedure for predict ing the compositions of the coexisting phases, for establishing tie lines, and for tracing the binodal and determining the critical point. These calcula tions were also based on the Flory-Huggins theory. A simple mathematical scheme can be developed to calculate the i n teraction parameters starting from the Flory-Huggins expression for the Gibbs free energy of mixing (AG ) for a system consisting of two polymers and one solvent: m
AG, m = n j l n ^ ! + ri£ In 02 + 3 l RT n
+
(Χ12Φ1Φ2 + Χ13Φ1Φ3 +
X (mitli + 7712712
+
m
Φ3
n
Χ23Φ2Φ3)
(i)
3 3) n
where n is the number of moles of ith component i n the mixture, φ is the volume fraction of ith component, χ · is the Flory-Huggins interaction pa rameter, and m is the ratio of the molar volume of i to that of the reference component. Subscripts 2 and 3 denote polymers 2 and 3, and 1 denotes the solvent. The chemical potentials of each component (8) can be obtained by differentiation of the Gibbs free energy of mixing with respect to η : {
ί
ί;
{
RT (2) Δμ «Γ
2
= 1ηφ +
1 - (—
2
Φ + (1 3
τη )φχ 2
(3) Δμ fiT
3
+ τη [χ (φ 3
3
2
+ φγ) + χ φ\ + χι Φι] 2
2
(4)
where Χι =
i
( Χ ΐ 2 + Χ13 ~ Χ 2 3 )
(5a)
Χ2 =
i
(Xl2 + Χ23 - Χ 1 3 )
(5b)
Χ3 = i (Xl3 + Χ23 ~ Χ 1 2 )
(5c)
and ra and m are the molar volume ratios of the polymers to the reference volume V . The reference volume V is the molar volume of solvent V\ ; mi is, therefore, equal to unity, and 2
3
0
0
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
6
POLYMER BLENDS A N D COMPOSITESIN MULTIPHASE SYSTEMS
m
2
= V iV
(6a)
m
3
= ν /ν
(6b)
2
x
3
λ
Equations 2, 3, and 4, mathematically independent, are equivalent to Equations 5a, 5b, and 5c of Ref. 6, respectively. At equilibrium the chemical potential of each component must be the same i n both phases. Denoting the two conjugate phases by single and dou ble primes, Αμ{ = Αμί', Δ μ = Α μ , and Δμ3 = Δμ3. Thus, Equation 2 w i l l give 2
ΧΐΚΦί
+ * 3 ) " (*!
+ Φ ί ) ] + Χ2(Φί
!
- Κ
£
)
2
2
+
~ Φί') + X j ( * 3
2
(» -
- «
" Φί*>
!
(> -
+
- «
7
Similarly, Equation 3 yields Χ [(03 + ΦΙ) - (03 + Φί') ] + Χ ( 0 3 " 03 ) + Χ ΐ ( Φ ί ~ Φ Γ ) 2
2
2
2
3
2
2
= — l n f é ' ) + ( — - — ) (Φ - Φ ) + ( — - ΐ ) (Φί'- Φί) ™2 \02/ \η» m / V?n / 2
v
3
3
3
(8)
2
and Equation 4 yields Χ [(Φ ' + Φ ί ) - (02 + Φ ί ' ) ] + Χ2(Φ ' " 02 ) + Χι (Φί - Φι" ) 3
2
2
2
2
2
2
2
2
= ^ Ι η β Ι ) + ( — - — ) ( * 2 " 02) + ( — - ΐ ) (Φί- Φί) ™ V 0 3 / \»"3 "»2/ \"»3 /
(9)
3
Subtracting Equation 7 from Equation 8 and simplifying yield (10)
2χ (Φ2 - Φί) + 2 ( φ ί ' - φί) = l n ( - | £ ) - ^ 1 η ( - * " V Φί / «Ι \ 02 2
2
Χ ι
2
Subtracting Equation 7 from Equation 9 and simplifying yield 2χ (Φ " - 0 ') + 2 ( 0 i - 0Γ) 3
3
3
X l
= ^ (|[) - (|i') ln
ln
("J
Substituting the values for χ , χ , and X3 from Equations 5a, 5b, and 5c yields for Equation 10 χ
2
Χ23ΚΦ2 ~ Φί) ~ (Φί ~ Φί)] - Χΐ3[(Φ2 - Φί)] + Χ12[(Φ2 ~ Φί) + (Φί " Φί)] = 1η(Φί/φί) - — 1η(φ£7φ ') m 2
2
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
(12)
1.
7
Gel Permeation Chromatography
NARASIMHAN ET AL.
and for Equation 11 X23[(03 ~ 03) + (Φί - 0 ί ) ] + Xl3[(03 ~ 03) " (0l" ~ 0l')] - Xi [(03
- 03) + (Φι - Φί)] = — ln(03 /03) - ln(0f/0i) (13) m Thus we have two equations i n terms of the concentrations of the con jugate solutions and the three interaction parameters. A d d i n g Equations 12 and 13 and simplifying yield 2
3
— =
In(03703)
~ l n ( 0 2 7 0 ' ) - (xi3 - Xia)(0i - Φι) 2
_^3
rri2
(03
^
~ 03' + 02 - 02 )
Calculating the interaction parameters from measured equilibrium concentrations by using the above equations w i l l be discussed i n the section entitled "Results and Discussion." Equations can also be developed for the interaction parameters when dealing w i t h polydisperse polymers. The free enthalpy of mixing function Ζ employed i n this development is given by Koningsveld et al. (43). AG Ζ = - j y ^ ; = 01 In 0i + Σ 0 ,*w~ j In 0 k
2
ι + Σ 0 ,m -- 1η0 ;' = i 3
3
1
2
M
+ ^(02,03, )
( )
T
3/
l 5
where φ is the interaction function given by Φ = Φ\2 + ^13
+
Φ23 = 0102X12(02, 03)
+ 0103X13(02, 03) + 0203X23(02, 03) and where Ν is the total number of moles; Δ G is the free enthalpy (Gibbs free energy) of mixing; R is the gas constant; Τ is the absolute temperature; 0! is the volume fraction of the low molecular weight solvent; 0 j is the volume fraction of species i i n polymer 2; m is the relative chain length of species i i n polymer 2; 0 · is the volume fraction of species / i n polymer 3; m - is the relative chain length of species / i n polymer 3; 02 = Σ φ^ι is the volume fraction of the whole polymer 2; 0 = Σ 0 · is the volume fraction of the whole polymer 3; and χ , χ , and χ 3 are the interaction parame ters. There are k and I components, respectively, i n polymer 2 and polymer 3 m
2?
2 i
3 ;
3 ;
3
1 2
1 3
3/
2
k ι Ν = ni + Σ n m + Σ nym i= I /=1 where η is the number of moles of component 1, 2, or 3. u
u
3j
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
(16)
P O L Y M E R B L E N D S A N DC O M P O S I T E S I N M U L T I P H A S E S Y S T E M S
As i n the ease of monodisperse polymers, the chemical potentials of each component can be obtained by differentiation of Equation 15 w i t h respect to n. Partial differentiation yields the following equations for cases in which polymer 2 alone is polydisperse and for cases i n w h i c h polymers 2 and 3 are both polydisperse. W h e n polymer 2 alone is polydisperse, we get Α μι
RT
Γ, /ιM = In Φι + + ι - — 02 + \m J
—
\m J_
2
X202 + X303
W*
2
iVrifi,
Αμ
2
~rt
/ ιM
1 -
Έ
=
Α
~
3
φ
2
2
2
3
Δμ
3
= 1η
Φ 3 0 ΐ ( ^ ( 1 - Φ \ "02 φ
3
+ 1 - [ B L
m
φ
I
2i
(17)
mn
T
2i
ί
- *2)
1
(18)
) - Φ 3 ^ ΟΦζ + m3[χ3(Φ2 + Φ ι )
3
φ
ι
φ
2
( ^
{
1
- φ) - φ 2
3
^ )
(1
02)
φ3
(19)
(1 - Φζ) ~ Φ3
When polymers 2 and 3 are both polydisperse,
HT
2
(^ "^ " ^)
+ Φ3 Φί (
Αμι
2i
+ *3 (»2 - 1 * — - —
- 0 3 ^ )
+ (1 - πι )φι
2
1?))
+ 03
Ί^
2
+ Χ2Φ1 + ΧΐΦΐ] + ^
+ φ2φ3
2
2
+ ΧιΦΪ} + Φζ\φιΦ*(^
+ ^ ( ^ ( 1 - 0 2 ) +
φ
mn\ 2i
2
3
3
*{
φ
^ 2 « η 2 » + Φι ("2 ~ ^
+ 0 { x ( 0 3 + Φι) + Χ Φ 2
+
/
1η
„ -
ς !
^)\
+ Φ )ΦΜ^(1 3φ
+ *pbc,»,
*
- 0) - 0 3 ^ ) 3φ 2
ν
2
3
«ι>
+
2
Χ,Φ',
+
χι·* )
+ 0203^(1 " 3φ
2
02)
2
for ρ Ψ q; ρ = 2, 3; and g = 2, 3 and where χ χ , and χ are given by Equation 5a, 5b, and 5c, respectively. From these equations, suitable expressions can be derived to calculate interaction parameters by using phase equilibrium data, as discussed ear lier for the monodisperse polymers, and by using information about the molecular weight distributions obtained by G P C . ΐ 9
2
3
Experimental The system of PS and PBD was selected to demonstrate the case of narrow molecu lar weight distributions. The importance of the two polymers in the polymer indus try and the commercial availability of these two polymers in narrow molecular weight distribution samples made this system a logical choice. The characteristics of the particular polymer samples employed are given in Table I. Toluene was se lected because it is a good mutual solvent and because the χγ values for PS-toluene are published (44). The procedure adopted for the sample preparation is given earlier (27). Equi librium was attained at 23 °C and 1 atm. A detailed discussion of GPC with sequential RI and U V detectors for the quantitative analysis of the conjugate phases of the incompatible system of PS and PBD with T H F as solvent is also given earlier (27, 28). 2
Results and Discussion The binodal curves, tie lines, and plait points for the two P S - P B D systems studied are given i n Figures 1 and 2. The polymer-solvent interaction parameter χ for PS-toluene was obtained from the work of Scholte (44) and used i n Equations 12 and 13 to solve simultaneously for the interaction parameters χ (PBD-toluene) and X (PS-PBD). The results are presented i n Tables II and III. 1 2
1 3
23
Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.
P O L Y M E R B L E N D S A N DC O M P O S I T E S I N MULTIPHASE SYSTEMS
10
Table I. Characteristics of Polymer Samples Χ ΙΟ"
Sample
M
PS 37,000 PS 100,000 P B D 170,000°
36.0 100.0 170.0 ± 17
w
M„ Χ
3
10-
3
33.0 100.0 135.0 ± 13
MJM„
