Aug 11, 1980 - ined and a quantitative prediction made of critical temperatures of solution. These ... calculated from the equations f*---- - -. T* = ---. (1). ~=1+ ~,T/3. I +a,T'. (2) ..... Points of binodal maxima were accepted as critical point, s,.
Polymer Science U.S.S.R. Vol. 23, No. 12, pp. 3018-3026, 1981 Printed in Poland
0032-3950/81/123018-09507.50/0 ©1982 Pergamon Press Ltd.
PHASE EQUILIBRIUM OF POLYMER SYSTEMS* S. A Vs~Ivl~ov and N. A. KOMOLOVA A. M. Gor'kii State University, Ural
(Received 11 August 1980) Temperature dependences of the parameter of interaction X~2of components calculated by the Flory and Patterson equations were compared with experimental resuits of the phase equilibriumofpolystyrene solutions in decalin, benzene, ebhylbenzerie, cyclohexane and of the polystyreno--polymethylmethacrylate system. A study was made in a wide range of temperatures of the phase equilibrium of the polystyrenepolymethyl methacrylate-ethylacetate system and using the Scott equation the temperature dependence of the parameter of interaction of polystyrene with polymethyl methacrylate in solutions of critical composition, determined. It was shown that the Flory and Patterson equations enable the temperature dependence X~ to be calculated accurately for the polymer-solvent and polymer-polymer systems examined and a quantitative prediction made of critical temperatures of solution. These values of Z1~cannot, however, be used for calculating the complete phase diagram (binodals and spinodals) without the concentration dependence of Xl~; boundary curves calculated are considerably displaced (in relation to experimental curves) to the range of lower polymer concentrations.
I ~ practice polymer systems are usually processed at high temperatures (~433-443°K), at which homogeneous systems are formed. Products made o f these systems are used at much lower temperatures and in m a n y cases the homogeneous systems formed during processing lose their stability and separate which shortens useful life. When forecasting the useful life of polymer compositions it is therefore essential to know the temperature relation dependence of interaction of components and phase diagrams of systems, which is not always simple t o determine experimentally. Theoretical calculations of the temperature dependence of the parameter of interaction of components are therefore gaining increasing significance. This par a m et er being known it is possible in m a n y cases to determine th e position of binodal and spinodal of the system. The F i e r y t heory [1, 2] f u r t h e r developed b y Patterson [3] is most extensively used for this purpose. Calculations mad e using this t heor y are described in several papers [4-7]. A study is made in this paper of results of a theoretical and experimental analysis of the phase equilibrium using PS solutions in four solvents and a PS-PMMA system. *Vysokomol. soyed. A28: No. 12, 2781-2787, 1981. 3018
Phase equilibrium of polymer systems
3019
P S so/utions. The temperature dependence of the parameter of interaction o f components was calculated from the equation [3] Z,3=-
Clv2
e l T2
+
1--~-+
2 ( ~~,-+ -- 1)
where cx is one third of the number of external degrees of freedom of the solvent molecule; v is the parameter characterizing the difference between cehesive energies of the polymer and solvent and dimensions of the polymer chain segment and solvent molecule; Vl is effective solvent volume; z--parameter characterizing the difference in free volume of components, which was calculated from the equation = 1 .=T**/T~*
Parameters of temperature reduction of the solvent T~* and polymer T* were calculated from the equations f*---- - -
T* = ---
(1)
~=1+ ~,T/3 I +a,T'
(2)
where ~ is the coefficient of volumetric expansion of the i component. The empirical Cowie equation [5] was also used to calculate T*, which relates this parameter with the critical temperature of the solvent Tcr T* = -- 17.491+ 74"774 × Tcr--61"085 × 10-aTc~r v1 was calculated from the equation [1] *
*
P, Vs M, c, =
RT~
'
where M, is the molecular weight of solvent; P * - - p a r a m e t e r of pressure reduction, which is
t'* = (oqT/Z,) ~,~ ,
(3)
1 V~ (aV,/aP)~ is the coefficient of isothermal compression of the solvent; V* is the parameter of volumetric reduction of the solvent, which is
where f l = -
L
V*-
V1 =
V1 -
~,T/3 ]3
[1+ 1--~I T J L
(4)
8026
S.A. Vsuav~ov and 1~'. A. Ko~o~ovA
T h e value of r ~ was calculated from the ratio v'=
where
P*= (sl and V{ being the surface and volume of the solvent molecule). For spherical molecules sa-~3/Rx, where R1 is the sphere radius. When calculating s~ the polymer molecule was simulated using a number o f cylinders. In this case i'
s2=2/R~ where R2 is the crystallographic radius of the molecule. According to theory [1], parameters of reduction calculated from equations (1)-(4) are constant. However, in the range of high temperatures aa and fll vary considerably therefore, parameters of reduction also vary, which we considered in our calculations. FABLE
1. M O L E C U L A R
PARAMETERS
OF P S
SOLUTIONS IN DIFFEI~ENT SOLVENTS AT
ACEP x Solvent
(31T2
Cl
Cly2
x 10 -6,
Y
AVIX 103,
0'10 0"19 0"22 0"34
0"10 0"15 0"24 0"14
J/m ~ Decalin Ethylbenzene Cyclohexane Benzene
1.27 1.15 t 1-1414] 1.37
0.054 0.087 0.10114] 0.151
0"0244 0"0217 0.021914] 0
0"37 0"31 0.26 0
68'1 28"0 65-2 -3"3
298°K
rn,~/kg
Calculated molecular parameters for PS solutions are shown in Table 1. These results were used for calculating the two terms of the interaction parameter Zl~ el T2
Cl v2 -
-
Q - 1-~1 -+
-
4~-+
Y--- 2(~vl - 1 ) "
Figure 1 shows calculated temperature dependences of parameters Q, Y and )CI~ of the systems examined. I t can be seen t h a t for all systems the value of Y dependent on the difference in free volumes of components, increases with temperature, while the value of Q dependent on the difference in cohesive energies of the polymer and solvent, decreases. Parameter Xlz which is the total of these: effects is equal to 0.5 (0 conditions) for the PS-decalin system in the low temperature range and for the PS-benzene system, in the high temperature range. For PS-ethylbenzene and PS-cyclohexane systems X12 twice assumes the value o f 0.5 at low and high temperatures. Consequently, according to theory [1-3], the PS-decalin system has an upper, while the PS-benzene system has a lower critical temperature of solution (UCTS and LCTS). PS--ethylbenzene and PS--cyelo-
302.~
Phase equilibriumof polymer systems
hexane systems have UCTS and LCTS. Similar calculations for the PS-cyolohexane system were carried out previously [4, 7]. Table 1 shows the effect on zls of parameters Q and Y. This Table presents values of A C E D = C E D ( P S ) - - C E D (solvent) and A V1-~ Vf (PS) --V! (solvent), where CED is cohesive energy density, V1--free volume. I t follows from Table 1 t h a t except for PS solutions in cyclohexane, a correlation is observed between Q and the difference in the cohesive energy density of components; an increase in ACED increases Q. a
0.5
C
-- -
xt~
0.1
i
-
-
t
~
I 9
i
I
"273
o.1~
t
i
373
L
~L
J
578
L
q73
L
37J
q73 T, g
Fio. 1. Temperature dependence of Q, y, xt2 for PA-decalin (a), PS-benzene (b), PS-ethylbenzene (c) and PS-cyclohexane (d) systems. Broken lines correspond to the value of Zt,= 0.5. With the exception of PS solutions in benzene, a correlation is observed between d Vf and Y: an increase in the difference in free volumes of components increases the effect of Y on Z12. The temperature dependence of Z1~was used to calculate critical parameters of 12 the system. The temperature corresponding to Xer-----0"5 ( l + r - * ) 2 was assumed to be the critical temperature, r - - t h e number of segments in the polymer molecule
T A B L E 2. C R I T I C A L PARAMETERS OF P S SOLUTIONS
(M~= 3-3 × 10') System PS-deealin (UCTS) PS-benzene (LCTS) PS-othylbenzene (UCTS) (LCTS) PS-cyclohexane (UCTS) (LCTS)
Tcr, K = calculated found 286 523 265 572 269 493
288 [12] 514 267 [13] 568 306 488
calculated 0-007 0.005 0.006 0.006 0.006 0.006
found 0.018 [12] 0.046
o.138 [13l 0-132 0.020 0-044
3022
S . A . Vs~vxov and N. A. Ko~or.ovA
Was considered to be equal to the ratio of molar volumes of the polymer and .solvent. Critical concentration was calculated from the equation [9]
Spinodals were calculated from the equation [9]
q~+
1
l (1--2Zl~--r-1)~+
--0
(5)
Figure 2 shows spinodals and binodals of the systems examined calculated flx)m equation (5) and determined experimentally. Curves of cluod points and :spinodals were determined b y methods previously described [10, 11]. It follows
a
7;,K
xl
,2
"287
52J
/
!
-"
.
283
I
I /
/
c
/
/
583 57J
/
~8
d >¢"
×.
~x.×_.... ~ ' ~ 30~
2
6
2
6
~p~,lO i
FIG. 2. Phase diagrams of PS-decalin (a), PS-benzene (b), PS-ethylbenzene (c), PS-cyclohexane (d) systems (broken lines showy calculated spinodals, continuous lines--experimental results): 1-- binodals, 2--spinodals. f r o m Fig. 2 and Table 2 that critical temperatures calculated using the F l o r y Patterson theory show satisfactory agreement with temperatures determined experimentally. A considerable discrepancy was observed for critical eon.cenCrations. Values of ~ r are 3-8 times lower than the ones determined experi-
Phase equilibrium of polymer systems
3023
mentally. This is, apparently, due to the concentration dependence of Xxz which has n o t been taken into account in the calculations. PS--polymethyl meJhacrylate~tstem. The Flory theory m a y be used, according to Patterson [14], also for polymer-polymer systems if one of the polymers is accepted conventionally as solvent. W e used PS as polymer solvent. An equation previously derived was used for calculation [1]
M, vl' =
'*/+
2(-~--1)
'
where x./M2V* is the parameter of interaction of PS (2) with PMMA (3) related to unit volume of PS, X . is the parameter characterizing the energy density of interaction of molecular segments of PS (2) and PMMA (3). The main form of interaction between macromolecular segments for this system is dispersion, for which Xss is ~ l0 TJ / m S ~ 100 arm [15].
~ r O~' =, !i
273
373 FIG. 3
q73 T,K FIG. 4
~Ifl. 3. Temperature dependence of the parameter of interaction between PS and PMMA: 1--Xas/MaV* (calculated using the Flory-Patterson equation), 2--Z23/V1 (determined accr cr cording to Scott), 3 and 4--critical values of Xzs/MsVi and Xss/V1. FIG. 4. Phase diagram of the PS--PMM~--ethyl acetate (E) system. The following parameters of reduction were used for the calculation: 112"= 0.810 × ×10-Sma/kg; T * : 7 2 4 0 K; P*-----5.47×10 s P a [16] for 298 K; V*~---0.837× × 1 0 -s mS/kg; T * = 8 7 0 0 K; P * = 5 . 0 0 × 1 0 s P a [16] for 473 K; T * = 7 0 5 0 K for 298 K [16]; V*-----0.73 × 10 -s mS/kg. Figure 3 shows the temperature dependence of the parameter of interaction of
3024
S. A. VsmvKov and N. A. KOMOLOV.¢
PS--PMMA and the critical value of this parameter calculated from the equation er
X~a
lr,M 2V,JJw½__~ M V+~-~ -r-~ a aj j
M2V.~L~.
~'or PS ~ = 3 . 3 × l0 s, for PMMA ~ 3 = 5 . 1 × l0 4. It can be seen that in the entire temperature range X~3/M2V~ is higher than the critical value, i.e. polymers are incompatible. In fact, at room temperature the mutual solubility of PS with PMMA is low, as shown by Kuleznev et al. [17]. O n increasing temperature X~/M2V~ decreases, which is evidence of improved interaction between PS and PMMA. Consequently, this system has UCTS, this temperature, however, is much higher than the breakdown temperature of the polymer and cannot be achieved, It was interesting to compare parameters of interaction between PS and PMMA with a similar parameter determined experimentally. A study was therefore made of the phase equilibrium of a PS-PMMA-ethylaeetate ternary system, It can be seen in Fig. 4 that the range of homogeneous solutions on increasing temperature first increases and then decreases. These results were used for calculating the parameter of interaction between PS and PMMA in solutions of critical composition using the equation proposed b y Scott [8]
X~a/V1=
½ [ V~ ½ -{- V3"t-]l ( 1 - - ~l,er) -1 ,
where V1, V2 and Va are the molecular volumes of ethyl acetate, PS and PMMA; q~, er is the volume fraction of the solvent in the critical point. This equation can be used when V~Va.~V~ [18]. For this system this condition holds. Points of binodal maxima were accepted as critical point, s, since ethyl acetate interacts in practically the same w a y with PS and PIVIMA [19] and this position is observed at all temperatures. This is confirmed b y t h e constancy of the weight ratio of PS and PMMA in solutions corresponding to maximum values of binodals at different temperatures. The temperature dependence of the parameter of interaction of PS with PMMA in solutions of critical composition is shown in Fig. 3. The value of Z2a/V~ at 298°K shows satisfactory agreement with results in the literature [20]. The same Figure shows the critical value of x2car/V1,calculated from the equation [18~
zg / v~ =
½ [(
v d v~)-~ + ( v~/ v~)-*]3
It can be seen that the critical parameters of interaction between PS and PMMA, calculated for the binary system of PS-PMMA and for a ternary system of P S PMMA-ethyl acetate coincide in practice; they are 12 and 10 m -3. The parameter of interaction of polymers in solution is lower by one order of magnitude than the parameter of interaction of PS with PMMA without a solvent. Consequently, in solutions of even critical composition the polymers interact more satisfactorily with each other than without a solvent.
8025.
Phase equilibrium of polymer systez~
On increasing temperature X2,/Vz varies according to a curve with a minimum, which conflicts with the temperature dependence of x,/MsV~. To explain the causes of this discrepancy a study was made of the phase equilibrium of binary systems: PS-ethyl acetate and PMMA-ethyl acetate. Results are shown in Fig. 5. It can be seen that both systems separate both during heating and cooling. Consequently, the thermodynamic affinity of ethyl acetate with PS and PMMA deteriorates both on increasing and reducing temperature. This results in the preferable interaction of polymers with each other in a ternary system, therefore, the parameter of interaction of PS and PMMA in solution at high and low temperatures approximates to the parameter of interaction of PS-PMMA without a solvent, i.e. increases (Fig. 3).
T,K. 483k
22j t~.× Z
CL
b
493 ~X"x~ x.~ 6'
2~
l
~
7 ]
la
x'-" x " ~ ' ~ - - - - - ' - - ~
-
I
N
x"
]
30 C,9/dl
FIG. 5. Phase diagraans of PS-ethyl acetate (a) and PMMA-ethyl acetate (b) systems. Results confirm that the Flory theory, further developed by Patterson enables the temperature dependence of the parameter of inte!action between bomponents of polymer'solvent and polymer-polymer systems to be calculated accurately and critical temperatures of solution predicted quantitatively. However, Xz=values obtained using this theory cannot be applied for calculating the entire phase diagram (binodals and spinodals). The difficulty in plotting full phase diagrams in this case is due to the fact that X12shows a complex dependence on a number of unrelated parameters: concentration, MW, MWD of the polymer, type of low molecular weight component and temperature. Consequently, Zl=: is not a function but a functional. Therefore, different points of the binodal and spinodal may correspond to the same value of XI~ for the same polymer-solvent system. This should be taken into account when analysing phase equilibria of" polymer systems. It is only by varying all parameters that determine Xn that a form of functional Xn may be found which is used in plotting binodals and spinodals. Boundary curves calculated are otherwise considerably displaced in relation to experimental ones to the range of low polymer concentrations. The authors are grateful to A. A. Tager for his attention and for valuable discussions. Tra~lat~ by E. S ~ . z
3026
S . A . YS~LLv~OVand N. A. KOMOLOVA REFERENCES
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