L. J. GARFIELD AND S. E. PETRIE
1760
cm. for the calculation. The sharp peaks are a result of the discontinuous change from el to e2 a t the boundaries. If our model had O varying continuously
between O1 and O2 then the barrier would have been rounded. The linearity of the sides of the barriers shows that the first term in eq. 11-11predominates.
Viscosity and Glass-Transition Behavior of Polymer-Diluent Systems’
by L. J. Garfield and S. E. Petrie Research Laboratories, Eastman Kodak Company, Rochester, N e w York
(Received November 6 , 1063)
Recent theories of Ferry and Stratton, Fujita and Kishimoto, and Kelley and Bueche, based on free-volume considerations, yield relationships describing the dependence of the glass-transition temperature and of the viscosity on concentration in polymer-diluent systems. These theories, which permit the prediction of these properties over wide ranges of concentration from a knowledge of the expansion coefficients and glass-transition temperatures of the pure polymer and of the diluent, are shown to be basically equivalent. n‘ew thermal and viscosity data are presented for a number of polymer-diluent systems which provide a direct experimental test of the relationships and thereby support, in general, the underlying assumptions of these theories. The glass-transition phenomenon in low-molecular-weight liquid substances, a somewhat unfamiliar concept, is shown to be as much a determinant of their rheological behavior as it is observed to be in the case of amorphous polymers. Glass transitions of several diluents have been detected by differential thermal analysis, and the temperatures a t which they were observed are given.
are particularly pertinent to this problem. Free Introduction volume is considered to be that part of the free space Recently, important advances have been made in a liquid which can be redistributed without energy towards an understanding of the glass-transition and change. It is the amount and the distribution of this viscosity behavior of polymer-diluent systems. The papers of Fujita and Kishimoto,2 Ferry and S t r a t t ~ n , ~ free volume which governs the mobility of the molecules of a liquid. Such concepts provide only a and Kelley and Bueche4have treated the effect of diluent qualitative picture of free volume and do not give it concentration on the glass temperature and viscous an operational definition. Nevertheless, they can be and viscoelastic properties of amorphous polymers on the basis of a relatively simple free-volume model of the (1) Presented in part at the Society of Rheology Meeting, Baltimore, liquid state. Prior to these contributions, the behavior Md., October, 1962. of polymer-diluent systems had been described only (2) (a) H. Fujita and A. Kishimoto, J . Polymer Sci., 28, 547 (1958); (b) J . Chem. Phys., 34, 393 (1961). qualitatively in the literature.6 The purpose of this (3) J. D. Ferry and R. A. Stratton, Kolloid Z.,171, 107 (1960). paper is to investigate further the basis of this free(4) F. N. Kelley and F. Bueche, J . Polymer Sei., 50, 549 (1961). volume theory, to present experimental verification of (5) T. G Fox, 9. Gratch, and S. Loshaek, “Rheology,” Vol. 1, some of the expressions developed by these authors, and F. R. Eirich, Ed., 1958, p. 459. to study the implications of these expressions. (6) M . H. Cohen and D. Turnbull, J . Chem. Phys., 3 1 , 1164 (1959). The free-volume concepts of Cohen and T ~ r n b u l l ~ (7) ~ ~M. H. Cohen and D. Turnbull, ibid., 34, 120 (1961). ~
The Journal of Physical Chemistry
~
~~
~
~~~~
~
VISCOSITY A N D GLASS-TRANSITIOS BEHAVIOR O F POLYMER-DILUENT SYSTEMS
used to describe the glass transition as occurring a t a temperature a t which the basic flow units of the liquid have sufficient thermal energy so that there can be local fluctuations of dlensity. Thus, a t various points in the liquid enough empty space is localized adjacent to a flow unit for it to move. Cohlen and Turnbull predict, on the basis of this theory, that all liquids, whether of low molecular weight or high, will become glasses if conditions aire such that there is no intervening first-order transition. In considering the viscosity and glass-transition behavior of polymer-diluent mixtures iit is convenient to apply first the free-volume concepts to the pure components in their unmixed states. Subsequent analysis of the mixtures is based on the assumption of simple additivity. The application of free-volume considerations to glass-forming liquids by Williams, Landel, and Ferrys led to the generalized expression for the temperature dependence of the viscosity. The WLF (WilliamsLandel-Ferry) relationship was derived from a modified Doolittleg equation. A necessary condition for the derivation from the free-volume model, of even the form of the WLF relationship, is that the fractional free volume of the liquid be a linear function of ternperature. That is fop) = JYT,)
+ 4 T - T,)
Vf,,(T,Vl) = V d T )
+ V,,(T)
(2)
or alternately, in terms of fractional free volume
f(T,Vl)
=
[fllT) - fi(T)I~l -tf2(T)
diluent, fl(T). This assumption is for all liquids. Substituting eq. 1 into eq. 3 for fl(T ) and fz(T ) assuming that
(4) Kelley and Bueche have derived the equation
The assumption involved in eq. 4,that the glass transition is an iso-free-volume state for the diluent as well as the polymer, does not require specific knowledge of the magnitudes of fl( T,J and fz(TgJ,but only that they be approximately equal. The Kelley-Bueche relationship, which gives T,,,, the glass-transition temperature of the polymerdiluent mixture, in terms of directly measurable quantities, has not been verified using measured glass temperatures of both the pure polymer and the diluent. Such a verification would support the assumption of the additivity of free volume for polymer-diluent systems (eq. 2). Fujita and Kishimoto2bhave applied these same freevolume concept,s to the concentration dependence of the viscosity. They have derived the relationship 1
(1)
where QI is the (differencebetween the macroscopic expansion coefficients above and below the glass-transition temperature T , Equation 1 may be expected to hold for all liquids, polymeiric or other, throughout the temperature range in which their viscosities obey the WLF relationship. According to the free-volume model, the effect of diluent on a polymer system is to modify the free volume. The common assumption of the t h e ~ r y ~is- ~ that the free volume contributed by the diluent, Vf,(T), can be added to the free volume of the polymer, Vf,(T), to give the free volume of the polymer-diluent system having u1 volume fraction of diluent, Vf12(T,v~).Thus
(3)
where f , ( T ) and f z ( T )are the fractionad free volumes of the pure diluent and polymer, respectively, in their unmixed states. The temperature dependence of the fractional free volume of the pure polymer, jz(T), js given by eq. 1. Kelley and Bueche4 assume that the same explicit expression holds for the temperature dependence of the fractional free volume of the pure
1751
where a, =
77
V*(l -
Ul>
77 is the viscosity of the system at a given temperature and volume fraction of diluent vl, and q* is the melt viscosity of the pure polymer a t the same temperature. Equation 6 applies only to solutions sufficiently concentrated in polymer to permit interchain entanglements. Fujita, et uZ.,2b,18have tested eq. 6 with concentration data for a variety of polymerdiluent systems. They have deduced some generalizations about the nature and temperature dependence of the parameter P(T). Ferry and Stratton3pointed out that if the assumption of the additivity of free volume (eq. 3) is made, then
(8) M. L. Williams, R. F. Landel, and J. D. Ferry, J . Am. Chem.. Soc., 77, 3701 (1956). (9) A . K. Doolittle, J . A p p l . Phys., 22, 1471 (1951).
Volume 68,Number 7
J u l y , 1964
1752
L. J. GARFIELD AND S. E. PETRIE
I n addition, they suggested that since f i ( T ) is usually an order of magnitude smaller thanfl(T), P(T) is nearly equal to .fi( 5”). In this paper we propose that p( T ) can be written P(T)
w(T
-
Tg,) - az(T - TgJ
(8)
if fi(7’) and f i ( T ) can be assumed to be simple linear functions of temperature (eq. l),and if the glass transition is approximately an iso-free-volume state. The a’s in eq. 8 have the same meaning as in eq. 1. This gives to the quantity P ( T ) , which Fujita and Kishjmot0 have called the “plasticizing efficiency” of a diluent, a simple and explicit definition that is based upon directly measurable thermal quantities.
Experimental Materials and Methods Of the diluents employed in this investigation, the dicyclohexyl phthalate the di(2-ethylhexyl) phthalate, and the diisodecyl phthalate were used as received from the Food Machinery and Chemical Corp. The diphenyl phthalate and the di-n-octyl phthalate were Eastman Practical Grade, whereas the diethyl phthalate and dibenzyl succinate were Eastman Grade. The Aroclor 1254 was obtained from Monsanto Chemical Go. and used as received. The viscosities of these diluents as a function of temperature were measured in Canon-Fenske-Ostwald thermostated capillary viscometers. The sample of polyneopentyl succinate (NPS), prepared by T. &I. Laakso, of these laboratories, was a whole polymer with a molecular weight of 33,000 as determined by light scattering. lo The isotactic polystyrene” was a whole polymer with an inherent viscosity of 1.04 in trichlorobenzene. The bisphenol A polycarbonate, supplied by G. P. Waugh, of these laboratories, was a whole polymer with an inherent viscosity of 0.83 in 1:1 phenol-chlorobenzene. The XPS-djluent systems were prepared by heating the mixtures in closed bottles a t 120’ for several hours, with occasional stirring. At this temperature, negligible degradation of the 1JPS was observed after 6 hr. The polystyrene and bisphenol ,4 polycarbonate-Aroclor solutions were made up by dissolving them in a comnion, volatile solvent and then removing the volatile solvent from a thin coating. The glass-transition data were obtained for quenched samples by differential thermal analysis.12 The differential thermal method detects the discontinuous change in the thermal diffusivity a t the glass transition.l3?l4 Quenching of the pure diluents was achieved by heating them above their melting points and then quickly plunging them into liquid nitrogen. The Journal of Php-ical Chemistry
Measurements of the steady-shear Kewtonian viscosities of the pure polymer and of the polymer-diluent systems were made on a Ferranti-Shirley cone and plate viscometer.1°
Results and Discussion Glass-Transition Temperatures of Pure Liquids. The glass-transition temperatures determined for a number of diluents are given in Table I, along with the glasstransition temperatures of the pure polymers employed in these studies. hll these transitions were sharply defined and similar in appearance. For glycerol, a glass transition of -84’ was observed. This value is in good agreement with the discontinuity observed in specific heat-temperature data.l5p16 Thus, there is support for the concept of the glass transition of a diluent. Flow Behavior o j Diluents. Since the glass-transition phenomenon is not peculiar to polymers, it seemed reasonable to analyze the flow behavior of the diluents with the WLF relationship by the same method that is commonly used for polymers in a temperature region as close to their Tg’s as possible. Data on the temperature dependence of the viscosity of some of the diluents are shown in Fig. 1. Here log aT, defined as the ratio of the flow time through the capillary a t a particular temperature to the flow time at 2 5 O , is plotted us. the temperature for two of the diluents. The experimental points in Fig. 1 fit a curve calculated from the WLF equation,8 a t least up to a temperature of about 100’ above the respective Tg’s. Data a t lower temperatures could not be obtained because the shear conditions induced crystallization, although samples were readily quenched to the glassy state for the DTA studies. Because of the limited range below T , 100, the exact values of the parameters f(T,) and a could not be obtained from a WLF analysis. The main conclusion to be drawn from these capillary-flow data is that, in view of the agreement of the data with the form of the WLF relationship, the viscous-flow behavior of monomeric liquids can be treated, in the appropriate temperature region, by the same free-volume model that has been so successfully applied to polymers. In addition, we have support for expressing the fractional free volume
+
(10) L. J . Garfield, S. E. Petrie, and D. W. Vanas, Trans. $oc. RheoZogy, 6 , 131 (1962). (11) J. R. L. Williams, J . Org. Chem., 23, 1206 (1958). (12) S.E. Petrie and D. W. Vanas, to be published. (13) J. Coste, Ind. Plastigzces Mod. (Paris), 9, 37 (1957). (14) J. J. Keavney and E. C. Eberlin, J . A p p l . Polymer Sci., 3, 47 (1960). (15) G. E. Gibson and W. F. Giauque, J . Am. Chem. Soc., 45, 93 (1923). (16) G. S. Parks and H . Hoffman, J. Phys. Chem., 31, 1842 (1927)
1753
VISCOSITYAND GLASS-TRANSITION BEHAVIOR OF POLYMER-DILUENT SYSTEMS
\
4-
\
i.2
- 2.0 -3 - 24 - 2.8 -3 2
100
1-emperature ("C) Volume f r a c t m of Aroclor
Figure 1. Temperature dependence of the viscosity of diluents: 0 , dibenzyl succinate; A, dicyclohexyl phthalate.
Figure 2. Glass-transition temperatures of polymer-Aroclor 1254 systems.
of a monomeric liquid by eq. 1to a t least 100' above its T,. Glass-Transition Temperatures of Polymer-Diluent Systems. I n Fig. 2 and 3, the glass-transition temperatures of the various polymer-diluent, systems investigated are plotted as functions of the corresponding diluent concentrations given in volume fractions. The solutions were made on a weight rather than on a volume basis, but the differences in densities were considered small and were ignored. The points represent the experimental values, and the curves were calculated from the Kelley-Rueche relationship, eq. 5 , using the glass-transition temperatures of Table I and expansion coefficients of the pure polymers and diluents. For the expansion coefficients of the polymers, the generally accepted value of 4.8 X was employed.* In
Te
t
Diluent
'C.
Diluent
OC.
Aroclor 1254 Dibenzyl succinate Diethyl phthalabe Di-n-octyl uhthalate
-- 24 - 58
Di( 2-ethylhexyl) phthalate Diisodecyl phthalate Diphenyl uhthalate Di&clo hexyl phthalate
- 82
--85 -- 87
Polymer
Polyneopentyl succinate Isotactic polystyrene Bisphenol A polycarbonate
- 16
95
0
P g -2s ... e
P
E
+
E .=
g
-50
NPS- Diphenyl phthalate
e
NPS-Dicyclohexyl phthalate NPS-Di ( 2 - ethylhexyl) phthalate
, w e
u1
0
-75 I
0.6
I
1.0
Volume fraction of diluent
Figure 3. Glass-transition temperatures of NPS-diluent systems.
Table I : Glass Temperatures of Diluents and Polymers Te!
I254
- 77 - 15
-33
the absence of specific expansion data for the diluents, cy = was assumed. Use of a value such as 8 X lo-* obtained by Birnboim" for the expansion coefficient of di(2-ethylhexyl) phthalate does not noticeably change the correspondence of the experimental and calculated T , data. However, more specific values will be used later in connection with the viscosity analysis, which is more sensitive to the expansion coefficient values than is eq. 6. The agreement of the experimental data with the calculated curves is satisfactory in all cases. In-
152 (17) M. H . Birnboim, Doctoral Dissertation, University of Wisconsin, 1961.
Volume 68, Number 7 Julg, 1964
1754
terestingly enough, the T,’s of the mixtures of polyneopentyl succinate with diphenyl phthalate are unchanged over the entire composition range. In this case the agreement between the calculated and the experimental data appears to be almost trivial. Kevertheless, the data as a whole adequately verify the Kelley-Bueche equation. The main conclusion drawn from this glass temperature-composition work is that, in view of the agreement of the experimental data with the theory, the underlying assumption of the simple additivity of free volume (eq. 2 ) is adequately substantiated for the systems discussed here. viscosity of Polymer-Diluent Mixtures. I n order to verify eq. 6 experimentally, using values of p(T) computed from eq. 8, mixtures of polyneopentyl succinate with various diluents were made. The viscosities of the pure polymer and of the mixtures were measured a t 50’. The In a, data are plotted us. volume fraction of diluent in Fig. 4 and are compared with the curves calculated from eq. 6. I n order to compute p ( T ) a t 50’ from eq, 8, glass-temperature data from Table I and values for al of 8 X 10-4, 7 X 10-4, and 8.5 X 10-4 for diphenyl phthalate, dicyclohexyl phthalate, and dibenzyl succinate, respectively, were used. The value of the fractional free volume of the pure polymer a t 50’, f2(T),used in eq. 6 was computed from eq. 1 using a value for f ( T g )of 0.037 and a value for a of 4.8 X 10-4.10
The agreement between the calculated and the experimental data in Fig. 4 is satisfactory in the concentration range covered here in view of the fact that the calculated curves were obtained from thermal data. The parameter p( T ) has been derived empirically for many polymer-diluent systems by Fujita, et nl., from viscous-flow data, I n the system polymethyl acrylate and diethyl phthalate, which Fujita and Maekawa18 studied extensively, they empirically obtain a value of 0.065 for P(T) a t 3’, the glass-transition temperature of the polymer. Using the expansion data for diethyl phthalate given in their paper and our measured value for the T , of diethyl phthalate, -85’) we obtain a value for p (3’) from eq. 8 of 0.06, which is in excellent agreement with their experimental value. Furthermore, Fujita and others have concluded, empirically, that the “plasticizing efficiency” of any
The Journal of Physical Chemietry
L. J. GARFIELD AND S. E. PETRIE
2.0,
NPS + Dicyclohexyl phtholote
0
0
/ I I
/.
/
NPS + Dibenzyl succinote
~~~~
0
0.1
0.2
0.3
0.4
0.5
“I
Figure 4. Viscosity behavior of polymer-diluent systems.
diluent is not only greatest a t the glass temperature of the polymer but also is only a function of the properties of the diluent a t that temperature. These conclusions are apparent from an examination of P(T) as given in eq. 8, except for those cases in which the glass temperature of the diluent is equal to or greater than that of the polymer. I n addition, it can be predicted from eq. 8 that, in general, the lower the glass temperature of the diluent the greater will be its “plasticizing efficiency.” When the glass-transition temperature of the pure polymer and that of the diluent are nearly the same, the lowering of the viscosity of the system by the diluent a t temperatures above T , can be explained by the difference in their expansion coefficients.
Acknowledgment. The authors wish to thank Mr. J. Saturno, who assisted with many of the measurements. (18) H . Fujita and E. Maekawa, J . Phys. Chem., 66, 1053 (1962).
