2 Glass Transition Temperature of Polymers Effect of Plasticizer, Chain Ends, and Comonomer M . C . S H E N a n d Α. V. T O B O L S K Y
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Princeton University, Princeton, N. J.
The a f f e c t o f low c o n c e n t r a t i o n s o f p l a s t i c i z e r , c h a i n e n d s , o r c o m o n o m e r on the glass tion
temperature
o f a polymer
transi
c a n b e de
scribed by a l i n e a r e q u a t i o n w i t h r e s p e c t t o t h e w e i g h t f r a c t i o n . O v e r the entire r a n g e o f c o m p o s i t i o n , t h e s e effects c a n be d e s c r i b e d approximately
b y a quadratic equation. The
i n t e r r e l a t i o n o f all of t h e s e e q u a t i o n s is dis cussed a n d c o n s i d e r e d .
incorporating a plasticizer into a polymer almost always leads to a lowering of the glass transition temperature of the polymer. Two empirical equations have been proposed relating the glass transition temperature depression to the diluent content. The first one (4) is : 1
T  T i  kw g
e
(1)
2
where T and T \ are the glass transition temperatures of plasticized and unplasticized polymers, respectively, w is the weight fraction of the diluent, and the coefficient, k, varies from 200° to 500° K . for different diluents in polystyrene. Equation 1 is valid at relatively low dilution if the diluent and polymer are com patible (3,10,11,13). g
0
2
However, when it is desirable to cover the entire region of diluent concentra tion, a parabolic function is required to account for the deviation from linearity at higher dilution. The second empirical equation (7) is as follows: Tg = Tgl Wl + Tg w + Kw w 2
2
x
2
(2)
In Equation 2, T is the glass transition temperature of the pure plasticizer, w\ is the weight fraction of the pure polymer, and Κ is an empirical constant that ranges from —100° to — 300°K. T \ and T are expressed in degrees Kelvin. For other polymers, for instance polymethyi methacrylate (7), values of Κ range from f58° to 300°K. g2
g
g2
27 In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
28
PLASTICIZATION A N D PLASTICIZER PROCESSES
By putting w =» 1 — w in Equation 2 and neglecting the term of order W2 , one obtains a linear equation which can be compared with Equation 1. This results in x
2
2
k  T  T gl
 Κ
g2
(3)
Values of k can be obtained directly from experimental results using Equation 1. Values of Κ are then computed from Equation 3. Both k and Κ for polystyrene and several piasticizers (7) are shown in Table I. T a b l e I. C o e f f i c i e n t s f o r T — W
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0
2
Correlation
0
k, °K. K, °K. c ' X ia< c χ 10* (cole.) (expt.) 220 105 8.5 6.3 280 130 9.1 9.5 280 138 9.3 7.7 350 161 9.5 9.7 280 (83) (4.3) 3.8 450 215 5.6 5.0 340 159 4.2 3.7  340 114 4.0 4.9 580 340 5.6 5.9 420 173 3.9 4.9 380 131 3.2 3.4 580 330 4.8 4.8 420 235 3.3 3.6 560 301 2.5 2.3
M, T °K. grams 385 243.5 368 217 335 208.5 272 169.5* 154 161.5 125 123.4 123 177.5 118 132.5 97 118.5 92 111.5 85 109.5 83 108.5 78 173.5 44 99.5
Plasticizer 0Naphthai salicylate Tricresyl phosphate Phenyl salicylate Methyl salicylate Carbon tetrachloride wButyl acetate Nitrobenzene Chloroform Ethyl acetate Toluene Dichloromethane Methyl acetate Benzene Carbon disulfide
2
oil
p
p
• Recalculated from (4). Toi in this study was 3 5 8 . 5 ° K . , indicating that its molecular weight was ~10*. Toi* is 3 7 3 ° K . for an "infinite" molecular weight polystyrene (13). ' T h i s value and subsequent ones in this column are given in (6), where they were obtained by an extrapo lation technique.
If we take Κ for polystyrene as having an average value of — 150°K., then Equation 3 is a somewhat useful equation for making a rough estimate of the glass transition temperature of a plasticized polymer: T
0
 T
 (Tg 
gl
Tg + 150)^2
X
(4)
2
Many theories (2, 0, 8, 17) have been proposed to explain the lowering of T by plasticizer. These all result in equations of form 1. Seemingly this would allow an a priori calculation of k in terms of molecular quantities. However, this is not the case; all the expressions involve quantities which cannot be evaluated without g
T a b l e II. Equations f o r G l a s s T r a n s i t i o n T e m p e r a t u r e o f P l a s t i c i z e d P o l y m e r s
Author
Approach
Fujita and Kishimoto
Isofree volume
Boyer and Spencer
I soviscous Tg
Equation T 0
Kauzmann Viscous and Eyring flow
T
Zhurkov
To
Active group
Tgi 
(β/α)ιν
2
BRT2
τ
W
a — bw

T + ol
2nRT uN
β/α*
(7)
(8, 10)
b
c
2
g
Lit. cited
BRT» 2E
~2T *
01
k
2&Γ* {n M + n M ) uNM x
x
2
2
d
(11)
t
• β — Contribution of diluent to the increase of free volume. » Β — Constant; R — gas constant; Γ — absolute temperature; Ε » activation energy for viscous flow. « a,b «• constants η *> number of diluent molecules; n\ — number of moles of polymer; nj « number of moles of diluent; k •» Boltzmann's constant; u — binding energy per node; Ν «• number of active groups. d
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
2.
Gloss Transition
SHEN A N D TOBOLSK Y
reference to the experimental facts. is given in Table II.
29
Temperature
A compilation of these proposed equations
Effect of Chain Ends It is interesting to compare Equation 1 with another equation, which was used to describe the decrease of T with decreasing molecular weight of the polymer. Based on isofree volume and viscous flow concepts, the following semiempirical equation was derived (5) : T  T *  c/Mn (5) g
g
where T \* is the T of an "infinite" polymer, M is the number average molecular weight of the polymer whose glass transition temperature is T , and c is a constant which depends on the polymer involved. In the case of polystyrene, c was originally estimated to be 1.8 Χ 10 . Later more extensive data (15) showed that g
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gl
g
n
g
5
c « 6.9 Χ 10
4
The quantity, c can be interpreted by the following equation (14) : c * 2 Ν φ/α
(6)
where Ν is the Avogadro's number, φ is the free volume per chain end, and a is the difference between the thermal expansion coefficients above and below the glass transition temperature. One of the major ideas concerning the lowering of glass transition temperature by adding plasticizer or by decreasing molecular weight is that both of these effects result from an increase in the fractional free volume. If so, it is interesting to treat these two effects in a uniform manner. Let us suppose that the molecules of low molecular weight plasticizer should be treated as part of the polymer mixture. The number average molecular weight of the plasticized mixture is defined as M , '> n v
Mn,
P
 l/Kwt/Mt) + (w /M )] 2
(7)
2
Substituting into Equation 5, we obtain
Where w again represents weight fraction; M is molecular weight; and subscripts 1 and 2 refer to polymer and plasticizer, respectively. We use the notation, c here to indicate that the constant may vary from one plasticizer to another. Figure 1 shows a plot of T vs. 1/ M for several polystyreneplasticizer systems. If M\ is very much larger than M2, Equation 7 becomes p
g
n
T  T *  c w /M2 g
gi
p
(9)
2
Hence for small values of w , comparison with Equations 2 and 3 yields 2
c
f
p
 M (T *  T  K) 2
gl
(10)
g%
Table I shows values of c computed from Equation 10 and c obtained experi mentally from Equation 8. The agreement is fairly good. To carry this analogy a step further, lowering the molecular weight of polymer can be regarded as adding chain ends as a "plasticizer" to create larger fractional p
p
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
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30
PLASTICIZATION A N D PLASTICIZER PROCESSES
100 ' 0
1 1
1 2
» 3
'/Μη,ρ * Ό
1
1
4
5
3
Figure 1. Glass transition temperature of plasticized polystyrene vs. reciprocal of the number average molecular weight of the mixture (6) free volume. Since each chain possesses two chain ends, M% « 2M where M is the molecular weight of the monomer. Looking at the effect of chain ends in this man ner, c in Equation 5 can be assigned the following definition by comparing Equa tions 5 and 10: 0t
c  2M (T *  T Q
ol
o2
0
 K)
(11)
f
where T 2 now represents the glass transition temperature of the dimer. K', which is the value of Κ appropriate to the chain ends, can be computed from Equation 11 taking c * 6.9 Χ ΙΟ ; Γ *  373°K.;and T  195°K. The value of K' is equal to  1 8 0 ° K . 0
4
g2
α1
We now utilize Equation 2 to describe the effect of chain ends on lowering T . Thus, g
Tg » Tgi* wi + T 2 w + Κ' W\ w 0
2
(12)
2
Values of Γ , obtained by using Equation 12 with K = — 180°K. and w equals weight fraction of chain ends, fits the observed values (13) of T vs. M very well α
r
2
g
n
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
2.
SHEN A N D TOBOLSKY
over the entire range
Glau Trantition
Température
31
for degrees of polymerization ranging from 2 t o 900 (Figure
2).
900 jxl
DEGREE OF POLYMERIZATION
25
I
1
1
80
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K= • 180°
°\
60 40 20
o \
T (°C)
0
g
20
\
ο
40
c
60 80 100
0.2
0.4
0.6
0.8
1.0
WEIGHT FRACTION OF CHAIN ENDS Figure 2. Glass transition temperature of polystyrene vs. weight fraction of chain ends (13) It i s w o r t h n o t i n g t h a t if w e substitute E q u a t i o n 6 i n t o E q u a t i o n 9, e q u a t i n g chain ends t o plasticizer molecules, we obtain
Tg  T
am
or
Τ
 T
β

gl
(13)
W
—
gl
2
2
(β/α)
(14a)
W2
where
β « 2Νφ/Μ
2
Now
the definition of φ
molecule. (6)
(14b)
= Νφ/Μο
i s g e n e r a l i z e d t o i n c l u d e t h e free v o l u m e
per plasticizer
E q u a t i o n 14ais equivalent t o the one proposed b y F u j i ta a n d K i s h i m o t o
f o r t h e effect of plasticizer o n T
g
as shown i n Table II.
Originally β w a s defined
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
32
PLASTICIZATION A N D PLASTICIZER PROCESSES
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WEIGHT PERCENT A C E N A P H T H Y L E N E 0.2 0.4 0.6 0.8 1.0 300r
0
0.2
0.4
0.6
WEIGHT PERCENT VINYLIDENE
0.8
1.0
FLUORIDE
Figure 3. Glass transition temperature of copolymers vs. weight fraction of monomer units (14) Ti is the temperature at which the 10second Young's modulus is 10 dynes/sq. cm. It is usually a few degrees higher than T
9
a
as "the contribution of diluent to the increase of free volume," in agreement with Equation 146. This consistency in results is gratifying but perhaps to be expected, since these theories are both based on free volume concepts.
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
2.
S H I N A N D TOBOLSKY
33
G i n s Transition Tmmpmranjre
Effect of Comonomer Units The following equation has been proposed as an empirical equation for the glass transition temperature of certain copolymers: Tg « Tgl
Wl
+ Tgl
(15)
W
2
where w's refer to weight fractions of corhonomers. Equation 15 is, of course, a special case of Equation 2 with Κ « 0. However, linearity in T copolymer com position curves are not always observed, hence Equation 2 should give a better fit to the experimental data. Values of Κ may range from positive to negative. Figure 3 shows a linear relationship for the styreneacenaphthylene copolymers (12) indicating a Κ value of zero. A value of — 270°C. was obtained for Κ in the vinylidene fluorideperfluoropropylene copolymer system, which seems reasonable in comparison with the Κ values of plasticized polymers. Positive values of Κ exist for copolymers but are quite rare (16). The use of Equation 2 for copolymer systems in analogy with plasticized systems is particularly apt for the "static mechanism" of plasticization (7), which suggests that diluents can be regarded as being attached to the polymer chain. Ideally speaking, the effect of this type of plasticization would be to convert the polymer molecule into a copolymer of plasticized and unplasticized segments. Conversely, change of chemical structure of the polymer chain by copolymerization can be regarded as changing T by "internal plasticization" (14). If we regard the free volume theory as being applicable to interpretation of glass transition tempera tures, it is not surprising that similar equations describe the effect of chain end concentration, effect of plasticizer, and effect of copolymerization. Probably the similarity in equations would result from other theoretical interpretations also.
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g
0
In summary, the equations proposed for lowering T in plasticized polymers and the lowering of T due to decreasing molecular weight show certain consistencies. Further analogies can also be drawn with the changing of T via copolymerization. From an empirical point of view the two very simple equations, 1 and 2, give a reasonably good and unified description for all these effects. For none of these cases, however, do we have a completely satisfying a priori theory. It would also be interesting to explore the consequences of Equation 2 for the various cases discussed here if weight fractions were replaced by volume fractions and by mole fractions. Recently Kovacs (9) suggested that a discontinuity in the dependence of T on composition is to be expected. This is not inherent in Equation 2, and an experi mental test of this suggestion is in order. g
0
g
g
Literature Cited (1) (2) (3) (4) (5) (6) (7) (8) (9)
Aiken, W., Alfrey, T. Jr., Janssen, Α., and Mark, H., J. Polymer Sci. 2, 178 (1947). Boyer, R. F., and Spencer, R. S., J. Polymer Sci. 2, 157 (1947). Breitman, L., J. Polymer Sci. 26, 1092 (1955). Ferry, J. D., "Viscoelastic Properties of Polymers," Wiley, New York, p. 367, 1961. Fox, T. G., and Flory, P. J., J. Appl. Phys. 21, 581 (1950). Fujita, H., and Kishimoto, Α., J. Polymer Sci. 28, 547 (1958). Jenckel, E., Heusch, R., KolloidZ. 130, 89 (1958). Kauzmann, W., and Eyring, H., J. Am. Chem. Soc. 62, 3113 (1940). Kovacs, Adv. Polymer Sci. 3, 487 (1964).
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.
34
PLASTICIZATION A N D PLASTICIZER PROCESSES
(10) Nielsen, L. E., Buchdahl R., and Leverault, R., J. Appl. Phys. 21, 607 (1950). (11) Richard, W. R., and Smith, P. A. S., J. Chem. Phys. 18, 230 (1950). (12) Schaffhauser, R. J., Shen, M. C., and Tobolsky, Α. V., J. Appl. Polymer Sci. 8, 2825 (1964). (13) Shen, M. C., Ph.D. thesis, Princeton University, 1963. (14) Simril, V. L., J. Polymer Sci. 2, 142 (1947). (15) Ueberreiter, K., and Kanig, G., J. Colloid Sci. 1, 569 (1952). (16) Wood, L. Α., J. Polymer Sci. 28, 319 (1958). (17) Zhurkov, S. N., Compt. rend. acad. sci. U.R.S.S. 47, 475 (1945).
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RECEIVED APRIL 22, 1964.
In Plasticization and Plasticizer Processes; Platzer, N.; Advances in Chemistry; American Chemical Society: Washington, DC, 1965.