The crystallization of fractions of partially isotactic poly(propylene oxide)

The crystallization of fractions of partially isotactic poly(propylene oxide) C. BOOTH, D. V. DODGSON and I. H. HILLIER The crystallization kinetics of fractions of poly(propylene oxide) have been studied dilatometrically and microscopically. The dilatometric results have been successfully analyzed in terms of a model involving an increase in density within the outlines of the spherulites growing with a constant radial rate. Nucleation theory has been adapted to take account of the stereoregular nature of the polymer, and the rate constants from both techniques have been analyzed using values of the thermodynamic melting point and number average isotactic sequence lengths obtained from melting point data for the polymer. STUDIES of the isothermal crystallization kinetics in bulk polymers are capable in principle of yielding information on the mechanism of crystallization, and its correlation with polymer morphology and the molecular structure of the polymer 1. Dilatometric data are usually interpreted in terms of the growth of macroscopic entities of particular geometry, and yield rate constants, the temperature dependence of which is interpreted using classical nucleation theory. Because these rate constants may depend upon the particular model used to interpret the dilatometric data, independent measurements of the growth rate of the crystalline entities (spherulites) (usually by optical microscopy) allow a more unequivocal interpretation of their temperature dependence. Analyses of dilatometric data for a number of polymers have shown that the Avrami equation e with an exponent of 3 or 4, corresponding to the growth of spherulites, is rarely adhered to; this has led 3 6 to explanations of the deviations in terms of phenomenological theories involving secondary crystallization. The nature of this is discussed in terms of mechanisms such as the thickening of lamellar crystals 7, or the crystallization of interfibrillar low molecular weight, or other less readily crystallizable material s . The temperature dependence of the crystallization rate constants, used to obtain estimates of the interfacial free energies in the nucleation process, is most commonly analyzed using equations which are strictly applicable only to homopolymers of infinite molecular weight. A more complete analysis of the data requires a consideration of the finite molecular weights, the molecular weight distribution, and, in the case of copolymers, the finite lengths and distribution of lengths of the crystallizable sequences. Therefore to fully interpret dilatometric and microscopic data it is necessary to use well characterised polymers. To this end we have studied the crystallization of fractions of partially isotactic poly(propylene oxide). The preparation of these fractions has already been described in detail u. Briefly, poly(propylene oxide) was prepared TM using the zinc diethyl and water catalyst and fractionated 11 by precipitation from dilute solution in iso-octane to yield fractions having identical wide molecular weight distributions but with differing degrees of isotacticity. The pertinent characteristics of the fractions are given in Table 1. The number-average degrees of polymerization (x,,) are obtained l¥om the 11

C. BOOTH, D. V. DODGSON a n d I. H. HILLIER Table 1 The characteristics of fractions of poly(propylene oxide) Fraction

Weight %

F1 F2 F3

6.4 3"8 2'5

xn

yn

X

5 5 5

55 22 15

0.98 0"95 0"93

(× 10 -3)

viscosity-average molecular weights and distribution widths reported earlier 9. The number-average isotactic sequence lengths (yn) and the thermodynamic melting point of isotactic poly(propylene oxide) (Tin ° = 82°C) are obtained from an analysis of the multiple melting transitions found in these fractions 9. The mole fractions of isotactic diads (X) are calculated by assuming a random distribution of non-isotactic diads along the polymer chain. The true values of X cannot exceed these estimates, and our own and other 12 estimates of X by n.m.r, spectroscopy show that X > 0.9.

EXPERrMENTAL

Glass dilatometers were used. Small samples ( < 100 rag), moulded in high vacuum, were placed in the dilatometers, outgassed and finally confined with mercury. The dilatometers were immersed in boiling water for 15 to 30 rain and then quickly transferred to a thermostat ( ± 0.01 °C). Thermal equilibrium was established within 2 rain. The contraction ( ~ 40 mm) was followed by means of a cathetometer (:~ 0.04 mm). Spherulite growth rates were measured on a hot stage 13 of a polarizing microscope. Thin films of the sample on a microscope slide were heated to 120°C for 2 to 5 min and then quickly transferred to the hot stage which was kept at a constant temperature (~z 0"02°C) by a proportional controller. Melting at 100°C resulted in a high density of nuclei with consequent difficulty in the resolution of individual spherulites. Spherulite sizes were recorded photographically. ANALYSIS OF RESULTS

Dilatometric data The dilatometric data have been analyzed by means of the Avrami equation 2 ho - - h t

x(t)

It0 -- ho~ -- X(c~) -- [1 -- exp (-- ztn)]

(1)

where h, is the dilatometer meniscus height at time t, x(t) the degree of crystallinity, and z and n the Avrami rate constant and exponent respectively. A value of n -- 3 or 4 corresponds to the constant radial growth of instantaneously or sporadically nucleated spher.ulites growing with a constant density. The results of least squares computer fitting of equation (1) with 12

CRYSTALLIZATIONOF POLY(PROPYLENEOXIDE) Table 2 Comparison of Avrami equation with crystallization isotherms Fraction

Temperature

z

n

Standard deviation (°/o)

(°C) FI

46"0 48"4 50"1 52.4 55.0

2"54 3-29 3-82 3-44 4.56

x × x "~ x

10 :~ 10 -'L 10 s 10 ~' 10 v

2-0 2"1 2"3 2.0 2.4

5"9 4"5 4"1 4.8 3.8

F2

40"l 43-8 46.4 49.9 52.4 54-8

7.39 x 7.86 x 3.44",x 3.20",: 6.78 x 1.34 <

10 ~ I0 s 10 s 10 -~ 10 ~ 10 7

2.1 2.2 2.0 2.0 2-0 2.0

4.0 3.6 4-5 3-7 3-2 4.7

F3

40.2 42"7 44.0 45.4 50.8

3.30 1"85 2.16 6-20 1.46

lO '~ 10 ~ 10 -s 10 a I0 7

2.4 2"6 2.0 2.0 2.0

4.4 3"5 5-3 4.7 5.0

/, )< × X ×

a d j u s t a b l e p a r a m e t e r s n a n d z are g i v e n in Table 2. T h e o r y a n d e x p e r i m e n t are f u r t h e r c o m p a r e d in Figure 1. E x a m i n a t i o n o f t h e fit o b t a i n e d reveals t h a t t h e m a j o r c o n t r i b u t i o n s t o t h e l a r g e s t a n d a r d d e v i a t i o n s h o w n in Table 2 o c c u r s t o w a r d s t h e e n d o f t h e c r y s t a l l i z a t i o n , w i t h a p p r o x i m a t e l y t h e first 6 0 % o f e a c h i s o t h e r m b e i n g well fitted b y t h e A v r a m i e q u a t i o n w i t h a n e x p o n e n t o f 3 (Figure 1). F u r t h e r m o r e , in o u r s a m p l e s o f p o l y ( p r o p y l e n e o x i d e ) w e o b s e r v e s p h e r u l i t e s g r o w i n g

1.0

0

t-

'o x: 0.5 X 'o

0'0

I

w

0-1

1

10

t/t~

Figure l Crystallization isotherm for sample F2 at 43.8°C. The curve is calculated from equation (1) with n ~ 3. 13

C. BOOTH, D. V. DODGSON And I. H. HILLIER w i t h a c o n s t a n t r a d i a l rate. T h e s e c o n s i d e r a t i o n s i n d i c a t e t h a t t h e large d e v i a t i o n s f r o m t h e A v r a m i e q u a t i o n , and the l o w e x p o n e n t values, arise f r o m an i n c r e a s i n g density o f the g r o w i n g spherulites similar to t h a t f o u n d e x p e r i m e n t a l l y for p o l y p r o p y l e n e 4. T o t a k e a c c o u n t p h e n o m e n o l o g i c a l l y o f this d e n s i t y i n c r e a s e we h a v e a s s u m e d , as in p r e v i o u s analyses s, t h a t this s e c o n d a r y c r y s t a l l i z a t i o n c a n be d e s c r i b e d by a first o r d e r p r o c e s s w h i c h leads to the e q u a t i o n ho -- ht ho --~h~

x(t) X(OO)

x(a, oe) X(Oe ) [1 - - exp ( - - zp t3)]

t

-V

x(s, ~)

X(O~)

f

zsJ

[1 - - exp ( - - zp 08)1exp [ - - zs (t - - 0)]

d0(2)

0

w h e r e X (a, oo) a n d X (s, oe) are the degrees o f crystallinity arising f r o m the p r i m a r y ( A v r a m i ) a n d s e c o n d a r y (first o r d e r ) p r o c e s s e s alone, t h e c o r r e s p o n d ing rate c o n s t a n t s b e i n g z~ a n d zs. T h e fit o f o u r d a t a to this e q u a t i o n is s h o w n in Table 3 a n d it can be seen t h a t the i n c l u s i o n o f an a d d i t i o n a l

Table 3 Comparison of equation (2) with crystallization isotherms Fraction

Temperature (~'C)

zp

zs

Fraction of secondary crystallization

Standard deviation (%)

FI

46.0 48.4 50.1 52.4 55.0

2.17 2.04 3.28 3'42 1-81

x x x × x

10 -4 10 -6 10 -8 10 -7 10 -8

7'08 5'58 3-30 1-40 4.45

x x x × x

10 -3 10 -s 10 -s 10 -s 10 4

0.22 0.21 0.18 0"19 0"17

1.9 0"9 1.0 1.0 1-3

F2

40"1 43'8 46'4 49-9 52"4 54-8

7"08 3"75 4-20 1'67 1'52 9"41

× × × × × x

10 -5 10 -6 10 .7 10 .8 10 .9 10 -11

2"88 × 9"47 x 4"58 x 1"20 x 6"33 x 2"44 x

10 -~ 10 -a 10 -3 10 -s 10 .4 10 -4

0"32 0"26 0'29 0'29 0"32 0'24

1'2 0-8 0"8 0'6 0"9 0'7

F3

40.2 42"7 44"0 45"4 50"8

3'70 3"64 1'10 2"62 1"01

x N X X x

10-" 10 -7 10 -7 10 -8 10-1°

2"39 x 4"87 X 8'16 × 3"78 X 1-53 ×

10 -s •0 -4 10 -4 10 4 10 4

0"19 0"•7 0"19 0"20 0'24

I'1 0-9 0'8 0"9 0'9

p a r a m e t e r yields a m u c h i m p r o v e d fit. T h i s p r o v i d e s e v i d e n c e t h a t e q u a t i o n (2) is m o r e a p p r o p r i a t e t h a n the simple A v r a m i p i c t u r e o f the c r y s t a l l i z a t i o n process. W e h a v e also c o n s i d e r e d t h e possibility o f a s l o w r a t e o f f o r m a t i o n o f nuclei, as r e p o r t e d by A g g a r w a l et al 6. T h e i n c l u s i o n in t h e rate e x p r e s s i o n o f this effect, w i t h a first o r d e r rate c o n s t a n t f o r n u c l e u s f o r m a t i o n as an a d d i t i o n a l a d j u s t a b l e p a r a m e t e r , d o e s n o t significantly i m p r o v e the fit o f 14

CRYSTALLIZATION OF POLY(PROPYLENE OXIDE)

the data. We conclude that in our experiments the rate of nucleus formation is rapid compared with the rate of crystallization. M i c r o s c o p i c data The constant radial growth rates obtained for the three fractions of polymer are listed in Table 4. Analysis of the temperature dependence of the rate constants (Tables 3 and 4) follows the discussion of the effect of finite molecular weight, and partial isotacticity, on the nucleation of poly(propylene oxide). Table 4 Spherulite growth rates Fraction

Temperature (°C)

(ram rain-1 :~: 103)

FI

42"3 46.3 47"6 51"0

19"55 7"95 2"35 I.O

F2

40"5 42"3

9'95 6"25

44"2 46-3

4.0 4.4

39-0 40.5 42.3 44.2

3.5 2.4 2-0 1-I

F3

NUCLEATION THEORY APPLIED TO PARTIALLY ISOTACTIC POLYMER

It is generally taken that at temperatures sufficiently far from the glass transition, the rate determining step in the linear spherulite growth rate is secondary nucleation. For such a model, the growth rate G is given 1 as G -- Go exp (-- 2xF*/RT)

(3)

where AF* is the free energy of formation of a nucleus of critical dimensions. For a homopolymer of infinite molecular weight this is given by AF* = 4c~eau/~.fu

(4)

for a two-dimensional nucleus, and AF* --- 8,~e+~/AJ;, "

(5)

for a three-dimensional cylindrical nucleus 1. Here (re and ¢ru are the end and lateral interfacial free energies of the nucleus, and 2xfu is the free energy of fusion per chain unit, all energies being expressed in units of cal mole 1. For the polymer investigated here, equations (4) and (5) must be modified to take account of the finite molecular weights and lengths of the isotactic sequences. We consider a model for poly(propylene oxide) in which both the molecules and isotactic sequences vary in length according to the most probable distribution, with number-average degree of polymerization x,~ and number-average isotactic sequence length y,. For a homopo!ymer with a 15

C. BOOTH, D. V. DODGSON

and

i. H. HILLIER

most probable distribution, and degree of crystallinity (1 -- A), the free energy o f fusion AFf at temperature T is given as 14 AFf --(1--

A)Afu+

RT{

xnlnh

+ (-l--A~)[lnD-t-(~--l)lnpx]}~

(6)

Here /~ is the length of the crystalline sequences, N the number of polymer molecules, and l n D = -- 2 a e / R T p x = [1 -- (1/Xn)]

For a polymer with a most probable distribution of isotactic sequences an additional entropy term 14 AS = -- R[(1 -- h)/~] [ l n X +

(~ -- 1) lnpv]

(7)

must be included in equation (6). Here X is the mole fraction of isotactic diads and py =

[1 -

(l/y.)]

The free energy of formation of a cylindrical nucleus, ~ units long and p units in cross section becomes, using equations (6) and (7) A F = 2 z r ~ p ' ~ u + 2pae - - ~pAfu - - R T {Nln[1 -- (p~/xnN)] + p [ l n X + (~ -- 1) l npxpu] }

(8)

As Xn is large ( ~ 5 000) and yn is of the order of 20 we may put (1/xn) ~ 0, and lnpu ~_ - - 1/yn. With these approximations, equation (8) becomes AF=

2~r~p~au -k 2pae - - ~ p A f u -k R T p [(~ -- 1)/yn] - - R T p l n X

(9)

The dimensions ~*, p*, of the critical nucleus may be obtained from equation (9) by setting c~p ] ~ - - ~ 0 a n d

p

giving O* '-" ~ 2~r~au/(Afu - - R T / y n )

U" = (4a, -- 2 R T I n X -

2RT/yn)/(Afu -- RT/yn)

(10) (11)

and AF* = 1r~p*~*au

(12)

For our polymer, X is near unity, and although a value of cre for poly(propylene oxide) has not been reported, a value for poly(ethylene oxide) is 1 500 cal mole -1 15 so that equation (11) may be approximated as ~* ~- 4ae/(Afu - - R T / y n )

giving AF* = 8zrauZCre/(Afu - - R T / y n ) z With the usual approximation Afu = (Tin ° - - T)Ahu/Tm ° = A TAhu/Tm °

16

(13)

CRYSTALLIZATION OF POLY(PROPYLENE OXIDE)

where Tm° is the thermodynamic melting point of the polymer and Ah+Lthe heat of fusion

, / ATAhu -i'nRT)+. A F * = S~ra,+ZC~e/t-rmo --

(14)

For a two-dimensional nucleus, ~ units long and p sequences in breadth. a similar argument yields g* = 2cre/(4fu -- RT/3',,) p* = 2cr~/(&A -- Rr/y,,) and

A F * = 4~++~e/

Tm o

.

..

Thus, a plot of l n G v s I/T i ATAhu

RT i" Zm ° 3',+ ! where m -- 1 for two-dimensional and 2 for three-dimensional nucleation, should yield a single straight line for all three fractions if the pre-exponential factor in equation (3) remains unchanged. In Figure 2 we plot the rate constants (z v and z,) and the spherulite growth rates (G) in terms of the conventional two-dimensional nucleation theory

-lOl C

L

.13

_¢ (.9

-20

I

2.6

2.8

3'0

3,2

3'4

t

I

3'OlTr~/T&T ) xlO 24"0

(T~ITAT) x 10 2

-,{

Figure 2. Temperature dependence of rate constants [equations (3), and (4)] for FI (A), F2 ( . ) and F3 (O). The lines are the least squares fit of the data

,,?

3"0 4"0 (r Fn/T&T) x 10 =

applicable to infinite molecular weight homopolymer [equations (3) and (4)]. For each fraction a distinct linear plot is obtained. In Figures 3 and 4 the data are analyzed, with a value for Ah~ of 2 000 cals/mole 16, by means of the theory developed here [equations (3), (14) and (15)] to take account of the B

17

C. BOOTH, D. V. DODGSON

and

I. H. HILLIER

variation in isotactic sequence lengths between the three fractions. Although the microscopic data are somewhat scattered, they can be satisfactorily represented for both two and three dimensional nucleation by single straight lines for all three fractions. A similar conclusion is reached for the rate constants obtained from the dilatometric data, although there is a tendency for the data for fraction F1 to be slightly displaced.

& .i.,,, i-

•

~°

&

k.

(.9 4"

I

I

15

I

I

17

I

19

[ T(&fu-RT/y n ) ] ' l x 10 ¢

16

t 18

J 20

i 22

t 24

26

[T( Af u-RT/y n )]-~× 10 ¢ Figure 3 Temperature dependence of rate constants [equations (3), and (15)] for F I (A),

F2 (m) and F3 (O) DISCUSSION The microscopic observations and the fit of the crystallization isotherms in the early stages by an Avrami equation with an exponent of 3 shows that crystallization proceeds by the constant radial growth of spherulites. The deviations from the Avrami description are postulated to be due to a slow increase in density (secondary crystallization) within the growing spherulites, which can be described by a first order process with rate constants given in Table 3. The extent of this secondary crystallization does not vary significantly with crystallization temperature or polymer fraction and accounts for 20 ~ of the total crystallinity developed. 18

CRYSTALLIZATION OF POLY(PROPYLENE OXIDE)

The rate of secondary crystallization has a similar temperature dependence to the rate of primary crystallization indicating that it is controlled in rate by a similar nucleation act. A choice between the two explanations of the secondary crystallization-either a slow thickening of lamellae, or crystallization of interfibrillar (less isotactic) material - cannot be made fromtheresultsreported in this paper. However, we have reported elsewhere 9 that there is no significant change in the melting points of the fractions with increasing degree of crystallization. This suggests that the secondary crystallization is due to slow interfibrillar crystallization. Table 5 Parameters for growth rates (units, cal and mole) .) ~e~Ttt -

(Te~Ttt

,nO:

]/T \ V,,o

y,,/i

InG vs

1/T t'AT/3hu R1)

Microscopic Dilatometric

3.3 • I0 z 2.6 ,: 10~

220 173

2

Microscopic Y,;__ Dilatometric

7"i,

crlt

4.6 x 10~; 3. I ;< I 0 ~

55 46

(D E

I

I

I

8

-I 0

•

_

~

=

I

I

10 [T "1(Afu-RT/yn) "2] x 10 8

v

12

_

.

N E

-20

I 8

I

10

I

I

12 14 [ T - V ( ~ f u - R T / y n ) - 2 ] x 10 8

I 16

t

18

F(gure 4 Temperature dependence of rate constants [equations (3), and (14)] for FI ( V ) ; F2 ( I ) ; and F3 ( 0 )

19

C. BOOTH, D. V. DODGSON and I. H. HILLIER

The temperature dependence of the growth rates can be satisfactorily analyzed in terms of classical nucleation theory if account is taken of the length and distribution of isotactic sequences in the polymer. The values of the thermodynamic melting point of the isotactic polymer and the numberaverage isotactic sequence lengths of the partially isotactic fractions, obtained from melting data for the three fractions, thus provide a consistent set of parameters for the samples. In common with other workers we cannot distinguish between two and three dimensional nucleation from our data. The parameters for growth rates from Figures 3 and 4 are summarized in Table 5. The values are in agreement with the 'universal' parameters found by Mandelkern et a117 from data for a number of polymers. The values for e,, are calculated on the assumption that ere = 1 500 cal mole 1. The value of eu for polytpropylene oxide) is unknown but, by comparison with the value for branched polyethylene18, might be expected to lie in the range 100-200 cal mole -1. The data thus suggest a two dimensional nucleation process. Our earlier analysis of the melting transitions in these fractions was based upon the predominance of three dimensional nucleation. This divergence between rate and melting data is not uncommon, and has been explained in terms of isothermal lamellar thickening 19 by a factor of two.

Department of Chemistry, University of Manchester.

(Received 22 August 1969)

REFERENCES 1 Mandlekern, L., 'Crystallization of Polymers', McGraw Hill, 1964. 2 Avrami, M. J. Chem. Phys. 1939, 7, 1103; J. Chem. Phys., 1940, 8, 212; J. Chem. Phys. 1941, 9, 177 3 Hillier, I. H., J. Polym. Sci., (A) 1965, 3, 3067 4 Hoshino, S., Meinecke. E., Powers, J., Stein R. S. and Newman S. J. Polym. Sci. (A) 1965, 3, 3041 5 Price, F. P. J. Polym. Sci. (A) 1965, 3, 3079 6 Aggarwal, S. L., Marker, L., Kollar, W. I. and Geroch, R., J. Polym. Sei. (A) 1966, 4, 715 7 Fischer, E. W. and Schmidt, G. F., Angew. Chem. (Int. Ed.), 1962, 1, 448 8 Keith, H. D., KolloidZeits. 1969, 231,421 9 Booth, C., Devoy, C. J., Dodgson, D. V. and Hillier, I. H. J. Polym. Sci., in press. l0 Booth, C., Higginson, W. C. E. and Powell, E., Polymer, Lond. 1964, 5, 479 11 Allen G., Booth, C. and Jones, M. N., Polymer, Lond., 1964, 5, 257 12 Tani, H., Ogumi, N. and Watanabe, S. J. Polym. Sci. (B) 1968, 6, 577 13 Hay, J. N. J. Sci. Instrum. 1964, 41, 456 14 Flory, P. J. J. Chem. Phys. 1949, 17, 223 15 Braun, W., Hellwege, K. H., and Knappe, W., KolloMZeits., 1967, 215, 10 16 Devoy, C. J., Ph.D. Thesis, University of Manchester, 1966 17 Mandelkern., K., Jain, N. L. and Kim, H., J. Polym. Sci., (A), 1968, 6, 165 18 Turnbull, D. and Cormia, R. L. J. Chem. Phys. 1961, 34, 820 19 Hoffman, J. D. SPE Trans. 1964, 4, number 4

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