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Effect of molecular weight on the rate of crystallization of polyethylene fractions at high undercooling
Effect of molecular weight on the rate of crystallization of polyethylene fractions at high undercooling d. M. Barrales-Rienda and d. M. G. Fatou Division of Polymer Physics, Instituto de Plasticos y Caucho, Madrid 6, Spain (Received 13 July 1971) Isothermal crystallization of polyethylene fractions by calorimetry at high undercoolings has been studied for a wide range of molecular weights between 1"55x106 and Cz6H74. The influence of molecular weight on crystallization rates as undercooling increases is very moderate. The crystallization temperature coefficient has been analysed according to threedimensional nucleation theory and it has been shown that the crystallization is described by a unique function of the free energy required for nucleation when the change of the interfacial free energy with molecular weight is considered. The interracial free energy reaches an asymptotic value at high molecular weights, decreases as the molecular weight is lowered and reaches an asymptotic value at the lower molecular weight.
INTRODUCTION The crystallization of polymers from the melt is characterized by a high negative temperature coefficient, which shows that nucleation of the system is the predominating factor in the transformation t, 2, independently of the mechanism involved. Moreover, the rate with which crystallization takes place from the melt can be obtained from the general expression for the kinetics of phase transformations 3, 4 and for very low levels of transformation, it can be assumed that the critical free energy for nucleus growth is approximately the same as that required for nucleation, the overall crystallization rate depending on two energetic terms: the activation energy for transport and the critical free energy for forming a nucleus. The latter can be calculated for finite molecular weights by the expressions developed by Mandelkern et ak s, 6 for tri- and bi-dimensional growth mechanisms and this treatment has recently been applied 7, 8 to the analysis of spherulitic growth for different polymers, showing that independently of the molecular weight, the growth rate is a unique function of the free energy for nucleation and that the excess interfacial free energy in the (001) face of the crystal is a function of the molecular weight. On the other hand, the variation in crystallization rate from the melted state at low undercoolings has been attributed to a transport effect so that, when the crystal-
lization temperature decreases and conditions for nucleation are less restricted, the viscosity of the medium is less important and there are no complicating effects by the energy of transport 2, 9 The main purpose of this work is, therefore, to analyse polyethylene crystallization kinetics at high undercoolings and to study the temperature coefficient as a function of the molecular weight and o f the thermodynamic parameters involved encompassing a molecular weight range from 1-55 x 106 to 1-8 × 103. The highly purified C36H74 paraffin has been included in the analysis for comparative purposes. EXPERIMENTAL Crystallization kinetics were studied using the calorimetric technique described in detail elsewhere 9. Eight polyethylene fractions with the following molecular weights were used" 1.55×106 , 4.25x105 , 1 . 0 x l 0 s , 2"0x104 , 12"5x103 , 5"3x103 , 3"3×103 and 1-8×103 . These samples were obtained using a column fractionation technique already described 1°, using linear polyethylenes. A highly purified C86I-I74 paraffin (Fluka, A.-G.) was the fraction with the lowest molecular weight. The crystallization was studied with a Perkin-Elmer, DSC-1B differential scanning calorimeter. The method used is the same as described by us in a previous paperL Briefly, it can be summarized as follows. The molten samples were undercooled at 64°C per minute until the
POLYMER, 1972, Vol 13, August
407
Crystallization of polyethylene fractions at high undercoofing : J. M. Barrales-Rienda and J. M. G. Fatou I.O a
,
O)
T 'T'~o
113
0.2
"lb
o
i!
°/
I
119
~ ~ %
o I
O.I
[
I
0"1
" 0 •I
Ii
IO
t (rain)
Figure 1 Typical isotherms for molecular weight fractions of (a) 3300,and (b) 20 000 linear polyethylene fractions and (c) paraffin (C36H74)at high undercoolings. Temperature (°C) of crystallization is indicated for each isotherm
desired crystallization temperature was reached. Crystallization heat was recorded as a function of time until crystallization was complete or the heat evolved was small enough to be detected. The percentage crystallization was obtained in the usual way3. AHu=6.8 cal/g was used for the indium and AH~=(960/14)cal/g was used for the high molecular weight polyethylene fractions (Mn = 2.0 x 104). The melting heat for the lowest molecular weights were calculated using the Flory and Vrij 11 equation modified for a value of AHu = 960 cal/mol z2, AHu = 960-- AC~A T - (2150/n)
(1)
where AHu is the melting enthalpy per -CH2- unit and n is the chain length, A T is the difference between the equilibrium melting temperature of an extended chain polyethylene crystal of infinite molecular weight in equilibrium, that is to say 145.5°C11,13 and the melting temperature for a chain with n methylene groups, and AC~ is the difference between the heat capacity of extended chain polyethylene crystaP4 and amorphous polyethylene15. The melting heat of 30 900cal/mollm 17 taken for the C46H74 paraffin corresponds to melting enthalpy of AHu= 853.3 cal/mol per -CHz-- group. Thus, the functions ( 1 - At) versus t were obtained for each undercooling. The value (1-At)oo was obtained by the usual method of taking the extrapolated value of the curve after two decades of time, because secondary crystallization was extremely slow. The usual form of the equation of Avrami ln0--kt n was obtained by calculating the function O from the following relation: (1 --
At)oo.(1
-- 0) = (1 -- At)
(2)
The most important fact noted is that at the highest crystallization temperatures (121-123°C) a maximum in the crystallization rate may be observed which corresponds as a minimum when plotting z0.5 against h#n. This minimum is not so well defined in each case and, as can be seen, as undercooling increases, it moves to much lower molecular weights and nearly completely disappears at the lowest undercooling. This minimum is more pronounced at the highest crystallization temperature, as may be noted for all crystallization temperatures used. Furthermore, with any molecular weight, crystallization rate increases when crystallization temperature decreases. It is a well established fact that this indicates a nucleation process in the crystallization is occurring. Another important fact is that at the lowest crystallization temperatures and in the range of the highest molecular weight, crystallization rate only seems to be slightly dependent on molecular weight. The nucleation theories developed5, 6 have shown that the free energy for stable nucleus formation is independent of molecular weight for chains with a molecular weight above 105 whatever the crystallization temperature and crystallization rate changes with molecular weight have been attributed to the transport term or rate of growth 2. When the crystallization temperature decreases, the critical nucleation conditions diminish and the viscosity of the medium is not of important influence. This has been shown in a previous work and it is strongly supported by our own results which cover a wider range of molecular weights. Crystallinity, at the end of the transformation at a given crystallization temperature, is a function of the molecular weight and decreases as this increases. Figure 3 represents this variation after crystallization at 115°C for fractions of molecular weight between 1.55 x 10~ and 1.8 x 103. For these limits, crystallinity varies from 35 ~o for the fraction of highest molecular weight up to 85 ~o for the fraction of lowest molecular weight. Similar results were previously reported z0 after crystallization at low undercoolings. On the other hand, the crystallization rate can be expressed as a function of time r0.1 in which 10~ of the transformation takes place by the equationa, 4: In (~'o.1 ) - 1
=
In 0-o.1)o I
__
EI#RTe - AF*/RTe
(3)
IOO
IO
l.n
k li9 ~21 123
J l
I
o
=-\~ i~O- o / V."n---.--.O_O_ . .
67"
Isotherms thus obtained for two of the fractions and paraffin C36H74 are shown in Figure 1.
~
/ = ~=.A / ~n
III o I09
X '~v
I13 o-I
I
I
I
I
l°3
i°4
!°S
l°~
l°7
RESULTS AND DISCUSSION The results of the crystallization rates, expressed in terms of time ~-0.5for 50 ~ of the transformation are given in Figure 2 for the various temperatures studied.
408
POLYMER, 1972, Vol 13, August
Double logarithmic plot of crystallization half-time, ~'0.5 (minutes) against number-average molecular weight. Isothermal crystallization temperatures (°C) indicated for each curve Figure 2
Crystallization of polyethylene fractions at high undercooling : d. M. Barrales-Rienda and J. M. G. Fatou where Te is the crystallization temperature, AF* the free energy for forming a stable nucleus and E 9 is the transport term. From equation (3) it can be deduced that a representation of In(r0.1) -1 against A F * / R T e should correspond to a straight line with a - 1 slope and independent of the molecular weight when the value of AF* is stipulated for the model of finite molecular weight in a tridimensional nucleation process. For the case of a cylindrical nucleus, it has been shown that 5: AF* =
~1/2~*p*1/2o'2
I.O
0.8
"-z ~, - 0.4
(4)
where ~:* and p* are the critical dimensions of the nucleus and au the lateral interfacial free energy per structural unit. For a finite chain with chain length x, the critical dimensions are related by the following expressions :
"a x
©.6
0.2
I
ic~
I
10 3
I
10 4
I
10 S
10 6
io ~
Figure 3 Degree of crystallinity (1 -At), as a function of molecular weight for linear polyethylene fractions after crystallization at 115°C
(~:*/2) [Afu - (RTc/x) + ( R T c / x - ~* + l) ] = 2 a e - R T c l n [ ( x - ~ * + l)/x]
(5)
and p.1/2 = 2~rl/Zau / [Afu - (RTc/x) - ( R T c / x - ~* + I) ]
(6)
4
where ae is the interfacial free energy per unit in the basal plane and Afu is the free melting energy per repeating unit at Tc which is given by the approximation A f u = A H u ( T ° - Tc)/T°
(7)
A H u being the melting enthalpy per unit and T°, the equilibrium melting temperature. In the limit of high molecular weights equations (4), (5) and (6) are reduced to the known expression :
AF* = 8rraea~( T°)2/ AH2u( T ° - To) 2
(8)
Analysis of the experimental data according to the above equations require exclusively the specification of T°, AHu and ~re. In the case of polyethylene, Flory and Vrij n have developed the theoretical relationship between the molecular weight and the equilibrium melting temperature. This theoretical treatment and suggested extrapolations 12, 13 give an equilibrium melting temperature for infinite molecular weight polyethylene of 145.5°C. Melting enthalpy is known by independent measurements and for high molecular weights corresponds to 960cal/mo11L If a value for ire is stipulated it is possible to calculate the ratio AF*/2a~ Te. The value chosen was that of ae=2400cal/mol, although the conclusions obtained do not depend on it if other values are selected. The representation of ln(z0.1) -1 against AF*/2a;" Tc is shown in Figure 4. All the analysed fractions give straight lines, with a common intersection, except those corresponding to the fraction of 1.8 x 103 and paraffin C36H74, which are not included. The slopes of these lines vary with the molecular weight and the lower the molecular weight, the lower the slopes are. However, the straight lines can be extrapolated to a common intercept, i.e. ln(~-0.1)oL Similar results have been described in bidimensional analysis of the spherulitic growth of different polymers 7, s Considered literally, variations in the slopes in Figure 4 represent a change in the values of ae or Cru or both, if AHu is a constant. If we assume that (ru is constant and
3
x 5
2
q I
\ 0
-I I
O
O-Ol
I
0"02
~'
" "
I
0"03
0.04
AF~/2o~rc Figure 4 Plot of In(~-o.1)-z against AF*/2a2uTc for linear polyethylene molecular weight fractions, ae=2400; T°=145"5°C. Numberaverage molecular weight of fractions: V , 3300; x, 5300; Q, 12 500; [~, 20 000; V , 100 000; A , 425 000; ©, 1 550 000
independent of the molecular weight, and a value of ~e is established for the fraction of the highest molecular weight (ae = 2400 cal/mol) and as a result au = 15 cal/mol, the value of ae can be determined for each molecular weight. These values are indicated in Figure 5, where ae has been represented against number-average molecular weight.
POLYMER, 1972, Vol 13, A u g u s t
409
Crystallization of polyethylene fractions at high undercoofing : J. M. Barrales-Rienda and J. M. G. Fatou 25 6
23 5
x N X
21
o/
19
? 0
3
17
2
__=__a.v + ~ o 15 10 2
4
i
i
I
I
I
103
104
105
106
107
I
108
AT.
0
Figure 5 Plot of oe, interfacial free energy, against numberaverage molecular weight
I
I
1500
I
I
1600
I
I
I 700
I
I
1800
O-e
Figure 7 Plot of AF* against interfacial free energy, ire, at various temperatures for paraffin C36H74. A, 67°C; B, 68°C; C, 69°C; D, 70°C
x
V
T
& X
k~
_c
0 A V x
-I
•
2
i
i
3
4
5
6
~F*IR Tc Figure 6 In(~-o.z)-x against AF*/RTc for samples studied utilizing interfacial free energies shown in Figure 5. ×, C86HTa; A , 1800; V , 3300; V', 5300; 0 , 12 500; R, 20 000; II, 100 000; A , 425 000; O, 1 550 000
Representation of the data for the fraction of lower molecular weight (1-8 x 103) lead to a curve. A value of the slope can be estimated by extrapolation of this curve at the previously established value of ln(z0.1)-1. The value of the slope has been used analytically according to equations (5) and (6) for calculating AF*.
410 POLYMER, 1972, Vol 13, August
The analysis of experimental data can now be made calculating AF* according to equations (4), (5) and (6) for each fraction and each crystallization temperature, with the corresponding ere values. The representation of ln(To.1)-z VS. AF*/RTc, as can be seen in Figure 6, give a unique straight line with slope - 1 according to equation (3). The case of C36H74 deserves special discussion. As has been pointed out, this paraffin presents three crystalline forms: monoclinic, orthorhombic and hexagonal with transition temperatures of 345.3 °, 347.0 ° and 349.1°C respectively16-18. The melting energy corresponding to the monoclinic to liquid form is 30 900 cal/mo116-18. In the experimental crystallization conditions (341°, 342 °, 343 °, and 344°C), the most stable form corresponds to the monoclinic form, although as has been pointed out by Ohlberg 19 and subsequently by Atkinson and Richardson 17, an intermediate component appears, as a precursor of the stable phase of high molecular weight paraffins. Representation of kinetic parameters as a function of AF*/2~r~Tc leads to curves in the range of large values of this quantity when oe=2400cal/mol is used. The extrapolation or determination of the slope at high undercoolings is experimentally impossible, owing to the high speed of crystallization, and analytically very risky. The correct treatment corresponds to the analysis of equations (5) and (6) for different values of ~e, for which these equations have real values. The variation of AF* as a function of ere, for cru=15cal/mol is indicated in Figure 7, on condition that p*= 1. For values of oe greater than those represented, the value of AF* tends towards infinity as Tin, melting temperature, is lower than crystallization temperature, and the solutions of equations (5) and (6) have values with no possible physical significance. If a similar value is stipulated to that of the fraction of 1-8×103, that is, cre=1660cal/mol, ~*=32"05 and p*=14"86 for Tc=343K, the values of AF*/RTc are according to equation (3), as shown in Figure 6. Small
Crystallization of polyethylene fractions at high undercooling : J. M. Barrales-Rienda and J. M. G. Fatou variations in the value assigned to ae do not alter this conclusion. This excellent concordance shows the validity of the model of nucleation for finite molecular weights according to the theories of Mandelkern et a l ) , 6 The variation in ere with the molecular weight as shown in the analysis of the spherulitic growth 7, 8 corresponds fundamentally to the region in which there is a change in morphology in the crystal, of folded chains to extended chains. The highest values of ere correspond to the areas of molecular weights where ~* is much less than x. When ¢* is comparable to x, the value of ~e is much smaller and also reaches a constant value. The region situated between these two asymptotic regions corresponds to the molecular weights in which there is a morphological change 7, s and the ratio ~*/x ranges in between the values of 0.001 and 0.900, where ~:* has been taken for a constant crystallization temperature of 115°C for fractions and 70°C for paraffin C36HTa. The results described in this work show that the crystallization process of polymers is governed by the nucleation of the system and that when the variation in the interfacial energy in the (001) face of the crystal with the molecular weight is stipulated, the phase transformation is described by a unique function of the free energy required for nucleation.
REFERENCES 1 Mandelkern, L. Polym. Eng. Sci. 1967, 7, 232 2 Mandelkern, L., Fatou, J. M. G. and Ohno, K. J. Polym. Sci. (B) 1968, 6, 615 3 Mandelkern, L. 'Crystallization of Polymers', McGraw-Hill, New York, 1964 4 Devoy, C., Mandelkern, L. and Bourland, L. J. Polym. Sci. (A-2) 1970, 8, 869 5 Mandelkern, L., Fatou, J. M. G. and Howard, C. J. Phys. Chem. 1964, 68, 3386 6 Mandelkern, L., Fatou, J. M. G. and Howard, C. J. Phys. Chem. 1965, 69, 956 7 Devoy, C. and Mandelkern, L. J. Polym. ScL (A-2) 1969, 7, 1883 8 Lovering, E. G. J. Polym. Sci. (A-2) 1970, 8, 747 9 Fatou, J. M. G. and Barrales-Rienda, J. M. J. Polym. Sci. (A-2) 1969, 7, 1755 10 Fatou, J. M. G. and Mandelkern, L. J. Phys. Chem. 1965, 69, 417 11 Flory, P. J. and Vrij, A. J. Am. Chem. Soc. 1963, 85, 3548 12 Quinn, F. A. Jr. and Mandelkern, L. J. Am. Chem. Soc. 1958, 80, 3178 13 Gopalan, M. R. and Mandelkern, L. J. Phys. Chem. 1967, 71, 3833 14 Wunderlich, B. Z Phys. Chem. 1965, 69, 2078 15 Wunderlich, B. J. Chem. Phys. 1962, 37, 1203 16 Schaerer, A. A., Busso, C. J., Smith, A. E. and Skinner, L. B. J. Am. Chem. Soc. 1955, 72, 2017 17 Atkinson, C. M. L. and Richardson, M. J. Trans. Faraday Soc. 1969, 65, 1749 18 Atkinson, C. M. L., Larkin J. A. and Richardson, M. J. J. Chem. Thermodyn. 1969, 1, 435 19 Ohlberg, S. M. J. Phys. Chem. 1959, 63, 248
POLYMER, 1972, Vol 13, August 411
INTRODUCTION The crystallization of polymers from the melt is characterized by a high negative temperature coefficient, which shows that nucleation of the system is the predominating factor in the transformation t, 2, independently of the mechanism involved. Moreover, the rate with which crystallization takes place from the melt can be obtained from the general expression for the kinetics of phase transformations 3, 4 and for very low levels of transformation, it can be assumed that the critical free energy for nucleus growth is approximately the same as that required for nucleation, the overall crystallization rate depending on two energetic terms: the activation energy for transport and the critical free energy for forming a nucleus. The latter can be calculated for finite molecular weights by the expressions developed by Mandelkern et ak s, 6 for tri- and bi-dimensional growth mechanisms and this treatment has recently been applied 7, 8 to the analysis of spherulitic growth for different polymers, showing that independently of the molecular weight, the growth rate is a unique function of the free energy for nucleation and that the excess interfacial free energy in the (001) face of the crystal is a function of the molecular weight. On the other hand, the variation in crystallization rate from the melted state at low undercoolings has been attributed to a transport effect so that, when the crystal-
lization temperature decreases and conditions for nucleation are less restricted, the viscosity of the medium is less important and there are no complicating effects by the energy of transport 2, 9 The main purpose of this work is, therefore, to analyse polyethylene crystallization kinetics at high undercoolings and to study the temperature coefficient as a function of the molecular weight and o f the thermodynamic parameters involved encompassing a molecular weight range from 1-55 x 106 to 1-8 × 103. The highly purified C36H74 paraffin has been included in the analysis for comparative purposes. EXPERIMENTAL Crystallization kinetics were studied using the calorimetric technique described in detail elsewhere 9. Eight polyethylene fractions with the following molecular weights were used" 1.55×106 , 4.25x105 , 1 . 0 x l 0 s , 2"0x104 , 12"5x103 , 5"3x103 , 3"3×103 and 1-8×103 . These samples were obtained using a column fractionation technique already described 1°, using linear polyethylenes. A highly purified C86I-I74 paraffin (Fluka, A.-G.) was the fraction with the lowest molecular weight. The crystallization was studied with a Perkin-Elmer, DSC-1B differential scanning calorimeter. The method used is the same as described by us in a previous paperL Briefly, it can be summarized as follows. The molten samples were undercooled at 64°C per minute until the
POLYMER, 1972, Vol 13, August
407
Crystallization of polyethylene fractions at high undercoofing : J. M. Barrales-Rienda and J. M. G. Fatou I.O a
,
O)
T 'T'~o
113
0.2
"lb
o
i!
°/
I
119
~ ~ %
o I
O.I
[
I
0"1
" 0 •I
Ii
IO
t (rain)
Figure 1 Typical isotherms for molecular weight fractions of (a) 3300,and (b) 20 000 linear polyethylene fractions and (c) paraffin (C36H74)at high undercoolings. Temperature (°C) of crystallization is indicated for each isotherm
desired crystallization temperature was reached. Crystallization heat was recorded as a function of time until crystallization was complete or the heat evolved was small enough to be detected. The percentage crystallization was obtained in the usual way3. AHu=6.8 cal/g was used for the indium and AH~=(960/14)cal/g was used for the high molecular weight polyethylene fractions (Mn = 2.0 x 104). The melting heat for the lowest molecular weights were calculated using the Flory and Vrij 11 equation modified for a value of AHu = 960 cal/mol z2, AHu = 960-- AC~A T - (2150/n)
(1)
where AHu is the melting enthalpy per -CH2- unit and n is the chain length, A T is the difference between the equilibrium melting temperature of an extended chain polyethylene crystal of infinite molecular weight in equilibrium, that is to say 145.5°C11,13 and the melting temperature for a chain with n methylene groups, and AC~ is the difference between the heat capacity of extended chain polyethylene crystaP4 and amorphous polyethylene15. The melting heat of 30 900cal/mollm 17 taken for the C46H74 paraffin corresponds to melting enthalpy of AHu= 853.3 cal/mol per -CHz-- group. Thus, the functions ( 1 - At) versus t were obtained for each undercooling. The value (1-At)oo was obtained by the usual method of taking the extrapolated value of the curve after two decades of time, because secondary crystallization was extremely slow. The usual form of the equation of Avrami ln0--kt n was obtained by calculating the function O from the following relation: (1 --
At)oo.(1
-- 0) = (1 -- At)
(2)
The most important fact noted is that at the highest crystallization temperatures (121-123°C) a maximum in the crystallization rate may be observed which corresponds as a minimum when plotting z0.5 against h#n. This minimum is not so well defined in each case and, as can be seen, as undercooling increases, it moves to much lower molecular weights and nearly completely disappears at the lowest undercooling. This minimum is more pronounced at the highest crystallization temperature, as may be noted for all crystallization temperatures used. Furthermore, with any molecular weight, crystallization rate increases when crystallization temperature decreases. It is a well established fact that this indicates a nucleation process in the crystallization is occurring. Another important fact is that at the lowest crystallization temperatures and in the range of the highest molecular weight, crystallization rate only seems to be slightly dependent on molecular weight. The nucleation theories developed5, 6 have shown that the free energy for stable nucleus formation is independent of molecular weight for chains with a molecular weight above 105 whatever the crystallization temperature and crystallization rate changes with molecular weight have been attributed to the transport term or rate of growth 2. When the crystallization temperature decreases, the critical nucleation conditions diminish and the viscosity of the medium is not of important influence. This has been shown in a previous work and it is strongly supported by our own results which cover a wider range of molecular weights. Crystallinity, at the end of the transformation at a given crystallization temperature, is a function of the molecular weight and decreases as this increases. Figure 3 represents this variation after crystallization at 115°C for fractions of molecular weight between 1.55 x 10~ and 1.8 x 103. For these limits, crystallinity varies from 35 ~o for the fraction of highest molecular weight up to 85 ~o for the fraction of lowest molecular weight. Similar results were previously reported z0 after crystallization at low undercoolings. On the other hand, the crystallization rate can be expressed as a function of time r0.1 in which 10~ of the transformation takes place by the equationa, 4: In (~'o.1 ) - 1
=
In 0-o.1)o I
__
EI#RTe - AF*/RTe
(3)
IOO
IO
l.n
k li9 ~21 123
J l
I
o
=-\~ i~O- o / V."n---.--.O_O_ . .
67"
Isotherms thus obtained for two of the fractions and paraffin C36H74 are shown in Figure 1.
~
/ = ~=.A / ~n
III o I09
X '~v
I13 o-I
I
I
I
I
l°3
i°4
!°S
l°~
l°7
RESULTS AND DISCUSSION The results of the crystallization rates, expressed in terms of time ~-0.5for 50 ~ of the transformation are given in Figure 2 for the various temperatures studied.
408
POLYMER, 1972, Vol 13, August
Double logarithmic plot of crystallization half-time, ~'0.5 (minutes) against number-average molecular weight. Isothermal crystallization temperatures (°C) indicated for each curve Figure 2
Crystallization of polyethylene fractions at high undercooling : d. M. Barrales-Rienda and J. M. G. Fatou where Te is the crystallization temperature, AF* the free energy for forming a stable nucleus and E 9 is the transport term. From equation (3) it can be deduced that a representation of In(r0.1) -1 against A F * / R T e should correspond to a straight line with a - 1 slope and independent of the molecular weight when the value of AF* is stipulated for the model of finite molecular weight in a tridimensional nucleation process. For the case of a cylindrical nucleus, it has been shown that 5: AF* =
~1/2~*p*1/2o'2
I.O
0.8
"-z ~, - 0.4
(4)
where ~:* and p* are the critical dimensions of the nucleus and au the lateral interfacial free energy per structural unit. For a finite chain with chain length x, the critical dimensions are related by the following expressions :
"a x
©.6
0.2
I
ic~
I
10 3
I
10 4
I
10 S
10 6
io ~
Figure 3 Degree of crystallinity (1 -At), as a function of molecular weight for linear polyethylene fractions after crystallization at 115°C
(~:*/2) [Afu - (RTc/x) + ( R T c / x - ~* + l) ] = 2 a e - R T c l n [ ( x - ~ * + l)/x]
(5)
and p.1/2 = 2~rl/Zau / [Afu - (RTc/x) - ( R T c / x - ~* + I) ]
(6)
4
where ae is the interfacial free energy per unit in the basal plane and Afu is the free melting energy per repeating unit at Tc which is given by the approximation A f u = A H u ( T ° - Tc)/T°
(7)
A H u being the melting enthalpy per unit and T°, the equilibrium melting temperature. In the limit of high molecular weights equations (4), (5) and (6) are reduced to the known expression :
AF* = 8rraea~( T°)2/ AH2u( T ° - To) 2
(8)
Analysis of the experimental data according to the above equations require exclusively the specification of T°, AHu and ~re. In the case of polyethylene, Flory and Vrij n have developed the theoretical relationship between the molecular weight and the equilibrium melting temperature. This theoretical treatment and suggested extrapolations 12, 13 give an equilibrium melting temperature for infinite molecular weight polyethylene of 145.5°C. Melting enthalpy is known by independent measurements and for high molecular weights corresponds to 960cal/mo11L If a value for ire is stipulated it is possible to calculate the ratio AF*/2a~ Te. The value chosen was that of ae=2400cal/mol, although the conclusions obtained do not depend on it if other values are selected. The representation of ln(z0.1) -1 against AF*/2a;" Tc is shown in Figure 4. All the analysed fractions give straight lines, with a common intersection, except those corresponding to the fraction of 1.8 x 103 and paraffin C36H74, which are not included. The slopes of these lines vary with the molecular weight and the lower the molecular weight, the lower the slopes are. However, the straight lines can be extrapolated to a common intercept, i.e. ln(~-0.1)oL Similar results have been described in bidimensional analysis of the spherulitic growth of different polymers 7, s Considered literally, variations in the slopes in Figure 4 represent a change in the values of ae or Cru or both, if AHu is a constant. If we assume that (ru is constant and
3
x 5
2
q I
\ 0
-I I
O
O-Ol
I
0"02
~'
" "
I
0"03
0.04
AF~/2o~rc Figure 4 Plot of In(~-o.1)-z against AF*/2a2uTc for linear polyethylene molecular weight fractions, ae=2400; T°=145"5°C. Numberaverage molecular weight of fractions: V , 3300; x, 5300; Q, 12 500; [~, 20 000; V , 100 000; A , 425 000; ©, 1 550 000
independent of the molecular weight, and a value of ~e is established for the fraction of the highest molecular weight (ae = 2400 cal/mol) and as a result au = 15 cal/mol, the value of ae can be determined for each molecular weight. These values are indicated in Figure 5, where ae has been represented against number-average molecular weight.
POLYMER, 1972, Vol 13, A u g u s t
409
Crystallization of polyethylene fractions at high undercoofing : J. M. Barrales-Rienda and J. M. G. Fatou 25 6
23 5
x N X
21
o/
19
? 0
3
17
2
__=__a.v + ~ o 15 10 2
4
i
i
I
I
I
103
104
105
106
107
I
108
AT.
0
Figure 5 Plot of oe, interfacial free energy, against numberaverage molecular weight
I
I
1500
I
I
1600
I
I
I 700
I
I
1800
O-e
Figure 7 Plot of AF* against interfacial free energy, ire, at various temperatures for paraffin C36H74. A, 67°C; B, 68°C; C, 69°C; D, 70°C
x
V
T
& X
k~
_c
0 A V x
-I
•
2
i
i
3
4
5
6
~F*IR Tc Figure 6 In(~-o.z)-x against AF*/RTc for samples studied utilizing interfacial free energies shown in Figure 5. ×, C86HTa; A , 1800; V , 3300; V', 5300; 0 , 12 500; R, 20 000; II, 100 000; A , 425 000; O, 1 550 000
Representation of the data for the fraction of lower molecular weight (1-8 x 103) lead to a curve. A value of the slope can be estimated by extrapolation of this curve at the previously established value of ln(z0.1)-1. The value of the slope has been used analytically according to equations (5) and (6) for calculating AF*.
410 POLYMER, 1972, Vol 13, August
The analysis of experimental data can now be made calculating AF* according to equations (4), (5) and (6) for each fraction and each crystallization temperature, with the corresponding ere values. The representation of ln(To.1)-z VS. AF*/RTc, as can be seen in Figure 6, give a unique straight line with slope - 1 according to equation (3). The case of C36H74 deserves special discussion. As has been pointed out, this paraffin presents three crystalline forms: monoclinic, orthorhombic and hexagonal with transition temperatures of 345.3 °, 347.0 ° and 349.1°C respectively16-18. The melting energy corresponding to the monoclinic to liquid form is 30 900 cal/mo116-18. In the experimental crystallization conditions (341°, 342 °, 343 °, and 344°C), the most stable form corresponds to the monoclinic form, although as has been pointed out by Ohlberg 19 and subsequently by Atkinson and Richardson 17, an intermediate component appears, as a precursor of the stable phase of high molecular weight paraffins. Representation of kinetic parameters as a function of AF*/2~r~Tc leads to curves in the range of large values of this quantity when oe=2400cal/mol is used. The extrapolation or determination of the slope at high undercoolings is experimentally impossible, owing to the high speed of crystallization, and analytically very risky. The correct treatment corresponds to the analysis of equations (5) and (6) for different values of ~e, for which these equations have real values. The variation of AF* as a function of ere, for cru=15cal/mol is indicated in Figure 7, on condition that p*= 1. For values of oe greater than those represented, the value of AF* tends towards infinity as Tin, melting temperature, is lower than crystallization temperature, and the solutions of equations (5) and (6) have values with no possible physical significance. If a similar value is stipulated to that of the fraction of 1-8×103, that is, cre=1660cal/mol, ~*=32"05 and p*=14"86 for Tc=343K, the values of AF*/RTc are according to equation (3), as shown in Figure 6. Small
Crystallization of polyethylene fractions at high undercooling : J. M. Barrales-Rienda and J. M. G. Fatou variations in the value assigned to ae do not alter this conclusion. This excellent concordance shows the validity of the model of nucleation for finite molecular weights according to the theories of Mandelkern et a l ) , 6 The variation in ere with the molecular weight as shown in the analysis of the spherulitic growth 7, 8 corresponds fundamentally to the region in which there is a change in morphology in the crystal, of folded chains to extended chains. The highest values of ere correspond to the areas of molecular weights where ~* is much less than x. When ¢* is comparable to x, the value of ~e is much smaller and also reaches a constant value. The region situated between these two asymptotic regions corresponds to the molecular weights in which there is a morphological change 7, s and the ratio ~*/x ranges in between the values of 0.001 and 0.900, where ~:* has been taken for a constant crystallization temperature of 115°C for fractions and 70°C for paraffin C36HTa. The results described in this work show that the crystallization process of polymers is governed by the nucleation of the system and that when the variation in the interfacial energy in the (001) face of the crystal with the molecular weight is stipulated, the phase transformation is described by a unique function of the free energy required for nucleation.
REFERENCES 1 Mandelkern, L. Polym. Eng. Sci. 1967, 7, 232 2 Mandelkern, L., Fatou, J. M. G. and Ohno, K. J. Polym. Sci. (B) 1968, 6, 615 3 Mandelkern, L. 'Crystallization of Polymers', McGraw-Hill, New York, 1964 4 Devoy, C., Mandelkern, L. and Bourland, L. J. Polym. Sci. (A-2) 1970, 8, 869 5 Mandelkern, L., Fatou, J. M. G. and Howard, C. J. Phys. Chem. 1964, 68, 3386 6 Mandelkern, L., Fatou, J. M. G. and Howard, C. J. Phys. Chem. 1965, 69, 956 7 Devoy, C. and Mandelkern, L. J. Polym. ScL (A-2) 1969, 7, 1883 8 Lovering, E. G. J. Polym. Sci. (A-2) 1970, 8, 747 9 Fatou, J. M. G. and Barrales-Rienda, J. M. J. Polym. Sci. (A-2) 1969, 7, 1755 10 Fatou, J. M. G. and Mandelkern, L. J. Phys. Chem. 1965, 69, 417 11 Flory, P. J. and Vrij, A. J. Am. Chem. Soc. 1963, 85, 3548 12 Quinn, F. A. Jr. and Mandelkern, L. J. Am. Chem. Soc. 1958, 80, 3178 13 Gopalan, M. R. and Mandelkern, L. J. Phys. Chem. 1967, 71, 3833 14 Wunderlich, B. Z Phys. Chem. 1965, 69, 2078 15 Wunderlich, B. J. Chem. Phys. 1962, 37, 1203 16 Schaerer, A. A., Busso, C. J., Smith, A. E. and Skinner, L. B. J. Am. Chem. Soc. 1955, 72, 2017 17 Atkinson, C. M. L. and Richardson, M. J. Trans. Faraday Soc. 1969, 65, 1749 18 Atkinson, C. M. L., Larkin J. A. and Richardson, M. J. J. Chem. Thermodyn. 1969, 1, 435 19 Ohlberg, S. M. J. Phys. Chem. 1959, 63, 248
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