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Mechanical γ and β relaxations in polyethylene—II. Physical models of the mechanical γ relaxation in polyethylene
Eur. Polym. J. Vol. 28, No. 8, pp. 949-953, 1992 Printed in Great Britain
0014-3057/92 $5.00 + 0.00 Pergamon Press Ltd
M E C H A N I C A L 7 A N D fl R E L A X A T I O N S IN P O L Y E T H Y L E N E - - I I . PHYSICAL M O D E L S OF THE M E C H A N I C A L 7 R E L A X A T I O N IN P O L Y E T H Y L E N E N. ALBEROLA,h2 J. Y. CAVAILLE2'3 and J. PEREZ2 iLaboratoire Materiaux Composites, ESIGEC, Universit6 de Savoie, BP 1104, 73011 Chambery Cedex, France 2Laboratoire GEMPPM, INSA, Bat. 502, Av. A. Einstein, 69621 Villeurbanne, France 3CERMAV, BP 53X, 38041 Grenoble Cedex, France (Received 6 November 1991)
Abstract--Mechanical spectrometry performed in the ), region of polyethylene has shown that this well-defined relaxation has some characteristics of relaxation related to glass transition (see Part I). In order to verify the validity of such an approach, this paper attempts to give a theoretical basis to an approach based on a physical model previously developed to predict the deformation of amorphous materials near Tg. As polyethylene is a semicrystallinepolymer, the behaviours of the two phases must be separated by using a model for composite systems. It was shown that the dynamic mechanicalbehaviour of the amorphous phase of polyethylenein the 7 region is well described by the physical model. Moreover, this physical model gives evidence for the crosslinking of the amorphous phase by crystallites acting as physical ties.
INTRODUCTION
In Part I, it was shown that the mechanical 7 and fl relaxations in polyethylene both have characteristics of mechanical relaxation related to glass transition of polymers. As the mechanical 7 relaxation is welldefined, an attempt is made to give a theoretical approach to such a mechanical relaxation by applying a physical model previously developed to describe the inelastic and plastic deformation of amorphous polymers n e a r Tg [1-3]. The purpose of such an approach is: (i) to point out the fundamental parameters which govern the characteristics of this relaxation in such a semicrystalline polymer and to attribute to them a clear physical meaning in terms of microstructure; and (ii) to verify the self consistency of an approach based on the relation between the 7 relaxation and rubbery glass transition. Before applying this physical model to the 7 relaxation of polyethylene, it is valuable to separate the dynamic mechanical behaviours of the crystalline and amorphous phases. This separation has been made by using the analysis of Halpin and Kardos [4] and its extension to the case of the viscoelastic behaviour of polypropylene [2]. In this paper, the experimental data used for modelling are those determined in Part I. SEPARATION BETWEEN THE BEHAVIOURS OF THE TWO PHASES
The aim of such a separation for this mechanical model, based on the lack of connection between the mechanical behaviour of the phases, is to remove
the reinforcement effect of the crystalline phase in such a composite material in order to keep only the crosslinking of the amorphous phase caused by the crystallites. The Halpin and Kardos analysis [4] includes a set of geometrical parameters eu related to the three dimensions of the crystalline lamellae. As a single dimension, the thickness, can be evaluated and, in order to avoid further approximations and sources or error, we have applied the simplified form of the Halpin and Kardos model, viz. the Halpin-Tsai equation including a single geometrical factor (two dimensional model). Crystalline lamellae are assumed to be platelets of diameter d and thickness 1: = (d/l) ~/3. Then, the complex modulus of the amorphous phase can be evaluated through the inversion of the following equation: G Ga
Gc(1+ ~Xc) + Ga(1 -- Xc)
Gc(l - ~X~) + Ga(~ + X~)
with G, the complex shear modulus of the composite material; X~ the crystallinity content; Ga , the complex modulus of the amorphous phase; Gc the crystalline modulus estimated as 3.5 (GPa) [5]. It is assumed that Gc is independent of frequency. This can be done because the y region is far away from temperature or frequency ranges where relaxations in the crystalline phase can occur. The morphological parameters considered for the mechanical modelization are reported in Table 1. Figures l(a) and l(b) show the dynamic mechanical behaviour vs temperature of the amorphous phase of the quenched and annealed samples, respectively. With increasing annealing temperature, the increase in the height of the rubbery plateau is due to increase of the crosslinking effect of the amorphous phase by the crystalline phase. 949
N. ALBEROLA et al.
950
Table I. Morphological parameters taken into account for the Halpin-Tsa~ modelling of the 7 relaxation System X~(%) d(pm) I(A) Quenched 40 0.5 110
G"
+annea&d at:
100'~ 110° 130°
50 52 65
0.5 0.5 0.5
130 140 220
PHYSICAL MODEL APPLIED TO THE MECHANICAL ? RELAXATION
The aim of such a modelling of the ? relaxation is to quantify the interactions between phases. According to the physical model previously developed in our laboratory [1-3], the structure of a glassy polymer consists of close packing repeat units in which there are randomly distributed high energy sites which correspond to frozen in density fluctuations. These sites designated as defects exhibit enthalpy (and entropy) excess. At T < Tg (isoconfigurational state), defect concentration CD(T) is constant: C o ( T ) = Co(Tg ). For temperatures above Tg, the defect concentration CD(T) increases. The determination of CD(T) requires knowledge of the enthalpy and entropy excesses which could be derived from calori-
(a) 1
• 0
0
•
ft.
o
/
/o 0 -180
~o
\o 5
(b) 9
(3.
c
-
.2
o~
...I
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G,u - GaR G* = GRa -4 1 + H(iO~Zmr)-h + (ia~Zmr) k'
(1)
Subscript a designates the amorphous phase and G~ and G R are the unrelaxed and relaxed modulus respectively. The relaxation time Zmr is expressed by:
zl = t0 exp U / R T
~• -o. o
-0-
I-
metric measurements. The application of a stress results in microstructural changes within the defect and the shear stress leads to nucleation of shear microdomains (so-called smd). The resulting deformation linearly increases with defect concentration and results in inelasticity (subglass transition) usually observed for vitreous systems. When the application time of the stress is long enough (or when local deformation increases), the size of the stud increases. The growth of smd is attributed to hierarchically constrained motions of the surrounding molecules. After a relatively long period of applied stress, growth of the smd leads to merging of the lines bordering neighbouring defect sites and flow occurs. Finally, the description of the main relaxation leads to an analytical expression of the complex modules G~* where all parameters have physical meaning:
z~ corresponds to the mean time for an elemental microscopic molecular movement. It was suggested to identify t] as the relaxation time corresponding to the first subglass relaxation and it can be expressed by:
T (°C)
1 -
Gu
G' Fig. 2. Determination of the parameters of the physical model for Cole~ole diagram.
tin, = ( ~ / t ' , - k ) ' / k I -20
-100
G r
-100
5
-20
T (°C) Fig. I. Dynamic mechanical behaviour of the amorphous phase calculated from the Halpin-Tsai modelling for (a) samples quenched from 170° (O) and 200° ( ) to room temperature, (b) samples annealed for 24 hr at 100° ( ), 110° ( 0 ) and 130° ( - - ) .
where z0~ 10-13-10-14sec and U is the activation energy of the subglass relaxation. The time tj is an adjusting parameter and may vary from t0 to Zl. The correlation parameter k (0 < k < l) characterizes the effectiveness of correlation effects required for the growth of smd. k increases with the defect density CD(T ). Thus, at T < Tg (isoconfigurational state) k is constant. In the metastable equilibrium state (T > Tg), k increases with increasing temperature because the cooperativity of motions decreases as the specific volume increases. The parameter h (0 < k < h < l) is related to the increase of correlation effects (i.e. the decrease of k) as the shear strain locally increases. As a matter of fact, the local shear that appears near the defect sites causes a decrease in the degree of freedom of the chain segments as the segments become locally extended. Thus, the presence of junction points i.e.
Mechanical 7 and fl relaxations in polyethylene--II Table 2. Characteristic values used for the modelling of the 7 relaxation System
(10~Pa)
(106Pa)
G~ /G~
k
h
1.3
4.5
220
0.27
0.68
1.3 1.3
5.5 21
180 47
0.27 0.27
0.60 0.51
Quenched +annealed at: 100 130
chemical or physical crosslinking ties hindering the deformation, leads to an increase in the correlation effect and therefore to a decrease in h. For a small molecule monomolecular amorphous system, h is equal to 1, i.e. k does not vary with duration of the applied stress (i.e. with local shear strain). For a macromolecular system showing physical [2] or chemical [1] crosslinking ties, h is smaller (0.60 < h < 0.80). Thus, h is a probe of the polymer structure: it is very sensitive to the obstacles hindering the extension of the local deformation. Parameters k and h are determined from the Cole-Cole diagram (Fig. 2). The theoretical description of isothermal mechanical behaviour of polyethylene in the 7 region can be calculated from equation (1) and compared with experimental master curves determined after having separated the behaviours of the two phases using the mechanical model. The Cole-Cole diagram gives
G~ ,G~ and the k and h values. The parameter H is adjusted in order to obtain a correct height in the Cole-Cole diagram, rmr is chosen in order to have good coincidence between calculated and experimental peaks of tan 4~ =f(frequency). The characteristic values used in the molecular modelling of the 7 relaxation are reported in Table 2. Theoretical and experimental curves (after separation between the two phases) are compared in Fig. 3. It can be seen that the molecular model well describes the mechanical behaviour of the amorphous phase of polyethylene near Tg. Analysis of the results leads to the two following conclusions: (i) for the analysed systems, the k value (0.27) is constant and almost identical to those found for polypropylene [2] and for chemically crosslinked polymers [1]; (ii) h values (0.514).68) are of the same order of magnitude as that of polypropylene [2]. With increasing annealing temperature, h decreases: correlation effects increase because of increase of the crosslinking of the amorphous phase by the crystalline phase. In order to show the self consistency of our results, it can be verified that the increase in correlation effects with increasing annealing temperature is accompanied by increase in the height of the rubbery modulus.
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2
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Fig. 3. Dynamic mechanical behaviour under isothermal conditions of the amorphous phase: O, calculated spectra from the Halpin Tsai equation; - - , calculated spectra from the physical model for deformation for samples, (a) quenched from 170° and annealed at (b) 100° and (c) 130°.
952
N. ALBEROLAet al.
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5
0 -180
O.o.O.O,o'O'O'IQ% . . . . oooe
-100
I -20
T (°C)
T (°C)
Fig. 4. Dynamic mechanical modulus of the amorphous phase vs temperature; O , points calculated from experimental results with Halpin-Tsai equation; - - - , calculated spectra from the physical model of deformation for samples, (a) quenched from 170 ° and annealed at (b) I00 °, (c) 110 ° and (d) 130 °.
The theoretical description of the isochronal behaviour of the amorphous phase of polyethylene in the y region can also be calculated from equation (1). Such a calculation requires not only knowledge of the parameters defined above but also evaluation of the change in defect concentration Co (T) with increasing temperature:
CD(T) =
1
[1 + e x p ( H e ~ c / R T ) e x p ( -
Sex¢/R)]
where Hexc and Sexc are the enthalpy and entropy excesses of the defects. These values can be evaluated from calorimetric data. As the activation energy U of the elementary mechanism (the subglass relaxation 72) cannot be determined because of the closeness of the primary relaxation, it can be considered that U is proportional to temperature in the Arrhenius type equation giving the relaxation time z,. U is thus evaluated as 35 k J/tool. In fact, as values of H~x~ and S,xc cannot be determined experimentally because no change in the slope of Cp is detected in the V region, the ~mr values are determined from experimental data at temperatures above and below Tg: - - i n the metastable equilibrium state, z~r is deduced from G" curves obtained under isothermal conditions: a~Zm,= 1 for G"m~x.
For curves with no maximum, ~mr is determined for each temperature from the translational factor log av, - - i n the isoconfigurational state, Zmr values are estimated from log a t data. Figure 4 shows the calculated curves log G' and tan ~ vs temperature and the corresponding experimental ones (after separation between the behaviours of the two phases). As for isothermal mechanical curves, it can be seen that there is good agreement between experimental and calculated isochronal behaviours of the amorphous phase. Thus, the agreement between the description of the ? relaxation by the molecular model and experimental curves supports the idea that this relaxation is due to the glass transition of polyethylene. Moreover, this model allows quantification of the crosslinking effect of the amorphous phase caused by the crystalline phase. This is described through a parameter h very sensitive to the microstructure and which could be determined experimentally from the Cole-Cole diagram. CONCLUSION
After separation of the mechanical behaviours of the two phases using a mechanical model based
Mechanical ~ and ~ relaxations in polyethylene--I1 on the composite character of polyethylene, the dynamic mechanical behaviour of the amorphous phase of polyethylene in the 7 region was well described by a physical model, previously developed to predict the mechanical behaviour of amorphous systems near Tg. This physical model gives, through an experimental parameter, h, quantitative evidence for crosslinking of the amorphous phase by the crystalline lamellae acting as physical ties.
953
REFERENCES
1. J. Y. Cavaille, J. Perez and G. P. Johari. Phys. Rev. 1339, 2411 (1989). 2. C. Jourdan, J. Y. Cavaille and J. Perez. J. Polym. Sci., B: Polym. Phys. 27, 2361 (1989). 3. J. Perez, J. Y. Cavaille, S. Etienne and C. Jourdan. Rev. Phys. Appl. 23, 125 (1988). 4. J. C. Halpin and J. L. Kardos. J. Appl. Phys. 43, 2235 (1972). 5. R. H. Boyd. Polymer 26, 1123 (1985).
0014-3057/92 $5.00 + 0.00 Pergamon Press Ltd
M E C H A N I C A L 7 A N D fl R E L A X A T I O N S IN P O L Y E T H Y L E N E - - I I . PHYSICAL M O D E L S OF THE M E C H A N I C A L 7 R E L A X A T I O N IN P O L Y E T H Y L E N E N. ALBEROLA,h2 J. Y. CAVAILLE2'3 and J. PEREZ2 iLaboratoire Materiaux Composites, ESIGEC, Universit6 de Savoie, BP 1104, 73011 Chambery Cedex, France 2Laboratoire GEMPPM, INSA, Bat. 502, Av. A. Einstein, 69621 Villeurbanne, France 3CERMAV, BP 53X, 38041 Grenoble Cedex, France (Received 6 November 1991)
Abstract--Mechanical spectrometry performed in the ), region of polyethylene has shown that this well-defined relaxation has some characteristics of relaxation related to glass transition (see Part I). In order to verify the validity of such an approach, this paper attempts to give a theoretical basis to an approach based on a physical model previously developed to predict the deformation of amorphous materials near Tg. As polyethylene is a semicrystallinepolymer, the behaviours of the two phases must be separated by using a model for composite systems. It was shown that the dynamic mechanicalbehaviour of the amorphous phase of polyethylenein the 7 region is well described by the physical model. Moreover, this physical model gives evidence for the crosslinking of the amorphous phase by crystallites acting as physical ties.
INTRODUCTION
In Part I, it was shown that the mechanical 7 and fl relaxations in polyethylene both have characteristics of mechanical relaxation related to glass transition of polymers. As the mechanical 7 relaxation is welldefined, an attempt is made to give a theoretical approach to such a mechanical relaxation by applying a physical model previously developed to describe the inelastic and plastic deformation of amorphous polymers n e a r Tg [1-3]. The purpose of such an approach is: (i) to point out the fundamental parameters which govern the characteristics of this relaxation in such a semicrystalline polymer and to attribute to them a clear physical meaning in terms of microstructure; and (ii) to verify the self consistency of an approach based on the relation between the 7 relaxation and rubbery glass transition. Before applying this physical model to the 7 relaxation of polyethylene, it is valuable to separate the dynamic mechanical behaviours of the crystalline and amorphous phases. This separation has been made by using the analysis of Halpin and Kardos [4] and its extension to the case of the viscoelastic behaviour of polypropylene [2]. In this paper, the experimental data used for modelling are those determined in Part I. SEPARATION BETWEEN THE BEHAVIOURS OF THE TWO PHASES
The aim of such a separation for this mechanical model, based on the lack of connection between the mechanical behaviour of the phases, is to remove
the reinforcement effect of the crystalline phase in such a composite material in order to keep only the crosslinking of the amorphous phase caused by the crystallites. The Halpin and Kardos analysis [4] includes a set of geometrical parameters eu related to the three dimensions of the crystalline lamellae. As a single dimension, the thickness, can be evaluated and, in order to avoid further approximations and sources or error, we have applied the simplified form of the Halpin and Kardos model, viz. the Halpin-Tsai equation including a single geometrical factor (two dimensional model). Crystalline lamellae are assumed to be platelets of diameter d and thickness 1: = (d/l) ~/3. Then, the complex modulus of the amorphous phase can be evaluated through the inversion of the following equation: G Ga
Gc(1+ ~Xc) + Ga(1 -- Xc)
Gc(l - ~X~) + Ga(~ + X~)
with G, the complex shear modulus of the composite material; X~ the crystallinity content; Ga , the complex modulus of the amorphous phase; Gc the crystalline modulus estimated as 3.5 (GPa) [5]. It is assumed that Gc is independent of frequency. This can be done because the y region is far away from temperature or frequency ranges where relaxations in the crystalline phase can occur. The morphological parameters considered for the mechanical modelization are reported in Table 1. Figures l(a) and l(b) show the dynamic mechanical behaviour vs temperature of the amorphous phase of the quenched and annealed samples, respectively. With increasing annealing temperature, the increase in the height of the rubbery plateau is due to increase of the crosslinking effect of the amorphous phase by the crystalline phase. 949
N. ALBEROLA et al.
950
Table I. Morphological parameters taken into account for the Halpin-Tsa~ modelling of the 7 relaxation System X~(%) d(pm) I(A) Quenched 40 0.5 110
G"
+annea&d at:
100'~ 110° 130°
50 52 65
0.5 0.5 0.5
130 140 220
PHYSICAL MODEL APPLIED TO THE MECHANICAL ? RELAXATION
The aim of such a modelling of the ? relaxation is to quantify the interactions between phases. According to the physical model previously developed in our laboratory [1-3], the structure of a glassy polymer consists of close packing repeat units in which there are randomly distributed high energy sites which correspond to frozen in density fluctuations. These sites designated as defects exhibit enthalpy (and entropy) excess. At T < Tg (isoconfigurational state), defect concentration CD(T) is constant: C o ( T ) = Co(Tg ). For temperatures above Tg, the defect concentration CD(T) increases. The determination of CD(T) requires knowledge of the enthalpy and entropy excesses which could be derived from calori-
(a) 1
• 0
0
•
ft.
o
/
/o 0 -180
~o
\o 5
(b) 9
(3.
c
-
.2
o~
...I
I -180
G,u - GaR G* = GRa -4 1 + H(iO~Zmr)-h + (ia~Zmr) k'
(1)
Subscript a designates the amorphous phase and G~ and G R are the unrelaxed and relaxed modulus respectively. The relaxation time Zmr is expressed by:
zl = t0 exp U / R T
~• -o. o
-0-
I-
metric measurements. The application of a stress results in microstructural changes within the defect and the shear stress leads to nucleation of shear microdomains (so-called smd). The resulting deformation linearly increases with defect concentration and results in inelasticity (subglass transition) usually observed for vitreous systems. When the application time of the stress is long enough (or when local deformation increases), the size of the stud increases. The growth of smd is attributed to hierarchically constrained motions of the surrounding molecules. After a relatively long period of applied stress, growth of the smd leads to merging of the lines bordering neighbouring defect sites and flow occurs. Finally, the description of the main relaxation leads to an analytical expression of the complex modules G~* where all parameters have physical meaning:
z~ corresponds to the mean time for an elemental microscopic molecular movement. It was suggested to identify t] as the relaxation time corresponding to the first subglass relaxation and it can be expressed by:
T (°C)
1 -
Gu
G' Fig. 2. Determination of the parameters of the physical model for Cole~ole diagram.
tin, = ( ~ / t ' , - k ) ' / k I -20
-100
G r
-100
5
-20
T (°C) Fig. I. Dynamic mechanical behaviour of the amorphous phase calculated from the Halpin-Tsai modelling for (a) samples quenched from 170° (O) and 200° ( ) to room temperature, (b) samples annealed for 24 hr at 100° ( ), 110° ( 0 ) and 130° ( - - ) .
where z0~ 10-13-10-14sec and U is the activation energy of the subglass relaxation. The time tj is an adjusting parameter and may vary from t0 to Zl. The correlation parameter k (0 < k < l) characterizes the effectiveness of correlation effects required for the growth of smd. k increases with the defect density CD(T ). Thus, at T < Tg (isoconfigurational state) k is constant. In the metastable equilibrium state (T > Tg), k increases with increasing temperature because the cooperativity of motions decreases as the specific volume increases. The parameter h (0 < k < h < l) is related to the increase of correlation effects (i.e. the decrease of k) as the shear strain locally increases. As a matter of fact, the local shear that appears near the defect sites causes a decrease in the degree of freedom of the chain segments as the segments become locally extended. Thus, the presence of junction points i.e.
Mechanical 7 and fl relaxations in polyethylene--II Table 2. Characteristic values used for the modelling of the 7 relaxation System
(10~Pa)
(106Pa)
G~ /G~
k
h
1.3
4.5
220
0.27
0.68
1.3 1.3
5.5 21
180 47
0.27 0.27
0.60 0.51
Quenched +annealed at: 100 130
chemical or physical crosslinking ties hindering the deformation, leads to an increase in the correlation effect and therefore to a decrease in h. For a small molecule monomolecular amorphous system, h is equal to 1, i.e. k does not vary with duration of the applied stress (i.e. with local shear strain). For a macromolecular system showing physical [2] or chemical [1] crosslinking ties, h is smaller (0.60 < h < 0.80). Thus, h is a probe of the polymer structure: it is very sensitive to the obstacles hindering the extension of the local deformation. Parameters k and h are determined from the Cole-Cole diagram (Fig. 2). The theoretical description of isothermal mechanical behaviour of polyethylene in the 7 region can be calculated from equation (1) and compared with experimental master curves determined after having separated the behaviours of the two phases using the mechanical model. The Cole-Cole diagram gives
G~ ,G~ and the k and h values. The parameter H is adjusted in order to obtain a correct height in the Cole-Cole diagram, rmr is chosen in order to have good coincidence between calculated and experimental peaks of tan 4~ =f(frequency). The characteristic values used in the molecular modelling of the 7 relaxation are reported in Table 2. Theoretical and experimental curves (after separation between the two phases) are compared in Fig. 3. It can be seen that the molecular model well describes the mechanical behaviour of the amorphous phase of polyethylene near Tg. Analysis of the results leads to the two following conclusions: (i) for the analysed systems, the k value (0.27) is constant and almost identical to those found for polypropylene [2] and for chemically crosslinked polymers [1]; (ii) h values (0.514).68) are of the same order of magnitude as that of polypropylene [2]. With increasing annealing temperature, h decreases: correlation effects increase because of increase of the crosslinking of the amorphous phase by the crystalline phase. In order to show the self consistency of our results, it can be verified that the increase in correlation effects with increasing annealing temperature is accompanied by increase in the height of the rubbery modulus.
(a)
(b) °° o • e ° Q ° ° °
_
::::
~.. /:~'"~%,,..
c I--
oo
o o o ,o" ,o •
•
•
•
•
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951
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I
I
-2
0
2
4
6
_.1
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0 -6
-4
Log (F/Hz)
-2
0
2
4
Log (F/Hz)
(c)
(d) oooO o ° ° °
/,~"
O e O 0 0 O01 O /
/0
1 t-
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n
O..
v O') O _J
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,,-I
l 6
-4
-2
0
Log (F/Hz)
2
4
%o
o -6
-4
-2
0
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Fig. 3. Dynamic mechanical behaviour under isothermal conditions of the amorphous phase: O, calculated spectra from the Halpin Tsai equation; - - , calculated spectra from the physical model for deformation for samples, (a) quenched from 170° and annealed at (b) 100° and (c) 130°.
952
N. ALBEROLAet al.
(b)
(a) -9
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-9
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¢-..
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0
80
-100
5
0 -180
O.o.O.O,o'O'O'IQ% . . . . oooe
-100
I -20
T (°C)
T (°C)
Fig. 4. Dynamic mechanical modulus of the amorphous phase vs temperature; O , points calculated from experimental results with Halpin-Tsai equation; - - - , calculated spectra from the physical model of deformation for samples, (a) quenched from 170 ° and annealed at (b) I00 °, (c) 110 ° and (d) 130 °.
The theoretical description of the isochronal behaviour of the amorphous phase of polyethylene in the y region can also be calculated from equation (1). Such a calculation requires not only knowledge of the parameters defined above but also evaluation of the change in defect concentration Co (T) with increasing temperature:
CD(T) =
1
[1 + e x p ( H e ~ c / R T ) e x p ( -
Sex¢/R)]
where Hexc and Sexc are the enthalpy and entropy excesses of the defects. These values can be evaluated from calorimetric data. As the activation energy U of the elementary mechanism (the subglass relaxation 72) cannot be determined because of the closeness of the primary relaxation, it can be considered that U is proportional to temperature in the Arrhenius type equation giving the relaxation time z,. U is thus evaluated as 35 k J/tool. In fact, as values of H~x~ and S,xc cannot be determined experimentally because no change in the slope of Cp is detected in the V region, the ~mr values are determined from experimental data at temperatures above and below Tg: - - i n the metastable equilibrium state, z~r is deduced from G" curves obtained under isothermal conditions: a~Zm,= 1 for G"m~x.
For curves with no maximum, ~mr is determined for each temperature from the translational factor log av, - - i n the isoconfigurational state, Zmr values are estimated from log a t data. Figure 4 shows the calculated curves log G' and tan ~ vs temperature and the corresponding experimental ones (after separation between the behaviours of the two phases). As for isothermal mechanical curves, it can be seen that there is good agreement between experimental and calculated isochronal behaviours of the amorphous phase. Thus, the agreement between the description of the ? relaxation by the molecular model and experimental curves supports the idea that this relaxation is due to the glass transition of polyethylene. Moreover, this model allows quantification of the crosslinking effect of the amorphous phase caused by the crystalline phase. This is described through a parameter h very sensitive to the microstructure and which could be determined experimentally from the Cole-Cole diagram. CONCLUSION
After separation of the mechanical behaviours of the two phases using a mechanical model based
Mechanical ~ and ~ relaxations in polyethylene--I1 on the composite character of polyethylene, the dynamic mechanical behaviour of the amorphous phase of polyethylene in the 7 region was well described by a physical model, previously developed to predict the mechanical behaviour of amorphous systems near Tg. This physical model gives, through an experimental parameter, h, quantitative evidence for crosslinking of the amorphous phase by the crystalline lamellae acting as physical ties.
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