2653
Macromolecules 1986, 19, 2653-2656
calculate Lhexis that the different chains do not overlap along the rod. This seems reasonable since if it was not true, the chains would make entangled fibers, a result which is not supported by the behavior of the solution. On the other hand, (2r)hexdepends upon the assumption of compact cylinders. From Table I it can be seen that Lhex is almost constant for both solutions if one discards the lowest concentrations, especially with ethanol. However, the absolute values of Lhex and (2r)hexraise a serious problem. Lhexand (2r)hexare almost equal for water and of the same order of magnitude for ethanol; the cylinders look rather like spheres than like rods. The value of Lhex in water is smaller than the distance D between nearest neighbors, and this is also true, although less pronounced, for ethanol. There is a contradiction between the shortness of the individual rods and their tendency to arrange end to end along lines with a separation D generally much greater than For this reason, although Lhex is rather constant, we think that its small value argues against the hexagonal lattice. 3. Cubic Phase of Rods. In this case, Figure 41°, the cubic lattice has a parameter a = d = 2 n / Q . The closest approach between orthogonal lines is a / 2 . The volume of the primitive cell is a3 and it contains a total length 3a of lines or rods. The volume of each rod within the primitive cell is given by d4z3/3 (a, or d). If one compares this volume with the volume V of one polymeric chain, one gets the length Lab covered by one chain ( n = 1) along the rod. T h e calculation is straightforward and the result can be expressed in terms of Lhex: Lcub = 2(3)'/' Lhel = 3.46Lhe1. As for the hexagonal lattice, Lab is independent on any assumption about the nature of the rod and supposes no rod overlapping; similarly, with the supplementary assumption of compacity, one can calculate the diameter of the cylinder (2r),b = (31/2/6)'/2(2r)hex = 0.54(2r),,, (Table I). This model keeps L constant throughout dilution except for the lowest concentrations, but gives absolute values of L that are much larger than the diameter, and thus compatible with a rod shape. It must also be noticed that L is larger than a / 2 over the entire concentration range except for the lowest concentration in water. So, among the three models, this model appears best. Comparison between the two sets of samples shows that the individual rod lengths are increased in going from water to ethanol from about 192 to 437 A, a factor of 2.28. Since the dielectric constants of water and ethanol are respectively 80 and 24, the results is not qualitatively surprising. In ethanol the electrostatic repulsion between charges is stronger than in water, and one expects the system to be more elongated; in these systems with no coions the Debye screening length is longer than L.l0 The ratio of rod lengths, 2.28, is presumably related to the ratio of the dielectric constants. A complete theory is still lacking; the condensation of the cations, the electrostatic interactions between the anions through both the solvent and the polymer, and the finite radius of the cylinder must also be taken into account. Here we provide a crude analysis based on what has been done on po1yelectrolytes."J2 If the force opposing elongation is the elasticity of chains, which varies as L2, the optimization of the sum of such a term added €-'I3 and to an electrostatic contribution l / t L would lead to L in the present case to a ratio of elongation Lethol/Lwater= (24/80)-'13 = 1.49, much smaller than the observed one. In these compounds, which have a rather low charge content and strong phobicity to solvents, one may also think that the term opposing electrostatic repulsion is the unfavorable enthalpy of interaction between solvent molecules and polymer. In other words, a n elongation of the rod increases the surface of contact 2nrL between the solvent and the polymer. If one calls y the interfacial energy and notes that for a compact cylinder of constant L-'I2, one has to add to the electrostatic energy a density r surface energy expressed by yL1l2. Optimization leads to L Unfortunately, y is a quantity to which we have no direct access. In principle, y could be estimated from the Hildebrand parameters of the solvents and the p ~ l y m e r . ' ~However, the simultaneous presence of different kinds of physical interactions (van der Waals, dipolar, dipolar induced, etc.) makes any such deduction rather speculative. In the absence of data we take the same y for both solvents and determine the ratio of the elongations to be 2.23, a value close to the experimental value 2.28. Although this agreement relies upon a hypothetical constancy of y, we
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0024-9297/86/2219-2653$01.50/0
believe that ow resulta favor the compact cylinder model, in which the solvent-polymer contact is at the surface of the "micelle", rather than the open coil model, in which the solvent-polymer contact is maintained all along the chain.
Acknowledgment. We acknowledge the LBon Brillouin Laboratory (Saclay, France) for providing neutron facilities and are especially grateful to Dr. J. P. Cotton and Dr. J. Teixera, staff of this laboratory, for their help i n the SANS e x p e r i m e n t s and data reduction. Registry No. Neutron, 12586-31-1; Nafion 117, 66796-30-3. References and Notes Roche. E. J Pineri. M.: Dudessix. R.: Levelut. A. M. J.Polvm. Sci., Polym. Phys: Ed: 1 9 h , 19,'l. ' Gierke, T. D.: Munn, G. E.: Wilson, F. C. J . Polym. Sci., Polym. Phys. Ed. 1981,' 19, 1687. Grot, W. G.; Munn, G. E.; Wamsley, P. N. Paper 154 presented at the Electrochemical Society Meeting, Houston, TX, May 7-11, 1972. Eisenberg, A.; Yeager, H. L., Eds. ACS Symp. Ser. 1982, No. 180.
E. I. du Pont de Nemours, Inc. Nafion Perfluorinated Membranes (product literature), Feb 1, 1984. Grot, W. G.; Chadds, F. European Patent 0066369, 1982. Martin. C. R.: Rhoades. T. A,: Fereuson. J. A. Anal. Chem. 1982,54, 1641. Covitch. M. J. American Chemical Societv Meeting. Philadelphia,' PA, Aug 1984. Ise, N.; Okubo, T.; Kunugi, S.; Matsuoka, H.; Yamamoto, K.; Ishii, Y. J. Chem. Phys. 1984,81, 3294. Matsuoka, H.; Ise, N.; Okubo, T.; Kunugi, S.; Tomiyama, H.; Yoshikawa, Y. J. Chem. Phys. 1985, 83, $78. de Gennes, P.-G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. (Paris) 1976, 37, 1461. Manning, G. S. J. Chem. Phys. 1969., 51, 924. Katchalsky, A. Pure Appl. Chem. 1971,26, 327. Barton, A. F. M. Chem. Rev. 1975, 75, 731. I
I
Elastic Modulus of Isotactic Polypropylene in the Crystal Chain Direction As Measured by X-ray Diffraction CHIE SAWATARI and MASARU MATSUO* Department of Clothing Science, Faculty of Home Economics, Nara Women's University, Nara 630, Japan. Received May 6, 1986
The ultimate value of the Young's modulus of polymeric m a t e r i a l s is well-known to be e q u i v a l e n t to the crystal lattice modulus in the direction of the polymer chain axes. The s t u d y of crystal lattice moduli has c o n c e n t r a t e d on polyethylene, and the values have been reported as measured b y X - r a y diffraction (235 GPa),' Raman spectrosc o p y (290 G P a ) , 2 and inelastic neutron s c a t t e r i n g (329 GPa).3 The Raman and neutron values are significantly higher than that o b t a i n e d b y X - r a y diffraction. T h i s difference m a y be due to the essential problem of determining the crystal lattice modulus b y Raman spectroscopy and inelastic neutron scattering; i.e., i n a d d i t i o n to the difficulty i n e s t i m a t i n g lamellar length b y small-angle X - r a y scattering, both methods c o n t a i n an unavoidable assumption concerning the frequencies of absorption bands i n a polymeric system. X-ray diffraction measurements were carried out by Sakurada et al.' using several k i n d s of oriented polyethylene samples having different crystallinities and molecular weights, w h i c h were prepared b y elongation of
* To whom correspondence should be addressed. 1986 American Chemical Society
melt-crystallized films. X-ray diffraction has the advantage of determining the crystal lattice modulus directly. However, an essential question has arisen concerning the interpretation of the X-ray data: Can the stress within a specimen be assumed to be everywhere the same as the external applied stress? In previous work: the crystal lattice modulus was measured by using ultradrawn polyethylene films with different draw ratios of 50,100,200, and 300 in order to check the homogeneous stress hypothesis. The resultant values were almost independent of the draw ratio and were in the range 213-229 GPa. This supporta the homogeneous stress hypothesis. The ultradrawn films were prepared by gelation/crystallization from dilute solution, and the detailed preparation method was described elsewhere.6 It is noteworthy that these values were lower than that obtained by Sakurada e t al.' This discrepancy can readily be explained as arising from differences in the degree of orientation of the molecular chains. The ultradrawn films have an advantage in comparison with films drawn by the melt-crystallization method in that further elongation, also termed creep, is hardly observable under constant applied stress. This is due to extremely high molecular orientation. If the molecular orientation and crystallinity are not sufficient to ensure compliance with the homogeneous stress hypothesis, as in the case of melt-crystallized films, the inner stress becomes slightly different from the external applied stress. This is because creep causes a decrease of the cross-section area, especially in the center of the specimen where the incident X-ray beam impinges. The present paper is concerned with the estimation by X-ray diffraction measurement of the crystal lattice modulus of polypropylene in the direction of the molecular chain axis. Ultradrawn fiis prepared by a method similar to that of Manley et al.6were used. It should be noted that there exists no detectable crystal plane whose reciprocal lattice vector is parallel t o the molecular chain axis. In the unit cell of polypropylene the reciprocal lattice vector of the (113) plane, among all the crystal planes, most closely parallels the crystal c axis. Hence, the apparent crystal modulus of the (113) crystal plane was measured. Using the apparent value, we estimate the real value of the crystal lattice modulus through a somewhat complicated mathematical treatment.
Experimental Section A linear polypropylene with high molecular weight (4.4 X lo6) was used as the specimen. The solvent was decalin. A decalin solution containing 0.5% (w/w) polypropylene and 0.1% (w/w) of the antioxidant di-tert-butyl-p-cresolwas prepared by heating the well-blended polymer/solvent mixture at 150 'C for 40 min under nitrogen. The hot homogenized solution was quenched by pouring into an aluminum tray which was surrounded by water, thus generating a gel. The decalin was allowed to evaporate from the gel under ambient conditions. The nearly dried gel film was immersed in an excess of ethanol and subsequently air-dried to remove residual traces of the decalin-ethanol mixture. The strips were clamped into a manual stretchingdevice in such a way that the length to be drawn was 15 mm. The specimen was placed in an oven at 165-170 O C and elongated manually to the desired draw ratio, which was less than h = 20. Immediately after stretching, the stretcher with the sample was quenched to room temperature. Draw ratios were determined in the usual way by measuring the displacement of ink marks placed 2 mm apart on the specimen prior to drawing. Elongation to draw ratios beyond X = 20 was done in a second stage. This was due to the physical restriction of the size of the oven. Thus the original specimen was first drawn to h = 20 and the drawn film was cut into strips of length 50-60 mm. These specimens,each clamped over a length of 10 mm at the end (the length to be stretched being 30-40 mm), were drawn to the desired ratio beyond h = 20 in a second stage,
*
Macromolecules, Vol. 19, No. 10, 1986
2654 Notes
XI
XP
Figure 1. Schematic diagram showing the relationship between the crystal c-axis direction (X3axis) acd the direction of the reciprocal lattice vector of the crystal (113) plane (U3axis), in which Euler angles &, @z, and & specify the orientation of the
U3axis with respect to the Cartesian coordinate O-X1X2X3 fixed within a film specimen.
-
xx3 =33
I
u3
Figure 2. Stress q3in the direction of the X3axis.
using the method described above. Sample strain was measured with a constant-tension Stretching apparatus. The specimen was mounted horizontally in the stretching clamps of the apparatus in such a way that the tilting angle between the film normal direction and the direction of the incident beam was about 10'. Details of the apparatus are not presented since they have already been de~cribed.~ The crystal lattice strain of the crystal (113) plane was observed by X-ray diffraction measurement with a scintillation counter and a 12-kW rotating-anode X-ray generator (Rigaku RDA-rA operated at 200 mA and 40 kV). The measurement was carried out in such a way that the incident beam was collimated by a collimator 2 mm in diameter. The diffracted beam was detected by a square slit 0.9 mm X 0.9 mm. The X-ray beam was monochromatized with a curved graphite monochromator. The intensity distribution was measured with a step-scanning device at a step interval of 0.lo, each at a fixed time of 100 s, in the range 41-45' (twice the Bragg angles).
Results and Discussion Figure 1 shows a schematic diagram defining the relationship between the reciprocal lattice vector of the (T13) plane and the crystal c axis. O-X,X2X3are Cartesian coordinates within the specimen, with the X 3 axis lying along the stretching direction and the X 2 X 3plane parallel to the surface of the specimen. When the crystallites orient perfectly in the stretching direction, the crystal c axis corresponds to the X 3 axis. The U , axis is taken in the direction of the reciprocal lattice vector of the crystal (113) plane. The U3axis is defined by polar angles C$3, C$2, and with respect to the X 3 , X 2 , and XI axes. The mechanical properties of a polypropylene crystal unit showing a monoclinic system can be described by Hook's law SI, s23 s23
s33
0 0 0
0 0 0
0 0 0 s44
0 0
0 0 0 0 s55
0
0 0
Notes 2655
Macromolecules, Vol. 19, No. 10, 1986 relating strain e to stress IS, where S, is the elastic compliance. The subscripts 3 and 2 are chosen in the direction of the X3and X2axes, respectively. When uniform stress CJ ( = u ~ is ~ )directed along the X3 axis as shown in Figure 2, eq 1 reduces to
@)=ti ;i_ / 413
Ec = 43GPa 0 (b) X.40
ell
855
;44
c12
Then the component of strain
s66
ti, may
€11
= S13U
€22
= S23a
€33
= s33c
0
be given by
€23 = €13 = 612 = 0 (3) According to the geometrical arrangement in Figure 1, the strain e331 in the direction of the U3axis may be given by €331 = (cos2 4 3 ) € 3 3 + (cos2 4 2 ) € 2 2 + (cos2 4 1 ) € 1 1 + (cos &)(cos &)e12 + (cos dl)(cOs 4 3 ) € 1 3 + (cos $2)(cOs &)e23 = (cos2 4 3 b 3 3 + (cos2 4 2 ) € 2 2 + (cos2 4 1 h 1 (4) Since the uniaxially drawn specimen has fiber symmetry, the U3axis exhibits a random orientation with respect to the X, axis and its own axis. Then cos2 $1 = cos2 $2 = (1- cos2 43)/2 (5)
Substituting eq 3 and 5 into eq 4, we can write eq 4 as €331
+ (1/2)(1 - cos2 4 3 ) ( € 1 1 + €22) 4 3 ) S 3 3 + (1/2)(1 - cos2 & ) ( S i 3 + S 2 3 ) I c
= (cos2 &)e33
= {(COS'
With the relationship S331=
(ea)
(6b) 6b is given by
+ (1/2)(1 - cos2 4 3 ) ( s 1 3 + s 2 3 )
(7) With the assumption that the mechanical properties of a polypropylene crystal unit are uniaxial around the crystal c axis because of the helical structure, eq 7 reduces to s 3 3 1 = (cos2 4 3 3 ) s 3 3 + (1 - cos2 4 3 ) s 1 3 = {cos24 3 3 + (1 - cos2 4 3 ) ( S i 3 / S 3 3 ) J S 3 3 (8) The crystal lattice modulus E," in the crystal c-axis direction is represented by using the apparent modulus E, of the crystal (113) plane. That is s33'
= (cos2 4 3 ) s 3 3
c ~ ~ / c Teq ,
E,' = {COS' $3 + (1 - cos2 4 3 ) ( S 1 3 / S 3 3 ) J E c (9) The ratio (S13/S33) in eq 9 can be written by using the elastic stiffness as follows: s13/s33
= -c13/(cll
+ c12)
(10)
Here it should be noted that there has been no report of the theoretical calculation of the elastic stiffness of a polypropylene crystal and so a t present it is impossible to estimate ( s 1 3 / s 3 3 ) exactly. The elastic stiffness of ultradrawn polypropylene film (draw ratio of 20) at 24 "C was measured by Leung and Choy using an ultrasonic techn i q ~ e .From ~ the relationship between the elastic stiffness and temperature (see Figure 10 in ref 7), we can read these values roughly. That is Cll = 5 GN/m2 C12 = 3 GN/m2 C13 =
4 GN/m2
Ec 43GPa
u)
(11)
E, = 45GPa 0
0.I Crystal Strain
(%I
Egure 3. Relationship between the crystal lattice strain of the (113) plane and the external stress for specimens drawn up to X = 60.
Assuming that the values for bulk polypropylene in eq 11 are almost equal to those of a crystal unit, we can determine the value of S13/S33 very roughly as follows: -1 /Z
- s13/533
