POLYMER ORDERAND POLYMER DEKSITY
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Polymer Order and Polymer Density
by Richard E. Robertson General Electric Research Laboratory, Schenectady, New York 12301 (Received October 2.9, 1964)
The high density of amorphous polymers, compared to their crystalline states, suggests that a large degree of order exists. Using a simple model, consisting of strings of cylindrical beads, we have attempted to develop a means of assessing the order in an amorphous polymer from its density.
Introduction How much molecular order exists in a typical glassy polymer such as polystyrene or in the amorphous regions of a crystalline polymer such as nylon? Kargin, et ~ l . , l - believe ~ that the polymer chains collect into bundles, even in an amorphous polymer, and they have attempted to prove this by X-ray and electron diffraction pictures. On the other hand, Flory, et u Z . , ~ - ~ O believe that their calculations and experiments on the mean end-to-end distance and its temperature dependence imply that there is no ordering of the polymer chains in the bulk. It is the assertion of this report that because of the relatively high density of amorphous polymers relative to their crystalline states, a fairly large amount of order exists in amorphous polymers. And, as will be indicated later, this conclusion need not be denied by the results of Flory, et al., on the mean end-to-end distance.
the volume vB = Z28 will be larger than v,. (The subscripts a = amorphous, c = crystalline.) To draw an analogy between these stiff cylindrical segments and a mass of spaghetti, let us assume that the segments are jammed together; ie., each segment is nominally touching its neighbors; however, the segments are not necessarily parallel. I n this state, what will be the average separation between neighboring chain axes? There are so many possible configurations of these strings of segments that it is diflicult to consider this question rigorously. Figure 1 has a configuration, though, which we believe to be fairly representative for calculating the average separation between the axes by considering the separation between centers. For the angle 8 between the neighboring segments, the separation between their centers is
a
+ (~/2)2]”~sin (4 + e/2) d cos 8/2 + I sin 8/2
= 2[(d/2)2
(1)
a =
(2)
A Model for an “Amorphous” Polymer
Since the angle 0 effectively represents the degree of
Amorphous polymers have frequently been likened to a mass of wet spaghetti in order to indicate the randomness inherent in an amorphous polymer. A real amorphous polymer is not that random. However, to introduce our model we will consider the specific volume of the ideally random mass of wet spaghetti. Let us consider a collection of strings of cylindrical segments, each segment of length 2. and diameter d. We will associate with each segment a rectangular cell with square ends of width (z and length 6. The cell represents the specific volume, per segment, of the collection. If the segments are packed perfectly, the dimensions of the cell are those of the segment, giving the volume v, = d2Z. If the segments are not packed perfectly, the dimensions of the cell will be such that
(1) V. A. Kargin, A. I. Kitaigorodskii, and G. L. Slonimskii, Kolloidn. Zh., 19, 131 (1957). (2) V. A. Kargin and G. S. Markova, Dokl. Akad. Naulc SSSR, 117, 427 (1957). (3) V. A. Kargin, N. F. Bakeev, and Kh. Vergin, ibid., 122, 97 (1958). (4) V. A. Kargin, J. Polymer Sci., 30, 247 (1958). ( 5 ) P. J. Flory, “Principles of Polymer Chemistry,” Cornel1 University Press,Ithaca, N. Y., 1953, pp. 426, 602. (6) P. J. Flory, Proc. Roy. Soe. (London), A234, 60 (1956). (7) A. Ciferri, C. A. J. Hoeve, and P. J. Flory, J . Am. Chem. Soc., 83, 1015 (1961). (8) P. J. Flory, A. Ciferri, and R. Chiang, ibid., 83, 1023 (1961). (9) C. A. J. Hoeve, J . Chem. Phys., 35, 1266 (1961); K. Nagai and T. Ishikawa, ibid., 37, 496 (1962). (10) C. A. J. Hoeve and M. K. O’Brien, J . Pobymei Sci., A l , 1947 (1963).
Volume 69, Number 6 M a y 1966
RICHARD E. ROBERTSON
1576
Figure 2. A model for calculating the effective length b of a segment.
which gives the average
6
= 1(2/2/3)(&
- 1)
(7) The ratio of the specific volumes of the perfectly packed state to this disordered state is then v,/v,
Figure 1. A model for calculating the mean separation a between two neighboring chains.
5 l/d
(3)
Let us now assume that, except for the restriction that each segment nominally touch its neighbors, the angle 0 is random. The average separation is then E =
[1’’ sin de]-’ s,””(d cos 0/2 + 0
I sin 6 / 2 ) sin 6 dB (4) = (4/3)[d(1/8
- 1) + I1
(5)
(The spherical weighting factor sin 0 is used in eq. 4 to indicate t>hatmore configurations are available as e increases toward n/2.) 6 is the average separation between the axes of two strings of segments. If we relax sufficiently the condition that the segments touch each of their neighbors, we can neglect the correlation between the average separation distances of a segment from its several neighbors. I n particular, we can assume that E is the average separation between the axis of a string of segments and each of its neighbors. The cell associated with each segment will then have a square cross section of width a. Within a cell of this cross section, though, the segment will not require the full length 1. If we let 0 be the deviation from collinearity, as in Figure 2, the separation between centers of segments adjacent along the string is then b = I
COS
The Journal of Physical Chemistry
e/2
azz/az6
= [0.1915(1.828
disorder in the collection of segments, the model has significance only as long as a increases with e. This is seen to limit e to tan 8/2
=
(6)
+ Z/d)2]-1
(8)
To evaluate this expression we need some value for the ratio l / d . In general, its value will be around 1. First of all, from the inequality in ( 3 ) we see that we must have l / d 2 1 for the integration up to 0 = a/2 to be valid in eq. 4; ie., we must have l / d 2. 1 if we are to replace the flexible, wet spaghetti, which is extremely diacult to treat analytically, by the string of segments. Secondly, of more importance later when we treat a polymer chain, is the fact that Z is essentially the ‘(length of independence” of a chain from the surrounding molecules. Also, it is proper to strip the chain of its pendant groups in estimating a diameter. For a vinyl chain, then, we would take as a minimum the 2.51 8.between next-neighbor carbon atoms as a length of independence, giving l/d 2 1. On the other hand, the ratio Z/d cannot be taken to be very much larger than 1. A physical interpretation of the model is that it describes the wedges of vacant space which result when neighboring chains are other than parallel and which are too small to be filled by other chains. Even if we consider tubes that are very stiff, because of the ability of another tube to partly fill the wedge left by the first two, the effective length for our model will still be of the order of a diameter. As a reasonably safe guess, then, let us assume that l / d = 1. Equation 8 then gives v,/v, = d21/a28 = 0.652. Although this ratio may represent an ideally random mass of spaghetti (which we have not tested, partly because of the difficulty in obtaining either the ideally random or the assumed perfect cubic states), it is much too small for the ratio of the crystalline and amorphous specific volumes (the reciprocal of the corresponding densities). Included in Table I are both polymers that tend to crystallize
POLYMER ORDERAND POLYMER DENSITY
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in a helical fashion (e > 0") and those in a planar zigzag fashion (e = 0'). The crystalline densities are taken from the unit cell obtained from X-ray work. I n all cases, the ratio of the specific volumes is seen to be in the range 0.85 to 0.95. Table I : A Comparison of the Amorphous and Crystalline Densities of Several Polymers Polymer
do
Polyethylene Polypropylene Polystyrene isotactic Polyvinyl alcohol Polyethylene terephthalate Bisphenol-A polycarbonate Nylon66(~~)
d
da/d,
( I = d ) 8, deg. Ref.
6*
b*
15.0' 8.4
0 >O
a b
1.111 1.054 0.947
4.8
>O
b
5=
1.345 1.269 0.943
5.2
o
c
d,
1.455 1.335 0.917
7.8
O
d
1.30 1.20 0.92 1.220 1.069 0.876
7.5 12.4
O
e
O
f
'
Polymer Order 21s. Polymer Density I n this section we will attempt to adapt the model described in the previous section as a means of measuring the order existing in amorphous polymers. Let us again consider a collection of strings of cylindrical segments, with the segments jammed together. Instead of allowing the angle between two neighboring segments to be randomly distributed between 0 and 90°, we will assume a cutoff angle, 8*, so that the angle is assumed to be randomly distributed only between 0" and 8*. From eq. 2 , the average separation a* is then
[16* lo* sin 0 de]-'
( d cos 8/2
+
1 sin 0/2) sin 8 de (9) = 2 ( 3 sin20 * / 2 ) - l [ d ( l
- cos38*/2)
+
1 sin3e*/2]
(10)
Xow by virtue of the restriction imposed on the divergence between neighboring segments, adjacent segments along the string will also have these restrictions. Again assuming a lack of correlation in the average separation distances of the axis of a string from that of its various neighbors, we can consider the
=
21(1 - COS^ 8*/2)/(3 sin2 e*/2)
(11)
If the chain would normally pack in a helical shape, with each succeeding segment at some fixed angle 8 from collinearity, we can take 8* as the limits of deviation from 8. For small values of e*, though, b* will not deviate far from b* = 1. The ratio of the specific volumes is then vc/v, = d,/d, = dZ2/a*%*, which, for chains that tend to pack in a planar zigzag fashion, gives
1.00 0.855 0.855 0.937 0.854 0.911
a S. Matsuoka. J . A p p l . Phys., 32, 2334 (1961). R. L. Miller and L. E. Nielaen, J . Pdymer Sci., 55, 643 (1961). S. Onogi, et al., ibid., 58, 1 (1962). G. Farrow and I. M. Ward, Polymer, 1, 330 (1960). A. Prietzschk, Kolloid-Z., 156, 8 (1958). H. W. Starkweather and R. E. Moynihan, J. Polymer Sci., 22, 363 (1956).
a* =
averaging of b independently of the averaging a. If the chain would normally pack in a crystal in a planar zigzag fashion, from eq. 6
(;)3{
[1 - cos3 e*/2 sin3e*/2
1]2(
+
i
1 - cos3 -
;*))--I
(12)
where d, and d, are the densities of the amorphous and crystalline phases. Assuming l/d = 1, for the reasons stated previously, we can use eq. 12 to find values of 8* corresponding to the experimental values of dJd,. Although we strip the chain of pendant groups in estimating l / d , we will assume that the packing efficiency of the pendant groups is similar to that of the main chain. With the help of a GE-225 computer we obtain the curve in Figure 3 for the correspondence be-
0.85
0.90
0.95
d,Jdc
Figure 3. The computed order parameter e* for several values of Z/d us. the measured ratio of amorphous to crystalline densities.
Volume 69,;\'umber
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RICHARD E. ROBERTSON
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tween 0* and (d,/d,) for l/d = 1 and for several other values of E/d in the vicinity of 1. For the polymers list'ed in Table I, the values of e* obtained for Z/d = 1 average around 10" with the largest value, 15.0", occurring only for polyethylene. These small values for e* indicate a fairly high degree of order. If the polymer chain tends to pack in a helix, the rat,ioof the densitliesis 3
d, =
(5)
2/
\[
1 - cos3e*/2 sin3 0*/2
zT
+ j,
sin2 S*/Y}-'
(13)
At the same value of dJd0 this equation will give a smaller value for e* even than eq. 12. However, because of the smallness of e* found above, the value of 6* given by this equation will not be significantly different.
A Paracrystalline Nature of Amorphous Polymers From density considerations we are led to the conclusion that a fairly high degree of order exists in amorphous polymers. This is apparently contrary to the work of Flory, et al., especially the work of Hoeve and O'Brien.10 The mean square, end-to-end distance of n segments of length E, with a randomly distributed divergence between adjacent segments along the chain up to the angle e*, is given by -
R2 = d 2 ( 1
+
__ COS
e)/(l
- KO)
where
or
R3
=
nzy3
+
COS
e*)/(i
- COS e*)
(16)
As an application o,f this equation we can assume t?* = 15" and E = 2.51 A. and compute the value of (@)'" we might expect from these considerations alone for = 1.56 X lo-' M"' cni. polyethylene: (p)1/2
The Journal of l'hglsical Chemistrti
This is about 15 times larger than the value given by Kurata and Stockmayerll a t 100" in a 6-solvent. One way out of this inconsistency is to assume that there is a certain, albeit small, probability that the polymer chain can reverse itself. In particular, let us assume that a t each juncture the next segment can be within either the cone 0" to e* or the cone (180" - e*) to 180". This would not affect, the density, but it would reduce the calculated root mean square, end-to-end distance to that of a randoin walk (about one-fourth as large as the measured value). -4llowing the chain effectively to reverse its direction periodically is not an unreasonable assumption. Since the configurations in the crystalline state can generally be assumed to be represented in the amorphous state, if tight chain folds are seen in the crystalline state, it is reasonable to expect them in the amorphous state. Our picture of the configuration of a polymer chain in the amorphous state is thus something that might result if a molecule in a 6-solvent were compressed in two dimensions. Our picture of the amorphous polymer thus approaches the paracrystalline state described by Hosemann12 or the defect-filled polymer crystals of Zaukelies.l 3 The order postulated, though, is strictly one-dimensional. The polymer chains tend to be parallel to one another, but there is no particular correlation betn-een neighboring chains about their axes or along their length. A cross section perpendicular to the chain axes, then, would give the random appearance of a liquid of small molecules. Acknowledgments. The author wishes to thank his colleagues for many helpful discussions on this subject, especially J. 34. O'Reilly, F. P. Price, and A. R. Shultz of the General Electric Research Laboratory and F. E. Burke of Canadian General Electric Co. (11) M. Kurata and W. H. Stockmayer. Fortschr. Hochpolym. Forsch., 3, 196 (1963). (12) R. Hosemann, J . App2. Phys., 34, 25 (1963). (13) D. A. Zaukelies, ibid., 33, 2797 (1962).
