454 RYONG-JOON ROEAND H. E. BAIR
TABLEI I RELATIVE CON CENT RATIO^
OF
Macromolecules
Polym cond MF,
7--
Sam- mol Conv, ple no. I11
--Relative concentration-of triads V , % 1IF
TABLE 111 COPOLYMER EQUATIONS
TRIADS
FII
VERIFICATION OF
--
FIF
IF1 ~
A B
C D E
F Gu
6.5 13 14 15 19 21 19
10 10 13 32 20 32 13
11.6 3.8 2.6 27.0 2.0 8.5 7.8
18.0 16.2 13.3 12.4 8.3 12.1 10.3
18.0 16.0 12.9 12.4 8.6 12.0 9.9
17.2 24.0 29.0 17.9 36.3 27.6 30.9
35.2 40.0 42.3 30.3 44.8 39.7 41.1
n Sample G was polymerized at 0'. while others were polymerized at 25 '.
composition ratios determined from a n elemental analysis and from nmr data according to eq 21. Agreement is fairly good except for the sample A, in which isobutylene content estimated from nmr data is lower than that from elemental analysis. Probably this disagreement results from the terminal group effect, which cannot be negligible in the copolymer A. The copolymer A is greaselike, and its molecular weight is not so high because it was polymerized from a monomer mixture with an extremely high isobutylene concentration. In the proton spectrum of copolymer A the resonances appearing at 9.0 ppm and at higher fields, which are assigned to the terminal methyl groups, are stronger in intensity than in the spectra of the other copolymers. I n Table 111 are given rIFIvalues estimsted from nmr data using eq 17. In principle it is possible to calculate
Sample no.
VlIFll
1.05 0.60 0.46 0.67
0.31
YIFI
1711h/??lF
VFlFi
1.05 0.67 0.46 0.69 0.19 0 44 0.33
1.84 1.50 2.30 1.23 1.52 1.43
5.66 1.64 1.38 2.23 1.30 1.63 1.56
0.07 0.10 0.07 0.12 0.05 0.11 0.08
Calculated from nmr data according to eq 21. termined from elemental analysis.
* De-
values of r l l I and r I I I from these data in accordance with eq 16, and the comparison of these values with r I F Imay lead to the order of the Markoff process by which the copolymerization reaction is expressed. It was, however, difficult to evaluate rIIIand rrII from the ratio V I l l / V l , ~ . One of the reasons is in the difficulty of the precise determination of VIII, because trace amounts of isobutylene homopolymer may be included in the copolymer. Since a large amount of polyisobutylene is detected in the crude copolymer obtained at -78' as described previously, it may be suspected that a homopolymerization of isobutylene takes place simultaneously to some extent at 25" ria a cationic mechanism. Further work is now in progress. Acknowledgment. The authors wish to express their grateful thanks to Mr. Makoto Noshiro of Asahi Glass Co. Ltd. for his cooperation in the course of this study.
Thermodynamic Study of Fold Surfaces of Polyethylene Single Crystals Ryong-Joon Roe and H. E. Bair Bell Telephone Luborutories, Inc., Murrcij, Hill, New Jersejs Receired Noceniber 6 , 1969
ABSTRACT: Calorimetric data obtained on single crystals of polyethylene are analyzed to derive information on the nature of fold surfaces. From the magnitudes of surface enthalpy and entropy, it is estimated that a fold, on the average, would consist of 20 or more segments and thus a substantial layer of disorder exists at the surface
of single crystals. The surface free energy value was determined by two methods: from the customary plot of melting temperature cs. reciprocal lamellar thickness and from the free energy of formation of crystals calculated from the observed values of specific heat. The results of these two methods agree well. The surface enthalpy value was determined from the plot of heat of fusion us. reciprocal lamellar thickness. The assumptions underlying these procedures have been analyzed and the conditions for proper evaluation of thermodynamic quantities are stated and followed. In particular, some of the crystals were cross-linked lightly in order to suppress lamellar thickening during measurements.
I. Introduction The nature of the surfaces of folded-chain single crystals of polyethylene and other polymers has been the subject of intensive study in recent years. Despite many efforts, there still remain important questions. One of the main points of argument is concerned with the size of the folds, i.e., the average number of seg-
ments that are participating in a fold. In this work we present new evidence on polyethylene single crystals, indicating that a fold would consist of 20 or more CHr units on the average. In other words, there appears to exist a substantial layer of disorder on the crystal surfaces amounting to an average depth of at least ten CH, units. This is obviously in conflict with the picture of
POLYETHYLENE SINGLE CRYSTALS455
V O ~3,. NO. 4, JU!),-August 1970
arrays of tight, regular folds which have been favored by many during the past decade. The evidence we present is obtained from calorimetric studies o n polyethylene single crystals prepared under a variety of conditions. We examined melting points and apparent heats of fusion as a function of the lamellar thickness. I n addition, specific heats have been measured through a range of temperatures o n one of the samples. F r o m analysis of these calorimetric data we determine the values of surface free energy and its enthalpy and entropy components. The magnitudes of surface enthalpy and entropy are surprisingly large. By comparing these with the enthalpy and entropy of fusion we are led to a n estimate of the lower limit to the average number of segments in a fold. 11. Thermodynamic Equations
The free-energy difference per segment between the crystal and the supercooled liquid is
AF(T,.r)
=
F(T,O - F,(T)
(1)
where subscript a refers to amorphous liquid, and < is the average number of segments in a section of the polymer chain running between the two lamellar surfaces. If we designate the quantity pertaining to a large extended chain crystal by a superscript zero, as in P ( T )= F(T, ), we then have
+ AFo(T)
AF(T,s’) = [F(T,{)- Fo(T)]
(2)
Here AFo(T), the free energy of formation of long extended chain crystals from melt, can be estimated approximately by expressions such as (3)
where AH, = -AHO(T,,O) is the heat of fusion per segment. The first term in (2), F(T,{) - Fo(T),represents the free-energy excess arising from the fact that the lamellar thickness is finite rather than infinite. T h e excess arises mostly from the perturbation at the fold surface, but may partly be contributed from internal defects or lattice distortions. As long as these latter effects originate from the propagation of strain from the fold surface, their contribution to the excess free energy would still be proportional to the surface area. We therefore write F(T,() - P ( T ) = 2ue(T)/i-
(4)
where u,(T) is the surface free energy per fold in the broader sense, including the small contributions from the surface-induced lattice defects, etc. (Note: A “fold” is here defined as the sequence of CH2units with both ends terminating at the crystalline lattice surface, and would thus occupy the area of two lattice sites.) Experimentally, the free energy of formation of crystals from supercooled liquid, AF(T,{), can be determined from the observed values of specific heat l (per segment) C,(T,{) by
AWL)
=
-LITn
(1) M. J. Richardson, Trans. Faraday Soc., 61, 1876 (1965).
*v3
We show later in the text that the values of ue obtained by these two methods agree with each other satisfactorily. Although the enthalpy qe and entropy component se of ue can, in principle, be calculated from the temperature dependence of ue, the experimental accuracy normally attainable does not warrant such a procedure. Instead, we have to look for a way of determining the surface enthalpy qe directly. For the average enthalpy difference per segment, A H(T,{), between the crystal and the supercooled liquid, one can write equations paralleling those given for AF(T,{). Thus AH(T,
