1530
L. H. TUNG AND S. BUCKSER
for K mica. Here then, there should be a possibility of evaluating polarizability effects on adsorption heats as has been done by Barrer and Riley. l 3 Finally, some comments on the B E T treatment of the adsorption datal can be made in the light of these considerations. Similar B E T results are obtained for all the measurements, in spite of the differences in the types of interactions a t low and moderate coverage. This apparently is due t o an energetic “smoothing” of the surface before the higher degrees of occupation a t which B E T plots are made (near the monolayer; cf. the previous paper1). This may be compared with the results of Steele and Astonl6 using pre-adsorbed gases. If the “resolving power” of the gas atoms is not sufficient t o distinguish the differences in the surface sites, the variations in heats and entropies as a (16) W. A. Steele and J. G. Aston, J . Chem. Phys., 83, 1547 (1955).
Vol. 62
function of coverage are smoothed out, and the isotherms may fit the BET theory over a wider range. This is of course an over-simplification in general, since we’have not considered localization and other simplifications inherent in the B E T theory. Many unresolved problems obviously remain in this area and much of the present discussion is of necessity highly hypothetical. The present results, however, suggest further interesting applications of mica to the experimental attack on the complexities of physical adsorption. For example, polar molecules should of course exhibit very different adsorptive properties with mica surfaces. How ion exchange will affect such systems is a most interesting question. Acknowledgment.-I am greatly indebted to Miss C. P. Rutkowski for assistance with the calculations.
THE EFFECT OF MOLECULAR WEIGHT O N THE CRYSTALLINITY OF POLYETHYLENE1 BY L. H. TUNG AND S. BUCKSER High Pressure Laboratory, Polycheinicals Research Department, The Dow Chemical Company, Midland, Michigan Received June 6 , 1968
The effect of molecular weight on the crystallinity of polyethylene was studied dilatometrically using fractions of various molecular weights. The crystalline contents of the fractions a t various temperatures were calculated from the measured specific volumes. The low crystalline contents of the higher molecular weight materials were found to persist up to temperatures in the rapid melting region. This observation ruled out the possibility that the phenomenon was a rate effect. A theory was proposed to explain this dependence of crystallinity on the molecular weight of polyethylene.
Introduction The effect of molecular weight on the crystallinity of polyethylene was first reported by Richards.2 He observed that samples of relatively low molecular weight polyethylene crystallized rather more easily than did samples of higher molecular weights. Later a similar effect of molecular weight on low pressure polyethylenes was reported by Burch and his c o - ~ o r k e r s . ~Recently one of the present aut h o r ~has ~ observed a more pronounced effect of molecular weight on the crystallinity of fractionated low pressure polyethylene. This decrease of density in high molecular weight fractions was mainly an effect of molecular weight since infrared spectroscopy revealed no structural factors which could account for this decrease. The effect of molecular weight on the crystalline melting point of polymers has been successfully treated by Flory5 and later studied by many other workers. The same effect on the degree of crystallinity has, however, received relatively little attention. Since many physical and mechanical properties of a crystalline polymer depend on the degree of crystallinity, a closer understanding of ( 1 ) Presented at the 133rd National Meeting of the American Chemical Society, San Francisco, Calif., April, 1958. (2) R. B. Richards, J . A p p l . Chem., I, 370 (1951). (3) G. N. B. Burch, G. B. Field, F. H. McTigue and H. M. Spurlin, S.P.E. Journal, 13, 34 (1957). (4) L. H. Tung, “Some Tensile Properties of Fractionated Low Pressure Polyethylenes,” Presented at SPE meeting, Detroit, 1958. (5) P. J. Flory, J . Chem. Phya., 17, 223 (1949).
this effect is desirable. I n the present work we have studied dilatometrically the effect of molecular weight on the crystallinity of polyethylene as a function of temperature. Experimental
.
The materials used in the present work were fractions of low pressure polyethylene. These fractions were relatively free of chain branches and their molecular weights were more precisely defined than the unfractionated polymer. The technique of fractionation has been described elsewhere,O and the molecular weights of these fractions were obtained from intrinsic viscosity measurements in tetralin a t 130’ based on an intrinsic-molecular weight relationship determined by osmometry in this L a b ~ r a t o r y . ~The dilatometer with a detachable 1mm. precision bore capillary was about 4 ml. in volume, and approximately one-gram polymer samples were used. Samples were centrifuged a t 160’ under vacuum in a gravitation field of 5,000 g for about 30 minutes to remove trapned air. The dilatometer with the sample in place was filled with mercury as the confining liquid by vacuum distillation. The entire dilatometer then was immersed in a bath controlled within a temperature During the measurement the temvariation of =kO.Ol’. perature was raised a t five degree intervals. The mercury level in the capillary was observed through a precision cathetometer and a change corresponding to 0.0001 ml. in volume could be detected accurately. The equilibrium volume a t any temperature was taken when no change of volume was observed for a t least 30 minutes.
Results and Discussion I Figure 1 shows the results plotted as specific volume versus temperature for three fractions from (6) L. H. Tung, J . P o l y . Sci., 20, 495 (1956). (7) L. H. Tung, ibid., 24, 333 (1967).
J
Dec., 1968
EFFECT OF MOLECULAR WEIGHTON CRYSTALLINITY OF POLYETHYLENE 1531
polyethylene A. The unfractionated polymer is I3200 also shown for comparison. Except for the lowest I29W molecular weight fraction, the difference of specific volume in these polymers persisted until the tem- I 2El 30 perature was very close to the melting point of the I 2000 samples. Figure 2 shows the results of four fractions from a different sample, polyethylene B. I n I 1700 all cases the specific volume changed only slightly I1400 until the polymers were in the fast melting tempera- I l l 0 0 ture region. Above the melting point the specific volumes of the fractions reversed their order with I oeoo I0500 respect to molecular weight. The melting points of these polymers agreed I020030 40 Jo with the theory of Flory.6 In the case of polyethylene A, the theory predicts that the higher molecular Fig. l.-Speci& weight fractions should melt a t about the same temperature and the 2,000 molecular weight fractions should melt seven degrees lower. The observed values were 130, 129 and 123". In the case of polyethylene B, the theory predicts a decrease of 15" for the melting point of the low molecular weight fraction. The observed decrease was about lo", 125 as compared to 135" for the other higher molecular weight fractions. Considering that' not too many points were taken just below the melting points of these fractions such an agreement was excellent. To convert the experimental results to per cent. crystallinity we used the method of Hunter and Oakes.8 The specific volume of the amorphous phase in the polymer was obtained by the extrapolation of the liquid state specific volume of the sam- Fig. 2.-Specific ple above its melting point. The specific volume of the crystalline phase was based on a reIationship used by N i e l ~ e nnamely ,~ specific volume = 1.025
€4
70
80
90
100
110
120
130
140
1x1
T,*C.
volume us. temperature for polyethylene A.
T,OC.
volume us. temperature for polyethylene
E.
+ 0.00037'
Figures 3 and 4 show the results plotted as per cent. crystallinity versus temperature. From these figures it can be seen that before the temperature reached the rapid melting region the differences in crystallinity between fractions of various molecular weights changed very little with respect to temperature. This observation ruled out the possibility that the low crystallinity of high molecular weight samples was a rate effect. Our time-scale of observation was of the same order at all temperatures, whereas the viscosity of polyethylene should increase rapidly with temperature. As a consequence, a gradual decrease in the difference in crystallinity would be observed if the full development of crystallinity was prevented by viscous forces. In order to explain this phenomenon we shall attempt t o put the hypothesis advanced by Burch and his eo-workers on a more quantitative basis. These authors assumed that the velocity of crystallization was related to the number of chain ends in the sample. If a chain had an end in the surrounding amorphous region, this chain end could readily diffuse t o the surface of the crystalline region, whereas a chain section with both ends attached to crystalline regions would be forced to coil up in a random fashion in order t o fit the available space and thus lead to a higher amorphous content of the ( 8 ) E. Hunter and W. G. Oakes, Trans. Faraday Soc., 41, 49 (1945). (9) L. E. Nielsen, J . A p p l . Phys., Z6, 1209 (1945).
0 T,'C.
cent. crystallinity us. temperature for poly,ethylene A.
Fig. 3.-Per
30
40
Fig. 4.-Per
50
60
70
80
90 T,%.
100
110
I20
I30
140
15'
cent. crystallinity us. temperature for polyethylene B.
*
L. H. TUNG AND S. BUCI~SER
1532
Vol. 62
length of y units is X- N =- 1 Y JfY
and Mark's model of chain section between crystallites.
Fig. B.-Alfrey
Of these, N sections contain the ends of the'chains resulting in 2 N of the free chain sections (ie., the sections with only one end attached t o a crystallite) of average length y/2. The total number of the sections with both ends attached to crystallites is
The fractional crystallinity Q of the polymer is the number of crystallized repeating units divided by the total number of repeating units
Here n
qe
is the equilibrium q demanded by equation
2. 'Ob
'
'
'
'
'
50
'
' YW
' I
'
100 I
'
'
'
'
'
I50
IV',
Fig. 6.-Crystallinity vs. mol. wt. of low pressure polyethylene fractions a t 23'.
sample having a smaller number of chain ends. We shall, however, treat each chain section from an equilibrium point of view. The model of Alfrey and Mark,l0 shown in Fig. 5 , will be used to describe the chain section with both ends attached to crystalline regions. We now let y = no. of repeating units in the chain section considered q = no. of repeating units in the crystalline phase D = total distance between the ends of the chain section in terms of the no. of lengths of a crystallized
repeating unit along the chain
The free energy of the entire chain section in question is where H O = enthalpy of the amorphous polymer AH = heat of crystallization per repeating unit 8;:: = entropy of an amorphous chain, having y - p units long and D - p unit distance between the ends
The entropy of the crystallized units has been neglected. The equilibrium value of q for such a chain section is then obtained by setting A F / A q = 0 as in equation 2 AH
-T
-
[&l(f::]) S5:qq] =
0
(2)
We shall assume that the chain section with a chain end free from attachment to a second crystallite is completely crystallized. This assumption is not unrealistic since the change of entropy of this type of chain section resulting from the crystallization of each repeating unit is constant. Thus, if the prevailing condition favors one unit to crystallize, the entire chain section is capable of entering into crystallization. Let then x = no. of repeating units in the polyethylene chain M = molecular weight of such a unit N = -2- = no. of chaina per gram of the polymer
XM The total number of chain sections with an average
I n order to compare equation 3 with observation, we let y = 400 and qe = 328. The value of y was selected because previous data4 at room' temperature showed that the density of the polymer remained constant below a molecular weight of about 5,000 or 6,000 and in equation 3, Q = 1 when x = y. The value of Le was selected rather arbitrarily, but it will be shown later that this value of qe can be obtained from equation 2 with a reasonable value of D. Figure 6 shows the comparison of equation 3 with observations at room temperature reported previously. The degree of crystallinity has been converted from density readings by using the density-crystallinity relationship reported by Sperati and his co-workers.ll The theory, though not in perfect agreement with the experimental curve, can account for most of the features of the crystallinity versus molecular weight curve. The initial slope of the curve calculated from equation 3 is steeper than that observed. This discrepancy could have resulted from the assumption that the free chain sections crystallized completely. The assumption of the constancy of y and D in all samples could also be a cause for the discrepancy between the two curves. One can also make a better fit of the two curves by adjusting the constants of the equations. Without a knowledge of what factors are influencing these constants, such steps do not seem to be warranted. It is of interest, however, to compare the temperature coefficient of crystallinity as calculated from equation 3 with the experimental data. I n order to do this, an expression for the entropy term in equation 2 has to be found. The configuration entropy of long chains has been calculated satisfactorily by Huggins12 and Flory13 from a liquid lattice model. For the convenience of calculation we shall assume a three dimensional cubic lattice. On this lattice there are six different types of steps to construct a chain having ends a known distance apart. Let n,, n-z, n,, n-v, n, and n-, denote the (10) T. Alfrey and H. Mark, THISJOURNAL, 46, 112 (1942). (11) C. A. Sperati, W. A. Franta and H. W. Starkweather, Jr., J . A m . Cham. Soc., 76,6127(1953). (12) M.L. Huggins, J . Chem. Phys., 8 , 181 (1940). (13) P. J. Flory, ibid., 10, 51 (1942).
.'
EFFECT OF MOLECULAR WEIGHTON CRYSTALLINITY OF POLYETHYLENE 1533
Dec., 1958
number of each type of steps used to connect the chain. If the net displacement of the chain section is in the direction of +x, then the restrictions for selecting the different types of steps are Zlti = nLz n-, nu = n. =
-
.
y =
-q D -p
(4)
n-y n-.
The total number of combinations satisfying the above restrictions is Y - D
c (i + 2
1)
i=O
This multiplied by the permutation for each combination gives the total permutation
Fig. 7.-ln
[&+l/(y
- q)&] as a function of q when y = 400 and D = 340.
Y-D
F _
i
2
L?b:Q0 =
(i i=O
+ 1) k = O
Factoring out the (y
- q) !, we have - p)! $%;:
(5)
i2bIQu= (y
where YZD 0 T?C,
i
1
k = O ( k ! ) 2 [(i
- k)!]Z
(6)
I n the above expressions, certain numbers of configurations a’ contain undesirable steps such as a ‘ I -x” step immediately following a “+x” step. To exclude such configurations an expression of a’ = dn -I- du 9. - dzu - dua d21. (7) has to be subtracted from a &:\. Here c$$ represents the number of configurations that contain a t least one pair of “2, -2” or “-x, 5’’steps; dzz, represents the number of configurations that contain a t least one pair of (‘2, -5’’steps and one pair of “y, -y” steps; etc. Equation 7 is quite cumbersome to evaluate numerically and is almost impossible if y becomes large. Since we are dealing with only .the ratio of &:\’s, we shall neglect this extra term and put
+
$2.
+
S = k In i25Yq
Upon substitution equation 2 now becomes
’
Figure 7 shows the calculated values of In [(#* + I ) / (y - q) as a function of q when y = 400 and D = 340. If we use a AH of 785 cal. per mole divided by Avogadro’s number, the qe a t 23” calculated from equation 8 using y = 400 and D = 340 is 328 which is the number we have used for calculating the curve in Fig. 6. The temperature coefficient of crystallinity calculated from equations 8 and 3 is very small. The decrease of crystallinity as a function of temperature is even smaller than the experi-
Fig. 8.-Comparison of variation of crystallinity with temperature from observation with that calculated from equations 8 and 3.
mental curves as shown in Fig. 8. The experimental curves are those taken from Fig. 3, but with an expanded ordinate of crystallinity. One feature of the experimental data predicted by the theory is that the higher molecular weight fractions showed slightly larger negative temperature coefficients of crystallinity. The discrepancy between the absolute values of experiments and theory in Fig. 8 is not surprising. Besides the uncertainties mentioned earlier, the extrapolated specific volume of the amorphous phase used to convert the experimental results to crystallinity was somewhat arbitrary. The assumption that the center-to-center distance D between crystallites is constant throughout the temperature span is also over-simplified. With these considerations in mind, we feel that the present postulations give an adequate explanation of the effect of molecular weight on crystallinity. The large deviation from theory of the lowest molecular weight fractions may be caused by: (1)negglecting the difference between a methyl end group and a methylene group in the body of the chain, (2) the relative ease of change of the center to center distance D between crystallites of the low molecular weight fractions. A few interesting observations may be made from the present treatment. (1) If the present assumed values of y and D are not far off, the average length of low pressure polyethylene crystallites a t room temperature should be about 400 A. This is the dimension which is difficult to measure by X-ray diffraction and we know of no direct experimental
1534
J. J. BIKERMAN
value. However, it has been reported by Smith14 that the width of such crystallites are from 360 to 390 A. The 400 A. length, therefore, does not seem to be unreasonable. (2) Again, if the present values of y and B are not far from reality the chains in the amorphous phase of polyethylene are highly oriented. Their configurations are much more extended than the random coil configuration. Thus the specific volume of the amorphous phase niay be quite different from the value extrapolated from the liquid state. This fact also supports the view on the growth of spherulites that the formation of a nucleus is induced by the growth of the first crystallite. (3) Equation 3 indicates that for polydispersed polymers the degree of crystallinity depends on the number average molecular weight. The number average molecular weights of commercial polyethylenes usually vary within a very narrow range.16 As a result, the crystallinity is not greatly affected by the molecular weight based on viscosity measurements. Acknowledgment.-The authors are indebted t o Dr. H. W. McCormick of the Physical Research Laboratory for his assistance in the centrifugation of polymer samples.
Vol. 62
counted in the term dXu,etc. I n order to count each type of undesirable configurations only once we arrived a t the present form of equation 7. To evaluate c $ ~ , one has to count the configurations containing groupings of different numbers of x, -x steps, such as “x, -x”, “x, -x, x”, etc., separately. Let qhXdenotes the number of configurations containing the largest grouping of x and -x steps. Then
,
The first part of 41%is the permutation within the x step grouping and the second part is the permutation outside this grouping. For the number of configurations & containing the next largest grouping of x and -x steps, the expression is 9% =
+
n, n-. - l)! [ ( ( n , - l)!n-.!
+
(n,
+ n-. - l)!
n,! (n-z
-
l)!l
I
The first bracket again is the permutation within the x step grouping. The first term in the second bracket is the permutation outside the x grouping and the second term which represents the configAppendix urations with another x step adjacent to the x The Evaluation of Q’.-In the derivation of grouping is subtracted from the first term. Thus equation 7, c $ ~4,~, , and c $ have ~ ~ ~ already been c $ is ~ the sum of all these @qx, dzX. . . . terms correcounted once in the term qjX; similarly 4xyzhas been sponding to all possible numbers of x step groupings. All the other 4 terms have to be evaluated similarly. (14) D. C. Smith, “Molecular Structure of Marlex Polymers,” It is quite obvious then when y becomes large 8’ is presented a t ACS meeting, Dallas, 1956. impossible to evaluate numerically. (15) H. Smith, J . Poly. Sei., 21, 563 (1956).
USE OF HYSTERESIS OF WETTING FOR MEASURING SURFACE TENSION1 BY J. J. BIKERMAN Department of Civil and Sanitary Engineering, Massachusetts Institute of Technology, Cambridge 39, Mass. Recewed June 16, 1968
The hysteresis of wetting observed when a artially wetted vertical slide is moved in and out of a liquid can be used to calculate the surface tension of the liquid. !&e vertical displacement of the liquid surface h2 - h1 = (2y/gp)’/e(M2:/a M l ’ / g ) , if A f 2 = 1-( 1 - F22/L2y2)‘/zand Ilfl = 1 - (I - F12/L2y2)’/2; F , and F2 are the capillary forces on the slide a t the two positions of the surface, L is the length of the 3-phase boundary, -1 is surface tension, g is acceleration due to gravity, and p the density of the liquid. The method was tested successfully on water and mercury and is suitable for the latter liquid.
I. Introduction I n the course of a study of the hysteresis of wetting it was noticed that the surface tension y of the liquid can be calculated from the hysteresis data. As a new method of measuring y , the procedure seems t o be advantageous for mercury only. Solids which give a zero contact angle with mercury are soluble in the latter and thus are apt to contaminate it (alt,hough the contamination may be insignificant in favorable instances2); thus the easier methods of measuring y cannot be used with safety, and those that can are not more convenient than the present one. At any rate it has a theoretical significance and thus deserves a description. (1) Presented a t the 134th National Meeting, American Chemical Society, Chicago, September, 1958. (2) E. A. Owen and A . P. Dufton, I’roc. Pliys. Soc. ( T m d o h ) , 38, 204 (1920).
11. Theory A vertical plate suspended in a liquid surface (see the continuous curve in Fig. 1) is subject to capillary force3 F, =
L ~ C O S ~ ~
~
(1)
L is the perimeter of the plate (that is the length of the 3-phase boundary line) and is the contact angle. If now the level of the liquid be raised t o the position indicated by the dashes while, because of the hysteresis of wetting, the 3-phase line retains its position, the contact angle increases to e2 and the capillary force diminishes t o F~ = L~
COS
e2
(2)
From F1 and Fz, y can be calculated without measuring el and e2. (3) E.Q.,J. J. Bikerman, “Suiface Chemistry.” 2nd edition, Aca. deniic Press, Inc., New York, N. Y . , 1958, p. 9.
,
