THEENTHALPY OF FUSION OF LINEARPOLYETHYLENE
309
The Enthalpy of Fusion of Linear Polyethylene’ by L. Mandelkern, A. L. Allou, Jr., and M. Gopalan DepaTtment of Chemistry and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida (Received August 10, 1867)
The enthalpy of fusion of molecular weight fractions of linear polyethylene, crystallized under a variety of controlled conditions, has been measured. For samples crystallized in the bulk, the enthalpy of fusion is very dependent on the molecular weight. For a given crystallization procedure, there is a monotonic decrease in this quantity with increasing molecular weight, so that a wide range of values are observed. A comparison of the degree of crystallinity calculated from enthalpy of fusion and density measurements indicates a very good agreement for samples which are never cooled below the initial crystallization temperature. However, for samples which are subsequently cooled to room temperature, the enthalpy measurements give lower values for the degree of crystallinity than does the density. This result is attributed to the small-size crystallites which form on cooling and the associated contribution from the interfacial enthalpy. I n contrast to the above results, the enthalpy of fusion of crystals formed from dilute solution is independent of molecular weight and depends only on the crystallite size. The interfacial enthalpy and entropy of such crystals, calculated from the experimental data, are discussed in terms of different morphological models that have been proposed. the properties of molecular weight fractions of high polymers, where the supply of the sample is invariably limited, can be studied. I n the present paper, we reThe melting of a collection of crystalline long-chain port the results and analysis of enthalpy of fusion molecules has been demonstrated to be a first-order measurement on fractions of linear polyethylene enphase transition.2 However, for a variety of morcompassing a very wide molecular weight range. The phological reasons, broad fusion curves and depressed were conducted under a variety of conmelting temperatures are characteristically ~ b s e r v e d . ~ ~crystallizations ~ trolled conditions from both the melt of the pure polyI n addition, there is usually a significant lowering in the mer and also from very dilute solutions. By these values of the pertinent thermodynamic quantities, such methods, it was possible to obtain samples which posas the density3 and enthalpy of f ~ s i o n . ~ JThe more sessed a range in crystallite sizes and a diversity of detailed reasons for these deviations from idealized interfacial structures and properties. A preliminary behavior can be attributed to the finite size of the crysreport of the results for bulk samples, crystallized at a tals in the chain directioq6 the molecular nature of the high temperature and subsequently cooled to room interfacial region,O and the conformational differences temperature, has been previously given.4 and associated thermodynamic properties of the chain units in the interzonal regions which connect crystalExperimental Section l i t e ~ . ~ , ’Thus it has been shown that thermodyWith one exception, molecular weight fractions of namiq3J mechanical,6and spectral properties8 are very linear polyethylene were used in this study. The fracsensitive t o morphological detail, so that a large range (1) This work was supported in part by the Army Research Office in properties can be observed, depending on the details (Durham) and the Division of Biology and Medicine, Atomic Energy of the crystallization process. Controlled changes in Commission. the crystallite size and interfacial region can be achieved (2) L. Mandelkern, “Crystallization of Polymers,” McGraw-Hill Book Co., Inc., New York, N . Y., 1964. by utilizing molecular weight fractions and varying the (3) J. G. Fatou and L. Mandelkern, J . Phys. Chem., 69, 417 (1965). crystallization conditions, particularly the crystalliza(4) L. Mandelkern, J. G. Fatou, R. Denison, and J. Justin, J . Polytion temperature. mer flci., B3,803 (1965). A detailed study of thermodynamic properties should (5) L. Mandelkern, ibid., C15, 129 (1966). enable the development of a better understanding and (6) L. Mandelkern, J. M. Price, M. Gopalan, and J. G. Fatou, ibid., A2-4, 385 (1966). insight into the molecular nature of the interfacial and (7) R. Chiang and P. J. Flory, J . Am. Chem. floc., 83, 2857 (1961). interzonal regions and how they depend on the mode of (8) T. Okada and L. Mandelkern, J . Polymer Sci., A2-5, 239 (1967). crystallization. Modern developments in instrumenta(9) E. S. Watson, M. J. Oneill, J. Justin, and N. Brenner, Anal. tion have allowed enthalpies of fusion to be measured Chem., 36, 1233 (1964). utilizing very small amounts of material.4~9~10Thus (10) A. P. Gray and K. Casey, J . Polymer Sci., B2, 381 (1964).
Introduction
Volume 78, Number 1 January 1968
310 tionation procedure and characterization of the fractions have been previously described in detaiL3 The viscosity average molecular weights ranged from 1.20 X lo4 to 1.47 X 106. To investigate, the influence of very high molecular weight on the properties of an unfractionated sample of M q = 7 X 106 was also studied. The specimens for study were crystallized either from the pure melt or from very dilute solutions. Two distinctly different kinds of procedures were utilized for the bulk crystallization. I n all cases, however, the samples were first melted at 155-160" for 20 min and then the crystallization procedure initiated. I n one kind of experiment, the samples were crystallized at a predetermined temperature for a given period of time and then cooled to room temperature. In the other category, the samples were crystallized in the calorimeter and, simultaneously, in a dilatometer at a given temperature for a predetermined time and never cooled below the crystallization temperature. The time for the crystallization was set so that the accelerated portion of the transformation was completed and the protracted crystallization was taking place.2 The resulting data, discussed below, indicate that a definite distinction must be made between these crystallization procedures. I n one set of experiments, involving bulk crystallization, the specimens were wrapped in aluminum foil and immediately transferred from the high-temperature thermostat to water at 23". For the crystallization at elevated temperature, which involved long times, the samples were crystallized isothermally in sealed evacuated tubes. One series was maintained at 130" for 30-40 days and cooled to room temperature over a 24-hr period. Other specimens were crystallized a t 120" for 5-7 days and then cooled t o room temperature. The crystallization at intermediate temperatures was carried out the same way. The times of crystallization were adjusted based on crystallization kinetic studies,2111so that no additional significant crystallization was expected at the crystallization temperatures. The samples, thus prepared, particularly those crystallized at 130" for long times, were monitored for possible oxidation by the examination of the infrared absorption spectra in the region 1709-1729 cm-l. Except in a few instances, where the specimens were discarded, the samples displayed no evidence of oxidation. Samples which were oxidized, either inadvertently or deliberately, are characterized by abnormally high densities and low enthalpies of fusion and thus become easily recognizable. Crystals were also prepared from dilute (0.05-0.1%) solutions of p-xylene. The crystallization was carried out, isothermally, in the temperature range 60-90" for periods of time varying from 3 days to 3 weeks, depending on the crystallization kinetics. The precipitate was separated from the supernatant either by cooling to room temperature and then filtering, or by filtering The Journal of Physical Chemistry
L. MANDELKERN, A. L. ALLOU,JR.,AND M. GOPALAN at the crystallization temperature and then cooling. The same results were obtained by either procedure. The samples were dried in O ~ C U Oat 45". For all the samples cooled to room temperature, the densities were determined at 25" in a toluene-dioxane gradient column, which was calibrated with glass floats. The densities could be measured to *0.0005 g/cc. As has been previously r e p ~ r t e d the , ~ densities of samples thus determined agree to better than 3 parts per 1000 with corresponding samples in a dilatometer which had undergone similar crystallization procedures. The densities of the samples, which were never cooled below the crystallization temperature, were determined from the dilatometer readings. The crystallite thickness, at room temperature, of the dilute-solution prepared specimens was determined from low-angle X-ray diffraction measurements using a Rigaku-Kenki camera. The mrell-defined maxima were converted to linear dimensions by means of Bragg's law. The experimental uncertainty is estimated to be rt4"A. The heats of fusion were measured with a PerkinElmer DSC-1 or DSC-1B differential scanning calorimeter.9 Typical examples of the calorimeter's performance and the resulting data, for both bulk and solution crystallized samples, have already been given in detai1.4J2 The operation of the instrument was adjusted so that the area under the fusion curve was always in the range of 10 in.2. To properly delineate the area to be measured, cognizance must be taken of the onset of fusion at a temperature below which the major portion of the melting occurs.12 At a heating rate of 5"/min, the measured enthalpy of fusion per gram was independent of the mass used in the range 3-17 mg. The uncertainty of the results under these conditions, for repetitive experiments utilizing the same sample, was well within the range of A 1 cal/g. An uncertainty in the base-line delineation exists when smaller size samples are used. Under these latter conditions, there is a larger experimental error. Sample sizes in the range of 3 mg were used in this work but in selected controlled experiments increasing sample weights were also studied. Within the range cited, identical values for the enthalpies of fusion were obtained with heating rates of either 5 or lO"/min. The value of the heat of fusion of the polymer sample was determined by comparing the area under its fusion curve with that obtained with a known weight of indium heated under the same conditions. The dilatometric procedures employed here have already been described in detail.3vl3J4 The density and (11) J. G. Fatou and L. Mandelkern, unpublished results. (12) L. Mandelkern and A. L. Allou, Jr., J. PoZymer Sci., B-4, 447 (1966). (13) P.J. Flory, L. Mandelkern, and H. K. Hall, J . Am. Chem. Soc., 73, 2532 (1951). (14) L. Mandelkern and P. J. Flory, ibid., 73, 3206 (1951).
THEENTHALPY OF FUSION OF LINEARPOLYETHYLENE specific volume measurements were converted to a degree of crystallinity by means of the relation and constants given by Chiang and Flory.7
Results and Discussion Bulk Crystallized Samples. The enthalpies of fusion and densities ,that were obtained for the samples crystallized isothermally from the pure melt for extended time periods and then cooled to room temperature are summarized in Figures 1 and 2, respectively. The quantities of interest are plotted against the logarithm of z, the number of CH2units per chain, for the different crystallization temperatures. The more extensive data that are given here for samples crystallized at 130" are very similar to the preliminary results previously reported4 and the trends discerned earlier are further substantiated. At this latter crystallization temperature, for z equal to or less than approximately 3 X lo3, the plotted densities and enthalpies of fusion are comparable to the values expected for a large macroscopic crystal. From the dimensions of the unit cell,16the density should be 1.00 at room temperature. An enthalpy of fusion 69 1 cal/g is also expected from independent analyses and experiment.16*17 Densities as high as 0.99 and enthalpies of fusion of 66 cal/g are directly observed in this molecular weight range. As the molecular weight is in-
*
T-
20 102
i
I
10'
io4
I lo5
I 106
x
Figure 1. Plot of enthalpy of fusion, AH* as a function of chain length z for samples initially crystallized a t indicated temperatures and cooled to room temperature; crystallization temperatures: 0,130'; A, 126"; 0, 124'; 0, 120'; and . , 23'. 1.00
-
I
I
I
h
8g 0%as-
a%09.
*: I
I
I
,
311
creased, however, these quantities decrease monotonically. For the highest molecular weight sample studied here, the density has been reduced to 0.94 and the enthalpy of fusion to 34 cal/g. Thus, despite the very stringent conditions of high temperature and long crystallization times, a Ride range in values, which depend on molecular weight, is observed for these thermodynamic quantities. At the lower temperatures of crystallization, which are characterized by more rapid rates of crystallization, there is also a systematic change in the density and enthalpy of fusion with molecular weight. For x 5 lo5, these quantities decrease with the lowering of the crystallization temperature at a given molecular weight. There is, therefore, a corresponding decrease with increasing chain length, at a fixed crystallization temperature. However, when 2 exceeds 2 X lo4, the change with molecular weight becomes very small so that the curve representing the data intersects and joins the one delineating the results for crystallization at 130". For the samples which were quickly transferred from the melt to room temperature, this intersection has not as yet occurred over the molecular weight range presently available for study. The trends in the data indicate that this intersection will occur for a molecular weight of the order of 10'. The density and enthalpy of fusion expected at this point are 0.925 g/cc and 25 callg, respectively. These latter values are actually observed for quenched specimens of the highest molecular weight sample and represent the lowest values that have been reported for linear polyethylene. These low values are obtained from a crystallization procedure wherein the samples spend a minimal time at the elevated temperature. These results must, therefore, be a consequence of the crystallization process and the morphology that results and cannot be attributed to any oxidation during crystallization. Hendus and Illersl* have recently reported enthalpy of fusion measurements for molecular weight fractions which were slowly cooled from the melt to room temperature. Although this is a relatively undefined crystallization procedure, the slow cooling process does insure that some crystallization occurs at elevated temperatures. Their results are qualitatively similar to those reported here. For M q 5 4 X lo4,the enthalpies of fusion are slightly greater than 60 cal/g, but not as high as those in Figure 1 for the 130" crystallization. As the molecular weight increases beyond this value, the enthalpy of fusion monotonically decreases. For M = 1 X lo6,the enthalpy of fusion has decreased to 35 cal/g. For the lower molecular weights, the enthalpy of fusion results correspond to those obtained (16) C . W. Bunn, Trans. Faraday SOC., 35,482 (1939). (16) F. A. Quinn, Jr., and L. Mandelkern, J . Am. Chem. Soc., 80, 3178 (1958); 81, 6533 (1958). (17) M. G. Broadhurst, J . Res. Natl. Bur. Std., 66A, 241 (1963). (18) H. Hendus and K. H. Illers, Kunstofe, 57, 193 (1967).
Volume 71,Number 1 January 1068
L. MANDELKERN, A. L. ALLOU,JB.,AND M. GOPALAN
312
Table I : Thermodynamic Parameters for Samples Crystallized in Calorimeter
__----------Crystallized in calorimeter--------------Mol wt
TC. OC
x
125
7
106
x
106
3.7
x
106
2
x
106
5.6
x
104
AH*,
cal/g
140 317 1087 20 100 22 1065 720 1140 960 900 2467 975
123
4.7
1,
min
120 125 123 128 127 128 128 128
18.8 20.7 21.5 15.6 17.1 25.0 35.7 38.0 37.4 37.6 46.04 49.1 52.6
(1
-
X)AH
0.27 0.29 0.30 0.22 0.24 0.35 0.50 0.54 0.52 0.53 0.65 0.69 0.74
(1
- X)dQ
0.28 0.29 0.30 0.22 0.26 0.34 0.53 0.53 0.53 0.55 0.67 0.70 0.76
Ratio
0.96 1.0 1.0 1.0 0.92 1.0 0.94 1.0 0.98 0.96 0.97 0.99 0.97
_-__ AH*
Cooled t o room temperature------(1
- MAH
(1
- X)d
30.5
0.43
0.60
0.72
41.5
0.59
0.71b
0.83
44.0
0.62
0.74
0.84
54.3 56.6
0.77 0.81
0.81 0.83
0.95 0.98
= Determined from dilatometer measurements under the same conditions of crystallization temperature and time. initially a t 120'.
by isothermal crystallization at 126" and subsequent cooling to room temperature. For the higher molecular weights, the results are similar to those obtained after crystallization at 130". Of major interest is the confirmation of the influence of molecular weight on the enthalpy of fusion and the further demonstration of the relatively low values obtained for the highest molecular weights studied. The data for the samples crystallized in the calorimeter and never cooled to room temperature are summarized in Table I. Also given in the table are the values obtained after cooling to room temperature over a 24-hr period. Because of the unduly long times that are involved, the crystallization in the calorimeter could not be conducted at 130". The densities are reported in terms of the degree of crystallinity, calculated according to the relation7 (1 -
X)d
=
PA
PA
-P -
b
Crystallized
prior to the cooling of the samples, indicates that the thermodynamic quantities depend on molecular weight in a manner qualitatively similar to that which has been cited above. The measured enthalpy of fusion decreases from 52.6 to 20 cal/g, under comparable crystallization conditions, as the molecular weight is increased. Similar changes are observed in the density. It is of interest to examine the enthalpy of fusion data in terms of the molecular nature of the crystalline state that is actually formed. A formal and general expression can be developed which describes the enthalpy difference between the crystalline polymer and the pure melt for various morphologies that can evolve. We consider a system of n crystallites, which are uniform in size and which contain p crystalline sequences each of length {. The degree of crystallinity 1 - X is defined as
(1)
Here BAis the specific volume of the completely amorphous polymer, Pc the corresponding quantity for the crystalline polymer, and P is the specific volume of the actual system. The values of AH* and the density are, as expected, much lower than those obtained under similar initial crystallization Conditions with subsequent The ~ ~ increases that cooling t o room t e m p e r a t ~ r e . ~ occur in these quantities upon cooling, which is a consequence of further crystallization, become more pronounced as the molecular weight is increased. This is in accord with a previous report of the densities obtained after isothermal crystallization at 130°.3 I n these latter experiments, for example, when M = 1.5 x lo6, the degree of crystallinity increases from 0.37 to 0.70 upon cooling. On the other hand, for M = 2 X lo4,the relatively small change of from only 0.82 to 0.96 is observed. An examination of the data, The Journal of Physical Chemistry
Ratio
where No is the total number of units in the system. If H is defined as the enthalpy per unit, prior to complete fusion, then NOH =
(NO- nSP)Ha
+ nTp(1 - a ) H c + anlPHd + 2npAHe
(3)
Here Ha is the enthalpy per unit of the noncrystalline units, which are identified with the enthalpy of a unit in the pure melt; H , is the corresponding enthalpy per unit in the completely crystalline state; a is the fraction of units per sequence, within the interior of the crystal which is defected, each such unit contributing an excess enthalpy Hd; and AHe is the enthalpy deficiency per sequence at the terminus of each crystalline sequence. It is equivalent to the enthalpic contribution to the interfacial free energy and as such contains
THEENTHALPY OF FUSION OF LINEAR POLYETHYLENE
313
concluded that the term (2AHe)/({AHU)must be negligibly small in these cases. This conclusion implies the existence of either a large value of { or a characteristically small value of the ratio AH,: AH,. I n order to pursue the matter in more detail, these latter quantities must be independently determined. Here AH* is the measured enthalpy of fusion per All the required information is not available at present. repeating unit, AH, is the corresponding quantity for There are, however, some data with respect to the molecfusion of a completely crystalline polymer, and AHd ular weight dependence of crystallite sizes, as deteris defined as H, - Hd. Equation 4 contains contribumined from electron microscope studies of replicas of tions from a variety of sources and indicates many fracture surf aces.6 These results are limited, however, reasons why the observed enthalpy of fusion can be 130" for long time periods for samples crystallized at less than the value expected for a large completely and then cooled to room temperature. Characteristic crystalline nondlefected substance. lamella-like crystallites are observed over the complete In the absence of internally defected structures, molecular weight range and the sizes in the chain direceq 4 reduces to tion were found to be dependent on chain length. The size analysis that was made6was restricted to the thickest or largest size grouping of lamella present. It was observed, however, that for M > 5.6 X lo4, thinner If the degree of crystallinity calculated from enthalpy lamella were also present, their profusion increasing measurements is defined in a manner analogous to that with molecular weight. For molecular weights correof eq 1, namely that sponding to z 2 1-2 X lo3, the average size { correAH* sponds very closely to the chain length z. Such crystal(1 - X)dH = lites have been termed extended chain crystals. As 2 AHu increases to 4 X lo3,the crystallite thickness increases then eq 5 becomes from about 600 to 900 units, so that the ratio of crystallite size to chain length decreases to about 0.2. With (7) a further increase in molecular weight to 5.7 X lo5 (the upper molecular weight limit of the electron microscope measurements), the crystallite thickness Inherent in the definitions of (1 - A), and (1 reaches an upper limit of about 1000-1200 CH2 units. that have been given above are the assumption The ratio of the crystallite size to extended chain length that the contributions of the ordered and nonordered thus becomes very small. The above represents the regions are additive. I n addition, any possible cononly available experimental data concerned with tribution from the interfacial region to the quantity in crystallite sizes that are pertinent to the problem at question is tacitly neglected. A similarly defined degree of crystallinity can be obtained from infrared abhand. A reasonable assumption can be made that the larger sorption*s19and from wide-angle X-ray diff ra~tion.~gJO sizes that were observed are representative of the It has recently been shown that for linear polyethylene crystallites formed at the elevated temperature of samples which encompass a very wide range in density, crystallization.2s21 Hence, for the higher molecular there is very good accord between the values of the deweights, since { is of the order of lo3,rather large values gree of crystallinity calculated from the density, wideof the ratio AHe/AH, can be tolerated and still allow angle X-ra~r,~SJoand infrared absorption*llS data. for concordance in the degree of crystallinity calculaFor these cases, therefore, there is either no significant tions. For example, if AH,/AH, = 10, (1 - A),/ contribution froim the interfacial regions or these con(1 - A), = 0.98, which is consistent with the experitributions are self-compensating. The latter circummental observation. Thus, even a value as high as stance would be a highly unlikely coincidence. The data reported here for the density and enthalpy 10,000 cal/mole of sequences for AH8 would not reflect of fusion under a variety of crystallization conditions itself in a disparity in the degree of crystallinity calculacan also be used as a basis of comparison between tions. For the lower molecular weights, when { be1 - A , as defined by eq 1 and 6. We consider first the comes comparable to z, there must be a change in data, summarized in Table I, for the samples never interfacial structure consistent with the extended chain crystals that are formed. A reduction in AH,, accooled below the crystallization temperature. Under these circumstances, as can be seen from columns 5 companying the lowered crystallite size, would allow and 6 in the table, there is excellent accord between the for the observed experimental agreement between the two quantities over a range of 1 - X from about 0.75 (19) H. Hendus and G. Sohnell, Kunstofe, 51, 69 (1960). to 0.20. I n this and subsequent calculations, AH, has (20) M.Gopalan and L. Mandelkern, J. Polymer Sci., 5B,925 (1967), been taken to be 70 cal/g. Consequently, it must be (21) M. R. Gopalan and SA. Mandelkern, J . Phys. Chem., in press. the contribution of the chain units which comprise the interfacial region. Equation 3 can be rewritten as
Volume 72, Number 1 January 1968
314
Figure 3. Plot of degree of crystallinity calculated from enthalpy measurements as a function of same quantity calculated from density for samples initially crystallized a t indicated temperature and cooled to room temperature; crystallization temperatures: 0, 130"; 0, 120'; and 0, 23".
calculated degrees of crystallinity in this molecularweight range. Since the exact crystallite sizes or size distributions under these conditions of crystallization are not known, a more detailed analysis cannot be given at present. However, the magnitude involved and the change suggested in AHe with molecular weight is consistent with the values for the interfacial free energy.6022 Two independent studiess*22 of the dependence of the melting temperature on crystallite size and molecular weight have yielded concordant results for the interfacial free energies to be assigned the mature crystallites. For molecular weights greater than 5 X lo4,an asymptotic value of between 8000 and 9000 cal/mole of crystalline sequences was obtained for the interfacial free energy of the crystallites formed. As the molecular weight decreased, with the concomitant increase in the ration {/z, the interfacial free energy decreases and reaches a value slightly less than 5000 cal/mole for the lowest molecular weight of interest in the present work. A comparison of the degrees of crystallinity for the samples crystallized at the elevated temperatures and then cooled to room temperature are summarized in Figure 3. It is clear from the figure that these results are quite different from those just discussed for specimens never cooled below the crystallization temperature. Except at the very high levels of crystallinity, which are obtained with the lowest molecular weights studied, there is a significant disparity between the two different methods of calculation the degree of crystallinity. The calculations based on the enthalpy of fusion measurements consistently yield lower values than those obtained from the density measurements utilizing the Chiang-Flory' specific volume of 1.17 cm3/g for the completely amorphous polymer at 25". Utilizing the slightly lower value 1.16 cm3/g that has been suggestedl8JQ for this specific volume reduces this difThe Journal of Physical Chemistry
L. MANDELKERN, A. L. ALLOU,JR.,AND M. GOPALAN
L
I
I
I
-102
IO'
IO' x
io5
106
Figure 4. Plot of quantity B as function of chain length x for samples initially crystallized to elevated temperature and then cooled to room temperature; crystallization temperatures: 0, 130'; 0, 120'; and I, 23'.
ference slightly. The divergence between the two sets of values becomes larger as the level of crystallinity becomes smaller. The concordance previously noted between the degrees of crystallinity obtained from density, infrared absorption, and wide-angle X-ray diffraction is for samples cooled to room temperature. The enthalpy of fusion measurements are thus unique at present in yielding lower values for the degree of crystallinity for this particular mode of crystallization. The differences between the two basically different kinds of crystallization procedures, as manifested by a comparison of the data in Table I and Figure 3, cannot be attributed either to the samples or the technique of measurements, since they are identical in both cases. Neither can the range in the level of crystallinity be involved, since there is a significant overlapping in these values. The difference must therefore reside in the crystallization procedure with a major contribution thus expected to come from the properties of the crystallites formed on cooling. For example, for the samples initially crystallized at 130",a much higher proportion of the crystallinity eventually developed is found in the higher molecular weight samples as compared to the lower ones. Concomitantly, the disparity in the calculations is greater for the higher molecular weight fractions. On cooling, much smaller size crystallites are expected, since the sizes are primarily controlled hy nucleation processes2I2l and thus on the crystallization temperature. Since extended chain crystals will now not be formed, except in the extreme of very low molecular weight, a compensation or lowering of the value of A H , would not be anticipated. Consequently, there could be a major contribution to the measured enthalpy of fusion from the term (2AHe)/{ = B. I n this event, the simple additivity law, as embodied in eq 6, would not be valid to calculate the degree of crystallinity. Values of the quantity B, calculated according to eq 5, are plotted in Figure 4 as a function (22) J. M. Sohulta, W. H. Robinson, and G . M. Pound, J . Polymer Sei., A2-5, 511 (1907).
THEENTHALPY OF FUSION OF LINEARPOLYETHYLENE
315
temperature for the higher molecular weights. This conclusion makes mandatory the presence of chain units in random conformation which connect crystallites.6~'~* From the above discussion, it follows that a plot of AH* against the specific volume does not necessarily have to be linear as has been supposed.1sJ8 Neither would it be expected to be represented by a unique function which encompasses all the data. This is seen analytically by combining eq 1and 5 to give
Specific Volume
Figure 5. Plot of enthalpy of fusion as function of specific volume; dashed straight lines (curves 1-4 from top to bottom, respectively) calculated according to eq 8 for B = 0, curve 1; B = 10, curve 2; B = 15, curve 3; and B = 25, curve 4. Crystallization conditions are the same as in preceding figures.
of molecular weight for each of the crystallization temperatures. Although the values of B are definitely dependent on the crystallization temperature, the qualitative dependence on molecular weight is the same in all cases. R,elatively constant values are calculated for the lower molecular weights, but a rapid increase occurs when 2 exceeds 3 X lo4. For 2 > lo6, an asymptotic value of B , which is characteristic of each crystallization temperature, appears to be reached. For a given molecular weight, the value of B decreases with an increase in the initial crystallization temperature. The quenched samples have the largest values of B consistent with the low crystallization temperature and the accompanying small crystallite sizes. For example, a value of B of 10 cal/g corresponds to a ratio of 0.86 for the calculated degrees of crystallinity. As this ratio belcomes smaller, the greater the molecular weight and the lower the crystallization temperature become. A consistent interpretation of the data can then be found in the contribution of the interfacial enthalpy of the crystallites formed on cooling to the measured enthalpy of fusion. A more quantitative analysis requiires more detailed information with respect to the crystallite sizes initially formed as well as those formedl on cooling, since the melting of all the crystallites in the system at room temperature contribute to the enthalpy of fusion. Although the degree of crystallinity data just described is unique in not being in agreement, it should be realized that the disparity is a consequence of a particular crystallization procedure which introduces small crystallites into the system. Processes, such as annealing or much slower cooling, which minimize the latter effect, should lead to more concordant results. It is clear, however, from a variety of methods, that relatively low levels of crystallinity are achieved a t room
A linear relation between AH* and P is only to be expected when B is a constant. Since the quantity 2 A H e / f depends on the crystallite morphology and size, the latter condition will not in general be fulfilled. The data in Figure 4 indicate that there will be a wide variation in B , depending on the crystallization conditions. I n Figure 5, plots of AH* as a function of 7 are given for the hypothetical situation of B being a constant. Curve 1 is for the case of B = 0 and the straight line drawn has the property that for the completely crystalline polymer P = 1.00, AH* = 70 cal/g, while for the completely amorphous polymer a t room temperature AH* = 0, P = 1.17. This straight line corresponds to the case where the calculated degrees of crystallinity are in agreement. The data for the samples never cooled below the crystallization temperature lie on this straight line. The other straight lines in the figure represent increasingly higher values of B. Although all the curves originate from the point AH* = 0, P = 1.17, the expected values of AH* corresponding to P = 1.00 decreases as B increases. The experimental results for the samples cooled to room temperature are also plotted in the figure. As can be anticipated from the data in Figure 4,there is no simple pattern to the plots. Each point in the plot occupies a position corresponding to the appropriate value of B. The data for samples crystallized at 120" or quenched to room temperature cannot be represented linearly. I n this grouping, based on the data a t the lower specific volumes, an exceptionally low value for AH*, corresponding to the pure crystal, is extrapolated. With increasing specific volume, the points correspond to increasing values of B. An extrapolation to AH* = 0, PA = 1.17 is suggested along the lines of B approximately equal to 20-25. On the other hand, the data for samples initially crystallized at 130" can be empirically represented by a straight line, which originates at AH* = 70 cal/g, P = 1.00, and yields an extrapolated value of V = 1.12 corresponding to AH* = 0. This specific volume is much less than the value of 1.17 that is expected for the completely amorphous polymer. This empirical linear relation is a result of the low, but not negligible, values of B that are char(23) E. W. Fischer and G. Hinrichsen, KoZloid-Z., 213, 93 (1966).
Volume 78, Number 1 January 1968
316
L. MANDELKERN, A. L. ALLOU,JR.,AND M. GOPALAN
acteristic of the lower specific volumes and the fact Table 11: Properties of Crystals Formed that the asymptotic value of B has not been reached from Dilute Xylene Solutions for a sufficient number of samples to indicate the change Density, TO. d, AH*, in slope and the approach to the ultimate value of M? OC A C d K K/QO AH* = 0, P = 1.17. The expectation that there is 19,000 60 99 48.3 0,968 not a common linear relation is thus confirmed by the 0.975 70 104 47.6 experimental data. The results just cited are another 85 130 53.3 0.983 manifestation of the effect of small-size crystals on yield90 148 53.7 0.985 ing differing values of the degree of crystallinity cal45,000 60 101 47.2 0.967 culated from density and enthalpy measurements. 70 106 49.2 0.975 85 129 52.2 0.9801 Crystals Formed from Dilute Solution. When crystal90 149 54.9 0.9837 lized from dilute solution, polymers display the char300,000 70 112 51.1 0.9698 acteristic platelet 0: lamella habit. The lamella are 70 112 52.2 0.9728 usually about 100 A thick and the width, or lateral 85 129 53.0 0.9730 dimension, is of the order of about a micron.24 It has 90 146 54.5 0.9774 0.9792 90 148 54.6 been well established that the chain axes are pref1,200,000 70 112 51.8 0.973 erentially oriented normal to the wide faces of the 90 148 54.7 0.9775 lamella.24 Hence, a given molecule must traverse a crystallite many times. Based primarily on the visual observation of the electron micrographs, it has been structures are being analyzed simultaneously. We presumed that this interface is comprised of regularly limit ourselves here to a discussion of the properties of folded chains,24with a minimum number of units concrystals which have not been annealed. sistent with the potentials hindering bond rotation The enthalpy of fusion data and other related propparticipating in the folded structure. For linear polyerties for the molecular weight fractions crystallized ethylene, this involves five bonds in gauche orientation.26 isothermally from dilute xylene solutions are sumOn the other hand, a wide variety of physical measuremarized in Table 11. ments indicates that about 15-20% of the chain units There are certain salient features in the data listed must be in nonordered conformati0ns.~6-~~The presin Table 11. I n striking contrast to samples crystalence of such a large number of noncrystalline chain lized from the melt, AH* is now essentially independent units leads to the conclusion that a disordered amorof molecular weight over the range of 1.9 X 106-1.2 x phous overlayer is p r e ~ e n t . ~ 8 1This ~ ~ latter conclusion lo6 that has been studied here. The values of AH* is also consistent with the electron micrographs that given in the table are about 10% lower than the differhave been presented. It, therefore, becomes a matter ential thermal analysis measurements reported by of interest to see if enthalpy of fusion measurements of Fischer and Hinrichsen2a for unfractionated polycrystals formed from dilute solution can help elucidate ethylene and two measurements involving fractions. this problem. This discrepancy may be due to the different experiA detailed report of the course of fusion of such crysor the inclusion of an additive mental methods used tals has already been given.12 The fusion process is constant to each measurement in the latter work. The relatively broad12 and characterized by partial melting validity of these enthalpy measurements is discussed and recrystallization before final melting O C C U T S . ~ ~ ~ ~ ~ below. The enthalpy of fusion is primarily dependent The onset of melting is detected at relatively low temon the crystallization temperature and increases as this peratures, ranging from 100 to 110", depending on the molecular weight and the crystallization temperature.12 temperature increases. Concomitant with an increased crystallization temperature, there is also a sigExtreme care must, therefore, be exercised in evaluating nificant increase in the crystallite thickness which is also the enthalpy of fusion from the experimental data to properly account for the contribution to the fusion (24) A. Keller, Phil. Mag., 2, 1171 (1967); A. Keller, Makromal. Chem., 34, 1 (1969). of this low-temperature region. It is also important (26) J. D.Hoffman, SOC.Plastic Eng., 4,315 (1964). that a distinction be made between isothermally (26) P. J. Flory, J . Am. Chem. SOC.,84, 2857 (1962). crystallized samples which have never been heated (27) E.W.Fischer and G. Schmidt, Angew. Chem., 74, 651 (1962). above the crystallization temperature, after separation (28) J. B.Jackson, P. J. Flory, and R. Chiang, Trans. Faraday SOC., from the mother liquor, and those which have been 59, 1906 (1963). heated or annealed at an elevated temperature. I n the (29) W.0.Statton and P. H. Geil, J. Appl. Polymer ScSCi., 3, 357 (1960). latter case, partial melting and recrystallization is (30) W.P. Slichter, J. Appl. Phys., 32, 2339 (1961). known to 0ccur.~2J7 The necessity for making this (31) A. Peterlin, G. Meinel, and H. G. Olf, J . Polymer Sci., B-4, distinction has not always been recognized in interpret399 (1966). ing experimental data.1*123133If this is not made, dif(32) T.Okada and L. Mandelkern, ibid., B-4,1043 (1966). ferent crystallite morphologies with different interfacial (33) V. F. Holland, J. Appl. Phys., 35, 59 (1964). The Journal of Physical Chemistry
THEENTHALPY OF FUSION OF LINEAR POLYETHYLENE I
I
I
I
I
317
1
cording to eq 9, is given in Figure 6a. Within the restricted values of (l/{)that are available, the data are well represented by a straight line. The intercept corresponding to 1/{ = 0 is 69 f 1 cal/gJ39 which is in good accord with the expected value for AH,. From the slope of the straight line A H , is found to be 10,500 f 1500 cal/mole of sequence. I n a purely formal manner, the enthalpy of fusion data are thus consistent with a regularly folded interface in that eq 9 is obeyed. However, the value deduced for the interfacial enL J thalpy must also be consistent with the model. For an interface comprised of regularly folded chains, AHe has been estimated to be 1500 cal/mole.26 Hence the experimental values are much larger than the theoretical value required to justify a surface of regularly folded chains. Although an admittedly very long L extrapolation is involved, the intercept is so close to the 4 0 ~ 20 60 100 140 theoretically expected value so as to rule out any major I/( x contribution from size independent internal defects. Figure 6. (a) Plot, of enthalpy of fusion, AH*, as a function The data of Fischer and HinrichsenJZ3previously reof l / S for molecular weight fractions crystallized from to, have a very similar functional form when ferred dilute xylene solutions: M = 1.2 X lo6, 0 ; M = 3 X lo6, 0; plotted in this manner. The straight line representing M = 4.5 X 104, U; and M = 1.9 X 104, 0 ; and (b) plot of quantity AH*/(X - X ) d as a function of l/.t for their data parallels the one in Figure 6a, but is displaced molecular weight fractions crystallized from dilute xylene upward. The intercept, corresponding to l/{ = 0, solutions: M = 1.2 X lo6, 0 ; M = 3 X lo6, 0; is equal to 77 cal/g. This is an abnormally high value M = 4.5 X lo4, U; and M = 1.9 X lo4, 0. for A H , which evades any theoretical explanation at present. It is a consequence of the higher experimental molecular weight independent in the range studied. values reported for AH* and must raise some concern This increase in crystallite thickness is well known, and in regard to these values. the size-temperature relation reported here agrees From solubility studies of mature crystallites formed very well with those given by o t h e r ~ . ~ J ~It- ~ ~in dilute solution, the interfacial free energy has been should be noted that the crystallite sizes are all in the found to be 1900 cal/moleZ8 of crystalline sequence. range of about 100 A. This result is independent of any model assumed for The density of the samples agrees very well with the the crystallite morphology. From the experimental range reported by the vast majority of other investiinterfacial enthalpy deduced above for the regularly g a t 0 r s . 2 3 ~ ~ 1 ~ ~ An 8 ~ ~increase 1 ~ ~ ~ in the density is obfolded model, the corresponding interfacial entropy is served with increasing crystallization temperature. 22 eu/mole of sequence. For a fold consisting of five There appears to be a small but detectable decrease in CH2units, more than 4 eu would have to be assigned to density for the two highest molecular weight fractions each chain unit in the interface. This appears to be an when compared with the lower molecular weights unduly high value since the entropy of fusion per CH2 for the same crystallization temperatures. Despite the unit is only 2.3 eu.2116 disparity in the enthalpy measurements, the densityThe enthalpy of fusion data can also be analyzed accrystallite size-molecular weight relations agree very cording to the model of a finite-size crystal with an well with those given by Fischer and Hinrichsen.28 amorphous or noncrystalline ~ v e r l a y e r . ~ ~I nJ ~this For a regularly structured interface, exemplified by case, the chain does not necessarily return to the crystal regularly folded chains, 1 - X is by necessity unity. with the sequences juxtaposed. The loops are of variEquation 5 can, .therefore, be rewritten as able length, and the chain units are in nonordered conformation. For this model, eq 5 is valid and the ap2AH, AH* = A H , (9) (34) V. F. Holland and P. Lindenmeyer, J . Polymer Sci., 57, 589
‘9’
-
5
Hence a plot of AH* against l/{ should be linear, giving an intercept equal to AHu and slope equal to 2AHe. I n making this analysis, it must be realized that only a limited range of values for [ is experimentally available. This limitation will obviously hamper any thermodynamic analysis wherein the crystallite size assumes a dominant role. The appropriate plot, ac-
(1962). (35) F. P. Price, J . Chem. Phys., 35, 1884 (1961). (36) A. Keller and A . O’Connor, Polymer, 1, 163 (1960). (37) E.W.Fischer and R. Lorena, Kolloid-Z., 189, 97 (1963). (38) G. hl. Martin and E. Passaglia, J . Res. Natl. Bur. Std., 70A, 221 (1966). (39) The significance of this intercept is only valid for the highest
molecular weights. For the lower molecular weight fractions, there would have to be a discontinuity for this model when becomes comparable with the extended chain length.
r
Volume 73,Number 1 January 1968
S. HARTLAND
318 propriate plots are given accordingly in Figure 6b.40 Here the degree of crystallinity is again calculated from the measured density. The data can again be represented by straight lines which extrapolate to 69 rf: 1 cal/g. The data for the two lowest molecularweight fractions are displaced slightly below those for the higher molecular weight. Although this slight divergence could be real, it might also very well be a reflection of the extreme sensitivity of the plotted data to the calculated degree of crystallinity. An interfacial enthalpy of 4500 h 1000 cal/mole is deduced from the plot in Figure 6b. It is significantly less than the corresponding value for the regular chain model. The interfacial entropy is now found to be about 6 eu per crystalline sequence. If the interface is comprised of about 10-20 units per sequence, then the interfacial entropy per unit is very small and consistent with a disordered interfacial structure. The enthalpy of fusion data, for crystals formed from dilute solution, are formally or algebraically consistent with either of the two major crystallite morphologies
considered. The choice from solely the point of view of the enthalpy measurements depends on the value of AHe and A s , appropriate to the particular molecular structure. The interfacial enthalpy and entropy that are deduced appear to be much too high to be reconcilable with a regular folded-chain structure. The parameters for an irregular structured interface are, however, acceptable ones. The data and this latter conclusion are thus compatible and consistent with the diversity of other physical measurements previously cited for similar constituted crystalline ~ y s t e m s . ~ ~ J ~ The degrees of crystallinity calculated from the density data are always higher than the corresponding value obtained from the enthalpy of fusion data with neglect of the interfacial contribution. This is expected, from the discussion in the previous section, because of the small crystal sizes and the value of the interfacial enthalpy that has been deduced. (40) There is a question as to the crystallite thickness to use here with respect to the low angle maxima. However, a reduction of the spacings by 5-10 A does not sensibly alter the analysis.
The Radius of the Draining Film beneath a Drop Approaching a Plane Interface by S. Hartland Chemical Engineering Department, The University of Nottingham, Nottingham, England Accepted and Transmitted by The Faraday Society (May 8, 1967)
The radius of the draining film beneath a drop approaching a plane interface through an immiscible fluid may be obtained from the Bashforth and Adams tables’ in terms of the size of the drop and its physical properties relative to the surrounding fluid. Derjaguin and Kussakov2 have obtained an expression relating these quantities when the drop size is very small. This agrees with the lower limit of the Bashforth and Adams relationship, but should not be used outside the region for which it was experimentally verified. When a drop of heavy fluid approaches a plane interface through an immiscible light fluid, the drop deforms as shown in Figure 1 so that a film of light fluid is trapped between the drop and the plane. (With a gas bubble in a liquid, Figure 1 would be inverted.) The rate of approach of the drop to the plane is controlled by the rate of drainage of the liquid in this film. When the film becomes sufficiently thin a t some point, it ruptures and the drop wets the plane. The final static-equilibrium position is then obtained, and this depends on the angle of contact between the fluids and The Journal of Physical Chemistry
the plane. The shape of such a static drop has been tabulated by Bashforth and Adams’ in terms of the physical properties and the radius of curvature a t the top of the drop. Only the period before rupture is considered here, and equilibrium refers to the conditions when the overall dimensions of the draining film become substantially constant. This dynamic-equilib(1) F.Bashforth and J. C. Adams, “An Attempt to Test the Theories of Capillary Action,” Cambridge University Press, London, England, 1883. (2) Derjaguin and Kussakov, Acta Physiochim. URSS, 10,25 (1939).
