Macromolecules 1989,22, 914-920
914
Equilibrium Dissolution Temperature of Low Molecular Weight Polyethylene Fractions in Dilute Solution Abaneshwar Prasad and Leo Mandelkern* Department of Chemistry and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306. Received May 3, 1988; Revised Manuscript Received July 8, 1988 ABSTRACT The melting temperaturecomposition relation applicable to dilute solutions is developed. The resulting expression is found to be identical, with minor approximation, with the conventional one which is restricted by theory to concentrated solutions by virtue of the use of the Flory-Huggins mixing expression. Departure from the limiting high molecular weight form is only expected for low molecular weight species at high dilution. Experimental determinations of the equilibrium melting temperature (dissolutiontemperature) of low molecular weight polyethylene fractions in p-xylene confirm the major theoretical expectations. The experimentally determined equilibrium melting temperatures are used to analyze already existing literature data on the crystallization kinetics from dilute solution.
Introduction The classical melting point-composition relation of polymer-diluent mixtures1i2has received widespread experimental ~erification.~ The melting temperature as a function of concentration can be expressed as
(&-$)= (&)($)[++
(1 - ;)ul
(1)
- xlu12]
Here TM is the melting temperature of the polymer-diluent mixtures, TMo is the melting temperature of the pure polymer, Vuand VI are the molar volumes of the polymer repeating unit and diluent, respectively, and x1 is the Flory-Huggins thermodynamic interaction parameter. The composition is expressed by u1 and u2, the volume fraction of diluent and polymer, respectively, and x is the number of segments per molecules as is defined by the Flory-Huggins expre~sion.~ In the limit of large molecular weight x m and eq 1 reduces to
-
(2) These equations are derived on the premise of phase equilibria with the crystalline phase assumed to be pure. The chemical potential of each of the species in the liquid state is taken from the Flory-Huggins expression for the free energy of mixing. Thus a t equilibrium (2) puc - p$
= p, - pL,o = -AG,
(3)
Here pL,o is the chemical potential of the repeating unit in the reference state; puc and wu are the chemical potentials of a repeating unit in the crystalline and liquid state, respectively. AG, represents the free energy of fusion per repeating unit. The chemical potential of a structural repeating unit in the disordered state is given by4 wll
- P"O =
RT(
$)[?
- (1 - ;)(1 - u2)
+ X'(1 - u J 2
I
(4)
Equation 1 then results from these conditions. The Flory-Huggins expression for the free energy of mixing requires that the chain segments be uniformly 0024-929718912222-0914$01,50/0
distributed throughout the liquid mixture. This condition is only fulfilled either by concentrated solutions or 0 solvents. Therefore, the applicability of eq 1 and 2 is limited. The success of eq 1and 2 to experimental data has been achieved in the concentrated range. In contrast to concentrated systems, dilute solutions are characterized by a nonuniform polymer segment distribution throughout the medium. Consequently, the use of the Flory-Huggins free energy function will not be valid in the dilute range, except in solvents where the molecules interpenetrate one another freely. For the same conditions of phase equilibria a more appropriate expression for the polymeric species, and thus p,, is needed to derive the melting temperature-composition relation in dilute solution. The starting point clearly involves expressing the chemical potential of the solvent species in virial form.5 The results of such an analysis for dilute solutions was presented many years ago.6 However, details of the derivation were not given. Only the broad outlines of the method were presented. At that time there was no experimental data available with which to test the equation that resulted. In the ensuing years, the results of several other theoretical analyses of the problem have appeared.'+ More recently, a set of experimental data, involving the melting temperature of low molecular weight fractions of linear polyethylene in dilute solution, has been reported.lO These data are well suited to be analyzed in terms of the original theory, a point which had been neglected. The analysis of these results, as will be shown subsequently, indicated the need for data covering a wider molecular weight range and with melting temperatures being determined in even more dilute solutions than had been studied.1° We report here a more complete set of experimental melting temperature (or dissolution temperature) data of low molecular weight fractions of linear polyethylene covering the complete concentration range. Major emphasis is given, however, to the very dilute region. The results are analyzed in terms of the relation originally given.6 In view of the renewed theoretical and experimental interest in this problem, we outline again the derivation of the appropriate melting point relation in dilute solution. It needs to be recognized at the outset that an equilibrium theory is involved. This conditions requires the use of extended chain crystals; i.e., the crystalline thickness must be comparable to the extended chain length.' At present this requirement can only be met with low molecular weights."J2 However, it turns out that this restraint does not impose any practical impediment to the present work. We will find, coincidentally, that only low 0 1989 American Chemical Society
Macromolecules, Vol. 22, No. 2, 1989
fractn A B
C D E
F
MN 575' 785' 9100 1055* 1675c 29Wd
Table I Mw MJMN
Dissolution Temperature of Polyethylene Fractions 915
TwoC 79.0 95.0
1142
1925 3100
1.08 1.15 1.07
103.0 110.5 122.0 131.0
a Fraction prepared for this work; MN determined by ' H NMR (see text). bFraction used in ref 13; characterized by GPC. CFractionused in ref 14; characterized by GPC. dFraction supplied by SNPA; characterized by GPC.
molecular weight species will be of theoretical interest. One of the consequences of this study is the determination of the equilibrium melting temperature at a given concentration. This melting temperature is an important parameter in analyzing crystallization kinetics. With these new results we have also examined the appropriate crystallization kinetic data that are available in the literature.
L 400
+Temp,
O K
b
Experimental Section Six low molecular weight polyethylene fractions, M N ranging from 575 to 2900, were used in this work. Their major characteristics are given in Table I. The melting temperatures, TM, given in the last column of the table are those obtained for the pure, undiluted polymer. They were taken as the temperature of the endothermic peak, extrapolated to zero heating rate and corrected for the dependence of melting temperature on crystallization temperature.ls (A detailed analysis of the melting temperature in the low molecular weight range will be given in a subsequent publication.) Fractions A, B, and C were specifically prepared for this work from a parent homopolymer. The other three samples were available from other sources. Fraction F was obtained from the SNPA at Lacq, France, and was characterized by GPC. Fraction E was obtained from the Petrolite Corp. and labeled Polywax 2000. Its characterization and properties have been previously described.'* Fraction D was used previously in an electron microscopy study of the structures found in dilute solution ~rystallization.'~ Fractions E and F appear to be identical with PE 3100 and PE 2000 of b u n g et al. (Table I of ref 10). They were obtained from the same source and have identical molecular weight characteristics. Fractions A, B, and C were obtained from a whole polymer supplied by Scientific Polymer Products. The manufacturer gave ~ a number average molecular weight of M N = 640 and M w / M = 1.10. A GPC analysis of this polymer gave M N = 660 and M,/MN = 1.17.'' This sample is designated PE 700. Hence, in terms of higher molecular weights this sample would be considered to be a very good fraction. However, in thii low molecular weight range, as we shall find, it is not an adequate fraction for the purpose at hand. It can be readily separated into many fractions. High-resolutioncarbon-13NMR (seebelow) established that there were two methyl end groups per molecule. The fractionation procedure that was adopted was based on the crystallization from dilute solution. Since there is a very marked dependence of melting temperature on the chain length in the low molecular weight range,'* the crystallization rates of the different species at relatively low undercoolingsare drastically affected. It is the major difference in crystallization rate that allowed for the fractionation that was accomplished. Even for what would be considered a narrow fraction in the high molecular weight range, a significant separation of species can be accomplished in the very low molecular weight range of interest here. The fractionation was carried out from a 1% polymer solution in p-xylene (133 mg of polymer) contained in a sealed evacuated tube filled with a wintered glass filter at its center. The polymer was initially dissolved at 130 "C with continuous shaking. The tube was then transferred to a thermostatic bath set at 72 "C which had been predetermined to be the temperature for the onset of crystallization. Precipitation occurred at this temperature and was allowed to continue for 4 days. The precipitate was separated at this temperature and found to be the order of 1 mg and was rejected. The remaining homogeneous solution was then kept
-+ Temp
OK
Figure 1. (a) Thermogram from differential scanning calorimetry of low molecular weight fractions of linear polyethylene. A, B, and C are for fraction described in Table I; Y is for fraction designated as PE 700(1)in text; X represents the whole polymer PE 700. (b) Normalized plot of thermograms from Figure la.
at 65 "C for 5 days, and the precipitate, amounting to 2 mg, was also separated at this temperature. This small amount of material designated PE 700(1)was used only for a DSC measurement of the pure polymer (see Figure 1). The polymer was allowed to completely precipitate from the remaining filtrate and then redissolved in a similar tube at 1% concentration. Crystallization was then repeated at 57 "C for 10 days and the precipitate separated at this temperature had a dry weight of 20 mg. This sample is designated as fraction C in Table I. The remaining filtrate was kept at room temperature for 3 days during which further precipitation occurred. Fraction B (90 mg) was obtained by this method. The addition of acetone to the remaining filtrate at room temperature yielded 15 mg of polymer designated as fraction A. We were able to recover 96% of the polymer by this method. Besides the determination of the molecular weights (cf. seq.) the efficacy of the fractionation can be discerned by examining the thermograms obtained by differential scanning calorimetry for each of the fractions and comparing them with the parent whole polymer. In Figure la the thermograms are given for each fraction of PE 700 as well as that of the whole polymer. The weight of each sample that was used in these thermograms was arbitrarily chosen. Despite the relatively narrow molecular weight distribution of the whole polymer, its thermogram is quite broad. The thermograms become much sharper for each of the fractions. The melting temperature (endothermic peak) increases substantially with the crystallization temperature of fractionation. There is a 30-deg difference between the melting temperature of the fractions indicating the inherent polydispersity of the parent sample for present purposes. Figure l b represents a set of normalized thermograms that were constructed from the data in Figure la. The thermogram for the whole polymer was taken as a reference in this figure. The thermograms of the fractions were scaled so as to represent their proportion (on a mass basis) of the
Macromolecules, Vol. 22, No. 2, 1989
916 Prasad and Mandelkern whole polymer. Within the experimental error, a reasonable representation can be made of the thermogram of the whole polymer. The reconstruction of the thermogram for the whole polymer indicates the need to work with very narrow fractions in studying crystallization of low molecular weight samples from dilute solution. Conventional fractions, which have been so illuminating in studying the crystallization behavior of high molecular weights, are inadequate in the low molecular weight range. The molecular fractionation that must inevitably take place with low molecular weights will seriously obscure the results that are obtained.
Molecular Weight Determination Utilizing the number-average molecular weight supplied by the manufacturer, or that determined by GPC, we were able to establish by high-resolution solution 13C NMR measurements that this polymer sample contained two methyl end groups per molecule. This observation then served as the basis for the determination of the number average molecular weight of the fractions. Carbon-13 NMR would be the best choice of methods by which to determine the methyl group concentration of the fractions. The large amount of material required for each measurement (about 200 mg) precluded the use of this method with the fractions. However, by comparing the 13C NMR results, for low molecular weight unfractionated polyethylenes as well as n-alkanes, we were able to establish that lH NMR could be used to determine the methyl group concentration. The molecular weights of the fractions were obtained by this method. The details of the NMR procedures adopted are as follows. The 13C NMR spectra was obtained a t 380 K by using a Bruker WP270 SY spectrometer operating at 67.93 mHz. Solution concentration was approximately 10-15% in 1,2,4-trichlorobenzene. A 90" pulse width with the delay times between pulses of 30 s was used. The chemical shifts were referenced internally to the main backbone methylene carbon resonance a t 29.99 ppm from Me4Si. The 13C chemical shifts assignments for the end CH3 and a-, p-, and y-carbons were made following the procedure suggested by Randall.lg Randall's assignments were found to be in close agreement with that obtained experimentally. The chemical shifts for CH3(end groups) and a-, p-, and y-carbons were at 14.10, 22.88, 32.18, and 29.58 ppm, respectively. Proton NMR spectra was obtained with the same spectrometer at 335 K. Polymer concentration was approximately 1% in CDC1,. The chemical shifts were referenced internally to the proton of CHC13 a t 1.455 ppm. A spectral width of 3000-5000 Hz/cm was used with a pulse width of 2-5 ps. A time constant of 0.1 Hz and an accumulation of 16-32 scans were done. The end-group analysis was done by taking the integrated area under the peak of 0.9 ppm (for end CH,). The integrated areas a t 1.1 and 1.2 ppm were taken for hydrogens at a- and pcarbons. The integrated area for the backbone -(CH2); was taken a t 1.28 ppm. For a comparison of the two NMR methods, a set of unfractionated low molecular weight whole polymers and two n-alkanes were studied. The results are summarized in Table 11. The whole polymers PE800 and PE600 were obtained from Scientific Polymer Products; PE700 was supplied by Polysciences, Inc. PE6OO-H is a fraction obtained from PE600. The manufacturer specificationswere as follows: PE600 ( M , = 595, MN = 535); PE700 ( M , = 700, MN = 636); PE800 ( M , = 794, MN = 691). The comparative results between the lH and 13CNMR for both the low molecular weight polymers and the n-hydrocarbons are in very good agreement for all the samples. The absolute values obtained for the n-alkanes are excellent.
Table I1 samples 1. PE 800 (whole) 2. PE 700 (wholey 3. PE 600 (whole) 4. PE 600-H (fraction) 5. CwHloz 6. CBOH122 a
M N by 'H NMR 650 725 560 800 690 840
MN by I3C NMR 640 710 530 800 700 850
Parent polymer subsequently fractionated.
Hence, we can use the molecular weight of the fractions obtained by IH NMR. The dissolution temperatures of the fractions were determined in p-xylene. The p-xylene used was of high spectroscopic grade, of 99.9% purity, supplied by the J.T. Baker Co. The polymers, in the form of extended chain crystals, were placed in a preweighed Perkin-Elmer large volume capsule (stainless steel) that had O-ring seals. The appropriate amount of p-xylene was transferred to these sealable capsules to make a volume fraction (uz) of polymer in the range 0.2-0.9. Each sample pan was weighed for any leakage, before and after scans. The sample pans were kept for 24 h at room temperature to ensure homogeneous mixing. The pans were then transferred directly into DSC and were heated to a t least 15-20 "C below the theoretically expected melting temperature at a rate of 10°/min. It was held a t this temperature for 20-30 min. Heating was then resumed at a rate of 1.25"-5.0" min. The range normally used was 2 or 5 mcal/s. The peak temperature was used to define the dissolution temperature. The temperature scale was calibrated with indium over the range of interest. For polymer concentrations u2 50.2, to concentrations as low as u2 = 0.001, the melting temperatures were determined by visual methods previously described by Jackson et a1.20 Both methods were used to determine the dissolution temperature in a series of experiments in the vicinity of u2 = 0.2. The agreement between the two methods in this overlapping concentration region was found to be very good. The experimental results and analysis that are described below clearly indicate that it is necessary to extend the dissolution temperature measurements to the very dilute range. The determination of the melting temperature in this concentration range is beyond the capability of the differential calorimeter. As has been discussed in the Introduction section, it is necessary, for purposes of present analysis, that the crystallites prior to dissolution be in extended form. All the samples for the dissolution experiments were prepared by crystallization in the bulk. For samples A-E the formation of extended chain crystallites is easily obtained as has been described in the 1 i t e r a t ~ r e . lSample ~ ~ ~ ~ E was crystallized a t 105 "C for 3 days. For a molecular weight sample similar to fraction F it has been shown that the extended crystallite form is only developed within a limited temperature range.15 Consequently, care must be taken in selecting the crystallization temperature. Sample F was crystallized a t 117.5 "C for 1 week. From the low-frequency Raman longitudinal acoustical mode (LAM) analyzed in the manner previously d e s ~ r i b e d the ~ ~ peak . ~ ~ in the ordered sequence length distribution was found to be 261 k. This value corresponds quite well with the expected ordered length of 266 A for the completely ordered chain. Results and Discussion Theoretical Development. To determine the melting temperature-concentration relation in the dilute range, it is necessary to have available an expression for the chemical potential of the polymeric species. This quantity is
Macromolecules, Vol. 22, No. 2, 1989
Dissolution Temperature of Polyethylene Fractions 917
obtained from the chemical potential of the solvent species
and the Gibbs-Duhem equation. The chemical potential of the solvent species can be expressed in virial We shall, for convenience, use the relations of Flory and K r i g b a ~ mwhere ~ ~ the chemical potential of the solvent can be expressed as u2
I'2u22
. .I
I',~LJ~~ + ~+ (5) P
Here, p1 and p t are the chemical potential of solvent and that of the pure liquid, respectively; R is the gas constant, T is the temperature; VI is the molar volume of solvent, M is the molecular weight of the polymer; P is the partial specific volume of the polymer; g is a factor whose value depends on the polymer-solvent system and is usually of the order of 0.25. r2is equal to the product of the second virial coefficient A2 and the molecular weight. The second virial coefficient can be expressed as
($)(
XI
-
i ) F ( X , = -yQ MA2
(6)
where y is the actual number of repeating units of polymer. The explicit expression for F ( X ) is given in ref 5. From the Gibbs-Duhem equation we obtain
a[
(yo) --In,; =--
+ (1 - :)
x
for the chemical potential of a repeating unit. By applying the conditions for phase equilibria, with the assumption that the crystalline phase remains pure, one obtains
(& &) (&)($)b1 (-&)(; + %)+ (-&)(;I -
=
- XIU?)
-
x
Analyses of the virial coefficient data of the polyethylenes in the literature in p-xylene and similar solvents2k27indicate that the bracketed terms will be negligibly small (2-3 orders of magnitude less) than the lead terms. Hence eq 8 can, to a negligible approximation, be given as
Equation 9 is, surprisingly, identical with eq 1 and the expression originally given for this problem.'P2 Thus the second and higher virial coefficients have a negligible influence on the melting point depression. We thus have the theoretical expectation that deviations from the infinite molecular weight limit (eq 2) will only occur a t very low
molecular weights a t high dilution. Other theoretical developments of this problem have also been given in the literature.'* Beech and Booth7 followed a similar method of analysis but did not have occasion to actually derive the melting point relation. Pennings8 also followed a similar procedure, but the higher virial coefficient was expressed in a different manner. Sanchez and DiMarzio, on the other hand, only offered an empirical resolution to the p r ~ b l e m . ~ We examine the consequences of eq 9 by the theoretical plot that is given in Figure 2a. The interest here is in the low molecular weight range y = 25 (MN = 350) t o y = lo00 (MN = 14000). For this model calculation we have arbitrarily assumed a value of x1 = 0 and other parameters that are applicable to polyethylene in p-xylene. In the theoretical melting temperature-concentration plot in Figure 2a it is convenient to take the curve for y = 1000 as a limiting reference plot. In the concentrated polymer range there is very little influence of molecular weight on composition except for the obvious dependence of the melting temperature on chain length. However, in the dilute range, u2 I0.2 a very marked effect is expected of the molecular weight on the dissolution temperature with decreasing molecular weight. What could be approximated by a linear relation in the concentrated range shows definite curvature in the dilute range. The curvature becomes more severe as the molecular weight is lowered. A very significant decrease in the melting temperature is expected in the lower molecular weight range relative to a linear extrapolation from the more concentrated region. Equation 9 is analyzed in a slightly different way in Figure 2b. Here the expected relative melting point depression (l/TM - l / T M o ) is plotted against up. The deviations from the limiting form with dilution and low molecular weight are accentuated in this type plot. The very significant expected change in melting temperature at a fixed concentration in the low molecular weight range can be clearly seen. At high molecular weights, y 2 500, an imperceptible change would be expected from the infinite molecular weight expression. In the next section we examine the experimental results in terms of these theoretical expectations. Experimental Results. Melting Temperatures. In order to assess the validity of the analysis that has been given here, as well as others that have been reported, we have made an extensive study of the melting temperature-concentration relation of very low molecular weight polyethylene fractions in p-xylene over the entire composition range. As has been pointed out above, there are two reasons why this work is restricted to low molecular weights. An equilibrium theory is involved, a condition which can only be achieved by the use of extended chain crystals. A t present this condition requires the use of low molecular weights. In addition, the major deviations from the limiting form are only predicted to occur a t low molecular weights. Thus there is the natural focus on the low molecular weight region. The actual dissolution temperatures in p-xylene that were obtained with the low molecular weight fractions over the complete concentration range are plotted in Figure 3. The solid curves in this figure represent the theoretical expectations. They were calculated according to eq 9 taking x1 = 0.2. This parameter was previously determined independently.20 Examining the results first from a qualitative viewpoint we note that the major theoretical expectations of eq 9 are fulfilled. In the dilute range, u2 = 0.001-0.05, deviations from the limiting form are found which are enhanced with
918 Prasad and Mandelkern
Macromolecules, Vol. 22, No. 2, 1989 I
I
I
I
I
I
I
1
1
I
I
400
300
360
y
c-€ I
340
:1-
v
320
30r 280
0.0
0.6
0.4
0.2
I0
0.0
lJ2
VZ
Figure 2. (a) Plot of theoretically expected melting temperatures, calculated from eq 9, as a function of u2 for indicated chain lengths. (b) Plot of l/Tm- l/Tmo,calculated from eq 9, as a function of u2 for indicated chain lengths.
32r
i
300
I
0
I
I
02
1
I
1
04
1
06
1
I
08
I
1
IO
r2
Figure 3. Plot of melting temperature against u2 for indicated number-average molecular weight. Solid curves theoretical expectations calculated from eq 9. Symbols actual experimental
results.
decreasing chain length. These deviations appear as downward curvature as the dilute range is approached. For example, there is a decrease of about 10 "C for the lowest molecular weight fraction in the range of u2 = 0.05-0.001 and about 5 "C for the highest molecular weight fraction studied in the same concentration range. This kind of result can be anticipated from the earlier work of Koningsveld and Pennings28 who, without concern for the dilute solution problem, used eq 1 as directly derived for concentrated systems. The experimental results quite clearly bring out another important point. Because of the
curvature of the T, - u2 plots in the dilute range it is incorrect to linearly extrapolate the data from u2 = 0.05 to u2 = 0.001. This extrapolation procedure was adapted to determine melting temperatures at lower concentrations when the actual experimental determinations were terminated at u2 = 0.05.1° The error introduced by this procedure depends on the molecular weight. The analysis of the crystallization kinetics from dilute solution, at concentrations less than u2 = 0.05, will be affected since it depends on a reliable value of the equilibrium melting temperature. From a quantitative point of view the plots in Figure 3 demonstrate that there is excellent agreement between theory, eq 9, solid curves and experiment for the four lower molecular weights over the entire composition range, from pure polymer to very dilute solution. For the two higher molecular weights there is also excellent agreement in the concentrated range. Only small discrepancies are found in the concentration range u2 I0.2. In fact the only deviations that are observed from theory are in this dilute range for the two highest molecular weight fractions. The possible experimental errors in molecular weight do not reduce this discrepancy. NakajimaB has suggested using the complete formulation for x1 (eq 10) in the melting point equation. 1
Xi =
2
+
K 1 =
1 2
--*I
*le +T
(10)
where \kl and K~ are the entropy and enthalpy parameters respectively. 0 is the theta temperature. Analyzing the melting point data according to the method he suggested did not help in removing the small disparity. It would appear, however, that the fault would be in the temperature and concentration dependence of xl,a quantity which still remains to be formulated in a completely precise manner. Irrespective of the small discrepancies in this region it is quite apparent from Figure 3 that by far almost all of the experimental results are in strikingly good accord with the theory that has been developed.
Macromolecules, Vol. 22, No. 2, 1989
Dissolution Temperature of Polyethylene Fractions 919
3601
t'I
I '
t
i
I t G"
Figure 4. Comparison of the observed melting (dissolution) temperature as a function of molecular weight with theoretical expectations: (A) uz = 0.001; (B) u2 = 0.05; A, 0, from ref 10; A, 0 from this work; solid curve, from eq 9, ---, from ref 9; - -, from ref 8.
-
I t is appropriate a t this point to compare the results obtained here with other data in the literature and other formulations of the melting temperature-composition relation in the dilute range!pg (The concentration scale used in ref 10 is ambiguous. We have assumed for present purposes that the concentration is expressed as weight percent of polymer.) We have made the comparison at two compositions, up = 0.001 and u2 = 0.05, which are presented in parts a and b, respectively, of Figure 4. The comparisons at the two different compositions lead to essentially the same conclusions. The small differences between the observed melting temperatures result from the extrapolation method and the concentration scale used.1° Although at the higher molecular weights there are only small differences in the melting temperatures, these differences become very important in analyzing the crystallization kinetics. As has been pointed out previously, the experimental data adhere quite well to the theory developed here. The equation given by Pennings appears to be the furthest removed from the experimental data. However, the general approach to the problem was the same as that given here. The only difference was the way the third virial coefficient was introduced and the molecular weight dependence of the second virial coefficient that was used.8 In contrast, Sanchez and DiMarziog presented an empirical relation which uses the Flory-Vrij18 values for pure nhydrocarbons and the Pennings relations. These results are represented by the dashed curve in Figure 4. The position of this curve is purely fortuitous. It attempts to raise the melting temperature calculated by Pennings by introducing those for the pure state of molecular crystals from the form calculated by Flory-Vrij.18 We conclude this section by observing that the classical melting temperature-composition relation is applicable in dilute solution resulting in a continuous function that is applicable over the complete composition range. Deviation from the limiting high molecular weight form is only observed in very dilute solution at low molecular weights. Crystallization Kinetics. Studies of the isothermal crystallization kinetics from dilute p-xylene solution, by measuring the rate of growth of the (110)faces, of four low molecular weight polyethylene fractions have been reported.30 The molecular weight range M , = 3100-11 600 was studied. For the higher molecular weight fractions the growth rate decreases continuously with crystallization
temperature in the usual and expected manner. However, for the two lower molecular weight fractions, M , = 3100 and M , = 4050, the growth curve shows a discontinuity, or notched, appearance with temperature. The analysis of these latter sets of data appeared to be quite complex and involved the introduction of some new concept^.^^,^^ However, in analyzing this type of crystallization kinetics data, a precise knowledge of the equilibrium melting or dissolution temperature is needed. We have shown in the present work that for this specific system that the dissolution temperature varies with concentration in the range of interest. The initial analysis that was presented of the crystallization kinetics data was performed with the equilibrium dissolution temperature being taken as independent of c o n ~ e n t r a t i o n .Consequently, ~~ it seemed advisable to reanalyze the kinetic data in the low molecular weight range using the newly determined equilibrium melting temperature at each concentration. We have already determined the required equilibrium melting temperature for the fraction M , = 3100 and MN = 2900 (fraction F in this study).which is one of the fractions where kinetics was studied. We are, therefore, in a position to analyze the crystallization kinetics of this fraction in detail. We have also attempted to analyze the crystallization kinetics of the fraction designated MN = 3900 and M , = 4050.30 However, during the isothermal crystallization, at a temperature to ensure formation of extended crystallites, molecular fractionation took place. Consequently, the equilibrium dissolution temperatures a t the appropriate concentrations could not be determined which thus prevented a reanalysis of the kinetic data. In analyzing the crystallization kinetic data for the fraction M , = 3100 and MN = 2900, for concentrations u2 = 5X and 5 X we have utilized the simplest expression for a nucleation controlled lamellar growth rate. The growth rate, G, is assumed to be controlled by coherent two dimensional nucleation theory. No a priori assumptions need to be made with respect to the type of nucleus.32 In fact, the expression given in eq 11 is equally applicable to polymers as well as low molecular weight systems. Accordingly, we can write
G = Go@exp ( R K Z T ) where Gois essentially a constant, @ is an exponential term representing segmental mobility, u and ue are respectively the lateral and longitudinal surface free energies for nucleation, T, is the crystallization temperature, and AT is the undercooling. The constant K specifies the type nucleation and the regime in which the crystallization is occurring. The data for the four concentrations of fraction MN = 2900 and M , = 3100, whose kinetics were studied, are plotted in Figure 5. The three lowest concentrations yield straight lines. For u2 = the straight line is excellent; for u2 = 5 X and there are small aberrations which can be attributed to very small experimental error. The slopes decrease continuously, from -41 400, -40 000, and -30 800, as the concentration decreases for the three sets of data. The results for u2 = 5 X are quite different. In this case the data are well represented by two intersecting straight lines (see insert in Figure 5). The slope in the low-temperature region is -42 800 and is very similar to that for u2 = and the trend established at the lower concentrations. The ratio of the slopes of the two straight lines that is drawn in the figure is 1.9. This ratio can be interpreted as representing a transition from regime I to 11.
Macromolecules, Vol. 22, No. 2, 1989
920 Prasad and Mandelkern
observed in the experimental growth-temperature plot vanish when the data are plotted according to classical nucleation theory (except for the regime change).
3'0 Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. L.M. wishes to acknowledge the very helpful and stimulating discussions with the late Professor P. J. Flory in the derivation that was originally presented.6 Registry No. PE, 9002-88-4. References a n d Notes
4 -o -L -
IO
I5
I x Tc (AT)
20
2 5
lo4
Figure 5. Plot of log of growth rate G against temperature function for different concentrations: up = 0.001 0;u2 = 0.005, A; v:, = 0.01, 0;~2 = 0.05, 0 . M , = 3100; M , = 2900.
We note that these results are quite different from that originally given for the same data plotted in an identical manner (see Figure 12 of ref 30). These differences are due to the choice of the equilibrium melting temperature. For u2 = 0.05 an opposite change in slope was given; Le., although represented by two intersecting straight lines the low-temperature slope was greater than the high temperature one. The plots for the three lower concentrations were also represented by two intersecting straight lines with the low-temperature range having a larger magnitude. This reanalysis of the kinetic data, with directly determined equilibrium melting temperatures, yields major differences and will require a reexamination of the original interpretati~n.~~ It is not our intent here to analyze in any further detail the crystallization kinetics of these low molecular weight systems. The results are obviously part of a more general problem and should not be considered out of context. However, some observations can be made. The importance of reliable values for the equilibrium melting temperature is quite evident. The results for this low molecular weight polyethylene diluent can be considered classical in a formal sense. More complex analysis is not necessary to explain these kinetic data.30i31 The system a t u2 = 0.05 goes through a regime 1-11 transition as developed by Hillig33 and Calvert and Uhlmann34 for low molecular weight systems and theoretically adopted to long-chain molecules by Sanchez and DiMarzio= and L a u r i t ~ e n .It~ ~has been observed in a large number of bulk crystallized systems. The transition from regime I to regime I1 has also been reported for high molecular weight polyethylene in dilute s~lution.~'The kinetics for the lower concentrations have been assigned to regime I1 by reason of the values of the slopes of their temperature coefficient. The role of concentration in determining this transition is obviously in need of further study. The true discontinuities that are
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