Molecular weight dependence of isothermal long period growth of polyethylene single crystals

Molecular We ,ht Dependence of Isothermal Long Period Growth of Polyethylene Single Crystals A. PETERLIN The long period growth L of polyethylene single crystals during annealing requires an increase in length of all straight sections of every single macromolecule between the fold planes, The material needed has to be transported from the free ends toward the centre of the macromolecule. In order to explain the temperature dependence of long period growth rate, the transport has to proceed by collective jumps of all monomers of one straight chain section between the fold containing surfaces. The number of such jumps needed for a given thickness increase in unidirectional motion or random diffusion is proportional to (M /L) 2 and (M /L) 3 respectively. The growth rate hence turns out to be strongly dependent on molecular weight M. In the most probable case of random diffusion the L versus log t curves are shifted by 3 log M to higher times. The growth rate extremely rapidly increases when the extended chain length is only two, three or even four times larger than the long period L. The transformation of the folded into extended chain crystal in such a case is achieved in a time which roughly equals the time needed for isothermal single crystal growth to Lext./4. This time, of course, is shorter the lower the molecular weight and the higher the temperature. Such a mechanism may eventually explain the formation o5 extended chain crystals observed in fractionated samples of low molecular weight isothermally crystallized from melt close to 130°C and in whole Marlex samples crystallized from melt under high pressure at 230°C.

TIaE experimentally established 1-° long period growth of polyethylene single crystals during annealing at constant temperature and especially its dependence on the logarithm of annealing time ~,~ can be written as 7 L = L * [1 + 2"303B log (t +C)] (1) L* the ordinate intercept of the asymptote (C=0) at t = 1 (see Figure 17), C = e x p [ ( L 0 / L * - 1 ) / B ] with L0 crystal thickness at t = 0 , 2"303L*B slope of the asymptote in the plot of L over log t. A theoretical explanation was suggested by considering the surface nucleation 5 and the mechanism of mass transport 7 to the fold planes from the free ends of the macromolecules. The latter may either proceed by successive individual jumps of a single dislocation or by collective jumps of aU the monomer units of one straight section of the macromolecule between the fold containing surfaces. It was shown in a previous pape r7 that collective jumps lead to a temperature dependence of L * B in agreement with experimental data so that only this case will be treated in that which follows. Some experiments with fractionated polyethylene by Takayanagis show a substantial dependence of crystal thickness growth on molecular weight. The smaller M the larger seems to be L * B . Such a dependence can be

25

A. P E T E R L I N

derived from our model by more detailed consideration of the transport mechanism, particularly of the number of jumps necessary for a thickness increase by d0=2'53A, the length of a monomer unit. This number depends strongly on molecular weight of the sample. With short molecules the transport ways from the ends through the centre of the molecule are relatively short so that the growth rate is larger than with long molecules. ACTIVATION

ENERGY

OF

CRYSTAL

THICKENING

The free energy increase for formation on the fold containing surface of a nucleus of lateral dimension x (quadratic cross section) which is by do thicker than the original crystal read# A F = 4xdo o" - 2x~doo', / L (2) where ~ and o'e are surface energy density at lateral and fold surface respectively, and L is the long period. For a circular cross section one would have ~ instead of 4 in the first and ~/2 instead of 2 in the second term. The uncertainty due to the special choice of cross section is actually negligible. The maximum of free energy increase AF~r=2d0Lo-2/o-, determines the lateral dimensions x = x * = o - L / o - , of the critical nucleus. It just represents the activation energy for surface nucleation by which the crystal increases its thickness. If the nucleation process is the slowest step in long period growth, the thickening rate turns out to be d L / d t = A exp ( - A F * / k T ) = A

exp ( - b ' L )

(3)

By comparison with equation (1) one obtains A = L * B exp (1/B) b'= 2doo2 /o-ekT = 1 / L * B

d L / d log t = 2- 303L~B = l'601o-ekT/d0 o-2 (4) With o-~=49, o-= 12.2 erg/cm 2 and d0=2-52 A suggested by Hoffman and Weeks9, one obtains for the slope of L versus log t curves 2"303L*B=O'208T [A], i.e. 83"6 A at 130°C. The nearly twice as large o-, value postulated recently by Mandelkern et al. 1°, Frank and Tosi 11 and by Wunderlich12 yields 0"4T, i.e. 160A at 130°C. The slopes according to equation (4) have the correct order of magnitude although they are markedly larger than the observed values, but completely fail to show the experimentally established strong temperature dependence of L * B which between 100 ° and 130 ° increases by a factor of more than ten (Figure 1). An additional increase in temperature dependence of L * B is produced by the mass transport mechanism from the free ends of the macromolecule to the fold planes. If one assumes that, in order to transport a monomer unit into the latter, the whole straight section of the chain between the fold planes jumps as an entity by do, the activation energy for such a collective jump is LE,~/do (where E,~ denotes the energy bander between consecutive equilibrium positions of the monomer and L/do is the number of monomers in the section). The growth rate hence, besides, the above mentioned activation energy for nucleation, also contains the activation energy for mass transport and reads 26

LONG PERIOD GROWTH

OF POLYETHYLENE

S I N G L E CRYSTALS

d L / dt = A exp ( - L E , , / dokT - 2d0Lo-2/o-ekT) = A exp ( - b L ) b =(Era~do + 2doo'2/°',)/kT

(5)

yielding the slope L * B = 1 / b = k T / (E,, / do + 2d0 o-2/o-,)

(6)

This expression differs from equation (4) by the barrier energy term E,,/do which not only reduces the too, high values of equation (4) but also provides, by its temperature dependence, an explanation for the experimental data as shown in Figure 1. T h e rapid increase of L * B when approaching the

120- 2'303L*B

]

9O

Figure / - - L o n g period growth rate coefficient 2"303LYrB=dL/dlog t from equation (1) as a function of tempera- 60 ture according to Fischer and Schmidt G. The broken line 0.208T is calculated from equation (4) with Hoffman and Weeks's data 9 for o-Jo-2=0.33 30

i

100

110

P

120 T, °C

I

I

130

14.0

melting point is a consequence of the rapid decrease of E,, in good agreement with different estimates of the lattice force field as a function of temperature 13. With increasing unit cell expansion, the energy barriers separating the equilibrium positions of monomers rapidly decrease and nearly vanish at the melting point. The activation energy E,, derived from experimental values L * B agree fairly well with data calculated from unit cell expansionL Equations (3) and (5) differ in the value of the coefficients /~" and b but are identical as far as the dependence on L and t is concerned. By introducing homogeneous coordinates X = b'L or b L respectively and r = A t both can be written as d ~ / d r = e -x (7) with the general solution e x = ~-+ %

(8)

By putting the integration constant % = 0 one obtains the asymptotic solution exp (X)=r or 2,=2.303 log ~- plotted in Figure 3. DETAILED

TRANSPORT

MECHANISM

The coefficient A in equation (5) contains all other factors which, besides the activation energy, affect the mass transport. A m o n g them the most important are the frequency ~ of longitudinal oscillation of the straight segment and the number of jumps needed for the transport of monomer units from the free ends to the fold planes. The frequency ~ given by the 27

A. P E T E R L I N

restoring force f and the mass M0 of the monomer as v=(flMo)~/2~ turns out to be independent of the length L of the straight section and of the degree of polymerization P = M / M o . It therefore does not affect the growth rate as far as its dependence on L and M is concerned. The number of jumps needed for an increase of long period by do, however, is smaller the shorter the chain and the longer the straight sections. A chain of length Pdo has n = P d o / L straight sections (Figure 2). In order to increase the thickness by do, one has to transport n monomers from the free ends into internal straight sections between the fold planes, n / 2 from each side. If the transport is unidirectional, i.e. every jump is in the right direction so that it transports monomers from the free ends to the internal folds, and if, for sake of simplicity, one assumes that the thickness increase is achieved by adding one monomer to each fold so that the crystal grows in thickness by do/2 at each fold containing surface, the extreme sections have to jump ( n - 1 ) / 2 times each (n odd), the next section ( n - 3 ) / 2 times and so on through the central section which does not have to jump at all. If n is

2

6

4

n-5 n-3 n-1

10

5

7

9

macromolecule in folded chain crystal

with n=Pdo/L straight sections and n - I folds. I,n order to increase the thickness by d~ (length of monomer), one has to add one monomer at every fold, i.e. altogether n/2 or n / 2 - 1 monomers have to be transported from each end of the macromolecule

L

J 1

Figure 2--Polyethylene

n-6 n-4n-2

11

even, one has to start on one side with n,/2 and on the other side with ( n / 2 - 1) jumps and then decrease the number by one at every subsequent section. Altogether one needs ( n - 1) (n+ 1)/8 N n2/8 =(Pdo/L)2/8 jumps from each end of the macromolecule. The growth rate hence reads

d L / d t = 8~ (L/Pdo) 2 e-bZ=aeL 2 e -bL a" = 8v/p2d2o

(9)

or in homogeneous coordinates ~ = b L , r = a r t / b = t r * d~t/dr =X 2 e -x

(10)

The solution turns out to be

Ei (h) -eX / X = r+ r o A

Ei (X)=

-

f

(e=/x) dx

(11)

0'3725

The integration constant % = C r * is given by the long period value L0 at 7 = t = 0 . The constant 0-3725 corresponds to the function tabulated by 28

LONG PERIOD GROWTH OF POLYETHYLENE SINGLE CRYSTALS Jahnke and E m d e 14. T h e asymptotic solution according to equation (11), plotted in Figure 3 as a function of log ~-, is situated above the values X = 2 3 0 3 log r of the asymptotic solution of equation (7). This is a consequence of the factor 3.2 in equation (10). However, since there is no force pulling the m o n o m e r s and the straight sections in the right direction, unidirectional motion is rather unlikely. One has to assume that the whole transport proceeds by random diffusion. In 25

20

Figure 3--Reduced long period (asymptotic solution, C=0) as a function of log r according to equation (7) [1], equation (11) [2] and equation (15) [3]. The growth of initial length X0 with time is also shown. From ~0 one obtains the integration constant r0. The ratio r0/r* determines how rapidly the crystal grows

32

155 I0

I

T

such a case the n u m b e r of jumps for transporting one m o n o m e r f r o m the end of the macromolecule to any fold is equal to half the square of unidirectional jumps needed for the same transport. Consequently, the total number of jumps f r o m one end of the macromolecule for increasing the thickness b y do turns out to be ( n - 1)n ( n + 1)/24 ~-, n~/48=(Pdo/L)~/48

(12)

The rate of thickness growth reads

d L / d t =- 48v (L/Pdo) ~ exp ( - b L ) = a L 3 exp ( - bL) a=48~ /l~d~o

(13)

or in homogeneous coordinates 3' = bL, ~"= at/b 2 d 3 ' / d r = 37 e -x

(14)

f (3")= ½ [Ei ( 3 ' ) - eX/3. - e x /3.2] = r + r o

(15)

with the solution (Figure 3)

T h e corresponding plot of the asymptotic solution as a function' of log r

(Figure 3) is situated above that f r o m equations (7) or (11). A comparison of the three curves shows only slight changes in slope due to the factors 3.2 and 3.~. W i t h the exception of the area at rather small r 29

A. PETERLIN between 1 and 102 where the curves 1 and 2 increase more rapidly, the slope is roughly 2-303 yielding 2"303/b for tho L versus log t plot. The details of the transport mechanism (unidirectional flow or random diffusion) only influence the constant a which shifts the log time scale against that of log ~" but in practice do not influence the slope. This result also applies when the mass transport is performed by individual jumps of Reneker's type dislocations. For unidirectional motion one obtains d ~ / d r =)t exp ( - ) t ) and for random diffusion )t2 exp ( - ) t ) with Z = b'L [see equation (3)]. In a )t versus log r plot the corresponding solutions ~-=Ei (A) and Ei ()t)-exp ()t)/)t have slopes practically identical with exp (A), i.e. 2"303/b'. Therefore, also with mass transport by individual jumps of dislocations the detailed consideration of the number of jumps needed for long period growth does not markedly change the much too small temperature dependence of the growth rate r. In order to obtain the time dependence of long period for a crystal with initial thickness )t0 at r = 0 one has to determine the integration constant ro = f ()to), i.e. the abscissa of the intercept of the asymptote with the horizontal line through the ordinate )t0. The complete solution )t as a function of (r + %) starts as a horizontal line with the ordinate intercept )t~. With approaching r = r o it gradually bends into the asymptote as shown in Figure 3. The transition is rather sharp so that the deviation from the horizontal part or from the asymptote is below one per cent when log r differs from log r 0 by + 1. The curves in Figure 3 explain very well the extremely rapid growth of long period when )to is smaller than )t*, the asymptotic reduced crystal thickness at t = 1, i.e. at r * = a / b 2 (random diffusion) or a'/b (unidirectional transport) respectively. In such a ease the constant C = b2ro/a or br.o/a( respectively is smaller than the time unit (for instance, 1 min) and the crystal grows from )to to )t* in a time smaller than the unit. If, however, the crystal has an initial thickness )to > )t* the constant C is large, the growth is drastically reduced and may be even unnoticeable during normal observation time. The value r 0, or still better r o/r*, determines the time scale for long period growth during isothermal annealing of crystals with initial length )to. The curves in homogeneous coordinates )t, r may be adapted easily to specific experimental data, i.e. to the intercept L * and the slope 2.303L*B at the intercept (t = 1) by considering the following relationship d)t/d lnr=~ l(d~ ldX)=rX3 /e ~ )t*/(d)t/d l n r ) * = L * / ( d L / d In t)* = L * / L * B = 1/B = exp)t~/r*)t .2

(16)

B = (dL / d log t)* / 2" 303L* r* = B eX~r/ )t ~r2 The last equation can be solved graphically by plotting, in addition to the curve )t as a function of log~-, the curve r = B e x p ) t / ) t 2 (Figure 4). The intersection yields )t* and 7- corresponding to L* and t = 1 respectively, and hence b = ) t * [ L * and a = r~rb2. 30

LONG PERIOD GROWTH OF POLYETHYLENE SINGLE CRYSTALS From the data of Fischer and Schmidt one obtains L * = 2 4 0 A and 2 . 3 0 3 L * B = 6 2 A for Ta=130°C yielding 1/2.303B=3"84 and l o g r = A / 2 . 3 0 3 - 2 1 o g X - 0 " 9 4 7 . The latter curve is nearly parallel to r = f (;~) so that graphically the intersection cannot be determined with high t ; ~ . ~ ~ I~

Figure 4--The curves ~ r = f 0Q and r----Bexp ~/A2. 10 The intersections 1., 2" and 3" determine r* and .~ ~* for 1/2"303B=L*/ (dL/d log t)-k= 2.84, 3 . 8 4 7 J and 4"84 respectively

S

~

r= f (x)

/

0

101

100

i

I

10 2

103

m

,/.

precision. One obtains )~*=12.36 and r * = 2 - 2 4 yielding b=0"0515 A -1 and a = 2"24 min- 1. In Figure 4 two additional curves with 1 / 2' 303B = 2 84 and 4.84 are plotted, leading to X*=10-16 and 14'55, r * = l ' 6 0 and 2"95 respectively. The higher B the more the r = B exp )~/A= curve is shifted to the right, leading to a smaller 2~* and r * and consequently to a smaller b and a. The molecular weight dependence of long period growth is contained in the factor a. According to equation (13), it is inversely proportional to the third power of molecular weight

aM=at (M~/M) 3

(17)

where a, corresponds to M0. The same applies to the homogeneous coordinate r:~ = r0 (M0/M) 3 (18) The )t curve consequently is shifted horizontally by 3 log (M/Mo) in the ;~ versus log r plot (Figure 5). The curve in the centre corresponds to annealing of Martex 50 at 130°C, the left and the right one to a polyethylene sample with ten times smaller and larger molecular weight respectively. With any initial long period L0 of polyethylene crystals one obtains a family of curves [equation (15)] corresponding to different molecular weights. The smaller P or M the sooner the horizontal section bends in the asymptote, the shorter the times for significant increase of long period. At any given time the crystal thickness, due to isothermal annealing, is markedly greater for low than for high molecular weight samples. The resulting change in long period growth rate is spectacular. Single crystals with 2 0 0 A initial length grow to 300 A thickness in 10 -2, 10 and 104 minutes, 400 A in 0-4, 4 x 10~ and 4 x 105 minutes, respectively. However, 31

A. P E T E R L I N

one must not forget that some modifications of our final results [equation (15)] are necessary as soon as the extended chain length Pdo is not a great many times larger than L. This correction has first to be applied to the left curve in Figure 5 with the smallest molecular weight. f 1 I

] O=r "k I

1

600

10 4 I

10 6

I

l

]

10 s I

I

[

/

.~/lO

-~30

400

~'300 a*

oo ,oo

0

I 10 -2

l

l 10 0

l

i 10 2 t. m i n

I

I

I

10 4

I /0 10 6

Figure 5 - - H o r i z o n t a l shift o f asymptotic ~ versus log r curves with degree o f polymerization o f polyethylene. The central curve corresponds to annealing at 130°C o f Marlex 50 single crystals (r~r=2-24, X ~ = 12"36)

With mass transport by individual jumps of Reneker type dislocations the molecular weight dependence is similar to that in equations (17) and (18) with the second instead of third power in (Mo/M). Consequently the time scale in h versus log ~- curves is shifted horizontally by 2 log M, the molecular weight dependence is a little smaller than in the above treated case of collective jumps. Because annealing also takes place during isothermal crystallization 15, the long period in the solidified sample will be larger for low than for high molecular weight polyethylene. The effect is more conspicuous the higher the chain mobility, i.e. the closer the crystallization temperature to the melting point. At lower temperatures the growth rate soon becomes so small that the increase of long period during isothermal crystallization is not very important. LONG

PERIOD

GROWTH

AT

SMALL

Pdo/L

The above considerations are a rather good approximation for large values of n=Pdo/L but they fail at small a because one has replaced the correct expression for the number of necessary jumps by the one term with the highest power of n [equation (12)]. Moreover, one completely neglected the fact that as a rule the sections containing the, free ends of the folded macromolecule are shorter than L and consequently, due to a smaller activation energy, can jump much faster than a full length section. As a consequence, with low n, e.g. 2, 3 or even 4, the long period growth during isothermal annealing is markedly accelerated a s compared with our data 32

LONG PERIOD GROWTH OF POLYETHYLENE SINGLE CRYSTALS for large n. In this case, only the sections containing the free ends of the macromolecule have to jump in order to increase the long period. The first jump at the most requires an activation energy EmL/do, the next one Em ( L / d o - 1 ) and so on. The long period growth proceeds faster and faster like an avalanche until such a crystal transforms into. an extended chain crystal. Let us assume that at the beginning one has Pdo =4L0, i.e. four straight sections are parts of a single macromolecule. For the sake of simplicity, the number of monomers in the three folds will be neglected. The length L' of the section containing the free end of the macromolecule is connected with the long period L by L" + L = 2 L o

(19)

The growth rate of crystal thickness turns out to be dL / dt =(~do/2) exp [-(L'E,~/ do + A F * ) / kT] =a" e -b'L a" = (~d°/2) exp ( - 2L.E,, / dokT)

(20)

b"=(E,,,/ do - 2d,,~r2/o-~)'/k T with the solution e-X=r0

- r

= b"L

3.o = b"Lo

r = a"b"t

ro = e -xo

(21)

The crystal redoubles its thickness ()t~=2X0) in % = r0 ( 1 - r0), i.e. in time t. where to = exp ( - b"Lo) [ 1 - exp ( - b"L0)] / a"b" = e x p (bLo) x [1 - e x p ( - b"L0)] 2kT/~do (E,,/do -2d~oo'2/o-~)

(22)

The main contribution is that of the first factor. If, in the brackets, one neglects the negative term, t2 roughly equals the time needed for crystal thickness to grow to Lo from any initial value lower than L0. This may be a very short time. A similar consideration can be applied to the last step by which the thickness increases from 2L0 to 4L0 and the folded chain crystal transforms into an extended chain crystal. Such a transformation may occur preferentially with low molecular weight samples where the extended chain length Pdo is not a great many times larger than L. In the case studied by Anderson l°, samples with M below 12 000, i.e. Pdo below 1 100 A, yielded extended chain crystals when slowly crystallized at nearly 130°C. At such a temperature L0 is between 200 and 300 A, and one has exactly the case of avalanche-like long period growth. At a lower crystallization temperature, however, the starting long period and, to a still higher degree, the mobility are so much smaller that folded chain crystals may persist and remain nearly unchanged during crystallization. This applies particularly when crystallization occurs during rapid cooling or quenching. Of course, the lowest temperature at which, during isothermal crystallization, the initially folded chain crystals are 33

A. P E T E R L I N

transformed into extended chain crystals depends on the molecular weight of the sample. It decreases with decreasing M. On the other side, with larger M one had to go so close to the melting point in order to obtain a sufficiently large long period L ~ Pdo/4 that such an experiment is very likely to, be extremely difficult to perform. Extended chain crystals obtained with unfractionated polyethylene at high temperature (230°C) and high pressure lr may very likely be explained in a similar way. Due to the high pressure the free energy difference between the crystal and the melt is reduced, the length L, of the minimum stable nucleus increased. If now this length becomes close to one quarter of the extended chain length of polyethylene molecules in the sample, the avalanche-like long period growth rapidly transforms the folded into extended chain crystals. The non-uniform length of these crystals is most likely due to the local fluctuation of average molecular weight.

The author gratefully acknowledges the financial support o[ this work by the Camille and Henry Drey[us Foundation. • Camille Dreylus Laboratory, Research Triangle Institute, P.O. Box 490, Durham, N.C. (Received April 1964) REFERENCES 1 KELLER, A. and O'CONNOR, A. Disc. Faraday Soc. 1958, 25, 114 2 RANBY, B. G. and BRUMBERGER,H. Polymer, Lond. 1960, 1, 399 a STATTON,W. O. and GEIL, P. H. J. appl. Polym. ScL 1960, 3, 357 4 ST^Trout, W. O. d. appl. Phys. 1961, 32, 2332 HIRAI, N., YAMASHIT^,Y., MATSUHATA,T. and TAMURA, Y. Rep. Res. Lab. Surface Sci., Okayama U. 1961, 2, 1 s FISCHER, E. W. and SCHMIDT,G..4ngew. Chem. 1962, 74, 551 7 PETERLIN,A. Polymer Letters, 1963, 1, 279 8 TAKAYANAGI,M. Rep. Progr. Polymer Phys. Japan, 1964, 7, 77 0 HOFFMAN,J. D. and WEEKS, J. J . / . chem. Phys. 1962, 37, 1723 10 MANDELKERN, L., POSNER, A. S., DORIO, A. F. and ROBERTS, D. E. J. appl. Phys. 1961, 32, 1509 11 FRANK, F. C. and TosI, M. Proc. Roy. Soc. A, 1961, 263, 323 13 WUNDERLICH,B. J. Polym. Sci..4, 1963, 1, 1245 13 PETERLIN, A., FISCrIER, E. W. and REINHOLD, CHR. J. Polym. Sci. 1962, 62, 559 PETERLIN, A., FISCHER,E. W. and REtNHOLD,Crm. J. chem. Phys. 1962, 37, 1403 ~* JAHNtO~, E. and EMDE, F. Tables of Functions. Dover: New York, 1945 15 WEEKS, J. J. J. Res. Nat. Bur. Stand. A, 1963, 67, 441 PETEV,LIN, A. J. appl. Phys. 1964, 35, 75 16 ANDEaSON, F. R. Paper BB4 in Electron Microscopy, edited by S. S. Breese, Jr. Academic Press : New York, 1962 ANDERS.ON,F. R. J. appl. Phys. 1964, 35, 64 x~ GEm, P. H., ANDEaSON, F. R., WU~ERLICH, B. and ARAKAWA, T. J. Polym. ScL A, 1964, 2, 3707

34