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A study of growth rates of polyethylene single crystals
Journal of Crystal Growth 18 (1973) I 11-123 ~) North-Holland Publishing Co.
A STUDY OF G R O W T H RATES OF POLYETHYLENE SINGLE CRYSTAI2~ A. K E L L E R and E. PEDEMONTE*
University of Bristol, If. H. Wills Physics Laborator); Royal Fort, Tyndall Avemte, Bristol BS8 ITS, En~lar~d Received 31 October 1972 "[ he lateral growth of polyethylene single crystals was followed by a prev4ously established elec~ronmicroscopic sampling technique. Having reconfirmed the linearity of this growth, the growth rates were determined as a function of concentration at various crystallization temperatures for different molecular weights and in two solvents, in agreement with earlier work the growth rate (G) was found to depend on concentration fc) as G occ J, where 0cis a constant. In the usual whole polymer and in the high molecular weight fraction the rates showed only slight, oncentration dependence (0t ~ I) hardly affected by the crystallization temperature whde in the lower tool, ,:ular weight material, particularly for sharp :'r~,ctions, the growth rates display(d a pronounced concent, ation dependence and increasingly so for the Mgher crystallization temperatures yielding 0t values up to 2. These observations could be satisfactorily accounted for when considering the mechanism of the folded deposition of the chains, and in particular the situation, unique for polymers, that a given chain can be partly attached to the crystals while parts of it remain in solution (thus forming cilia). Further, the growth rate values in themselves allowed inferences to be made on "entropy tension' during the chain deposition and the temperature dependence of these rates permitted deductions to be made as regards the surface energies involved which are broadly in line with expectation from the kinetic theories of chain folding.
1. Introduction
The rate of growth ts an tmportapt aspect tn crystellizatton studies in general. In polymers the crystal growth rate ts of additional slgmficance because in view of chatn folding, which according to present beliefs is a ktnettcally controlled phenomenon, crystal growth and the structure of the resulting crystal are closely Itnked. The term "structure" here includes all features associated with chain folding such as the fold length, the nature of the fold and irregularities along the fold surface. Further, because of the long chain nature of the molecules their deposi,.ion occurs in a sequence of steps which creates a situation distinct from anything else encountered in crystal growth (see reviews !, 2). 1"he growth of individual crystals can only be followed tn crystalhzation phenomena from solutton. Even so the following of the growth of any one unit prese.n.ts great di.ffi__o_,!ties.The practical so!ut,on therefore ~s to ensure that all crystals originate at the same t~me ~n which case the total crystal populauon can be representatively sampled by one, or by very few c~stals at any stage of growth. This con&tion can be realized b~ the sell" seeding method which is a way to introduce * Presen! address: Islituto Di Chimica Industriale. Universit~t GLnova. Via A Pastore 3. Genova, Italy
heterogeneous nucleation centres, in this case constituted by a stabilized form of the same polymer, which all start growing into observable crystals simultaneousiy3- s). For purposes of growth rate studies the first most important recognition is the fact that at constant temperature the lateral growth of the tabular crystals ~s hnear w~th time al all except the late stages of growth when the solution is nearing exhaustion6'7). Thus the isothermal lateral growth rate can be defined by one rate value. In view of the fact that the monolayer crystals in this case do not thicken during growth this defines the growth of the crystal as a whole. The growth rate thus determined can be studied as a function of different variables. The foremost of these is the crystallization temperature. The first work along these hnes was carried out by Holland and Lindenmeyer 6) on polyethylene samples ~ hich as we now know mustsimilar works (see review 2)the temperature dependence of the growth rate conformed to the kinetic theories of chain folded crystal growth. The concern of the present wor~ is the dependence of crystal growth rate on concentration. In this we shall follow up the preliminary studies by Blundell and Keller 7) on polyethylene. The principal finding of these 111
112
A. KFLt.ER AND E. PEDEMONTE
Fig. !. Electron micrographs of typical crystals used for the present work. They were obtained by self seeding which ensures :miform monolayer crystal population. The isothermal crystal1; -.ation was interrupted at the chosen time by sudden quenching. i t.~ size reached at that stage is defined by the step (arrowed). Th,' crystal portion outside the step has arisen during cooling from the intended crystallization temperature.
studies was a surprisingly weak concentration dependence of the growth rate, in fact the rate increases era3 with the cube root of the oncentration. Essential ~ the same conclusion has been reached (with only shght dtfferences m the numerical values) by at least tv, o further stu&es involving also another polymerS'9). Thus the phenomenon in questmn appears to be general Biundell and Keller suggest that the weak concenu'auon dependence must be assocmted w~th the shieldmg of the crystal face by chains which are still m solution but accumulated along the face. Such chains could be full molecules or portions of molecules where the rest is already attached to the crystal. The ~ork by Blundell and Keller 7) invited obvious extension. In particular, work on fractions is called for because molecular weight dependence of the effect was anticipated and fractionafion during crystal growth from the ur_,fractionated samples was to be expected as a consec uence. Secondly, the work only covered a small temperature range which obviously needed extending. Th~s was undertaken in the work now to oe repolted. in addmen, the new study was extend~ ~1 also 'o a s~ond solvent. 2. MaCerial~ The following materials were used throughout this
Unfractionated linear polyethylene, Marie× 6009, to be denoted M, with Mn = 6700, Mw = i0 s and
Mw/M, = 15. Molecular weigbt distributit~n as obtained with G.P.C. is shown in fig. 2. A high molecular weight fract,on of Marlex 6009, to be denoted FH, obtmned by precipitation from stirred solution by the method of Pennings, provided by Dr. F. M. Wtllmouth from these laboratories. The prec~pltation was from a 2t % solution at 102 "C. Here the stirrer induced crystallization has removed the low molecular weight end of the distribution. M, = 2.5 x 105. (G.P.C. curve of this particular sample is not available. Chromatograms of similar samples arc shown in ref. 10.) Several low molecular weight fractions were kindly supplied by Dr. Pennings, Dutch State Mines Laboratory, denoted F2, F4, and FI0. As seen from figs 2b-2d the fractions are of different polydispersity. FI0 has the sharpest molecular weight distribution with M, = 10000, M,,, = 16000, and Mw/Mn = 1.5. ~'--ror F2, M, = 2600 and M,~/M. is also 1.5. Nevertheless mspect~,~n of hg. 2d shows that the distnbutmn ~s not ~: simple o n e and there is a wide spread in the I Q ~ molecular weight side of the p~ak. in fact a seco@~::~ peak is indicated with a maximum at ~,~ = !700~o~(~ mainpeak being at M = 4000). F4 ha. a single ~ but - broader molecular weight distribution a s ~
113
A S T U D Y OF G R O W T H RATES OF P O L Y E T H Y L E N E SINGLE C R Y S T A L S
G, ,ulh
.. a
I,) !
3
~
S
?
i '
Ibl
,o
: F2/xylene
i
l!
I
3
.j
L
/,
:-
:
5
]
t.
01
s
iti ilii
//
I .
.
.
.
.
.
.
,/~,o to~o
t . . . . . . . . . . . . . 1 0. 2
1 1 0 "1
C,%
Fig. 3. Linear growth rate (pm/hr) versus concentration of crystals from F2 grown from xylene at dtfferent temperatures (r©).
(dl
J
3
l,
5 Log 14
-
~-J, .........
6
1
?
__
['lg. 2. Molecular wetght distributions as obtained from G.P.C. chroma'ograms (a) Sample M (ortginal Marlex), (b), It) and (d) are fracttons FI0, F4 and F2 respectively
directly from lig. 2c. Hcrc M. = 3250, Mw = 10000 hcnce Mw/ M. ~ 3 The solvents were commercial xylene and octane, distilled before use.
3. Experimental The experimental method adopted was idcnttcal to that used by Blundell and Keller?). Ftrst, the self seeding nuclei were obtained in the usual manner 3'4) and were used to intttate crystal growth at the desired concentration and temperature. The cD'stal population was sampled at any requtred stage ol growth by removing a few drops from the crystallizing solution and transferring them to a lower temperature.. The stage, where the original isothermal growth has been interrupted, ts marked by a discontinuous downward step in the essentially monolayer crystal ?) (fig. I). The linear crysta! dimensions up to that step were measured and
plotted as a function of time of the corresponding crystallization. The linear growtb "-,~tn time, at least during the initial stages of growth, was reconfirmed in all cases. The slope of the crystal size against time plot then yielded the growth rate which was then explored as a function of the other variables.
4. Results The primary results are represented by plots oi" linear growth rate (6;) against concentration (c) expressed in wt 9/o. These are presented for materials F2, F4, Fl0, M and FH in the two solvents, xylene and octane in figs. 3-11. Each of these figures contains G versus c plots on a logarithmic scale at different temperatures of crystallization (To). As seen the growth rate as a function of temperature define straight lines on a logarithmic plot (except for M, fig. 6, which defines two such lines) hence confirming the relation G oc c',
(l)
where ~ is the slope The slopes as a fanct~on of crystallization temperature are shown in figs. 12 and 13 for the t - o solvents in the case of each material.
114
A. K E L L E R
AND
E. P E D E H O N T E
f
t00 O F - -
FIO I xy lene
, IT, : t
; 6,p~h
¢
186.2
1,'o/084
~
b
100~
, l,-SlO-
t
f
/889
/
~ 140
/
/
..~
/ /
i
~c~o~-
,o F d
1001~
F4/xylene
,
/
/
lo,s
b 18a2 lOS8 ¢ ]845
105.5
i
O01L__
.L-
1
~0 ~ ....
I0 "3
10"2
C,%
10"1
1='18. 4. Linear 8~)wth rate (l~m/hr) versus concentration of crystals from F4 grown from ~.ylene at different temperatures (Tc).
104
C,'/,
Fig. 5. Linear growth rate (gm/hr) versus concentration of crystals from FIO grown from xylene at different temperatures
(r=)
0 28
:.lope G
1
I0"~
I ........
.
~
a
,u/h
~ J 10_
o,L
00t -
/ Mar!ex/xy lene
b c d
877 i 027 88 95 '10 30 90 5 i 0 335
g h
93 q I 0 36 942 0 44 g5 2 049
9
=/
/J"
t
I
~. . . . . . .
10"4 2 . . . . ~'.
~°. q ~ m / h r ;
~ _
__ 4__ .
. . . . . . . . . . .
10"3
t .
.
lC~'~
.
.
.
.
.
.
.
.
L..
10"1
c. %
vcrs~s concentration ofcrystals iron- sample M t~rowrt from x3,len© at different teml~ratu,r¢~ (~r~|
A S T U D Y OF GROWTH RATES OF POLYETHYLENE SINGLE CRYSTALS'
I|5
G. p l h G, p/h
10
FH/xylene TC = 8g OeC slope: 0,25
01 . . . . . . . . . .
1
_-
~°'4
10"a
Fi~. 7.
Linear gro vth rate p m / h r )
. L
10.1
. . . . . . .
I04
c,v.
J
~0"I
versus concentration of
10
crystals for sample F H grown' rom xylene at 89.0 °C.
~.,u/h
01 ~_A=_ . . . . . . . . 10" 2
± 10"1
. . . . . . .
I 10
C.%
Fig. 9. Linear growth rate (pm/hr) versus concentration of crystals for sample FI0 grown from octane at different tempera-
tures(T,). I00-
G.~/h
d
10
F4/n
Mope
b
93 ~ 94 5
c
96 3
0 25 0 37 0 73
a
d 011 10"3
; I0"2
I 10"I
- octane
I Tc ,'(
[i g ? 5
lO0;-
I
084
b
i J
( .%
Fig. 8. Linear growth rate (pm/hr) versus concentration of crystals for sample F4 grown from octane at different temperatures (T~).
1 off-
Marlex/n-octane
~
"
~
,, I-s 102,~ , I,o,, Io-
5. Discussion 5. !. SLOPES As a statdng point we take the. slopes of the log G versus log c plots, i.e. values of the exponent a in eq. (1) as expressed in figs. 12 and 13. We see that 0t can vary from 0. t 8 to 2.0. Let us first consider the meaning of a. First we fecal' the argument in ref. 7 that growth is not diffusion con trolled: the observed rat=s are by orders of magnitude slower than expected from diffusion eontrollled behav-
lTc."c L"~,
01L. .........
.........
..L_ 10 "~"
! 10 "I
- =_ C.%
Fig. 10. Linear growth rate Om/hr) versus concentration for sampie M grown from octane at dirt'-trenttemperatures (To),
iour. It ibllows that the molecules have a chance to reach the crystal face on the time scale of the growth of the crystal and accordingly what determines this rate is not the arrival time of the molecule but ,the efficiency by which these molecules become incorporated in the
!!6
A, X E L L E R A N D E. P E D E M O N T E
_-
--q n - octane
FHln- octane ! ,'~.'c ~op, .
.
.
.
.
,
/
b 11O05 018 1101 5 0 15 i
slope ot~ the IogGII~jC plot (~ 8 6
.
4 2 10
i.
.
.
.
.
.....
.
t.
. .
_. . . . .
~o"2
c,'/,
02-
M,~rlex 6 0 0 9
°°t 2I
j
t
t
t
90
92
94
9~
~
t
96
. . . . .
a,- ....
1oo
~
. . . . .
lO2
Tc,~
Fig. 13. Slopes of the (log growth rate)-(Iog concentration) plots, ex in eq. (1), for crystals grown from octane (from figs 8-9) versus crystallization temperature.
simultaneous arrival of several a t o m s the growth rate of the face will be a correspondingly higher power function of the concentrauon. Thus ~f the event of attachment ~nvolves two atoms ~ will be equal to 2 (bl-atomlc reaction) 1t ~ ~s smaller than ! we have a situation where the rate determining step is insensitive to the number of atoms in the solution. In tl~e extreme case that ~ = 0 ~.he growth rate ,~ ~ndepcndcnt on the concentration altogether. This would be the case if the face were blocked to arriving atoms in a way which is independent of conccntratton. In simple substances this would ' occur physically if the fa,-e were covered say by an. absorption layer due to some foreign matter. Here the rate of growth is determined by the event of the growth
xylene
plot
t~ /
"2
,/
~
•
O4
'1 4
08
F
L-. . . . . . . . .
c~sta!. Hence the rate determining steps will be some stage ~n th~s latter process. For visualising the meaning of ~ let us for a m o m e n t c.~nsider the example of a simple atomic crystal where there is only one stage o f attachment. An atom at the face becomes either attached to it or not with a fimte probab~hty. Similarly, an a t o m becoming attached to the fao.~ v, lll have a finite probab~hty ef s~aymg there or rctLrmng to the soluhon. The latter detachment ,.ate v,~ll be largely dependent only on the interaction :-ct~ee~ ~urface and atom: ~t w~ll not depend on con.:~trat~,.m On the other hand the attachments formed ~n untt u m e wall depei~d o~, the ntambcr 6f,zto~l~,s colhd tng a'~th the surface, hence will depend on the concentrauon If the attachment involves only one a t o m at a t me - as it usually ~s in crystal growth - the attachment rate wdl be proportional to the number of atoms, l,ence w~ll depend linearly on concentration. Thus should be umty. If, say, the attachment requires the
slope o~ the l o g G l l o g £
FIO
~ F4 • FH
~o"~
Fig. I I L i n e a r g r o w t h rate (lam/hr) versus, c o n c e n t r a t i o n o," ¢~'stals f o r s a m p l e F H g r o w n f r o m o c t a n e at different t e m p e r a tures (T~).
6
/
/
[
~o-3
0
/
I ~
~1 F 2 A F4
0 f10
....o~.....c~
t
J
:
rc °c "6 "r~ 80 8L~ ~4 8'6 8B t~ m 9,*' 94 96 g8 of ~.h: ~}og growth ra~e)--(Iog concentration) plots, ~x in eq. (I), for crystals grown from xylone (from fi~, 3 , , ~ Cr}~'~'Jlll~7<"t~lGO temperature.
,~
, ~',"
~pc~
A STUDY OF GROWTH RATES OF POLYETHYLENE SINGLE CRYSTALS
atom replacing an absorbed atom at the face. In case the obstruction of the face were not complete or penetration ot the obstruction showed some concentration dependence ~ would be between 0 and 1. We see that we have a general scheme by which all our observations can be discu:~d. For this the above principles have to be applied "c the specific case ot crystallization of long chains in the chain folded manner in question. First we need to define the rate determining steps. As we shall see the experiments indicate that there is no such step which is unique over all our experimental conditions. Even so the existing theories of crystallization provide a starting point I t- t4). According to these theories the rate determining step is the formation of a chain folded strip of stable size along a smooth crystal face, hence a two-dimensional nucleus capable ot further growth. Once this has occurred the deposition of further chain folded segments proceeds rapidly covering the whole face. However, as pointed out by Sanchez and Di Marzio I s.t6) the frequency of such a nucleation event per face ought to be proportional to the h.ngth of the face, hence the growth rate should incre tse expone-tially with the size of the crystal which is definitely not cbserved. The independence of the growth rate on the crystal size implies that th twodimensional growth which ema ,ates from a given 'tucl, us must be hmited to a certain constant length alon3 a face. As laid out by Sanchez and Di Marzio, imperfections of various kinds along the face would i~tterrupt the chain folded deposition defining statistically such a constant length. Our experiments cover a range of materials, both of different molecular weight and polydispersity and involve two solvents. It will be stated at the outset that we shall assign different priorities to different data in the discussion. The emergence of certain clearly defined physical l~rinciples should serve as a justification of 3ur procedure. Thus we shall consid:r FH and F I0 as the best defined materials from our point of view which d~splay both the most consistent and most extreme behaviour. To begin with we shall take both solvents together. We see that FH desplays the lowest • (0.18 in octane and 0.22 in xylene) which at least in octane, where we have sufficient poims, is temperature independent. FlO exlubtts the highest slopes (~ up to 2.0) with the
117
strongest tf mperature dependence. We shall discuss these two cases in detail first. As stated, a b o ~ ~isnmlle~:than-1 im#ies a , b a ~ e r t6~%~ invoked fin rffl ~7 where, the baXaer was consideredto :'-~ consist of the c~hain molecules themselves as here inn contrast to simple substances the situation can be envisaged that parts of the molecules are in the crystal and parts in the solution. A concrete structural description of this situation is contained by the theory of Sanchez and Di Marzio in terms of ciliale). The existence of cilia is a necessary attribute of folded deposition of chains of finite length, It arises as a natural conse,luence of the deposition of one chain coming to an end and that of another one aommencing in the way proposed in ref. 16 and shown in fig. 16. In addition, it c~a arise at any stage When the deposition of one chain is interrupted by the deposition of a segment belonging to another chain (fig. 16). Sanchez and Di Marzio ~s'~6) now consi¢ er that these cilia will cover up part of the growing surface and will initiate nucleation themselves. In particular, they start off the next chain-folded layer in competition with molecules fully in the solution. Denoting cilia and solution nucleation rates by S ~ and S' respectively it is shown that high S': is favoured by long molecules and low crystallization temperatures. For long enough molecules S ¢ becomes independent of solute concentration. Thus if S ¢ alone is the rate determining step for crystal growth 0c should be zero. Structurally this is to be visualised as follows: cilia emanating from the crystal commence growth along new layers. Chains from the solution will add to the deloosition commenced by the cilia contributing to the growth of the crystal and at the same time generating new cilia which will perpetuate the process. (It is obvious that if all the rate determining steps come from the cilia alone the growth of the crystal will be unaffected by solute concentration.) The high molecular weight fraction FH closely approximates the above situation which is in agreement with expectation. The small but finite value of ~ (0.2) however indicates that S ~ is nevertheless not entirely negligible. FI0 however shows ~,n entirely different behaviour. Even at its lowest (~ = 0.6) the molecules from the solution have a noticeably larger contribution than the cdia Furthermore, significantly the contribution of S s
l lS
~.. K E L L E R
AND
E. P E D E M O N T E
100 01 B,~/h G, ~ t / h
•
FH - Tc=Sgo*c-slope 0 25
O Flt3-To=Beg*c-slope 140 F 10 - re -83 9"C -slope 0 71
100
~OOr-
f
.._..I-'I /
V
I0~
. ~ , , ~ '~''''4~'~ •,
-octane
0 F 10- Tc"93 4"C I
•
01~
I
10"2 Fig.
FH - Tt : 101 5"C
14b.
Comparison
of
growth
lo "1 r:tte
versus
C. '/.
concentration
plok, of crystals grown from ociane for samples FH and F10. o~011
..... l
~o-4
10"3
L. . . . . .
~0"z
,~ ---
~o"~
c. "~
Fig. laa. C o m p a r i s o n o f g r o w t h rate versus c c n c e n t r a t t o n p l o t s of" cr3s a]s g r o w n f r o m ×yicne f o r samples F H and F i O
~ncre~ ses dramaticallv w~th crystallization temperature. Th~s again is to be expected for two interconnected reason~ F~rstly, the fold length becomes longer and the c~ha length x~,~tl be ~nsuffiment to imtmte a nucleus on ~ts o~n. Secondly, n o t . only the length of ~he fold rut the number of folds reqmred to reach a chain fok-ied str,p of stable size " 2; ~.crease with decreasing supercoohng and length of chain required to form ~t may become comparable with the length of a full molecule. In this case the cdia contribution becomes neghgible a,.:d S ~ alone will be the rate determining factor giving x = 1. Increas;ng the crystallization temperature further even a full chain may become inadequate to form a stable nucleus and cooperation of solute mol:cules ~s required. In th~s way ~ will exceed !. The h~ghest value ot 2, reached at the lowest reahsable supercoohng ~n octane, therefore =replies that ~,, c, ,.hams are revolved ~q the lnmat|on of the growth, of a new chain folded layer. 5 "~ Rares
So far only the slopes of the log G versus log c curves
were considered. Some interesting conclusions emerged from the comparison of the rates themselves. Thus compare G for F H and FI0 at the same temperature whmh is possible for crystalhzation at 89 ~C from xylene (fig. 14a). For low concentrations the rate for FI0 is lower which is to be expected as 89 °C corresponds to a much reduced supercoohng for FlO (accordIng to ~anchez and Dm Marzm's collection of data by at least l0 °C). However, with increasing concentration this difference is reduced and in fact around c = l °/~ois elilainated. According to all indications at still higher concentration G for F l 0 should exceed G for FH. Owin[~ to lack of mnterial this crossing over was not verified Nevertheless we see from fig. 14a that for a crystaiPi:,.atlon temperature of 84 °C the rates for FIO are hsg)~er than those for FH at 89 °C for all concentration measured. At 84 °C the supercooling for FI0 is still significantly lower than tor F H at 89 °C. A ~ +~,:,~ ~+,~,~,~ " .~,~,~A~
~r~QII;,~ that
t h ~ r ~ t ~ r~f n~trqr~u_
tmn is given by
G = G0exp - k
exp - k
'
(2)
where A~b is the ~vork in~ol~ed ,ii ' ,,~. ' " - l'ormat~on oI the critical nucleus while AF is the activation free energy
A S T U D Y OF G R O W T H
R A T E S OF P O L Y E T H Y L E N E
SINGLE CRYSTALS
119
G, ~l.II h
I
/ /o- /
0o
,~,,~/.,~
/
/
/ , F10 Xyl¢~ll
001
1~2---- "
lo~
"
~
nIt
C.~'o
Fig. 15. Superposition c f growth rate versus concentration plots of crystals grown at different temperatures for samples FH and FI0.
Fig. 16. Schematic representation of the origin of fold surface looseness due to cilia (from ref. 18).
for interracial transport t t). Aq~ depends inversely on supercoohng while AF does not depend on it m an cxphc~t manner. In fact the higher growth rate of FI0 at lower supercoolings therctore ~mplies that at the high concentrations in question it is not A~. the work required for a critical nucleus but AF, i.e. the facility with which a segment becomes attached, which is the rate determining factor. The above results imply that fur identical thermodynamic driving force the deposition rate per segment length is faster for the shorter molecule and therefore the overall growth rate will be higher. The reason %r this can be readily perceived. Looking at it from the pc,int of the long molecule the fixing of a given segment length Zong the crystal face wdl affect the configuration of the whole iong chain still in solution which will oppose, hence slow down, this deposltton. In other words a large coil still in solution will impede the reeling in process as opposed to the much smaller resistence due to a shorter chain portion. In a quantitive form this is expressed by Sanchez and Di Marzio under the concept of localisa-
t~on free energy and has been invoked long ago hv Stuait 17) as "entropy tension". We see that our experiments provide support for its existence. Next confine ourselves to sample F10 and compare the rates at different temperatures as a function of concentration (fig. 15). Consider first th.= curves for crystallization from octane. We see that they converge for higher concentrations. As a crossing over is most unlikely (the crystal growth rate of th,," same material cannot become higher at lower supercoclings) the curves must all tend to the same limiting growth rate. This means that at a sufficiently high concentration the growth rate becomes independent of temperature. This is just in line with the point made above, namely, that it is not the "'t n ~ l - i 1 llk~lUt~ ' 1 ll~l,l.l i i1,¢ • ,,,'a-:'',,,.k.., uul, ,i,..,,,...,.-..--°**<="hment rate ot the segments which becomes the rate controlling factor. At sufficiently high concentration the molecules needed for the growth are all available at the crystal face hence it is not the frequency of the molecular collisions with the face, or even their simul.. taneous collision, but the time any segment takes to
A. KELLER /LND E. PEDEMONTE
stant. We have seen before that lower molecular weight both increases a and its de~ndence on temperature which according to oar scheme means increasing contribution of S ~ compared to that of S c. As M contains shorter molecules the observed behaviour falls within this scheme. In molecular terms there will b¢ fewer and in average shorter cilia per unit length of growth face which at sufficiently high T¢ will significantly reduce ~.he contributioa of S ¢ in M as compared to FH. As regards the ratr~ we see that in case of similar a these are compar~ bte for the two samples in the same concentration range. However, in cases where the measurements could be extended to extremely low concentrations in the case of M the rates dropped more rapidly below t = 10 -3 giving a kink in the log G versus log e curves thus a change in a towards a larger value (fig. 6). No such discontinuity was observed for the fraction FH (T¢ = 89.0 °C in xylene) (fig. 7). This behaviour is likely to be associated with fractionation for sample M during the initial stage of growth to which the measurements refer, As we have no information on this point the effect cannot be discussed further. 5.3. EFFECTOF SOLVENT Here merely attention will be drawn to the importance Comparison of rates in the different solvents is of working with sufficiently well defined fractions when most straightforward for the FH fraction owing to the details of growth rate effects are to be interpreted. As closel3 stm~lar concentration dependence. When con- seen elimination of extremes mn the molecular weight s~dertng the rates at ~d:nttcal supercoolings on the d~stnbutmn has led to v considerable simplification of \,,lene-octane solvent pair a dtfferencc ~n super- the effects observed and it is on these grounds that prmnties In thts dlse.'~,sion have been ass,shed. coohng of i0 C for sdemlcal crystalhzatmn temperatures ~s rehabl~ indicated t ~) we see that thc crysta!hza 5.5. LOWEST-FRACTIONS t~on rate Is faster in octane (figs. 7, 11) Samples F4 and F2 will only be given brief mention. At the,. stage a comment on the effect of solvents may Both cases fall m between FII and FI0 with a being be oppo-tune. Xylene is a better solvent than octane. It follo~s that the molecule in xylene is more highly larger than for F H and increasing with F~ but not to solvated and the coil more expanded. From this alone the same ex:ent as for FI0. On applying the same argument as used for FI0, extended to still lower molecular the faster rate m octane would qualitatively follow. weight, one might have expected the trend FH ~ FI0 5.4. POLYDISPERSITY to continue, i.e. a further increase in 0c values. The fact For the effect of polydispe,sity co~apare samples M that this is not the case (we see that 0t approaches but and FH. in view of the tact that FH was obtained by does not exceed 1) may be attributable to the following -emoval ot the low molecular weight material from M two causes. (1) Polydispersity of the samples. The molecular ~t ~s constdered as a high molecular eveight portion of weight distribution of F4 we see is particularly broad Consider again the slopes. We se,: from figs 12 and and has a relatively larger proportion cf long chain 13 that at the lower crystallization temperat'.tres 0t is consituents than 310 (fig. 2c). By the trend deduced v~rtually equal for the two but for increasing F,, 0c in- above this would shift the bdut~iotir pattern in the creased slgmficantly for M while for FH it stays con- direction of sample FH which is broadly the observa-
become attached which determines the rate. Or looking at it more formally, in view o| the fact that with 0t > 1 the rate itself becomes ,-apidly so high with concentration that other processes which had been faster at lower concentrations, but are less dependt nt on concentration become the rate determining steps. At this stage ~ will have to drop to the level appropriate to the coacemration depeadence of this new rate determining process which is seen to be the case (the slopes all d~rease at the convergence point). That the segment a'tachment rate stdl shows some concentration dep.'ndence (fimte slope at the convergence points) could be due to the fact that the attachment rate depends on the availability of segments in the most appropriate orientation and this will depend on the number of segments available for deposition at a given site For crystallization from xylene a similar converging trend is d~scernible but this is much less pronounced. Thts means that conditions for temperature independent growth rate would only be achieved at a much higher, in fact unrealistically high, concentration.
121
A S T U D Y OF G R O W T H R A T E S OF P O L Y E T H Y L E N , : S I N G L E CRYSTAI_~ 9
10
11
12
13
1,~(T¢ AT)xlUS
G,u/h
tl
'
10I
~
~
¢:10% SlOpe=-O41=~ 5
~ •
F l o • xyl,,.t==~o'c;~=s,
10(~
C=0.1Y, mlm=-o,t?=10s , qr~=n0
.
.
.
.
.
.
~
,
.
C=O 1%
1 ~olL
O1
Oo. , .~' '" ~ c _ ,
L
¢=~m,&
. g
.
slope:-O 77x10:) (upper stale I ~ = 103,5
0.001
I C: O~01~/= 5 stoPe=-O 46 xlO
N.rlr,,,.~..,e 1 ""
Tm= 1111=C(Nakajlma) ~ 16
~ L
1
....
[
_
_.._.j
..|.
~
~
.
_£
Fig. 17. Linear crystal growth rotes plotted against I/(T=AT) for sample M.
don. The polydispersity of sample F2 seems to be small by the criteria of' M,,,/M. value. Nevertheless the molecular weight distribution is not a simple one as a bimodal distribution is apparent (fig. 2d) the consequences of which we cannot assess. (2) The rate determining step may be different. The crystallization mechanism of very short polymer molecules is not well understood. Particularly for F2 some chains may fold only once whlJe others may not fold at all. It is questionable therefore whether the chain folded nucleation mechanism as established for longer chains applies, and in particular whether it can be invoked as a rate determining step. [ f instead we consider the deposition of chains at existing steps along the face as the rate d;termining steps, as it might be for the growth ot'a paraffin, then = = ! would be expected. Cilia might still occur as unattached portions of a molecule only one stem long or less, and this together with any form of obstruction (absorption layer) would lower = below 1. In any case = = I seems to be an upper
limit which is at least consistent with this suggestion and inc~icates an approach to the crystallization behaviour of simple substances which is in line with the fact that with F2 we are approaching paraffins. 5.6.
FOLD SURFACE FREE ENERGIES
The term A,~ in eq. (2) is usually expressed as a~
4boaocT"°m =
AH AT kT
19
20
21
22
23
I/(TcAT)XlOs
. . . . . . . .
(3)
Fig. 18. Linear crystal growth rates plotted against I/(T=AT) for sample FI0.
in the kinetic treatments of chain foldinglt). If the formation of the stable chain-folded strips is the rate determining factor then the plot of log G against I/TAT should be a straight line with a slope of 4bo~cT°/AHk, where bo is a cross sectional dimension of the chain, T° the melting point of the infinite crystal, AH the heat of fusion, ~ the side surface free energy and ¢= the end surface free energ~ which involves the work required to make a fold. As all except the surface free energies are known quantities the slope serves to define ¢¢=. a is approximately ob':ainable for data on paraffin (it is about 10 erg/cm2), thus a=, which is one of the most important parameters of the chain folded system, can be estimated. Growth rate data, more often in the case of spherulites from the melt than for single crystals from solution, are being extensively used for this purpose (see reviews 2 and 20). In view of the wide spread interest in this particular topic we subjected our data to this ~-ind of analysis. As data over an appreciable temperature range are needed sample FH could not be evaluated in this manner. We chose therefore M as the representative material for a typical high molecular weight polymer and F10 for a sharp low molecular weight fraction. The io8 G versus I/T ATplots for M and F10 are given by figs. 17 and 18 respectively. Taking fig. 17 first we see that the points define stra:ght lines. This agrees with al~ preceding works 2o,2), and is consistent with ideas based on kinetic theori~ in the sense stated above. We see further that a hundred
122
A. KELLER A N D E. P E D E M O N T E
fold variation in concentration has only a very small effect on the slopes: the slope is l l ~ larger for c = 0.01 °'.o than for c = 1 ~ . This near equality of the slopes shows that the surface free energies of the fold surfaces are very similar over the whole concentration range. Consequently, it is justified to consider the crystals as structurally essentially invariant entities over the whole range of experimental conditions ~nv,)lved in this work. For ~nterpretmg the small differences in trtr© it would need to be known what the variation in trtr~ (which in thts case amounts to variations in tr=) really me.an in structural terms. More specifically, does a loose surface possess a larger or=than a more regular one consisting of sharper folds or vice versa ? Although there is a considerable amount of material in the literature (e.g. ref. 20) the ~ssue is not a priori clear to us even in the quahtative terms formulated above. Tighter folds might be expected to increase tr= owing to strained bonds. On the other hand in the case of more looseness the amount of material in the form of the energetically higher noncB'stalhne phase is increased which again would raise ~¢ The optmaisation of these factors clearly leads to the controversml ,ssue of what is the stablest configuration of the fotd w=th all its wider ~mphcatlons \~, =thout a defimt=ve answer to this quest=on we refrain from 'aterprettng the m~nor varmtmns in crcro now obserxed The ac+.ual numer:cal value of crtr~ (the small ~,ariat~ons apart) depends on the choice ot the other pmameter~ ln~ol';'ed m Aq~. The greatest uncertainty lies in 2,T o ~ n g to the uncertainty in T°. Two choices are shox~n in fig 17 The value of 111.1 °C is close to what has been generally used in the past2~). Nevertheless the posslbd~ty of values as high as 118 °C have been lately indicated'Z). In the first case trcr¢ is n the rapge of 600 in the latter m the range of 1000 (erg/cm2) '~. Taking cr = I0 erg/cm 2 from paraffin da,',a this would give a range of 60-100 erg/cm 2 for cr~ wbieh iq certainly acceptable, and spans the values usually quoted on the ha;Is of other hnes of studtes. F~g. 18 for FI0 is more problematic. Firstly, the po,ms do not define such good straight lines, in fact the departures at the lowest concentrations at the smallest AT values would make the justification of an assign.. ment of a single slope questionable. The overall slopes . r~. ~ ,~,~, c~,~: ~e defined are tess steep then for
sample M, also there are some minor variations with concentration, the lower concentration giving a smaller
slope. When comparing sample FIO with M in respect o f o , it needs remembering that there will be more chain ends in the former. It has been shown by comparing folded and extended crystals in poly(oxyethylene) that trc is significantly lower for the extended chainsZZ). It follows that an increasing number of ends at the fold surface should lower ire c which is in line with what we now observe. While the above argument appears reasonable it cannot be made more precise at this stage for the following reasons. Firstly, the actual value of the slopes is uncertain owing to the much greater uncertainty in T° for the low molecular weight fraction particularly in the case of octane. (The estimate for T° in xylene quoted in ref. 15 and used for fig. 18 may well be on the low side and so wilJ ~tr= derived from it. There is an additional uncertainty for the octane value which was obtained by adding the same 10 °C to the xylene value which was found appropriate for the high molecular weight polymer.) Secondly, the actual application of the usual consideration based on log G versus I/TAT plots to this material ,s questionable. In the foregoings the large variations in both the growth rates and Jn have been Interpreted as due to changes ir~ the mechanIsm of the crystal growth both with the crystaihzatlon temperature and concentration. Expectation of a unique slope in the plot in question may therefore be unjustified. Acknowledgement We are greatly indebted to Professor F. C. Frank for constructive discussions and for critical comments on the manu,,cript. References I) A. Keller, ReDt. Progr. Phys. 32 t~art 2 (1968) 623. 2) A. Keller, in" M.T.P. lnternatu.nal Review o/ Science, Phy~:cal £hem:stry, series 1, vol. 8, Macromolecular Series, Ed, C E H Bawn,(1972}p 105 3) D. I. Bltmdell, A. Keller and A. J. Kovacs, J. Polymer Sci, B 4 (! 966) 48 I. 4) D. J. Blundell and A. Keller, 3. Macromol. Chem. B 2 t1967) 337. 5) A..I. Kovacs and J. A. Manson, Kol)oid Z. Z. Polymere 214 (1966) I. 6) V, F. J-iolland and P. H. Lindenm .'yet, J. Polymer Sci, 5"7 1965) 589.
A STUDY OF G R O W T H RATES OF POLYETHYLENE SINGLE CRYSTALS 7) D. J. Blundell and A. Keller, J. Polymer Sci. B 6 (1968) 433. 8) T. Seto and N. Mori. Rcpt. Progr. Pol)arter Phys. Japan 12 (1969) 157. 9) V. Johnsen and J. Lehmannn, Koiloid Z. Z.~Polymere 230 (1969) 317. 10) A. Keller and F. M. Willmouth, J. Macromol. Sci. B 6 (1972) 539. I I) J. I. Lauritzen and J. D. Hoffman, J. Res. Natl. Bur. Std. A 64 (1960) 73. 12) F. C. Frank and M. Tosi, Proc. Roy. Soc. (London) A (1961) 323. 13) F. P. Price, J. Chem. Phys. 35 (1961) 1884. 14) J. I. Lauritzen and E. Passaglia, J. Res. Natl. Bur. Std. A 71 (1967) 261.
123
15) I. C. Sanchez and E. A. Di Marzio, Macromolecules 4 (1971) 677; and private communication. 16) I. C. Sanchez and E. A. Di Marzio, J. Chem. Phys. SS (I 971) 893.
l~) H. A. Stuart;Konoid Z, l~(l~SS) 3. 18) A. Keller, Kolloid Z. Z. Polymere 231 (1969) 386. 19) T. Kawai and A, Keller, Phil. Mag. 8 (1965) i203. 20) IF. P, Price, in: Nucleotion, Ed. A. C. Zettelmoyer (Wiley, New York, 1969) p. 405. 21) A. Nakajima, S. Hayashi, T. Korenago and T. Sumida, Kolloid Z. Z. Polymere 7.7,2 (1968) 128. 22) J. F. Jackson, L. Mandelkern and O. C. Long, Makromolecules 1 (1968) 218. 23) A.J. Kovacs and A. Gonthier, Koi,~oid Z. Z. Polymere, 250 (1972) 530; and private communication.
A STUDY OF G R O W T H RATES OF POLYETHYLENE SINGLE CRYSTAI2~ A. K E L L E R and E. PEDEMONTE*
University of Bristol, If. H. Wills Physics Laborator); Royal Fort, Tyndall Avemte, Bristol BS8 ITS, En~lar~d Received 31 October 1972 "[ he lateral growth of polyethylene single crystals was followed by a prev4ously established elec~ronmicroscopic sampling technique. Having reconfirmed the linearity of this growth, the growth rates were determined as a function of concentration at various crystallization temperatures for different molecular weights and in two solvents, in agreement with earlier work the growth rate (G) was found to depend on concentration fc) as G occ J, where 0cis a constant. In the usual whole polymer and in the high molecular weight fraction the rates showed only slight, oncentration dependence (0t ~ I) hardly affected by the crystallization temperature whde in the lower tool, ,:ular weight material, particularly for sharp :'r~,ctions, the growth rates display(d a pronounced concent, ation dependence and increasingly so for the Mgher crystallization temperatures yielding 0t values up to 2. These observations could be satisfactorily accounted for when considering the mechanism of the folded deposition of the chains, and in particular the situation, unique for polymers, that a given chain can be partly attached to the crystals while parts of it remain in solution (thus forming cilia). Further, the growth rate values in themselves allowed inferences to be made on "entropy tension' during the chain deposition and the temperature dependence of these rates permitted deductions to be made as regards the surface energies involved which are broadly in line with expectation from the kinetic theories of chain folding.
1. Introduction
The rate of growth ts an tmportapt aspect tn crystellizatton studies in general. In polymers the crystal growth rate ts of additional slgmficance because in view of chatn folding, which according to present beliefs is a ktnettcally controlled phenomenon, crystal growth and the structure of the resulting crystal are closely Itnked. The term "structure" here includes all features associated with chain folding such as the fold length, the nature of the fold and irregularities along the fold surface. Further, because of the long chain nature of the molecules their deposi,.ion occurs in a sequence of steps which creates a situation distinct from anything else encountered in crystal growth (see reviews !, 2). 1"he growth of individual crystals can only be followed tn crystalhzation phenomena from solutton. Even so the following of the growth of any one unit prese.n.ts great di.ffi__o_,!ties.The practical so!ut,on therefore ~s to ensure that all crystals originate at the same t~me ~n which case the total crystal populauon can be representatively sampled by one, or by very few c~stals at any stage of growth. This con&tion can be realized b~ the sell" seeding method which is a way to introduce * Presen! address: Islituto Di Chimica Industriale. Universit~t GLnova. Via A Pastore 3. Genova, Italy
heterogeneous nucleation centres, in this case constituted by a stabilized form of the same polymer, which all start growing into observable crystals simultaneousiy3- s). For purposes of growth rate studies the first most important recognition is the fact that at constant temperature the lateral growth of the tabular crystals ~s hnear w~th time al all except the late stages of growth when the solution is nearing exhaustion6'7). Thus the isothermal lateral growth rate can be defined by one rate value. In view of the fact that the monolayer crystals in this case do not thicken during growth this defines the growth of the crystal as a whole. The growth rate thus determined can be studied as a function of different variables. The foremost of these is the crystallization temperature. The first work along these hnes was carried out by Holland and Lindenmeyer 6) on polyethylene samples ~ hich as we now know mustsimilar works (see review 2)the temperature dependence of the growth rate conformed to the kinetic theories of chain folded crystal growth. The concern of the present wor~ is the dependence of crystal growth rate on concentration. In this we shall follow up the preliminary studies by Blundell and Keller 7) on polyethylene. The principal finding of these 111
112
A. KFLt.ER AND E. PEDEMONTE
Fig. !. Electron micrographs of typical crystals used for the present work. They were obtained by self seeding which ensures :miform monolayer crystal population. The isothermal crystal1; -.ation was interrupted at the chosen time by sudden quenching. i t.~ size reached at that stage is defined by the step (arrowed). Th,' crystal portion outside the step has arisen during cooling from the intended crystallization temperature.
studies was a surprisingly weak concentration dependence of the growth rate, in fact the rate increases era3 with the cube root of the oncentration. Essential ~ the same conclusion has been reached (with only shght dtfferences m the numerical values) by at least tv, o further stu&es involving also another polymerS'9). Thus the phenomenon in questmn appears to be general Biundell and Keller suggest that the weak concenu'auon dependence must be assocmted w~th the shieldmg of the crystal face by chains which are still m solution but accumulated along the face. Such chains could be full molecules or portions of molecules where the rest is already attached to the crystal. The ~ork by Blundell and Keller 7) invited obvious extension. In particular, work on fractions is called for because molecular weight dependence of the effect was anticipated and fractionafion during crystal growth from the ur_,fractionated samples was to be expected as a consec uence. Secondly, the work only covered a small temperature range which obviously needed extending. Th~s was undertaken in the work now to oe repolted. in addmen, the new study was extend~ ~1 also 'o a s~ond solvent. 2. MaCerial~ The following materials were used throughout this
Unfractionated linear polyethylene, Marie× 6009, to be denoted M, with Mn = 6700, Mw = i0 s and
Mw/M, = 15. Molecular weigbt distributit~n as obtained with G.P.C. is shown in fig. 2. A high molecular weight fract,on of Marlex 6009, to be denoted FH, obtmned by precipitation from stirred solution by the method of Pennings, provided by Dr. F. M. Wtllmouth from these laboratories. The prec~pltation was from a 2t % solution at 102 "C. Here the stirrer induced crystallization has removed the low molecular weight end of the distribution. M, = 2.5 x 105. (G.P.C. curve of this particular sample is not available. Chromatograms of similar samples arc shown in ref. 10.) Several low molecular weight fractions were kindly supplied by Dr. Pennings, Dutch State Mines Laboratory, denoted F2, F4, and FI0. As seen from figs 2b-2d the fractions are of different polydispersity. FI0 has the sharpest molecular weight distribution with M, = 10000, M,,, = 16000, and Mw/Mn = 1.5. ~'--ror F2, M, = 2600 and M,~/M. is also 1.5. Nevertheless mspect~,~n of hg. 2d shows that the distnbutmn ~s not ~: simple o n e and there is a wide spread in the I Q ~ molecular weight side of the p~ak. in fact a seco@~::~ peak is indicated with a maximum at ~,~ = !700~o~(~ mainpeak being at M = 4000). F4 ha. a single ~ but - broader molecular weight distribution a s ~
113
A S T U D Y OF G R O W T H RATES OF P O L Y E T H Y L E N E SINGLE C R Y S T A L S
G, ,ulh
.. a
I,) !
3
~
S
?
i '
Ibl
,o
: F2/xylene
i
l!
I
3
.j
L
/,
:-
:
5
]
t.
01
s
iti ilii
//
I .
.
.
.
.
.
.
,/~,o to~o
t . . . . . . . . . . . . . 1 0. 2
1 1 0 "1
C,%
Fig. 3. Linear growth rate (pm/hr) versus concentration of crystals from F2 grown from xylene at dtfferent temperatures (r©).
(dl
J
3
l,
5 Log 14
-
~-J, .........
6
1
?
__
['lg. 2. Molecular wetght distributions as obtained from G.P.C. chroma'ograms (a) Sample M (ortginal Marlex), (b), It) and (d) are fracttons FI0, F4 and F2 respectively
directly from lig. 2c. Hcrc M. = 3250, Mw = 10000 hcnce Mw/ M. ~ 3 The solvents were commercial xylene and octane, distilled before use.
3. Experimental The experimental method adopted was idcnttcal to that used by Blundell and Keller?). Ftrst, the self seeding nuclei were obtained in the usual manner 3'4) and were used to intttate crystal growth at the desired concentration and temperature. The cD'stal population was sampled at any requtred stage ol growth by removing a few drops from the crystallizing solution and transferring them to a lower temperature.. The stage, where the original isothermal growth has been interrupted, ts marked by a discontinuous downward step in the essentially monolayer crystal ?) (fig. I). The linear crysta! dimensions up to that step were measured and
plotted as a function of time of the corresponding crystallization. The linear growtb "-,~tn time, at least during the initial stages of growth, was reconfirmed in all cases. The slope of the crystal size against time plot then yielded the growth rate which was then explored as a function of the other variables.
4. Results The primary results are represented by plots oi" linear growth rate (6;) against concentration (c) expressed in wt 9/o. These are presented for materials F2, F4, Fl0, M and FH in the two solvents, xylene and octane in figs. 3-11. Each of these figures contains G versus c plots on a logarithmic scale at different temperatures of crystallization (To). As seen the growth rate as a function of temperature define straight lines on a logarithmic plot (except for M, fig. 6, which defines two such lines) hence confirming the relation G oc c',
(l)
where ~ is the slope The slopes as a fanct~on of crystallization temperature are shown in figs. 12 and 13 for the t - o solvents in the case of each material.
114
A. K E L L E R
AND
E. P E D E H O N T E
f
t00 O F - -
FIO I xy lene
, IT, : t
; 6,p~h
¢
186.2
1,'o/084
~
b
100~
, l,-SlO-
t
f
/889
/
~ 140
/
/
..~
/ /
i
~c~o~-
,o F d
1001~
F4/xylene
,
/
/
lo,s
b 18a2 lOS8 ¢ ]845
105.5
i
O01L__
.L-
1
~0 ~ ....
I0 "3
10"2
C,%
10"1
1='18. 4. Linear 8~)wth rate (l~m/hr) versus concentration of crystals from F4 grown from ~.ylene at different temperatures (Tc).
104
C,'/,
Fig. 5. Linear growth rate (gm/hr) versus concentration of crystals from FIO grown from xylene at different temperatures
(r=)
0 28
:.lope G
1
I0"~
I ........
.
~
a
,u/h
~ J 10_
o,L
00t -
/ Mar!ex/xy lene
b c d
877 i 027 88 95 '10 30 90 5 i 0 335
g h
93 q I 0 36 942 0 44 g5 2 049
9
=/
/J"
t
I
~. . . . . . .
10"4 2 . . . . ~'.
~°. q ~ m / h r ;
~ _
__ 4__ .
. . . . . . . . . . .
10"3
t .
.
lC~'~
.
.
.
.
.
.
.
.
L..
10"1
c. %
vcrs~s concentration ofcrystals iron- sample M t~rowrt from x3,len© at different teml~ratu,r¢~ (~r~|
A S T U D Y OF GROWTH RATES OF POLYETHYLENE SINGLE CRYSTALS'
I|5
G. p l h G, p/h
10
FH/xylene TC = 8g OeC slope: 0,25
01 . . . . . . . . . .
1
_-
~°'4
10"a
Fi~. 7.
Linear gro vth rate p m / h r )
. L
10.1
. . . . . . .
I04
c,v.
J
~0"I
versus concentration of
10
crystals for sample F H grown' rom xylene at 89.0 °C.
~.,u/h
01 ~_A=_ . . . . . . . . 10" 2
± 10"1
. . . . . . .
I 10
C.%
Fig. 9. Linear growth rate (pm/hr) versus concentration of crystals for sample FI0 grown from octane at different tempera-
tures(T,). I00-
G.~/h
d
10
F4/n
Mope
b
93 ~ 94 5
c
96 3
0 25 0 37 0 73
a
d 011 10"3
; I0"2
I 10"I
- octane
I Tc ,'(
[i g ? 5
lO0;-
I
084
b
i J
( .%
Fig. 8. Linear growth rate (pm/hr) versus concentration of crystals for sample F4 grown from octane at different temperatures (T~).
1 off-
Marlex/n-octane
~
"
~
,, I-s 102,~ , I,o,, Io-
5. Discussion 5. !. SLOPES As a statdng point we take the. slopes of the log G versus log c plots, i.e. values of the exponent a in eq. (1) as expressed in figs. 12 and 13. We see that 0t can vary from 0. t 8 to 2.0. Let us first consider the meaning of a. First we fecal' the argument in ref. 7 that growth is not diffusion con trolled: the observed rat=s are by orders of magnitude slower than expected from diffusion eontrollled behav-
lTc."c L"~,
01L. .........
.........
..L_ 10 "~"
! 10 "I
- =_ C.%
Fig. 10. Linear growth rate Om/hr) versus concentration for sampie M grown from octane at dirt'-trenttemperatures (To),
iour. It ibllows that the molecules have a chance to reach the crystal face on the time scale of the growth of the crystal and accordingly what determines this rate is not the arrival time of the molecule but ,the efficiency by which these molecules become incorporated in the
!!6
A, X E L L E R A N D E. P E D E M O N T E
_-
--q n - octane
FHln- octane ! ,'~.'c ~op, .
.
.
.
.
,
/
b 11O05 018 1101 5 0 15 i
slope ot~ the IogGII~jC plot (~ 8 6
.
4 2 10
i.
.
.
.
.
.....
.
t.
. .
_. . . . .
~o"2
c,'/,
02-
M,~rlex 6 0 0 9
°°t 2I
j
t
t
t
90
92
94
9~
~
t
96
. . . . .
a,- ....
1oo
~
. . . . .
lO2
Tc,~
Fig. 13. Slopes of the (log growth rate)-(Iog concentration) plots, ex in eq. (1), for crystals grown from octane (from figs 8-9) versus crystallization temperature.
simultaneous arrival of several a t o m s the growth rate of the face will be a correspondingly higher power function of the concentrauon. Thus ~f the event of attachment ~nvolves two atoms ~ will be equal to 2 (bl-atomlc reaction) 1t ~ ~s smaller than ! we have a situation where the rate determining step is insensitive to the number of atoms in the solution. In tl~e extreme case that ~ = 0 ~.he growth rate ,~ ~ndepcndcnt on the concentration altogether. This would be the case if the face were blocked to arriving atoms in a way which is independent of conccntratton. In simple substances this would ' occur physically if the fa,-e were covered say by an. absorption layer due to some foreign matter. Here the rate of growth is determined by the event of the growth
xylene
plot
t~ /
"2
,/
~
•
O4
'1 4
08
F
L-. . . . . . . . .
c~sta!. Hence the rate determining steps will be some stage ~n th~s latter process. For visualising the meaning of ~ let us for a m o m e n t c.~nsider the example of a simple atomic crystal where there is only one stage o f attachment. An atom at the face becomes either attached to it or not with a fimte probab~hty. Similarly, an a t o m becoming attached to the fao.~ v, lll have a finite probab~hty ef s~aymg there or rctLrmng to the soluhon. The latter detachment ,.ate v,~ll be largely dependent only on the interaction :-ct~ee~ ~urface and atom: ~t w~ll not depend on con.:~trat~,.m On the other hand the attachments formed ~n untt u m e wall depei~d o~, the ntambcr 6f,zto~l~,s colhd tng a'~th the surface, hence will depend on the concentrauon If the attachment involves only one a t o m at a t me - as it usually ~s in crystal growth - the attachment rate wdl be proportional to the number of atoms, l,ence w~ll depend linearly on concentration. Thus should be umty. If, say, the attachment requires the
slope o~ the l o g G l l o g £
FIO
~ F4 • FH
~o"~
Fig. I I L i n e a r g r o w t h rate (lam/hr) versus, c o n c e n t r a t i o n o," ¢~'stals f o r s a m p l e F H g r o w n f r o m o c t a n e at different t e m p e r a tures (T~).
6
/
/
[
~o-3
0
/
I ~
~1 F 2 A F4
0 f10
....o~.....c~
t
J
:
rc °c "6 "r~ 80 8L~ ~4 8'6 8B t~ m 9,*' 94 96 g8 of ~.h: ~}og growth ra~e)--(Iog concentration) plots, ~x in eq. (I), for crystals grown from xylone (from fi~, 3 , , ~ Cr}~'~'Jlll~7<"t~lGO temperature.
,~
, ~',"
~pc~
A STUDY OF GROWTH RATES OF POLYETHYLENE SINGLE CRYSTALS
atom replacing an absorbed atom at the face. In case the obstruction of the face were not complete or penetration ot the obstruction showed some concentration dependence ~ would be between 0 and 1. We see that we have a general scheme by which all our observations can be discu:~d. For this the above principles have to be applied "c the specific case ot crystallization of long chains in the chain folded manner in question. First we need to define the rate determining steps. As we shall see the experiments indicate that there is no such step which is unique over all our experimental conditions. Even so the existing theories of crystallization provide a starting point I t- t4). According to these theories the rate determining step is the formation of a chain folded strip of stable size along a smooth crystal face, hence a two-dimensional nucleus capable ot further growth. Once this has occurred the deposition of further chain folded segments proceeds rapidly covering the whole face. However, as pointed out by Sanchez and Di Marzio I s.t6) the frequency of such a nucleation event per face ought to be proportional to the h.ngth of the face, hence the growth rate should incre tse expone-tially with the size of the crystal which is definitely not cbserved. The independence of the growth rate on the crystal size implies that th twodimensional growth which ema ,ates from a given 'tucl, us must be hmited to a certain constant length alon3 a face. As laid out by Sanchez and Di Marzio, imperfections of various kinds along the face would i~tterrupt the chain folded deposition defining statistically such a constant length. Our experiments cover a range of materials, both of different molecular weight and polydispersity and involve two solvents. It will be stated at the outset that we shall assign different priorities to different data in the discussion. The emergence of certain clearly defined physical l~rinciples should serve as a justification of 3ur procedure. Thus we shall consid:r FH and F I0 as the best defined materials from our point of view which d~splay both the most consistent and most extreme behaviour. To begin with we shall take both solvents together. We see that FH desplays the lowest • (0.18 in octane and 0.22 in xylene) which at least in octane, where we have sufficient poims, is temperature independent. FlO exlubtts the highest slopes (~ up to 2.0) with the
117
strongest tf mperature dependence. We shall discuss these two cases in detail first. As stated, a b o ~ ~isnmlle~:than-1 im#ies a , b a ~ e r t6~%~ invoked fin rffl ~7 where, the baXaer was consideredto :'-~ consist of the c~hain molecules themselves as here inn contrast to simple substances the situation can be envisaged that parts of the molecules are in the crystal and parts in the solution. A concrete structural description of this situation is contained by the theory of Sanchez and Di Marzio in terms of ciliale). The existence of cilia is a necessary attribute of folded deposition of chains of finite length, It arises as a natural conse,luence of the deposition of one chain coming to an end and that of another one aommencing in the way proposed in ref. 16 and shown in fig. 16. In addition, it c~a arise at any stage When the deposition of one chain is interrupted by the deposition of a segment belonging to another chain (fig. 16). Sanchez and Di Marzio ~s'~6) now consi¢ er that these cilia will cover up part of the growing surface and will initiate nucleation themselves. In particular, they start off the next chain-folded layer in competition with molecules fully in the solution. Denoting cilia and solution nucleation rates by S ~ and S' respectively it is shown that high S': is favoured by long molecules and low crystallization temperatures. For long enough molecules S ¢ becomes independent of solute concentration. Thus if S ¢ alone is the rate determining step for crystal growth 0c should be zero. Structurally this is to be visualised as follows: cilia emanating from the crystal commence growth along new layers. Chains from the solution will add to the deloosition commenced by the cilia contributing to the growth of the crystal and at the same time generating new cilia which will perpetuate the process. (It is obvious that if all the rate determining steps come from the cilia alone the growth of the crystal will be unaffected by solute concentration.) The high molecular weight fraction FH closely approximates the above situation which is in agreement with expectation. The small but finite value of ~ (0.2) however indicates that S ~ is nevertheless not entirely negligible. FI0 however shows ~,n entirely different behaviour. Even at its lowest (~ = 0.6) the molecules from the solution have a noticeably larger contribution than the cdia Furthermore, significantly the contribution of S s
l lS
~.. K E L L E R
AND
E. P E D E M O N T E
100 01 B,~/h G, ~ t / h
•
FH - Tc=Sgo*c-slope 0 25
O Flt3-To=Beg*c-slope 140 F 10 - re -83 9"C -slope 0 71
100
~OOr-
f
.._..I-'I /
V
I0~
. ~ , , ~ '~''''4~'~ •,
-octane
0 F 10- Tc"93 4"C I
•
01~
I
10"2 Fig.
FH - Tt : 101 5"C
14b.
Comparison
of
growth
lo "1 r:tte
versus
C. '/.
concentration
plok, of crystals grown from ociane for samples FH and F10. o~011
..... l
~o-4
10"3
L. . . . . .
~0"z
,~ ---
~o"~
c. "~
Fig. laa. C o m p a r i s o n o f g r o w t h rate versus c c n c e n t r a t t o n p l o t s of" cr3s a]s g r o w n f r o m ×yicne f o r samples F H and F i O
~ncre~ ses dramaticallv w~th crystallization temperature. Th~s again is to be expected for two interconnected reason~ F~rstly, the fold length becomes longer and the c~ha length x~,~tl be ~nsuffiment to imtmte a nucleus on ~ts o~n. Secondly, n o t . only the length of ~he fold rut the number of folds reqmred to reach a chain fok-ied str,p of stable size " 2; ~.crease with decreasing supercoohng and length of chain required to form ~t may become comparable with the length of a full molecule. In this case the cdia contribution becomes neghgible a,.:d S ~ alone will be the rate determining factor giving x = 1. Increas;ng the crystallization temperature further even a full chain may become inadequate to form a stable nucleus and cooperation of solute mol:cules ~s required. In th~s way ~ will exceed !. The h~ghest value ot 2, reached at the lowest reahsable supercoohng ~n octane, therefore =replies that ~,, c, ,.hams are revolved ~q the lnmat|on of the growth, of a new chain folded layer. 5 "~ Rares
So far only the slopes of the log G versus log c curves
were considered. Some interesting conclusions emerged from the comparison of the rates themselves. Thus compare G for F H and FI0 at the same temperature whmh is possible for crystalhzation at 89 ~C from xylene (fig. 14a). For low concentrations the rate for FI0 is lower which is to be expected as 89 °C corresponds to a much reduced supercoohng for FlO (accordIng to ~anchez and Dm Marzm's collection of data by at least l0 °C). However, with increasing concentration this difference is reduced and in fact around c = l °/~ois elilainated. According to all indications at still higher concentration G for F l 0 should exceed G for FH. Owin[~ to lack of mnterial this crossing over was not verified Nevertheless we see from fig. 14a that for a crystaiPi:,.atlon temperature of 84 °C the rates for FIO are hsg)~er than those for FH at 89 °C for all concentration measured. At 84 °C the supercooling for FI0 is still significantly lower than tor F H at 89 °C. A ~ +~,:,~ ~+,~,~,~ " .~,~,~A~
~r~QII;,~ that
t h ~ r ~ t ~ r~f n~trqr~u_
tmn is given by
G = G0exp - k
exp - k
'
(2)
where A~b is the ~vork in~ol~ed ,ii ' ,,~. ' " - l'ormat~on oI the critical nucleus while AF is the activation free energy
A S T U D Y OF G R O W T H
R A T E S OF P O L Y E T H Y L E N E
SINGLE CRYSTALS
119
G, ~l.II h
I
/ /o- /
0o
,~,,~/.,~
/
/
/ , F10 Xyl¢~ll
001
1~2---- "
lo~
"
~
nIt
C.~'o
Fig. 15. Superposition c f growth rate versus concentration plots of crystals grown at different temperatures for samples FH and FI0.
Fig. 16. Schematic representation of the origin of fold surface looseness due to cilia (from ref. 18).
for interracial transport t t). Aq~ depends inversely on supercoohng while AF does not depend on it m an cxphc~t manner. In fact the higher growth rate of FI0 at lower supercoolings therctore ~mplies that at the high concentrations in question it is not A~. the work required for a critical nucleus but AF, i.e. the facility with which a segment becomes attached, which is the rate determining factor. The above results imply that fur identical thermodynamic driving force the deposition rate per segment length is faster for the shorter molecule and therefore the overall growth rate will be higher. The reason %r this can be readily perceived. Looking at it from the pc,int of the long molecule the fixing of a given segment length Zong the crystal face wdl affect the configuration of the whole iong chain still in solution which will oppose, hence slow down, this deposltton. In other words a large coil still in solution will impede the reeling in process as opposed to the much smaller resistence due to a shorter chain portion. In a quantitive form this is expressed by Sanchez and Di Marzio under the concept of localisa-
t~on free energy and has been invoked long ago hv Stuait 17) as "entropy tension". We see that our experiments provide support for its existence. Next confine ourselves to sample F10 and compare the rates at different temperatures as a function of concentration (fig. 15). Consider first th.= curves for crystallization from octane. We see that they converge for higher concentrations. As a crossing over is most unlikely (the crystal growth rate of th,," same material cannot become higher at lower supercoclings) the curves must all tend to the same limiting growth rate. This means that at a sufficiently high concentration the growth rate becomes independent of temperature. This is just in line with the point made above, namely, that it is not the "'t n ~ l - i 1 llk~lUt~ ' 1 ll~l,l.l i i1,¢ • ,,,'a-:'',,,.k.., uul, ,i,..,,,...,.-..--°**<="hment rate ot the segments which becomes the rate controlling factor. At sufficiently high concentration the molecules needed for the growth are all available at the crystal face hence it is not the frequency of the molecular collisions with the face, or even their simul.. taneous collision, but the time any segment takes to
A. KELLER /LND E. PEDEMONTE
stant. We have seen before that lower molecular weight both increases a and its de~ndence on temperature which according to oar scheme means increasing contribution of S ~ compared to that of S c. As M contains shorter molecules the observed behaviour falls within this scheme. In molecular terms there will b¢ fewer and in average shorter cilia per unit length of growth face which at sufficiently high T¢ will significantly reduce ~.he contributioa of S ¢ in M as compared to FH. As regards the ratr~ we see that in case of similar a these are compar~ bte for the two samples in the same concentration range. However, in cases where the measurements could be extended to extremely low concentrations in the case of M the rates dropped more rapidly below t = 10 -3 giving a kink in the log G versus log e curves thus a change in a towards a larger value (fig. 6). No such discontinuity was observed for the fraction FH (T¢ = 89.0 °C in xylene) (fig. 7). This behaviour is likely to be associated with fractionation for sample M during the initial stage of growth to which the measurements refer, As we have no information on this point the effect cannot be discussed further. 5.3. EFFECTOF SOLVENT Here merely attention will be drawn to the importance Comparison of rates in the different solvents is of working with sufficiently well defined fractions when most straightforward for the FH fraction owing to the details of growth rate effects are to be interpreted. As closel3 stm~lar concentration dependence. When con- seen elimination of extremes mn the molecular weight s~dertng the rates at ~d:nttcal supercoolings on the d~stnbutmn has led to v considerable simplification of \,,lene-octane solvent pair a dtfferencc ~n super- the effects observed and it is on these grounds that prmnties In thts dlse.'~,sion have been ass,shed. coohng of i0 C for sdemlcal crystalhzatmn temperatures ~s rehabl~ indicated t ~) we see that thc crysta!hza 5.5. LOWEST-FRACTIONS t~on rate Is faster in octane (figs. 7, 11) Samples F4 and F2 will only be given brief mention. At the,. stage a comment on the effect of solvents may Both cases fall m between FII and FI0 with a being be oppo-tune. Xylene is a better solvent than octane. It follo~s that the molecule in xylene is more highly larger than for F H and increasing with F~ but not to solvated and the coil more expanded. From this alone the same ex:ent as for FI0. On applying the same argument as used for FI0, extended to still lower molecular the faster rate m octane would qualitatively follow. weight, one might have expected the trend FH ~ FI0 5.4. POLYDISPERSITY to continue, i.e. a further increase in 0c values. The fact For the effect of polydispe,sity co~apare samples M that this is not the case (we see that 0t approaches but and FH. in view of the tact that FH was obtained by does not exceed 1) may be attributable to the following -emoval ot the low molecular weight material from M two causes. (1) Polydispersity of the samples. The molecular ~t ~s constdered as a high molecular eveight portion of weight distribution of F4 we see is particularly broad Consider again the slopes. We se,: from figs 12 and and has a relatively larger proportion cf long chain 13 that at the lower crystallization temperat'.tres 0t is consituents than 310 (fig. 2c). By the trend deduced v~rtually equal for the two but for increasing F,, 0c in- above this would shift the bdut~iotir pattern in the creased slgmficantly for M while for FH it stays con- direction of sample FH which is broadly the observa-
become attached which determines the rate. Or looking at it more formally, in view o| the fact that with 0t > 1 the rate itself becomes ,-apidly so high with concentration that other processes which had been faster at lower concentrations, but are less dependt nt on concentration become the rate determining steps. At this stage ~ will have to drop to the level appropriate to the coacemration depeadence of this new rate determining process which is seen to be the case (the slopes all d~rease at the convergence point). That the segment a'tachment rate stdl shows some concentration dep.'ndence (fimte slope at the convergence points) could be due to the fact that the attachment rate depends on the availability of segments in the most appropriate orientation and this will depend on the number of segments available for deposition at a given site For crystallization from xylene a similar converging trend is d~scernible but this is much less pronounced. Thts means that conditions for temperature independent growth rate would only be achieved at a much higher, in fact unrealistically high, concentration.
121
A S T U D Y OF G R O W T H R A T E S OF P O L Y E T H Y L E N , : S I N G L E CRYSTAI_~ 9
10
11
12
13
1,~(T¢ AT)xlUS
G,u/h
tl
'
10I
~
~
¢:10% SlOpe=-O41=~ 5
~ •
F l o • xyl,,.t==~o'c;~=s,
10(~
C=0.1Y, mlm=-o,t?=10s , qr~=n0
.
.
.
.
.
.
~
,
.
C=O 1%
1 ~olL
O1
Oo. , .~' '" ~ c _ ,
L
¢=~m,&
. g
.
slope:-O 77x10:) (upper stale I ~ = 103,5
0.001
I C: O~01~/= 5 stoPe=-O 46 xlO
N.rlr,,,.~..,e 1 ""
Tm= 1111=C(Nakajlma) ~ 16
~ L
1
....
[
_
_.._.j
..|.
~
~
.
_£
Fig. 17. Linear crystal growth rotes plotted against I/(T=AT) for sample M.
don. The polydispersity of sample F2 seems to be small by the criteria of' M,,,/M. value. Nevertheless the molecular weight distribution is not a simple one as a bimodal distribution is apparent (fig. 2d) the consequences of which we cannot assess. (2) The rate determining step may be different. The crystallization mechanism of very short polymer molecules is not well understood. Particularly for F2 some chains may fold only once whlJe others may not fold at all. It is questionable therefore whether the chain folded nucleation mechanism as established for longer chains applies, and in particular whether it can be invoked as a rate determining step. [ f instead we consider the deposition of chains at existing steps along the face as the rate d;termining steps, as it might be for the growth ot'a paraffin, then = = ! would be expected. Cilia might still occur as unattached portions of a molecule only one stem long or less, and this together with any form of obstruction (absorption layer) would lower = below 1. In any case = = I seems to be an upper
limit which is at least consistent with this suggestion and inc~icates an approach to the crystallization behaviour of simple substances which is in line with the fact that with F2 we are approaching paraffins. 5.6.
FOLD SURFACE FREE ENERGIES
The term A,~ in eq. (2) is usually expressed as a~
4boaocT"°m =
AH AT kT
19
20
21
22
23
I/(TcAT)XlOs
. . . . . . . .
(3)
Fig. 18. Linear crystal growth rates plotted against I/(T=AT) for sample FI0.
in the kinetic treatments of chain foldinglt). If the formation of the stable chain-folded strips is the rate determining factor then the plot of log G against I/TAT should be a straight line with a slope of 4bo~cT°/AHk, where bo is a cross sectional dimension of the chain, T° the melting point of the infinite crystal, AH the heat of fusion, ~ the side surface free energy and ¢= the end surface free energ~ which involves the work required to make a fold. As all except the surface free energies are known quantities the slope serves to define ¢¢=. a is approximately ob':ainable for data on paraffin (it is about 10 erg/cm2), thus a=, which is one of the most important parameters of the chain folded system, can be estimated. Growth rate data, more often in the case of spherulites from the melt than for single crystals from solution, are being extensively used for this purpose (see reviews 2 and 20). In view of the wide spread interest in this particular topic we subjected our data to this ~-ind of analysis. As data over an appreciable temperature range are needed sample FH could not be evaluated in this manner. We chose therefore M as the representative material for a typical high molecular weight polymer and F10 for a sharp low molecular weight fraction. The io8 G versus I/T ATplots for M and F10 are given by figs. 17 and 18 respectively. Taking fig. 17 first we see that the points define stra:ght lines. This agrees with al~ preceding works 2o,2), and is consistent with ideas based on kinetic theori~ in the sense stated above. We see further that a hundred
122
A. KELLER A N D E. P E D E M O N T E
fold variation in concentration has only a very small effect on the slopes: the slope is l l ~ larger for c = 0.01 °'.o than for c = 1 ~ . This near equality of the slopes shows that the surface free energies of the fold surfaces are very similar over the whole concentration range. Consequently, it is justified to consider the crystals as structurally essentially invariant entities over the whole range of experimental conditions ~nv,)lved in this work. For ~nterpretmg the small differences in trtr© it would need to be known what the variation in trtr~ (which in thts case amounts to variations in tr=) really me.an in structural terms. More specifically, does a loose surface possess a larger or=than a more regular one consisting of sharper folds or vice versa ? Although there is a considerable amount of material in the literature (e.g. ref. 20) the ~ssue is not a priori clear to us even in the quahtative terms formulated above. Tighter folds might be expected to increase tr= owing to strained bonds. On the other hand in the case of more looseness the amount of material in the form of the energetically higher noncB'stalhne phase is increased which again would raise ~¢ The optmaisation of these factors clearly leads to the controversml ,ssue of what is the stablest configuration of the fotd w=th all its wider ~mphcatlons \~, =thout a defimt=ve answer to this quest=on we refrain from 'aterprettng the m~nor varmtmns in crcro now obserxed The ac+.ual numer:cal value of crtr~ (the small ~,ariat~ons apart) depends on the choice ot the other pmameter~ ln~ol';'ed m Aq~. The greatest uncertainty lies in 2,T o ~ n g to the uncertainty in T°. Two choices are shox~n in fig 17 The value of 111.1 °C is close to what has been generally used in the past2~). Nevertheless the posslbd~ty of values as high as 118 °C have been lately indicated'Z). In the first case trcr¢ is n the rapge of 600 in the latter m the range of 1000 (erg/cm2) '~. Taking cr = I0 erg/cm 2 from paraffin da,',a this would give a range of 60-100 erg/cm 2 for cr~ wbieh iq certainly acceptable, and spans the values usually quoted on the ha;Is of other hnes of studtes. F~g. 18 for FI0 is more problematic. Firstly, the po,ms do not define such good straight lines, in fact the departures at the lowest concentrations at the smallest AT values would make the justification of an assign.. ment of a single slope questionable. The overall slopes . r~. ~ ,~,~, c~,~: ~e defined are tess steep then for
sample M, also there are some minor variations with concentration, the lower concentration giving a smaller
slope. When comparing sample FIO with M in respect o f o , it needs remembering that there will be more chain ends in the former. It has been shown by comparing folded and extended crystals in poly(oxyethylene) that trc is significantly lower for the extended chainsZZ). It follows that an increasing number of ends at the fold surface should lower ire c which is in line with what we now observe. While the above argument appears reasonable it cannot be made more precise at this stage for the following reasons. Firstly, the actual value of the slopes is uncertain owing to the much greater uncertainty in T° for the low molecular weight fraction particularly in the case of octane. (The estimate for T° in xylene quoted in ref. 15 and used for fig. 18 may well be on the low side and so wilJ ~tr= derived from it. There is an additional uncertainty for the octane value which was obtained by adding the same 10 °C to the xylene value which was found appropriate for the high molecular weight polymer.) Secondly, the actual application of the usual consideration based on log G versus I/TAT plots to this material ,s questionable. In the foregoings the large variations in both the growth rates and Jn have been Interpreted as due to changes ir~ the mechanIsm of the crystal growth both with the crystaihzatlon temperature and concentration. Expectation of a unique slope in the plot in question may therefore be unjustified. Acknowledgement We are greatly indebted to Professor F. C. Frank for constructive discussions and for critical comments on the manu,,cript. References I) A. Keller, ReDt. Progr. Phys. 32 t~art 2 (1968) 623. 2) A. Keller, in" M.T.P. lnternatu.nal Review o/ Science, Phy~:cal £hem:stry, series 1, vol. 8, Macromolecular Series, Ed, C E H Bawn,(1972}p 105 3) D. I. Bltmdell, A. Keller and A. J. Kovacs, J. Polymer Sci, B 4 (! 966) 48 I. 4) D. J. Blundell and A. Keller, 3. Macromol. Chem. B 2 t1967) 337. 5) A..I. Kovacs and J. A. Manson, Kol)oid Z. Z. Polymere 214 (1966) I. 6) V, F. J-iolland and P. H. Lindenm .'yet, J. Polymer Sci, 5"7 1965) 589.
A STUDY OF G R O W T H RATES OF POLYETHYLENE SINGLE CRYSTALS 7) D. J. Blundell and A. Keller, J. Polymer Sci. B 6 (1968) 433. 8) T. Seto and N. Mori. Rcpt. Progr. Pol)arter Phys. Japan 12 (1969) 157. 9) V. Johnsen and J. Lehmannn, Koiloid Z. Z.~Polymere 230 (1969) 317. 10) A. Keller and F. M. Willmouth, J. Macromol. Sci. B 6 (1972) 539. I I) J. I. Lauritzen and J. D. Hoffman, J. Res. Natl. Bur. Std. A 64 (1960) 73. 12) F. C. Frank and M. Tosi, Proc. Roy. Soc. (London) A (1961) 323. 13) F. P. Price, J. Chem. Phys. 35 (1961) 1884. 14) J. I. Lauritzen and E. Passaglia, J. Res. Natl. Bur. Std. A 71 (1967) 261.
123
15) I. C. Sanchez and E. A. Di Marzio, Macromolecules 4 (1971) 677; and private communication. 16) I. C. Sanchez and E. A. Di Marzio, J. Chem. Phys. SS (I 971) 893.
l~) H. A. Stuart;Konoid Z, l~(l~SS) 3. 18) A. Keller, Kolloid Z. Z. Polymere 231 (1969) 386. 19) T. Kawai and A, Keller, Phil. Mag. 8 (1965) i203. 20) IF. P, Price, in: Nucleotion, Ed. A. C. Zettelmoyer (Wiley, New York, 1969) p. 405. 21) A. Nakajima, S. Hayashi, T. Korenago and T. Sumida, Kolloid Z. Z. Polymere 7.7,2 (1968) 128. 22) J. F. Jackson, L. Mandelkern and O. C. Long, Makromolecules 1 (1968) 218. 23) A.J. Kovacs and A. Gonthier, Koi,~oid Z. Z. Polymere, 250 (1972) 530; and private communication.