A detailed interpretation of the kinetics of homogeneous nucleation and growth of crysLals or a linear homopolymer from dilute solution is given. The probability of forming both nuclei with fold ed chains, and conventional bundlelike nuclei, from dilute solution is analyzed. It is predicted that at sufficiently high dilution, critical nuclei of length 1: will be formed from single polym er molecules by sharp folding of the chain backbone. The step height of the nucleus is given approximately by 1:=4O' ,/A f. Here fI , is the fr ee energy requ ired to form a unit area of the loop-containing end surfaces, and Af is the free energy d ifference per unit volum e of crystal between the crystalline and solution states. The quantity Af is approxim ately proportional to the degree of s upercooling AT. The growth of thesc nuclci is then analyzcd. After growth, the resulting crystal is flat and platelike, the loops formed by the chain folds being on the upper and lower surfaces Kinetic factors d etermine that the distance between the flat slll'faces in the grown crystal will vary over only a narrow range about a value that is in the vicinityofl *=4O' ./Af. (NeglectingefTects du e to edge free encrgies, the theoretical upper and lower limits are l *= 40' ,/ Af and I *= 2O' ,/Af, respccLi vely.) In so me cases the predicted temperature dependence of the step height of the grown crystal, I*= const./ AT, may be modified by the existence of a constant tcrm resulting from thc presence of an edge fr ce cnergy E". A grown loop-type cry tal is predicted to be stable in comparison with a bundlclike crystal of the same shape and volume in a sufficiently diluLe olution. The logarithm of the nucleation rate is approximately proportional to 1/ (A T )2 near the m elting point. The cxponent n in the free growth rate law is predicted under various ass umption s. To the extent that compariso n is possibl e, the predictions givcn agree with the experimental r es ults obtained by K eIJ er and O'Connor an d others on single crystals of unbranched polyethylene grown from dilute solution.A su rvey is given of homogeneo us ll!..! cleaiion in bulk polymers, where the conventional bundlelik e nucleus containing segmenLs from many different molecules is v alid, and the essential results comparcd with thosc calculated for the dilu te solution case.The theory given for loop nuclei is both general and precise enough at the critical points to s uggest that, on crystallization from sufficiently dilute solution, crystals of a definite step height arc commonly to be expect()d for other crystalli zable linear polymers than polyethylene, provid ed loop formation is sterically possible.

The kinetic theory of the rate of growth G and the initial lamellar thickness lg* of chain-folded crystals is extended so that it is applicable at high undercoolings. Attention is centered on the details of how the first step element and the first fold are put down on the substrate. A parameter φ that varies between zero and unity, which apportions the free energy of attachment of the step element between the forward and backward reactions, is used to denote variations in this process. Expressions for G are derived from flux equations for two limiting cases: regime I, single surface nucleation act with rapid substrate completion and regime II, numerous surface nucleation acts with very slow substrate completion. Data from the literature on G for isotactic polystyrene (regime II) and polyethylene single crystals (regime I) are analyzed to obtain surface free energies, and these are used with the revised theory for lg* to predict the lamellar thickness of these polymers. Good agreement between lg* and published data is found for 1 > φ > 0. Values of φ below unity imply that the molecules are physically adsorbed onto the substrate prior to actual crystallographic attachment. A discussion is given of the segmental transport effects that dominate the behavior of G at high undercoolings in bulk polymers.

A syst ematic study of t h e problem of spherulitic growth in linear polymers in bulk has been carried out. A calculation of the radial growth of polymer spherulites is given for four models. These concern growth where the surface nuclei t hat control th e r a te a re (1) bundlelike and coherent, (2) chain fold ed and coherent, (3) chain fold ed and noncoherent, a nd (4) bundlelike a nd non coher en t. The required mod ifi cations of nucleation theory fir e given . Then the radial growth rate laws are derived for each model, and the type of "spherulite" t hat would be formed discussed .The model with chain fo lded and coherent growth nuclei leads to a typical lamellar spherulite. The properti es of the ind ividual chain folded lamellae that form the spherulite are predicted, including the change of step h eight with growth t emperature, m elting behavior, and the behavior on r ecrys talliza t ion . (Chain folded lamellae may a lso occur in sp ecimens t hat are not obviously spherulitic.) Under certain condition , the noncohcrent model with chain folds can lead to a mod ified la m ellar spherulite. None of t he bundlelike models will lead to a typical lamellar spheru li te, though a spherical microcryst alline obje ct might be formed . It is concluded that lamellar spherulites consist largely of chain fold ed structures.The fa ctors tha t could cause chain folded crystals to appear in profus ion in bulk polymers are discussed. The case of homogeneous initiation is considered first. Homogeneous initiation of chain folded nuclei in bulk will prev ail if the end s urface free energy of the bundJelike nucleus exceeds that of the folded . It is shown that the end s urface free energy of the bundlelike nuc le us, as calculated with a den sity grad ient model, will be larger t han h a d bee n s upposed previously. It is therefore considered to be theore t ically possible th a t the end s urface free energy of the bundlelike nucle us may in some cases exceed t hat of the folded nucle us. Attention is given to the possibility th a t folded structures appear in large numbers beca use cumulative strain or large chain ends prevent th e growth of bund lelik e nuclei to la rge si ze, e ven when the latter type of nucle us is energetically favored when small. H e terogen eous initia tion of fold ed structures is then considered .Other topics m entioned include: (1) Condi tions that might lead to nonlamellar or nonspherulitic crystalli zation in bulk, (2) t he origin of the twi t that is frequ ently exhibited by t he lamellae in spherulites, (3) the transitions that may sometimes occur in the rad ial growth r a te law, and (4) interla mellar link.

Th e growth rate C of th e cr ys tallin e bodi es a pp ea rin g in eac h of a se t of 35 c haracte rize d po ly. e th ylen e fracti ons ra ngin g from 3600 to 807 ,000 in molec ular we ight ha s been meas ured as a fun ction of th e unde rcoolin g tJ.T. In isoth e rmal c rys tallization, only ax ial ites we re found from M",= 3600 to 18 ,000.(For th ese run s, tJ.T < 17.5 °C.) From M".= 18 ,000 to M". "" 115,000 coa rse·grain ed non ·band e d s phe ru · lit es we re fo und for tJ.T > 17. 5 °C , and axialites for tJ.T < 17.5 °C : a rathe r s ha rp brea k occ urre d in th e log .o C ve r sus T data at tJ.T "" 17.5 0c. Th e morphologi ca l changes we re more gradual. Above Mil' "" 115,-000, only nea rly s tru ctureless " irregular" s ph erulit es we re found a t all und e rcoolin gs corres pondin g to isothermal growth. Typica l rin ged sphe rulites were o btain ed only on quenching. Wid e·a ng le x·ray data s howed that th e usu al orthorhombi c subce ll predominat ed in all th e morph ologies e ncount e re d. Lo w· angle x·ray data showed tha t th e s pec ime ns ex hibit e d lam e llar crystallization irres pectiv e of th e particular gross morphology involv ed. Th e growth rat e data on eac h fra c tion we re analyzed us in g /T (tJ.T )/]where / "" I to obtain va lu es of K" a nd Co. Th e va lue of Y in K,,= Ybuue/ (tJ.hj )k was obta ined fo r eac h morphology by a ppl ying th e "Z" test of La uritze n. Y = 4 for regim e I cr ys talli zation (s in gle s urface nucl eus leads to co mpl etio n of s ubs trat e) and Y =2 for regim e II cr ys talli za tio n (num e rous s urface nu c lei involve d in s ubstra te co mp letion). It wa s found that the axia lites obeyed regim e I kin e tics (Y=4), th e coarse·grain e d s phe rulit es regime II kin etics (Y = 2), and th e irregular s ph e r ulit es " mix e d " kineti cs (Y -3). Th e ass umpti on that th e sub strat e le ngth L in Laurit ze n's regim e theory was -5 I'-m led to the prediction of a rath e r s harp regim e 1 ..... regime II tran sition (corresponding to a break in the 10glO C versus T dat a) a t tJ.T "" 17 .5 °C, in accord with ex pe riment. Th e UUe value ca lc ulat ed fro m K" and Y for M". ;;. 20 ,000 was ap proxim ate ly co nst a nt with molec ular we ight and indep end e nt of morphology; th e limiting value of uU e from kineti c meas ure me nt s was about 1285 e rg'/c m 4 • corre· s ponding to Ue(x)=90.5 erg/e m ' and u= 14.2 erg/c m ' . (This valu e of Ut'{x) co mpare s favorably with u dcQ)= 93 ± 8 e rg/ cm ' from me ltin g point expe riment s.) T he in c rease of uUe and U e that too k place at low mol ec ul ar we ight s on up to -20 ,000 was treat e d usin g an expression given by Hoffman , viz , Ue=Udx) [(,.,+/3;) / (,.,+1)] where v=number of fo ld s per molecule, /3; = U"c;lIum)/Ud x) , I ntermittent hi gh and low values of U e were found ex p erim entally in thi s regio n , s howin g that j3 ; varied with in creas· in g mol ec ular we ight between 0.15 and -0.7. Theore tic al estimates of these upper and lowe r bound s for /3; are give n . Th e ...

Theory of Copolymer Crystallization 631 polymer concentrations and why the effect should be independent of the chain length of the added polymer. At this stage, more is to be gained by a candid expression of our lack of understanding than by the proposal of an in-terpretation which may fit only part of the data.Acknowledgment. We are grateful to the National Institutes of Health for financial support of this work under Grant GM-05811.