A systematic study of the 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 that control the rate are (1) bundlelike and coherent, (2) chain folded and coherent, (3) chain folded and noncoherent, and (4) bundlelike and noncoherent. The required modifications of nucleation theory are given. Then the radial growth rate laws are derived for each model, and the type of “spherulite” that would be formed discussed.The model with chain folded and coherent growth nuclei leads to a typical lamellar spherulite. The properties of the individual chain folded lamellae that form the spherulite are predicted, including the change of step height with growth temperature, melting behavior, and the behavior on recrystallization.  Under certain conditions, the noncoherent model with chain folds can lead to a modified lamellar spherulite. None of the bundlelike models will lead to a typical lamellar spherulite, though a spherical microcrystalline object might be formed. It is concluded that lamellar spherulites consist largely of chain folded structures.The factors that could cause chain folded crystals to appear in profusion in bulk polymers are discussed. The case of homogeneous initiation is considered first. Homogeneous initiation of chain folded nuclei in bulk will prevail if the end surface free energy of the bundlelike nucleus exceeds that of the folded. It is shown that the end surface free energy of the bundlelike nucleus, as calculated with a density gradient model, will be larger than had been supposed previously. It is therefore considered to be theoretically possible that the end surface free energy of the bundlelike nucleus may in some cases exceed that of the folded nucleus. Attention is given to the possibility that folded structures appear in large numbers because cumulative strain or large chain ends prevent the growth of bundlelike nuclei to large size, even when the latter type of nucleus is energetically favored when small. Heterogeneous initiation of folded structures is then considered.Other topics mentioned include: (1) Conditions that might lead to nonlamellar or nonspherulitic crystallization in bulk, (2) the origin of the twist that is frequently exhibited by the lamellae in spherulites, (3) the transitions that may sometimes occur in the radial growth rate law, and (4) interlamellar links. In recent years there has been much interest in the growth of spherulites in bulk polymers. These objects are the principal site of the crystallization in a number of highly crystallizable linear polymers, at least under certain conditions. Further, the mechanical, optical, and dielectric properties of such polymers are known to be affected by the presence of spherulitic crystallinity. Hence the rate of spherulitic growth, and the type of crystals existing in the spherulites, are of prime importance in connection with any attempt to understand the physical properties of these systems. Except where specifically noted otherwise, this paper is confined to crystallization from the unoriented melt. Attention is directed mainly to polymers that may be represented as systems of flexible linear chains which can in time achieve a high degree of crystallinity.chain folds. Stacks of thin platelets resembling a terraced pyramid are frequently formed in dilute solution preparations, often around what appears to be a spiral dislocation. In such cases, each terrace step corresponds to the thickness or step height of a single crystal. A theory describing the formation of single polymer crystals from dilute solution has been given by Lauritzen and Hoffman At and below bmin(amineσ+lminσ) is denoted as ΔH**.  This (ΔT)+1 steady state nucleation rate expression appears to be new in nucleation and growth theory. It must immediately be pointed out, however, that it is very unlikely that such a rate law would be observed over any substantial range of temperature because of the effect of nonsteady-state nucleation near and below cT. In the case of such small nuclei forming by steady state nucleation in a strongly supercooled polymer, the additional effect of pre-existing embryos of minimal size must be taken into account.9where the constant term 2T1 above the melting point of the polymer, the free energy of formation of an embryo always increases as its size increases. This is in contrast to the case of embryos in the supercooled liquid state, where the free energy of formation goes through a maximum at a saddle point so that embryos can become nuclei, and eventually stable crystallites. Nevertheless, numerous small embryos will exist in the normal liquid above mT, and the population of such embryos can be estimated by straightforward methods.  Now when a polymer specimen is rapidly cooled from a temperature T1 that is above mT to a temperature T2 in the strongly supercooled state, a number of these pre-existing embryos will be found to be of the critical size relevant to T2. The number of such pre-existing embryos that correspond to nuclei of critical size will be negligible if the stable nuclei arc large, as will be the case at low or moderate supercooling, but near and below cT, where the nuclei are of the minimal dimensions amin, bmin, lmin, the number of pre-existing embryos that correspond to nuclei of critical size will be high. Thus, near and below cT, this transport of bundlelike embryos from the normal melt to the strongly supercooled state will greatly increase the rate of injection of nuclei above that predicted by cT as to cause a very rapid and fine-grained crystallization to occur that might aptly be described as a “nucleative collapse” of the supercooled liquid state. (This effect may deter glass formation as noted in section 8.2.)At any temperature H*/kT). Depending on the values of ΔH* and the various surface free energies involved, this could happen in regions A or B.At sufficiently low temperatures, the rate of injection will eventually fall because of the increasing importance of the interfacial jump rate term, exp (−ΔH*/kT), with its positive temperature coefficient, will commonly overcome the strongly negative temperature coefficient of the nucleation term, exp cT, the excess nucleation rate characteristic of nonsteady state nucleation in region B may be seen.The overall picture of the homogeneous nucleation rate for bundlelike nuclei is shown in chain folds are denoted by bold face symbols. Note especially thatQuantities that are closely related or specific to The specific type of folded nucleus to be discussed is shown in σ for the folded nucleus refers to an abrupt phase boundary, since no polymer molecules pass through this surface and connect the supercooled liquid and crystalline phases. The quantity σ is thus similar in general character to the quantity σ for the bundlelike system, i.e., σ≅σ, and σ will usually fall between 5 and 25 erg cm−2.The lateral surface free energy the folded nucleus has abrupt phase boundaries on all its faces. As a consequence, the treatment of chain folded nuclei can be approached with more certainty than can bundlelike nuclei under like circumstances. Both σ and σe may to a good approximation be assumed to be independent of temperature and other variables.The end of the chain folded nucleus, unlike the end of the bundlelike nucleus, has a well defined phase boundary. Hence, σe for a folded nucleus is related to the work required to form a fold  for normal values of the parameters. This dependence on temperature is completely negligible compared to the other two exponential terms in G0(1). Then we have as a good approximation the expressionA′, which extends from near mT down to cT (see below). The term ϵ/b0 will not exceed σe in magnitude, and may be considerably smaller.On inserting a and l, setting the results equal to zero to get l* =2σe/(Δf)+2ϵ/b0(Δf) and a* = 2σ/(Δf), inserting these into (30) to get Δϕ*=4(b0σσe+σϵ)/(Δf), and thence to ϵ=0 7awhere lT′m (obs.), will thus refer to the larger of these. We must expect the step height at the observed melting point to reflect the mean step height of the growth or primary nucleus, whose l value varies as 1/(ΔT). Suppose that the observed melting point corresponds to a step height β times larger  than the mean step height of a primary nucleus. Then β is to a first approximation assumed to be constant. For this simple illustrative example one findsm.T This may prove useful in determining the equilibrium melting temperature. However, the main points here are that the observed melting point will tend to increase as the crystallization temperature is increased, and that it is entirely possible that such melting points may either all be above the line β>1, or all below as for β<1. If the edge free energy is large, and the distribution fairly broad, β may be in the vicinity of 1.5 or slightly more. On the other hand, if the edge free energy is negligible and the distribution narrow, β will tend to fall in the range 0.6 to perhaps 1.0. Experiments showing that β was in the range 0.6 to say 1.5 or 2.0 would provide strong evidence for the retention of the step height of the crystal near that of the nucleus length.The melting point of a polymer is commonly understood to refer to the temperature where the last detectible trace of crystallinity disappears. Lamellae will ordinarily exhibit a certain distribution of step heights, and the “observed” melting point, mT, and any subsequent recrystallization at this higher temperature would form higher-melting lamellae (see below). Also, a certain increase in step height may sometimes arise from lengthwise diffusion effects in the chain folded crystal. Either way, β will increase, and m.T Values of β as high as 5 or so may still be regarded as being consistent with chain folded growth, but the most clear cut case is that where β lies in the range of about 0.6 to roughly 1.5 or 2.0.The above results apply to the case where the lamellae maintain their original step heights as they are rewarmed. Small lamellae will melt out well below β values [Some recent experiments indicate that a satisfactory explanation of the melting of certain bulk polymers of the lamellar type is given by β values .mT than the thinner ones produced at lower temperatures. To this approximation, the breadth of the melting process is proportional to (ΔT). This result will be modified somewhat when the distribution of step heights within a given lamella is considered.Combination of xT(1) will melt out at the xT(1) to xT(2), a small step height will tend to be replaced by a larger one. This effect should not be confused with the slow and monotonic increase of step height due to lengthwise diffusion of the chains in the crystal that may sometimes take place. The latter process, if it occurs, should be identifiable, since it will lead to a slow and isothermal increase of step height at the original crystallization temperature.19Recrystallization may occur on rewarming. The thinner lamellae formed in the original crystallization at a and b directions than the patch originally melted out. The recrystallization process may be promoted by the presence of loop type crystals in the immediate vicinity. Something rather like what has been described may occur for a lamella on a heterogeneous surface, but here the problem is more complicated because of the interaction with the foreign substrate.Since the step height increases on recrystallization, the new lamellae formed in this process will be smaller in the not to be regarded as a “limiting” step height from a theoretical point of view.Growth of the step height to increasingly higher values by recrystallizing at successively higher temperatures through slow warming cannot be carried out indefinitely, since kinetic considerations will at some temperature near the melting point prevent such recrystallization from taking place at a sensible rate. The large step height achieved by recrystallization at the highest practical temperature is σ=10 erg cm−2, T= 400 °K, and b0=5×10−8 cm, the quantity We turn now to the question of what the model implies about the distribution of step heights of various lamellae. To a good approximation, it has been shown that the mean square deviation of the step heights of different lamellae about the mean value given by 〈(l−l¯)〉If the edge free energy is greater than zero on the face of the crystal, and falls to a lower value at the edge, then the distribution of step heights of different lamellae will be even less than that given by l*  can take on only discrete values of l, it was shown that the step height would maintain itself slightly above the value l=2σe/(Δf) for the case ϵ=0 from near mT down to temperatures corresponding to a rather high degree of supercooling. Though some minor differences exist, the more rudimentary theory given in this paper, where l* is assumed to remain constant after the initial surface nucleation act, is verified on the main points.σe/(Δf) without becoming unstable, and does not tend to become substantially larger than this value because it is expensive from an energetic standpoint to allow a loop to protrude much out of the fold plane. This causes the lamella to have the maximum steady-state growth rate when its step height is l* (cf. sec. 5.1).The question of the constancy of the step height within a given coherently growing lamella subsequent to the initial surface nucleation act has been treated in some detail by Lauritzen .20 AllowInsofar as the disorder of a folded crystal is a result of the distribution of step heights, i.e., irregularities in the fold plane, Lauritzen’s calculations indicate that crystals formed at low supercooling should be more ordered than those produced at high supercooling, where fluctuations are more pronounced.Growth by coherent nucleation of folded nuclei can lead to the formation of a spherical object that in most important respects strongly resembles a lamellar spherulite.σe planes of the lamellae (loop containing surfaces) will be more or less parallel to the spherulite radius, and the polymer chains will be essentially normal to the radius.Consider first the case of a heterogeneously initiated spherulite. Here stacks of lamellae of the general type shown in σe faces of those already in existence. Second, spiral dislocations can occur. The axis of these dislocations, s−s′, will be at a right angle to the σe plane, i.e., normal to the spherulite radius, and “radial” growth will occur from the edges of such dislocations. Certain types of branching are also possible as secondary nucleation events. For example, noncoherent nucleation may occur on the faces or edges of a lamella.The spherulite will tend to fill out and become a three-dimensional object by virtue of one or more of several processes. First, new lamellae will tend to form and grow along the mT prior to crystallization (see sec. 8.4.).It is evident from the above that this model leads in a natural way to the lamellar and polymer chain orientation common in spherulites, and it is also clear that a three-dimensional spherical object can be formed. If the number of active heterogeneous nuclei is very large, easily recognizable spherulitse may not form because of early impingement, but the polymer texture will still be basically lamellar. This effect may be greatly reduced by ridding the system of the heterogeneities, which is difficult, or somewhat reduced by destroying some of the embryos in the fissures on the surface of the heterogeneities by heating the sample further above Homogeneous nucleation can also lead to a lamellar spherulite. Here the nucleus will first grow a small platelet. Then by the mechanisms mentioned above, a three-dimensional object can form from it. Early in its life, the spherulite may be fan or sheaffike. (The same would be true of a heterogeneously nucleated spherulite if only a few nucleation sites on each heterogeneity were active.) The spiral dislocation mechanism is probably important in creating a three-dimensional spherulite, but the radial growth itself is not due to the formation of such spiral ramps. At sufficient supercooling, the number of homogeneous nuclei will become very large even early in the crystallization process, so that obvious spherulites may not appear despite the fact that the texture is actually lamellar.T)−1 radial growth rate law. Geil [An essential point to bear in mind in the case of ideal coherent growth with chain folds is that a number of the lamellae should extend outward from the nucleation center without interruption a substantial fraction of the way to the boundary of the spherulite. It should be possible to obtain information on this point by examining electron micrographs of a cross section of a spherulite. Evidence that a single lamella without any obvious disruption extended over a substantial fraction of the spherulite radius would be consistent with the coherent growth with chain folds mechanism, and at the same time give reason for anticipating a , van der Waals interactions comprise the main forces that must be overcome to separate the lamellae. Steps of the type illustrated in y—y′. This may be done to a certain extent without seriously affecting the step height of the folds or the rate expressions. Even the fairly frequent inclusion of such chains will not tend to incur the cumulative strain effects that may be associated with the formation of purely bundlelike structures. Also, groups of lamellae will tend to grow outward from the nucleation center together. The situation should resemble that shown in A fairly large number of molecules may be incorporated bundle-fashion into a chain folded lamella growing in the bulk supercooled liquid, as illustrated in Polymer molecules emanating bundle-fashion from the fold plane may sometimes return to the same crystal at a position well removed from the original point of exit. This type of “folding” may be denoted as “nonadjacent re-entry”, as opposed to the usual type of chain folding, which exhibits adjacent re-entry. However, it seems quite unlikely that the basic structure of a well-defined lamellar polymer crystal formed at low to moderate supercooling consists of folds that largely or entirely exhibit nonadjacent re-entry.perpendicular to the direction of stress. Such a result would provide strong support for the existence of lamellae with chain folds. The existence of too many interlamellar links, local melting caused by rapid stress, or excessive elongation, could easily lead to the opposite result.  Examples where the tipping of crystals at low draw ratios has been observed may be found in the literature [In unoriented bulk polymers crystallized under conditions where the number of interlamellar links is low, elongation of the specimens by cold stretching in one direction should cause the lamellae to tend to aline in the direction of stretch; in such a case, the polymer molecules in the crystals would tend to be oriented more or less terature , 32, 33.terature  to origiThe existence of interlamellar links in polymer crystallized in bulk may cause such material to differ from masses of chain folded polymer platelets filtered from dilute solution preparations. In the latter case, interlamellar links should be virtually absent. If numerous, the interlamellar links in the bulk crystallized polymer might cause disturbances in the solubility, mechanical properties, and melting behavior.dθ as one proceeds along the radius of the spherulite. The heavy arrow marked “G” indicates the direction of radial growth. The leading face of the lamella has been drawn wedgeshaped to correspond to growth in a preferred crystallographic plane  in many real spherulites are twisted, and it remains to be seen if coherently grown lamellae with chain folds could exhibit such an effect. The type of twist under consideration here is shown schematically in see also .The twist of the lamellae can arise from stress in the plane of the chain folds. One way in which the required type of stress can arise will now be outlined.σe surfaces is not the same as that characteristic of the interior of the crystal, and that if the folds are larger than a certain size, an anisotropic surface stress is certain to result. . This will, of course, tend to be balanced by forces due to the internal packing arrangement. Second, the chain folds may tend to become staggered, causing the folded surface to resemble a terra cotta roof. The staggered configuration corresponds to that where the polymer chain axes are no longer perpendicular to the plane of the chain folds.x, y, and l* possesses a surface stress f in dynes cm−2 that is distributed over a surface layer of thickness γl*, where y is a small fraction. The situation on the opposite side of the parallelepiped is identical to that just described.Now consider the problem of the origin of the twist in a body with surface stress of the type that may occur in polymer lamellae. The particular type of surface stress arising from the model shown in f exceeds a certain value which isS is the shear modulus. Numerical estimates for the extremely thin lamellae characteristic of spherulitic crystallization indicate that twist can arise for surprisingly low values of cf/S if l* is in the usual range for such crystals. It is therefore considered entirely reasonable to propose that surface repulsions of the type described can lead to the twist of the lamellae. Surface stress arising from other causes could also lead to a similar effect.From the theory of the buckling of plates,l* the crystal is already twisted, the twist will tend to become more prominent if l* is reduced further. This assumes, of course, that x does not fall off as rapidly as l*. The variation of γ with l* is unimportant in most applications.If for some value of dθ will tend to increase rapidly as the growth temperature is increased and at some temperature become extremely large. In practical terms this means that the concentric bands observed in many spherulites when viewed with crossed nicol prisms will become separated more and more as the growth temperature is increased, finally disappearing altogether. The lattice forces in the interior of a lamella tend to keep it in the untwisted state, while the surface stress promotes twist. Hence, if the step height l* increases as it does with an increase of growth temperature, the twist will rapidly become less pronounced at the same time.Perhaps the most important prediction that arises from the present conception of the origin of twist in coherently grown lamellae is that l* is reached , the twist will disappear completely. Examples where the twist diminishes rapidly with increasing growth temperature are well known in lamellar spherulites. It is conceivable that the situation described above could be reversed in some cases because x increases more rapidly than l* with rising growth temperature.When a certain value of Considering the proposed mechanism leading to twist, it would not be surprising to find that lamellae with a marked warp or twist were also characterized by the polymer chains being not exactly perpendicular to the plane of the chain folds.dθ will be the same for both right and left handed sectors. If the polymer is heated above Tm, embryos preserving the sense of twist may be retained in cracks or pores in a heterogeneity, with the result that a spherulite similar to the original one will frequently be regenerated at the same place on subcooling. A boundary will exist between right and left handed sectors. Since the probability of polymer chain connections between these sectors is small, spherulites should frequently fracture under shear at such boundaries.A heterogeneously nucleated spherulite may have sectors with both right and left handed twist. At any given nucleation site on a heterogeneity, lamellae with a given sense of twist will be generated, and the sector that grows out from this site will tend to preserve this particular twist (see below). The quantity If the lamellae growing out from a nucleation center are either densely packed, or grow out together as a stack, the sense of the twist in the sector under consideration will be preserved. (This is the “cooperative effect” of Keith and Padden .) HoweveThe foregoing development indicates that the coherent growth mechanism with chain folds can reproduce many of the structural features of lamellar spherulites. The existence of the lamellae themselves, and the behavior of the step height that characterizes them is explained. The orientation of the lamellae, and the orientation of the polymer chains in the lamellae, is in the correct relation with respect to the radius of the spherulite. Moreover, the twist frequently exhibited by stacks of the lamellae can be accommodated by the theory, thus removing one of the objections that had heretofore been leveled at coherent growth mechanism generally.T)−1 radial growth rate law, is worthy of serious consideration in the analysis of the behavior of lamellar spherulites.Our conclusion is that the coherent growth mechanisms with chain folds, with its  that is larger than σe, which is sufficient to practically completely arrest coherent growth. The value of σ on the coherently growing face will probably also take on a higher value which may be denoted σ(s). The subscript (s) denotes strain.It is conceivable that coherent growth of the kind described in the previous section will be hampered by certain effects so that such growth practically ceases after a time. For example, continued coherent growth may in some cases lead to noncoherent surface nucleus on the presumably still wettable face of the stunted lamella. The rate laws for such growth will be derived, and the conditions under which the formation of a noncoherent surface nucleus will be the rate determining step in the radial growth of a spherulite determined. The effect of such nucleation on spherulitic structure will be mentioned.The problem that will be studied in this section is whether or not it is possible that radial growth can be reactivated in such a case by the formation of a The concept that noncoherent surface nucleation may be involved in spherulitic growth is due to Price . He sugga, b, and 1 is assumed to form the face of the stunted lamella The noncoherent nucleus has normal “unstrained” values of σ, σe, and Δf.h The substrate lamella bears the strained values of the surface free energies σ(s) and σe(s), but the volume contribution, Δfh, has its normal value. The noncoherent surface nucleus attaches to the substrate lamellar face at an angle, ψ or 2π — ψ, that is consistent with minimizing strain. Thus, there is a true interface between the surface nucleus and the strained substrate crystal. This means that the noncoherent surface nucleus will behave in a three-dimensional manner, provided that neither a, b, nor 1 falls to minimal dimensions. Observe that the term “noncoherent” does not mean that the surface nucleus is not attached to the substrate.The model shown in “G”. Then the surface strain gradually accumulates, and the lamella is stunted after growing a distance λ. After a time, a noncoherent surface nucleus forms, and the process is repeated. Under appropriate circumstances, the formation of the three-dimensional noncoherent surface nucleus may be the rate determining step in the radial growth process.Once a noncoherent surface nucleus of stable size is attained, coherent growth begins again, and the spherulite radius at first grows rapidly in the direction indicated by the heavy arrow marked a, b, and l may be regarded as variables in the sense that they have not reached minimal values, the free energy of formation of the unstrained noncoherent surface nucleus illustrated in δ is defined by the relationσinterface is the total interfacial free energy between the surface nucleus and the lamella to which it is attached; σ(s) is the surface free energy on the face of the strained lamella. The other quantities have the same significance as in section 5. The edge free energy is omitted for simplicity.Assuming that σ is a measure of the interaction of the noncoherent surface nucleus with the strained substrate lamella. It takes on a value of zero if there is no interaction between the two objects. In this case Δϕ is identical to that for a primary nucleus with chain folds. On the other hand, if σ(s)→σ and δ→2σ, cf.ϵ→0). Thus δ will in general have a value between zero and 2σ.The quantity δ thus indicates strong interaction of the noncoherent surface nucleus with the substrate (high degree of wettability), and a small value of δ indicates weak interaction (low wettability).A large value of a, b, and l may be regarded as variables, and it is therefore proper to calculate ϕΔ* on the basis that the nucleus is formed from many step elements. Using procedures of the type outlined in sections 2.3 and 2.5, it is found thatb* had not fallen to 0b because of too large supercooling and (2) noncoherent surface nucleation was actually the rate determining step.Near the melting point, Tδ, the nucleus is monomolecular, i.e., 0b=b, and the (ΔT)−2 law does not apply. Further, the conditionΔTδ, and need not be considered further. The right hand inequality states that noncoherent nucleation must be more rapid than “strained” coherent growth, so that noncoherent nucleation will be the rate determining step in the radial growth process.Let us now consider the range of validity of a = a0 isTδ, b0 (Δf) will be considerably larger than (2σ−δ), and the radial growth rate may be approximated asA′.At a degree of supercooling equal to or exceeding that given in cT≅4σmT/(Δhf)amin. There and below, the rate of injection of nuclei in the surrounding medium will increase as described under Region B′ of section 5.2.The rate law described by δ, say in the vicinity of 3σ/2, will still give a (ΔT)−2 radial growth rate region that extends from near the melting point on down to growth temperatures corresponding to a rather high degree of supercooling. For such values of δ, σ(s)σe(s) need be only several times larger than σσe at ordinary supercooling to satisfy T)−2 to a (ΔT)−1 law may occur at Tδ, as noted in Tδ is large, the rate laws will be mixed to the point that they cannot be clearly differentiated.A schematic diagram of the growth rate behavior of this model is shown in In broad aspect, a spherulite built up with chain folds by the process (rapid coherent growth→arrested “strained” coherent growth →noncoherent nucleation) will resemble one built on the coherent loop pattern. The spherulite will be basically lamellar, the chain axes will be normal to the radius, and the plane of the chain folds will tend to lie along the spherulite radius. Also, the coherently grown part of the lamellae will behave in a manner similar to that described in the previous section. However, there will be certain differences that may distinguish coherently grown from noncoherently grown loop type spherulites.ψ or 2π-ψ at intervals of λ. For a completely isolated lamella, the noncoherent nucleus will attach to the substrate with equal probability at ψ or 2π-ψ, so that such a lamella might show disturbances in its sense of twist at intervals of λ. Also, a distribution of ψ values may exist.During the growth process, the lamellae will be disoriented with respect to the substrate at an angle of λ becomes sufficiently large, spasmodic radial growth may be observed. Each rapid increase in radial growth would correspond to the coherent growth that follows upon the formation of a noncoherent nucleus. The distance traveled in each pulse would correspond to λ. If sufficiently small, the repeat distance λ may lead to characteristic X-ray spacings, but these would probably be very difficult to identify or observe.Perhaps the most unusual feature of spherulitic growth on this pattern that may become observable in an optical microscope is that if σe increases because of cumulative strain as coherent growth proceeds, the characteristic step height l* will tend to increase somewhat as the coherent growth step takes place. This is shown schematically in Since s−s' in The extension of the interface between the strained crystallites should provide possible sites for the initiation of spiral dislocations −2 law. This rate law is to be anticipated in lamellar spherulites that grow in a spasmodic manner near the melting point, since the coherent model with chain folds will not lead to such an effect.26The noncoherent loop type growth model is realistic enough to at least warrant testing spherulite growth rate data to see if they accord with a  and eσ(s) which are larger than the normal values σ and e.σ The latter refer to the surface free energies of relatively small bundlelike structures as yet unaffected by chain ends.The model employed is shown schematically in a and b directions caused σ and especially eσ to increase.Alternatively, it could be assumed that cumulative strain resulting from growth in the It is considered that if a large spherical object is to be built at all on the bundlelike pattern, the present model is, at least in the beginning, a more reasonable one than the strictly coherent bundlelike model discussed in section 4, where the possible restrictions imposed on crystallite size by large chain ends or strain were arbitrarily neglected.a, b, and l forms on the strained substrate crystal. This noncoherent bundlelike surface nucleus has normal values of eσ and σ. The noncoherent surface nucleus is assumed to form by virtue of the wettability of some portion of the substrate crystal by a normal crystal. Once it reaches critical size, this noncoherent surface nucleus will lead to coherent growth of a new crystallite of limited size. Then the process will be repeated. It is emphasized that the noncoherent surface nucleus is actually attached to the substrate, and that there is a true interface between the two objects.We will assume that a noncoherent surface nucleus in the form of a parallelepiped with the dimensions a, b, and l may be regarded as unrestricted by any minimal molecular dimensions. The free energy of formation of the noncoherent bundlelike surface nucleus shown in δ is defined in a manner analogous to l* is not a step height in the case of a bundlelike nucleus, since the nucleus will grow to at least a certain extent in the l direction.Sufficiently near the melting point, δ/2σ), and this results from the assumption that an unstrained surface nucleus can wet a strained crystallite of the same polymer to some extent.By comparison of b approaches b0. This will occur at a degree of supercoolingTδ, the rate law may be approximated ascΔT=4mσT/(fΔh) amin, bundlelike embryos of small size will be transported from the superheated state above mT into the supercooled state by nonsteady state nucleation where they will become nuclei of stable size. As in previous cases, this will lower the radial growth rate of any spherulites born at or near t=0.At a growth temperature corresponding to a degree of supercooling of T)−2 law will appear near and somewhat below the melting point because of the three-dimensional character of the noncoherent surface nucleus in that region. A transition to a (ΔT)−1 rate law may occur if the δ is fairly large. This transition will not be abrupt.A diagram of the radial growth rate behavior possible with this model is shown in There is nothing in the present model that suggests the existence of lamellae of large extent that have a uniform step height.l direction.If large chain ends that cannot be assimilated into the crystal are taken to be the cause of the limitation on lengthwise growth of each of the individual crystallites in the coherent growth process, a large distribution of lengths, i.e., “lamellar thicknesses” would result. The mean length of the individual crystallites would depend on the molecular weight and its distribution, rather than on the degree of supercooling. With small chain ends that can occasionally enter the crystal, there is still no reason to suspect anything but a large distribution of such lengths, though the mean length would be larger and in any event not be of the correct magnitude to correspond to the thickness of a lamella. It is in our view extremely improbable that a collection of bodies of varying radii and length would aggregate in such a manner as to form slabs of uniform thickness with flat faces. We therefore conclude that the asumption that chain ends limit lengthwise growth is not consistent with the prediction of typical lamellae. Even if one considers bundlelike crystals before such a mean “equilibrium” length is attained, one must contend with the fact that large bundlelike crystals will tend to be ellipsoidal in order to minimize the total surface free energy and strain , 26, andIf cumulative strain at the bundle ends is taken as the cause of the cessation of coherent growth, the “length” of the crystals might be quite uniform. However, the existence of such strain would clearly tend to force the individual bundlelike crystallites to have curved end surfaces, which is inconsistent with lamellar structure. Here again the mean length of the ellipsoidal crystallites would not depend strongly on the degree of supercooling.In both of the cases mentioned above, the introduction of the noncoherent nucleation step allows the continuation of radial growth, but in no way suggests that true lamellae could be produced from the decidedly irregular or ellipsoidal microcrystals produced in the coherent step.l direction as noted in section 4.2 until chain ends stopped them, but then they would possess a distribution of lengths as noted previously. Alternatively, if cumulative strain were not relieved by the tilting, each crystallite would have curved ends, and therefore not be able to form a lamella. The latter objection also holds for the model shown in No contradiction to the concept that bundlelike crystallization will not lead to typical lamellar structures arises when the noncoherent variations of the special models depicted in The bundlelike microcrystals will tend to melt close to the crystallization temperature because of the accumulated strain.The bundlelike noncoherent growth model might lead to a more or less spherical and semicrystalline aggregate composed of a vast number of small crystallites of nonuniform size.  If such an object is identified in some bulk polymer by electron microscopy or other methods, it would be reasonable to attempt to treat its radial growth with this model. However, it would seem more appropriate to treat a typical lamellar spherulite in terms of the coherent or noncoherent chain fold models.If noncoherent nucleation is impossible because no suitable surface is presented by the coherently grown crystal , small bundlelike crystallites may appear more or less at random in the system. These aborted structures might coexist with folded structures that were meanwhile growing to large size.The most important laws describing the rate of radial growth of spherulites derived in this paper may be summarized in the general form:Another law was mentioned, but it is probably of little importance and is included mainly for the sake of completeness:A convenient summary of the rate laws as they arise in the various models is given in T)+1 radial growth rate law holds when the rate determining surface nucleus has no dimensions that may be regarded as variables in the expression for the free energy of formation of the nucleus. The nucleus associated with this model, which is of the size aminbminlmin, may therefore be described as zero-dimensional.The (ΔT)−1 radial growth rate law bears the general meaning that the rate determining step involves a surface nucleus that has one fixed and two variable dimensions.  This meaning holds true for both the (ΔT)−1 law that appears beginning at the melting point in the coherent growth models, and the (ΔT)−1 law that arises at moderate or strong supercooling for the noncoherent growth models.The (ΔT)−2 radial growth rate law arises when the rate determining step is the formation of a surface nucleus that has three dimensions that are to be regarded as variables in the expression that describes the free energy of formation ; such a body is commonly referred to as a three-dimensional surface nucleus. In the present conception, such a nucleus could be formed if the polymer molecules were deposited on the polymer substrate in such a manner that they were no longer colinear with those in the substrate, thus forming an interface between the two bodies. In this event, the surface nucleus takes on the three-dimensional character of a primary nucleus as regards temperature dependence, but is energetically preferred to the corresponding primary nucleus by virtue of the wettability of the substrate by the noncoherent surface nucleus. In our view, then, the observation of a (ΔT)−2 radial growth rate law would not only mean that the rate determining step was the formation of a three-dimensional surface nucleus, but would also imply that a noncoherent surface nucleus was involved.The +1 rate law will be frequently encountered in experimental studies. This rate law will generally be obscured or distorted by nonsteady state nucleation effects. If it did appear, it would do so at moderate to strong supercooling where the jump rate term would make it difficult to identify.It is considered highly improbable that the (ΔT)−1 radial growth rate law would appear to deserve strong consideration in the analysis of data, especially in the case of obviously lamellar spherulites. We refer here specifically to the (ΔT)−1 law arising from the coherent model with chain folds. This model is capable of predicting many of the details of a lamellar spherulite, and coherent growth must be regarded as a probable mechanism, particularly if there is any reason to believe that a small and efficiently packed chain fold that is consistent with the interior lattice structure can be formed. On the grounds that studies on solution-grown single crystals appear to indicate that such folds can be formed in a number of cases, and the fact that lamellae of considerable extension have been observed in spherulitic bulk polymer samples, we believe that it is reasonable to suppose that a (ΔT)−1 radial growth rate law arising from coherent growth with chain folds should be found experimentally. Linear polymers with no bulky side groups or effective atactic sections are doubtless promising systems for such studies.The (ΔT)−1 growth rate law, but we do not consider this a likely source of such a rate law. It cannot produce a lamellar spherulite at all. Under a variety of conditions, the model does not even lead in a straightforward way to any kind of spherical object.The coherent bundlelike model also leads to a (ΔT)−2 radial growth rate law deserves consideration in the analysis of data, most particularly in connection with the noncoherent chain fold model. It seems conceivable that in some polymers the chain folds may be too bulky to allow continued coherent growth, and in such cases it is possible that noncoherent three-dimensional surface nuclei may form, and constitute the rate determining step in the radial growth mechanism.  If spasmodic growth of the spherulite radius is observed near the melting point, a (ΔT)−2 law becomes a good possibility.The (ΔT)−2 law arising from the noncoherent bundlelike model need only be considered for nonlamellar spherulites.The . The second type of transition postulated in the radial growth rate is not due to any change in the surface growth nucleus itself; bundle or loop type nuclei where all three dimensions are of minimal size appear in large numbers as a result of nonsteady state nucleation in the surrounding medium, and interfere with the growth of well developed spherulites (transition in surrounding medium).Two different types of rate transition have been postulated. In the first, the transition is due to some dimension of the surface nucleus approaching a minimal value, causing a different radial growth rate law to exist at lower temperatures (A″ (upper) → A″ (lower) and A″ (upper) → A″ (lower). Such transitions occur only for noncoherent surface nuclei −2 law near the melting point and a (ΔT)−1 law beginning somewhat below it is found in a lamellar spherulite, the noncoherent model with chain folds would be indicated. A similar situation in a nonlamellar spherulite would point to the noncoherent bundlelike model.The coherent folded nucleus model does not exhibit this type of transition, but the noncoherent folded nucleus model does. Therefore, if a rate transition involving a  nuclei in the medium surrounding a spherulite that may interfere with its growth.Consideration will now be given to the transition at cT transition can be relatively abrupt, occurring over a temperature range as little as several degrees. It will be accompanied by a noticeable fall in G. A transition fitting this general description has been found by Takayanagi [mT, has been found by Hoffman and Weeks [cT, even though G falls. This effect is particularly striking in the case of poly (chlorotrifluoroethylene).It is to be expected that the kayanagi  in poly nd Weeks . AnothercT transition as such is certainly not to be expected in radial growth rate studies in all polymers. For many polymeric substances, σ will be around 10 erg cm−2, and amin will be perhaps 10 A. Then the cT transition would appear only at a degree of supercooling of about 150 °C or more if (Δfh) lies in the usual range. This will usually be below gT, where the radial growth rate has already been greatly lowered by the jump rate effect. Even if cT is nearer to the melting point than this, it is rather likely to be obscured by experimental difficulties arising from extremely rapid growth. The cT transition is most apt to be observed in materials where it is nearest to the melting point, i.e., those with a low lateral surface free energy and a large molecular diameter.The mT, rises to a maximum below mT, and then falls again without any obvious discontinuities anywhere is the one to be most commonly expected. However, it is important to bear in mind the possibility that rate transitions can occur when analyzing radial growth rate data for polymeric systems.On the basis of the above remarks, the situation where the radial growth rate is low near cT falls well above the glass transition temperature. Then the supercooled liquid (which would otherwise form a glass on cooling below gT) would tend to nucleate and crystallize rapidly at or somewhat below c.T At the very least, such a “glass” would contain a large number of “frozen in” embryos, nuclei, or crystallites in addition to any truly amorphous glassy material. The ordinary homogeneous mechanism (or a heterogeneous one) could, of course, cause crystallization to become very rapid above c.T In any event, the failure of many linear polymers to easily form truly amorphous glasses may involve the “nucleative collapse” effect beginning at c.TGlass formation might be practically impossible, even with the most rapid quenching, if cT transition, namely, transport of a large number of nuclei of minimal size from the melt to the supercooled state, can be employed to explain certain effects associated with prequenching on spherulitic growth. In the case of the theory discussed so far, it has been assumed that we were dealing with specimens that were cooled directly from well above mT to the growth temperature, and the rate of radial growth observed under these conditions. Suppose, however, that experiments were carried out by first quenching the polymer to some low temperature, and then observing radial growth after the specimen is rewarmed to a certain growth temperature. Then at this growth temperature, G will often be lower than if the value of G had been obtained in the normal way at the same growth temperature. Results of this type are known, and may be ascribed to the incursion of small nuclei during the quenching process. The difference might be negligible if cT is near or below g.T Growth rate data obtained on prequenched and reheated specimens should be regarded with caution from an interpretive standpoint, since the retardations to radial growth will involve factors that have not been considered in deriving the expressions for G.29The basic effect that causes the The objective of this section is to emphasize the possibility that (1) crystallization without chain folds may occur, and (2) that crystallization with chain folds but without any obvious spherulites may exist.eσ<σe for both nuclei and crystals, and assuming that loop type structures are not initiated at heterogeneities, bundlelike crystallites might occur on a considerable scale. If noncoherent surface nucleation did not take place, the  crystallites formed would be scattered throughout the medium, i.e., no spherulites would form, but if noncoherent nucleation did take place, a spherical and microcrystalline but distinctly nonlamellar “spherulite” might form (see sections 4 and 7). The condition eσ>σe will lead to chain folded structures, but under certain conditions these may not be in a readily identifiable spherulite (see below).In the event that mT that lamellar spherulites are formed, and that the principal site of the crystallization is in the spherulites. Typical lamellar spherulitic crystallization frequently occurs under these conditions. We now ask what might change this state of affairs.Suppose it is known for a certain polymer that when it is crystallized by cooling the unoriented melt to a temperature slightly below mT, and that this system is then supercooled. The possibility then exists that bundlelike rather than loop type crystals will form. In this event, the sample would contain neither lamellae or lamellar spherulites. A distinct long X-ray spacing may arise from such bundlelike crystallites if they have a fairly uniform size in the l direction due to strain or other factors. This spacing will almost certainly not vary as 1/(ΔT) as will the spacings from chain folded crystals formed from the unoriented melt, and care should be taken not to confuse the spacings obtained on strongly oriented and unoriented specimens. In moderately oriented systems, chain folded and bundlelike crystals may coexist.Assume now that the polymer liquid is in some manner caused to be in a highly oriented state above cT, i.e., in region B. This will often lead to a hierarchy of very tiny crystallites, most so small they scatter but little visible light.  It is reasonable to suppose that eσ<σe might hold for nuclei of minimal size even when eσ>σe for larger nuclei and crystals. This is implied by the concept that cumulative strain may exist at the bundle ends. In such a case, chain folded lamellar spherulites or other lamellar structures would form above CT, while many of the small crystals and embryos formed below cT would be bundlelike. The fine-grained crystallization produced below cT will tend to be mixed with spherulites formed above cT during the subcooling process.Another possible source of nonspherulitic crystallization in a polymer whose macroscopically observable growth form at low to moderate supercooling is of the lamellar spherulitic type is rapid crystallization well below eσ/σe. Conditions may exist where bundlelike and chain folded structures occur together.In summary, the presence (or absence) of typical lamellar spherulites could depend on the previous orientation of the melt, the temperature of crystallization, the number of heterogeneities present, and the ratio mT, a knowledge of AI as a function of temperature would allow the product σ2σe to be determined by σ2σe” values that are significantly low. These effects are discussed below, partly with the objective of indicating why the radial growth rate of spherulites, rather than their “homogeneous” injection rate, was stressed in the paper.One important reason for attempting to obtain the homogeneous injection rate of spherulites as a function of temperature lies in the fact that the temperature dependence of this quantity determines certain products involving the surface free energies that are different from those obtained from the radial growth rate. Thus, for a lamellar spherulite near heterogeneous nucleation of spherulites. Turnbull [t=0 after the material is supercooled. The number of such active embryos is strongly dependent on the temperature T1 above mT to which the system is initially heated if a crack-size distribution exists. By sufficient superheating, the embryos in the pores or cavities can be melted out, thus rendering them inactive as nucleation centers in a subsequent crystallization. Turnbull’s theory shows that, other things being equal, the embryos in the larger pores are melted out first as T1 is increased. In cases where the cavities are small, and where the heterogeneity is rather strongly wetted by the crystalline phase, the embryos may persist hundreds of degrees above the bulk melting point.Consider first Turnbull  has showt=0, and their number per unit volume is markedly dependent on T1—increasing T1 substantially reduces this number.  It is therefore clear that spherulite producing structures can be, and frequently are, maintained in cracks or fissures in heterogeneities.Spherulites in many cases are well known to follow this pattern of heterogeneous initiation. In such a case they are all born at or near pseudohomogeneous nucleation of spherulites. Heterogeneities may contain flat but wettable regions in addition to pores or cracks. No embryos will persist on these flat surfaces above the bulk melting point. However, when a system containing such heterogeneities is supercooled, nuclei will preferentially appear on these flat surfaces by virtue of the wettability of these surfaces by the polymer crystal. If the number of heterogeneities with flat surfaces is large, the resulting crystals can appear in the supercooled system essentially sporadically in time instead of at t=0 or t=i.τ Under these conditions the crystals will appear not only sporadically in time, but also  randomly in space. For convenience we have denoted this as “pseudohomogeneous” nucleation.  SpT1 or where a hierarchy of spherulites born at t=0 also appears together with those born sporadically. Such effects indicate the presence of numerous wettable heterogeneities. Some of the studies reported in the literature, where the polymer spherulites were believed to be of truly homogeneous origin for the reason that they appeared more or less sporadically in time and randomly in space, may actually refer to the pseudohomogeneous category.Spherulites are sometimes seen to appear nearly sporadically in time and space in polymers. Pseudohomogeneous initiation should be suspected in any sample where the number of spherulites formed per unit volume in unit time for a given growth temperature depends on Homogeneous nucleation refers to the process where crystallization centers are spontaneously formed at random positions in the pure (homophase) mother phase by thermal fluctuations. Such a process is characterized by a rate of production of nuclei per unit volume of mother phase that is, after the establishment of the steady state, truly constant in time. In short, the nuclei appear sporadically in time and randomly in space in the supercooled liquid. This process will be independent of T1 for a wide range of T1 values. True homogeneous nucleation of a bulk phase is an ideal situation not easily achieved experimentally in any bulk system, heterogeneous or pseudohomogeneous nucleation undoubtedly being much more common.t=0; (b) the spherulites appear at random positions in space, and show no memory of previous position in space in successive experiments;T1.Spherulites of strictly homogeneous origin may appear in polymers, but a proof that this is true would be quite difficult. The following conditions are consistent with homogeneous initiation: (a) The spherulites appear sporadically in time with no excess population at or near any bulk material. One would hardly expect polymers to be an exception. Only work with logs [σ2eσ” values quoted in the literature on the basis of the assumption of homogeneous nucleation in the bulk phase are too low.It should be realized that true homogeneous nucleation has rarely been achieved or proved even with very carefully prepared specimens of ordinary size of ith logs  or dispeith logs , where tσ2σe values using dispersions of polymers in a manner analogous to that used by Turnbull for the n-paraffins [It would be of great interest to obtain reliable araffins .An alternative theory for the existence of a “step height” in polymer crystals has been advanced by Peterlin and Fischer . They prambıent temperature rather than the growth temperature, the latter being the case for the theory presented in this paper. There is no evidence suggesting, for example, that the lamellae in bulk become thinner with lowering ambient temperature.Entirely apart from the question of whether or not such a concept is correct in principle, the following comments are relevant: (1) The Peterlin-Fischer theory does not predict, or even in the mode analysis take account of, the existence of chain folds in bulk or in dilute solution. (2) To the extent that their work may be interpreted as referring to the bundlelike system with chains normal to the bundle ends, the objection illustrated in The theory presented here and in an earlier paper shows how chain folded crystals can come into being, and provides a reasonably detailed picture of the properties of systems crystallizing in this pattern.We now give a brief summary of some major points that have been brought out concerning spherulitic crystallization in bulk polymers.eσ>σe, then homogeneous nucleation of chain folded structures will prevail in bulk. It was noted that heterogeneous nucleation is much more probable in real polymer systems, but that if eσ>σe, this type of nucleation will in all likelihood still initiate chain folded structures in bulk. It was then shown that coherent surface nucleation with chain folds will lead to structures possessing a considerable number of physical features commonly associated with lamellar spherulites.  It was also indicated that the assumption of noncoherent nucleation with chain folds could produce a modified lamellar spherulite. In each case, emphasis was placed on predicting the radial growth rate of the spherulite as a function of the crystallization temperature. This property follows a different law for coherent and noncoherent growth, and a differentiation of the two is amenable to careful experiment. Also, the behavior of the step height of a chain folded lamella in bulk was discussed. This included the step height as a function of growth temperature, the behavior on rewarming , and the uniformity of the step height. The predictions given are mostly subject to experimental verification, and some are known to be at least qualitatively correct. It is highly significant that a kinetic theory of crystal growth, wherein a nucleus length maintains itself during growth because of chain folds, can reproduce so many of the known features of spherulitic crystallization in bulk.It was demonstrated that if one assumes that the end surface free energy of a bundlelike nucleus is larger than the corresponding quantity for a folded nucleus, i.e., σe plane is sufficient to create the required type of surface stress. Others have previously shown that such twist can explain many aspects of the optical extinction patterns of spherulites, most particularly the complex rings that are often seen with a polarizing microscope.Given the basic lamellar structure produced by the kinetic crystal growth theory with chain folds, it was shown how surface stress could cause the lamellae to warp or twist. A slight and noncumulative repulsion of the chain folds in the eσ might exceed σe. It was concluded that the value of σe will be considerably larger than had been supposed heretofore, because of the fact that a bundlelike nucleus clearly must possess a density gradient of considerable size at the bundle ends: calculations with a simple model were given showing that even the minimal value of σe would be quite large because of the work required to construct this density gradient region. When considered together with the theoretical and experimental estimates for σe, the result is that it is certainly not implausible to suppose that eσ>σe, at least in some situations. In the one specific case that was considered (polyethylene), it was clear enough that this condition might indeed apply. If eσ>σe for a polymer, chain folded growth is to be regarded as an intrinsic mode of crystallization in the bulk phase.An effort was made to indicate the nature of the reasons that eσ<σe . In sufficiently dilute solution, chain folded platelets will form, no matter whether eσ>σe or eσ<σe, because of entropy considerations.Certain circumstances were mentioned whereby chain folded structures might be prevalent, though in competition with numerous small bundlelike crystallites, in a bulk polymer even if eσ>σe does exist, at least for fairly large crystals, we regard the most probable cause of this condition to be cumulative or noncumulative strain at the ends of the bundlelike nucleus or crystal. Such strain arises ultimately from the fact that for a bundlelike system, the crystal and liquid phases are “connected” through covalent bonds, a situation that does not occur to a significant extent in the folded system. Again we emphasize the fact that the condition eσ>σe virtually assures the predominance of the chain folded growth mechanism, whatever the type of initiation.Assuming that the condition Considerable attention was directed toward a critical examination of whether the classical bundlelike model of polymer crystal growth, or certain variations of it, could lead to a lamellar spherulite. It was concluded that this was highly improbable.eσ-type faces. The bundlelike nucleus with cumulative strain will not grow to large size, and will definitely have rounded ends. (3) The assumption that the exclusion of large chain ends from the crystal ultimately causes the cessation of lengthwise growth of the bundles (on an equilibrium basis) leads to a wide distribution of crystallite lengths that is not consistent with either the uniform thickness, or the surface smoothness of a typical lamella. (4) The tipping of polymer crystals at low draw ratios is not readily understood in terms of bundlelike crystals. (5) It seems improbable that assemblies of strictly bundlelike “lamellae” would cleave along the required planes.In general, the bundlelike models suffered one or more of the following drawbacks: (1) The bundlelike nucleus with noncumulative strain is found to grow in the polymer chain direction, thus destroying any semblance of the stability or uniformity of the step height as observed experimentally in spherulites. (2) Bundlelike crystallites will have a tendency to exhibit rounded ends because of considerations based on strain or minimization of total surface free energy, or both, and this is not consistent with the existence of lamellae with large and flat The deficiencies of the bundlelike models, as contrasted with the ability of the chain fold models to reproduce many of the significant structural feaures of lamellar spherulites, leads to the conclusion that it is highly probable that lamellar spherulites formed in bulk consist of structures that are built on a basically chain folded pattern.To this it must be added that some bundlelike character, in the form of interlamellar links, or chains protruding from the fold plane, must be expected in lamellar spherulites. Further, the existence of a microcrystalline but nonlamellar spherulite built on the bundlelike pattern by noncoherent nucleation is by no means excluded. Therefore, the possiblilty exists that there is more than one basic scheme for the construction of spherulites in bulk despite the evidence that a number that have been carefully studied are lamellar.Finally, some limitations that may exist on spherulitic growth with chain folds were noted. Exclusion of chain ends from the crystal for reasons of large size may hinder fold formation at low supercooling where the step height is large. At high supercooling, nonsteady state nucleation may occur in the medium surrounding a spherulite, and seriously hinder its growth.


Coherence properties of different light sources and how they affect the image quality of holographic display are investigated. Temporal coherence is related to the intrinsic spectrum bandwidth of the light source, while spatial coherence can be affected by the size of the light source and propagation distance in use. These two coherence properties are measured for various light sources of diode-pumped solid-state (DPSS) laser, laser diode (LD), light emitting diode (LED), super luminescent light emitting diode (sLED) and micro light emitting diode (mLED) in different settings, together with the quality of the holographic reconstructed images. Although the image sharpness and speckle are related to both coherence parameters, our results and subsequent analysis show that the spatial coherence can be linked directly to the image sharpness and the temporal coherence to the speckle. This will provide a quantitative way not only to optimize the image quality between uniformity and sharpness but also to determine the safety power level for different light sources when viewing the produced images by human eyes directly. Light source plays a critical role in holographic displays, and the conventional requirement is a high degree of coherence for achieving sharp reconstructed images. Lasers are normally used in holographic displays because they have high spatial and temporal coherence. However, the high degree of coherence also brings in significant speckle14 in the reconstructed images, which affects the image quality greatly. Several techniques have been reported19 to tackle the speckle issue, such as time-averaged superposition of the same reconstructed image with uncorrelated initial random phases or different sub reconstructed images consisting of selected points of the same target image and applying phase grating or diffusers. However, all these techniques either increase the complexity of the system or increase the computation costs and decrease the bandwidth of the reconstructed images.Holographic displays can reconstruct three-dimensional (3D) images with full wavefront information22. The low temporal coherence of LEDs could reduce the speckle, but the low spatial coherence of LEDs requires additional spatial filter such as pinholes or microscopic objectives to be used to select the emitted light from a local area of an LED in order to increase the spatial coherence and obtain reconstructed images with acceptable sharpness. As a result, the spatial filter will decrease the energy efficiency significantly. We could see that the coherence properties of the light source directly influence the quality of holographic reconstructed images on both image sharpness and speckle. However, how temporal and spatial coherence affect the quality of the holographic reconstructed images especially quantitatively remains to be investigated.In the past, holographic displays based on partially coherent light sources, such as the light emitting diodes (LEDs), has been reportedIn this work, we study the properties of spatial coherence and temporal coherence of different light sources and the resultant qualities of holographic reconstructed images. Two ways to increase spatial coherence for a partially coherent source have been introduced and the experimental results are in good agreement with the theoretical values. Subsequently we analyze quantitatively the influence of the two coherence properties on the sharpness of the reconstructed images and speckle. Finally, we conclude with some general requirements for optimized holographic image quality when using different coherence light sources, which can be applied to both digital and optical holographic displays including those viewed by eyes directly.Theoretically a holographic display is based on the use of an ideal coherent light source. However, most of the light sources used in practice are not fully coherent. To characterize the coherence property of a light source, temporal coherence and spatial coherence are generally evaluated.t. If t = 0, there will be no delay and the two signals are fully correlated. The coherence time tC is defined as the delay time needed when the correlation drops to 0, while the coherence length Lc is defined as the distance which the wave travels within this period of time. Temporal coherence describes how monochromatic a light source is. For a light source with a Gaussian emission spectrum, the coherence length is given by23:c is the speed of light, n is the refractive index of the medium, λ is the central wavelength and Δλ is the full width half maximum(FWHM) of the emission peak in wavelength spectrum. For the light source in the visible range used for displays, the value of λ ranges from 400 nm to 700 nm, but the spectrum width varies greatly for different light sources. The light sources with a small Δλ such as lasers are highly temporally coherent, while the light sources with a large Δλ such as white light lamps are generally temporally incoherent.Temporal coherence measures the averaging correlation of the light signals at any pair of time moments between a wave from a light source and itself delayed by a time period of 27 and describe the spatial coherence as:x1, y1) and  are the coordinates for the two points in space, r1 = x1i + y1j, r2 = x2i + y2j, S the light source size and I  the intensity distribution of the light source. If we use a uniform square LED source of a side length a and place a set of double slits of pitch b after the source at a distance z, we can treat I  = 1 within the emitting area and simply Eq. Spatial coherence describes the correlation between two points in the space and the ability of the two points to interfere in extent of wave in averaged time. To calculate the degree of spatial coherence, we apply Van Cittert-Zernike theoremWe can see that for a fixed separation between the two slits, the degree of spatial coherence depends on both the size of the light source and the distance between the light source and the double slits. In the following, we will use Eq. 30 are loaded on a spatial light modulator (SLM) such as a LCOS device31 for the reconstruction of holographic images. The CGH itself works as a diffractive optical element (DOE) to diffract incident light wave to form target patterns. Although such a DOE may appear to be complicated, its working principle remains to be the same as that of a grating, or in an even simpler case a set of double slits which could be regarded as a representation of two spatially separated pixels of a CGH in forming the interference patterns as shown in Fig. In digital holography, computer generated holograms (CGHs)W1 of one wavelength, it will form an interference pattern on the observation plane. For the wave W2 with a longer wavelength, the corresponding interference pattern will have a longer peak-to-peak distance. As the two patterns are of different wavelengths, they will add up on the observation plane in intensity forming a new pattern with an imposed envelope curve so that the intensity of higher orders will decrease more quickly.The effect of temporal coherence is discussed in Fig. S1, then the light wave from it travels to a set of double slits where it will be separated into two new waves. The two new waves will interfere with each other and form interference patterns at the observation plane. For another point source S2 of the same wavelength but slighted displaced from S1 with the same distance from the double slits, it will generate almost identical but shifted interference patterns on the observation plane (shown as the dashed line). These two interference patterns will add up in amplitude and often result in a drop in the contrast of the superposed interference pattern in terms of intensity.The effect of spatial coherence on image sharpness is illustrated in Fig. Comparing the effects of spatial coherence and temporal coherence on the sharpness of reconstructed images showing in Fig. S{}, the light source or the system input as gin  and the reconstructed image at the system output as gout , where  and  refer to the coordinates in the input plane and reconstruction plane, respectively, we can have:In the system, the wave from an ideal point source is collimated first by a collimation lens and then imposed on with phase information by a phase hologram. After that, this information carrying wave is focused by a Fourier transform lens to its focal plane, where we could see the reconstructed image. If we denote the transformation of the whole display system as S{} consists of four free space Fresnel propagations, two lens phase factors and one phase hologram, all these are linear phase kernels, thus the whole system obeys:The system gin  as an extended light source, then it can be expressed as the integration of many point sources. For each point source at the position  on the input plane, it can be regarded as a δ-function, and according to the sifting property32 of the δ-function, gin  can be expressed as:Therefore, we can see that the holographic display system is a linear system. If we consider the system input gout  becomes:gin  can be regarded as a weighing factor applied to δ. Then according to the property of linear system as expressed in Eq. By substituting Eq. h to denote the impulse response function of the system, whose input is the δ-function at , it can be described as:If we use δ-function represents an ideal point source and the impulse response function h of the system is the reconstructed image of it. If the point source moves its position on the input plane, the center position of the output or the reconstructed image will move accordingly but the distribution of the reconstructed image stays the same. In other words, h depends only on the distance differences of x2 − ξ and y2 − η. In this case, the holographic display system is a space-invariant system. For a space-invariant system, h can be written asHere the With Eq. Gout, Gin and H represent the Fourier transform for gout, gin and h respectively. Since h can be regarded as the reconstructed image in a holographic display system, H will be its representation in the frequency domain. If the input image is a point source, Gin will be the Fourier transform of a δ-function, which will have Gin = F{gin} = F{δ} = 1, where F{} denotes a Fourier transformation. This means that using a point source of light the full information of the reconstructed image in the frequency domain will be retained at the output.It shows that in this system the output image is the convolution of the input image with the impulse response function:29, which is then multiplied with the representation of the reconstructed image in the frequency domain. As a result, the Fourier transform of the light source shape will directly “filter” the frequency information of the reconstructed image and hence its quality for the output. As discussed in above, the size/shape of an extended light source is directly related to its spatial coherence, which will in turn affect the sharpness of the reconstructed image at the output through modifying its frequency information.For an extended light source in the shape of a square or a circle, the corresponding Fourier transform is a 2D sinc function or a 2D Bessel functionThe representation of a reconstructed image in the frequency domain from its center to outside corresponds to the frequency components from low to high frequencies. For a 2D sinc or a 2D Bessel function, most of the energy is distributed in the central order. The FWHM of the central order depends on the size of the light source: the larger size, the smaller FWHM. Consequently, using a large size light source will result in a significant loss of the frequency information of the reconstructed image at the output.Simulation results in Fig. The foregoing results are based on the assumption that the light source is monochromatic, i.e. fully temporally coherent. However, in practice, many light sources such as LEDs or even some lasers are not really monochromatic. In general, it is more often the case that a light source is polychromatic.32. However, at different moments, the wavelength of the wave emitting from the source may be different, thus we need to consider the factor of time. As a result, for a light source with a narrow spectral bandwidth, a time-averaged impulse response function For a light source with a narrow spectral bandwidth such as laser, it can still be considered as a highly coherent sourcementclasspt{minimaIin and Iout are the intensity distributions of the light source and the reconstructed image, respectively.For a light source with a broad spectral bandwidth such as LED, it should be regarded as an incoherent source. In this situation, the system is no longer a linear system in amplitude, instead, it becomes a linear system in intensity. Thus Eq. To conclude, both the temporal and the spatial coherence of the light source affect the sharpness of the reconstructed image, but the spatial coherence may contribute more. Reducing the temporal coherence of the light source will change the impulse response function of the display system, which deforms the target reconstructed image with an imposed envelope curve. While reducing the spatial coherence of the light source results in a loss of frequency information, especially the high frequency part, which represents the details of the reconstructed image.There are usually two origins for speckle in digital holographic display: reflection or refraction from rough surfaces and interference between adjacent spatial frequency pixels.Firstly, we consider a wave emitting from a point source and there is a rough surface on its propagation path. The wave reflected by such a rough surface is the sum of secondary waves from many independent scattering areas. For one specific point on the observation plane, the secondary waves travel different light paths on arrival, thus these dephased secondary waves will interfere at the point. If we choose another point on the observation plane, the corresponding interference may vary. All these points on the observation plane form the speckle. The speckle can be observed in the form of granularity, where the bright part indicates strongly constructive interference, the dark part indicates strongly destructive interference, and the grey part is between these two extremes. It is not limited to reflection scenario, refraction from optical components with rough surfaces such as lens or polarizer could also cause speckle. The condition that two secondary waves can interfere is that their light path difference is shorter than the coherent length, while the coherent length is directly associated with temporal coherence. Low temporal coherence with shorter coherence length will help to reduce this kind of speckle. Furthermore, for an extended light source, we consider two points emitting waves with the same wavelength. If the two waves are spatially incoherent, their interference at the same point on the observation plane will add up on intensity basis. Therefore, low spatial coherence also contributes to reduce speckle.29. For example, if the overall aperture of the hologram is square, then the profile of one spatial frequency pixel is 2D sinc function. As a consequence, between two adjacent spatial frequency pixels, the higher oscillation orders of the sinc function will interfere, and cause speckle19, which is illustrated in Fig. Secondly, in digital holographic display system, the overall intensity profile in the replay plane is decided by the aperture of the hologram pixel, while the intensity profile of one spatial frequency pixel in the replay plane is decided by the overall aperture of the hologramσ and the spectrum bandwidth W:C is the new speckle contrast and C0 is the old speckle contrast, and we see that for a light source with broader spectrum bandwidth, the speckle contrast will decrease. In the second case, one point source will generate one pattern of speckle in the replay field, while the other point source will generate another one following the same principle. However, since these two sources are in the same wavelength and have fixed phase differences, the two different patterns of speckle will add up on basis of complex form and generate a new pattern of speckle.To investigate the effect of coherence properties on speckle in this situation, we consider the two cases that shown in Fig. Therefore, it can be concluded that both temporal and spatial coherence contribute to the formation of speckle for a holographic display system. However, for the speckle formed by interference between adjacent spatial frequency pixels, the change of spatial coherence only changes the pattern of speckle, while the decrease of temporal coherence could efficiently reduce the speckle contrast. Note that the spatial coherence needs to be high to deliver sharp holographic images, hence reducing temporal coherence is more often used to reduce speckle in practice.35 from Plessey, a super luminescent LED (sLED) (λ = 662 nm)38 from EXALOS, a laser diode (LD) (λ = 649 nm) and a DPSS laser (λ = 671 nm). The measured intensity spectrum and calculated temporal coherence length according to Eq. We select some typical or potential light sources for digital holographic displays, including a light emitting diode (LED) (λ = 632 nm), a mircro LED (mLED) (λ = 477 nm)er λ = 67 nm. The From the results, we could see that DPSS laser and LD have long temporal coherence lengths and the three different types of LED all have short temporal coherence lengths. Thus, in terms of temporal coherence, lasers are good for digital holographic displays in achieving sharp reconstructed images.z is set to be 1 m. We measure the degree of spatial coherence for LED , mLED, sLED and DPSS laser. The results are shown in Fig. To measure the spatial coherence of different light sources, we use double-slits experiment setup as shown in Fig. Imax is the central peak value and Imin is the neighbouring valley value. The value of µ ranges from 0 to 1, with 0 for completely spatially incoherent and 1 for fully spatially coherent. The spatial coherence values for the different light sources are LED 0.02, mLED 0.49, SLED 0.67 and DPSS laser 0.88, respectively, as shown in Fig. The average cross-section intensity distribution of the interference patterns are shown in Fig. From the experimental results, we can see that the LED has a low spatial coherence value and we can barely see any interference patterns at the observation plane. The mLED has a square emitting area, whose side length is about 300 µm. Due to its small size, the spatial coherence value of mLED is high and we can clearly see the interference patterns at the observation plane. sLED is one kind of source with a low temporal coherence length Fig.  but a hiThe results indicate that if we directly use an LED in a holographic display, we cannot see a clearly reconstructed image. However, we could add a spatial filter to increase the spatial coherence value of the LED and obtain a clear image. If we use an sLED or a DPSS laser, we may find some speckle within the fringe lines, since these light sources have high spatial coherence values.The next thing we do is to analyze the spatial coherence values for an LED source with different sizes of emitting areas. Firstly, we measure the spectrum of the same LED source filtered by different pinholes and the result is as shown in Fig. z = 80 cm and the size of the pinholes changes from 1 mm to 0.1 mm. The theoretical and experimental values are shown in Fig. Secondly, we measure the spatial coherence using the same double slits setup as shown in Fig. From the results, we can find out that the higher the spatial coherence value is, the sharper the interference patterns are. Meanwhile, the experimental spatial coherence values are in good agreement with the theoretical ones, and both of them are not monotone decreasing when we increase the emitting size of the light source. Besides, the interference fringes is regarded as sharp when the pinhole size is smaller than 0.3 mm, or the spatial coherence value is higher than 0.5. Therefore, we consider it as feasible to use LEDs in digital holographic displays to achieve sharp reconstructed images, if the size of its emitting area is smaller than 0.3 mm.Apart from limiting the emitting size of an LED, we can also increase its spatial coherence by increasing the distance of its light propagation. For a uniform light source propagating in free space, we can calculate the theoretical spatial coherence values shown in Eq. From Fig. On the other hand, for the same propagation distance but different emitting sizes, the spatial coherence values show a similar trend as that in Fig. In this section, we study the corresponding holographic reconstructed images for the change in image sharpness. The experimental setup is shown in Fig. The zero order locates in the center of the replay field, which is due to unmodulated light. It is the imaging of the emitting area of the LED, which can be observed in Fig. To investigate the relations between coherence properties and image sharpness as well as speckle, we analyze the holographic reconstructed images based on DPSS laser, LD, sLED, LED and mLED. We have measured the spatial coherence of these light sources and normalized them based on the value of DPSS laser, which is 0.88. The reconstructed images of the same target image and the analysis results are shown in Fig. N are the row and column values of the area of interest, Ii,j is the intensity value at a particular position  and Ῑ is the mean value of intensity in the area of interest. As shown in Fig. In Fig. On image sharpness, the features in the reconstructed images are all distinguishable for the different light sources in use. To make the results comparable, LED sources have been spatially filtered and all the images are normalized to the same brightness level. To evaluate the image sharpness, we choose the same area of interest containing one edge of the square pattern, as shown in Fig. As we discussed previously, the sharpness of the reconstructed images in holographic displays are mostly influenced by the spatial coherence, while the speckle are mostly influenced by the temporal coherence. According to the data in TableW~Δλ/λ2 = α/Lc . In this case, if we take a look at Fig. Coherence property of a light source can be characterized by its temporal coherence and spatial coherence values, respectively. Light sources such as DPSS laser, LD, LED, sLED and mLED have been characterized and the corresponding holographic images displayed.Image sharpness and speckle are influenced by both temporal coherence and spatial coherence of the light source in use. It is found that the image sharpness value is linear proportional to spatial coherence value, while the speckle contrast value is linear proportional to the temporal coherence value.Temporal coherence is decided by the intrinsic spectrum bandwidth of the light source and it can be improved by filtering the spectrum of the light source. On the other hand, spatial coherence is influenced by the size of the light source and the propagation distance in use, it can be improved by changing the size of the utilized light emitting area or the light propagation distance. For example, in an LED based holographic display system, a spatial filter is often applied to reduce the utilized size and increase the spatial coherence of the LED source. The results showed that a pinhole smaller than 300 µm is enough to obtain a sharp holographic image even for a short propagation distance. However, improving either the temporal coherence or the spatial coherence of a light source by external means is at the cost of the reduced light intensity hence the reduced light efficiency.Consequently, a light source with high spatial coherence and low temporal coherence is ideal for a holographic display in order to obtain high quality images with good sharpness and minimum speckle. sLEDs and mLEDs, which are not commonly used in holographic displays, are suitable light sources for this purpose. LEDs with a broad spectrum can also be used to reconstruct holographic images with less speckle, but it has to be spatially filtered for reconstructive sharp images. Otherwise there will be significant reductions in the energy efficiency and brightness of the reconstructed images.By selecting suitable levels of temporal and spatial coherences of a light source respectively, it is possible to optimize the produced image quality between uniformity and sharpness quantitatively. The corresponding speckle level can further help to determine the range of intensity variation due to the light coherence and hence the safety power level for a given light sources when viewing the produced images by human eyes directly.40.For future work, we will investigate the ways to improve the spatial coherence of an LED while maintaining its good light efficiency. We will also consider investigating the impact of different coherence properties on the quality of star imaging for better positioning and tracking