1. IntroductionThe acknowledged versatility of polymeric

materials, which are widely used in the form ofplastics, films, coatings and fibers, arises fromthe complex structural organization in thesematerials. X-ray diffraction (XRD) has long beensuccessfully used to study various aspects ofthese structures in semicrystalline polymers,which includes thermoplastics, thermoplasticelastomers and liquid crystalline polymers.While many of the well established methods forthe determination of molecular structure, evalu-ation of crystallinity and analysis of texture,continue to be improved to enhance the speedand precision of these measurements, newtechniques are also being continually intro-duced. With the availability of intense X-raysources, high-speed detectors and faster meth-ods of analyzing the data, it is now possible toexamine the structure at higher spatial resolu-tion and smaller time scales. For instance, it ispossible to follow development of structure in aprocessing line, such as in spinning and draw-ing operations, or in extrusion of films with atime resolution of less than one second [1–4], orexamine the structural inhomogeneities that in-

duced over distances of a few micrometers bythe temperature and stress gradients that existduring processing [5–7]. These developmentsalso permit combining XRD analysis with othercharacterization techniques such as thermalanalysis [8], rheology, spectroscopy and mi-croscopy [9]. We will discuss some of these de-velopments that are useful in understanding thehierarchical structure and inhomogeneity inpolymers, and the analyses of the 2-D data thatare now commonly obtained during these mea-surements

2. Structure in Semicrystalline PolymersPolymers exhibit structural hierarchy at multi-

ple length scales (Figure 1) [10, 11]. As a poly-mer is cooled from the melt, a fraction of thepolymer chains (0.5 nm dia.) crystallizes intolamellae or small crystallites (10 nm in size), andanother fraction remains amorphous as itfreezes in its molten state. The organization ofthese crystals into the next level of hierarchydepends on the external constraints. In fibers,these lamellae are organized into fibrils (100 nmin length), which in turn form filaments (5 mmdia) that are then made into fibers (0.5 mm dia).

THE RIGAKU JOURNALVOL. 21 / NO. 1 / 2004, 15–24

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RECENT DEVELOPMENTS IN POLYMER CHARACTERIZATIONUSING X-RAY DIFFRACTION

N. SANJEEVA MURTHY

Physics Department, University of Vermont, Burlington, VT 05405

An introduction to polymer structure and morphology will be followed by a brief review ofthe structural parameters that are commonly measured using X-ray scattering techniques. Newmeasurement capabilities and their relevance for understanding the structural basis of perfor-mance will be discussed. Two techniques, microdiffraction and surface-enhanced scattering,two areas of investigation, crystallization kinetics and polymer deformation, and two examplesof analysis of data, complete analysis of 2-D wide-angle pattern and analysis of 2-D small-angleX-ray scattering patterns in elliptical coordinates will be highlighted to illustrate the recent de-velopments in polymer X-ray diffraction.

Fig. 1. A model for the hierarchical structure that is expected to be present in an orientedsemicrystalline polymer.

In the absence of an orientational force, thelamellae organize into spherulites (1–10 mm indiameter). X-ray scattering can be used to ob-tain structural information at three lengthscales—1, 10 and 100 nm—using scattering atwide-, small- and ultra small-angles, respec-tively.

A continuum of structures between the ex-tremes of what are generally regarded as amor-phous and crystalline phases are present in areal polymer, and these entities have complexorganization. But, a model that describes thesemicrystalline polymers in terms of twophases, an average amorphous and an averagecrystalline phase, has been found to be ade-quate for many practical purposes. The fractionof the material that is crystalline, the crys-tallinity or crystalline index, is an important pa-rameter in the two-phase model. Crystallinitycan be determined from a wide-angle X-ray dif-fraction (WAXD) scan by comparing the areaunder the crystalline peaks to the total scatteredintensity [12]. The accuracy and the precision ofthese measurements can be improved by draw-ing a proper base-line, using an appropriateamorphous template, and by carefully choosingthe crystalline peaks [13, 14]. The disorder inthe crystalline domains can be evaluated bymeasuring the crystallite sizes which are relatedto the radial widths D(2q) of the reflections at ascattering angle 2q by the Scherrer equation. Inreality, there are two contributions to the width:one is the size and the other is the paracrys-tallinity or microstrain [15, 16]. A more detailedanalysis based on the Warren-Averbach methodis widely used in metals and ceramics, but lessso in polymers [17]. The disorder in the crys-talline domains is also reflected in the unit celldimensions. But, calculation of the unit cell pa-rameters requires an accurate measurement ofthe positions of many crystalline peaks, whichcan be difficult. Therefore, in practice, relativepositions of selected crystalline peaks are usedas accurate measures of the changes unit cellparameters [18, 19].

Structures at length scales larger than a unitcell (�10 nm instead of �1 nm) can be investi-gated using small-angle X-ray scattering(SAXS). The methodology for these analysis isnow highly developed and can be found in anystandard literature [9, 20–24]. While WAXD isused to study the orientation of the crystals,and the packing of the chains within these crys-tals, SAXS is used to study the electron densityfluctuations that occur over larger distances asa result of structural inhomogeneities. SAXS is

widely used to study the lamellar structure bymeasuring parameters such as lamellar spac-ing, height and diameter of the lamellar stacks,and thickness of the transition layer betweenthe crystalline and amorphous domains. In theanalysis of fibers, SAXS can provide informa-tion about the details of fibrillar morphologysuch as fibril diameter and orientation, andlarge scale inhomogeneity such as microporesand cracks. This information is somewhat simi-lar to that obtained from a transmission elec-tron micrograph, with one important difference:SAXS requires no sample preparation, and thedata is averaged over the area (typically �0.1mm2) of illumination. SAXS is also used forstudying conformation, size and dynamics ofpolymers in solutions and in gels.

3. New Methods to Study PolymerStructure

The two-phase model for the polymer hasbeen quite useful in providing a qualitative un-derstanding of the polymer properties in termsof its structure, but is not adequate for quantita-tive prediction of the polymer properties. Forthis purpose, a detailed knowledge of the char-acteristics and distribution of soft (amorphous)and hard (crystalline) domains, and the interac-tions between these domains is necessary. Newtechniques that have been introduced duringthe past decade provide precisely this informa-tion. Some of these techniques will be dis-cussed here.

3.1. Microbeam DiffractionMicrobeam diffraction, or microdiffraction,

has been used in semiconductor industry forover 25 years [25]. It is now being used to ex-amine polymeric materials. In most routinecharacterization of polymers, it is assumed thatthe structure is homogeneous. But, this is notalways the case. Temperature gradients are pre-sent during injection molding, and both temper-ature and stress gradients are present duringextrusion and drawing. These gradients intro-duce structural inhomogeneities that influencepolymer performance. Even filaments that areonly 10 mm in diameter show variations in ori-entation and density across the cross section [5,26]. These structural gradients, and the changesin these gradients during deformation can nowbe studied at spatial resolutions as small as1 mm using microbeam diffraction [26]. An ex-ample of the typical structural gradients presentin a shown in Figure 2 [6]. This diffractogramwas obtained from KevlarTM fiber with a 3 mm

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diameter X-ray beam at the ESRF (EuropeanSynchrotron Radiation Facility) synchrotronsource. The data show that the Herman’s orien-tation function of the crystalline domains in this12 mm diameter fiber increases from 0.955 atthe center to 0.980 at the surface of the fiber.The higher orientation of the skin layer is obvi-ously due to large shear stresses at the spin-neret, extensional forces in the air-gap and thesolidification in the coagulation bath. Such a

structural gradient implies that the modulus de-creases from the skin to core. It is interesting tonote that these inhomogeneities gradually de-crease and disappear under uniaxial stress.

Microbeam techniques have reached a levelof sophistication that it is now possible to focusX-rays on a micron size crystal and follow thechanges in the structure from one crystal to thenext within a spherulite [27]. Figure 3 shows aseries of hundred patterns registered from a

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Fig. 2. (a) Wide-angle X-ray diffraction data from a Kevlar fiber using 3 mm diameter beam(l�0.08 nm). (b) Plot of the Herman’s orientation function fc (courtesy of C. Riekel, ESRF).

Fig. 3. A series of wide-angle X-ray diffraction photographs from crystals located along thevertical line within a spherulite of poly(hydroxyl butarate) shown in the left inset. The enlargeddiffractograms are from three areas separated by 60 mm as shown in the optical micrograph(courtesy of C. Riekel, ESRF).

spherulite of Poly(hydroxyl butarate). The pho-tographs show the changes in texture as thebeam was stepped in increments of 3 mm alongthe vertical line drawn in the polarized opticalmicrograph of the spherulite shown on the left.The three photographs in the foreground showthe differences in the texture of three crystals60 mm apart.

In other experiments, liquid crystal molecularalignment of the memory state in polymer dis-persed liquid crystal (PDLC) used for lightvalves and displays has been investigated [28].Microbeam diffraction, both SAXS and WAXD,has been used to measure the misalignment ofthe crystalline cellulose microfibrils with respectto the fiber axis from the azimuthal broadeningsboth of the equatorial small-angle scatteringstreak and of Bragg reflections, and the diffusescattering on the layer lines and the equator ofthe fiber diffraction diagram is used to identifythe presence of disordered cellulose betweenthe cellulose microfibrils and of defects insidethe crystallites [29]. Microbeam diffraction isalso useful in understanding the interfaces, forinstance the morphology of transcrystalline re-gions [30].

Although the experiments described above,and other to be described later, were carried outat synchrotron sources, it should be noted thatit is possible to carry out somewhat less de-manding, but equally important measurementsusing in-house facilities. Microfocus X-raybeams from glass capillaries are now used toexamine areas as small as 50 mm using sealedor rotating anode generators

3.2. Grazing Incidence DiffractionInhomogeneities in materials can be explored

from an entirely different perspective usinggrazing incidence diffraction (GID), also knownas glancing-angle diffraction or surface-en-hanced scattering [31–33]. X-rays have a refrac-tive index of slightly less than 1 in a solid andhence undergo total external reflection for an-gles of incidence (a) less than a critical angle acwhich is typically 0.2°. This totally-reflectedbeam penetrates only the top 50 Å at the sur-face. A small fraction of this beam will be dif-fracted giving a weak diffraction pattern fromthe surface region alone. For a�ac, we get dif-fraction from layers below �50 Å from the sur-face of the film. Comparison of the two scansshows how the effect of the surface on the poly-mer structure. By keeping the angle a (see insetto Figure 4) that the X-ray beam makes a withthe sample surface small (�1°), it is possible

limit the penetration depth of the X-rays intothe sample, thus reducing background scatter-ing from the substrate or the bulk of the poly-mer. By varying a , one can change the penetra-tion depth of the X-rays from several nm up totypically several 100 nm (determined by the ab-sorption length). Depending on the length scaleof the structure of interest, the exit angle can beeither small (�10 nm structures, GISAXS) orlarge (0.1 nm structures, GIXRD). This techniqueis useful for studying skin-core structural gradi-ents at the surfaces of flat samples at depthsfrom a fraction of a mm to more than a millime-ter, as well structures near the surface at depthsas small as a few nm [34–36]. The glancingangle technique is especially well suited if thepolymers are heavy absorbers, e.g., fluoropoly-mers, in which the penetration depth of X-raysis as little as 5 mm at even large incidence an-gles of 1°. In industrial laboratories, these mea-surements are useful in analyzing multilayerfilms and in measuring the changes from skinto core in injection molded plastics, for instance

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Fig. 4. Grazing incidence angle X-ray diffractionscans obtained at several incidence angles (a) from athree-layer laminate of polyethylene (PE), nylon 6(N6) and polyethylene (PE).

in assessing the performance and dimensionalstability of engineering plastics.

The utility of this technique is illustrated inFigure 4 which shows GID scans obtained atseveral incidence angles (a) from a multilayerpackaging film [32]. The peak at 2q�23.4° in thefirst scan (a�0.25°) is due to the crystals ofpolyethylene (PE) at the surface of the top PElayer. The peak at 2q=22.8° in the second scan(a=0.5°) is due to the PE crystals beneath thesurface in the top PE layer. As a is increasedfurther, we begin to see the 25° peak from thebiaxially oriented nylon layer. These measure-ments clearly demonstrate the ability of the GIDtechnique for examining the surfaces, interfacesas well as depth-profiling the structure in poly-meric materials.

GID technique is useful in many areas thatdeal with surfaces and interfaces includingpaints and coatings, adhesives, polymer-basedelectronic devices, and biocompatible materials.GID is currently used extensively to studynanostructured surfaces and the structure at air-polymer and polymer-substrate interfaces inpolymer films deposited onto a substrate. Ex-amples include the use of GID to assess thestructure the structure formation in multicom-ponent ultrathin polymer blend films at andbelow the surface [37], and the study of orienta-tion, conformation and packing modes of thechains near a substrate [38].

3.3. Crystallization KineticsCrystallization kinetics in polymers has been

traditionally investigated using optical mi-croscopy. But recently XRD has been used tofurther elucidate the development of hierarchi-cal structure in polymers. Whereas optical mi-crograph show the crystal nucleation eventsand growth usually at the level of spherulitesand fibrils, XRD results can show the growth ofstructures at smaller length scale, i.e., lamellaeand crystals. These data can help us in deter-mining whether crystallization occurs via spin-odal decomposition or by nucleation, reveal thepresence of mesophases and different poly-morphs that are typically formed under differ-ent cooling conditions, and enable the study oftransformation from one crystalline polymorphto another due to external conditions such asheat and stress.

Figure 5 shows typical results of the investi-gation using combined small- and wide-angle X-ray scattering from polyethylene during isother-mal crystallization [9]. The data show that dur-ing primary crystallization, both average longperiod (L) and lamellar thickness (lc) decreasesignificantly. These values are much smaller,approximately linear with log time, and de-crease over a longer period of time during sec-ondary crystallization. The decrease in L and lcis attributed to formation or insertion of thinnerlamellae within existing stacks. Similar experi-

19

Fig. 5. Evolution of lamellar parameters (long period (L), crystal thickness (lc), amorphouslayer thickness (la) and invariant (Q) and crystallinity (Fc) extracted from time-resolved 1DSAXS/WAXD study of PE isothermal crystallization at 115°C. Left panel is the model of that de-scribes the data show on the right (courtesy of B. Hsiao, SUNY, Stony Brook).

ments have been carried out to study thecomonomers and plasticizers that slow the rateof crystallization, and nucleating agents as wellprior mechanical history such as shear that in-crease the rate of crystallization [39].

Crystallization mechanism other than nucle-ation and growth are being investigated usingX-ray techniques. It has been suspected forquite some time that crystallization in orientedamorphous polymers is somehow differentthan in unoriented polymers [40]. These modesof crystallization are now attributed to spinodalmechanisms [41], and are being intensely inves-tigated. In some recent experiments on polymermelts quenched below the melting temperature,spinodal kinetics are observed in small-angle X-ray scattering before the emergence of Braggpeaks at wide angles [28, 42]. It is proposed thata coupling between density and secondaryorder parameters such as chain orientationgives rise to a liquid–liquid binodal buriedwithin the equilibrium liquid–crystal coexis-tence region. Orientation can enhance the ki-netic role of this hidden binodal [43].

3.4. Polymer DeformationThe macroscopic modulus in inorganic mate-

rials is about the same order of magnitude asthat calculated from the crystal modulus. Inthese materials, when the macroscopic modu-lus is far less than the crystal modulus, the rea-

sons are clearly understood in terms of defects,dislocations and other well studied structuralfeatures. In contrast, the macroscopic modulusin many polymers is only a small fraction(about 1/50) of the crystal modulus. Reasons forthis large difference between the maximumachievable modulus (single chain property) andthe observed modulus (bulk property) are notclearly understood. This and other related ques-tions can be answered by studying the defor-mation of structures at various length scales asthe polymer is deformed in situ. Many resultshave been published in this area [44–50]. Theinstrument of choice is usually a synchrotron[51], although interesting experiments can becarried out on-in-house equipment as well. Theusefulness is illustrated with an example below.

Figure 6 (left panel) is the combined wide-and small angle X-ray diffraction from a poly(ethylene terephthalate) fiber [52]. By measur-ing changes in both the wide- and small-angleregions, simultaneously and in real time, it ispossible to deformation in structures at threedifferent length scales, the crystals, the lamellaeand the fibrils. Figure 6 (right panels) shows thesequence of SAXS patterns as the fiber isstretched, starting from a fiber with no load(Figure a), at two levels of stress (Figures b andc), and after the load was removed (Figure d).The pattern from the relaxed fiber is same asthat of the starting fiber indicating that the plas-

20

Fig. 6. Left Panel: Combined small- and wide-angle X-ray pattern obtained from a poly(eth-ylene terephthalate) fiber. The origin is x�50 and y�250 pixels. The SAXS data near the originextends to about 2.5° (l�0.0919 nm). Right panels: A sequence of SAXS patterns starting from(a) unstretched fiber, (b) at some initial load (b) at a load close to the breaking strength and (d)after removing the load. The x and y scales are give in pixels. 2q (rad)�(distance in pixels fromthe origin)*0.0816/515.

tic deformation is minimal even though thefiber was taken close to the breaking strength ofthe fiber. The relaxed fibers have four reflec-tions whereas the fibers under load have tworeflections. This is attributed to the surface-nor-mal of the lamellae being tilted with respect tothe fiber axis in the relaxed fibers, and beingoriented along the fiber-axis in the stretchedfibers. The changes from the four-point in theSAXS pattern of the relaxed fiber to the two-points in the fibers under stress suggests thatstructures at 10–50 nm length scales undergolarge scale, reversible reorganization understress. The results have shown that while thesingle-chain modulus is �200 GPa, as measuredby the increase in the chain-axis repeat of thechains within the crystals, the modulus of thelamellar structures is �5 GPa, same as that ofthe bulk polymeric fiber. This indicates that re-organization of crystallites, as evidenced by thetilting of the lamellae, could be one of the rea-sons for the fiber modulus being a small frac-tion of the chain-modulus in semicrystallinepolymers.

3.5. Analysis of Diffraction PatternsLeast-squares analysis of the complete 1-D

scan is now commonly used to obtain the rele-vant structural parameters such as crystallinity,crystallite size and unit cell dimensions [53]. Ri-etveld refinement is also used when high qual-ity data is available and when the structure ofthe polymer is known [54, 55]. In addition tothese two techniques, full-pattern analysis ofthe 2-D pattern also provides the most reliablestructural parameters. Because polymers aredisordered, and considering that proper analy-sis of even a 1-D scans is difficult, progress inanalysis of the 2-D data from polymers hasbeen slow.

One parameter that can be obtained onlyfrom 2-D analysis of the diffraction scans is theamorphous orientation. While crystalline orien-tation can be determined from 1-D azimuthal in-tensity distribution of the crystalline reflections,and expressed in terms of Herman’s orientationfunction [12], the evaluation of amorphous ori-entation is not straight forward even in amor-phous polymers. It might appear that in com-pletely amorphous polymers, such as rubber orpolycarbonate, the azimuthal width of amor-phous halo, in analogy with the crystalline ori-entation, is a measure of amorphous orienta-tion. But unlike the crystalline peaks, the amor-phous halo has a large non-zero base-line (Fig-ure 7a). This is attributable to a large fraction of

unoriented amorphous chain segments. Notethat this base-line drops as the fiber is drawn in-dicating that a large fraction of the amorphouschain segments are being oriented during thedrawing process. To fully describe the amor-phous orientation, we need two parameters:one is the width of the amorphous peak and theother is the ratio of the area of the amorphouspeak above the base line to the entire amor-phous intensity. The first parameter representsthe degree of orientation of the oriented amor-phous components (foa), and the second (Foa)describes the fraction of the oriented amor-

21

Fig. 7. (a) Azimuthal intensity distribution in theamorphous halo from a poly(ethylene terephthalate)fiber. The full lines are from an undrawn, as spunfiber. The dotted lines are from a drawn fiber. (b) 2-Dcontour map of the diffraction pattern from an ori-ented PET fiber (c) 2-D fit using a two functions.

phous phase [19, 56]. Such analysis can be effi-ciently accomplished by 2-D fitting of the crys-talline reflections and amorphous halo [57, 58],This is illustrated in Figures 6b and 6c thatshows, respectively, the raw data and fitted datafor an oriented amorphous PET [59]. These datawere fitted using just two functions, one to ac-count for the uniform base-line intensity fromthe unoriented amorphous, and the other to ac-count for the Gaussian intensity distributionfrom the oriented amorphous segments. A simi-lar 2-D analysis of the diffraction pattern fromsemicrystalline polymers, with both crystallineand amorphous reflections can be used torapidly determine all the parameters that char-acterize the polymer structure such as crys-

talline and amorphous orientation, disorder,size, crystallinity and unit cell dimensions [58].

In analyzing the 2-D data in the small-angleregion, one is faced with a new problem. Thedata cannot be described conveniently in eitherthe Cartesian or the polar coordinates [60, 61].A detailed investigation of the contours of theSAXS pattern has shown that, because of themanner in which the lamellar lattice is de-formed, the small-angle data requires ellipticalcoordinates shown in Figure 8a. In the figurethe lamellar reflections are overlaid on an ellip-tical u–v coordinate system. This is equivalentto the Cartesian x–y or polar r–q coordinates.That the shape of the scattering is indeed ellipti-cal can be seen in a plot of the (Lf)2 vs. tan2 f ,

22

Fig. 8. Elliptical fitting of SAXS data. (a) Illustration of the elliptical coordinates. (b) A twopoint pattern used to illustrate the validity of the elliptical nature of scattering. (c) A straightline fit to (Lf)2 vs. tan2 f curve as expected from an elliptical curve. (d) Raw data from a nylon 6fiber. (e) Simulation of the raw data using two functions, one to describe the equatorial streakand the other to describe the lamellar reflection.

which is straight line out to azimuthal angles�70° as predicted for an elliptical curve (Figure8c) [61]. Here, f is the angle of a point along themaximum intensity contour line shown in figure(a), and Lf is the Bragg-spacing correspondingto this point measured along the y-axis. An ex-ample of the fit to the SAXS data from nylon 6fibers is shown in Figures 8d and 8e [11]. Thisfit was achieved using just two functions, one todescribe the equatorial streak, and the other todescribe the lamellar reflections. In addition torapid analysis of the data, such a procedureprovides complete information on the lamellarorganization such as the lamellar spacings(from the position of the reflections in 2q), theangle of tilt of the crystals with respect to thefiber-axis (from the azimuthal separation of thereflections), the diameter and the height of thelamellar stacks (from the width of the lamellarreflections in the x- and y-directions). Additionalparameters concerning the fibrils and voids canbe obtained from the equatorial streaks.

AcknowledgmentI wish to thank Dr. C. Riekel (ESRF) for giving

permission to use Figures 2 and 3, and Prof. B.Hsiao for providing Figure 5.

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