saxs data analysis of a lamellar two–phase system. layer

Colloid and Polymer Science 1007

Original Contributions

SAXS data analysis of a lamellar two–phase system. Layer statisticsand compansion1

N. Stribeck

Institut für Technische und Makromolekulare Chemie der Universität Hamburg, Germany

Abstract: An analysis of small angle x–ray scattering (SAXS) data from threeinjection molded poly(ethylene terephthalate (PET) samples is carried out.Two of the samples are annealed at different temperatures. The chosen con-cept of data analysis is that of Ruland’s interface distribution function (IDF)of lamellar two–phase systems. The IDF can be expanded into a series of dis-tance distributions, containing the information on the topological properties ofthe ensemble of lamellar stacks in the semicrystalline sample.

The paper describes the stepwise refinement of the topological model. Thefinal model is described by only few parameters of physical meaning. It unifiesthe well–known concepts of an ensemble of non–uniform stacks, finite stacksize and one–dimensional paracrystalline disorder in an analytical expression.In order to deduce this expression, the concept of inhomogeneity is (imaginea variation of the long period from stack to stack) is generally treated in termsof “compansion”, a suggested superposition principle. Its mathematical equiv-alent in one dimension is the Mellin convolution.Key words: SAXS — two–phase system — finite lamellar stacks — stackingstatistics — PET

1 Introduction

A frequently studied type of superstructure in poly-mers is the lamellar two–phase system. Transmissionelectron microscopy (TEM) as well as small anglex–ray scattering (SAXS) are common methods of re-search in this field. While TEM offers the advantageof a local visual impression of the structure, the SAXSmethod requires appropriate evaluation methods andstructure modeling in order to gain information onglobal parameters which characterize the structure.

Even the restricted objective of only determiningthe “long period” from the reflection maxima in theSAXS curve is not a simple task, as Reinhold, Fis-cher and Peterlin [1] have shown. In order to explaindeviations from Bragg’s law, the authors suggest aspecial type of asymmetric function to describe thefrequency distribution of long periods. Asymmetricdistributions of lamellar thicknesses or long periods

have several times been considered for lamellar [2–4]as well as for general [5] two–phase systems.

Additionally, taking into account the SAXS linewidths, Strobl and Müller [6] conclude that the en-semble of lamellar stacks shows features of an in-homogeneity in the sense that the long period variesfrom stack to stack. This model has been consideredby others [7, 8] and is similar to the concept of linebroadening due to a “homogeneous tension distribu-tion” in a polycrystalline solid [9, 10].

If one considers even more features of the scat-tering curve, the obtainable results become more de-tailed [11–15]. However, their significance stronglydepends on the validity of the structural modeladopted for data analysis.

The data evaluation with the least a priori assump-tions has been developed by Vonk and Kortleve [16],who study the one–dimensional correlation function,

. From , however, it is not a simple task1typos corrected by the author and reprinted from Colloid Polym Sci 271:1007–1023 (1993)

1008 Colloid and Polymer Science, Vol. 271 No. 11 (1993)

to extract the essential parameters of the lamellar sys-tem (e.g. thickness of the transition zone between thephases, thickness distributions of the lamellae, the sta-tistical law governing the stacking of the lamellae).

A more advanced method has been developed byRuland [17], who describes the determination of theinterface distribution function (IDF), , by meansof a Fourier transform of an interference function,

or . It is somewhat laborious to ob-tain from the measured scattering intensity,whereas for the correlation function method it is, ingeneral, sufficient to subtract a constant backgroundfrom the measured curve before a Fourier transformis performed. Despite the additional effort associatedwith the determination of , this procedure hastwo advantages. Firstly, the parameters describing thedeviations of a distorted two–phase system from anideal one have been determined [18], and their effecthas been “peeled off”, i.e. eliminated in the remain-ing data set, paying attention to the validity of Porod’slaw. Secondly, , the Fourier transform of the in-terference function, can easily be expressed in termsof the questioned frequency distributions of the crys-talline and amorphous layer thicknesses of an idealtwo–phase system. Such an ideal system is definedby only two constant phase densities and sharp do-main boundaries.

The above mentioned information peeling princi-ple is the fundamental tool in the present study. Itshall be applied repeatedly, until the lamellar super-structure appears to be resolved. Thus a model is suc-cessively generated during several steps of data analy-sis and a theoretical consideration. The question aftereach step will be, are we able to recognize the modelstatistics of lamellar arrangement along the stack? Ifthis is not the case, we will have to peel off anothershell of information first.

2 Experimental and results from as-sociated measurements

2.1 Samples and density measurements

Three injection molded PET samples have been pre-pared. After molding, two of the samples have beenannealed for 4 h at different temperatures ( oC and

oC).The sample dimensions are 40 mm 10 mm

2 mm. Thickness variation of the individual sam-ple is small.

The densities of the samples have been measuredin a density gradient tube. For the determination,small piecelets of polymer have been cut away, be-ginning from the upper edge, resulting in nine layersof test pieces covering the whole height of the sam-ple. This procedure has been carried out twice forevery sample in two vertical bands, approx. 1 cmfrom each end of the platelet. The unannealed sam-ple shows an increase of density towards the centralaxis of the platelet where the SAXS study has beencarried out previously. The volume crystallinity, ,increases from 0.37 at the edges to 0.42 in the cen-ter of the sample. The annealed samples are ratherbrittle, so for these samples it has been impossible tocut intact piecelets ranging from the front to the backside of the sample. This might in part be a reasonfor the poor statistics and only faint indication of adensity increase towards the center, as compared tothe unannealed sample. For every sample the highestmeasured density along the center line of the sampleplatelet has been used to compute the mean volumecrystallinity . Its difference to the lowest densityreading along this line is assumed to give the error ofdetermination (cf. Table 1).

Table 1: Injection molded PET samples and their charac-terization. Annealing temperature , thickness , density

and volume crystallinity , as determined from den-sity measurement using g cm and

g cm[C] — 240 248

[mm]g cm

2.2 X–ray measurements

For all measurements Ni–filtered Cu–K radiationhas been used. With a pinhole camera, wide–angleX–ray patterns as well as small–angle X–ray patternshave been recorded on photographic film. The record-ings from both methods show perfect isotropy and along period of approx. 10 nm for all three samples.

For quantitative SAXS measurements a KratkyCompact Camera, equipped with proportional counterand energy discriminator has been used. The receiv-ing slit length, , has been set to an integral lengthof nm in units of the modulus of the scatter-

Stribeck, SAXS of a two–phase system. Layer statistics and compansion 1009

ing vector (for definition cf. section 4). At eachposition of the scattering curve 20000 counts havebeen accumulated in order to keep the statistical errorsufficiently small. Normalization to absolute inten-sity [19] has been carried out by means of the movingslit method.

Every scattering curve is composed of two sets,recorded with different entrance slit heights (75 mand 130 m). Slit–height desmearing does not changethe chape of the individual sets.

3 Data evaluation tools

All data evaluation is carried out on IBM compatiblepersonal computers. Some programs have been devel-oped by the author himself by means of Turbo Pascal6.0. The sources of these programs are documentedin German language.

3.1 TOPAS

TOPAS is a computer program for the processing ofSAXS data from two–phase systems. Its ultimatecapabilities are computation of correlation functions,chord distributions or IDFs. A small handbook inEnglish, and an English and German user interfaceare available. The minimum hardware platform is a80286–computer. Hercules graphics or VGA are nec-essary.

3.2 Nonlinear regression analysis

Interface distribution functions are analyzed by meansof the Simplex algorithm of Caceci and Cacheris [20].To estimate the quality of the fit, the program moduleis extended by several procedures which have beendescribed by Draper and Smith [21]

Computation of the asymptotic correlation ma-trix, yielding information on parameter correla-tion (to avoid overparametrization);

Computation of the asymptotic intervals of con-fidence for each parameter value;

Plot of the estimation error (to avoid under-parametrization);

Plots of data and fitted curve.

Since the computation of some special model func-tions applied in the present study is time consum-ing, an 80486 with 33 MHz is recommended. Inorder to use the different programs successfully, theuser must be able to understand and modify object–oriented Turbo–Pascal programs which make exten-sive use of pointer structures.

3.3 Commercial software

Rapid prototyping and visualization in data analysisas well as in theoretical development has been sup-ported by MathCAD 2.5. Mathematica has been usedfor the verification of theoretical deductions.

4 Non–ideal and global aspects of thetwo–phase system

The extraction of the outer shell of information fromthe SAXS data sets starts from the absolute, slit–smeared 1D–interference function or, aftertransformation from reciprocal to physical space,from the IDF . The latter functions contain in-formation only on the topology of an ideal two–phasesystem along the axis normal to the surfaces of thelamellae. is the modulus of the scattering vector

, with being defined as half of thescattering angle. nm is the wavelengthof radiation. is the irradiated volume of the sample.According to Ruland [17,18] is defined by

(1)

with

erfc

Here, , is related to , the width of thetransition zone between the phases. is the densityfluctuation background, which is assumed to be a con-stant. , the Porod asymptote for the slit–smearedSAXS curve, is the constant governing Porod’s law.The term corrects for deviations under theslit–smearing operation, caused from a phase bound-ary which is not infinitesimally sharp, but smeared bya Gaussian distribution with standard deviation .

(1) is exact, but cannot be used to obtain andeasily from experimental data. Under the condition

that is “not too big” in the region where Porod’s


law is valid, Koberstein, Morra and Stein [22] haveobtained an approximate solution

(2)

which often can serve as a tool to obtain good ap-proximate values for and directly from a plot

vs. . The procedure toobtain from the scattering intensity first uses(2) to obtain approximate values and second (1). Themethod has been described elsewhere in detail [23],and can be carried out with the author’s computer pro-gram TOPAS.

0 0.1 0.2 0.3s [nm-1]

0

104

2.104

J(s)

/V [e

.u./n

m4 ]

not annealedTa = 240 CTa = 248 C

Figure 1: Absolute, smeared SAXS intensity, ,of three semicrystalline injection molded PET samplesrecorded with a Kratky camera. Annealing temperatures,

, are indicated.

0 0.1 0.2 0.3s [nm-1]

-3

-2

-1

0

1

2

3

G1(

s) [e

.u./n

m7 ]


raw data, not annealed

~

Figure 2: Smeared interference functions, , of threeinjection molded PET samples obtained from af-ter correction for density fluctuations and phase transitionzone between the amorphous and crystalline layers. Thepresented curves have been smoothed by spline functions.One set of raw data is shown in addition.

Fig. 1 shows the absolute SAXS intensityof the three samples. In Fig. 2 the corre-

sponding interference functions, , are plotted.

The curves are smoothed by means of spline func-tions. In order to judge the reliability of the splinesat large values, the raw data fro one of the curveshave been supplied in addition. Density fluctuationbackground, , as well as the dimension of the tran-sition zone, , describe the deviations from the idealtwo–phase system. The values for these parametersare given in Tab. 2. has been computed from themeasured smeared background using the equation

( cf. section 2.2). Compared to therather small long periods, the obtained values ofare remarkably high. This means that the approx-imations which lead to the Koberstein–Morra–Steinplot [22] are not valid. Thus the fluctuation back-ground in the tail of the scattering curve cannot bedetermined straightforwardly from this plot. The factbecomes evident, as soon as one computes the inter-ference function with the values for andestimated from the approximative plot and (1). Thecorrection which has to be applied to the estimatedvalues is not small.

Table 2: Injection molded PET samples. Parameters de-scribing the deviations from an ideal two–phase system.Density fluctuation background and dimension ofthe transition zone between the phases, as determined fromdeviations from Porod’s law.

[C] — 240 248el nm

nm

Table 3: Injection molded PET samples. Invariants of theideal two–phase system. , the asymptote of Porod’s lawin the slit–smeared SAXS curve. , the scattering powerof the SAXS with respect to the irradiated volume. , themean chord length as computed from and

[C] — 240 248el nm

el nmnm

Fortunately, in the present case, the IDF isfound to be very sensitive to the chosen fluctuationbackground. Only within a small range of values for

no artificial oscillations can be observed in theIDF. Thus, in Tab. 2, those parameter values are listedwhich minimize these artifacts in the interface dis-tribution function. It must be noted that the chosenmethod of a constant fluctuation background subtrac-tion may lead to a systematic error for the determinedvalues of and the scattering power. As has been


demonstrated by Wiegand and Ruland [24], the shapeof the density fluctuation background should moregenerally be approximated by a polynomial in evenpowers of . Moreover, Siemann and Ruland [25]have shown that even small changes in the fluctuationbackground for a system of SIS block copolymers canresult in considerable variation in the Porod asymp-tote .

One observes that the density fluctuationwithin the phases decreases as a function of anneal-ing temperature , whereas the transition zonebetween the phases is significantly enlarged for thesample annealed at o .

Tab. 4 shows the invariants of the ideal two–phasesystem, , and , the scat-tering power. The mean chord length is computedfrom and by means of the equation

.

Table 4: Structural model fitting on IDFs from injectionmolded PET samples. Topological structure parameters ofthe lamellar stacks according to the fits with free–runningvariances of the distance distributions (i.e. on as-sumptions on the statistical model of lamellar arrangement,but all distance distributions assumed to be Gaussian dis-tributions). For the values of for higher cf. Fig. 6.

[C] — 240 248el nm

[nm][nm]

According to Ruland [17] the interface distribu-tion is computed from by the Fourier–Bessel transform

(3)

with the kernel being expressed in Bessel func-tions of the first kind

J J J J

The resulting interface distributions of the three sam-ples are shown in Fig. 3. One should bear in mindthat the first two peaks (i.e. the first two distance dis-tributions) in the IDF are the distributions of the amor-phous and crystalline thicknesses. The curves appar-ently show that the crystallinity within the lamellarstacks varies as a function of annealing temperature.

In order to quantify this result, a model fit on the IDFshall be carried out.

0 10 20 30r [nm]

-0.5

0

0.5

1

1.5

2. g1(

r) [e

.u./n

m8 ] not annealed

Ta = 240 CTa = 248 C

Figure 3: Interface distributions, , of three injectionmolded PET samples obtained by Fourier–Bessel transfor-mation of

5 IDF analysis without assumptionson stacking statistics

5.1 Considerations concerning the modelingof the IDF

As Ruland [8] has shown, the IDF of a lamel-lar two–phase system can be expanded into a series ofdistance distributions

(4)

with for mod , and formod . (4) is valid for infinite stack height.

Let us number the distance distributions in as-cending order with respect to their centers of gravity,

. If we consider a semicrystalline lamellar systemwith a linear crystallinity , the first distancedistribution will reflect the distribu-tion of the amorphous layers in the stack. Its center ofgravity, , can be characterized as the averagedistance between the density alterations at both thesurfaces of any amorphous layer. In the same man-ner can be identified as the distri-bution of crystalline thicknesses, where isnow the average thickness of the crystalline layers.

corresponds to the distribution of “long peri-ods”, i.e. the distances from the beginning of a crys-talline layer to the end of the neighboring amorphous


one (index: ), which should be equal to the distri-bution of distances from the beginning of an amor-phous layer to the end of the neighboring crystallineone (index: ). Thus, ,and is the long period.

The subsequent distance distributions can be iden-tified as , ,

, and . Theseseven distance distributions describe the scatteringof two crystalline and three amorphous layers fromthe infinite stack in a consistent manner. If thesub–stack height is increased by a single crystallinelamella, three more distance distribution have to betaken into account. Thus, it is reasonable for the up-per limit of summation in the model to choose

Trivially, the center of grav-ity for every with can be computed fromthe centers of gravity and of the first two dis-tance distributions by successive addition.

If one intends to fit the IDF by a model, at least thegeneral type of distance distributions must be fixed.Several studies [1–4] have shown that from experi-mental findings there may be good reason to assumethat the distributions of crystalline thicknesses andlong periods are skewed. To the knowledge of the au-thor, the only analytical asymmetric function that hasbeen applied to SAXS data is the Reinhold distribu-tion [1]. Its skew is controlled by its center of grav-ity and a second parameter . A further asymmetricdistribution, which has been proposed by Hanna andWindle [5], is defined by a rather unwieldy product offunctions. Moreover, this distribution is not definedfor real number arguments, and there is no analyti-cal expression for its center of gravity. Both func-tion types are shown in Fig. 4. The Reinhold distribu-tion shows a typical cut–off at its steep side. If thesefunctions were to be considered as basic functions formodeling the IDF, traces of the cut–off should be ob-servable in IDFs computed from measured data. Suchtraces have never been found. Thus, if an asymmet-ric distribution should be considered, it should morelikely look similar to the Hanna–Windle distribution.

x

f(x)

Reinhold distribution

Hanna-Windle distribution

Figure 4: Examples for shapes of asymmetric distribu-tions of thicknesses, as discussed by Reinhold, Fischer andPeterlin [1] as well as Hanna and Windle [5]

If, on the other hand, one tends to assume the dis-tance distribution to be symmetrical about their cen-ters of gravity, it is convenient to propose Gaussiannormal distributions as a prototype ofthe distance distributions

(5)Thus, as a first approximation, the model has been

generated from (4) and (5) with and the sym-metric Gaussian functions. This model has the dis-advantage of a large parameter set, but the advantagethat no pre–determinations on the statistics of layerstacking have yet been introduced. Due to the largeparameter set, one should be cautious with the inter-pretation of its results. Nevertheless, the author hasfrequently used it for the determination of an “infinitelinear crystallinity”, , which assumes the irradi-ated volume to be filled with lamellar stacks of infiniteheight, defined by

(6)

as well as for the determination of the widths of thedistributions and . It would probably bequestionable to discuss the widths of the higher dis-tance distributions in detail. Results on the basis ofsuch a model of “free–running– ” have been pub-lished in several preceding papers [26–28].

Despite the above reservation, the general courseof the whole set of parameters shall be used to hint atthe mechanism which is most probably determiningthe statistical arrangement of the ensemble of layerstacks and the lamellae within.


One–dimensional statistics of distances are fre-quently treated in terms of a model of paracrystallinedisorder. If one considers a two–phase system, onecan distinguish between a paracrystalline stacking ofboth types of layer thicknesses and a paracrystallinelattice model, where the lattice constant is subject toparacrystalline disorder, whereas one type of layersdecorates the distorted lattice. If one assumes eitherof these variants to be valid, the number of free widthparameters, is reduced to two. The constrainingequations for the computation of the other havebeen presented by Ruland [8]. Here, it is only impor-tant that for both paracrystalline models the relation

const holds.

A different model for one–dimensional statisticsis that of an “inhomogeneous system with exact lat-tice” (Kratky, cited in Porod [7]), where each stack isassumed to be a perfect lattice with the lattice constantvarying from stack to stack. Ruland [8] named thismodel “homogeneous long period distribution” (“hL–distribution”, cf. [9, 10]. For this model the increaseof the is limited by the relation const , anda single width parameter determines all other .

5.2 Results

For the “free–running– model” a part of the pa-rameter set and the corresponding best–fit values areshown in Tab. 4. The specified “asymptotic intervalsof confidence” of every parameter value are computedfrom the parameter correlation matrix according toDraper and Smith [21]. is the fitted integral ofthe IDF, which should be close to

the Porod asymptote of the unsmearedscattering curve (of the ideal two–phase system).The following rows in the table specify the averagethicknesses of the crystalline and the amorphous lay-ers, respectively, and . is identified by meansof , the crystallinity from density measurement.At the table bottom the relative standard deviations,

, for the first three Gaussian distance distribu-tions, , are listed. The width parameters of thefollowing distance distributions will be presented inFig. 6.

Fig. 5 shows an example for the model fit and thedecomposition of the corresponding IDF into the se-ries of Gaussian distance distributions.

0 10 20 30r [nm]

-0.5

0

0.5

1

1.5

g 1(r

) [e.

u./n

m8 ]

Figure 5: Decomposition of into Gaussian–typedistance distributions for the unannealed injection moldedPET sample. The width of each Gaussian has been fittedindividually by nonlinear regression (Simplex algorithm)

Table 5: Injection molded PET samples. Deducedstructure parameters, computed by means of the resultsof the fit by free–running– and the invariants fromTab. 3. , linear crystallinity;

, the contrast between amorphousand crystalline phase.

[C] — 240 2480.42 0.65 0.68

el nm 43.2 49.5 47.8Tab. 5 shows values for deduced parameters,

which have been computed in order to discuss andvalidate the results of the fits. The crystallinity fromdensity, (cf. Tab. 1), and from IDF analysis, ,agree only for the unannealed sample. From bulkcrystallinity and scattering power , one computesthe electron density difference, , between thephases by means of the equation

(7)

Assuming that were the correct bulk crystallinity,one obtains the values presented in the second row ofTab. 5. These values have to be compared to the the-oretical contrast, el nm , which is com-puted from the mass density difference between theamorphous and crystalline phases (cf. Tab. 1). In par-ticular, the value for the sample annealed at oCis much too high. Thus, , in general, cannot beidentified with the bulk crystallinity. A similar resulthas been obtained in a previous study [23] on differ-ent PET samples. In the paper mentioned, we drewthe conclusion that “the results favor the view that thelamellar stacks consist of only 3—6 coherently scat-tering crystal lamellae and there exist amorphous re-


gions outside the lamellar stacks. . . ”.Let us now search for a complete model for the

SAXS which has a small set of parameters with phys-ical meaning. Fig. 6 shows the sets of width param-eter values, , as obtained from the free–running–model fit. The presented plot vs. appearsto be appropriate to hint at the strongest principle oflamellar stacking statistics. As mentioned above, anyparacrystalline statistics would result in a trend curvewith horizontal asymptote. An hL–distribution typeof statistics, on the other hand, would be indicatedby values following an inclined straight line throughthe origin — at least for the values from ,which correspond to distances of multiples of the longperiod.

0 10 20 30 40di [nm]

0

2

4

6

8

10

σ i2

/ di

[nm

]

10

20

30

axis

for a

nnea

led

sam

ples

as inj. moulded

L

2L

3L

4Lannealed 240 Cannealed 248 C

Figure 6: Relative variances of the Gaussian–typedistance distributions from a decomposition in fitsaccording to Fig. 5 and Tab. 4. Data are plotted as afunction of , the position of the –th distribution. Solidlines indicate trends for the unannealed (straight line) andthe annealed (increasing curve) PET samples. This plotserves the purpose to discuss probable principles of lamel-lar stacking statistics

The left y–axis refers to the unannealed sam-ple (filled circles). The trend for this sample ap-pears to warrant further work on the generalizationof th hL–distribution’s principle. The right y–axisrefers to the data of the annealed samples. Here, theloss of correlation is even worse than for the unan-nealed sample. Such a behavior could be caused fromlamellar stacks of finite height The author has plottedwidth data for several other samples in such test dia-grams. A paracrystalline trend has never been found.Sometimes there has been the indication of an hL–distribution, but mostly the trend has been unclear orit has hinted at the presence of finite stacks.

The ideal hL–distribution is completely definedby a single width parameter. But the plots suggest that

at least two independent widths should always be in-corporated in the model. Therefore, let us try to gen-eralize the principle of the hL–model and peel off thecorresponding shell of information from SAXS data.

6 Theoretical description of the com-pansion principle

The ideal hL–model describes the observable struc-ture as a superpositon of perfect 1D–lattices. It isassumed that there exist regions in the sample vol-ume where a common lattice constant can be defined.From region to region this lattice constant is subjectto such a variation that every local structure can begenerated from an “average local reference structure”by an affine compression or expansion, respectively.This superposition principle of the homogeneous longperiod distribution ma easily be generalized in orderto describe distorted reference structures.

Let be defined as the IDF of the referencestack, which shall describe the average local structurefrom the ensemble of lamellar stacks. Let be afrequency distribution of compression/expansion fac-tors , which have to be applied to the reference stackin order to generate the ensemble of scattering stacksas a whole. Then describes the heterogeneityof the sample, and the observed IDF, , is givenby the superposition

(8)

A well–known integral transform, describing an-other kind of superposition, is the convolution. It de-scribes the superposition of a function with itself bysuccessive translation in the space of its definition.The convolution is controlled by a second functionwhich can be considered as the frequency distribu-tion of translation factors. (8) describes a different su-perposition principle in which a function is superim-posed by compressed/expanded images of itself. Letus name this principle “compansion”. For the presentwork it is sufficient to consider “1D–compansion”, acompansion in one dimension.


6.1 Problem reduction. Definition and evi-dent properties of the compansion

Since, according to (4) , every IDF can be expandedinto the series of the distance distributions , theintegral transform may be carried out term by term

(9)

The frequency distribution , by definition, isnormalized to 1 and its center of gravity is at .By a simple transformation, the same properties canbe guaranteed for . The necessary transforma-tion combines an affine compression of the –axis anda multiplication by a scalar factor. The remnant prob-lem is of a purely mathematical nature, and involvesthe study of the integral transform

(10)

for the class of functions with norm and mean equalto unity (for definitions also cf. section 6.2), i.e.,

(11)

(12)

and , the –th moment of the function , beingdefined by (cf. Zelen and Severo [29, chapter 26])

(13)

In mathematical literature (10) is known as knownas “Mellin convolution”2. The relation to the ordinaryconvolution is readily established by substituting thevariables in (10) by their logarithms.

In the same way as the ordinary convolution inte-gral reduces to a simple multiplication upon Fourieror Laplace transformation, a Mellin transformation of(10) results in the product of Mellin transforms

(14)where the Mellin transformation is defined as

(15)

(see, e.g. Titchmarsh [30, p. 53]. For a complete dis-cussion of the integral (1) see Marichev [31].)

Commutativity ofthe compansion can be verified by variable substitu-tion in (10). Thus, frequency distributionand structure distribution are, in principle, indistin-guishable.

6.2 Moment series expansion of the com-panded function

Consider the series of moments of the compandedfunction , as well as that of the operand functions

and . Again, following [29], the –th central mo-ment of a function is defined by

(16)

Here, is called the mean of the function. is called its variance. is

the skewness of the function and describes its asym-metry. According to Oberhettinger [32, Theorem 4],the Fourier transformed function of , canbe expanded into a series of the moments of

O (17)

(17) is valid under the premise of the symmetrical def-inition

e (18)

of the 1D–Fourier transformation. According to [29],the central moment can be expressed by a series ofthe moments about origin,

(19)

Rearrangement of (19) yields the inverted relation

(20)

2The following part of this section cites from the referee’s comments to the manuscript, which are gratefully acknowledged by theauthor.


6.3 Transformation of moments under com-pansion

Apply the moment definition (13) to the definition ofcompansion, (10) and substitute . One ob-tains the theorem

(21)

i.e., the moments about origin multiply under Mellinconvolution. Thus, it follows from (11) and (12) thateven the companded function belongs to the class offunctions with norm and mean equal to unity. Thistheorem can as well be deduced from (14) due to theobvious relation between the Mellin transformationand the moments of a distribution.

6.4 The second moment of the compandedfunction

Using (20) and (21) for functions from the mentionedclass one obtains for the second moment of theMellin convoluted function

(22)

and thus for the second central moment, , of

(23)

6.5 The third moment of the compandedfunction. Asymmetry

For the computation of the third moment, , let theoperand functions and be symmetric, i.e., theirthird central moment shall vanish

Then for , it follows from Theorem (21)

(24)

and with (20) and (23) we obtain for the third centralmoment of the companded function

(25)

Thus, , the compansion of the symmetric func-tion with the symmetric function is asymmetric,in general. For its skewness it follows

(26)

6.6 Gaussian distributions under compan-sion

Now assume that as well as are normalizedGaussian distributions according to (5). The centralmoments of Gaussians are well–known definite inte-grals

for

for

Using the notation of the “double factorial”the –function can be simpli-

fied to yield

for (27)

With this background, one may recursively computeall the moments of the compansion of two Gaussiansand thus obtain the power series of (cf. (17)).

6.7 Fourier transformation of the com-panded function

The Mellin convolution (1D–compansion) from twoGaussians does not lead to a simple analytical solu-tion. In general, extensive tables of Mellin transforms[31, 33] can be used in order to calculate Mellin con-volutions or to choose model distributions for which asimple analytical solution exists. But the properties ofcompansion are such that it appears to be promisingto search for an analytical , the Fourier transformof the compansion.

Let be the 1D–Fourier transformation of thecompanded function (cf. (18)), then from thereciprocity theorem of Fourier transformation theory,

(28)

it follows that

(29)


Here, designates the 1D–Fourier transform of; and correspond in analogous man-

ner. Now let and be represented by Gaus-sians with norm and mean equal to unity. We thenobtain for the Mellin convolution in reciprocal space

This integral is soluble. After simplification, com-pletion of the quadratic form, and variable substitu-tion, the problem can be reduced to the integration of

, and one finally obtains

(30)with

(31)

(32)

Compansion’s commutativity is reflected in the result.It has therefore been demonstrated that the com-

pansion may be solved analytically in reciprocalspace. Since no analytical expression has been foundin real space, reciprocal space data have to be trans-formed to real space via numerical Fourier transfor-mation. A high accuracy algorithm is the “fast Fouriertransformation” (FFT), which can be fed with conve-nient reciprocal space data, so that it produces the re-sults at precisely the positions where they are needed.

For application in SAXS data analysis, one mayidentify , the heterogeneity of the ensem-ble of stacks. In this case is the relativestandard deviation of the –th distance distribution.

Fig. 7 shows examples for the compansion of twoGaussians. The heterogeneity parameter is variedfrom 0 to 1. The upper drawing shows compansion inreal space. The pure Gaussian for and theincreasing asymmetry as heterogeneity grows shallbe pointed out. The lower drawing shows the samecurves before Fourier transformation. In reciprocalspace the companded functions are presentedwith negative sign, in order to make the relation to the

unsmeared interference function more sugges-tive. Only the branch for positive is shown, since ourproblem is dealing with even functions only. Thus forapplication in SAXS data analysis, we omit the imag-inary part in (30), but double the weight of .

0 1 2 3r0

0.05

0.1

0.15f(r) σH = 0.0 σg = 0.3

σH = 0.2

σH = 0.4

σH = 0.6

σH = 1.0

a)

0 1 2s-2

-1

0

1- F(s) σg = 0.3σH = 0.0

σH = 0.2

σH = 0.4

σH = 0.6

σH = 1.0

b)

Figure 7: Examples for the Mellin convolution (1D–compansion) of two Gaussians with mean and norm equalto unity. The width parameter of the first Gaussian is

. Only the width parameter of the second Gaussian, ,takes different values between 0 and 1. a) the compansion

in physical space. b) the negative Fouriertransform of the compansion . Functions in both dia-grams are even

A computer program for the computation ofcompanded functions on the basis of two Gaussianoperands is available from the author.

7 Results from compansion basedIDF analysis

Having the compansion approach at hand, it wouldbe easier to fit the unsmeared interference function

than to fit , because for the first func-tion all terms are analytical. In this case one wouldavoid repeated computation of the FFT in each regres-sion step. But since the author is not able to handle a


desmearing which will result in a confidential ,the following results have been obtained from fits onthe IDF as well.

Let us repeat a free–running– model fit, wherethe distance distributions are not simple Gaussiansany more, but companded distributions from a Gaus-sian heterogeneity distribution and mean localreference distributions . Again, the intention isto obtain information on the statistics of layer stack-ing — now for the assumed local stacks.

0 10 20 30 40r [nm]

-0.5

0

0.5

1

1.5

2. g1(

r) [e

.u./n

m8 ]

Figure 8: Decomposition of into compansion–typedistance distributions for the injection molded PET sam-ple annealed at oC. Heterogeneity, , as well as thewidth of each local Gaussian, . have been fitted individ-ually by nonlinear regression (Simplex algorithm)

The results for the annealed samples show a re-markable property, which is presented in Fig. 8. Thediagram illustrates the corresponding decompositioninto the global distance distributions, , which arecompanded functions. On now observes the asymme-try in the negative peaks, which are the multiples ofthe long periods. But, more striking, the distance dis-tribution which range between outer edges of crystal-lites are almost symmetric (thin solid lines; ,

, ). In other words, the lo-cal distributions of these distances are rather narrow.Since the distance distributions in between are con-siderably wide, we have to conclude that the annealedsamples contain lamellar stacks of finite height.

8 Determination of an appropriatemodel for finite stacks by means ofnonlinear regression analysis

If one intends to deal with finite stack systems, threequestions arise. First, it is important to ask for the

distribution of cluster heights. Second, one wouldlike to find an appropriate model for the descriptionof the cluster’s border zone, in which the correlationis lost. And third, it is necessary to determine a suit-able model for the statistics of lamellar arrangementwithin the clusters. These questions have been stud-ied by model variation, fit on the experimental IDFs,and comparison of the fit quality.

8.1 The distribution of cluster heights

For the study of the distribution of cluster heights avast model has been composed. Both paracrystallinestacking as well as a lattice with paracrystalline dis-order have been checked in parallel fits. Both modelsassume that every cluster height from (crystallinelamella without correlation) to a cluster of crys-talline lamellae is present with individual weights.The result of this test is that the weights are a func-tion of the chosen , but all weights except 3 are neg-ligible. These 3 weights always belong to adjacentcluster heights. Thus in the next step the complexityof the model is reduced by assuming that the distribu-tion of cluster heights can be described by a Gaussianenvelope. This model results an an extremely narrowGaussian envelope. Thus it should be allowed to con-sider stacks of a single stack height only.

8.2 Models for the cluster border

All the preceding modeling has been carried out un-der the assumption that no border zone contributes tothe scattering. Now, with the finite stack model of asingle stack height, 4 model variants have been tested.

Sketches of these models are shown in Fig. 9.Fig. 9a shows a stack with the outer correlated zonesbeing crystalline. In Fig. 9b the outer correlated zonesof the stack are amorphous. For all samples model (b)fits better than model (a) (by at least 20% what theresidual sum of squares (RSS) is concerned).


r

ρ(r)

r

ρ(r)

r

ρ(r)

r

ρ(r)

finite stacks without border zone

finite stacks with border zone

a) b)

c) d)

ρc

ρa

Figure 9: Sketches of models for finite stacks without andwith border zone. The best fits on result from modeld), a stack composed of crystalline, amorphouslayers and some diffuse border zone

The border zone for the models in the lowerrow has been introduced by defining an additionallayer with average thickness, thickness distributionand contrast of its own. Model (c) with some excesscontrast shows severe convergence problems. Model(d) proves to be the best fitting model of all. But evenhere fit quality estimation of the regression analysistells that only one of the additional parameters ( ,the standard deviation of the border zone thickness) isa parameter of some evidence, whereas average thick-ness and contrast obtain values without confidence.As a consequence the border zones thickness has beenfixed at a constant value of nm, and the contrast hasbeen given half the contrast between and . Sincethese settings for the border zone are artificial, the ef-fect of the border zone term to the scattering curveshall be given particular attention in section 9. It willturn out that, indeed, this term is indeterminate do tothe inherent insensitivity of the interference functionat small .

8.3 Statistics with paracrystalline disorder:Stack vs. lattice

Tab. 6 shows the parameter set of the model fit withthis finals model for paracrystalline stacking statis-tics, which is the best choice for all samples. Theheterogeneity, , of the system of lamellar stacksis almost constant for the three samples. The av-erage crystalline thickness increases with annealingtemperature, whereas its standard deviation with re-spect to the local stacks vanishes. Thus, in the an-nealed samples the few crystalline lamellae within afinite local stack are of approximately the same thick-

ness, while the thicknesses of the amorphous interlay-ers from these local stacks vary by 40% and more.

Table 6: Structural model fitting on IDFs from injectionmolded PET samples. Topological structure parameters ofthe lamellar stacks according to the results of the fits by thefinal compound model

[C] — 240 248el nm

[nm][nm]

[nm]

The average thickness of the amorphous layersthemselves decreases with increasing annealing tem-perature. Due to compansion, the global distributionof amorphous layers is rather asymmetric, whereasfor the crystalline lamellae this is only true for theunannealed sample. For the latter, the stack height

is rather high. is the number of complete longperiods which belong to a single stack of correlatedlamellae. With a stack height of between 5 and 15crystalline lamellae, a model of infinitely extendedstacks should result in a proper fit as well. This is notthe case with the annealed samples. Here, the stackheight is truly finite. For the computation of the lin-ear crystallinity, , this means that the contributionof the additional amorphous layer “from the other endof the stack” is not negligible. Thus, for finite stacks,

must be computed from the equation

(33)

The corresponding values are given in the bottom rowof Tab. 6, and now the agreement with the crystallinityfrom density, , is good. Thus, for the present sam-ples it has been demonstrated that the discrepanciesbetween bulk crystallinity and linear crystallinity haveto be ascribed to the existence of lamellar stacks offinite height which are embedded in the amorphousmatrix.

Since for the unannealed sample the agreementbetween the bulk and the infinite linear crystallinityhas been perfect, and, moreover, the stack height ishigh, fits with different infinite models have been car-ried out as well. The results are presented in Fig. 10.The worst fit shown is that with purely paracrystalline


stacking statistics which does not consider compan-sion. If compansion is considered in addition, RSS isdecreased by a factor of 31. The fit which considersa finite stack height (but is not shown in the figure) isonly slightly better (RSS = 0.0038). Assuming com-panded paracrystalline lattice statistics, the fit qualitydecreases by almost a factor of two as compared toparacrystalline stacking statistics. This result is repre-sentative for all three samples. Thus lattice statisticscan be excluded.

0 10 20 30r [nm]

-0.5

0

0.5

1

1.5

2. g1(

r) [e

.u./n

m8 ]

Paracryst. stack (RSS = 0.136)Companded paracryst. stack (RSS = 0.0044)Compand. paracryst. lattice (RSS = 0.0079)

Figure 10: Results from fits with different statistical mod-els for the unannealed PET sample, for which the stackheight is close to infinity. RSS is the residual sum ofsquares. Solid line: Measured interface distribution, .Fine dashed line: Paracrystalline stacking with infinitestack height. Dash–dotted line: Paracrystalline stacks ofinfinite height under compansion. Dash–dot–dotted line:Paracrystalline lattices of infinite height under compan-sion, with the crystalline layers decorating the distorted lat-tice

9 Synthesis of the scattering curvesfrom the model data

Compansion is an analytical function in reciprocalspace. This means that for the suggested model thedesmeared interference function, , is analyticaland can be expressed by

(34)

where characterizes the contribution of theborder zones, and is the main contributionoriginating from the correlation among the crystallineand amorphous layers of the finite stacks

(35)

Here is a weighting factor and will be explainedin the following. is the Fourier trans-form of a companded function, which is related to adistance between two phase boundaries in the finitestack. These phase boundaries enclose a sub–stackof one or more layers. is the number of layers inthe sub–stack. The accumulation for concernsthose sub–stacks beginning with a layer of “type–1”,and for the sub–stacks starting from a layer of“type–2” are accumulated. Type 2 is the phase whichforms both the outer layers of the finite stack. Withrespect to the present study, it can be identified as thephase of the amorphous layers. is the total num-ber of layers which exist in a finite stack built fromlayers of type–1, thus, round . is thefrequency with which a sub–stack is found inside thefinite stack. This sub–stack begins with type– andcontains layers. Sub–stacks with an odd number ofparticipants obtain negative sign

div

The operators round and div return the integer frac-tions of rounding and division, respectively. isthe standard deviation of the local Gaussian distancedistribution computed from width parameters and

, according to the paracrystalline stacking statis-tics. Here is the standard deviation of thelocal distribution of crystalline (i.e. type–1) layers.

is identified with and for the variances of thesub–stack distance distributions we obtain

div div

The average distance between bottom and top of thesub–stack, follows from

div div

where assignment and is madein analogy to the preceding definition. The Fouriertransform of the compansion is computed using theheterogeneity and applying the transfor-mation rules from section 6.1. In these rules we set

, the relative standard deviation, and com-press the –scale by the factor . It thus follows that

which is double the real part from (30).


In a similar way the border term has been definedby

(36)Here, is related to the form factor of the totalfinite stack including the border zones, but with onlyhalf the contrast. The variables in the sum are nowdefined differently by

div div

div div

with nm fixed and small. is themodel parameter for the standard deviation of the lo-cal border zone.

Fig. 11 shows the decomposition of intothe terms and for the unannealed sam-ple. The thin dashed curves demonstrate the com-pansion of the global distribution of crystallinethicknesses and , which in similar manner cor-responds to the amorphous layer distribution.

0 0.1 0.2 0.3 0.4s [nm-1]

-2

0

2

G1(

s) [e

.u./n

m7 ]

Hc(s)Ha(s)G1b(s)G1s(s)

Figure 11: Components of the model from the best fit ofthe unannealed PET sample, presented in terms of the un-smeared interference function, , the Fourier trans-form of with negative sign. , : Formfactors of the crystalline lamellae and from the amor-phous lamellae, respectively. : Stack–border termin . : Stack–without–border term of the finitestack in

Obviously, the –term is only significant forsmall values of , where the interference function israther insensitive due to the multiplication of the scat-tering intensity by . Thus, the effect of the borderterm should only be regarded as some correction termwith low physical meaning.

The next step in data synthesis is the multipli-cation of by in order to obtainthe desmeared scattering curve, , of the idealtwo–phase system. The synthesized curves are pre-sented in Fig. 12. Utilizing the equation

0 0.1 0.2s [nm-1]

0

2.105

4.105

6.105

I(s)/V

[e.u

./nm

3 ]


Figure 12: Synthesis of the unsmeared scattering curves,, from the synthetic curves computed from

the parameter sets retrieved from the best model fits on theIDFs from measurement.

(37)one can approximately generate the smeared scatter-ing intensity by numerical integration and compare itto the raw measured data. nm is theupper limit, up to which the model function has beencomputed. The computation neglects the effect of afinite width of the phase transition zone, , which,with respect to the chosen plot, will lead to very smallsystematic deviations in the tail of the curves. Fig. 13shows the results.

0.1 0.2 0.3s [nm-1]0

104

2.104

J(s)

/V [e

.u./n

m4 ]


Figure 13: Smeared scattering curves syn-thesized from IDF model fits obtained by numeri-cal smearing of the curves in Fig. 12 and subsequent


background addition compared to the initial measureddata from 3 injection molded PET samples

From a lower limit of nm themodel functions fit the original data. According tothe preceding considerations, the region formay be dominated by the nature of the loss of correla-tion in the border zone of the finite lamellar stacks. Tostudy this zone, fits of the interference function or onthe IDF are not appropriate, since these functions areinsensitive to considerable changes of the raw data atsmall values of , the modulus of the scattering vector.

10 Conclusion

This study shows that the frequently discussed mod-els for the statistics in an ensemble of lamellar stacksmay be superimposed. This superposition results ina data set which, in fact, agrees with none of thepure models. The novel compound model proposedin this study considers the principles of compansion,finite stack height, and one–dimensional paracrys-talline disorder. It is able to describe a wide range ofdistorted two–phase structures. The model comprisesa small parameter set of only eight parameters, sevenof which have physical meaning. As has been shown,the SAXS curve can be synthesized by means of theanalyzed parameter values from the model fit. The de-viations between measured intensity and model inten-sity at the beginning of the curve may be related to anot yet understood principle which describes the lossof correlation at the border of finite stacks of lamellae.Some of the deviations between both curves may, inaddition, result from the coarse approach to synthe-sis of scattering intensity since integration has beencarried out by summation over a finite range only.

Asymmetric distributions of long periods or thick-nesses of lamellae have been discussed in the litera-ture. This study has shown that such asymmetry isinherent to the compansion principle, which has beenproposed here in a theoretical treatment. In the de-duced form, the compansion may be applied to struc-tures which are of one–dimensional nature, such asthe analyzed lamellar systems or fibrillar two–phasesystems, e.g. certain thermoplastic elastomers in theelongated state. A generalization of the compansionintegral transform to three dimensions has not beenattempted.

Compansion may not be a general principle of

distorted two–phase systems, but it should be takeninto consideration. Its presence may be investigate bymeans of the presented data evaluation method.

Acknowledgments. I am indebted to Prof. F. J. BaltáCalleja, Madrid, and to Prof. L. Mandelkern, Tallahassee,for different sets of different samples which gave the lastimpulse for the development of compansion theory. I grate-fully acknowledge the preparation of the PET test samplesby Dr. R. K. Bayer, Kassel and the stimulation of thisstudy by Prof. H. G. Zachmann. Special thanks to Ms.U. Dambeck for carrying out measurements with the pin-hole camera and for preparing the density gradient.

References

[1] Reinhold C, Fischer EW, Peterlin A (1964) J ApplPhys 35:71

[2] Hall IH, Mahmoud EA, Carr PD, Geng YD (1987)Colloid Polym Sci 265:383

[3] Strobl GR, Schneider MJ, Voigt-Martin IG (1980) JPolym Sci Part B Polym Phys B18:1361

[4] Voigt-Martin IG, Mandelkern L (1989) J Polym SciPart B Polym Phys B27:967

[5] Hanna S, Windle AH (1988) Polymer 29:207

[6] Strobl GR, Müller N (1973) J Polym Sci Part BPolym Phys 11:1219

[7] Porod G (1951) Kolloid Z 124:83

[8] Stribeck N, Ruland W (1978) J Appl Cryst 11:535

[9] Dehlinger U, Kochendörfer A (1939) Z Kristallograf101:134

[10] Kochendörfer A (1944) Z Kristallograf 105:438

[11] Tsvankin DY, Zubov A, Kitaigorodskii AI (1968) JPolym Sci Part C 16:4081

[12] Strobl GR (1973) J Appl Cryst 6:365

[13] Brämer R (1974) Colloid Polym Sci 252:504

[14] Kilian HG, Wenig W (1974) J Macromol Sci PhysB9:463

[15] Shibayama M, Hashimoto T (1986) Macromolecules19:740

[16] Vonk CG, Kortleve G (1967) Kolloid Z u Z Polymere220:19

[17] Ruland W (1977) Colloid Polym Sci 255:417

[18] Ruland W (1971) J Appl Cryst 4:71

[19] Polizzi S, Stribeck N, Zachmann HG, Bordeianu R(1989) Colloid Polym Sci 267:281


[20] Caceci MS, Cacheris WP (1984) Byte 1984:340

[21] Draper NR, Smith H (1980) Applied RegressionAnalysis, Second Edition. John Wiley & Sons, NewYork

[22] Koberstein JT, Morra B, Stein RS (1980) J Appl Cryst13:34

[23] Santa Cruz C, Stribeck N, Zachmann HG,Baltà Calleja FJ (1991) Macromolecules 24:5980

[24] Wiegand W, Ruland W (1979) Progr Colloid PolymSci 66:355

[25] Siemann U, Ruland W (1982) Colloid Polym Sci260:999

[26] Stribeck N (1989) Colloid Polym Sci 267:301

[27] Stribeck N, Bösecke P, Polizzi S (1989) ColloidPolym Sci 267:687

[28] Stribeck N (1992) Colloid Polym Sci 270:9

[29] Abramowitz M, Stegun IA (eds.) (1968) Handbookof Mathematical Functions. Dover Publications, NewYork

[30] Titchmarsh EC (1948) Introduction to the theory ofFourier integrals. Clarendon Press, Oxford

[31] Marichev OI (1983) Handbook of Integral Trans-forms of Higher Transcendental Functions. Ellis Hor-wood Ltd., Chichester

[32] Oberhettinger F (1973) Fourier Transforms of Distri-butions and Their Inverses – A Collection of Tables.Academic Press, New York

[33] Oberhettinger F (1974) Tables of Mellin Transforms.Springer, Berlin

Received September 30, 1992;accepted December 2, 1992

Author’s address:Dr. N. StribeckUniversität HamburgInstitut TMCBundesstr. 45D–20146 HamburgGermany

saxs data analysis of a lamellar two–phase system. layer

Documents