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X-Ray Small-Angle Scattering of Bulk Polyethylene1

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X-Ray Small-Angle Scattering of Bulk Polyethylene1 Vonk and Kortleve, X-Ray Small-Angle Scattering o] Bulk Polyethylene, I I 19 From the Central Laboratory, N.V. Nederlandse Staatsmi]nen, Geleen (The Netherlands) X-Ray Small-Angle Scattering of Bulk Polyethylene II. Analyses of the Scattering Curve By...

X-Ray Small-Angle Scattering of Bulk Polyethylene1
Vonk and Kortleve, X-Ray Small-Angle Scattering o] Bulk Polyethylene, I I 19 From the Central Laboratory, N.V. Nederlandse Staatsmi]nen, Geleen (The Netherlands) X-Ray Small-Angle Scattering of Bulk Polyethylene II. Analyses of the Scattering Curve By C. G. Vonlc and G. Kor t leve With 6 figures in 7 details (I~oeeivod January IS, 1967) Electron-microscopic studies have shown that melt-crystallized polymers have a layer- like structure. This was first indicated in 1956 by Claver, Buchdahl and Mil ler (1), who compared the free surface of a bulk polyethylene sample to a fanned deck of cards. Later, the occurrence of layers was directly confirmed by electron mierographs, both of fracture surfaces (2) and of ultra- microtome sections (3), and also by electron microscopic examination of the debris ob- tained by oxidation of polyethelene with fuming nitric acid (4). The relation which these layers bear to the spherulites in the bulk polymer on the one hand and to the single crystals obtained fl'om solutions on the other, seems to be established in a qualitative way. There is still much contro- versy, however, about the question how the crystalline and amorphous regions are ar- ranged within the layers. Useful information on this problem can be obtained from a detailed analysis of the X-ray small-angle scattering. In principle, such an analysis can be performed by calculating the scattering curve of a reason- able model of the structure. I f the model is correct, its parameters can be varied to make the calculated scattering curve fit the experimental one. This procedure is often applied in the study of colloidal systems. In the study of high polymers, however, it has appeared advantageous to include an inter- mediate step. Calculation of the Four ier transform of the experimental scattering curve yields the so-called correlation func- tion. This function was introduced by Debije (5) and is closely related to the "Persistenzfunktion" used by Porod (6). As the correlation function is, moreover, proportional to the autoeonvolution of the electron density variations in the sample, it can be calculated for various models. These calculated functions can be compared with that derived from the experimental scattering curve. The advantage of this approach is that if the model is essentially correct, differences between the calculated and the experimental correlation functions can be much more readily interpreted in terms of the model than in the event the scattering curve itself is used. In this second part of the present series 1) the mathematical relations needed in the application of this procedure will be derived. The Experimental Correlation Function Throughout the following we shall assume that within the range of distances, covered by our experiments (10-800 •) the layers are essentially flat and parallel. From diffraction theory it follows that the intensity of the small angle scattering from a single region of parallel layers can be represen- ted by an intensity function in reciprocal space; this function is confined to a central line perpendicular to the layers. Its value i (s), expressed in electron units, on this line at a distance s from the origin, is given by co i(s) = 2 V f ~' (x) cos 2 ~ x s dx [1] 0 where and co r'(x) = S v(~ - x) v(~) d~ [2l 0 2sin0 8- - 2 V = irradiated volume of the region under con- sideration x = coordinate perpendicular to the layers 0 = haft angle of diffraction = wavelength of X-rays ~/(~) = local fluctuation of the electron density around the average value. 1) Par t I, dealing with sl i t -height correction, was presented earlier (10). 2* 20 Kolloid-Zeitschrifl und Zeitschri/t /i~r Polymere, Band 220. Heft I From [21 it follows that y' (0) is equal to <@>, the average of the square of the electron density fluctuations. Following De- bile, Anderson and Brumberger (7), we intro- dace the correlation function, which in our case is one-dimensional r (x) = 7' (x)/@~>. [3] The one dimensional-correlation function can be visualized as follows: according to Chaltceley et al. (8), one considers a measuring rod A B of length x perpendicular to the layers, which moves in the x-direction through the layers. In each position the product of the electron density deviations at A and B is determined, and the values thus found are averaged over all positions. The correlation function is obtained by multi- plying the average by 1/, where is the average obtained for x = 0. Hence, the value of y (0) is + 1, and - 1 < y (x) < + 1. y' (x) and i (s ) are Four ie r transforms of each other, so that ~i(s) cos 2 [4] ~'(x) = 2 V ~ ~xsds u and r ~'(o) = <~> = ~ f ~(~)d~. [5] 0 In an actual sample all orientations of the layers are present, and the intensity func- tion I (s), which corresponds to the measured intensity, shows spherical symmetry. I (s) is related to i(s) by I(s) ~-~ i(s)/4 ~ s ~ . [6] Combining [3], [4], [5] and [6], we find the relation between y(x) and the experimental scattering curve: co s ~ I (s) cos 2 z xs ds r (~) = o U] oo o The general features of the one-dimensio- nal correlation function as obtained from [7] for polyethylene samples are shown in fig. 1. The function shows a number of minima and maxima of decreasing height and finally becomes zero at large vMues of x. This be- haviour suggests a periodicity in the struc- ture; the position of the first maximum then corresponds to the average value of the distance of periodicity. It must be stressed here that the correlation function can be obtained from relative intensity values; if the intensity scale has been calibrated with the aid of samples of known electron density distribution, may be calculated separately from [5] and [6]. This provides independent and highly interesting information; however, we have no provisions as yet for performing the necessary measurements on our equipment. The calculation of the correlation func- tion involves some practical problems. When applying [7], one must know the entire small- angle scattering curve I (s). In practice it is measured from a lower limit (corresponding to a Bragg-d is tance of 800~ on our in- strument) and extended to s = 0 by extra- polation. It was found experimentally that the method of extrapolation has very little effect on the relevant part of the correlation function; neither the position nor the height of the first maximum is seriously affected, not even if absurd methods of extrapolation are used. 1Y[ore care must be observed in deter- mining the tail of the experimental scatter- ing curve. In the case of polyethylene the intensity does not drop to zero, as the tails of the small-angle scattering and the wide angle scattering curves overlap to some extent. In our experiments the base line was shifted upward until it became tangent to the experimental curve at the point of mini- mum intensity. Probably, this procedure is not quite correct, but it can be shown that the correlation function is affected seriously only at small values of x. In fig. 1, this part of the curve is therefore indicated by a broken line. The Correlation Function Calculated from a Model The experimental correlation function must be compared with the correlation func- tion calculated by means of relations [2] and [3] for a model of the layer structure. In choosing the model, we assumed the structure to consist of alternating layers of high and low electron density. These are designated as crystalline and amorphous layers, though the use of these names implies neither that the crystalline layers are in fact entirely crystal- line nor that the amorphous layers are en- tirely amorphous. The thickness of the individual layers is xc and xa respectively; the thicknesses are assumed to be distributed around the average values c and cb according to the normalized functions Pc (xc) and Pa (Xa). c is called the The quantity q}-c§ crystallinity by volume. Furthermore, pet denotes the probability that, if end A of the Vonk and Kortleve, X-Ray Small-Angle Scattering o~ Bulk Polyethylene, I i 21 measuring rod is in a crystalline layer, end B is also inside a crystalline layer. By means of the relations [9]-[17] for the analogous quantities P00, P01 etc. given in the paper by Debije, Anderson and Brumberger, the fol- lowing expression for the correlation func- tion can be easily derived: 7(x) pe~(x)-- 1 - - q) [8] ~(x) i 0,5 ' , , - ~ > - ~ i ~ 100% x(s I O,5- Fig. 1. One dimensional correlation function of a sample of bulk polyethylene poe(X) can be considered as the sum of the probabilities q: fee(X) = qc(x) -l- qeac(X) + qeacac(X)... [9] where the subscripts of q indicate the types of neighbouring layers traversed by the measuring rod AB. Thus qe(x) denotes the probability that, with A in a crystalline layer, B will be in the same layer (see fig~ 2). Clearly, this can only be so if A is inside the region xe -x in the left-hand part of the layer. For this particular layer, qc would be equal to (xe -x ) /xc , and for all layers qc is found from r I (xe - x) Pc (x~) dXe %(x) - co I xcPdz~) dx~ 0 c<) _ t ~ (ze - x) Pe (xe) dxe [10] r + a) In order to make a contribution to qcac(X), the measuring rod must be situated as indicated in fig. 3. In this case it is composed of three vectors xl, x~ and xe. According to a well-known theorem [see for example lit. (9)] the distri- bution of such Vasum of vectors is given by the normalized distribution function Pcac(X) = Q (x~:) P(xa) (2 (xz) [ i ] ] Ix o B I • • I x t I X C I Am ICr Am I Fig. 2. Posibion of ~he measuring rod AB, which con- tributes to qc where Q(xl), P(x~) and Q(x~) are the nor- malized distribution functions for x 1, xa and x 2 respectively, and the sign ~ indicates the convolution process. Clearly, P(xa) is equal to the distribution function Pa(xa) defined before, and Q @1) = Q (x~). To find Q @1), it is argued that a vector of length x 1 can occur only in those layers for which Xc > xl; this is the fraction co f Pc(Xe) ~xo. After normalization, Q is found to be: c~ 1 ~ Pc(xe) dxc [12] Q @1) ~)(~ + a) xl and Pea ~ (x) may be calculated from equ. [11]. I ] 1 Am Cr Am Cr Am Fig. 3. Position of the measuring rod AB, which contributes ~o qeac To facilitate the argumentation, we assume in the following that the measuring rod A B can occupy only N discrete positions when A is travelling over one unit of length in the x-direction. Then qeae(X) is related to the probability Pcae (x) as follows: ~e~e(X) Te~e(x) Re(x) and Teae(x) -- Pcae(X) " Seae N where Tcac(x) is the number of positions of a measuring rod of length x with A and B in neighbouring crystalline layers, and Re(x) the number of positions of a measuring rod of length x with A in a crystalline layer and B in either a crystalline or an amorphous layer. Re(x) is equal to CN per unit lengbh. Scac is the total number of vectors of the type cac. This number is equal to r ~ (c + a) 22 Kolloid-Zeitschri/t and Zeitschri[t /i~r Polymere, Band 220. Heft 1 per unit length. The relations can be com- bined to yield: q~,~ = ~(c + a) P~,~. [la] In an entirely similar way, one obtains: qcacac = qS(c + a) Pcacac [13a] where pcacac = QcPaPcPaQc etc. Combinat ion of (8), (9), (I0), (13), (13a), etc., and simultaneous transformation of the unit length in the x-direction into c + a, yields the expression from which the cor- relation function can be calculated: X "-~ (xc--x)Pc(xe) dxc +Pcac + Pcacac + . . . - - 1 [14] In practice, only the first four terms of the expression in brackets were calculated. So far, no assumptions have been made on the form of Pa or Pc. It is to be expected, however, that the correlation function is much more sensitive to the width of the distributions than to their exact form. For this reason, the choice of Pc and Pa is not very critical; for practical reasons log-normal distributions with variable widths were chosen. Fitting the Calculated to the Experimental Correlation Functions In fitting the calculated y (x)-curve to the experimental one, the unit of length in the experimental curve is first made equal to the value of x at the first maximmn. Three para- meters must then be adjusted: r Bc and Ba, where Bc and Ba represent the widths of P~ and Pa respectively. This is done in steps: first, ~ can be estimated from the fact that at narrow distributions of -Pc and Pa, the distance OA (fig. 1) is equal to ~b(1- ~5) and BC (fig. 1) equals (1 - ~)/~. Next, the width of the distribution of the thicker layers is varied to fit the first maximum in the exper imentM curve ; in this stage the thinner layers are supposed to be equally thick. The procedure is repeated to obtain the best values for ~5 and Bc (or Ba). For a large number of polyethylene samples (detailed results of which will be reported in part I I I) almost the entire experimental cor- relation function could be adequately re- produced in this way. By way of example, the curves for a Marlex sample are shown in fig. 4. However, small deviations were generally found in the neighbourhood of the first minimum, as can be seen from the curves a and b in fig. 4. For this reason the possibility of distributing the thicknesses of the thinner layers was introduced. In this way these discrepancies were largely eliminated, as is shown by the curve c in fig. 4. 1,o ~'(x) l ,7 0,5 1 L, / - ~ ' , o ~ %,,: _c ' ' /5 / ' 560 ~ ' - - " " b o Fig. 4. Experimental (a) and calculated (b and c) correlation functions of a Martex sample, cooled at the rage of 0.02~ from the melt As interchanging of the electron densities of the two types of layers would not affect the scattering curve, it follows that X-ray scattering data alone fail to reveal which of the two types of layers has the higher electron density. Usually, however, the answer to the question follows quite obviously from other data of the sample. Deviations from the Ideal Model An important question regarding the results obtained by this method concerns the extent to which the results depend on the specific model adopted. All conclusions are based on the assumption of parallel layers; if this condition should not be fulfilled, the parameter values would be meaningless. It can, however, be shown by argumentation that the correlation function is not influenced by buckling of the layers as long as these remain concentric and the radius of curvature is larger than about twice the identity period. This means that the twisting of the layers observed in polyethylene probably has no effect on the conclusions. The effect of layers of irregular thickness, (fig. 5), has been found to equal the effect of an increase in width of Pc and Pa. Hence, our experiments cannot decide whether the variations in layer thickness exist within each layer and/or between the various layers Theoretically this point might be cleared up with the aid of the slope of the correlation function at the origin; however in the present Vonk and Kortleve, X-Ray Small-Angle Scattering o] Bulk Polyethylene, I I 23 exper iments the value of this slope is too uncertain for this purpose. Finally, consideration must be given to the effect of the assumption that the system is a two-phase one. For this purpose let us suppose, that there is a third phase of volume Am ~ m (~/ /~ Am Fig. 5. Layers of irregular thickness fract ion a, for which the electron density is equal to the average electron density and which is assumed to consist of layers inter- posed between the amorphous and crystall ine layers (fig. 6a). Then, all vectors A B for which either A or B is within the third phase would not contr ibute to the correlation function for this three-phase model. I t can be shown that, in consequence, the absolute value of the cor- relation function is lowered by a factor decreasing from 1 for x = 0 to 1 -~ for x --> oo. The effect is similar to that of an increase in the width of the distr ibution of the layer thicknesses and would not be distinguishable f rom it. The three-phase model can be considered as a rough approximat ion of a model with continuous density variations, as is indicated in fig. 6b). I t is therefore to be expected that this latter model would modi fy the two- phase correlation function in the same way as the three-phase model. Conclusions The present method of interpretation of the X-ray small-angle scattering curves of bulk polyethylene gives meaningful results, provided the basic assumption that the material is built up of parallel (or concentric) layers differing in electron density is fulfilled. This assumption itself cannot easily be tested with the aid of X - ray small-angle scattering, but other exper iments (electron microscopy) make it a very probable one. The presence of layers of irregular thickness, or of more than two phases will influence pr imari ly the parameters Bc and Ba, and only to a minor extent the value of 4. The values of c and a will in all eases where the basic assumption is fulfilled, correspond tO the mean thicknesses of alternat ing layers of different densities, even though there is not a sharp transit ion between these regions. Though ~b is not necessari ly equal to the crystal l inity by volume in the usual sense of the word, a good agreement between these two quantit ies is found in samples of unbranched polyethylene, as will be described in a following paper. We feel that this agreement supports the correctness of the present method. Cr Am Cr Am Fig. 6. a) 3-Phase model; b) Model with continuous density variations. The line AB represents the average density We are indebted to Dr. A. Di]kstra, who suggested the use of the onedimensional correlation function, and who carried out preliminary calculations. Furthermore we are indebted to Mr. Keuning for writing the programs for the computer calculations. Summary On the basis of the assumption that bulk poly- ethylene is built up of parallel layers differing in electron density, the correlation function for the direction perpendicular to the layers is calculated from the experimental scattering curve at small angles. This function is compared with that calculated for a model consisting of alternating crystalline and amorphous layers. One scale factor and three parameters have to be adjusted to obtain good agreement. The scale factor accounts for the average periodicity in the direction perpendicular to the layers. The parameters are the crystalline fraction and the width of the distributions of the thicknesses of the crystalline and amorphous layers respectively. It is furthermore shown that especially the erystallinity thus found is not appreciably influenced by even relatively large de- viations from the primary model. Zusammen/assung Ausgehend yon der Annahme, dab aus der Sehmelze erstarrtes I)oly~thylen aus parallelen Schichten unter- schiedlicher Elektronendichte zusammengesetzt ist, wird die Korrelationsfunktion ftir die Richtung, senk- reeht zu den Schiehten, aus der experimentellen Klein- winkel-Streuungskurve berechnet. Diese Funktion wird mit derjenigen Funktion verglichen, welche ffir ein weehselweise aus kristallinen und amorphen Schichten bestehendes Modell berechnet worden ist. Es muBten zur Erreichung einer guten ~-bereinstimmung ein Skalenfaktor und drei Parameter eingestellt werden. Im Skalenfaktor gelangt die mittlere Periodizit~t in de
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