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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