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The scattering of light from systems with nonrandom orientation correlations
The Scattering of Light from Systems with Nonrandom Orientation Correlations J. J. V A N A A R T S E N Akzo Research Laboratories Arnhem, Corporate Research Department, Arnhem, The Netherlands
Received September 17, 1971; accepted November 29, 1971 Theories for calculating the scattering of light from solid polymeric materials can be divided into two classes. The particle scattering theories using models of the scattering entities and treating the total scattering in a "dilute solution" approximation form one of these. The statistical theories utilizing correlation functions constitute the other class. They are better in principle but cannot be easily connected with the known physical structure of the material. Moreover, the existing theories are either too simple or too complicated to be of practical use. In the present article a new theory is developed which takes the correlation function approach, but in such a way that the resulting expressions can be applied to obtain meaningful parameters from experimental data. The derivation is accomplished by expansion of the general correlation function G(r) into a series of spherical harmonies and introducing a number of rather general assumptions. It is also pointed out that cross correlations are intractable. This new theory is particularly suited to compare the two types of light scattering theories as applied to maeroscopically isotropic materials. It also shows that the determination of the usual scattering components is not enough to obtain the maximum amount of information possible.
INTRODUCTION I n recent years there has been an increasing interest in light scattering as a tool in the characterization of solid polymeric materials. This has led to the development of new and more accurate photometers, such as the Bleeker Small Angle Scattering Photometer (1), as well as to a method of correcting the measured intensities (2, 3). Such correction is necessary because there are a number of optical effects which influence the measurements. These corrections will not be discussed here, because they can be found in the literature (2, 25). Nevertheless, the exact definition of the so-called vector O, which
is essential in order to take into account the nonideality of polarizer plates, has to be considered rather carefully (3, 4). The interpretation of the experimental Copyright@1972by AcademLePress, Inc.
scattering curves in terms of the structure of the material has been accomplished using two different approaches. First the correlation function approach was developed. Following the original theory of Debye and Bueche (5), who considered fluctuations in average local polarizability only, Goldstein and Miehalik (6) published their well-known general scattering theory. However, this theory has never found application, because its general character makes it extremely complicated, while, moreover, it does not allow the experimentM nonideality of the polarizer plates to be taken into account. Instead, Stein and coworkem tried to develop a theory on the basis of ao simplified model, but one which would leadL to expressions that could be used in practice. This so-called random orientation correlation approach was introduced b y Stein and Journal of Colloidand Inter/ace Science, Vol.39, No. 3, June 1972
583
584
VAN AARTSEN"
Wilson (7) and later extended by Stein and Hotta (8). Although this theory was used rather successfully in a number of cases (9-12), and in a more detailed form proved to be able to describe a number of experiments rather well (11-13), it was realized that the Stein-Wilson theory had one serious drawback. Because of the assumptions used by these authors the scattering should be symmetrical around the incident beam and, therefore, dependent on the scattering angle 0 only. Photographs of the scattering pattern by means of a laser showed that for a number of samples this is not the case, and so the assumptions of Stein and Wilson do not hold any longer (14-16). A second approach is to regard the sample as a collection of individual scattering entities with a specified optical structure. This kind of model calculations was used by Stein and Rhodes (14) to describe the scattering of light from spherulitic semicrystalline polymers, while the present author utilized a model of a random collection of cylinders with finite length and width for the interpretation of light scattering measurements on certain hydrogels (17-19). Also the group of Frenkel and coworkers frequently uses this kind of model calculation (20). This particle scattering approach presupposes, however, that the scattering system can be regarded as dilute in an optical sense. Interparticle interference effects are neglected, which can be a serious drawback. A more sophisticated version of the scattering theory of Stein and Rhodes, which does take interparticle interference effects into account, was presented by Picot and Stein (21). In view of this situation it appeared interesting to investigate what assumptions are necessary in order to reduce the general correlation function approach of Goldstein and 5~[ichalik (6) to a theory which can be used in practice while being yet more general than the random orientation correlation of Stein and Wilson. This new Journal of Colloid and Interface Science, Vol. 39, No. 3, June 1972
theory should describe scattering of light which is not symmetrical around the incident beam and should offer the possibility of incorporating the consequences of nonideality of the polarizer plates. A first attempt to formulate a theory for nonrandom orientation fluctuations was made by Stein and coworkers (22), but this was a two-dimensional approach only. Moreover the resulting equations are rather cumbersome. A different approach was made by Xeijzers et al. (3), who interpreted their measurements on semicrystalline polypropylene and polystyrene on the basis of a linear combination of the random orientation correlation approach (7) and the spherulite model of Stein and Rhodes (14). Although this worked rather well in practice, the combination of two fundamentally different kinds of theory for one sample is not entirely satisfactory. Nioreover the new theory presented here enables one to see more clearly the connection between the usual particle scattering calculations and the statistical correlation function approach. This aspect was dealt with more fully at a recent symposium (23). THEORY
It should be remembered throughout the following derivation that the whole theory is only valid for systems which are macroscopically isotropic. Samples possessing some kind of preferred orientation should be treated in an entirely different way, as has been done by the author for stretched spherulitie samples (24). In order to derive the theory in its most general form it is necessary to use three different coordinate systems. These will be described first. The first system (I), being the coordinates of the laboratory experiment, is given in Fig. 1. The incident beam is propagated in the x direction and the scattering is measured in the direction s ~. A unit vector in the direction so - s' = s
NONRANDOM ORIENTATION OOt~RELATIONS
585
scattered beam
t-1
eamp[e
.
~r
j
/i
F~G. 1. The coordinates of the laboratory experiment. P and A are polarizer and analyzerpolaroid plates, both perpendicular to the incident beam (so = i). is called h, so t h a t s = 2 sin O/2h
[I]
and h = sin (0/2)i -
-
cos (0/2) sin pj
cos
(e/2)
cos
[2] ,k
T h e second coordinate system ( I I ) is chosen in such a way tha~ one direction corresponds with h. The other two mutually perpendicular directions are given b y the unit vectors p, perpendicular to h and in the plane through h and k, and q, perpendicular to both h and p. I n this system a vector r~. is defined as
where A~ and A j are the amplitudes of light scattered from the ith and jth volume elements, k = 2~/X,X is the wavelength of the light in the medium, and r~i = r~ - r~. is the vector separation of the ith and jth volume elements. T h e vector s = 2 sin 0/211 has already been defined b y Eqs.
[~] and [2]. T h e amplitude A~ is proportional to the component of the dipole m o m e n t ml induced b y the incident field E ~ in the ith volume element, which is perpendicular to the direction of propagation s' of the scattered ray and which is passed through a polarizer in the scattered light p a t h . This
rlj = rr = r [cos a h -~ sin a cos ~tp
[3] ~-
sin
a
sin
~q]
£
/
T h e third system ( I I I ) is t h e n given b y the set of unit vectors r, d ~nd e, where d = --sin ah -b cos a cos ~p
[4] +
cos a sin flq
and
e
'\
\
e = + s i n ~p -- cos f~q.
I
\ "
\
[5]
The systems I I and I I I are shown in Fig. 2. T h e intensity of scattering is given b y
I = EE i
y
AiA~ cos [k(r,j.s)]
I
[6]
"L i
FIG. 2. The second and third coordinate systems in relation to each other as defined in texg. Journal of ColToid and Interface Science, Vol.
39, Xo. 3, J u n e 1979
586
VAN AARTSEN
can be accounted for by Ai = C t [m~.O] 0
[7]
where
A~A j = C2Eo202[&hs(E.a~) (ai.O)
c = (4~/~o ~)
• (E.a~) (aj.O)
and 0 is a factor to take into account the nonideality of the analyser. We shall assume that the anisotropic volume elements have cylindrical symmetry a n d m a y be described by two polarizabilities, {au )~ in the principal direction defined by *he unit vector ai (see Fig. 3) and ( a , ) i perpendicular to this direction. The anisotropy of the ith volume element is specified by ~ = (all -- a ~ ) i . The dipole moment induced by the effective incident field Ei~ is then m,
=
&(Ei~-a0a~
+
(a.~),Ei~ .
[8]
Introducing the average polarizability of the ith volume element ai as ~i
Substitution of Eq. [10] into Eq. [7] leads to the result
=
(1/3)[(a,)i
+
2(a~)i]
[9]
and E~ = EoE, where E is a unit vector along the polarization direction of the incident beam, we can also write mi = E0[&(E.a~)a~ + (hi - I~&)E]
+ (~,as + ~ 6 ~ . ) ( E . O ) 2 + -~/E.
o)
• { (E.a~) ( h i . O )
aihs (E" 0 ) • { ( E . a j ) ( a s . O ) -- 1/~(E.O)
+
ai~i(E" O) •{(E.a,) (a,.O)
-- 1/~(E.O)}]
The unit vector ai along the principal polarizability direction of the ith volume element is defined in the coordinate system I I I as (see Fig. 3) a~ = cos 3r + sin 3 cos 3,d [12] +
[10]
and an analogous equation holds for m3.
[11]
+ (E.a3)(aj.O)} +
sin 3 sin 3' e
Now the principal polarizability direction in the j t h volume element, defined by the unit vector a j , is given by aj = cos 0~#i + sin O~j cos ~o-a{ +
sin
0~j sin ~oa{'
[13]
where hi' an a{' form a left-handed cartesian coordinate system together with a~. If it is assumed t h a t at constant 0~j- the angle q~o" takes on all values with equal probability, then
/'
N
(E .a~) (hi. O)
\ N \
= F{(E.a,)(a,.O)
\ N
FIG. 3. The coordinate s y s t e m used to describe the o r i e n t a t i o n of the optic ~xis a j in volume element j w i t h respect to the optic axis a~ in volume element i and the vector ri3" connecting b o t h elements. Journal of Colloid and Interface Science, Vol. 39, No. 3, ffune 1972
--
I~(E-O)}
+
[14]
~(E.o)
In this equation F stands for F = 1/~(3 cos 2 0~j -- 1)
[15]
Using Eq. [14], the product A~A~ can now
NONRANDOM ORIENTATION CORRELATIONS be written as
where
A ~ A j = C 2 E o 2 0 2 [ ~ j F ( E • a~)2(a~ • O)2
P ~ ( c o s ~) = (sin ~ ~),,~/2
+ (~i~j + ~ / 6 ~ T ) ( E . O ) ~ + ' d(cos ~)'~
- (2/3)~,~T (E. O) • (E. ad (a,. o )
+ (~T • { (E.ai)
-
[16a]
587
[P,,,(cos ~)]
1 2,~n !
P d c o s ~) -
+ ~d(E.O) dn [ _ sin 2 $]~ " d(cos ~)'~
(a~.O)
(1/3)(E.o)I].
The last term in Eq. [16a] is caused by cross correlations between ai and ~ a n d / o r between aj and ~j. This type of correlations is assumed to be absent (see, however, the Appendix). Because a strictly homogeneous sample does not give rise to any scattering of light at all, only deviations of ai and % from the mean polarizability a play a role. By writing n~ = a~ -- a and ni = ~ -- a it is evident t h a t the product a~a~- in Eq. [16a] m a y be replaced by n~n¢. Denoting ~5~.F by G gives
and the coefficients A~,~ are functions of r only• The intensity of scattered light can now be calculated b y writing Eq. [6] in integral form and averaging over all possible positions of the vector al at constant ri¢, or ~o
7r
27c
z=Jofo£ =0
20
• =0
=0 ~r
2~
•cos [hr cos a] sin a d~ dar ~ dr AiA ~ = C~Eo~&[G(E.a~)~(a,'O) ~
[16]
+ (~w + ½G)(E'O) ~ +
w
2~r
f0 f0
- ( 2 / 3 ) a ( E . O ) ( E .al)(a~. O)1 • ~ For a sample with nonrandom orientation fluctuations and without macroscopic orientation, G will be a function of fl, 3, and r (see Fig. 3). In general this function iS not known, b u t it will be assumed that G can be developed in a series of spherical harmonics so that ~ G = < ~ ( 3 cos~ 0~.j = ~ ~
ra~O
[~
~r~O
1)~. >
A ..... ( r ) P ~ ( c o s f l )
[17]
• COS m~l
It should be noted that possible correlations in the magnitude of ~ and ~i are incorporated alSO.
AiA3. s i n ~ d T d
• cos(hr cos a)sin a d~ dar 2 dr
[18]
In order to carry out the required calculations we start with Eq. [16] and express the vector products (E. al)(a~. O) ancI (E.a~) (a~.O) in terms of our third coordinate system. The next step is to express the vector products which appear in the result of the first step in terms of coordinate system II, remembering that terms contMning odd powers of trigonometric functions of ¢~, 7, a or f~ will average to zero or lead to purely imaginary results. In the third step the integration over the angle a is carried out. This may be done because the phase factor, cos (hr cos a), does not depend upon Journal of Colloid and Interface Science, V o l . 39, :No. 3, J u n e 1972
588
VAN AARTSEN
~. After rearrangement of ~he different terms ghe final result is given by I = C 2 0 2 ~ 0 2 `
2 H2
-
[H2°P2° -[- 3H2 P2 ]
4- -~ H4°P4° -Jr"~ H4 P~ + H~4P4~ + Y
Ho°Po° +
H~° - 4~2
+
•
4-
1
i fo
H 4 4 - 4v~2 1 f o ' f o 2= G ~1 sin 4 fl cos 43' 8 •sin/3 d'y d~ = ~ A4,4(r)
,,}
+
The P's (not to be confused with spherical harmonics P O ) are given b y ( J being a Bessel function of the first kind) 7r[2
Po° = [
In this general formula the different symbols have the following meaning
= 2 \-fir/
H O_
1 4~r~2 ~r
1 (3 cos 2
1) • cos
= G sin ~ d3' df~ = Ao,o(r)
J1/:(hr)
~r/2
p: = f
1 4~r~ dO f2~ d0
cos (hr cos a) s i n a d a
J0
fo~jl 2~ ~ ~s
•sin f~ d~, df~] poor~ dr
•f
G~
1
+ 91HoOpoO(E. 0)21 r ~ dr
Hoo _
- ~ cos f~ +
• cos 23' sin fl d~, df~ = 5 A2,4(r)
[19]
4- (h.E)(h.O)[H2°P~ ° + 3H~P21
+ 4~r(E. 0 ) 5 j o ' " [ 1~
[20]
4r~ 2
- - 21( E 'HO ) (° ( E°' OP) ( ° ° 3 1 [H20p2o _}_ 3H22t°2]2 ) 3
cos 4 ~ -
H42 _
+
•
+ ~ H~2P~2 + H4~P4~
G
1 • sin ~ d',/d~ = ~ Ao,4(r)
[H~°P~° + 3H~ P2 ]
L3aH~°P~° 4- ~ H~ P~ + H~4P~~
~_
G ~ sm /~
2 • cos 27 d~, dfl = ~ A2,2(r)
4~r f0'° ~2 [ ( X - Y)
• ~ Ho°Po° --
, ji-ji
4~r~2
p2 2
( 2 7 r ~ 1/2 -
1/2 \~/ 0 'r/2
(hr cos
a) sin o~ d~
Jsi2(hr)
1/2 sin 2 a cos (hr cos a) sin a da
P 2 = f0 ~r/2 1/8(35 cos 4 a - 30 cos 2 a 4- 3) 2~
1
• cos (hr cos a) sin a da 1
•sin ~ d3, d~ = ~ Ao,2(r) Journal of Gollold and Interface Science,
Vol. 39, No. 3 June 1972
ly .F J,/,(hr)
= -2 \-~r]
[211
NONRANDOM
p2 =
f ,~m 1 sin 2 ~(7 cos2 a
-
ORIENTATION
~ sin 4 a cos (hr cos a) sin a da.
while the symbols X, Y and Z denote the influence of the experimental circumstances X = 1 + 2(E.O) 2 Y = ( h . O ) 2 ~- ( h . E ) 2
[22]
+
The consequences of the general scattering equation [19] will first be discussed with regard to the kind of information that is obtained from light-scattering measurements. If we first focus our attention on the last term of Eq. [19], we see t h a t from the socalled density fluctuations it is only possible to obtain the random orientation part. 2 This appears from the fact that, if we call the average polarizability per unit volume of the sample a, and vi = a i - -
Z = The value of the vector h is given in Eq. [2]; E and O depend upon polarizer and analyzer settings. For the case t h a t the primary beam direction is along the x axis and the plane of the analyzer-polaroid is perpendicular to this axis, Keyzers et al. have sho~m t h a t with t~ = - s i n i j ~ cos ~/k
589
DISCUSSION
1)
• cos (hr cos a) sin a da P2 =
CORRELATIONS
~
while
vj = a j -
Idons = C20~E0~4~(E • O) ~ •
~
~ W sin ~ d7 d~
and taking measurements in the horizontal (x, y) plane, the vector O is given by
• {sin0sin@ i -- sin0sin@ j
[24]
[27]
the only term leading to a nonzero scattering effect arising from the product aiaj will be v~n~, as mentioned before. So the density fluctuations give rise to a scattering effect given by
[23]
O = [sin2@ ~- cos~0p * cos2~b]-1'~
~
[28]
• PoOr2 dr.
Now, assuming that v ~ is a function of fl, and r and using an analogous approach as in Eq. [17], we may write this product as
÷ cos@* cos¢~ k} The angle Op* depends upon the real scattering angle 0 (inside the sample!) but shown in (3) not to be identical with it. The ease where the analyzer is perpendicular to the scattering direction (s') is treated by Gouda and Prins (4). Using the same symbols it is found t h a t t~ = sin e ( - sin 0~i + cos 0 j ) + cos ~ok
+ cos o~k}
n=0
] 2 B~,n( °)Pn (COS~) cosmv]
[291
m~0
Substituting Eq. [29] into Eq. [28] and performing the integrations results in =
C
O
Eo
(E • 1.
-
sin
• Jo Bo,o(r) ~
(hr)
[30] r 2 dr
[25]
and so O = / - s i n 0 sin M -4- cos 0 sin wj
= P
[26]
which is seen to be identical to their equation [7] if it is realized t h a t these authors use the symbol O * for the same effective vector which is denoted by OO in this article.
and so it is seen that the coefficient Bo,o(r) is the only one from the series expansion in Eq. [29] which shows up in the scattering equation. Equation [30] is equivalent to the usual expression for scattering b y density fluctuations (5, 7, 9, 10) where B0,0(r) is It is stressed again that this conclusion only holds for samples without macroscopic orientation: isotropic samples[ Journal of Colloid and fnterface Science, Vol. 39, No. 3, J u n e 1972
590
VAN AARTSEN
called a correlation function usually denoted by ~/(r). Now by combining Eq. [30] with Eq. [19] the general scattering formula can also be written as
In the third place, the relation used by Stein et al. (7) and Keyzers et al. (3, 11) to obtain the intensity resulting from density correlations (id = W, for y = z = O) also breaks down and now
I = 4rrC202Eo 2 I~2(x -- 2y + z)
4
W = il - ~ i4 = 4~rC2Eo 2
+ (E.o) 2 ~
+ 5 1 ~2( x -
"y(r)Po°r2 d~
2y-t-z) + 1~2(4g +
5Z)}
• ~
[31]
"y(r)Po°r 2 dr + 3
[33] + {(h-O) 2 + (h.E)2}~2(3y -- 5z)
• {(4y + 5z) - 4 cos2 0/2(3y - 5z)} 1
-- 2(E.O)(h.E)(h.O)~2(y + 10z)
= id + (4~/3)c2Eo2~ 2
+ 35(h. E)2(h • O)2~2z|
• {(4y + 5z) -- 4 cos 2 0 / 2 ( 3 y -- 5z) }.
where x =
~-~ Ho°Po°r 2 dr
[32a]
y = fo
~ [H20P2 0 + 3H2 2P2 2]r 2 dr
[32b]
3-5 [H4°P4° + 20H42P42
[32c]
z =
fo
+ 35H4~P44]r 2 dr
From this formulation it can be seen that for random orientation fluctuations (where y = z = 0) Eq. [31] properly reduces to the equations given by Stein et al. in various publications (7, 9, 22). Nevertheless Eq. [31] is still more general, because it incorporates every polarizer and analyzer setting and is not restricted to the usual scattering components V~, H~, Vh, and H h . For the general case, where y # 0 and z # 0, it can be shown that the corrected intensities V~ = il, H , = i4, V~ = i5 and Hh = i2 obey the following relations (see also Keyzers et al. (3), who use the same symbols). In the first place, Q = i~ still holds• Secondly, the relation/2 = il c o s 2 0 -[- Q sin 2 0 does not hold any more and one finds T = i2--
ilcos 2 0 -
i~sin 20
= 4~rC2Eo2~2 cos2 0/2[(1 + cos 2 0)
[32] • (3y -- 5z) + 35 cos2(O/2)z -- 2 cos O(y + 10z)]. Journal of Colloidand InterfaceScience,Vol.
39, No. 3, June 1972
Our final remark concerns the way in which Keyzers et al. (3) treated their data and tried to separate them into a random orientation part and a perfect spherulitic part. Using exactly the same symbols and assuming t h a t the same experimental details hold, it can be shown t h a t the parameters given in their Eqs. [37] and [38] m a y be written as KA 2 = 4~rC2Eo2~2135z
+ 4 sin 2 0 / 2 ( 3 y -- 5z)] dB
= 4~rC2Eo2~2 cos 0[cos O( 3y - 5z) -- ( y +
10z)]
[34] [35]
So it turns out t h a t both these parameters are functions of higher-order moments of the general correlation function as defined by Eq. [17]. The part used to calculate the spherulite dimensions (KA2) is mainly determined by z. It is now possible to use values of ii, i2, i3 and i4 (where i3 is the corrected intensity measured with crossed polarizer and analyzer whose transmittance directions are at 45 ° to the vertical) and calculate from them x, y, z and id aS a function of the scattering parameter h = (4r/~) sin (0/2). From these results the values of ~ and ~2 as well as the functions ~(r) and Ao.o(r) can be obtained by Fourier transformations (10, 11). If it is also assumed that the gen-
NONII.ANDOM ORIENTATION CORRELATIONS erM correlation function of Eq. [17] depends on fl only, it may even be possible to obtain Ao,2(r) and Ao,4(r) from Eqs. [32b] and [32c], where now H2~ --- H , ~ = H~~ = 0, via a Hankel transformation. All this can only be achieved by assuming t h a t cross correlations do not occur. In the Appendix it is shown how the presence of such correlations can be formally accounted for, Mthough it is not possible to evaluate experimental measurements in such eases. APPENDIX Retaining the last term of Eq. [16a] and carrying out the cMeulations required by Eq. [18] one obtains an additional term in Eq. [19] with the following structure Ic .... = 47rC'E'O 2
• { ( h O ) ( h . ~ ) ( E . O) -
(~. O)'
.
591
Even this assumption does not lead to any marked improvement. The only way in which an evaluation of the experimental results is rendered possible is to assume that ~i6]F = \ ~ ]
G
[A31
This assumption then leads to I¢ .... = 4~'C'Eo20 '
"I (h' O ) ( h ' E ) ( E ' O ) _ 31 (E.O). 1 [A41 "\9]
l-, I12 r dr
Only with the additional assumption 3,@) = Ao,o(r) can Eq. [19] be solved for the four unknown quantities x ( h ) , y(h), z(h) and
(n'/~'). In view of the assumptions that were
[7070" .
1 (3 cos' ~ -- 1) sin ~ d3, d¢~| r ' dr l 2 I-" wf'Tr
",'kl o
+
,
[~&l]
+-,',',
" 2 sin'3 fi cos 23, d3, dJ
Ir) r' d
Generally, (ai~ + a ; ~ F ) ~Jll be a function of r, fl and 3' which is different from G, and the intensity contribution caused by cross correlation effects cannot be separated from the other terms of Eq. [19] A number of assumptions about the functional form of (asSi + a~SjF) can be made, one of the most drastic being that there is no -/-dependence and that ajS~ depends on the distance r only. i n that ease
necessary to reach this possibility one realizes that in fact a model has been obtained, in which n and ~ vary in the same way. This result can also be achieved by starting from an explicitly defined model, such as given by the author for a random collection of cylinders of finite length and radius (17). This was discussed in some detail at a recent symposium (24). REFERENCES 1. Described by Keijzers et al. (3) and commercially available from Bleeker N.V., Zeist, The Netherlands or from IMASS, Accord (Hingham) MA. 02018. 2. K~IjZERS,A. E. M., Light Scattering by Crystalline Polystyrene and Polypropylene. Thesis, Delft, 1967. 3. KEIJZERS, A. E. M,, VAN AARTSEN, J. 5., AND PRINS, W . , J . A m . Chem. Soc. 90,
3107 (1968). 4. GOUDA.,J. H., &ND PRINS, W., 3r, Polymer Sei.
i ..... = 4~rC2Eo'O 2
A-2, 8, 2029 (1970).
- [ ( h . O)(h.E)(E. O)
_ ~1 (E. 0). l
5. DEBYE', P. W., AND BUECHE, A. M., Or. Appl.
Phys. 20,518 (1949). 0. GOLDSTEIN, ~I., AND M1CHALIK, E. R., 26,
" 2-1i
P2o
~i~3
• _1 (3 cos' ¢~ -- 1) sin ~ d~/ r ' dr 2 _J
1450 (1955). 7. STEIN, R. S., A~) WILSON, P. R., Or. Appl. Phys. 33, 1914 (1962). 8. STEIN, R. S., AND HOTTA, W., Or. Appl. Phys. 35, 2237 (1964). Journal of Colloid and Interface Science, VoL 39, No. 3, June 1972
592
VAN AARTSEN
9. STEIN, R. S., in, Proceedings of the International Conference on Electromag. Scattering" (M. Kerker, ed.), p. 430. Pergamon Press, New York, 1963. 10. STEIN, R. S., KEANE, J. J., NORRIS, F. H., BETTELHEIM, F. A., AND WILSON, P. g . , Ann. N. Y. Acad. Sci. 83, 37 (1959). 11. KEIJZERS, A. E. M., VAN AARTSEN, J. J., AND PnlNS, W., J. Appl. Phys. 36, 2874 (1956). 12. STEIN, R. S., WILSON, P. R., AN9 STIDHAM, S. N., J. Appl. Phys. 34, 46 (1963). 13. COALSON, R. L., MA~CHESSAULT, 1%. H., AND PETEaL1N, A., J. Polymer Sci. C13, 123 (1966). 14. STEIN, R. S., AND RHOnES, M. B., J. Appl. Phys. 31, 1873 (1960). 15. R~ZODES, M. B., AND STEIN, ]~. S., ASTM Special Techn. Pub. No. 348 (1963). 16. FRENKEL, S. YA., VOLI-:OV, W. J., BARANOV, V. G., AND SHALTYKO, L. G., Polym. Sci. USSR 7,942 (1965). 17. VAN AARTSEN, J. J., Eur. Polym. J. 6, 1095 (1970).
Journal o/Colloid and Interface Science,
Vol. 39, No. 3. June 1972
18. DONKERSLOOT, M. C. A., GOUDA, J. H., VAN AARTSEN, J. J., AND PRIES, W., Rec. Tray. Chim. 86,321 (1967). 19. GOUDA, J. H., Structural Characterization of noncrystalline hydrogels. Thesis, Delft 1969. 20. GASPARJAN,K. A., GOLOUBEE, Ja., BARANOV, V. G., AND FRENE:EL, S. ya., Polym. Sci. USSR 10, 96 (1968). 21. PICOT, C., AND STEIN, R. S., IUPAC Int. Symp. Leiden 1970, Paper IV-15. 22. STEIN, R. S., ERt/ARDT, P., CLOUGH, S., AND VAN AARTSEN, J. J., in, "Proceedings of The Second International Conference on Electromagnetic Scattering" (1~. L. Rowel and R. S. Stein, eds.), p. 339. Gordon & Breach, New York, 1967. 23. VAN AAaTSEN, J. J., in, " Polymer Networks" (A. J. Chompff and S. Newman eds.) p. 307. Plenum Press, New York, 1971. 24. VAN AARTSEN, J. Y., AND STEIN, R. S., J. Polymer Sci. A-2, 9,295 (1971). 25. STEIN, •. S., AND KEANE, J. J., J. Polymer Sci. 17, 21 (1955).
Received September 17, 1971; accepted November 29, 1971 Theories for calculating the scattering of light from solid polymeric materials can be divided into two classes. The particle scattering theories using models of the scattering entities and treating the total scattering in a "dilute solution" approximation form one of these. The statistical theories utilizing correlation functions constitute the other class. They are better in principle but cannot be easily connected with the known physical structure of the material. Moreover, the existing theories are either too simple or too complicated to be of practical use. In the present article a new theory is developed which takes the correlation function approach, but in such a way that the resulting expressions can be applied to obtain meaningful parameters from experimental data. The derivation is accomplished by expansion of the general correlation function G(r) into a series of spherical harmonies and introducing a number of rather general assumptions. It is also pointed out that cross correlations are intractable. This new theory is particularly suited to compare the two types of light scattering theories as applied to maeroscopically isotropic materials. It also shows that the determination of the usual scattering components is not enough to obtain the maximum amount of information possible.
INTRODUCTION I n recent years there has been an increasing interest in light scattering as a tool in the characterization of solid polymeric materials. This has led to the development of new and more accurate photometers, such as the Bleeker Small Angle Scattering Photometer (1), as well as to a method of correcting the measured intensities (2, 3). Such correction is necessary because there are a number of optical effects which influence the measurements. These corrections will not be discussed here, because they can be found in the literature (2, 25). Nevertheless, the exact definition of the so-called vector O, which
is essential in order to take into account the nonideality of polarizer plates, has to be considered rather carefully (3, 4). The interpretation of the experimental Copyright@1972by AcademLePress, Inc.
scattering curves in terms of the structure of the material has been accomplished using two different approaches. First the correlation function approach was developed. Following the original theory of Debye and Bueche (5), who considered fluctuations in average local polarizability only, Goldstein and Miehalik (6) published their well-known general scattering theory. However, this theory has never found application, because its general character makes it extremely complicated, while, moreover, it does not allow the experimentM nonideality of the polarizer plates to be taken into account. Instead, Stein and coworkem tried to develop a theory on the basis of ao simplified model, but one which would leadL to expressions that could be used in practice. This so-called random orientation correlation approach was introduced b y Stein and Journal of Colloidand Inter/ace Science, Vol.39, No. 3, June 1972
583
584
VAN AARTSEN"
Wilson (7) and later extended by Stein and Hotta (8). Although this theory was used rather successfully in a number of cases (9-12), and in a more detailed form proved to be able to describe a number of experiments rather well (11-13), it was realized that the Stein-Wilson theory had one serious drawback. Because of the assumptions used by these authors the scattering should be symmetrical around the incident beam and, therefore, dependent on the scattering angle 0 only. Photographs of the scattering pattern by means of a laser showed that for a number of samples this is not the case, and so the assumptions of Stein and Wilson do not hold any longer (14-16). A second approach is to regard the sample as a collection of individual scattering entities with a specified optical structure. This kind of model calculations was used by Stein and Rhodes (14) to describe the scattering of light from spherulitic semicrystalline polymers, while the present author utilized a model of a random collection of cylinders with finite length and width for the interpretation of light scattering measurements on certain hydrogels (17-19). Also the group of Frenkel and coworkers frequently uses this kind of model calculation (20). This particle scattering approach presupposes, however, that the scattering system can be regarded as dilute in an optical sense. Interparticle interference effects are neglected, which can be a serious drawback. A more sophisticated version of the scattering theory of Stein and Rhodes, which does take interparticle interference effects into account, was presented by Picot and Stein (21). In view of this situation it appeared interesting to investigate what assumptions are necessary in order to reduce the general correlation function approach of Goldstein and 5~[ichalik (6) to a theory which can be used in practice while being yet more general than the random orientation correlation of Stein and Wilson. This new Journal of Colloid and Interface Science, Vol. 39, No. 3, June 1972
theory should describe scattering of light which is not symmetrical around the incident beam and should offer the possibility of incorporating the consequences of nonideality of the polarizer plates. A first attempt to formulate a theory for nonrandom orientation fluctuations was made by Stein and coworkers (22), but this was a two-dimensional approach only. Moreover the resulting equations are rather cumbersome. A different approach was made by Xeijzers et al. (3), who interpreted their measurements on semicrystalline polypropylene and polystyrene on the basis of a linear combination of the random orientation correlation approach (7) and the spherulite model of Stein and Rhodes (14). Although this worked rather well in practice, the combination of two fundamentally different kinds of theory for one sample is not entirely satisfactory. Nioreover the new theory presented here enables one to see more clearly the connection between the usual particle scattering calculations and the statistical correlation function approach. This aspect was dealt with more fully at a recent symposium (23). THEORY
It should be remembered throughout the following derivation that the whole theory is only valid for systems which are macroscopically isotropic. Samples possessing some kind of preferred orientation should be treated in an entirely different way, as has been done by the author for stretched spherulitie samples (24). In order to derive the theory in its most general form it is necessary to use three different coordinate systems. These will be described first. The first system (I), being the coordinates of the laboratory experiment, is given in Fig. 1. The incident beam is propagated in the x direction and the scattering is measured in the direction s ~. A unit vector in the direction so - s' = s
NONRANDOM ORIENTATION OOt~RELATIONS
585
scattered beam
t-1
eamp[e
.
~r
j
/i
F~G. 1. The coordinates of the laboratory experiment. P and A are polarizer and analyzerpolaroid plates, both perpendicular to the incident beam (so = i). is called h, so t h a t s = 2 sin O/2h
[I]
and h = sin (0/2)i -
-
cos (0/2) sin pj
cos
(e/2)
cos
[2] ,k
T h e second coordinate system ( I I ) is chosen in such a way tha~ one direction corresponds with h. The other two mutually perpendicular directions are given b y the unit vectors p, perpendicular to h and in the plane through h and k, and q, perpendicular to both h and p. I n this system a vector r~. is defined as
where A~ and A j are the amplitudes of light scattered from the ith and jth volume elements, k = 2~/X,X is the wavelength of the light in the medium, and r~i = r~ - r~. is the vector separation of the ith and jth volume elements. T h e vector s = 2 sin 0/211 has already been defined b y Eqs.
[~] and [2]. T h e amplitude A~ is proportional to the component of the dipole m o m e n t ml induced b y the incident field E ~ in the ith volume element, which is perpendicular to the direction of propagation s' of the scattered ray and which is passed through a polarizer in the scattered light p a t h . This
rlj = rr = r [cos a h -~ sin a cos ~tp
[3] ~-
sin
a
sin
~q]
£
/
T h e third system ( I I I ) is t h e n given b y the set of unit vectors r, d ~nd e, where d = --sin ah -b cos a cos ~p
[4] +
cos a sin flq
and
e
'\
\
e = + s i n ~p -- cos f~q.
I
\ "
\
[5]
The systems I I and I I I are shown in Fig. 2. T h e intensity of scattering is given b y
I = EE i
y
AiA~ cos [k(r,j.s)]
I
[6]
"L i
FIG. 2. The second and third coordinate systems in relation to each other as defined in texg. Journal of ColToid and Interface Science, Vol.
39, Xo. 3, J u n e 1979
586
VAN AARTSEN
can be accounted for by Ai = C t [m~.O] 0
[7]
where
A~A j = C2Eo202[&hs(E.a~) (ai.O)
c = (4~/~o ~)
• (E.a~) (aj.O)
and 0 is a factor to take into account the nonideality of the analyser. We shall assume that the anisotropic volume elements have cylindrical symmetry a n d m a y be described by two polarizabilities, {au )~ in the principal direction defined by *he unit vector ai (see Fig. 3) and ( a , ) i perpendicular to this direction. The anisotropy of the ith volume element is specified by ~ = (all -- a ~ ) i . The dipole moment induced by the effective incident field Ei~ is then m,
=
&(Ei~-a0a~
+
(a.~),Ei~ .
[8]
Introducing the average polarizability of the ith volume element ai as ~i
Substitution of Eq. [10] into Eq. [7] leads to the result
=
(1/3)[(a,)i
+
2(a~)i]
[9]
and E~ = EoE, where E is a unit vector along the polarization direction of the incident beam, we can also write mi = E0[&(E.a~)a~ + (hi - I~&)E]
+ (~,as + ~ 6 ~ . ) ( E . O ) 2 + -~/E.
o)
• { (E.a~) ( h i . O )
aihs (E" 0 ) • { ( E . a j ) ( a s . O ) -- 1/~(E.O)
+
ai~i(E" O) •{(E.a,) (a,.O)
-- 1/~(E.O)}]
The unit vector ai along the principal polarizability direction of the ith volume element is defined in the coordinate system I I I as (see Fig. 3) a~ = cos 3r + sin 3 cos 3,d [12] +
[10]
and an analogous equation holds for m3.
[11]
+ (E.a3)(aj.O)} +
sin 3 sin 3' e
Now the principal polarizability direction in the j t h volume element, defined by the unit vector a j , is given by aj = cos 0~#i + sin O~j cos ~o-a{ +
sin
0~j sin ~oa{'
[13]
where hi' an a{' form a left-handed cartesian coordinate system together with a~. If it is assumed t h a t at constant 0~j- the angle q~o" takes on all values with equal probability, then
/'
N
(E .a~) (hi. O)
\ N \
= F{(E.a,)(a,.O)
\ N
FIG. 3. The coordinate s y s t e m used to describe the o r i e n t a t i o n of the optic ~xis a j in volume element j w i t h respect to the optic axis a~ in volume element i and the vector ri3" connecting b o t h elements. Journal of Colloid and Interface Science, Vol. 39, No. 3, ffune 1972
--
I~(E-O)}
+
[14]
~(E.o)
In this equation F stands for F = 1/~(3 cos 2 0~j -- 1)
[15]
Using Eq. [14], the product A~A~ can now
NONRANDOM ORIENTATION CORRELATIONS be written as
where
A ~ A j = C 2 E o 2 0 2 [ ~ j F ( E • a~)2(a~ • O)2
P ~ ( c o s ~) = (sin ~ ~),,~/2
+ (~i~j + ~ / 6 ~ T ) ( E . O ) ~ + ' d(cos ~)'~
- (2/3)~,~T (E. O) • (E. ad (a,. o )
+ (~T • { (E.ai)
-
[16a]
587
[P,,,(cos ~)]
1 2,~n !
P d c o s ~) -
+ ~d(E.O) dn [ _ sin 2 $]~ " d(cos ~)'~
(a~.O)
(1/3)(E.o)I].
The last term in Eq. [16a] is caused by cross correlations between ai and ~ a n d / o r between aj and ~j. This type of correlations is assumed to be absent (see, however, the Appendix). Because a strictly homogeneous sample does not give rise to any scattering of light at all, only deviations of ai and % from the mean polarizability a play a role. By writing n~ = a~ -- a and ni = ~ -- a it is evident t h a t the product a~a~- in Eq. [16a] m a y be replaced by n~n¢. Denoting ~5~.F by G gives
and the coefficients A~,~ are functions of r only• The intensity of scattered light can now be calculated b y writing Eq. [6] in integral form and averaging over all possible positions of the vector al at constant ri¢, or ~o
7r
27c
z=Jofo£ =0
20
• =0
=0 ~r
2~
•cos [hr cos a] sin a d~ dar ~ dr AiA ~ = C~Eo~&[G(E.a~)~(a,'O) ~
[16]
+ (~w + ½G)(E'O) ~ +
w
2~r
f0 f0
- ( 2 / 3 ) a ( E . O ) ( E .al)(a~. O)1 • ~ For a sample with nonrandom orientation fluctuations and without macroscopic orientation, G will be a function of fl, 3, and r (see Fig. 3). In general this function iS not known, b u t it will be assumed that G can be developed in a series of spherical harmonics so that ~ G = < ~ ( 3 cos~ 0~.j = ~ ~
ra~O
[~
~r~O
1)~. >
A ..... ( r ) P ~ ( c o s f l )
[17]
• COS m~l
It should be noted that possible correlations in the magnitude of ~ and ~i are incorporated alSO.
AiA3. s i n ~ d T d
• cos(hr cos a)sin a d~ dar 2 dr
[18]
In order to carry out the required calculations we start with Eq. [16] and express the vector products (E. al)(a~. O) ancI (E.a~) (a~.O) in terms of our third coordinate system. The next step is to express the vector products which appear in the result of the first step in terms of coordinate system II, remembering that terms contMning odd powers of trigonometric functions of ¢~, 7, a or f~ will average to zero or lead to purely imaginary results. In the third step the integration over the angle a is carried out. This may be done because the phase factor, cos (hr cos a), does not depend upon Journal of Colloid and Interface Science, V o l . 39, :No. 3, J u n e 1972
588
VAN AARTSEN
~. After rearrangement of ~he different terms ghe final result is given by I = C 2 0 2 ~ 0 2 `
2 H2
-
[H2°P2° -[- 3H2 P2 ]
4- -~ H4°P4° -Jr"~ H4 P~ + H~4P4~ + Y
Ho°Po° +
H~° - 4~2
+
•
4-
1
i fo
H 4 4 - 4v~2 1 f o ' f o 2= G ~1 sin 4 fl cos 43' 8 •sin/3 d'y d~ = ~ A4,4(r)
,,}
+
The P's (not to be confused with spherical harmonics P O ) are given b y ( J being a Bessel function of the first kind) 7r[2
Po° = [
In this general formula the different symbols have the following meaning
= 2 \-fir/
H O_
1 4~r~2 ~r
1 (3 cos 2
1) • cos
= G sin ~ d3' df~ = Ao,o(r)
J1/:(hr)
~r/2
p: = f
1 4~r~ dO f2~ d0
cos (hr cos a) s i n a d a
J0
fo~jl 2~ ~ ~s
•sin f~ d~, df~] poor~ dr
•f
G~
1
+ 91HoOpoO(E. 0)21 r ~ dr
Hoo _
- ~ cos f~ +
• cos 23' sin fl d~, df~ = 5 A2,4(r)
[19]
4- (h.E)(h.O)[H2°P~ ° + 3H~P21
+ 4~r(E. 0 ) 5 j o ' " [ 1~
[20]
4r~ 2
- - 21( E 'HO ) (° ( E°' OP) ( ° ° 3 1 [H20p2o _}_ 3H22t°2]2 ) 3
cos 4 ~ -
H42 _
+
•
+ ~ H~2P~2 + H4~P4~
G
1 • sin ~ d',/d~ = ~ Ao,4(r)
[H~°P~° + 3H~ P2 ]
L3aH~°P~° 4- ~ H~ P~ + H~4P~~
~_
G ~ sm /~
2 • cos 27 d~, dfl = ~ A2,2(r)
4~r f0'° ~2 [ ( X - Y)
• ~ Ho°Po° --
, ji-ji
4~r~2
p2 2
( 2 7 r ~ 1/2 -
1/2 \~/ 0 'r/2
(hr cos
a) sin o~ d~
Jsi2(hr)
1/2 sin 2 a cos (hr cos a) sin a da
P 2 = f0 ~r/2 1/8(35 cos 4 a - 30 cos 2 a 4- 3) 2~
1
• cos (hr cos a) sin a da 1
•sin ~ d3, d~ = ~ Ao,2(r) Journal of Gollold and Interface Science,
Vol. 39, No. 3 June 1972
ly .F J,/,(hr)
= -2 \-~r]
[211
NONRANDOM
p2 =
f ,~m 1 sin 2 ~(7 cos2 a
-
ORIENTATION
~ sin 4 a cos (hr cos a) sin a da.
while the symbols X, Y and Z denote the influence of the experimental circumstances X = 1 + 2(E.O) 2 Y = ( h . O ) 2 ~- ( h . E ) 2
[22]
+
The consequences of the general scattering equation [19] will first be discussed with regard to the kind of information that is obtained from light-scattering measurements. If we first focus our attention on the last term of Eq. [19], we see t h a t from the socalled density fluctuations it is only possible to obtain the random orientation part. 2 This appears from the fact that, if we call the average polarizability per unit volume of the sample a, and vi = a i - -
Z = The value of the vector h is given in Eq. [2]; E and O depend upon polarizer and analyzer settings. For the case t h a t the primary beam direction is along the x axis and the plane of the analyzer-polaroid is perpendicular to this axis, Keyzers et al. have sho~m t h a t with t~ = - s i n i j ~ cos ~/k
589
DISCUSSION
1)
• cos (hr cos a) sin a da P2 =
CORRELATIONS
~
while
vj = a j -
Idons = C20~E0~4~(E • O) ~ •
~
~ W sin ~ d7 d~
and taking measurements in the horizontal (x, y) plane, the vector O is given by
• {sin0sin@ i -- sin0sin@ j
[24]
[27]
the only term leading to a nonzero scattering effect arising from the product aiaj will be v~n~, as mentioned before. So the density fluctuations give rise to a scattering effect given by
[23]
O = [sin2@ ~- cos~0p * cos2~b]-1'~
~
[28]
• PoOr2 dr.
Now, assuming that v ~ is a function of fl, and r and using an analogous approach as in Eq. [17], we may write this product as
÷ cos@* cos¢~ k} The angle Op* depends upon the real scattering angle 0 (inside the sample!) but shown in (3) not to be identical with it. The ease where the analyzer is perpendicular to the scattering direction (s') is treated by Gouda and Prins (4). Using the same symbols it is found t h a t t~ = sin e ( - sin 0~i + cos 0 j ) + cos ~ok
+ cos o~k}
n=0
] 2 B~,n( °)Pn (COS~) cosmv]
[291
m~0
Substituting Eq. [29] into Eq. [28] and performing the integrations results in =
C
O
Eo
(E • 1.
-
sin
• Jo Bo,o(r) ~
(hr)
[30] r 2 dr
[25]
and so O = / - s i n 0 sin M -4- cos 0 sin wj
= P
[26]
which is seen to be identical to their equation [7] if it is realized t h a t these authors use the symbol O * for the same effective vector which is denoted by OO in this article.
and so it is seen that the coefficient Bo,o(r) is the only one from the series expansion in Eq. [29] which shows up in the scattering equation. Equation [30] is equivalent to the usual expression for scattering b y density fluctuations (5, 7, 9, 10) where B0,0(r) is It is stressed again that this conclusion only holds for samples without macroscopic orientation: isotropic samples[ Journal of Colloid and fnterface Science, Vol. 39, No. 3, J u n e 1972
590
VAN AARTSEN
called a correlation function usually denoted by ~/(r). Now by combining Eq. [30] with Eq. [19] the general scattering formula can also be written as
In the third place, the relation used by Stein et al. (7) and Keyzers et al. (3, 11) to obtain the intensity resulting from density correlations (id = W, for y = z = O) also breaks down and now
I = 4rrC202Eo 2 I~2(x -- 2y + z)
4
W = il - ~ i4 = 4~rC2Eo 2
+ (E.o) 2 ~
+ 5 1 ~2( x -
"y(r)Po°r2 d~
2y-t-z) + 1~2(4g +
5Z)}
• ~
[31]
"y(r)Po°r 2 dr + 3
[33] + {(h-O) 2 + (h.E)2}~2(3y -- 5z)
• {(4y + 5z) - 4 cos2 0/2(3y - 5z)} 1
-- 2(E.O)(h.E)(h.O)~2(y + 10z)
= id + (4~/3)c2Eo2~ 2
+ 35(h. E)2(h • O)2~2z|
• {(4y + 5z) -- 4 cos 2 0 / 2 ( 3 y -- 5z) }.
where x =
~-~ Ho°Po°r 2 dr
[32a]
y = fo
~ [H20P2 0 + 3H2 2P2 2]r 2 dr
[32b]
3-5 [H4°P4° + 20H42P42
[32c]
z =
fo
+ 35H4~P44]r 2 dr
From this formulation it can be seen that for random orientation fluctuations (where y = z = 0) Eq. [31] properly reduces to the equations given by Stein et al. in various publications (7, 9, 22). Nevertheless Eq. [31] is still more general, because it incorporates every polarizer and analyzer setting and is not restricted to the usual scattering components V~, H~, Vh, and H h . For the general case, where y # 0 and z # 0, it can be shown that the corrected intensities V~ = il, H , = i4, V~ = i5 and Hh = i2 obey the following relations (see also Keyzers et al. (3), who use the same symbols). In the first place, Q = i~ still holds• Secondly, the relation/2 = il c o s 2 0 -[- Q sin 2 0 does not hold any more and one finds T = i2--
ilcos 2 0 -
i~sin 20
= 4~rC2Eo2~2 cos2 0/2[(1 + cos 2 0)
[32] • (3y -- 5z) + 35 cos2(O/2)z -- 2 cos O(y + 10z)]. Journal of Colloidand InterfaceScience,Vol.
39, No. 3, June 1972
Our final remark concerns the way in which Keyzers et al. (3) treated their data and tried to separate them into a random orientation part and a perfect spherulitic part. Using exactly the same symbols and assuming t h a t the same experimental details hold, it can be shown t h a t the parameters given in their Eqs. [37] and [38] m a y be written as KA 2 = 4~rC2Eo2~2135z
+ 4 sin 2 0 / 2 ( 3 y -- 5z)] dB
= 4~rC2Eo2~2 cos 0[cos O( 3y - 5z) -- ( y +
10z)]
[34] [35]
So it turns out t h a t both these parameters are functions of higher-order moments of the general correlation function as defined by Eq. [17]. The part used to calculate the spherulite dimensions (KA2) is mainly determined by z. It is now possible to use values of ii, i2, i3 and i4 (where i3 is the corrected intensity measured with crossed polarizer and analyzer whose transmittance directions are at 45 ° to the vertical) and calculate from them x, y, z and id aS a function of the scattering parameter h = (4r/~) sin (0/2). From these results the values of ~ and ~2 as well as the functions ~(r) and Ao.o(r) can be obtained by Fourier transformations (10, 11). If it is also assumed that the gen-
NONII.ANDOM ORIENTATION CORRELATIONS erM correlation function of Eq. [17] depends on fl only, it may even be possible to obtain Ao,2(r) and Ao,4(r) from Eqs. [32b] and [32c], where now H2~ --- H , ~ = H~~ = 0, via a Hankel transformation. All this can only be achieved by assuming t h a t cross correlations do not occur. In the Appendix it is shown how the presence of such correlations can be formally accounted for, Mthough it is not possible to evaluate experimental measurements in such eases. APPENDIX Retaining the last term of Eq. [16a] and carrying out the cMeulations required by Eq. [18] one obtains an additional term in Eq. [19] with the following structure Ic .... = 47rC'E'O 2
• { ( h O ) ( h . ~ ) ( E . O) -
(~. O)'
.
591
Even this assumption does not lead to any marked improvement. The only way in which an evaluation of the experimental results is rendered possible is to assume that ~i6]F = \ ~ ]
G
[A31
This assumption then leads to I¢ .... = 4~'C'Eo20 '
"I (h' O ) ( h ' E ) ( E ' O ) _ 31 (E.O). 1 [A41 "\9]
l-, I12 r dr
Only with the additional assumption 3,@) = Ao,o(r) can Eq. [19] be solved for the four unknown quantities x ( h ) , y(h), z(h) and
(n'/~'). In view of the assumptions that were
[7070" .
1 (3 cos' ~ -- 1) sin ~ d3, d¢~| r ' dr l 2 I-" wf'Tr
",'kl o
+
,
[~&l]
+-,',',
" 2 sin'3 fi cos 23, d3, dJ
Ir) r' d
Generally, (ai~ + a ; ~ F ) ~Jll be a function of r, fl and 3' which is different from G, and the intensity contribution caused by cross correlation effects cannot be separated from the other terms of Eq. [19] A number of assumptions about the functional form of (asSi + a~SjF) can be made, one of the most drastic being that there is no -/-dependence and that ajS~ depends on the distance r only. i n that ease
necessary to reach this possibility one realizes that in fact a model has been obtained, in which n and ~ vary in the same way. This result can also be achieved by starting from an explicitly defined model, such as given by the author for a random collection of cylinders of finite length and radius (17). This was discussed in some detail at a recent symposium (24). REFERENCES 1. Described by Keijzers et al. (3) and commercially available from Bleeker N.V., Zeist, The Netherlands or from IMASS, Accord (Hingham) MA. 02018. 2. K~IjZERS,A. E. M., Light Scattering by Crystalline Polystyrene and Polypropylene. Thesis, Delft, 1967. 3. KEIJZERS, A. E. M,, VAN AARTSEN, J. 5., AND PRINS, W . , J . A m . Chem. Soc. 90,
3107 (1968). 4. GOUDA.,J. H., &ND PRINS, W., 3r, Polymer Sei.
i ..... = 4~rC2Eo'O 2
A-2, 8, 2029 (1970).
- [ ( h . O)(h.E)(E. O)
_ ~1 (E. 0). l
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" 2-1i
P2o
~i~3
• _1 (3 cos' ¢~ -- 1) sin ~ d~/ r ' dr 2 _J
1450 (1955). 7. STEIN, R. S., A~) WILSON, P. R., Or. Appl. Phys. 33, 1914 (1962). 8. STEIN, R. S., AND HOTTA, W., Or. Appl. Phys. 35, 2237 (1964). Journal of Colloid and Interface Science, VoL 39, No. 3, June 1972
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9. STEIN, R. S., in, Proceedings of the International Conference on Electromag. Scattering" (M. Kerker, ed.), p. 430. Pergamon Press, New York, 1963. 10. STEIN, R. S., KEANE, J. J., NORRIS, F. H., BETTELHEIM, F. A., AND WILSON, P. g . , Ann. N. Y. Acad. Sci. 83, 37 (1959). 11. KEIJZERS, A. E. M., VAN AARTSEN, J. J., AND PnlNS, W., J. Appl. Phys. 36, 2874 (1956). 12. STEIN, R. S., WILSON, P. R., AN9 STIDHAM, S. N., J. Appl. Phys. 34, 46 (1963). 13. COALSON, R. L., MA~CHESSAULT, 1%. H., AND PETEaL1N, A., J. Polymer Sci. C13, 123 (1966). 14. STEIN, R. S., AND RHOnES, M. B., J. Appl. Phys. 31, 1873 (1960). 15. R~ZODES, M. B., AND STEIN, ]~. S., ASTM Special Techn. Pub. No. 348 (1963). 16. FRENKEL, S. YA., VOLI-:OV, W. J., BARANOV, V. G., AND SHALTYKO, L. G., Polym. Sci. USSR 7,942 (1965). 17. VAN AARTSEN, J. J., Eur. Polym. J. 6, 1095 (1970).
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18. DONKERSLOOT, M. C. A., GOUDA, J. H., VAN AARTSEN, J. J., AND PRIES, W., Rec. Tray. Chim. 86,321 (1967). 19. GOUDA, J. H., Structural Characterization of noncrystalline hydrogels. Thesis, Delft 1969. 20. GASPARJAN,K. A., GOLOUBEE, Ja., BARANOV, V. G., AND FRENE:EL, S. ya., Polym. Sci. USSR 10, 96 (1968). 21. PICOT, C., AND STEIN, R. S., IUPAC Int. Symp. Leiden 1970, Paper IV-15. 22. STEIN, R. S., ERt/ARDT, P., CLOUGH, S., AND VAN AARTSEN, J. J., in, "Proceedings of The Second International Conference on Electromagnetic Scattering" (1~. L. Rowel and R. S. Stein, eds.), p. 339. Gordon & Breach, New York, 1967. 23. VAN AAaTSEN, J. J., in, " Polymer Networks" (A. J. Chompff and S. Newman eds.) p. 307. Plenum Press, New York, 1971. 24. VAN AARTSEN, J. Y., AND STEIN, R. S., J. Polymer Sci. A-2, 9,295 (1971). 25. STEIN, •. S., AND KEANE, J. J., J. Polymer Sci. 17, 21 (1955).