J . Phys. Chem. 1985,89. 2671-2675 the right-hand Cartesian coordinate system. Ti is the orthogonal matrix that transfofms a vector 3 i n reference frame i 1 to its representation I/' in frame i
+
3/ = T,.p
of the micelle (or polyion). Due to the cylindrical symmetry, all 4i values will have the same probability. Thus
P(ei,...te/'l;4i,...,4bl) = P(8i),..P(8j-l)P(~i)...P(4j-l)(A71
(-41)
p(ei) =
with 8, and 4i defined as in Figure 1. The matrix Ti is COS
[6
sin Oi
ei
2671
0
Ti = sin B i cos @i - cos Oi cos @i sin @i sin 0 i sin @i - cos 0 i sin @i - cos @ i
]
... = p(ej-l)
P ( k ) = (.. = ~(4j-1)= (-42)
The step vector is in its local frame given by the column vector
('48) 1
('49)
The statistical independence of different steps ensures that the average product of the matrices is equal to the product of the average of the matrix for each step, which furthermore is equal for all steps. j- 1
(T,...Tj-l)= n ( T k ) = ((T)Y-' k=i
= lZ(Ti...Tj-l)l,l
(A 10)
('44)
where superscript T denotes transpose and subscript 1,l the 1,l element of the matrix. The averages to be evaluated are
where, formally
1 ...S
27
7
(Ti...TFl)=
0
0
,...
d4i...d4bl d8 dej-, sin Bi...sin 8j-l X
p ( 8,...,8j;4j,...,4j-i)Tj...Ti_j(A6)
The probability density function for the exit direction (e,&) at step i is assumed to be independent of the previous steps and the same at each step. On a particular step the exit function is determined by the diffusion of the ion over the spherical surface
For 0 < i < j C n the double sum in (A13) can be simplified by collecting terms with equal power k = j - i of cos 8, giving finally P I
~ ( 7 ~ =2 P~ C) (-~k)(cos e ) k
i
