Cluster Size

At low concentrations we are interested in the cluster size distribu­ tion as the existence of a maximum and a minimum in the distribution has been suggested as an effective indicator of a micellar region.

At higher concentrations, such as in the bilayer region, the cluster size distribution is very narrow. The mean cluster size is of more interest here. Plotting contours of the ratio of the mean cluster size to the total number of amphiphiles, N^, on a temperature-composition diagram has helped determine the phase boundary of a lamellar region.

The number-average cluster size, n, and the root-mean-square devia-

80

tion, a, of the cluster size distribution are given by 00 1 i=l n (4.13) 00 “ "1 i=l and 00 i = l o2 _ (4.14) 00

where n^ is the total number of clusters containing i amphiphiles and a is a measure of the width of the cluster size distribution.

Assuming each observed cluster is an independent sample taken from some underlying cluster size distribution, the standard error, 6^, in n is given by

Following Care (1987b), the uncertainty of the Monte Carlo results may be estimated by comparing the standard error, 6j, with the error, 62* obtained by dividing the data collecting MC steps into p blocks of q steps such that pq is the total number of MC steps at each tempera­ ture. Typically p=500 and q=50. The number average cluster size after each q steps is and an alternative estimate of the number-average cluster size is given by n ’, where

o

(4.15)

2

CD

(4.16) P

and the error, 62j in n ’ is

00 ₂

2 (nj - n ’ )2

62 = j=l (4.17)

p(p-l)

The values of the errors in n and n ’ should be similar provided that

to the persistence of clusters through several configurations.

The size of the choice of the subblock, p, is discussed by Bishop & Frinks (1987). They conclude statistical efficiency cannot be calcu­ lated accurately for either small or large values of p. The subblocks are correlated when p is too small and when p is too large there are too few subblocks for accurate analysis.

4.3.10 Principal Moments Of Inertia

The size and shape of aggregates are often described in terms of the principal components of the moments of the inertia tensor I. The inertia tensor is found to have diagonal components

the n'1 are independent. Failure of this condition indicates that there is strong correlation between successive n*^ and corresponds physically

00

i = l <* = x,y, z (4.18)

the subscript i referring to the mass m^. The diagonal components are given by the products of inertia

CD

■<*13 = ~.2 m iociR i = 1 (3oc’ * = x >y>z

i = l 13 = x,y,z (4.19)

oc ^ (3

For any point in a rigid body, a set of Cartesian axes exist for which the inertia tensor will be diagonal. These axes are the principal axes and the corresponding diagonal elements are the principal moments of inertia, Ix , I and I z. The principal moments of inertia are the eigenvalues of I, and are found by solving

*xx ” *x

*xy

*xy *yy ” *y

*xz *yz

The summations Ixx, I , Ixz etc. are done in the cluster counting routine, (see section 4.3.8). In cases where the cluster completely spans the lattice in one or more directions, periodic boundary condi­ tions mean there is no ’edge’ to the cluster. The cluster perimeter is determined by the order in which sites are found by the ’ant in a labyrinth’ algorithm.

We are interested in distinguishing between spheres, cylinders and planes and consider the principal moments of inertia for a sphere, a thin rod and a circular disk

xz yz *zz *z

= 0 (4.20)

Ix = I y = T z = (2/5)MR2 : sphere

Ix = I = 0, Iz = (1/12)ML2 : rod (4.21) lx = Iy = (1/4)MR2 , Iz = (1/2)MR2 : disk

Measuring the principal moments of inertia from the simulation in­ volves averaging over many clusters of different sizes and orienta­ tions. We therefore sort the moments of each cluster into ascending order and normalise them to sum to unity. Thus

!L + + !S = h * * Js <4 -2 2 >

where IL , IM , Ig are the largest, middle and smallest principal mo­ ments of inertia of a cluster in the ensemble.

The mean principal moments, T^, 1^ and Ig, are weighted by the cluster size, i 00 2 il i = l Oci oc = L,M,S (4.23) GO 2 I i = l Oci

Similarly to 4.14 and 4.15 the root-mean-square deviation and standard error are given by

o = 2 i2I i=l oci 00 2 n: i=l (4.24)

84

and

a 2

00 _(4.25)

For the normalised principal moments of inertia, equations (4.21) become

The ratio of the smallest to the largest moment (sphere=l, rod=0, disk=0.5) are plotted on contour diagrams to show change in structure with respect to temperature and concentration. These ratios were used as a measure of aggregate shape fluctuation in the molecular dynamics simulation of micelles of Woods et al (1986).