LOGARITHMIC DISTRIBUTION FUNCTIONS FOR COLLOIDAL PARTICLES
interesting to note that if a structure in which the sodium is bridged to the iron in FeC14- through interaction with two chlorines is considered, the average energy for the six bonds in the same as the average of the eight Fe-Cl bonds in Fe2Clfi. The standard heat of formation of NaFeC14(g) at 298°K.) calculated using -60 f 3 kcal. as AN’
3093
for FeC13(g)5t12and -98.2 kcal. for NaC1,’G is - 154 4 kcal. mole-’.
f
Acknowledgment. This work was supported in part by grants from the National Science Foundation and from the U. S. Army Research Office (Durham), which we acknowledge with thanks.
Logarithmic Distribution Functions for Colloidal Particles’&
by W. F. Espenscheid,IbM. Kerker, and E. MatijeviC Chemistry Department, Clarkson College of Technology, Potsdam, N e w York
(Received J u l y I S , 1964)
The distribution function used in earlier light-scattering studies had been erroneously termed a logarithmic normal distribution when, in fact, this was a new distribution function with different parameters and properties. This new function, called a zeroth-order logarithmic distribution, is described herein. Since one of the parameters is the modal value of the variable, this new function permits exploration of the effect of changing the breadth of the distribution while keeping the mode invariant. A generalized logarithmic function is described which permits selection of other moments of the distribution as the size parameter appearing explicitly in the distribution function.
In a recent series of papers from this laboratory, the particle size distribution of a variety of colloids has been determined by comparison of the polarization of the scattered light with theoretical calculations. These calculations assumed that the particle sizes could be represented by a two-parameter distribution function which we had called a logarithmic normal distribution. 2-4 For each particular system, a “solution” consisted in determining the values of the distribution parameters which gave theoretical light-scattering functions in agreement with the experimental data. This approach has also been followed1 by Heller and his collaborators with different experiments and a different distribution function than utilized lby us.6 It has become apparent to us that the distribution function which we had used is actually a new distribution function, rather than the logarithmic normal distribution, and that the parameters in question stand
for physical quantities different from those stated. We would like to clarify the matter in this paper by describing in detail some of the properties of this new distribution and by comparing it with the logarithmic normal distribution. We will then write the expression for a generalized logarithmic distribution function which may be reduced to the logarithmic normal distribution or to the one used by us. For reasons which (1) (a) Supported in part by research grant AP-0048 from the Division of Air Pollution of the Public Health Service: (b) SoconyMobil Fellow: part of a Ph.D. Thesis by W. F. Espenscheid. (2) (a) 34. Kerker, E . Daby, G. L.Cohen, J. P. Kratohvil, and E . Matijevib, J. Phys. Chem., 67, 2105 (1963); (b) M. Kerker, E. Matijevi6, W. F. Espenscheid, W. A. Farone, and S.Kitani, J. Colloid Sci., 19, 213 (1964). (3) E. Matijevi’, S. Kitani, and M. Kerker, ibid., 19, 223 (1964). (4) W. F. Espenscheid, E. Matijevii., and M. Kerker, J . Phys. Chem., 68, 2831 (1964). (5) W. Heller and M. L. Wallach, ibid., 67, 2577 (1963); H. L. Bhatnagor and W. Heller, J . Chem. Phys., 40, 480 (1964).
Volume 68, Number 1 1
November, 1964
W. F. ESPENSCHEID, M. KERKER,AND E. MATIJEVI~
3094
will become obvious, we call this a zeroth-order logarithm i c distribution. Other workers in colloid science may find this distribution advantageous for their purposes, so we are hopeful that, in addition to correcting misstatements in our earlier work, this paper may also serve a more general purpose. A distribution or frequency function, p(r), will be defined by
P(r)
= Jvri-Ar
p(r)dr
where P ( r ) gives the fraction of the population with values of the parameter r between r and r Ar. The function is normalized if the value of the integral over all possible values of r is unity. Undoubtedly, the best known distribution function is the normal distribution
Thus, In rm is the mean value of In r and rm is, in this case, both the median and the geometric mean value of r r, = rm = ( r 1 ’ r z . r 3 . . . . r , )l/n (8)
The second parameter of this distribution, ug, is the standard deviation of In r. Its antilogarithm is the geometric mean standard deviation. The mode, the median, and the mean are related to the geometric mean by
+
where the two parameters, ii and U, are the mean value and the standard deviation of r , defined by T =
$-”, rp(r)dr
In
rM =
In r, -
In rh
=
In rg
In r
=
In r,
(9)
ug2
(10)
+ ug2/2
(11)
Figure 1 shows the frequency curves for three logarithmic normal distributions for which rm = r, = 3.0 and u, = 0.1, 0.2, 0.3, and 0.5. All three curves have the same values of rm. The skewness of the distribution depends upon ug, and, for sufficiently small values of ug,there is so little skewness that the frequency curve
(3) 2.0
and =
UZ
$--
( r - P)p(r)dr
(4)
Here, the exponential factor in the distribution describes a Gaussian curve, and the pre-exponential factor normalizes the expression. Because of the symmetry of this distribution, two other parameters of interest, the modal value, r h f , and the median, rm,are identical with the mean. The mode is the value of r at the maximum frequency; the median is the value below which 50y0 of the population falls. Obviously, a normal distribution cannot represent a distribution of particle sizes because it admits negative values of r. In addition, unlike the symmetrical normal distribution, naturally occurring populations are frequently positively skewed, A satisfactory representative of many such populations is the logarithmic normal distribution p(r)
=
1
-(In r - In r,)2
v‘27rugr exp
2u,2
sorn
and The Journal of Physical Chemistry
In rp(r)dr
P(r) I.o
0.5
(5)
In this distribution, it is In r rather than r which is normally distributed, so that In rm =
1.5
C
I
r
Figure 1. Distribution curves for the logarithmic normal distribution, eq. 5, with T~ = T~ = 3.0 and ug = 0.1, 0.2, 0.3, and 0.5. I n the inset, the frequency is plotted us. In T for one case.
3095
LOGARITHMIC DISTRIBUTION FUNCTIONS FOR COLLOIDAL PARTICLES
can be closely approximated by a normal distribution. As a general rule for practical work, both r and log T can be considered to be normally distributed as long as ug < 0.14.6 Another feature of this distribution is the movement of the modal value of r toward smaller values as ug increased for a fixed value of rm. Although we had termed the distribution used in our earlier cited work, a logarithmic normal distribution, in actual fact, we had used the frequency function [ p ( r ) ]=
-1
4% TU0
exp
-(In r - In r M ) 2 (12) 2u02
Neehan and Beattie, who also claimed to have utilized a logarithmic normal distribution for analysis of particle size by light scattering, have made a similar error.’ The above expression will lead to the same value of p ( r ) as a logarithmic normal distribution only if it is integrated with respect to In r , rather than with respect to r 8s indicated by eq. 1. As it now stands, the expression given by eq. 12 must be normalized in order to convert it into a true frequency function so that
sorn
exp- (In r - In rM)2/(2a02) p(r) = exp-((In r - In rM)2/(2q02)dr exp-((In r - In r y ) 2 / ( 2 a 0 2 ) (13) dGaorM exp(d/2) The evaluation of thle integral in the denominator io described in the Appendix. This new distribution will be called a zeroth-order logarithmic distribution. It i s defined by two parameters, TM, which is the modal value of r , and (TO, which is ti measure of the width and the skewness of the distribution, This latter parameter is related to the standard deviation in a manner which will be described below. It will be termed the zeroth-order logarithmic standard deviation. The frequency function for the zeroth-order logarithmic distribution is plotted in Fig. 2 for TM = 3.0 and u0 = 0.1, 0.2, 0.3, and 0.5. Perhaps the most interesting feature here is that the modal value of r isJixed, while the width of the distribution changes. It offers the interesting possibility of exploring the effect of changing the breadth of the distribution while keeping the mode invariant. The relation between the modal value of r and the mean value can be shown to be (see Appendix)
+
In I =: In TM 3u02/2 (14) The standard deviation is given by (see Appendix) 4uoz - e30b2]’/z u = rM[e
2.0
1.5
in r
I\
Pk) 1.0
0.5
0
A I
r Figure 2. Distribution curves for the zeroth-order logarithmic distribution, eq. 13, with rM = 3.0 and uo = 0.1, 0.2, 0.3, and 0.5. I n the inset, the frequency is plotted us. In r for one case.
so that for u0
