and then inoculated on aluminum foil, they failed to be sterilized with ethylene oxide. Likewise, Bacillus subtilis var. niger spores suspended in… [632154]

436
and then inoculated on aluminum foil, they failed to
be sterilized with ethylene oxide. Likewise, Bacillus
subtilis var. niger spores suspended in peptone and
dried on aluminum foil with a contamination level of
only 10 spores/carrier were very difficult to sterilize
with ethylene oxide. This would be similar to the
protective effect found by previous investigators (6),
i.e., occlusion of the spores in crystals.
It is also possible to produce artificial conditions
which would seem to make the organisms extremely
susceptible to ethylene oxide. Spores suspended in
2% glycerin and then dried on polystyrene were less
resistant to ethylene oxide than spores suspended in
distilled water (7). Bacillus subtilis var. niger
spores were suspended in 10% glycerin, inoculated
on chromatography paper strips, and dried at room
temperature. After drying at room conditions, the
glycerin will contain a certain amount of water de-
pending on the room relative humidity, for it will
gain or lose moisture depending on its surroundings.
At 130"F., 1,200 mg./L. ethylene oxide, these strips
showed the same resistance (3 min. for inactivation)
at 40% RH as at 10% RH even though the strips at
40% RH would contain a greater amount of moisture
after equilibration. A similar situation would indi-
cate to the unwary that no optimum RH was re-
quired. However, clean spores inoculated from
distilled water required 1 hr. at 10% RH and 3 min.
at 40% RH for inactivation.
SUMMARY
The evaluation of the efficiency of an ethylene
oxide process depends a great deal on the type of
bacteriological control used. It is therefore recom-
mended that an evaluation be made of the type of
load to be sterilized, so that the bacteriological con-
trols chosen will accurately monitor the process. If Journal of Pharmaceutical Sciences
one uses unwashed spores for preparation of sterility
monitors when sterilizing clean materials, then he
will likely obtain erratic indications of nonsterility
rather than sterility. If, however, one does attempt
to sterilize dirty materials, contaminated with dirt,
blood, feces, ctc., he must realize that the process at
its best will sterilize occasionally and should only be
considered a decontamination procedure. The only
dependable method of testing a sterilization process
is to determine if it kills living microorganisms. The
microbial control chosen should simulate actual con-
ditions of the materials being sterilized.
REFERENCES
(1) Smith, N. R., Gordon, R. B., and Clark, F. E., U. S.
(2) Doyle, J. E.: and Erst, R. R., Appl. Microbiol., 15, Depl. Agy. Monogr. No. 16 (1952).
79firiafi7) .-"\–".,.
(3) Ernst, R. R., and Shull, J. J., ibid., 10, 337(1962).
(4) Phillips, C. R., "Recent Developments in the Steriliza- tion of Sureical Materials." The Pharmaceutical Press.
London, Endand, 196l:pp. 57-761-~
R. K., and Phillips, C. R., Appl. dlicrobiol., 12, 496(1964).
14 ~7~riafii) (5) Gilbert, G. L., Gembill, V. M., Spiner, D. R., Hoffman,
(6) Royce, A., and Bowler, C., J. Pharm. Pharmacol., -.., -. – \-""-,.
(7) Opfell. J. P.. Hohmann, J. P., and Latham, A. B..
J. Am. Pharm. Assoc., Sci. Ed., 48, 617(1959).
Keyphrases
Ethylene oxide sterilization
B. subtilis spores-ethylene oxide effect
Staph. aureus-ethylene oxide resistant
Sterilization, ethylene oxide-variables af-
fecting
Preferential Aggregation and Coalescence in
Heterodispersed Systems
By NORMAN F. H. HO and W. I. HIGUCHI
A theoretical study of preferential coalescence and aggregation of small particles in
heterodispersed systems has been carried out where moderate electrical barriers
exist between the particles. Equations based on the concepts?of Derjaguin, Verwey,
and Overbeek were employed. Computation over a wide range of conditions has
shown that small particles may aggregate (or coalesce) with themselves or with
larger particles at rates that are 10 to 50 orders of magnitude faster than particles 10
times larger. These findings may explain (a) the relatively narrow particle size
distributions observed in certain emulsions and flocculated suspensions and (6) the
limited flocculation and coalescence behavior observed in certain instances.
HERE ARE many situations involving sus- T pensions and emulsions where with time the
particles or droplets of the dispersed phase simul-
taneously increase in size and narrow in their
relative size distributions, and then later became
Received April 14 1967 from the College of Pharmacy,
University of Michigan. An'n Arbor, MI 48104
Accepted for publication October 3, 1967.
This investigation was supported by fellowship 5-FI-GM-
24,039 from the Institute of General Medical Sciences,
National Institutes of Health, U. S. Public Health Service,
Bethesda. Md. quite stable kinetically. If the dispersed phase
is soluble (or miscible) enough in the solvent,
then the phenomenon may be accounted for by
molecular diffusion (1) or Ostwald ripening.
However, there are many examples in the litera-
ture (2-5) where the changes appear to occur
primarily through particle-particle aggregation
or droplet-droplet coalescence.
The authors have recently observed that urea-

Vol. 57, No. 3, March 1968
denatured ovalbumin aggregates precipitated in
buffered saline near the isoelectric point are
often made up of rather uniform particles of
around 1 p diameter (2). Sherman (3) found
that emulsions pass through an initial phase of
rapid coalescence followed by a second period of
slow coalescence, and then the emulsions became
relatively stable. All of his emulsions showed
a relatively narrow size distribution in the later
stages. Wiley (4) and Gillespie (5) have also
studied the problem of limited coalescence when
droplets in oil-in-water emulsions are stabilized by
finely divided particles.
Although the effects of particle size on the
stability of colloids have been recognized, quan-
titation has been limited to the case involving the
potential energy barriers for particles of equal
sizes (G) or for particles of different sizes in which
the mean size is used (3). While the equations
for the repulsive potential energy between two
unequal spheres have already been derived, an
expression for the attractive potential energy
between two unequal spheres with retardation
factors has not been considered. The present
authors are not aware of quantitative com-
putations involving preferential collision of
differing particle sizes in the presence of a repul-
sive force. It is the purpose of this paper to
examine the theory as it relates to preferential
and limited aggregation (or coalescence). As
will be seen, some of the theoretical conclusions
consistently describe many of the findings of
investigators in the field. 437
particles, and W is the stability factor, which will be
later discussed in detail. We may also write:
THEORY
Rate of Preferential Aggregation or Coalescence-
Let us assume that there are equal concentrations
of spherical particles of three sizes, that is, spheres
.TI, SZ, and Sa of radii a,, UZ, and a3, respectively,
where a1 < a2 < u3. The dispersion is in an electro-
lyte solution and the particles are allowed to aggre-
gate (coalesce) at 25". The following question is
asked: what are the probabilities of S, colliding with
S,, SZ, or Ss, of SZ with SZ or Sa, and of S3 with S3?
By means of theory let us determine the initial
rates of aggregation or coalescence for the various
size pairs.
It will be assumed following the Derjaguin-
Verwey-Overbeek concepts, the interparticle inter-
action results from electrical repulsion and the Lon-
don-van der Waals attraction. According to the
theories of Smoluchowski and Fuchs (7), the rate
of collision between spherical particles of sizes i
and j is given by:
where G is the initial rate, D is the Stokes-Einstein
diffusion coefficient, R is the distance between
centers of the spheres, N is the concentration of RT 1
6~p (ai aj) DijRij = – – + – (ai + aj) (Eq. 2)
where k is the Boltzman constant, T is the absolute
temperature, and p is the viscosity of the suspending
medium. Then the initial rate is:
The factor Wij accounts for the energy barrier to
aggregation and is given by:
where s = R/d and d = (ai + aj)/2. The W,j
usually ranges from 1 5 W < and takes into
account the double-layer potential, the concentra-
tion of electrolytes, London-van der Waal disper-
sion forces, and the particle sizes of the two spheres.
Potential Energy of Interaction-In Eq. 4 the
V~(i,j) is the total potential energy of interaction
between two particles and is given by the sum of the
repulsive energy, V~(i,j), and the attraction energy,
VA (%,j ) ; thus :
vT(i,j) = vR(i,j) f VA(i,j) (Eq. 5)
The repulsive energy between interacting spheres of
sizes i and j can be calculated by the approximate
expression derived by Hogg, Healy, and Fuerstenau
(8):
In [l + exp. (-KH)] (Eq. 6) eaiaj\l/02
vR(i.j) = ___ (ai + ail
where e is the dielectric constant of the medium,
J.0 is the surface potential, K is the reciprocal of the
Debye-Huckel thickness of the double layer, and H
is the shortest distance between the surfaces of the
spheres. The equation is valid for Kai and Kuj >> 1.
The attraction energy of particles is more difficult
to determine because the force between particles is
considerably reduced by the retardation between
atoms. Since the retardation effect is a function of
distance, a single expression for the attraction
energy is not available. When ai. aj >> Hand H 5
150" A, the attraction interaction energy can be
calculated by the expression:
0%. 7)
where A is the Hamaker constant, and Xis the charac-
teristic wavelength of the atoms.
For H 2 150" A,
VA(IA(i3i) =
Aa,aj -2.45X +L- 2 17X2 K) [m 360?r2H3 1680n3H4
(Eq. 8)
One should refer to the Appendix for the discus-
sion of the retardation factor and the derivation of
Eqs. 7 and 8.
CALCULATIONS
Computations employing Eqs. 3-8 were carried out
for a wide range of conditions with the aid of the
IBM 7090 digital computer. $0 values from 0 to

Journal of Pharmaceutical Sciences
-10kt Setondmry Minimum
–
!
100 mv. and various radius combinations were
taken at different K. Because the Hamaker con-
stant is estimated (9, 10) to be somewhere in the
neighborhood of 1 to 5 X 10-ls erg for most organic
substances in water, calculations were made using
the two values, 1 X and 5 X The
characteristic wavelength, A, was taken to be lows
cm. Some plots of TIT versus H are shown in Figs.
Table I presents-some of the significant results.
Calculated values of V‘T~~~., the maximum in the VT
versus H curves, are shown. The rate constant, G’,
is defined as: 1-3.
DISCUSSION
Where K is large, i.e., K >” 0.4 to 2 X lo’, or
when do N 0, it is seen (Table I) that, for essentially
all particle sizes in the range of 0.1 to 5.0 p radius,
the rates of aggregation approach the Smoluchowski
rates. Thus, in 1% NaCl solutions, maximum
rates may be expected. Figure 4 shows the maxi-
mum limiting rate constants under these conditions
for which W,j equals unity.l The well-known pref-
erential aggregation relationship due to Muller
1 Actually the limiting Wii values may be somewhat less
than unity when all repulsive barriers are removed because
VT will be negative (see Eq. 4). Fig. l-In$uence of #ar-
ticle sizes on the total po-
tential energy of interac-
tion. A = 5 X erg,
$0 = 25 mu., and K =
2 x loE cm.-1.
(11) can be seen here. However, under such
“rapid” aggregation conditions, the rate constant
for a 0.1 p-1.0 p pair is only about three times
greater than that for particles of the same size.
Let us now direct our attention to the region of
kinetic stability where the rates may be regarded
as neither very rapid nor very slow. In this inter-
mediate stability region we note a dramatic change
in importance of particle size. This can be seen
in Table I and in Fig. 5. In Fig. 5 it can be seen
how electrolyte concentration may markedly in-
crease the preference for the aggregation (or coales-
cence) of small particles with each other or with
large particles. Thus, when K = 2 to 4 X 10’
for rL0 IU_ 25 mv., it can be seen (Table I or Fig. 5)
that the rate of aggregation (or coalescence) of
0.1-p particles with itself or larger particles may be
10 to 30 orders of magnitude greater than that for
two 0.5-p particles. Comparing Fig. 5 with Fig. 6,
we note that the rates for various particle sizes are
highly sensitive to the Hamaker constant taken as
1 X 10-13 and 5 X erg.
The above conclusions depend upon the assump-
tion that other contributions to particleparticle
repulsion are absent. Therefore, in the case of
emulsion coalescence, strongly adsorbed barrier-
forming interfacial films are assumed to be absent.
In practice one would expect that when surface
coverage of ionic emulsifiers is low, say about 5 to
10% of maximum coverage, the repulsion between
two oil droplets in an aqueous medium would be

VoZ. 57, No. 3, March 1965
lOkt 439
– 40kt –
30kt 1 r OI = 0.1
a2= 0.5 P
50 100 150 0 50 100 150
INTERPARTICLE DISTANCE, a.
Fig. 2-InjZuence of electrolyte concentratwn on the total potential energy of interaction for two particle sizes.
A = 1 X erg and J.,, = 26 mv.
Q, =25
~
200 400 600 200 400 600
Fig. 3-Influence of surface potential on the total potential energy of interaction for two particle sizes. A =
primarily electrical. These findings are consistent
with the observations (3) that there is frequently
an initial period of rapid coalescence with freshly
prepared emulsions. Usually this is explained on
the basis of the lack of sufficient stabilizer to pro-
vide a complete monolayer for the droplets. How-
ever, this explanation alone does not account for the
relatively narrow size distributions that often result.
The absence of small particles indicates that they
preferentially disappear rapidly.
It is proposed that in a freshly prepared emulsion 5 X erg, +e = 26 mu., and K = 2 X 106 cm.-'.
the shortage of surfactant prevents the formation of
complete interfacial adsorbed films. Consequently,
the rupture or displacement of the adsorbed mole-
cules may not he rate determining in coalescence.
Then if ILo is low (225 mv.) or if K is large, the
rapid and selective disappearance of small droplets
through coalescence would occur. The surviving
larger droplets coalesce at a much slower rate as
Wij is extremely sensitive to particle size. Thus, a
relatively narrow distribution would result if Brown-
ian motion alone is inducing the coalescence.

X
v) -x 9" 3.0
1
8 Li
c.
0% u3 N – Journal of Pharmaceutical Sciences
CI / I 5 d $1 2.0 /'
/"
/'/
/' in Z
0 O I //' ."I 1.0
12 4 6 8 10
RATIO OF PARTICLE SIZES, ai/aj
Fig. &Probability of rapid aggregation between
purticles of different sizes when the stability factor Wij
equals unity.
Gillespie (5) has discussed limited coalescence of
oil-in-water emulsions stabilized by finely divided
solids. He proposed a theory based on the rate of
accretion of the solid stabilizer by the oil droplets.
Furthermore, he found that there was an optimum
concentration range for an ionic surfactant additive
that produced the most uniform stable drops.
This concentration range of the ionic surfactant
appeared to produce a moderate or small electrical
repulsive barrier. These findings are consistent with
the theoretical results.
While the findings are only tentative, the pre-
cipitation of denatured proteins (2) near the iso-
electric point appears to involve the formation of
relatively uniform primary particles of around 1 p
diameter. Such narrow distributions are consistent
with the above theory that selective aggregation of
small particles with large ones take place. The
extreme situation where only single denatured
protein molecules deposit upon the larger particles
would lead to the formation of relatively compact
spherical particles of relatively narrow size ranges.
The authors believe that the results presented
provide a much better understanding of the role
of particle size in the mechanisms of aggregation and
coalescence. It provides a possible scientific basis
for tailor-making dispersion formulations of con-
trolled size and size distributions.
APPENDIX
Attractive Energy between Unequal Spheres and
the Effect of Retardation-Hamaker (12) was the
first to describe the potential energy of attraction
between unequal spheres, V~(s~/s~,, of similar con-
stitution. For al, a2 >> H, the limiting expression is:
where Hamaker's constant A = 7r2q2,3, and q is the
number of atoms per ~m.~, p is the London constant,
al, a2, and H are defined as before.
Schenkel and Kitchener (10) made a thorough
analysis of the retardation effect. Using the equa-
tions for f(p) as a check that are presented in several
references (10, 13), they derived empirical expres-
sions that gave the best fit and that would give an
analytic solution for interaction energies with modi-
fications for the retardation effect. Accordingly,

Val. 57, No. 3, March 1968
1 I I I I
1 2 4 6 8 10 20 40 60 80
CONCN.. rnrnoles/L. I0
Fig. 5-Rate constant (G') as a function of concentration of 1-1 electrolyte for various particle sizes. A =
1 X erg and $0 = 26 my.
– 60
b
B s
-40
-2J ::1':5 \\
c–– 0.11 0.5
0.110.1
I I I I I I I1 I
2 4 6 8 10 20 40 60 80 100
CONCN.. rnrnoles/L.
Fig. 6-Rate constant (G') as a function of concentration of 1-1 electrolyte for various particle sizes. A =
6 X erg and $0 = 26 mv.
for the range of 0.5 < p < m,
2.45 0 59 f(p) N – – 2L7 + A
P PZ P3 (Eq. 2a)
For comparatively large distances, Casimir and
Polder (14) showed that the London energy be-
tween atoms decays very rapidly due to retardation
effects. Consequently, the London interaction
energy of a pair of atoms of r distance apart, length of the atoms and B is the London constant.
Equation 4a may be rewritten:
In deriving the expressions for the retarded inter-
action energies for equal spheres, Schenkel and
Kitchener used Derjaguin's method (15, 16)
for calculating the interaction of spheres. It was
B pointed out that the algebra was simpler than for
(Eq' 3a) Hamaker's method (12) and that the results obtained
were the same by both methods, provided it is only
(Eq. 4a) the limiting expressions that are required. In a
similar manner, the method is applied to the inter-
action between unequal spheres. vA(London) = – r6
must be corrected so that
V'A(at./at.) = vA(London)f(P)
where p = 27rr/X, and X is the characteristic wave-

442 Journal of Pharmaceutical Sciences
Applying Eq. 4a to the geometry in Fig. 7, the
interaction energy between an atom and all of the
atoms in an infinitely large flat plate, V‘A(~~./~),
may be described by the superposition of their
retarded attractive energies, i.e.,
V’A(~~./~) = V”, + V”’A + V””A (Eq. 6~)
where
2rp4pdpdx
V”A = – 161 [(R + x)2 + p2]7/2 (Eq. 7a)
and + = 2.45Xp/2~; B = 2.17X2fi/(2~)z; and 2 =
0.59X3fl/(2~)3. Also, the Hamaker constant A =
rZq2p. It follows that the interaction between two
flat plates, V’A(~~~), is
V‘~(p/p) = qV’a(st./p)dR 0%- 10~)
SRm
If the surface of two spheres is thought to be built
up of parallel pairs of infinitesimally small rings
with radius It (8), the interaction of unequal spheres,
V‘A(S,IS,), is expressed by:
(Eq. lla)
The edge effects are neglected. Finally, the limit-
ing expression for the interaction energy for un-
equal spheres is:
V’A(Si/Sn) =
Aaiaz 2.451 2.17X2 0.59X3
-2 I] 60zH2 + 3m – m41
(Eq. l%)
(for H << al, az; 0.5 < PO < a).
= 2rH/X and X is usually taken as 10-6
cm., Eq. 12a should not be used for approximately
H < 150” A. It is noteworthy that the equation
reduces to the identical expression2 obtained by Since
2 See equation (I) in the appendix of Reference 10. Fig. 7-Geometry of the London-van der
Waul inleraction between an atom and an
infinitely large plate of thickness 6. The
plate is built up of injinitesimal rings
of diameter p, cross-section dpdx at a dis-
tance (R + x) from the atom (17).
Schenkel and Kitchener for interaction of equal
spheres, i.e., a, = at.
For the range below po = 0.5, the retardation
effect is still signiiicant until the interparticle dis-
tance is comparatively very small, in which case
Eq. la will apply. Based on extrapolation of the
attractive potential energy curve, an empirical
expression that represents the region 0 < #O < 2,
approximately H < 300” A, is given by Eq. 7.
REFERENCES
(1) Higuchi, W. I., and Misra, J., J. Pharm. Sci., 51,
459(1962).
(2) Ho, N. F. H., and Higuchi, W. I., ibid., 56, 248(1967). (3) Sherman, P., J. Phys. Chem., 67, 2531(1963).
(4) Wiley R. M. J. Collord Scr., 9 426(1954).
(5) Gillesbie, T., ’J. Phys. Chem., 6i, 1303(1958).
(6) Verwey, E. J. W., and Overbeek, J. T. G., “Theory
of the Stability of Lyophobic Collolds Elsevier Publishing
Co., Amsterdam, The Netherlands, 1948, p. 176.
(7) Kruyt H. R., “Colloid Science vol. I Elsevier Pub-
lishing Co., Amsterdam, The Netherlabds, 1955, chap. 7.
(8) Hogg R. Healy T. W. and Fuerstenau, D. W.,
Trans. Faraday sot., 62,’1638(1986).
(9) Srivastava, S. N., and Haydon. D. A., {bid., 60,
971(1964).
(lo) Schenkel, J. H., and Kitchener, J. A., ibid.. 56, 161
(1960).
(11) Kruyt H. R. “Colloid Science” vol. I Elsevier
Publishing Cd., Amsteidam, The Netherlands, 1955, p. 287.
(12) Hamaker, H. C., Physics,,? 1058(1937).
(13) Overbeek, J. T. G., in 6olloid Science,” vol. I,
Kruyt, H. R., ed., Elsevier Publishing Co., Amsterdam. The
Netherlands, 1952, p. 266.
(14) Casimir. H. B. G., and Polder, D., Phys. Rer., 73,
360(1948).
(15) Derjaguin B. Kolloid Z. 69 155(1934).
(16) Verwey, E. J.’W., and dverbeek, J. T. G. “Theory
of the Stability of Lyophobic Colloids,” Elsevier, Publishing
Co., Amsterdam, The Netherlands, 1948, p. 137.
(17) Ibid., p. 102.
Keyph rases
Heterodispersed systems
Coalescence, aggregation-small particles
Electrical barrier, effect-coalescence, ag-
Energy of interaction potential-small par-
Spheres, unequal-attractive energy, re- gregation
ticles
tardation