Introduction. Dispersing of oil in water or in the solution of an emulsifier, results in polydisperse emul- sions with a certain distribution of particle size.
Kolloid-Z. u. Z. Polymere 244, 324-332 (1971 )
Kolloide l~rom Faculty o/Technology o/ University at Novi Sad (Yugoslavia)
Investigation on the particle size distribution of O/W emulsions B y L. D ] a k o v i d ,
P. D o k i d , and P. R a d i v o j e v i d
With 6 figures and 6 tables
1. Introduction Dispersing of oil in water or in the solution of an emulsifier, results in polydisperse emulsions with a certain distribution of particle size. Polydispersity of emulsion droplets is the consequence of random events occurring inside the liquid, during emulsification. Droplet size distribution is influenced by many circumstances as surface tension of the emulsifier solution and the dispersed oil, i.e. tension at the oil/emulsifier solution interface, medium viscosity, mixing intensity, and others (1, 2, 3). It is not easy to subdue particle subdivision process to a consistent theoretical treatment, for it is very difficult to accept and explain certain regularity of droplet subdivision in the process of homogenisation. Coarser droplets probably do not disintegrate neither to give particles of equal sizes at any given condition (as is the ease at polymer destruction to monomer molecules), nor one may accept that one droplet may be subdivided to two or more droplets of equal sizes. Consequently, the process of emulsion homogenis~tion should be considered to be a statistical one, being a consequence of the events b y chance. The conclusion has been reached by many investigators (4-10) and many trials have been made in order to find out a general probability distribution function of the particle sizes. The approach to the emulsification processes b y means of the statistical methods has many advantages in respect to other possibilities. For example the parameters of statistical probability relations have certain physical meaning, and can be determined directly and simply b y experimentally measured values of the given distribution. A more general equation of size frequency distribution would make possible to follow
(Received July 13, 1970)
the changes of emulsion particle size distribution in an exact way, which may be of great practical interest. As an example we may mention the aging process of stable emulsions when droplets coalesce very slowly. In this case, it is more difficult to establish efficiency of the emulsifier b y simple observations, because stable emulsions apparently remain unchanged for a longer interval, although coalescence takes place permanently. The trend and the intensity of the particle size distribution changes in an emulsion could be determined b y means of sensitive and exact criteria, i.e. b y means of the eharacteristic parameters of given distribution function. It would make possible to decide about emulsifier efficiency, and would lead to a better understanding of the essential processes inside emulsions. During homogenization and aging processes, distribution of the emulsion droplet sizes changes gradually, depending on the given conditions. There is no doubt that in both eases the properties and the concentration of the emulsifier are of fundamental importance (11, 12). If the rest of the conditions are kept constant, than, observing the changes of the droplet size distribution during the homogenization process, one may conclude about the emusifier properties and find a method of simple and rapid characterization of the emulsifier efficiency. Furthermore, droplet size distribution of disperse phase may affect the viscosity, i.e. the theological behaviour, especially at coneentrated emulsions (13, 14, 15), as well as the other properties. The particle size distribution function may also be of interest from this point of view. The laws governing the emulsion droplet size distribution have been researched by many investigators. There are different
D]akovid et al., Investigation on the particle size distribution o] O/W emulsions
mathematical expressions, some of them being derived even b y introducing some theoretical assumptions (1, 4-10, ]6, 17, 18). From the practical point of view, a more general distribution equation, being va]id in broader variety of different emulsion particle size distribution, would be of the greatest importance. For example, if particle size distribution is affected by changes of different conditions of emulsion preparation (if emulsifier agent or dispersed oil, time of homogenization, type of homogenizer, aging time, concentrations and so on are altered), then, b y following the parameter changes of the distribution equation, it would be possible to conclude the emulsifier action on the physical and other properties of emulsions, and relate them to the given particle size distribution. The aim of the undertaken investigations is to find out a more general function of emulsion particle size distribution that would be easily applicable to practical problems, especially to particle size distribution changes during the homogenization process.
2. Theoretical
Besides the fact that a great number of different conditions act on final particle size distribution in a stable emulsion, one may suppose some regularities b y which the emulsification process should be governed. It is possible, for example, to suppose that when homogenization process is long enough, droplets tend to take some characteristic diameter and become monodisperse in size. This characteristic size of droplets should be identical to the most frequently represented diameters, formed up in the emulsion during the homogenization process. In other words, drops tend to take most probable size, during homogenization process. This conclusion leads to another. During homogenization process, both bigger drops as well as the smallest ones, arisen from them, disappear, giving thus droplets of the diameter occurring most frequently. Consequently, drops subdivide and coalesce simultaneously during the homogenization process, giving finally particles of most probable diameter which depends on different conditions (as temperature and viscosity of phases, intensity of mixing, constructive characteristics of the homogenizer), as well as on the physieochemieal properties of emulsion components.
325
The assumption that some drops may tend to take a definite characteristic minimum or maximum diameter is not quite acceptable, as these quantities are also a consequence of the events by chance. The problem has been solved by taking that diameter of smallest drops is close to zero, and that of the largest ones are of infinite sizes (19). It may happen, for example, in slightly homogenized emulsions of poor emulsifier agents, i.e., when the emulsion is quite polydisperse (7, 8, 17). Assuming, therefore, that 2 opposite processes are taking place simultaneously in an emulsion during its preparation, one, leading to the particle disintegration (owing to the mechanical action of the homogenizer), the other, causing the coalescence of droplets, the outeoming distribution of droplet sizes is the consequence of these two random events. Going no further into the essence and the chemismus of these two processes, it may be assumed that in the simplest case, the increment of drop number dn during disintegration is directly proportional to the number of drops n, existing in the emulsion, and inversely proportional to its diameter x. The smaller the size of drops in the disintegration process, the greater their number (if the total volume of drops remain unchanged). Drop number increment dn, related to the unity of diameter increment, or "the rate of change of the number of particles with their diameter" (10), i.e. dn/dx, depends also on an "ability of the droplet size change", which is, in turn, related to the most probable particle diameter. I f a drop of the diameter x, during the homogenization process, tends to take some characteristic diameter X, than, at any other stage of the homogenization process it will possess a certain "ability" of changing its size, which is proportional to the difference (x - x). If ( X - - x ) = O, any additional homogenization has no further effect, and the disintegration and the coalescence are in the equilibrium. The greater the absolute value of the differenee ( X - - x ) , the more intensive and the more rapid is the relative change of increment dn in respect to the change of the diameter increment dx. So the final relation may be set up as : dn
n
d~- = a x (X - x)
[11
where n - is the number of drops having diameter x, X - most frequently occurring dia-
326
K o l l o i d - Z e i t s c h r i / t u n d Z e i t s c h r i / t /iir P o l y m e r e , B a n d 2 4 4 9 H e f t 2
meter, and a - t h e p r o p o r t i o n a l i t y coefficient or the homogenization rate constant. As the disintegration process is the consequence of the events b y chance, the eq. [1] is a statistical one. I t m u s t fulfill conditions: co
d x = l, and / ( x ) > 0. After
H(x)
integra-
0
tion, deviding b o t h sides b y the t o t a l n u m b e r of drops N, the relation [1] gives: n N
_
a aX + 1 F ( a X + 1)
[2]
x a X . e- a x
which is now the n u m b e r average diameter distribution function. Quantities t h a t e x a c t l y characterize given particle size distribution m a y be found b y d e t e r m i n a t i o n of the dist r i b u t i o n moments (20). The m e a n diameter 2 is the first m o m e n t a b o u t the zero (Vl), and variance a 2 is the second m o m e n t a b o u t the mean (#2). Skewness ill, curtosis ill, and its coefficients, Yl and Y2 respectively, characterizing the given distribution, can be calculated by use of parameters a and X of the eq. [2]. The distribution function [2] gives: Mean diameter: 2 = X "-- I/a or Most frequently occurring diameter x = 2 - ]la. Variance 1#2 (X)I = #2 is given b y (r2_ -
diameter. This is in a g r e e m e n t with the conclusion t h a t all the particles t e n d to t a k e the most probable d i a m e t e r at an unlimited homogenization. The relation [2] gives the numerical freq u e n c y distMbution function. Distribution functions of the diameter, of the surface area and of the v o l u m e averages (7, 8, 10) are given b y the following equations, respectively: n i xi
n i di
~.. n i xi i
D
ni:nxi~
_
F~nizxi ~
aa x § 2
__
l"(aX +
nipi
_
2)
x a X + 1 . e-aX
[3]
x aX+2.e-ax
[4]
aaX +3
Y ( a X + 3)
P
i niaxia/6 _ n i ~ xia/6
nivi V
_
aaX + r xaX + a.e-aX[5] J F ( a X -t- 4)
i
The characteristic p a r a m e t e r s of these distributions are also defined b y their m o m e n t s and differ from each other if different averages are t a k e n into account. F o r example, dispersion is the smallest at numerical dia m e t e r f r e q u e n c y distribution, and it is the largest at volume average d i a m e t e r distribution, as illustrated in fig. 1.
aX + i a 2
and the s t a n d a r d deviation = V a X + ]la.
The skewness or a s y m m e t r y is the square of the third m o m e n t a b o u t the mean #a 2 d e r i d e d 4
b y / ~ a , i.e. fll = #s2/#a~ - - a X + 1 Coefficient of skewness 7 1 -
~~ a
_
0
1 2 3 /-, 5 6 7 8 9 10 1'1 1'2 13 1/-, 15 1'6 17 18 19 2'0 + PARTICLE DIAMETER[J~]
l / ~ , in the 2 distriFig. 1. Curves of the equations [2-5] for numerical ( 9 diameter (x), surface area (E]) and volume (A) frequency distributions, for 30O/o of sunflower oil in water emulsion stabilized by PDBS, homogenized 8 m i n
b u t i o n given b y eq. [2] is Yl -- Vh~ + 1 3(aX
+
3)
Curtosis f 1 2 - #~'~ ' or f l ~ - a X + 1 and coefficient of excess Y2 defined 6 a X + 1"
by
(/~4/tt22)-3 is Y 2 - I f fll > 0 the distribution has positive skewness and the longer p a r t of the curve is on the right side in respect to its m a x i m u m X. The curtosis is the measure of degree of the "flattening" near the center of distribution. It is interesting to note that in eq. [i] x = X stands for dn/dx = 0, i.e. when the particles become quite uniform and the disintegration process is finished, their d i a m e t e r is equal to the most f r e q u e n t l y occurring
As the exponents of the x are changed regularly, diameters calculated and expressed as different averages are m u t u a l l y related, so t h a t 2 n = X ~ , 2~ = X~,, 2~ = X v. I t is also interesting to note, t h a t p a r a m e t e r a remains constant. Measure of the developed surface area in an emulsion m a y be the quotient b e t w e e n the total surface area of the dispersed particles P, and the t o t a l volume of the dispersed phase V, so t h a t the specific surface area S m a y be given b y S = P / V . Specific surface area S m a y be derived b y dividing eq. [5]
D]alcovi6 et al., Investigation on the particle 8ize distribution o/O/W emulsions
327
b y eq. [4], so t h a t S-
6c~
23
aX+3 o
3. E x p e r i m e n t a l
Q
3.1. Experimental techniques (materials) Emulsions prepared by means of the homogenizer Zenith type V III (DDP@ The agent-in-water method (2) of the emulsion preparation used9 Emulsification temperature kept constant at approximately 20 ~ by use of an ultrathermostat. Experimental measurement of the particle diameters taken by microscopic method (21). Emulsion has been diluted by 400/0 glycerole solution and placed in the Thomas chamber. Pictures were taken by a Zeiss-Jena microscope 10 min later. Magnifying of the photos made measurement of the droplet diameters possible. Measured values were classified into 1 # wide intervals (09 1.5-2.5, 2.5-3.5#, etc.). The diameter size distribution has been determined at O/W emulsions of different compositions, changing the type of the emulsifier and the oil, the homogenization time, the concentration of the oil phase, and the aging time. The following systems were investigated: 1. Emulsions of the 70% paraffin oil in the 3% solution of the "Tween 80" in the destilled water. The samples for particle size determinations had been taken out of the emulsion during the homogenization, within 5, 12, 20, 45, 83 and 120 rain intervals, and the photographs were taken after about 30-40 rain. 2. 70%, 50% and 30% of commercial sunflower oil (Vital, Vrbas) had been added into the 1% solution of Na-paradodecil-benzenesulfonate (PDBS), and samples were taken out after 8, 12, 20, 60 and 120 min. 3. 70%, 50% and 30% of the sunflower oil dispersed in the 1% casein solution (Brit. Drug Co) and gelatine (Difco) had been homogenizing for 20 min and the particle size distribution determined. All the concentrations given above have been done in the weight %. Concentrations of the emulsifier agents have been calculated in respect to oil weights.
4~
9
g
r
~q
& r
~q
3.2. Experimental results and discussion T h e a i m of t h e e x p e r i m e n t a l i n v e s t i g a t i o n has b e e n t h e v e r i f i c a t i o n o f v a l i d i t y o f the derived expressions. The validity of the derived expressions had been checked not o n l y at e m u l s i o n s w h i c h differ in c o m p o s i t i o n , b u t also for d i s t r i b u t i o n c h a n g e s arisen in different s t a g e s o f t h e h o m o g e n i z a t i o n a n d aging processes. T h e e q u a t i o n fitness to e x p e r i m e n t a l d a t a h a d b e e n c h e c k e d b y an e l e c t r o n i c d i g i t a l c o m p u t e r ( E l i o t t 803), g r a p h i c a l l y a n d b y comparing calculated and experimental values. C u r v e (2) has b e e n t r a n s f o r m e d t o a linear f o r m [ w h e r e ( n / N ) 100 = y] a n d t h e c o m p u t e r p r o g r a m w a s b a s e d on t h e f o l l o w i n g r e l a t i o n : Y = A X + BZ where Z =
Y=
ln(y~/y~),
(x~ - - x ~ ) .
X=
[6] ln(x~/%),
and
DO
"r. 9 O
Kolloid-Zeitschri/t und Zeitschri/t /i~r Polymere, Band 244. He/t 2
328
n.100 &.
~'-~.100 20~
20 18 "~"-36 20
A
,=
4~ 2 o 1
2 3
4
,
5
1
9
16t /
~ 4
,4t
~
,0508
2
2 .
8
4
6
7
8
7
8
~
)
9 10 11 12 13 14 15 16xr~ 3
,~
6
3
6
"6= 20,25 a• 1,515
lO
2
5
,,
81
1
4
16
lO I
0
3
I -f --100
~'= 1561 aX= 1.408
~
2
:
i
i
0.598
q
I
,
9 10 11 12 13 14 15 16x[~]
Fig. 2. H i s t o g r a m s and correspondent curves f o u n d for different emulsions b y means o f the eq. [2]
700/0 30o/0 300/o 300/0
of paraffin oil in w a t e r stabilized b y Tween 80 of sunflower oil in water stabilized b y casein of sunflower oil in water emulsion stabilized of sunflower oil in w a t e r emulsion stabilized
Parameter A = a X , and B = - - a . Constants A and B had been found by the least square method and the characteristic parameters of the given distribution function, had been calculated as the most frequently occuring diameter X, mean diameter 2, dispersion ax=, standard deviation a,, and specific surface area S. The values calculated by the computer have been compared to the values found experimentally. Calculated and correspondent experimentally measured quantities may be seen in table 1. Fitting of the curve (2) into the experimental results has been checked b y a drawing of the curves together with their histograms, which gave satisfactory results. Fig. 2 shows the characteristic examples. Parameters a and X used for calculation of the points (Yi, xi) given in the fig. 2, have been found in a linear diagram (3 log y) to (A log x). The right line has been drawn through the points by personal estimation.
(a) (b) b y gelatine (e) b y P D B S (d)
The above procedure of the parameter determinations may be simplified by calculating one of the parameters by means of experimental data (for example parameter a X ) may be calculated out of the mean diameter 2 and dispersion, a2 i.e. a X - - ~
Z~2
and its
more exact value, and the other parameter, m a y be found in the first approximation of the linear graph. The procedure makes it possible to form the linear graph simply and quickly and to control the fitting of the given equation. If both parameters a and X have been determined out of the experimental values (X is the most frequently occurring diameter) the equation of the given distribution may be formed without any other operations. Experimental control of eq. [2] by means of linear diagrams (log y - a X log x) to (x), where a X had been calculated directly from the experimentally measured values, have shown very good fitting of curve (2) in dif-
Djakovid et al., Investigation on the particle size distribution o/O/W emulsions
~
og y -aX kx:j x
x~
329
ferent emulsions. The examples are given in fig. 3. Diagrams in fig. 4 represent curve (2) at an emulsion homogenized during different time intervals, and fig. 5 shows changes of droplet sizes caused b y the aging process [experimental data for fig. 5 have been taken from the paper of Harkins and Beeman (22)].
-1-
\
log y -
a Xlog x
-2-
-3-
-5
i l',l
-6
"~
Fig. 3. Linear graphs of the eq. [2] at different emulsions of sunflower oil: (O) - 30O/o emulsion stabilized by gelatine, homogenized 20 min ( • ) - 30% emulsion stabilized by PDBS, homogenized 120 min, (V) - 50O/o emulsion stabilized by PDBS, homogenized 120 min, ([-~) - 70~ emulsion stabilized by PDBS, homogenized 20 min, and that of paraffin oil: (A) - 70O/o emulsion stabilized by "Tween 80", homogenized 83min
l"~og y - a.X log x
o~ i \ \ ~ ~ 8
9 lo 11 12 13 1.~
1days
Fig. 5. Changes of the emulsion particle size distribution during the aging
Experimental checking has confirmed the assumption that all droplets tend to take the most probable diameter X, during homogenization process, and at constant homogenization conditions their number fraction rises only. Fraction of the particles of mean diameter 2 rises during the homogenization process too, differing in the 2 value which 28- ~J-.lO0
26-
-~
days
homogenized o 5 rain
24-
12 min v 20 min = /,5 rain v 120 min
222018
12
loi 6-
"J"
120 rain 8252nn 0
Fig. 4. The emulsion particle size frequency distribution in linear graph at 70% paraffin oil in "Tween 80" water solution, homogenized 5, 12, 20, 45, 83 and 120 min
t
2
3
4
5
6
7
8
9 10 11 12 13 14 15 x[j~]
Fig. 6. Influence of the homogenization time on the particle size frequency distribution (eq. [2]), at 700/o emulsion of paraffin oil in water solution of "Tween 80"
330
Kolloid.ZeitschriJt und Zeitschri/t /i~r Polymere, Band 244 9 Heft 2
Table 2. The values of X and x and its frequences calculated by eq. [2] at emulsions homogenized during different time intervals Time of homogenization (rain)
5
12
20
45
83
120
Most probable diameter X(#) Frequency of X Mean diameter ~ (/~) Frequency of ~ Standard deviation ~x
3.22 15.51 5.05 12.60 3.04
3.28 17.26 4.78 14.45 2.69
3.45 18.63 4.72 16.08 2.44
3.18 25.74 3.90 23.31 1.68
3.30 26.01 4.00 23.73 1.66
3.22 38.21 3.84 35.34 1.55
does n o t r e m a i n c o n s t a n t during homogenization b u t g r a d u a l l y decreases. T a b l e 2 lists calculated a n d m e a s u r e d values of X a n d 2, in 7 0 % emulsion of paraffin oil in w a t e r solution of " T w e e n 80" (3 w. ~ of the emulsifier to t h e oil weight). The fact t h a t during h o m o g e n i z a t i o n protess droplet fraction of the m o s t p r o b a b l e dia m e t e r X rises on the a c c o u n t of b o t h the larger (x > X) a n d smaller (x < X) particles, m a y be recognized f r o m the g r a p h s given in fig. 6. W i t h r e g a r d to the fact t h a t p a r a m e t e r a r e m a i n s a c o n s t a n t at all averages of d i a m e t e r f r e q u e n c y distributions (eq. [2-5]), a n d t h a t t h e e x p o n e n t of x is regularly c h a n g e d b y addition of integers, the rest of p a r a m e t e r s
can easily be calculated if the p a r a m e t e r of one of these relations is known. T h e m e a n i n g of the p a r a m e t e r s of eq. [2-5] h a v e been checked e x p e r i m e n t a l l y . T h e p a r a m e t e r s h a d b e e n d e t e r m i n e d b y a digital c o m p u t e r (Eliott 803) b y reducing the equations [2-5] to a more general m a t h e m a t i c a l form, g i v e n b y eq. [7]: y = 7 x c~e x p ( - a x ) . [7] T h e eq. [7] was t a k e n into the c o m p u t e r p r o g r a m in linear f o r m g i v e n b y eq. [6] altering A for ~. The e x p e r i m e n t a l d a t a h a v e been used in the c o m p u t e r to calculate the p a r a m e ters of t h e eq. [7]. T h e results listed in t a b l e 3 confirm t h a t the e x p o n e n t s of eq. [2-5] do really change as m a t h e m a t i c a l l y derived.
Table 3. Parameters of eqs. [2-5] calculated in the computer by means of more general distribution eq. [7] for emulsions of sunflower oil, stabilized by PDBS 30% emulsion equation a a X O'x~ fll fl~ Yl Y2
n [2]
d [3]
p [4]
v [5]
n [2]
d [3]
io [4]
v [5]
3.68 1.39 2.65 3.37 2.43 0.853 4.281 0.923 1.281
4.69 1.39 3.37 4.09 2.94 0.703 4.053 0.839 1.053
5.70 1.39 4.07 4.78 3.41 0.597 3.896 0.772 0.896
6.72 1.39 4.81 5.53 3.96 0.518 3.777 0.718 0.777
3.28 2.00 1.64 2.14 1.07 0.935 4.400 0.966 1.400
4.28 2.00 2.14 2.65 1.32 0.757 4.136 0.869 1.136
5.28 1.99 2.65 3.15 1.58 0.637 3.955 0.797 0.955
6.28 2.00 3.15 3.65 1.83 0.549 3.824 0.740 0.824
T a b l e 3 shows t h a t e x p e r i m e n t a l l y f o u n d values of the p a r a m e t e r a, do n o t c h a n g e at different a v e r a g e d i a m e t e r frequences, a n d t h a t e x p o n e n t of x, i, e. ~, is c h a n g e d b y integers (according to eq. [2-5]), so t h a t c% = a X , a~ = a X @ 1, c~!o = a X @ 2 a n d ~v = a X @ 3. Dispersion ax ~ indicates t h a t polidispersity of v o l u m e a v e r a g e d i a m e t e r of the particles is the highest one, as illustrated in d i a g r a m s o f fig. 1. F u r t h e r m o r e , the results given in t a b l e 3 m a k e clear t h a t 2 n = X~, 2a = X~, ~
70% emulsion
= X v.
T h e parameter.s of d i a m e t e r f r e q u e n c y dist r i b u t i o n equation, m a y be used to follow
a n d characterize d i s t r i b u t i o n changes of emulsion droplets in a n e x a c t w a y a n d to i n v e s t i g a t e t h e effects of different circumstances, as, for e x a m p l e , composition of a n emulsion (nature a n d c o n c e n t r a t i o n of oil a n d emulsifier used), effect of physical conditions ( t e m p e r a t u r e , m e c h a n i c a l action, etc.), hom o g e n i z a t i o n duration, aging effect, a n d so on. T a b l e 4 lists p a r a m e t e r s of statistical distribution eq. [2] a n d [4], o b t a i n e d for emulsions h o m o g e n i z e d within different t i m e intervMs. I n t h e first case, d a t a calculated by the surface average diameter function (eq. [4]) for 3 0 % sunflower oil emulsion in
331
D]alcovi5 et al., Investigation on the particle size distribution o / O / W emulsions
Table 4. Parameters of statistical distribution eqs. [2] and [4] for emulsions homogenized during different time intervals Time of homoge~ nization (rain) Sunflower oil (4)
8 12 20 60 120 5 12 20 45 83 120
Paraffin oil (2)
a
3.06 3.515 4.99 5.70 5.83 1.76 2.17 2.74 4.39 4.78 5.14
a
X
~
ax~
S
fll
Yl
f12
~
0.407 0.596 0.823 1.40 1.60 0.546 0.662 0.794 1.379 1.445 1.603
7.53 5.91 6.07 4.07 3.64 3.22 3.28 3.45 3.18 3.30 3.22
9.99 7.58 7.28 4.78 4.26 5.05 4.78 4.72 3.90 4.09 3.84
24.50 12.83 8.88 3.42 2.66 9.23 7.23 5.93 2.83 2.77 2.40
0.599 0.792 0.824 1.246 1.402 0.682 0.768 0.834 1.120 1.120 1.180
0.987 0.887 0.667 0.597 0.586 1.45 1.26 1.07 0.732 0.692 0.651
0.993 0.941 0.817 0.773 0.765 1.20 1.12 1.03 0.856 0.832 0.807
4.480 4.330 4.001 3.896 3.878 5.20 4.89 4.61 4.12 4.03 3.98
1.480 1.330 1.001 0.896 0.878 2.20 1.89 1.61 1.12 1.03 0.98
Table 5. Parameters of the number average diameter distribution at emulsion of different composition Emulsion No.
Oil
1. 2. 3. 4. 5.
Emulsifier
Sunflower oil 30~o Sunflower oil 70~ Sunflower oil 30~o Sunflower oil 70~o Paraffin oil 70~o
aX
l~/o PDBS l~o PDBS 1~o Gelatine 1~o Gelatine 3~o Tween 80
P D B S w a t e r s o l u t i o n (1 w. % o f P D B S t o t h e oil w e i g h t ) , a r e g i v e n . I n t h e n e x t case, t h e parameters of the number average diameter d i s t r i b u t i o n eq. [2], for t h e e m u l s i o n o f 7 0 % p a r a f f i n oil in " T w e e n 80" w a t e r s o l u t i o n (3 w . % o f " T w e e n 80" t o t h e oil w e i g h t ) , a r e given. I t m a y b e o b s e r v e d in t a b l e r t h a t p a r a m e t e r s a, X a n d 2 a r e b e i n g r e g u l a r l y changed under the influence of homogenization time. The most probable diameter X a t n u m b e r a v e r a g e d i a m e t e r d i s t r i b u t i o n is not affected by homogenization and remains c o n s t a n t ( a t s u r f a c e a v e r a g e d i a m e t e r dist r i b u t i o n i t b e c o m e s s m a l l e r ) . P a r a m e t e r s fll a n d 71 i n d i c a t e t h e f o l l o w i n g r e l a t i o n : t h e longer the homogenization, the more symmetric the distribution.
2.98 2.81 1.408 3.625 2.74
a
0.821 1.54 0.507 1.185 0.794
X
3.63 1.82 2.78 3.06 3.45
"2
(~x2
S
4.84 2.47 4.76 3.90 4.72
5.91 1.60 7.36 3.30 5.93
0.824 1.59 0.691 1.073 0.834
Table 5 contains the parameters of the number average diameter distribution funct i o n for e m u l s i o n s w h i c h differ f r o m e a c h o t h e r in c o n c e n t r a t i o n o f oil p h a s e '(cases 1 a n d 2), in n a t u r e o f t h e e m u l s i f i e r (cases 1 a n d 3 ) , a n d in c a s e 5 b o t h t h e e m u l s i f i e d oil and the emulsifier agent are changed. The t i m e o f h o m o g e n i z a t i o n h a s b e e n k e p t cons t a n t (20 m i n ) . T a b l e 6 s h o w s r e g u l a r i t i e s in t h e d i s t r i b u tion parameter changes at aging and points out to the investigation possibiilites of the aging processes at emulsions [experimental data have been taken from the paper of H a r l c i n s a n d B e e m a n (22)]. I f all t h e c o n d i t i o n s a t e m u l s i o n p r e p a r a tion are held constant, except the kind and concentration of an emulsifier, the distribu-
Table 6. Parameters of the particle size distribution equation for the emulsions of different ages Emulsi- Age tier (days) Na01 CsO]
1 3 7 1 3 7
aX
a
X
"2
a~x
S
fll
f12
2.003 3.469 4.022 1.584 2.735 3.882
1.984 1.562 1.205 1.818 1.330 1.111
1.010 2.220 3.338 0.871 2.057 3.494
].512 2.863 4.170 1.421 2.808 4.396
0.761 1.833 3.463 0.782 2.111 3.958
2.380 1.449 1.029 2.380 1.391 0.969
1.322 0.895 0.796 1.548 1.071 0.819
4.998 4.343 4.195 5.322 4.606 4.299
Kolloid-Zeitschri]t und Zeitsehri/t /i~r Polymere, Band 2d~l 9
332
tion equation parameters will reflect influences of the emulsifier on the homogenization process. In this way it might be possible to develop a fast method of emulsifier characterization and judging about the emulsifying properties of an emulsifier.
Acknowledgement This work has been supported by the Federal Fund for Supporting Scientific Works (Yugoslavia).
Summary The aim of this research has been the investigation of a general equation of the particle size distribution in emulsions, applicable to different emulsions, especially to particle size distribution at homogenization processes.
I t has been supposed that at homogenization particles tend to assume a characteristic, most probable diameter X, due to different conditions, so that bigger particles X disintegrate, while smaller ones coalesce. It may be assumed that the "ability" of altering particles of a certain diameter x depends, in that case, on the difference (X -- x). The speed of particle number alterations at their diameter, i.e. dn/dx may be presented by the eq. [1], which when integrated gives a statistic distribution eq. [2] having a function form, i.e.:
n N
aaX'+ 1 x a X e -ax F ( a X + 1)
where a and X are the constants, n - the particle number of diameter x, and N - the total particle number. Experimental checkings proved satisfactory regarding the emulsions O/W, which differ in their characteristics and composition. The equation has also proved ~atisfactory for the emulsions of identical composition, homogenized for different periods of time, as well as for the ones that suffered certain particle distribution due to aging.
Zusammen]assung Ziel der Forschung war, eine allgemeine Gleichung des Verteilens der Tr6pfchengr6~e in den Emulsionen zu finden. Diese Gleichung sollte ffir die verschiedenen Emulsionen, besonders bei der Verteilung der Tr6pfchengr513e in dem ProzeB der Homogenisierung, gelten. Es ist dargestellt, dal3 TrSpfchen wghrend der Homogenisierungszeit streben, einen eharakteristischen wahrseheinliehen Durchmesser X zu bekommen, was in der Abhgngigkeit yon den versehiedenen Bedingungen steht, so dal3 im Proze] der Homogenisierung alle grSBeren Tr5pfchen yon X zerfallen, und alle kleineren yon X koaleszieren werden. Man kann aunehmen, dad die ,,Fghigkeit" des Wechselns der TrSpfehengrSBe mit irgendeinem Durehmesser x, in diesem Zufall yon dem Unterschied IX -- x I abhgngig ist. Die Schnelligkeit des Wechselns der Tr5pfchenzahl mit ihren Durchmesser, d. h. dn/dx, kann durch die Abhgngigkeit, die mit der G1. [1] gegeben ist, dargestellt werden. Diese G1. [1]
2
gibt, nach der Integration, die statistische Gleichung des Verl~eilens [2], die eine Gammafunktion besitzt, d. h. 7,~ __
N
aaX~r 1
x aX
c-ax
I'(aX + 1
wo a und X Konstante sind; n - - ist die Zah] der Tr6pfchen mit dem Durchmesser x; und N - - ist die gesamte Zahl der Tr6pfchen. Die Experimentalkontrolle zeigte, dal3 die ausgefiihrte Gleichung einen Effekt bei der Emulsion des Typs O/W, die sich voneinander durch die Eigenschaften und die Zusammensetzung unterscheiden, hat. Es wurde aueh gezeigt, dab diese Gleichung bei den Emulsionen der gleiehen Zusammensetzung, die eine verschiedene Zeit homogenisiert wurden, oder bei denen es zur Vergnderung in der Verteilung der Tr6pfchengr6Be folgens des Alterns gekommen ist, effektvoll befriedigt. f~e/erence8
1) Sumner, C. G., Clayton's Theory of Emulsions and their Technical Treatment (London 1954). 2) Becher, P., Emulsions: Theory and Practice (New York 1957). 3) Richardson, s G., J. Colloid Sci. 5, 404 (1950). 4) Rossi, C., Gazz. Chim. Ital. 63, 190 (1933). 5) Cooper, F. A., J. Soc. Chem. Ind. (London) 56, 447 T (1937). 6) Ra]agopal, E. S., Kolloid-Z. 162, 85 (1959), 7) Schwarz, N. und C. Bezemer, Kolloid-Z. 146, 139 (1956). 8) Bezemer, C. und N. Schwarz, Kolloid-Z. 146, 145 (1956). 9) Radvel, A. A., 2V. A. Novikova und V. P. Derevyagina, Kolloidn Zh. 28, No 2, 258 (1966). 10) Jellinek, H. H. G., J. Soe. Chem. Ind. (London) 69, 225 (1950). 11) Fischer, E . K . and W.D. Harkins, J. Phys. Chem. 36, 98 (1932). 12) Reh/eld, S . J . , J. Coll. Interface Sei. 24, 358 to 365 (1967). 13) Richardson, E. G., J. Colloid Sci. 5, 404 (1950); 8, 367 (1967). 14) Roscoe, R., Brit. J. Appl. Phys. 3, 267 (1952). 15) D]akoviS, Lj. and P. Doki5, The Collection of Works of the Faculty of Technology at Novi Sad 1, 5-17 (1967). 16) Dobrowsky, A., Kolloid-Z. 95, 286 296 (1941). 17) Mugele, R. A. and H. D. Evans, Ind. Eng. Chem. 43, 1317-1324 (1951). 18) Troesch, H. A., Chem. Ing. Tech. 26, 311 (1954); Kolloid-Z. 143, 50 (1955). 19) Riyad R. Irani and Clayton F. Callis, Particle Size: Measurement, Interpretation and Application (New York 1963). 20) Burrington, R. S. and D. S. May, Handbook of Probability and Statistics (Ohio 1958). 21) Levius, H . P . and E. G. Drommond, J. Pharm. Pharmacol. 5, 743, 755 (1953). 22) Harkins, W . D . and N. Beeman, J. Amer. Chem. Soc. 51, 1674 (1929). Author's address : Prof. Dr. L]. Djakovid et al. University of Novi Sad Faculty of Technology, St. Musida 66, 21 OO0 Novi Sad (Yugoslavia)
