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The use of the coulter counter for investigating the coagulation kinetics of mixed monodisperse particulate systems
The Use of the Coulter Counter for Investigating the Coagulation Kinetics of Mixed Monodisperse Particulate Systems~ B. A. M A T T H E W S A~N~DC. T. R H O D E S The
Physical Pharmacy Department, School of Pharmacy, Portsmouth College of Technology, Portsmouth~ Hants, England Received May 9, 1968 The Coulter counter has been used to investigate the particle size distributions of polystyrene and polyvinyl toluene latices. The distributions have been shown to be closely log normal over the maior part of their extent. Mixtures of two latex species have been examined in cases where overlap of the distributions does and does not occur. It has been shown that, in agreement with the conclusions of earlier workers, the measured distributions are broader than would be expected from the manufacturer's figures. The change in particle size profile of mixtures has been examined when coagulation occurs, aud this has been compared to computer-predicted profiles. The significance of the results in terms of the coagulation behavior of supracolloidal suspensions is discussed. INTRODUCTION
A recent series of papers involving the use of the Coulter counter (1-6) with monodisperse polymer latices has demonstrated the value of this technique for investigating the coagulation kinetic theory of monodisperse systems. Although most suspensions of materials such as drugs are heterodisperse, it has been shown (6) t h a t a comparison can be made between coagulation in suspensions and in model systems. A mixture of two monodisperse particulate systems of different mean particle diameters can be considered as an interim state between mono- and heterodispersity. An examination of the coagulation behavior of such mixtures is likely to give valuable information as to the relative coagulation behavior of smaJ1 and large particles. I n the present paper the authors have used the Coulter counter to examine such systems and to test the kinetic theory for mixtures of two mono-
disperse particulate region of 1-2 ~.
suspensions,
in
the
THEORY The kinetics of coagulation of a lyophobic colloid was considered by Von Smoluchowski (7) in his theory of rapid coagulation. The process was considered as a diffusion phenomenon, caused b y Browian motion with no repulsion between the particles (8). H e derived the following equation to describe the rate of change of concentration of any particle Nk with time t (9).
No ( t ~ k-1 N~ =
\T~/~!
[1]
1 • T1/2! where No is the number of primary particles originally present and T1/2 is the time taken for the total number of particles to decrease by one half. Equation [t] has been tested experimentally on a gold sol by Turkevich (10), who used gelatin to halt the coagulation at
This paper was presented to the Conference on "General Applications of the Coulter Counter", London, March 6th, 1968. 71 Journal of Colloid and Interface
Ncience, Vol. 28, No. 1, Sep%ember 1968
72
MATTHEWS AND RHODES
different intervals and counted each species under an electron microscope, and by Higuchi et al. (1), who used the Coulter counter to differentiate the relative amounts of each species, up to lo = 3, for a uniform polystyrene latex. The present authors (5) have also used Eq. [1] for the singlet to investigate the extent of departure from the Smoluehowski rate for rapid coagulation. Most colloidal and supracolloidal disperse systems are polydispersed and likely to show considerable variations from the Smoluchowski theory, since it has been known for many years (11) that, in such systems, the smaller particles disappear much more rapidly than the larger ones. To account for this, Miiller (12) extended the Smoluchowski theory and derived the following equation for the coagulation of a mixture of large and small particles.
No
[2]
VR 2
(VR2N~-t-1)(1
+ T1/~!
1
where n is the total number of small particles at time t, No is the initial number of large particles; no is the initial number of small particles; NR = No~no ; VR = R / r = the ratio of the radii of the two primary species; and T1/2 is the coagulation time of the large particles. This theory has been extensively tested for true colloids by Wiegner and Tuorila (13, 14), but little information is available on how mixtures of supracolloid~l particles will behave. The growth of high polymer latex technology (15) and the advent of the Coulter counter (16) here made such a study possible. The ability to distinguish between each combination, during the coagulation of a mixture of 2 monodisperse particles, is likely to be dependent upon the following four factors. 1. Choice of O p t i m u m Particle Size Ratio.
The course of a coagulation reaction between two species can be predicted from the kinetic
theory outlined above and from probability calculations. As the reaction proceeds, the number of possible combinations becomes extremely great. By restricting the study t 9 the early stages (up to the Tlj2) and neglecting species higher than the triplet, the process may however be simplified. The number of possible combinations, the probability of formation of each multiplet, and the relative theoretical proportions of each, at t = 0 and t = TI/2, are shown in Table I. The differences of probability for collisions between species of different sizes have been neglected although this can be computed from the Mfiller theory. The relative probabilities of each species have been calculated, on the assumption that only binary reactions occur. This assumption is probably valid in considering the coagulation of latices in the concentration range of 107-10s particles per milliliter (1, 5), where the coagulation time is 4.70.47 hours. However, it is possible to obtain true colloids with a concentration greater than 10I4 particles per milliliter (9), when the coagulation time is about ~000 sec. In this case many higher order reactions will occur. The response of the Coulter counter between its working limits is proportional to the volume of the particle measured (17), and for each species to be detectable, it is desirable that there should be no overlap between the volume ranges of successive particle combinations. In selecting the most suitable particle size ratios for fulfilling this criterion, it is evident that there are two possible types of overlap. a) When the volume of the larger species B is equal to, or less than three times the volume of, the smaller species A. b) When the volume of the larger species B is greater than three times the volume of A. For type (a), there is the possibility of direct overlap, e.g., when the volume of B = twice the volume of A, it will be impossible to distinguish the AA doublet from the B singlet. A volume ratio of 3/1 is equivalent to a diameter ratio of only 1.442/ 1, and with two species of such close particle sizes, the Mfiller theory would predict very little difference in collision rates be-
Journal of Colloid and Interface Science, Vo1.28, No. 1, September 1968
USE OF THE COULTER COUNTER
73
TABLE I R E L A T I V E PROPORTION OF EACH ~PECIES AS A FUNCTION OF T I M E IN THE COAGULATION OF A M I X T U R E OF T w o
SUSPENSIONS Relative proportions at different times
Species
Ratios
Probability
t = 0 Total
Singlet Doublet Triplet
A1 B 1 AA 1 AB 2 BB 1 AAA 1 AAB 3 ABB 3 BBB 1
0.5 0.5 0.25 0.5 0.25 0.125 0.375 0.375 0.125
tween the A-A reaction and A-B reaction. However, if two particle sizes in the range can be selected with little or no overlap, it would provide a means of checking this theory. I n the case of type (b) the most important overlap will occur between mixtures of various proportions of A and B, and pure B aggregates. This will be determinedlargely b y the degree of monodispersity of A and B. T h e singlets, doublets, triplets, and quadruplets of the pure species should be readily distinguishable.
t = Tl/e
Individual
0.50 0.50 -----_ --
1.0 --
--
Total
0.25 0.125
0.0625
TABLE
"DOT"
Individual
0.125 0.125 0.03125 0.0625 0.03125 0.0078125 0.0234375 0.0234375 0.0078125
II
LATICES USED IN THE COAGULATION
EXPERIMENTS Latex
Polystyrene Polystyrene Polystyrene Polystyrene Polyvinyl toluene 5 Polyvinyl toluene 6
Particle diameter (#)
1 2 3 4
Standard deviation
Relative S,D. (%)
0.714 0.81 1.099 1.305 i. 857
0.0053 0.0063 0.0059 0.0158 0.007
0.74 0.78 0.54 I. 21 0.38
3.49
0.017
0.49
2. The Particle Size Spread of the Two Species. The particles selected for this study are the polystyrene and polyvinyl toluene latices produced by the D o T Chemical C o m p a n y ? The particle diameters and standard deviations as stated b y the manufacturers are shown in Table II. The fact that standard deviations are quoted for these latices would seem to predicate a normal Gaussian distribution. Wales (18) examined polyisoprene latex samples b y means of the Coulter counter and found t h e m quite closely log normal. Parfitt (19) has quoted a significant positive skewness for "DOT" monodisperse latices when measured on the Coulter counter. Wachtel and La M e t (20) found that the size distribution of a 1.17 t~ polystyrene latex, when measured on the Coulter counter, was broader t h a n would be ex2 Marketed in Europe by Serva EntwicMungs Labor, Heidelberg, Germany.
petted from the standard deviation obtained from the electron microscope. Similar results using flow ultramieroseopy were reported by Davidson et al. (21). We have assumed that, up to the triplet state, the volume of each multiplet is equal to the sum of the volumes of each constituent particle. Above the quadruplet, different configurations and the occluded space in the interiors of the clusters is likely to nullify this assumption. However, Ho and Higuchi (4) found that, using a multiehatmel analyzer to record the pulses, they could distinguish oetuplets of a monodisperse latex. With the triplets, quadrupulets, and above, different configurations are possible, and, for example, a linear quadrupulet m a y have a different Coulter volume from a compact one. Allen (16) has shown t h a t for a rod-shaped particle totally enclosed
Journal of Colloid and Interface Science, VoI. 28, No. 1, September 1968
74
MATTHEWS AND RHODES
within the orifice, the Coulter size may be 7.8% large when the equivalent spherical diameter of the particle is 40 % of that of the orifice. When it is only 10% of the orifice diameter, the error is about 0.5 %. Since all our measurements were made at about this percentage, or less, the different configurations are not likely to affect results significantly.
3. The Proportion of Each Species Present Initially. Equation [2] can be used to calculate the theoretical course of the eoagulation for any ratio of No/no, but the ability of the Coulter counter to detect the species present in smaller number must be considered in selecting this ratio. 4. Resolution of the Coulter Counter. The resolution of the Coulter counter is likely to be very critical in governing whether species with volumes very close together can be distinguished. Ho and Higuchi (4) reported a resolution using the model A of 4-0.1-0.2 tL in diameters of 1.17-2.05 ~. MATERIALS AND METHODS
Apparatus. A Coulter counter model B 8 was used in conjunction with a 30 ~ orifice. The electrolyte was 3 % NaC1, and this was filtered through a 0.45 t~ Gelman 4 membrane filter and then through a 0.45 ~/0.1 u merebrahe filter sandwich. In order to disperse the latices and to obtain the highest possible proportion of singlet species, an ultrasonic disintegrator 5 was used. An Elliot 803 digital computer, programreed in Algol 60, was used to determine the normal distributions from the manufacturer's standard deviations and to predict particulate profiles at the TI/~, from log normal distributions approximating the Coulter results (see later). Materials. The " D o w " latices were used as received from the manufacturers. Suspensions were prepared in distilled water which had been freshly distilled from an Marketed by Coulter Electronics Ltd., Dunstable, Bedfordshire, England. 4 Marketed by Fisons Scientific Ltd., Loughborough, Leicestershire, England. 5M.S.E. Ultrasonic Disintegrator model 60 w, marketed by Measuring and Scientific Equipmerit Ltd., London, England.
TABLE III CALIBRATION FACTORS OBTAINED WITH D I F F E R E N T LATICES Latex
Polystyrene 1 Polystyrene 3 Polyvinyl toluene 5 Polyvinyltoluene 6
Manufacturers diameter (#)
Calibration constant K
0.714 1.099 1.857 3.49
0.325 0.342 0.403 0.449
all-glass still. The aluminum chloride 6 was reagent grade A1C13.6H20 which was assayed prior to use by the method given b y Vogel (22). Apparatus. All glassware used in the coagulation experiments was cleaned with chromic acid and rinsed six times in distilled water and twice in filtered distilled water. EXPERIMENTAL PROCEDURES
Calibration. The instrument was calibrated in the usual manner (23), using latices 1, 3, 5, and 6 in 3 % NaC1, the calibration constant decrease(i with the size of the latex. The results are given in Table I I I (means of two replicate determinations). Because of the differences in calibration constant, threshold settings corresponding to diameters between those of the means of the latices were calculated by drawing a graph of calibration constant against diameter and reading off intermediate K values. Prineen (24) has also reported different calibration constants for 19 ~ and 30 t~ orifice tubes, from different latices, but in this author's results there appears to be no definite trend. With the use of the calibration factor obtained from latex 3, latex 4 had a diameter of 1.17 ~. No difference in pulse height could be observed between latiees 1 and 2, and so the latter was not used further. Electron mierographs were taken of latices 1, 3, and 4, and these gave mean diameters of 0.71 t~, 1.10 t~, and 1.20 t~, respectively. The electron mierographs of latices 3 and 4 confirmed the Coulter results. In selecting the intervals at which counts should be taken, it was estimated that the error in setting the lower threshold scale using the ~Marketed by British Drug Houses Ltd., Poole, Dorset, England.
Journal of Colloid and Interface Science, Vo[. 28, No. 1, September 1968
USE OF THE COULTER COUNTER vernier was ±0.1 unit. A lower threshold value of ±0.1 unit corresponded to a diameter of ±0.002 t~, and the intervals selected were therefore 0.02 ~ for examining mixtures of particles of similar mean diameters. When mixtures of particles with ]arger mean diameter ratios were examined intervMs of 0.03 were used. I t was realized that talcing counts at such small intervals would probably mean a greater variation in the counts at any setting, but since it was desired to examine two latices, whose modes differed only by 0.07 ~, channels of 0.02 ~ were thought necessary to distinguish the peaks adequately. It was found that when differential counts were taken using the upper threshold setting, only approximately 75% of the total number of particles was counted. Cumulative counts when converted to differential gave greater than 95 % of the total, and these were employed. This also enabled coincidence corrections to be made easily. Coincidence Corrections. The working eoneentration of 2.5 X 107 particles per milliliter is equivalent to 1,250,000 particles in 0.05 ml, whieh is the manometer volume employed with the 30 u orifice tube. The instruction manual (23) permits a maximum count of 148,150 particles for 10% coincidence. Prineen and Kwolek (25) have reported that the coincidence correction factor for this combination is much too low and ~11 introduce large errors at high particle counts. They have suggested an alternative method of correction. Wales and Wilson (26) have also suggested an alternative correction procedure. The Coulter counter is unable to distinguish between multiplets and the equivalent number of discrete particles in coincident passage through the orifice tube, and this could give erroneous results. In the coagulation experiments, the mixture was diluted in filtered electrolyte to give a count of approximately 20,000, and coincidence corrections were applied by the Coulter method. Prineen and Kwolek (25) and Lines (27) have shown that at this concentration, differences in the various methods of coincidence correction are not significant..
75
)
7G
0 0 ~ •
00
99
o
o:,
o:2
0.3
o'-4
o'.s
de
d~
[JOGe DKAMETER
Fro. 1. Plot of cumulative probability per cent below diameter against log~ diameter for latices 3, 4, 5. - - O - - latex 3; - - @ - latex 4; - - e - - latex 5
Examination of Particle Size Distribution. Cumulative size distributions were obtained for latices 3, 4, and 5. A log probability plot of the results, as sho~m in Fig. 1, indicates a definite skewness toward larger particle sizes. An excellent straight-line fit is obtained for all particles below 85% probability, and the positive skew above t ~ s is probably due to higher species present in the suspension initially. Higuehi et al. (1) and the present authors (5) have previously noted that it is difficult to achieve greater than about 90 % of the singlet species. In order to determine how the distributions measured on the Coulter counter compared with those expected from the standard deviations as quoted by the manufacturers, histogram plots were drawn of particle frequency as a function of diameter, for the Coulter results and for normal distributions, calculated by computer, using a program based on a standard normal distribution procedure (28). The results for latices 3 and 4 are given in Fig. 2, and in the ease of the Coulter results the two curves are superimposed. Smaller size divisions are used
Journal of Colloid and Interface Science, Vol. 28, No. I, September 1968
76
MATTHEWS AND RHODES
o.3=
r'--
3
|,
COULTER 4 // // // // // // // //
0 -21
----,
/ / // // // //
I •
3
~
THEORY
// //// ////
2:
//
//
~ / / / t
ul
¢ U.
I I
0-11
'
I
// // /~//
/~/
, -,
// //] ," / / A // //I
["
/
/ / / / I . , ' t
II
/z
I/1
/-/ / /
o
I i.o
-
I'
7"7" i-i
i,2
DIAMETER ~$
i-a
,;io
a:,s
i.~o
FIG. 2. Theoretical and Coulter size distribution plots of particle frequency as a function of diameter D for latices 3 and 4.
for the theoretical results to keep the frequencies on the same scale. These results indicate a much broader distribution than the standard deviations would suggest. It may be seen that a monodisperse system with a mean of 1.10 ~ and a standard deviation of 0.006 should be almost completely distinguishable from one with a mean of 1.17 ~ and a standard deviation of 0.016, whereas in practice there is considerable overlap. Latex 5 was also found to have a broader distribution than expected from the quoted standard deviation, but despite this, it may be completely distinguished from latex 3, as Fig. 1 would indicate. Parfitt (19) has suggested that similar wide distributions obtained with the Coulter counter using a 30 orifice are associated with the geometry of small orifice tubes. Coagulation Investigations. Two coagulation experiments were performed using equal
proportions of latex 3 and 4 and equal proportions of latex 3 and 5. The total particulate concentration was 2.5 X 10v particles per milliliter and the p H was adjusted to 3.0 with filtered HC1. The flasks containing the suspensions were kept in a water bath at 25 ° 4- 0.1°C in order to reduce convection currents, which can accelerate coagulation of large particles. The method used was similar to that described previously (5). Coagulation was initiated by adding a concentrated filtered solution of aluminum chloride at pH 3.0 to give a working concentration of 0.1 M. Counts were taken periodically of the total concentration of particles and of the total singlet concentration. The experiment was halted at the T1/2 (when the singlet concentration had fallen by 3/4 and the total concentration by 1/~), b y dilution in filtered electrolyte. I t was found by theoretical calculations and by practical observation that sedimentation
Journal of Colloid and Interface Science, ¥o1. 28, No. 1, September 1968
USE OF THE COULTER COUNTER
77
Z
0-3,
0-2 =
72
g:
2 ii
LL
O,[i
O, I'0
I'1
DIAMETER
;1"2
1
11
1'3
i
Fro. 3. Comparative plots of particle frequency as a function of diameter D, for Coulter results and log normal distributions used in computer predictions, for latices 3 and 4. Shaded histogram, computer results; open histogram, Coulter results. could be ignored. The measured coagulation time was approximately twice that which would be expected from the Smoluchowski theory. This is in agreement with earlier results (5), and the siglnificanee of this has been also discussed in the earlier paper. Computer Programs. The program used to determine the normal distribution results was modified to evaluate log normal distribution. This was done using the equations given b y Keeping (29) for transforming the mean and standard deviations from normal to log normal, viz., = log~ ,u~ -- I~o-u2, /
O-x2)
% = /1//loge (1 ~- ttz - -2 ,
[3] [4]
where u is the mean and ¢ is the standard deviation, and the suffixes x and y refer
to the normal and log normal distributions, respectively. The distributions were adjusted about the mode which was calculated using the formula (30) Mode = antilog ( ~ - 2.3026 ~2).
[5]
A comparison of log normal distributions for two particles with modes of 1.10 t~ and 1.17 ~ and ¢~ values of 0.025, with the results of the particle size analysis of equal concentrations of the latex species is shown in Fig. 3. The shaded curve represents the computed results. The program does not take into account the proportion of large particles which are off the log normal distribution and appear on the right of the histogram. A computer program was written to evaluate the theoretical particulate profile at the T~/~ in terms of singlets, doublets, and triplets. This was achieved b y dividing
Journal of Colloid and Interface Science, ¥ol. 28, No. 1, September 1968
78
MATTtIEWS AND RHODES
800 COULTER 600 N
40<
COMPUTER 8O(
N 6O{
-I AB
40¢
BB
TRIPLETS
20(
~.o
,'~
,.~
Dp
,.~
FIG. 4. Plots of number of particles N as a function of diameter D for computer predictions and Coulter results in the coagulation of a mixture of 10,000 particles of latex 3 and 10,000 particles of latex 4 per unit volume, at the Tll~. the singlet distributions into small units and combining each unit of A with that of A again to give AA, with B to give AB, and similarly combining B with A and with B. The diameter of each species was calculated from the formula (shown for the AB doublet) Da~ = (DA3 + D~3)1/3.
[6]
The predicted particulate frequency was c~lculated by multiplying the frequency of e a c h u n i t of A with t h a t of each unit of B, with the appropriate figure in Table I. This frequency was then allocated into the size
fraction belonging to the resultan~ new diameter. It was realized that there are inevitable inaccuracies introduced by dividing the distributions into fractions since the mean fraction diameter was used for the whole fraction. B y using a small unit (in any case considerably smaller than the width of the band measured by the Coulter), these inaccuracies can be minimized. The triplet distributions were similarly calculated, by combining all the singlets fractions with all the doublets. Limitations of the computer prevented continuing to the quad-
Journal of Colloid and Interface Science, Vol. 28, No. 1, S e p t e m b e r 1968
USE OF T H E C O U L T E R C O U N T E R
79
rl
800
COULTER
N 6OC
40(
'°, L
i
•
A B
800
COMPUTER
N 600 AB 400
20O
I.o
L.a
1:6
I
D p
FIG. 5. P l o t s Of n u m b e r of p a r t i c l e s N as
I%
L 2:2
2"5
a f u n c t i o n of d i a m e t e r D for c o m p u t e r p r e d i c t i o n s
and Coulter results in the coagulation of a mixture of 10,003 particles of latex 3 and 10,000 particles of latex 5 per unit volume, at the Tt/2. ruplet stage but the principles are identical except that for the quadrupulets, singlet/ triplet and doublet/doublet reactions must be considered. Calculations at stages in the coagulation, other than the T1/2, can be similarly performed using Eq. [11, but when t becomes >TI/2 the errors involved in ignoring species above triplet and quadruplet become progressively larger.
is taken of positive skew clearly evident at the right of the A, B Coulter histogram. Because of the considerable overlap of the two species, no distinction can be made between the respective doublets and triplets, hut the peaks of the computer-predicted distributions show good agreement with the Coulter results. The doublet and triplet peaks of the Coulter histograms tend to be broader than predicted, cud this RESULTS AND DISCUSSION may be due partly to the particles outside The results of computer predictions and the limits of the computer histogram and Coulter counter readings at the T~/~ are partly to inaccuracies in the additive volume given in Fig. 4 for the 1.10 ~ and 1.17 t~ assumption. It could also be due to coinmixture and in Fig. 5 for the 110 ~ and 1.86 cidence, although at this concentration, it is mixture. considered unlikely. 1.10 tz and 1.17 tL Latices. The computer Despite the fact that the combined singlet predictions show good agreement with the distribution is broader than would be the Coulter counter results, when consideration ease for a pure species, it is still possible Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
80
MATTHEWS AND RHODES
both in theory and practice to separate the singlets and doublets almost entirely. This provides further confirmation of the technique described earlier (5), for evaluating stability ratios from the singlet depletion rate. 1.10 ~ and 1.86 t~ Latices. The computer predictions show that despite partial overlap between AAB, AB, and B, and between ABB and BB, the peaks should be definitely distinguishable. An examination of the Coulter counter results shows that, although some rounding of the peaks occurs, this is true and that the agreement between the two curves is again good. The total number of each doublet and triplet species present at the T1/2 seems greater than predicted; this is probably due to higher species present at the start of the coagulation experiment. No attempt was made in this experiment to correct for these higher species. The unevenness of the Coulter counter distribution is probably due to the very narrow channel limits selected for these experiments, and it is suggested that a multiehannel analyzer, such as was used by Ho and Higuehi (4), would probably give better results. In order to evaluate comparative reaction constants for the A-A, A-B, and B-B reactions, it is necessary to be able to detect the AA, AB, and BB doublets. Of these only AA can be detected with no overlap. AB overlaps the B singlet and the AAB triplet, and BB overlaps the ABB triplet. It would seem from from these results that, unless particles with a narrower size distribution can be obtained, there is no possibility of determining these comparative constants. A comparison between the computer and Coulter results suggests a greater proportion of the A singlet remaining at the T1/2 than the B. This would suggest that the larger particles are aggregating more rapidly than the smaller. In order to confirm this, the relative proportions of the two singlet species were determined at different fractions of the T1/2 • The results are given in Table IV. They indicate, at first inspection, an initial higher depletion rate of the A species and an ultimate higher depletion rate of the B species. However, in measuring B,
TABLE THE
VARIOUS
I~ELATIVE
IV PROPORTIONS
OF
EACH
SINGLET SPECIES AS A FUNCTION OF T I M E IN THE COAGULATION OF
LATICES 3 AND 5
Latex 3
t/Tl/~
0 0.36 0.43 0.91 1.00
Latex 5
A slnglet
B singlet + • some AB doublet
0.500 0.497 0.472 0.527 0.542
0.500 0.503 0.528 0.473 0.458
there is the unavoidable overlap with the AB doublet. Initially the concentration of AB doublets is zero but, if Eq. [1] is evaluated for the doublet, it may be seen that the concentration of doublets reaches a maximum at t/T1/2 = 0.5, and then falls. This would explain the initial apparently slower depletion rate of the B singlets. The influence of particle size on stability has been reviewed by Verwey and Overbeek (31). The general rule of increasing stability with increasing dimensions is not always obeyed when the particle size is large in comparison to the dimensions of the double layer, i.e., in high electrolyte concentrations. This may afford an explanation of the results obtained in this experiment although the effect may be due to the differences of chemical constitution of the particles or of stabilizers present. Further experiments are planned to investigate the effect of particle size on the coagulation behavior of supracolloidal suspensions. SUMMARY It has been demonstrated that the size distributions of polystyrene and polyvinyl toluene latices are closely log normal when measured on the Coulter counter but that the particle size spread is much greater than would be expected from normal distributions, with the standard deviations quoted by the manufacturer. It has been shown that peaks of the size distributions of a mixture of two species, with modal Coulter diameters of 1.10 t~ and 1.17 ~, can be detected, but that considerable overlap of the distributions occurs. The change in particle size profile during
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
USE OF THE COULTER COUNTER the coagulation of this mixture and of a mixture of two latices of wider diameter ratios, has been c o m p a r e d with computerpredicted profiles and good agreement was found. Evidence was found that. the large particles coagulated more rapidly than the smaller. Because of the overlap of the successive size ranges of the different combinations, it has not been possible to evaluate the reaction constants for coagulation between large and small particles. This paper has d e m o n s t r a t e d the utility of the Coulter counter in investigating the coagulation behavior of supraeolloidal monodisperse suspensions. I t is suggested t h a t the resolution of the instrument could be i m p r o v e d b y the use of multichannel pulse analyzer equipment and, if monodisperse systems can be found with narrower Coulter counter distributions, the possibility of detecting the various combinations will be greatly enhanced. ACKNOWLEDGMENTS We thank Mr. S. V. Lincoln of the Department of Mathematics, Portsmouth College of Technology, for statistical advice and the staff of the computer section of the Department of Mathematics for helpful discussions on the computer programs. We thank Mr. J. A. Waters, of I.C.I. Limited, Wexham Road, Slough, Bucks, for the electron mierographs. REFEI~ENCES 1. HIGUCHI, W. I., OKADA, P~., STELTER, G. A., AND LEMBERGER, A. P., J. Pharm. Sci., 52, 49 (1963). 2. S'WIFT, D. L., AND FRIEDLANDEII;, S. K., J. Colloid Sci. 19, 621 (1964). 3. HIGUCHI,W. I., RHEE, W. O., AND FLANAGHAN, D. R., J. Pharm. Sei. 54, 510 (1965). 4. Ho, N. F. H., AND HIGUCHI, W . I., J. Pharrn. Sci. 56, 248 (1967). 5. MATTHEVVS, B. A., AND RHODES, C. T., J. Pharm. Sci. 57, 557 (1968). 6. MATTHEV,rS, B. A., AND RHODES, C. T., d. Pharra. Sci. 57, 569 (1968). 7. VON SMOLUCHOVVSKI,M., Z. Physih. Chem. (Leipzig) 92, 129 (1917). 8. OVERBEEK, J. Th. G., In H. R. Kruyt, ed.,
9. 10. 11.
12.
$1
"Colloid Science", Vol. I., p. 278. Elsevier, Amsterdam, 1952. 0VEI~B~EI;, J. Th. G., In H. R. Kruyt, ed., "Colloid Science," Vol. I, p. 282. Elsevier, Amsterdam, 1952. TemI
13. WIEGNER, G., AND TUORILA, P., Kolloid Z. 38, 3 (1926). 14. T~:ORIL.%P., Kolloidchem. Beih. 22, 191 (1926). 15. BLACKLEY, ]). C., "High Polymer Latices", Vols. I and II. Maclaren and Sons Ltd., London, 1966. 16. ALLEN, W., In "Particle Size Analysis Con~ ference, 1966", p. 110. Society for Analytical Chemistry, London, 1967. 17. KUBITSCHEK, H. E., Nature, 182, 234 (July 26, 1958). 18. WALES, M., J. Phys. Chem. 66, 1768 (1962). 19. PARFITT, G. D.,Discussions Faraday Soc. 42, 142 (1967). 20. WACItTEL, R. E., AND L& MER, r . K., J. Colloid Sci. 17, 531 (1962). 21. DAVIDSON, J. A., MACOSEO, C. W., AND COLLINS, E. A., J. Colloid and Interface Sci. 25, 381 (1967). 22. VOGEL, A. I., "Quantitative Inorganic Analysis" 3rd ed., p. 436. Longmans, London, 1961. 23. "Coulter Counter Model B Instruction Man'ual". Coulter Electronics Limited, Dunstable, Bedfordshire, England. 24. PRINCEN,L. H., Rev. Sci. Instr. 37, 1416 (1966). 25. PRINCEN, L. H., AND KVVOLEI(., W. F., Rev. Sei. Instr. 36, 646 (1965). 26. WALES,M., ANDWILSON, J. N., Rev. Sci. Instr.
32, 1132 (1962). 27. LINES, 1~. W., Conference on General Applications of the Coulter Counter, London, 1968. 28. ALGORITH~'I226 "Collected Algorithms from C.A.C.M." 29. KEEPING, F. S., "Statistical Inference," p. 89. Van Nostrand, New Jersey, 1962. 30. DIEM, K., ed., "Documenta Geigy", 6th ed., p. 164. Geigy Pharmaceutical Company Limited, Manchester, 1962. 31. VER~,VEY,E. J. W., AND OVERBEEK, J. Th. G., "Theory of the StabilRy of Lyophobic Colloids," p. 177. Elsevier, Amsterdam, 1948.
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
Physical Pharmacy Department, School of Pharmacy, Portsmouth College of Technology, Portsmouth~ Hants, England Received May 9, 1968 The Coulter counter has been used to investigate the particle size distributions of polystyrene and polyvinyl toluene latices. The distributions have been shown to be closely log normal over the maior part of their extent. Mixtures of two latex species have been examined in cases where overlap of the distributions does and does not occur. It has been shown that, in agreement with the conclusions of earlier workers, the measured distributions are broader than would be expected from the manufacturer's figures. The change in particle size profile of mixtures has been examined when coagulation occurs, aud this has been compared to computer-predicted profiles. The significance of the results in terms of the coagulation behavior of supracolloidal suspensions is discussed. INTRODUCTION
A recent series of papers involving the use of the Coulter counter (1-6) with monodisperse polymer latices has demonstrated the value of this technique for investigating the coagulation kinetic theory of monodisperse systems. Although most suspensions of materials such as drugs are heterodisperse, it has been shown (6) t h a t a comparison can be made between coagulation in suspensions and in model systems. A mixture of two monodisperse particulate systems of different mean particle diameters can be considered as an interim state between mono- and heterodispersity. An examination of the coagulation behavior of such mixtures is likely to give valuable information as to the relative coagulation behavior of smaJ1 and large particles. I n the present paper the authors have used the Coulter counter to examine such systems and to test the kinetic theory for mixtures of two mono-
disperse particulate region of 1-2 ~.
suspensions,
in
the
THEORY The kinetics of coagulation of a lyophobic colloid was considered by Von Smoluchowski (7) in his theory of rapid coagulation. The process was considered as a diffusion phenomenon, caused b y Browian motion with no repulsion between the particles (8). H e derived the following equation to describe the rate of change of concentration of any particle Nk with time t (9).
No ( t ~ k-1 N~ =
\T~/~!
[1]
1 • T1/2! where No is the number of primary particles originally present and T1/2 is the time taken for the total number of particles to decrease by one half. Equation [t] has been tested experimentally on a gold sol by Turkevich (10), who used gelatin to halt the coagulation at
This paper was presented to the Conference on "General Applications of the Coulter Counter", London, March 6th, 1968. 71 Journal of Colloid and Interface
Ncience, Vol. 28, No. 1, Sep%ember 1968
72
MATTHEWS AND RHODES
different intervals and counted each species under an electron microscope, and by Higuchi et al. (1), who used the Coulter counter to differentiate the relative amounts of each species, up to lo = 3, for a uniform polystyrene latex. The present authors (5) have also used Eq. [1] for the singlet to investigate the extent of departure from the Smoluehowski rate for rapid coagulation. Most colloidal and supracolloidal disperse systems are polydispersed and likely to show considerable variations from the Smoluchowski theory, since it has been known for many years (11) that, in such systems, the smaller particles disappear much more rapidly than the larger ones. To account for this, Miiller (12) extended the Smoluchowski theory and derived the following equation for the coagulation of a mixture of large and small particles.
No
[2]
VR 2
(VR2N~-t-1)(1
+ T1/~!
1
where n is the total number of small particles at time t, No is the initial number of large particles; no is the initial number of small particles; NR = No~no ; VR = R / r = the ratio of the radii of the two primary species; and T1/2 is the coagulation time of the large particles. This theory has been extensively tested for true colloids by Wiegner and Tuorila (13, 14), but little information is available on how mixtures of supracolloid~l particles will behave. The growth of high polymer latex technology (15) and the advent of the Coulter counter (16) here made such a study possible. The ability to distinguish between each combination, during the coagulation of a mixture of 2 monodisperse particles, is likely to be dependent upon the following four factors. 1. Choice of O p t i m u m Particle Size Ratio.
The course of a coagulation reaction between two species can be predicted from the kinetic
theory outlined above and from probability calculations. As the reaction proceeds, the number of possible combinations becomes extremely great. By restricting the study t 9 the early stages (up to the Tlj2) and neglecting species higher than the triplet, the process may however be simplified. The number of possible combinations, the probability of formation of each multiplet, and the relative theoretical proportions of each, at t = 0 and t = TI/2, are shown in Table I. The differences of probability for collisions between species of different sizes have been neglected although this can be computed from the Mfiller theory. The relative probabilities of each species have been calculated, on the assumption that only binary reactions occur. This assumption is probably valid in considering the coagulation of latices in the concentration range of 107-10s particles per milliliter (1, 5), where the coagulation time is 4.70.47 hours. However, it is possible to obtain true colloids with a concentration greater than 10I4 particles per milliliter (9), when the coagulation time is about ~000 sec. In this case many higher order reactions will occur. The response of the Coulter counter between its working limits is proportional to the volume of the particle measured (17), and for each species to be detectable, it is desirable that there should be no overlap between the volume ranges of successive particle combinations. In selecting the most suitable particle size ratios for fulfilling this criterion, it is evident that there are two possible types of overlap. a) When the volume of the larger species B is equal to, or less than three times the volume of, the smaller species A. b) When the volume of the larger species B is greater than three times the volume of A. For type (a), there is the possibility of direct overlap, e.g., when the volume of B = twice the volume of A, it will be impossible to distinguish the AA doublet from the B singlet. A volume ratio of 3/1 is equivalent to a diameter ratio of only 1.442/ 1, and with two species of such close particle sizes, the Mfiller theory would predict very little difference in collision rates be-
Journal of Colloid and Interface Science, Vo1.28, No. 1, September 1968
USE OF THE COULTER COUNTER
73
TABLE I R E L A T I V E PROPORTION OF EACH ~PECIES AS A FUNCTION OF T I M E IN THE COAGULATION OF A M I X T U R E OF T w o
SUSPENSIONS Relative proportions at different times
Species
Ratios
Probability
t = 0 Total
Singlet Doublet Triplet
A1 B 1 AA 1 AB 2 BB 1 AAA 1 AAB 3 ABB 3 BBB 1
0.5 0.5 0.25 0.5 0.25 0.125 0.375 0.375 0.125
tween the A-A reaction and A-B reaction. However, if two particle sizes in the range can be selected with little or no overlap, it would provide a means of checking this theory. I n the case of type (b) the most important overlap will occur between mixtures of various proportions of A and B, and pure B aggregates. This will be determinedlargely b y the degree of monodispersity of A and B. T h e singlets, doublets, triplets, and quadruplets of the pure species should be readily distinguishable.
t = Tl/e
Individual
0.50 0.50 -----_ --
1.0 --
--
Total
0.25 0.125
0.0625
TABLE
"DOT"
Individual
0.125 0.125 0.03125 0.0625 0.03125 0.0078125 0.0234375 0.0234375 0.0078125
II
LATICES USED IN THE COAGULATION
EXPERIMENTS Latex
Polystyrene Polystyrene Polystyrene Polystyrene Polyvinyl toluene 5 Polyvinyl toluene 6
Particle diameter (#)
1 2 3 4
Standard deviation
Relative S,D. (%)
0.714 0.81 1.099 1.305 i. 857
0.0053 0.0063 0.0059 0.0158 0.007
0.74 0.78 0.54 I. 21 0.38
3.49
0.017
0.49
2. The Particle Size Spread of the Two Species. The particles selected for this study are the polystyrene and polyvinyl toluene latices produced by the D o T Chemical C o m p a n y ? The particle diameters and standard deviations as stated b y the manufacturers are shown in Table II. The fact that standard deviations are quoted for these latices would seem to predicate a normal Gaussian distribution. Wales (18) examined polyisoprene latex samples b y means of the Coulter counter and found t h e m quite closely log normal. Parfitt (19) has quoted a significant positive skewness for "DOT" monodisperse latices when measured on the Coulter counter. Wachtel and La M e t (20) found that the size distribution of a 1.17 t~ polystyrene latex, when measured on the Coulter counter, was broader t h a n would be ex2 Marketed in Europe by Serva EntwicMungs Labor, Heidelberg, Germany.
petted from the standard deviation obtained from the electron microscope. Similar results using flow ultramieroseopy were reported by Davidson et al. (21). We have assumed that, up to the triplet state, the volume of each multiplet is equal to the sum of the volumes of each constituent particle. Above the quadruplet, different configurations and the occluded space in the interiors of the clusters is likely to nullify this assumption. However, Ho and Higuchi (4) found that, using a multiehatmel analyzer to record the pulses, they could distinguish oetuplets of a monodisperse latex. With the triplets, quadrupulets, and above, different configurations are possible, and, for example, a linear quadrupulet m a y have a different Coulter volume from a compact one. Allen (16) has shown t h a t for a rod-shaped particle totally enclosed
Journal of Colloid and Interface Science, VoI. 28, No. 1, September 1968
74
MATTHEWS AND RHODES
within the orifice, the Coulter size may be 7.8% large when the equivalent spherical diameter of the particle is 40 % of that of the orifice. When it is only 10% of the orifice diameter, the error is about 0.5 %. Since all our measurements were made at about this percentage, or less, the different configurations are not likely to affect results significantly.
3. The Proportion of Each Species Present Initially. Equation [2] can be used to calculate the theoretical course of the eoagulation for any ratio of No/no, but the ability of the Coulter counter to detect the species present in smaller number must be considered in selecting this ratio. 4. Resolution of the Coulter Counter. The resolution of the Coulter counter is likely to be very critical in governing whether species with volumes very close together can be distinguished. Ho and Higuchi (4) reported a resolution using the model A of 4-0.1-0.2 tL in diameters of 1.17-2.05 ~. MATERIALS AND METHODS
Apparatus. A Coulter counter model B 8 was used in conjunction with a 30 ~ orifice. The electrolyte was 3 % NaC1, and this was filtered through a 0.45 t~ Gelman 4 membrane filter and then through a 0.45 ~/0.1 u merebrahe filter sandwich. In order to disperse the latices and to obtain the highest possible proportion of singlet species, an ultrasonic disintegrator 5 was used. An Elliot 803 digital computer, programreed in Algol 60, was used to determine the normal distributions from the manufacturer's standard deviations and to predict particulate profiles at the TI/~, from log normal distributions approximating the Coulter results (see later). Materials. The " D o w " latices were used as received from the manufacturers. Suspensions were prepared in distilled water which had been freshly distilled from an Marketed by Coulter Electronics Ltd., Dunstable, Bedfordshire, England. 4 Marketed by Fisons Scientific Ltd., Loughborough, Leicestershire, England. 5M.S.E. Ultrasonic Disintegrator model 60 w, marketed by Measuring and Scientific Equipmerit Ltd., London, England.
TABLE III CALIBRATION FACTORS OBTAINED WITH D I F F E R E N T LATICES Latex
Polystyrene 1 Polystyrene 3 Polyvinyl toluene 5 Polyvinyltoluene 6
Manufacturers diameter (#)
Calibration constant K
0.714 1.099 1.857 3.49
0.325 0.342 0.403 0.449
all-glass still. The aluminum chloride 6 was reagent grade A1C13.6H20 which was assayed prior to use by the method given b y Vogel (22). Apparatus. All glassware used in the coagulation experiments was cleaned with chromic acid and rinsed six times in distilled water and twice in filtered distilled water. EXPERIMENTAL PROCEDURES
Calibration. The instrument was calibrated in the usual manner (23), using latices 1, 3, 5, and 6 in 3 % NaC1, the calibration constant decrease(i with the size of the latex. The results are given in Table I I I (means of two replicate determinations). Because of the differences in calibration constant, threshold settings corresponding to diameters between those of the means of the latices were calculated by drawing a graph of calibration constant against diameter and reading off intermediate K values. Prineen (24) has also reported different calibration constants for 19 ~ and 30 t~ orifice tubes, from different latices, but in this author's results there appears to be no definite trend. With the use of the calibration factor obtained from latex 3, latex 4 had a diameter of 1.17 ~. No difference in pulse height could be observed between latiees 1 and 2, and so the latter was not used further. Electron mierographs were taken of latices 1, 3, and 4, and these gave mean diameters of 0.71 t~, 1.10 t~, and 1.20 t~, respectively. The electron mierographs of latices 3 and 4 confirmed the Coulter results. In selecting the intervals at which counts should be taken, it was estimated that the error in setting the lower threshold scale using the ~Marketed by British Drug Houses Ltd., Poole, Dorset, England.
Journal of Colloid and Interface Science, Vo[. 28, No. 1, September 1968
USE OF THE COULTER COUNTER vernier was ±0.1 unit. A lower threshold value of ±0.1 unit corresponded to a diameter of ±0.002 t~, and the intervals selected were therefore 0.02 ~ for examining mixtures of particles of similar mean diameters. When mixtures of particles with ]arger mean diameter ratios were examined intervMs of 0.03 were used. I t was realized that talcing counts at such small intervals would probably mean a greater variation in the counts at any setting, but since it was desired to examine two latices, whose modes differed only by 0.07 ~, channels of 0.02 ~ were thought necessary to distinguish the peaks adequately. It was found that when differential counts were taken using the upper threshold setting, only approximately 75% of the total number of particles was counted. Cumulative counts when converted to differential gave greater than 95 % of the total, and these were employed. This also enabled coincidence corrections to be made easily. Coincidence Corrections. The working eoneentration of 2.5 X 107 particles per milliliter is equivalent to 1,250,000 particles in 0.05 ml, whieh is the manometer volume employed with the 30 u orifice tube. The instruction manual (23) permits a maximum count of 148,150 particles for 10% coincidence. Prineen and Kwolek (25) have reported that the coincidence correction factor for this combination is much too low and ~11 introduce large errors at high particle counts. They have suggested an alternative method of correction. Wales and Wilson (26) have also suggested an alternative correction procedure. The Coulter counter is unable to distinguish between multiplets and the equivalent number of discrete particles in coincident passage through the orifice tube, and this could give erroneous results. In the coagulation experiments, the mixture was diluted in filtered electrolyte to give a count of approximately 20,000, and coincidence corrections were applied by the Coulter method. Prineen and Kwolek (25) and Lines (27) have shown that at this concentration, differences in the various methods of coincidence correction are not significant..
75
)
7G
0 0 ~ •
00
99
o
o:,
o:2
0.3
o'-4
o'.s
de
d~
[JOGe DKAMETER
Fro. 1. Plot of cumulative probability per cent below diameter against log~ diameter for latices 3, 4, 5. - - O - - latex 3; - - @ - latex 4; - - e - - latex 5
Examination of Particle Size Distribution. Cumulative size distributions were obtained for latices 3, 4, and 5. A log probability plot of the results, as sho~m in Fig. 1, indicates a definite skewness toward larger particle sizes. An excellent straight-line fit is obtained for all particles below 85% probability, and the positive skew above t ~ s is probably due to higher species present in the suspension initially. Higuehi et al. (1) and the present authors (5) have previously noted that it is difficult to achieve greater than about 90 % of the singlet species. In order to determine how the distributions measured on the Coulter counter compared with those expected from the standard deviations as quoted by the manufacturers, histogram plots were drawn of particle frequency as a function of diameter, for the Coulter results and for normal distributions, calculated by computer, using a program based on a standard normal distribution procedure (28). The results for latices 3 and 4 are given in Fig. 2, and in the ease of the Coulter results the two curves are superimposed. Smaller size divisions are used
Journal of Colloid and Interface Science, Vol. 28, No. I, September 1968
76
MATTHEWS AND RHODES
o.3=
r'--
3
|,
COULTER 4 // // // // // // // //
0 -21
----,
/ / // // // //
I •
3
~
THEORY
// //// ////
2:
//
//
~ / / / t
ul
¢ U.
I I
0-11
'
I
// // /~//
/~/
, -,
// //] ," / / A // //I
["
/
/ / / / I . , ' t
II
/z
I/1
/-/ / /
o
I i.o
-
I'
7"7" i-i
i,2
DIAMETER ~$
i-a
,;io
a:,s
i.~o
FIG. 2. Theoretical and Coulter size distribution plots of particle frequency as a function of diameter D for latices 3 and 4.
for the theoretical results to keep the frequencies on the same scale. These results indicate a much broader distribution than the standard deviations would suggest. It may be seen that a monodisperse system with a mean of 1.10 ~ and a standard deviation of 0.006 should be almost completely distinguishable from one with a mean of 1.17 ~ and a standard deviation of 0.016, whereas in practice there is considerable overlap. Latex 5 was also found to have a broader distribution than expected from the quoted standard deviation, but despite this, it may be completely distinguished from latex 3, as Fig. 1 would indicate. Parfitt (19) has suggested that similar wide distributions obtained with the Coulter counter using a 30 orifice are associated with the geometry of small orifice tubes. Coagulation Investigations. Two coagulation experiments were performed using equal
proportions of latex 3 and 4 and equal proportions of latex 3 and 5. The total particulate concentration was 2.5 X 10v particles per milliliter and the p H was adjusted to 3.0 with filtered HC1. The flasks containing the suspensions were kept in a water bath at 25 ° 4- 0.1°C in order to reduce convection currents, which can accelerate coagulation of large particles. The method used was similar to that described previously (5). Coagulation was initiated by adding a concentrated filtered solution of aluminum chloride at pH 3.0 to give a working concentration of 0.1 M. Counts were taken periodically of the total concentration of particles and of the total singlet concentration. The experiment was halted at the T1/2 (when the singlet concentration had fallen by 3/4 and the total concentration by 1/~), b y dilution in filtered electrolyte. I t was found by theoretical calculations and by practical observation that sedimentation
Journal of Colloid and Interface Science, ¥o1. 28, No. 1, September 1968
USE OF THE COULTER COUNTER
77
Z
0-3,
0-2 =
72
g:
2 ii
LL
O,[i
O, I'0
I'1
DIAMETER
;1"2
1
11
1'3
i
Fro. 3. Comparative plots of particle frequency as a function of diameter D, for Coulter results and log normal distributions used in computer predictions, for latices 3 and 4. Shaded histogram, computer results; open histogram, Coulter results. could be ignored. The measured coagulation time was approximately twice that which would be expected from the Smoluchowski theory. This is in agreement with earlier results (5), and the siglnificanee of this has been also discussed in the earlier paper. Computer Programs. The program used to determine the normal distribution results was modified to evaluate log normal distribution. This was done using the equations given b y Keeping (29) for transforming the mean and standard deviations from normal to log normal, viz., = log~ ,u~ -- I~o-u2, /
O-x2)
% = /1//loge (1 ~- ttz - -2 ,
[3] [4]
where u is the mean and ¢ is the standard deviation, and the suffixes x and y refer
to the normal and log normal distributions, respectively. The distributions were adjusted about the mode which was calculated using the formula (30) Mode = antilog ( ~ - 2.3026 ~2).
[5]
A comparison of log normal distributions for two particles with modes of 1.10 t~ and 1.17 ~ and ¢~ values of 0.025, with the results of the particle size analysis of equal concentrations of the latex species is shown in Fig. 3. The shaded curve represents the computed results. The program does not take into account the proportion of large particles which are off the log normal distribution and appear on the right of the histogram. A computer program was written to evaluate the theoretical particulate profile at the T~/~ in terms of singlets, doublets, and triplets. This was achieved b y dividing
Journal of Colloid and Interface Science, ¥ol. 28, No. 1, September 1968
78
MATTtIEWS AND RHODES
800 COULTER 600 N
40<
COMPUTER 8O(
N 6O{
-I AB
40¢
BB
TRIPLETS
20(
~.o
,'~
,.~
Dp
,.~
FIG. 4. Plots of number of particles N as a function of diameter D for computer predictions and Coulter results in the coagulation of a mixture of 10,000 particles of latex 3 and 10,000 particles of latex 4 per unit volume, at the Tll~. the singlet distributions into small units and combining each unit of A with that of A again to give AA, with B to give AB, and similarly combining B with A and with B. The diameter of each species was calculated from the formula (shown for the AB doublet) Da~ = (DA3 + D~3)1/3.
[6]
The predicted particulate frequency was c~lculated by multiplying the frequency of e a c h u n i t of A with t h a t of each unit of B, with the appropriate figure in Table I. This frequency was then allocated into the size
fraction belonging to the resultan~ new diameter. It was realized that there are inevitable inaccuracies introduced by dividing the distributions into fractions since the mean fraction diameter was used for the whole fraction. B y using a small unit (in any case considerably smaller than the width of the band measured by the Coulter), these inaccuracies can be minimized. The triplet distributions were similarly calculated, by combining all the singlets fractions with all the doublets. Limitations of the computer prevented continuing to the quad-
Journal of Colloid and Interface Science, Vol. 28, No. 1, S e p t e m b e r 1968
USE OF T H E C O U L T E R C O U N T E R
79
rl
800
COULTER
N 6OC
40(
'°, L
i
•
A B
800
COMPUTER
N 600 AB 400
20O
I.o
L.a
1:6
I
D p
FIG. 5. P l o t s Of n u m b e r of p a r t i c l e s N as
I%
L 2:2
2"5
a f u n c t i o n of d i a m e t e r D for c o m p u t e r p r e d i c t i o n s
and Coulter results in the coagulation of a mixture of 10,003 particles of latex 3 and 10,000 particles of latex 5 per unit volume, at the Tt/2. ruplet stage but the principles are identical except that for the quadrupulets, singlet/ triplet and doublet/doublet reactions must be considered. Calculations at stages in the coagulation, other than the T1/2, can be similarly performed using Eq. [11, but when t becomes >TI/2 the errors involved in ignoring species above triplet and quadruplet become progressively larger.
is taken of positive skew clearly evident at the right of the A, B Coulter histogram. Because of the considerable overlap of the two species, no distinction can be made between the respective doublets and triplets, hut the peaks of the computer-predicted distributions show good agreement with the Coulter results. The doublet and triplet peaks of the Coulter histograms tend to be broader than predicted, cud this RESULTS AND DISCUSSION may be due partly to the particles outside The results of computer predictions and the limits of the computer histogram and Coulter counter readings at the T~/~ are partly to inaccuracies in the additive volume given in Fig. 4 for the 1.10 ~ and 1.17 t~ assumption. It could also be due to coinmixture and in Fig. 5 for the 110 ~ and 1.86 cidence, although at this concentration, it is mixture. considered unlikely. 1.10 tz and 1.17 tL Latices. The computer Despite the fact that the combined singlet predictions show good agreement with the distribution is broader than would be the Coulter counter results, when consideration ease for a pure species, it is still possible Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
80
MATTHEWS AND RHODES
both in theory and practice to separate the singlets and doublets almost entirely. This provides further confirmation of the technique described earlier (5), for evaluating stability ratios from the singlet depletion rate. 1.10 ~ and 1.86 t~ Latices. The computer predictions show that despite partial overlap between AAB, AB, and B, and between ABB and BB, the peaks should be definitely distinguishable. An examination of the Coulter counter results shows that, although some rounding of the peaks occurs, this is true and that the agreement between the two curves is again good. The total number of each doublet and triplet species present at the T1/2 seems greater than predicted; this is probably due to higher species present at the start of the coagulation experiment. No attempt was made in this experiment to correct for these higher species. The unevenness of the Coulter counter distribution is probably due to the very narrow channel limits selected for these experiments, and it is suggested that a multiehannel analyzer, such as was used by Ho and Higuehi (4), would probably give better results. In order to evaluate comparative reaction constants for the A-A, A-B, and B-B reactions, it is necessary to be able to detect the AA, AB, and BB doublets. Of these only AA can be detected with no overlap. AB overlaps the B singlet and the AAB triplet, and BB overlaps the ABB triplet. It would seem from from these results that, unless particles with a narrower size distribution can be obtained, there is no possibility of determining these comparative constants. A comparison between the computer and Coulter results suggests a greater proportion of the A singlet remaining at the T1/2 than the B. This would suggest that the larger particles are aggregating more rapidly than the smaller. In order to confirm this, the relative proportions of the two singlet species were determined at different fractions of the T1/2 • The results are given in Table IV. They indicate, at first inspection, an initial higher depletion rate of the A species and an ultimate higher depletion rate of the B species. However, in measuring B,
TABLE THE
VARIOUS
I~ELATIVE
IV PROPORTIONS
OF
EACH
SINGLET SPECIES AS A FUNCTION OF T I M E IN THE COAGULATION OF
LATICES 3 AND 5
Latex 3
t/Tl/~
0 0.36 0.43 0.91 1.00
Latex 5
A slnglet
B singlet + • some AB doublet
0.500 0.497 0.472 0.527 0.542
0.500 0.503 0.528 0.473 0.458
there is the unavoidable overlap with the AB doublet. Initially the concentration of AB doublets is zero but, if Eq. [1] is evaluated for the doublet, it may be seen that the concentration of doublets reaches a maximum at t/T1/2 = 0.5, and then falls. This would explain the initial apparently slower depletion rate of the B singlets. The influence of particle size on stability has been reviewed by Verwey and Overbeek (31). The general rule of increasing stability with increasing dimensions is not always obeyed when the particle size is large in comparison to the dimensions of the double layer, i.e., in high electrolyte concentrations. This may afford an explanation of the results obtained in this experiment although the effect may be due to the differences of chemical constitution of the particles or of stabilizers present. Further experiments are planned to investigate the effect of particle size on the coagulation behavior of supracolloidal suspensions. SUMMARY It has been demonstrated that the size distributions of polystyrene and polyvinyl toluene latices are closely log normal when measured on the Coulter counter but that the particle size spread is much greater than would be expected from normal distributions, with the standard deviations quoted by the manufacturer. It has been shown that peaks of the size distributions of a mixture of two species, with modal Coulter diameters of 1.10 t~ and 1.17 ~, can be detected, but that considerable overlap of the distributions occurs. The change in particle size profile during
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
USE OF THE COULTER COUNTER the coagulation of this mixture and of a mixture of two latices of wider diameter ratios, has been c o m p a r e d with computerpredicted profiles and good agreement was found. Evidence was found that. the large particles coagulated more rapidly than the smaller. Because of the overlap of the successive size ranges of the different combinations, it has not been possible to evaluate the reaction constants for coagulation between large and small particles. This paper has d e m o n s t r a t e d the utility of the Coulter counter in investigating the coagulation behavior of supraeolloidal monodisperse suspensions. I t is suggested t h a t the resolution of the instrument could be i m p r o v e d b y the use of multichannel pulse analyzer equipment and, if monodisperse systems can be found with narrower Coulter counter distributions, the possibility of detecting the various combinations will be greatly enhanced. ACKNOWLEDGMENTS We thank Mr. S. V. Lincoln of the Department of Mathematics, Portsmouth College of Technology, for statistical advice and the staff of the computer section of the Department of Mathematics for helpful discussions on the computer programs. We thank Mr. J. A. Waters, of I.C.I. Limited, Wexham Road, Slough, Bucks, for the electron mierographs. REFEI~ENCES 1. HIGUCHI, W. I., OKADA, P~., STELTER, G. A., AND LEMBERGER, A. P., J. Pharm. Sci., 52, 49 (1963). 2. S'WIFT, D. L., AND FRIEDLANDEII;, S. K., J. Colloid Sci. 19, 621 (1964). 3. HIGUCHI,W. I., RHEE, W. O., AND FLANAGHAN, D. R., J. Pharm. Sei. 54, 510 (1965). 4. Ho, N. F. H., AND HIGUCHI, W . I., J. Pharrn. Sci. 56, 248 (1967). 5. MATTHEVVS, B. A., AND RHODES, C. T., J. Pharm. Sci. 57, 557 (1968). 6. MATTHEV,rS, B. A., AND RHODES, C. T., d. Pharra. Sci. 57, 569 (1968). 7. VON SMOLUCHOVVSKI,M., Z. Physih. Chem. (Leipzig) 92, 129 (1917). 8. OVERBEEK, J. Th. G., In H. R. Kruyt, ed.,
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$1
"Colloid Science", Vol. I., p. 278. Elsevier, Amsterdam, 1952. 0VEI~B~EI;, J. Th. G., In H. R. Kruyt, ed., "Colloid Science," Vol. I, p. 282. Elsevier, Amsterdam, 1952. TemI
13. WIEGNER, G., AND TUORILA, P., Kolloid Z. 38, 3 (1926). 14. T~:ORIL.%P., Kolloidchem. Beih. 22, 191 (1926). 15. BLACKLEY, ]). C., "High Polymer Latices", Vols. I and II. Maclaren and Sons Ltd., London, 1966. 16. ALLEN, W., In "Particle Size Analysis Con~ ference, 1966", p. 110. Society for Analytical Chemistry, London, 1967. 17. KUBITSCHEK, H. E., Nature, 182, 234 (July 26, 1958). 18. WALES, M., J. Phys. Chem. 66, 1768 (1962). 19. PARFITT, G. D.,Discussions Faraday Soc. 42, 142 (1967). 20. WACItTEL, R. E., AND L& MER, r . K., J. Colloid Sci. 17, 531 (1962). 21. DAVIDSON, J. A., MACOSEO, C. W., AND COLLINS, E. A., J. Colloid and Interface Sci. 25, 381 (1967). 22. VOGEL, A. I., "Quantitative Inorganic Analysis" 3rd ed., p. 436. Longmans, London, 1961. 23. "Coulter Counter Model B Instruction Man'ual". Coulter Electronics Limited, Dunstable, Bedfordshire, England. 24. PRINCEN,L. H., Rev. Sci. Instr. 37, 1416 (1966). 25. PRINCEN, L. H., AND KVVOLEI(., W. F., Rev. Sei. Instr. 36, 646 (1965). 26. WALES,M., ANDWILSON, J. N., Rev. Sci. Instr.
32, 1132 (1962). 27. LINES, 1~. W., Conference on General Applications of the Coulter Counter, London, 1968. 28. ALGORITH~'I226 "Collected Algorithms from C.A.C.M." 29. KEEPING, F. S., "Statistical Inference," p. 89. Van Nostrand, New Jersey, 1962. 30. DIEM, K., ed., "Documenta Geigy", 6th ed., p. 164. Geigy Pharmaceutical Company Limited, Manchester, 1962. 31. VER~,VEY,E. J. W., AND OVERBEEK, J. Th. G., "Theory of the StabilRy of Lyophobic Colloids," p. 177. Elsevier, Amsterdam, 1948.
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968