ON THE COAGULATION O F COLLOIDS BY ARNE WESTGREN AND JOSEP REITSTOTTER
One of the chief objects of colloid chemistry is certainly to find out the conditions of the stability of colloids. To attain this aim it must evidently be of great importance to know what happens, when a colloid for some reason suddenly loses its stability. The process of coagulation has therefore been subject to a great number of investigations and as a multitude of more or less well-founded hypotheses has been proposed in order to explain this phenomenon, it is certainly no easy task to give a well-posed and objective survey of this field of research. It is not our intention to make any atttempt in this direction, since M. Smoluchowski (1, 2, 3) a few years ago has given an excellent summary of the coagulation theories and K. Zsigmondy (4) has published a critical estimate of the measurements and speculations on the coagulation of colloids produced by electrolytic admixtures. The purpose of this paper is only to give a summary sketch of Smoluchowski's kinetic theory of coagulation which has made an exact treatment and experimental examination of the entire coagulation problem possible and also to give a short survey of the experimental work performed in order to try the new theory. Smoluchowski, the ingenious pioneer, who lived to see the first experimental confirmation of his work, was carried off b y an epidemic war-disease at Cracow on September 5 , 1917. If a deep-red colloidal gold-solution is mixed with a diluted solution of sodium chloride, its colour turns more or less quickly into violet or blue. This conspicuous change of colour is a macroscopic characteristic of the coagulation of the colloidal gold-solution, the particles of which unite to larger complexes, and after a certain time sink to the bottom, while the liquid becomes almost colourless. All attempts made hitherto to explain the respective laws
538
A r n e Westgren and Josep Reitstotter
of this process in an empirico-inductive, way, have been rather unsuccessful (S. Miyazawa, N. Ishizaka, H. Freundlich, I. A. Gann, etc.), and even certain formulae and laws framed in the beginning had to be withdrawn after more accurate researches (Freundlich and Gann), wherefore Smoluchowski, having recourse to a deduction method, stated some new principles for studies in this branch of colloid physics. According to Smoluchowski, the reason why the empirical methods have proved a failure, is, that in all previous works certain factors (viscosity, the relative quantity of the substance dissolving under determinate conditions, transparency, etc.) were considered as measures of coagulation, whereas in reality there is no exact measure for it, as coagulation cannot be expressed by one variable quantity alone. In spite of all pains bestowed on these researches, a precise cognition of the acting causes on which stability or coagulation are depending, was still wanting. Smoluchowski arrived at the assumption that, when sufficiently near together, the particles attract one another in consequence of the capillary effect, but that a combination does not take place under normal circumstances, which is to be attributed to a protective effect of the electric double-layer of the particles. An electrolyte being added, a partial or complete discharge of this double-layer takes place, owing to the adsorption of the ions demonstrated by Freundlich ; thus the protective effect is diminished, and, from a certain concentration i t does not suffice to prevent the particles from joining together. Still another influence must be taken into consideration: i t causes a collision of the particles, but, at the same time, checks their permanent Combination, viz., the molecular forces which, among other things, manifest themselves in the Brownian movement. Before entering on the calculations made by Smoluchowski, we must still premise some general reflections on coagulation. In order that coagulation may take place, a certain electrolytic concentration must be attained, “Bodlander’s
The Coagulation of Colloids
539
limit” (Schwelleizweut), below which there is no coagulation at all. This limit, however, is not sharply defined. Zsigmondy has proved by experiments, that small electrolytic admixtures exert no influence on a fine-grained colloidal gold-solution. A gradual increase of the electrolytic concentration leads into a domain of slow coagulation, through a zone rather limited with respect to the electrolytic concentration. I n this zone small changes of the electrolytic concentration produce very great changes in the velocity of coagulation. It has thus been proved, that the velocity of coagulation is a continuous function of the electrolytic concentration, i. e., coagulation does not suddenly occur at a certain degree of the latter; but when it begins after a gradually increased electrolytic’ admixture its velocity augments, and finally attains a maximum which is not exceeded even if the concentration is very considerable. Hence i t follows, that the notion of “coagulation-limit” so current in the literature of colloids, in reality has no explicit counterpart. It is therefore more proper to speak with Zsigmondy of a “limit-zone” within which the velocity of coagulation varies with the electrolytic concentration. This zone passes continuously into the adjoining concentrationregions, into the region where no coagulation takes place, in the one direction and into the region where the velocity of coagulation becomes a maximum in the other. If we use the term “coagulation-limit,” we must do as Zsigmondy and Reitstotter have done, i. e., connect i t with a certain velocity of coagulation within the limit-zone. That the particles of a colloid do not agglutinate is evidently due to powers of repulsion manifesting themselves a t their mutual immediate approach. These powers must be connected with a protective effect of an electrically charged double-layer. The presence of these repelling powers has been shown by J. Perrin and R. C0stantin.l If we mix the colloid with increasing quantities of an Comptes rendus, 158, 1171 (1914).
540
Arne Westgren and Josep Reitstotter
electrolyte, the potential of the double-layer decreases, as we have already shown. At a certain standard (Bodlander’s limit), the limit of stability is exceeded, and then there is a tendency towards coagulation which rapidly increases with the increase of the electrolytic admixture; in other words, the protective effect of the charge of the double-layer is in part neutralized by a rising effect of attraction. These considerations form the foundation of Zsigmondy’s doctrine, according to which the maximum of the coagulationvelocity, designated as “quick” coagulation in the following lines, corresponds to a discharge of the double-layer down to a certain potential. I n this state the attraction forces of the particles are fully developed. According t o Smoluchowski, each particle of ‘the colloid is, from the moment of the admixture of the electrolyte, surrounded by a sphere of attraction of the radius R, within which there is so intense an attraction, that any other particle is retained in it, forming an indissoluble combination with the first, as soon as its centre gets into that sphere. This view of the coagulation-process has the advantage of not meddling with the question of the structure of the double-layer, its connection with the nature and number of the ions, etc. Besides, it enables us mathematically to formulate the kinetics of coagulation. Smoluchowski considers, that the main object for a mathematical treatment of the problem must be to calculate the numbers vl, vZ, v 3 , . . . of the simple, double, treble . . . particles from the factors characterizing the whole system, viz., the original number of particles vo, the dimension of the radius of action R, and the constant of velocity D of the Brownian movement. The problem therefore, is: If no particles exist in the unit of space a t the time=O, how great is, at the time t , the average number n of those particles, whose centres until then have not passed into the sphere of action R ? Instead of considering all the particles in the unit of space, we select a single one. The probability PI ( t ) of its not having been touched by another until the time t ,
The Coagulation of Colloids
541
is expressed by the percentage of the single particles still remaining free :
To make the remaining calculation easier we assume, that this special particle is at rest, that it is retained and that only this one possesses a sphere of attraction. Therefore all the other particles can join only with it, but not with each other. We thus assume, that the spherical surface of the radius R retains each arriving particle, and that i t acts absolutely adsorbing; in other words, that the concentration on the spherical surface always is 0. I n order to calculate the amount of the substance diffused on to the sphere, we have only to determine the distribution of a substance that at first has uniformly filled space (initial concentration = c) but, from the moment t = 0, is diffusing towards the sphere of the radius R, to whose surface it fully adheres , thus making the concentration immediately outside the sphere constantly u = 0. From the general equation of diffusion' it results, that, on 1
Ht9
1 -
b(ru) b2(ru) -=Dbt drz
D = ; H is the gas-constant 83.19 X lo8;t9 the absolute temperature; N 6m7a N the constant of Avogadro 60, 6.1OZ2;7 the viscosity of the system, and a the radius of the particle. u = c when t = 0, r > R'and u = Owhenr = R,t> 0 r-R
2
x
P
0
Hence it follows that the amount diffused o n to the sphere R in the time t dt is
+
I
Tdt = 4rrDR2 - dt br au R
=
4rDRc
.. , t
542
A r n e Westgren and Josep Reitstotter
the stipulated conditions, the amount diffused on to the dt is sphere R in the space of time t . . . t
+
and the total quantity disengaged from the beginning is M
=
4aDR c
[ +7$] t
To simplify the calculation we neglect the root quantity, that is to say, we consider the process of coagulation at a stage, in which t is great compared to R2/D. The probability, that a certain particle existing at first anywhere in the space V joins the sphere R in the time t . . . .
t
+ dt
1
(whenc = ); is: Pt
=
4nDR dt
V
The coagulation-time in which, on an average, a particle sticks to the one selected, is
the concentration ‘c having been substituted by the number of the particles v o in the unit of space; the number of the particles during the unit of time sticking to the selected adsorption-nucleus is 4aRDva. But now we must take into consideration, that the selected particle R is not at rest, but subjected to the Brownian movement; its relative motion, therefore, comes into consideration for coagulation. But as the relative motion of two particles which, independently of each other, are subjected to Brownian molecular movement characterized by the diffusion-constants D1, I),, is likewise a Brownian molecular move-. ment, viz., such a one as is characterized by a diffusion-con-
The Coagulation of Colloids
543
+
stant D1,2= D1 Dz,we need only double the coefficient D in the present case.l The decrease of the number of the simple primary particles is given by Vl
=
yo
1
vo
+ 8nDRvot l +2tr =-
But here we have taken into account only the uniting of simple particles into double particles ; whereas, in reality, also the formation of multiple particles is to be considered. The double and treble particles already formed act in their turn as nuclei of coagulation, unfortunately in such a manner as cannot be exactly calculated, the form of the multiple particles being in all probability not spherical. Also here exist similar relations as before in the simple case, and the total decrease of the primary particles may be represented by the equation
In an analogous manner we get the reaction-equations for the different categories of multiple particles. By integration we get the total number of particles ~ = v 1 + v ? : + v 3 +. . . = YO 1
+
Pt
1 -.. As P ( x ) d x =e - 4D1 d x , 2di~Dt the probability that a particle at the end of the time t has been displaced E. . . E dE out of its original position is evidently the product of the two probabilities, independent of each other, that one particle has been dislocated x and another particle 4 x: I
.
+
+
+m
--m
544
A r n e Westgren and Josep Reitsutter
where p = 4 DR vl. The number of the primary particles then is: vo
= [l
+
Pt]’
that of the double particles
and that of the n-fold particles: v, =
(Pi) n-
+
PO
[l /3t]n+l. Introducing the diffusion-constant into the formula of coagulation-time, we get from
T=-
1
4~DRvn
and the coagulation-time
This formula, however, applies only to quick coagulation, discussed in the beginning of this paper, when the electrolytic admixture is considerable, the sphere of attraction being already independent of the electrolytic concentration. If the electrolytic admixture is small, we may imagine, that, of the immediate collisions of the particles, only a fraction, dependent on the electrolytic concentration, leads to the joining of particles. It is evident, that this causes only a retardation of the process, the inner progress of the same remaining unchanged. Smoluchowski is of opinion, that in the case of an incomplete discharge of the double-layer, the attractive powers of the sphere of action are not fully developed and, because of that, only a certain fraction of the immediate collisions of the particles causes their instantaneous combination. In the formula, already given, E Pt must then be put in the place of fit. IS a = 4nRD,
The Coagulation of Colloids
545
it follows that
where e represents a coefficient corresponding to that fraction. The coagulation-curves obtained from the different concentration series of the colloid and the electrolyte must therefore be similar. By a corresponding change of the measure of time, they can be brought to coincide. The above theory of coagulation-kinetics was worked out, in a more deductive manner, by Smoluchowski a t the instigation of R. Zsigmondy, and was brought before the public in a course of lectures delivered at Gottingen on the 20th, 21st and 22nd of June, 1916. Smoluchowski’s calculations were first confirmed by R. Zsigmondy’s experiments (4,5 , 6) on colloidal gold-solutions, which were brought to quick coagulation by a considerable electrolytic admixture. By adding a strong protective colloid (solution of gum arabic) Zsigmondy and Reitstotter, after a certain time interrupted the process of coagulation, and counted the remaining primary particles by means of the ultramicroscope. Equally satisfactory and corroborative for the theory are the results obtained by Westgren and Reitstotter (7), who, instead of the primary particles ( v I ) determined the total number ( e Y ) of the particles. A few years ago, by investigating coarse-grained gold h j drosols, Westgren (8), has also found, that Smoluchowski’s view concerning slow coagulation is likewise correct, that the radius of action, when fully developed is in all probability exactly twice as great as that of the particle, and that the particles thus exercise no effect on each other before colliding or at least not until they come very near each other. In a theoretical work H. Freundlich (9) has made a very interesting attempt to explain the mechanism of the slow coagulation. He assumes, that the partially discharged particles in a slowly coagulating colloid do not join together, unless their power o€ collision exceeds a minimum value. It
546
’
Arne Westgren and Josep Reitstotter
would be of a certain interest mathematically to investigate whether this hypothesis is compatible with the experimental results, gained by Westgren (S), that uniform gold-sols of different particle-size, that are coagulated with the same electrolyte show identical curves of coagulation. It seems, a t least a t a superficial estimation, that this can hardly be the case. If the violence of collision is of decisive importance for the aggregation, not only the relative velocity of the particles, but also their mass should have some influence on the progress of coagulation. The change in colour of Congo ruby hydrosols, mixed with an electrolyte, has been studied by H. Liiers (lo), who found Smoluchowski’s formula applicable to this phenomenon. In connection with this, he has also gained some interesting results concerning the influence of protective colloids on the said change in colour. Quite recently H. R. Kruyt and A. E. van Arkel (11, 12, 13) have experimentally contributed to the question whether Zsigmondy’s and Smoluchowski’s opinion on the mechanism of coagulation holds good. These scientists have carried out a series of investigations on the coagulation of selenium hydrosols. Their results differ to a certain degree from those obtained formerly by the examination of coagulating gold-sols. In several experiments the total number of particles (Zv) was found to decrease during the coagulation in a way not a t all consistent with Smoluchowski’s formula, and in the cases, where agreement was established, the values obtained for the ratio of the attraction-radius to the particle-radius, R/a, were varying from 0.5 to 1.0. Thus only part of the particlecollisions should lead to aggregation. Kruyt and van Arkel have tried different ways of explaining these divergences from Smoluchowski’s theory. A source of error, however, to which they seem to have paid no attention, is the possibility, that their selenium sols have contained not only a main quantity of almost equally big submicroscopic grains, but also a considerable fraction of a microscopic selenium-particles. If the colloids have had such a composition the total number of par-
The Coagulation of Colloids
547
ticles visible in the ultramicroscope must evidently change in a complicated manner during coagulation. Selenium sols, which have not by centrifuging been freed from eventual amicroscopic particles and electrolytes cannot be said to be as well defined and suitable material for investigations of this art, as coarse-grained gold sols, that have been prepared by repeated sedimentation of the particles and pouring off of the supernatant liquid. From the above short survey of the experimental examination of Smoluchowski’s valuable theory i t may be evident, that there still remains very much to do before colloid chemistry has made the most of i t as a leading principle on this sphere of research. If this short review can cause any scientist of colloids to perform a new investigation of the coagulation process based on the said theory, the present writers consider the purpose of their paper fulfilled. List of works quoted 1. M. Smoluchowski: “Drei Vortrage uber Diffusion, Brownsche Molekular-
2.
3.
4. 5. 6.
7.
8. 9.
10.
bewegung und Koagulation von Kolloidteilchen.” Physikalische Zeitschrift, 17, 557, 586 (1917). M. Smoluchowski: “Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Losungen.” Zeitschrift fur physikalische Chemie, 92, 129 (1917). M. Smoluchowski: “Grundriss der Koagulationskinetik kolloider Losungen.” Kolloid-Zeitschrift, 21, 98 (1917). R. Zsigmondy : “Uber Koagulation und Teilchenattraktion.” Nachr. der Akademie der Wissensch. Gottingen 1917, Heft 1, p. 1. R. Zsigrnondy: “Uber Koagulation und Teilchenatiraktion.” Zeitschrift fur physikalische Chemie, 92, 600 (1917). R. Zsigmondy : “Uber Koagulation.” Zeitschrift fur Elektrochemie, 23, 148 (1917). A. Westgren and J. Reitstotter: “Zur Koagulation grobdisperser Goldhydrosole.” Zeitschrift fur physikalische Chemie, 92,750 (1917). A. Westgren: “Zur Kenntnis der Koagulation.” Arkiv for kemi, mineralogi och geologi, utg. a v K. Svenska Vetenskapsakad., Stockholm, 1918, Bd 7, Nr 6, p. 1. H. Freundlich: “Zur Theorie der Koagulationsgeschwindigheit.” Kolloid-Zeitschrift, 23, 163 (1918). H. Lders: “Der zeitliche Verlauf des Kongorubin-Farbenurnschlagsunter tiem Einfluss von Elektrolyten und Schutzkolloiden.” Kolloid-Zeitschrift, 27, 123 (1920).
548
A r n e Westgren and Josep Reitstotter
11. H. R. Kruyt and 9. E. van Arkel: “La vitesse de floculation du sol de s6lCnium, lPre communication: Floculation B l’aide de chlorure de potassium.” Recueil des travaux chimiques des Pays-Bas, (4) 39,656 (1920). 12. A. E. van Arkel: “Uitvlokkingssnelheid van het seleensol.” Diss. Utrecht 1920. 13. H. R. Kruyt and A. E. van Arkel: “La vitesse de floculation du sol de sCl6nium. 11: Floculation a u moyen de chlorure de baryum.” Recueil des travaux chimiques des Pays-Bas, (4) 40, 916 (1920).
‘
