Coagulation Kinetics A Laboratory Experiment Waldemar Nowicki and Grazyna Nowicka A. Mickiewicz University, Poznan, Poland
Because of small size, colloidal particles in lyophobic sols undereo Brownian motion that results in collisions between the p&ticles. Such collisions may lead to particle agyregation. The effectiveness of the aggregation depends on the re1atix.e magnitudes of the forces of attraction and repulsion between the approaching particles. According to DLVO theory (1,2),the interparticle torcescan be regarded as the sum of two contributions. These are the electrostatic repulsions resulting from the overlapping of electrical double layers of particles and the van der Waals attraction, in particular its dispersion component. A very important difference between these two forces is their different dependence on the interparticle distance. T o a first approximation, the repulsion is an exponential, whereas the attraction is a hyperbolic function of the distance. As aresult of the combination of the two functions, the attraction is dominant a t large and small distances, hut in between an energy barrier may occur, as shown in Figure 1. If the barrier is high enough to he unsurpassable by the kinetic energy, then the sol is stable against aggregation. The decrease in the barrier height caused, for instance, by the introduction of an electrolyte to the dispersion medium, involves the rise in the fraction of particle encounters leading to permanent contacts or, in other words, to coagulation. The introduction of electrolyte to a sol practically does not affect the van der Waals attraction between the particles (as long as the solution is dilute); however, the electrostatic repulsion strongly depends on concentration, valences, and, to some extent, on the nature of the dissolved electrolytes. Addition of an indifferent electrolvte. " . i.e... whose ions do not adsorb specifically on the particle surface, causes compression of the diffuse part of the double layer and a reduction of the Stern potential, thus reducing the energy harrier. This, in turn, involves the increase in the fraction of collisions resulting in aggregation of particles. Thus, in kinetic terms, the increase in the ionic strength brings about the increase in the rate of coagulation. As a rule, however, for a given electrolyte there is a limiting concentration above which the coagulation rate remains constant, that is, independent of the increasing ionic strength of the medium. In principle, this limiting concentration, known as the critical coagulation concentration, c.c.c., is the electrolyte concentration, which just makes the potential energy barrier disappear. Coaeulation kinetics is classified as beiue " r a ~ i d "if everv collisi& leads to aggregation and "slow" if does nor. Aver;, simple theow of coa.eulation was eiven hv Smoluchowski (3. 4) who assumed t h a t t h e disappearance of primary can be considered a diffusion-controlled second-order reaction. The light scattering technique is one of the most suitable methodiof observingthe coagulation process, since the experimental turhidities can be interpreted in terms of the number and the size of the scattering centers. The theory of light scattering was developed by Rayleigh for systems containing monodisperse, nonabsorhing, spherical colloidal particles, the dimensions of which are small compared with the wavelength of the light.
By combining ~moluehowski'sand Rayleigh's theories, one can derive the followina dependence between the intensity of light scattered by ahlloidal system and the time of coagulation:
where In-the total intensitv of liaht scattered bv the unit volume of colloidal system, &-theintensity of thk incident light. K-the constant d e ~ e n d e non t the waveleneth of the incident light and on the refractive indices of particles and of dispersion medium, vn-original number of particles, Voth;volume of prim& particles, t-the time bfcoaguiation, T-half-period of coagulation, i.e., the time of coagulation in which the original number of particles is just halved. For low particle concentrations, the combination of eq 1 with the Lambert's law of light absorption lets us relate the apparent absorbance of the colloidal system, A, and the time of coagulation:
where 1 is a sample thickness which the light traverses. 10
V [lom J] .
Born repulsion
8 -i
0
6
10
16
20
26
30
36
Distance [nm] Figure I.lllustrstion of the potential energy of imeraction vs. the surface to surface distance lor spherical panicles system (particlediameter-50 nm. 1:l electrolyte concentration-0.030 M. Stern potential-50 mV, Hamaker constant-5r 10-20J, temperature-297 K).
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Number 6 June 1991
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Because of the restrictions imposed by the assumptions of Smoluchowski and Rayleigh theories, relations 1and 2 describe well only the initial stage of the coagulation process. At higher t values, a negative deviation of the measured IR = f(t) and A = f(t) dependenciesfrom thelinearity is observed. Thus. in order t o determine a kinetic narameter of coaeulation (for instance Tvalue) an experimentally obtained IR (or A) versus t deoendence has to be extraoolated to t = 0. as italso results from the accepted assumptions, the exoeriment should be performed with dilute susoensions, Ghichmakes it possibie to neglect the effect of secbndarv . lieht - scatterine on the results of spectro~hotometric measurements.
-
oreo over,
Experimental The analysis of the kinetics of coagulation of a colloidal suspension induced by an indifferent electrolyte is the purpose of the experiment described. Silver iodide hydrosol is used as the colloidal svstem. whereas ootassium nitrate is taken as the coagulating electrolyte. ~ i analysis e involves a determination of half-periods of coagulation occurring a t different electrolyte concentrations in the dispersion medium, and, hence the determination of the critical coagulation concentration (c.c.c.) of the electrolyte employed, i.e., the lowest electrolyte concentration inducing a rapid coagulation. Preparation of Silver Iodide Hydrosol T o a 150- or 200-mL beaker containing 50 mL of a 0.88 mM aqueous solution of potassium iodide, 50 mL of a 0.80 mM aqueous solution of silver nitrate are introduced in drops from a buret. The reaction has to be performed in dark and under permanent intensive stirring. The obtained sol is then left for about 30 min. Measurements of the Rate of the Coagulation A series of 10 potassium nitrate solutions in concentration ranee between 0.100 and 0.400 M is ureoared. Then 5 mL of as fast as possible with 5 mL of a ~ ~ I " h ~ d r o sisomixed l
KN03 solution, and the intensity of light scattered or the apparent absorbance of the system at 420 nm is measured at I -min intemals. The measurements can be performed on any nephelometer or spectrophotometer using a 1-cm cell and water as reference. I t is also advised to conduct the experiment a t a controlled constant temperature. Data Analysls I t was proved that the parabolic equation is a good apomximation of the exoerimentallv obtained dependencies IR = f(t) or A = f(t), as is shown in Figure 2 representing sets of A and t experimental data. (Further considerations are limited to theiesults of absorption measurements only. The treatment of the scattered light intensity data is parallel.) Thus, first the coefficients of the quadratic eq 3 have to be calculated: I t can be realized both by employing a computer program that fits coefficients of the second-dearee polvnomial to the data from the experiment or by adjusi,ng band r coefficients of a parabola with a constant "0" term. In order to ar~plvthe :amlust method of calculations, the shsorbanre A , of a ple diluted with water, instead of electrolyte solution, in rhe proper ratio, has to be measured. The ~0;alue corresponds, with a great accuracy, to the term "a" of eq 3. Then, by employing the Cramer's method, the band c coefficients can be found from equations:
where
o-tlj-p D,= (A - A,)?
(5)
- (A? - ~ , t ) l i
(6)
The shaded determinants indicate mean values. In the next stage of calculations, the equation of a tangent to the obtained parabola a t the point (Ao, t = 0) has to be found. The differentiation of the eq 3 at t = 0 gives the tangent equation with a gradient b: It is recommended to represent graphically the expcrimenrallv obtained de~endenciesA = f(1I at different electrolvte co&entrations and corresponding&ngents to the appr&~ o i n tas . it is illustrated at mated curves a t the (Ao. . -. t = 0) . . Figure 2. A comparison of eas 2 and 8 leads to the followine " relations: Ku,V$ = a = A o
0
1
2
3
4
6
6
7
8
0 1 0 1 1 1 2
time [min] Figure 2. A = f(O exDerimental dependencies end curves calculated from eqs 3 and 8.
524
Journal of Chemical Education
(9)
The relations permit an easy calculation of the half-periods T of coagulation occurring a t different electrolyte concentrations. Since above the C.C.C. the coagulation rate attains the maximum value, independently of increasing electrolyte concentration, Tshould be also constant in this range of salt concentrations and have minimum value, T,i,. Thus, the half-period of rapid coagulation, T,i,, can be found as an arithmetic mean of T values obtained a t a certain number of the highest electrolyte concentrations used in the experiment. Verwey and Overbeek (2) have shown that in the slow coagulation domain (low electrolyte concentration) there is a linear dependence between log (klk,,,) and log CE, where k and k,, denote rate constants of slow and fast coagulation,
Figure 3. Determinationof c.c.c. from a log (TIT,.) versus log q dependence.
respectively, whereas CE is the electrolyte concentration. I t is a consequence of the Smoluchowski theory, hut it is also intuitively obvious that T is an inverse function of k as well as T,i, is proportional to llk,.,. Thus, the dependence between log (TIT,i.) and log c~ should also be linear for CE < C.C.C.
Plotting the dependence log (T/T,i,) versus log CE in the sufficientlv broad range of electrolvte concentrations, one can estimate the c.c.c.;alue, interpieted as the lowest electrolyte concentration at which the coagulation rate reaches its maximum value, or, in other words; T attains T,i. The procedure of C.C.C. determination is illustrated in Figure 3. he C.C.C. is determined by the intersection of thesteep, linear part of the log (T/T,i,) = f(1og CE) curve with its horizontal part lying on the abscissa axis. The coefficients of the steep part of the curve should he estimated using the least-squares method. The C.C.C. value for the system described occurs between 0.10 and 0.15 M KN03. The ~ u r ~ o of s ethe ex~erimentdescribed is to acauaint the studenis with the problems of colloid stability, although the elements of chemical kinetics and es~eciallvof the technique of spectrophotometric kinetic measurements are also included. For more advanced students, the experiment can be extended to the verification of the Schulze-Hardy rule. T o this effect the C.C.C. values should be also determined for nitrates of di- and trivalent cations. Moreover, the stabilizing effect of macromolecular substances toward colloidal suspensions, the so-called protective action, can be also demonstrated. In this case, an additional series of kinetic measurements has to be performed on the A d sol stabilized by introduction of synthetic or natural (e.g., poly(vinyl alcohol), polyacrylamide, or gelatine). Literature Cited 1. Deryagin, 8.V.; Landau. L. Aclo Physicochim. URSS 1941,14,633462. 2. Verwey, E. J. W.; Overbeek, J. Th. G.Thaorv o l the Stability of Lyophobic Colloids: Elrevier: New York, 1948. 3. Smoluehowski. M.PhysicZ 1916,17,557-565. 4. Smoluchowski. M.E. Physic. Chem. 1917.92. 129-168. 5. Mirnik. M . C r m t Cham.Arfo. 1988.61.81-101. 6. Kun,H. ColioidScisnce: Elsevier: New York, 1952;vbl. 1.
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