Chapter 7 Kinetics and Structure of Colloidal Aggregates 7.1 Diffusion Limited Cluster Aggregation – DLCA 7.1.1 Aggregation Rate Constant – DLCA In the case where no repulsive barrier exists between the particles, i.e., VT(D) is a monotonously increasing function with no maximum, the rate of particle aggregation is entirely controlled by Brownian motion. Let us compute the flow rate F of particles aggregating on a single ref- erence particle. For this, we note that the flow rate, F of identical particles diffusing through a sphere centered around a given particle, is given by: dN F = 4πr2 D with N = N at r = 1 (bulk) (7.1) 11 dr 0 where N is the particle concentration, r the sphere radius and D11 the mutual diffusion coeffi- cient, which since both colliding particles undergo Brownian motion is equal to twice the self diffusion coefficient, i.e., D11 = 2D. At steady state, i.e., F = const:, the particle concentration profile is given by: F 1 N = N0 − (7.2) 4πD11 r If we now assume that upon contact the second particle disappears (because of coagulation), that is N = 0 at r = R11 where R11 = 2a is the collision radius, we have: F = 4πD11R11N0 (7.3) which is the flow rate of particles aggregating on the single particle under consideration. If we now consider only the beginning of the aggregation process, we can compute the rate of 95 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES N N0 r 2a 0 decrease of the particle concentration in bulk due to the aggregation process, Ragg as follows: 1 1 1 R0 = − N F = − 4πDR N2 = − β N2 (7.4) agg 2 0 2 11 0 2 11 0 where the factor 1=2 is needed to count each event only once. In the above expression we have introduced the aggregation rate constant β11 = 4πD11R11, which plays the same role as the reaction rate constant in a second order kinetics. In the case of two particles of equal size, a this reduces to: β11 = 16πDa (7.5) and using the Stokes-Einstein equation to compute D = kT=(6πηa), where η is the dynamic viscosity of the medium, we obtain: 8 kT β = (7.6) 11 3 η which is size independent. In the case of two particles of different radius, Ri and R j, we have: βi j = 4π Di + D j Ri + R j ! 2 kT 1 1 = Ri + R j + (7.7) 3 η Ri R j It is interesting to investigate the behavior of the aggregation rate constants for particles of different sizes. 96 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES β a a i j i j 4πDa a a 4 ) β is size independent 1 2 a · 3 a a=2 a + a a + 2 = 3 2 = 4:5 1 4 a · 5 a a=4 a + a a + 4 = 5 4 = 6:25 1 n a (n+1) (n+1)2 a a=n a + a a + n = (n + 1) n = n (n+1)2 limn!1 n = 1 From the values in the table above it is seen that small/large collisions are more effective than small/small or large/large ones. 7.1.2 Cluster Mass Distribution – DLCA In order to derive the Cluster Mass Distribution (CMD) of the aggregates containing k primary particles (or of mass k) we need to consider the following population balance equation, also referred to as Smoluchowski equation: dN 1 Xk−1 X1 k = β N N − N β N (7.8) dt 2 i;k−i i k−i k ik i i=1 i=1 where Nk is the number concentration of aggregates of size k, and the first term on the r.h.s. rep- resents all possible collisions leading to the formation of an aggregate of mass k, while the second is the rate of disappearance of the aggregates of mass k due to aggregation with ag- gregates of any mass. If we now assume that the aggregation rate constant is constant with size, i.e., βi j = β11 and we sum both sides of the equation above to compute the total aggregate concentration of any mass, that is X1 N = Nk (7.9) k=1 we obtain: dN 1 1 = β N2 − β N2 = − β N2 (7.10) dt 2 11 11 2 11 which integrated with the I.C.: N(t = 0) = N0 leads to: 1 1 1 = + β11t (7.11) N N0 2 97 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES N/N0 1 0.5 0 tRC t If we define the characteristic time of coagulation or of rapid coagulation, tRC the time needed to half the initial value of aggregates we have: 2 1 tRC = = (7.12) β11N0 2πD11R11N0 which for particles of equal size reduces to: 3η 2 × 1011 tRC = ≈ [s] (7.13) 4kT N0 N0 where a water suspension at room temperature has been considered in the latter equation, −3 with N0 in cm . As an example, for concentrated colloids of a few percent in solid volume, 14 −3 N0 = 2 × 10 cm the tRC is in the range of milliseconds. The CMD computed from the Smoluchowski equation is given by: k−1 N0 (t/τ) Nk = (7.14) (1 + t/τ)k+1 where time has been made dimensionless using the rapid coagulation time, i.e., τ = tRC. 7.1.3 Role of Aggregate Morphology – DLCA In order to solve the Smoluchowski equation in the general case where the aggregation rate constant is size dependent, that is: ! 2 kT 1 1 βi j = Ri + R j + (7.15) 3 η Ri R j 98 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES we need to correlate βi j directly to the masses i and j of the two colloiding clusters. Accord- ingly, we need to derive a relation between the size Ri of an aggregate and its mass i, that is to say something about the structure of the aggregate. If we consider a solid sphere this relation is simple: 4 i = ρ πR3 ; ρ = density (7.16) 3 i which reported on a log-log plot leads to a straight line with slope 1=3. This expression can be used in the case of coalescence, where smaller droplets (e.g. of a liquid) aggregate to form a larger drop. In this case, by substituting it in the above equation we get the following expression for the coalescence rate constant: 2 kT β = i1=3 + j1=3 i−1=3 + j−1=3 (7.17) i j 3 η Using Monte Carlo simulations and light scattering measurements it has been shown that a similar straight line is obtained also for the polymer aggregates if we define as aggregate size, Ri the radius of the smallest sphere enveloping the entire cluster. This is the size that we have tacitely used above in computing the self diffusion coefficient of the fractal using Stokes- Einstein equation. The difference with respect to the solid sphere is that in this case the slope of the straight line is much larger due to the many voids present in the fractal structure, and equal to 1=df, where df is defined as the fractal dimension. 99 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES ln Ri 1/df 1/3 ln i Ri In typical applications df ranges from 1:5 to 3:0. If we now accept that our aggregates have a fractal structure, then we use the following scaling of their size with mass 1=df Ri / i (7.18) which leads to the following expression for the aggregation rate constant: 2 kT β = i1=df + j1=df i−1=df + j−1=df (7.19) i j 3 η which can be used to solve numerically the Smoluchowski equation in diffusion limited con- ditions. 7.2 Reaction Limited Cluster Aggregation – RLCA 7.2.1 Aggregation Rate Constant – RLCA In the case where a repulsive energy barrier is present between particles, i.e., VT(D) exhibits a maximum value, then the aggregation process becomes slower than the diffusion limited described above and becomes controlled by the presence of the interparticle potential. This is often referred to as reaction limited cluster aggregation, RLCA. In this case, in computing the particle flow rate, F entering a sphere of radius, r centered on a given particle, we have 100 CHAPTER 7. KINETICS AND STRUCTURE OF COLLOIDAL AGGREGATES to consider not only the particle concentration gradient, i.e., brownian diffusion, but also the potential gradient. The particules are diffusing toward each other while interacting in the potential field VT(r) whose gradient is producing a force: dV F = − T (7.20) T dr which is contrasted by the friction force given by Stokes law, Ff = Bu, where B = 6πηa represents the friction coefficient for a sphere of radius, a in a medium with dynamic viscosity, η. The two forces equilibrate each other once the particle reaches the relative terminal velocity, ud so that FT = Ff. This implies a convective flux of particles given by: N dV J = u N = − T (7.21) c d B dr which superimposes to the diffusion flux: dN J = −(D ) (7.22) d 11 dr where D11 is the mutual diffusion coefficient of the particles. The flow rate of particles aggregating on one single particle, F, which at steady state conditions is constant, is given by: ! dN N dV F = − 4πr2 (J + J ) = 4πr2 D + T (7.23) c d 11 dr kT dr where using the Stokes-Einstein equation the friction factor can be represented in terms of the diffusion coefficient by B = kT=D11.