Chapter 6 Thermodynamics of Surfaces and Interfaces and Some Consequences 153
6.7 OSTWALD RIPENING BY SURFACE REACTION CONTROL
Because the solubility of a particle depends on its radius, if there is a distribution of particle sizes of B as shown in Figure 6.9, this is an unstable configuration and the smaller particles will dissolve and the larger particles will grow. As a result, the average particle size will increase with time: particle coarsening or Ostwald ripening. For particles of B to grow, several series steps are required: the smaller ones must dissolve, B must diffuse to the larger particles and then B atoms must attach themselves
to the larger particles. The three kinetic steps in series are (1) surface reaction-controlled dissolution of smaller particles; (2) solid-state or liquid diffusion of B; and (3) surface reaction- controlled growth of larger B particles. Diffusion control of the process will be examined later. For now, assume that growth of the larger particles by reaction at their surfaces controls the rate of particle growth, step 3.
As shown earlier, the rate of growth—or shrinkage—of the particles can be obtained from the molar flux to—or from—the surface of the particles. With the assumption of spherical particles, where r = particle radius and a first-order reaction with reaction rate constant k (m/s):
J k B=
( [ ]
r−[ ]
Br)
(6.17)where:
r is the mean particle size of the distribution Br
is the concentration of B in the solution (mol/cm3) determined by the mean particle size Br
is the equilibrium concentration of B (mol/cm3) at the surface of a particle or radius r.
Equation 6.17 implies that particles with radii less than r will shrink and those with radii greater than r will grow. This will lead to an increase in the average particle radius, r, with time. All pow- ders, particles, and precipitates show a range of particle size that can be described by some sort of particle size distribution. If the all of the particles are changing size, the distribution and the mean particle size are also changing. That makes modeling particle size changes with time a very difficult problem. This complex problem was solved over half a century ago, and the complicated part was developing the equation for the steady-state particle size distribution (Wagner 1961). Several other solutions with different particle size distributions based on volume constraints have been developed since then (Ratke and Voorhees 2002).
The messy mathematical details of developing the appropriate particle size distribution will not be attempted here. Nonetheless, insight into the solution is realized by examining the behavior of a simple two-size distribution: one in which there are some very large particles all of the same radius and also some very small particles, again, all with the same smaller radius. An example of this might be fog particles over a lake where there is only one large particle, the lake with a radius of curvature of infinity.
This is not an unreasonable distribution. If a composition, x0, in Figure 6.10 were cooled from the liquid phase, solid B would form when the temperature reached the liquid + B phase boundary. Because these B particles are growing from the liquid, they could be “large” because diffusion in the liquid is rapid.
When the eutectic temperature is reached, all of the remaining liquid freezes to a eutectic mixture of alpha + B. If the rate of cooling were fast enough, these eutectic B particles could be much smaller than those formed earlier. As a result, a two-particle size distribution is produced. If this composition were reheated to some temperature below the eutectic temperature, then the small particles would dissolve and B would deposit on the large particles, increasing their size. Because the large B particles are much larger than the smaller particles, essentially then ≅ Br B∞ , so Equation 6.17 becomes
J k B=
( [ ]
∞−[ ]
B r)
. (6.18) Working on the left-hand side of the flux equation for the change in radius with time:J M A dV
dt M r
d r
dt M
dr
= ρ = ρ
( )
= dtπ
π ρ
1 1
4
4 3
2
3
(6.19) so, combining Equations 6.18 and 6.19
dr dt
Mk B B r
= ρ
(
− ∞)
(6.20)where:
M is the molecular weight of B (g/mol) ρ is the density of B (g/cm3)
k is the first-order reaction rate constant (cm/s)
[B]r is the concentration of B at the surface of a particle of radius r (mol/cm3) [B]∞ is the concentration of B over a flat surface.
Now, M ρ =VB so that the rate of particle growth becomes dr
dt =kVB
( [ ]
B∞−[ ]
Br)
dr
dt V k B B e V k B V
B rRT
V
rRTB B B
= −
≅ − −
∞ ∞ ∞
2
1 1 2
γ γ
= −V k B ∞ V
B 2γrRTB
(6.21)
and leads to
rdr dt
V k B RT
= −2 B2
[ ]
∞ γ(6.22) and is integrated to give
r r V k B
RT t At
2 B 02
4 2
− = −
[ ]
∞γ = −(6.23) where r0 is the initial particle size at time = 0. This result basically says that the smaller particles will continue to get smaller and finally disappear. Wagner’s (1961) result for his particle size distribution is
r r V k B
RT t A t
2 B 02
2 2
8
− = 9
[ ]
∞γ = ′(6.24) where the mean particle size increases with time. The result for a two-size distribution, Equation 6.23, and that for a complex particle size distribution, Equation 6.24, have the same terms with two exceptions: in the complete solution, r2 is r2, the mean particle radius squared, and instead of the 4 in Equation 6.23, the more accurate solution in Equation 6.24 has this factor as (8/9)2. The exact solution takes into account a particular particle size distribution produced by the particle coarsening and that the rate of growth of a particle of a given radius depends on the difference between its solubility and that of the average particle size in the distribution, which changes with time. Determination of the particle size distribution produced by the coarsening makes model development much more complex. But notice that all the other parameters and predicted time dependence are exactly the same in Equations 6.23 and 6.24! The main point is that the particle size grows or shrinks roughly as the square root of time and is directly proportional to the thermodynamic solubility (over a flat surface), [B]∞, the interfacial energy, γ, and the reaction rate constant, k.
In wet chemistry, Ostwald ripening or coarsening is used to “age” precipitates by waiting a while after precipitation before filtering to a get a larger particle size, making it easier to filter out the solid par- ticles. This was one of the original observations examined by Ostwald. In addition, particle coarsening can occur via transport through the vapor phase because small particles have a higher vapor pressure—
or reaction product pressure—than larger particles. For example, studies on the effect of atmospheres
on sintering have shown that Ostwald ripening occurs when the product gas pressures over solids are enhanced in a reactive atmosphere. Figure 6.11a shows this for iron oxide in HCl (Lee 1984)
Fe2O3(s) + 6 HCl(g) = 2 FeCl3(g) + 3 H2O(g)
whereas Figure 6.11b shows the same for Al2O3 (Marc Ritland, unpublished research) Al2O3(s) + 6 HCl(g) = 2 AlCl3(g) + 3 H2O(g).
It is the equilibrium metal chloride pressures—FeCl3 and AlCl3—for these two materials that are enhanced by the curvature: a higher pressure over small particles compared with that over large par- ticles. Note, particularly for the largest particle size, there are crystal faces or facets on the surfaces of the aluminum oxide grains, but the iron oxide grains are uniformly rounded except at the fracture surfaces, where particles were bonded together—grain boundaries. This shows that the surface ten- sion of Al2O3 is still quite anisotropic at 1500°C, whereas that of Fe2O3 is isotropic at 1200°C. The anisotropy in surface tension would be expected to decrease with temperature simply because G = H − TS holds for surface energies as well. The main thing of interest about this coarsening is that no shrinkage or densification takes place while heating in the reactive HCl atmosphere. If the same starting compacted powders—with about 45% initial porosity—were held at the same temperatures and times in air, the resulting sintered powders would achieve densities on the order of 95% of the theoretical crystalline density—or only about 5% residual porosity by volume. The main reason that densification is significantly decreased in a reactive atmosphere is that the particles grow rapidly and reduce the surface tension, the driving force for sintering, which, of course, is the same driving force for particle growth. This process of sintering solids in a reactive atmosphere can be used to make porous materials of controlled porosity (amount: 0% to about 50% by volume) and pore size (propor- tional to the grain size) for filter applications.