Langmuir 2004, 20, 2975-2976
On the Critical Radius in Ostwald Ripening Robert Finsy* Department of Physical and Colloid Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Received October 22, 2003. In Final Form: January 28, 2004
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(1) At any instant of the ripening process, there exists a critical radius ac. Particles with a larger radius grow, and smaller particles shrink. If the radius equals ac, the size of the particle does not change. During the ripening process, ac increases with time. (2) In the long time limit, a stationary regime is reached for which: (a) The Ostwald ripening rate v, defined as daN3/dt, is constant and is given by
Introduction Ostwald ripening is the process by which larger particles grow at the expense of smaller ones. This process is a direct consequence of the Kelvin effect,1 that is, the higher solubility of small particles, which quantitatively reads
C(a) ) C(∞) exp(R/a)
(1)
where C(a) is the solubility of a dispersed phase particle with radius a. The bulk solubility C(∞) corresponds to the solubility of a particle with infinite radius, that is, to the solubility of the dispersed phase when it has a flat surface or the bulk solubility; R is called the capillary length and is given by
R)
2γVm RT
Vm is the molar volume of the dispersed phase, γ is the interfacial tension, R is the universal gas constant, and T is the absolute temperature. From eq 1, it can be seen that small particles are more soluble than large ones. Thus smaller particles tend to lose their molecules and these molecules diffuse through the continuous phase and reprecipitate onto larger particles. This leads to an increase of average particle size with time. A quantitative description of Ostwald ripening in a twophase system is given by the Lifshitz-Slyozof-Wagner (LSW) theory.2-6 This theory is based on the following assumptions:4 (1) The particles are fixed in space. (2) The system is infinitely dilute (implying the absence of interparticle interactions). (3) The concentration of internal phase molecules is the same throughout the whole external phase, except in the direct neighborhood of the particles (a peal with radii a and 2a, where a is the particle radius). (4) Single internal phase molecules are transported by molecular diffusion from one particle to another. With these assumptions and by using Fick’s first law,7 the mass balance, and the continuity equation for the particle size distribution, the following major results are derived: * Telephone: +32 2 629 3485. Fax: +32 2 629 3320. E-mail:
[email protected]. (1) Adamson, A. W.; Gast, A. P. Physical chemistry of surfaces; Wiley: New York, 1997; p 53. (2) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (3) Wagner, C. Z. Electrochem. 1961, 65, 581. (4) Kahlweit, M. Physical Chemistry; Eyring, E., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Volume 10, [13a], p 719, [13b], p 754. (5) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69. (6) Taylor, P. Adv. Colloid Interface Sci. 1998, 75, 107. (7) Vold, R. D.; Vold, M. J. Colloid and interface chemistry; AddisonWesley: London, 1983; p 186.
v)
daN3 4RDmC(∞) ) dt 9
(2)
where Dm is the molecular diffusion coefficient of the internal phase, and aN is the number-average particle radius. (b) The particle size distribution W becomes self-similar, when the sizes are scaled to ac:
W(a/ac) ) W(µ) )
81eµ2 exp(1/(2µ/3 - 1))
x3 32(µ + 3)7/3(1.5 - µ)11/3 (0 e µ e 1.5) (3)
W(µ) ) 0
(µ > 1.5)
(c) The critical radius ac is equal to the number-average radius aN of the limiting self-similar size distribution. Note that the limiting self-similar size distribution given by eq 3 is skewed with a tail toward the smaller particle sizes and predicts that no particles with sizes larger than 1.5 times the average size are present. Such a distribution is however not realistic for many freshly prepared dispersions or emulsions. For such systems, the size distribution is skewed with a tail toward the larger particle sizes and mostly a substantial fraction of particles with sizes larger than 1.5aN are present. It has also been shown by simulation8 that the transition from the initial size distribution to the limiting one is a long-lasting process due to the initial presence of particles with sizes larger than 1.5aN. Since the Ostwald ripening process is determined by the value of the critical radius ac and since the critical radius cannot be measured experimentally, the question arises of what the relation is between ac and aN for more realistic size distributions. In this contribution, a short proof is given that the relation ac ) aN is a direct consequence of the growth rule used in the LSW theory combined with the mass balance. Hence this relation is independent of the actual form of the size distribution and is also valid even before the limiting distribution (eq 3) and the long time limit have been attained. Proof That ac ) aN Whatever the Form of the Size Distribution In the derivation of the Ostwald ripening rate, the growth rate of a particle is given by Fick’s first law. In particular, the increase of the number of molecules n of a particle with radius a is given by (8) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884.
10.1021/la035966d CCC: $27.50 © 2004 American Chemical Society Published on Web 02/28/2004
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Langmuir, Vol. 20, No. 7, 2004
Notes
dn dC ) AJ ) 4πa2J ) -4πa2Dm dt dr
(4)
where A is the interfacial area of the particle, J is the flux of molecular dispersed material, and r is the radial distance from the particle. In the LSW theory, the concentration of the internal phase molecules in the external phase is assumed to be constant throughout the system. All particles are surrounded by the same concentration of internal phase molecules Cm at a distance from the particle surface equal to their own radius. Hence dC/dr is estimated as [C(a) - Cm]/a yielding
dn ) 4πDm(Cm - C(a))a dt
(5)
Obviously when C(a) ) Cm, the particle will not grow nor shrink. Hence the critical radius ac satisfies the relation C(ac) ) Cm. Using Kelvin’s law to relate the solubility to the particle size yields
()
Cm ≡ C(ac) ) C(∞) exp
R ac
(6)
(
)
(7)
In eq 7, the factor (a/ac - 1) essentially embodies the growth law. Clearly when a > ac the number of molecules in a particle increases, that is, the particle grows. If a < ac, the particle shrinks. The number of molecules in a spherical particle with radius a is given by
4πa3NA n(a) ) 3Vm
(8)
where NA is Avogadro’s number. Differentiating eq 8 yields
dn )
4π NA 3 da 3 Vm
(9)
Hence eq 7 becomes
(
Vm a da3 ) 3DmRC(∞) -1 dt NA a c
)
(
)
Vm t′ ) 3DmRC(∞) t NA
(11)
simplifies the notation of eq 10 to
(
a da3 ) -1 dt′ ac
)
(12)
The increase of the particles with sizes larger than ac goes at the expense of the decrease of particles with sizes smaller than ac. The law of mass conservation implies that at any instant of the growth process the increase in volume of the larger particles is equal to the decrease in volume of the smaller ones. Hence for a discrete set of N particles with radii ai N
∑ i)1
N
dai3 )
( ) ai
∑ i)1 a
- 1 dt′ ) 0
(13)
c
yielding
1
In the case R , a and R , ac, the exponential factor in eq 6 is approximated by the two first terms of its Taylor series. That way, eq 5 becomes
a dn ) 4πDmC(∞) -1 R dt ac
Introducing a new variable
N
∑ ai ) a c
N i)1
(14)
The left-hand side of eq 14 is by definition the numberaverage size aN. In the case of a continuous distribution W of particle radii a, eq 13 comes down to
∫0∞ W(a,t′)a da ) 1
1 ac
(15)
The integral in eq 15 is by definition aN; thus aN ) ac, whatever the form of W. Hence, as long as the mass of disperse phase material is conserved, the critical radius in the Ostwald ripening process equals the number-average radius, independent of whether the limiting stationary growth regime has been obtained. Only when the mass of dispersed material is not conserved, such as in the case of Ostwald ripening in the presence of a sink,9 the critical radius and the numberaverage radius become different. Finally, note that the proof implicitly assumes that the amount of the dissolved material in the medium is much smaller than the amount of material in the particles. This condition is fulfilled in most of the practical cases. LA035966D
(10) (9) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1999, 15, 6745.
