Page 83 - Book Hosokawa Nanoparticle Technology Handbook

2.2 PARTICLE SIZE FUNDAMENTALS
*
(radius; r ) when the precursor monomers (atoms or As stated above, homogeneous nucleation is sup-
*
molecules) reach a certain critical concentration C . pressed whenever C(b,t)/ t 0, i.e., whenever the
When coagulation of nuclei does not take place, each last term within brackets in equation (2.2.11) is posi-
nucleus then grows in size as other monomers in its tive. This gives us the necessary condition for homo-
vicinity diffuse toward its surface and nucleate onto it. geneous nucleation to occur:
The attainable number concentration of nucleated parti-
*
cles n , is determined from the balance between the 3 DC r
**
0
monomer generation rate and the rate of monomer G b 3 (2.2.12)
depletion by diffusion toward previously formed nuclei.
In regions where the nuclei concentration is so low The constant cell radius b can be expressed as a func-
that the monomer generation rate is larger than the tion of the number concentration of nucleated parti-
diffusion depletion rate, the monomer concentration cles n as 4 b n */3 1, so that the critical monomer
3
*
0
C increases with time until the critical concentration generation rate G , i.e., the generation rate above
*
0
*
is reached (C C ) and new nuclei can be formed. As which homogeneous nucleation occurs and below
the nuclei concentration increases, the depletion rate which nucleation is suppressed, can be written as:
of monomers, resulting from diffusion toward their
surfaces, also increases, until a time is reached at * * * *
which the depletion rate balances the generation rate. G 4 r DC n 0 (2.2.13)
0
From this time onward, the monomer concentration
decreases gradually and no new nuclei can be formed. The number concentration of nucleated particles
This local process occurs everywhere in the system can be predicted using equation (2.2.13), the mean
and, hence, the concentration of nucleated particles volume diameter of nucleated particles d can be esti-
v
attains a uniform value n . Considering the system mated from a mass balance:
*
0
overall, it can be stated that nucleation stops at a time 13 /
when the concentration of nucleated particles attains ⎛ 6 MC f ⎞
* ⎟
*
its maximum possible value n , beyond which the d ⎜ ⎝ n ⎠ (2.2.14)
v
0
monomer concentration decreases ( C/ t 0). p 0
The process of monomer diffusion toward the sur-
face of a nucleus is modeled assuming that each where C is the final monomer concentration without
f
nucleus is in the center of a spherical cell of constant nucleation, the particle density, and M the molecu-
p
radius b. The so-called cell model has been exten- lar weight. *
sively used in the description of other phenomena, The critical nucleus radius r can, in principle, be
such as evaporation, condensation, and melting. The estimated by the Gibbs–Thompson equation.
diffusion of monomers toward the nucleus can be However, this equation contains parameters whose
expressed as: actual values are only known approximately. The crit-
ical radius is usually taken as 1 nm. The monomer
C 2 diffusion coefficient D can be estimated using the
Gt D C (2.2.9) Wilke–Chang equation or the King equation. The
()
t *
monomer generation rate G can be estimated as
0
*
*
G ( C/ t) t t *, where t is the nucleation time. The
where G(t) is the monomer generation rate and D its 0 *
diffusion coefficient. generation rate of monomer G can be approximated
0
*
* *
With a constant monomer generation rate, taking as G C /t . Therefore, equation (2.2.13) can be
0
*
t 0 as the time at which C C , assuming r written using the attainable number concentration of
*
*
constant b, and using the initial and boundary nucleated particles n as:
0
*
*
conditions: C(r,0) C , C(r ,t) 0, ( C/ t) r b 0, 1
the approximate solution of equation (2.2.9) can be n (2.2.15)
*
0
*
written as: 4 rDt *
This simple relation can be used to predict the num-
⎛ 1 1⎞ ⎡ ⎛ Gb ⎞ Gb ⎤ ber concentration of nucleated particles even if the
3
3
Cr t)
⎜ ⎝ r * r ⎠ ⎟ e ⎢ ⎜ ⎝ C r 3 D ⎠ ⎟ 3 D , ⎥ (2.2.10) reaction rate expression and the critical supersatura-
**
(,
⎣
⎦
tion concentration are unknown.
Figure 2.2.6 shows a comparison between theoreti-
*
3
where: 3r Dt/b . cal (equation (2.2.15)) and experimental particle
Differentiating this equation with respect to time number concentrations. We used two types of particle
yields: formation methods. One is that of silver particles gen-
erated by chemical reduction, and the other is that of
⎞
*
3
Cr t) ⎛ 1 1 3 rD ⎛ Gb ⎞ zinc sulfide particles generated by homogeneous pre-
(,
**
⎜ ⎟ e ⎜ C r ⎟ cipitation. As seen, the calculated values are in good
t ⎝ r * r ⎠ b 3 ⎝ 3 D ⎠ (2.2.11) agreement with the experimental data.
59