A R T I C L E S
Talapin et al.
surface disorder) of nanocrystalssin this paper, we use this
general term to describe a smooth and defect-free surface (to a
smaller or larger extent) being free of traps. The difference in
the surface quality of nanocrystals in an ensemble originates
from the growth conditions and is not caused by the size-
selective precipitation after preparation. The latter is only a tool
to isolate fractions of nanocrystals with different surface quality
and size.
describe the evolution of an ensemble of nanoparticles in
colloidal solution, several processes occurring simultaneously
have to be considered: the kinetics of the addition or the
removal of monomer from the nanocrystals, changes of the
particle concentration with time due to nucleation and dissolution
processes, temporal evolution of the monomer concentration,
etc. A theory of Lifshitz, Slyozov, and Wagner (LSW)45,46
provides an asymptotic solution for the evolution of an ensemble
of particles during Ostwald ripening. However, this approach
as well as further analytical models and numerical simulations
of Ostwald ripening47-55 fails in the description of ensembles
of nanometer-sized particles as only two terms of the expansion
of the Gibbs-Thompson equation are used:
Why does the surface quality vary in an ensemble of growing
nanocrystals systematically with the nanocrystal size? To answer
this question we have to consider the parameters governing the
surface quality or surface roughness of crystalline solids during
their growth. The surface roughness of bulk crystals depends
strongly on the growth conditions, mainly on the ratio between
the growth temperature and the characteristic temperature (TR)
of the so-called “roughening transition”.38-41 Below TR, the
surface of the growing crystal is microscopically smooth, above
it is rough. For most semiconductors, TR is above 1000 °C, being
slightly dependent on the crystallographic plane.39 In principle,
for nanocrystalline matter, TR can be much lower, as the melting
point depends on the particle size.42 However, even if we assume
that TR is size-dependent, it would not allow us to explain similar
distributions of surface quality within ensembles with different
mean particle size. Therefore, it seems more probably that the
observed distribution of PL efficiencies can be explained in
terms of “kinetic roughening”, the phenomenon well known for
the growth of bulk crystalline solids:38,43,44 when growth rate
of a crystal increases, the critical size of two-dimensional (2D)
nucleation sites at the surface becomes so small that the 2D
nucleation barrier vanishes. As a result, the crystal surface
becomes microscopically rough and macroscopically rounded.
In an ensemble of nanocrystals in colloidal solution, the
particle growth rate is strongly size-dependent: the main concept
of the Ostwald ripening implies that larger particles in an
ensemble grow at the expense of monomer released by the
dissolution of smaller particles. As the Ostwald ripening is a
self-similar process with an unique limiting particle size
distribution,45-55 the particle growth rate distribution has to be
similar at all stages of the particle growth. To explain the
influence of the growth mechanism on the distribution of PL
efficiency in any ensemble of growing colloidal nanoparticles,
we developed a model describing the evolution of a nanocrystal
ensemble growing in a solution of monomer.
2γVm
2γVm
C(r) ) C0flat exp
≈ C0flat 1 +
(1)
(
)
[
]
rRT
rRt
where C(r) and C0flat are the solubilities of a particle with radius
r and of the bulk material, respectively, γ is the surface tension,
and Vm is the molar volume of the solid.
The coefficient 2γVm/(RT) called “capillary length” is usually
of the order of 1 nm,51,52 and eq 1 satisfactorily describes the
solubility of colloidal particles with radius larger than ∼20 nm.
For nanocrystals with r ) 1-5 nm, the value of the capillary
length is approaching the particle radius, and the particle
solubility becomes strongly nonlinear against r-1. On the other
hand, the Gibbs-Thompson equation in its exact form is,
probably, valid for very small particles: it was reported that
the surface tension remains nearly constant for many interfaces
even if the particle size is as small as ∼1 nm.56
Recently, we proposed a model of particle growth and
dissolution which is based on the dependence of activation
energies on the particle size.23 This model allows us to describe
the size-dependent evolution of a nanoparticle under reaction,
diffusion, and mixed reaction-diffusion control. Thus, the
evolution of a single particle of radius r in a solution of
monomer with constant concentration [M] is given by the
following equation (see details in ref 23):
S - exp[1/r*]
dr*
dτ
)
(2)
r* + K exp[R/r*]
0
where r* ) (RT/2γVm)r and τ ) (R2T2DC /4γ Vm)t are
2
flat
Theoretical Description of the Dynamic Growth Rate
Distribution within the Ensemble of Nanocrystals. To
dimensionless particle radius and time, respectively.
K ) (RT/2γVm)(D/kflat) is the dimensionless parameter de-
g
scribing the type of the process involved, i.e., the ratio between
the rates of a purely diffusion-controlled process (D is the
diffusion coefficient of the monomer) and a purely reaction-
controlled one (kfglat is a first-order reaction rate constant for
the process of addition of a monomer to a flat interface; see ref
23 for details). Thus, a value of K < 0.01 corresponds to the
almost pure diffusion-controlled process, K > 100 corresponds
to the reaction-controlled one, and the range 0.01 < K < 100
corresponds to the regime of mixed control with comparable
contributions of both processes.
(38) Liu, X. Y.; Bennema, P. J. Cryst. Growth 1994, 139, 179.
(39) Heyraud, J. C.; Me´tois, J. J.; Bermond, J. M. Surf. Sci. 1999, 425, 48.
(40) Jackson, K. A.; Miller, C. E. J. Cryst. Growth 1977, 40, 169.
(41) Vakarin, E. V.; Badiali, J. P. Phys. ReV. B 1999, 60, 2064.
(42) Goldstein, A. N.; Echer, C. M.; Alivisatos, A. P. Science 1992, 256, 1425.
(43) Scneidman, V. A.; Jackson, K. A.; Beatty, K. M. J. Cryst. Growth 2000,
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(44) Gilmer, G. H. J. Cryst. Growth 1976, 35, 15.
(45) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35.
(46) Wagner, C. Z. Elektrochem. 1961, 65, 581.
(47) Marqusee, J.; Ross, J. J. Chem. Phys. 1984, 80, 536.
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(49) Yao, J. H.; Elder, K. R.; Guo Hong; Grant, M. Phys. ReV. B 1993, 47,
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The dimensionless parameter S ) [M]bulk/C0 describes the
(50) Venzl, G. Phys. ReV. A 1985, 31, 3431.
flat
(51) Kabalnov, A. S.; Shchukin, E. D. AdV. Colloid Interface Sci. 1992, 38, 69.
(52) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884.
(53) Carlow, G. R.; Zinke-Allmang, M. Phys. ReV. Lett. 1997, 78, 4601.
(54) De Smet, Y.; Deriemaeker, L.; Parloo, E.; Finsy, R. Langmuir 1999, 15,
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oversaturation of the monomer in solution; R is the transfer
coefficient of the activated complex (0 < R < 1).
(55) Egelhaaf, S.; Olsson, U.; Schurtenberger, P.; Morris, J.; Wennerstro¨m, H.
Phys. ReV. E 1999, 60, 5681.
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5788 J. AM. CHEM. SOC. VOL. 124, NO. 20, 2002