31. The MWNT E values were calculated with the as-
sumption that the nanotubes are solid cylinders.
TEM imaging of our MWNTs showed that they have
hollow cores with diameters #2 nm. Because the
moment of inertia scales as r
4
, this assumption
produces only a very small error on the order of
0.01%.
32. O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B.
Spence, T. Weng, J. Appl. Phys. 41, 3373 (1970).
33. D. H. Robertson, D. W. Brenner, J. W. Mintmire,
Phys. Rev. B 45, 12592 (1992).
34. G. Overney, W. Zhong, D. Tomanek, Z. Phys. D 27,
93 (1993).
35. R. S. Ruoff and D. C. Lorentz, Carbon 33, 925
(1995).
36. J. P. Lu, Phys. Rev. Lett. 79, 1297 (1997).
37. The discontinuity in F-d is not due to a discontinuity
in the topography. First, high-resolution images
demonstrate that the MoS
2
substrate is atomically
flat in the region where the force discontinuity is
observed. Furthermore, the topographic signal,
which was recorded at the same time as F, is con-
stant across the region of force discontinuity.
38. Mechanical deformation of metal whiskers ultimately
leads to a decrease in the initial linear (elastic) F-d
behavior. This decrease is, however, due to plastic
deformations (4).
39. D. Hull and T. W. Clyne, An Introduction to Compos-
ite Materials (Cambridge Univ. Press, Cambridge,
1996).
40. The bending strength is relevant to composites
formed with randomly oriented nanotubes or NRs.
In a composite made with oriented nanotubes or
NRs, the tensile and compressive strengths should
also be considered. The tensile and bending
strengths are comparable for SiC whiskers (6, 7)
and are also expected to be similar for SiC
NRs. The tensile strength of a carbon nanotube is,
however, expected to be substantially larger than
the nanotube bending strength and the strength of
SiC NRs.
41. We thank J. W. Hutchinson and F. Spaepen for
helpful discussions and S. Shepard for assistance
with SiO deposition. C.M.L. acknowledges partial
support of this work by the NSF Division of Materi-
als Research and the Air Force Office of Scientific
Research.
13 June 1997; accepted 19 August 1997
Enhancement of Protein Crystal Nucleation by
Critical Density Fluctuations
Pieter Rein ten Wolde and Daan Frenkel*
Numerical simulations of homogeneous crystal nucleation with a model for globular
proteins with short-range attractive interactions showed that the presence of a meta-
stable fluid-fluid critical point drastically changes the pathway for the formation of a
crystal nucleus. Close to this critical point, the free-energy barrier for crystal nucleation
is strongly reduced and hence, the crystal nucleation rate increases by many orders of
magnitude. Because the location of the metastable critical point can be controlled by
changing the composition of the solvent, the present work suggests a systematic ap-
proach to promote protein crystallization.
As a result of rapid developments in bio-
technology, there has been an explosive
growth in the number of proteins that can
be isolated. However, the determination of
the three-dimensional structures of proteins
by x-ray crystallography remains a time-
consuming process. One bottleneck is the
difficulty of growing protein crystals good
enough for analysis. In his book on this
subject, McPherson wrote, “The problem of
crystallization is less approachable from a
classical analytical standpoint, contains a
substantial component of trial and error,
and draws more from the collective experi-
ence of the past century....Itismuch like
prospecting for gold” (1, p. 6). The experi-
ments clearly indicate that the success of
protein crystallization depends sensitively
on the physical conditions of the initial
solution (1, 2). It is therefore crucial to
understand the physical factors that deter-
mine whether a given solution is likely to
produce good crystals.
Studies have shown that not just the
strength but also the range of the interac-
tions between protein molecules is crucial
for crystal nucleation. In 1994, George and
Wilson (3) demonstrated that the success of
crystallization appears to correlate with the
value of B
2
, the second osmotic virial coef-
ficient of the protein solution.
The second virial coefficient describes
the lowest order correction to the van’t
Hoff law for the osmotic pressure P:
P/rk
B
T 5 1 1 B
2
r1(terms of order r
2
)
(1)
where r is the number density of the dis-
solved molecules, k
B
is Boltzmann’s con-
stant, and T is the absolute temperature.
For macromolecules, B
2
can be determined
from static light-scattering experiments (4).
Its value depends on the effective interac-
tion between a pair of macromolecules in
solution (5).
George and Wilson measured B
2
for a
number of proteins in various solvents.
They found that for those solvent condi-
tions that are known to promote crystalli-
zation, B
2
was restricted to a narrow range
of small negative values. For large positive
values of B
2
, crystallization did not occur at
all, whereas for large negative values of B
2
,
protein aggregation rather than crystalliza-
tion took place.
Rosenbaum, Zamora, and Zukoski (6, 7)
established a link between the work of
George and Wilson and earlier studies of
the phase behavior of spherical, uncharged
colloids (8–11). Since the theoretical work
of Gast, Russel, and Hall (8), it has been
known that the range of attraction between
spherical colloids has a drastic effect on the
appearance of the phase diagram. If the
range of attraction is long in comparison
with the diameter of the colloids, the phase
diagram of the colloidal suspension resem-
bles that of an atomic substance, such as
argon. Depending on the temperature and
density of the suspension, the colloids can
occur in three phases (Fig. 1A): a dilute
colloidal fluid (analogous to the vapor
phase), a dense colloidal fluid (analogous to
the liquid phase), and a colloidal crystal
phase. However, when the range of the
attraction is reduced, the fluid-fluid critical
point moves toward the triple point, where
the solid coexists with the dilute and dense
fluid phases. If the range of attraction is
made even shorter (less than 25% of the
colloid diameter), two stable phases remain,
one fluid and one solid (Fig. 1B). However,
the fluid-fluid coexistence curve survives in
the metastable regime below the fluid-solid
coexistence curve (Fig. 1B). This is indeed
found in experiments (11, 12) and simula-
tions (10). This observation is relevant for
solutions of globular proteins, because they
often have short-range attractive interac-
tions. A series of studies (13–15) showed
that the phase diagram of a variety of pro-
teins is of the kind shown in Fig. 1B. More-
over, the range of the effective interactions
between proteins can be changed by the
addition of nonadsorbing polymer (such as
polyethylene glycol) (11, 16) or by chang-
ing the pH or salt concentration of the
solvent (1, 2).
Rosenbaum, Zamora, and Zukoski (6, 7)
observed that the conditions under which
a large number of globular proteins can be
made to crystallize map onto a narrow
temperature range, or more precisely, a
narrow range in the value of the osmotic
second virial coefficient of the computed
fluid-solid coexistence curve of colloids
with short-range attraction (10). Several
authors had already noted that a similar
crystallization window exists for colloidal
suspensions (17). Here our aim was to use
simulation to gain insight into the physi-
cal mechanism responsible for the en-
FOM Institute for Atomic and Molecular Physics, Kruis-
laan 407, 1098 SJ Amsterdam, Netherlands.
*To whom correspondence should be addressed.
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hanced crystal nucleation. We found that
the presence of a metastable fluid-fluid
critical point is essential.
The rate-limiting step in crystal nucle-
ation is the crossing of a free-energy barrier.
In our simulations we computed the barrier
for homogeneous crystal nucleation for a
model globular protein. Our model system
has some of the essential features needed to
get a proteinlike phase diagram of the type
shown in Fig. 1B—the particles repel
strongly at short distances and attract at
slightly larger distances.
For the interaction between the parti-
cles, we chose a suitable generalization of
the Lennard-Jones potential:
n~r!
5
4ε
a
2
5
1
F
S
r
s
D
2
21
G
6
2a
1
F
S
r
s
D
2
21
G
3
6
(2)
where s denotes the hard-core diameter of
the particles, r is the interparticle distance,
and ε is the depth of the potential well. The
potential in Eq. 2 provides a simplified de-
scription of the effective interaction be-
tween real proteins in solution: It accounts
both for direct and for solvent-induced in-
teractions between the globular proteins.
The width of the attractive well can be
adjusted by varying the parameter a.We
found that for a'50, the system repro-
duced the phase behavior of protein solu-
tions studied in (13–15), that is, the fluid-
fluid coexistence curve was located in the
metastable region ;20 % below the equi-
librium crystallization curve.
Conventional molecular dynamics sim-
ulations cannot be used to study crystal
nucleation under realistic conditions. The
reason is that the formation of a critical
nucleus is a rare event; crystallization in real
protein solutions may take days or weeks. In
a simulation, the situation would be worse
because the volume that can be studied by
simulation is orders of magnitude smaller,
and the probability of crystal formation is
decreased by the same amount. Moreover,
the computational cost of molecular dy-
namics simulations that cover more than
10
28
s becomes prohibitive. Hence, the
problem has to be approached in a different
way. In the present work, we used an ap-
proach (18) based on the Bennett-Chan-
dler numerical scheme to compute the rate
of activated processes (19). The rate G can
be written as the product of two factors:
G5exp(2DG*/k
B
T) 3n (3)
where DG* is the height of the (Gibbs)
free-energy barrier separating the metasta-
ble fluid from the crystal phase. The factor
exp(2DG*/k
B
T) denotes the probability
that a spontaneous fluctuation will result in
the formation of a critical nucleus; it de-
pends strongly on the degree of supercool-
ing. The kinetic prefactor n is a measure of
the rate at which critical nuclei, once
formed, transform into larger crystallites. It
depends only weakly on supercooling, un-
less the system is close to gelation (11).
Below, we discuss simulations under condi-
tions away from the gelation curve. Hence,
the variation of the nucleation rate is dom-
inated by the variation in the barrier height
DG*. A rough estimate of DG* is given by
classical nucleation theory (20),
DG*/k
B
T 5 16pg
3
/(3k
B
Tr
2
Dm
2
) (4)
where g is the free-energy density per unit
area of the solid-fluid interface, r is the num-
ber density of the solid phase, and Dm is the
difference in chemical potential between the
fluid and the solid. It is the thermodynamic
driving force for crystallization.
To compute the free-energy barrier that
the system must overcome to form a critical
crystal nucleus, we used the umbrella-sam-
pling Monte Carlo scheme of Valleau (18,
21). This is a biased Monte Carlo scheme
that makes it possible to concentrate the
sampling in the region of the free-energy
barrier. Without the bias, this region of
configuration space is effectively inaccessi-
ble. To compute the free energy as a func-
tion of the size of the incipient crystal nu-
cleus, we must be able to identify which
particles belong to the crystal nucleus. As
discussed in (18), each particle can be clas-
sified as either solidlike or liquidlike by
analyzing the local symmetry of its sur-
roundings. If the distance between two sol-
idlike particles is less than 1.5s, they are
considered to be connected and belong to
the same cluster. We denote the number of
solidlike particles belonging to a given crys-
tal nucleus by N
crys
. Crystallization near the
metastable fluid-fluid critical point is
strongly influenced by the large density
fluctuations that occur in the vicinity of
such a critical point; therefore, the free-
energy barrier associated with formation of
dense, liquidlike droplets should also be
considered. Hence, for every particle, we
determined not just the structure of its sur-
roundings but also the local density. In the
same way, the size of a high-density cluster
(be it solid or liquidlike) can be defined as
the number of connected particles, N
r
, that
have a significantly denser local environ-
ment than particles in the remainder of the
system. In our simulations, the free-energy
“landscape” of the system was determined as
a function of the two coordinates N
crys
and
N
r
. In a crystal nucleation event, the be-
ginning is from the homogeneous liquid
(N
crys
' N
r
' 0). The free energy then
increases until it reaches a saddle-point (the
critical nucleus). From there on, the crystal
will grow spontaneously.
We first computed the phase diagram
shown in Fig. 1B using the techniques de-
scribed in (10) and then studied the free-
energy barrier for crystal nucleation at four
different points in the phase diagram: one
well above the metastable critical point
(T 5 2.23 T
c
), one at T
c
, and the remain-
ing two at 0.89 T
c
and 1.09 T
c
. The degree
Fig. 1. (A) Typical phase diagram of a molecular substance with a relatively long-range attractive
interaction. This phase diagram corresponds to the Lennard-Jones 6-12 potential {v(r) 5 4«[(s/r)
12
–
(s/r)
6
], solid curve in insert} (24). The dashed line indicates the triple point. (B) Typical phase diagram of
colloids with short-range attraction. The phase diagram was computed for the potential given in Eq. 2
(solid curve in insert), with a550. In both figures, the temperature is in units of the critical temperature
T
c
, and the number density is in units s
–3
, where s, the effective diameter of the particles, is defined in
the expression for v(r). The diamonds indicate the fluid-fluid critical points. In (A) and (B) the solid lines
indicate the equilibrium coexistence curves. The dashed curve in (B) indicates the metastable fluid-fluid
coexistence. Crystal-nucleation barriers were computed for the points denoted by open squares.
SCIENCE
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www.sciencemag.org1976
of supercooling was chosen such that clas-
sical nucleation theory would predict the
same value of DG*/(k
B
T) for all systems. To
estimate DG* from Eq. 4, we used
Dm'DH(T
m
2T)/T
m
(5)
where DH is the enthalpy of melting at the
coexistence temperature T
m
. Turnbull’s
empirical rule was used to estimate the
surface free-energy g, which states that g
is proportional to DH (20). For all points,
we studied the free-energy landscape and
the lowest free-energy path to the critical
nucleus.
We found that away from T
c
(above and
below), the path of lowest free energy is one
where the increase in N
r
is proportional to
the increase in N
crys
(Fig. 2A). Such behav-
ior is expected if the incipient nucleus is a
small crystallite. However, around T
c
, crit-
ical density fluctuations lead to a striking
change in the free-energy landscape (Fig.
2B). First, the route to the critical nucleus
leads through a region where N
r
increases
while N
crys
is still essentially zero. In other
words, the first step toward the critical nu-
cleus is the formation of a liquidlike droplet.
Then, beyond a certain critical size, the
increase in N
r
is proportional to N
crys
, that
is, a crystalline nucleus forms inside the
liquidlike droplet.
Clearly, the presence of large density
fluctuations close to a fluid-fluid critical
point effects the route to crystal nucleation.
But, more importantly, the nucleation bar-
rier close to T
c
is much lower than at either
higher or lower temperatures (Fig. 3). The
observed reduction in DG* near T
c
by
;30k
B
T corresponds to an increase the nu-
cleation rate by a factor 10
13
. One interpre-
tation of this observation is that near the
fluid-fluid critical point, the wetting of the
crystal nucleus by a liquidlike layer results
in a value of the interfacial free energy g,
and therefore of the barrier height DG*,
that is much lower than would be estimated
on the basis of Turnbull’s rule. In fact, Haas
and Drenth (22) noted that the experimen-
tally determined interfacial free energy of
small protein crystals (23) is much smaller
than the value predicted on the basis of
their version of Turnbull’s rule.
The conventional way to lower the crys-
tal nucleation barrier is to prepare a more
supersaturated solution. However, highly
supersaturated solutions tend to form aggre-
gates rather than crystals (3, 6, 7, 11, 12,
15). Moreover, in such a solution, the ther-
modynamic driving force for crystallization
(m
liq
2m
cryst
) is also enhanced. As a con-
sequence, the crystallites that nucleate will
grow rapidly and far from perfectly (2). One
of the implications of our finding that the
crystal nucleation barrier is reduced near I
c
is that one can selectively speed up the rate
of crystal nucleation, without increasing the
rate of crystal growth, or the rate at which
amorphous aggregates form. This can be
achieved by adjusting the solvent condi-
tions (for instance, by the addition of non-
ionic polymer) and thereby changing the
range of interaction, such that a metastable
fluid-fluid critical point is located just be-
low the sublimation curve.
Our description of the early stages of
protein crystallization is highly simplified,
yet we suggest that the mechanism for
enhanced crystal nucleation described
here is quite general. The phase diagram
shown in Fig. 1B is likely to be the rule
rather than the exception for compact
macromolecules. Moreover, it occurs both
in the bulk and in (quasi) two-dimension-
al systems (such as membranes). It is
therefore tempting to speculate that na-
ture already makes extensive use of critical
density fluctuations to facilitate the for-
mation of ordered structures.
REFERENCES AND NOTES
___________________________
1. A. McPherson, Preparation and Analysis of Protein
Crystals (Krieger, Malabar, FL, 1982).
2. S. D. Durbin and G. Feher, Annu. Rev. Phys. Chem.
47, 171 (1996); F. Rosenberger, J. Cryst. Growth
166, 40 (1996).
3. A. George and W. W. Wilson, Acta Crystallogr. D 50,
361 (1994).
4. E. G. Richards, An Introduction to Physical Proper-
ties of Large Molecules in Solution (Cambridge Univ.
Press, Cambridge, 1980).
5. See, for example, T. L. Hill, An Introduction to Statis-
tical Thermodynamics (Dover, New York, 1986),
chap. 15.
6. D. Rosenbaum, P. C. Zamora, C. F. Zukoski, Phys.
Rev. Lett. 76, 150 (1996).
7. D. F. Rosenbaum and C. F. Zukoski, J. Cryst.
Growth 169, 752 (1996).
8. A. P. Gast, W. B. Russell, C. K. Hall, J. Colloid Inter-
face Sci. 96, 251 (1983); 109, 161 (1986).
9. H. N. W. Lekkerkerker et al., Europhys. Lett. 20, 559
(1992).
10. M. H. J. Hagen and D. Frenkel, J. Chem. Phys. 101,
4093 (1994).
11. S. M. Ilett, A. Orrock, W. C. K. Poon, P. N. Pusey,
Phys. Rev. E 51, 1344 (1995).
12. W. C. K. Poon, A. D. Pirie, P. N. Pusey, Faraday
Discuss. 101, 65 (1995); W. C. K. Poon, Phys. Rev.
E 55, 3762 (1997).
13. C. R. Berland et al., Proc. Natl. Acad. Sci. U.S.A. 89,
1214 (1992); N. Asherie, A. Lomakin, G. B. Benedek,
Phys. Rev. Lett. 77, 4832 (1996).
14. M. L. Broide, T. M. Tominc, M. D. Saxowsky, Phys.
Rev. E 53, 6325 (1996).
15. M. Muschol and F. Rosenberger, J. Chem. Phys.
107, 1953 (1997).
16. A. McPherson, J. Biol. Chem. 251, 6300 (1976).
17. A. Kose and S. Hachisu, J. Colloid Interface Sci. 55,
487 (1976); C. Smits et al., Phase Transitions 21,
157 (1990).
18. P. R. ten Wolde, M. J. Ruiz-Montero, D. Frenkel,
Phys. Rev. Lett. 75, 2714 (1995); J. Chem. Phys.
104, 9932 (1996).
Fig. 2. Contour plots of
the free-energy land-
scape along the path
from the metastable fluid
to the critical crystal nu-
cleus for our system of
spherical particles with
short-range attraction.
The curves of constant
free energy are drawn as
a function of N
r
and N
crys
and are separated by
5k
B
T.(A) The free-ener-
gy landscape well below
the critical temperature
(T/T
c
5 0.89). The lowest
free-energy path to the
critical nucleus is indicat-
ed by a dashed curve.
This curve corresponds
to the formation and
growth of a highly crys-
talline cluster. (B) As (A),
but for T 5 T
c
. In this case, the free-energy valley (dashed curve) first runs parallel to the N
r
axis
(formation of a liquidlike droplet), and then moves toward a structure with a higher crystallinity (crystallite
embedded in a liquidlike droplet). The free-energy barrier for this route is much lower than the one in (A).
Fig. 3. Variation of the free-energy barrier for ho-
mogeneous crystal nucleation, as a function of
T/T
c
, in the vicinity of the critical temperature. The
solid curve is a guide to the eye. The nucleation
barrier at T 5 2.23T
c
is 128k
B
T and is not shown
in this figure. If Turnbull’s phenomenological rule
for g would hold (20), Eq. 4 would predict a con-
stant nucleation barrier. But the simulations show
that the nucleation barrier goes through a mini-
mum around the metastable critical point.
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26 SEPTEMBER 1997 1977
19. C. H. Bennett, in Diffusion in Solids: Recent Devel-
opments, A. S. Nowick and J. J. Burton, Eds. (Aca-
demic Press, New York, 1975), pp. 73–113; D.
Chandler, J. Chem. Phys. 68, 2959 (1978).
20. See, for example, K. F. Kelton, in Crystal Nucleation
in Liquids and Glasses, H. Ehrenreich and D. Turn-
bull, Eds. (Academic Press, Boston, 1991), vol. 45,
pp. 75–177. As shown in this reference, the con-
stant of proportionality is system-specific.
21. G. M. Torrie and J. P. Valleau, Chem. Phys. Lett. 28,
578 (1974).
22. C. Haas and J. Drenth, J. Cryst. Growth 154, 126
(1995).
23. A. J. Malkin and A. McPherson, ibid. 128, 1232
(1992); ibid. 133, 29 (1993).
24. J. P. Hansen and L. Verlet, Phys. Rev. 184, 151
(1969).
25. We thank E. J. Meijer for computing part of the phase
diagram in Fig. 1B, and A. van Blaaderen, M. Dog-
terom, J. Drenth, J. M. W. Frenken, S. Fraden, C.
Haas, H. N. W. Lekkerkerker, B. Smit, R. Sear,
J. T. M. Walraven, and C. Zukoski for critical reading
of the manuscript. Supported by Scheikundig
Onderzoek Nederland and by Fundamenteel Onder-
zoek der Materie (FOM) with financial aid from Ned-
erlandse Organisatie voor Wetenschappelijk Onder-
zoek. Computer time was provided by Nationale
Computer Faciliteiten.
9 June 1997; accepted 11 August 1997
Reversible Tuning of Silver Quantum Dot
Monolayers Through the Metal-Insulator
Transition
C. P. Collier, R. J. Saykally, J. J. Shiang, S. E. Henrichs,
J. R. Heath*
The linear and nonlinear (x
(2)
) optical responses of Langmuir monolayers of organically
functionalized silver quantum dots were measured as a continuous function of inter-
particle separation under near-ambient conditions. As the distance between metal
surfaces was decreased from 12 to ;5 angstroms, both quantum and classical effects
were observed in the optical signals. When the separation was less than 5 angstroms,
the optical second-harmonic generation (SHG) response exhibited a sharp disconti-
nuity, and the linear reflectance and absorbance began to resemble those of a thin
metallic film, indicating that an insulator-to-metal transition occurred. This transition
was reversible.
Recent developments in chemical tech-
niques for producing narrow size distribu-
tions of various metal (1) and semiconduc-
tor (2–4) quantum dots, and for fabricating
ordered superlattices from these nanocrys-
tals, have sparked much interest in the pos-
sibility of forming solids that have electrical
and optical properties that could be tuned
through chemical control over particle size,
stoichiometry, and interparticle separation
(5–10). In principle, control over interpar-
ticle separation provides a means for con-
trolling both quantum and classical cou-
pling interactions. This is of practical im-
portance because control of quantum inter-
actions would provide a route to preparing a
solid of quantum dot “atoms” and precisely
tuning the electronic properties of that sol-
id by controlling the wave function overlap
between adjacent particles. Although re-
cent reports have documented classical (en-
ergy-transfer) coupling (11, 12) between
semiconductor quantum dots (13, 14),
quantum coupling has not been observed.
Nor has quantum coupling been document-
ed in coinage metal quantum dots, even
though classical coupling [in the form of
color changes (15)] and percolation phe-
nomena (16) have long been observed in
such systems.
Here, we describe in situ measurements
of both the linear and nonlinear optical
properties of organically functionalized sil-
ver nanocrystal Langmuir monolayers as a
continuous function of interparticle separa-
tion distance. As the monolayer is com-
pressed from an average separation between
the surfaces of the metal cores (d)of12
(62)Åto;5(62) Å, the linear and
nonlinear optical properties reveal evidence
of both classical and quantum interparticle
coupling phenomena. Below d;5Å,evi-
dence for a sharp insulator-to-metal transi-
tion is observed in both optical signals. The
nonlinear optical response abruptly decreas-
es to a nearly constant value, and the linear
reflectance drops precipitously until it
matches that of a continuous metallic film.
This transition is reversible: The particles
can be redissolved (as a colloid), or, if the
trough barriers are opened, the film is again
characterized by the optical properties of
near-isolated silver nanocrystals.
Silver quantum dots are expected to ex-
hibit strong linear and nonlinear optical
responses (17). The linear response is dom-
inated by the surface plasmon resonance
v
sp
. This transition, which has no molecu-
lar analog, is characterized by an oscillator
strength resembling the physical dimen-
sions of the particle (18). v
sp
can be mod-
eled as the free (valence) electrons in the
particle responding to an applied electro-
magnetic field, with the positive nuclear
cores providing a restoring potential. For
particles in the size range considered here
(2 to 5 nm), v
sp
is determined by the free
electron density in the particle and by the
dielectric surrounding the particle surface.
The resonance width is determined by the
time scale for electron scattering at the
particle boundaries and is dependent on
particle size. The physical picture describing
the nonlinear optical response is somewhat
different. Consider an electron cloud asso-
ciated with a nanoparticle. x
(1)
is a mea-
surement of the polarizability of the cloud,
which scales as the volume of the particle.
x
(2)
, the second-order susceptibility, is a
measurement of how x
(1)
changes with re-
spect to an applied electric field. A require-
ment for a nonzero x
(2)
is a lack of inversion
symmetry, a condition that is met by a
supported monolayer of nanocrystals. x
(2)
is
almost exclusively sensitive only to the op-
tical response of the crystallites and is a
near background-free measurement.
Both the linear and nonlinear optical
responses should provide excellent probes
of interparticle coupling. When particles
are brought into close proximity to one
another, the dielectric surrounding a single
particle is strongly modified by the presence
of adjacent (conducting) spheres, resulting
in shifts (to lower energy) of v
sp
. For quan-
tum mechanical coupling, delocalization of
charge carriers over multiple particles in-
creases the scattering lifetime and leads to
narrower plasmon linewidths. In the limit-
ing case of complete electron delocalization
between particles (that is, the formation of
a continuous metal film), strong carrier ab-
sorption at energies below v
sp
and low re-
flectivity are expected. For example, con-
sider the familiar case of thin metal films,
which tend to be darkly colored. Only when
a metal film is substantially thicker than the
optical skin depth does it become reflective.
x
(2)
should also be an excellent probe of
interparticle coupling. A nonlinear optical
response is enhanced if the resonant state
has both high oscillator strength and large
volume (19–21), criteria that are satisfied by
the particles discussed here. For classical cou-
pling, local field effects should modify x
(2)
according to multipole coupling models
(22). In contrast, quantum mechanical cou-
C. P. Collier and R. J. Saykally, Department of Chemistry,
University of California, Berkeley, CA 94720, USA.
J. J. Shiang, S. E. Henrichs, J. R. Heath, Department of
Chemistry and Biochemistry, University of California, 405
Hilgard Avenue, Los Angeles, CA 90095, USA.
*To whom correspondence should be addressed.
SCIENCE
z
VOL. 277
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26 SEPTEMBER 1997
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www.sciencemag.org1978
![Fig. 1. (A) Typical phase diagram of a molecular substance with a relatively long-range attractive interaction. This phase diagram corresponds to the Lennard-Jones 6-12 potential {v(r) 5 4«[(s/r)12 – (s/r)6], solid curve in insert} (24). The dashed line indicates the triple point. (B) Typical phase diagram of colloids with short-range attraction. The phase diagram was computed for the potential given in Eq. 2 (solid curve in insert), with a 5 50. In both figures, the temperature is in units of the critical temperature Tc, and the number density is in units s –3, where s, the effective diameter of the particles, is defined in the expression for v(r). The diamonds indicate the fluid-fluid critical points. In (A) and (B) the solid lines indicate the equilibrium coexistence curves. The dashed curve in (B) indicates the metastable fluid-fluid coexistence. Crystal-nucleation barriers were computed for the points denoted by open squares.](/figures/fig-1-a-typical-phase-diagram-of-a-molecular-substance-with-17a0jvwy.webp)