letters to nature
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22. Schnitzer, I., Yablonovitch, E., Caneau, C., Gmitter, T. J. & Scherer, A. 30% external quantum
efficiency from surface textured, thin-film light-emitting diodes. Appl. Phys. Lett. 63, 2174–2176
(1993).
23. Fricke, J., Yang, B., Brandt, O. & Ploog, K. Patterning of cubic and hexagonal GaN by Cl
2
/N
2
-based
reactive ion etching. Appl. Phys. Lett. 74, 3471–3473 (1999).
Acknowledgements
We thank O. Mayrock and H.-J. Wu
¨
nsche for help in the calculation of single-particle
wavefunctions and exciton binding energies. This work was supported in part by the
Volkswagen-Stiftung.
Correspondence and requests for materials should be addressed to P.W.
(e-mail: walter@pdi-berlin.de).
.................................................................
Liquid crystal phase transitions in
suspensions of polydisperse
plate-like particles
Felix M. van der Kooij, Katerina Kassapidou & Henk N. W. Lekkerkerker
Van’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute,
Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands
.................................. ......................... ......................... ......................... ......................... ........
Colloidal suspensions that form periodic self-assembling struc-
tures on sub-micrometre scales are of potential technological
interest; for example, three-dimensional arrangements of spheres
in colloidal crystals
1
might serve as photonic materials
2
, intended
to manipulate light. Colloidal particles with non-spherical shapes
(such as rods and plates) are of particular interest because of their
ability to form liquid crystals. Nematic liquid crystals possess
orientational order; smectic and columnar liquid crystals addi-
tionally exhibit positional order (in one or two dimensions
respectively). However, such positional ordering
3,4
may be inhib-
ited in polydisperse colloidal suspensions. Here we describe a
suspension of plate-like colloids that shows isotropic, nematic and
columnar phases on increasing the particle concentration. We find
that the columnar two-dimensional crystal persists for a poly-
dispersity of up to 25%, with a cross-over to smectic-like ordering
at very high particle concentrations. Our results imply that liquid
crystalline order in synthetic mesoscopic materials may be easier
to achieve than previously thought.
Suspensions of (almost) monodisperse colloidal particles are
used as mesoscopic models of the phase behaviour of atomic and
molecular systems. In this respect, the fluid–crystal transition in
suspensions of spheres
1
and the liquid crystal phase transitions in
suspensions of non-spherical particles
5,6
have attracted considerable
attention. However, colloidal particles are hardly ever truly mono-
disperse but often exhibit a size distribution of finite width. The
consequences of this inherent polydispersity are of substantial
fundamental and industrial interest. In particular, the inhibitive
role of polydispersity in the crystallization process of hard spheres is
a contentious issue that deals with the value
3,7–9
and even the very
existence
10,11
of a so-called terminal polydispersity j
t
, above which
no crystallization can occur.
Moreover, one may wonder how polydispersity affects positional
ordering if such ordering exists in just one or two dimensions, that
is, in the case of smectic or columnar liquid crystals (sketched in
Fig. 1). Driven by a gain in excluded volume entropy, these liquid
crystal phases may be formed in concentrated suspensions of hard-
body rod or plate-like particles
12,13
. For dense systems of rods,
computer simulation predicts a stable smectic phase up to a
polydispersity in rod length of 18%, while for higher polydisper-
sities the smectic phase is pre-empted by a columnar phase
4
.
Accordingly, in experiments, almost monodisperse rod-like virus
particles show a smectic phase
6
while polydisperse solutions of
DNA-rods show a columnar phase instead
14
.
The phase behaviour of hard plate-like particles, on the other
hand, has received considerably less attention, largely because
suitable experimental model systems have been developed only
recently
15,16
. One of these model systems, comprising fairly mono-
disperse platelets of low aspect ratio, exhibits an isotropic and
columnar phase
17
; but, computer simulation for monodisperse
platelets of sufficiently large aspect ratio predicts an isotropic (I),
Figure 1 Structure of the three main classes of liquid crystals. The nematic phase (N), the
columnar phase (C), and the smectic phase (S) are schematically depicted here for the
case of plate-like particles. While each of these phases exhibits long-range orientational
order, they differ by the positional correlations between the particles. In the nematic phase
long-range positional order is absent. The columnar phase has a two-dimensional lattice
of columns, which are constituted of liquid-like stacks of particles. The smectic phase is
characterized by a one-dimensional periodic array of layers of particles.
Figure 2 Transmission electron microscopy images. a, The parent suspension
(j
D
¼ 25%); and b, the fractionated suspension (j
D
¼ 17%). The platelets are gibbsite
(that is, Al(OH
3
)) colloids whose surface is grafted with a modified polyisobutylene
(
M
w
< 1;000 g per mole). Owing to this steric stabilization layer, the particles interact
through an approximately hard-core (that is, short-range repulsive) interaction potential
when dispersed in apolar solvents (in this case, toluene)
15,23
. The reduction of the
particles’ polydispersity in diameter to 17% is achieved by fractionation of part of the
parent system, using a scheme which resembles the method of depletion-enhanced
crystallization fractionation of emulsion droplets as described
24
. The suspension is
submitted to I–N phase separation, using 62gl
−1
added non-adsorbing polymer to
enhance fractionation, yielding roughly 80% nematic phase in coexistence with 20%
isotropic phase. The isotropic upper phase is subsequently removed. By dilution of the
remaining nematic phase (using polymer solution) back to the I–N region and repeating
this scheme twice, the suspension thus obtained has a lower polydispersity than the
original system. The non-adsorbing polymer is subsequently removed by redispersioning
in polymer-free solvent after sedimentation. The number-average diameter h
D
i of the
grafted platelets (Table 2) is based on the diameter of the core, determined from
transmission electron microscopy (TEM) micrographs, plus twice the estimated thickness
(4 nm) of the grafted polymer layer
25
. We define the diameter of the hexagons by the
diameter of a circle of equal area, with a relative standard deviation
j
D
¼ðh
D
2
i 2 h
D
i
2
Þ
1=2
=h
D
i. The number-average thickness h
L
i of the plates (including the
grafted polymer layer) is roughly 14 nm, with a standard deviation which is experimentally
not readily accessible but probably lower than that in diameter
26
.
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a nematic (N), and a columnar (C) phase upon increasing the
concentration of platelets
13,18–21
. The isotropic to nematic (I–N)
phase transition, which does not involve positional order (see
Fig. 1), has recently been observed in experiments on suspensions
of gibbsite platelets
15
. In the present study we explore the liquid
crystal phase behaviour of these fairly polydisperse suspensions at
higher densities, where simulations predict a N–C transition in the
monodisperse case. We consider two systems which differ by the
degree of polydispersity in diameter of the constituent platelets (17
and 25%, respectively, see Fig. 2). A dimension-based speculation
on the terminal polydispersity for columnar ordering of plates,
which can be considered as two-dimensional crystallization, would
yield a value in the range 5–20%, falling between the values for
three-dimensional crystallization of spheres and for one-dimen-
sional smectic ordering of rods.
This raises the following questions. Do suspensions of plates,
particularly in the studied range of polydispersity, show a nematic–
columnar phase transition as predicted for monodisperse plates? Or
do they, in order to circumvent the effect of polydispersity in
diameter, form a smectic phase instead? What is the role and
extent of particle size partitioning between coexisting phases?
The phase behaviour of the platelet suspensions as a function of
the plate volume fraction is presented in Fig. 3. Isotropic–nematic
phase coexistence is observed just below f ¼ 0:2, yielding an
isotropic upper phase and a birefringent nematic bottom phase.
Macroscopic phase separation is complete within 12 hours. The
width of the biphasic region ∆f
IN
depends strongly on the poly-
dispersity in diameter j
D
, ∆f
IN
being twice as broad for the
suspension with j
D
¼ 25% compared to that with j
D
¼ 17%
(Fig. 4). This observation is consistent with the relation ∆f
IN
~ j
2
as found in computer simulations for polydisperse disks
20
.
Upon increasing the plate volume fraction to roughly twice the I–
N coexistence density, f < 0:4, both the suspension of 17 and 25%
polydispersity enter a biphasic region where a nematic upper phase
coexists with a more concentrated birefringent bottom phase. A
columnar signature of the lower phase is suggested by unequivocal
Bragg reflections when illuminated by white light (Fig. 3). The fact
that these reflections appear for wavelengths of visible light demon-
strates that the crystalline order pertains to a periodicity on a length
scale of the plate diameter (characteristic of columnar ordering)
rather than the much smaller plate thickness (as in smectic
ordering). By applying Bragg’s law to the angle of reflection
measured for different wavelengths of light, we identify the char-
acteristic spacing (of the (100) reflection, see small-angle X-ray
scattering (SAXS) results) as 219 6 5 nm and 214 6 5 nm for the
parent and fractionated suspensions, respectively. This corresponds
to a typical distance between the centres of the columns of 253 6 6
and 247 6 6 nm. A comparison of the experimentally observed I–N
and N–C transition densities to computer simulations for mono-
disperse hard disks
13
, as depicted in Fig. 4, shows that the transitions
in the experiment are shifted to slightly lower densities. The
difference in shape of the platelets studied (hexagonal in the
experiment versus circular in the simulation) we expect to be a
major contribution to this shift, as shown recently for the I–N
transition
21
. Macroscopic phase separation in the N–C biphasic gap
is complete within 2 weeks.
Figure 3 Tubes containing suspensions at varying concentrations. They are
photographed between crossed polarizers to distinguish between isotropic (dark) and the
birefringent nematic and columnar liquid crystalline phases. The suspensions depicted
here comprise platelets with 17% polydispersity in diameter. From left to right, the
concentration ranges from f ¼ 0:19 (I þ N), 0.28 (N), 0.41 (N þ C), to 0.47 (C). The
tube to the far right depicts the monophasic columnar sample at f ¼ 0:45 as observed
without polarizers but illuminated by white light. The colours of its Bragg reflections
(visible as small bright spots) vary from yellow to green, as the angle between the incident
light and viewing direction is in the range 50–708. The particle volume fraction f (which
includes the solvent immobilized in the grafted polymer layer) is calculated as the mass
concentration (determined by drying a known amount of dispersion to constant weight at
75 8C) divided by the effective mass density of the grafted particles. Following ref. 27, the
latter is derived from the TEM particle dimensions, the estimated thickness of the
stabilizing layer
25
, the polymer mass fraction from elemental analysis and the mass
density of gibbsite
28
. This yields a mass density of about 1.3 g cm
−3
, with an estimated
error of 10% that is due to the uncertainty in polymer layer thickness.
0.1
10
0
q (nm
–1
)
0.05
0.5
0.02
d
c
b
a
10
6
10
2
10
4
(100)
(001)
(002)
(110)
(210)
Scattered intensity
Figure 5 Small angle X-ray scattering patterns for samples varying in polydispersity and
volume fraction. a, j
D
¼ 17% at f ¼ 0:45, b–d, j
D
¼ 25% at f ¼ 0:45, 0.5 and 0.6
respectively. Curves a and b correspond to a columnar phase, curve c to smectic-like
ordering at densities above columnar stability, and curve d depicts the glassy state
encountered upon increasing the density even further. Curves shown are radially
averaged scattered intensities, in arbitrary units, as a function of the scattering vector
q
¼ 4p=lsinðv=2Þ, with v the scattering angle. Experiments are performed at the
DUBBLE beam line at the European Synchroton Radiation Facility, France.
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
0
25
50
75
25
50
75
100
Simulation
I+N
C
I
N
N+C
100% N
(Volume % N) (Volume % C)
C
N+C
N
I+N
I
φ
plate
Figure 4 Phase diagram of the suspensions. The relative volume of the nematic and
columnar phase are depicted after phase separation as a function of the platelet volume
fraction f. Results apply to j
D
¼ 17% (triangles) and j
D
¼ 25% (circles), respectively.
The points marked by a cross and a star belong to the latter system but at densities
beyond columnar stability, corresponding to curves c and d, respectively, in Fig. 5. The
dotted lines indicate the boundaries of the coexistence regions of the suspension with
j
D
¼ 17%. Results from computer simulation
13
for monodisperse hard disks, extra-
polated to the current aspect ratio h
D
i/h
L
i of roughly 13, are included for comparison.
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SAXS experiments provide further proof for the columnar
signature of the dense liquid crystal phase. Curves a and b in
Fig. 5 correspond to the columnar phase in the case of 17 and 25%
polydispersity, respectively. These curves show almost identical
features. In the small q regime, where the spacing d ¼ 2p=q is of
the order of the diameter of the plates, we can distinguish one
major peak and two additional peaks. The q values of these three
peaks (Table 1), whose q ratio is like 1 :
3
p
:
7
p
, reveals that
ordering in the plane of the plate diameters is hexagonal, with the
peaks corresponding to the (100), (110) and (210) reflections. The
(200) reflection seems to be relatively weak compared to its
neighbouring (110) and (210) peaks, such that it is not (clearly)
resolved in the scattering profile. From the q values of the (100),
(110) and (210) peaks we obtain the typical distance between the
centres of the columns to be 251 6 4 and 258 6 1 nm in the
suspension with j
D
¼ 25% and 17%, respectively, in good agree-
ment with the values found with light scattering. This distance is
hence almost equal to ð1 þ j
D
Þ 3 D for both suspensions. The two
peaks at much larger q correspond to 1 and 0.5 times a spacing of
roughly the plate thickness, such that we identify them as (001) and
(002) reflections. These peaks may therefore relate to (liquid-like)
order between the plates along the z-axis of a column of plates.
Although the observed scattering patterns are consistent with a
columnar structure, they could also stem from a structure in which
particles are hexagonally ordered in layers without lateral correla-
tions between adjacent layers. These structures can be distinguished
by orienting a single crystal in the X-ray beam with the beam parallel
to the plate normals
17,22
. If the structure is columnar, tilting the
sample with respect to the beam will not result in a change in q
values of the scattering peaks, similar to the behaviour of a three-
dimensional crystal of spheres. Tilting a sample in a 0.2-mm flat
capillary (approximating a single crystal) indeed does not lead to a
significant (.1%) shift in the q positions of the scattering peaks.
Apart from the unlikely (though not excluded) possibility of a three-
dimensional crystal structure, the tilting experiment thus confirms
that the dense liquid crystal phase is a columnar phase.
A remaining question concerns the origin of the observed
columnar stability in these systems where polydispersity must be
an important factor. In analogy with computer simulations’ pre-
dictions for crystallization of polydisperse hard spheres, stabiliza-
tion of the ordered phase may emanate from fractionation, lowering
the polydispersity in the ordered phase at the expense of the
polydispersity in a coexisting disordered phase. In the present
study we can determine the extent of fractionation experimentally,
by examination (using transmission electron microscopy, TEM) of
small samples of a coexisting nematic and columnar phase. Frac-
tionation indeed gives rise to a reduction of the polydispersity in the
columnar phase, as we find j
D
¼ 18 and 14% in the columnar phase
of the two systems studied (Table 2). This indicates that at least the
pre-fractionated system (j
D
¼ 17%) should be able to enter a
monophasic columnar state beyond the N–C coexistence region.
In fact, the fully columnar state is observed for both suspensions.
This demonstrates that even in the case of j
D
¼ 25% fractionation
is not an absolute condition for columnar ordering, and that this
polydispersity in diameter is therefore still below the terminal value.
One may wonder, however, whether the columnar structure will
persist upon increasing the volume fraction of a fully columnar
sample, that is, if the spacing between the columns decreases such
that the disruptive effect of polydispersity in diameter becomes
more pronounced. For f ¼ 0:50 (curve c in Fig. 5), the columnar
(100), (110) and (210) peaks in a sample of j
D
¼ 25% become
markedly suppressed, whereas, at the same time, the (001) and (002)
peaks become more distinct. Bragg reflections for visible light
(which pertain to the (100) reflection) almost completely disappear.
We speculate that the suppression of the columnar peaks and the
simultaneous structuring with a periodicity of the order of the plate
thickness is indicative of a cross-over to smectic-like ordering.
Unlike the columnar phase, a smectic phase is not sensitive to
polydispersity in diameter, as ordering within the smectic layers is
liquid-like. Instead, the smectic requires a low polydispersity in
thickness. The relative stability of the nematic, columnar and
smectic phases is therefore determined not only by the volume
fraction of the suspension and the polydispersity in diameter, but
also by polydispersity in thickness. Our observation that a system
with j
D
as high as 25% forms a columnar phase, and its cross-over
to a smectic-like structure at higher f, are therefore probably
connected with the fact that the platelets have a polydispersity in
thickness too. Indeed, in the case of j
D
¼ 17%, the columnar phase
is still stable at f ¼ 0:53 as demonstrated by strong Bragg reflections
for visible light.
The stability of a columnar phase in these polydisperse suspen-
sions of plates seems remarkable in the light of predictions for the
terminal polydispersity for the crystal phase of hard spheres and
smectic phase for hard rods, although in the latter only polydis-
persity in length is taken into account. Further insight into the
stability of the nematic, columnar and smectic phase in systems of
polydisperse plates therefore requires study by computer simula-
tion, addressing the role of polydispersity in both diameter and
thickness.
M
Received 26 April; accepted 4 July 2000.
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dispersions. Langmuir 14, 3129–3132 (1998).
Table 1 q values of the scattering peaks in the SAXS pattern
q (10
−2
nm
−1
)
Sample j
D
(%) f (100) (110) (210) (001) (002)
.............................................................................................................................................................................
a 17 0.45 2.82 4.88 7.40 27.8 55.5
b 25 0.45 2.89 5.00 7.44 25.6
c 25 0.5 (2.88) 32.3 64.4
.............................................................................................................................................................................
Table 2 Fractionation in the N–C coexistence region
System Overall N-phase C-phase
hDi (nm) j
D
(%) hDi (nm) j
D
(%) hDi (nm) j
D
(%)
.............................................................................................................................................................................
I 198 25 196 26 200 18
II 212 17 209 18 220 14
.............................................................................................................................................................................
The diameter distributions of the parent (I) and the pre-fractionated system (II), before and after N–C
phase separation. Values are determined from TEM micrographs by measuring about 200 particles
in each case.
© 2000 Macmillan Magazines Ltd
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New York, 1994).
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Acknowledgements
We thank A. R. Rennie for discussions concerning the columnar phase, I. Dolbnya and
W. Bras for technical support at DUBBLE, and D. Frenkel for a critical reading of the
manuscript. This work was supported by the Foundation for Fundamental Research on
Matter (FOM) and the Netherlands Organization for the Advancement of Research
(NWO).
Correspondence should be addressed to H.L. (e-mail: H.N.W.Lekkerkerker@chem.uu.nl).
.................................................................
Multiscale modelling of plastic flow
localization in irradiated materials
Tomas Diaz de la Rubia*, Hussein M. Zbib
†
, Tariq A. Khraishi
†
,
Brian D. Wirth*, Max Victoria
‡
& Maria Jose Caturla*
* Lawrence Livermore National Laboratory, 7000 East Avenue, L-353, Livermore,
California 94550, USA
†
Washington State University, School of Mechanical & Materials Engineering,
Pullman, Washington 99164, USA
‡
EPFL-CRPP-Fusion Technology Materials, 5232 Villigen PSI, Switzerland
.................................. ......................... ......................... ......................... ......................... ........
The irradiation of metals by energetic particles causes signifi-
cant degradation of the mechanical properties
1,2
, most notably
an increased yield stress and decreased ductility, often accom-
panied by plastic flow localization. Such effects limit the life-
time of pressure vessels in nuclear power plants
3
, and constrain
the choice of materials for fusion-based alternative energy
sources
4
. Although these phenomena have been known for
many years
1
, the underlying fundamental mechanisms and
their relation to the irradiation field have not been clearly
demonstrated. Here we use three-dimensional multiscale simu-
lations of irradiated metals to reveal the mechanisms under-
lying plastic flow localization in defect-free channels. We
observe dislocation pinning by irradiation-induced clusters of
defects, subsequent unpinning as defects are absorbed by the
dislocations, and cross-slip of the latter as the stress is increased.
The width of the plastic flow channels is limited by the interaction
among opposing dislocation dipole segments and the remaining
defect clusters.
The microstructure of irradiated materials evolves over a
wide range of length and time scales, making radiation damage
an inherently multiscale phenomenon. At the shortest scales
(nanometres and picoseconds), recoil-induced cascades of energetic
atomic displacements give rise to a highly non-equilibrium con-
centration of point defects and point defect clusters
5
. Over macro-
scopic length and time scales these defects can migrate and alter the
chemistry and microstructure, often inducing significant degrada-
tion of mechanical and other properties
1,2
. In metals, the main
features of neutron or ion beam irradiation-induced mechanical
behaviour can be summarized as
2
: (1) a sharp increase in yield stress
with irradiation dose; (2) the appearance of a yield point followed
by a yield drop in f.c.c. metals; and (3) an instability that results in
plastic flow localization within ‘dislocation channels’ and leads to
loss of ductility and premature failure. An example of flow localiza-
tion is shown in Fig. 1 (ref. 6). The transmission electron micro-
scopy image shows a channel where all visible irradiation-induced
point defect clusters are absent and where uniform large shear
(about 100%) has taken place. The channels are 100 to 200 nm wide.
Plastic flow localization is responsible for the observed loss of
ductility.
Early theories of irradiation hardening focused on various
source and dispersed barrier mechanisms
7,8
. Recently
9,10
, the
analytical cascade-induced source hardening (CISH) model
was proposed
9,10
. This model uses insights from molecular
dynamics simulations
11–14
and accounts for some of the
recent experimental observations. In the model, it is postulated
that interstitial clusters produced in displacement cascades
form glissile dislocation loops that migrate in one dimension
by thermally activated glide, and decorate dislocations, thereby
pinning them. However, while this model provides a rational
explanation for the observed increase in yield stress, it does
not account for plastic flow localization and the development
of plastic instabilities.
Here we couple these experimental and atomistic simulation
results
11–18
to a three-dimensional dislocation dynamics (DD)
simulation to investigate the relation between the irradiation field
and mechanical behaviour. We consider two cases, Cu and Pd,
which exhibit different damage morphologies under irradiation.
Experiments have shown that vacancy stacking-fault tetrahedra
(SFT) are the predominant defect type in low stacking fault
energy Cu
6,19
, whereas in high stacking fault energy Pd, self-inter-
stitial atom Frank sessile loops constitute the majority of observed
defects
20,21
. Our DD simulation box is a cube 5 mm in size that
contains an initial density of Frank–Read dislocation sources
distributed at random on {111} planes. In our DD simulation
22–24
,
the plastic deformation of a single crystal is obtained by explicit
accounting of the dislocation evolution history, that is, their motion
and structure.
The motion and interaction of an ensemble of dislocations in a
three-dimensional crystal is marched in time. Dislocations are
discretized into straight-line segments of mixed character. The
Peach–Koehler force F acting on a dislocation segment inside the
computational cell is calculated from the stress fields that are caused
by immediate neighbouring segments, all other dislocations seg-
ments, all defect clusters and the applied stress. The result is used to
advance the dislocation segment based on a linear mobility model,
v
gi
¼ M
gi
F
gi
, where v
gi
is the glide velocity of the dislocation
segment, M
gi
is the dislocation mobility, and F
gi
is the glide
component of the Peach–Koehler force minus the Peierl’s friction.
On the basis of the history of dislocation motion, we obtain a
measure for the macroscopic plastic strain rate. To ensure con-
tinuity of dislocation lines across the boundaries, we apply reflec-
tion boundary conditions
22
. Due to the long-range character of the
dislocation stress field, long-range interactions are computed
explicitly
22
.
In the simulation, segments that are on the verge of experiencing
short-range interactions are identified. Based on a set of physical
rules, such reactions may result in the formation of junctions, jogs,
dipoles, and so on. The dislocations multiply by a variety of
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