Coarse-grained simulation studies of a liquid crystal dendrimer: towardscomputational predictions of nanoscale structure through microphaseseparation
Zak E. Hughes,a Mark R. Wilson*a and Lorna M. Stimsonb
Received 3rd August 2005, Accepted 23rd September 2005
First published as an Advance Article on the web 17th October 2005
DOI: 10.1039/b511082c
Coarse-grained simulations are described in which the behaviour of a system of model liquid
crystalline dendrimer molecules is studied in both liquid and smectic-A liquid crystalline phases.
The model system is based on a third generation carbosilane dendrimer, which is functionalised at
the surface by short polymeric chains terminated in mesogenic units. The design of the coarse-
grained model is based on initial Monte Carlo studies of a single carbosilane molecule at an
atomistic level, which yield structural data. The coarse-grained dendrimer is represented in terms
of a combination of spherical sites representing the dendrimer core and polymer chains, and
spherocylinders representing the mesogenic groups. A strong coupling is seen between internal
molecular structure and molecular environment, with individual dendrimer molecules undergoing
a remarkable transition from spherical to rod-shaped at the isotropic–smectic phase transition.
The driving force for mesophase formation is provided by nanoscale microphase separation of
mesogens and the dendrimer core.
1 Introduction
Liquid crystal dendrimers (LCDrs) are unusual materials. The
underlying branched structure of a dendrimer should lead to
molecular architectures with spherical symmetry. This would
appear to rule out the formation of liquid crystalline phases,
which normally require the presence of anisotropy in mole-
cular shape and in molecular interactions. However, synthetic
chemists have produced a number of interesting architectures,
which effectively combine a dendritic structure with liquid
crystallinity. For example, mesogenic groups may be incorpo-
rated into the dendrimer between branching points, leading to
the dendrimer molecules themselves becoming anisotropic and
able to form calamitic nematic and smectic thermotropic
phases.1 Alternatively, mesogenic groups can be bonded to
the ‘‘surface’’ of a dendrimer, either directly or by using
short ‘‘decoupling’’ chains as shown by Richardson and
co-workers.2–4 Here, conformational changes allow the
dendrimer to rearrange the distribution of its terminal
mesogenic groups providing a mechanism for formation of
liquid crystal phases.
Wilson and co-workers5 have recently simulated a model for
a third generation carbosilane liquid crystalline dendrimer in
solution in isotropic, nematic and smectic-A solvents. The
results show an interesting coupling between dendrimer
structure and molecular environment. In the isotropic solvent,
the dendrimer itself forms an isotropic sphere with the meso-
genic groups randomly arranged on the periphery. In the
nematic solvent, the dendrimer rearranges its molecular
structure, so that the mesogenic groups can lie, on average,
parallel to the nematic director with an average order para-
meter approaching that of the solvent. In the smectic solvent,
the dendrimer rearranges further, so that mesogens can lie
commensurate with the smectic layers.
These simulations were stimulated by recent synthetic and
X-ray diffraction studies of carbosilane LCDrs.2,3 Here it
seems that mesophase formation can be influenced by the
generation number. In the lower generations, the formation of
rods (as seen in the simulations discussed above) leads to
smectic behaviour characterised in the X-ray measurements by
a smectic layer spacing. In generation 5, the dendrimers can
also form a columnar phase. Here it is likely that conforma-
tional changes can lead to the dendrimer rearranging structure
in a second way to form discs, which can stack to form
columns, which in turn pack to form columnar phases.
The purpose of the current paper is to design a coarse-
grained (CG) model to study carbosilane LCDrs in the bulk
and to look at the structure of the phases formed. This is a
difficult task. Unlike simpler macromolecules, such as linear
homopolymers (which can be coarse-grained by designing a
suitable model for a single monomer unit) a CG model of the
dendrimer must account for a range of different interaction
sites. Moreover, it must be able to represent a complicated
molecular shape, which can change structure through con-
formational rearrangement, yet be sufficiently ‘‘cheap’’ to
simulate a relatively large number of molecules over long
simulation times. The design of the simulation model is
described in section 2. Simulation results from long simulation
runs are described in section 3, and we point to some con-
clusions in section 4 and mention the possible extension of this
type of model to other problems in soft matter chemistry
aDepartment of Chemistry, University of Durham, South Road, Durham,DH1 3LE, UK. E-mail: [email protected] and Statistical Mechanics Group, Laboratory ofComputational Engineering, Helsinki University of Technology,P.O. Box 9203, 02015-HUT, Finland. E-mail: [email protected]
PAPER www.rsc.org/softmatter | Soft Matter
436 | Soft Matter, 2005, 1, 436–443 This journal is � The Royal Society of Chemistry 2005
involving microphase separation and the formation of well-
ordered nanostructures.
2 Simulation model
The simulation model aims at providing a simplified descrip-
tion of the third generation carbosilane LCDr2,3 shown in
Fig. 1. In initial work, Monte Carlo (MC) simulations of a
single dendrimer molecule were carried out in the gas phase
using the internal coordinate Monte Carlo approach of
Wilson6 employing the the OPLS force field7,8 with additional
parameters for mesogenic groups taken from ref. 9. After
energy minimisation and equilibration, production run lengths
of approximately 20 6 106 MC steps were carried at 400 K to
obtain structural information for a dendrimer molecule in the
gas phase. Several key observations arise from the initial all-
atom gas phase simulations:
N the inner dendritic core (from the central Si atom out to the
third generation branching Si atom) forms a fairly rigid
scaffold, which is approximately spherical in shape and is
largely decoupled from the outer parts of the molecule. The
degree of penetration of the outer parts of the molecule into
the spaces within this inner core is small.
N the flexible chains (from the third generation branching Si
to the mesogenic units) exhibit considerable conformational
freedom and in the gas phase the dendrimer is able to undergo
major changes in structure by rearrangement of the chains.
N in the gas phase the dendrimer is, on average, spherical in
structure but conformational changes can lead to fluctuations
in which the overall dendrimer is instantaneously prolate or
oblate in shape.
N in the gas phase there is no ordering of the terminal meso-
genic units and the mesogens are able to sample all directions
in response to changes in conformation of the flexible chains.
However, application of a small ordering potential, Uext 5 2eext
P2 (cos h), of strength eext 5 10 kJ mol21 coupling the CMN
bond axes to a mean field director leads to an average
order parameter of ,S2. # 0.55 for the mesogenic groups
by preferentially promoting conformations in which the
mesogenic groups lie parallel/anti-parallel to the mean field
director. Application of the ordering potential also leads to the
dendrimer becoming more ‘‘rod-shaped’’ overall.
Based on these observations the coarse-grained (CG) model
shown in Fig. 2 was designed, using a central sphere to
represent the spherical inner core, coarse-graining each short
chain via four spheres and representing the mesogenic groups
by spherocylinders. In mapping real atoms to the coarse-
grained sites we initially maintained the correct excluded
volume for the individual parts of the dendrimer, as measured
from the atomistic simulations. Here, we make use of the
Monte Carlo calculations to give an ensemble averaged
volume per CG-site, as determined from a Connelly surface
with probe radius 0.1 A, to obtain the radii, R1, R2, R3
(Table 1). The equilibrium bond distances, a–d, (Table 1)
for the CG-sites were obtained also as ensemble average
values from the Monte Carlo calculations. Determination
of appropriate L and D parameters for the spherocylinders
was more problematic. The initial Monte Carlo simula-
tions yielded a mean mesogen length of 16.14 A (L + D)
including van der Waals radii. The volume of the mesogenic
group,
V~4
3p
D
2
� �3
zLpD
2
� �2
,
therefore yields a value of L/D 5 2.84, with ssc 5 3.74 A.
However, we know already from earlier simulations of spher-
ocylinders that we do not expect a liquid crystal phase for such a
small aspect ratio. Therefore we used an alternative approach
fixing the value of D 5 ssc 5 3.74 A, which is appropriate for the
chemical structure but extending L to a value of 6, which we know
from earlier simulations yields a large nematic region in the
spherocylinder phase diagram. (For a soft particle model, it would
be possible to use a smaller aspect ratio relying on anisotropic
attractive interactions to reduce the length/breadth ratio required
for mesophase formation.10) We also carried out simulations with
a longer L/D 5 8 spherocylinder mesogen.
For simplicity and in common with many CG-polymer
models11,12 purely repulsive potentials were used. For spherical
sites the cut and shifted form of the Lennard-Jones potential
was employed
ULJ~ 4es
rij
� �12
{s
rij
� �6
z1
4
" #, rijvrcut, rij§rcut, 0,
8><>: (1)
where ULJ is the Lennard-Jones potential at a position rij and e
and s are respectively the well depth and distance of closest
approach, which characterize the potential. The potential goes
smoothly to zero at rcut 5 21/6s. For the mesogenic units the
cut and shifted form of the spherocylindrical potential, i.e., theFig. 1 Chemical structure of a third generation carbosilane
dendrimer.
This journal is � The Royal Society of Chemistry 2005 Soft Matter, 2005, 1, 436–443 | 437
soft repulsive spherocylinder (SRS) potential,13,14 was used.
Here, the values of rij and rcut in eqn. (1) were respectively
replaced by the minimum distance of separation of the two
spherocylinders, dij13 and the appropriate cutoff, dcut. This
potential can also be used to describe the interaction of a
spherocylinder with a Lennard-Jones site, by considering dij as
the minimum separation between the centre point of the sphere
and the line that runs along the centre of the spherocylinder.
(This is equivalent to considering a sphere as a spherocylinder
where the length of the spherocylinder, L, is equal to zero.) For
this study e for each CG-site was taken as 0.247 6 10220 J
based on experience with earlier work on hybrid Lennard-
Jones/Gay–Berne models.10,15
In CG-polymer models it is common to use FENE16
potentials. Here we use a similar, but slightly simpler, type
of bonding potential
Ubond rij
� �~
1
2kbond rij{l0
� �12, (2)
where the bond length is given by rij, the force constant is
given by kbond, and l0 values correspond to the equilibrium
bond lengths a, b, c or d of Table 1. Eqn. (2), has a shallow
bottom and strongly repulsive walls like the FENE
potential and is therefore restrictive in terms of bond distances
allowed. However, it avoids the discontinuity of the FENE,
making it easier to use in CG-molecular dynamics. Using
eqn. (2) means excluding all bonded pairs of CG-sites from
the sum of nonbonded terms. Finally, appropriate masses for
each CG-site were taken, along with a moment of inertia of
5.9930 6 10224 kg m2 for the mesogenic particle.
All simulations used molecular dynamics, employing cubic
periodic boundary conditions in the constant-NpT and
constant-NVT ensembles with 100 dendrimer molecules
(12 900 spherical sites and 3200 spherocylinders). We used
the parallel domain decomposition program, GBMOL_DD, of
Ilnytskyi and Wilson,17–19 using a Verlet algorithm suitable for
anisotropic particles,17–19 and employed time-steps of between
5.0–7.5 6 10215 s. The latter was checked as the system
changed density to ensure excellent conservation of the
extended Hamiltonian in the appropriate ensemble. Phase
changes with density are classically considered to be repre-
sentative of lyotropic liquid crystals. However, as the
potentials described in this study are purely repulsive,
temperature is less important than density and the phase
behaviour of the system is considered in terms of changes in
the latter. Initially, the system was compressed from a gas
Fig. 2 Schematic diagram showing the dimensions and coarse-
grained mapping for the model dendrimer. In the top part of the
diagram bold lines around the spheres are drawn to scale for
appropriate s values and dotted lines indicate the potential cutoff
(21/6s). Only 8 of the 32 chains are shown for clarity.
Table 1 Parameters used in the definition of the CG-simulationmodel
Parameter Measurement Value
L Length of spherocylinder 6D, 8DD 5 ssc Breadth of spherocylinder 3.74 A
R1~2
16 ssmall
2
Radius of large sphere 10.6853 A
R2~2
16 smedium
2
Radius of medium sphere 3.1167 A
R3~2
16 slarge
2
Radius of small sphere 2.2955 A
a Large sphere–medium spherebond length
14.901 A
b Medium sphere–small spherebond length
3.5973 A
c Small sphere–small spherebond length
3.617 A
d Small sphere–spherocylinderbond length
2.979 A
kbond Bond force constant 60 6 10220 J A212
438 | Soft Matter, 2005, 1, 436–443 This journal is � The Royal Society of Chemistry 2005
phase low density lattice with no particle overlaps using NpT
simulations. After the required densities were obtained via
NpT, each state point was equilibrated using NVT runs (NpT
runs were used exclusively at high density), while monitoring
changes in energy, structure and order. Typical equilibration
times in the liquid phase were 750 ps, with runs lengths
extending 35.75 ns at high density in ordered phases.
3 Results
The pressure–density phase diagram for the two model
dendrimers is shown in Fig. 3. For each state point in Fig. 3
we checked the evolution of nematic and smectic ordering,
following the general procedures of references.10,20,15 Firstly,
the average orientational order was obtained by the order
parameter, ,S2. 5 ,P2 (cos h)., where h is the angle
between the long axis of a spherocylinder and the director, n,
of the system; and the transition to smectic ordering was
monitored by the behaviour of the site–site radial distribution
function, g(r); and its components parallel, g||(r), and
perpendicular, gH(r), to the director. For the L/D 5 6 system
the three highest density state points in Fig. 3 show a slight
increase in ,S2.. For these three densities, observation of
snapshots from the system indicate spontaneous formation of
microphase segregated domains, accompanied by a rise in the
first peak of the radial distribution function for the spher-
ocylinders. A further increase in the density leads to the
formation of liquid crystal ordering within these domains as
illustrated by the snapshot in Fig. 4b. Despite run lengths of up
to 7.5 ns, these domains did not merge to form an overall
smectic phase. To speed up possible slow annealing of this
phase, we applied an external potential, Uext 5 2eextP2(cos h),
to the spherocylinders. This led to uniform alignment of the
local directors within each domain within 1.0 ns. However, the
high viscosity of the phase allowed only slow movement of
molecules within the phase, and domains were not able to
merge during the time scale of the simulations. Removal of the
field led to a slow relaxation of the domain directors over a
period of 1–2 ns.
For the L/D 5 8 system the four highest density state points
also showed spontaneous microphase separation. Here, the use
of longer spherocylinders led to microphase separation at a
lower density (when the fluid has a lower viscosity) and the
Fig. 3 Pressure–density phase diagram for the model dendrimer. The
bold line is for the system with L/D 5 6 mesogens and the dashed line
for the system with L/D 5 8 mesogens.
Fig. 4 Snapshots from simulations showing the changing bulk
structure of the model dendrimer fluid. Top: (a) Isotropic phase for
the L/D 5 6 system at a density of 174.5 kg m23. Middle: (b)
Microphase segregated structure of for the L/D 5 6 system at a density
of 1440.6 kg m23. Bottom: (c) Smectic-A structure for the L/D 5 8
system at a density of 1022.23 kg m23. Spherical sites are shown in
white. Spherocylinders are colour coded with green along the director
and red/green/blue colours corresponding to mutually perpendicular
directions.
This journal is � The Royal Society of Chemistry 2005 Soft Matter, 2005, 1, 436–443 | 439
spontaneous formation of larger liquid crystalline domains
over runs of up to 35.75 ns. Application of a small aligning
field of strength eext 5 2 kJ mol21, leads to the domains
merging and the formation of a smectic liquid crystal phase, as
shown in Fig. 4c. This phase is completely stable upon removal
of the applied field. Simulations of .5 ns were carried out to
check that there was no decay of orientational or positional
order over this period on removal of the field. Application of a
field in this way does not guarentee the thermodynamic
stability of the phase. However, given that (a) we see no decay
of orientational order on removal of the field, (b) microphase
separation leads spontaneously to the formation of smectic
domains, (c) the system is less viscous than the L/D 5 6 and
the decay of the pair-wise orientational correlation for
function for spherocylinders is more rapid in the L/D 5 8
system; it seems likely that the smectic phase is the equilibrium
structure.
The transition to a smectic phase can be monitored by the
strong growth in the first peak of g(r) for the mesogenic
groups, accompanied by an abrupt change from no structure
in g||(r) to clearly defined peaks associated with the mesogenic
layers, as shown in the radial distribution functions plotted in
Fig. 5. Liquid-like structure is seen in gH(r), with only one
sharp peak in the smectic phase and a secondary smaller peak
at approximately twice the distance of the first peak, indicating
that this phase is a fluid smectic, i.e., smectic-A or smectic-C.
In Fig. 5 the distance between the peaks in g||(r) corresponds to
the layer spacing of the smectic phase. This is approximately
d 5 53.5 A, a value which is obtained also from an
independent calculation of the structure factor, S(k), which
has a maximum value for the wavevector kmax 5 2p (0/L, 0/L,
3/L) with kmax 5 |kmax| 5 0.1175 for our cubic box of length
L 5 160.4 A, at a density of 1022.23 kg m23. The inter-layer
distance corresponds approximately to the distance across a
dendrimer core plus a single spherocylinder. The mesogen
layers are therefore fully interdigitated with mesogens from the
layer above mixing completely with those from the layer
below. From the direction of the wavevector, kmax, it is
possible to define a layer normal vector, p, and to calculate the
mean tilt angle, ,h. 5 ,p?n., for the system. We have also
calculated the tilt angle for each layer individually, replacing n
by a separate director for each layer, nlayer. ,h. for the system
is equal to 0.4u ¡ 0.2 in the smectic phase and the maximum
tilt angle detected for an individual layer during the course of
the simulation was 2.7u. These data indicate that the
mesophase is smectic-A rather than smectic-C. However, we
note also from the snapshot in Fig. 4c the presence of local
domains where layers are not uniformly parallel to kmax and
there is local tilting of mesogenic groups relative to n. For
flexible molecules, we expect such fluctuations in both the
position of layers and the local tilt. In real systems this leads to
a smearing out of Bragg peaks corresponding to the layer
spacing as measured by X-ray diffraction.
The results tie in well with the interpretation of X-ray
diffraction measurements from third generation dendrimers
by Ponomarenko and co-workers.2 The latter are consistent
with a morphology corresponding to alternating mesogen/
dendrimer core layers as shown in Fig. 4c. Moreover, from
the X-ray layer spacing in the Ponomarenko et al. work it is
evident that each mesogenic layer contains spherocylinders
from the dendrimer above and below the layer with consider-
able interdigitation of mesogens in the generation 3 dendrimer.
In order for the interdigitated layer structure to form, the
dendrimers must undergo a major change in shape from the
spherical structure seen both in the gas and liquid phases
(Fig. 6). To illustrate this we plot the site distribution function,
r(r), in Fig. 7 for the liquid phase, microphase separated
region and the smectic mesophase. Here the distances, r, for
sites within the dendrimer are measured relative to the position
Fig. 5 Radial distribution functions calculated for the isotropic
(density 132.6 kg m23), phase separated (density 891.3 kg m23) and
smectic phase (density 1022.23 kg m23) of the L/D 5 8 dendrimer.
440 | Soft Matter, 2005, 1, 436–443 This journal is � The Royal Society of Chemistry 2005
of the central sphere in each molecule. In addition in Fig. 8, we
show the partial distribution functions r||(r) and rH(r) in the
smectic mesophase, calculated for cone-like sectors limited by
the angles h 5 p/6 around the head and tail directions of the
director for r||(r); and within the sector given by h s [p/3, 2p/3]
for rH(r). The r(r) curves demonstrate that the radially
averaged structure of the dendrimer does not change much
across the phases. There is a small increase in the density of
chains and spherocylinders close to the core in the microphase
separated and smectic regimes arising from the increase in the
bulk density of the system. However, there are major
differences in the behaviour of partial distribution functions
between phases. While r||(r) and rH(r) in the isotropic phase
take the same form as r(r) in Fig. 7a (and are therefore not
plotted), the form of r||(r) and rH(r) in the smectic phase
(Fig. 8) are radically different. There is a dramatic increase in
the spherocylinder density in r||(r) relative to rH(r) and an
increase in the peaks corresponding to the chain atoms in
rH(r) relative to r||(r). These changes indicate that the
mesogenic groups become aggregated at the two ends of the
dendrimer. This shape change is mediated by conformational
changes in the chain, which lead to it being confined around
the equatorial region of the core. This change arises from a
strong coupling of internal molecular structure to the bulk
structure of the phase itself. To illustrate this shape change,
Fig. 9 shows a snapshot representation of a typical molecular
conformation in the smectic phase taken from the configura-
tion shown in Fig. 4c. We note that shape changes of the type
seen in Fig. 4c have been predicted by Vanakaras and Photinos
using molecular theory.21
While the conformational changes seen in this system are
extreme, a similar coupling of internal structure and molecular
environment has been noted in other systems. In particular, in
side chain liquid crystal polymers the formation of well-
ordered smectic phases are accompanied by an elongation of
the polymer backbone, as evidenced by changes in the radius
of gyration of the polymer backbone, parallel and perpendi-
cular to the director, measured by small angle neutron
scattering (SANS).22 Moreover, recent simulations have shown
explicitly that a flexible polymer backbone is excluded from
smectic layers and becomes stretched as it is confined between
layers composed of mesogenic groups.20
4 Conclusions
A coarse-grained simulation study of a model carbosilane
liquid crystal dendrimer has been carried out. The simulations
show the formation of a smectic liquid crystal phase in
which individual layers of mesogens are separated by layers
Fig. 6 Snapshot showing the three-dimensional structure of a single
coarse-grained dendrimer molecule using spherocylinders with an
aspect ratio of L/D 5 8. Each molecule contains 32 spherocylinders,
1 large sphere, 32 medium sized spheres and 96 small spheres.
Fig. 7 The distribution function, r(r), in arbitrary units, for separate
parts of the L/D 5 8 dendrimer relative to the central core site. Bold
line: medium sized sphere, dotted line: small spheres, dashed line:
spherocylinders. The spherocylinder has been treated as 5 sites to get a
better picture of the spatial arrangement distribution for this site.
Top: (a) Isotropic phase at a density of 132.6 kg m23. Middle: (b)
Microphase separated phase at a density of 891.3 kg m23. Bottom: (c)
Smectic-A phase at a density of 1022.23 kg m23.
This journal is � The Royal Society of Chemistry 2005 Soft Matter, 2005, 1, 436–443 | 441
containing the dendrimer cores. Mesogens from a single
dendrimer contribute to smectic layers above and below the
dendrimer core. The simulations demonstrate a strong
coupling between internal molecular structure and molecular
environment. This leads to the mesogenic groups becoming
aggregated at the two ends of the dendrimer. Consequently,
the simulations show that individual dendrimer molecules
change from being ‘‘spherical in the liquid phase’’ to ‘‘rod-
shaped’’ in the liquid crystal phase: a remarkable result.
The dendrimer studied here is part of a broader class of
polyphilic mesomorphic materials, in which separate building
blocks with different types of interaction site are combined
(for example aliphatic, aromatic, fluoro, and siloxane-based
segments).23 In addition to dendritic molecules,24 such systems
include rod–coil molecules,25 low molecular weight polyphilic
molecules23,26 and ternary block-copolymers.27 Microphase
separation in these materials opens up the opportunity to self-
assemble well-defined nanostructures for potential molecular
electronics, photonics or biomimetic applications.28,29 The
CG-simulation methodology presented here (possibly modified
to include anisotropic attractive interactions) may provide a
route towards the simulation of these systems. As such it could
be a useful tool in helping design novel nanoscale structures
for future applications.
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Fig. 8 The distribution functions, r||(r) and rH(r), in arbitrary units,
for separate parts of the L/D 5 8 dendrimer relative to the central core,
plotted for the smectic-A phase at a density of 1022.23 kg m23. Bold
line: medium sized sphere, dotted line: small spheres, dashed line:
spherocylinders.
Fig. 9 Snapshot showing the structure of a representative molecule
from the L/D 5 8 system in the smectic-A phase at a density of
1022.23 kg m23.
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