Coarse-grained simulation studies of a liquid crystal dendrimer: towards computational predictions...

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


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.


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.


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.


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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29 V. Percec, A. E. Dulcey, V. S. K. Balagurusamy, Y. Miura,J. Smidrkal, M. Peterca, S. Nummelin, U. Edlund, S. D. Hudson,P. A. Heiney, D. A. Hu, S. N. Magonov and S. A. Vinogradov,Nature, 2004, 430, 764.


Coarse-grained simulation studies of a liquid crystal dendrimer: towards computational predictions...

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