Molecular Dynamics Simulation of the Mesophase
Behaviour of a Model Bolaamphiphilic Liquid Crystal
with a Lateral Flexible Chain
Andrew J. Crane1, Francisco J. Martínez-Veracoechea1,2, Fernando A. Escobedo2, Erich
A. Müller1,*
1) Department of Chemical Engineering, Imperial College London, UK
2) Department of Chemical and Biomolecular Engineering, Cornell University, Ithaca,
NY
* Author to whom correspondence should be addressed, e.muller@imperial.ac.uk
1
Table of Contents entry
Simulations detail the dynamical and structural aspects of the self-assembly of Tshaped polyphilic liquid crystal molecules. Smectic, columnar and lamellar mesophases
are formed by quenching from an isotropic phase.
2
Abstract
We present coarse-grained simulations of a model of a bolaamphiphile liquid crystal
molecule with a grafted flexible side chain. The coarse-graining approach employed is
based on minimising the attractions present in the system, on the premise that the most
important features of the liquid structure stem from the balance between the close range
repulsions and the strong directional forces typical of hydrogen bonding and
association. The model consists of six fused rigid spheres, where the two end spheres
have a significant attraction amongst themselves while the rest are repulsive in nature.
A weakly self-attracting lateral chain consisting of fully flexible tangently bonded
spheres is attached to one of the central spheres. A parametric study is made of the
configurations of collectives of these molecules at temperatures that span from the
isotropic fluid range down to the onset of crystallisation. The underlying rigid core
molecules (with no side chain) are set up to exhibit a smectic liquid crystal behaviour.
Upon increasing the number of spheres in the lateral chains from 1 to 12, the liquid
regions exhibit a rich variety of self-assembled structures; for small number of lateral
spheres columnar arrays of different cross sections (triangular, square, rectangular,
hexagonal) are obtained and for the longer chains lamellar structures of different
interlayer spacing are observed. We showcase and give a rational physical explanation
for the global phase behaviour of the model, based on pertinent order parameters and
apparent diffusivities of the several regimes encountered. Although no attempt has
been made to fit the parameters of the model to real molecules, the model is inspired in
the reported synthesis of a family of T-shaped polyphilic molecules (C. Tschierske,
Chem. Soc. Rev., 2007, 36, 1930-1970) where some of the above mentioned phases
have been inferred from experimental measurements.
3
1
Introduction
Recent literature is scattered with examples of “designer assemblies” i,ii where molecules
are specifically synthesized with the appropriate building blocks so they can selfassemble in pre-determined ways. The range of uses of these self-assembling materials
is enormous, from optical and electronic devices to more unconventional biomimetic
applications and smart fluids applications in industry, to name just a few. The synthesis
of these new soft materials is a chemical tour de force, and ways to understand and
predict the phase behaviour and stability from the chemicals constituents are much
welcome. Computer simulation can play an important role in this aspect, as most of the
larger and complex mesostructured phases found in biological and soft matter have
roots in the intricate interplay of relatively simple interactions between like and unlike
molecular segments. Unfortunately, the capabilities of present day computers do not
allow the study of macromolecules on a level in which atomic detail is present. The
timescales and system sizes required for self-assembly, make studies prohibitively large
for currently available and foreseeable computers (typically microsecond simulations
on the order of 105 atoms). Detailed studies are thus rare and have only been performed
for some “smaller” liquid crystals (LC), oligomers or biomolecules. A recent review by
Wilsoniii highlights the difficulties involved in the detailed atomistic approaches to
simulate these highly structured liquids.
In spite of the above, most of the mesoscopic behaviour of soft matter can be captured
in an accurate way by appropriately coarse-graining the atomistic level detail, whilst
retaining overall size, energy and connectivity details that are relevantiv,v. Reaching the
4
appropriate level of abstraction required is a subtle matter, as an undue simplification
could mask the underlying physical chemistry. We use here an approach based on
minimising the attractions in the systems, that is, basing our models on a soft repulsive
interaction, on the premise that the more important details of the fluid structure stem
from close range repulsions. In this context, the strong directional interactions, arising
from hydrogen bonding and electrostatics are treated as simple attractions. Further
refinement can be obtained by including the dispersion interactions as a perturbation.
This broad approach, as applied to simple fluids, has lead to successful treatments of
small organic mixtures, typical of the chemical and petrochemical industry and is
exemplified in the historical development of the SAFT equation of state. vi For larger
and more complex macromolecules, an analogous detailed theoretical framework is not
yet available, and one must rely on a combination of experimental observations
combined with suitable molecular simulations to elucidate and interpret the
experimental results.
This type of approach has been successfully used in
understanding phase segregation in related systems, such as polymer diblock
moleculesvii,viii, liquid crystalsiii, asphaltenesix, and biological systems.x
A recent reviewxi of advances in the experimental synthesis and characterisation of
functional liquid crystal assemblies highlighted the importance of a novel family of
polyphilic molecules that consist of a rigid polyphenyl core with incompatible end and
side groups. Typically the end groups are polar, with the ability to form multiple
intermolecular hydrogen bonds. Hence the combined core and end groups are in
essence a bolaamphiphilic unit.xii
A recent series of papers by Tschierske et al.
showcase the synthesis and characterisation of these bolaamphiphiles, with grafted side
chains composed of alkanes,xiii partially and totally perfluorinated chainsxiv,xv or
5
complex poly ethoxy chainsxvi,xvii (for a review see ref. xviii). An example of such a
molecule is the grafted 4,4’-bis (2,3-dihydroxypropoxy-biphenyl) depicted in Figure 1.
These unique T-shaped molecules all share the same broad aspects: a liquid crystal core
where the end groups have been substituted, and a side chain of variable length grafted
to it. The relative incompatibility between each molecular component cause these
compounds to have rich phase behaviour, even as pure substances.
Experimental
analysis, through polarized light optical microscopy, differential scanning calorimetry
and X-ray scattering, has indicated at least 11 unique self-assembled liquid structures
that form depending on the length and chemical properties of the grafted chains. The
phases observed are typically columnar phases where the lateral chains segregate into
honey-combed cylinders of bolaamphiphilic rigid units networks, or lamellar phases
composed of lateral chain rich and bolaamphiphilic unit rich layers.
It has been
suggested that the geometry of these phases is driven by both micro-segregation of
incompatible units and minimisation of free volume. Current descriptions are however
inferred from indirect experimental observations, and as such would benefit from
confirmation and further characterisation within a physically robust framework. In this
sense, this work presents a molecular dynamics study of coarse-grained grafted liquid
crystal bolaamphiphile analogues, in an attempt to elucidate the principles that originate
complex mesophase behaviour, as well as gain an improved picture of their global
phase diagrams.
2
2.1
Methodology
Molecular model
A simple coarse-grained model for a bolaamphiphilic rod with flexible lateral chain is
hereby implemented. In this model, six beads are kept in a rigid linear configuration
6
with an inter-bead distance of 0.7, where  is a characteristic length of the model,
defined to be roughly the diameter of a bead. This particular configuration resembles
that used to model mesogenic molecules, where the aspect ratio of 4.5 is elongated
enough to guarantee LC behaviour by itself.xix,xx,xxi,xxii,xxiii A flexible chain of length Nflx
beads is attached to the third rigid bead of the rigid unit. In this work Nflx is varied
from zero to twelve. The basic topology of this model molecule is depicted in Figure 2.
Bonded interactions between beads in the flexible chain and in the link between the
flexible chain and the rigid unit are represented using a harmonic potential of the form
1
2
U har  k sp r  ro 
2
(1)
where r is the distance separating the beads, ro is the equilibrium separation and ksp is

the spring constant. In the model an equilibrium distance, ro= and spring constant,
ksp=50(2), were used. Here  is a characteristic energy of the model, defined in
terms of the pair potentials between beads, described latter. While this represents a
completely flexible configuration, the reader is reminded that this is a coarse-grained
representation, where each bead corresponds to a group of atoms. From this point of
view, the flexibility shown by this model is analogous to the coarse-grained model
typically used for polymer systems.xxiv
In order to mimic bolaamphiphilic behaviour, the two end spheres of the rigid unit are
defined to attract each other in a preferential way. The lateral chain beads are also
defined to have self-attraction, although the strength of this is less than the end bead
counterparts. Hence the beads in the model may be categorised into three distinct types.
Type 1 beads, located at each extreme of the rigid unit, seek to represent strongly
7
interacting hydrogen-bond-like site. Type 2 beads, constituting the remainder of the
rigid unit, represent weakly interacting sites typical of the polyphenyl core. Finally,
type 3 beads, located in the lateral chain, represent medium-strength interaction site,
typical of alkane or perfluoroalkane chains. In line with the coarse-grained nature of
our model, the non-bonded inter bead interactions, are either soft short-range attractions
or soft repulsions. To this end, all six bead pair potential energies were defined through
the Lennard Jones cut and shifted potential (LJCS)
U
LJCS
AB
LJ
LJ
U AB
(r)  U AB
(CAB )
(r;Cij )  
0
for
for
r  CAB
r  CAB
(2)
with

 12  6 
LJ
UAB
(r)  4AB AB    AB  
 r   r  
(3)
where, A and B define the bead type, r is the bead separation, AB is an energy

parameter defining the potential well depth, AB is the length parameter defining the
range of the potential, and CAB is the cut and shifted distance.
Type 1 beads,
representing the hydrogen bonding groups, are assigned an LJCS potential self
interaction with 11=11= and C11=2. No intrinsic saturation associated with these
“hydrogen bonding” beads has been built in here. A molecule like that depicted in
Figure 1 shows a large hydrogen bonding capability, as each end group compromises
two O–H bonds and one –O– group, allowing in principle more than one hydrogen
bond per end group. Steric hindrance will most likely limit the maximum number and
type of these bonds in both the real molecule and the model. Type 3 beads have a selfinteraction with 33=33= and C33=2, where 33 was chosen to reflect the
8
weaker nature of this attraction. Finally, the type 2 bead self-interaction and all cross
interactions are modelled with an LJCS potential with AB=  
AB= and
1
CAB  2 6  This latter cut-off corresponds to the length at which the standard
Lennard-Jones potential is a minimum, consequently this potential represents a purely

repulsive interaction (typically known as the Weeks-Chandler-Anderson potential).
While the rather small value of C11 is an attempt to capture with a very simple model
the behaviour of strongly interacting hydrogen-bond forming site, it has also been used
for computer efficiency and to avoid the introduction of an additional parameter. All
beads in the system are chosen to have the same mass.
In principle, each bead accounts for a group of atoms. Consequently, if an accurate
description of a particular model were desired, the properties of these beads would have
to be mapped to experimental physical properties (e.g. radial distribution functions,
densities, etc). This has not been attempted in this work, as we are interested in
presenting a proof-of-concept rather than a particular application. We wish to make the
statement that the use of a simplistic, coarse-grained model helps to highlight the
“crucial” characteristics that allow for the experimentally observed complex mesophase
behaviour. By decreasing proportionally the strength of the interactions of all segments
of the molecule, we can increase substantially the efficiency of the calculation, without
changing the basic structure of the fluid phase, which will depend more on the interplay
between the repulsive cores of the molecules and the directional and specific nature of
the hydrogen bonds.
The temperature, T*=Tk/, where k is Boltzmann’s constant, and pressure, P*=P3/,
9
are expressed throughout this work in the standard reduced form that reflect the energy
and length scales of the model.
Simulations were performed in continuum space via Molecular Dynamics using the
simulation suite DL_POLYxxv. The following ensembles were used: the MicroCanonical (NVE) ensemble where the number of particles (N), the total volume (V),
and the total energy (E) are fixed; the Canonical (NVT) ensemble where N, V, and, the
temperature T are fixed; the Isobaric-Isothermal ensemble (NPT) where N, T, and the
pressure (P) are fixed; and the Isotension-Isothermal ensemble (NT) where N,T and
the components of the stress tensor () are fixed. In the NT ensemble both the size
and shape of the simulation box are allowed to change in order to satisfy the constraints
imposed. Thus, the NT ensemble has the advantage that it can ensure the equality of
pressure in the three directions. Simulation details are provided in the ESI, section S3.
Phase structures and fluidities were studied through the computation of order
parameters and mobility coefficients from simulation configuration data. This allowed
a quantitative characterisation of the different phases and improved location of phase
transitions temperatures. A description of these quantities and the methods used to
calculate them are outlined in the following two sections. Hysteresis in the systems was
investigated through running an additional series of NσT simulations over phase
transition temperature ranges. Whilst there was noticeable hysteresis, it is likely that
this is due to the system sizes and simulation times. Phase transitions reported in this
work should be taken as a rough guide to the actual transition temperatures expected.
2.2
Metrics
10
2.2.1 Order Parameter
To calculate order parameters, that provide a measure of structural order, an axis within
the system that characterises the direction of order is required. For columnar/lamellar
phases, this director may be viewed as the vector orthogonal to their layers. For our
model, the set of unit vector, {u1, u2,..., uN}, associated with the bolaamphiphilic rigid
unit axes were used to define the orientation of each molecule in the system. With this
arrangement, the problem of determining the director was equivalent to finding a unit
vector, n, that is maximally orthogonal to this set of molecular orientation vectors.
Concisely this may be written as
min  Un  subject to n  1
where U represents the N by 3 matrix containing the molecule orientation unit vectors,
ui. The solution to this total least square problem is obtained through determining the
Singular Value Decomposition (SVD) factorisation of matrix U. An outline of SVD
terminology and methodology may be found in standard linear algebra textbooks; it is
however sufficient to state here that the director, n, is found to be the right singular
vector of U corresponding to the smallest singular value. We note that n does not
correspond to the nematic director, but rather to a vector perpendicular to the planes
formed by the unit vectors. With this director determined a range of order parameters
become available.
The planar order parameter and the planar orientational order
parameters were used in this study and are defined below.
2.2.2 The Planar Order Parameter
The planar order parameter, S2, measures how orthogonal the molecular orientation
11
vectors are to the director vector, and is defined by the equation
N
 3sin 
2
S2 
j
2
j1
N
(4)
where j is the angle between the director, n, and the orientation vector, uj, of the jth
 molecular rigid unit. This order parameter is analogous to the P2 order parameter that is
frequently used to determine the orientation order of rod-like liquid crystalline
systems.xxvi For an isotropic system where there is no correlation between the director
and orientation vectors, S2 is defined such that it takes a value of 0. Conversely for an
ordered system where all the molecular orientation vectors lie in the plane orthogonal to
the director, S2 takes a value of 1.
2.2.3 The Planar Orientational Order Parameter
To monitor the geometry of the columnar phases as seen from an observer looking into
the director vector (i.e. triangular/square/hexagonal columns), the planar orientational
order parameter, k, is defined through the equation
1 N
 k   exp ik j 
N j1
(5)
where j is the angle between the vector given by the projection of the orientation

vector, uj, onto the director orthogonal plane, and a fixed arbitrary axis orthogonal to
the director. The coefficient k takes the value 2, 4 and 6 for the linear, square and
hexagonal planar orientational order parameters, respectively. As with the planar order
12
parameter, it is designed so that values of the planar orientation order parameter close to
0 indicate low order, whilst values close to 1 indicate a high level of order.
2.2.4 The mobility coefficient
To quantify the mobility of the bolaamphiphile molecules in the various phases, mean
squared bead displacements were calculated from the NVE simulations previously
described. For a given number of time steps, s, each of time the mean squared bead
displacement over time s, Rs, was calculated through the equation
N 6+N flx ts
   r
is ,i
R
s

j1
k1 i 0
N6 + N flx t  s
j,k 2
(6)
where r(i+s),ij,k is the displacement of the kth bead in the jth molecule, between time

i and (i+s), and t is the total number of time steps in the simulation.
By
determining the rate of change of this displacement with time, a quantity defined as the
mobility coefficient was obtained. As the simulations were performed on coarsegrained models the reduced time does not have a direct link with real time, therefore
this quantity is related in a non-trivial way to the diffusion coefficient of the system.
Nevertheless it remains useful as a measure of the relative molecular mobility in the
different phases. In determining the mobility coefficient it is important to realise that
the mean squared displacement against time chart is composed of two distinct regions.
At times less than the period of vibration of the beads, there is a steep increase in the
mean squared bead displacement, due to localised vibrations of the beads. At longer
times, the change of mean squared bead displacement with time is due to non-local
movement.
At high temperatures non-local bead movement dominates over any
13
localised vibrations, consequently it is not possible to delineate these regions. At lower
temperatures, however, where the mobility of beads is less, these two regions become
clear. The gradient of the mean bead displacement in the longer time region is the
appropriate one to use for the mobility coefficient, since non-local movement was of
interest. As is seen from equation 6, on increasing the interval length s, the sample of
bead displacements used in the averaging procedure decreases. Hence for the longer
time intervals, the value obtained for the mean squared displacement is less accurate,
due to poorer statistics. To systematically determine the appropriate gradient in the
longer time region, a least squares approach was used to fit a linear trend to a fixed time
range of the mean squared displacement against time data, starting at a specific time.
This process was repeated for different starting times, maintaining the same time range.
To assess the quality of each linear regression the coefficient of determination was
found (this is equivalent to the square of the Pearson correlation coefficient of the data
set). The mobility coefficient was then assigned as the gradient of the linear regression
with the maximum coefficient of determination.
3
3.1
Results
Global self-assembling structures
As previously outlined the results are discussed in terms of Nflx the number of lateral
chain beads that characterise each particular molecule. A global phase diagram of the
phases encountered when varying temperature and number of lateral chain beads is
presented in Figure 3. Further discussion on the observed phases is given below.
3.2
Bolaamphiphile
For the case of a system with no lateral grafted chain, Nflx=0, the system behaves as a
14
bolaamphiphilic liquid crystal. For T*≥1.33 the observed phase is isotropic (Figure a).
For 0.67≤T*≤1.00 we observe the smectic phase (Figure b). For T*<0.33 the system
crystallises as the mobility coefficient decreases several orders of magnitude below the
isotropic value. The inclusion of attractions on the end segments favours the alignment
in a smectic like pattern, precluding the possibility of a nematic phase, for which we
found no evidence.
3.3
Columnar phases
If side chains are grafted to the bolaamphiphilic molecular core, their presence severely
disrupts the parallel rod ordering. These side chains are moderately attractive to each
other, so they will have a tendency of agglomerating in the same regions of space.
Compounded with that propensity, the terminal beads of each bolaamphiphilic core will
have the tendency of maintaining themselves close to each other. The overall result is
the possibility of formation of columnar structures, where the walls of these structures
are delimited by the cores and the inner part of the columns are “filled” with the side
chains. Figure 5 showcases some examples of the types of columnar structures we have
encountered in our simulations, all of which echo experimentally observed
conformations.
The experimental behaviour of these systems has been described empirically with
reference to the volume fraction of lateral chains with respect to the whole system. As
this definition can be ambiguous for soft matter systems, below we define the volume
fraction of lateral chains as f=Vflx/Vmol. The conventions for the volumes and packing
fractions used are detailed in the ESI, section S3.
3.3.1 Nflx=1
15
The simplest possible perturbation to the bolaamphiphilic core is the addition of a
single side bead. For this scenario, where Nflx=1, equivalent to f=0.14, a triangular
section columnar phase is observed for T*=0.50 (see Figure a). If the attachment of
lateral chain is changed from the third position to second position of the rigid body, that
is, to a bead adjacent to a type 1 bead, an imperfect smectic phase is observed. The
triangular shaped columnar phase has also been experimentally confirmed for a similar
T-shaped polyphilic molecule xvii.
3.3.2 Nflx=2
A richer phase behaviour is observed when considering Nflx=2, equivalent to f=0.24.
The results for the Nflx=2 are summarised in Figure 6. Reviewing the diagram from
high to low temperature, at T*=0.77 the transition from the isotropic state to that of a
square columnar phase (Figure 5b), is clearly indicated by an increase in the S 2 and 4
order parameters.
The value of 6 remains close to zero during this transition,
indicating, as expected, that there is little hexagonal symmetry to the columnar
structures. On decreasing the temperature further the values of S2, 4 and 6 remain
relatively constant until around T*=0.43.
At this point the values of 6 starts
increasing, whilst 4 starts decreasing. Inspections of simulation movies reveal that
below T*=0.43, the square columnar structures start to slant into a rhombus columnar
structure. This behaviour clearly supports the trends in the order parameters observed.
At these low temperatures, through deformation, the cross section areas of the columns
are reduced whilst retaining the same cross sectional perimeter. This allows the lateral
flexible chain particles to get closer in space, whilst the structure preserves the strongly
bonded rigid rod network topology. With a further reduction of temperature below
16
T*=0.17 the values of S2, 4 and 6 remain constant at 0.87, 0.73 and 0.70, respectively.
Further information on the mobility of these phases may be garnered through looking at
of the mobility coefficient, also displayed in Figure . Between T*=0.13 and T*=0.47
the value of the mobility coefficient remains close to zero suggesting a solid crystalline
phase. Fluctuations of the mobility coefficient, evident at lower temperatures, may be
attributed to defects in the crystal structures. Inspection of the absolute displacements
of individual molecules during a simulation provides evidence for this. Displayed in
Figure a is the absolute displacement, in units of , for a typical molecule in a
simulation at T*=0.47 (in the vicinity of the solid-LC transition). The oscillation here
may be attributed to the molecule vibrating locally in the columnar structure. For a
different molecule in the same simulation, a similar behaviour is shown in Figure b,
however here at t*=7000 the molecule undergoes a large displacement before settling
back into a local oscillation. Isolation of this molecule in the simulation movie (see
supplementary material S1) clearly shows it shifting into a vacant site in the crystalline
columnar structure at t*=7000. Lastly, Figure c displays the behaviour of a molecule
from the T*=0.47 simulation whose absolute displacement oscillates with a large
amplitude. In the aforementioned movie this molecule is seen to be located outside the
main columnar lattice, consequently the rigid rod has greater freedom to move due to
weaker bonding of its type 1 beads (hydrogen bonding). The previous two examples
represent typical defects that contribute to the non-zero mobility coefficient at low
temperatures. To reduce the fluctuations observed in the mobility coefficient at lower
temperatures, larger simulations or longer times could be used. This would allow the
number of defects observed to reflect the true average expected at a given temperature.
For this investigation however these results were deemed adequate, as the crystalline
17
nature of the structures were not of primary interest. Between T*=0.47 and T*=0.77
the mobility coefficient increases linearly with increasing temperature. This provides
clear evidence that there exist molecular mobility in this temperature range. Inspection
of absolute displacements for molecules at T*=0.73 for example, show that whilst some
molecules remain local in the liquid crystal structure, as shown in Figure a, there is a
significant proportion of molecules that are mobile in spite of the structured nature of
the fluid phase, as displayed in Figure b. Above T*=0.77 there is a steep increase in the
mobility coefficient, associated with the transition to an isotropic phase, followed by a
linear increase with temperature.
In summary, the order parameter and mobility
coefficient data suggest three phases present for the Nflx=2 compound; a crystalline
rhomboidal columnar phase at temperatures below T*=0.47, a liquid crystalline square
columnar phase at temperatures between T*=0.47 and T*=0.77, and finally an isotropic
phase above T*=0.77.
3.3.3 Nflx=3,4
For systems in which there are three or four beads in the lateral chain, only defective
columnar shapes are formed.
For example, with Nflx=4, equivalent to f=0.39, at
T*=0.67, the columns in the LC region present cross sections of varying polygonal
shapes wherein pentagons and hexagon are among the most regular ones observed (see
Figure c). In this case there is an obvious competition between the pentagonal and
hexagonal columns as regular pentagons themselves cannot tessellate a plane and
hexagonal columns provide too much void space to be filled by the side chains, thus
packing frustration is evident.
3.3.4 Nflx=5
18
For the case of a five bead lateral chain, Nflx=5, equivalent to f=0.44, the system
exhibits an LC phase where hexagonal columns are abundant. The results for the Nflx=5
are summarised in Figure 3. On reviewing the diagram passing from high to low
temperature, the transition from the isotropic to hexagonal columnar phase (figure 5d)
is clearly delineated by a jump from near zero values of S2 and 6 to values of around
0.72 and 0.36 respectively at T*=0.87. This hexagonal ordering has been inferred from
experimental observations of T-shaped bolaamphiphilesxviii. As supplementary material
(S2) we include a movie of the quenching process for this molecule. Two things are
readily apparent from the movie. In the first place, it is seen from the slideshow how
precipitously the system achieves a general ordering. The system relaxes, finding
minimum energy configurations as hexagonal-columnar structures. It is also clear from
the movie that the system is, at this temperature, a liquid phase, as the movement of
individual molecules is evident. On decreasing the temperature further both, S2 and 6,
slowly increase until around T*=0.60, where there is a steep increase in 4. Below this
temperature the 4 parameter slowly increases, whilst S2 and 6 slowly decrease.
Visual inspection of the molecular dynamics simulation (not included in the
supplementary material) indicates that this result is due to a flattening of the hexagonal
columns into a more rectangular shape. Figure shows the variation of the j and j
population distribution (see equations 4 and 5) with temperature, and provide a good
indicator of structure.
The j population distribution shows changes that would
otherwise be difficult to pick out from inspection of the order parameters alone. The
angle j spans between the rigid core of each individual molecule and the system
director, that is generally the vector aligned with the columnar axes. At isotropic
conditions, the orientation of the bolaamphiphilic molecular cores is essentially random,
19
the director is therefore ill-defined, and the distribution of j is sinusoidal, reflecting the
lack of orientational correlation between the bolaamphiphilic cores and director. At
temperatures below the isotropic-LC transition, T*=0.87, there is a clear indication that
a large proportion of the molecules are, at any given time, at 90° from the director, i.e.
aligned in planes which are roughly orthogonal to the director. Since the phase is a
liquid one, the peak shown is broad. The transition to the solid phase is characterised
by a narrowing of the distribution, as is evident below T*=0.60, corresponding to the
LC-solid transition. At T*=0.43 the single peak at 90° splits into three peaks centred at
approximately 72°, 90°, 108°, indicating the structure relaxes from a state having
columns formed by parallel stacked hexagons to a form where a proportion of the rigid
rods are not directed in the plane orthogonal to the director. The outcome of this
change is to retain the hydrogen bonding network but effectively reduce the cross
sectional area of the column in the plane orthogonal to the director. This reduction of
cross sectional area allows the lateral chain units to arrange closer in space and hence
bond more effectively. Inspection of the population distribution for the angles j
between the molecules in a given plane orthogonal to the director also shows the
changes associated with these abovementioned phase transitions.
In the isotropic
region, as stated before the orientation of the bolaamphiphilic molecular cores is
essentially random, therefore the director is ill-defined, and the distribution of j is
shows no angular dependence. However, in the LC region, six distinct equispaced
peaks are shown with a period of 60°, corresponding to the angles expected in the
regular hexagonal structure. Upon crystallisation, below T*=0.60, the peaks sharpen
and at the lowest temperature range studied, corresponding to the secondary crystalline
structure, two of the peaks tend to merge. This, along with the j distribution, suggests
20
a structure that is collapsing towards a herringbone flattened rectangular columnar
structure, whose layers have resemblance to the pages of a partially open book.
As with the Nflx=2, further insight into the phase behaviour of the system may be
obtained by looking at the variation of the mobility coefficient with temperature. At
temperatures below T*=0.60 the mobility coefficient is very close to zero.
The
fluctuations at lower temperature may be rationalised in the same way as previously
discussed. On increasing the temperature above T*=0.60, the mobility coefficient
begins a roughly linear increase with temperature, until the isotropic transition at
T*=0.87. The non-zero values of the mobility coefficient in this region clearly indicate
that the phases have some fluidity, and hence point towards liquid crystalline behaviour.
Inspection of the j population distribution in this temperature range (Figure b), where
the distribution is noticeably diffuse on comparison to temperatures below T*=0.60,
support the assertion that this phase is relatively mobile (and also rationalise the
relatively low value of 6). The population of rigid rod units at angles intermediate to
those associated with the hexagonal column, may be due to molecule flipping between
hexagonal column sides. At T*=0.87 the onset of the isotropic phase is associated with
a significant jump in the mobility coefficient, followed in the isotropic phase by a linear
increase with temperature. To summarise, the information here indicates the presence
of four distinct phases, an isotropic phase above T*=0.87, a LC phase between T*=0.60
and T*=0.87, and two crystalline type phases, one between T*=0.43 and T*=0.60 and
one below T*=0.43.
Similar columnar behaviour is also observed for Nflx=6 and Nflx=7 but the structures
21
formed are riddled with increasing number of defects.
3.4
Lamellar phases
For side chains of eight or more beads long, the columnar structure breaks down and the
system presents a lamellar structure.
This is in synchronicity with the results of
experiments on related T-shaped molecules, where lamella are formed for systems with
large side substituentsxviii. Unique from the simulations is the observation that the
lamella are bridged by LC cores. For Nflx=8, equivalent to f=0.56, at T*=0.66 a
lamellar phase with bridges one-molecule long is observed (Figure 4a). We report that
for Nflx≥8 the systems dynamics slow down and equilibrating phases become
increasingly difficult. We have not observed regular columnar structures for higher
values of Nflx (only defective phases with no periodicity). We do not preclude the
existence of other structures, and note that it would require longer runs and bigger unit
cells than the ones used here. For the even longer chain length Nflx=11, equivalent to
f=0.64, at T*=0.50 a lamellar phase with bridges two-molecules long is observed
(Figure 4b). When the flexible chains are removed the system shows resemblance with
the “extended hexagons” proposed in the experimental paperxviii. However, not all the
“hexagons” are of the same size and they present perforations in the lateral walls.
3.4.1 Nflx=12
During the simulations for the twelve bead system, Nflx=12, equivalent to f=0.66, at
T*=0.83, a lamellar phase was observed composed of a flexible chain rich layer and a
rigid unit rich layers.
In this phase the rigid units were distributed more or less
isotropically within the plane of the rigid-unit domain (see Figure 5a). When the
temperature was lowered to 0.5≤T*≤0.73, the lamellar structure persisted. However, in
22
the rigid-unit rich domain a transition was observed, where smectic order was observed
within the layers (as can be seen in Figure 5b). Additionally, a few molecules were
observed to lie perpendicular to the rest. Equilibration from direct quenching was
relatively slow, making it in some cases necessary to run the system up to 10 7
integration steps in order to see the formation of the lamellar phase.
Following the initial simulations, a study of the variation of temperature on the twelve
bead system was undertaken, as summarised in Figure . For this system the S2, 4 and
2 order parameters were found to be the most effective at monitoring the temperature
dependence of the phase behaviour. Above T*=0.90 the values of S2 and 2 are close to
zero in the isotropic phase. As the temperature is reduced below T*=0.90 , S2 increases
to a value of around 0.56, whilst 4 and 2 remains close to zero. On inspection of the
simulation movie (not included) the phase is clearly seen to exhibit a lamellar structure
with isotropic arrangement of rods in the lamellar plane, as previously discussed. A
further reduction in the temperature below T*=0.80 leads to a further increase of S 2 to
around 0.9, and an increase of 4 and 2 to 0.66 and 0.73, respectively. The change in
2 is associated with alignment of the rigid rod units in the lamellar plane, inspection of
the simulation snapshots reveal this to be the in-plane smectic. The fact that 2 is not
closer to 1, and 4 is relatively close to 1, is due to the presence of defects where the
rigid rod units of the bolaamphiphile are perpendicular to the in-lamellar-plane smectic
director between the smectic layering. Inspection of the j population distribution, in
Figure clearly demonstrates this to be the case, with two weak peaks found to be
exactly mid-way between the main 180 separated peaks.
As the temperature is
reduced to T*=0.47, 2 is found to dip slightly before returning to 0.67, whilst S2 and
23
4 slowly increase to 0.96 and 0.91, respectively.
Below T*=0.47 all the order
parameters stay the same. Studying the mobility coefficient reveal four main regions.
Below T*=0.47 the mobility coefficient is very close to zero, indicating a crystalline
structure. Between T*=0.47 and T*=0.80, there is a slight positive gradient in the
mobility coefficient with increasing temperature indicating some fluidity to the phase.
Inspection of Figure supports this with a clear transition from sharply defined j
population distribution below T*=0.47 to a more spread out distribution above this
temperature. On going above T*=0.80 there is a jump in the mobility coefficient, and
then the gradient of mobility coefficient with temperature, increases from that in the
T*=0.47 to T*=0.80 range. Above T*=0.90, with the onset of the isotropic phase there
is another increase in the gradient. To summarise, for this chain length, the order
parameter and mobility coefficient analysis point towards four distinct phases; a
crystalline lamellar phase below T*=0.47, a liquid crystalline lamellar phase with inlamellar-plane smectic ordering of the bolaamphiphiles rigid rods between T*=0.47 and
T*=0.80, a liquid crystalline lamellar phase with isotropic ordering of the
bolaamphiphiles rigid rods between T*=0.80 and T*=0.90, and an isotropic phase
above T*=0.90.
The temperature and the mesophase have a strong influence on the conformations of the
lateral chains. Here we have studied only infinitely flexible chains, however, it is
expected that the actual flexibility will depend on the chemical nature of the chains (e.g.
alkane-like, perfluorinated or semi-perfluorinated). A detailed description of the
observed configurations of these lateral chains is included in the ESI (section S4)
4
Conclusions
24
We have shown here, as a matter of proof-of-concept, that one may successfully
employ coarse-graining techniques that reduce the overall attraction of the components
of the molecules and simplify the overall geometry to model very complex fluid selfassembly. The key aspect of the method is to retain in the CG model the key physical
elements of the original molecule, in terms of rigidity and flexibility, relative volume
fractions and relative intramolecular characteristic energies. The success of such a
minimalist model in qualitatively representing the experimentally observed phase
behaviour of the T-shaped amphiphiles, strongly indicates that the complex behaviour
observed is fairly independent of the specific details of the liquid crystal molecule and
rests on larger scale issues that seem to induce the formation of complex mesophases:
1) the presence of a rigid section with strongly attractive sites at both ends 2) a lateral
chain, with unlike interactions towards the aforementioned rigid part, hence inducing,
as a consequence of packing frustration considerations, the formation of complex
structures.
Although no effort was made in this work to map the parameters of the molecules to the
actual prototype T-shaped amphiphiles, we were able to account for most of the
reported experimentally observed phases and rationalise the formation of them in terms
of simple metrics and physical insight. If a more direct mapping were required for a
particular molecule, the model present has a sufficient independent variables (sizes,
number and energy of the different segments) to be able to give quantitative
information.
Acknowledgements
Partial financial support for this work has been given by the U.K. Engineering and
25
Physical Sciences Research Council (EPSRC), grant EP/E016340, “Molecular Systems
Engineering”, and U.S. Department of Energy, grant no. DE-FG02-05ER15682.
Fruitful discussions with Prof. George Jackson are gratefully acknowledged.
26
Figure 1
OH
HO
OH
O
O
OH
R
Structure of the core bolaamphiphile molecule that served as prototype for this study.
R corresponds to a substitution of an alkane,
perfluoroalkane or mixed alkane-perfluoroalkane chain. Adapted from ref. xiv
27
Figure 2
0.7

Cartoon of the molecules studied. Type 1 beads, corresponding to hydrogen-bond-like sites, are coloured orange; Type 2 beads, corresponding
to the polyphenyl core of the molecules are coloured green; Type 3 beads, corresponding to the grafted side chains are coloured blue. All cross
interaction are repulsive and all beads are the same diameter, , and mass.
28
Figure 3
Graphical representation of the phases encountered upon varying the reduced
temperature for the different molecular conformations studied (varying the number of
flexible grafted side chains, Nflx). Phase description: (A) Isotropic (B) Smectic (C)
Crystal (D) Triangular Columnar Crystal (E) Square Columnar Liquid Crystal (F)
Rhomboid Columnar Crystal (G) Hexagonal Columnar Liquid Crystal (H) Rectangular
Columnar Crystal (I) Out-of-plane Bent rectangular Columnar Crystal (J) Lamellar Inplane Isotropic Liquid Crystal (K) Lamellar In-plane Smectic Liquid Crystal (L)
Lamellar Crystal.
29
Figure 4
(a)
(b)
Snapshots of equilibrium configurations for molecules with no grafted chains (Nflx=0). (a) isotropic phase at T*=1.3; (b) smectic phase at
T*=1.0. Orange spheres represent the type 1 beads (hydrogen bonding), green spheres represent the type 2 beads (biphenyl core).
30
Figure 5
(b)
(a)
31
(d)
(c)
Snapshots of equilibrium configurations for liquid crystal columnar phases. Orange spheres represent the type 1 beads (hydrogen bonding),
green spheres represent the type 2 beads (biphenyl core), blue spheres represent the type 3 beads (perfluoroalkane grafted chains). (a) to (d)
corresponds to fluid phases with Nflx= 1, 2, 4 and 5 respectively.
32
Figure 6
Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx=2 system as a function of temperature. Green line is the
planar order parameter, S2 ; yellow line is the square columnar order parameter, 4 ; purple line is the hexagonal columnar order parameter, 6 ;
red line is the mobility coefficient.
33
Figure 7
Absolute displacements of selected individual molecules as a function of time for a system of Nflx=2 at T*=0.47, close to the LC-solid transition.
34
Figure 8
Absolute displacements of selected individual molecules as a function of time for a system of Nflx=2 at T*=0.73, close to the isotropic-LC
transition.
35
Figure 3
Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx=5 system as a function of temperature. Symbols as in
Figure .
36
Figure 10
Population plots as a function of temperature for the system Nflx=5 for the (a) variation of the angle I between the molecules’ rigid core and the
column director n; (b) variation of the angle I between the molecules’ rigid core and a random (fixed) vector in the plane orthogonal to the
column director n.
37
Figure 4
(b)
(a)
Snapshots of equilibrium configurations for lamellar phases. Colouring as in Figure . (a) Nflx=8; (b) Nflx= 11.
38
Figure 5
(a)
(b)
Snapshots of equilibrium configurations for lamellar phases for Nflx=12. Details of the lateral chains have been omitted for clarity. The
snapshots look into the layers of the lamella from the side-chain rich mesophase, i.e. a view orthogonal to that of figure 11. (a) Isotropic phase at
T*=0.83; (b) smectic phase at T*=0.6.
39
Figure 13
Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx=12 system as a function of temperature. green line is the
planar order parameter, S2 ; purple line is the linear columnar order parameter, 2 ; orange line is the square columnar order parameter, 4 ; red
line is the mobility coefficient.
40
Figure 14
Population plots as a function of temperature for the system Nflx=12 for the (a) variation of the angle I between the molecules’ rigid core and
the director n, in this case, orthogonal to the lamella; (b) variation of the angle I between the molecules’ rigid core and a random (fixed) vector
in the plane of the lamella.
41
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