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A model for rod-coil block copolymers

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ISSN 18112382, Polymer Science, Ser. C, 2013, Vol. 55, No. 1, pp. 70–73. © Pleiades Publishing, Ltd., 2013.
70
1
INTRODUCTIONThe study of nanostructureforming materials ishighly relevant from a technological point of view. For instance, the functionality of lightemitting or photo voltaic devices depends critically on the existence of nanoscale domain structures with large interface areasand optimized domain sizes that are, at least in onedimension, comparable to the exciton diffusionlength. In practice, thin multilayer films are mostcommonly used. However, these are nonequilibriumstates, whose structure depends critically on details of the process of fabrication (e.g. deposition), and whoselifetimes at higher temperatures are limited by interdiffusion. Alternative systems where the desired nanostructures form spontaneously as equilibrium statesare clearly of interest. Promising candidates are block copolymers with two covalently linked components [1,10–18]. If the stiffness disparity of the blocks is high(rodcoil block copolymer), they will microphaseseparate already at relatively low molecular weight intoa variety of nanostructures with morphologies that can be tuned, e.g., by varying the chemistry and themolecular weight of the coil component. This can beexploited to design functional materials with tunableproperties.The phase behavior of rodcoil block copolymer systems is much richer than that of conventional flexible block copolymers, and far less understood [1, 2].The phase diversity results from the interplay betweenthe conformational entropy of the coils, the “conventional” incompatibility between chemically differentmonomers, and local packing effects which lead to liquid crystalline or even (semi)crystalline order in therod domains. A large number of selfassembled structures has been observed experimentally in systems of rodcoil copolymers, such as tilted and untilted smec
1
The article is published in the srcinal.
tic phases, wavy lamellar, zigzag, arrowhead, perforated lamellar, hockey puck, hexagonal strip, and nanorod phases [1]. Some, but not all of these phases have been predicted or reproduced in theoretical studies(see. e.g., [1, 2, 19, 20] and references therein). Computer simulations are still scarce [3–9, 21–24]. Tosimulate these materials, structures on very differentlength scales have to be taken into account. Thisclearly calls for a hybrid multiscale treatment.MODEL In the present paper, we propose a simple model for the study of rodcoil block copolymer selfassembly.To capture the relevant physics, the models mustaccount for the possibility of local orientational andcrystalline order in the block domains as well as for theconformational degrees of freedom of the coildomains. Therefore, we treat the rod and the coil blocks at different coarsegraining levels. The rods arerepresented as elongated impenetrable spherocylinders. The coils are modeled as flexible chains, whosemonomer interactions are defined in terms of anEdwards Hamiltonian that depends on local densities[25–27]. The interactions between rods and coils isdefined in terms of a density functional and willexplicitly account for the extended size of the rods.The system is built up of
N
rodcoil block copolymers. Each rodcoil block copolymer consists of onerod which is connected to a polymeric coil with
n
monomers (see Fig. 1). Each coil part consists of
n
– 1Gaussian springs and is linked to its associated rodtotaling in
n
springs for each rodcoil block copolymer. The rods are hard spherocylinders with length
L
and diameter
D
. The total energy of a threedimensional system with
N
rodcoil block copolymers comprises four different contributions
(1)
β = β +β +β +β .
G CC CR RR
A Model for RodCoil Block Copolymers
1
Stefan Dolezel*, Hans Behringer, and Friederike Schmid**
Institut für Physik, Johannes GutenbergUniversität Mainz, Staudinger Weg 7D–55128 Mainz, Germany*email: dolezel@unimainz.de** email: friederike.schmid@unimainz.de
Abstract
—A coarsegrained model for studying the phase behavior of rodcoil block copolymer systems onmesoscopic length scales is proposed. The polymers are represented on a particle level (monomers, rods) whereas the interactions between the system’s constituents are formulated in terms of local densities. Thisconversion to density fields allows an efficient Monte Carlo sampling of the phase space. We demonstrate theapplicability of the model and of the simulation approach by illustrating the formation of typical microphaseseparated configurations for exemplary model parameters.
DOI:
10.1134/S1811238213060015
POLYMER SCIENCESeries C
Vol. 55
No. 1
2013
A MODEL FOR RODCOIL BLOCK COPOLYMERS71
Here denotes the inverse temperature, i.e. with being Boltzmann’s constant. The first term
(2)
models the Gaussian chain statistics of the polymer coils and provides the connectivity [25, 26]. Eachpolymer chain consists of
n
“pointlike” monomers atpositions
,
with the polymer index . The monomer positions can takeany value in the threedimensional volume (simulation box, offlattice model). The monomer with index is associated with a “virtual” monomer, fixed toone end of the spherocylindrical rod of the givencopolymer (see Fig. 1). This monomer only contributes to (via bond/spring ) in Eq. (1), butnot to the monomer density which determines and (see below). The strength of the Gaussian bond energy is given by the spring constant
k
which isrelated to the averaged bond length
b
by inthree space dimensions.The second term in Eq. (1) models the excluded volume interactions of the monomers and is given by the first term in the corresponding virial expansion
(3)
with denoting the local monomer density relatedto a given configuration [25, 26]. The parameter needs to be positive to have a stable polymer configuration. The rodcoil interaction is modeled by hard body constraints (a monomer may not enter a spherocylinder) and an additional densitydependent interaction which describes soft interactions of longer range.
(4)
The field denotes the rod density and the parameter the strength of the monomerrod interaction.
β
β =
1
k T
B
/
k
B
α, − α,α= =
β = −
∑∑
G
211 1
( )2
N ni i i
k
R R
α,
i
R
= , , ,
1 2
i … n
α = , , ,
1 2
… N
α,
i
R
V
=
0
i
β
G
→
0 1
β
CC
β
CR
=
2
3
k b
/
χβ = ρ
∫
CCCC C
2
( ( ))2
V
d
r r
ρ
( )
r
C
χ
CC
∞ :β = χ ρ ρ :
∫
CR CR C R
monomers enter rodsotherwise
( ) ( )
V
d
r r r
ρ
( )
r
R
χ
CR
Finally, the rodrod interactions are pure hardbody interactions between spherocylinders.
(5)
We note that the expression does not contain anexplicit temperature dependence. The connectivity isprovided by chemical bonds which are basically unbreakable for the temperature range of interest. Theremaining energy parameters and are effectiveparameters that stem from integrating out microscopicdegrees of freedom. They consequently exhibit a temperature dependence which is however (assumed to be) weak in the range of interest. We will thereforeinvestigate the phase behavior in dependence of theseparameters instead of looking explicitly at the temperature.The phase behavior is investigated by means of Monte Carlo simulations. A new polymer conformation is generated by a set of trial moves. These comprise simple translational moves for the monomers of the coil parts whereas new rod configurations are proposed in a twofold manner, namely by spatial translations and by rotations. We will now briefly discuss howa discretized version of the energy terms (3) and (4)can be generated for the use in such a particlebasedMonte Carlo simulation [28–30]. To this end the configuration of monomers and rods has to be convertedinto density fields and . The conversion is basedon the introduction of a regular rectangular grid of cells so that for each cell a corresponding discretizeddensity can be computed. For a cell with center node the local monomer density for all points inside isapproximated by with beingthe number of monomers in the cell . Here, denotes the volume of cell accessible to the monomers (see below, all cells have the same form and volume with being the discretization sizeof the mesh in
j
direction). The excluded volumeenergy of the polymers is then given by
(6)
Let us just remark that the conversion of the monomer positions to a local density field introduces a further model parameter via the discretization length or the
∞ :β = :
RR
rods overlapotherwise
0
β
χ
CC
χ
CR
χ
ρ
C
ρ
R
I
I
r
,
ρ = ρ =
( )
I I I
C V
r
C C I
/
I
C I
I
V I
= δ δ δ
□
x y z
V
δ
j
( )
χ χ.ρ
∑∫
□
≃
CC CCC
22
( )2 2
I I I V
C d V V
r r
δ
j
LD
u
012
nn
−
1
Fig. 1.
Cartoon of our model. Rod of diameter
D
, length
L
and unit vector
u
and the connected coil consisting of
n
monomersconcatenated by springs.
72
POLYMER SCIENCESeries C
Vol. 55
No. 1
2013
STEFAN
DOLEZEL et al.
discretization volume . These parameters can beinterpreted as effective monomer size or volume,respectively. The contribution of a given monomer tothe local field is here obtained by a simple “binning”procedure. We note, however, that higher order schemes can be applied which lead to a smoother assignment to several neighboring cell nodes, for details see e.g. [30].The local rod density associated with a given cell is just the volume fraction occupied by rods. This volume
fraction
is determined by means of a MonteCarlo integration scheme: For a cell a number of random positions is generated, the fraction of randompositions inside rods is then taken to be the volumefraction, i.e. the rod density of cell is just . Itis not necessary to generate the distribution of randompoints in a cell for each Monte Carlo move of a rod.However, to avoid artifacts the set of random positionshas to be regenerated regularly.The hard body constraint between monomers androds has the consequence that the volume per cellaccessible to monomers is reduced by the rods and is just given by . The monomerrod interaction (4) in discretized form thus reads
(7)
To conclude we summarize the model parameters which define the rodcoil block copolymer system onthe mesoscale. The system contains
N
rodcoil block copolymers in a rectangular box of sizes along the
j
□
V I
Φ
I
,
ρ
I
R
I
Φ
I
= −Φ
□
(1 )
I I
V V
χ ρ ρ χ Φ .
∑∫
□
≃
CR C R CR
)
( ( )
I I I I V
C d V V
r r r
j
s
direction defining the box volume and with periodic boundary conditions.The grid for the local density assignment is determined by the cell sizes corresponding to an effectivemonomer size. The spherocylindrical rods are characterized by the length
L
and diameter
D
. Each polymer consists of one rod connected to
n
Gaussian bonds with a spring constant
k
. Finally the interactions aredefined by the energy parameters and .To illustrate the model, we show two examples of rodcoil block copolymer systems (Fig. 2). Here wehave used the parameters
, ,
=0.05. The reduced density is comparable in both cases, where is the rod density given by and
(8)
is the density of close packing of spherocylinders. Atthis reduced density , a system of bare spherocylinders with aspect ratio is in a disordered isotropic state [31]. This disordered structure is alsoassumed by the rodcoil system with after equilibration from an srcinally layered state. At, the layered state remains stable. Hence theattached coils are found to stabilize layered structures.In summary, we have described a coarsegrainedmodel system designed for the investigation of thephase behavior of rodcoil block copolymer systemson mesoscopic length scales. The model captures therelevant physics by providing packing of rods throughhard core interactions, in addition, the conforma
=
S x y z
V s s s
δ
j
χ
CC
χ
CR
=
1
D
=
12
k
χ = χ
CC CR
ρ = ρ ρ
cp
* /
ρ = .
*
0 59
ρ
S
N V
/
ρ = +
1 ( 2 3 )
L D
cp
/ /
ρ
*
<
3
L D
/
=
2
L D
/
=
3
L D
/
(a)(b)
Fig. 2.
Snapshots of rodcoil block copolymer systems in different states. The (dark grey) coil parts are displayed in a tuberepresentation of the bonds. The reduced density in both cases is . (a) Rods with aspect ratio attached to chains of
n
= 40 monomers. The cell size is . (b) Rods with aspect ratio attached to chains of
n
= 60 monomers. The cellsize is .
ρ = .
*
0 59
=
/
3
L D
δ = .
1 5
j
=
/
2
L D
δ =
2
j
POLYMER SCIENCESeries C
Vol. 55
No. 1
2013
A MODEL FOR RODCOIL BLOCK COPOLYMERS73
tional freedom of the flexible coils is explicitly takeninto account as well as an energetic repulsion betweencoils and rods. The model together with its efficientnumerical treatment within a Monte Carlo simulationis particulary suited to investigate how entropic andenergetic contributions of coils alter ordering effects of rods for different packing fractions. In addition, we will analyze in future works to which degree the modelcan reproduce spontaneously formed microphases of rodcoil systems which have been reported in the literature. ACKNOWLEDGMENTSThis work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the InternationalResearch Training Group 1404
Selforganized materials for optoelectronics.
Computing time on the lc2 cluster at the ZDV Mainz is also acknowledged.REFERENCES
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