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A model for rod-coil block copolymers

A model for rod-coil block copolymers
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  ISSN 1811󰀭2382, Polymer Science, Ser. C, 2013, Vol. 55, No. 1, pp. 70–73. © Pleiades Publishing, Ltd., 2013. 70 1 INTRODUCTIONThe study of nanostructure󰀭forming materials ishighly relevant from a technological point of view. For instance, the functionality of light󰀭emitting 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 non󰀭equilibriumstates, whose structure depends critically on details of the process of fabrication (e.g. deposition), and whoselifetimes at higher temperatures are limited by inter󰀭diffusion. Alternative systems where the desired nano󰀭structures 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(rod󰀭coil block copolymer), they will microphase󰀭separate 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 rod󰀭coil block copolymer systems is much richer than that of conventional flex󰀭ible block copolymers, and far less understood [1, 2].The phase diversity results from the interplay betweenthe conformational entropy of the coils, the “conven󰀭tional” incompatibility between chemically differentmonomers, and local packing effects which lead to liq󰀭uid crystalline or even (semi)crystalline order in therod domains. A large number of self󰀭assembled struc󰀭tures has been observed experimentally in systems of rod󰀭coil copolymers, such as tilted and untilted smec󰀭 1 The article is published in the srcinal. tic phases, wavy lamellar, zigzag, arrowhead, perfo󰀭rated lamellar, hockey puck, hexagonal strip, and nan󰀭orod 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). Com󰀭puter 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 rod󰀭coil block copolymer self󰀭assembly.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 coarse󰀭graining levels. The rods arerepresented as elongated impenetrable spherocylin󰀭ders. 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   rod󰀭coil block copoly󰀭mers. Each rod󰀭coil 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 rod󰀭coil block copoly󰀭mer. The rods are hard spherocylinders with length L and diameter D  . The total energy of a three󰀭dimen󰀭sional system with N   rod󰀭coil block copolymers com󰀭prises four different contributions (1) β = β +β +β +β . G CC CR RR        A Model for Rod󰀭Coil Block Copolymers 1 Stefan Dolezel*, Hans Behringer, and Friederike Schmid** Institut für Physik, Johannes Gutenberg󰀭Universität Mainz, Staudinger Weg 7D–55128 Mainz, Germany*e󰀭mail: dolezel@uni󰀭** e󰀭mail: friederike.schmid@uni󰀭  Abstract —A coarse󰀭grained model for studying the phase behavior of rod󰀭coil 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 micro󰀭phaseseparated configurations for exemplary model parameters. DOI: 10.1134/S1811238213060015  POLYMER SCIENCESeries C    Vol. 55   No. 1   2013  A MODEL FOR ROD󰀭COIL 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  “point󰀭like” monomers atpositions ,  with the polymer index . The monomer positions can takeany value in the three󰀭dimensional volume (simula󰀭tion box, off󰀭lattice 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 contrib󰀭utes 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 configu󰀭ration. The rod󰀭coil interaction is modeled by hard󰀭 body constraints (a monomer may not enter a sphero󰀭cylinder) and an additional density󰀭dependent inter󰀭action which describes soft interactions of longer range. (4) The field denotes the rod density and the param󰀭eter the strength of the monomer󰀭rod 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 rod󰀭rod interactions are pure hard󰀭body 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 tem󰀭perature dependence which is however (assumed to be) weak in the range of interest. We will thereforeinvestigate the phase behavior in dependence of these󰀭parameters instead of looking explicitly at the tem󰀭perature.The phase behavior is investigated by means of Monte Carlo simulations. A new polymer conforma󰀭tion is generated by a set of trial moves. These com󰀭prise simple translational moves for the monomers of the coil parts whereas new rod configurations are pro󰀭posed in a two󰀭fold manner, namely by spatial transla󰀭tions 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 particle󰀭basedMonte Carlo simulation [28–30]. To this end the con󰀭figuration 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 mono󰀭mers (see below, all cells have the same form and vol󰀭ume 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 vol󰀭ume fraction is determined by means of a MonteCarlo integration scheme: For a cell a number of ran󰀭dom 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 re󰀭generated 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 monomer󰀭rod inter󰀭action (4) in discretized form thus reads (7) To conclude we summarize the model parameters which define the rod󰀭coil block copolymer system onthe mesoscale. The system contains N   rod󰀭coil 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 deter󰀭mined by the cell sizes corresponding to an effectivemonomer size. The spherocylindrical rods are charac󰀭terized 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 rod󰀭coil 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 spherocylin󰀭ders with aspect ratio is in a disordered isotro󰀭pic state [31]. This disordered structure is alsoassumed by the rod󰀭coil 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 coarse󰀭grainedmodel system designed for the investigation of thephase behavior of rod󰀭coil 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 rod󰀭coil block copolymer systems in different states. The (dark grey) coil parts are displayed in a tube󰀭repre󰀭sentation 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 ROD󰀭COIL 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 micro󰀭phases of rod󰀭coil systems which have been reported in the liter󰀭ature. ACKNOWLEDGMENTSThis work was supported by the Deutsche Fors󰀭chungsgemeinschaft (DFG) via the InternationalResearch Training Group 1404 Self󰀭organized materi󰀭als for optoelectronics.  Computing time on the lc2 clus󰀭ter at the ZDV Mainz is also acknowledged.REFERENCES 1.M. 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