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Molecular dynamics study of the intercalation of diblock copolymers intolayered silicatesJae Youn Lee, Arlette R. C. Baljon, Dotsevi Y. Sogah, and Roger F. Loring Citation: J. Chem. Phys. 112, 9112 (2000); doi: 10.1063/1.481538 View online: http://dx.doi.org/10.1063/1.481538 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i20 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 20 22 MAY 2000
Molecular dynamics study of the intercalation of diblock copolymersinto layered silicates
Jae Youn Lee, Arlette R. C. Baljon,a) Dotsevi Y. Sogah, and Roger F. LoringDepartment of Chemistry and Chemical Biology, Baker Laboratory, Cornell University,Ithaca, New York 14853
~Received 3 December 1999; accepted 28 February 2000!
Polymer-layered silicate nanocomposites may be formed by annealing layered silicate particles witha polymer melt. Polymer molecules flow from a bulk melt into the galleries between silicate sheets,swelling the silicate structure. The use of an amphiphilic intercalant raises possibilities of formingnovel structures and enhancing the intercalation kinetics relative to the case of homopolymerintercalants. We perform molecular dynamics simulations of the flow of a symmetric diblockcopolymer from a bulk melt into a slit whose surfaces are modified by grafted surfactant chains, andwhose walls are maintained at a constant pressure to permit the slit to open as polymer intercalates.Intercalation kinetics are examined for a variety of polymer–surface and interblock interactions andfor thermodynamic states in which the bulk polymer occupies either a lamellar or disordered phase.Comparison to previous simulations of homopolymer intercalation demonstrates that diblockcopolymers may be used to intercalate a block that would not spontaneously intercalate as ahomopolymer. ©2000 American Institute of Physics.@S0021-9606~00!50120-2#
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I. INTRODUCTION
A variety of nanostructures are based on the combinaof layered silicate minerals and polymers.1–21 These includeboth layered structures in which a silicate particle is swolby polymer, but retains its layered morphology, and exfoated structures in which individual silicate sheets are dpersed in a polymer matrix.1–3 Such polymer-inorganicnanocomposites offer the potential of improved mechanproperties, barrier properties, and resistance to heat distorelative to conventional composites composed of macscopic polymer and inorganic phases.1–3 However, the for-mation of these structures generally requires the mixinghydrophobic polymer and hydrophilic silicate. Synthestrategies for these materials require schemes for circventing these unfavorable interactions. One such strategto modify the silicate surfaces by intercalating alkylammnium salts into the silicate, producing short alkyl chains achored to the silicate by Coulomb interactions.1–3 This ap-proach has permitted the intercalation of high molecuweight polystyrene into fluorohectorite to form layerestructures.5,6 A second strategy, which has produced exfoated structures, involves intercalation of polymer precursinto the silicate, followed byin situ polymerization.17,18 Athird potential strategy is to employ amphiphilic intercalansuch as block copolymers, in which one portion of the mecule has a greater affinity for the silicate surface thanother. This strategy is currently being pursued by Soget al.,22 and Gianneliset al.23
The intercalation of block copolymers into layered sicates produces thin films of these molecules. The genphase diagram of block copolymers in the bulk is w
a!Permanent address: Department of Physics, San Diego State UniveSan Diego, CA 92182.
9110021-9606/2000/112(20)/9112/8/$17.00
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established,24–27 and attention has turned recently to thmodification of this behavior by the confining surfaces inthin film.28–32 In a confined film of block copolymers, thconfining surfaces may dictate a structure different from tof the most stable bulk phase.28,29 Laboratory28,29 andsimulation30–32studies of the structure of confined block cpolymer films have focused on symmetric diblocks, whomost stable bulk phase at temperatures below an orddisorder transition is lamellar. For the case in which oblock has a preferential affinity for the confining surfacrelative to the other block, lamellae parallel to the confinisurfaces, whose spacing may be perturbed from the bvalue, have been observed.28 For the case in which the surface affinities of both blocks are comparable, lamellae ppendicular to the confining surfaces have been identifie29
Both structures have also been observed in recent MoCarlo studies.30–32
In this work, we treat the flow of symmetric dibloccopolymers from an ordered or disordered bulk phase inconfining slit whose walls represent the sheets in a layesilicate. Oligomer chains are grafted to the surfaces to moorganically modified silicates.1–3 These simulations represena system that embodies two of the three strategies descrabove for making polymer and silicate thermodynamicacompatible: surface modification and the use of amphiphpolymers. The walls are maintained at constant pressurelowing the slit spacing to increase as polymer flows.20 Sur-factants and polymers are represented with the model of Kmer and Grest33 as Lennard-Jones spheres connectedanharmonic springs. The phase behavior of symmediblock copolymers within this model has been exploredGrestet al.34 and by Muratet al.35 Our model contains fourcomponents: the surfaces, the surfactants, and two bloThe purpose of this study is to investigate the effectsinteractions among those components on the dynamic
ity,
2 © 2000 American Institute of Physics
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9113J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Intercalation of diblock copolymers
formation of intercalated structures. In particular, we focon the question of whether the use of amphiphilic intercants can produce structures containing material that wonot spontaneously intercalate as a homopolymer. The mis specified and the simulation algorithm described in SecResults are presented and discussed in Sec. III. The prdure used to equilibrate bulk lamellar phases at conspressure is described in the Appendix.
II. MODEL AND SIMULATION
The model employed here is a generalization to the cof block copolymers of that used in our previous studyhomopolymer intercalation,20 and is depicted in Fig. 1. Thesimulation cell is divided into three compartments. The ctral compartment, which represents one gallery within a scate particle, contains two rigid walls whose surfacesdecorated with grafted surfactants, and which are subjecconstant external pressure,pex. The outer compartmentcontain reservoirs of bulk polymer melt, maintained at fixtemperature and pressure. The direction of polymer flowdesignated as thex direction, with they andz directions alsodefined in Fig. 1. Periodic boundary conditions are appliedthe y andz directions, and the reservoirs are bounded inx direction by structureless walls maintained at the sameternal pressurepex as applied to the slit walls.
Polymers and surfactants are represented as chainsoft spheres connected by the anharmonic FENE potentiathe model of Kremer and Grest.33–35Each surfactant is composed of 12 beads,20 and each polymer is a symmetrdiblock composed ofN524 beads. Each slit wall is forme
FIG. 1. The simulation cell is shown prior to intercalation in panel~a! andduring intercalation in panel~b!. The two outer compartments contain a budiblock copolymer melt, in either an ordered or disordered phase, at contemperature and pressure and the inner compartment contains a slitwalls at constant pressure whose surfaces have grafted short-chain stants.
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of a rigid fcc lattice of spheres, with each lattice orientwith ~111! planes parallel to thexy plane. Thex and y di-mensions of the slit are 21.3sp and 7.4sp , respectively,with sp the polymer bead diameter. The end beads ofsurfactant chains are rigidly fixed at grafting sites, which alocated in a regular array at the interstitial sites of the~111!surface at a surface density7,20 of 0.105ss
22 , correspondingto a total of 46 chains per surface. The diameter of a surtant bead is denotedss . Each silicate sheet is treated asrigid body of mass 100mp , with mp the mass of a polymebead of either block, that may only move in thez direction.This choice of slit wall mass and the imposition of boundaconditions in thez dimension for a slit of variable width arediscussed in Ref. 20.
There are four types of particles in this model: polymbeads composing one of the blocks, labeledA, polymerbeads composing the second block, labeledB, surfactantbeads, labeleds, and lattice particles making up the slwalls, labeledl. Properties common to both types of polymbeads are labeledp. We must define nine pairwise interactions among these particles. There is nol l interaction, sincethe slit walls are taken to be rigid lattices. All of the nobonded interactions have the form of a Lennard-Jones~LJ!potential that has been truncated at a cutoff distancer c andshifted upward in energy to vanish atr 5r c :
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9114 J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Lee et al.
surfactant bead and a lattice particle is identical to thattween two surfactant beads. Polymer beads of typesA andBare distinguished only by the difference in intrablock ainterblock potential energies and by differing interactiowith the slit surfaces. Three parameters in Table I,eAB , eAl ,andeBl , are not assigned numerical values, as these areparameters that will be varied. Increasing the magnitudeeAB increases the interblockrepulsion, while increasing themagnitude ofeAl or eBl increases the polymer–surfaceat-traction. Polymer–surface attractive interactions are actbetween the interior slit walls and polymer beads. The inaction between beads in the reservoirs and the exteriortion of the slit walls in contact with the reservoirs is purerepulsive, in order that polymer in the reservoir not adherethese exterior walls. Polymer beads interact with the strtureless surfaces bounding the reservoirs in thex dimensionwith a purely repulsive potential obtained by integrating tintraspecies polymer–polymer potential over thexy plane.19
The temperature is maintained atT51 with a Brownianthermostat.33 During the intercalation process, the equatioof motion are integrated with the RESPA algorithm,36 withthe forces on the lighter surfactant beads updated everystep of 0.0025, and the forces on the more massive polybeads updated every alternate time step.
At high values of the interblock repulsion, the polymein the reservoir form an ordered, lamellar phase. In this cawe prepare the bulk polymer in the reservoirs with a mofication of the algorithm of Grestet al.34,35 First, a lamellacontaining one species of bead is constructed by grafpolymer brushes of lengthN/2 facing each other on twoplanar surfaces, each parallel to theyz plane. This lamella isthen equilibrated at constant temperature and pressure,periodic boundary conditions applied in they and z dimen-sions. Then, the system is replicated along thex dimension toform the desired number of lamellae. The planar surfacesthen removed, and the grafted beads are joined to fdiblock copolymers. The potential energies are altered toflect the identities of the blocks. Since the lamellar spacappropriate to a given thermodynamic state is not knowapriori , the dimensions of the simulation cell parallel and ppendicular to the lamellae must be allowed to fluctuateconstant pressure to permit the system to seek the apprate lamellar thickness.
At this point, our procedure differs from that of Greet al.,34 who carried out this equilibration for a bulk bloc
TABLE I. Parameters for nonbonded interactions in Eq.~1!. Polymer beadsare labeledA andB, lattice particles are labeledl, and surfactant beads arlabeleds.
r c e s
AA 2 1/6 1 1AB 21/6 eAB 1As 21/6 ~0.8! 1 0.8Al 2.2~0.8! eAl 0.8BB 21/6 1 1Bs 21/6 ~0.8! 1 0.8Bl 2.2 ~0.8!, 21/6 ~0.8! eBl 0.8ss 21/6 ~0.6! 1 0.6sl 21/6 ~0.6! 1 0.6
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copolymer melt with a modification of the constant-pressmethod of Andersen37 that permits independent fluctuationof the dimensions of an orthogonal simulation cell. In tpresent model, periodic boundary conditions in thex dimen-sion would not be appropriate, and hence this dimensiobounded by a physical wall at constant external pressuredescribed in the Appendix, we implemented the Andersmethod in thez dimension only. During this stage of thequilibration, they dimension is fixed to be equal to that othe slit, l y57.4, while the cell dimensions inx andz, l x andl z , fluctuate independently. All calculations employ an eternal pressure ofpex55.2ep /sp
3 . During the equilibration ofa lamellar phase in the reservoirs, the method of the Appdix is applied to updatel x andl z at each simulation time stepIn a typical equilibration,l z decreased from an initial valuof 10.4 to an equilibrium value of 6.5 in a time interval o2500 LJ time units, corresponding to an average changel z
per time step on the order of 1025sp . A typical root-mean-squared pressure fluctuation associated with forces in tzdimension is 6.2531023ep /sp
3 . After the lamellar spacingl x , and l z have been determined, the Andersen constapressure mechanism in thez dimension is turned off, and thesystem is re-equilibrated with constant external pressureplied to the bounding walls in thex direction, but with theyandz dimensions fixed.
As the order–disorder transition is approached fromordered phase by decreasing the interblock repulsion at fitemperatureT51, the equilibration procedure just describebecomes impractical because of large fluctuations in syssize. Our present goal is not to map out the phase diagrathis model, which has been treated atT51 by Grestet al.34,35 Some of the pitfalls associated with that exercissuch as finite size effects, are discussed there.34,35Rather, ourpurpose is to compare intercalation dynamics from ordeand disordered bulk states. We therefore prepare disordstates by starting with an equilibrium configuration for buhomopolymer, instantaneously turning on the interblockpulsive interactions, and equilibrating under constant exnal pressure in thex dimension, with periodic boundary conditions in y and z. For sufficiently weak interblockrepulsions, this procedure produces a disordered state, wfor stronger interblock interactions, phase segregation isdent. Before intercalation, each reservoir contains 90 pomer chains of lengthN524 for the lamellar phase or 3chains for the disordered phase at an approximate bead nber density of 0.85.
We test this equilibration procedure by simulating bumelts of diblock copolymers and comparing the resultsthose of Grestet al.,34,35who demonstrated that for symmeric diblocks within the model described above, transitiofrom disordered to lamellar states could be observed bying T51 and varying the interblock repulsioneAB . The val-ues of eAB at the order–disorder transition for chainslength (N) were estimated to be 3.9~20!, 1.85 ~40!, 1.28~100!, and 1.20~200!, with the value forN520 representinga lower bound. For consistency with our previous studieshomopolymer intercalation forN512,19–21 we employ hereN524, a value not treated by Grestet al.34,35Figure 2 showsthe order parameter, c(x)[@rA(x)2rB(x)#/@rA(x)
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9115J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Intercalation of diblock copolymers
1rB(x)# for N524, T51, and two values of the interblocrepulsion:eAB55 andeAB53. The number density of beadof type n, averaged over a slab parallel to theyz plane, isdenotedrn(x). At eAB55, c(x) shows well-defined lamellae. Lowering the interblock repulsion toeAB53 leads toless distinct layering. AteAB52, the system is characterizeby large volume fluctuations, as discussed above, andhypothesize, guided by the findings of Grestet al.,34,35 thatthe most stable phase is disordered. These results suthat the order-disorder transition forN524 andT51 takesplace in the range 2.0,eAB
ODT,3.0, which is consistent withthe bounds 1.85,eAB
ODT,3.9 implied by the work of Greset al.34,35 These calculations identify thermodynamic stacorresponding to bulk ordered and disordered structuwhich we will use in the intercalation studies describedlow.
III. RESULTS AND DISCUSSION
To place into context our studies of intercalation froordered and disordered block copolymer melts, we firstview our previous investigation of the intercalation of hmopolymers, which focused on the role of attractipolymer–surface interactions in determining the spontanof intercalation.20 For N512, a degree of polymerizatioequal to the block length in the present work, and the savalues of temperature, pressure, surfactant chain lengthgrafting density, polymer–surfactant interactions, andtraspecies polymer–polymer interactions employed herevaried the polymer–slit wall potential energy. This potentwas taken to have an attractive well as in Eq.~1! with r c
52.2spl , and LJ energy scaleepl . For epl51, no spontane-ous intercalation was observed on the time scale of the silations. Polymers intercalated spontaneously forepl52,3,4.For epl52,3, the slit opened in two steps, consistent withlayering of polymers confined between parallel surfaces.epl54, only a single such step was observed on the sim
FIG. 2. The structure of bulk melts of symmetric diblock copolymers wNA5NB512 is characterized by the order parameter,c(x)5@rA(x)2rB(x)#/@rA(x)1rB(x)#, with r j (x) the number density of beads of typejaveraged over a slab parallel to theyz plane. The temperature is fixed aT51 and the interblock repulsioneAB is varied as shown.
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tion time scale, suggesting that intercalation may slow ifpolymer–surface interactions are too strongly attractive.
Figure 3 shows the time dependence of the slit spacindin the left-hand panel and the number of each type of beathe slit in the right-hand panel foreAB55, eAl53, andeBl
51. The polymer-surface interaction includes an attractwell, while all other interactions are purely repulsive, as dscribed in the previous section. The bulk melt has a lamestructure with layers perpendicular to the long dimensionthe slit, and in each reservoir theA block is located nearest tothe slit opening. In this thermodynamic state, a homopomer melt of pureA with chains of length 12 would spontaneously intercalate, while a melt of pureB chains of the samelength would not.20 The A beads, which are more stronglattracted to the slit walls and which are initially adjacentthe slit openings in the reservoirs, enter the slit first, evenally transportingB blocks into the slit. Thus, theB blocksthat would not spontaneously intercalate under these cotions as homopolymers are brought into the slit by theAblocks. Figure 4~c! of Ref. 20 shows the intercalation ofhomopolymer melt of chains identical to theA block in Fig.3 in chain length and in polymer–polymer, polymersurfactant, and polymer–slit wall interactions at the savalues of temperature and pressure employed here. Inhomopolymer case the slit spacing increases fromd'3 tod'4 in two approximately equal steps, the first att,1000and the second att'7000, which we have associated withe formation of layers of polymer beads within the slit.20 Bycontrast, Fig. 3 shows an increase in slit spacing of comrable magnitude that takes place in a single step, and wis complete byt'500. This difference in kinetics betweethe diblock and homopolymer cases reflects differencesthe structures formed within the slit in the two cases.
In Fig. 4, the calculation of Fig. 3 is repeated with idetical parameter values, but with the ordering of the bulamellae shifted so that theB blocks are initially located nexto the slit entrances in the reservoirs. As in Fig. 3, boblocks intercalate, with the block initially adjacent to the sintercalating first, although in the case of Fig. 4, the blothat is present at highest concentration at the latest time s
FIG. 3. Intercalation from a lamellar bulk phase is depicted throughtime-dependent increase in slit spacingd in the left-hand panel, and througthe number of intercalated beads of typesA andB in the right-hand panel.The polymer–polymer, polymer–surfactant, surfactant–surfactant,surfactant–surface interactions are purely repulsive, and only the polymsurface interactions contain an attractive well. Here,T51, eAB55, eAl53,eBl51, N524. A homopolymer melt of pureA with N512 would interca-late spontaneously, while a melt of pureB with N512 would not intercalateon the simulation time scale. The bulk melt in each reservoir is configuwith A blocks in contact with the slit openings.
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9116 J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Lee et al.
ied is that with the less favorable surface interactions. Fig4 suggests that the repulsive interblock interaction is msignificant than the difference in surface affinities in drivithis intercalation process.~That notion is confirmed below inFig. 7.! The results of Fig. 4 show that the intercalation ospecies as a block that would not spontaneously intercaas a homopolymer is not limited to the especially advangeous initial conditions of Fig. 3.
The structure of the intercalated polymer for the procdepicted in Fig. 3 is shown in Fig. 5. This figure showssnapshot of the locations of the intercalated beadst52500, which is the latest time shown in Fig. 3. The upppanel shows the locations of beads adsorbed to one of thwalls, and the lower panel shows the configuration of beadsorbed to the other slit wall. We define adsorbed beadhave centers that are withinsp (1.25spl) of the plane ofcenters of the particles forming a slit surface. The midpanel shows the locations of all beads that are not adsoto a slit surface by this definition. Beads of typeA are rep-resented by filled squares and those of typeB are shown by
FIG. 4. The calculation of Fig. 3 is repeated with the only difference thatordering of the lamellae in the reservoirs is shifted so thatB blocks ratherthanA blocks are in contact with the slit openings. Even though theB blocksare less strongly attracted to the slit surfaces than are theA blocks, andwould not spontaneously intercalate as a homopolymer, there is spontanflow of both blocks into the slit.
FIG. 5. The structure of intercalated polymer is shown for the procesFig. 3. Beads adsorbed to one slit wall are shown in the upper panel,beads adsorbed to the other wall in the lower panel. Beads not adsorbeither wall are shown in the middle panel. Beads of typeA are representedas filled squares and those of typeB are represented by open circles. Sufactants are not shown, resulting in the apparent void spaces. 37% ofBbeads are not adsorbed, compared with 4% of theA beads.
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open circles. For clarity, the surfactant molecules areshown, resulting in the apparent voids in Fig. 5. AlthoughAbeads have stronger attractive interactions with the surfathan B beads, both species can adsorb to a surface. Mbeads in this intercalated phase are adsorbed to a slit wand hence are shown in the upper and lower panels. Hever, 96% of theA beads are adsorbed, while only 63%theB beads are adsorbed. The configuration shown in Figdisplays phase separation both in thexy plane and in thezdimension. The structures that result from intercalationthese simulations are not, in general, equilibrium structuof a thin film. However, it should be noted that phase seration in the plane of the confining surfaces has beenserved for thicker confined films of diblock copolymers boin the laboratory29 and in simulation.30–32
In Fig. 6, the interaction between theA block and the slitwalls has the same form as in Fig. 3, andeBl51 as in Fig. 3,but here the LJ interaction between wall particles andBbeads has been truncated at its minimum to produce a purepulsive interaction. In this case, the slit opens slightlyaccommodateA beads, but noB beads intercalate on thsimulation time scale, producing a structure in whichAblocks are intercalated with the accompanyingB blocks re-maining in the reservoirs. In this case, the interactiontweenB and the surfaces is sufficiently unfavorable thattercalation is not induced by theA block.
In the scenario of Fig. 3, one block would spontaneouintercalate as a homopolymer and the other would not. Fure 7 treats a case in which neither block would intercalspontaneously as a homopolymer:eAl5eBl51,eAB55 at T51. As in Figs. 3–5, the polymer-surface interactions cotain an attractive well, while all polymer–polymer interations are purely repulsive. In this situation, the diblock intecalates rapidly. Given the unfavorable interaction betwedifferent blocks, the diblock apparently decreases its fenergy by exchanging the weakly repulsive intrablopolymer–polymer interactions for the weakly attractipolymer-surface interactions.
Comparison of Figs. 3 and 4 shows that for a bulk lamlar phase in the reservoirs, the identity of the block adjacto the slit opening does not determine the spontaneity
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FIG. 6. The calculation of Fig. 3 is repeated, but with the weakly attractinteraction of blockB with the slit walls replaced by a purely repulsive onby truncating the LJ potential at its minimum. As in Fig. 3,eAB55, eAl
53, eBl51. In this case, theA block would intercalate spontaneously ashomopolymer, while theB block would not. In contrast to the process oFig. 3, theBl interaction is sufficiently unfavorable that theB beads do notintercalate. The right-hand panel shows that whileA beads near the slitopening enter the slit, theB blocks remain in the reservoirs.
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9117J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Intercalation of diblock copolymers
intercalation. This finding suggests that intercalationdiblocks should be possible from a disordered bulk stateall of the parameters of Fig. 3 are retained except thatinterblock repulsion is decreased fromeAB55 to eAB52, thereservoirs contain bulk polymer in a disordered phase. Incalation from this phase is shown in Fig. 8, with the left-hapanel showing the time dependence of the slit widthd, andthe right-hand panel illustrating the time dependence ofnumbers ofA andB beads in the slit. The slit widthd has at50 value of less than 3.0, as shown in Figs. 3, 4, 6, anThe time dependence ofd shows an initial rapid increasefollowed by a slower rise. The time scale in Fig. 8 was chsen to depict the slower expansion, so the rapid initial risnot visible in the plot ofd. The associated rapid entry opolymer beads of both species is visible in the right-hapanel. As for the case of a lamellar bulk phase, beads of bspecies intercalate, although theB block would not sponta-neously intercalate as a homopolymer. This phenomenonot limited to a microphase separated initial condition. Coparison of Fig. 8 to Figs. 3 and 4 shows that the time scalintercalation is longer for Fig. 8, and that the slit has nopened as much at its plateau value. Since the interbrepulsion has been reduced in Fig. 8 relative to the othercalculations, this comparison supports the interpretationfor this case, interblock repulsion drives the intercalatprocess. The structure of the intercalated material in
FIG. 7. Intercalation dynamics of symmetric diblocks from a bulk lamephase are shown foreAB55, eAl51, and eBl51. All polymer–polymerinteractions are purely repulsive, while polymer–silicate interactions inclan attractive well. In this case, neither block would intercalate spontaously as a homopolymer in this thermodynamic state, yet the diblock flspontaneously into the slit.
FIG. 8. Intercalation dynamics from a disordered bulk phase are shown.time dependence of the slit spacing is shown in the left-hand panel, andof the numbers of intercalated beads in the right-hand panel. The paramof Fig. 3 are retained, but with the interblock repulsion decreased feAB55 to eAB52. As in Fig. 3, theA block would intercalate as a homopolymer, but theB block would not. Here, both blocks intercalate, shoing that an ordered phase in the bulk is not required for one block tomote the intercalation of another.
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is-oftckoatne
simulation of Fig. 8, which is not depicted here, shows phseparation similar to that in Fig. 5, although the bulk phasdisordered.
The interpretation that interblock repulsion is more snificant than differing polymer–surface affinities in drivinthe intercalation processes discussed here is supporteanother simulation, not shown here, in which the parameof Fig. 8 are retained, except for the interblock repulsiowhich is further reduced toeAB51. In this limit, the inter-block repulsion is identical to the intrablock repulsion, athe polymers in the bulk behave as homopolymers. The bphase is necessarily disordered. The two blocks differ onltheir interactions with the slit surfaces:eAl53, eBl51. Thismodel represents the inverse of the situation examinedFig. 7, in which the two blocks have identical surface affinties, but a strong interblock repulsion. In this case, a simution of durationt5104, 25% longer than that shown in Fig8, produces a structure in whichA beads are intercalated tosimilar extent as in Fig. 8, but theB blocks remain outside othe slit. The number ofB beads in the slit fluctuates aboutvalue less than 10. In this case, the differential surface afity of the two blocks, in the absence of interblock repulsiodoes not suffice to induce intercalation of theB species onthe simulation time scale.
The calculations presented here suggest that amphipintercalants may be used to create nanostructures fromterials that as homopolymers are not compatible withinorganic phase. To directly transfer the results of this stuto the laboratory, a method to map laboratory diblock cpolymers onto the coarse-grained model used here woulneeded. Kremer and Grest have proposed for the homopmer version of this model a mapping of laboratory polymeonto model polymers, which is based on the identificationthe entanglement molecular weight of the model melt.33 Thismapping could be applied separately to the blocks inmodel used here. There are two potential difficulties in aplying that procedure to the present model. First, the validof the procedure has not been tested for polymers confion the small length scales considered here, in which the bentanglement molecular weight may not be relevant. Secothe generalization of this procedure to block copolymersdetermine the magnitude of the energy scale of interblinteractions,eAB , is not obvious. We also note that comptational constraints required the use of rather short blockN512. Since the activation energy for anA block to cross aB microdomain increases with increasing block length,38 thefree energy barrier for such a rearrangement may be lowethe present calculations than in a typical laboratory situatiIn the absence of a direct procedure for evaluating moparameters for laboratory polymers, we must regard oursults as illustrations of qualitative trends. Further measuments of the dynamics of confined block copolymers mprovide the insight to construct such a procedure.
Future theoretical investigations of these phenommust focus on two areas. First, a combination of theory aMonte Carlo simulation31,32 must be used to determine ththermodynamically stable structures of these systems, sthe present dynamics simulations need not produce structcharacteristic of confined films at thermodynamic equil
ee-s
heaters
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9118 J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Lee et al.
rium. Second, dynamic simulations must focus on asymmric diblock copolymers to determine the block length rathat is required to induce spontaneous intercalation odiblock for which one or both blocks would not intercalaspontaneously as homopolymers.
ACKNOWLEDGMENTS
We acknowledge support from the National ScienFoundation, Division of Materials Research, throughCornell Center for Materials Research. We thank ProfesE. P. Giannelis for discussions of unpublished data.
APPENDIX
We describe here the procedure by which we allowthe x and z dimensions of the simulation cell to fluctuaduring equilibration of the bulk lamellar phases in the resvoirs. This issue was addressed for this polymer modelGrestet al.34,35 who employed a modification of Andersenisobaric molecular dynamics algorithm.37 In the originalAndersen approach, the system volume is treated as anamical variable and is allowed to fluctuate isotropicalGrestet al. modified this strategy to permit thex, y, and zdimensions of the cell to fluctuate independently, whmaintaining periodic boundary conditions.34,35 In the presentstudy, periodic boundary conditions are not applied in thxdimension, which is bounded by physical walls maintainedconstant external pressure. Periodic boundary conditionsapplied in they dimension, which is fixed at the magnitudof they dimension of the silicate surfaces. To permit dimesional fluctuations in thez direction, we implement theAndersen algorithm in this dimension only. Thus, fluctutions inx arise from interactions with a physical wall at fixeexternal pressure, while those inz arise from the fictitiousinternal piston of the Andersen method. The derivation hfollows the development of Andersen,37 but with the z di-mension of the cell,l z , treated as a dynamical variable, athe remaining dimensions,l x and l y, treated as parameters
The Cartesian coordinates of beadj, (xj ,yj ,zj ) are ex-pressed in variables scaled by the cell dimensions:37
qjx5xj
l x, ~A1!
qjy5yj
l y, ~A2!
qjz5zj
l z. ~A3!
For constantl x and l y and time-varyingl z , the AndersenLagrangian is
L~q,q,l z , l z!5mp
2 S l x2(
jq jx
2 1 l y2(
jq jy
2 1 l z2(
jq jz
2 D2(
i , jf i j 1
1
2M l z
2l x2l y
22plxl yl z . ~A4!
The bead mass ismp , f i j is the potential energy of interaction between beadsi and j, M is the ‘‘mass’’ @dimensions of
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t-
a
eeor
d
-y
y-.
tre
-
-
e
mass~length!24# of the fictitious piston, andp is the desiredpressure. We have adopted the valueMl x
2l y2510mp .34,35The
momenta conjugate toq and l z are
P5]L
] l z
5M l zl x2l y
2 , ~A5!
pnx5mpl x2qnx , ~A6!
pny5mpl y2qny , ~A7!
pnz5mpl z2qnz . ~A8!
The corresponding Hamiltonian is
H51
2mp(
nS pnx
2
l x2
1pny
2
l y2
1pnz
2
l z2 D
1P2
2Ml x2l y
21(
i , jf i j 1plxl yl z . ~A9!
The Hamiltonian equations of motion are
qna5]H
]pna5
pna
mpl a2
, ~A10!
pna52]H
]qna5 l aFna , ~A11!
l z5]H
]P5
P
Ml x2l y
2, ~A12!
P52]H
] l z5
1
mpl z3 (
npnz
2 2plxl y2(i , j
]f i j
] l z, ~A13!
where Fna is the a component of the force on particlen.Equation~A12! is differentiated with respect to time to obtain the equation of motion forl z
M l zl x2l y
25mpl z(n
qnz2 2plxl y2(
i , j
]f i j
] l z. ~A14!
The equations of motion for each particle in Cartescoordinates can be obtained through the correspondence37
xj~ t !5 l xqjx~ t !, ~A15!
yj~ t !5 l yqjy~ t !, ~A16!
zj~ t !5 l zqjz~ t !, ~A17!
pjx5p jx / l x , ~A18!
pjy5p jy / l y , ~A19!
pjz5p jz / l z . ~A20!
Equations~A15!, ~A16!, and~A17! are differentiated to yield
x j5F jx
mp, ~A21!
y j5F jy
mp, ~A22!
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f-
e
edneu
o-
-
. P.
ol-
E.
-
i-
.
m.
W.
J.
Rev.
.
ell,
P.
hys.
9119J. Chem. Phys., Vol. 112, No. 20, 22 May 2000 Intercalation of diblock copolymers
zj5F jz
mp1
l z
l zzj . ~A23!
As expected, thex andy coordinates obey the usual form oNewton’s second law, whilez is coupled to the cell dimension. The reduced variableqnz in Eq. ~A14! is replaced byunscaled variables to get the equation of motion forl z
M l z5Fmp
l z(
nS zn2
l z
l zznD 2
2(i , j
1
l z
]f
]zi jzi j 2plxl yG Y l x
2l y2 . ~A24!
In our simulations,l y is fixed, but l x fluctuates through in-teractions with a physical wall under constant external prsure. Thus, the instantaneous value ofl x at t is used in Eq.~A24!.
The velocity form of the Verlet algorithm cannot be usto integrate these equations of motion as the acceleratiothe box dimension in Eq.~A24! has an explicit dependencon the velocityl z . These equations can be integrated as sgested by Fox and Andersen.39 For simplicity of notation,consider a single degree of freedomz coupled to the celldimensionl z according to:
z~ t !5 f @z~ t !, z~ t !, l z~ t !, l z~ t !#, ~A25!
l z~ t !5g@z~ t !, z~ t !, l z~ t !, l z~ t !#. ~A26!
The coordinates at timet1dt are, to orderdt2,
z~ t1dt !5z~ t !1dt z~ t !1 12 dt2f @z~ t !, z~ t !, l z~ t !, l z~ t !#
~A27!
and
l z~ t1dt !5 l z~ t !1dt l z~ t !
1 12 dt2g@z~ t !, z~ t !, l z~ t !, l z~ t !#. ~A28!
The velocities att1dt are then approximated using the cordinates att1dt and the velocities att
zapp~ t1dt !5 z~ t !1 12 dt@ f ~z~ t !, z~ t !, l z~ t !, l z~ t !!
1 f ~z~ t1dt !, z~ t !, l z~ t1dt !, l z~ t !!#,
~A29!
l zapp~ t1dt !5 l z~ t !1 12 dt@g~z~ t !, z~ t !, l z~ t !, l z~ t !!
1g~z~ t1dt !, z~ t !, l z~ t1dt !, l z~ t !!#.
~A30!
Equations~A29! and ~A30! are then further applied to produce the working approximations to the velocities att1dt:
z~ t1dt !5 z~ t !1 12 @ f ~z~ t !, z~ t !, l z~ t !, l z~ t !!
1 f ~z~ t1dt !, zapp~ t1dt !,
l z~ t1dt !, l zapp~ t1dt !!#, ~A31!
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s-
of
g-
l z~ t1dt !5 l z~ t !1 12 @g~z~ t !, z~ t !, l z~ t !, l z~ t !!
1g~z~ t1dt !, zapp~ t1dt !,
l z~ t1dt !, l zapp~ t1dt !!#. ~A32!
These relations are then substituted into Eqs.~A27! and~A28! to connect the forces at a given timet1dt to coordi-nates at that time and to velocities at an earlier time,t.
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