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J. Phys. Chem. B 2007, 111, 11756-11764
Self-Organization Process of Ordered Structures in Linear and Star Poly(styrene)-Poly(isoprene) Block Copolymers: Gaussian Models and Mesoscopic Parameters of Polymeric Systems Ce´ sar Soto-Figueroa,*,†,‡ Luis Vicente,‡ Jose´ -Manuel Martı´nez-Magada´ n,†,§ and Marı´a-del-Rosario Rodrı´guez-Hidalgo† Departamento de Ciencias Quı´micas, Facultad de Estudios Superiores Cuautitla´ n, UniVersidad Nacional Auto´ noma de Me´ xico AV. 1° de Mayo s/n Campo 1. Cuautitla´ n Izcallı´, 54740 Estado de Me´ xico, Me´ xico, Departamento de Fı´sica y Quı´mica Teo´ rica, Facultad de Quı´mica, UniVersidad Nacional Auto´ noma de Me´ xico, Me´ xico 04510, D.F. Me´ xico, and Programa de Ingenierı´a Molecular, Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, Me´ xico 07730, D.F. Me´ xico ReceiVed: May 28, 2007; In Final Form: July 26, 2007
Mesoscopic simulations of linear and 3-arm star poly(styrene)-poly(isoprene) block copolymers was performed using a representation of the polymeric molecular structures by means of Gaussian models. The systems were represented by a group of spherical beads connected by harmonic springs; each bead corresponds to a segment of the block chain. The quantitative estimation for the bead-bead interaction of each system was calculated using a Flory-Huggins modified thermodynamical model. The Gaussian models together with dissipative particle dynamics (DPD) were employed to explore the self-organization process of ordered structures in these polymeric systems. These mesoscopic simulations for linear and 3-arm star block copolymers predict microphase separation, order-disorder transition, and self-assembly of the ordered structures with specific morphologies such as body-centered-cubic (BCC), hexagonal packed cylinders (HPC), hexagonal perforated layers (HPL), alternating lamellar (LAM), and ordered bicontinuous double diamond (OBDD) phases. The agreement between our simulations and experimental results validate the Gaussian chain models and mesoscopic parameters used for these polymers and allow describing complex macromolecular structures of soft condensed matter with large molecular weight at the statistical segment level.
1. Introduction Nowadays, mesoscale models such as freely jointed segment, general random flight, and Gaussian chain models are receiving great attention because they can characterize many different types of systems, including liquids, solids, surfaces, and clusters. The combination of these statistical models together with dynamics simulation techniques has made possible the routine study of the microscopic details of chemical processes in the condensed phase where macroscopic properties can be predicted.1,2 In the polymers research field, accurate statistical models for the prediction of material properties are crucial for the improvement of existing polymeric materials, the design of novel materials with tailored-made properties for given applications, the optimization of production processes, and finally, the reduction of production costs.3 Flexible polymers built from a variety of thermoplastic elastomers or thermoplastic olefins can take up an enormous number of configurations by rotations of chemicals bonds, and the shape of the polymers can therefore only be usefully described statistically. The polymers are designed to have length scales ranging from tens to thousands of nanometers. Such structures consist of thousands of atoms, and an atomic level description of its dynamics becomes prohibitively expensive. One way to circumvent this problem is to coarse-grain a group of atoms into a single bead and to * Corresponding author. † Departamento de Ciencias Quı´micas. ‡ Departamento de Fı´sica y Quı´mica Teo ´ rica. § Instituto Mexicano del Petro ´ leo.
replace the molecular interactions by simpler bead-bead interactions. In practice, the theoretical study of neat polymers, polymer blends, and polymers in dilute solutions has been based upon coarse-grained mesoscale models, which are used instead of fully atomistic representations to extend the time and length scales accessible to simulation within reasonable computing times.4 In this statistical representation, each polymeric molecule is composed of N segments that conform a mechanical system constituent by spherical beads connected by harmonic springs.5 In these models, it is necessary to establish the relation between the real molecular structure and the mesoscopic parameters; this feature is key for the description of the physical behavior into the coarse-grained or the mesoscale models and where the large scales play an important role. Crawshaw, J. and Windle, A. H.6 have establish that the measurable properties of polymers depend on a structural hierarchy which ranges from chemical details to single chains to aggregates of chains, up to continuum phenomena, and therefore, to a complete description of a polymer requiring a model encompassing a wide range of length and time scales. The mesoscale models describe macromolecules at the statistical segment level through parameters such as persistence length, characteristic ratio, mean squared end-toend distance, and Kuhn statistical segment length, where the details of molecular structures are ignored. At this point, it is imperative to emphasize that the mesoscale models can incorporate some degrees of freedom of the conformational structure or molecular symmetry, introducing important information of
10.1021/jp074122q CCC: $37.00 © 2007 American Chemical Society Published on Web 09/15/2007
Star Poly(styrene)-Poly(isoprene) Block Copolymers the relationship between microscopic structure and macroscopic properties. Nevertheless, the actual mesoscale model only requires some parameters such as, for example, compressibility and chain length, as well as corresponding effective parameters of monomer-monomer and chain-chain interaction and in some process of dilute solutions between polymers and solvent, the chain-solvent interaction. The application of mesoscale models in the nanotechnological field is feasible; theoretical studies of block copolymers7-17 apply these statistical models to describe the self-assemble processes of nanomaterials exhibiting fascinating periodically ordered microphases, for instance, body-centered-cubic (BCC), hexagonal packed cylinders (HPC), hexagonal perforated layers (HPL), lamellar (LAM), and the ordered bicontinuous double diamond (OBDD) phases via the microphase separation process.18 Srinivas et al.15,57 establish a coarse-grain model to represent the PEO-PEE diblock copolymer structure in watery solutions, which are based on atomistic structures of the corresponding diblock copolymer. Nevertheless, unfortunately, majority of theoretical studies that involve mesoscale models do not take into account explicitly the real molecular structure and the mesoscopic parameters that describe the degrees of freedom; instead, they are arbitrarily adjusted to the experimental outcome, generating in this way some uncertainty in the mesoscale models and in the employed mesoscopic parameters. Xu et al.19 reported the microphase separation of star diblock copolymer melts; they employed a Gaussian chain model to describe two types of star diblock copolymers ((A)4(B)4 and (AB)4). Star copolymers are constructed by joining n identical linear diblock copolymers. In these models, the molecular structure and the employed statistical segment level of each system is not mentioned, and the mesoscopic parameters that characterize the interaction between A an B blocks are chosen arbitrarily. Lately, Fraaije and co-workers.20 apply a coarsegrained Gaussian model of type E3P9E3 to describe the linear block copolymer (ethylene oxide)13(propylene oxide)30 (ethylene oxide)13; the parametrization of this model and mesoscopic parameters were obtained from experimental outcomes in an empirical fashion. However, they establish that the choice of the Gaussian chain parameters is an important aspect to simulate the phase separation dynamics and describe this block copolymer system. It is the purpose of this work to describe a methodology for mapping copolymeric systems with large chain length and different molecular architecture into Gaussian chain models under a statistical segment level and evaluate representative mesoscopic parameters to each specific system. The linear and 3-arm star poly(styrene)-poly(isoprene) block copolymers have been selected in this work because of the academic and industrial significance that they represent. Linear-shape and star-shape block copolymers possess the ability to exhibit different morphologies in polymer solutions, in the bulk state as well as in thin layers at surfaces and interfaces because of their complex architecture. This unique property allows them to become potential materials for many modern applications, for example, nanocarriers for drug delivery, additives for multigrade oil, membranes, and fabrications of smart polymers films. In this way, the molecular structures of poly(styrene)-poly(isoprene) block copolymers with linear and star architecture and their mesoscopic parameters together with dissipative particle dynamics (DPD) are employed to explore the microphase separation process, order-disorder transition, and ordered structures formation via self-organization processes.
J. Phys. Chem. B, Vol. 111, No. 40, 2007 11757 This paper is organized as follows. In section 2, we briefly present the material models and mesoscale simulation methods employed in this work. Section 3 contains a summary of our main results and discussions of the Gaussian chain models, mesoscopic parameters, and predictions obtained for the poly(styrene)-poly(isoprene) block copolymers. This section is divided into two parts; in section 3.1, the molecular structures of the poly(styrene)-poly(isoprene) diblock copolymers with linear and star architecture were mapped into Gaussian chain models in the characteristic ratio statistical segment level. The main mesoscopic parameters that display the bead-bead interaction in these statistical models are obtained from styreneisoprene molecular interaction via Monte Carlo simulations, and they are advantageous to particularly represent the poly(styrene)-poly(isoprene) block copolymer systems into DPD simulations. In the second part, section 3.2, we put on view the detailed results and discussion of the mesoscopic simulations of linear and 3-arm star block copolymers, where we analyze the ordered structures evolution via the self-organization process. In this section, we compared our results with some interesting experimental results previously reported by Khandpur et al.21 In the final section (4), our conclusions are offered. 2. Material Models and Mesoscale Simulation Methods 2.1. Material Models. The molecular structure of poly(styrene)-poly(isoprene) diblock copolymer was built using the Polymer Builder Module.22 The molecule contains a total of 200 repetitive units in the diblock chain, and its linear architecture is characteristic for this system; see Figure 1a. The molecular weight of the linear diblock copolymer presents an interval of 13985-20466. The 3-arm star block copolymer has three poly(styrene)-poly(isoprene) diblock chains linked through a central molecule; see Figure 1b. The star copolymer contains 200 repetitive units per diblock chain. The molecular weight of the 3-arm star block copolymer has an interval of 4196561386. The conformational properties of linear and 3-arm star copolymers were calculated using the RIS Metropolis Monte Carlo (RMMC) and COMPASS (condensed-phase optimized molecular potentials for atomistic simulation studies) force field.23,24 All single and partial double bonds in the block chains of these copolymers are allowed to rotate during the molecular simulation except those involving bonds of the rings of the polystyrene fragment. The block copolymers can take up an enormous number of configurations by the rotation of chemical bonds. The shape of these macromolecules was described by means of the Gaussian chain models constituted by beads connected across harmonic springs,25 where each spherical bead represents a segment with a statistical distribution of the block copolymer. In this statistical model, block copolymers are represented by chains consisting of N statistical segments, each one with an effective bond length b. The conformation of the Gaussian chain is symbolize by the set of (N + 1) position vectors {Rn} ≡ (R0...RN) of the joints or alternatively by the set of bond vectors {rn} ≡ (r0...rN) where
rn ) Rn - Rn-1
n ) 1, 2, ..., N
(1)
Since the bond vectors rn are independent of each other, consider a chain whose effective bond length has a Gaussian distribution function of type
ψ(r) )
[ ] ( ) 3 2πb2
3/2
exp -
3r2 2b2
(2)
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Soto-Figueroa et al.
Figure 1. Schematic representations of bead-spring chain models of the poly(styrene)-poly(isoprene) block copolymers (a) linear block copolymer and (b) 3-arm star block copolymer. These mechanical models consider three experimentally controllable factors: (i) the overall degree of polymerization N, (ii) architectural constrains, and (iii) diblock chain composition.
where ψ(r) denotes the random distribution of a vector of constant effective bond length b so that
〈r2〉 ) b2
(3)
The conformational distribution function of Gaussian chain can be expressed as N
Ψ({rn}) )
∏ n)1
[ ] [ ] [ ] [∑ 3
3rn2
3/2
exp -
2πb2
)
2b2
3N/2
3
3(Rn - Rn-1)2
n)1
2b2
exp -
2πb2
]
N
(4)
N
3
kbT
2
2b
(Rn - Rn-1)2 ∑ n)1
(5)
where U0 is the potential energy and kb is the Boltzmann constant. An essential property of the Gaussian chain is that the distribution of vectors Rn - Rm between any two segments n and m is Gaussian type, being given by
Φ(Rn - Rm,n - m) )
[
3 2πb2|n - m|
] [ 3/2
exp -
]
3(Rn - Rm)2 2|n - m|b2
Cn )
〈R2〉0 Nl2
Lp ) (1/li)
(6)
In order to apply the Gaussian chain model to realistic block copolymers, a number of structural and conformational parameters such as polymer molecular weight, persistence length,
∑j
〈li.lj〉
aK )
〈R2〉 Rmax
(7)
where Cn is characteristic ratio, Lp is persistence length, and aK is statistical Kuhn segment length. These conformational parameters describe different statistical segment levels, and they are correlated with the statistical segment Gaussian distributions of eqs 4 and 6. The conformational parameters were obtained directly by Monte Carlo simulations. These statistical parameters go together with the following expression5
CSGD )
The Gaussian chain therefore is depicted by a mechanical model constituent by spherical beads coupled by harmonic springs whose potential energy is given by
U0({Rn}) )
characteristic ratio, mean squared end-to-end distance, and statistical Kuhn segment length, among others, can be used.
Mp Mm(SSL)
(8)
where CSGD are bead numbers (statistical segments) with Gaussian distribution, Mp is the molar mass of the polymer, Mm is the molar mass of a repeat unit, and SSL means statistical segment level (Cn, Lp , aK, etc.). In previous papers,26,27 we show that the characteristic ratio gives reliable results to describe the poly(styrene)-poly(isoprene) systems so we employ this statistical segment level for mapping the molecular structures of linear and more complex star block copolymers into Gaussian chain models. The Cn, as compared with Lp and aK, displays a less number of beads in the Gaussian chain representation. From a practical numerical point of view, it is desirable to use fewer beads in the linear and 3-arm star block copolymer description.27 The quantitative estimate of the bead-bead interaction was calculated from the monomer-monomer interaction using the statistical thermodynamics model of the Flory-Huggins theory.28,29 In this theory, the miscibility is governed by mixing Gibbs free energy
∆G )
φ2 φ1 ln φ1 + φ2 + χφ1φ2 X1 X2
(9)
Star Poly(styrene)-Poly(isoprene) Block Copolymers
J. Phys. Chem. B, Vol. 111, No. 40, 2007 11759
where ∆G denotes the Gibbs free energy of mixing per mole, φ1 and φ2 are volume fractions for components 1 and 2, X1 and X2 are the degree of polymerization, and χ is the Flory-Huggins interaction parameter defined as
χ)
Z∆E12 RT
(10)
where Z is the coordination number of model lattice, and E12 is the energy of interaction of unlike pair
1 1 ∆E12 ) (E12 + E21) - (E11 + E22) 2 2
(11)
In this model, each repeating unit is defined as occupying a single lattice site. The parameters ∆G, Z, ∆E12, and χ were evaluated by means of molecular simulations using the FloryHuggins modified model by Fan et al.30 In this extension of the Flory-Huggins model, the molecules are not arranged on a planar lattice, as in the original Flory-Huggins theory; they are arranged off-lattice (three-dimensional). The coordination number Z is explicitly calculated for each of the possible molecular pairs using Monte Carlo molecular simulations. This numerical procedure involves generated clusters in which nearest neighbors are packed around the central molecule. Average calculated Z values were employed in the temperature dependence expression of the interaction parameter. The ∆G(T) is obtained from the configurationally interaction energies and coordination numbers as follows
∆G(T) )
[Z12E12(T) + Z21E21(T) - Z11E11(T) + Z22E22(T)] 2 (12)
The temperature dependence of the interaction parameters, χ(T), is calculated as a function of ∆G(T) and Z as
∆G(T) RT
(13)
(Z12E12 + Z21E21 - Z11E11 + Z22E22) 2RT
(14)
χ(T) ) χ(T) )
This methodology has been successful employed in calculating the properties of different macromolecules.31 2.2. Mesoscale Method: Dissipative Particle Dynamics. The microphase separation behavior and ordered structures generation of linear and 3-arm star block copolymers in the melt via a self-organization process have been studied by means of a mesoscopic method known as DPD. Because DPD methodology has been amply exhibited in many works, below, we give only a short summary. The DPD method, introduced by Hoogerbrugge and Koelman,32,33 is a mesoscale simulation technique that combines some of the detailed description of molecular dynamics (MD) but allows the simulation of hydrodynamic behavior in much larger and more complex systems constituted by particles. Espan˜ol and Warren34-36 have identified the link between the DPD algorithm and an underlying stochastic differential equation for particle motion, thereby establishing DPD as a valid method for the simulation of the dynamics of mesoscopic particles. Groot et al.37-39 have related the DPD method with the solutions of the Flory-Huggins theory, thus, allowing one to study large molecular weight systems under industrial operation conditions. Furthermore, they have demonstrated that this approach is consistent with the mean field theory. In a DPD simulation, a particle having mass mi
represents a small segment of copolymer chains moving together in a coherent manner. These DPD particles are subject to soft potentials and governed by predefined collision rules. Like MD, the DPD particles obey Newton’s equation of motion In the poly(styrene)-poly(isoprene) block copolymers, there are two different species of DPD particles, poly(styrene) and poly(isoprene). Each particle is subject to soft interactions with its neighbors via three forces: conservatives (FCij ), dissipative (FDij ), and random forces (FRij ); all the forces between particles i and j vanish beyond some cutoff radius rc and are given by:
FCij )
[
aij(1 - rij)rˆ ij (if |rij| < 1) (if |rij| g 1) 0
(15)
FDij ) -γωD(rij)(rˆ ij.vij)rˆ ij
(16)
FRij ) σωR(rij)ζ∆t-1/2rˆ ij
(17)
where rij ) ri - rj, rij ) |rij|, rˆ ij ) rij/|rij|, γ is the dissipation strength, σ is the noise strength, ωD and ωR are r-dependent weight functions, vij ) vi - vj, aij is a maximum repulsion strength between particles i and j and ζij is a Gaussian noise term with the following properties: ζij ) ζji, 〈ζij(t)〉 ) 0,〈ζij(t)ζkl(t′)〉 ) (δikδjl + δilδjk)δ(t - t′). The parameter aij, henceforth referred to as bead-bead repulsion parameters or simply as DPD interaction parameter, depends on the underlying atomistic interactions and is related to the parameter χ of section 2.1 through
aij ≈ Rij + 3.21χij
for
F)3
(18)
In this way, we connect the molecular character of the styreneisoprene interaction with the DPD system. The dissipative and random forces have two effects: they act as a thermostat (an alternative DPD thermostat has been proposed by Lowe40), and they allow transport properties such as the viscosity to be tuned without altering the equilibrium thermodynamics. All the forces are additive pairwise, central, and satisfy Newton’s third law thus conserving linear and angular momentum. The forces depend only on relative positions and velocities, making the Galilean model-invariant. All forces between particles i and j vanish beyond some cutoff radius (rc). In the dynamical simulation of linear and 3-arm star block copolymers, the behavior of the system is followed by integration of the equations of motion using a modified version of the Verlet algorithm,41,42 where each particle is defined by its position ri and momentum pi, which are calculated at each time step. The relative magnitudes of the three forces evolve to a steady state that keeps up a correspondence to the Gibbs canonical ensemble, NVT. Integration of the equations of motion for the poly(styrene)-poly(isoprene) block copolymer with linear and 3-arm star architectures generates a trajectory through the system’s phase from which thermodynamic observables may be constructed from a suitable average. From this information, the microphase separation process and ordered structures generation can be observed. For a more thorough account on DPD, see, for example, refs. 32-39. 3. Simulation Results and Discussion 3.1. Gaussian Chain Model Development of Linear and 3-Arm Star Block Copolymers. Just as was mentioned above, the linear and star block copolymer structures were subjected to a study of conformational properties by means of Monte Carlo simulation. Table 1 summarizes the conformational properties
11760 J. Phys. Chem. B, Vol. 111, No. 40, 2007
Soto-Figueroa et al.
TABLE 1: Structural and Conformational Parameters of Poly(styrene) and Poly(isoprene) Block Chains structural and conformational parametersa
poly(styrene) block (PS)
poly(isoprene) block (PI)
molar mass molar mass per repeat unit no. rotatable bonds in system mean squared end-to-end distance 〈r2〉 characteristic ratio 〈Cn〉 mean squared radius of gyration 〈S2〉 ratio 〈r2〉/〈s2〉 persistence length, projection on first bond 〈a1〉 persistence length, mean projection on all bonds 〈a2〉
10417 104.17 299 4917.8 ( 437 Å2 10.49 ( 0.49 825.85 ( 6.24 Å2 6.23 4.72 ( 0.66 Å 9.36 ( 0.16 Å
6814 68.139 399 7198.1 ( 528 Å2 9.03 ( 0.37 1243.6 ( 10.1 Å2 6.099 10.35 ( 0.82 Å 7.37 ( 0.13 Å
a Conformational parameters of poly(styrene) and poly(isoprene) chains were obtained by averaging a large number of generated configurations, typically 750 000; the COMPASS force field was used in the RIS Metropolis Monte Carlo simulations.
of the poly(isoprene) and poly(styrene) block chains. To develop the characteristic parameters of the Gaussian chain models, we need target observables, and these are taken from specific conformational properties of the block copolymer chains, from molecular structures and from all diblock chain molecular simulations. The number of beads in each statistical model was determined using the molar mass of each block copolymer, the molar mass of a repeat unit, the degree of polymerization, and the (Cn) characteristic ratio of each polymeric system. The linear diblock copolymer is represented by a mechanical model constituted by 20 spherical beads denoted by [PS]n - [PI]m, whereas the 3-arm star block copolymer is expressed by a model of 60 beads, [PSn - PIm]3 , as sketched schematically in Figure 1a,b, respectively. As shown in this figure, each block copolymer model is constructed by identical lines of (PSn - PIm) diblock fragments. In this way, the linear diblock copolymer is embodied by one linear chain, while the star block copolymer is constructed by three identical linear (PSn - PIm) diblock chains, freely jointed together through a central particle C which is considered to be similar to either of the blocks. The Gaussian chain models developed for the linear and 3-arm star block copolymers describe the real molecular structures at the characteristic ratio statistical segment level where the atomistic details are ignored. The [PS]n - [PI]m and [PSn - PIm]3 block copolymers are modeled according to sketches confirmed in Figure 1, for the subsequent dynamical mesoscopic simulations. The bead-bead interaction parameter between different segments in these Gaussian statistical models is given by the magnitude of repulsion between different repetitive units. In this work, we evaluated the styrene-isoprene molecular interactions by the use of a combination of the Flory-Huggins theory with a Monte Carlo simulation. As was mention before (section 2.1) the Monte Carlo method, which includes constraints arising from the excluded volume, provides an efficient algorithm for sampling relative orientations of the two molecules. The fundamental parameters that include the heat of mixing associated with styrene-isoprene molecular interactions and the numbers of possible interaction partners, that is, coordination number, Z12, and interaction energies, ∆E12, were obtained by averaging a large number of generated configurations (typically 500 000). The bead-bead interaction between different segments is given by the magnitude of repulsion between different repetitive units. For example, at temperature of 300 K the styrene-isoprene (monomer-monomer) molecular interaction parameter obtained from simulation is χ ) 0.3955; the average repetitive units of PS and PI beads is 10 times per bead for both polymers; then, the bead-bead interaction parameter is χ ) 3.955. Additionally, temperature dependence of the interaction parameter, χ(T), was obtained by eq 12. Figure 2 shows this temperature dependence and the values in the range 200-500 K. These molecular interaction parameters obtained by Monte
Figure 2. Molecular interaction parameter for the styrene-isoprene system obtained by Monte Carlo molecular simulation.
Carlo simulation are comparable with those achieved from experimental data (solubility parameters)43,44 using the FloryHuggins relation.45,46 More details of these solubility parameters and experimental data are given in the Soto-Figueroa et al.26 reference. It is important to keep in mind that a reliable and realistic value of χ(T) is necessary seeing that it is used as an input in eq 18. The styrene and isoprene molecules are incompatible in different degrees, and the microphase separation occurs with temperature lowering. A small incompatibility between individual styrene-isoprene molecules is amplified in giant molecules as block copolymers; this fact has been exploited in setting up bead-bead interaction between different segments of the statistical models. The election of the appropriate mesoscopic parameters for the account of bead-bead interaction of [PS]n - [PI]m and [PSn - PIm]3 Gaussian models is an important aspect that must be fulfilled for the subsequent mesoscopic simulations of the self-organization process of linear and 3-arm star block copolymers. All different beads in mechanical models are assumed to have equal volume in the mesoscopic simulations; this is a necessary hypothesis in order for them to obey the rules of the Flory-Huggins theory and DPD model.47 All mesoscopic simulations were carried out in a cubic box of (10rc ,10rc, 10rc) size, containing a total of 150 Gaussian chain models per system, spring constant, C ) 4, and a density, F ) 3. DPD simulations were made at temperature kBT ) 1, this allows a reasonable and efficient relaxation for each poly(styrene)-poly(isoprene) block copolymer system. A total of 105 time steps with step size of ∆t ) 0.05 in DPD reduced
Star Poly(styrene)-Poly(isoprene) Block Copolymers units are performed for equilibration. The number of beads representatives for each Gaussians chain model were assumed to be constant and the composition intervals analyzed are from 0.1-0.9 (volume fraction) of poly(styrene) with increments of 0.05. In order to verify that the morphologies are not influenced by the size of the box, selected calculations with a box of (20rc, 20rc, 20rc) size containing a total of 2400 Gaussian chains were performed. Our conclusion is that the morphologies are validated to be stable under increasing system size. 3.2. Self-Organization Process and Microphase Behavior in Linear and 3-Arm Star Block Copolymers. A remarkable property of block copolymers is their ability of self-organization or self-assembly in melt or in solution into a variety of ordered structures with characteristic dimensions in the range of a few nanometers up to several micrometers. The self-organization process of poly(styrene)-poly(isoprene) block copolymers is governed by the microphase separation thermodynamics because of the repulsive interaction of several chemically dissimilar components and the chemical incompatibility between homopoly(styrene) and homopoly(isoprene) chains and is driven by the enthalpy and entropy of demixing of the constituent components of the block copolymers. The enthalpy of demixing is proportional to the Flory-Huggins segmental interaction parameter, χ, while entropic effects are associated with configuration, conformation, and translation of different block chains. The enthalpic-entropic balance governs the structural self-organization via the microphase separation process in these materials;48,49 this fact has been exploited in subsequent dynamics simulations of Gaussian chain models. Specifically, in linear and 3-arm star block copolymers, the immiscibility between poly(styrene) and poly(isoprene) blocks induces microphase separation and self-assembly into a variety of characteristic ordered structures. All poly(styrene)-poly(isoprene) block copolymers start from a random disordered state in the mesoscopic simulation, where the polymers are in a homogeneous melted state; during the temperature relaxation, we observe the microphase separation process and ordered structures generation into nanoscale domains. In the ordered state, the poly(styrene) and poly(isoprene) block chains segregate into the respective domains to form microdomains in an ordered lattice. A transition from a homogeneous melt of Gaussian chains to a heterogeneous melt of ordered microphase-separated domains is defined to be an order-disorder transition (ODT) or microphase separation transition (MST). The sketch of each copolymer block (linear and 3-arm star) generates a coarse-grained system sufficiently large to identify the ordered structures with defined morphologies via the self-organization process; the resulting morphologies show rich microdomains of a single type of homopolymer chain separated by interfaces. One of the most important factors determining the phase morphology in linear and 3-arm star block copolymers is their composition and architecture.50 The shape of the poly(styrene)/poly(isoprene) interface varies with the relative chain length of component homopolymer chains, that is to say, with the weight percent of poly(styrene) and poly(isoprene) in the linear diblock copolymer. As the composition of linear poly(styrene)-poly(isoprene) copolymer varies, mesoscopic simulation generates, via the self-organization process, a large number of ordered microphases. The DPD simulations of [PS]n - [PI]m linear diblock copolymer display a rich variety of ordered structures with defined morphologies such as body-centered-cubic (BCC), hexagonal packed cylinders (HPC), alternating lamellar (LAM), and the ordered bicontinuous double diamond (OBDD) phases
J. Phys. Chem. B, Vol. 111, No. 40, 2007 11761 TABLE 2: Evolution of Ordered Structures with Defined Morphologies of [PS]n - [PI]m Linear Diblock Copolymers diblock copolymer [PS]n - [PI]m composition (volume fraction) PS/PI
phase morphology (ordered structures)
0.1/0.90 0.15/0.85 0.2/0.80 0.25/0.75 0.30/0.70 0.35/0.65 0.40/0.60 0.45/0.55 0.50/0.50 0.55/0.45 0.60/0.40 0.65/0.35 0.70/0.30 0.75/0.25 0.80/0.20 0.85/0.15 0.90/0.1
body-centered-cubic body-centered-cubic hexagonal packed cylinders hexagonal packed cylinders ordered bicontinuous double lamellar lamellar lamellar lamellar lamellar lamellar lamellar ordered bicontinuous double hexagonal packed cylinders hexagonal packed cylinders body-centered-cubic body-centered-cubic
as the composition is varied in the analyzed interval, as shown in Table 2 and Figure 3a-d. In accordance with previous results, morphologies are validated under system size increases; we constructed an assembly of 216 boxes in order to compare them with the transmission electron micrograph (TEM) image. Otherwise, simulations with boxes of (150rc, 150rc, 150rc) extent would need to be performed, which is not possible at the present level of computer facilities. The morphologies obtained by mesoscopic simulation for the linear [PS]n - [PI]m system can be compared positively with the TEM morphologies reported by Khandpur et al.;21 see Figure 3e-h. For the case of linear diblock copolymer of symmetric composition in their main chain (i.e., when volume fractions of both blocks chains are equivalent), it exhibits a lamellar morphology consisting of alternating layers of the poly(styrene) and poly(isoprene) components, as shown in Figure 3a. When the volume fraction of the poly(styrene) component increases in relation to the other component (poly(isoprene)), the interface between poly(styrene) and poly(isoprene) microdomains tends to become bent because the poly(styrene) chains are more extended than the poly(isoprene) chains, allowing then the formation of curve interfaces. Thermodynamically, the conformational entropy loss of the majority component is too high. Therefore, in order to gain the needed conformational entropy, the poly(styrene) chains have a tendency to expand along the direction parallel to interface under the condition that segment densities of both block chains keep constant and be the same as homopolymer bulk densities. As a result, the poly(styrene)/ poly(isoprene) interface becomes convex toward the minority component. The effect of that interface curvature in the [PS]n - [PI]m linear copolymer is more pronounced when the composition between poly(styrene) and poly(isoprene) is more asymmetric; hence, ordered structures (morphological variations with composition) such as OBDD (composition 0.30/0.70) Figure 3c, HPC (composition 0.25/0.75) Figure 3b, and BCC (composition 0.2/0.8) Figure 3a are obtained. In the specific case of bicontinuous double diamond microphases, these exist only in a narrow range of composition poly(styrene)/poly(isoprene) between regimes of cylindrical and lamellar packing; this behavior is valid for the [PS]n - [PI]m and [PSn - PIm]3 block copolymers with linear and 3-arm star architecture. By way of increasing volume fraction of the poly(styrene) component (now majority component) in the [PS]n - [PI]m copolymer, the morphology appears in reversed order.
11762 J. Phys. Chem. B, Vol. 111, No. 40, 2007
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Figure 4. Equilibrium morphologies of ordered microphases of [PSn - PIm]3 3-arm star block copolymer obtained as the composition poly(styrene)/poly(isoprene) is changed (see Table 3) by mesoscopic simulation. The orange and gray regions are for PS and PI microdomains, respectively: (a) BCC, (b) BCC (irregular spheres), (c) HPC, (d) HPL, (e) OBDD, and (f) LAM.
TABLE 3: Evolution of Ordered Structures with Defined Morphologies of [PSn - PIm]3 3-Arm Star Block Copolymers
Figure 3. Representative morphologies of the [PS]n - [PI]m linear diblock copolymer obtained as the composition poly(styrene)/poly(isoprene) is changed (see Table 2) by DPD simulation. (a) LAM (gray or green are PS or PI microdomains), (b) OBDD (the gray regions are PI density surfaces), (c) HPC (orange regions are the caps of PS cylinders, gray regions are the PI cylinder density surfaces), and (d) BCC (red regions and gray regions are PS and PI microdomains, respectively). Transmission electron micrograph image of ordered structures of poly(styrene)-poly(isoprene) system taken from ref 19 (e) LAM, (f) OBDD, (g) HPC, and (h) BCC.
The self-organization process of 3-arm star block copolymer in bulk also has been investigated through DPD methodology. For mesoscopic dynamic simulation of [PSn - PIm]3, we observe the microphase segregation process during the temperature relaxation and ordered microphases arrangement into nanoscale domains, this behavior is in agreement with the reported experimental observation for star block copolymers.51 The [PSn - PIm]3 3-arm star copolymer shows a microphase separation transition similar to that of [PS]n - [PI]m linear copolymer. We have obtained a series of ordered nanostructures by varying the composition of poly(styrene) block in the (PSn - PIm) 3-arm of star block copolymer. The [PSn - PIm]3 block copolymer generated ordered microphases of body-centered-cubic (BCC), hexagonal packed cylinders (HPC), ordered bicontinuous double diamond (OBDD), hexagonally perforated lamellar (HPL), and alternating lamellar (LAM) phases, as it is shown in Figure 4af. These figures are also constituted by an an assembly of 216
star block copolymer [PSn - PIm]3 composition (volume fraction) PS/PI 0.10/0.90 0.15/0.85 0.20/0.80 0.25/0.75 0.30/0.70 0.35/0.65 0.40/0.60 0.45/0.55 0.50/0.50 0.55/0.45 0.60/0.40 0.65/0.35 0.70/0.30 0.75/0.25 0.80/0.20 0.85/0.15 0.90/0.10
phase morphology (ordered-structures)
body-centered-cubic (irregular spheres) body-centered-cubic (outstretched spheres) irregular cylinders hexagonal packed cylinders ordered bicontinuous double hexagonal perforated layers lamellar lamellar lamellar lamellar lamellar hexagonal perforated layers ordered bicontinuous double hexagonal packed cylinders irregular cylinders body-centered-cubic (outstretched spheres) body-centered-cubic (irregular spheres)
boxes. Table 3 summarizes the details characterization of ordered microphases obtained for the [PSn - PIm]3 star block copolymer. The ordered structures evolution and their morphologies in the star block copolymer can be understood in terms of the effects of the composition, architecture (three arm star block), packing density, and entropic interactions (conformational, configurational, and translational) on microdomains space between (PSn - PIm) blocks. In the [PSn - PIm]3 system, the enthalpic interactions between poly(styrene) and poly(isoprene) chains are more complex because of the architecture of (PSn -
Star Poly(styrene)-Poly(isoprene) Block Copolymers PIm) 3-arm star. The microphase segregation behavior during the temperature relaxation is more restricted by architecture effect due to packing frustration, because the (PSn - PIm) diblock chains emanate from a common junction point, in the [PSn - PIm]3 star block copolymer; the chains tend to avoid chain stretching (and achieve maximum possible entropy) by relaxing and forming a curvature interface, and therefore, a rich variety of ordered microphases is generated. In this way, the 3-arm star copolymers are easier to succeed microphase separation under the same conditions that linear block copolymer. The morphologies obtained by mesoscopic simulation for the [PSn - PIm]3 3-star block copolymer can also be compared positively with the morphologies reported by Bi and Fetters, Alward et al., Herman et al., and Tselikas et al.52-55 It is worthwhile to mention that the BCC, HPC, OBDD, HPL, and LAM ordered microphases of general block copolymers by mesoscopic simulations have been recently published by Warren and Groot39 and Sheng et al.56 In these studies, they propose direct mechanical models of beads without any a priori specification of any molecular structure. On the other hand, the Gaussian models for the linear and 3-arm star systems used in the present work were obtained from the real molecular structure of the block copolymers. The mesoscopic parameters are acquired from the monomer-monomer molecular interaction by molecular simulation without any approach from experimental outcomes. Then, the use of different monomers should totally modify the mesoscopic models, bead-bead interaction parameters, and consequently the final morphology. As mentioned before, a similar approximation was proposed by Srinivas et al.15,57 for PEO-PEE diblock copolymer structure in watery solutions. They consider coarse-grain models which are based on atomistic structures of the corresponding diblock copolymer. The comparisons with the experimentally observed morphologies is feasible, from a quantitative point of view and, considering the reach of DPD, is viable evaluate lamellar spacing, cylinder diameters, and interfacial tension;58,59 these evaluations are not considered in the aim of this work. 4. Conclusions The main goal of this research is to provide a methodology and general ideas for mapping macromolecular systems in the statistical segment level for mesoscopic dynamic simulations. On the basis of molecular simulations and considering a great number of atoms, we find it is reasonable to represent these systems by suitable statistical models for studying microscopic details of chemical processes in the condensed phase where macroscopic properties can be predicted. The simplicity of the methodology allows us to probe copolymers self-assembly as it occurs locally on the nanosecond to microsecond time scale. In this way, the molecular structures of poly(styrene)-poly(isoprene) copolymers with linear and 3-arm star architecture were represented by Gaussian chain models into characteristic ratio statistical segment level. The mesoscopic parameters that display the bead-bead interaction in these models were obtained from styrene-isoprene molecular interaction, and they are advantageous in order to represent these copolymeric systems into DPD simulations. The mesoscopic simulation of Gaussian chain models of two types of poly(styrene)-poly(isoprene) block copolymers of dissimilar architecture allowed for the exploration of the microphase separation process, orderdisorder transition, and ordered structures formation with defined morphologies via the self-organization process. One of the most important factors determining the phase morphology in the linear and 3-arm star block copolymers is
J. Phys. Chem. B, Vol. 111, No. 40, 2007 11763 their composition and architecture. When we varied the composition of the component homopolymer chains in each system, we found ordered structures of body-centered-cubic, hexagonal packed cylinders, ordered bicontinuous double diamond, hexagonal perforated layers, and alternating lamellar phase. The results were compared with those obtained by the experimental work for the same materials. The agreement between our dynamic simulations and experimental outcomes validate the Gaussian chain models and mesoscopic parameters of poly(styrene)-poly(isoprene) diblock copolymer; these are well-suited for the description of microphase separation phenomena, and they predict ordered microphases. Previous knowledge can be extended to an ample variety of complex macromolecular structures of soft condensed matter with wide length range. Acknowledgment. C.S.-F. acknowledges to Universidad Nacional Auto´noma de Me´xico (UNAM) and Progama de Ingenierı´a Molecular of Instituto Mexicano del Petro´leo (IMP) for their economic and technical support for the development of this work. References and Notes (1) Economou, I. G. Ind. Eng. Chem. Res. 2002, 41, 953. (2) Tuckerman, M. E.; Martyna, G. J. J. Phys. Chem. B 2000, 104, 159. (3) Forster, S.; Konrad, M. J. Mater. Chem. 2003, 13, 2671. (4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091. (5) Strobl, G. The physics of Polymers, 2nd ed., Springer: Berlin, 1997. (6) Crawshaw, J.; Windle, A. H. Fibre Diffraction ReView 2003, 11, 52. (7) Zhang, K.; Manke, C. W. Comput. Phys. Commun. 2000, 129, 275. (8) Yu-Jane, S.; Chih-Hsiung, N. J. Phys. Chem. B 2006, 110, 21643. (9) Fraaije, J. G. E. M.; van Vlimmeren, B. A. C.; Maurits, N. M.; Postma, M.; Evers, O. A.; Hoffman, C.; Altevogt, P.; Goldbeck-Wood, G. J. Chem. Phys. 1997, 106, 4260. (10) Morita, H.; Kawakatsu, T.; Doi, M. G.; Yamaguchi, D.; Takenaka, M.; Hashimoto, T. Macromolecules 2002, 35, 7473. (11) Fermeglia, M.; Pricl, S. AIChE J. 1999, 45, 2619. (12) Hoffmann, H.; Somer, J. U.; Blumen, A. J. Chem. Phys. 1997, 106, 6709. (13) Zvelindovsky, A. V.; Sevink, G. J.; van Vlimmeren, B. A.; Maurits, N. M.; Fraaije, J. G. E. M. Phys. ReV. E 1998, 57, 4879. (14) Kim, S. H.; Jo, W. H.; Kim, J. Macromolecules 1996, 29, 6933. (15) Srinivas, G.; Shelley, J. C.; Nielsen, S. O.; Discher, D. E.; Klein, M. L. J. Phys. Chem. B 2004, 108, 8153. (16) Cao, X.; Xu, G.; Li, Y.; Zhang, Z. J. Phys. Chem. A 2005, 109, 10418. (17) Qian, H.-J.; Lu, Z.-Y.; Chen, L.-J.; Li, Z.-S.; Sun, C.-C. Macromolecules 2005, 38, 1395. (18) Hamley, I. W. The physics of block copolymers; Oxford University Press, Oxford, U.K., 1998. (19) Xu, Y.; Feng, J.; Liu, H.; Hu, Y. Mol. Simul. 2006, 32, 375. (20) van Vlimmeren, B. A. C.; Maurits, N. M.; Zvelindovsky, A. V.; Sevink, G. J. A.; Fraaije, J. G. E. M. Macromolecules 1999, 32, 646. (21) Khandpur, A. K.; Forster, S.; Bates, F. S.; Hamley, I. H.; Ryan, A. J.; Bras, W.; Almadal, K.; Mortensen, K. Macromolecules 1995, 28, 8796. (22) Accelrys, Material Studio Release, Notes Release 4.1; Accelrys software Inc.: San Diego, 2006. (23) Sun, H.; Ren, P.; Fried, J. R. Computational and Theoretical Polymer Science 1998, 8, 229. (24) Sun, H. J. Phys. Chem. 1998, 102, 7338. (25) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: Oxford U.K., 1998. (26) Soto-Figueroa, C.; Rodrı´guez-Hidalgo, M. R.; Martı´nez-Magada´n, J. M. Polymer 2005, 46, 7485. (27) Soto-Figueroa, C.; Vicente, L.; Martı´nez-Magada´n, J. M.; Rodrı´guez-Hidalgo, M. R. Polymer doi:10.1016/j.polymer.2007.04.043. (28) Flory, P. J. J. Chem. Phys. 1941, 9, 660. (29) Huggins, M. L. J. Chem. Phys. 1941, 9, 440. (30) Fan, C. F.; Olafson, B. D.; Blanco, M. Macromolecules 1992, 25, 3667. (31) Luı´s-Vicente.; Soto-Figueroa, C.; Martı´nez-Magada´n, J. M. Fluid Phase Equilib. 2006, 239, 100. (32) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Europhys. Lett. 1992, 19, 155.
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