Dissipative particle dynamics of triblock copolymer melts: A midblock conformational study at moderate segregation Syamal S. Tallury, Richard J. Spontak, and Melissa A. Pasquinelli
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Citation: The Journal of Chemical Physics 141, 244911 (2014); doi: 10.1063/1.4904388 View online: http://dx.doi.org/10.1063/1.4904388 View Table of Contents: http://aip.scitation.org/toc/jcp/141/24 Published by the American Institute of Physics
Articles you may be interested in Communication: Molecular-level insights into asymmetric triblock copolymers: Network and phase development The Journal of Chemical Physics 141, 121103121103 (2014); 10.1063/1.4896612 Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation The Journal of Chemical Physics 107, (1998); 10.1063/1.474784
THE JOURNAL OF CHEMICAL PHYSICS 141, 244911 (2014)
Dissipative particle dynamics of triblock copolymer melts: A midblock conformational study at moderate segregation Syamal S. Tallury,1,2 Richard J. Spontak,1,3 and Melissa A. Pasquinelli2,a)
1
Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA 2 Fiber and Polymer Science Program, North Carolina State University, Raleigh, North Carolina 27695, USA 3 Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA
(Received 29 September 2014; accepted 1 December 2014; published online 31 December 2014) As thermoplastic elastomers, triblock copolymers constitute an immensely important class of shapememory soft materials due to their unique ability to form molecular networks stabilized by physical, rather than chemical, cross-links. The extent to which such networks develop in triblock and higherorder multiblock copolymers is sensitive to the formation of midblock bridges, which serve to connect neighboring microdomains. In addition to bridges, copolymer molecules can likewise form loops and dangling ends upon microphase separation or they can remain unsegregated. While prior theoretical and simulation studies have elucidated the midblock bridging fraction in triblock copolymer melts, most have only considered strongly segregated systems wherein dangling ends and unsegregated chains become relatively insignificant. In this study, simulations based on dissipative particle dynamics are performed to examine the self-assembly and networkability of moderately segregated triblock copolymers. Utilizing a density-based cluster-recognition algorithm, we demonstrate how the simulations can be analyzed to extract information about microdomain formation and permit explicit quantitation of the midblock bridging, looping, dangling, and unsegregated fractions for linear triblock copolymers varying in chain length, molecular composition, and segregation level. We show that midblock conformations can be sensitive to variations in chain length, molecular composition, and bead repulsion, and that a systematic investigation can be used to identify the onset of strong segregation where the presence of dangling and unsegregated fractions are minimal. In addition, because this clustering approach is robust, it can be used with any particle-based simulation method to quantify network formation of different morphologies for a wide range of triblock and higher-order multiblock copolymer systems. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904388]
INTRODUCTION
Due to their unique ability to spontaneously undergo microphase separation and self-assemble into a rich variety of soft, periodic nanostructures as a consequence of thermodynamic incompatibility,1,2 block copolymers continue to capture the attention of numerous fundamental studies and technological innovations.3–5 Triblock copolymers composed of hard (glassy or semicrystalline) A endblocks and a soft (rubbery) B midblock (generally referred to as ABA copolymers) are of particular widespread interest from a commercial standpoint because they constitute thermoplastic elastomers (TPEs),6 a class of elastic materials that rely on the formation of molecular networks stabilized by physical, rather than chemical, cross-links (discussed in further detail below). Since their initial commercialization, TPEs have been commonly used in the large-scale manufacture of consumer products, traditionally ranging from automotive parts to sporting and household/office goods. More recent efforts have demonstrated that these versatile materials can be broadly incorpoa)Author to whom correspondence should be addressed. Electronic mail:
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rated into microfluidic devices,7 vibration-dampening media, soft electroactuators,8–10 hot-melt/pressure-sensitive adhesives,11,12 shape-memory materials,13 and large-strain flexible electronics.14,15 Unlike their chemically cross-linked analogs, TPEs can also be recycled and reprocessed through the use of either solvent or thermal means to avoid contributing to solid-waste pollution.15 From a fundamental viewpoint, ABA triblock copolymers, alone or in the presence of one or more small-molecule or polymeric additives, are interesting because they can self-organize into the same nanostructures16–20 (as well as others21) observed in simpler AB diblock copolymer systems. The primary difference between the two copolymer genres lies in the ability of ABA copolymer molecules to connect neighboring microdomains and thus form physically cross-linked molecular networks with remarkable and physically tunable mechanical properties (e.g., timecomposition superpositioning22,23). While the bulk thermomechanical properties of ABA copolymers have been extensively studied and wellestablished since the initial commercialization of these materials, relatively recent experimental studies24,25 have sought to elucidate the onset of network formation by exploring the transition from AB diblock to ABA triblock copolymer
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FIG. 1. Schematic illustration of the possible chain conformations of a microphase-ordered ABA triblock copolymer exhibiting a dispersed (spherical or cylindrical) morphology. The bead repulsion parameter between A and B blocks (α AB) is also indicated.
behavior through the use of molecularly asymmetric A1BA2 triblock copolymers grown from a parent A1B diblock copolymer. These efforts have demonstrated that the phase behavior, ordered nanostructure, and mechanical properties exhibit noticeably abrupt transitions at a critical size of the progressively grown A2 endblock. Complementary theoretical and simulation considerations26 reveal that this transition coincides with the formation of an equilibrium network, thereby establishing a direct connection between diblock and triblock copolymer archetypes. While this strategy has been developed to follow the network evolution in molecularly asymmetric triblock copolymers, only molecularly symmetric ABA copolymers (with MA1 = MA2, where M denotes block mass) are considered further in the present work. Under strong-segregation conditions wherein the A and B blocks are highly incompatible, an ABA copolymer molecule typically self-assembles in one of two ways (cf. Figure 1) by either depositing (i) both endblocks in a single microdomain, thereby causing the midblock to adopt a looped conformation, or (ii) each endblock in a neighboring microdomain, thereby resulting in a bridged midblock. Independent experimental27,28 and theoretical29,30 studies have sought to quantify the fractions associated with these arrangements so that molecular conformations can be directly related to macroscopic (specifically mechanical) properties. In moderately segregated systems, however, other midblock conformations, such as dangling ends (hereafter referred to as “dangles”) and unsegregated (“mixed”) chains (both of which are depicted in Figure 1), must be included in the chain population analysis. In this vein, the seminal study of Mattice and co-workers31 has demonstrated that Monte Carlo simulations can be used to ascertain the fractions of bridges, loops, dangles, and mixed chains (denoted here as f L , f B, f D , and f M , respectively, where f L + f B + f D + f M = 1) in ABA triblock copolymer systems. In the present work, we quantify these fractions in moderately and strongly segregated ABA copolymer melts through the use of dissipative particle dynamics (DPD), a simulation approach that is similar to molecular dynamics and that combines a conservative force with viscous and random forces to permit emulation of viscous matter.32 While DPD simulations have been previously used to investigate the self-assembly of
diblock copolymers,33–35 their recent application to networkforming triblock copolymer melts has been largely limited to strongly segregated systems36,37 for which f B can be discerned explicitly from chain end-to-end distributions. Here, we demonstrate that a density-based clustering algorithm (DBSCAN)38 can be applied to analyze segmental trajectories from DPD simulations in order to identify microdomains and extract the midblock conformational fractions for ABA triblock copolymers varying in segregation level, chain length, and molecular composition. COMPUTATIONAL METHODS DPD simulation details
The coarse-grained particles (or beads) in DPD simulations are compressible and thus behave as soft spheres, which are linked together to replicate polymer chains as beadspring equivalents. The net force (fi ) experienced by each bead i at position ri during a DPD simulation corresponds to a summation of the forces that account for all pairwise interactions of the ith bead with all of the other beads (designated as j) within a cutoff radius (Rc , which is commonly set to unity), and is given by ( ) D R S fi = FC (1) ij + Fij + Fij + Fij . j,i
The first term in the summation in Eq. (1) represents the conservative (repulsive) force (FC ij ) and is written as ) ( r ij αij 1 − rˆ ij for (r ij < Rc ) C R , (2) Fij = c 0 for (r ≥ R ) ij c where αij signifies the repulsion potential between beads i and j, rˆ ij = rij/|rij|, and r ij = rij corresponds to the distance between the beads. Note that FC ij in Eq. (2) reduces to zero if r ij > Rc . The dissipative (FijD ) and random (FijR ) forces in Eq. (1) are defined as FijD = −γW D r ij rˆ ij · vij rˆ ij (3) and FijR = σW R r ij θ ijrˆ ij,
(4)
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respectively, where γ represents the viscous drag parameter, vij = vi − v j denotes the relative velocity between two beads, and θ ij is a randomly fluctuating term that obeys Gaussian statistics. The weight functions (W D and W R ) are expressed as 2 R 2 1 −r ij/Rc r ij < Rc D , (5) W r ij = W r ij = r ij > Rc 0 and the dissipative and random forces are related through the fluctuation-dissipation theorem39 by σ 2 = 2γk BT, where k B is the Boltzmann constant and T denotes absolute temperature. (Note that this theorem naturally enforces a system thermostat.) Finally, the spring force (Fijs ) in Eq. (1) accounts for bead connectivity and is often represented by a Fraenkel spring, given by40 Fijs = −k s r ij −r 0 rˆ ij, (6) where k s is the spring force constant and r 0 is the equilibrium bond length. In contrast to the other forces in Eq. (1), Fijs does not vanish when r ij ≥ Rc so that bead connectivity is maintained throughout the simulation. A reduced system of units is typically employed in DPD simulations, and the unit of length in our simulations translates to 1.73 nm since the edge of the smallest cubic cell employed here measured 39.5 nm. All simulations were performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software package41 using a microcanonical ensemble (possessing constant mass, volume, and energy) with the system thermostat set to 373 K, at which temperature k BT was assigned unit thermal energy. To ensure coarse-grain consistency with previous DPD studies of diblock35,42 and cyclic43 block copolymers, the spring parameters were set to k s = 8.0 and r 0 = 0.8. The number bead density, ρ, ¯ and γ were held constant at 3.0 and 4.5, respectively, and the pair-repulsion parameters were calculated for the specific case of a triblock copolymer composed of styrene (A) and ethylene-co-butylene (B) moieties. According to Groot and co-workers,44 the values of the like pair-repulsion parameters (αAA and αBB) are each 25.0, as calculated from 75k BT/ ρ. ¯ The pair repulsion parameter between dissimilar beads (αAB) is commonly computed from αAA + 3.27 1 + 3.9/N 0.51 χAB, where N is the number of beads per chain and χAB is the FloryHuggins interaction parameter, which can be estimated45 from V (δ A − δ B)2/k BT, where V is the bead volume and δ i represents the solubility parameter of species i (where i = A or B). These pure-component properties are accurately approximated for nonpolar species from group contribution methods46 to be δ A = 21.59 MPa1/2 and δ B = 16.97 MPa1/2. For the desired level of coarse graining, V corresponded to about 10 styrenic units (1000 cm3/mol), or about 1.725 nm3, which is consistent with the DPD results reported earlier by Maly and co-workers.42 Thus, for moderately segregated chains composed of 50 beads, αAB was determined to be 48.3. Random chain structures incorporating these parameters were initially generated from the chain.py routine in the LAMMPS computational package41 and then subjected to equilibration runs of 100 000 steps with periodic boundary conditions. Subsequent production runs were performed for 500 000 steps with a time step of about 2 ps in real time,
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thereby yielding a total run time of ≈1 µs. All simulations were run on the Henry2 cluster at the NC State High Performance Computing Center. Simulations were conducted for systems containing copolymer chains systematically differing in length (N), segregation level (αAB), and molecular composition (Φ A), where Φ A corresponds to the fraction of A beads in a copolymer chain and is varied to examine the role of morphology on network formation: spherical (Φ A = 0.2), cylindrical (Φ A = 0.3), and lamellar (Φ A = 0.5). Post-simulation clustering strategy
To systematically identify copolymer microdomains from the simulation trajectories, we have elected to use DBSCAN38 since single- and complete-linkage hierarchical clustering methods are either too sensitive to background noise or insufficiently sensitive to a broad range of existing clusters. This pattern-recognition algorithm ensures reachability while enforcing a minimum cluster size, without presuming a particular cluster shape (such as spherical) a priori. This algorithm is robust and versatile, as evidenced by its use in a variety of contemporary applications, such as identifying broadband internet user time preference,47 lipid rafts in mixed lipid bilayers,48 and objects in noisy video footage,49 as well as elucidating chromatin interactions in genes.50 In essence, the DBSCAN protocol initially defines each bead to lie in a different cluster. Two independent parameters are then used to establish the existence of an actual cluster: Pmin, the minimum number of observations required to classify a grouping as a legitimate cluster, and ε, the reachability that defines the span over which cluster-based linkages persist. If beads meet the conditions required by both parameters, they are allocated to the core of a cluster. If the conditions set by ε are satisfied but those of Pmin are not, the corresponding beads are assigned to a cluster boundary. Beads that do not meet either condition are considered noise and are rejected from being part of any of the clusters. In terms of triblock copolymer melts, clusters correspond to endblock-rich microdomains, whereas the noise from snapshots of DPD simulations identifies beads belonging to the midblock-rich matrix (depending on the copolymer composition, since the morphology might be lamellar). Our adaptation of DBSCAN forces microdomains to obey periodic boundary conditions in all three dimensions, in which case bonded beads along an endblock that crosses a periodic wall in the simulation box must be unwrapped and are identified as beads located in the same cluster (microdomain). In this study, the DBSCAN cluster analysis employs the flexible procedures for clustering (fpc) package within the R statistical software program51 on a Windows 64-bit configuration machine.
RESULTS AND DISCUSSION Density-based clustering parameterization
A nontrivial challenge associated with using the DBSCAN to systematically identify copolymer microdomains lies in identifying sensible values of Pmin and ε, which tend to be specific for the intended application. For instance, in
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FIG. 2. Radial pair distribution function for DPD systems containing 500 copolymer chains varying in length (N ) at constant copolymer composition (Φ A = 0.2). The color-coded legend for N is provided in the inset, which is an enlargement of the boxed region to highlight the intersections indicated by arrows. Also included for the case of N = 50 is a DPD system containing 2000 chains (bold dashed black line). The solid lines serve to connect the data.
a recent study50 wherein DBSCAN was used to elucidate chromatin interactions in a genome, the optimal parameter values selected for identifying clusters have been set equal to the number of detected intra-chromosomal paired-end-tag clusters that are significant at a false discovery rate of 70. It is interesting to note that f B becomes independent of αAB when αAB ≈ 50. Since f L , f D , and f M continue to evolve with increasing αAB beyond this value, we propose that unsegregated chains and dangles prefer to transform into loops rather than bridges for body-centered cubic micelles. Comparison of Figures 5 and 6 also confirms that the transition to strong segregation in ABA melts is preceded by nonlinearly decaying f D and f M , as well as a corresponding increase in f L . Since αAB is related to χAB and χAB generally depends on reciprocal
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FIG. 6. Dependence of the midblock fractions (labeled and color-coded) on the pair repulsion parameter (α AB) at N = 50 and Φ A = 0.2 for DPD systems containing 500 chains. The error bars correspond to the standard error, and the solid lines serve to connect the data.
temperature, it stands to reason that all three midblock conformations should change in an opposite fashion from what is observed in Figure 6 as the temperature is increased. An increase in T ∗ introduces more random forces (Eq. (4)) and decreases the conservative forces (Eq. (2)), thereby reducing the net driving force for microphase separation, especially as the order-disorder transition is approached. This behavior is qualitatively similar to that obtained from on-lattice Monte Carlo simulations of molecularly asymmetric A1BA2 triblock copolymers.26 Although many triblock copolymers intended for commercial use as TPEs possess a relatively small endblock fraction to ensure the formation of a dispersed morphology (i.e., spherical or cylindrical micelles) within a continuous rubbery matrix, copolymer composition constitute an important consideration since changes in morphology relate directly to variations in interfacial area. That is, the spherical micellar morphology possesses the highest interfacial area density of the classical morphologies. With N held constant, interfacial area density monotonically decreases for the cylindrical and lamellar morphologies as Φ A is increased, in which case we anticipate a systematic variation in midblock conformations. To test this hypothesis, values of f B, f L , f D , and f M extracted from simulations conducted at αAB = 48.3 for 500 chains with N = 50 are displayed as functions of Φ A in Figure 6. Included in this figure are ranges of f B reported from SCFT,29,30 MC simulations,31 and experimental studies.27,28 While the trends observed in Figure 6 are consistent with expectation and are certainly comparable, the DPD results significantly underestimate the value of f B evaluated for dispersed morphologies. Once again, though, we attribute this difference primarily to the presence of dangles, which account for about 10% of the midblock conformations in the spherical and cylindrical morphologies. Renormalization of f B and f L to eliminate f D and f M (so that f B + f L = 1) yields results that are in closer quantitative agreement with previous findings (Figure 6). As anticipated on the basis of considerations such as the interfacial area density, both f B and f M decrease, whereas f L increases, monotonically with increasing Φ A. Note that
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the onset of strong segregation, wherein the fraction of dangles and unsegregated chains becomes negligible. Quantitative comparison of the results obtained here with previous results reported for the midblock bridging fraction illustrates some discrepancy for the case of the dispersed (spherical and cylindrical) morphologies, which is most likely due to the exclusion of dangles in theoretical formalisms derived on the basis of equilibrium thermodynamics. Because the clustering methodology employed here is systematic and robust, it can be applied to a wide range of triblock and higher-order multiblock copolymer systems in which network formation is of considerable interest. ACKNOWLEDGMENTS FIG. 7. Dependence of the midblock fractions (labeled and color-coded) on Φ A at N = 50 for DPD systems containing 500 chains. Bridging and looping data renormalized to exclude f D and f M (so that f B + f L = 1) are included (O) for comparison with previous theoretical,29,30 simulation,31 and experimental27,28 results (gray regions). The error bars correspond to the standard error, and the solid lines serve to connect the data.
f D does not change very much with increasing Φ A relative to the maximum standard error, whereas the corresponding reduction in f M most likely reflects more complete microphase segregation as Φ A is increased. Another morphological aspect that must be considered in the interpretation of the midblock fractions is the system dimensionality, since the diffusion of endblocks to microdomains is facilitated in three dimensions (spherical micelles) relative to two dimensions (cylinders) or one dimension (lamellae). While results similar to those evident in Figure 7 have also been acquired by increasing N to 75 and 100, f M becomes increasingly negligible in those cases, which supports the renormalization-based comparison discussed above. CONCLUSIONS
Since thermoplastic elastomers constitute a fundamentally and technologically important class of soft materials, the objective of this study is to elucidate the midblock conformations of microphase-ordered ABA triblock copolymers varying in size, segregation, and composition through the use of DPD simulations and a density-based cluster algorithm. We have found that the parameters employed during cluster recognition can be conveniently estimated from the radial pair distribution function of microdomain-forming beads, in which case no prior knowledge of the cluster size or number is required. The DPD simulations performed here yield classical copolymer morphologies (i.e., body-centered cubic spherical micelles, hexagonally packed cylinders and lamellae), which are then analyzed in terms of bead clustering to extract the fractions associated with the possible midblock conformations: bridges, loops, dangles, and unsegregated chains. These fractions are evaluated as functions of several important material/simulation considerations, namely, chain length, system size, bead repulsion, and copolymer composition. While system size is determined to have relatively little effect on the midblock conformations in all the cases examined, systematic variations in chain length and bead repulsion reveal
This study was supported by the Nonwovens Institute at North Carolina State University, Grant No. 10-127. The authors thank Professor A. Motsinger-Reif and Mr. J. Majikes for technical assistance, and Dr. H. Chintakunta for insightful discussions with regard to machine learning. 1I.
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,
Citation: The Journal of Chemical Physics 141, 244911 (2014); doi: 10.1063/1.4904388 View online: http://dx.doi.org/10.1063/1.4904388 View Table of Contents: http://aip.scitation.org/toc/jcp/141/24 Published by the American Institute of Physics
Articles you may be interested in Communication: Molecular-level insights into asymmetric triblock copolymers: Network and phase development The Journal of Chemical Physics 141, 121103121103 (2014); 10.1063/1.4896612 Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation The Journal of Chemical Physics 107, (1998); 10.1063/1.474784
THE JOURNAL OF CHEMICAL PHYSICS 141, 244911 (2014)
Dissipative particle dynamics of triblock copolymer melts: A midblock conformational study at moderate segregation Syamal S. Tallury,1,2 Richard J. Spontak,1,3 and Melissa A. Pasquinelli2,a)
1
Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA 2 Fiber and Polymer Science Program, North Carolina State University, Raleigh, North Carolina 27695, USA 3 Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA
(Received 29 September 2014; accepted 1 December 2014; published online 31 December 2014) As thermoplastic elastomers, triblock copolymers constitute an immensely important class of shapememory soft materials due to their unique ability to form molecular networks stabilized by physical, rather than chemical, cross-links. The extent to which such networks develop in triblock and higherorder multiblock copolymers is sensitive to the formation of midblock bridges, which serve to connect neighboring microdomains. In addition to bridges, copolymer molecules can likewise form loops and dangling ends upon microphase separation or they can remain unsegregated. While prior theoretical and simulation studies have elucidated the midblock bridging fraction in triblock copolymer melts, most have only considered strongly segregated systems wherein dangling ends and unsegregated chains become relatively insignificant. In this study, simulations based on dissipative particle dynamics are performed to examine the self-assembly and networkability of moderately segregated triblock copolymers. Utilizing a density-based cluster-recognition algorithm, we demonstrate how the simulations can be analyzed to extract information about microdomain formation and permit explicit quantitation of the midblock bridging, looping, dangling, and unsegregated fractions for linear triblock copolymers varying in chain length, molecular composition, and segregation level. We show that midblock conformations can be sensitive to variations in chain length, molecular composition, and bead repulsion, and that a systematic investigation can be used to identify the onset of strong segregation where the presence of dangling and unsegregated fractions are minimal. In addition, because this clustering approach is robust, it can be used with any particle-based simulation method to quantify network formation of different morphologies for a wide range of triblock and higher-order multiblock copolymer systems. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904388]
INTRODUCTION
Due to their unique ability to spontaneously undergo microphase separation and self-assemble into a rich variety of soft, periodic nanostructures as a consequence of thermodynamic incompatibility,1,2 block copolymers continue to capture the attention of numerous fundamental studies and technological innovations.3–5 Triblock copolymers composed of hard (glassy or semicrystalline) A endblocks and a soft (rubbery) B midblock (generally referred to as ABA copolymers) are of particular widespread interest from a commercial standpoint because they constitute thermoplastic elastomers (TPEs),6 a class of elastic materials that rely on the formation of molecular networks stabilized by physical, rather than chemical, cross-links (discussed in further detail below). Since their initial commercialization, TPEs have been commonly used in the large-scale manufacture of consumer products, traditionally ranging from automotive parts to sporting and household/office goods. More recent efforts have demonstrated that these versatile materials can be broadly incorpoa)Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/141(24)/244911/9/$30.00
rated into microfluidic devices,7 vibration-dampening media, soft electroactuators,8–10 hot-melt/pressure-sensitive adhesives,11,12 shape-memory materials,13 and large-strain flexible electronics.14,15 Unlike their chemically cross-linked analogs, TPEs can also be recycled and reprocessed through the use of either solvent or thermal means to avoid contributing to solid-waste pollution.15 From a fundamental viewpoint, ABA triblock copolymers, alone or in the presence of one or more small-molecule or polymeric additives, are interesting because they can self-organize into the same nanostructures16–20 (as well as others21) observed in simpler AB diblock copolymer systems. The primary difference between the two copolymer genres lies in the ability of ABA copolymer molecules to connect neighboring microdomains and thus form physically cross-linked molecular networks with remarkable and physically tunable mechanical properties (e.g., timecomposition superpositioning22,23). While the bulk thermomechanical properties of ABA copolymers have been extensively studied and wellestablished since the initial commercialization of these materials, relatively recent experimental studies24,25 have sought to elucidate the onset of network formation by exploring the transition from AB diblock to ABA triblock copolymer
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FIG. 1. Schematic illustration of the possible chain conformations of a microphase-ordered ABA triblock copolymer exhibiting a dispersed (spherical or cylindrical) morphology. The bead repulsion parameter between A and B blocks (α AB) is also indicated.
behavior through the use of molecularly asymmetric A1BA2 triblock copolymers grown from a parent A1B diblock copolymer. These efforts have demonstrated that the phase behavior, ordered nanostructure, and mechanical properties exhibit noticeably abrupt transitions at a critical size of the progressively grown A2 endblock. Complementary theoretical and simulation considerations26 reveal that this transition coincides with the formation of an equilibrium network, thereby establishing a direct connection between diblock and triblock copolymer archetypes. While this strategy has been developed to follow the network evolution in molecularly asymmetric triblock copolymers, only molecularly symmetric ABA copolymers (with MA1 = MA2, where M denotes block mass) are considered further in the present work. Under strong-segregation conditions wherein the A and B blocks are highly incompatible, an ABA copolymer molecule typically self-assembles in one of two ways (cf. Figure 1) by either depositing (i) both endblocks in a single microdomain, thereby causing the midblock to adopt a looped conformation, or (ii) each endblock in a neighboring microdomain, thereby resulting in a bridged midblock. Independent experimental27,28 and theoretical29,30 studies have sought to quantify the fractions associated with these arrangements so that molecular conformations can be directly related to macroscopic (specifically mechanical) properties. In moderately segregated systems, however, other midblock conformations, such as dangling ends (hereafter referred to as “dangles”) and unsegregated (“mixed”) chains (both of which are depicted in Figure 1), must be included in the chain population analysis. In this vein, the seminal study of Mattice and co-workers31 has demonstrated that Monte Carlo simulations can be used to ascertain the fractions of bridges, loops, dangles, and mixed chains (denoted here as f L , f B, f D , and f M , respectively, where f L + f B + f D + f M = 1) in ABA triblock copolymer systems. In the present work, we quantify these fractions in moderately and strongly segregated ABA copolymer melts through the use of dissipative particle dynamics (DPD), a simulation approach that is similar to molecular dynamics and that combines a conservative force with viscous and random forces to permit emulation of viscous matter.32 While DPD simulations have been previously used to investigate the self-assembly of
diblock copolymers,33–35 their recent application to networkforming triblock copolymer melts has been largely limited to strongly segregated systems36,37 for which f B can be discerned explicitly from chain end-to-end distributions. Here, we demonstrate that a density-based clustering algorithm (DBSCAN)38 can be applied to analyze segmental trajectories from DPD simulations in order to identify microdomains and extract the midblock conformational fractions for ABA triblock copolymers varying in segregation level, chain length, and molecular composition. COMPUTATIONAL METHODS DPD simulation details
The coarse-grained particles (or beads) in DPD simulations are compressible and thus behave as soft spheres, which are linked together to replicate polymer chains as beadspring equivalents. The net force (fi ) experienced by each bead i at position ri during a DPD simulation corresponds to a summation of the forces that account for all pairwise interactions of the ith bead with all of the other beads (designated as j) within a cutoff radius (Rc , which is commonly set to unity), and is given by ( ) D R S fi = FC (1) ij + Fij + Fij + Fij . j,i
The first term in the summation in Eq. (1) represents the conservative (repulsive) force (FC ij ) and is written as ) ( r ij αij 1 − rˆ ij for (r ij < Rc ) C R , (2) Fij = c 0 for (r ≥ R ) ij c where αij signifies the repulsion potential between beads i and j, rˆ ij = rij/|rij|, and r ij = rij corresponds to the distance between the beads. Note that FC ij in Eq. (2) reduces to zero if r ij > Rc . The dissipative (FijD ) and random (FijR ) forces in Eq. (1) are defined as FijD = −γW D r ij rˆ ij · vij rˆ ij (3) and FijR = σW R r ij θ ijrˆ ij,
(4)
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respectively, where γ represents the viscous drag parameter, vij = vi − v j denotes the relative velocity between two beads, and θ ij is a randomly fluctuating term that obeys Gaussian statistics. The weight functions (W D and W R ) are expressed as 2 R 2 1 −r ij/Rc r ij < Rc D , (5) W r ij = W r ij = r ij > Rc 0 and the dissipative and random forces are related through the fluctuation-dissipation theorem39 by σ 2 = 2γk BT, where k B is the Boltzmann constant and T denotes absolute temperature. (Note that this theorem naturally enforces a system thermostat.) Finally, the spring force (Fijs ) in Eq. (1) accounts for bead connectivity and is often represented by a Fraenkel spring, given by40 Fijs = −k s r ij −r 0 rˆ ij, (6) where k s is the spring force constant and r 0 is the equilibrium bond length. In contrast to the other forces in Eq. (1), Fijs does not vanish when r ij ≥ Rc so that bead connectivity is maintained throughout the simulation. A reduced system of units is typically employed in DPD simulations, and the unit of length in our simulations translates to 1.73 nm since the edge of the smallest cubic cell employed here measured 39.5 nm. All simulations were performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software package41 using a microcanonical ensemble (possessing constant mass, volume, and energy) with the system thermostat set to 373 K, at which temperature k BT was assigned unit thermal energy. To ensure coarse-grain consistency with previous DPD studies of diblock35,42 and cyclic43 block copolymers, the spring parameters were set to k s = 8.0 and r 0 = 0.8. The number bead density, ρ, ¯ and γ were held constant at 3.0 and 4.5, respectively, and the pair-repulsion parameters were calculated for the specific case of a triblock copolymer composed of styrene (A) and ethylene-co-butylene (B) moieties. According to Groot and co-workers,44 the values of the like pair-repulsion parameters (αAA and αBB) are each 25.0, as calculated from 75k BT/ ρ. ¯ The pair repulsion parameter between dissimilar beads (αAB) is commonly computed from αAA + 3.27 1 + 3.9/N 0.51 χAB, where N is the number of beads per chain and χAB is the FloryHuggins interaction parameter, which can be estimated45 from V (δ A − δ B)2/k BT, where V is the bead volume and δ i represents the solubility parameter of species i (where i = A or B). These pure-component properties are accurately approximated for nonpolar species from group contribution methods46 to be δ A = 21.59 MPa1/2 and δ B = 16.97 MPa1/2. For the desired level of coarse graining, V corresponded to about 10 styrenic units (1000 cm3/mol), or about 1.725 nm3, which is consistent with the DPD results reported earlier by Maly and co-workers.42 Thus, for moderately segregated chains composed of 50 beads, αAB was determined to be 48.3. Random chain structures incorporating these parameters were initially generated from the chain.py routine in the LAMMPS computational package41 and then subjected to equilibration runs of 100 000 steps with periodic boundary conditions. Subsequent production runs were performed for 500 000 steps with a time step of about 2 ps in real time,
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thereby yielding a total run time of ≈1 µs. All simulations were run on the Henry2 cluster at the NC State High Performance Computing Center. Simulations were conducted for systems containing copolymer chains systematically differing in length (N), segregation level (αAB), and molecular composition (Φ A), where Φ A corresponds to the fraction of A beads in a copolymer chain and is varied to examine the role of morphology on network formation: spherical (Φ A = 0.2), cylindrical (Φ A = 0.3), and lamellar (Φ A = 0.5). Post-simulation clustering strategy
To systematically identify copolymer microdomains from the simulation trajectories, we have elected to use DBSCAN38 since single- and complete-linkage hierarchical clustering methods are either too sensitive to background noise or insufficiently sensitive to a broad range of existing clusters. This pattern-recognition algorithm ensures reachability while enforcing a minimum cluster size, without presuming a particular cluster shape (such as spherical) a priori. This algorithm is robust and versatile, as evidenced by its use in a variety of contemporary applications, such as identifying broadband internet user time preference,47 lipid rafts in mixed lipid bilayers,48 and objects in noisy video footage,49 as well as elucidating chromatin interactions in genes.50 In essence, the DBSCAN protocol initially defines each bead to lie in a different cluster. Two independent parameters are then used to establish the existence of an actual cluster: Pmin, the minimum number of observations required to classify a grouping as a legitimate cluster, and ε, the reachability that defines the span over which cluster-based linkages persist. If beads meet the conditions required by both parameters, they are allocated to the core of a cluster. If the conditions set by ε are satisfied but those of Pmin are not, the corresponding beads are assigned to a cluster boundary. Beads that do not meet either condition are considered noise and are rejected from being part of any of the clusters. In terms of triblock copolymer melts, clusters correspond to endblock-rich microdomains, whereas the noise from snapshots of DPD simulations identifies beads belonging to the midblock-rich matrix (depending on the copolymer composition, since the morphology might be lamellar). Our adaptation of DBSCAN forces microdomains to obey periodic boundary conditions in all three dimensions, in which case bonded beads along an endblock that crosses a periodic wall in the simulation box must be unwrapped and are identified as beads located in the same cluster (microdomain). In this study, the DBSCAN cluster analysis employs the flexible procedures for clustering (fpc) package within the R statistical software program51 on a Windows 64-bit configuration machine.
RESULTS AND DISCUSSION Density-based clustering parameterization
A nontrivial challenge associated with using the DBSCAN to systematically identify copolymer microdomains lies in identifying sensible values of Pmin and ε, which tend to be specific for the intended application. For instance, in
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FIG. 2. Radial pair distribution function for DPD systems containing 500 copolymer chains varying in length (N ) at constant copolymer composition (Φ A = 0.2). The color-coded legend for N is provided in the inset, which is an enlargement of the boxed region to highlight the intersections indicated by arrows. Also included for the case of N = 50 is a DPD system containing 2000 chains (bold dashed black line). The solid lines serve to connect the data.
a recent study50 wherein DBSCAN was used to elucidate chromatin interactions in a genome, the optimal parameter values selected for identifying clusters have been set equal to the number of detected intra-chromosomal paired-end-tag clusters that are significant at a false discovery rate of 70. It is interesting to note that f B becomes independent of αAB when αAB ≈ 50. Since f L , f D , and f M continue to evolve with increasing αAB beyond this value, we propose that unsegregated chains and dangles prefer to transform into loops rather than bridges for body-centered cubic micelles. Comparison of Figures 5 and 6 also confirms that the transition to strong segregation in ABA melts is preceded by nonlinearly decaying f D and f M , as well as a corresponding increase in f L . Since αAB is related to χAB and χAB generally depends on reciprocal
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FIG. 6. Dependence of the midblock fractions (labeled and color-coded) on the pair repulsion parameter (α AB) at N = 50 and Φ A = 0.2 for DPD systems containing 500 chains. The error bars correspond to the standard error, and the solid lines serve to connect the data.
temperature, it stands to reason that all three midblock conformations should change in an opposite fashion from what is observed in Figure 6 as the temperature is increased. An increase in T ∗ introduces more random forces (Eq. (4)) and decreases the conservative forces (Eq. (2)), thereby reducing the net driving force for microphase separation, especially as the order-disorder transition is approached. This behavior is qualitatively similar to that obtained from on-lattice Monte Carlo simulations of molecularly asymmetric A1BA2 triblock copolymers.26 Although many triblock copolymers intended for commercial use as TPEs possess a relatively small endblock fraction to ensure the formation of a dispersed morphology (i.e., spherical or cylindrical micelles) within a continuous rubbery matrix, copolymer composition constitute an important consideration since changes in morphology relate directly to variations in interfacial area. That is, the spherical micellar morphology possesses the highest interfacial area density of the classical morphologies. With N held constant, interfacial area density monotonically decreases for the cylindrical and lamellar morphologies as Φ A is increased, in which case we anticipate a systematic variation in midblock conformations. To test this hypothesis, values of f B, f L , f D , and f M extracted from simulations conducted at αAB = 48.3 for 500 chains with N = 50 are displayed as functions of Φ A in Figure 6. Included in this figure are ranges of f B reported from SCFT,29,30 MC simulations,31 and experimental studies.27,28 While the trends observed in Figure 6 are consistent with expectation and are certainly comparable, the DPD results significantly underestimate the value of f B evaluated for dispersed morphologies. Once again, though, we attribute this difference primarily to the presence of dangles, which account for about 10% of the midblock conformations in the spherical and cylindrical morphologies. Renormalization of f B and f L to eliminate f D and f M (so that f B + f L = 1) yields results that are in closer quantitative agreement with previous findings (Figure 6). As anticipated on the basis of considerations such as the interfacial area density, both f B and f M decrease, whereas f L increases, monotonically with increasing Φ A. Note that
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the onset of strong segregation, wherein the fraction of dangles and unsegregated chains becomes negligible. Quantitative comparison of the results obtained here with previous results reported for the midblock bridging fraction illustrates some discrepancy for the case of the dispersed (spherical and cylindrical) morphologies, which is most likely due to the exclusion of dangles in theoretical formalisms derived on the basis of equilibrium thermodynamics. Because the clustering methodology employed here is systematic and robust, it can be applied to a wide range of triblock and higher-order multiblock copolymer systems in which network formation is of considerable interest. ACKNOWLEDGMENTS FIG. 7. Dependence of the midblock fractions (labeled and color-coded) on Φ A at N = 50 for DPD systems containing 500 chains. Bridging and looping data renormalized to exclude f D and f M (so that f B + f L = 1) are included (O) for comparison with previous theoretical,29,30 simulation,31 and experimental27,28 results (gray regions). The error bars correspond to the standard error, and the solid lines serve to connect the data.
f D does not change very much with increasing Φ A relative to the maximum standard error, whereas the corresponding reduction in f M most likely reflects more complete microphase segregation as Φ A is increased. Another morphological aspect that must be considered in the interpretation of the midblock fractions is the system dimensionality, since the diffusion of endblocks to microdomains is facilitated in three dimensions (spherical micelles) relative to two dimensions (cylinders) or one dimension (lamellae). While results similar to those evident in Figure 7 have also been acquired by increasing N to 75 and 100, f M becomes increasingly negligible in those cases, which supports the renormalization-based comparison discussed above. CONCLUSIONS
Since thermoplastic elastomers constitute a fundamentally and technologically important class of soft materials, the objective of this study is to elucidate the midblock conformations of microphase-ordered ABA triblock copolymers varying in size, segregation, and composition through the use of DPD simulations and a density-based cluster algorithm. We have found that the parameters employed during cluster recognition can be conveniently estimated from the radial pair distribution function of microdomain-forming beads, in which case no prior knowledge of the cluster size or number is required. The DPD simulations performed here yield classical copolymer morphologies (i.e., body-centered cubic spherical micelles, hexagonally packed cylinders and lamellae), which are then analyzed in terms of bead clustering to extract the fractions associated with the possible midblock conformations: bridges, loops, dangles, and unsegregated chains. These fractions are evaluated as functions of several important material/simulation considerations, namely, chain length, system size, bead repulsion, and copolymer composition. While system size is determined to have relatively little effect on the midblock conformations in all the cases examined, systematic variations in chain length and bead repulsion reveal
This study was supported by the Nonwovens Institute at North Carolina State University, Grant No. 10-127. The authors thank Professor A. Motsinger-Reif and Mr. J. Majikes for technical assistance, and Dr. H. Chintakunta for insightful discussions with regard to machine learning. 1I.
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