Molecular dynamics simulations are used to study the phase behavior of a single linear multiblock copolymer with blocks of A- and B-type monomers under poor solvent conditions, varying the block length $N$, number of blocks $n$, and the solvent quality (by variation of the temperature $T$). The fraction $f$ of A-type monomers is kept constant and equal to 0.5, and always the lengths of A and B blocks were equal ($N_{A}=N_{B}=N$), as well as the number of blocks ($n_{A}=n_{B}=n$). We identify the three following regimes where: (i) full microphase separation between blocks of different type occurs (all blocks of A-type monomers form a single cluster, while all blocks of B-type monomers form another), (ii) full microphase separation is observed with a certain probability, and (iii) full microphase separation can not take place. For very high number of blocks $n$ and very high $N$ (not accessible to our simulations) further investigation is needed.
We studied structures formed by a single $(AB)_k$ multiblock copolymer chain, in which interaction between A-type beads was purely repulsive, and B-type beads tended to aggregate. We studied how attraction between A-type beads and B-type beads affected the structure of the chain. We discovered formation of an equilibrium globular structure, which had unique heterogeneous checkerboard-like distribution of contact density. Unlike the structures usually formed by a single $(AB)_k$ multiblock copolymer chain, this structure had contact enrichment at the boundaries of A and B blocks. This structure…
Microphase separation in random multiblock copolymers is studied with mean-field theory assuming that long blocks of a copolymer are strongly segregated, whereas short blocks are able to penetrate into "alien" domains and exchange between the domains and interfacial layer. A bidisperse copolymer with blocks of only two sizes (long and short) is considered as a model of multiblock copolymers with high polydispersity in the block size. Short blocks of the copolymer play an important role in microphase separation. First, their penetration into the "alien" domains leads to the formation of joint l…
We use a standard bead-spring model and molecular dynamics simulations to study the static properties of symmetric linear multiblock copolymer chains and their blocks under poor solvent conditions in a dilute solution from the regime close to theta conditions, where the chains adopt a coil-like formation, to the poorer solvent regime where the chains collapse obtaining a globular formation and phase separation between the blocks occurs. We choose interaction parameters as is done for a standard model, i.e., the Lennard-Jones fluid and we consider symmetric chains, i.e., the multiblock copolyme…
Molecular dynamics simulations are used to study the phase behavior of linear multiblock copolymers with two types of monomers, A and B, where the length of the polymer blocks $N_{A}$ and $N_{B}$ ($N_{A}=N_{B}=N$), the number of the blocks $n_{A}$ and $n_{B}$ ($n_{A}=n_{B}=n$), and the solvent quality varies. The fraction $f$ of A-type monomers is kept constant and equal to 0.5. Whereas at high enough temperatures these macromolecules form coil structures, where each block A or B forms rather individual clusters, at low enough temperatures A and B monomers from different blocks can join togeth…
We use molecular dynamics simulations to study the static properties of a single linear multiblock copolymer chain under poor solvent conditions varying the block length $N$, the number of blocks $n$, and the solvent quality by variation of the temperature $T$. We study the most symmetrical case, where the number of blocks of monomers of type A, $n_{A}$, equals that of monomers B, $n_{B}$ ($n_{A}=n_{B}=n/2$), the length of all blocks is the same irrespective of their type, and potential parameters are also chosen symmetrically, as for a standard Lennard-Jones fluid. Under poor solvent conditio…
The phase behaviour of amphiphilic multiblock copolymers with a large number of blocks in semidilute solutions is studied by lattice Monte Carlo simulations. The influence on the resulting structures of the concentration, the solvent quality and the ratio of hydrophobic to hydrophilic monomers in the chains has been assessed explicitely. Several distinct regimes are put in evidence. For poorly substituted (mainly hydrophilic) copolymers formation of micelles is observed, either isolated or connected by the hydrophilic moieties, depending on concentration and chain length. For more highly subst…
The extension elasticity of rod-coil mutliblock copolymers is analyzed for two experimentally accessible situations. In the quenched case, when the architecture is fixed by the synthesis, the force law is distinguished by a sharp change in the slope. In the annealed case, where interconversion between rod and coil states is possible, the resulting force law is sigmoid with a pronounced plateau. This last case is realized, for example, when homopolypeptides capable of undergoing a helix-coil transition are extended from a coil state. Both scenarios are relevant to the analysis and design of exp…
We investigate the localization of a hydrophobic - polar (HP) - regular copolymer at a selective solvent-solvent interface with emphasis on the impact of block length $M$ on the copolymer behavior. The considerations are based on simple scaling arguments and use the mapping of the problem onto a homopolymer adsorption problem. The resulting scaling relations treat the gyration radius of the copolymer chain perpendicular and parallel to the interface in terms of chain length N and block size M, as well as the selectivity parameter χ. The scaling relations differ for the case of weak and strong …
Multiblock copolymer chains in implicit nonselective solvents are studied by Monte Carlo method which employs a parallel tempering algorithm. Chains consisting of 120 $A$ and 120 $B$ monomers, arranged in three distinct microarchitectures: $(10-10)_{12}$, $(6-6)_{20}$, and $(3-3)_{40}$, collapse to globular states upon cooling, as expected. By varying both the reduced temperature $T^*$ and compatibility between monomers $ω$, numerous intra-globular structures are obtained: diclusters (handshake, spiral, torus with a core, etc.), triclusters, and $n$-clusters with $n>3$ (lamellar and other), wh…
The self-consistent field theory is a popular and highly successful theoretical framework for studying equilibrium (co)polymer systems at the mesoscopic level. Dynamic density functionals allow one to use this framework for studying dynamical processes in the diffusive, non-inertial regime. The central quantity in these approaches is the mobility function, which describes the effect of chain connectivity on the nonlocal response of monomers to thermodynamic driving fields. In a recent study [Mantha et al, Macromolecules 53, 3409 (2020)], we have developed a method to systematically construct m…
Polymeric micelles are used in a variety of applications, with the micelle's shape often playing an important role. Consequently, a scheme to design micelles of arbitrary shape is desirable. In this paper, we consider micelles formed from a single, linear, multiblock copolymer, and we study how easily the micelle's shape can be controlled by altering the copolymer block lengths. Using a rational design scheme, we identify a few aspects of the multiblock composition that are expected to have a well-behaved, predictable effect on micelle shape. Starting from a reference micelle composition, itse…
An extensive study of single block copolymer knots containing two kinds of monomers $A$ and $B$ is presented. The knots are in a solution and their monomers are subjected to short range interactions that can be attractive or repulsive. In view of possible applications in medicine and the construction of intelligent materials, it is shown that several features of copolymer knots can be tuned by changing the monomer configuration. A very fast and abrupt swelling with increasing temperature is obtained in certain multiblock copolymers, while the size and the swelling behavior at high temperatures…
We present a lattice Monte Carlo simulation for a multiblock copolymer chain of length N=240 and microarchitecture $(10-10)_{12}$.The simulation was performed using the Monte Carlo method with the Metropolis algorithm. We measured average energy, heat capacity, the mean squared radius of gyration, and the histogram of cluster count distribution. Those quantities were investigated as a function of temperature and incompatibility between segments, quantified by parameter ω. We determined the temperature of the coil-globule transition and constructed the phase diagram exhibiting a variety of patc…
We perform off-lattice, canonical ensemble molecular dynamics simulations of the self-assembly of long segmented copolymers consisting of alternating, tunably attractive and hydrophobic {\em binder} domains, connected by hydrophilic {\em linker} chains whose length may be separately controlled. In such systems, the molecular design of the molecule directly determines the balance between energetic and entropic tendencies. We determine the structural phase diagram of this system, which shows collapsed states (dominated by the attractive linkers' energies), swollen states (dominated by the random…
We consider the adsorption kinetics of a regular block-copolymer of total length $N$ and block size $M$ at a selective liquid-liquid interface in the limit of strong localization. We propose a simple analytic theory based on scaling considerations which describes the relaxation of the initial coil into a flat-shaped layer. The characteristic times for attaining equilibrium values of the gyration radius components perpendicular and parallel to the interface are predicted to scale with chain length $N$ and block length $M$ as $τ_{\perp} \propto M^{1+2ν}$ (here $ν\approx 0.6$ is the Flory exponen…
Composition fluctuations in disordered melts of symmetric diblock copolymers are studied by Monte Carlo simulation over a range of chain lengths and interaction strengths. Results are used to test three theories: (1) the random phase approximation (RPA), (2) the Fredrickson-Helfand (FH) theory, which was designed to describe large fluctuations near an order-disorder transition (ODT), and (3) a more recent renormalized one-loop (ROL) theory, which reduces to FH theory near the ODT, but which is found to be accurate over a much wider range of parameters.
Adsorption of a symmetric (AB) random copolymer (RC) onto a symmetric (ab) random heterogeneous surface (RS) is studied in the annealed approximation by using a two-dimensional partially directed walk model of the polymer. We show that in the symmetric case, the expected a posteriori compositions of the RC and the RS have correct values (corresponding to their a priori probabilities) and do not change with the temperature, whereas second moments of monomers and sites distributions in the RC and RS change. This indicates that monomers and sites do not interconvert but only rearrange in order to…
The localization kinetics of a regular block-copolymer of total length $N$ and block size $M$ at a selective liquid-liquid interface is studied in the limit of strong segregation between hydrophobic and polar segments in the chain. We propose a simple analytic theory based on scaling arguments which describes the relaxation of the initial coil into a flat-shaped layer for the cases of both Rouse and Zimm dynamics. For Rouse dynamics the characteristic times for attaining equilibrium values of the gyration radius components perpendicular and parallel to the interface are predicted to scale with…
Using fast off-lattice Monte Carlo simulations with experimentally accessible fluctuations, we report the first systematic study unambiguously quantifying the shift of the order-disorder transition (ODT) $χ^*$ of symmetric diblock copolymers from the mean-field prediction $χ^*_{\rm MF}$. Our simulations are performed in a canonical ensemble with variable box lengths to eliminate the restriction of periodic boundary conditions on the lamellar period. Exactly the same model system (Hamiltonian) is used in both our simulations and mean-field theory; the ODT shift is therefore due to the fluctuati…
We analyze the energetics of sphere-like micellar phases in diblock copolymers in terms of well-studied, geometric quantities for their lattices. We argue that the A15 lattice with Pm3n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find thatthe A15 phase is stable.
The effects of the length of each hydrophobic end block N_{st} and polymer concentration \barϕ_{P} on the transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent field lattice model. When the system is cooled, micelles are observed, i.e.,the homogenous solution (unimer)-micelle transition occurs. When N_{st} is increased, at fixed \barϕ_{P}, micelles occur at higher temperature, and the temperature-dependent range of micellar aggregation and half-width of specific heat peak for unimer-micelle transition increase monotonously. Compare…
Surface Block Identity Controls Transport of Symmetric Diblock Copolymer Through Nanopores
University of Pennsylvania（宾夕法尼亚大学）；Université de Lyon, CNRS, Université Claude Bernard Lyon 1, INSA Lyon, Université Jean Monnet（里昂大学、法国国家科学研究中心、里昂第一大学、里昂国立应用科学学院、让·莫内大学）；KU-KIST Graduate School of Converging Science and Technology, Korea University（高丽大学-韩国科学技术院融合科学与技术研究生院）
We present a model addressing the orientation transition of symmetric block copolymers such as PS/PMMA on smooth and rough surfaces. The distortion free energy of parallel and perpendicular lamellar phases in contact with a rough solid surface is calculated as function of the surface roughness amplitude and wavelength, as well as the polymer lamellar periodicity (molecular weight). We find an analytical expression for the orientation transition. This expression is compared and agrees well with recent experiments done with six different polymer molecular weights and surface preparations.
We propose a systematic coarse-grained representation of block copolymers, whereby each block is reduced to a single ``soft blob'' and effective intra- as well as intermolecular interactions act between centres of mass of the blocks. The coarse-graining approach is applied to simple athermal lattice models of symmetric AB diblock copolymers, in particular to a Widom-Rowlinson-like model where blocks of the same species behave as ideal polymers (i.e. freely interpenetrate), while blocks of opposite species are mutually avoiding walks. This incompatibility drives microphase separation for copoly…
Simulations of five different coarse-grained models of symmetric diblock copolymer melts are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy on the invariant degree of polymerization $\overline{N}$, and to study universal properties of the order-disorder transition (ODT). The ODT appears to exhibit two regimes: Systems of very long chains ($\overline{N} \gtrsim 10^{4}$) are well described by the Fredrickson-Helfand theory, which assumes weak segregation near the ODT. Systems of smaller but experimentally relevant values, $\overline{N} \lesssim 10^4$,…
The structure of lamellar phases of symmetric $AB$ diblock copolymers in a thin film is investigated. We quantitatively compare the composition profiles and profiles of individual segments in self-consistent field calculations with Monte Carlo simulations in the bond fluctuation model for chain length N=32 and $χN=30$. Three film thicknesses are investigated, corresponding to parallel oriented lamellae with 2 and 4 interfaces and a perpendicular oriented morphology. Taking account of capillary waves, we find good quantitative agreement between the Monte Carlo simulations and the self-consisten…
The Leibler weak segregation theory in molten diblock copolymers is generalized with due regard for the 2nd shell harmonics contributions defined in the paper and the phase diagrams are built for the linear and miktoarm ternary ABC triblock copolymers. The symmetric linear copolymers with the middle block non-selective with respect to the side ones are shown to undergo the continuous ODT not only into the lamellar phase but also into various non-conventional cubic phases (depending on the middle block composition it could be the simple cubic, face-centered cubic or non-centrosymmetric phase re…
We generalize the Fredrickson-Helfand theory of the microphase separation in symmetric diblock copolymer melts by taking into account the influence of a time-independent homogeneous electric field on the composition fluctuations within the self-consistent Hartree approximation. We predict that electric fields suppress composition fluctuations, and consequently weaken the first-order transition. In the presence of an electric field the critical temperature of the order-disorder transition is shifted towards its mean-field value. The collective structure factor in the disordered phase becomes an…
We investigate the phase behavior of symmetric AB diblock copolymers confined into a thin film. The film boundaries are parallel, impenetrable and attract the A component of the diblock copolymer. Using a self-consistent field technique [M.W. Matsen, J.Chem.Phys. {\bf 106}, 7781 (1997)], we study the ordered phases as a function of incompatibility $χ$ and film thickness in the framework of the Gaussian chain model. For large film thickness and small incompatibility, we find first order transitions between phases with different number of lamellae which are parallel oriented to the film boundari…
A nonequilibrium molecular dynamics computer simulation on microsegregated solutions of symmetrical diblock copolymers is reported. As the polymer concentration increases, the system undergoes phase transitions in the following order: body centered cubic (BCC) micelles, hexagonal (HEX) cylinders, gyroid (GYR) bicontinuous networks, and lamellae (L), which are the same morphologies that have been reported for block copolymer melts. Structural classification is based on the patterns of the anisotropic static structure factor and characteristic 3-dimensional images. The systems in the BCC micella…
Using the self-consistent field lattice model, polymer concentration $\barϕ_{P}$ and chain length $N$ (keeping the length ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When chain length is increased, at fixed $\barϕ_{P}$, micelles occur at higher temperature. The variations of average volume fraction of stickers $\barϕ_\textrm{co}^\textrm{s}$ and the lattice site numbers $N_\textrm{co}^\textrm{ls}$ at the micellar cores with temperature are dependent on $N$ a…
We examine the phase transition of polymer adsorption as well as the underlying kinetics of polymer binding from dilute solutions on a structureless solid surface. The emphasis is put on the properties of regular multiblock copolymers, characterized by block size M and total length N as well as on random copolymers with quenched composition p of sticky and neutral segments. The macromolecules are modeled as coarse-grained bead-spring chains subject to a short-ranged surface adhesive potential. Phase diagrams, showing the variation of the critical threshold for single chain adsorption in terms …
Coarse-grained theories of dense polymer liquids such as block copolymer melts predict a universal dependence of equilibrium properties on a few dimensionless parameters. For symmetric diblock copolymer melts, such theories predict a universal dependence on only chi N and Nbar, where chi is an effective interaction parameter, N is a degree of polymerization, and Nbar is a measure of overlap. We test whether simulation results for the structure factor S(q) obtained from several different simulation models are consistent with this two-parameter scaling hypothesis. We compare results from three m…
In this work we used dissipative particle dynamics simulations to study the copolymerization process in the presence of spatial heterogeneities caused by incompatibility between polymerizing monomers. The polymer sequence details as well as the resulting system spatial structure in the case if phase segregation occurs during the chain growth can be predicted using the method. We performed the model verification with the available literature data on styrene-acrylic acid copolymerization in the bulk and a very good agreement between experimental and simulated data for both chain average composit…
In the present work, we study morphologies arising due to competing substrate interaction, electric field and confinement effects on a symmetric diblock copolymer. We employ a coarse grained non-local Cahn-Hilliard phenomenological model taking into account the appropriate contributions of substrate interaction and electrostatic field. The proposed model couples the Ohta-Kawasaki functional with Maxwell equation of electrostatics, thus alleviating the need for any approximate solution used in previous studies. We calculate the phase diagram in electric field-substrate strength space for differ…
We study a model of flexible block copolymers (BCPs) in which there is an enlthalpic preference for local alignment, among like-block segments. We describe a generalization of the self-consistent field theory (SCFT) of flexible BCPs to include inter-segment orientational interactions via a Landau-DeGennes free energy associated with a polar or nematic order parameter for segments of one component of a diblock copolymer. We study the equilibrium states of this model numerically, using a pseudo-spectral approach to solve for chain conformation statistics in the presence of a self-consistent torq…
Diffusion of symmetric diblock copolymer chains in macroscopically oriented lamellar block copolymers are studied in a molecular dynamics simulation. Results for diffusion constant both parallel $D_\parallel$ and perpendicular $D_\perp$ to the lamellar planes are compared to results in the disordered one phase region. For diblocks of length N=40 (1.1 N_e) and $100 (3N_e)$, where $N_e$ is the entanglement length of a homopolymer melt at the same density, $D_\parallel$ is nearly independent of $χ$, while $D_\perp$ is strongly suppressed as $χ$ is increased. These results are in agreement with th…
A coarse-graining strategy for dilute and semi-dilute solutions of interacting polymers, and of colloid polymer mixtures is briefly described. Monomer degrees of freedom are traced out to derive an effective, state dependent pair potential between the polymer centres of mass. The cross-over between good and poor solvent conditions is discussed within a scaling analysis. The method is extended to block copolymers represented as "necklaces" of soft "blobs", and its success is illustrated here in the case of a symmetric diblock copolymer which exhibits microphase separation.
We inquire about the possible coexistence of macroscopic and microstructured phases in random Q-block copolymers built of incompatible monomer types A and B with equal average concentrations. In our microscopic model, one block comprises M identical monomers. The block-type sequence distribution is Markovian and characterized by the correlation λ. Upon increasing the incompatibility χ (by decreasing temperature) in the disordered state, the known ordered phases form: for λ > λ_c, two coexisting macroscopic A- and B-rich phases, for λ < λ_c, a microstructured (lamellar) phase with wave number k…
Nanocomposite materials made of block copolymer and nanoparticles display properties which can be different from the purely polymeric matrix. The resulting material is a crossover of the original properties of the block copolymer and the presence of the assembled nanoparticles. We propose a mesoscopic study via cell dynamic simulations to quantitatively assert the properties of such hybrid materials. The most relevant parameters are identified to be the fraction of nanoparticles in the system and its chemical affinity, while the nanoparticle size with respect to the block copolymer length scal…
Using the self-consistent field approach, the effect of asymmetry of the coil block on the microphase separation is focused in ABC coil-rod-coil triblock copolymers. For different fractions of the rod block $f_{\text B}$, some stable structures are observed, i.e., lamellae, cylinders, gyroid, and core-shell hexagonal lattice, and the phase diagrams are constructed. The calculated results show that the effect of the coil block fraction $f_{\text A}$ is dependent on $f_{\text B}$. When $f_{\text B}=0.2$, the effect of asymmetry of the coil block is similar to that of the ABC flexible triblock co…
A microscopic theory for coarse graining diblock copolymers into dumbbells of interacting soft colloidal particles has been developed, based on the solution of liquid-state integral equations. The Ornstein-Zernike equation is solved to provide a mesoscopic description of the diblock copolymer system at the level of block centers of mass, and at the level of polymer centers of mass. Analytical forms of the total correlation functions for block-block, block-monomer, and center-of-mass pairs are obtained for a liquid of structurally symmetric diblock copolymers as a function of temperature, densi…
A twin boundary (TB) is a common low energy planar defect in crystals including those with the atomic diamond structure (C, Si, Ge, etc.). We study twins in a self-assembled soft matter block copolymer (BCP) supramolecular crystal having the double diamond (DD) structure, consisting of 2 translationally shifted, interpenetrating diamond networks of the minority polydimethyl siloxane block embedded in a polystyrene block matrix. The coherent, low energy, mirror-symmetric double tubular network twin has one minority block network with its nodes offset from the (222) TB plane while nodes of the s…
We present a hybrid numerical method to introduce hydrodynamics in dynamic self-consistent field (SCF) studies of inhomogeneous polymer systems. It solves a set of coupled dynamical equations: The Navier-Stokes equations for the fluid flow, and SCF-based convection-diffusion equations for the evolution of the local monomer compositions. The Navier-Stokes equaitons are simulated by the lattice Boltzmann method and the dynamic self-consistent equations are solved by a finite difference scheme. Two applications are presented: First, we study microphase separation in symmetric and asymmetric block…
We study the effect of a random Flory-Huggins parameter in a symmetric diblock copolymer melt which is expected to occur in a copolymer where one block is near its structural glass transition. In the clean limit the microphase segregation between the two blocks causes a weak, fluctuation induced first order transition to a lamellar state. Using a renormalization group approach combined with the replica trick to treat the quenched disorder, we show that beyond a critical disorder strength, that depends on the length of the polymer chain, the character of the transition is changed. The system be…
Reactions at a strongly segregated interface of a symmetric binary polymer blend are investigated via Monte Carlo simulations. End functionalized homopolymers of different species interact at the interface instantaneously and irreversibly to form diblock copolymers. The simulations, in the framework of the bond fluctuation model, determine the time dependence of the copolymer production in the initial and intermediate time regime for small reactant concentration $ρ_0 R_g^3=0.163 ... 0.0406$. The results are compared to recent theories and simulation data of a simple reaction diffusion model. F…
A lattice model for a symmetrical copolymer "bottle brush" molecule, where two types (A,B) of flexible side chains are grafted with one chain end to a rigid backbone, is studied by a variant of the pruned-enriched Rosenbluth method (PERM), allowing for simultaneous growth of all side chains in the Monte Carlo sampling. Choosing repulsive binary interactions between unlike monomers and varying the solvent quality, it is found that phase separation into an $A$-rich part of the cylindrical molecule and a $B$-rich part can occur only locally. Long range order (in the direction of the backbone) doe…
By means of classical density functional theory and its dynamical extension, we consider a colloidal fluid with spherically-symmetric competing interactions, which are well known to exhibit a rich bulk phase behavior. This includes complex three-dimensional periodically ordered cluster phases such as lamellae, two-dimensional hexagonally packed cylinders, gyroid structures or spherical micelles. While the bulk phase behavior has been studied extensively in earlier work, in this paper we focus on such structures confined between planar repulsive walls under shear flow. For sufficiently high she…
We present numerical calculations of lamellar phases of block copolymers at patterned surfaces. We model symmetric di-block copolymer films forming lamellar phases and the effect of geometrical and chemical surface patterning on the alignment and orientation of lamellar phases. The calculations are done within self-consistent field theory (SCFT), where the semi-implicit relaxation scheme is used to solve the diffusion equation. Two specific set-ups, motivated by recent experiments, are investigated. In the first, the film is placed on top of a surface imprinted with long chemical stripes. The …
Co-assembly of inorganic nanoparticles (NPs) and nanostructured polymer matrix represents an intricate interplay of enthalpic or entropic forces. Particle size largely affects the phase behavior of the nanocomposite. Theoretical studies indicate that new morphologies would emerge when the particles become comparable to the soft matrix's size, but this has rarely been supported experimentally. By designing a multicomponent blend composed of NPs, block copolymer-based supramolecules, and small molecules, a 3-D ordered lattice beyond the native BCP's morphology was recently reported when the part…
