Monte Carlo Field-Theoretic Simulations for Melts of Symmetric Diblock Copolymer

Stasiak, Pawel; Matsen, Mark W.

doi:10.1021/ma401687j

Cited by 39 publications

(77 citation statements)

References 30 publications

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“…It is reasonable to expect the saddle-point approximation for W + (r) to be accurate, and indeed there are a couple studies indicating this to be true. 21,24 Stasiak and Matsen 25 have managed to perform 3D MC-FTS down to N̅ = 10 4 , which is tantalizingly close to the experimental regime. In doing so, however, they encountered an ultraviolet (UV) divergence, where increasing the mesh resolution (thereby allowing shorter wavelength fluctuations) tends to disorder the melt.…”

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confidence: 73%

“…Rather than integrating over fluctuations in the pressure field, MC-FTS applies the saddle-point approximation (i.e., W + → w + ), 21,25 whereby the partition function…”

Section: −10mentioning

confidence: 99%

“…Details of the numerical techniques are given in ref 25. In short, there are two computational challenges to deal with.…”

Section: −10mentioning

confidence: 99%

“…The fast Fourier transforms for our smaller system sizes (e.g., m ≲ 24) are calculated on central-processing units (CPUs), while for larger systems we use graphics processor units (GPUs). 29 The other challenge is locating the saddle point, w + (r), for which we employ the Anderson mixing scheme in ref 25. In the current study, however, a more stringent error tolerance of ϵ = 10 −4 is imposed.…”

Section: −10mentioning

confidence: 99%

“…Ideally, FTS results would become independent of the spatial grid at a sufficiently fine resolution, but this is prevented by the UV divergence mentioned above. In ref 25, the divergence was compensated for by expressing results in terms of an effective interaction parameter originally derived by Olvera de la Cruz et al…”

Section: −10mentioning

confidence: 99%

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Field-Theoretic Simulation of Block Copolymers at Experimentally Relevant Molecular Weights

2015

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Field-theoretic simulation (FTS) offers an efficient means of predicting the equilibrium behavior of high-molecular-weight structured polymers, provided one is able to deal with the strong ultraviolet (UV) divergence that occurs at realistic molecular weights. Here melts of lamellar-forming diblock copolymer are studied using a Monte Carlo version (MC-FTS), where the composition field fluctuates while the pressure field follows the mean-field approximation. We are able to control the UV divergence by introducing a new effective Flory− Huggins interaction parameter, χ e , thereby permitting MC-FTS for molecular weights extending down to values characteristic of experiment. Results for the disordered-state structure function, the layer spacing and compressibility of the ordered lamellar phase, and the position of the order−disorder transition (ODT) show excellent agreement with recent particle-based simulation. Given the immense versatility of FTS, this opens up the opportunity for quantitative studies on a wide range of more complicated block copolymer systems.

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mentioning

confidence: 73%

“…Rather than integrating over fluctuations in the pressure field, MC-FTS applies the saddle-point approximation (i.e., W + → w + ), 21,25 whereby the partition function…”

Section: −10mentioning

confidence: 99%

“…Details of the numerical techniques are given in ref 25. In short, there are two computational challenges to deal with.…”

Section: −10mentioning

confidence: 99%

Section: −10mentioning

confidence: 99%

Section: −10mentioning

confidence: 99%

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Field-Theoretic Simulation of Block Copolymers at Experimentally Relevant Molecular Weights

2015

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Continuous Thermodynamic Integration in Field‐Theoretic Simulations of Structured Polymers

Spencer

Vorselaars

Matsen

2017

Macro Theory & Simulations

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transitions, which can take prohibitively long to occur. The use of TI in field-based polymer simulations was pioneered by Lennon et al., who used it to calculate the phase diagram for a diblock copolymer melt. [2] Their method was adapted from the particle-based method of Frenkel and Ladd, [3] where free energies are computed by integrating from an Einstein crystal (EC) reference state. Calculating the free energy relative to a reference state with known free energy allows for the evaluation of absolute, not merely relative, free energies. Using a reference state also facilitates the examination of order-order transitions, unlike methods that rely on external ordering fields to create closed loops in configuration space. [4] The work of Lennon et al. was extended by Delaney and Fredrickson to more carefully evaluate the phase diagram, as well as the compression modulus of the lamellar phase (LAM) and the behavior of homopolymers. [5] In the traditional TI method, employed in the work described above, [2,5] a number of simulations are conducted over the range of the integration parameter. Each simulation calculates a thermodynamic average, which gives the derivative of the free energy at that point. These averages are then used to integrate the free energy. This can take many simulations and is generally computationally expensive. It also leads to complicated considerations for evaluating and managing statistical uncertainties. One must decide how long to equilibrate simulations before collecting statistics to reduce a systematic equilibration error; how long to collect statistics for averages, which contain a random error; and how many steps in parameter space are required for an acceptable finite-step error. The random error in thermodynamic averages propagates through the integral and is quantifiable using standard techniques. The equilibration and finite-step errors are more difficult to quantify.Here, we adopt an alternative TI scheme, involving only one simulation per integral. The integration variable is incremented slowly, throughout the simulation, and the integral is performed using instantaneous quantities, rather than their thermodynamic averages, as free energy derivatives. This continuous method is also known as adiabatic switching, slow growth, or single-configuration TI, and is well established in particle-based simulations. [6][7][8][9] In addition to its speed and simplicity, continuous TI also offers a simple way of quantifying and minimizing errors. The large number of steps, and thus small step size, renders finite-step errors negligible. Random errors are quantifiable and are typically small. [6] The main source of error is systematic and is due to the system being Field-Theoretic SimulationsThis work explores the use of continuous thermodynamic integration in fieldtheoretic simulations of a symmetric diblock copolymer melt. Free energies of the lamellar and disorder phases are evaluated by thermodynamic integration from a reference state (an Einstein crystal, λ = 0) to a diblock copolymer (...

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Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions

2016

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We review the latest developments in computational methods for direct simulation of fully fluctuating field theories of polymeric assemblies. In this context, we describe a newly developed theoretical and computational framework for accurately computing fluctuation-corrected phase diagrams of mesostructured polymer systems and report the first such complete phase diagram for a diblock copolymer melt. The method is based on complex Langevin sampling of a UV regularized field-theoretic model, with Helmholtz free energies computed using thermodynamic integration. UV regularization ensures that the free energies do not have an arbitrary reference; they can be compared between incommensurate phases, permitting for the first time the computation of order-order transitions with fluctuation corrections. We further demonstrate that computed free energies are accurate in the disordered phase by comparison to perturbation theory on the one-loop level. Importantly, we note that our method uses no uncontrolled approximations beyond the initial definition of a coarse-grained molecular model for the polymer melt or solution. The method can be applied straightforwardly to melts and solutions containing multiple species with diverse polymer architectures.

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Monte Carlo Field-Theoretic Simulations for Melts of Symmetric Diblock Copolymer

Cited by 39 publications

References 30 publications

Field-Theoretic Simulation of Block Copolymers at Experimentally Relevant Molecular Weights

Field-Theoretic Simulation of Block Copolymers at Experimentally Relevant Molecular Weights

Continuous Thermodynamic Integration in Field‐Theoretic Simulations of Structured Polymers

Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions

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