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

Vorselaars, Bart; Stasiak, Pawel; Matsen, Mark W.

doi:10.1021/acs.macromol.5b02286

Cited by 36 publications

(61 citation statements)

References 54 publications

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“…This is because large systems take longer to equilibrate, effectively increasing the work required to drag the system through parameter space and thereby increasing the systematic error. Our results are consistent with those found by Vorselaars et al, who bracketed the ODT by finding a range of χ b N over which DIS and LAM were metastable throughout long simulations . The thermodynamic integration approach was able to go to a larger system size than the metastability interval approach.…”

Section: Resultssupporting

confidence: 92%

“…It is simpler, however, to find the equilibrium spacing and fix the box dimensions. We therefore find the ODT at the spacing determined by Vorselaars et al, then find the LAM spacing using the box move, followed by a second determination of the ODT. In each case, the difference in ODT and in LAM spacing between the first and the second determination was small.…”

Section: Resultsmentioning

confidence: 99%

“…On the contrary, decreasing Δ leads to an increasing contribution to the free energy, which acts to disorder the melt. This ultraviolet (UV) divergence is explored more thoroughly in previous work . In order to compare with experiments, or even similar simulations with a different Δ, the diverging contribution to the free energy is absorbed into the monomer interaction energy, thereby renormalizing the bare interaction parameter, χ b , to obtain χ e .…”

Section: Theorymentioning

confidence: 99%

“…Subsequent work has concluded that once the simulation box is commensurate with the underlying microstructure, results are consistent between simulations of different size . This may not always be the case as nonmonotonic changes in the ODT with system size have been suggested in simulations, even when the box lengths can adopt the preferred domain size . That ODT determination was, however, not precise enough to be definitive for experimentally relevant molecular weights.…”

Section: Introductionmentioning

confidence: 99%

“…Calculating free energies using TI allows one to locate first‐order phase transitions without having to simulate structural transformations between phases. This is a major improvement over other simulation methods, such as metastability interval calculations, particularly for strongly first‐order 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 .…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Resultssupporting

confidence: 92%

Section: Resultsmentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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