Continuous Thermodynamic Integration in Field‐Theoretic Simulations of Structured Polymers

Abstract

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 (...

Fluctuation Effects in Semiflexible Diblock Copolymers

“…However, experiments observe an elevated order–disorder transition χ ODT over the mean-field predictions. Such quantitative discrepancy is partially corrected by incorporating finite molecular weight effects with concentration fluctuations. − Molecular simulations and experiments show agreement with the fluctuation-corrected theories at relatively high molecular weights. ,,,− However, nonuniversal phase behavior at low molecular weights is also noted by recent experiments …”

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation

“…Another related issue is how to adjust the size of the simulation box to be commensurate with the equilibrium period of an ordered phase. For the lamellar and cylinder phases, the period can determined by minimizing the free energy with respect to the aspect ratio of the simulation box subject to a constant volume constraint [ 50 , 66 ]. Unfortunately, this strategy cannot be extended to triply-periodic phases.…”

“…Beardsley and Matsen [ 48 ] located the ODT of symmetric diblocks using a slight variation of the technique where the ensemble average and integration are performed simultaneously [ 66 ]. Figure 3 shows the resulting free energy difference between the lamellar and disordered phases, , for diblocks of and in a cubic simulation box of size .…”

Computationally Efficient Field-Theoretic Simulations for Block Copolymer Melts

“…This is done using continuous thermodynamic integration, 36 where the integration and ensemble averaging is performed simultaneously by incrementing λ between successive Langevin steps. Figure 7a shows F (λ) obtained from MC-FTS and L-FTS (solid curves), using the same relatively large step size of dλ = 10 −3 .…”

“…The fact that the slopes, obtained from continuous thermodynamic integrations of the same phase in opposite directions, are so close to zero testifies to the high accuracy of the χ b integrations. 36 There are, nevertheless, a couple of instances where the slope for the two lamellar phases is slightly positive or negative, but this is only because they happen to have different periods.…”