Model and methods

This protocol is extracted from research article:

Shear-solvo defect annihilation of diblock copolymer thin films over a large area

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Sci Adv**,
Jun 14, 2019;
DOI:
10.1126/sciadv.aaw3974

Shear-solvo defect annihilation of diblock copolymer thin films over a large area

Procedure

Advanced rare-event sampling technique named as the string method was used to investigate the kinetic energy variation during the defect annihilation in BCP thin films, in combination with the TICG model. The TICG model takes advantage of having explicit polymer chains expressed with Gaussian bead spring model, while nonbonded Hamiltonian is evaluated based on polymeric field theoretic form: ${H}_{\text{nb}}=\frac{{\mathrm{\rho}}_{0}{k}_{\mathrm{B}}T}{N}{\int}_{V}\mathit{dr}(\mathrm{\chi}N{\mathrm{\varphi}}_{\mathrm{A}}(r){\mathrm{\varphi}}_{\mathrm{B}}(r)+\frac{\mathrm{\kappa}N}{2}{({\mathrm{\varphi}}_{\mathrm{A}}(r)+{\mathrm{\varphi}}_{\mathrm{B}}(r)-1)}^{2})$, where ${\mathrm{\varphi}}_{\mathrm{\alpha}}=\frac{{\mathrm{\rho}}_{\mathrm{\alpha}}}{{\mathrm{\rho}}_{0}}$ denotes the normalized number density of species α and ρ_{0}, χ, and κ are the average bulk number density of beads, Flory parameter, and the inverse compressibility, respectively. A string connecting defective and defect-free structures with a contour variable (reaction coordinate α) varying from 0 and 1 was defined on a multidimensional collective variable (CV) space constructed with density fields on a spatial grid over the simulation box. Each component of CV **m** is defined by normalized order parameter $\frac{{\mathrm{\varphi}}_{\mathrm{A}}({r}_{\mathrm{i}})-{\mathrm{\varphi}}_{\mathrm{B}}({r}_{\mathrm{i}})}{{\mathrm{\varphi}}_{\mathrm{A}}({r}_{\mathrm{i}})+{\mathrm{\varphi}}_{\mathrm{B}}({r}_{\mathrm{i}})}$, where ϕ_{A}(*r*_{i}) and ϕ_{B}(*r*_{i}) are density values of A and B segments at the grid of *r*_{i}. Among many possible strings connecting two states, the MFEP was defined with a string along which the gradient of free energy disappears. At each iteration of updating the string, a free energy gradient at 128 nodes uniformly distributed along the string was evaluated by umbrella sampling restraining TICG Monte Carlo simulations to be around corresponding CV **m** by adding the additional potential, ${H}_{\mathrm{c}}({r}^{\mathit{nN}}\mid \mathbf{m})=\frac{\mathrm{\lambda}{k}_{\mathrm{B}}T}{2}\mathrm{\Delta}{L}^{3}{[\mathbf{m}-\widehat{\mathbf{m}}]}^{2}$. $\widehat{\mathbf{m}}$ is the CV value for particle coordinates, *r ^{nN}*. Then, the free energy gradient can be estimated by $\mathrm{\lambda}\mathrm{\Delta}{L}^{3}{k}_{\mathrm{B}}T[\mathbf{m}-{\u3008\widehat{\mathbf{m}}\u3009}_{c}]\stackrel{\mathrm{\lambda}\to \infty}{\to}\frac{\mathcal{F}}{\partial \mathbf{m}}$. The string was updated using Euler technique, ${\mathbf{m}\mid}^{1}={\mathbf{m}\mid}^{0}-\mathrm{\tau}\mathrm{\lambda}[{\mathbf{m}\mid}^{0}-{\u3008\widehat{\mathbf{m}}\u3009}_{c}]$, at every node. Once the string was converged, free energy values at every node were estimated from $\Delta \mathcal{F}[\mathrm{\alpha}]={\int}_{0}^{\mathrm{\alpha}}\mathit{ds}{\frac{\mathrm{\partial}\mathcal{F}}{\mathrm{\partial}\mathbf{m}}\mid}_{\mathbf{m}(s)}\cdot \frac{d\mathbf{m}(s)}{\mathit{ds}}$. Our simulations were conducted for lamellar-forming symmetric BCP thin films, which is shown to have qualitatively similar defect annihilation behavior as cylinder-forming cases (

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