# (305d) Free Energy Contributions to Template-Assisted Self-Assembly of Sub-10 Nm Particles from Steered Molecular Dynamics Simulations

#### AIChE Annual Meeting

#### 2022

#### 2022 Annual Meeting

#### Engineering Sciences and Fundamentals

#### Computational Studies of Early-Stage and Low-Dimensional Self-Assembly

#### Tuesday, November 15, 2022 - 1:15pm to 1:30pm

Calculation of free energy change between two states, in the liquid film and on the templated surface, for a self-assembled nanoparticle will be immensely aided in locating the optimal yield. Various molecular-simulation-based approaches such as probability ratio method (PRM) in molecular dynamics (MD) studies aimed at estimating the free energy have been reported. PRM tracks the positions of neighboring molecules as a function of time, which are transformed into a probability distribution function with respect to the states of tracked nanoparticle. The probability distribution function is then used to calculate the relative free energy. Nevertheless, reports indicate that PRM may suffer from insufficient sampling, in addition to the unreliable estimates of the probability density distribution across the surface. An alternative method to the PRM is the thermodynamic integration (TI) in which the position-dependent potential of mean force, as a representative of the free energy, acting on the molecule of interest at different positions is integrated over the distances of constraint forces. Both PRM and TI are equilibrium methods that require the simulation environment to be at equilibrium during the transition of tracked molecule. This means the simulation needs to regain an equilibrium state after every movement of tracked molecule. As such, aforementioned techniques seem not suitable for measuring free energy contributions to the DSA-n, considering that the process is continuously out of equilibrium. The lengthy equilibration processes can therefore lead to a high computational cost.

Enlightened by the Jarzynskiâ€™s equality in statistical mechanics, the steered molecular dynamics (SMD) extends the realm of free energy calculation to non-equilibrium simulations at a reasonable computational cost. SMD simulations have been widely applied in revealing mechanical process of protein binding, folding, and stretching. Recently some studies have also attempted to apply SMD simulations to calculate the free energy by employing Jarzynskiâ€™s equality. Jarzynskiâ€™s equality is a relation that explains the free energy between two equilibrium states, âˆ†E12, in terms of the external work done through a non-equilibrium process between above-mentioned two states, W12. The work is obtained from an ensemble of finite-time measurements of external work performed on the system during the transition, i.e.

exp(âˆ’Î²âˆ†E12) = âŸ¨exp(âˆ’Î²W12)âŸ© (1)

where Î² = 1/(k_{B}T), k_{B} is the Boltzmann constant, T is the temperature, and âŸ¨.âŸ© represents averaging over independent simulations. Applying Jarzynskiâ€™s equality, SMD simulations enable free energy calculations from non-equilibrium processes. To obtain an accurate estimate of free energy, one needs to collect enough samples as the average of exponential term in Equation 1 requires a large number of independent SMD simulations.

The purpose of this work is to utilize Jarzynskiâ€™s equality to calculate the free energy contributions to the DSA-n into templated surfaces from SMD simulations. For SMD simulations, we use many-body dissipative particle dynamics (MDPD). As coarse-gained molecular dynamics (MD) simulations, MDPD simulations are capable of combining both microscopic and mesoscopic interactions while capturing the properties of fluid interfaces at much larger scales than traditional MD simulations, such as those in evaporation-mediated DSA-n. Without loss of generality, we focus on directed self-assembly of a spherical nanoparticle into a circular nanocavity, which is etched out of a substrate with an otherwise flat surface. The self-assembly is observed in two cases: from (1) a stagnant bulk thin liquid film, and (2) a flowing thin liquid film with a receding interface, to the nanocavity. In the second case, to generate the flow field and receding interface in the thin liquid film, we model a hydrophilic surface with a downward moving plate. The transition of states is induced by attaching one end of a spring to a probe nanoparticle while pulling the other end towards the nanocavity in the substrate, mimicking the DSA-n as shown in Figure 1. Free energy variation is calculated as a function of pulling distance. All MDPD simulations are performed using LAMMPS package at a constant temperature in a manner that satisfy the Markov property and detailed balance. The initial states, which depends on the positions and velocities of the nanoparticles, liquid, and the probe nanoparticle are sampled from the canonical ensemble corresponding to the Hamiltonian of the particle system.

Our results suggest that overall the free energy contribution to DSA-n decreases with the nanoparticle density until a critical density after which the free energy increases. Moreover, we find that the free energy is minimum at this critical density whether the liquid film is stagnant or moving. The results indicate that at low nanoparticle density, it is entropically less favorable for the nanoparticle to be deposited in the nanocavity; however, at the critical density, we find that there is no change in the entropy as the nanoparticle leaves the bulk liquid and enters the nanocavity. Once the nanoparticle density goes beyond the critical density, there will be an energetic barrier for the nanoparticle deposition due to a favorable interaction with other nanoparticles in the bulk, which increases the free energy of DSA-n.

Figure 1. Panel A presents a top view of a liquid film with suspended nanoparticles (yellow beads) going over a templated surface used in SMD simulations. Side view of SMD simulations at 3 stages (from B to D) show the progression of a probe nanoparticle (red bead) deposition guided by a spring while the liquid film (not shown for clarity) is receding downward.