(525f) Free Energy Contributions to Template-Assisted Self-Assembly of Sub-10 Nm Particles from Steered Molecular Dynamics Simulations

Directed self-assembly of nanoparticles (DSA-n) over a templated surface has extensive applications in miniaturization of nanodevices, including semiconducting, plasmonic,and photovoltaic devices. The pre-designed templated surface is able to selectively capture nanoparticles, induce interaction-based orientation and even organize them into a complex structure using a bottom-up fabrication process. Taking advantage of the great potential in template-assisted DSA-n, many studies have utilized a thin liquid film sliding over a templated surface to realize the assembly process, which allows high yield of nanoparticle deposition with specific ordering or alignment. In DSA-n with sub-10 nm particles, the thin liquid film provides a restriction of long-range molecular interactions from multiple nanoparticle layers; that in turns facilitates the positioning of nanoparticles in DSA-n. Although the fabrication strategy of template-assisted DSA-n is able to achieve ordered arrays of nanoparticles over large areas of the templated surface, the yield of self-assembly is sensitive to the nanoparticle density among other parameters. This variation of yield at different nanoparticle density indicates that the density is one of the structural determinants of the efficiency of DSA-n, and plays a key role in the energetics of the complex self-assembly process. In this process, computational modeling has come illustrative to aid with explaining the contributions to the associated free energy.
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 each motion 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 Jarzynski’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 Jarzynski’s equality. Jarzynski’s equality is a relation that explains the free energy between two equilibrium states, ∆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(−β∆E12) = ⟨exp(−βW12)⟩ (1)
where β = (kBT)−1, kB is the Boltzmann constant, T is the temperature, and ⟨.⟩ represents averaging over independent simulations. Applying Jarzynski’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 Jarzynski’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 thin liquid film with flow field and 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 piston. The transition of states is induced by attaching one end of a spring to a tracer 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 tracer particle 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 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 tracer nanoparticle deposition guided by a spring while the liquid film is receding downward.