A new contact model for the discrete element method simulation of $\hbox {TiO}_2$ nanoparticle films under mechanical load

Abstract

We develop a novel coarse-grained contact model for Discrete Element Method simulations of $\hbox {TiO}_2$ nanoparticle films subjected to mechanical stress. All model elements and parameters are derived in a self-consistent and physically sound way from all-atom Molecular Dynamics simulations of interacting particles and surfaces. In particular, the nature of atomic-scale friction and dissipation effects is taken into account by explicit modelling of the surface features and water adsorbate layers that strongly mediate the particle-particle interactions. The quantitative accuracy of the coarse-grained model is validated against all-atom simulations of $\hbox {TiO}_2$ nanoparticle agglomerates under tensile stress. Moreover, its predictive power is demonstrated with calculations of force-displacement curves of entire nanoparticle films probed with force spectroscopy. The simulation results are compared with Atomic Force Microscopy and Transmission Electron Microscopy experiments.

Appendix

In this Appendix we summarise the experimental as well as all computational and methodological details concerning the performed MD and DEM simulations.

1.1 AFM experiments

Highly porous $\hbox {TiO}_2$ nanoparticle films on mica substrates are produced using a flame spray reactor as described in detail in our previous works [13, 14]. Force-displacement curves for the detachment of single agglomerates from the film are measured with an Atomic Force Microscope (Nanowizard 3 from JPK) under ambient conditions (relative humidity about 50%, temperature about 21 $^\circ \hbox {C}$), using $\hbox {Si}_3\hbox {N}_4$ cantilever (DNPS from Bruker) with a spring constant of approximately 0.16 N/m, as determined from the thermal noise method [57]. The AFM is placed on a vibration isolation table (i4 from Accurion) inside an acoustic hood. To account for statistical significance, several force curves with a cantilever speed of 2 $\upmu \hbox {m/s}$ and a setpoint of 2.5 nN are collected by force mapping with 64 measurement points in an area of 2 $\upmu \hbox {m}^2$. The dynamical agglomerate behaviour under strain is imaged using a combined AFM/TEM set-up with an $\hbox {Si}_3\hbox {N}_4$ cantilever of spring constant 5.3 N/m. These in-situ investigations are performed inside a Phillips CM 200 FG transmission electron microscope equipped with an AFM/TEM holder (Nanofactory Instruments AB) and with a column vacuum of approximately $10^{-6}$ mbar.

1.2 All-atom MD simulations

All-atom $\hbox {TiO}_2$ nanoparticle models of different sizes, shapes and with surface-adsorbed water corresponding to a relative humidity of 50% are created as described in detail in refs. [10,11,12]. This results in nanoparticles preserving the $\hbox {TiO}_2$ stoichiometry, but terminated with OH groups as a consequence of a thought-process of dissociative water adsorption, which saturates under-coordinated Ti and O sites with OH and H, respectively. An extended $\hbox {TiO}_2$ surface slab model with nanometer-scale roughness and size of $16 \times 14$ $\hbox {nm}^2$ is created via cleavage of an infinite $\hbox {TiO}_2$ single rutile crystal in the $\langle 213 \rangle$ lattice direction, subsequent surface relaxation, annealing and hydroxylation, similarly as for the particle models. The choice of the $\langle 213 \rangle$ direction is arbitrary; any combination of relatively large Miller indices would be suitable, as long as it leads, after annealing and termination with OH and H, to a non-periodic series of asperities and grooves on the size-scale of 0.1 to 1 nm. The Ti-O interactions are described by a modified Matsui/Akaogi force field [53], water molecules are described by the TIP3P model [58], and the interactions between water and $\hbox {TiO}_2$ by the force field of Schneider et al. [54]. All MD simulations are performed using LAMMPS [59] within the NVT ensemble at 300 K, using a Berendsen Thermostat, a timestep of 1 fs, a force cutoff of 12 Å for the van-der-Waals interactions and a particle-particle particle-mesh long-range solver for the Coulomb interactions.

In the MD simulations of single particles dragged tangentially over the $\hbox {TiO}_2$ surface, the particle/surface systems are first relaxed for 1 ns, then a harmonic constraint with a spring-stiffness of 160 N/m is applied to their centres of mass and moved at constant speed (0.1 to 1 m/s in different simulations) over a distance of 5 nm, while the atom positions of the slab are kept fixed, except for the terminal OH atoms and the adsorbed water molecules, which are free to move. The dragging force is recorded every $10^{-6}$ nm and averaged over $10^4$ datapoints, giving a single force value every 0.01 nm. The angular momentum acting on the particle is set to zero at every time step in the sliding simulations, while it is unconstrained in the rolling simulations. The largest local maxima of the force-displacement curves obtained in these simulations are selected as in other works [30, 33]. Namely, first the as-recorded curve is smoothed using a low-pass smoothing function with a cut-off frequency of $f_{cut}$ = 2.8 $\hbox {nm}^{-1}$. The peak forces and peak positions of the smoothed function are then determined analytically. Finally, the maxima of the pristine curve nearest to the positions of the maxima of the smoothed forces are located. Averaging the force peak values and the spacings between the peaks leads the mean values of $F_{t,max}^S$ and $F_{t,max}^R$, as well as the period $\lambda _t$.

In the all-atom MD simulations of the nanoparticle agglomerate, 14 particles of different sizes are placed in contact and relaxed within 2 ns, keeping fixed the positions of the four outermost particles. The particles are created as (irregular) hollow spheres with a shell thickness of 1 nm, which considerably decreases the computational effort with no effect on the interparticle contact forces. The pulling force is exerted via a harmonic constraint with a spring-stiffness of 160 N/m moving at the constant speed of 5 m/s, recorded at every time step and averaged over $10^4$ steps before plotting the force-displacement curves.

1.3 Generation of DEM particle film models

Nanoparticle aggregates are built from primary particles following a discretized log-normal distribution, as measured from TEM image analyses [14]. The primary particle diameter ranges from 3 to 23 nm, with a stepsize of 1 nm (median 9.0 nm, $\sigma _G = 1.45$). These primary particles are combined using a Sequential Algorithm (SA) combined with a Cluster-Cluster Aggregation (CCA) as presented in Ref. [60]. This procedure generates random aggregates that follow the fractal distribution $N_p = k_f \cdot \left( \frac{d_g}{\bar{d}_s}\right) ^{D_f}$, which is frequently used to describe FSP-synthesized particles [49, 61,62,63]. Here, the amount of particles $N_p$ in an aggregate is predicted using the diameter of gyration $d_g$, the Sauter diameter $\bar{d}_s$ (defined as $\bar{d}_s = 6 V_{p,tot}/A_{p,tot}$, with $V_{p,tot}$ the total particle volume and $A_{p,tot}$ the total surface area), as well as the fractal prefactor $k_f$ and the fractal dimension $D_f$. $k_f$ and $D_f$ are 1.0 and 1.8, respectively, for FSP-synthesized particles [61]. The aggregate generation algorithm combines 6 particles to one cluster using the SA. Subsequently, these clusters are combined using the CCA. The number distribution of rigid particle aggregates is estimated from Disc Centrifuge experiments [14]. The equation $N_p = k_{m} \cdot \left( \frac{d_m}{\bar{d}_s}\right) ^{D_{fm}}$ is used according to  [49]. In this equation $d_m$ describes the mobility diameter from Disc Centrifuge experiments and $k_{m}$ and $D_{fm}$ are the mass mobility prefactor and dimension, respectively, that are 1.0 and 2.15 for FSP-synthesized particles. The number distribution ranges from 1 to 128 primary particles with a median of 36 particles and an arithmetic standard deviation $\sigma _s$ of 0.5. The rigid aggregates are deposited individually on the bottom surface of a tetragonal simulation box according to diffusion and a thermophoretic velocity as discribed elsewhere  [47]. The contact of the aggregate with the bottom surface or a previously deposited aggregate marks it as deposited. A deposited aggregate remains static during the ongoing film formation. The coefficient of diffusion for polydispersed nanoparticle aggregates is calculated according to Zhang et al. [64]. The program zeno is used to determine the hydrodynamic radius [65] and 56 angles are used to calculate the mean projected area PA of the deposited agglomerates. The friction factor f follows:

$$\begin{aligned} f = \frac{6\pi \mu R_H}{C_C \left( \frac{\lambda \pi R_H}{PA} \right) } \end{aligned}$$

(14)

with the fluid viscosity $\mu$, mean free path $\lambda$, the hydrodynamic radius $R_H$ and the Cunningham slip correction $C_C$. The application of a thermophoretic velocity of 0.1 m/s [48] results in a film with a porosity of 98 %, which matches very well the typical experimental value [47, 48]. The box presents a side length of 140 $\bar{d}_s$, which corresponds to a box with geometric sizes of $1.5814\times 1.5814 \times 3.3580~\upmu \hbox {m}^3$.

1.4 DEM simulations

All DEM simulations are performed using the LIGGGHTS package [66], in which we have implemented the here-developed contact model, using a timestep $\varDelta t = 0.5$ ps. Capillary forces and solvation forces are implemented using a tabulated potential to reduce computational cost. Initial relaxation of each simulated system is performed using a viscosity term acting on all particles (viscosity of 10 $\hbox {pN/}(\hbox {ms}^{-1})$). The action of this ‘environmental viscosity’, together with the friction terms in the contact model, guarantees that the systems evolve quasi-statically, with sufficient dissipation of the kinetic energy that builds up as a consequence of the nanoparticle rearrangements under mechanical stress. The AFM tip with a side length of 0.8 $\upmu \hbox {m}$, represented by non-interacting and rigid primary particles with 20 nm radius, is placed above the film without any initial contact to the film. The dynamical simulations are then carried out in the microcanonical ensemble, with the sole energy dissipation of the DEM contact model able to maintain the system under quasi-static conditions. The tip is moved with a constant velocityof −0.5 m/s along the z direction and the forces acting on the tip particles are averaged over periods of 10 ns, equaling a tip displacement of 5 nm. After reaching a repulsive force of 2 nN, the tip velocity is slowly decreased, reversed and then again increased to +0.5m/s within a period of 0.2 $\upmu \hbox {s}$. Periodic boundary conditions are applied to the simulation box, with the bottom layer of aggregates kept fixed in order to mimic an underlying sample holder.