Determination of dynamics and stability of particles can be seen an introduction to the study of bodies in motion as applied to engineering systems and structures. Scientists usually study the dynamics of particle motion and bodies in rigid planar motion, which will consist of both the kinematics and kinetics of motion. Kinematics deals with the geometrical aspects of motion describing position, velocity, and acceleration, all as a function of time. Kinetics is the study of forces acting on these bodies and how it affects their motion. Generally speaking, particle stability depends on its size-dependent characteristics. All particle structures have thermodynamic properties and energy disadvantages relative to the volume state, their stability can therefore only be maintained for a limited time.

Figure 1. Real-time dynamic in-situ ESTEM-HAADF images of practical Pt nanoparticles on carbon, in hydrogen gas as a function of operating temperature, revealing single-atom dynamics. (Martin, T.E.; *et al*. 2020)

### Density Functional Theory (DFT)

At Alfa Chemistry, we have applied DFT-based techniques to study the structure, dynamics and stability of nanoparticles. Total energy calculations and DFT molecular dynamics are available for investigating systems containing multi-shell and disordered nanoparticles. Our well-designed molecular dynamics simulations have been successfully carried out at a wide range of temperature.

### Bi-variant Dynamic Model of Continuous Mixed-Suspension

Our teams use two-particle coordinates (agglomerate size and number of primary crystals) population balance approaches to create a bivariate dynamic model with agglomerated continuous mixed suspension and mixed product removal (MSMPR) precipitation. Several finite difference methods are compared and implemented to solve the numerical value of the simplified model equation. Our experts select the implicit forward time center space as the best method to perform the accurate dynamic particle size distribution (PSD) prediction. In addition, our technique has proven to be applicable in the study of agglomeration.

### Derjaguin-Landau-Verwey-Overbeek (DLVO) Theory

We use Derjaguin-Landau-Verwey-Overbeek method to calculate the total interaction potential between the separation distance (s) and the radius of curvature (r) between two objects. The stability of particles in solution phase depends on the energetics of collision and potential energy. As the probability of elastic collision increases, the stability of the nanoparticles is improved. We use the collision theory and DLVO theory to estimate the energy of the solution phase structure, thereby determining the stability of the particles.

### Gravitational Search Algorithm (GSA)

The GSA is a swarm optimization algorithm which is developed based on the Newtonian laws of gravity and motion. At Alfa Chemistry, we have established and optimized the GSA method to perform particle dynamics and stability analysis. Our accurate GSA can be used to solve the optimization problem of various nonlinear functions. Our analysis process is as follows:

- First, particle dynamics are represented as a nonlinear feedback controlled system.
- Second, we apply the Lyapunov stability theory that involves with randomness and time varying parameters to analyze the stability of the particle dynamic system. Moreover, the stability conditions we use are derived by treating the particle dynamics in the sense of the Lyapunov stability.

Alfa Chemistry's modified algorithm is evaluated based on various benchmark functions including unimodal test functions, multimodal test functions, and multimodal test functions with fixed dimensions. The performance of the modified GSA is also tested. In addition, stability conditions have been investigated for two categories of examples: following and violating stability conditions.

### Our Capabilities

- We are capable of studying the stability in finite times of the trajectories of interacting particles.
- Apart from particle dynamics and stability analysis, we can also apply the convergence analysis of the GSA and improved GSA algorithms to demonstrate each particle converges.

Our determination of dynamics and stability of particles services remarkably reduce the cost, promote further experiments, and enhance the understanding of chemical process for customers worldwide. Our personalized and all-around services will satisfy your innovative study demands. If you are interested in our services, please don't hesitate to contact us. We are glad to cooperate with you and witness your success!

Reference

- Martin, T.E.;
*et al*. Atom-by-atom analysis of sintering dynamics and stability of Pt nanoparticle catalysts in chemical reactions.*Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences*. 2020, 378(2186): 20190597.