# Movement Simulation

## Purpose of simulations

In our simulation, we intend to confirm correlation between radius of polystyrene particle and movement of JET.
We are now producing 10um size of JET now, however, we are also thinking about production of smaller size of JET in the future. It is still difficult to confirm the completion of less than 1um size of JET in practical experiment. So, we observe movement of smaller JET by 2D simulation. Theoretically speaking, nano-size or micro-size particles are largely affected by Brownian movement; therefore, radius of particles becomes key parameter which determines movement of JET. For example, movement of JET is more likely to be affected by Brownian motion as the radius of particle is smaller.

## Principle of simulations

Our simulator is based on off lattice simulation of random walk. This program is written in MATLAB.
At each time step, we calculate the x-coordinate and y-cordinate displacement of the particle.Polystyrene beads mainly move by two forces, the reaction force caused by the detachment of the bubbles and Brownian movement. Moreover, we put two types of Brownian movement in this simulation, rotator Brownian movement and translational Brownian movement.

## Methods of simulations

### Growth and detachment of the bubbles and reaction force

To estimate the bubble average detachment rate we use following bubble growth model.
At each time step, we calculate bubble growth, and when the radius of bubble reached fixed maxR, the bubble detaches.
Bubble radius growth is estimated by following formula.
Substituting value of radius, we can get the average rate of mass change.
Finally, velocity of JET causes by detachment of horizontal speed of detached bubbles as following formula.

### Rotatory Movement

In previous section, we got a value of velocity.
Next, we calculate the angle from the +y axis to the vertex of Cr hemisphere heads.
Angle of the vertex from the +y axis will change at each time step by rotator Brownian movement,
and the mean amount of changes is shown as following equation.
And also, the variance σ of Brownian rotator changes ∆φ is calculated as following equation.
We get value of rotation using function whose name is “normrnd” in MATLAB. Function “normrnd” returns normal random numbers if we input the value of mean and variance.
At each time step, we calculate the angle φ by accumulating this amount of change.

Next, we calculate angle ψ which bubbles start to grow.
The range of ψ is hemispherical surface where Pt particles are conjugated by DNA.
We use uniform random number to determine the value of ψ. Following equation shows that bubbles can grow from anywhere on the hemispherical surface at the same possibility.

Finally, using these angles, we can find the angle of movement θ from y-axis as following equation.

### Translational Movement

We can calculate displacement of x-coordinate and y-coordinate from value of velocity and the angle of movement from y-axis.
JET mainly moves these two displacements at each time step.
We are now taking another look at parameters and considering how to deal with these displacements and 2D Brownian translational movement simultaneously.