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{{Titech/Nano-Jugglers/HEAD}}
{{Titech/Nano-Jugglers/HEAD}}
__NOTOC__
=Simulation Models=
::[[Image:Simulation models.png|800px]]
==Physical principles for simulations==
:We confirm the movement of rocket on 2D plots in simulation.


 
:We assumed that movement of biomolecular rocket is affected by following four forces and dynamics in simulation.
=Simulation=
 
==1. Driving forces from Bubble detachment==
==Purpose of simulations==
===1.1. Calculation for Speed===
::In our simulation, we intend to confirm correlation between radius of polystyrene particle and movement of JET.
:'''Bubbles detachment helps Biomolecular Rocket go straightforward.'''
 
:The Biomolecular Rocket is accelerated by a single bubble detachment every Δt<sub>d</sub> seconds .
::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.
:Bubbles detachments occur when fixed time Δt<sub>d</sub> passed.
 
:We defined radius changes of bubbles with time as following formula.
 
::[[Image:TNJFormula12.png|250px]]
==Principle of simulations==
:Δt<sub>d</sub> is defined as the time which is required bubbles to reach its detachment radius R<sub>d</sub>.
::Our simulator is based on off lattice simulation of random walk. This program is written in MATLAB.
::[[Image:TNJFormula13.png|250px]]
 
:We defined velocity v<sub>i</sub> produced by single detachment and Δt<sub>d</sub> as following formula.
::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==
::[[Image:TNJFormula1.png|250px]]
===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.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim1.png‎]]
|-
|
|
::<math>R(t)</math> : the bubble radius at t seconds
::[[Image:TNJconstant1.png|250px]]
::<math>R_0</math> : the initial bubble radius
::<math>R_g</math> : the universal gas constant
::<math>T</math> : temperature
::<math>P</math> : pressure of the bubble
::<math>k</math> : the catalytic reaction rate constant
::<math>\alpha</math> : the Langmuir adsorption constant
::<math>c</math> : hydrogen peroxide concentrations
|}
::Substituting value of radius, we can get the average rate of mass change.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim2.png‎]]
|-
|-
|
|
::<math>\Delta m</math> : the mass change induced by a single bubble
::[[Image:TNJFormula10.png|250px]]
::<math>\rho_O2</math> : the density of oxygen
|}
::Finally, velocity of JET causes by detachment of horizontal speed of detached bubbles as following formula.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim3.png‎]]
|-
|
|
::<math>N</math> : the number of bubbles
::[[Image:TNJ constant2.png|275px]]
::<math>v_0</math> : the initial horizontal speed of a detached bubble
::<math>\mu</math> : the viscosity of the liquid
::<math>a</math> : the radius of polystyrene beads
|}
|}


===Rotatory Movement===
===1.2. Directional Calculation===
::In previous section, we got a value of velocity.
:'''Where bubbles generation occured is determined randomly on the hemisphere surface with catalytic engine.'''
::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.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim4.png‎]]


[[Image:TechJug-sim10.png]]
{|
|-
|
:We defined angle θ as bubbles detachment direction.
:θ is determined by uniformed numbers in the area where catalytic engines are still attacched. 
:Bubbles detachment supply the Biomolecular Rocket velocity of opposite direciton.
|width="30px"|
<br>
|
|
::<math>D_R</math> : Rotatory diffusion coefficient
[[Image:Directional.jpg|180px]]
::<math>k_B</math> : Boltzmann constant
::<math>\eta</math> : viscosity
|}
|}
:And also, the variance σ of Brownian rotator changes ∆φ is calculated as following equation.
 
{|style="margin-left:80px"
==2. Fluid resistance==
|-
:'''Fluid resistance decreases speed of the Biomolecular Rocket.'''
|[[ Image:TechJug-sim9.png‎]]
:Fluid resistance depends on the velocity of the Biomolecular Rocket and viscosity of solution.
:Resistance is defined as
::[[Image:TNJFormula9.png|200px]]
:Therefore, acceleration of the Biomolecular Rocket is
{|
|
::[[Image:TNJFormula4.png|200px]]
|
::[[Image:TNJConstant5.png|300px]]
|}
|}
::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.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim5.png‎]]
|}
::At each time step, we calculate the angle φ by accumulating this amount of change.


 
==3. Translational Brownian displacement==
::Next, we calculate angle ψ which bubbles start to grow.  
:'''Translational Brownian movement prevents Biomolecular Rocket from going straight forward.'''
::The range of ψ is hemispherical surface where Pt particles are conjugated by DNA.
:This is because body of the Biomolecular Rocket is so small and smaller particles can't be controlled under Brownian Movement.
::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.
:Translational displacement by Brownian movement is described as
{|style="margin-left:80px"
{|
|-
|
|[[ Image:TechJug-sim6.png‎]]
::[[Image:TNJFormula7.png|200px]]
|-
|-
|
|
<math>rand()</math> : uniform random number
::[[Image:TNJFormula8.png|200px]]
|
::[[Image:TNJconstant3.png|300px]]
|}
|}


 
==4. Rotatory Brownian changes==
::Finally, using these angles, we can find the angle of movement θ from y-axis as following equation.
:'''Rotatory Brownian movement decreases the directional controllability of Biomolecular Rocket.'''
{|style="margin-left:80px"
:Movement of Biomolecular Rocket is also much influenced by Rotatory Brownian Movement
:Rotatory changes by Brownian movement is described as  
{|
|
::[[Image:TNJFormula5.png|200px]]
|-
|-
|[[ Image:TechJug-sim11.jpg‎]]
|
|[[ Image:TechJug-sim7.png]]
::[[Image:TNJFormula6.png|200px]]
|}
|
===Translational Movement===
::[[Image:TNJconstant4.png|400px]]
::We can calculate displacement of x-coordinate and y-coordinate from value of velocity and the angle of movement from y-axis.
{|style="margin-left:80px"
|-
|[[ Image:TechJug-sim8.png‎]]
|}
|}
::JET mainly moves these two displacements at each time step.
::::::>back to [[Biomod/2012/Titech/Nano-Jugglers/Results#2.2._Numerical_estimation_of_the_speed_of_the_Biomolecular_Rocket|Results 2.2. Numerical estimation of the speed of the Biomolecular Rocket]]
::We are now taking another look at parameters and considering how to deal with these displacements and 2D Brownian translational movement simultaneously.
::::::>back to [[Biomod/2012/Titech/Nano-Jugglers/Results#3.3._Directional_control_of_the_Biomolecular_Rocket_by_the_photo-switchable_DNA_system|Results 3.3 Directional control of Biomolecular Rocket by the photo-switchable DNA system]]
 
==Results of Simulation==


==Reference==
=Tools=
*Scilab
=References=
*J. G. Gibbs and Y.-P. Zhao (2009) ''Autonomously motile catalytic nanomotors by bubble propulsion.'' University of Georgia, Athens, Georgia 30602, USA, American Institute of Physics.
*V. A. KiriUov and V. P. Patskov (1979) ''SOME REGULARITIES OF BUBBLE GROWTH UNDER CHEMICAL REACTION.'' Institute of Catalysis, Novosibirsk, USSR, React. Kinet. Catal. Lett., Vol. 11, No. 1, 15-19 (1979)

Latest revision as of 01:29, 28 October 2012


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Simulation Models

Physical principles for simulations

We confirm the movement of rocket on 2D plots in simulation.
We assumed that movement of biomolecular rocket is affected by following four forces and dynamics in simulation.

1. Driving forces from Bubble detachment

1.1. Calculation for Speed

Bubbles detachment helps Biomolecular Rocket go straightforward.
The Biomolecular Rocket is accelerated by a single bubble detachment every Δtd seconds .
Bubbles detachments occur when fixed time Δtd passed.
We defined radius changes of bubbles with time as following formula.
Δtd is defined as the time which is required bubbles to reach its detachment radius Rd.
We defined velocity vi produced by single detachment and Δtd as following formula.

1.2. Directional Calculation

Where bubbles generation occured is determined randomly on the hemisphere surface with catalytic engine.
We defined angle θ as bubbles detachment direction.
θ is determined by uniformed numbers in the area where catalytic engines are still attacched.
Bubbles detachment supply the Biomolecular Rocket velocity of opposite direciton.


2. Fluid resistance

Fluid resistance decreases speed of the Biomolecular Rocket.
Fluid resistance depends on the velocity of the Biomolecular Rocket and viscosity of solution.
Resistance is defined as
Therefore, acceleration of the Biomolecular Rocket is

3. Translational Brownian displacement

Translational Brownian movement prevents Biomolecular Rocket from going straight forward.
This is because body of the Biomolecular Rocket is so small and smaller particles can't be controlled under Brownian Movement.
Translational displacement by Brownian movement is described as

4. Rotatory Brownian changes

Rotatory Brownian movement decreases the directional controllability of Biomolecular Rocket.
Movement of Biomolecular Rocket is also much influenced by Rotatory Brownian Movement
Rotatory changes by Brownian movement is described as
>back to Results 2.2. Numerical estimation of the speed of the Biomolecular Rocket
>back to Results 3.3 Directional control of Biomolecular Rocket by the photo-switchable DNA system

Tools

  • Scilab

References

  • J. G. Gibbs and Y.-P. Zhao (2009) Autonomously motile catalytic nanomotors by bubble propulsion. University of Georgia, Athens, Georgia 30602, USA, American Institute of Physics.
  • V. A. KiriUov and V. P. Patskov (1979) SOME REGULARITIES OF BUBBLE GROWTH UNDER CHEMICAL REACTION. Institute of Catalysis, Novosibirsk, USSR, React. Kinet. Catal. Lett., Vol. 11, No. 1, 15-19 (1979)
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