In [1]:

%matplotlib inline
from scipy.optimize import minimize
import numpy as np
import matplotlib.pyplot as plt
from scipy.constants import pi, Boltzmann



# Relation 1

The form of the damping is given as:

$$\Gamma_0 = \dfrac{6 \pi \eta_{air} r}{m} \dfrac{0.619}{0.619 + K_n} (1+ c_k)$$

(Li et al. 2011 - https://arxiv.org/pdf/1101.1283.pdf)

Where:

• $\eta_{air}$ is the viscosity of air
• $r$ is the radius of the silica nanoparticles
• $m$ is the mass of the silica nanoparticles
• $K_n$ is the Knudsen number $\dfrac{s}{r}$ where $s$ is the mean free path of the air particles
• $c_k$ is a small positive function of $K_n$ which takes the form $(0.31K_n)/(0.785+1.152K_n+K_n^2)$

The mean free path is dependant upon the pressure of the system. The mathematical form the mean free path, is dependant upon whether the particles under study are con- sidered to be hard like spheres colliding or as “soft” spheres following Lennard-Jones Potential. In this case assuming the gas particles to be hard spheres yields the following form,

$$s = \dfrac{k_B T_0}{ \sqrt{2} \pi d_{gas}^2 P_{gas}}$$

(Muddassar - Thesis - Cooling and Squeezing in Levitated Optomechanics 2016)

Where:

• $d_{gas}$ is the diameter |of the gas particles
• $T_0$ is the temperature of the gas
• $P_{gas}$ is the pressure of the gas


In [179]:

# constants
k_B = Boltzmann
eta_air = 18.27e-6 # Pa # (J.T.R.Watson (1995)).
d_gas = 0.372e-9 #m #(Sone (2007)), ρSiO2
rho_SiO2 = 1800 # #kg/m^3 - Number told to us by

T0 = 300

R = 50e-9 # m




In [180]:

def mfp(P_gas):
mfp_val = k_B*T0/(2**0.5*pi*d_gas**2*P_gas)
return mfp_val



Alternativity one can use:

$$s = \dfrac{\eta_{air}}{P_{gas}} \sqrt{\dfrac{\pi k_B T_0}{2m}}$$

#### this produces the same result as the previous form

https://en.wikipedia.org/wiki/Mean_free_path

Where

• $\eta_{air}$ is the viscosity of air
• $m$ is the molecualar mass of air
• $T_0$ is the temperature of the gas
• $P_{gas}$ is the pressure of the gas

molecular mass of air is $28.97 g/mol$ and the number of molecules in a mole is Avogadro's Number $6.0221409e^{23}$ therefore we get the molecular mass of air to be $4.81e^{-26} Kg$



In [181]:

m_gas = 4.81e-26




In [182]:

def mfp_2(P_gas):
mfp_val = eta_air/P_gas * (pi*k_B*T0/(2*m_gas))**0.5
return mfp_val




In [183]:

s = mfp(300) # 3mbar = 300 Pascals
print(s)




2.2455999057589032e-05




In [184]:

s2 = mfp_2(300) # 3mbar = 300 Pascals
print(s2)




2.2397878589354806e-05




In [185]:

mass = rho_SiO2 * 4/3*pi*radius**3
Pressure_pascals = 100*Pressure_mbar
s = mfp(Pressure_pascals)
c_K = 0.31*K_n/(0.785 + 1.152*K_n + K_n**2)
Gamma_0 = 6*pi*eta_air*radius/mass * 0.619/(0.619 + K_n) * (1+c_K)
return Gamma_0




In [186]:

Gamma_env(R, 3)




Out[186]:

25163.29798430737



Muddassar and Gieseler's simplified formula for the environmental damping is:

$$\Gamma_0 = 0.619 \dfrac{9 \pi}{\sqrt{2}} \dfrac{\eta_{air}d_{gas}^2}{\rho_{SiO_2} k_B T_0} \dfrac{P_{gas}}{r}$$

This produces the same result as the full unsimplified form for all pressures in range of interest.

Where:

• $\eta_{air}$ is the viscosity of air
• $d_{gas}$ is the diameter of the gas particles
• $\rho_{SiO_2}$ is the density of the silica nanoparticles
• $r$ is the radius of the silica nanoparticles
• $T_0$ is the temperature of the gas
• $P_{gas}$ is the pressure of the gas


In [187]:

Pressure_pascals = 100*Pressure_mbar
return Gamma_0




In [ ]:




In [188]:

Gamma_env_simple(R, 3)




Out[188]:

25180.643201394465



# Relation 2

In Gieseler's Thermal Nonlinearities paper he has the following equation for $\Gamma_0$

$$\Gamma_0 = \dfrac{64a^2}{3m\bar{v}}P$$

https://www.nature.com/nphys/journal/v9/n12/full/nphys2798.html

This appears to be incorrect as it is exactly double that which you get with Chang's formula and James Millen's formula

Where:

• $a$ is the radius of the particle
• $m$ is the mass of the particle
• $\bar{v}$ is the average verlocity of the gas particles

Where we can use the following formula for $\bar{v}$

$$\bar{v} = \sqrt{\dfrac{8k_B T_0}{\pi \mu}}$$

Where:

• $T_0$ is the temperature of the gas
• $\mu$ is the mass of the air molecules


In [189]:

Pressure = 100*Pressure_mbar
ave_velocity = (8*k_B*T0/(pi*m_gas))**0.5
return Gamma0




In [190]:

Gamma_alternative(R, 3)




Out[190]:

36253.43341158769



# Relation 3

In Chang et al. paper "Cavity opto-mechanics using an optically levitated nanosphere"

They have $\Gamma_0 = \dfrac{\gamma_g}{2} = \dfrac{8}{\pi}\dfrac{P}{\bar{v}r\rho}$

Where

• $\rho$ is the density of the nanoparticle
• $P$ is the pressure of the gas
• $\bar{v}$ is the mean speed of the gas particles
• $r$ is the radius of the nanoparticle


In [191]:

ave_velocity = (8*k_B*T0/(pi*m_gas))**0.5




In [192]:

ave_velocity




Out[192]:

468.2736410204197




In [193]:

Pressure = 100*Pressure_mbar
ave_velocity = (8*k_B*T0/(pi*m_gas))**0.5
return 2*Gamma0




In [194]:

Gamma_chang(R, 3)




Out[194]:

18126.71670579385



# Also relation 3 (different derivation by Millen et al.)

James Millen derives the following form of the damping due to impinging particles:

$$\Gamma^{imp} = \dfrac{4\pi}{3}\dfrac{mNr^2 \bar{v}_{T_{imp}}}{M}$$

However in their earlier paper http://iopscience.iop.org/article/10.1088/1367-2630/15/1/015001/meta they get double this, which is what Gieseler gets in his thermal non-linearities paper.

Where:

• $m$ is the molecular mass of the gas
• $N$ is the particle density of the gas
• $r$ is the radius of the nanoparticle
• $M$ is the mass of the nanoparticle
• $\bar{v}_{T_{imp}}$ is the mean thermal velocity $\sqrt{\dfrac{8 k_B T^{imp}}{\pi m}}$

Using the ideal gas equation $P = R\rho T$ and $N= \dfrac{\rho}{m}$ with $R=\dfrac{k_B}{m}$ we get $N = \dfrac{P}{k_BT}$



In [195]:

Pressure = 100*Pressure_mbar
ave_velocity = (8*k_B*T0/(pi*m_gas))**0.5
N = Pressure/(k_B*T0)
return Gamma0




In [196]:

Gamma_Millen_imp(R, 3)




Out[196]:

18126.716705793842



### This agrees exactly with Chang's result



In [197]:

Gamma_chang(R, 3)




Out[197]:

18126.71670579385



# Relation 3+ (more damping due to considering emerging particles)

James Millen derives the following form of the damping due to emerging particles:

$\Gamma^{em} = \dfrac{mNr^2\pi^{\frac{3}{2}}}{3\sqrt{h'}M}$

Where:

• $m$ is the molecular mass of the gas
• $N$ is the particle density of the gas
• $r$ is the radius of the nanoparticle
• $h'$ is $\dfrac{m}{2k_B T_0}$ where $T_0$ is the temperature of the gas
• $M$ is the mass of the nanoparticle

Using the ideal gas equation $P = R\rho T$ and $N= \dfrac{\rho}{m}$ with $R=\dfrac{k_B}{m}$ we get $N = \dfrac{P}{k_BT}$

He also says that this leads to $\Gamma^{em} = \dfrac{\pi}{8}\dfrac{T^{em}}{T^{imp}}$

From this you get the total effective damping rate is $$\Gamma_0 = \Gamma^{em} + \Gamma^{imp} = \dfrac{\pi}{8}\sqrt{\dfrac{T^{em}}{T^{imp}}}\Gamma^{imp} + \Gamma^{imp}$$

### Therefore damping rate is higher if you consider this



In [198]:

def Gamma_Millen_em(radius, Pressure_mbar, T_em):
Pressure = 100*Pressure_mbar
h_prime = m_gas/(k_B*T_em)
N = Pressure/(k_B*T_em)
return Gamma0




In [199]:

def calc_surface_temp_Millen(T_em, T_imp=300):
accomodation_coef = 0.777 # accomodation coefficient of silica (from Nanoscale temp measurement paper)
T_surf = T_imp + (T_em + T_imp)/accomodation_coef
return T_surf



# Plot of all 3 relations and measured data



In [200]:




In [201]:

P_G_Dict = dict(zip(P_exp, Gamma_exp))




In [202]:

r = np.linspace(5e-9, 1000e-9, 1000)
P = 3.6 # mbar
alpha=0.5

plt.figure(figsize=[10, 10])
plt.loglog(r, Gamma_env_simple(r, P), 'k', label="Rashid/Gieseler Full form", alpha=alpha)
#plt.semilogy(r, Gamma_env_simple(r, P), 'grey', label="Rashid/Gieseler simplfied form", alpha=alpha)
plt.loglog(r, Gamma_alternative(r, P), label="Gieseler Thermal Non-linearities form", alpha=alpha)
plt.loglog(r, Gamma_chang(r, P), label="Chang form", alpha=alpha)
plt.loglog(r, Gamma_Millen_imp(r, P), label="Millen (imp) form", alpha=alpha)
plt.legend(loc='best')
plt.show()







In [203]:

r = 50e-9
P = np.linspace(1e-2, 1000, 1000)

plt.figure(figsize=[10, 10])
plt.loglog(P, Gamma_env_simple(r, P), 'k', label="Rashid/Gieseler Full form", alpha=alpha)
#plt.loglog(P, Gamma_env_simple(r, P), 'grey', label="Rashid/Gieseler simplfied form", alpha=alpha)
plt.loglog(P, Gamma_alternative(r, P), label="Gieseler Thermal Non-linearities form", alpha=alpha)
plt.loglog(P, Gamma_chang(r, P), label="Chang form", alpha=alpha)
plt.loglog(P, Gamma_Millen_imp(r, P), label="Millen (imp) form", alpha=alpha)
plt.loglog(P_exp, Gamma_exp, label="Experiment", alpha=alpha)
plt.xlabel("P (mbar)")
plt.legend(loc='best')
plt.show()







In [ ]: