Particle Sedimentation Behaviors in a Density Gradient

Abstract

Density gradient centrifugation, as an efficient separation method, is widely used in the purification of nanomaterials including zero, one-, and two-dimensional nanomaterials, such as FeCo@C nanoparticles, gold nanoparticles, gold nanobar, graphene, carbon nanotubes, hydrotalcite, zeolite nanometer sheet (the examples can be found in Chap. 5). Each system needs separation parameter optimization, which comes from tremendous research experiments. When particles are put on the top of density gradient medium, they will have a definite settling rate under centrifugal force (F_c) [1], which is influenced by their net density, size, and shape. In a sufficiently intense centrifugal field, the particle motion held quietly free from gravity and vibration [2]. This is the principle of the density gradient ultracentrifuge. Based on the above principle, we discussed the particle sedimentation behaviors and built the kinetic equation in a density gradient media. The kinetic equation could apply to zero, one-, and two-dimensional nanomaterials, within its variation form accordingly. We found that the separation parameters could be optimized based on the kinetic equation. A MATLAB program was further developed to simulate and optimize the separation parameters. The calculated best parameters could be deployed in practice to separate given nanoparticles successfully.

Appendix: MATLAB Program for the Computational Mathematical Optimization of Spherical Nanoparticles

% model assumptions:

%1. All the nanoparticles are sphere; if not, use morphology factor f to modify the model;

%2. The nanoparticles have a solvation layer;

%3. Using linear density gradient; if not, modify the gradient function;

%4. The ideal diameter distribution of nanoparticles is linear distribution; the objective function is G;

%5. Nanoparticles do not react with the medium;

%6. Optimization variables: linear acceleration time (T₀), the time of constant speed (T₁), linear deceleration time T₂, (the total time is T = T₀ + T₁ + T₂), the angular velocity in constant speed (omega), c and d are the coefficient of linear density gradient; a and b are the coefficient of the ideal linear distribution; the objective function is G(T₀, T₁, T₂, omega, c, d, a, b) = c₁ ^ 2 * (x₀−(a * r₀ + b)) ^ 2 + sum((x(r_j,T)−(a * r_j + b)) ^ 2)(j = 0:m) + c₂ ^ 2*(x_m−(a * r_m + b)) ^ 2, (c₁ and c₂ are appropriate constant); among them, r_j = r₀ + j*(r_m–r₀)/m, r₀, r_m are the minimum and the maximum radius of particles, respectively; x₀ is the distance between the top of the centrifuge tube and the center of rotation, x_m is the distance between the bottom of the centrifuge tube and the center of rotation.

%7. The movement of NPs (X(t)) follows the following equation: x” + 9 * ita(p_m(x))/(2 *p_p(r) * r ^ 2) * x’ + (p_m(x) − p_p(r))*omega(t)/p_p(r) * x = 0; ita is the viscosity of the medium solution, the ideal density gradient: p_m(x) = c + d * x, the density of nanoparticle: p_p(r); the angular velocity: omega(t) = omega * t/T₀ (0 < t<T₀); omega (T₀ < t<T₀ + T₁); omega*(T₀ + T₁ + T₂−t)/T₂ (T₀ + T₁ < t<T₀ + T₁ + T₂);

% Using the Lsqnonlin in MATLAB to solve the optimization problem

global r x₀ x_m

m = 10; % the number of output dots

x₀ = 6.5; x_m = 11.8; r₀ = 1.3e-7; r_m = 3.6e-7; % the unit is centimeter, might be different for different rotors; x₀ and x_m is the distances between center of rotation and the top and bottom of the centrifuge tube, respectively; r₀ and r_m is the size of particles.

h = (r_m–r₀) / m; r = r₀:h:r_m;

y₀ = [60,3600,60,4000,0.6,0.5,2e7,4]; % the initial value of the optimization calculation

l_b = [30,1200,30,1000,0,0,0,−1e8]; % the minimum bounds of optimization variables

u_b = [1000,100000,1000,10000,5,5,1e10,1e10]; % the maximum bounds of optimization variables

options = optimset(‘LargeScale’,’on’,’Display’,’iter’,’TolX’,1e-30, ‘MaxIter’, 200, ‘MaxFunEvals’, 5000, ‘TolFun’, 1e-10);

[y, resnorm] = lsqnonlin(@objfun, y₀, l_b, u_b,options) % optimization calculation

a = y(7); b = y(8);

arb = a * r + b; % the ideal linear distribution

Fm = objfun(y); xt = Fm(2:end-1)’ + arb;

plot(xt, 2 * r,’o’, arb, 2 * r) % output comparison chart

title(‘Optimized distribution and the ideal distribution comparison chart’)

xlabel(‘The distance from the NPs to the top of the centrifuge tube(cm)’)

ylabel(‘Particle diameter(cm)’)

legend(‘Optimized distribution’,’the ideal distribution’)

function F = objfun(y) % the objective function

global r x₀ x_m

T₀ = y(1); T₁ = y(2); T₂ = y(3); omega = y(4); c = y(5); d = y(6); a = y(7); b = y(8);

T = T₀ + T₁ + T₂;

tspan = [0,T];

xt = [];

m = length(r);

for i = 1:m

ri = r(i); [t,x] = ode15s(@odefun,tspan,[x₀,0],[],y,ri);

xti = x(end,1);

xt = [xt,xti];

end

arb = a * r + b;

c₁ = 10000; c₂ = 10000;

F = [c₁ * (x₀-arb(1)), xt−arb, c₂ * (x_m−arb(end))];

F = F’;

function xp = odefun(t,x,y,ri)

p_c = 6; h = 2e-7; p_m = 0.9; % p_c is the density of core, p_m is the density of shell, the unit is g/cm³; h is the thickness of the shell, the unit is centimeter.

p_p = p_m + (p_c−p_m) * (1−h/ri) ^ 3;

T₀ = y(1); T₁ = y(2); T₂ = y(3);

c = y(5); d = y(6); omega = y(4);

if t >= 0 & t < T₀

omegat = omega * t / T₀;

elseif t >= T₀ & t <= T₀ + T₁

omegat = omega;

elseif t > T₀ + T₁ & t <= T₀ + T₁ + T₂

omegat = omega * (T₀ + T₁ + T₂−t)/T₂;

else

omegat = 0;

end

(c + d * x(1)) > 0.5 & (c + d * x(1)) < 2; % the minimum and maximum bounds of the density gradient

ita = ita(p_m); % viscosity is relate with the density of liquid medium, the unit of viscosity is mPa.s

x_p = [x(2);-9 * ita * x(2) / (2 * ri * ri * p_p) + (p_p−(c + d * x(1))) * omega * omega * x(1) / p_p]; % the kinetic equation of spherical nanoparticles