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langevin_random.m
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langevin_random.m
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% %
% ///// / %
% ///// // %
% ,,,,,. //// ,,,,, /// %
% ,,,,, #### /// ,,,, ,,,, //// %
% ,,,,, ### /// ,,, ,,, ///// %
% ,,,, ### // ,,, ,, ////// %
% ,,, ,,, ## / ,, , ////// %
% ////// ,,,,, ,, # / , , /////// ///// %
% //////// ,,,, , ( / , //////// ////// %
% ////// ,, ///////// //// %
% */// , ////// ///// / %
% ### / ///, ,///// / ,,,,, %
% ######### /////////////// ,,,,,,,,, %
% *#//////////////// %
% ,,,,,,,,,,,,,, ###***###//////////// ,,,,,,,,,,,,,, %
% ,,,,,,,,,,, ############///////// ,,,,,,,,,, %
% ###############(// # %
% ,,,,,,,,, ############### ######### %
% ,,, ## ############ // ## %
% // ###### ####### //// %
% //// ######### , ,, /////// %
% /////// ######## , / , ,,, //////// %
% //// ####### , , / # ,, ,,,, ///// %
% ###### ., ,, / ### ,, ,,, %
% ##### ,, ,,, // ### ,,, %
% ##### ,,, ,,,, /// #### ,,,, %
% #### ,,,, ,,, /// ### ,,,,, %
% ### ,,,, //// ,,,,, %
% ## ///// %
% # ///// %
% ( %
% ________ ________ ________ ___ ___ ________ _______ ________ %
% |\ ____\|\ __ \|\ __ \|\ \|\ \|\ __ \ / ___ \|\ ___ \ %
% \ \ \___|\ \ \|\ \ \ \|\ \ \ \\\ \ \ \|\ \/__/|_/ /\ \____ \ %
% \ \ \ __\ \ _ _\ \ \\\ \ \ \\\ \ \ ____\__|// / /\|____|\ \ %
% \ \ \|\ \ \ \\ \\ \ \\\ \ \ \\\ \ \ \___| / /_/__ __\_\ \ %
% \ \_______\ \__\\ _\\ \_______\ \_______\ \__\ |\________\|\_______\ %
% \|_______|\|__|\|__|\|_______|\|_______|\|__| \|_______|\|_______| %
% %
% Authors: Elvira Kazimova; %
% Giovanni Madella; %
% Andrea Somma; %
% Giovanni Tomaciello; %
% Sabrina Ulivelli; %
% %
% Pickering emulsions; Paper D; Applied Physical Chemistry (2022-2023); %
% Politecnico of Milan. %
clc; clear; close all
%-----------------------------------------------------------------------------------------------------------------------
%% data
%-----------------------------------------------------------------------------------------------------------------------
%simulation limits ------------------------------------------------------------------------------------------[# - s - K]
max_silica = 2000;
max_magnetite = 800;
tot_part = max_magnetite + max_silica;
oil_p = 2;
time_end = 15e-6;
succ_tsteps = 100000;
dt = time_end/succ_tsteps;
T = 273.15 + 25;
% simulation box ----------------------------------------------------------------------------------------------------[m]
box = [1850 -1850 1850 -1850 1850 -1850].*1e-9; % front - back - east - west - north - south
ext_box = [1900 -1900 1900 -1900 1900 -1900].*1e-9; % front - back - east - west - north - south
% particles radius --------------------------------------------------------------------------------------------------[m]
oil_r = [270 300].*1e-9;
silica_r_mean = 10e-9;
silica_r_error = 1e-9;
silica_r = silica_r_mean.*ones(1,max_silica) + ...
silica_r_error*randn(max_silica,1)';
magnetite_r_mean = 35e-9;
magnetite_r_error = 2e-9;
magnetite_r = magnetite_r_mean.*ones(1,max_magnetite) + ...
magnetite_r_error*randn(max_magnetite,1)';
% particles mass ----------------------------------------------------------------------------------------------[Kg - m3]
oil_rho = 870;
silica_rho = 2650;
magnetite_rho = 5170;
oil_mass = 4/3*pi.*(oil_r).^3.*oil_rho;
silica_mass = 4/3*pi.*(silica_r).^3.*silica_rho;
magnetite_mass = 4/3*pi.*(magnetite_r).^3.*magnetite_rho;
% initial oil position -------------------------------------------------------------------------------------------------
oil1 = [0.1e-9 -350e-9 0.2e-9];
oil2 = [0.3e-9 400e-9 0.4e-9];
% particle type -----------------------------------------------------------------------------------------------------[m]
oil = 1;
silica = 2;
magnetite = 3;
% generation control ---------------------------------------------------------------------------------------------------
gen_increase = 1e-5;
gen_prob = tot_part/succ_tsteps + gen_increase;
% saving control -------------------------------------------------------------------------------------------------------
s_mat = 2000; % matrix saving interval, based on time-step
s_vect = 500; % vectors saving interval, based on time-step
% ----------------------------------------------------------------------------------------------------------------------
%% solution
% ----------------------------------------------------------------------------------------------------------------------
% memory allocation ----------------------------------------------------------------------------------------------------
oil_dist = zeros(1,succ_tsteps/s_vect);
attached_p = zeros(1,succ_tsteps/s_vect);
% [p_type position(x,y,z) velocity(x,y,z), radius, mass, p_ident]
particle_sheet(1,1:10) = [oil, oil1, zeros(1,3), oil_r(1), oil_mass(1), 1];
particle_sheet(2,1:10) = [oil, oil2, zeros(1,3), oil_r(2), oil_mass(2), 2];
n_particles = 2;
p_ident = 2;
generated_silica = 1;
generated_magnetite = 1;
% data retrieving ------------------------------------------------------------------------------------------------------
positions = particle_sheet(:,3:5);
r = particle_sheet(:,8);
for t = 1 : succ_tsteps
% particle generation ----------------------------------------------------------------------------------------------
if rand(1) < gen_prob && p_ident < tot_part + 2
n_particles = n_particles + 1;
p_ident = p_ident + 1;
good = 0;
if rand(1) < max_silica/tot_part && generated_silica <= max_silica
while good == 0
% checking new generation tolerance < 10nm (surface surface)
newp_position = [box(1) box(3) box(5)].*(rand(1,3)-0.5)*2;
dist = sqrt(sum((positions - newp_position).^2,2));
r_new = silica_r(generated_silica);
radsum = r + r_new;
dist_surf = dist - radsum;
m_new = silica_mass(generated_silica);
if all(dist_surf > 10e-9)
good = 1;
end
end
generated_silica = generated_silica + 1;
particle_sheet = [particle_sheet; silica, newp_position, v_rand_init(r_new,m_new,dt,T), r_new, m_new, p_ident];
elseif generated_magnetite <= max_magnetite
while good == 0
% checking new generation tolerance < 10nm (surface surface)
newp_position = [box(1) box(3) box(5)].*(rand(1,3)-0.5)*2;
dist = sqrt(sum((positions - newp_position).^2,2));
r_new = magnetite_r(generated_magnetite);
radsum = r + r_new;
dist_surf = dist - radsum;
m_new = magnetite_mass(generated_magnetite);
if all(dist_surf > 10e-9)
good = 1;
end
end
generated_magnetite = generated_magnetite + 1;
particle_sheet = [particle_sheet; magnetite, newp_position, v_rand_init(r_new,m_new,dt,T), r_new, m_new, p_ident];
end
end
% data retrieving --------------------------------------------------------------------------------------------------
positions = particle_sheet(:,2:4);
r = particle_sheet(:,8);
m = particle_sheet(:,9);
v_0 = particle_sheet(:,5:7);
type = particle_sheet(:,1);
p_ident_v = particle_sheet(:,10);
% step 1 -----------------------------------------------------------------------------------------------------------
% compute new inter-particle forces
[Fatx, Faty, Fatz] = Fat(positions, r, type);
v1_2 = v_0 + 0.5.*[Fatx, Faty, Fatz]./m.*dt;
%step 2 ------------------------------------------------------------------------------------------------------------
% preparing while cycle variables
fixing = 0;
vi = 8.90e-4; % Pa*s
kb = 1.38e-23;
friction = 6*pi*vi.*r;
[R1, R2] = R(r,m,dt);
% calculate positions for vn+1/2dt
pos_prime = positions + m./friction.*(1 - exp(-friction./m.*dt)).*v1_2 + 1./friction.*sqrt(2*kb*T*friction).*R2;
% calculate desired distance
dij = r + r';
dij = dij - dij.*eye(size(dij));
% calculate relative shift
snx = pos_prime(:,1) - pos_prime(:,1)';
sny = pos_prime(:,2) - pos_prime(:,2)';
snz = pos_prime(:,3) - pos_prime(:,3)';
sn = sqrt(snx.^2 + sny.^2 + snz.^2);
% evalueting tolerance
omega = 5;
toll = (sum(sum(sn)))/(numel(sn)*omega)^2;
% calculate overlapping
olap = sn.^2 - dij.^2;
% evalueting random values
rand1 = abs(rand(size(positions,1),3));
while any(any(abs(olap + toll.*eye(size(olap))) > toll)) && fixing < 50
for i = 1:size(positions,1)
for k = 1:size(positions,1)
if olap(i,k) > toll && i ~= k
% evalueting position correction coefficients
betanx = olap(i,k) ./ ( 2.*snx(i,k).*(positions(i,1)-positions(k,1)).*( (1-exp(-rand1(i,1)./m(i).*dt))./rand1(i,1) + ...
((1-exp(-rand1(k,1)./m(k).*dt))./rand1(k,1)) ) );
betany = olap(i,k) ./ ( 2.*sny(i,k).*(positions(i,2)-positions(k,2)).*( (1-exp(-rand1(i,2)./m(i).*dt))./rand1(i,2) + ...
((1-exp(-rand1(k,2)./m(k).*dt))./rand1(k,2)) ) );
betanz = olap(i,k) ./ ( 2.*snz(i,k).*(positions(i,3)-positions(k,3)).*( (1-exp(-rand1(i,3)./m(i).*dt))./rand1(i,3) + ...
((1-exp(-rand1(k,3)./m(k).*dt))./rand1(k,3)) ) );
% new vel along x
v1_2(i,1) = v1_2(i,1) - betanx * positions(i,1)-positions(k,1);
v1_2(k,1) = v1_2(k,1) + betanx * positions(i,1)-positions(k,1);
% new vel along y
v1_2(i,2) = v1_2(i,2) - betany * positions(i,2)-positions(k,2);
v1_2(k,2) = v1_2(k,2) + betany * positions(i,2)-positions(k,2);
% new vel along z
v1_2(i,3) = v1_2(i,3) - betanz * positions(i,3)-positions(k,3);
v1_2(k,3) = v1_2(k,3) + betanz * positions(i,3)-positions(k,3);
end
end
end
fixing = fixing + 1;
% recalculate half positions
pos_prime = positions + m./friction.*(1 - exp(-friction./m.*dt)).*v1_2 + 1./friction.*sqrt(2*kb*T*friction).*R2;
% recalculate relative shift
snx = pos_prime(:,1) - pos_prime(:,1)';
sny = pos_prime(:,2) - pos_prime(:,2)';
snz = pos_prime(:,3) - pos_prime(:,3)';
sn = sqrt(snx.^2 + sny.^2 + snz.^2);
% recalculate overlapping
olap = sn.^2 - dij.^2;
end
% step 3 -----------------------------------------------------------------------------------------------------------
old_pos = positions;
% fixing instabilities
displ = pos_prime - old_pos;
displ(displ > 7e-9) = 7e-9;
displ(displ < -7e-9) = -7e-9;
positions = old_pos + displ;
% step 4-5 ---------------------------------------------------------------------------------------------------------
% calculate new acting forces
[Fatx, Faty, Fatz] = Fat(positions, r, type);
% calculating new velocities
v_dt = v1_2.*exp(-friction./m.*dt) + 1./m.*sqrt(2*kb*T*friction).*R1 + [Fatx, Faty, Fatz]./2./m.*dt;
% step 6 -----------------------------------------------------------------------------------------------------------
% calculating velocities magnitude difference
diffvx = v_dt(:,1) - v_dt(:,1)';
diffvy = v_dt(:,2) - v_dt(:,2)';
diffvz = v_dt(:,3) - v_dt(:,3)';
diffvij = sqrt(diffvx.^2 + diffvy.^2 + diffvz.^2);
% tolerance and preparing for the while cycle
tollv = 1e-6;
fixing = 0;
fixedv = 0;
% calculate initial
olapv = diffvij .* sn;
while any(olapv > tollv & olapv > 0,"all") && fixing < 50
for i = 1:size(positions,1)
for k = 1:size(positions,1)
if olapv(i,k) > toll && olapv(i,k) > 0 && i ~= k
% evalueting position correction coefficients
sigmax = (v_dt(i,1) - v_dt(k,1))*(positions(i,1) - positions(k,1))/dij(i,k)/(1/m(i) + 1/m(k));
sigmay = (v_dt(i,2) - v_dt(k,2))*(positions(i,2) - positions(k,2))/dij(i,k)/(1/m(i) + 1/m(k));
sigmaz = (v_dt(i,3) - v_dt(k,3))*(positions(i,3) - positions(k,3))/dij(i,k)/(1/m(i) + 1/m(k));
% calculating new velocities
v_dt(i,1) = v_dt(i,1) - sigmax * (positions(i,1) - positions(k,1)) / m(i);
v_dt(k,1) = v_dt(k,1) + sigmax * (positions(i,1) - positions(k,1)) / m(k);
v_dt(i,2) = v_dt(i,2) - sigmay * (positions(i,2) - positions(k,2)) / m(i);
v_dt(k,2) = v_dt(k,2) + sigmay * (positions(i,2) - positions(k,2)) / m(k);
v_dt(i,3) = v_dt(i,3) - sigmaz * (positions(i,3) - positions(k,3)) / m(i);
v_dt(k,3) = v_dt(k,3) + sigmaz * (positions(i,3) - positions(k,3)) / m(k);
end
end
end
% calculating velocities magnitude difference
diffvx = v_dt(:,1) - v_dt(:,1)';
diffvy = v_dt(:,2) - v_dt(:,2)';
diffvz = v_dt(:,3) - v_dt(:,3)';
diffvij = sqrt(diffvx.^2 + diffvy.^2 + diffvz.^2);
olapv = diffvij .* sn;
fixing = fixing + 1;
end
% step 7 -----------------------------------------------------------------------------------------------------------
% particle snapping on oil
% fixing the zeros on the diagonal
sn = sn + eye(size(sn));
% approx snapping first particle
if any( (sn(:,1)-dij(:,1)) < 1e-9 ) || exist("attached1","var")== 1
if exist("attached1","var")== 0
attached1 = p_ident_v((sn(:,1)-dij(:,1)) < 1e-9);
end
attached1 = unique([attached1 ; p_ident_v((sn(:,1)-dij(:,1)) < 1e-9)]);
versorx = snx(:,1)./sn(:,1);
versory = sny(:,1)./sn(:,1);
versorz = snz(:,1)./sn(:,1);
mov_x = versorx.*(dij(:,1) + 0.01e-9);
mov_y = versory.*(dij(:,1) + 0.01e-9);
mov_z = versorz.*(dij(:,1) + 0.01e-9);
if length(positions(:,1)) >= 3
for i = 3:length(positions(:,1))
if any(p_ident_v(i) == attached1)
positions(i,1) = mov_x(i) + positions(1,1);
positions(i,2) = mov_y(i) + positions(1,2);
positions(i,3) = mov_z(i) + positions(1,3);
end
end
end
end
% recalculate particle position
snx = positions(:,1) - positions(:,1)';
sny = positions(:,2) - positions(:,2)';
snz = positions(:,3) - positions(:,3)';
sn = sqrt(snx.^2 + sny.^2 + snz.^2);
sn = sn + eye(size(sn));
% approx snapping second particle
if any( (sn(:,2)-dij(:,2)) < 1e-9 ) || exist("attached2","var")== 1
if exist("attached2","var")== 0
attached2 = p_ident_v((sn(:,2)-dij(:,2)) < 1e-9);
end
attached2 = unique([attached2 ; p_ident_v((sn(:,2)-dij(:,2)) < 1e-9)]);
versorx = snx(:,2)./sn(:,2);
versory = sny(:,2)./sn(:,2);
versorz = snz(:,2)./sn(:,2);
mov_x = versorx.*(dij(:,2) + 0.01e-9);
mov_y = versory.*(dij(:,2) + 0.01e-9);
mov_z = versorz.*(dij(:,2) + 0.01e-9);
if length(positions(:,1)) >= 3
for i = 3:length(positions(:,1))
if any(p_ident_v(i) == attached2)
positions(i,1) = mov_x(i) + positions(2,1);
positions(i,2) = mov_y(i) + positions(2,2);
positions(i,3) = mov_z(i) + positions(2,3);
end
end
end
end
% step 8 -----------------------------------------------------------------------------------------------------------
% delete external particles and updating the sheet
particle_sheet(:,2:7) = [positions, v_dt];
ie = any(positions < [ext_box(2), ext_box(4), ext_box(6)] | positions > [ext_box(1), ext_box(3), ext_box(5)],2);
if ie(1) == 1 || ie(2) == 1
warning("broken simulation")
end
if any(ie,"all")
size_before = size(particle_sheet,1);
particle_sheet(ie,:) = [];
size_after = size(particle_sheet,1);
eliminated_part = size_before - size_after;
n_particles = n_particles - eliminated_part;
if exist("tot_eliminated_part","var") == 0
tot_eliminated_part = 0;
end
tot_eliminated_part = tot_eliminated_part + eliminated_part;
clc; fprintf("%g eliminated particle/s\n", tot_eliminated_part)
end
% cheking again before saving
if any(isnan(particle_sheet),"all")
error("nan in particle sheet, restart the simulation")
end
% step 9 -----------------------------------------------------------------------------------------------------------
% saving the particle sheet every n iterations
if (mod(t,s_mat)==0)
t_str = num2str(t);
writematrix(particle_sheet,strcat("./results/langevin.",t_str,'.txt'),'Delimiter',',')
end
if (mod(t,s_vect)==0)
if exist("attached1","var") == 0
long1 = 0;
else
long1 = numel(attached1);
end
if exist("attached2","var") == 0
long2 = 0;
else
long2 = numel(attached2);
end
attached_p(t/s_vect) = long1 + long2;
oil_dist(t/s_vect) = sqrt(sum((positions(1,:) - positions(2,:)).^2));
scatter(t*dt,oil_dist(t/s_vect),"blue",".");
hold on
if t == s_vect
mkdir("results")
ylim([300e-9 950e-9])
xlim([0 time_end])
ylabel("inter-particle distance [m]")
xlabel("time [s]")
end
message = sprintf('time=%d\nn-part=%d\nattached-particles=%d\n', t*dt, length(particle_sheet),attached_p(t/s_vect));
time = annotation('textbox',[0.15 0.78 0.15 0.15],'String',message,'EdgeColor','none');
drawnow
delete(time)
end
end
% ----------------------------------------------------------------------------------------------------------------------
%% functions
% ----------------------------------------------------------------------------------------------------------------------
% compute random velocities for new particles --------------------------------------------------------------------------
function out = v_rand_init(r,m,dt,T)
kb = 1.38e-23;
vi = 8.90e-4; % Pa*s
friction = 6*pi*vi*r;
tau2 = m/friction*(1 - exp(-friction/m*dt));
R1 = sqrt(tau2).*randn(1,3);
out = R1./m.*sqrt(2*kb*T*friction);
end
% computeforce ---------------------------------------------------------------------------------------------------------
function [Fatx, Faty, Fatz] = Fat(positions, r, type)
% Hamaker constants
A = [3.9e-21 4.6e-21 5.52e-19]; % oil silica magnetite
% fixed inverse debye lenght
kdb = 1./400e-9;
% zeta potential
zeta = [-10e-3 50e-3 -40e-3]; % oil silica magnetite
% memory allocation
Fatxs = zeros(size(positions,1),size(positions,1));
Fatys = zeros(size(positions,1),size(positions,1));
Fatzs = zeros(size(positions,1),size(positions,1));
% evalueting distances
dist_m = sqrt( (positions(:,1)-positions(:,1)').^2 + (positions(:,2)-positions(:,2)').^2 ...
+(positions(:,3)-positions(:,3)').^2);
% triangular matrix eval
for k = 2:size(positions,1) % rows
for i = 1:k-1 % columns
p1_pos = positions(k,:);
p2_pos = positions(i,:);
r1 = r(k);
r2 = r(i);
zeta1 = zeta(type(k));
zeta2 = zeta(type(i));
distance = dist_m(k,i);
H = distance - (r1 + r2);
% attraction force -----------------------------------------------------------------------------------------
top = r1*r1*r1 * r2*r2*r2 * distance;
bottom = H^2 * (H + 2*r1).^2 * (H + 2*r2)^2 *(H + 2*(r1 + r2))^2;
Fa = -32/3*top/bottom*A(type(k)); % module
% repulsion force ------------------------------------------------------------------------------------------
eps = 81; eps0 = 8.85419e-12;
Frep = pi*eps*eps0*r1*r2/(r1+r2)*kdb*( (zeta1-zeta2)^2 *exp(-kdb*H)/(exp(-kdb*H)-1) + (zeta1 + zeta2)^2 ...
*exp(-kdb*H)/(exp(-kdb*H) + 1) );
% resultants -----------------------------------------------------------------------------------------------
direction = p2_pos-p1_pos;
norm_d = norm(direction);
% limiting resultant to stabilize the sys
resultant = min(Fa-Frep,1e-8);
resultant = max(Fa-Frep,-1e-8);
% filling matrix
Fatxs(k,i) = (resultant)*direction(1)/norm_d;
Fatys(k,i) = (resultant)*direction(2)/norm_d;
Fatzs(k,i) = (resultant)*direction(3)/norm_d;
end
end
% Forces computation
Fatx = (sum(Fatxs)' - sum(Fatxs,2));
Faty = (sum(Fatys)' - sum(Fatys,2));
Fatz = (sum(Fatzs)' - sum(Fatzs,2));
if any(isnan(Fatx))
error("nan forces")
end
end
% compute v_rand -------------------------------------------------------------------------------------------------------
function [R1, R2] = R(r,m,dt)
% preparing the random array
size_array = size(r,1);
vi = 8.90e-4; % Pa*s
friction = 6*pi*vi.*r;
tau1 = m./friction.*(1 - exp(-friction./m.*dt));
tau2 = m./friction.*(1 - exp(-2.*friction./m.*dt));
rand1 = randn(size_array,3);
rand2 = randn(size_array,3);
R1 = sqrt(tau2).*rand1;
R2 = (tau1 - tau2)./sqrt(tau2).*rand1 + sqrt(dt - tau1.*tau1./tau2).*rand2;
if any(isnan(R1),"all") || any(isnan(R2),"all")
error("nan R1 | R2")
end
end