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AFM_scan_3.m
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AFM_scan_3.m
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function [tpos, surface_heights, tot_current_density, sim_time, a_contact] = ...
AFM_scan_3(x_pos, y_pos, z_pos, N_nodes, scan_ts, relax_time_ts, actual_ts, ...
dt, tip, sub, L, v_tip, eapp_base, ecorr_base, alpha0, i0_growth_base, E2_0, ...
i0, i0Me, i0_monolayer_base, i0_passive_base, cutoff, v_act)
%=====================================================================
% This function models a simple AFM tip scan
%=====================================================================
% Define a few physical constants
%=====================================================================
Faraday_Constant = 96485; %coul/mol
R = 8.314; %J/mol K
T = 298; %K
kb = 1.38e-23;
%=====================================================================
% Define some new variables from the parameters passed to the function
%=====================================================================
velocity_tip = v_tip;
dy = y_pos(end) - y_pos(end-1);
dx = x_pos(end) - x_pos(end-1);
relax_ts = relax_time_ts;
mz = min(z_pos);
% Set the minimum oxide thickness as 10% of the minimum oxide thickness
% in the z_pos array
zmin1 = 0.1* min(mz);
b_cutoff = 0.01; %0.25;
%=====================================================================
%=====================================================================
% Define material parameters for different simulation cases
%=====================================================================
switch tip
case 'Diamond'
nu_tip = 0.10;
E_tip = 1053e9; %Pa
R_AFM_tip = 35.0e-9; %70.0e-9; %m
otherwise
nu_tip = 0.10;
E_tip = 1053e9; %Pa
R_AFM_tip = 35.0e-9; %70.0e-9; %m
end
switch sub
case 'Cr2O3'
nu_substrate = 0.25; %Cr2O3
E_substrate = 125e9; %Pa Cr2O3
% H_substrate = (0.009807 * 8.25) * 1.0e9; %Pa - HV from Wikipedia for C2O3 and formula from gordonengland.co.uk
H_substrate = (0.009807 * 8.25) * 1.0e9; %Pa - HV from Wikipedia for Cr2O3 and formula from gordonengland.co.uk
K_archard = 40.0e-6; % 5.0e-5; %5.72e-5; %1.7e-5; % For a ferritic stainless steel in Wikipedia
k_film = 504; %nm.cm2/A.s
z = 1;
alpha_stress_modifier = 1;
rho = 5.22 *(1/(0.01 * 0.01 * 0.01)); %g/cm3 -> g/m3
M = 151.9904; %g/mol
Vm = M/rho; % m3/mol
case 'UNS S32750'
nu_substrate = 0.27; % UNS S32750 0.25; %Cr2O3
E_substrate = 210e9; % UNS S32750 Ps 125e9; %Pa Cr2O3
H_substrate = (0.009807 * 250) * 1.0e9; %Pa - HV from Wikipedia for 2507 and formula from gordonengland.co.uk
K_archard = 5.0e-5; %1.7e-5; %1.7e-5; % For a ferritic stainless steel in Wikipedia
k_film = 504;%nm.cm2/A.s %
z = 1;
alpha_stress_modifier = 1;
otherwise
nu_substrate = 0.2; % UNS S32750 0.25; %Cr2O3
E_substrate = 100e9; % UNS S32750 Ps 125e9; %Pa Cr2O3
H_substrate = (0.009807 * 10) * 1.0e9; %Pa - HV from Wikipedia for 2507 and formula from gordonengland.co.uk
K_archard = 1.0e-3; %1.7e-5; % For a ferritic stainless steel in Wikipedia
k_film = 504; %nm.cm2/A.s
z = 1;
alpha_stress_modifier = 1;
end
%=====================================================================
% Calculate Hertzian contact parameters
%=====================================================================
[r_damage_m, depth_m, p_max] = Hertzian_Contact(E_tip, nu_tip, E_substrate, nu_substrate, L, R_AFM_tip);
r_damage_nm = r_damage_m * 1.0e9;
depth_nm = depth_m * 1.0e9;
% This conditional check puts limits the damage radius to the AFM tip
% radius. This restriction may be lifted after further testing
if r_damage_nm > (R_AFM_tip * 1.0e9)
r_damage_nm = R_AFM_tip * 1.0e9;
end
%=====================================================================
%=====================================================================
% This section to be used to model changes in $\alpha^{+}$ as a
% function of stress
%=====================================================================
%=====================================================================
% This section to be used to model changes in $i_{0,field}$ as a
% function of stress
%=====================================================================
i0_model = i0_growth_base;
%=====================================================================
% Determine the number of y nodes to shift during scanning. Currently,
% the simulation provides realistic values only if the y-shift is 1
%=====================================================================
check_y_nodes_shift = round((R_AFM_tip * 1.0e9)/dy);
if check_y_nodes_shift < 1
num_y_nodes_shift = 1;
else
num_y_nodes_shift = check_y_nodes_shift;
end
%=====================================================================
%=====================================================================
% Create an oxide sructure to keep track of damaged and undamaged areas
%=====================================================================
node_counter = 1;
for i = 1:(N_nodes(2))
for j = 1:(N_nodes(1))
oxide(node_counter).grid = [x_pos(j), y_pos(i)];
oxide(node_counter).nodes = [j,i];
oxide(node_counter).height = z_pos(j,i);
oxide(node_counter).base_height = z_pos(j,i); %zmax; %2.5; %
oxide(node_counter).rebuild_height = 0.0;
oxide(node_counter).old_current = 0.0;
oxide(node_counter).new_current = 0.0;
oxide(node_counter).i0_growth = i0_growth_base;
oxide(node_counter).alpha_node = 0.0;
oxide(node_counter).k_film = k_film;
oxide(node_counter).num = 0.0;
oxide(node_counter).denom = 0.0;
oxide(node_counter).damage_current = 0.0;
oxide(node_counter).initiation_time = 0.0;
oxide(node_counter).rebuild_time = 0.0;
oxide(node_counter).has_damage = 0;
oxide(node_counter).cutoff_state = 0; %Off = 0, On = 1
oxide(node_counter).cutoff_current = 0.0;
oxide(node_counter).cutoff_time = 0.0;
node_counter = node_counter + 1;
end
end
num_nodes = node_counter - 1;
surface_heights = zeros(num_nodes,1);
%=====================================================================
%=====================================================================
% Start time evolution
%=====================================================================
tpos = zeros(actual_ts,4);
tot_current_density = zeros(actual_ts,1);
sim_time = zeros(actual_ts,1);
tip_pos_x = 0.0; %0.0:(v_tip*dt):(v_tip * total_time);
tip_pos_y = 0.0; %(ymax/2.0) - dy; %tip_pos_x;
direction_of_x_travel = 1; % 1 = forward, -1 = backward
direction_of_y_travel = 1; % 1 = up, -1 = bacvkward
total_pause = 1; %10;
pause_counter = 1;
another_eta_adjuster = 0.15;
%=====================================================================
%=====================================================================
% Initialize the damaged area calculation
area_sum = 0.0;
%=====================================================================
%=====================================================================
% Time iterations for AFM scanning and relaxation
%=====================================================================
tip_ts = 1;
for idx_time = 1:actual_ts
sim_time(idx_time,1) = (idx_time-1)*dt; %s
% if sim_time(idx_time,1) >= 80
% disp(sim_time(idx_time,1))
% end
if tip_ts >= (scan_ts+1)
% Stop the AFM tip once the scanning duration reached
velocity_tip = 0.0;
end
% This conditional only necessary if the total number of time-steps
% allows for forward and reverse scans of the AFM tip
if direction_of_y_travel > 0
if tip_pos_y >= y_pos(end)
% AFM tip has reached the upper edge of the computational
% cell
if pause_counter <= total_pause
direction_of_x_travel = 0;
% direction_of_y_travel = 0;
velocity_tip = 0.0;
pause_counter = pause_counter + 1;
else
direction_of_x_travel = 1;
direction_of_y_travel = -1;
velocity_tip = v_tip;
pause_counter = 1;
end
end
elseif direction_of_y_travel < 0
if tip_pos_y <= 0.0
% AFM tip has reached the lower edge of the computational
% cell
if pause_counter <= total_pause
direction_of_x_travel = 0;
% direction_of_y_travel = 0;
velocity_tip = 0.0;
pause_counter = pause_counter + 1;
else
direction_of_x_travel = 1;
direction_of_y_travel = 1;
velocity_tip = v_tip;
pause_counter = 1;
end
end
end
% Determine the x,y-position of the AFM tip and if it is scanning
% forward or backward
if direction_of_y_travel > 0
if tip_pos_x >= x_pos(end) && direction_of_x_travel > 0
direction_of_x_travel = -1;
tip_pos_y = tip_pos_y + (num_y_nodes_shift * dy);
elseif tip_pos_x <= (0.0) && direction_of_x_travel < 0
direction_of_x_travel = 1;
tip_pos_y = tip_pos_y + (num_y_nodes_shift * dy);
end
elseif direction_of_y_travel < 0
if tip_pos_x >= x_pos(end) && direction_of_x_travel > 0
direction_of_x_travel = -1;
tip_pos_y = tip_pos_y - (num_y_nodes_shift * dy);
elseif tip_pos_x <= (0.0) && direction_of_x_travel < 0
direction_of_x_travel = 1;
tip_pos_y = tip_pos_y - (num_y_nodes_shift * dy);
end
end
if velocity_tip > 0.0
%=====================================================================
% AFM tip scanning
%=====================================================================
for j = 1:num_nodes
testy = oxide(j).grid(2);
testx = oxide(j).grid(1);
testr = sqrt((testx - tip_pos_x)^2 + (testy - tip_pos_y)^2);
check_node_height = oxide(j).height;
check_damage = oxide(j).has_damage;
node_base_height = oxide(j).base_height;
if (testr < r_damage_nm)
ar_ratio = (testr^2)/(r_damage_nm^2);
p_node = p_max * sqrt(1 - (ar_ratio));
if abs(imag(p_node)) > 0.0
disp(num)
end
%======================================================
% Checks to see if the test node is inside the damage
% area
%======================================================
if (check_damage < 0.5) && ( check_node_height > zmin1)
% The test node is undamaged but is inside the
% damage area
oxide(j).has_damage = 1;
init_time = sim_time(idx_time,1);
oxide(j).initiation_time = init_time;
oxide(j).rebuild_time = init_time; % + dt
delta_t = sim_time(idx_time,1) - oxide(j).initiation_time;
delta_h_m = (K_archard/H_substrate)*p_node*dt*(velocity_tip*1.0e-9); %m
delta_h_nm = delta_h_m * 1.0e9; %nm
oxide(j).height = check_node_height - delta_h_nm;
node_height_nm = oxide(j).height; % nm
oxide(j).rebuild_height = node_height_nm;
E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% E0f = E2; %1.225/node_height_nm;
height_ratio = node_height_nm/node_base_height;
eta_modifier = another_eta_adjuster*(p_node/1.259e10); %E_substrate
eta_adj = eta_modifier *(2*p_node*Vm)/(3*z*Faraday_Constant); %
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj; %
if height_ratio < 1.0
alpha = alpha0;
i0_growth = i0_model;
i0_monolayer = i0_monolayer_base;
else
alpha = alpha0;
% eta_adj = 0.0;
i0_growth = 0.0;
i0_monolayer = 0.0;
end
elseif (check_damage > 0.5) && ( check_node_height > zmin1)
%======================================================
% The test node has already been damaged and is
% still inside the damage area
%======================================================
init_time = oxide(j).initiation_time;
delta_t = sim_time(idx_time,1) - init_time;
delta_h_m = (K_archard/H_substrate)*p_node*dt*(velocity_tip*1.0e-9); %m
delta_h_nm = delta_h_m * 1.0e9; %nm
oxide(j).height = check_node_height - delta_h_nm;
oxide(j).rebuild_height = oxide(j).rebuild_height - delta_h_nm;
node_height_nm = oxide(j).height; % nm
E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% E0f = E2; %1.225/node_height_nm;
height_ratio = node_height_nm/node_base_height;
eta_modifier = another_eta_adjuster*(p_node/1.259e10); %E_substrate
eta_adj = eta_modifier *(2*p_node*Vm)/(3*z*Faraday_Constant); %
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj;
if height_ratio < 1.0
alpha = alpha0;
i0_growth = i0_model; %i0_slope * p_node + i0_min;
i0_monolayer = i0_monolayer_base;
else
alpha = alpha0;
eta_adj = 0.0;
i0_growth = 0.0;
i0_monolayer = 0.0;
end
% continue
elseif (check_damage < 0.5) && ( check_node_height <= zmin1)
%======================================================
% Test node is undamaged but the asperity height is
% zero or negative so set it to the minimum
%======================================================
oxide(j).has_damage = 0.0;
oxide(j).height = zmin1;
surface_heights(j) = oxide(j).height;
% continue
elseif (check_damage > 0.5) && ( check_node_height <= zmin1)
%======================================================
% Test node has damage that has reduced its height
% to zmin or below. Set height to zmin and keep
% monitoring the corrosion current
%======================================================
oxide(j).height = zmin1;
oxide(j).rebuild_height = zmin1;
node_height_nm = oxide(j).height;
init_time = oxide(j).initiation_time;
delta_t = sim_time(idx_time,1) - init_time;
height_ratio = node_height_nm/node_base_height;
eta_modifier = another_eta_adjuster*(p_node/1.259e10); %E_substrate
eta_adj = eta_modifier *(2*p_node*Vm)/(3*z*Faraday_Constant); %
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj;
E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% E0f = E2; %1.225/node_height_nm;
if height_ratio < 1.0
alpha = alpha0;
i0_growth = i0_model;
i0_monolayer = i0_monolayer_base;
else
alpha = alpha0;
eta_adj = 0.0;
i0_growth = 0.0;
i0_monolayer = 0.0;
end
% continue
end
else
%======================================================
% The test node is outside the damage area from the AFM
% tip but, if it was previously damaged, keep
% monitoring the oxide re-formation current
%======================================================
if check_damage > 0.5
init_time = oxide(j).initiation_time;
delta_t = sim_time(idx_time,1) - init_time;
node_height_nm = oxide(j).height;
height_ratio = node_height_nm/node_base_height;
eta_modifier = another_eta_adjuster*(p_node/1.259e10); %E_substrate
eta_adj = eta_modifier *(2*p_node*Vm)/(3*z*Faraday_Constant); %
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj; %
E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% E0f = E2; %1.225/node_height_nm;
% delta_h_nm = oxide(j).base_height - node_height_nm;
if height_ratio < 1.0
eta_adj = 0.0;
alpha = alpha0;
i0_growth = i0_model; %i0_slope * p_node + i0_min;
i0_monolayer = i0_monolayer_base;
else
alpha = alpha0;
eta_adj = 0.0;
i0_growth = 0.0;
i0_monolayer = 0.0;
end
else
% Node is outside the damage area and is undamaged
node_height_nm = oxide(j).height;
i0_growth = 0.0;
alpha = alpha0;
i0_monolayer = 0.0;
E0f = 0.0;
oxide(j).damage_current = 0.0;
continue;
end
end
%=====================================================================
% $g^{+} = \alpha^{+}F/RT$
%=====================================================================
g_plus = (alpha * Faraday_Constant)/ (R * T);
stress_effect = 0.0; %(p_node * v_act)/(kb*T);
%=====================================================================
% eta_adj = 0.0;
%=====================================================================
% eapp_base, ecorr_base
eta = (E2_0 * oxide(j).base_height) - (E2 * node_height_nm);
% + ((eapp_base - ecorr_base) - E2_0) + eta_adj;
% eta = eapp_base - (ecorr_base - eta_adj); %(eapp_base + eta_adj;
%=====================================================================
%=====================================================================
% Check to see if the simulation time for the node is still
% within the node's rebuilding the oxide window. If it is,
% keep going. If it is now, transition to the passive
% background current for the node
%=====================================================================
if delta_t <= cutoff
[temp__interface_current,num,denom] = i_growth(k_film, E2_0, i0_growth, g_plus, stress_effect, eta + eta_me, delta_t); %
if (testr < r_damage_nm)
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj; %
temp_mono_current = monolayer_model(eta_me,i0Me);
else
temp_mono_current = 0.0;
end
temp_pass_current = 0.0;
if idx_time == 1
oxide(j).old_current = 0.0;
oxide(j).new_current = temp__interface_current;
else
oxide(j).old_current = oxide(j).new_current;
oxide(j).new_current = temp__interface_current;
end
else
[temp__interface_current,num,denom] = i_growth(k_film, E2_0, i0_growth, g_plus, stress_effect, eta, delta_t); %delta_t
if oxide(j).cutoff_state < 0.5
oxide(j).cutoff_state = 1;
oxide(j).cutoff_current = temp__interface_current;
oxide(j).cutoff_time = sim_time(idx_time,1);
end
temp_mono_current = 0.0;
i_pass_t = passive_model(delta_t, i0_growth, i0_passive_base);
if temp__interface_current >= i_pass_t
delta_ct = sim_time(idx_time,1) - oxide(j).cutoff_time;
temp_pass_current = (oxide(j).cutoff_current - i_pass_t) * exp(-delta_ct/b_cutoff) + i_pass_t;
temp__interface_current = 0.0;
else
temp_pass_current = i_pass_t;
temp__interface_current = 0.0;
end
end
%=====================================================================
oxide(j).damage_current = temp__interface_current + temp_mono_current + temp_pass_current;
% oxide(j).i0_growth = temp__interface_current; %i0_growth;
oxide(j).alpha_node = alpha;
oxide(j).num = num;
oxide(j).denom = denom;
surface_heights(j) = oxide(j).height;
end
else
%=====================================================================
% AFM tip stopped - monitor relaxation current
%=====================================================================
for j = 1:num_nodes
if oxide(j).has_damage > 0.5
init_time = oxide(j).initiation_time;
delta_t = sim_time(idx_time,1) - init_time;
node_height_nm = oxide(j).height;% * 1.0e-3; nm
height_ratio = node_height_nm/oxide(j).base_height;
eta_modifier = another_eta_adjuster*(p_node/1.259e10); %E_substrate
eta_adj = eta_modifier *(2*p_node*Vm)/(3*z*Faraday_Constant); %
eta_base1 = eapp_base - ecorr_base;
eta_base2 = eta_base1 - E2_0;
eta_me = eta_base2 - eta_adj; %
E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% E0f = E2; %1.225/node_height_nm;
% delta_h_nm = oxide(j).base_height - node_height_nm;
if height_ratio < 1.0
eta_adj = 0.0;
alpha = alpha0;
i0_growth = i0_growth_base; %i0_slope_2 * height_ratio + i0_max;
i0_monolayer = i0_monolayer_base;
else
alpha = alpha0;
eta_adj = 0.0;
i0_growth = 0.0;
i0_monolayer = 0.0;
end
%=====================================================================
% $g^{+} = \alpha^{+}zF/RT$
%=====================================================================
g_plus = (alpha * Faraday_Constant)/ (R * T);
stress_effect = 0.0;
%=====================================================================
% Check to see if the simulation time for the node is still
% within the node's rebuilding the oxide window. If it is,
% keep going. If it is now, transition to the passive
% background current for the node
%=====================================================================
eta = (E2_0 * oxide(j).base_height) - (E2 * node_height_nm);
% eta = eapp_base - (ecorr_base - eta_adj); %(eapp_base + eta_adj;
if delta_t <= cutoff
[temp__interface_current,num,denom] = i_growth(k_film, E2_0, i0_growth, g_plus, stress_effect, eta + eta_me, delta_t);
temp_mono_current = 0.0;
temp_pass_current = 0.0;
if idx_time == 1
oxide(j).old_current = 0.0;
oxide(j).new_current = temp__interface_current;
else
oxide(j).old_current = oxide(j).new_current;
oxide(j).new_current = temp__interface_current;
end
else
[temp__interface_current,num,denom] = i_growth(k_film, E2_0, i0_growth, g_plus, stress_effect, eta, delta_t);
if oxide(j).cutoff_state < 0.5
oxide(j).cutoff_state = 1;
oxide(j).cutoff_current = temp__interface_current;
oxide(j).cutoff_time = sim_time(idx_time,1);
end
temp_mono_current = 0.0;
i_pass_t = passive_model(delta_t, i0_growth, i0_passive_base);
if temp__interface_current >= i_pass_t
delta_ct = sim_time(idx_time,1) - oxide(j).cutoff_time;
temp_pass_current = (oxide(j).cutoff_current - i_pass_t) * exp(-delta_ct/b_cutoff) + i_pass_t;
temp__interface_current = 0.0;
else
temp_pass_current = i_pass_t;
temp__interface_current = 0.0;
end
end
%=====================================================================
% Sum the current from all contributions
oxide(j).damage_current = temp__interface_current + temp_mono_current + temp_pass_current;
% oxide(j).i0_growth = temp__interface_current; %i0_growth;
oxide(j).alpha_node = alpha;
oxide(j).num = num;
oxide(j).denom = denom;
surface_heights(j) = oxide(j).height;
end
surface_heights(j) = oxide(j).height;
end
end
tpos(idx_time,1) = tip_pos_x;
tpos(idx_time,2) = tip_pos_y;
tpos(idx_time,3) = tip_pos_y + r_damage_nm;
tpos(idx_time,4) = tip_pos_y - r_damage_nm;
tip_pos_x = tip_pos_x + (direction_of_x_travel * dx);
area_sum = area_sum + (((2*r_damage_nm) * dy)*1.0e-7*1.0e-7); %cm2
tip_ts = tip_ts + 1;
%=====================================================================
% Sum current from all nodes and rebuild the damaged oxide film by
% calculating the overpotential for oxide formation and the amount
% of oxide added per time-step
%=====================================================================
temp_tot_current = 0.0;
for j = 1:num_nodes
i_t = oxide(j).damage_current; %* dx*1.0e-7 * dy*1.0e-7
% init_time = oxide(j).initiation_time;
% delta_t = sim_time(idx_time,1) - init_time;
node_height_nm = oxide(j).height;% * 1.0e-3; nm
%
if oxide(j).rebuild_height < oxide(j).base_height && oxide(j).has_damage > 0.5
% E2 = abs(mdl_ef(node_height_nm*1.0e-9, i0, i0Me));
% b_height = oxide(j).base_height;
% igrowth = oxide(j).i0_growth;
% % eta = E2_0 - E2;
% eta = (E2_0 * b_height) - (E2 * node_height_nm);
% add_height = ox_growth(igrowth, alpha, E2_0, k_film, eta, delta_t); %oxide(j).i0_growth
add_height = ((oxide(j).new_current + oxide(j).old_current)/2.0) * dt * k_film;
new_height = oxide(j).rebuild_height + add_height;
oxide(j).rebuild_height = new_height;
elseif oxide(j).rebuild_height >= oxide(j).base_height && oxide(j).has_damage > 0.5
% new_height = 0.0;
oxide(j).height = oxide(j).base_height;
oxide(j).has_damage = 0.0;
end
temp_tot_current = temp_tot_current + i_t; %converted to A
end
tot_current_density(idx_time,1) = temp_tot_current/num_nodes;
a_contact = area_sum;
end
%=====================================================================
end