Replicating a figure of an article in MATLAB - matlab

Question

I want to replicate a figure from this article. More specifically, I want to replicate Figure number 4, which I believe is the representation of Equation 9.
So far I have come up with this code:
% implementing equation 9 and figure 4
step = 0.01; t = 1:step:3600;
d = 3; % dimension
N = 8000; % number of molecules
H = 0.01; % H = [0.01,0.1,1] is in mol/micrometer^3
H = H*6.02214078^5; % hence I scaled the Avogadro's number (right or wrong?)
D = 10; % diffusion coefficient in micrometer^2/sec
u(1) = 1./(1.^(d/2)); % inner function in equation 9; first pulse
for i = 2:numel(t)/1000
u(i) = u(i-1)+(1./(i.^(d/2))); % u-> the pulse number
lmda(i) = (1/(4*pi*D))*((N/(H)).*sum(u)).^(2/d);
end
figure;plot(lmda)
But I am not able to replicate it.
Equation 9
For details on the parameters, refer to the above code. The authors did mention that the summation in equation 9 is a Reimann Zeta series. Wonder if that has anything to do with the result?
Figure 4, which I am trying to replicate:
Could someone kindly tell me the mistake I am making?
P.s: This is not a homework.

Answer 1

Problem 1: You think you are scaling by Avogadro's number on this line
H = H*6.02214078^5;
In fact, you're scaling by approximately 7920=6.022^5. If you wanted to scale by the Avogadro number then you should do:
H = H * 6.02214078e23 % = 6.02214078 * 10^23 : the Avogadro number
Problem 2: You aren't plotting against t, you're plotting against the sample number which doesn't really make sense (unless your t happened to be in integer seconds). Remove the /1000 from your loop
for i = 2:numel(t)
% ...
end
% Then plot
plot(t, lmda)
At this stage we can see something is really wrong. Now that we're scaling by the correct Avo number, the orders of magnitude are way out. I suggest that you trust the H in figure 4 and the H in equation 9 are the same H, it would be very confusing if the author intended anything different!
On that basis, I would suggest you are using the wrong D, N, or time between pulses. I've set up the pulse timing a bit clearer in my code below. I've also streamlined your loop somewhat using vectorisation, and removed the H scaling.
If you tweak it so dtPulses=100 as well as D=100, then the plots are almost identical. You perhaps need to consider how these two numbers affect the result...
% implementing equation 9 and figure 4
d = 3; % dimension
N = 8000; % number of molecules
D = 100; % diffusion coefficient in micrometer^2/sec
dtPulses = 10; % Seconds between pulses
tPulses = 1:dtPulses:3600; % Time array to plot against
nt = numel(tPulses);
i = 1:nt; % pulse numbers
u = 1 ./ (i.^(d/2)); % inner function in equation 9: individual pulse
for k = 2:nt % Running sum
u(k) = u(k-1)+u(k);
end
% Now plot for different H (mol/micrometer^3)
H = [0.01, 0.1, 1];
figure; hold on; linestyles = {':k', '--k', '-k'};
for nH = 1:3
lmda = ((1/(4*pi*D))*(N/H(nH)).*u).^(2/d);
plot(tPulses, lmda, linestyles{nH}, 'linewidth', 2)
end