function Z = PB_code(phi,radius,M,charge)
%Online Supplemental Material to accompany: Goehring, Li and Kiatkirakajorn, 
%Drying paint: from micro-scale dynamics to mechanical instabilities, 
%Philosophical Transactions A, DOI:10.1098/rsta.2016.0161 


%Section 1.1: Define material properties.  Change as required
%Example values for ludox HS
%radius = 8; %Particle radius, in nm
%M = 0.005; %Concentration of background electrolyte, i.e. salt (in Donnan equilibrium)
%charge = 0.5; %Surface charge density, electrons per nm^2 (bare or titrated charge)

%Section 1.2: Define fundamental constants
B = 7.0*10^-10; %Bjerrum length, m
kT = (1.38*10^-23)*293; %Boltzmann energy, J
A = 6.02*10^23; %Avagadro's number
e0 = 1.6*10^-19; %fundamental charge, C

%During calculation all length scales are normalised by Bjerrum length
%The labels refer to the values inside first set of brackets.  Other terms normalise.
r0 = (radius*10^-9)/B; %radius of particle, m
n0 = (M)*B^3*A*1000; %concentration of background electrolyte (outside dialysis bag)
q = (charge)*10^18*B^2; %bare charge density, e/nm^2;

R = r0/(phi)^(1/3); %radius of neutral-charged cell

%Section 2
%Solve the nonlinear PB model
x = linspace(r0,R,100);
solinit = bvpinit(x,@mat4init);

sol = bvp4c(@twoode,@twobc,solinit);
y = deval(sol,x);

%the PDE solver itself is below
%define differntial equations
    function dxdy = twoode(r,y)
        dxdy = [ y(2)
                 -2/r*y(2)-4*pi*n0*(exp(-y(1))-exp(y(1)))];
    end

%and boundary conditions
    function res = twobc(ya,yb)
        res = [ya(2) + 4*pi*q
               yb(2) - 0];
    end

%and initial conditions for the fitting algorithm
    function yinit = mat4init(x)
        yinit = [ 4
                  0];
    end


%Section 3: define various outputs

%calculate effective Debye length 
%using Trizac et al. Langmuir 19, 4027 (2003), Eq. 6
pr = y(1,end);
k_PB = sqrt(8*pi*n0*cosh(pr));
D_eff = B/k_PB; %effective or screened debye length

%calculate effective charge 
%using Trizac et al. Langmuir 19, 4027 (2003), Eq. 16
Q = 4*pi*q*r0^2;
Q_eff = tanh(pr)/(k_PB)*( (k_PB^2*r0*R-1)*sinh(k_PB*(R-r0)) + (k_PB*(R-r0))*cosh(k_PB*(R-r0)) ); 
%Q_eff is the effective, or screened, or renormalized surface charge; 
%Q is the bare charge

%calculate osmotic pressure
%See Bonnet-Gonnet, Belloni, and Cabane, Langmuir 10, 4012 (1994) Eq. 4
P = kT*n0/B^3*(exp(-y(1,end))+exp(y(1,end))-2);

%from pressure, calculate compressibility
Z = P/(kT)*4/3*pi*(radius*10^-9)^3/phi;

%plots show the ion distribution (as in Fig. 2(b))
%figure(3)
%plot(x/r0,exp(-y(1,:)),'-r',x/r0,exp(y(1,:)),'-k');
%axis([1,R/r0,0,5]);
    
   
end