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- Math Models
Problem-Solving Example #11
Modeling a Nanostructured Solar Cell
Problem: How to develop solar cells with a new (higher) efficiency; grätzel cells.
There are many things said about what’s most important for the solar cell. So what they need is a model to know what’s the rate is limits for the whole system. By then they can choose what combination of parameters will give the best solar cell. The model shown is a one-dimensional non-steady state model; a start to compare it with the Laser experiments. The laser experiments are one of the things they use to predict the efficiency.
But without a model, does experiments really tell one anything? This model is only for one excitation from a laser beam and to analyze how the decay of all species are. There Is a model done for steady state, but its not really working very good in practice. Simulating the
non-steady state model for some time should converge to the steady state solution when there is equilibrium in the system. This means when the change of all species are zero over the film. This could be interesting to compare with other steady-state models.
When we are talking about efficiency, it should be for simulation of the whole system. Then we have to add certain things. There are continuous excitations of electrons which is the starting conditions in this model for the electrons and the excited dye. There are a few more reactions and we have to consider the other part of the solar cell which isn’t contained by nanostuctrured TiO2.
The main thing about the efficiency is that we want as many electrons leaving the back contact which is at x=0. In the reality the electrons will go out in an outer circuit to make a full circuit. But in the Laser experiments this does not happen because the outer circuit is open. In reality we will get out a current dependent on the incident light. There are many ways to measure the efficiency. IPCE(\lambda) incident-photon-to-current efficiency says how much of the incident light was converted to external current.
Title: Modeling a Nanostructured Solar Cell
Short review of the system:
We have a dye sensitizer attached to nano-structured Titanium dioxide (TiO2) film. The nano-structured particles are in a dye which transports the electrons from the electrode to the dye sensitizer. Incident light at a certain wavelength excites electrons in the dye sensitizer. So what happens to this electron after the excitation? A very fast process in nanosecond scale injects the electron to the TiO2 and its making a random walk (that’s what most people think its doing) to the back contact. A new electron from the dye is put in the place of the injected
electron. The electron’s goes through the nano-structured film to a back contact to the outer circuit and we have a total circuit.
But there are other reactions involved in the process. The excited electron can travel other ways then to the back contact like reacting with the dye or dye sensitizer. These reactions are limiting the efficiency of the cell.
Thus I thought it would be a good idea for the model to set up rate constants for all these reactions. Make a discretization along x which is the distance to the back contact. And the step through time and see how the kinetics, diffusion and the electric field is changing the concentration of the species along x for different times.
A macroscopic model for the concentration of s ( the dye sensitizer) could look something like this:
kinetics:
ds(x,t)/dt =
-k_3*s(x,t)*e(x,t)-k_4*s(x,t)*i(x,t)
(k_3 and k_4 rate constants e = electron concentration, i = iodine conc.)
diffusion
ds(x,t) / dt = D*d^2(s(x,t))/dx^2
(D = diffusion constant)
electric field E(x,t):
ds(x,t) / dt = my*ds(x,t) /dt*dE(x,t)/dt
(my = mobility for the species)
The electric field we get from integrating concentrations of all the charged species along x.
Explanations of each colour:
  = the dye which is the charge carrier, giving new electrons to the dyes and get new one at the anode. It is a redox couple of Iodine. It can also react with the excited electrons which gives a less good efficiency. There are also other leakage’s that contribute to decline
 = Dye molecules, the electrons of those are excited at incident light of certain wavelengths
 = The nanostructured semi conductor, most used is TiO2, the electrons diffuse in this medium towards the back contact.
 = back contact (x=0), where the electrons go to get to outer circuit, anode.
 = end of the nanostructured film, x=8*10^(-6)
 = the “entrance” for the electrons from outer circuit, the cathode.
We start with the species in the solar cell
S+ = excited dye
S = dye
I- = Iodine
I3- = three iodine
I0 = iodine radical
I02- = di iodine radical
There are some reactions between the species during simulation with reaction rates k1..k6
{S+} + {e-} -> {S} k_1
{S+} + {I-} -> {I0} k_2
{I0} + {I-} -> {I02-} k_3
2{I02-} -> {I3-} + {I-} k_4
{I02-} + {e-} -> 2{I-} k_5
{I3-} + 2{e-} -> 3{I-} k_6
The concentrations of each species is defined as
s(x,t) = {s+} i(x,t) = {I-}
e(x,t) = {e-} w(x,t) = {I02-}
q(x,t) = {I3-} z(x,t) = {I0}
The starting conditions are ( after a laser pulse there is excitation of the dyes s(x,0) and we look at the relaxation of all species after that)
s(x,0) = 360*10^(-9)*0.34*10^6*0.1*exp(-0.34*10^6*x)
i(x,t) = 0.5 e(x,t) = s(x,0)
w(x,t) = 0.0 q(x,t) = 0.05
z(x,t) = 0.0
The differential equations with electric field diffusion and reactions with the diffusion constants: Di, De, Dw, Dq, Dz; and mobility constants: my_s, my_i, my_e, my_w, my_q we set the constants
diffusion:
Ds, Di, Dq, Dw = 1.5*10^(-9)
De = 200*10^(-9)
mobility:
my_s, my_i, my_w, my_q = 1.5*10^(-9)*1.602*10^(-19)/(1.38*10^(-23)*273)
my_e = 200*10^(-9)*1.602*10^(-19)/(1.38*10^(-23)*273)
rate constants:
k_1 = 1.0 * 10^(-6)
k_2 = 3.0 * k_1 k_3 = 4.0 * k_1
k_4 = 5.0 * k_1 k_5 = 6.0 * k_1
k_6 = 9.0 * k_1
movement from diffusion:
ds(x,t)/dt = 0.0 ( stationary )
di(x,t)/dt = Di*d^2(i(x,t))/dx^2
de(x,t)/dt = De*d^2(e(x,t))/dx^2
dw(x,t)/dt = Dw*d^2(w(x,t))/dx^2
dq(x,t)/dt = Dq*d^2(q(x,t))/dx^2
dz(x,t)/dt = Dz*d^2(z(x,t))/dx^2
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movement
from electric force:
ds(x,t)/dt =
my_s*s(x,t)*dE(x,t)/dx+my_s*E(x,t)*ds(x,t)/dx
di(x,t)/dt =
my_i*i(x,t)*dE(x,t)/dx+my_i*E(x,t)*di(x,t)/dx
de(x,t)/dt = my_e*e(x,t)*dE(x,t)/dx+my_e*E(x,t)*de(x,t)/dx
dw(x,t)/dt =
my_w*w(x,t)*dE(x,t)/dx+my_w*E(x,t)*dw(x,t)/dx
dq(x,t)/dt =
my_q*q(x,t)*dE(x,t)/dx+my_q*E(x,t)*dq(x,t)/dx
dz(x,t)/dt = 0 ( not charged )
! next comes from Poisson’s equation
dE(x,t)/dx =
26.19925089*(s(x,t)-i(x,t)-e(x,t)-w(x,t)-q(x,t))
That makes the Electric field E=0 over the whole film at
time t=0, (sum of all charges in the simulation cell will always be zero).
kinetics:
ds(x,t)/dt = - k_1*s(x,t)*e(x,t) - k_2*s(x,t)*i(x,t)
di(x,t)/dt = -
k_1*s(x,t)*e(x,t) - k_3*i(x,t)*z(x,t)
de(x,t)/dt = - k_1*s(x,t)*e(x,t) - k_5*w(x,t)*e(x,t) -k_6*q(x,t)*e(x,t)
dw(x,t)/dt = - k_5*w(x,t)*e(x,t) + k_3*i(x,t)*z(x,t) - 2*k_4*w(x,t)^2
dq(x,t)/dt =
k_4*w(x,t)^2 -
k_6*q(x,t)*e(x,t)
dz(x,t)/dt = -
k_3*i(x,t)*z(x,t) + k_2*s(x,t)*i(x,t)
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x will be between 0 and 8*10^(-6) which is the thickness of the film where those reactions are.
At the boundaries we need to approximate the derivatives each time step for the diffusion and the electric field. There is no flow of particles out of these boundaries so we could set the concentrations change to zero at the boundaries; i.e. for the boundaries:
di(x,t)/dx=0
de(x,t)/dx=0
dw(x,t)/dx=0 dq(x,t)/dx=0
dz(x,t)/dx=0
So far, this is a initial value problem with six coupled nonlinear differential equations.
Future:
Making a full scaled 3 dimensional model and optimize the parameters for a optimal solar cell. Most of the parameters are adjustable, they are all dependent of the materials used. There are many different things said about what the cell efficiency really depends on.
Some even say that the important thing is to have as good a cathode as possible and others say its something completely different. A model is needed to guide the research for a better solar cell.
Calculus Programming Code:
Problem SolarCel
include 'SolarCel.inc'
C Ok we start with the species in the solar cell:
C ----------------
C S+ = excited dye S = dye I02- = di idodine radical
C I- = Ioidine I3- = three iodine I0 = iodine radical
C we have some reaktions between the species during simulation with
C reaction rates k1...k6:
C ----------------
C {S+} + {e-} -> {S} k1
C {S+} + {I-} -> {I0} k2
C {I0} + {I-} -> {I02-} k3
C 2{I02-} -> {I3-} + {I-} k4
C {I02-} + {e-} -> 2{I-} k5
C {I3-} + 2{e-} -> 3{I-} k6
C the concentrations of each species is defined as:
C ----------------
C s(x,t) = {s+} i(x,t) = {I-} e(x,t) = {e-}
C w(x,t) = {I02-} q(x,t) = {I3-} z(x,t) = {I0}
C The differential equations with electric field diffusion and reactions
C with the diffusion constants: Di, De, Dw, Dq, Dz and mobility
C constants: mys, myi, mye, myw, myq we set the constants:
C diffusion:
constDs = 1.5*1.e-9 : constDe = 200*1.e-9
constDi = constDs : constDq = constDs : constDw = constDs
print *,'Const.',constDs,constDi,constDe,constDw,constDq,constDz
C mobility:
mys = 1.5*1.e-9*1.602*1.e-19/(1.38*1.e-23*273)
myi = mys: myw = mys: myq = mys
mye = 200*1.e-9*1.602*1.e-19/(1.38*1.e-23*273)
print *, 'My.', mys, myi, mye, myw, myq, myz
C rate constants:
k1 = 1.0 * 1.e-6: k2 = 3 * k1: k3 = 4 * k1: k4 = 5 * k1
k5 = 6 * k1: k6 = 9 * k1
print *, 'Ks.', k1, k2, k3, k4, k5, k6
C x will be between 0 and 8*1.e-6 which is the thickness of the
C film where those reactions are:
xfinal = 8*1.e-6: xprint = xfinal / 100: dx = xprint / 10
tfinal = 1.e2: tprint = tfinal / 100: dt = tprint / 10
C At the boundaries I suppose we need to approximate the
C derivativeseach time step for the diffusion and the electric field.
C There is no flow of particles out of these boundaries
C so we could set the concentrations change to zero at
C the boundaries; i.e for the boundaries.
didx=0: dedx=0: dwdx=0: dqdx=0: dzdx=0
C the starting conditions are ( after a laser pulse there is
C excitation of the dyes s(x,0) and we look at the relaxation
C of all species after that):
initiate JANUS; for distance; equations
* dsdx/x, d2idx/didx, didx/x, d2edx/dedx, dedx/x, d2wdx/dwdx,
* dwdx/x, d2qdx/dqdx, dqdx/x, d2zdx/dzdx, dzdx/x, dEsumdx/Esum;
* of x; step dx; to xf;
print *,' TIME DSDT S DIDT I'
xf=xprint
do while (xf .le. xfinal)
integrate distance; by JANUS
print '(7(1pg13.5))', x, s, i, e, w, q, z
C @curves('plot')
xf=xf+xprint
end do
C @show('plot')
end
model distance
include 'SolarCel.inc'
s = 360*1.e-9*0.34*10**6*0.1*exp(-0.34*10**6*x)
i = 0.5 : e = s : w = 0. : q = 0.05 : z = 0.
C movement from diffusion:
dsdt = 0. ! stationary
didt = constDi * d2idx : dedt = constDe * d2edx
dwdt = constDw * d2wdx : dqdt = constDq * d2qdx
dzdt = constDz * d2zdx
initiate ATHENA; for ide; equations dsdt/t, didt/t,
* dedt/t, dwdt/t, dqdt/t, dzdt/t; of t; step dt; to tf;
print *,' X TIME DSDT S DIDT I'
tf=tp
do while (tf .le. tfinal)
integrate ide; by ATHENA
print '(6(1pg13.5))', x, t, dsdt, s, didt, i
tf=tf+tp
end do
find dsdx, didx, dedx, dwdx, dqdx, dzdx, dEsumdx;
* in eForce; by AJAX( cntrl1);
* to match xs, xi, xe, xw, xq, xz, xEsum
C @show('plot')
end
model ide ! Implicit Differential Equations
include 'SolarCel.inc'
find dsdt, didt, dedt, dwdt, dqdt, dzdt;
* in kinetics; by AJAX( cntrl1);
* to match ts, ti, te, tw, tq, tz
end
model kinetics
include 'SolarCel.inc'
C kinetics:
ts = dsdt - (- k1 * s * e - k2 * s * I)
ti = didt - (- k1 * s * e - k3 * i * z)
te = dedt - (- k1 * s * e - k5 * w * e -k6 * q * e)
tw = dwdt - (- k5 * w * e + k3 * i * z - 2 * k4 * w**2)
tq = dqdt - ( k4 * w**2 - k6 * q * e)
tz = dzdt - (- k3 * i * z + k2 * s * i)
end
model eForce
include 'SolarCel.inc'
C movement from electric force:
xs = dsdt - (mys * s * dEsumdx + mys * Esum * dsdx)
xi = didt - (myi * i * dEsumdx + myi * Esum * didx)
xe = dedt - (mye * e * dEsumdx + mye * Esum * dedx)
xw = dwdt - (myw * w * dEsumdx + myw * Esum * dwdx)
xq = dqdt - (myq * q * dEsumdx + myq * Esum * dqdx)
xz = dzdt - 0 ! not charged
! next comes from poissons equation
xEsum = dEsumdx - (s + i + e + w + q) ! is Objective xEsum = 0 ?
C I guess this is a initial value problem with six coupled nonlinear
C differential equations. /Jarl
end
controller cntrl1( AJAX)
summary=0
end
See Also
Problem-Solving Application Examples include:
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CurvFit: a curve fitting program with Lorentzian, Sine, Exponential and Power series are available models to match your data.
ODEcalc: an Ordinary Differential Equation Calculator! Solves BVP & IVP.
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Match-n-Freq: a Matched Filter program used to filter signals and slim pulses.
Robot4: Robotic Arm Movement; determines how to get from a point to another point.
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Industry Problem-Solving Descriptions include:
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AC Motor Design: a simulation program for A.C. motor design that was reapplied as a constrained optimization problem with 12 unknown parameters and 7 constraints.
Body Plasma Chemistry: determine the concentration of a Therapeutic treatment drug that is in the body over a period of time.
Efficient Solar Cells: Modeling a Nanostructured Solar Cell. Problem: How to develop solar cells with a new (higher) efficiency; grätzel cells.
Pulse Slimming to minimize InterSymbol Interference: via Arbitrary Equalization with Simple LC Structures to reduce errors.
Voice Coil Motor: basically an electromagnetic transducer in which a coil placed in a magnetic pole gap experiences a force proportional to the current passing through the coil.
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Heat Transfer Boundary Value Problem: Solves second order Differential Equation for temperature distribution in a tapered fin.
Electrical Filter Design: find the transfer function's poles & zeros; H(s) = Yout(s) / Yin(s).
Digitized Signal from Magnetic Recording: Magnetic recording of transitions written onto a computer disc drive may produce an isolated pulse as shown.
PharmacoKinetics: an open-two- compartment model with first order absorption into elimination from central compartment is presented here.
Rocket Feed System: illustrates solving implicit differential equations that model a liquid propellant rocket feed system in the presence of a longitudinal vibration.
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