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Interface_GOx_model.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>Interface_GOx_model-colnew</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
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<h1 id="Numerical-model-of-a-mediated-Glucose-Oxidase-electrode">Numerical model of a mediated Glucose Oxidase electrode<a class="anchor-link" href="#Numerical-model-of-a-mediated-Glucose-Oxidase-electrode">¶</a></h1>
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<p>Here we will solve the material balance for an electrode comprising the enzyme glucose oxidase co-immobilized in a hydrogel with the redox polymer mediator poly{N-vinylimidazole [Os(bipyridine)<sub>2</sub>Cl]<sup>+/2+</sup>}. The governing equations and boundary conditions are adapted from <a href="http://dx.doi.org/10.1016/0022-0728(95)04236-7">Bartlett and Pratt, 1995</a>:</p>
$$ -\frac{dN_M}{dx}= D_M\frac{d^2M}{dx^2}=\frac{\nu_M k_M k_{cat} E M S}{k_M M (K_S+S) + k_{cat}S} $$<p><br>
$$ -\frac{dN_S}{dx}= D_S\frac{d^2S}{dx^2}=\frac{\nu_S k_M k_{cat} E M S}{k_M M (K_S+S) + k_{cat}S} $$</p>
<p>at x=0: $V=U+\frac{nF}{RT}log\frac{M}{M_{tot}-M}$, $\frac{dS}{dx}=0$</p>
<p>at x=L: $\frac{dM}{dx}=0$, $S=S_0$</p>
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<p><span style="color:green">Please send comments or suggestions to scb at msu dot edu.</span></p>
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<div class=" highlight hl-ipython2"><pre><span class="o">%</span><span class="k">pylab</span> inline
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<pre>Populating the interactive namespace from numpy and matplotlib
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">scikits.bvp1lg</span> <span class="kn">as</span> <span class="nn">bvp</span>
<span class="kn">from</span> <span class="nn">scipy.interpolate</span> <span class="kn">import</span> <span class="n">interp1d</span>
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<h3 id="Parameters">Parameters<a class="anchor-link" href="#Parameters">¶</a></h3>
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<p>Parameters are taken from Ref. 1 and 2 and are typical of that system.</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">DM</span> <span class="o">=</span> <span class="mf">2.18e-9</span> <span class="c"># Mediator diffusivity,cm^s2/s</span>
<span class="n">DS</span> <span class="o">=</span> <span class="mf">7e-6</span> <span class="c"># Substrate (glucose) diffusivity, cm^2/s [Calvo 2006]</span>
<span class="n">ET</span> <span class="o">=</span> <span class="mf">17.96e-6</span> <span class="c"># Total enzyme concentration, mol/cm^3 </span>
<span class="n">ST</span> <span class="o">=</span> <span class="mf">50e-6</span> <span class="c"># Bulk substrate concentration, mol/cm^3</span>
<span class="n">Mtot</span> <span class="o">=</span> <span class="mf">659.7e-6</span> <span class="c"># Total mediator concentration, mol/cm^3</span>
<span class="n">kcat</span> <span class="o">=</span> <span class="mf">17.3</span> <span class="c"># Enzyme Turnover, s^-1</span>
<span class="n">kM</span> <span class="o">=</span> <span class="mf">212e3</span> <span class="c"># second order rate constant for mediator, cm^3/mol/s </span>
<span class="n">KS</span> <span class="o">=</span> <span class="mf">12.3e-6</span> <span class="c"># Michaelis constant for substrate, mol/cm^3</span>
<span class="n">vm</span> <span class="o">=</span> <span class="mf">1.</span> <span class="c"># Stoichiometric coeff. of mediator</span>
<span class="n">vs</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c"># Stoichiometric coeff. of substrate</span>
<span class="n">U</span> <span class="o">=</span> <span class="mf">0.43</span> <span class="c"># Mediator redox potential, V re:SHE</span>
<span class="n">V</span> <span class="o">=</span> <span class="mf">0.6</span> <span class="c"># Electrode potential, V re:SHE</span>
<span class="n">n</span> <span class="o">=</span> <span class="mf">1.</span> <span class="c"># Electron transfer coefficient </span>
<span class="n">R</span> <span class="o">=</span> <span class="mf">8.314</span> <span class="c"># Gas constant, J/mol/K</span>
<span class="n">F</span> <span class="o">=</span> <span class="mf">96485.4</span> <span class="c"># Faraday's constant, C/mol</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">313</span> <span class="c"># Temperature, K</span>
<span class="n">L</span> <span class="o">=</span> <span class="mf">1e-4</span> <span class="c"># Hydrogel film thickness, cm</span>
<span class="n">p</span> <span class="o">=</span> <span class="mf">1.</span> <span class="c"># Substrate hydrogel-solution partition coefficient</span>
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<h3 id="Nondimensionalization">Nondimensionalization<a class="anchor-link" href="#Nondimensionalization">¶</a></h3>
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<p>We nondimensionalize the system according to the scheme of <a href="http://dx.doi.org/10.1016/0022-0728(95)04236-7">Bartlett and Pratt, 1995</a>:</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">kappa</span><span class="o">=</span><span class="n">L</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">vm</span><span class="o">*</span><span class="n">kM</span><span class="o">*</span><span class="n">ET</span><span class="o">/</span><span class="n">DM</span><span class="p">)</span>
<span class="n">eta</span><span class="o">=</span><span class="p">(</span><span class="n">vm</span><span class="o">*</span><span class="n">DS</span><span class="o">*</span><span class="n">kM</span><span class="o">*</span><span class="n">KS</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">vs</span><span class="o">*</span><span class="n">DM</span><span class="o">*</span><span class="n">kcat</span><span class="p">)</span>
<span class="n">gamma</span><span class="o">=</span><span class="p">(</span><span class="n">kM</span><span class="o">*</span><span class="n">Mtot</span><span class="o">*</span><span class="n">KS</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">kcat</span><span class="o">*</span><span class="n">ST</span><span class="p">)</span>
<span class="n">mu</span><span class="o">=</span><span class="p">(</span><span class="n">ST</span><span class="o">/</span><span class="n">KS</span><span class="p">)</span>
<span class="n">eps</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">F</span> <span class="o">*</span> <span class="p">(</span><span class="n">V</span><span class="o">-</span><span class="n">U</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">R</span><span class="o">*</span><span class="n">T</span><span class="p">)</span> <span class="c"># From the Nernst equation</span>
<span class="n">ae</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">eps</span><span class="p">))</span> <span class="c"># Dimensionless potential boundary condition at the electrode surface from the Nernst eqn</span>
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<h3 id="Numerical-Solution">Numerical Solution<a class="anchor-link" href="#Numerical-Solution">¶</a></h3>
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<p>We will solve this problem numerically using the python package <a href="https://pypi.python.org/pypi/scikits.bvp1lg">bvp_1lg</a></p>
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<div class=" highlight hl-ipython2"><pre><span class="n">degrees</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">deq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">z</span><span class="p">):</span> <span class="c"># z=[m(x) vm(x) s(x) vs(x)];</span>
<span class="n">m</span><span class="p">,</span> <span class="n">vm</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vs</span> <span class="o">=</span> <span class="n">z</span>
<span class="n">Ra</span><span class="o">=</span> <span class="n">kappa</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">s</span><span class="o">/</span><span class="p">(</span><span class="n">gamma</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">mu</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">+</span><span class="n">s</span><span class="p">)</span>
<span class="n">dvmdx</span><span class="o">=</span> <span class="n">Ra</span>
<span class="n">dmdx</span><span class="o">=</span> <span class="n">vm</span>
<span class="n">dvsdx</span><span class="o">=</span> <span class="n">gamma</span><span class="o">/</span><span class="n">eta</span><span class="o">*</span><span class="n">Ra</span>
<span class="n">dsdx</span><span class="o">=</span> <span class="n">vs</span>
<span class="n">dzdx</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">dmdx</span><span class="p">,</span><span class="n">dvmdx</span><span class="p">,</span><span class="n">dsdx</span><span class="p">,</span><span class="n">dvsdx</span><span class="p">]);</span> <span class="k">return</span> <span class="n">dzdx</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">boundary_points</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">bc</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
<span class="n">a</span><span class="p">,</span> <span class="n">va</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vs</span> <span class="o">=</span> <span class="n">z</span>
<span class="n">res</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">eps</span><span class="p">)),</span> <span class="c"># Nernst BC on a at electrode-film interface</span>
<span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="c"># Zero flux BC on s at electrode-film interface</span>
<span class="n">va</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="c"># Zero flux BC on a at film-solution interface</span>
<span class="n">s</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span> <span class="p">)</span> <span class="c"># Concentration BC on s at film-solution interface</span>
<span class="k">return</span> <span class="n">res</span>
</pre></div>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">guess</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">dv</span><span class="o">=</span> <span class="n">zeros_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">u</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">v</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">dv</span><span class="p">])</span>
<span class="n">du</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">])</span>
<span class="k">return</span> <span class="n">u</span><span class="p">,</span> <span class="n">du</span>
</pre></div>
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<div class="prompt input_prompt">In [8]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">tol</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">]</span>
<span class="n">solution</span> <span class="o">=</span> <span class="n">bvp</span><span class="o">.</span><span class="n">colnew</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span>
<span class="n">boundary_points</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">deq</span><span class="p">,</span> <span class="n">bc</span><span class="p">,</span>
<span class="n">tolerances</span><span class="o">=</span><span class="n">tol</span><span class="p">,</span>
<span class="n">vectorized</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">maximum_mesh_size</span><span class="o">=</span><span class="mi">300</span><span class="p">,</span> <span class="n">initial_guess</span><span class="o">=</span><span class="n">guess</span><span class="p">)</span>
</pre></div>
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<div class="prompt input_prompt">In [9]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">x</span><span class="o">=</span><span class="n">solution</span><span class="o">.</span><span class="n">mesh</span>
<span class="n">y</span><span class="o">=</span><span class="n">solution</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">m</span><span class="o">=</span><span class="n">y</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
<span class="n">s</span><span class="o">=</span><span class="n">y</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
<span class="n">Ra</span><span class="o">=</span> <span class="n">kappa</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">s</span><span class="o">/</span><span class="p">(</span><span class="n">gamma</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">mu</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">+</span><span class="n">s</span><span class="p">)</span>
<span class="n">Ra</span><span class="o">=</span><span class="n">Ra</span><span class="o">/</span><span class="n">amax</span><span class="p">(</span><span class="n">Ra</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="s">'-og'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">s</span><span class="p">,</span><span class="s">'-vr'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">Ra</span><span class="p">,</span><span class="s">'b'</span><span class="p">)</span>
<span class="n">legend</span><span class="p">([</span><span class="s">'m'</span><span class="p">,</span><span class="s">'s'</span><span class="p">,</span><span class="s">'Ra'</span><span class="p">],</span><span class="n">loc</span><span class="o">=</span><span class="s">'center right'</span><span class="p">)</span>
<span class="n">xlabel</span><span class="p">(</span><span class="s">'x / L'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s">'Dimensionless Concentration'</span><span class="p">)</span>
<span class="n">ylim</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span><span class="mf">1.1</span><span class="p">))</span>
<span class="n">show</span><span class="p">()</span>
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<p>We can see that, for this case, the substrate concentration (s) is uniform, and there is only mass transfer limitations for the mediator (m). The reaction rate, Ra, varies nonlinearly with mediator concentration.</p>
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<h2 id="Polarization-Curve">Polarization Curve<a class="anchor-link" href="#Polarization-Curve">¶</a></h2>
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<p>The above numerical treatment can be enclosed in a function to calculate a polarization curve</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">polcurve</span><span class="p">(</span><span class="n">V</span><span class="p">,</span><span class="n">ST</span><span class="p">,</span><span class="n">L</span><span class="p">):</span>
<span class="n">kappa</span><span class="o">=</span><span class="n">L</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">vm</span><span class="o">*</span><span class="n">kM</span><span class="o">*</span><span class="n">ET</span><span class="o">/</span><span class="n">DM</span><span class="p">)</span>
<span class="n">eta</span><span class="o">=</span><span class="p">(</span><span class="n">vm</span><span class="o">*</span><span class="n">DS</span><span class="o">*</span><span class="n">kM</span><span class="o">*</span><span class="n">KS</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">vs</span><span class="o">*</span><span class="n">DM</span><span class="o">*</span><span class="n">kcat</span><span class="p">)</span>
<span class="n">gamma</span><span class="o">=</span><span class="p">(</span><span class="n">kM</span><span class="o">*</span><span class="n">Mtot</span><span class="o">*</span><span class="n">KS</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">kcat</span><span class="o">*</span><span class="n">ST</span><span class="p">)</span>
<span class="n">mu</span><span class="o">=</span><span class="p">(</span><span class="n">ST</span><span class="o">/</span><span class="n">KS</span><span class="p">)</span>
<span class="n">eps</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">F</span> <span class="o">*</span> <span class="p">(</span><span class="n">V</span><span class="o">-</span><span class="n">U</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">R</span><span class="o">*</span><span class="n">T</span><span class="p">)</span> <span class="c"># From the Nernst equation</span>
<span class="n">ae</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">eps</span><span class="p">))</span> <span class="c"># Dimensionless potential boundary condition at the electrode surface from the Nernst eqn</span>
<span class="n">degrees</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">deq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">z</span><span class="p">):</span> <span class="c"># z=[a(x) va(x) s(x) vs(x)];</span>
<span class="n">a</span><span class="p">,</span> <span class="n">va</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vs</span> <span class="o">=</span> <span class="n">z</span>
<span class="n">Ra</span><span class="o">=</span> <span class="n">kappa</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">s</span><span class="o">/</span><span class="p">(</span><span class="n">gamma</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">mu</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">+</span><span class="n">s</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">dvadx</span><span class="o">=</span> <span class="n">Ra</span>
<span class="n">dadx</span><span class="o">=</span> <span class="n">va</span>
<span class="n">dvsdx</span><span class="o">=</span> <span class="n">gamma</span><span class="o">/</span><span class="n">eta</span><span class="o">*</span><span class="n">Ra</span>
<span class="n">dsdx</span><span class="o">=</span> <span class="n">vs</span>
<span class="n">dzdx</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">dadx</span><span class="p">,</span><span class="n">dvadx</span><span class="p">,</span><span class="n">dsdx</span><span class="p">,</span><span class="n">dvsdx</span><span class="p">]);</span> <span class="k">return</span> <span class="n">dzdx</span>
<span class="n">boundary_points</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">bc</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
<span class="n">a</span><span class="p">,</span> <span class="n">va</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vs</span> <span class="o">=</span> <span class="n">z</span>
<span class="n">res</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">eps</span><span class="p">)),</span> <span class="c"># Nernst BC on a at electrode-film interface</span>
<span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="c"># Zero flux BC on s at electrode-film interface</span>
<span class="n">va</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="c"># Zero flux BC on a at film-solution interface</span>
<span class="n">s</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span> <span class="p">)</span> <span class="c"># Concentration BC on s at film-solution interface</span>
<span class="k">return</span> <span class="n">res</span>
<span class="k">def</span> <span class="nf">guess</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">dv</span><span class="o">=</span> <span class="n">zeros_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">u</span><span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">v</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">dv</span><span class="p">])</span>
<span class="n">du</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">dv</span><span class="p">])</span>
<span class="k">return</span> <span class="n">u</span><span class="p">,</span> <span class="n">du</span>
<span class="n">tol</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">,</span> <span class="mf">1e-6</span><span class="p">]</span>
<span class="n">solution</span> <span class="o">=</span> <span class="n">bvp</span><span class="o">.</span><span class="n">colnew</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">boundary_points</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">deq</span><span class="p">,</span> <span class="n">bc</span><span class="p">,</span>
<span class="n">tolerances</span><span class="o">=</span><span class="n">tol</span><span class="p">,</span>
<span class="n">vectorized</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
<span class="n">maximum_mesh_size</span><span class="o">=</span><span class="mi">300</span><span class="p">,</span>
<span class="n">initial_guess</span><span class="o">=</span><span class="n">guess</span><span class="p">)</span>
<span class="n">x</span><span class="o">=</span><span class="n">solution</span><span class="o">.</span><span class="n">mesh</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">solution</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">icd</span><span class="o">=-</span><span class="n">n</span><span class="o">*</span><span class="n">F</span><span class="o">/</span><span class="n">vm</span><span class="o">*</span><span class="n">DM</span><span class="o">*</span><span class="n">Mtot</span><span class="o">/</span><span class="n">L</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
<span class="k">return</span> <span class="n">icd</span>
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<p>We now can calculate polarization curves for three different film thicknesses (L):</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">V</span><span class="o">=</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">80</span><span class="p">)</span>
<span class="n">pv</span><span class="o">=</span><span class="n">vectorize</span><span class="p">(</span><span class="n">polcurve</span><span class="p">)</span>
<span class="n">icdp5</span><span class="o">=</span><span class="n">pv</span><span class="p">(</span><span class="n">V</span><span class="p">,</span><span class="n">ST</span><span class="p">,</span><span class="mf">0.5e-4</span><span class="p">)</span>
<span class="n">icd1</span><span class="o">=</span><span class="n">pv</span><span class="p">(</span><span class="n">V</span><span class="p">,</span><span class="n">ST</span><span class="p">,</span><span class="mf">1e-4</span><span class="p">)</span>
<span class="n">icd2</span><span class="o">=</span><span class="n">pv</span><span class="p">(</span><span class="n">V</span><span class="p">,</span><span class="n">ST</span><span class="p">,</span><span class="mf">2e-4</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">fp5</span><span class="p">,</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">icdp5</span><span class="o">*</span><span class="mi">1000</span><span class="p">,</span><span class="s">'-or'</span> <span class="p">)</span>
<span class="n">f1</span><span class="p">,</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">icd1</span> <span class="o">*</span><span class="mi">1000</span><span class="p">,</span><span class="s">'-og'</span> <span class="p">)</span>
<span class="n">f2</span><span class="p">,</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">icd2</span> <span class="o">*</span><span class="mi">1000</span><span class="p">,</span><span class="s">'-ob'</span> <span class="p">)</span>
<span class="n">xlabel</span><span class="p">(</span><span class="s">'Potential / V vs. SHE'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s">'Current Density / mA cm$^{-2}$'</span><span class="p">)</span>
<span class="n">legend</span><span class="p">([</span><span class="n">fp5</span><span class="p">,</span> <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">],[</span><span class="s">'L=0.5 $\mu$m'</span><span class="p">,</span><span class="s">'1 $\mu$m'</span><span class="p">,</span><span class="s">'2 $\mu$m'</span><span class="p">],</span><span class="n">loc</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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