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<title>Effective diffusion coefficient</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Effective diffusion coefficient</span></span>
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<p>The <b>effective diffusion coefficient</b> of a <span class="cleanup-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">diffusant</span> in <a href="Atomic_diffusion" title="Atomic diffusion">atomic diffusion</a> of solid <a href="Polycrystalline" class="mw-redirect" title="Polycrystalline">polycrystalline</a> materials like <a href="Metal_alloy" class="mw-redirect" title="Metal alloy">metal alloys</a> is often represented as a <a href="Weighted_average" title="Weighted average">weighted average</a> of the <a href="Grain_boundary_diffusion_coefficient" title="Grain boundary diffusion coefficient">grain boundary diffusion coefficient</a> and the <a href="Lattice_diffusion_coefficient" title="Lattice diffusion coefficient">lattice diffusion coefficient</a>.<sup id="cite_ref-Heitjans_1-0" class="reference"><a href="#cite_note-Heitjans-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Diffusion along both the grain boundary and in the lattice may be modeled with an <a href="Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a>. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4–0.6, so as temperature is lowered, the grain boundary diffusion component increases.<sup id="cite_ref-Heitjans_1-1" class="reference"><a href="#cite_note-Heitjans-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8 T<sub>melt</sub> (of an alloy), the grain boundary component can be neglected.
</p>
<div class="mw-heading mw-heading2"><h2 id="Modeling">Modeling</h2></div>
<p>The effective diffusion coefficient can be modeled using Hart's equation when lattice diffusion is dominant (type A kinetics):
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\text{eff}}=fD_{\text{gb}}+(1-f)D_{\ell }}">
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<annotation encoding="application/x-tex">{\displaystyle D_{\text{eff}}=fD_{\text{gb}}+(1-f)D_{\ell }}</annotation>
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</math></span><img src="./4b41edb8ab69474905eaba078506c536d667ef1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.936ex; height:3.009ex;" alt="{\displaystyle D_{\text{eff}}=fD_{\text{gb}}+(1-f)D_{\ell }}" loading="lazy"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\text{eff}}={}}">
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<annotation encoding="application/x-tex">{\displaystyle D_{\text{eff}}={}}</annotation>
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</math></span><img src="./e7c1fed903731c0e4053f61d5aa319dc8dc8515c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.992ex; height:2.509ex;" alt="{\displaystyle D_{\text{eff}}={}}" loading="lazy"></span>effective diffusion coefficient</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\text{gb}}={}}">
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<annotation encoding="application/x-tex">{\displaystyle D_{\text{gb}}={}}</annotation>
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</math></span><img src="./4c61f156a266bb6faa9835e0f35298f8572f073a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.991ex; height:2.843ex;" alt="{\displaystyle D_{\text{gb}}={}}" loading="lazy"></span>grain boundary diffusion coefficient</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\ell }={}}">
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<annotation encoding="application/x-tex">{\displaystyle D_{\ell }={}}</annotation>
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<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {q\delta }{d}}}">
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<annotation encoding="application/x-tex">{\displaystyle f={\frac {q\delta }{d}}}</annotation>
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<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={}}">
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</math></span><img src="./8319ff96f795b110daa65b3214c9ddc3997c3a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.168ex; height:2.009ex;" alt="{\displaystyle q={}}" loading="lazy"></span>value based on grain shape, 1 for parallel grains, 3 for square grains</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={}}">
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<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={}}">
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</math></span><img src="./d44a7dafab528e01ac032e45275dc55872810efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.147ex; height:2.343ex;" alt="{\displaystyle \delta ={}}" loading="lazy"></span>grain boundary width, often assumed to be 0.5 nm</dd></dl>
<p>Grain boundary diffusion is significant in <a href="Face-centered_cubic" class="mw-redirect" title="Face-centered cubic">face-centered cubic</a> metals below about 0.8 T<sub>melt</sub> (Absolute). Line dislocations and other <a href="Crystallographic_defect" title="Crystallographic defect">crystalline defects</a> can become significant below ~0.4 T<sub>melt</sub> in FCC metals.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Kirkendall_effect" title="Kirkendall effect">Kirkendall effect</a></li>
<li><a href="Mass_diffusivity" title="Mass diffusivity">Mass diffusivity</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-Heitjans-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Heitjans_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Heitjans_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.</span>
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