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Lattice diffusion coefficient

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In <a href="Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, lattice diffusion (also called bulk or volume diffusion) refers to <a href="Atomic_diffusion" title="Atomic diffusion">atomic diffusion</a> within a <a href="Crystalline_lattice" class="mw-redirect" title="Crystalline lattice">crystalline lattice</a>,<a href="#cite_note-1">[1]</a> which occurs by either <a href="Interstitial_defect" title="Interstitial defect">interstitial</a> or <a href="Substitutional_defect" class="mw-redirect" title="Substitutional defect">substitutional</a> mechanisms. In interstitial lattice diffusion, a diffusant (such as <a href="Carbon" title="Carbon">carbon</a> in an <a href="Iron_alloys" class="mw-redirect" title="Iron alloys">iron alloy</a>), will diffuse in between the lattice structure of another crystalline element. In substitutional lattice diffusion (self-diffusion for example), the atom can only move by switching places with another atom. Substitutional lattice diffusion is often contingent upon the availability of <a href="Point_defect" class="mw-redirect" title="Point defect">point vacancies</a> throughout the crystal lattice. Diffusing particles migrate from point vacancy to point vacancy by the rapid, essentially <a href="Random" class="mw-redirect" title="Random">random</a> jumping about (<a href="Jump_diffusion#In_physics" title="Jump diffusion">jump diffusion</a>). Since the prevalence of point vacancies increases in accordance with the <a href="Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a>, the rate of crystal solid state diffusion increases with <a href="Temperature" title="Temperature">temperature</a>. For a single atom in a <a href="Perfect_crystal" title="Perfect crystal">defect-free crystal</a>, the movement can be described by the "<a href="Random_walk" title="Random walk">random walk</a>" model.

<div class="mw-heading mw-heading2"><h2 id="Diffusion_Coefficient_for_Interstitial_Diffusion">Diffusion Coefficient for Interstitial Diffusion</h2></div>

An atom diffuses in the <a href="Interstitial_defect" title="Interstitial defect">interstitial</a> mechanism by passing from one interstitial site to one of its nearest neighboring interstitial sites. The movement of atoms can be described as jumps, and the interstitial <a href="Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a> depends on the jump frequency. The jump <a href="Frequency" title="Frequency">frequency</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">, is given by:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =zv\exp \left({\frac {-\Delta G_{m}}{RT}}\right)}</annotation>

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</math><img src="./00f0b0b3f8e90be4b725df337b35d67d1affa9ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.21ex; height:6.176ex;" alt="{\displaystyle \Gamma =zv\exp \left({\frac {-\Delta G_{m}}{RT}}\right)}" loading="lazy">

where

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> is the number of nearest neighboring interstitial sites.</li>

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is vibration frequency of the interstitial atom due to <a href="Thermal_energy" title="Thermal energy">thermal energy</a>.</li>

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{m}}</annotation>

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</math><img src="./b156fd5869714370699c8375dd5458d794feef2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle \Delta G_{m}}" loading="lazy"> is the <a href="Activation_energy" title="Activation energy">activation energy</a> for the migration of the interstitial atom between sites.</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{m}}</annotation>

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</math><img src="./b156fd5869714370699c8375dd5458d794feef2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle \Delta G_{m}}" loading="lazy"> can be expressed as the sum of activation <a href="Enthalpy" title="Enthalpy">enthalpy</a> term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta H_{m}}</annotation>

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</math><img src="./a4c9b327f440a16d464cfad4658a9f063e855355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.542ex; height:2.509ex;" alt="{\displaystyle \Delta H_{m}}" loading="lazy"> and the activation <a href="Entropy" title="Entropy">entropy</a> term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -T\Delta S_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle -T\Delta S_{m}}</annotation>

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</math><img src="./844fc14d50d4227a4084662a7fbb9dde961e0dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.48ex; height:2.509ex;" alt="{\displaystyle -T\Delta S_{m}}" loading="lazy">, which gives the diffusion coefficient as:

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<annotation encoding="application/x-tex">{\displaystyle D=\left[{\frac {1}{z}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\right]\exp {\frac {-\Delta H_{m}}{RT}}}</annotation>

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</math><img src="./4c66198b7742e154612c22f646ecbcd4a7f82e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.945ex; height:6.176ex;" alt="{\displaystyle D=\left[{\frac {1}{z}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\right]\exp {\frac {-\Delta H_{m}}{RT}}}" loading="lazy">

where

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is the jump distance.</li></ul>

The diffusion coefficient can be simplified to an <a href="Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a> form:

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<annotation encoding="application/x-tex">{\displaystyle D=D_{0}\exp {\frac {-Q_{I}}{RT}}}</annotation>

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</math><img src="./03171e14441ba3eff1e1f808bd2e11cc2b54dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.871ex; height:5.509ex;" alt="{\displaystyle D=D_{0}\exp {\frac {-Q_{I}}{RT}}}" loading="lazy">

where

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<annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation>

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</math><img src="./b943acf3ade50743482c4ee4570ad3de335d90d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{0}}" loading="lazy"> is a temperature-independent material constant. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}={\tfrac {1}{z}}\alpha ^{2}zv\exp {\tfrac {\Delta S_{m}}{R}}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{0}={\tfrac {1}{z}}\alpha ^{2}zv\exp {\tfrac {\Delta S_{m}}{R}}}</annotation>

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</math><img src="./77826aa0fc2d0d00e6d15d4fd38755a50c821660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.367ex; height:4.009ex;" alt="{\displaystyle D_{0}={\tfrac {1}{z}}\alpha ^{2}zv\exp {\tfrac {\Delta S_{m}}{R}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle Q_{I}}</annotation>

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</math><img src="./361979f4c239a5c97f4c2093ca6dcc672a680a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.899ex; height:2.509ex;" alt="{\displaystyle Q_{I}}" loading="lazy"> is the activation enthalpy. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{I}=\Delta H_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{I}=\Delta H_{m}}</annotation>

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</math><img src="./6a1d0d7536d6b3b6af584675d67ee665a6509523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.54ex; height:2.509ex;" alt="{\displaystyle Q_{I}=\Delta H_{m}}" loading="lazy"></li></ul>

In the case of interstitial diffusion, the activation enthalpy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{I}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{I}}</annotation>

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</math><img src="./361979f4c239a5c97f4c2093ca6dcc672a680a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.899ex; height:2.509ex;" alt="{\displaystyle Q_{I}}" loading="lazy"> is only dependent on the activation energy barrier to the movement of interstitial atoms from one site to another. The diffusion coefficient increases <a href="Exponential_growth" title="Exponential growth">exponentially</a> with temperature at a rate determined by the activation enthalpy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{I}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{I}}</annotation>

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</math><img src="./361979f4c239a5c97f4c2093ca6dcc672a680a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.899ex; height:2.509ex;" alt="{\displaystyle Q_{I}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Diffusion_Coefficient_for_Substitution_Diffusion">Diffusion Coefficient for Substitution Diffusion</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Self-Diffusion">Self-Diffusion</h3></div>

The rate of <a href="Self-diffusion" title="Self-diffusion">self-diffusion</a> can be measured experimentally by introducing <a href="Radioactive_tracer" title="Radioactive tracer">radioactive</a> A atoms (A*) into pure A and measuring the rate at which penetration occurs at various temperatures. A* and A atoms have approximately identical jump frequencies since they are chemically identical. The diffusion coefficient of A* and A can be related to the jump frequency and expressed as:

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<annotation encoding="application/x-tex">{\displaystyle D_{A}^{*}=D_{A}={\frac {1}{6}}\alpha ^{2}\Gamma }</annotation>

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</math><img src="./14fb60caee1d1a5c7baee6e7d3bfef7bd18d76ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.968ex; height:5.176ex;" alt="{\displaystyle D_{A}^{*}=D_{A}={\frac {1}{6}}\alpha ^{2}\Gamma }" loading="lazy">

where

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<annotation encoding="application/x-tex">{\displaystyle D_{A}^{*}}</annotation>

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</math><img src="./a1ed38b5c435a2e4feb2e589e373067102fdcf94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.389ex; height:2.843ex;" alt="{\displaystyle D_{A}^{*}}" loading="lazy"> is the diffusion coefficient of radioactive A atoms in pure A.</li>

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<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

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</math><img src="./b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> is the diffusion coefficient of A atoms in pure A.</li>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> is the jump frequency for both the A* and A atoms.</li>

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is the jump distance.</li></ul>

An atom can make a successful jump when there are vacancies nearby and when it has enough thermal energy to overcome the energy barrier to migration. The number of successful jumps an atom will make in one second, or the jump frequency, can be expressed as:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =zvX_{v}\exp {\frac {-\Delta G_{m}}{RT}}}</annotation>

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</math><img src="./03349fc456044dac2dc3933c069bc66dface65cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.129ex; height:5.509ex;" alt="{\displaystyle \Gamma =zvX_{v}\exp {\frac {-\Delta G_{m}}{RT}}}" loading="lazy">

where

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> is the number of nearest neighbors.</li>

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is the frequency of temperature-independent atomic vibration.</li>

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<annotation encoding="application/x-tex">{\displaystyle X_{v}}</annotation>

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</math><img src="./0046abbf61c76adf16ed633e4822101af2b5ef14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.954ex; height:2.509ex;" alt="{\displaystyle X_{v}}" loading="lazy"> is the vacancy fraction of the lattice.</li>

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{m}}</annotation>

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</math><img src="./b156fd5869714370699c8375dd5458d794feef2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle \Delta G_{m}}" loading="lazy"> is the activation energy barrier to atomic migration.</li></ul>

In <a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>,

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<annotation encoding="application/x-tex">{\displaystyle X_{v}=X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}</annotation>

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</math><img src="./cc8e3db3a14355454076215dc2a02681b9bc6c74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.522ex; height:5.509ex;" alt="{\displaystyle X_{v}=X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{v}}</annotation>

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</math><img src="./ace80e59586c98825c2fc1cd626d71946d272768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.792ex; height:2.509ex;" alt="{\displaystyle \Delta G_{v}}" loading="lazy"> is the free energy of vacancy formation for a single vacancy.

The diffusion coefficient in thermodynamic equilibrium can be expressed with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{m}}</annotation>

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</math><img src="./b156fd5869714370699c8375dd5458d794feef2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle \Delta G_{m}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta G_{v}}</annotation>

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</math><img src="./ace80e59586c98825c2fc1cd626d71946d272768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.792ex; height:2.509ex;" alt="{\displaystyle \Delta G_{v}}" loading="lazy">, giving:

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<annotation encoding="application/x-tex">{\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {-(\Delta G_{m}+\Delta G_{v})}{RT}}}</annotation>

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</math><img src="./6c91fc4140450204358cfda5bd66051b89f93d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.094ex; height:5.843ex;" alt="{\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {-(\Delta G_{m}+\Delta G_{v})}{RT}}}" loading="lazy">

Substituting ΔG = ΔH – TΔS gives:

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<annotation encoding="application/x-tex">{\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}\exp {\frac {-(\Delta H_{m}+\Delta H_{v})}{RT}}}</annotation>

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</math><img src="./3a996822b91dfbf77eb8cfc4f3f66cbd8f031da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.733ex; height:5.843ex;" alt="{\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}\exp {\frac {-(\Delta H_{m}+\Delta H_{v})}{RT}}}" loading="lazy">

The diffusion coefficient can be simplified to an Arrhenius equation form:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=D_{0}\exp {\frac {-Q_{S}}{RT}}}</annotation>

</semantics>

</math><img src="./b8a3928055a5791333cff4bede20b8c25f5b4568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.568ex; height:5.509ex;" alt="{\displaystyle D_{A}=D_{0}\exp {\frac {-Q_{S}}{RT}}}" loading="lazy">

where

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation>

</semantics>

</math><img src="./b943acf3ade50743482c4ee4570ad3de335d90d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{0}}" loading="lazy"> is approximately a constant. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}}</annotation>

</semantics>

</math><img src="./735ef47f747ede6612edfc64dab3c4bfa472c364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.263ex; height:5.509ex;" alt="{\displaystyle D_{0}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}}" loading="lazy"></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{S}}</annotation>

</semantics>

</math><img src="./8117f46ccbb24f836a9c48ebbae67ad457a347d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.131ex; height:2.509ex;" alt="{\displaystyle Q_{S}}" loading="lazy"> is the activation enthalpy. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{S}=\Delta H_{m}+\Delta H_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{S}=\Delta H_{m}+\Delta H_{v}}</annotation>

</semantics>

</math><img src="./53b3da38d122024d6a8a5db8923ba16c03e1b32f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.508ex; height:2.509ex;" alt="{\displaystyle Q_{S}=\Delta H_{m}+\Delta H_{v}}" loading="lazy"></li></ul>

Compared to that of interstitial diffusion, the activation energy for self-diffusion has an extra term (ΔHv). Since self-diffusion requires the presence of vacancies whose concentration depends on ΔHv.

<div class="mw-heading mw-heading3"><h3 id="Vacancy_Diffusion">Vacancy Diffusion</h3></div>

Diffusion of a <a href="Vacancy_defect" title="Vacancy defect">vacancy</a> can be viewed as the jumping of a vacancy onto an atom site. It is the same process as the jumping of an atom into a vacant site but without the need to consider the probability of vacancy presence, since a vacancy is usually always surrounded by atom sites to which it can jump. A vacancy can have its own diffusion coefficient that is expressed as:

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}\Gamma _{v}}</annotation>

</semantics>

</math><img src="./8edb3817fa74632be997009c4468acf82a15ff6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.075ex; height:5.176ex;" alt="{\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}\Gamma _{v}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{v}}</annotation>

</semantics>

</math><img src="./4550578819e51c28d5fcadccaf8fef26e7c02c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.482ex; height:2.509ex;" alt="{\displaystyle \Gamma _{v}}" loading="lazy"> is the jump frequency of a vacancy.

The diffusion coefficient can also be expressed in terms of enthalpy of migration (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H_{m}}</annotation>

</semantics>

</math><img src="./a4c9b327f440a16d464cfad4658a9f063e855355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.542ex; height:2.509ex;" alt="{\displaystyle \Delta H_{m}}" loading="lazy">) and entropy of migration (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta S_{m}}</annotation>

</semantics>

</math><img src="./e5310bae6e0aa3b00001561bcb9017efaea3c390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.036ex; height:2.509ex;" alt="{\displaystyle \Delta S_{m}}" loading="lazy">) of a vacancy, which are the same as for the migration of a substitutional atom:

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\exp {\frac {-\Delta H_{m}}{RT}}}</annotation>

</semantics>

</math><img src="./7d1a8dd99bc56f58c59482ab246bb67226cc9f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.52ex; height:5.509ex;" alt="{\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\exp {\frac {-\Delta H_{m}}{RT}}}" loading="lazy">

Comparing the diffusion coefficient between self-diffusion and vacancy diffusion gives:

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}={\frac {D_{A}}{X_{v}^{e}}}}</annotation>

</semantics>

</math><img src="./a5ffcd6d448e8d35b3be393758d9d69fbacc7131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.278ex; height:5.843ex;" alt="{\displaystyle D_{v}={\frac {D_{A}}{X_{v}^{e}}}}" loading="lazy">

where the equilibrium vacancy fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}</annotation>

</semantics>

</math><img src="./bfde4573499e810ff7df1733895b31d6d779e8a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.47ex; height:5.509ex;" alt="{\displaystyle X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Diffusion_in_a_Binary_System">Diffusion in a Binary System</h3></div>

In a system with multiple components (e.g. a <a href="Binary_alloy" class="mw-redirect" title="Binary alloy">binary alloy</a>), the <a href="Solvent" title="Solvent">solvent</a> (A) and the <a href="Solution_(chemistry)" title="Solution (chemistry)">solute</a> atoms (B) will not move in an equal rate. Each atomic species can be given its own intrinsic diffusion coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {D}}_{A}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {D}}_{A}}</annotation>

</semantics>

</math><img src="./45aced721d29b693e22794adcb4b40706fb90592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:3.009ex;" alt="{\displaystyle {\tilde {D}}_{A}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {D}}_{B}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {D}}_{B}}</annotation>

</semantics>

</math><img src="./f9f1dd736989122a9ad732b5f1f2de22b4d969d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.404ex; height:3.009ex;" alt="{\displaystyle {\tilde {D}}_{B}}" loading="lazy">, expressing the diffusion of a certain species in the whole system. The interdiffusion coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {D}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {D}}}</annotation>

</semantics>

</math><img src="./151fff560f471c0ce3c382dc71b908567e09ac6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\tilde {D}}}" loading="lazy"> is defined by the <a href="Darken's_equations" title="Darken's equations">Darken's equation</a> as:

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {D}}={\tilde {D}}_{A}X_{B}+{\tilde {D}}_{B}X_{A}}</annotation>

</semantics>

</math><img src="./f307cdb4240a539557f76812d34677da7e4b72dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.449ex; height:3.009ex;" alt="{\displaystyle {\tilde {D}}={\tilde {D}}_{A}X_{B}+{\tilde {D}}_{B}X_{A}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

</semantics>

</math><img src="./0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{B}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{B}}</annotation>

</semantics>

</math><img src="./e93fabcf27b168d679d6469c36fe4c1d460b96d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.404ex; height:2.509ex;" alt="{\displaystyle X_{B}}" loading="lazy"> are the <a href="Amount_fraction" class="mw-redirect" title="Amount fraction">amount fractions</a> of species A and B, respectively.

<ul><li><a href="Kirkendall_effect" title="Kirkendall effect">Kirkendall effect</a></li>

<li>Phase transformations in solids</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-1"><a href="#cite_ref-1">^</a> P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/index.html">Classical and nanoscale diffusion (with figures and animations)</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-04-16" href="https://en.wikipedia.org/wiki/?title=Lattice_diffusion_coefficient&oldid=1150048057">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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