Hume-Rothery rules

<h2 id="substitutional-solid-solution-rules">Substitutional solid solution rules</h2>
<p>For substitutional solid solutions, the Hume-Rothery rules are as follows:
</p>
<ol><li>The <a href="/facts/Atomic_radius/0mQaIJX9">atomic radius</a> of the solute and <a href="/facts/Solvent/LoMPMz8S">solvent</a> atoms must differ by no more than 15%:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>

%
        
           difference
        
        =
        
          (
          
            
              
                
                  r
                  
                    solute
                  
                
                −
                
                  r
                  
                    solvent
                  
                
              
              
                r
                
                  solvent
                
              
            
          
          )
        
        ×
        100
        %
        ≤
        15
        %
        .
      
    
    {\displaystyle \%{\text{ difference}}=\left({\frac {r_{\text{solute}}-r_{\text{solvent}}}{r_{\text{solvent}}}}\right)\times 100\%\leq 15\%.}
  
</li>
<li>The <a href="/facts/Crystal_structure/bJ4bCeHd">crystal structures</a> of solute and solvent must be similar.</li>
<li>Complete <a href="/facts/Solubility/kLE9h3Jb">solubility</a> occurs when the solvent and solute have the same <a href="/facts/Valence_(chemistry)/ck7cHByj">valency</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  A metal is more likely to dissolve a metal of higher valency, than vice versa.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li>
<li>The solute and solvent should have  similar <a href="/facts/Electronegativity/H7Vq7Uah">electronegativity</a>. If the electronegativity difference is too great, the metals tend to form <a href="/facts/Intermetallics/nnzksayl">intermetallic compounds</a> instead of solid solutions.</li></ol>
<h2 id="interstitial-solid-solution-rules">Interstitial solid solution rules</h2>
<p>For <a href="/facts/Interstitial_defect/3kvKwQer">interstitial</a> solid solutions, the Hume-Rothery Rules are:
</p>
<ol><li>Solute atoms should have a smaller radius than 59% of the radius of solvent atoms.  <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li>The solute and solvent should have similar <a href="/facts/Electronegativity/H7Vq7Uah">electronegativity</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li>Valency factor: two elements should have the same valence. The greater the difference in valence between solute and solvent atoms, the lower the solubility.</li></ol>
<h2 id="solid-solution-rules-for-multicomponent-systems">Solid solution rules for multicomponent systems</h2>
<p>Fundamentally, the Hume-Rothery rules are restricted to binary systems that form either substitutional or interstitial solid solutions. However, this approach limits assessing advanced alloys which are commonly multicomponent systems. Free energy diagrams (or <a href="/facts/Phase_diagrams/rcemFgW0">phase diagrams</a>) offer in-depth knowledge of equilibrium restraints in complex systems. In essence the Hume-Rothery rules (and <a href="/facts/Pauling%27s_rules/y9hfXjhw">Pauling's rules</a>) are based on geometrical restraints. Likewise are the advancements being done to the Hume-Rothery rules. Where they are being considered as critical contact criterion describable with <a href="/facts/Voronoi_diagrams/AhQY5LD8">Voronoi diagrams</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> This could ease the theoretical phase diagram generation of multicomponent systems.
</p><p>For <a href="/facts/Alloy/I7w7WQWR">alloys</a> containing <a href="/facts/Transition_metal/0slGWoLQ">transition metal</a> elements there is a difficulty in interpretation of the Hume-Rothery electron concentration rule, as the values of e/a values (number of itinerant electrons per atom) for transition metals have been quite controversial for a long time, and no satisfactory solutions have yet emerged.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/CALPHAD/mxyznjvc">CALPHAD</a></li>
<li><a href="/facts/Enthalpy_of_mixing/qgucNGGn">Enthalpy of mixing</a></li>
<li><a href="/facts/Gibbs_energy/bvenDuzh">Gibbs energy</a></li>
<li><a href="/facts/Phase_diagram/rcemFgW0">Phase diagram</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Harry_Bhadeshia/jkQnWdvc">Bhadeshia, H. K. D. H.</a> <a href="http://www.msm.cam.ac.uk/phase-trans/2004/titanium/hume.rothery.html">"Solid Solutions: The Hume-Rothery Rules"</a>. Retrieved 2007-11-24.</li>
<li>Mizutani, Uichiro (2010). <i>Hume-Rothery Rules for Structurally Complex Alloy Phases</i>. Taylor & Francis. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4200-9058-1.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Callister, William D.; Rethwisch, David G. (January 2018). Materials Science and Engineering: An Introduction (10th ed.). Wiley. p. 992. ISBN 978-1-119-40549-8. <a href="978-1-119-40549-8" target="_blank">978-1-119-40549-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Foundations of Materials Science and Engineering, 4th ed., W. Smith and J. Hashemi, pp.139-140 (2006). <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Callister, William D.; Rethwisch, David G. (January 2018). Materials Science and Engineering: An Introduction (10th ed.). Wiley. p. 992. ISBN 978-1-119-40549-8. <a href="978-1-119-40549-8" target="_blank">978-1-119-40549-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Goodman, D. A.; Bennett, L. H.; Watson, R. E. (January 1983). "Valency effects and relative solubilities in transition metal alloys". Scripta Metallurgica. 17 (1): 91–96. doi:10.1016/0036-9748(83)90077-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Bhadeshia, K. D. H. "Solid Solutions: The Hume-Rothery Rules". PT (Phase Transformations) Group Home. University of Cambridge. Retrieved 16 August 2024. <a href="https://www.phase-trans.msm.cam.ac.uk/2004/titanium/hume.rothery.html" target="_blank">https://www.phase-trans.msm.cam.ac.uk/2004/titanium/hume.rothery.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"The Structure of Metals - PPT video online download". <a href="http://slideplayer.com/slide/7666224/" target="_blank">http://slideplayer.com/slide/7666224/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>"Introduction the properties and behavior of metals (And alloys) depend on their: Structure Processing history and Composition Engr PPT video online download". <a href="http://slideplayer.com/slide/5866092/" target="_blank">http://slideplayer.com/slide/5866092/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>"Hume-Rothery Rules". Van Nostrand's Scientific Encyclopedia. John Wiley & Sons, Inc. 2002. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Wang, Zhijun; Huang, Yunhao; Liu, C. T.; Li, Junjie; Wang, Jincheng (4 Apr 2019), "Atomic packing and size effect on the Hume-Rothery rule", Intermetallics, 109: 139–144, doi:10.1016/j.intermet.2019.04.001, S2CID 141214774 <a href="https://scholars.cityu.edu.hk/en/publications/publication(b2f047b9-0158-4222-8e81-466e100550b3).html" target="_blank">https://scholars.cityu.edu.hk/en/publications/publication(b2f047b9-0158-4222-8e81-466e100550b3).html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Gao, Michael C (2016). High-Entropy Alloys: Fundamentals and Applications. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Mizutani, Uichiro; Sato, Hirokazu (January 2017). "The Physics of the Hume-Rothery Electron Concentration Rule". Crystals. 7 (1): 9. doi:10.3390/cryst7010009. ISSN 2073-4352. <a href="https://doi.org/10.3390%2Fcryst7010009" target="_blank">https://doi.org/10.3390%2Fcryst7010009</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
</ol>

Hume-Rothery rules open-in-new

Hume-Rothery rules