Rigid clusters in shear-thickening suspensions: A nonequilibrium critical transition

The onset and growth of rigid clusters in a two-dimensional (2D) suspension in shear flow are studied by numerical simulations. The suspension exhibits the lubricated-to-frictional rheology transition, but the key results here are for stresses above the levels that cause extreme shear thickening. At large solid fraction, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>ϕ</a:mi></a:math>, but below the stress-dependent jamming fraction, we find a critical <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>ϕ</b:mi><b:mi>c</b:mi></b:msub><b:mrow><b:mo>(</b:mo><b:mi>σ</b:mi><b:mo>,</b:mo><b:mi>μ</b:mi><b:mo>)</b:mo></b:mrow></b:mrow></b:math> where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>σ</c:mi></c:math> is a dimensionless shear stress and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>μ</d:mi></d:math> is the interparticle friction coefficient. For <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>ϕ</e:mi><e:mo>&gt;</e:mo><e:msub><e:mi>ϕ</e:mi><e:mi>c</e:mi></e:msub></e:mrow></e:math>, the proportion of particles in rigid clusters grows sharply, as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:msub><f:mi>f</f:mi><f:mi>rig</f:mi></f:msub><f:mo>∼</f:mo><f:msup><f:mrow><f:mo>|</f:mo><f:mi>ϕ</f:mi><f:mo>−</f:mo><f:msub><f:mi>ϕ</f:mi><f:mi>c</f:mi></f:msub><f:mo>|</f:mo></f:mrow><f:mi>β</f:mi></f:msup></f:mrow></f:math> with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>β</g:mi><g:mo>=</g:mo><g:mn>1</g:mn><g:mo>/</g:mo><g:mn>8</g:mn></g:mrow></g:math>. The fluctuations in the fraction of particles in rigid clusters yield a susceptibility measure <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msub><h:mi>χ</h:mi><h:mi>rig</h:mi></h:msub><h:mo>∼</h:mo><h:msup><h:mrow><h:mo>|</h:mo><h:mi>ϕ</h:mi><h:mo>−</h:mo><h:msub><h:mi>ϕ</h:mi><h:mi>c</h:mi></h:msub><h:mo>|</h:mo></h:mrow><h:mrow><h:mo>−</h:mo><h:mi>γ</h:mi></h:mrow></h:msup></h:mrow></h:math> with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>γ</i:mi><i:mo>=</i:mo><i:mn>7</i:mn><i:mo>/</i:mo><i:mn>4</i:mn></i:mrow></i:math>. The system is thus found to exhibit criticality. The results are shown to depend on an effective field <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>h</j:mi><j:mo>(</j:mo><j:mi>μ</j:mi><j:mo>)</j:mo></j:mrow></j:math>, which provides data collapse near <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:msub><k:mi>ϕ</k:mi><k:mi>c</k:mi></k:msub></k:math> for both <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msub><l:mi>f</l:mi><l:mi>rig</l:mi></l:msub></l:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>χ</m:mi><m:mi>rig</m:mi></m:msub></m:math>. This behavior occurs over a range of stresses, with <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:msub><n:mi>ϕ</n:mi><n:mi>c</n:mi></n:msub><n:mrow><n:mo>(</n:mo><n:mi>σ</n:mi><n:mo>,</n:mo><n:mi>μ</n:mi><n:mo>)</n:mo></n:mrow></n:mrow></n:math> increasing as the stress decreases.