Newtonian and non-Newtonian thin films create finite-time filaments: Experiments and theory

The sticky fluids found in pitcher plant traps leave behind dendritic, treelike filaments when the liquid dewets from a substrate. To understand the origin of these filaments, we investigate the retreating thin-film dynamics of aqueous glycerol and polyethylene oxide solutions on partially wetting polydimethyl siloxane substrates, conditionally generating regularly spaced filaments. We show that during the early-stage dynamics of these films, there is a critical average film thickness, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>h</a:mi><a:mi>f</a:mi></a:msub></a:math> (governed by the interplay between van der Waals and capillary forces), below which the film will be unstable, giving rise to filaments. Conversely, filaments are not formed when the average film thickness <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mover accent="true"><b:mi>h</b:mi><b:mo>¯</b:mo></b:mover></b:math> exceeds <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>h</d:mi><d:mi>f</d:mi></d:msub></d:math>. Experiments probing the conditions when filaments are formed and not formed show good agreement with the theory. After the early stage, filament dynamics enter an intermediate stage governed by a Plateau-Rayleigh instability. In this stage, the key scaling relation is <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>λ</e:mi><e:mi>f</e:mi></e:msub><e:mi>η</e:mi><e:mo>/</e:mo><e:mi>γ</e:mi><e:mo>∝</e:mo><e:msup><e:mover accent="true"><e:mtext>Ca</e:mtext><e:mo>̂</e:mo></e:mover><e:mi>s</e:mi></e:msup></e:mrow></e:math> with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>s</g:mi><g:mo>≈</g:mo><g:mn>1</g:mn></g:mrow></g:math>, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>λ</h:mi><h:mi>f</h:mi></h:msub></h:math> is the average spacing between filaments, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>η</i:mi></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>γ</j:mi></j:math> the viscosity and surface tension of the fluid, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mover accent="true"><k:mtext>Ca</k:mtext><k:mo>̂</k:mo></k:mover></k:math> is the capillary number (given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>η</m:mi><m:mi>U</m:mi><m:mo>/</m:mo><m:mi>γ</m:mi></m:mrow></m:math>; <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mi>U</n:mi></n:math> is the receding contact line velocity). The evolution of the thin-film shape is modeled numerically to show that the formation of filaments arises because the thin-film equation numerically hints at a singular solution after certain time <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mi>t</o:mi><o:mi>f</o:mi></o:msub></o:math> and when the thin-film thickness is below <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:msub><p:mi>h</p:mi><p:mi>f</p:mi></p:msub></p:math>.