Effects of Polydispersity and Concentration on Elastocapillary Thinning of Dilute Polymer Solutions

The self-thinning of liquid bridges under the action of capillarity occurs in widespread processes like jetting, dripping, and spraying and gives rise to a strong extensional flow capable of stretching dissolved polymers. If the resulting elastic stress exceeds the viscous stress, an exponential “elastocapillary” (EC) thinning regime arises, yielding a timescale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>τ</a:mi><a:mrow><a:mi>EC</a:mi></a:mrow></a:msub></a:math> that is commonly considered equivalent to the longest relaxation time of the polymer <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>λ</c:mi></c:math>. A long-standing question is why <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>τ</e:mi><e:mrow><e:mi>EC</e:mi></e:mrow></e:msub></e:math> depends strongly on the polymer concentration, even at high dilutions where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>λ</g:mi></g:math> should be constant in theory. To date, this is understood in terms of intermolecular interactions that arise due to “self-concentration” effects as polymers stretch. However, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>λ</i:mi></i:math> depends on the polymer molecular weight <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>M</k:mi></k:math>, and we show how the concentration dependence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>τ</m:mi><m:mrow><m:mi>EC</m:mi></m:mrow></m:msub></m:math> can be explained by considering the molecular weight distribution (MWD) inherent in real polymer samples, without the need to invoke self-concentration. We demonstrate this by blending low-<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>M</o:mi></o:math> and high-<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>M</q:mi></q:math> polymer samples with narrow MWDs at dilute concentrations and in different proportions and by measuring <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>τ</s:mi><s:mrow><s:mi>EC</s:mi></s:mrow></s:msub></s:math> for each blend in capillary thinning experiments. Through a simple model that qualitatively reproduces the experimental results, we show how elastic stresses generated by the polymer build up prior to the EC regime due to the sequential stretching of progressively decreasing molecular weight species in the MWD. Since the elastic stress generated by each species depends on its concentration, the fraction of the MWD that is required to stretch in order to induce the EC regime depends on the total polymer concentration <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>c</u:mi></u:math> in the solution. For higher <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>c</w:mi></w:math>, the EC regime is induced by stretching of a higher-<y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>M</y:mi></y:math> (longer <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>λ</ab:mi></ab:math>) fraction of the MWD and results in a longer measurement of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>τ</cb:mi><cb:mrow><cb:mi>EC</cb:mi></cb:mrow></cb:msub></cb:math>. Our results have significant implications for the application of capillary thinning measurements to extensional rheometry, for the interpretation of such measurements, and for the understanding of elastocapillary thinning dynamics in general.