Nonlinear response in oligomers across chain lengths

We present third-order and first-order dielectric measurements on a series of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>N</a:mi> </a:math> -propylene glycol ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>N</b:mi> </b:math> -PG) oligomers, focusing on the role of molecular length <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>N</c:mi> </c:math> in the glass transition. To avoid polymeric effects, we restricted the study to short chains ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>N</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> <d:mo>–</d:mo> <d:mn>20</d:mn> </d:math> ). Linear dielectric susceptibility <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi mathvariant="normal">Δ</e:mi> <e:msub> <e:mi>χ</e:mi> <e:mn>1</e:mn> </e:msub> </e:mrow> </e:math> scales as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mfrac> <g:mn>1</g:mn> <g:mi>N</g:mi> </g:mfrac> </g:math> , suggesting that only the terminal hydroxyl (–OH) groups contribute significantly to the dipolar response, while the flexible backbone units remain largely inactive. Third-order susceptibilities <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msubsup> <h:mi>χ</h:mi> <h:mn>3</h:mn> <h:mrow> <h:mo>(</h:mo> <h:mn>3</h:mn> <h:mo>)</h:mo> </h:mrow> </h:msubsup> </h:math> exhibit a growing hump upon cooling for all <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>N</i:mi> </i:math> -PG, consistent with the Bouchaud-Biroli prediction. This allows us to extract the relative evolution of the number of dynamically correlated molecules <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi>N</j:mi> <j:mi>corr</j:mi> </j:msub> </j:math> , which is as expected increasing upon cooling. But surprisingly, this evolution is identical across all oligomers, regardless of chain length, even though the size of the largest molecules likely exceeds that of the dynamic heterogeneities. This robustness of the relative evolution of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:msub> <k:mi>N</k:mi> <k:mi>corr</k:mi> </k:msub> <k:mrow> <k:mo>(</k:mo> <k:msub> <k:mi>τ</k:mi> <k:mi>α</k:mi> </k:msub> <k:mo>)</k:mo> </k:mrow> </k:mrow> </k:math> allows to deduce that it is insensitive to the backbone linking the monomers, the fragility, the OH-bond density, or the strength of the dipole-dipole interactions among oligomers, standard criteria to describe the phenomenon. These findings challenge kinetically constrained models, which fail to account for the observed behavior, while supporting the predictions of the random first-order transition theory.