Proceedings of the National Academy of Sciences of the United States of America

Recently, in polymeric liquids, unexpected solid-like shear elasticity has been discovered, which gave rise to a controversial discussion about its origin (12–3). The observed solid-like shear modulus $$ G$$ depends strongly on the distance $$ L$$ between the plates of the rheometer according to a power law $$ G\propto L^{-p}$$ with a nonuniversal exponent ranging between $$ p=2$$ and $$ p=3$$.

Zaccone and Trachenko (4) have published an article in which they claim to explain these findings by a nonaffine contribution to the liquid shear modulus. The latter is represented as

where $$ {\omega }_{p,L}(k)$$ and $$ {\omega }_{p,T}(k)$$ are the longitudinal (L) and transverse (T) phonon dispersions, and $$ \nu $$ is a sound attenuation coefficient.

From this, the authors (4) obtain a $$ \mathrm{\Delta }G\propto L^{-3}$$ behavior by 1) observing that, for small frequencies, the $$ \omega $$-dependent terms are negligible, and, consequently, the nominator cancels against the denominator, from which follows that the nonaffine contribution becomes just a mode sum MS = $$ \frac{1}{V}{\sum }_{\mathbf{k}}1$$; 2) converting the k sum $$ \frac{1}{V}{\sum }_{\mathbf{k}}$$ to an integral over k; and 3) representing the confinement of the sample by restricting the k integral to values $$ |\mathbf{k}|\ge L^{-1}$$.

However, the authors (4) disregard the fact that the liquid is not confined inside a sphere of diameter $$ L$$, but between two plates of the rheometer with gap distance $$ L$$. This means that we are dealing with a slab geometry, in which the sample boundaries $$ L_x$$ and $$ L_y$$ in $$ x$$ and $$ y$$ directions are much larger than the confinement $$ L$$ in the $$ z$$ direction.

Let us assume periodic boundary conditions with respect to $$ L_x,L_y$$ and $$ L$$. In the limit of $$ L_x=L_y\to \infty $$, the k sum for MS becomes

The $$ k_z$$ sum runs over discrete values labeled as $$ k_z^{(n)}=2\pi n/L$$. One can now order the summation as $$ n=0,\pm 1,\pm 2\dots $$ and convert the sum $$ \frac{1}{L}{\sum }_{k_z}$$ for $$ n\ne 0$$ into a $$ k_z$$ integral from $$ k_z^{(1)}=2\pi /L$$ to $$ k_{\text{max}}$$. This gives a $$ \mathrm{\Delta }G$$ contribution proportional to $$ L^{-1}$$ instead of $$ L^{-3}$$.

Apart from the fact that the claimed $$ L^{-3}$$ prediction is at variance with the nonuniversal exponent $$ p$$, we find that its derivation is in error. We feel that the origin of the observed solid-like properties of confined liquids is still elusive.