Do we understand the solid-like elastic properties of confined liquids?

Recently, in polymeric liquids, unexpected solid-like shear elasticity has been discovered, which gave rise to a controversial discussion about its origin (1–3). The observed solid-like shear modulus depends strongly on the distance between the plates of the rheometer according to a power law with a nonuniversal exponent ranging between and .

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 aswhere and are the longitudinal (L) and transverse (T) phonon dispersions, and is a sound attenuation coefficient.

From this, the authors (4) obtain a behavior by 1) observing that, for small frequencies, the -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 = ; 2) converting the k sum to an integral over k; and 3) representing the confinement of the sample by restricting the k integral to values .

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

Let us assume periodic boundary conditions with respect to and . In the limit of , the k sum for MS becomesThe sum runs over discrete values labeled as . One can now order the summation as and convert the sum for into a integral from to . This gives a contribution proportional to instead of .

Apart from the fact that the claimed prediction is at variance with the nonuniversal exponent , 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.