# The Kramers theorem, excluded volume, and scaling in droplet clusters

Scaling in physical chemistry is a fascinating issue since it combines temporal, spatial, and temperature domains. For a random walk or Brownian motion (the Einstein-Smoluchowski diffusion), the distance is proportional to power 1/2 of the path, R ~ sqr(L). There are numerous exceptions, known as anomalous diffusion (subdiffusion or superdiffusion) due to molecular crowding and other interesting effects, such as the non-ergodicity.

The same applies to long molecules, such as polymers, which form random chains in space. An ideal chain has an average end-to-end distance proportional to the square root of the number of segments, i.e., the scaling exponent of 1/2. An important issue is a chain without self-intersection or “with excluded volume”. Such chains have scaling exponents of 3/5 (or 0.588 based on elaborated scaling arguments) in the 3D case and of 3/4 in the 2D case. Interactions between monomers (such as the attraction or repulsion) and with other dissolved molecules (“the molecular crowding”) can also affect the scaling exponents.

In addition, for branched chains, there is the so-called Kramers theorem, which simplifies the calculation of the radius of gyration of a branched molecule.

In our new work, we show that branched droplet chains in levitating microdroplet clusters follow the Kramers theorem’s prediction. They also demonstrate scaling exponents close to 3/4 (namely, 0.76). The excluded volume effect is similar to the effect of the bonding angle limitation by 120 degrees. A simplified insight on that may be obtained from considering a chain with three segments A-A-A. An ideal chain (with self-intersections) would have an angle between branches varying between -180 and +180 degrees, with the average absolute value of 90, the straight triangle, and the total end-to-end distance equal to the square root of two times the length of the stretched molecule (of two segments), R = 1.41 L. With the angle limitation of between -120 and +120 degrees, the average value of the cosine is 0.4125 and the law of cosines yields R = 1.6814 L; the coefficient 1.6814 is very close to 2^0.75 = 1.6818.

M. Frenkel, A. A. Fedorets, D. V. Shcherbakov, L. A. Dombrovsky, M. Nosonovsky, and E. Bormashenko, 2022, “Branched droplet clusters and the Kramers theorem” Phys. Rev. E 105, 055104 (2022)
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.105.055104 DOI: 10.1103/PhysRevE.105.055104 