Branched droplet clusters and the Kramers theorem

Scaling laws inherent for polymer molecules are checked for the linear and branched chains constituting two-dimensional (2D) levitating microdroplet clusters condensed above the locally heated layer of water. We demonstrate that the dimensionless averaged end-to-end distance of the droplet chain $\overline{r}$ normalized by the averaged distance between centers of the adjacent droplets $\overline{l}$ scales as $\overline{r}/\overline{l}\sim n^{0.76}$, where $n$ is the number of links in the chain, which is close to the power exponent ¾, predicted for 2D polymer chains with excluded volume in the dilution limit. The values of the dimensionless Kuhn length $\tilde{b}\cong 2.12\pm 0.015$ and of the averaged absolute value of the bond angle of the droplet chains $\overline{\left|\theta \right|}=22.0\pm 0.5^0$ are determined. Using these values we demonstrate that the predictions of the Kramers theorem for the gyration radius of branched polymers are valid also for the branched droplets' chains. We discuss physical interactions that explain both the high value of the power exponent and the applicability of the Kramers theorem including the effects of the excluded volume, surrounding droplet monomers, and the prohibition of extreme values of the bond angle.

Figure 1

Branched droplet chains within the droplet cluster.

Figure 2

The double-log plot of the nondimensional end-to-end distance in nonbranched droplet chains vs the number of links in the chain. The linear fitting supplies the scaling law $\tilde{r}\sim n^{0.76}$ (the squared standard deviation $R^2=0.9878$). The standard deviation is growing with the length of the chain because the number of small chains increases. The results summarize the analysis of 823 droplet chains.

Figure 3

Schematic of the polymer chain; $l$ is the distance between monomers (droplets); θ is the bond angle, and $R_{\max }$ is the contour length.

Figure 4

(a) The distribution of the values of the bond angle θ; the average absolute values of the bond angle is $\overline{\left|\theta \right|}={22}^{\circ }\pm 0.5^{\circ }$ and $\langle cos\frac{\left|\theta \right|}{2}\rangle =0.9665$ based on 1443 measurements (734 positive values (red line) and 709 negative values (blue line)). (b) Bond angle θ distribution density is depicted. $\langle \theta \rangle \cong 0$ takes place.

Figure 5

The branched droplet chain built of 101 droplets (a fragment of cluster 1). The scale bar is 0.1 mm.

Figure 6

The ratio $\frac{R_{g2}}{R_{g1}}$ vs the number of droplets calculated for various branched droplet chains for three branched droplet clusters. The dashed line depicts the value of the ratio averaged across four clusters $\langle \frac{R_{g2}}{R_{g1}}\rangle =0.91\pm 0.08$ as established for 36 branched chains. Solid blue line depicts the value of the ratio averaged across group of twelve “perfectly branched” chains $\langle \frac{R_{g2}}{R_{g1}}\rangle =0.975\pm 0.045.$

Figure 7

A polymer chain.

Figure 8

(a) The linear chain and all of the possibilities of its dividing into two branches are depicted. (b) Branched chain with the single center of branching and various possibilities of its dividing are depicted.