Atomistic study of macroscopic analogs to short-chain molecules

We use a bath of chaotic surface waves in water to mechanically and macroscopically mimic the thermal behavior of a short articulated chain with only nearest-neighbor interactions. The chaotic waves provide isotropic and random agitation to which a temperature can be ascribed, allowing the chain to passively explore its degrees of freedom in analogy to thermal motion. We track the chain in real time and infer end-to-end potentials using Boltzmann statistics. We extrapolate our results, by using Monte Carlo simulations of self-avoiding polymers, to lengths not accessible in our system. In the long-chain limit we demonstrate universal scaling of the statistical parameters of all chains in agreement with well-known predictions for self-avoiding walks. However, we find that the behavior of chains below a characteristic length scale fundamentally differs. We find that short chains have much greater compressional stiffness than would be expected. However, chains rapidly soften as length increases to meet with expected scalings.

Figure 1

(a) The shaker assembly. (b) Three-dimensional schematic of one for the buoyant particles in the chain, showing the definition of restricting angle $\alpha$. (c) A link showing fixed length $R$. (d) A chain demonstrating the measurement of link angle ${\theta }_i$. Bulk rotations are eliminated from our coordinate system by measuring all ${\theta }_i$ off of the link most proximal to the anchor. This allows us to work exclusively in terms of the relative coordinates of members of the chain.

Figure 2

Average bond-angle velocity autocorrelation shown in red (thickest curve) for a chain with $\alpha =\pi /6$. Nearest, second-nearest, and third-nearest bond-angle velocity neighbor correlations shown in blue, purple, and magenta (second, third, and fourth thickest curves), respectively. The nearest-neighbor bond-angle velocities are initially anticorrelated due to buckling in the interconnected chain, i.e., as one bond angle is kicked its neighbor will react in the opposite direction. All correlations die off by $\sim 0.3$ s of lag time. The 10-Hz beat frequency results from the noncommensurate dish oscillation frequency (40 Hz) and imaging frequency (30 Hz).

Figure 3

Effective potentials for total bond winding angle ${\varphi }_{i,j}$ (top row) and interparticle distance $r_{i,j+1}$ (bottom row) with example subchains shown as inset diagrams. Potentials are determined from distributions using Eqs. (3) and (4). The blue (dark) curves are calculated directly from empirical data and the red (light) curves are calculated using the distribution for ${\varphi }_{i,i+1}$ to generate self-avoiding walks.

Figure 4

Master curves for the scaling of winding-angle variance ${\sigma }_{\varphi }^2$. The data are plotted in both log-log (top) and semilog (bottom) to illustrate power law and logarithmic scaling regimes, respectively. The shape of the markers indicates the probability distribution of nearest-neighbor link angles; stars for experimental, circles for uniform, and diamonds for bidisperse. The marker color goes from light to dark as the variance of the nearest-neighbor probability distribution increases. Equation (5) is plotted over the data. Inset: The critical length $N_{*}$ is inversely proportional to ${\sigma }_{\varphi ,0}^2$, the variance of the nearest-neighbor link angle for a given chain.

Figure 5

Scaling of variance in end-to-end distance ${\sigma }_r^2$ (lower curve) and squared mean of end-to-end distance ${\langle r\rangle }^2$ (upper curve). The shape and color of the markers indicates the probability distribution of nearest-neighbor bond angles as in Fig. 4. Both curves are scaled by a factor of $(C_1+C_2)N_{*}^{3/2}$, where $N_{*}$ is the same critical length from Fig. 4, and collapse to $N^{3/2}$ power laws for $N>>N_{*}$. Inset: The sum of scaling coefficients $C_1$ and $C_2$ plotted against critical length $N_{*}$.