Calculation of a fluctuating entropic force by phase space sampling

A polymer chain pinned in space exerts a fluctuating force on the pin point in thermal equilibrium. The average of such fluctuating force is well understood from statistical mechanics as an entropic force, but little is known about the underlying force distribution. Here, we introduce two phase space sampling methods that can produce the equilibrium distribution of instantaneous forces exerted by a terminally pinned polymer. In these methods, both the positions and momenta of mass points representing a freely jointed chain are perturbed in accordance with the spatial constraints and the Boltzmann distribution of total energy. The constraint force for each conformation and momentum is calculated using Lagrangian dynamics. Using terminally pinned chains in space and on a surface, we show that the force distribution is highly asymmetric with both tensile and compressive forces. Most importantly, the mean of the distribution, which is equal to the entropic force, is not the most probable force even for long chains. Our work provides insights into the mechanistic origin of entropic forces, and an efficient computational tool for unbiased sampling of the phase space of a constrained system.

Figure 1

The origin of force by a constrained chain. A freely jointed chain (blue balls and sticks) is terminally pinned to a point (crosshair) and can move in full space or half space (dashed horizontal line). (a) Constraint (centrifugal) force. When freely jointed mass points of the chain move with some velocities (red arrows), the pin point will experience a force. The mass points can also exert force on the pin point when they recoil from the wall. (b) Thermodynamic (entropic) force. If the chain is pinned through a longer tether, more conformations become available. For example, the first mass point can explore a larger surface (compare blue and gray circles). Some conformations prohibited by the wall boundary can become acceptable as well when the tether is lengthened. This increase in conformational space leads to an increase in entropy, and thus a net force away from the surface. (c) Generalized coordinates in the global frame. The first three mass points of the chain are shown as black balls. The first three Cartesian coordinates $(x_1,y_1,z_1)$ define the position of the first mass point with respect to the origin. The tether length is $r_1$. The positions of the rest are defined by global zenith $\left(\theta \right)$ and azimuthal $\left(\varphi \right)$ angles.

Figure 2

Chain pinned in full space. All forces are nondimensionalized in units of $k_{B}T/a$, and all lengths in units of $a$. (a) Distribution of kinetic energy at different points along the chain. The first mass point after the pinning position has less energy than average, and the last one has slightly more. The mean kinetic energy remains $N{k}_{B}T$ in all cases. Shown are results from different chain lengths $(N=10$, 20, and 30). (b) Distribution of radial constraint forces. While the mean constraint force equals that due to a single mass point, additional links in the polymer chain increase likelihood of strong pulling forces and even introduce the possibility of compressive forces. The most probable force is near $-1$ shown by the vertical dotted line. Distributions are computed for different chain lengths $(N=2$, 9, and 29) from 150 K samples. (c) Mean constraint force and entropic force. The average force (black circles) is independent of length and matches the predicted entropic force (red line), which results from the first mass point confined to the surface of a sphere moving to a larger radius. (d) Standard deviation of constraint forces. The standard deviation is on the same order of magnitude as the mean force and quickly saturates as chain length increases.

Figure 3

Origin of compressive forces. Some fraction of conformations (blue) and velocities (red) produce compressive forces along the first link in the chain, in contrast to the expectation from a single link acting as a pendulum. These occur when the chain is sharply bent, and the downstream mass points are moving more rapidly than those closer to the pin point $\left(v_2>v_1\right)$.

Figure 4

Chain pinned to plane. All forces are nondimensionalized in units of $k_{B}T/a$, and all lengths in units of $a$. (a) Distribution of forces for phase-space sampled chains. The surface effect shifts the mean of the distribution slightly, but the shape is not changed relative to the free case. The most probable force is near $-1$ shown by the vertical dotted line. Distributions are computed for different chain lengths $(N=2$, 9, and 29), from 150 K samples. (b) Mean force from chains sampled via phase space Monte Carlo (circles) and thermodynamic force from conformation space partition function (dashed line). The average force lies above the $2k_{B}T/a$ value of the free chain case, as additional entropy is gained by pulling away from the surface. The mean force exhibits some length dependence not seen in the free chain case; however, it quickly saturates around 10 links in the polymer chain. (c) Standard deviation of sampled forces. As in the free chain case, the deviation is on the same order of magnitude as the mean force. It grows slightly as a function of length, quickly saturating around 5–7 links. (d) Distribution of forces for conformational space sampled chains. The same distribution observed in (a) persists as we consider longer chains $(N=2$, 9, and 99). The most probable force is near $-1$ shown by the vertical dotted line. The curves in the figure represent 300 K samples. (e) Mean force from equilibrium sampled chains, before applying the Fixman correction $(\times )$ and after $(\circ )$, displayed alongside the partition function prediction (dashed line). The mean force remains constant as we explore larger lengths than were accessible in (b). (f) Standard deviation for force from equilibrium sampled chains, with $(\circ )$ and without $(\times )$ the Fixman correction.