Accurate estimation of dynamical quantities for nonequilibrium nanoscale systems

Fluctuations of dynamical quantities are fundamental and inevitable. For the booming research in nanotechnology, huge relative fluctuation comes with the reduction of system size, leading to large uncertainty for the estimates of dynamical quantities. Thus, increasing statistical efficiency, i.e., reducing the number of samples required to achieve a given accuracy, is of great significance for accurate estimation. Here we propose a theory as a fundamental solution for such problem by constructing auxiliary path for each real path. The states on auxiliary paths constitute canonical ensemble and share the same macroscopic properties (NVT) with the initial states of the real path. By implementing the theory in molecular dynamics simulations, we obtain a nanoscale Couette flow field with an accuracy of $0.2\mu \mathrm{m}/\mathrm{s}$ with relative standard error $<0.1$. The required number of samples is reduced by 12 orders compared to conventional method. The predicted thermolubric behavior of water sliding on a self-assembled surface is directly validated by experiment under the same velocity. This theory only assumes the system is initially in thermal equilibrium, then driven from that equilibrium by an external perturbation. It could serve as a general approach for extracting the accurate estimate of dynamical quantities from large fluctuations to provide insights on atomic level under experimental conditions, and benefit the studies on mass transport through (biological) nanochannels and fluid film lubrication of nanometer thickness.

Figure 1

Number of samples needed to estimate the flow rate of the nanoconfined water. (a) Relative fluctuation (RF) as a function of the number of particles $N$ in the system. Arrow shows the recent experimental systems from micro- [40], nano- [41], to angstrom [42] scale, and the dashed line indicates the number of water molecules in a cube with a side length of 100 nm. (b) Number of independent samples $n$ needed to estimate the velocity $v_{\mathrm{com}}$ with given relative standard error (RSE). Values of $v_{\mathrm{com}}$ for nanoconfined water in existing molecular simulations (0.03 to 24.3 m/s) and this work $\left(3.36\times {10}^{-6}\mathrm{to}3\mathrm{m}/\mathrm{s}\right)$ are plotted as different symbols.

Figure 2

Schematic of a specific protocol for APM. (a) The three axes are coordinate $r$, momentum $p$, and time $t$. The blue solid line stands for a real path in the nonequilibrium path ensemble with external perturbation and the green solid line is the corresponding auxiliary path. The black arrows reveal the origin of the middle states and the red arrow represents a relaxation with a duration of δt. (b) Schematic of the error analysis for the specific protocol. Two parts of error are introduced. One is the bias of ${\langle B\rangle }_t+{\langle A\rangle }_{\mathrm{EC}}$ (blue curve) from ${\langle A\rangle }_t$ (black curve) caused by the finite $\lambda$. The other is the error caused by the estimation of ${\langle B\rangle }_t+{\langle A\rangle }_{\mathrm{EC}}$ (blue curve) with ${\overline{B}}_m\left(t\right)+{\langle A\rangle }_{\mathrm{EC}}$ (red triangles).

Figure 3

Application of the specific protocol to nanoconfined water flow with homogeneous surfaces. (a) Side view of a typical studied system for water sheared by two double-layer graphene sheets. (b) Velocity profile of water flow obtained from APM with $V_{\mathrm{wall}}=100\mu \mathrm{m}/\mathrm{s}$. Blue dotted lines are the average position of the innermost walls of graphene sheets. (c) Comparison between the standard error of velocity profiles obtained by the conventional method and APM with the same number of samples. (d) Ratio γ defined as $\mathrm{S}{\mathrm{E}}_{\mathrm{APM}}/\mathrm{S}{\mathrm{E}}_{\mathrm{con}}$. (e) Local viscosity η of water at different distance away from the innermost walls of graphene sheets. (f) Density of water ρ and the number density of hydrogen bond $n_{\mathrm{hb}}$ at different distance away from the innermost walls of graphene sheets. (g) Slip length ($l_s$) vs $V_{\mathrm{wall}}$. The $x$ axis is composed of linear and logarithmic parts, separated by the break line. The only deviation is $l_s$ based on the conventional method at $V_{\mathrm{wall}}=20\mathrm{m}/\mathrm{s}$, where the velocity profile is covered by the thermal noise. (h) Comparison of the flow velocity ranges studied in previous experiments $\left(1\times {10}^{\text{–}6}\mathrm{to}4\times {10}^{\text{–}2}\mathrm{m}/\mathrm{s}\right)$ [9], previous simulations $\left(3\times {10}^{\text{–}2}\mathrm{to}1000\mathrm{m}/\mathrm{s}\right)$, [9, 35, 36, 37, 38, 39], and the present work $\left(3.36\times {10}^{-6}\mathrm{to}3\mathrm{m}/\mathrm{s}\right)$.

Figure 4

Comparison of APM and conventional method for Poiseuille flow. (a) Side view of a typical studied system for water flow generated by a pressure drop $\mathrm{\Delta }P$. (b) Velocity profile of water flow obtained from APM with $\mathrm{\Delta }P=0.2\mathrm{Pa}/\mathrm{nm}$. Blue dotted lines are the average position of the innermost walls of graphene sheets. (c) Center-of-mass velocity $v_{\mathrm{com}}$ vs time $t$. (d) Velocity profile of water flow obtained from APM with $\mathrm{\Delta }P=2\times {10}^6\mathrm{Pa}/\mathrm{nm}$. Blue dotted lines are the average position of the innermost walls of graphene sheets. Inset includes the standard error of velocity profile.

Figure 5

Direct comparison between the temperature dependence of friction coefficient measured in experiments and predicted by APM. (a) Schematic of the setup to measure friction coefficient at water-FDTS interface in experiments (left) and MD simulations (right). Area highlighted by the dotted red rectangle is the system simulations considered. (b) Morphology characterization of the FDTS monolayer surface, with root-mean-square (rms) roughness of 0.08 nm for a $10\times 10\mathrm{n}{\mathrm{m}}^2$ area. (c) Morphology characterization of the microsphere surface, with rms roughness of 1.12 nm for a $400\times 400\mathrm{n}{\mathrm{m}}^2$ area. (d), (e) Temperature dependence of friction coefficient normalized by the value at 298 K for MD simulations (d) and experiments (e) under the same shearing velocity $\left(v_s\right)$ range, i.e., 30 to $54\mu \mathrm{m}/\mathrm{s}$.

Figure 6

Evolution of the flow field for water sheared by graphene. (a)–(c) Flow fields shown in the first three rows were obtained from the APM with $V_{\mathrm{wall}}={10}^{-4}$, ${10}^{-3}$, or 100 m/s. (d) Flow fields shown in the last row were obtained from the conventional method with $V_{\mathrm{wall}}=100\mathrm{m}/\mathrm{s}$. Streamlines which are a family of curves that are instantaneously tangent to the velocity vector of the flow were computed by the velocity field with a grid size of $0.1\mathrm{nm}\times 0.1\mathrm{nm}$ (see SM, Sec. 5 for details) [18]. Movies S1–S4 corresponding to (a)–(d) are also provided as SM.