Shear-stress function approach of hydration layer based on the Green-Kubo formula

We present the analytic expression of the stress correlation (SC) function for the ubiquitous hydration water layer (HWL) using the Green-Kubo equation and the shear modulus of HWL. The SC function is then experimentally obtained by measuring the viscoelastic properties of HWL using shear-mode dynamic force spectroscopy. Interestingly, the SC changes sign from positive to negative as the HWL thickness increases, where the shear stresses acting on the HWLs bound to two nearby surfaces are out of phase. We also suggest that the repulsive hydration force originates from the SC of HWL. Our results provide the first demonstration of the microscopic understanding of the HWL viscoelasticity and may allow a deeper insight on the HWL dynamics as well as the complex liquids.

Figure 1

Experimental results of the elastic and damping coefficients of HWL obtained during slow approach and retraction of the tip (at about 1 nm/s speed). The contact point $\left(z_{\mathrm{c}}\right)$ is determined as the position where the damping motion of the tip is abruptly suppressed as the tip contacts the substrate. The inset is the experimental schematic that studies the shear dynamics of the HWL formed between the flattened fused quartz tip (with its SEM image) and the mica substrate.

Figure 2

Comparison between dynamic force experiment and viscoelastic HWL theory. (a) The measured effective elasticity $\left(k_{\mathrm{AFM}}\right)$ and the calculated elasticity from the HWL stress tensor $\left(k_{\mathrm{h}}\right)$. (b) The measured damping coefficient $\left(b_{\mathrm{AFM}}\right)$ and the calculated damping coefficient of HWL from the stress tensor $\left(b_{\mathrm{h}}\right)$. Inset: The presented data correspond to those shown in Fig. 1. The decay length increases up to $\sim 8$ nm for $z$ above $\sim 4$ nm, where the capillary effects dominate. The blue dashed line represents the expo-nential behavior of $e^{-z/\lambda }$ with $\lambda \sim 8$ nm that describes the capillary effects.

Figure 3

(a) Relaxation time of HWL. The empty diamond-shaped box and the red solid line show the mechanical relaxation time that is measured by AM-AFM $\left({\tau }_{\mathrm{AFM}}\right)$ and calculated by the HWL stress tensor $\left({\tau }_{\mathrm{h}}\right)$, respectively. Its value of $\sim 6 \mu \mathrm{s}$ is much longer than the water molecular dipole-autocorrelation relaxation time of HWL (10 ps $\sim$ 1 ns), indicating that the mechanical motion of HWL is much more sluggish. (b) Shear modulus of HWL. The black dots represent the measured shear modulus $\left(G\right)$, which can be compared to the shear modulus of HWL calculated by the general HWL stress tensor $\left(G_{\mathrm{h}}\right)$.

Figure 4

(a) Normalized SC fuction of HWL versus $z$, given by ${\mathrm{C}}_{\mathrm{n}}=\langle \sigma (0,0)\sigma (z,0)\rangle /\langle \sigma {(0,0)}^2\rangle$. The red solid curve shows the theoretically calculated SC function of HWL [Eq. (12)], while the red empty circles represent the experimental results that are converted from the measured $k_{\mathrm{AFM}}$ and $b_{\mathrm{AFM}}$ using Eq. (13). (The red empty circles represent the data shown in Figs. 1, 2, and 3 while the black cross the data given in Ref. [12].) The water molecules are tightly bound on the hydrophilic surfaces of mica substrate and silica tip at $z<1.4$ nm and contribute to the positive SC function. The SC function becomes negative for $z>1.4$ nm, which results from the loosely bound HWL consisting of the weakly hydrated water molecules that exist away from the hydrophilic surface. When $z$ is larger than $\sim 2$.5 nm, the data deviate from the theoretical values due to the dominant capillary effect. The blue dashed line represents the value of ${\mathrm{C}}_{\mathrm{n}}$ that includes qualitatively the capillary effects. (b) Schematic spatial distribution of the hydrated water molecules in the HWLs for $z<3$ nm and the corresponding SC function.