Microfluidics and Nanofluidics

, Volume 14, Issue 1, pp 299–308

Slip in nanoscale shear flow: mechanisms of interfacial friction

Authors

  • Xin Yong
    • Department of Mechanical, Aerospace and Nuclear EngineeringRensselaer Polytechnic Institute
    • Department of Chemical and Petroleum EngineeringUniversity of Pittsburgh
    • Department of Mechanical, Aerospace and Nuclear EngineeringRensselaer Polytechnic Institute
Research Paper

DOI: 10.1007/s10404-012-1048-x

Cite this article as:
Yong, X. & Zhang, L.T. Microfluid Nanofluid (2013) 14: 299. doi:10.1007/s10404-012-1048-x

Abstract

The atomistic mechanism of fluid–solid interfacial friction as the basis of slip is still not fully understood. This study explores the interfacial friction mechanisms and their interplay with the nanoscale slip behavior using non-equilibrium molecular dynamics simulations. Our results show that there is an abrupt jump of slip length at a critical shear rate, corresponding to the transition from “defect slip” at low shear rates to “collective slip” at high shear rates. Here, we identified two mechanisms of interfacial friction: surface potential and collision mechanisms. Their impacts on slip are elaborated through a quantitative scaling estimation and our results show that both mechanisms contribute to the defect slip at low shear rates, while the collision mechanism dominates the collective slip at high shear rates. We also verify the importance of the bulk viscous heating via a comparison among different thermostat strategies.

Keyword

SlipFrictionNanofluidicsMolecular dynamics

1 Introduction

The flow boundary condition at fluid–solid interfaces has been a subject haunting the fluid dynamics community for hundreds of years. Until recent decades, the boundary condition has merely been an assumption based on the observations at macroscale due to the lack of precise measuring techniques at the interface. The most common assumption of the hydrodynamic velocity boundary condition made at continuum scale is the no-slip condition, i.e. all three components of the fluid velocity at the immediate vicinity of the solid surface are equal to the respective velocity components of the surface. The inquiry of slip between fluid and solid has been under meticulous investigation for decades (Lauga et al. 2007; Neto et al. 2005; Bocquet and Barrat 2007). Recent controlled experiments have demonstrated apparent violations of the no-slip condition. Intense research on slip was fostered by the prosperous applications in micro- and nanofluidics, and enormous practical applications rely on understanding the nature of slip. For example, the permeability of channels and porous materials is tremendously reduced as the channel size decreases to micrometer or nanometer scale under no-slip condition. To bypass the stringent tradeoff between flow rate and channel size, and achieve good permeability for nanochannels, allowing the fluid to slip against the solid is crucial. Revealing the underlying physics of slip and further manipulating slip at nanoscale can profoundly impact the future of single-molecule analysis, nanotribology, energy conversion, and desalination.

For flow past an ideal smooth solid surface, the Navier or slip boundary condition has been proposed in the early days to describe the slip behavior as
$$ V_{\rm s}=L_{\rm s}\partial_nV_t, $$
(1)
with n and t pointing to the normal and tangential directions of the surface, Vt is the tangential fluid velocity, Ls is the slip length, and Vs is the slip velocity, which is defined as the difference between the solid surface velocity U0 and the fluid velocity at the wall Vtw as Vs = U0 − Vtw. This boundary condition assumes that the slip length is constant and independent of local shear rate ∂nVt. This linear assumption was verified for simple fluids within experimentally accessible shear rates (Cottin-Bizonne et al. 2005), but Thompson and Troian (1997) reported the limitation of this linear relation in numerical simulations at shear rates of order 10−1 τ−1, e.g. \(\approx10^{10}\, {\rm s}^{-1}\) for argon.

Experimental studies on slip at nanoscale face a few significant challenges associated with the manufacture of controlled nanoscale roughness and accurate measurement techniques in the vicinity of fluid–solid interface. Therefore, molecular dynamics (MD) simulations have been widely performed to shed light on underlying physical principles of slip and to advance our understanding of it. Fruitful studies focus on three major questions regarding slip. The first question is the slip dependence on shear-rate. Several groups consistently observed that the slip length diverges unboundedly at higher shear rates (Thompson and Troian 1997; Priezjev and Troian 2004; Voronov et al. 2007), while contradictory results of bounded behavior of slip at high shear rates were also reported (Martini et al. 2008a, b; Pahlavan and Freund 2011). Recent work by Martini et al. (2008a, b) initiated the study of solving the contradiction between bounded and unbounded slip. They obtained bounded slip at high shear rates by the variable-density Frenkel–Kontorova (VDFK) model and a Navier–Stokes based continuum calculation. However, because of the complex bulk thermal effects still remain in their MD results, their conclusion in attributing the unbounded slip to the rigid wall model is dubious. As reported previously for systems at high shear rates, the generated viscous heating can cause drastic temperature changes comparing to low-shear-rate systems both at interfaces and within bulk fluid (Khare et al. 1997; Bernardi et al. 2010). A simple but straightforward work by Pahlavan and Freund (2011) verified that the thermal wall, or more specifically the oscillation frequency of solid atoms is not responsible for the unbounded slip. Instead, it is the thermostating mechanism that dominates the slip behaviors at high shear rates. The bounded and unbounded behaviors are attributed to the temperature difference stemming from the thermostating mechanism.

The second question is a quantitative relation between slip and temperature, pressure, wall stiffness, and wall hydrophobicity (Voronov et al. 2006, 2007; Barrat and Bocquet 1999; Priezjev 2007a; Asproulis and Drikakis 2011). These parameters are directly influenced by the fluid–solid interfacial properties. A detailed analysis based on a Kubo-like formula of a phenomenological interfacial friction coefficient showed that the slip length depends on the contact density, temperature, and the structure factor of the first fluid layer adjacent to the surface at equilibrium (Barrat and Bocquet 1999; Sendner et al. 2009). The Priezjev group has validated similar nonlinear scaling relations for not only simple fluids but also polymer melts in moderated sheared systems (Priezjev 2007b, 2009, 2010; Niavarani and Priezjev 2008).

The third fundamental question of slip is related to physical mechanisms of atomistic fluid–solid interfacial friction which is the basis of slip. As pointed out in Ref. (Martini et al. 2008b; Barrat and Bocquet 1999), the substrate potential induces forces to resist the relative motion between the adsorbed fluid layers and the substrate, which is dependent on the effective roughness of the potential surface undulations. This mechanism was also readily interpreted from another perspective as the energy dissipation arising from anharmonic coupling between phonon modes and substrate-induced modulations in the adsorbed layers (Smith and Robbins 1996). Another source of interfacial friction is the momentum transfer between fluid and solid atoms due to collision (Martini et al. 2008a; Pahlavan and Freund 2011; Asproulis and Drikakis 2011). Although the collision contribution of friction can be easily understood conceptually, its quantitative relation on slip is complex and not yet well established.

The primary mission of this work is to provide a deeper understanding of the physical mechanisms of interfacial friction. Our simulations are designed to address the interplay between the two mechanisms mentioned above and slip at different shear rates. Our simulations attempt to isolate the dynamics from the viscous heating by utilizing isothermal systems similar to Ref. (Thompson and Troian 1997; Priezjev 2007b). Although applying a thermostat directly on the fluid is not physical at high shear rates (Pahlavan and Freund 2011; Khare et al. 1997; Bernardi et al. 2010), it helps elucidate the interfacial friction mechanisms because the drastic temperature change due to bulk viscous heating could bring complexities and difficulties in interpreting the results. Our results demonstrate different behaviors of slip as the shear rate changes in low- and high-shear-rate regimes. Different interfacial friction mechanisms are identified and their respective contributions to slip are revealed. The transition between low- and high-shear-rate regimes is correlated to the mechanisms. Finally, our study confirms that different thermostat strategies lead to the contradictory results of bounded and unbounded slip in the MD simulations.

2 Simulation method

We construct a nanochannel by a fluid–wall sandwich system, in which a fluid slab is confined between two smooth solid walls. Both fluid and wall atoms interact with each other by the truncated Lennard–Jones (LJ) potential as
$$ V_{{ij}} = \left\{ {\begin{array}{ll} {4\varepsilon _{{ij}} \left[ {\left( {\frac{{\sigma _{{ij}} }}{{r_{{ij}} }}} \right)^{{12}} - \left( {\frac{{\sigma _{{ij}} }}{{r_{{ij}} }}} \right)^{6} } \right],} & {r < r_{\rm c} } \\ {0,} & {r \geq r_{\rm c}} \\ \end{array} } \right. $$
(2)
where \(r_{ij}=\left|{\bf r}_i-{\bf r}_j\right|\) is the distance between atom i and atom j. \(\varepsilon_{ij}\) and σij are the characteristic energy and length scale of the potential, respectively. Subscripts i and j are indices representing the species of atoms: f for fluid and w for wall. rc = 2.5 σ is a cutoff radius. All the physical units are expressed in the LJ units defined by the intrinsic properties of the fluid: mf\(\varepsilon_{\rm ff}\) and σff. Furthermore, all subscripts of m, \(\varepsilon\) and σ are dropped when expressed as units.
Unless otherwise specified, the affinity of walls is considered as very hydrophilic (Priezjev 2007b; Yong and Zhang 2010) with \(\varepsilon_{\rm wf}=1.0\ \varepsilon\) and σwf = 1.0 σ. The self-interaction between wall atoms is excluded to reduce computational time. Each wall consists of atoms forming two layers of [001] fcc lattice with 1.01 σ as the lattice constant. The wall and the fluid densities are ρw = 3.90 σ−3 and ρf≈ 0.844 σ−3, respectively. This density ratio represents a high incommensurability in size between fluid atom and wall structure (Thompson and Robbins 1990a). The channel spans h ≈ 40 σ along the z axis with lateral dimensions Lx = Ly = 15.13 σ. There are two models of solid wall applied in this work: one is the frozen wall, in which the wall atoms are fixed at their equilibrium lattice sites without peculiar velocity, i.e. no thermal motion; the other one is the thermal wall, in which each wall atoms is tethered to its equilibrium lattice site via a harmonic potential:
$$ V_{\rm H}\left(\left|{\mathbf{r}}-{\mathbf{r}}_{\rm L}\right|\right)=\frac{1}{2}k_{\rm w} \left|{\mathbf{r}}-{\mathbf{r}}_{\rm L}\right|^2. $$
(3)
For the thermal wall, wall atoms are free to vibrate around their equilibrium positions, providing the ability to conduct heat. The spring constant kw represents the rigidity of the wall. The range of kw in this study follows two rules: it has to be large enough to satisfy the Lindemann criterion for melting (Barrat and Hansen 2003) and small enough so that the dynamics of wall atoms can be accurately solved with the MD integration time step (Priezjev 2007a). The mass of each wall atom is assigned as ten times of the mass of the fluid atom to optimize the heat conduction between the fluid and walls. Couette flow is induced in the fluid system by translating the position of the wall particles or their equilibrium lattice sites along the x direction with uniform shearing velocity U0. In this work, only upper wall is imposed with nonzero U0 and lower wall is kept as stationary. Periodic boundary conditions are applied in the flow direction x and neutral direction y.
Langevin (LGV) thermostat is applied on wall atoms to dissipate heat (Thompson and Troian 1997; Priezjev 2007a, Thompson and Robbins 1990b). The LGV thermostat represents a heat bath with a viscous damping term and a stochastic noise term as
$$ \dot{{\mathbf{p}}}_i = {\mathbf{F}}_i-\Upgamma \frac{{\mathbf{p}}_i}{m_i}+{\mathbf{f}}_i, $$
(4)
where the friction coefficient \(\Upgamma\) regulates the heat flux from the system and the Gaussian random force \(\mathbf{f}_i\) has a zero mean and satisfies the fluctuation-dissipation theorem.

Equation (4) is applied to all three components of the equations of motion after considering a bias from shear velocity U0. The temperature of the wall is set at \(T_{\rm ws}=1.1\ \varepsilon/k_B\) with large friction coefficients \(\Upgamma=20\) and 100 τ−1 depending on the thermostat being used.

To isolate the fluid dynamics from the bulk thermal effects, the dissipative particle dynamics (DPD) thermostat is applied to the fluid to create an isothermal system because of its proven adequacy for nonequilibrium systems (Español and Warren 1995; Soddemann et al. 2003; Pastorino et al. 2007). The DPD thermostat could reproduce the correct hydrodynamic behaviors (Español and Warren 1995; Soddemann et al. 2003; Hoogerbrugge and Keolman 1992) and it is profile-unbiased by construction. The DPD thermostat is superior to the LGV thermostat by conserving the momentum and thus fulfills Galilean invariance. The simple remedy introduced by the DPD thermostat is replacing the friction and noise terms that depend on single particle velocity with terms that depend on relative velocity \(\mathbf{v}_{ij}=\mathbf{v}_i-\mathbf{v}_j\) between short-range particle pairs.
$$ \dot{{\mathbf{p}}}_i = {\mathbf{F}}_i+\sum_{j\neq i}{\mathbf{F}}_{ij}^D+\sum_{j\neq i}{\mathbf{F}}_{ij}^R, $$
(5)
The pairwise friction and random forces are given by
$$ {\mathbf{F}}_{ij}^D=-\gamma \omega^D\left(r_{ij}\right)\left(\hat{{\mathbf{r}}}_{ij}\cdot {\mathbf{v}}_{ij}\right)\hat{{\mathbf{r}}}_{ij} $$
(6)
and
$$ {\mathbf{F}}_{ij}^R=\sqrt{2k_BT\gamma} \omega^R\left(r_{ij}\right)\theta_{ij}\hat{{\mathbf{r}}}_{ij}. $$
(7)
Here, \(\hat{\mathbf{r}}_{ij}=\left(\mathbf{r}_i-\mathbf{r}_j\right)/r_{ij}\) is the unit vector from particle j to particle i. γ is the friction coefficient. θij is the pairwise form of the Gaussian white noise. Two weight functions used in our simulations are
$$ \left[ {\omega ^{R} } \right]^{2} = \omega ^{D} = \left\{ \begin{array}{ll} \left( {1 - {r/ {r_{\rm c}^{\prime } }}}\right)^{2}, & {r < r_{\rm c}^{\prime } } \\ {0,} & {r \ge r_{\rm c}^{\prime } } \\ \end{array} , \right. $$
(8)
where the cutoff radius \(r^{\prime}_{\rm c}\) for thermostat is not necessarily the same as the cutoff radius of pairwise conservative forces rc. Since all forces are pairwise, Newton’s third law is strictly fulfilled, which leads to the conservation of momentum. Unless otherwise specified, the fluid temperature is maintained at \(T_{\rm fs}=1.1\ \varepsilon/k_B\) with thermostat parameter γ = 1.0 mτ−1 and thermostat cutoff radius rc = 2.5 σ.

Thermostats are coupled to the equations of motion of the system. The equations of motions are integrated using the velocity-Verlet algorithm by LAMMPS (Plimpton 1995). The DPD thermostat could be applied also on the solid wall, but it is much more expensive comparing to the Langevin thermostat. Since the dynamics of the wall atoms only has minor influences on the system, we apply different thermostats on the solid wall and the fluid to reduce computational cost. A time step of \(\Updelta t=0.002\ \tau\) is used for all simulations, where \(\tau=\sqrt{m\sigma^2/\varepsilon}\) is the characteristic time of the LJ potential. After an equilibration period of 5 × 105 time steps, a typical run requires at least 2 × 106 time steps to reach steady state. The measured quantities of interest are averaged for at least another 1 × 106 time steps.

3 Results and discussion

3.1 Slip length versus shear rates

We introduce the imposed shear rate as a controllable variable in this study, which is defined by the wall shearing velocity U0 and the channel height h as \(\dot{\gamma}_i=U_0/h. \) Noting that \(\dot{\gamma}_i\) only indicates the strength of the imposed shear. The real fluid shear rate \(\dot{\gamma}\) could be different from \(\dot{\gamma}_i; \) it is determined by fitting a straight line to the average velocity profile excluding the portions near both walls. The slip velocity Vs is measured as the difference between U0 and the velocity at the wall extrapolated from \(\dot{\gamma}. \) The slip length Ls is further obtained by Eq. (1). This slip length is also considered as an apparent slip length Lsapp because we do not focus on the intrinsic slip existing in the molecular scale adjacent to the walls (Lauga et al. 2007). The imposed shear rate in this study covers a large range (corresponding to \(10^8\sim10^{11}\ \mathrm{s}^{-1}\) for Argon), but the system still remains in the laminar condition.

We first examine the relation between Lsapp and \(\dot{\gamma}_i\) for a range of wall rigidity from \(k_{\rm w}=400\,\varepsilon/\sigma^2\) to completely rigid or frozen (see Fig. 1).
https://static-content.springer.com/image/art%3A10.1007%2Fs10404-012-1048-x/MediaObjects/10404_2012_1048_Fig1_HTML.gif
Fig. 1

Apparent slip length as a function of the imposed shear rate for the tabulated models of wall with various rigidity (\(\varepsilon/\sigma^2\)). The walls are very hydrophilic with \(\varepsilon_{\rm wf}=1.0\ \varepsilon. \) The solid lines are just a guide to the eye

Two regimes appear, which are separated by a jump in the slip length. They are the low-shear-rate regime and the high-shear-rate regime, which are shown independently in Fig. 2a, b. The slip length in the low-shear-rate regime shows monotonic increase as \(\dot{\gamma}_i\) increases, as shown in Fig. 2a, which is consistent for the frozen wall and the thermal walls with different rigidity. The softer wall yields a smaller slip at the same \(\dot{\gamma}_i\) because of additional roughness induced by the thermal motions of the wall atoms. When \(\dot{\gamma}_i\) reaches a critical value, the slip length exhibits a drastic increase. The critical value is higher for softer wall. In the high-shear-rate regime (Fig. 2b), Lsapp continues to increase sharply and proportionally to \(\dot{\gamma}_i, \) resulting in unbounded slip. The slope of the linear Lsapp\(\dot{\gamma}_i\) relation rises as the wall becomes stiffer. For the frozen wall, the slip length diverges vertically at the critical value and the fluid slab becomes close to stagnant, which is reminiscent of the oversheared regime in the previous study (Yong and Zhang 2010).
https://static-content.springer.com/image/art%3A10.1007%2Fs10404-012-1048-x/MediaObjects/10404_2012_1048_Fig2_HTML.gif
Fig. 2

Apparent slip length as a function of the imposed shear rate for the tabulated wall rigidity (\(\varepsilon/\sigma^2\)) in the a low-shear-rate regime and b high-shear-rate regime. The walls are very hydrophilic with \(\varepsilon_{\rm wf}=1.0\ \varepsilon. \) The dashed lines are linear fitting to the data points, with corresponding numbers representing slopes

We also investigate the detailed fluid velocity profiles, which maintain good linearity in both the low-shear-rate and high-shear-rate regimes. The linearity of the velocity profiles validates the slip length and slip velocity measurements we used. In the low-shear-rate regime, the slip velocity is negligible comparing to the fluid thermal velocity at zero-shear limit. But it then becomes comparable and larger than the fluid thermal velocity and the speed of sound when the shear rate approaches the critical shear rate. In the high-shear-rate regime, the slip velocity is typically one order of magnitude larger than the thermal velocity. No matter whether in the low-shear-rate or high-shear-rate regime, the fluid thermal velocity is always smaller than the speed of sound.

Although it is practical to use the imposed shear rate, it is ultimately the real fluid shear rate that we need to characterize the slip behavior. Unlike the imposed shear rate \(\dot{\gamma}_i\) which is proportional to U0 by definition, the fluid shear rate \(\dot{\gamma}\) does not show a monotonic trend with increasing shearing velocity U0, see Fig. 3a. The fluid shear rate increases in the low-shear-rate regime and reaches a maximum before the transition to the high-shear-rate regime. Following a dramatic drop associated with the jump in the slip length, it actually decreases with increasing U0 in the high-shear-rate regime. This behavior is correspondingly reflected in the slip length in Fig. 3b, where the slip length versus fluid shear rate curve separates into two branches. For a particular fluid shear rate, there exist two possible slip lengths located on each branch representing the two regimes.
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Fig. 3

a The relation between the real fluid shear rate and the shear velocity. b Apparent slip length as a function of the fluid shear rate for the tabulated models of wall with various rigidity (\(\varepsilon/\sigma^2\)). The walls are very hydrophilic with \(\varepsilon_{\rm wf}=1.0\ \varepsilon\)

3.2 Interfacial friction mechanisms

The discontinuity in the slip length indicates the change of underlying physical picture from low-shear-rate regime to high-shear-rate regime. In the low-shear-rate regime, the fluid atoms experience the strong interaction potential from the adjacent solid wall, which can be described as a Fourier series (Steele 1973) as functions of position. The corrugated potential surface of a perfect [001] fcc lattice can be represented by a simplified form that includes many local minima by only considering dominant contribution from the top layer of wall atoms at the shortest reciprocal lattice vector \(\mathbf{q}_{\parallel}\) (Barrat and Bocquet 1999) as
$$ V_{\rm sf}\left(x,y,z\right)=V_0\left(z\right)+V_1\left(z\right) \left[\cos\left(q_{\parallel}x\right)+\cos\left(q_{\parallel}y\right) \right], $$
(9)
where z and \(\mathbf{x}\equiv\left(x,y\right)\) represent the vertical distance of the fluid particles above the wall and the position within the xy plane, respectively. \(V_0\left(z\right)\) denotes the q = 0 term in the Fourier series and \(V_1\left(z\right)\) corresponds to the additional contribution from the lateral arrangement of wall atoms.
For the low-shear-rate regime, the concept of “defect slip” is well introduced by Martini et al. (2008b), which describes the motion of interfacial fluid atoms against the adjacent solid wall. Part of the first-fluid-layer atoms are trapped in the lower level of the surface potential. They can overcome energy barriers and hop between potential minima, as illustrated in Fig. 4. This transport behavior of fluid atoms against solid wall is analogous to the dislocation movement in crystalline solids. The constraint from the surface potential due to the wall contributes to the interfacial friction therefore affects slip. This is what we call the “surface potential mechanism” for the interfacial friction. When the hops are two-dimensionally isotropic with respect to the wall plane, collectively, there is no net relative motion between the interfacial fluid and the wall. The no-slip condition is recovered. Slip happens when a portion of hops becomes directional (along or against the shearing direction). The overall hops create collective diffusion, i.e. anisotropic diffusion of a group of atoms and cause net relative motion with respect to the wall. The magnitude of slip depends on the average number, frequency of the hops, and the average length covered by each hop in the shearing direction, which is dependent on the topology of the potential-energy corrugations.
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Fig. 4

Schematics of the first-fluid-layer atoms interacting with the solid wall: the hops of trapped atoms and the collisions between the impinging fluid atoms and wall atoms. Note that the presenting corrugated potential surface corresponds to the frozen wall. For the thermal wall, the undulation is irregular due to the thermal motion of wall atoms

Besides the surface potential mechanism stemming from constraining the hopping of the trapped atoms, the interfacial friction also manifests itself when other fluid atoms in the first layer interact with the wall through collision (Fig. 4) as described in the Maxwell interpretation of slip (Maxwell 1879), which we call it the “collision mechanism”. Generally, when a fluid atom collides with the wall, it can be reflected diffusely or specularly. The fluid atoms undergoing diffuse reflection with wall atoms reduce slip corresponding to larger friction, whereas the ones undergoing specular reflection enhance slip corresponding to smaller friction. Note that the collision we are discussing here represents more of an in-plane feature, where the velocity change happening in the direction normal to the plane (z direction) is not taken into account.

Extensive efforts have been made on the quantitative analysis of slip based on the surface potential mechanism (Barrat and Bocquet 1999; Priezjev 2007b; Thompson and Robbins 1990a). Utilizing the surface potential Eq. (9), the slip length is meticulously shown to be dependent on the contact density, ρc, temperature of fluid, Tc, as well as the structure factor of the first fluid layer, \(S\left(q_{\parallel}\right):\) (Barrat and Bocquet 1999; Sendner et al. 2009)
$$ L_{\rm s}\sim T_{\rm c}^2\left[S\left(q_{\parallel}\right)\rho_{\rm c}\right]^{-1}. $$
(10)

This correlation is originally proposed in the framework of linear response theory which is valid for the zero-shear limit. But it has been proven to be valid for shear rates up to \(\dot{\gamma}\sim 0.1\;\tau^{-1}\) where the slip length starts to exhibit shear rate dependence (Priezjev 2007b). An increase in shear rate represents a monotonic increase in the combined ratio. A breakdown of Eq. (10) is expected at higher shear rates where the nonlinear response dominates. Another important note to remember is that the aforementioned scaling relationship is only for rigid and perfect [001] fcc surface. A thermal wall also introduce atomistic displacements, which further induces additional roughness in the z direction and in-plane disorder into surface potential. The doubt in the validity of the scaling relation in the presence of the thermal roughness is also mentioned in Ref. (Priezjev 2007a). Therefore, Eq. (10) is only a crude approximation, in which the additional surface roughness contribution induced by thermal motion of wall atoms is missing and only valid for near zero-shear rates.

To address the limitations of the scaling relation, we first examine the slip length as a function of the combined ratio \(T_c^2\left[S\left(q_{\parallel}\right)\rho_{\rm c}\right]^{-1}\) for different wall affinity \(\varepsilon_{\rm wf}\) with fixed wall rigidity \(k_{\rm w}=1,600\ \varepsilon/\sigma^2\) in the low-shear-rate regime. Figure 5a shows the correlation between the slip length and the combined function of interfacial quantities remains good linearity near the zero-shear limit. The data within the linear segments for different \(\varepsilon_{\rm wf}\) can be collapsed onto a single master curve. The master curve shows that the different \(\varepsilon_{\rm wf}\) contribute the same amount in the scaling relation regardless of the shear rate that has been applied. The wall affinity is captured accurately in the scaling relation. This fact further indicates that the surface potential mechanism, which is represented well by the scaling relation, is the dominating mechanism at the near zero-shear limit. Nevertheless, as the shear rates increase away from the zero-shear limit the individual curve for each \(\varepsilon_{\rm wf}\) demonstrates consistent deviation away from the linear master curve. These deviations indicate the onset of the collision mechanism. More evidence can be shown in the following analysis.
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Fig. 5

Scaling relation of the slip length as a function of the combined ratio of the contact temperature squared to the product of the structure factor and the contact density in the low-shear-rate regime, for a different wall affinity but fixed rigidity \(k_{\rm w}=1,600\ \varepsilon/\sigma^2\) The lines are just a guide to the eye to exhibit the linearity and b the tabulated models of wall with various rigidity (\(\varepsilon/\sigma^2\)) but fixed affinity \(\varepsilon_{\rm wf}=1.0\ \varepsilon \)

We now examine the scaling relation for different wall rigidity kw with fixed affinity \(\varepsilon_{\rm wf}=1.0\ \varepsilon. \) The wall rigidity is expected to influence the slip length through these interfacial quantities in a complex and convoluted way (Asproulis and Drikakis 2011). As we mentioned earlier, the scaling relation does not account for the additional contribution due to thermal roughness. Figure 5b manifests two notable behaviors: in the near zero-shear limit different wall rigidity produces different slopes and y-interceptions through extrapolation in the linear segments; and for each individual curve the linear trend breaks down. The different slopes and interceptions among the linear segments for different wall rigidity near the zero-shear limit prevent them forming a master curve. It is due to the missing contribution from the thermal roughness in the correlation. Different thermal roughness induces varying amount of friction force, e.g. smaller wall rigidity leads to larger extra atomistic roughness and effectively to larger friction force and less slip, which then results in smaller slope.

As the shear rate increases, away from near zero-shear limit, the breakdown of the linear trend appearing in both Figs. 5a, b is attributed to the onset of the collision mechanism. As the shear rate increases, there is more significant relative motion between the fluid and the solid, which leads to the increasing contribution of the interfacial friction through collision. The percentage of fluid atoms participating in the collective diffusion increases. In extreme situations, the isotropic diffusion is completely replaced by the anisotropic collective diffusion. All fluid atoms consistently move parallel to the shearing direction with a uniform slip velocity. When this collective slip appears, the first-fluid-layer atoms do not intrude into and get trapped in the potential minima. Instead, they skim over the corrugated potential surface with large slip velocity, and the experienced effective roughness gradually disappears. Since the scaling relation only represents the surface potential mechanism, the onset of the collision mechanism is the cause for the deviations from the linear trend of the scaling relation as the shear rates start to increase away from zero-shear limit.

In the high-shear-rate regime, the interfacial friction from the wall potential vanishes and the only mechanism contributing to the interfacial friction is the collision between fluid and wall atoms, resulting in the drastic jump in the slip length during the transition to the high-shear-rate regime, as shown in the previous Section. The total slip is dependent on the probability of specular reflection. The stiffer the wall is, the smoother the surface effectively becomes, leading to a larger amount of fluid atoms being reflected specularly, and the friction is reduced. For the frozen wall where wall atoms are all fixed, the high surface density results in a “quasi-smooth” plane (Bernardi et al. 2010), on which most of fluid atoms experience specular reflection. This explains why the frozen wall yields approximately perfect slip in the high-shear-rate regime. Remember that we observe the increasing slip with the increasing shear rate in this regime. This means that the phenomenological friction coefficient at the fluid–wall interface β = η/Ls decreases as \(\dot{\gamma}_i\) increase (as shown in Fig. 6a), where η is the fluid viscosity. This trend can be presented with a better physical insight as the relationship between β and the slip velocity Vs. Figure 6b shows that the friction becomes weak as the slip velocity increases. This behavior can be explained such that large slip velocity leads to a small chance for the fluid atoms to be reflected diffusely, because larger slip velocity corresponds to smoother substrate which the first-fluid-layer atoms experience.
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Fig. 6

Phenomenological friction coefficient at fluid–wall interface as a function of the a imposed shear rate and b slip velocity for the tabulated wall rigidity (\(\varepsilon/\sigma^2\))

We end this section by summarizing the influences of wall rigidity and wall affinity on the transition from the low-shear-rate to high-shear-rate regime. Generally, lower rigidity and higher affinity delay the transition to the high-shear-rate regimes as represented by the shift of critical shear rate towards larger values in Fig. 7. Low rigidity and high affinity contribute to stronger interfacial friction from the wall potential as discussed in previous paragraphs and thus prevent the occurrence of the collective slip at relatively low shear rates. This dependence could be useful for future computational studies to estimate the limit of validity for the low-shear-rate regime.
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Fig. 7

Dependence of the critical shear rate as a function of the inverse of wall rigidity at \(\varepsilon_{\rm wf}=1.0\ \varepsilon\) and wall affinity at \(k_{\rm w}=1,600\ \varepsilon/\sigma^2 \)

3.3 Viscous heating effect

Our MD results have elucidated the two mechanisms of fluid–solid interfacial friction and how they are associated with the measured slip. But the analysis is conducted under the stringent or somewhat unphysical condition that the bulk temperature of the fluid is uniform and fixed with a thermostat. Although this condition successfully simplifies the system and leads to lucid results shown above, the important viscous heating effect is excluded. In reality, the only way to dissipate the heat in the fluid is through natural conduction between the fluid and the walls. The same thermostat strategy in MD simulations was proven to generate dramatic fluid temperature rise due to viscous heating (Pahlavan and Freund 2011; Khare et al. 1997; Liem et al. 1992) , even though such high shear rates and temperatures might not be realistic. The temperature has shown to be as a dominant physical parameter for slip (Pahlavan and Freund 2011), especially at high shear rates.

To elucidate this viscous heating effect, we design a comparison of slip behaviors among three thermally controlled systems: (1) the no fluid thermostat (NOF) system in which only walls are thermostated. The NOF system attempts to reproduce similar systems implemented in the previous studies with bulk thermal effects; (2) the DPD with varying temperature (DPDVT) system in which there is a DPD thermostat on the fluid set with varying thermostated temperature Tfs corresponding to the average temperature of the NOF system at the same prescribed imposed shear rate. The system is designed to artificially impose partial bulk thermal effects (i.e. the bulk temperature rise) on the original DPD system, the same as those in the NOF system; (3) the original DPD system with Tfs always equals \(1.1 \varepsilon/k_B. \) All the three systems have thermal walls with the same rigidity and the same assigned thermostat parameters. Figure 8a provides a lucid impression on the differences in the temperature profiles among the three systems. Because of drastic viscous heating, the NOF system yields a parabolic shape with much higher average temperature comparing to the original DPD system. The DPDVT system reproduces the same average temperature as the NOF system, but with entirely different temperature profile. The DPDVT system has much lower center temperature and higher contact temperature than the NOF system, because the thermostat on the fluid attempts to maintain the isothermal condition. From the temperature profiles, we stress that the unphysical DPDVT system is merely constructed to provide an intermediate thermodynamic state for analyzing the effect of thermostat.
https://static-content.springer.com/image/art%3A10.1007%2Fs10404-012-1048-x/MediaObjects/10404_2012_1048_Fig8_HTML.gif
Fig. 8

a Comparison of the fluid temperature profiles at \(\dot{\gamma}_i=0.41\ \tau^{-1}\) among three thermally different systems. Data points are shown as blue circles, green squares, and red prisms for the NOF, DPDVT and DPD systems, respectively. b Apparent slip length as a function of the imposed shear rate of the three systems

The different temperature profiles lead to corresponding results of the slip length, which is exhibited in Fig. 8b. The NOF system produces bounded slip that the slip length approaches a constant as the shear rate increases, which is the same as the previous studies (Martini et al. 2008a; Khare et al. 1997). We observe a slight decreasing in slip length in a small range of low-shear-rates, which is consistent with Ref. (Pahlavan and Freund 2011). The slip length then increases with increasing shear rate. Because of the high bulk temperature imposed artificially on the DPDVT system, the slip length of the DPDVT system has a trend very similar to the NOF system, reproducing bounded slip. There is no sign of transition to the high-shear-rate regime in the DPDVT system within the range of shear rate simulated. The high bulk temperature is associated with high pressure in this fix volume simulation, which yields high contact density and the shift of density peak towards the walls similar to the previous study (Pahlavan and Freund 2011). The high contact density indicates that the high temperature eventually forces the fluid atoms to remain deeply in the corrugate potential minima even at high shear rates, which prevents the transition and results in bounded slip. The difference in the slip length between the NOF and DPDVT systems is likely caused by the difference in their temperature profiles. Conversely, the DPD system shows the transition and unbounded slip is observed. The comparison proves that the bulk fluid temperature, both the average temperature and the distribution play important roles in the slip behavior. We confirm that the high bulk fluid temperature is the core reason of the bounded slip at high shear rates and the diverging slip is attributed to the unphysical fluid thermostat.

4 Conclusions

Our results in this work provide a deeper understanding of the atomistic interfacial friction and its relation to slip in nanoscale Couette flow. Our MD simulations successfully verify the prediction from previous studies that the discontinuity in the slip length is associated with the transition from the defect slip to the collective slip. The thermal walls with increasing rigidity exhibit slip behaviors asymptotically approaching the one of a frozen wall that produces unbounded slip. The wall affinity also has no substantial influence on the main feature of the slip length versus shear rates curve. The two mechanisms of interfacial friction, solid surface potential mechanism and fluid–solid collision mechanism, are thoroughly investigated. We confirm that both mechanisms contribute to the low-shear-rate regime with the defect slip. The contribution from the collision mechanism is reflected in the discrepancy of the microscopic scaling estimation which only accounts for the surface potential contribution. The collision dominates the interfacial friction for the collective slip in the high-shear-rate regime. The quantitative analysis of the scaling relation for varying wall affinity and rigidity indicates that the affinity primarily influences the surface potential contribution while the rigidity governs both the surface potential and the collision contribution.

Finally, the comparison among three thermally controlled systems verifies that the bulk viscous heating is ultimately responsible for the bounded slip at high shear rates. We hope this study of interfacial friction mechanism will not only benefit future MD simulations of nanoscale slip, but also cast light on the optimal designs and practical applications of nanofluidics.

Acknowledgments

This work was partially supported by NRC (NRC-38-09-954) and NSF (CMMI-0928448) and utilized the Rensselaer Polytechnic Institute Computational Center for Nanotechnology Innovations Blue Gene/L. We gratefully acknowledge the discussion with Dr. Mark O. Robbins.

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© Springer-Verlag 2012