Research Article

Austin Chem Eng. 2016; 3(2): 1031.

# On the Comparability of Chemical Structure and Roughness of Nanochannels in Altering Fluid Slippage

Misra CA and Bakli C*

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India

***Corresponding author: **Chirodeep Bakli, Department
of Mechanical Engineering, Indian Institute of
Technology, Kharagpur, India

**Received: **April 08, 2016; **Accepted: **May 10, 2016; **Published: **May 13, 2016

## Abstract

Interfacial hydrodynamic slippage of water depends on both on surface
chemistry and roughness. This study tries to connect the effect of chemical
property and the physical structure of the surface on the interfacial slippage of
water. By performing Molecular Dynamics Simulations (MDS) of Couette flow
of water molecules over a reduced Lennard-Jones (LJ) surface, the velocity
profile is obtained and extrapolated to get the slip lengths. The slip lengths are
measured for various surface-fluid interactions. These interactions are varied by
changing the wettability of the surface (characterized by the static contact angle)
and its roughness. The slip length variation with the static contact angle as is
observed. However, it is also observed that the presence of surface roughness
always reduces the slip length and it is proposed that the slip length varies with
non-dimensionalized average surface roughness as (1+a*)^{-2}. Thus a relation
between the chemical wettability and the physical roughness is established and
their coupled interactions modifying slip length is probed.

**Keywords:** Wettability; Slip length; Roughness

## Abbreviations

NEMS: Nano Electro Mechanical Systems; LJ: Lennard-Jones; MDS: Molecular Dynamics Simulations; fcc: Face Centered Cubic; SPC/E: Simple Point Charge/ Extended

## Introduction

The flow physics of fluid transport through nanochannels is extremely interesting and intriguing. The surface forces overweigh the volumetric forces at such small length scales, leading to an important role played by the interfacial effects. Thus modifying the surface characteristics of substrates of such channels over nanometer dimensions provides the option to engineer such flows. Study of these flow variations over sub-micron scale is warranted by the advent of research in the areas of Nano Electro Mechanical Systems (NEMS) and Nanoporous Energy Absorption System (NEAS) and also by the urge to study flow of natural fluids in biological membranes. Through extensive molecular dynamics simulations, researchers have explored various aspects of the transport processes and surface interactions concerned with micro and nano scale flows. They have studied the effects of the interface wettability on flow structure, fluid dynamics in surface-nanostructured channels [1], effects of wall lattice-fluid interactions on the density and velocity profiles [2], perturbations in fully developed pressure driven flows [3], slip behavior on substrates with patterned wettability [4], and effects of surface roughness and interface wettability on nanoscale flows [5].

Understanding fluid flow on nano-scale is crucial for designing
microfluidic devices, modern developments of nanotechnology like
the lab-on-a-chip [6,7], as well as for various applications of porous
materials, fluid flow through pores in bio-membranes etc., [8,9]. These
works have shown that contrary to the macroscopic hydrodynamic
theory, the no slip boundary condition might not necessarily hold true
for nano-scale channels. In such cases, the fluid velocity at the surface is more aptly described by a partial slip boundary condition that relates
the fluid velocity at the surface to its gradient *∂v/ ∂z* in the direction
normal to the surface by *v=b(∂v/ ∂z*), where b is the slip length [10-
12]. This interfacial slip has immense practical applications in micro
fluidics, bio-fluid dynamics, and lubrication etc. by virtue of the fact
that it reduces viscous friction at the surfaces and amplifies flow rates
in pressure driven flows and electro-osmotic flows [13,14]. This also
provides potential to generate power from nano-scale devices [15,16].
Thus it becomes imperative to estimate or measure slip length and
study its dependence on interfacial parameters. However, there is a
lot of disparity between the experimental data reported for the slip
length of typical hydrophobic surfaces. The experimental values of
slip lengths have been measured ranging from nanometers [17,18]
to micrometers [19]. Researchers have explained larger values of slip
length to be due to nano-bubble formation at the interface [20], and
molecular dynamics simulations of a model LJ system have shown an
increase in slip due to this phenomenon. In an attempt to resolve the
controversy in the experimental literature, Huang et al have reported
a quasi universal dependence of the slip length on the contact angle
*θc* by performing MDS [21]. Although Huang et al have considered
rough surfaces in their study they have not analyzed the dependence
of slip length on surface roughness. In this work, it is aimed to
establish a mathematical relationship between slip length and average
surface roughness by performing non-equilibrium MDS of Couette
flow of SPC/E rigid simple point charge model of water between
two parallel plates made of reduced LJ atoms. By applying a shear
boundary condition, the velocity profile is obtained and extrapolated
to get the slip length. In order to mimic the atomically rough surface,
the attractive part of LJ interaction potential between fluid and
surface [22] is modified. To estimate the contact angle, we perform
an equilibrium MDS of a droplet of water on the same surface as used
in Couette flow simulation. As a result of this work, it is observed
that the slip length varies with non-dimensionalized average surface roughness as *b* = (1+a*) ^{-2}.* On the basis of this work, a correction in
the formula presented by Huang et al [21] to account for variation in
surface roughness is proposed.

## Problem Formulation

The simulation system, for Couette flow simulation, is similar to
that used by Huang et al [21] and is shown in Figure 1. A channel of
height 5nm, bounded by walls made of four layers of reduced LJ atoms
arranged in close packed fcc (face-centered cubic) lattice oriented
such that (100) face is in contact with water is considered. The lateral
dimensions of the walls are *5nm x 5nm*. Wall atoms interact with each
other through a LJ potential. The LJ interaction parameter between
wall atoms σ_{ww} is taken to be 0.35*nm* while ε_{ww} is varied to change
the wall fluid interaction and hence the contact angle. The channel
is filled with 3584 water molecules SPC/E model (Simple Point
Charge/ Extended). This number ensures that the density of water is
1000kg/*m³* for the particular size of the channel considered. Hetero
nuclear LJ interactions are determined by standard Lorentz-Berthelot
combining rules [23]. Roughness is introduced by modifying the
attractive part of the LJ potential as shown in Eq.(1).

${U}_{LJ}=4{\u03f5}_{wl}\left[{\left(\frac{{\sigma}_{wl}}{r}\right)}^{12}-{\left(\frac{{\sigma}_{wl}}{r}\right)}^{6}f(x,y)\right]\text{(1)}$

**Figure 1:**Three dimensional view of the simulation system for (a) Couette Flow (N=2, ${\epsilon}_{ww}=0.4$ ) and (b) Contact angle estimation (Smooth, ${\epsilon}_{ww}=0.4$ ).

Figure 1:Three dimensional view of the simulation system for (a) Couette Flow (N=2, ${\epsilon}_{ww}=0.4$ ) and (b) Contact angle estimation (Smooth, ${\epsilon}_{ww}=0.4$ ).

Where σ_{wi} and ε_{wi} are the LJ potential parameters between wall
and liquid, r is the interparticle distance and f (x,y) is given by Eq.(2).

$f(x,y)=1+\frac{1}{3N}{\displaystyle \sum _{K=1}^{3}\frac{1}{K}}\left[cos\left(\frac{4\pi Kx}{b}\right)+cos\left(\frac{4\pi K(x+y\sqrt{3})}{2b}\right)+cos\left(\frac{4\pi K(-x+y\sqrt{3})}{2b}\right)\right]\text{(2)}$

The parameter N controls the amplitude of the surface roughness. Smaller the value of N rougher is the surface. The parameter K represents the wave number of the surface characteristics whereas parameter b controls the periodicity of the function (Figure 2).

**Figure 1:**Simulated surface roughness ($N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4;\text{\hspace{0.17em}}d=10$): The grid represents the geometrical location of wall atoms; Surface represents the corrugations introduced by modification of LJ potential.

Figure 2:Simulated surface roughness ($N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4;\text{\hspace{0.17em}}d=10$): The grid represents the geometrical location of wall atoms; Surface represents the corrugations introduced by modification of LJ potential.

Unlike the previous studies, choice of modified LJ potentials in
this way not only allows a simultaneous control over the topographical
and wettability characteristics of the confining boundaries through
the pertinent interaction forces, but also implicates an explicit
coupling between these two immensely consequential interfacial
features. In these simulations, N is varied to get various degrees of
surface roughness and fix *d* such that an integral number of periods of
roughness function are present in a simulation domain. This ensures
continuity of roughness function across several simulation domains.

The simulation system used for contact angle estimation is shown in Figure 1b. In order to determine the contact angle, the surface of the droplet by joining all those points at which the density is half of the bulk density is defined and a circle is fitted through these points as shown in Figure 3a, 3b. The angle between the surface and the tangent to the circle, at the intersection point of circle and surface, is the contact angle.

**Figure 3:**Droplet surface as obtained by (a) Simulation ( $N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4$ ) and (b) Calculated iso-density curve and fitted circle ( $N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4$

Figure 3:Droplet surface as obtained by (a) Simulation ( $N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4$ ) and (b) Calculated iso-density curve and fitted circle ( $N=2;\text{\hspace{0.17em}}{\epsilon}_{ww}=0.4$ ).

The simulations are performed using GROMACS MD package [24]. Production run for couette flow consisted of 2,000,000 steps with a time step of 0.001ps. Leap-frog algorithm was used for integration. Bond length and angle constraints were enforced with the SHAKE algorithm and a constant temperature of was maintained with a Nose-Hoover thermostat. Periodic boundary conditions were used in XYZ directions.

## Results

Figure 4a shows the variation of slip length as a function of static contact angle. It is clear from the graph that the results of this work match closely with those obtained by Huang et al [21], confirming their proposition that slip length varies with the static contact angle as shown in Eq. (3)

$b\propto {(1+cos\theta )}^{-2}\text{(3)}$

**Figure 4:**Slip length as a function of static contact angle for (a) Smooth LJ Surface and (b) Rough LJ Surfaces.

Figure 4:Slip length as a function of static contact angle for (a) Smooth LJ Surface and (b) Rough LJ Surfaces.

Where is the slip length and is the contact angle. Figure 4b shows
the variation of slip length as a function of static contact angle for
various degrees of surface roughness as obtained in the simulations. It
is evident from the figure that for each value of roughness amplitude
N, slip length shows the same variation with contact angle as suggested
by Eq.(3). It is true that all the points, even for different surface rough
nesses, do seem to lie on a universal curve, however, a general trend
that slip length decreases with increasing surface roughness can clearly be observed, which warrants further investigation. For this
purpose, the slip length b is non-dimensionalized by the smooth
surface slip length b_{smooth}, both obtained for the same value of wallfluid
interaction Σ_{ww}. The value of roughness function f (x,y) not only
represents the modification factor in LJ potential, but physically it is
equivalent to the height (depth) of crests (troughs) on the surface.
Mean value of $f\left(x,y\right)$ $\left(\overline{f(x,y}\right)$
represents the height of the effective
mean surface over the actual geometric surface. Therefore, one can
define non-dimensional average surface roughness parameter a* as
shown in Eq.(4). a* characterizes the roughness of a surface and varies
only with N.

${\alpha}^{*}=\frac{\alpha}{{\sigma}_{wl}}=\frac{\sum \left|f(x,y)-\overline{f(x,y)}\right|}{N}\text{(4)}$

Non-dimensionalization of the slip length is done by dividing it
by the hydraulic diameter of the channel. Fig. 5(a) shows this nondimensionalized
slip length as a function of non-dimensionalized
average surface roughness for different values of wall-fluid
interaction. Decreasing trend of slip length with increasing surface
roughness can be clearly seen from the graph. Further, almost parallel
curves indicate that qualitative dependence of slip length on nondimensional
average surface roughness is independent of wall-fluid
interaction parameter Σ_{ww}.

Another length scale that can be used to non-dimensionalize the
slip lengths obtained by varying the roughness for a particular value
of Σ_{ww} is the slip length obtained between fluid and smooth surface
for that Σ_{ww}. Figure 5b shows this non-dimensionalized slip length
as a function of non-dimensionalized average surface roughness for
different values of wall-fluid interaction (Σ_{ww}). Figure 5a shows that
irrespective of the value of Σ_{ww}, the non-dimensional slip length scales
with non-dimensional average surface roughness as given by the
Eq.(5). Deviation of curve for Σ_{ww} = 1 from Eq.(5), may be attributed to the hydrophilic nature of the wall seen at this value.

${b}^{*}=\frac{b}{{b}_{smooth}}=\frac{1}{{(1+{\alpha}^{*})}^{2}}\text{(5)}$

**Figure 5:**Non-dimensional slip length as a function of (a) Non-dimensional average surface roughness and (b) Contact angle.

Figure 5:Non-dimensional slip length as a function of (a) Non-dimensional average surface roughness and (b) Contact angle.

Huang et al [21], present a mathematical analysis through which they justify their universal scaling formula given in Eq.(3). Through this work, a correction to this scaling formula is proposed in order to calculate slip lengths for rough surfaces with more accuracy. The corrected formula is as shown in Eq.(6). Here k is the constant of proportionality which is same for both Eqns. (3) and (6).

$b(\theta ,{\alpha}^{*})=\frac{k}{{(1+cos\theta )}^{2}\times {(1+{\alpha}^{*})}^{2}}\text{(6)}$

One can verify the correctness of the formula presented in Eq.(5) through a simple mathematical analysis as shown below.

${c}_{wl}^{6}=4{\u03f5}_{wl}{\sigma}_{wl}^{6}\Rightarrow {c}_{wl}^{6}\propto {\u03f5}_{wl}\text{(7)}$

${c}_{wl}^{6}$ being the attractive LJ interaction term and from Eq.(1),

${c}_{wl,rough}^{6}=4{\u03f5}_{wl}{\sigma}_{wl}^{6}\times f(x,y)={c}_{wl,smooth}^{6}\times f(x,y)\text{(8)}$

$b\propto \frac{1}{{\u03f5}_{wl}^{2}}\text{(9)}$

Therefore, combining Eqns. (7), (8) and (9)

$\frac{{b}_{smooth}}{{b}_{rough}}={\left(\frac{{\u03f5}_{wl,rough}}{{\u03f5}_{wl,smooth}}\right)}^{2}={\left(\frac{{c}_{wl,rough}^{6}}{{c}_{wl,smooth}^{6}}\right)}^{2}=f{(x,y)}^{2}\text{(10)}$

Since f (x,y) varies over the surface, its average value is taken to represent the surface roughness. This average value, in nondimensional form, is nothing but a* as defined in Eq.(4) . For a smooth surface f (x,y) = 1 for all values of x and y, therefore a* for a smooth surface is equal to 0. Since this value is calculated with respect to the mean value of f (x,y), which is equal to 1 for all values of N (due to the symmetrical nature of f (x,y) about the x-y plane), therefore 1 is added to a* in order to obtain a true measure of f (x,y). By substituting this measure of in Eq.(10), the formula shown in Eq.(5) is obtained.

## Conclusion

Through molecular dynamics simulations of a realistic water model, the slippage of water over smooth and rough surfaces is studied. By measuring slip lengths and contact angles under various conditions and studying their variations, a mathematical relationship between slip length, static contact angle, and average surface roughness is proposed. Finally, through a simple mathematical analysis, the theoretical basis of this two-way coupling of the chemical structure and physical roughness and the effect of the same on interfacial slip is established.

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