Microfluidics and Nanofluidics

, Volume 17, Issue 6, pp 1011–1023

A quasi-continuum multi-scale theory for self-diffusion and fluid ordering in nanochannel flows


  • Antonios E. Giannakopoulos
    • Department of Civil Engineering, School of EngineeringUniversity of Thessaly
  • Filippos Sofos
    • Department of Civil Engineering, School of EngineeringUniversity of Thessaly
    • Department of Civil Engineering, School of EngineeringUniversity of Thessaly
  • Antonios Liakopoulos
    • Department of Civil Engineering, School of EngineeringUniversity of Thessaly
Research Paper

DOI: 10.1007/s10404-014-1390-2

Cite this article as:
Giannakopoulos, A.E., Sofos, F., Karakasidis, T.E. et al. Microfluid Nanofluid (2014) 17: 1011. doi:10.1007/s10404-014-1390-2


We present a quasi-continuum self-diffusion theory that can capture the ordering effects and the density variations that are predicted by non-equilibrium molecular dynamics (NEMD) in nanochannel flows. A number of properties that affect fluid ordering in NEMD simulations are extracted and compared with the quasi-continuum predictions. The proposed diffusion equation requires the classic diffusion coefficient D and a micro structural internal length g that relates directly to the shape of the molecular potential of the NEMD calculations. The quasi-continuum self-diffusion theory comes as an alternative to atomistic simulation, bridging the gap between continuum and atomistic behavior with classical hydrodynamic relations and reduces the computational burden as compared with fully atomistic simulations.


Quasi-continuum theoryMolecular dynamicsSelf-diffusion equationNanochannel flowsFluid orderingDensity profile oscillations

List of symbols


Constants determined by BC

A, B

Constants for inhomogeneous diffusion

Ai, Bi

Airy functions




Real constants


Bulk diffusion coefficient


Apparent diffusion coefficient


Diffusion functional


Apparent diffusion functional


Hypergeometric function


Magnitude of external driving force




Gibbs free energy


Gibbs free energy at equilibrium


Boundary value for concentration


Hermitian polyomial


Channel height


Diffusional flux


Gradient energy coefficient


Spring constant


Length of the computational domain in the x-direction


Length of the computational domain in the y-direction


Length of the computational domain in the z-direction


Diffusional mobility


Integer number, n = 0, 1, 2…




Boundary value for the non-classic flux term


Position of a wall atom on fcc lattice site


Position vector of atom i


Distance vector between ith and jth atom






LJ potential of atom i with atom j




Boundary condition for the normal flux


Normalized distances in the z-direction

Greek letters


Energy parameter in the LJ potential


Local chemical potential


Fluid density


Length parameter in the LJ potential

1 Introduction

Nanofluidics has emerged as an important subfield of nanotechnology with important applications in biomedical science where minute amount of fluids move through microtubules. Drug delivery investigations stand to profit enormously from studies of nanochannel flows. For such flows, non-equilibrium molecular dynamics (NEMD) offers an effective simulation method, as has been recently undertaken by many researchers (Binder et al. 2004; Hartkamp et al. 2012; Sofos et al. 2009a, b; Li and Liu 2012). Within this framework, simulation of planar Poiseuille flow in a channel offers the best way of establishing the range of applicability of continuum-based theories. Density profiles reveal strong oscillations in the number of fluid atoms at layers adjacent to the walls. This inhomogeneity in the fluid concentration in the channel cannot be explained by the classical diffusion theory. Nanofluidics is not the only field where ordering of the mater takes place. Magnetization, martensitic transformation and dislocations also show ordering of mater (Stanley 1971).

Somers and Davis (1991) presented an early full analysis of density profiles for channel widths from 0.6 to 2.35 nm. They observed that fluid atoms are ordered in distinct layers, symmetrical with respect to the midplane of the channel. These layers are not affected by the magnitude of the external driving force (Nagayama and Cheng 2004). The fluid density peak values decreases as temperature increases. The stiffness of the walls affects the fluid ordering for channels of very small widths (Priezjev 2007). Average fluid density also affects fluid ordering in a significant way. Iwai et al. (1996) performed Monte Carlo calculations and showed variation in the radial distribution function of supercritical carbon dioxide around OH groups of xylenol isomers at 308.15 K and 13 MPa.

In the present paper, we will develop a new quasi-continuum self-diffusion theory that can capture the ordering effects and the density variations that are predicted by non-equilibrium molecular dynamics (NEMD) in nanochannel flows from Sofos et al. (2009a). A new diffusion equation is proposed, together with new boundary conditions. The solution of the new diffusion equation provides a better approximation to the density variations. The flux constitutive equation requires the classic self-diffusion coefficient, D, and, in addition, a micro structural internal length, g, that relates directly to the shape of the molecular potential that is used in the NEMD calculations. There are several benefits for choosing a quasi-continuum self-diffusion theory instead of NEMD. Aside from the smaller computational cost, the quasi-continuum theory can easily describe complex two and three dimensional domains, include various boundary conditions and incorporate convective terms, if combined with other classic hydrodynamic numerical schemes.

1.1 A quasi-continuum theory of diffusion in fluids

Diffusional flux J is spontaneous and leads to the decrease in the Gibbs free energy G of a non-reacting fluid. Note that we use G in case of constant temperature, as in the examples that we will study. Our analysis has some similarities with the spinodal decomposition of a binary solid system (Cahn and Hilliard 1958); however, the present theory is not the same.

The basic constitutive equation concerns the driving force for diffusion and is taken as the gradient of the local chemical potential μ (Darken 1948):
$$\underline{J} = - M\underline{\nabla } \mu$$
where M is the diffusional mobility and is positive. The above relation implies isotropy (true for most fluids) and reminds Ohm’s law of electric conductivity. Eq. (1) implies that we will ignore the effect of the Brownian motion. Cook (1970) included the influence of the thermal fluctuations, adding the gradient of a fluctuating field.
In the absence of an external field (e.g., gravity), the local chemical potential is related to the free energy density as1
$$\mu = \frac{{{\partial }G}}{\partial c}$$
at constant pressure (p = const) and temperature (T = const). G is the free energy per unit molecule volume, and we can express it as (Cahn and Hilliard 1959)
$$G = G_{o} (c) + K(\underline{\nabla } c)^{2}$$
where c is the concentration (c = c(x, t) is a function of position x and time t,[c] = [molecules/m3]) and K is the gradient energy coefficient (K > 0). The energy term K(c)2 is a measure of the increase in energy due to the non-uniform environment of atoms in a concentration gradient and implies a surface tension due to the gradient of the concentration. The constant K is approximately independent of composition and temperature. K depends on the number of molecules per unit volume, the critical temperature and the square of the intermolecular distance. Note that Eq. (3) implies that G changes from point to point in the control volume of a fluid in thermodynamic equilibrium. In the equations, 2 is the Laplacian operator (2 = ∂2/∂x12 + ∂2/∂x22 + ∂2/∂x32, for Cartesian coordinates) and is the gradient vector ( = {∂/∂x1, ∂/∂x2, ∂/∂x3}, for Cartesian coordinates).
Combining Eqs. (2) and (3), we obtain
$$\mu = \frac{{\partial G_{o} }}{\partial c} + 2K\nabla^{2} c$$
Then, the Fick’s 2nd law can be found from Eqs. (1) and (4)
$$\underline{J} = - M\left\{ {\frac{{\partial^{2} G_{o} }}{{\partial c^{2} }}\underline{\nabla } c + 2K\underline{\nabla } \left( {\nabla^{2} c} \right)} \right\}$$
Define as the bulk diffusion D the positive quantity
$$D = M\frac{{\partial^{2} G_{o} }}{{\partial c^{2} }}$$
Note that, if D is a fluid constant parameter, Eq. (6) implies that \(\frac{{\partial^{2} G_{o} }}{{\partial c^{2} }}\) is a positive constant. Therefore,
$$\underline{J} = - D\left\{ {\underline{\nabla } c + \frac{2MK}{D}\underline{\nabla } (\nabla^{2} c)} \right\}$$
is the new 1st Fick’s law. In the case where ∇2c ≈ const, or for K ≈ 0, we obtain the classic Fick’s law
$$\underline{J}_{\text{classic}} = - D\underline{\nabla } c$$
Taking the conservation of mass, in the absence of convective (flow) and source (reactions, radiation, etc.) terms:
$$\frac{\partial c}{\partial t} = - \underline{\nabla } \underline{J}$$
Combining Eqs. (7) and (9), we obtain the augmented 2nd Fick’s law
$$\frac{\partial c}{\partial t} = D\nabla^{2} c + 2MK\nabla^{4} c$$
Note that the constant
$$g^{2} = \frac{2KM}{D}$$
has dimensions [g2] = [L2]. This will provide a dominant wave length of order g and a wave number 2π/g. Therefore, for t ≥ 0,
$$\frac{\partial c}{\partial t} = D\left\{ {\nabla^{2} c + g^{2} \nabla^{4} c} \right\}$$
We again observe that in case ∇2c ≈ const or K ≈ 0, we obtain the classic 2nd Fick’s law
$$\left. {\frac{\partial c}{\partial t}} \right|_{\text{classic}} = D\nabla^{2} c$$

Equation 12 describes a quasi-continuum theory of self-diffusion and can be solved provided we describe appropriate initial and boundary condition.

The Cahn–Hilliard spinodal decomposition diffusion law (Cahn 1961) uses the integral form of Eq. (3) as a total free energy functional, leading to a negative diffusion coefficient D and a similar form with Eq. (12).

1.2 Initial and boundary conditions

In order to solve Eq. (12) with respect to the concentration c as a function of the coordinates z and the time t, we need to establish initial conditions at a certain time t = t*
$$c_{ * } (\underline{z} ) = c(\underline{z} ,t_{ * } )$$
To establish boundary conditions, we formulate the steady state version of Eq. (12)
$$D\left( {\nabla^{2} c - g^{2} \nabla^{4} c} \right) = 0$$
We construct a functional F of the form
$$F(\nabla^{2} c) = \frac{D}{2}\iiint\limits_{\text{V}} {\left( {\nabla^{2} c} \right)(}\nabla^{2} c + g^{2} \nabla^{4} c){\text{d}}V$$
With the volume integral covering the volume V of the fluid that diffusion takes place. Using Green’s theorem (Courant and Hilbert 1953)
$$\frac{2}{D}F(\nabla^{2} c) = \iiint_{V} {\left\{ {(\nabla^{2} c)^{2} - g^{2} \underline{\nabla } (\nabla^{2} c) \cdot \underline{\nabla } (\nabla^{2} c)} \right\}}{\text{d}}V + g^{2} \iint_{\partial v} {\left( {\nabla^{2} c} \right)\left( {\underline{\nabla } (\nabla^{2} c) \cdot \underline{n} \, } \right)}{\text{d}}S$$
where n is the outer unit normal vector to the boundary surface \(\partial V\) of the volume V. We can then immediately observe the boundary condition (BC):
$$g^{2} \underline{\nabla } (\nabla^{2} c) \cdot \underline{n} = q\quad {\text{on}}\quad\partial V$$
The functions F and q will be considered as known functions of \(\underline{z} \in \partial {\text{V}}\) (surface coordinates) and of t ≥ to (time).
From the BC (18) and the variation of F(∇2c) with respect to ∇2c, we obtain the Euler–Lagrange equation which is exactly as the steady state Eq. (15). In order to have minimum for F(∇2c), we must have the additional Legendre condition
$$(\nabla^{2} c)^{2} - g^{2} \underline{\nabla } (\nabla^{2} c) \cdot \underline{\nabla } (\nabla^{2} c) \ge 0$$
In addition to the non-classic BC (18), we have the classic BC:
$$c = h\quad {\text{or}}\quad \, \underline{\nabla } c \cdot n = f\quad{\text{ on }}\quad\partial V$$
From Eq. (20), we also combine the non-classic BC in the form that is physically meaningful
$$\underline{n} \cdot \underline{J} (\underline{z} ,t) = - D\left\{ {\underline{\nabla } c + g^{2} \underline{\nabla } (\nabla^{2} c)} \right\} \cdot \underline{n} = - D({\text{f}} + {\text{q)}} = w\quad{\text{ on }}\quad\partial {\text{V}}$$
where h and w are known functions of \(\underline{z} \in \partial {\text{V}}\) (surface points) and of time t ≥ to.

Note that the BC (21) reflects directly to the chemical potential at the boundary (Eq. (1)), which also relates to the roughness of the boundary walls (Priezjev 2007) and to the hydrophilic/hydrophobic wall–fluid interaction (Voronov et al. 2006). Wetting condition could correspond to c·n = 0, so q = 0 in Eq. (18). Local conservation of mass supposes that w = 0 and combined with q = 0 gives f = 0.

Suppose that an apparent diffusion coefficient Dap can be related to the classic diffusion equation as
$$\frac{\partial c}{\partial t} = D_{\text{ap}} \nabla^{2} c$$
and the corresponding steady state equation
$$D_{\text{ap}} \nabla^{2} c = 0$$
We can construct a functional Fap
$$F_{\text{ap}} (\nabla^{2} c) = \frac{{D{}_{\text{ap}}}}{2}\iiint\limits_{\text{V}} {(\nabla^{2} c)^{2} {\text{d}}V}$$

The minimization of Fap leads to Eq. (23) (Crank 1975). In this case, we can use the BC from Eq. (20).

Using the enclosure theorem (Courant and Hilbert 1953) and the condition of Eq. (19), we have
$$\hbox{min} F(\nabla^{2} c) \ge \hbox{min} F_{\text{ap}} (\nabla^{2} c)$$
Using the minimization function ∇2c for both F(∇2c) and Fap(∇2c), we obtain
$$\frac{{D_{\text{ap}} }}{D} \le 1 - g^{2} \frac{{\iint\limits_{V} {\underline{\nabla } (\nabla^{2} c) \cdot \underline{\nabla } (\nabla^{2} c){\text{d}}V}}}{{\iint\limits_{V} {(\nabla^{2} c)^{2} {\text{d}}V}}} \le 1$$
This result states that the apparent diffusion constant Dap is less than the “true” (intrinsic) bulk diffusion constant D.

Note that F(∇2c) can be thought of as an evolutionary criterion toward the steady state, since the transient solution ∇2c (without convection) is an admissible function for the minimum of F(∇2c) (Prigogine and Glansdorff 1965).

Let us formulate a restricted variational principle: Define the local potential
$$F(\nabla^{2} c,\nabla^{2} c_{o} ) = \int\limits_{0}^{t} {} \iiint\limits_{\text{V}} {\left\{ {\frac{D}{2}\left[ {(\nabla^{2} c)^{2} - g^{2} \underline{\nabla } (\nabla^{2} c) \cdot \underline{\nabla } (\nabla^{2} c)} \right]} \right\} - \nabla^{2} c\left. {\frac{{\partial c_{o} }}{\partial t}} \right\}{\text{d}}V{\text{ d}}t}$$
with known additional initial condition ∇2c(z, t = 0) = 0. We seek to make the local potential (Eq. 27) stationary with respect to variations in ∇2c, holding co constant. After the variation, we set c = co (c is the true solution) and we obtain
$$D(\nabla^{2} c + g^{2} \nabla^{4} c) - \frac{\partial c}{\partial t} = 0$$
$$g^{2} \underline{\nabla } (\nabla^{2} c) \cdot n = 0$$

Clearly, Eq. (28) is the dynamic form of Fick’s 2nd law and Eq. (28) provides the new boundary condition for the problem.

1.3 The 1D steady state example

It is instructive to solve the 1D steady state problem:
$$\frac{{{\text{d}}^{2} c}}{{{\text{d}}z^{2} }} + g^{2} \frac{{{\text{d}}^{4} c}}{{{\text{d}}z^{4} }} = 0$$
The solution of Eq. (30) is detailed in Appendix 1. We have
$$\frac{c}{{c_{o} }} = \frac{{\frac{{\bar{c}}}{{c_{o} }} - \frac{2g}{{h_{\text{ch}} }}\sin \frac{{h_{\text{ch}} }}{2g}}}{{1 - \frac{2g}{{h_{\text{ch}} }}\sin \frac{{h_{\text{ch}} }}{2g}}} + \frac{{1 - \frac{{\bar{c}}}{{c_{o} }}}}{{1 - \frac{2g}{{h{}_{\text{ch}}}}\sin \frac{{h_{\text{ch}} }}{2g}}}\cos \left( {\frac{{\frac{{h_{\text{ch}} }}{2} - z}}{g}} \right)$$
Note that for hch ≥ 2 g, c(z) is positive for all z, provided that \(\bar{c}/c_{o} \ge 1\).

If \(\frac{g}{h} \to 0\), \({\rm A}_{1} \to \bar{c}\). For c ≥ 0, for all z, \(\frac{c}{{c_{o} }} \to \frac{{\bar{c}}}{{c_{o} }} \to 1\) and this is the classic solution \((c = \bar{c} = c_{o} )\). In other words, as the channel becomes large compared to g, the classic solution is recovered. The result of Eq. (31) was found to be in agreement with the molecular dynamics results of Sofos et al. (2009a) with a dominant wavelength g ≈ 0.9σ, where σ is the basic length that controls the Lennard-Jones potential (σ = 0.3405 nm for the Ar liquid used in our molecular dynamics computations).

2 The case of inhomogeneous diffusion mobility

In Sect. 2, we treated the diffusion D as a positive constant. However, this may not be true in cases of confinement of the fluid in thin channels or other devices. The solid walls usually decrease the diffusion close to the walls, but can under certain circumstances increase it:
$$D = D({\text{z}}) > 0\quad \underline{z} \in {\text{V}}$$
$${ \hbox{max} }D({\text{z}}){\text{ or min}}D({\text{z}})\quad \underline{z} \in \partial {\text{V}}$$
In this case, Eq. (11) describes a length that depends on z
$$g = g(\underline{z} )$$
To fix ideas [as indicated by the molecular dynamics calculation in Sofos et al. (2009b)], let us take the steady state case in one dimension with the symmetric inhomogeneous diffusion distribution (−hch/2 ≤ z ≤ hch/2)
$$D = D_{0} \left( {1 - A^{2} z^{2} } \right)$$
where D0 the bulk diffusion and A constant with 1 ≥ A2hch2/4. Molecular dynamics results confirm the above distribution for channels that have symmetric wall conditions. In Fig. 1, for an hch = 18.58σ (hch = 6.4 nm) channel, we divide the channel in five layers (L1–L5), each one of width about 3.7σ and calculate the local average diffusion coefficient, in each layer. These results are then used to draw the 2nd order polynomial fit shown in Fig. 1. We observe a good agreement of calculated data to the proposed fit and, moreover, calculated local diffusion coefficient values seem to be in agreement with values obtained using Eq. (35). All the necessary diffusion coefficient values are taken from the molecular dynamics simulation results of Sofos et al. (2009b).
Fig. 1

Comparison of diffusion coefficient calculated from Eq. (35) with MD results, for an hch = 6.4 nm channel. MD values shown are taken from Sofos et al. (2009b) at T = 150 K and are fitted to a 2nd order polynomial

The particular form of Eq. (35) can be explained by a potential that incorporates the wall interactions to include wetting conditions (Marko 1993; Lipowsky and Fisher 1986; Lipowsky and Huse 1986).

Note that both A and D0 depend on the channel size, where −hch/2 ≤ z ≤ hch/2. Clearly, as we approach the channel walls (z = ± hch/2), diffusion decreases.

$$g^{2} = \frac{2KM}{{D_{o} \left( {1 - A^{2} z^{2} } \right)}} = \frac{{g_{o}^{2} }}{{1 - A^{2} z^{2} }}$$
It must be recalled that go < hch/2. The diffusion Eq. (30) becomes
$$\frac{{{\text{d}}^{4} c}}{{{\text{d}}z^{4} }} + \frac{{1 - A^{2} z^{2} }}{{g_{o}^{2} }}\frac{{{\text{d}}^{2} c}}{{{\text{d}}z^{2} }} = 0$$
The solution of Eq. (37) is shown in Appendix 2. In a more general case, we can use the approximate method of Wentzel, Kramers and Brillouin (WKB) for solving the stationary Schrodinger’s equation that resembles up to a sign Eq. (37). We can write
$$g^{2} = g_{o}^{2} /P^{2} (z)$$
Then the approximate solution of Eq. (37) becomes
$$\frac{{d^{2} c}}{{{\text{d}}z^{2} }} \approx \frac{{c_{1} }}{\sqrt P }\sin \left( {\frac{1}{{g_{o} }}\int {P(z){\text{d}}z + c_{2} } } \right)$$
For P(z) = (1 − A2z2)1/2
$$\int {P(z){\text{d}}z = \frac{z}{2}\sqrt {1 - A^{2} z^{2} } + \frac{\arcsin (Az)}{2A}}$$
A “quantum” condition can be stated by observing that an increase in phase \(\frac{1}{{g_{o} }}\int {P{\text{d}}z + c_{2} }\) in a complete rotation would have to be an integer multiple of 2π. Let the boundaries z = ± hch/2 be the two turning points. Then
$$\int\limits_{{ - h_{\text{ch}} /2}}^{{h_{\text{ch}} /2}} {P(z){\text{d}}z = 2n\pi \, g_{o} ,} \quad n = 0,1,2, \ldots$$
In the present example, Eq. (40) gives
$$\left( {\sqrt {1 - A^{2} \left( {\frac{{h_{\text{ch}} }}{2}} \right)^{2} } + \frac{{\arcsin \left( {A \, h_{\text{ch}} /2} \right)}}{{A \, h_{\text{ch}} /2}}} \right) = 2\pi \frac{{2g_{o} }}{{h_{\text{ch}} }}n$$
If we know any two of hch, go and A, we can estimate the third one, provided we can estimate the integer n (A ≠ 0 in this case).

For the case of asymmetric wall conditions, refer to Appendix 3.

3 Molecular system modeling

3.1 System details

The flow system simulated by non-equilibrium molecular dynamics (NEMD) is consisted of two infinite plates with argon flowing between them (equivalent to Poiseuille flow), as shown in Fig. 2. The LJ potential is used here:
$$u^{\text{LJ}} (r_{ij} = 4\varepsilon \left( {\left( {\sigma /r_{ij} } \right)^{1/2} - \left( {\sigma /r_{ij} } \right)^{6} } \right)$$
where, for liquid argon, we employ σ = 0.3405 nm, ɛfluid/kB = 119.8 K, cutoff radius for the potential rc = 2.5σ, constant mean fluid density (ρ = 1,078 kg/m3, or ρ* = 0.642σ−3).
Fig. 2

The MD system under examination

MD simulation is performed in a simulation window of (Lx × Ly × Lz) dimensions. Periodic boundary conditions are used along the x- and y-directions, while the distance between the two plates in the z-direction corresponds to the channel width, hch. In this way, we construct a Poiseuille flow system, with two infinite plates in the xy-plane, separated by a distance hch. Based on the fundamental formulation of MD (e.g., Allen and Tildesley 1987), due to periodic boundary conditions, each particle that exits the simulation window on the right enters the channel form the left, while particles approaching the walls are bounced back toward the interior of the channel.

An external force Fext is applied to the x-direction to every fluid particle to drive the flow. Wall atoms are kept bound around their original fcc lattice positions by an elastic string force \({\mathbf{F}} = - K^{*} ({\mathbf{r}}(t) - r_{\text{eq}} )\), where \({\mathbf{r}}(t)\) is the position vector of an atom at time t, req is its initial lattice position vector, and K* = 57.15(ɛ/σ2) is the spring constant.

Wall atoms absorb the increase in kinetic energy of the fluid atoms, which is caused by the application of the external force, and Nose–Hoover thermostats are applied at the thermal walls in order to keep the system’s temperature constant (Sofos et al. 2009a; Evans and Holian 1985; Holian and Voter 1995). We employ two independent thermostats one for the upper wall and another for the lower wall in order to achieve better thermalization of the wall atoms (see Appendix 4 for details). More details on system parameters can be found in Sofos et al. (2009b).

The NEMD computed self-diffusion coefficient for liquid argon compares with the macroscopic measurements as suggested by the present approach. For T = 100 K, Sofos et al. (2009b) computed a self-diffusion coefficient D = 5 × 10−5 cm2/s. Experiments of Gini-Constagnoli and Ricci (1960) give D = 1.53 × 10−5 cm2/s for T = 84.56 K. Using molecular dynamics, Thomas and McGaughey (2007) report D = 4.03 × 10−5 cm2/s for T = 90 K.

3.2 Density profiles

In order to calculate values that consist the density profile, the nanochannel was partitioned in m computational domains (bins) along the z-direction, each of volume \(V_{\text{bin}} = L_{x} L_{y} h_{\text{bin}}\), where hbin = hch/m. At each time instant, the density of the liquid N*(z) is the number of atoms located inside each bin. The number of atoms is calculated during the whole time of the calculations and the average is
$$N(z) = \frac{{\left\langle {N_{\text{bin}} (z,z + h_{\text{bin}} )} \right\rangle }}{{N_{f} }}$$
To check the molecular dynamic model (LJ potential), the cutoff radius rc was allowed to take the values 2.5σ, 3σ and 3.5σ. Minor influence was observed to the results (an increase in rc reduces slightly the peaks of the calculated density distributions).
We examine the effect of temperature, T, on the density profile presented in Fig. 3a. For hch = 2.65σ (or, hch = 0.9 nm), we observe that fluid atoms are ordered in two distinct layers, symmetrical with respect to the channel mid-plane. The fluid density peak value decreases as temperature increases. Similar ordering in the fluid has been reported in Somers and Davis (1991) for hch = 2.5σ and 2.75σ. Although the peaks magnitude decreases, there is a clear periodicity for their appearance at about 0.9σ, as shown in Fig. 3a. This trend remains in all density profiles presented in this study, for all channel widths. The effect of the magnitude of the external applied force, Fext, to density profiles is negligible, at least in the range examined here (Fig. 3b). This is in agreement with the results reported in Nagayama and Cheng (2004). We remind that the force range studied here does not result in system non-linearities (Binder et al. 2004).
Fig. 3

Density profiles extracted from MD model, shown from channel middlepoint to the wall (symmetric channels) a for various T (hch = 7.9σ), b for various magnitudes of Fext (hch = 7.9σ), c for various channel widths, d for various K* values (hch = 4.42σ), e for various wall–fluid interaction values (hch = 2.65σ) and f various average fluid densities (hch = 2.65σ). For presentation reasons, since we refer to channels of various width, we normalized z-direction to 0 ≤ z* ≤ 1

In Fig. 3c we investigate the effect of channel width, hch, on the density profile. In small channels (2σ ≤ hch ≤ 8σ) the strong influence of the walls extends over all or most of the fluid atoms and this fact results in oscillations on the density profile. It is clear that as hch increases (10σ ≤ hch ≤ 20σ) homogeneity is induced in the interior of the channel, while there is always a region of fluid non-homogeneity region becomes non-significant, i.e., for hch ≈ 20σ the non-homogeneity region is less than 10 % of the available channel width.

The wall spring constant K* is an indication of wall atoms stiffness (Asproulis and Drikakis 2011), i.e., the walls become stiffer when K* value is greater. In the diagram of Fig. 3d, we observe that density peaks are broader and of smaller amplitude as K* decreases, as also found in Priezjev (2007), and we attribute this to the fact that, for smaller K* values, wall atoms oscillate more and fluid atoms are more possible to localize closer to the walls.

We examine the effect of various wall–fluid interactions ɛw/ɛf on density profile in Fig. 3e. As the ratio ɛw/ɛf increases, fluid atoms are attracted to the walls and wall surface becomes hydrophilic, while as ɛw/ɛf decreases, fluid atoms are less attracted to the walls and wall surface becomes hydrophilic (Voronov et al. 2006). This approach was also used in Sofos et al. (2012), where potential energy contours near the walls show that a large ɛw/ɛf ratio leads in increased fluid atom presence near the walls (hydrophilic wall), while a small ɛw/ɛf ratio leads in decreased fluid atom presence near the walls (hydrophobic wall). Wall wettability depends on the fluid contact angle on the surface, which, in turn, depends on the ɛw/ɛf ratio (Voronov et al. 2006).

Average fluid density, ρ* also affects fluid ordering in a significant way (Fig. 3f). We observe that as average fluid density decreases, the amplitudes of two peaks at the density profile increase significantly and fluid atoms are ordered near the walls, while strong inhomogeneity is induced.

Having studied all parameters (T, Fext, K*, ɛwall/ɛfluid and ρ*) for every channel width hch (we have examined all cases, but we do not present all diagrams here), we come to the conclusion that every parameter has a different impact on the density profile.

To sum up with results shown here, we first note that as system temperature increases, fluid ordering near the walls is decreased. This is attributed to increased fluid particle mobility in higher temperatures. As a result, an increase in system temperature leads to smoother density profiles. The flow is driven by a body force acting on all fluid particles equivalent to a pressure difference in Poiseuille flow. By changing the magnitude of the external force, we obtain no significant effect on fluid ordering near the walls. When the average fluid density decreases in the channels studied here, an increase of fluid inhomogeneity is observed. This can be attributed to the fact that for low density flows, particles are attracted by the wall atoms and tend to “stick” close to the walls, as they do not encounter strong attractive forces from other fluid atoms. For denser fluids, there is a significant number of fluid atoms in the channel that interact with each other and this fact helps them spread over the whole extend of the channel.

On the other hand, two properties that characterize the wall behavior, i.e., the wall spring constant K* and the ratio ɛwall/ɛfluid, affect only channels of smaller widths (h ≤ 10σ). For hch ≥ 10σ, the effect of these two parameters extends only in a small region close to the walls (about 1–1.5σ) and is negligible in the remaining of the channel.

4 Comparison of the quasi-continuum theory with MD simulations

The molecular dynamics results (Sofos et al. 2009a) suggest that the parameter A in Eq. (35) is almost invariant with temperature T, but decreases to zero as the channel width hch increases. From the density profiles shown in Fig. 3a–f, it is clear that an oscillatory density profile appears that is exponentially decreasing from the walls to the interior of the channel. In all cases studied, a high density peak appears near the wall and consecutive peaks of smaller height appear as the profile approaches the middle of the channel. The exponential decay weakens as the channel width increases and saturates to constant fluid density at widths about hch > 10σ.

It is also of great interest to keep in mind that the oscillation periodicity gives an almost constant wavelength of about 0.9σ (≈0.3 nm in the argon case), as shown in Fig. 3a. This wavelength is constant, no matter what property is altered, e.g., temperature, external force, channels width, etc. Even when the density peaks weaken to nearly constant, this wavelength does not change.

The above results can be captured very well by the quasi-continuum model. Excluding the results for the very small density (ρ* = 0.321), one can identify by go ≈ 0.9σ the wave length of the quasi-continuum theory. By fitting the d2c/dz2 as predicted by Eq. (56) (see Appendix 2) to MD calculations, we obtain the values of Ago for different channel widths, as shown in Fig. 4.
Fig. 4

Calculation of d2c/dz2 from Eq. (48). The values of Ago were fitted to minimize the difference with the molecular dynamic results of Fig. 3. The coordinate z scales with 0.2go(go = 0.3 nm). aAgo = 0.12, n = 1 (\(\frac{{c^{\prime\prime}}}{{c_{2} }} - \frac{z}{{0.2g_{o} }}\)) and h = 0.9 nm bAg0 = 0.045, n = 1 \(\left( {\frac{{c{\prime \prime }}}{{c_{2} }} - \frac{z}{{0.2g_{o} }}} \right)\) and h = 1.5 nm cAg0 = 0.024, n = 1 (\(\frac{{c^{\prime\prime}}}{{c_{2} }} - \frac{z}{{0.2g_{o} }}\)) and h = 2.7 nm

Figure 4a corresponds to the smallest channel studied (hch = 0.9 nm), Fig. 4b to hch = 1.5 nm and Fig. 4c to hch = 2.7 nm. In all cases, we have n = 1. Equation (42) can now be used in order to extract the go value for the quasi-continuum model. We mentioned before that if we know any two of hch, go and A, we can estimate the third one, provided we can estimate the integer n. The results are shown in Table 1, and the accuracy of the quasi-continuum model seems to be very good.
Table 1

Analysis of the molecular dynamic calculations by the quasi-continuum model


Molecular dynamics





go(nm), Eq. (53)
















Numerical integration of d2c/dz2 and symmetry condition give
$$c(z) = c(z = 0) + c_{2} \int {{}_{1}F_{1} } {\text{d}}z .$$
Now, we can plot the results for c(z) in Fig. 5a–c. Figure 5a is analogous to the MD extracted density profile for the hch = 0.9 nm channel. Calculations for the hch = 1.5 nm channel are presented in Fig. 5b. Here, the density peak oscillation is not so obvious, but still this is close to the respective MD extracted profile. Interesting results come from the hch = 2.7 nm channel, where the similarity with MD simulation density profile is clear.
Fig. 5

Numerical integration of d2c/dz2 shown in Fig. 4. The results scale with the density profile at z* = 0.5 (middle of the channel) found by molecular dynamics of Fig. 3. The coordinate scales with 0.2 go (go = 0.3 nm). a Ago = 0.12 and hch = 0.9 nm, \(0.0 2 5+ \frac{0.045}{2F}\) (average = 0.0385), b Ago = 0.045 and hch = 1.5 nm, \(0.0 2 5+ \frac{0.035}{2.1F}\) (average = 0.0420) and c Ago = 0.024 and hch = 2.7 nm, \(0.0 3 8 4+ \frac{0.00375}{6.2F}\) (average = 0.0428)

One could observe that the variation of D along the channel width plays the most important role in explaining the uneven peaks observed by the NEMD calculations. Strong curvatures of the D(z) result in high local peaks of c(z).

5 Conclusions

We have presented a quasi-continuum self-diffusion approach which is able to reproduce results from non-equilibrium molecular dynamics simulations of planar Poiseuille liquid argon nanoflow. If diffusion coefficient values are known, in addition with a micro structural internal length, g, the model can predict the ordering effects caused by atomic-scale flow conditions. In nanoflows, the effect of system temperature, applied driving force, wall spring constant, wall–fluid interaction ratio and average fluid density on the distribution of fluid density becomes important, leading to behavior different than expected from classical continuum theory.

MD simulations reveal strong fluid ordering through number density profiles for hch ≤ 2 nm, while for hch ≥ 2 nm profiles are uniform in most of the core channel area but ordering persists very close to the wall. Fluid ordering near the wall increases when system temperature decreases, the spring constant (wall stiffness) increases, wall hydrophilicity increases, and average fluid density decreases, while external forces do not affect number density profiles significantly.

To capture the MD calculated fluid distribution densities, we constructed a quasi-continuum model for the self-diffusion equation. This model was based on the concept of the spinodal decomposition of solid diffusion. A characteristic length g appeared in the new differential equation that gives rise to oscillations in the steady state distribution of the fluid atoms. Furthermore, a wave number g/hch can be constructed using the channel width hch. It is found that the inhomogeneity of the diffusion coefficient plays a significant role in the details of the fluid density distribution by exponentially decreasing the peaks of the density oscillations close to the channel boundaries. In this way, the fluid–wall interaction is fully introduced in the quasi-continuum approach. The resulting quasi-continuum equation is of Schrodinger type and a “quantum” condition can be formulated, connecting g, hch and ∂2D/∂z2.

Either the atomistic model or the quasi-continuum model presented in this work seems capable of reproducing the Poiseuille nanoflow characteristics regarding the ordering of the fluid in the channel. Of particular, importance is the inhomogeneity of the diffusion coefficient in capturing the density profiles with the quasi-continuum theory. The variation of the diffusion coefficient across the channel width (decreasing toward the channel walls) is related to wall smoothness.

The only shortcoming of the present scheme is that it necessitates an MD simulation realization in order to extract fluid parameters (diffusion coefficient variation and characteristic length g). However, if one has to simulate a complex flow domain, consisting of various channel parts to be simulated by fully atomistic MD methods, it would require extensive computational resources. In the frame of the proposed scheme, all parts could be modeled using the quasi-continuum model. We emphasize that the same equations can be employed in a multi-scale problem covering length scales from nano to macro.


If a gravitational field \(\phi\) is present, we replace \(\mu\) with \(\mu + m\phi\) (\(m\) is the mass per mole). For \(\phi = {\text{const}}\), the present results will still hold true.



This project was implemented under the “ARISTEIA II” Action of the “OPERATIONAL PROGRAMME EDUCATION AND LIFELONG LEARNING” and is co-founded by the European Social Fund (ESF) and National Resources.

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