A quasi-continuum multi-scale theory for self-diffusion and fluid ordering in nanochannel flows
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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
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Abstract
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.
Keywords
Quasi-continuum theoryMolecular dynamicsSelf-diffusion equationNanochannel flowsFluid orderingDensity profile oscillationsList of symbols
- A_{1–4}
Constants determined by BC
- A, B
Constants for inhomogeneous diffusion
- A_{i}, B_{i}
Airy functions
- c
Concentration
- c_{1–3}
Real constants
- D
Bulk diffusion coefficient
- D_{ap}
Apparent diffusion coefficient
- F
Diffusion functional
- F_{ap}
Apparent diffusion functional
- _{1}F_{1}
Hypergeometric function
- F_{ext}
Magnitude of external driving force
- g
Wavelength
- G
Gibbs free energy
- G_{0}
Gibbs free energy at equilibrium
- h
Boundary value for concentration
- H
Hermitian polyomial
- h_{ch}
Channel height
- J
Diffusional flux
- K
Gradient energy coefficient
- K^{*}
Spring constant
- L_{x}
Length of the computational domain in the x-direction
- L_{y}
Length of the computational domain in the y-direction
- L_{z}
Length of the computational domain in the z-direction
- M
Diffusional mobility
- n
Integer number, n = 0, 1, 2…
- p
Pressure
- q
Boundary value for the non-classic flux term
- r_{eq}
Position of a wall atom on fcc lattice site
- r_{i}
Position vector of atom i
- r_{ij}
Distance vector between ith and jth atom
- T
Temperature
- S
Area
- u(r_{ij})
LJ potential of atom i with atom j
- V
Volume
- w
Boundary condition for the normal flux
- z^{*}
Normalized distances in the z-direction
Greek letters
- ε
Energy parameter in the LJ potential
- μ
Local chemical potential
- ρ
Fluid density
- σ
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.
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
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.
The minimization of F_{ap} leads to Eq. (23) (Crank 1975). In this case, we can use the BC from Eq. (20).
Note that F(∇^{2}c) can be thought of as an evolutionary criterion toward the steady state, since the transient solution ∇^{2}c (without convection) is an admissible function for the minimum of F(∇^{2}c) (Prigogine and Glansdorff 1965).
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
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
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 D_{0} depend on the channel size, where −h_{ch}/2 ≤ z ≤ h_{ch}/2. Clearly, as we approach the channel walls (z = ± h_{ch}/2), diffusion decreases.
For the case of asymmetric wall conditions, refer to Appendix 3.
3 Molecular system modeling
3.1 System details
MD simulation is performed in a simulation window of (L_{x} × L_{y} × L_{z}) 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, h_{ch}. In this way, we construct a Poiseuille flow system, with two infinite plates in the xy-plane, separated by a distance h_{ch}. 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 F_{ext} 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, r_{eq} is its initial lattice position vector, and K^{*} = 57.15(ɛ/σ^{2}) 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} cm^{2}/s. Experiments of Gini-Constagnoli and Ricci (1960) give D = 1.53 × 10^{−5} cm^{2}/s for T = 84.56 K. Using molecular dynamics, Thomas and McGaughey (2007) report D = 4.03 × 10^{−5} cm^{2}/s for T = 90 K.
3.2 Density profiles
In Fig. 3c we investigate the effect of channel width, h_{ch}, on the density profile. In small channels (2σ ≤ h_{ch} ≤ 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 h_{ch} increases (10σ ≤ h_{ch} ≤ 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 h_{ch} ≈ 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 ɛ_{w}/ɛ_{f} on density profile in Fig. 3e. As the ratio ɛ_{w}/ɛ_{f} increases, fluid atoms are attracted to the walls and wall surface becomes hydrophilic, while as ɛ_{w}/ɛ_{f} 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 ɛ_{w}/ɛ_{f} ratio leads in increased fluid atom presence near the walls (hydrophilic wall), while a small ɛ_{w}/ɛ_{f} 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 ɛ_{w}/ɛ_{f} 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, F_{ext}, K^{*}, ɛ_{wall}/ɛ_{fluid} and ρ^{*}) for every channel width h_{ch} (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 ɛ_{wall}/ɛ_{fluid}, affect only channels of smaller widths (h ≤ 10σ). For h_{ch} ≥ 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 h_{ch} 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 h_{ch} > 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.
Analysis of the molecular dynamic calculations by the quasi-continuum model
h_{ch}(nm) | Molecular dynamics | Quasi-continuum | ||
---|---|---|---|---|
g_{o}(nm) | A(nm^{−1}) | Ah_{ch}/2 | g_{o}(nm), Eq. (53) | |
0.9 | 0.3 | 0.6667 | 0.3000 | 0.141 |
1.5 | 0.3 | 0.2500 | 0.1875 | 0.237 |
2.7 | 0.3 | 0.1333 | 0.1800 | 0.427 |
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 h_{ch} ≤ 2 nm, while for h_{ch} ≥ 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/h_{ch} can be constructed using the channel width h_{ch}. 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, h_{ch} and ∂^{2}D/∂z^{2}.
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.
Acknowledgments
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.