Darcy-Brinkman law and nanotechnology: toward an effective medium theory of systems at the nano-scale

Since their discovery in 1991 [1], carbon nanotubes (CNTs) have attracted unabated attention among the scientific community because of their remarkable combination of mechanical properties, such as exceptionally high elastic moduli, reversible bending and buckling characteristics, and super-plasticity. These properties ensure that complex interactions between turbulent/laminar fluid flow and patterned nanostructures composed of CNTs play an important role in a variety of applications, including mechanical actuators, chemical filters, and flow sensors. When placed on a body’s exterior, CNT “forests” can act as super-hydrophobic surfaces that significantly reduce drag [2]. Observations of fluid flow past CNTs suggest the potential use of CNT forests as sensors of tactile and shear forces. Scanning electron microscope images of CNTs deflection under turbulent flow are showed in Fig.1.

The predictive and diagnostic capabilities of nanosensors and other nanoforest-covered surfaces are generally hampered by the relative lack of quantitative understanding of their response to hydro- or aerodynamic loading. Most experiments dealing with these phenomena assemble CNTs into macroscopic sheets or forests. Until recently, there was no theoretical model able to account for so-called crowding effects on the mechanical response of CNTs forest to fluid-induced shear. The ‘crowding effect’ refers to the impact of the collective dynamics/motion of a population/assembly on any single member of the ensemble (Fig. 1). While molecular dynamics simulations impose a prohibitive computational burden, attempts to account for crowding effects by modifying the drag coefficient of each CNT were essentially phenomenological and treated CNTs as infinite cylinders.

We have recently showed that the classical theory of flow through porous media, i.e. the Darcy-Brinkman equation [3], can be successfully employed to model flow inside a CNT forest, and to predict the elastic bending of single CNTs to ambient laminar and turbulent fluid flows, while accounting for crowding effects [4].

We treat the CNT forest (the region occupied by CNTs) as a porous medium with constant porosity and permeability (see Fig. 2), and employ the Darcy-Brinkman equation to describe the distribution of the horizontal component of the intrinsic (spatially averaged) velocity in this region. We then couple the Darcy-Brinkman equation to Reynolds equation for turbulent flow over the CNT forest. A closed-form expression for the intrinsic velocity is obtained, and used to analytically determine the average drag exerted by the flow on a single CNT, and its maximum deflection under turbulent aerodynamic shear.

Figure 3 shows experimental (symbols) and predicted (lines) deflections of the CNT tip, X (μm), in response to dynamic loading by the turbulent flows of argon (squares) and air (circles) for a range of the bulk velocity values  . CNTs flexural rigidity, EI, computed from experiments with Argon, is the sole fitting parameter of the model (solid line). Its value is then used to make a parameter-free prediction (dashed line) for the CNTs deflection under turbulent airflow. The figure reveals good agreement between theory and experiment over a wide range of flow velocities.

In addition, intermediate asymptotic analysis in the low permeability limit reveals the existence of a self-similar solution for the bending profile of CNTs [5]. This self-similar solution is successfully used to estimate EI by linear fit of properly rescaled quantities, predict   and collapse, previously scattered, data from the same CNT sample onto a single line (see Fig. 4(a)-(b)). Self-similarity of appropriately rescaled quantities (i.e. intrinsic velocity, shear stress, and deflection) is achieved in the bulk of the porous medium (CNTs forest) and close to the interface separating the porous layer from the pure fluid [5].

The previous results suggest that Darcy’s law (and the Darcy-Brinkman equations), can provide a general and robust effective medium framework to describe flow at the nano-scale, and can be successfully used as a powerful analytical alternative to computationally intensive molecular dynamics simulations to model system dynamics at such scales. Also, the self-similar behavior of many dynamical quantities allows reducing the parameter-space even further.

Robustness and applicability regimes of porous medium advection-reaction-dispersion equations [6,7] to study transport processes at the nano- and micro-scale is object of current investigations.

References

[1]    ‘Helical microtubules of graphitic carbon’, S. Iijima, Nature 354, 56-58, 1991.

[2]    ‘Superhydrophobic Carbon Nanotube Forests’, K. K. S. Lau, J. Bico, K. B. K. Teo, M. Chhowalla, G. A. J. Amaratunga, W. I. Milne, G. H. McKinley, K. K. Gleason, Nano Lett., 3, 12, pp. 1701-1705, 2003.

[3]    ‘A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles’, H. C. Brinkman, Appl. Sci. Res., A1, pp. 27-34, 1947.

[4]    ‘Elastic Response of Carbon Nanotube Forests to Aerodynamic Stresses’, I. Battiato, P. R. Bandaru, D. M. Tartakovsky, Phys. Rev. Lett., 144504, 2010.

[5]    ‘Self-similarity in coupled Brinkman/Navier-Stokes flows’, I. Battiato, J. Fluid Mech., 699, pp.94-114, 2012.

[6]    ‘Applicability regimes for macroscopic models of reactive transport in porous media’, I. Battiato, D. M. Tartakovsky, J. Contam. Hydrol., 120-121, pp. 18-26, 2011.

[7]    ‘On Breakdown of Macroscopic Models of Mixing-Controlled Heterogenous Reactions in Porous Media’, I. Battiato, D. M. Tartakovsky, A. M. Tartakovsky, T. D. Scheibe, Adv. Water Resour., 32, pp.1664-1673, 2009.