Page 89 - Nanotechnology an introduction

(7.27)
where η is the dynamic viscosity of the solvent, and the thermal velocity depends on both the particle size and density ρ:

(7.28)
Substituting these expressions into equation (7.26) gives

(7.29)
3
−3
from which it follows that under any practical conditions (e.g., water at 20°C has a density of 1 g/cm and a viscosity of 10 Pa s) the particle size
exceeds the mean free path, hence there can be no ballistic transport and hence no unique feature at the nanoscale. In other words, although in
analogy to the case of electron conduction (Section 7.4.1) a nanofluidic device could be one for which the mean free path of particles moving within
it is greater than the characteristic size of the internal channels, this condition could never be realistically met. If some way could be found of
9
3
working with nanoparticles of material from a white dwarf star (mass density ~10 g/cm ), this conclusion would no longer be valid!
For devices that depend on thermal conductivity, it is limited by phonon scattering, but analogously to electron conduction (Section 7.4.1), if the
device is small enough ballistic thermal transport can be observed, with the quantum of thermal conductance being .
Another approach to defining a nanofluidics scale could be based on the fact that fluidics devices are fundamentally mixers. For reactions in minute
subcellular compartments (or taking place between highly diluted reagents) (see Section 2.4 for the notation). The nanoscale in
fluidics could be reasonably taken to be that size in which differs significantly from zero. The scale depends on the concentration of the
active molecules (A and B).
This calculation ignores any effect of turbulence on the mixing, for which the characteristic scale would be the Kolmogorov length ℓ :
K

(7.30)
where κ is the kinematic viscosity and ε is the average rate of energy dissipation per unit mass. But, this length is of micrometer rather than
nanometer size. Furthermore, it is essentially impossible to achieve turbulent flow in nanochannels; even microfluidics is characterized by extremely
low Reynolds numbers (below unity).

Fluids are often moved in miniature reactors using electroösmosis, in which an electric field is applied parallel to the desired rate of flow. If the walls
of the device are ionizable; for example, most metal oxides are hydroxylated at the interface with water, and hence can be protonated or
deprotonated according to

(7.31)
hence, the interface will be charged and the counterions will be dispersed in a diffuse (Gouy–Chapman) double layer, their concentration decaying
with a characteristic Debye length 1/κ, given by (for monovalent ions)

(7.32)
where n is the concentration of the ith ion bearing z elementary charges. The decay length is typically of the order of nanometers or tens of
i
i
nanometers; upon application of the electric field the counterions will move, entraining solvent and engendering its flow. If the radius of the channel
is less than 1/κ, electroösmosis will be less efficient. This length could therefore be taken as determining the nanoscale in fluidics devices
(provided some electrokinetic phenomenon is present), because below this scale bulk behavior is not followed. Electroösmosis is not of course
useful in a macroscopic channel, because only a minute fraction of the total volume of solvent will be shifted. Note the analogy between the
semiconductor space charge (cf. equation 7.3) and the Gouy–Chapman ion distribution.
The capillary length,

(7.33)
(confusingly, an alternative definition is current in macroscopic fluid systems, in which surface tension is compared with sedimentation)—the
surface tension γ is that between the solid wall and the fluid—gives an indication of the predominance of surface effects (cf. Section 2.2). It is
typically of the order of nanometers. There is no discontinuity in behavior as the characteristic size of a fluidic device falls to or below l cap .

In summary, as the radius of a fluid channel becomes less than a characteristic length such as 1/κ or l cap , in effect the properties of the channel wall
influence the entire volume of the fluid within the channel. The implications of this are far from having been thoroughly worked out. Especially in the
case of aqueous systems, the structure of the water itself (especially the average strength of its hydrogen bonds) is also affected by wall
interactions, doubtless with further far-reaching implications. But most discontinuities due to miniaturization of fluidic devices seem to already
manifest themselves at the microscale.

7.9.1. Mixers and Reactors

Miniaturizing mixers has been very successful at the microscale, as can be deduced from the huge proliferation of lab-on-a-chip devices for
analytical and preparative work in chemistry and biochemistry. Particular advantages are the superior control over flow compared with