To analyze the effect of nano-confinement on the self-diffusion coefficient, a

dynamical model was proposed (Tankeshwar and Srivastava 2007) wherein the

effect of confinement on molecular motion was analysed. The model was built up

on the model of a microscopic (local) self-diffusion coefficient varying as a function

of distance taken from the walls of the channel.

16.6.1 Model for Many-Body Liquid Problem

The configuration space of a many-body fluid system could be thought of as being

divided into a number of cells where each cell may be described by a fixed

configuration corresponding to the local minima on the potential energy hyper-

surface of the system. Inside the cell, motion of the liquid configuration is harmonic

around a local minima characterized by a well-defined frequency or band of

frequencies. The system could jump from one cell to other with a specific jump

frequency τ
¹. Use has been made of basic definitions and expressions of diffusion

and velocity autocorrelation functions which can be expressed as follows:

D ¼ ^kBT

m

Z 1

0

V tð Þdt and V tð Þ ¼ ¹

3

X

i, α

viα tð Þviα 0

ð Þ

h

i

ð16:7Þ

where viα(t) is the αth component of velocity for a ith particle at time t and such that

hviα(0)i² ¼ 3 kBT/m. Calculation of the velocity autocorrelation function (VACF)

for any real system is probable only through a simplified picture within the frame-

work of many-body theory.

16.6.2 Various Cases of Microscopic Model

1. Effect of Confinement: Solution is taken in the form of harmonic oscillator in

nonconfined state with motion along z-axis characterized by amplitude A and

frequency ω expressed as z(t) ¼ A sin (ωt). When width of the channel is of the

order of nano-/microscale, particles would inevitably experience a compression-

like situation which would eventually decrease its amplitude by d (as shown in

Fig. 16.9) and also change its frequency to Ω which becomes a function of z

z t1

ð Þ ¼ A
d ¼ A sin ωt1

ð

Þ

ð16:8Þ

where t1 ¼ ¹

ω ^{sin
1 1
d=A}

ð

Þ, leading to a new frequency of motion

Ω ¼ ^π

2t1 ^¼

πω

2 sin
¹ 1
d=A

ð

Þ

ð16:9Þ

whose corresponding curve is depicted in Fig. 16.10. Expressing ratio d/A as a

function of z, i.e.

288

K. Tankeshwar and S. Srivastava