Formation mechanism
The formation of AlOOH nanostructures from the reagents of aluminum and water can be explained by the following facile reaction:
2Al+6H2OΔ→2Al(OH)3+3H2(gas)![]()
Now, our primary emphasis is to analyze the formation of Boehmite nanostructures for that we would use tensor notations and Green’s functions [
14
]. As the conditions are inert, the effect due to impurities can be neglected and the mixture can be treated as a binary mixture of aluminum particles and de-ionized water. Firstly, on heating this mixture at a temperature of
240∘C
for 3 h leads to the formation of homogenous mixture of aluminum particles and DI water. Secondly, in this mixture, the aluminum crystals are broken down to nanosize under high pressure. Thirdly, due to difference in concentration and temperature there arise concentration and temperature gradients leading to convection, due to which a cell structure occurs which remains even after drying DI water [
14
,
15
]. Now, for a ternary mixture, we have
where
C′
is the concentration of impurities. Since the conditions are inert we have
C′=0
and hence for a binary mixture we can write
where
ρ
is the density of homogenous mixture of almunium particles and de-ionized water (which depends on both the temperature of this mixture and concentration of aluminum particles in de-ionized water),
Bc=αcΔC
,
Bt=αtΔT
and
αc
and
αt
are the solute and thermal expansion coefficients, To write momentum balance equation, we first define
Dvi=∂tvi+vj∂jvi
(with
∂ivi=0
, i.e., divergenceless vector field), called substantial derivative, which represents the time rate of change of a physical quantity subjected to space- and time-dependant velocity fluid in continuum mechanics. Now, momentum balance equation can be written as:
where
μ
is a constant viscosity and other symbols have their usual meanings. we also require equations
DT=κ∂2T
and
DC=d11∂2C
, where
d11
is the aluminum diffusion coefficient and
κ
is the thermal diffusivity of the fluid.
Using transformation
xi→lxi,∂i→l-1∂i,vi→kl-1vi,t→l2k-1t,
and
p→pμkl-2+ρ0g(l-λiri)
, satisfying the condition
∂ivi=0
, results in non-dimensional momentum conservation and continuity equations as under:
Pr-1Dvi=-∂ip+∂2vi+RcCλi+RtTλi,DC=Le11-1∂2C,DT=∂2T,![]()
where
Pr=kρ0μ-1
is the Prandtl number,
Rc=μ-1k-1Bcρ0gl3
is the concentration Rayleigh number,
Rt=μ-1k-1Btρ0gl3
is the temperature Rayleigh number, and
Le11=kd11-1
are the Lewis numbers.
Now, using the boundary conditions
λivi=0
,
λiΛj∂ivj=0
, where
Λiλi=0
, and
C(0)=T(0)=1
,
C(1)=T(1)=0
,
p(1)=0
,
λi=(0,0,1)
and
ri=(x,y,z)
, and adding small perturbations to steady-state solutions and defining
Dba=a-1∂t-b-1∂2
, we get
D1Prui+∂iϕ-(Rcθc+Rtθt)λi=0,DLe111θc-λiui=0,D11θt-λiui=0.![]()
Now,
where
E
is the operator taking twice a curl of any vector field
ai
and as we know the gradient of a scalar field vanishes when its curl is taken twice. Further,
E∂iϕ=0
, as
ui
is divergence-less, and
λi(Eλiϕ)=λi∂iλi∂jϕ-∂2ϕ=∂~2ϕ
. Now, acting by
E
, contracting by
λi
, using Green’s function and applying the boundary conditions (derivatives of
λi∂i
vanishes on
λiui
) we arrive at
D11DLe1D1Pr∂2λiui=Rc∂~2D11λiui+Rt∂~2DLe1λiui(r′).![]()
and its solution
λiui=Asinnπexp[i(kxx+kyy)+σt],![]()
Thus, we have
where
C(kn,k,σ)=(σ+kn2Le-1)(Pr-1σ+kn2)(σ+kn2)kn2-(σ+kn2Le-1)k2Rt-(σ+kn2)k2Rc.![]()
By substituting
σ=0
, the critical behavior under temperature dependance is given by
where
C(kn,k,0)=kn6k-2Le-1-Le-1Rt-Rc.![]()
Now, the effective Rayleigh number is
Hence,
R=kn6k-2.
For lowest value
n=1
, the instability starts at
which gives
k2=π2/2
, and the value of
R
will be
R=657.5
. Thus, by maintaining the rate of evaporation to an extent wherein
R>657.5 657.5$$\end{document}]]>
, the formation of nanostructures takes place.
