Download Polarizability affecting nucleation of water vapour condensation and

Indian Journal of Radio & Space Physics
Vol. 33, February 2004, pp. 43-49
Polarizability affecting nucleation of water vapour condensation and ice glaciation
in presence of external electric field
N Singh & Devendra Singh
Department of Physics, Nehru College, Chhibramau 209 721 (UP), India
Received 26 November 2002; revised 19 February 2003; accepted 24 December 2003
Using the modified value of polarizability based on the present calculations, it is found that the critical nuclei have
much less radii in the presence of electric field . The nucleation rate in presence of electric field is greater than in its absence.
The enhancement factor to the nucleation rate in presence of electric tield is several orders higher than in the absence of an
electric field. The polarizability of water vapour molecules in presence of external electric field plays an important role in
the nucleation rate of water vapour condensation and ice glaciation. The increase in polarizability results in the increase in
Gibb's free energy, and hence the increase in nucleation rate, but at the same time there is a decrease in relaxation time. The
15
polarizability enhances the nucleation rate and equilibrium concentration, e.g. for a given relaxation time 5xi0- s, the
9
9
radius of critical nucleus is found to increase from 27.72xl0- em to 81.07xl0- em when electric field increases from 10
esu to 50 esu, whereas Gibb's free energy increases from 3.22xl0- 15 erg to 27.54xl0- 15 erg for same increase in electric
field .
Keywords: Polarizability; Nucleation ; Water vapour nucleation ; Condensation; Ice glaciation; Gibb's free energy
PACS No.: 92.60.Jq; 92.60.Nv
1 Introduction
1
Murino pointed out the effect of an electric field
on the condensation of water vapour and concluded
that under a constant temperature a bigger sized drop
can be produced in a given time. Water is a strongly
polarizable medium. The polarizability a plays a
dominant role in inducing the electric dipole on the
water vapour molecules. Murino assumed a as
5x I o-23 cm 3 from the nucleation phenomenon. The
effective role of ions in thundercloud electrification
2
was considered by Sapkota and Varshneya ·3 .
Recently the value of polarizability a has been
2
modified as
a+~. It is found that the critical
3kT
nuclei have much less radii than in absence of electric
field. Singh et a/. 4 considered the resultant effect due
to an external electric field and the field due to central
dipole. They found that the critical nucleus is attained
in a time less than in the absence of electric field .
Sharma et a/. 5 suggested that a small value of electric
field is equivalent to very high supersaturation ratio to
get a nucleus of given size under similar conditions of
temperature.
Moore et al. 6 observed an abrupt increase in rate of
rainfall, sometimes shortly after a stroke of lightning
and proposed an explanation for the close association
between lightning and rainfall. Mathpal and
Varshneya7 explained
that
the
thundercloud
electrification by the combined precipitative and
convective charging mechanism play an important
role in the electrification of cloud. Szymanski et a!. 8
observed the precipitation intensification associated
with lightning. Kumar and Mukku 9 assumed that the
electric field grows inside Montana cumulonimbus,
using a parallel plate capacitor model. It was found
that the results of laboratory charge transfer
experiments of ice crystal, hailstone non-inductive
mechanism can explain the observed electric field
growth. Bartlett et a/. 10 as well as Maybank and
Barthakur 11 reported the tendency of generation of a
few ice splinters during ice crystal growth in strong
electric fields. Doolittle and Vali 12 considered the
effect of an electric field on samples of water at
13
different time-temperature cycles. Smironova et a/.
presented a model of lightning generated electric
fields . Boteler and Pirjola 14 estimated the electric and
magnetic fields produced in technological systems on
the earth's surface by a wide electrojet. The mass
dependent effects of parallel electric fields in ion
15
conic production has been discussed by Lund •
16
Jalaluddin and Sinha , and Parmar and
Jalaluddin 17 ' 18 have pointed out that the electric field
19
induces nucleation rate in a liquid. Pruppacher and
20
Salt observed the freezing of water due to pa~sage of
electric discharge through it. Evans 21 experimentally
INDIAN J RADIO & SPACE PHYS, FEBRUARY 2004
44
demonstrated the effect of an electric field on the
production of ice crystals in cloud chambers and
argued that the accelerated charged water molecules
move to the crystal tips, thereby increasing the
nucleation rate. Sharma et al. 5 evaluated that the
nucleation of water droplets in electric field is very
effective. Electric field is more efficient than
supersaturation. It has been shown from equivalence
of supersaturation ratio with electric field. The
purpose of the present work is to estimate the
modified value of polarizability (Ucrr). Using this
value the moment induced on water vapour molecules
in presence of external electric field is increased and
hence the increased values of nucleation rate and
equilibrium concentration of critical nuclei are
observed; whereas the size of nuclei and Gibb's free
energy for formation of critical nucleus are decreased.
The calculations have been made for these parameters
as the function of external electric field and also as
the relaxation time at given temperature.
condensation on nucleation makes a droplet. The rate
of change of mass of droplet is
••
000
... (l)
where, rw is the radius of water embryo, and E the
induced electric field .
The moment induced on water vapour molecule in
thundercloud is
. .. (2)
where, a is polarizability.
According to Langevin's formula
(6)
.
1
t o - 5 em, mw = I o-23 g, <Xerr = 7 .9 x t o -23 em 3 ,
usmg
11. =
T = 273 K, where T is the relaxation time.
The nucleation rate l w has been calculated using the
relation given by Pruppacher and Klett 22 as
000
The moment induced on a droplet m an external
electric field is
(5)
where, mw is the mass of water vapour molecule.
The critical radius of water nucl eus is
2 Theoretical consideration
M'=aE
0
(7)
where, esar.w is saturated vapour pressure over water,
a.: the condensation coefficient, P w the density of
water, NAthe Avogadro's number, Mw the molecular
weight of water, crwtv the surface tension between
water and vapour interface, Sv.w the supersaturation
ratio, R the universal gas constant, T the temperature
and k the Boltzmann constant.
This calculation becomes more tedious; hence
logarithmic value of l w is calculated instead.
Similarly, the equilibrium concentration of critical
nuclei is given as
C[n:] = C(l) 0 exp[ -fiG: I kT]
00
0
(8)
2
a
elf
=a+~
3kT
... (3)
where, Po is dipole moment of water molecule; k the
Boltzmann constant and T the temperature.
The electric field generated by embryo dipole at a
point distant rp from embryo is
where, C(l )0 is concentration of monomers (water
the number of water
vapour molecules), and
vapour molecules in a critically sized nucleus.
Similar calculations can be made for ice glaciation
(water vapour ---7 ice) replacing rw ---7 r; , P w ---7 p;, Sv. w
---7 S v,;. crwtv ---7 cr;tv and e sar.w ---7 esar.i·
n:
3 Results and discussion
... (4)
Due to condensation of water vapour molecule, the
embryo acquires a critical size, called nucleus. Further
Lightning is a common phenomenon, one is
familiar with. This occurs due to generation of intense
electric fields in thunderclouds, which are convective
cumulonimbus cloud with vigorous updrafts and
downdrafts. Each thunderstorm is constituted of
SINGH & SINGH: POLARIZABILITY & NUCLEATION OF WATER Y APOUR CONDENSATION & ICE GLACIATION
several such cells. The mean life time of one active
thunderstorm cell is of the order of 30 min. The
electrification of thundercloud is essentially the result
of a micro-scale charge separation and then
subsequent macro-scale separation of opposite
charges by air and/or by particle motion. There exists
an electric field inside thundercloud. The electric field
induces an electric dipole moment on both water
droplet and surrounding water vapour molecules. The
polarizability is temperature dependent.
The polarizability is an important parameter in
electric field induced nucleation phenomenon.
Neglecting the vibrational motion, the value of
polarizability a has been taken to be Sxl0-23 cm 3 at
273 K. Introducing the vibrational motion, the
polarizability may be written as
2
a
=a+J!.g_
... (9)
3kT
cff
where, p0 is the dipole moment, k the Boltzmann
constant and T the temperature.
The calculated values of Ucrr varying with
temperature are shown in Table 1. It is evident from
the Table that Uerr decreases inversely with increase in
temperature.
At a constant electric field, the values of radius r~
and Gibb's free energy ~c: of a critical nucleus are
found to increase with increase in time (Table 2). For
example, for a time-increase from lOxl0-6 s to
Table 1--Calculated values of <Xerr for different temperature using
ao = 5xl0- 23 cm 3, Boltzmann constant k = 1.381 x l0- 16 erg K- 1,
electric dipole moment p = 1.81 x l0- 18 esu
T(K)
243
253
263
273
283
20xl0-6 s the critical radius increases from
81.07xl0-9 em to 128.69xl0-9 , whereas critical
Gibb's free energy increases from 27.54xl0- 15 erg to
69.39xl0- 15erg. But, as the time advances, the
increase becomes slower. Thus, for time-increase
from 40xl0-6 s to 50xl0-6 s, the critical radius
increases from 204.29xi0-9 em to 237.06x l0-9 em,
whereas critical free energy increases from
174.87xl0- 15 erg to 235.47xlo- ' 5 erg.
Similarly, for a fixed time, the. radius r~ and
Gibb's free energy ~c: of a critical nucleus are
found to increase with increase in electric field
(Table 3). For example, for a given relaxation time
5xl o-6 s with the increase in electric field from 10 esu
to 50 esu, the radius of critical nucleus increases from
27.72xl0-9 em to 81.07xl0-9 em, whereas the Gibb's
free energy for the formation of critical nucleus
increases from 3.22xl0- 15 erg to 27.54xl0- 15 erg.
The variation of critical radius r~ with relaxation
time as the function of temperature and the effective
polarizability has been shown in Fig. I. The radius
has been found to increase with increase in time, but
rate of increase is faster in the beginning and slower
at a later time.
3.1 Relation between Gibb's free energy for the formation of
critical embryo (or nucleus), with nucleation
rate/relaxation time
The nucleation rate is the rate of formation of
critical nuclei, i.e. the number of nuclei forming per
unit volume per unit time.
The Gibb's free energy for the formation of an
water embryo is due to the phase change from vapour
to liquid. The Gibb's free energy for phase change for
a spherical water embryo is given by
4
3
2
~Gw =-3nrw~G, +4nrwawlv
8.254
8.125
8.006
7.896
7.794
45
... (10)
where, rw is the radius of water embryo, crw/v the
interfacial surface free energy or macroscopic surface
Table 2--Calcu lated values of radius ~~ and Gibb's free energy,
Table 3-Calculated values of radius r~ and Gibb's free energy,
~G:
~G:, of a critical nucleus at relaxation time 5x l0-6 s as the
of a critical nucleus at external electric field, £=10 esu, as
the function of time at 273 K
function of time at 273 K
t (xi0-6) s
r~ (xl0-9 em)
~G: (xl0- 15 erg)
E (esu)
r~ (xi0-9 em)
15
~G: (x 10- erg)
10
20
30
40
50
81.07
128.69
168.64
204.29
237.06
27.54
69.39
119.16
174.87
235.47
10
20
30
40
50
27.72
44.01
57.67
69.86
81.07
3.22
8.11
13.93
20.45
27.54
INDIAN J RADIO & SPACE PHYS, FEBRUARY 2004
46
250 ~----------------------------,
... (13)
T= 263 K
200
150
where, Mw is the molecular weight of water.
Corresponding Gibb's free energy for the formation
of critical nucleus is
•'"i 100
... (14)
50
This is the classical estimate of the amount of
energy, which must be supplied by fluctuations in the
metastable motherphase in order for nucleation to
occur. This represents the energy barrier to
nucleation.
The nucleation rate is expressed as
eu
"b
....
)(
..
0
0
20
40
60
•( x 10-<;)s
Fig. I -Variation of radius
r; or critical nucleus wit h relaxation
time as the function of effective polarizabillity, acrr and the
temperature, T, at constant electric field I 0 esu
tension at water vapour interface and
free energy per unit volume and
= PwRT
jj,G
v
In sv.w
/j,Gv
the Gibb's
... (15)
where, W -1 is the flux of water molecule to the
nucleus, and C(n:) the nuclei per unit volume.
This expression assumes the Gaussian distribution.
but in steady state the exact expression for nucleation
rate may be written as
... (11)
1 w = ZC(n: )W
where, Pw is the densi ty of water, R the universal gas
constant, T the temperature and S v.w the
supersaturation ratio of water vapour over the water
surface.
The first term in Eq. (1 0) represents the free energy
released due to phase change from water vapour to
liquid water called volume free energy, while second
term is the energy supplied to the embryo for the
formation of its surface, hence called surface free
energy.
The variation of Gibb's free energy for the
formation of a water embryo with its radius represents
a Gaussian curve. At a certain value of radius rw = r~,
-1 Q
... (16)
where, Z is the Zeldovitch factor.
The Zeldovitch factor is obtained by expanding
C(nw) about the minimum in a Taylor series through
terms of the second order in nw. This produces a
Gaussian
approximation
to
curve
in
that
neighbourhood, given by
... (17)
The Zeldovitch factor for spherical water nucleus is
given by
the change in Gibb's free energy (fj,G) becomes
maximum and one can have, at this radius,
Z
=(
~~:~w
1/ 2
)
= ~: ( ~;v
1/2
)
.. . (18)
. .. (12)
where
This radius is called the radius of the critically
sized embryo (or nucleus).
The radius of the critical nucleus is given by
. .. (19)
SINGH & SINGH: POLARIZABIUTY & NUCLEATION OF WATER VAPOUR CONDENSATION & ICE GLACIATION
Collecting results 22 , the nucleation rate of water nuclei
from vapour may be expressed as
. .. (20)
Other versions for spherical germs (nuclei) include
J w= c satw
.
w j, vw [crw/v ]1/2exp[- ~c:]
kT
kT
. . . (21)
and
J
w
=~(
2N~M7r wcrw,v ]''2( e,"t.w
J2Sv.wex p[- ~c:]
Pw
RT
kT
... (22)
This is same as Eq. (7).
In the kinetics of new phase formation, a
substantial supersaturation of motherphase is required
in order to provide the nuclei necessary for initiating
the growth of new phase. The supersaturation is
necessary, because at saturation, the nuclei are
unstable with respect to the decomposition into simple
molecules. Volmer and Weber 23 originally pointed out
that the rate of nucleation at any given supersaturation
could be related through the thermodynamic
considerations to the free energy of critical nuclei
formation (~c: ) . The free energy of formation of the
critical nucleus can be calculated approximately using
values of the macroscopic surface tension of the bulk
new phase (water).
Volmer and Flood24 and others have verified the
theory for a number of nucleation processes. Becker
and Doring 25 have refined the theory by use of the
kinetic theory to calculate the condensation rate onto
the embryo, the thermodynamics to calculate the rate
of evaporation of molecules from it.
Both the treatments of the nucleation process are
based on the steady state assumption. The agreement
between theory and experiment in numerous
applications indicates that the steady state assumption
is ordinarily adequate. However, data for some very
rapid processes involving the formation of aerosols
indicate that, at supersaturations sufficient to provide
nucleation according to the steady state theories of
Volmer26 and Becker-Doring25 , there may be a
measurable delay in nucleation. It further appears that
a similar delay may be important in those crystal
nucleation processes, where the viscous flow process
47
is slow or where very low accommodation
coefficients for the accretion of molecules are
involved.
Probstein27 , Kantrowitz28 , Farle/9 , Christiansen 30
and Wakeshima31 have shown that such delays in
nucleation may be anticipated on the basis of the
relaxation time required for the attainment of the
steady state concentrations of the embryos of the
various sizes, upon which the classical calculations
are based .
The exact calculations of the relaxation time,
however, turns out to be prohibitively difficult and
each of the above investigators has introduced
differing simplifying approximations and methods of
calculation. Kantrowitz 26 , for example, neglects the
thermodynamic barrier due to the free energy of
formation of the nucleus and calculates the time
dependent nucleation rate on the basis of the kinetic
theory alone. However, the final results of the several
methods of calculation have the common properly
that an approximate relaxation time may be found ,
which is independent of the free energy of formation
of the nucleus.
The expression for the relaxation time has been
derived by Collins 32 , as
T =
*2/3
'2
97r kT /l w f JlwCf w/v
{3
.. . (23)
w
where, n~v is the number of water molecules in a
critical sized nucleus. Here
Pw
is the frequency of
250
a.rr= 8.006 x 10-23 cm3
T=263 K
200
Ci
...
Gl
150
"'-b
....
)(
:--.
C)
100
<J
50
0 +------,--- - -- -,--- - - ---!
0
20
40
60
't( X 1 ~)5
Fig. 2 -
Variation of 6G: Gibb' s free energy with relaxation
time as the function of effective polarizabi lity, CX.:rr and the
temperature, T. at constant electric field I 0 esu
INDIAN 1 RADIO & SPACE PHYS, FEBRUARY 2004
48
collision of single water molecules per unit area,
given by
2/3
. 1
-4- 7 r
3mw
J
-4n Pw
(
w
. .. (25)
)
.. . (24)
where, n 1 is the concentration of water vapour
molecules, and mw the molecular mass and
where, Pw is the density of water.
Similar variation has been found between Gibb's
free energy and relaxation time (Fig. 2) and also, the
variation of nucleation rate and the relaxation time as
30~------------------------------,
7
6
23
5
25
-+- a.rr = 8.006 x 10'21 em3 , T = 263 K
...._ a.rr = 7.896 x 10' em
3
,
T= 263 K
T = 273 K
.,.._ a.rr = 7.794 x 10-'3 em3 , T
z
Ci
... 20
283 K
.."'...
Cl)
4
0
~
3
15
.•
C)
<l
2
10
5
1
0
-10
10
10
0
50
30
20
30
40
50
E(esu)
Fig. 3 - Variation of nucleation rate loge l w with relaxation time
as the function of effective polarizability, <l;,rr and the temperature,
T, at constant electric field lO esu
)(
Variation of radius ~c: Gibb's free energy with
electric field as the function of effective polarizability, CXerr and the
temperature, T, at constant relaxation time'! == 2xl0-6 s
90,-------------------------~
0.8 . - - - - - - - - - - - - - - - - - ,
80
0.7
.
50
...,J
40
Cll
a.rr = 8.006 x 10"23 cm3 , T = 263 K
...._ a.rr= 7.896 x 1<f 3 em3 , T = 273 K
....,. a.rr= 7.794 x 10'23 em 3 , T = 283 K
0.6
60
0.5
0.4
.2
0.3
30
20
0.2
10
0.1
0
0+---~---~---r-----~--~~
0
10
20
30
40
50
E (esu)
Fig. 4 -
-+-
T=263 K
70
eu
.b
...
:--...
Fig. 5 -
Variation of radius r~ of critical nucleus with electric
field as the function of effective polarizability, <lerr and the
temperature, T, at constant relaxation time'!= 2xl0-6 s
0
10
20
30
40
50
E(esu)
Fig. 6 - Variation of nucleation rate loge lw with electric field as
the function of effective polarizability, CXerr and the temperature, T.
at constant time'!= 2xl0-6 s
SINGH & SINGH: POLARIZABILITY & NUCLEATION OF WATER VAPOUR CON DEN SATION & ICE GLACIATION
the function of <lerr and the temperature T (Fig. 3). In
Fig. 3, the variation has been sho~n for loge lw and
not lw. Since the calculations for lw values are
tedious, values of loge lw = -~c: I kT have been
evaluated instead. Similar variations have been shown
in Figs 4, 5 and 6 between
and E, ~c: and E, loge
r:
J w and E as the function of <lerr and temperature T at
relaxation time 2xl0--6 s, respectively.
4 Conclusions
It is evident from the above study that in the
presence of an external electric field, the radius and
Gibb's free energy of formation for a critical nucleus
have been found to increase with increase in electric
field and relaxation time. Also, the nucleation rate and
equilibrium concentration of the critically sized nuclei
increase with the increase in electric field and
relaxation time.
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