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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. References I Murino G, S Afr J Phys (S Africa), 2 (1979) 113. 2 Sapkota B K & Varshneya N C, Meteorol Atmos Phys (USA), 39 (1988) 213. 3 Sapkota B K & Varshneya N C, Indian J Radio & Space Phys, 18(1989)251. 4 Singh N, Rai J & Varshneya N C, Ann Geophys (France), 4B (1986) 37. 5 Sharma A R, Singh N & Pandey S D, Indian J Radio & Space Phys, 21 (1992) 218. 6 Moore C B. 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