Ogiwara Sekiji

Ogiwara, Sekiji

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On the Solid Condensation Nucleus which is not Soluble in Water (Studies on the Formation of Cloud and Rain III)

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On the Solid Condensation Nucleus which is notSoluble in Water (Studies on the Formation ofCloud and Rain III)著者 Ogiwara Sekiji雑誌名 Science reports of the Tohoku University. Ser.5, Geophysics巻 2号 3ページ 158-162発行年 1950-11URL http://hdl.handle.net/10097/44446     ON THE SOLID CONDENSATION NUCLEUS        WHICH IS NOT SOLUBLE IN WATER (STUDIES ON THE FORMATION OF CLOUD AND RAIN                        Sekiji OGIWARA             Institute of Geophysics, Facultyof Science,  TOlioku University                             (Received April 15, 1950)III) § 1. Introduction.  It is considered that the condensation nuclei on which cloud particles are formed are hy-groscopic particles either in solid or in liquid state by solving in water. In fact it is nearly beyond question that those produce fog par-ticles in industrial areas are hygroscopic solu-tion which are produced by combustion. So far solid nuclei which are insoluble in water have been treated as same as water drops of the same size in condensation  processes"), and it has been considered that such solid nucleus cannot act as condensation nucleus in the atmosphere, as the maximum water vap-our pressure on it is higher than that on the hygroscopic nucleus. However, the result of experiment which was made by  juNGFP) using the cloud chamber showed that the non-hygroscopic solid nucleus can also act as con-densation nucleus even though it was requir-ed somewhat higher supersaturation than hy-groscopic nucleus. Moreover  KRAsTANoW(3) dealing with the formation of ice crystals at low temperatures by the aid of the theory of  BECKER and  DOMING showed that the non-hygroscopic solid nucleus is very important in sublimation processes.  When such nucleus is wetted fairly well by water the adhesive force of water to the solid nucleus is larger than the cohesive forceof water, hence the maximum water vapour pressure on the solid nucleus is lower than that on the water drop of the same size. Therefore it is clear that such nucleus can act as condensation nucleus more effectively than the water drop of the same size.  In this paper the author investigated thermodynamically the stability of the thin water film formed on such a solid nucleus and studied its role in condensation processes. §  2. Stability of the Thin Water Film    Formed on the Solid Nucleus.  Consider a spherical solid nucleus with a radius R. We shall first calculate the potential energy of a water molecule at P with respect to this nucleus  (Fig- 1). If we denote the                         mutual  poten-                       tial energy be-               QB                         tween the water    X A  P'  Y molecule and a                      molecule of the                        nucleus at a dis-                       tance  r apart by 012  (r)  , then the        Fig. potential energy     PA=f,  PY—ro of the water  0Q=0Y=R,  RS==h• molecule at P with respect to the entire solid nucleus is equal to  1112(R,  x)= f f  27'192r  sin  d sin 00i209dfSOLID                    DJ-F2Rfro          =27r p2fdf r012(r)dr,  r where  p2 is the molecular density of the nuc-leus (number of molecules per unit volume) and indices 1 and 2  defiote the water and the nucleus respectively. •  In the case of most substance the mutual potential energy of two molecules depends on the VAN DER  WA.A,Ls attractive force and in this case  012 (r) will be expressed as  012 (r)  Ci2   , then         irP2c123 1 1    U12(R, x)  =             12 1  R+  x( x2  (2R+  x)2  +2(11   (1)                (2RA-x)3 — x3  Therefore when the solid nucleus is covered with water film the potential energy of the water molecule at P can be expressed as    U(R,  x).=  Uli(R+  x,  811)  —Uil(R ,  x)+ 1.1-12(R, x),   (2) where  On is the distance between two neigh-bouring water molecules.  The stability or instability of the equilib-rium between the water film and the sur-rounding water vapour can be inferred from the fact that the thermodynamic potential  0 of the system constituted by them, considered as a  function of the thickness of the film  h, for a fixed value of the total number of mole-cules in two phases has a minimum in the former case and a maximum in the latter (just as in the case of the equilibrium be-tween a liquid drop and its supersaturated vapour).   The potential  0  . can be calculated as fol-lows. When the film is a part of a large mass of water its evaporation energy is equal to  -4-  n.{(  R +  h)3  R3}  Pi  U.., where  U., is the evaporation energy of a large mass of water referred to one  molecule. The difference be-tween this energy and the energy of the filmCONDENSATIONNUCLEUS          isthecorrection'10 .N NUCLEUS 359  the c rrection to the normal  valtie of the evaporation energy due to the  small thick-ness and the spherical shell shape of the film. Thus   W(R,  {(R+  h)3  R31  —  f 47rPi(R-Fx)2U(R,  dx-  • (3)  .1 can be defined as the corresponding correction to the potential energy of the film. In the above formula the integrand of the second term of the right-hand side is as follows. For the term corresponding to the first term of the formula (2) which enters in the integra-tion of the second term of (3) it is from 0 to  h, for the second term  from  811 to  h, and the third term from  612 to h.   Denoting the total number of molecules of water and water vapour by  N  (fixed) we thus obtain the following expression for 0 as a function of  h,    0=C/V—4-7rPi{(R+ h)3 —R3)-)0 A+—4rPt  33 •  X  {(R+11)3—R3}0B+  W(R,  h),.....-  •  •  •  (4) where  OA and  $B are the chemical potential  referred to one molecule of the water vapour and that of the liquid water respectively. The thermodynamic equilibrium of the system under consideration is determined by the condition  80=0, which, in  connection with  N=const., leads to the equation  013-0A±  h)  =  O.  (5)   Now we shall calculate the potential U(R,h).                    47Picu(R+h)3     Un(R+h, 811)                   3 842k -F2h-l- 8103            ( 3a„           68L 2(R+h)     Un(R,hPiciiR3)= — 3h3(2R+h)3                     47rP2ci2R3      0.12(R,h)=-- 3h3(2R-4-  h)3 •   However as un(0.1811)(=—nPlCll)is the                     66h i60 S. OGIWARA Ejotential energy of a large mass of water referred to one molecule, it is nothing but the evaporation energy referred to one mol-ecule or U... Again, U12 (R,  h)/Un (R, h)  (=  P2C12/PICii) s the ratio between the adhesive force of water to the solid nucleus and the cohesive force of water, and we shall denote it by e.  Moreover, the product of the surface area of one molecule S and the surface energy of water Us is approximately equal to the half of the evaporation energy referred to one molecule  U... Thus we have SUs= 2—1U,.. If we denote the volume which one water mol-ecule occupies by  vB, then we have  vB=-17r                     u Y,  x( au2v  VBP1  =1 and S---,47r ( a23aB n  \ Therefore we get U.4  vB="Us and 3  On             a on  2  R+h    2vBUs  . As the surface energy Usis  = (R+h)U.. nearly equal to the surface tension  c, we ob-tain the following expression for U(R, h).  U(R, h) =  U.(1 — 2vBa   (R+  h)U.                         Rs+8(e -1)U .   h3(2R+  h  )3 Then equation (5) reduces to    2VBC 34  013R+h    —OA+—8(e-1)U. h3(2RR+ h )3= 0 .  (6)  If we denote the maximum water vapour pressure on the film by and that on the plane water surface at the same tempera-ture T by  P.., then  OB  —OA  —kT  log PR'lb  (7)                   P. where k is the  BOLTZMANN's constant. Put-ting (7) into (6) we have           PRIt2aVB kT log—P.'=R+  —8(e---1)UQ,  Wei   (8)  113(2R+  h)3This is the formula which gives the maxi-mum water vapour pressure on the water film which is formed on the solid nucleus. § 3. Role of the Solid Nucleus in Conden-     sation Processes.  In the above equation if we put  e=1 and  11=0, we have the formula which gives the maximum water vapour pressure on the water drop with a  radius. R, namely      kT toPR  2avB               R'  g and this is nothing but  THOMSON's  formula.  Again if we put  R=  00, the maximum water vapour pressure on the water film which cov-ers the plane solid body will be  obtainedm as  Plt  kT  log  p c:  =  —(e  --1)U..   le  - From  these  special  cases  the  physical mean-ing of the equation (8) will be easily  under-stood. When the nucleus is wetted fairly well by water then and the contact angle between them is zero, when  0  <6.  <1 the nucleus is poorly wetted and the contact angle is  0°,---180", and  when e 0 it is not wetted at all and the contact angle is  180°. Therefore when the solid nucleus is wetted fairly well the water vapour pressure on it will be lowered than that on the water drop of the same size by the second term of the  right-hand side of the equation (8). However when it is poorly wetted the vapour pressure on it is higher than that on the water drop, so it is difficult to act as a condensation nucleus.   In Fig. 2 the relation between the relative humidity  (PR,h/P.) and the thickness of the film h was illustrated. In order to compare with the water drop, relative humidity  requir-ed for condensation on it was also illustrat-ed (Curve I in Fig. 2-a), its radius beingtaken on abscissa the film. 25SOLID  CONDENSATRW  NUCLÉtfginstead of the thickness of 20 ),?v1.5  1.Z .8   1.2  1,1  I  10   0.9  01  07  0.6  as-7  -6 Fig. 2 (a)-s                                       161      h, 3(e—uu.a?a?2                        vE  Fig. 2 is extremely valuable as it illustrates the changes in the thickness of the film or the size of the nucleus as the relative humid-ity changes, and its explanation will be given quite similarly with the case of the conden-sation on the hygroscopic  nucleus"). In Fig. 3 which is reproduced from Fig. 2, if we start-7=a1  1 1101001 1 1,0110,  I 1 744Ilog ,^^••41.•     -7  -6  -5                 Fig. 2 (b)   Relative Humidity over Water  'Minis on Solid  Nuclei, (a) in relation to their Radii  R  and (b)  to  e-1.   Curve  I in (a) illustrates the  relative humidity   over water  drc•, the abscissa of which denotes   its radius instead of the thickness of the film.  When .e>1 each curve has a maximum and the thickness of the film corresponding to it h* is given in Table 1  for the case  .6-1=1 and  T=273.  Table  it  4, 'n.  B a  A0 II411 R PR (cm)  0-4 10-5 10-6 0.17 10-7   0.56  5.6 X  10-8 X  l/09  tie h* /R  h*(  cm) 0.018 1.8  x  10-60.054 5.4  X  10-7  1.7  X  10-7 When be given pressionthe nucleus is not too small h* can approximately by the following ex-                   Fig. 3 with  the relative humidity reprsented by the point A, water film with a thickness repre-sented by H will be formed on the nucleus . If the relative humidity increases the drop will grow by condensation, the change in the thick-ness of the film with change in humidity being represented by the curve in the figure . If the relative humidity is further increased rising over saturation so that supersatura-tion sets in, the drop does not grow indefi-nitely but increases in size, remaining in equi-librium with the supersaturated atmosphere at each point on the curve until Q is reached . For thickness greater than h* the equilibrium relative humidity decreases as the drop grows bigger. This is an unstable state , for the larger the drop grows the smaller in the relative humidity with which it is in  equilib-rium. Thus, if the relative humidity is  main-                                           , tained or increased the drop will grow in-definitely and will form a cloud particle.  From the above consideration it is easily )62 S. OGTWARAunderstood that the role of the solid nucleus in condensation process depends upon the relative humidity which corresponds to the critical thickness  h*. The relative humidity referred above was tabulated below.                  Table 2  10-4  10-5  i  10-6  e-    0 100.12 101.2 113.0 330     1 100.12 101.1 110.4 200     10 100.11 101.1 108.7 166    100 100.11 101.0 107.1 141   1,000 100.11 100.9 105.1 126  10,000 100.10 100.7 103.5  116  tO   I  Am  1.1111  v ILIVES'.   2  1111111111 0 ,osiimairsi    -2   -4  1111111111111111111111  -60  2  4  6  S  1  0  ,  12  0-  60m)    Fig. 4 Relative Humidity which is Required to        Condensation on the Solid Nuclei in Relation        to their Radii R and  log(6-1).  The relative humidity corresponding to h* depends both on  e and R, so the relation between them was illustrated in Fig. 4  • for the humidity from 101 to 105  % for every  1  %.   From the above table and Fig. 4 we can   see that the solid nucleus which is wetted   by water lowers considerably the maximum   water vapour pressure on it and can act as   a condensation nucleus in the atmosphere.     It is our regret, however, that the value   of  e has been measured for no substances,   therefore we cannot decide what kind of sub-  stance is most effective in condensation. As   most solid nuclei contain small cracks and   fissures it is easily presumed that they can   act as condensation nuclei in the atmosphere  even when the value of  e is not so  large.   For instance, with rapidly ascending air, as   in cumulo-nimbus or in thunderstorms, super- satuation of over 5 % may occur and in these  cases the solid nuclei will be able to act as   condensation nuclei.    Finally solid nucleus for which r is less  than 1 it is poorly wetted or is not wetted  at all by water and the water vapour pres- sure on such a nucleus is higher than that  on the water drop of the same size , therefore  it is clear that such particle cannot act as a  condensation nucleus at all. •   § 4.  Acknowledgment.    It is noted with thanks that the financial  support of the Department of Education was   offered to the present investigation .                  References  (1)  O. C.  SIMPSON Quart. Journ. Roy. Met. Soc.,        67  99  (1941  ).     (2) C.  JUNGE Beitr. z. Gecphys., 46 108  (1936).     (3) L.  KRASTANOW  : Met. ZS., 57 357 (1940).     (4)  J.  FRENKEL  : Kinetic  Theory of Liquids, Oxford        (1947).

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On the Solid Condensation Nucleus which is not Soluble in Water (Studies on the Formation of Cloud and Rain III)

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