<^cl\rt^-m NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1368 SUPERSATURATION IN THE SPONTANEOUS FORMATION OF NUCLEI IN WATER VAPOR By Adolf Sander and Gerhard Damkohler Translation of " Ubersattigung bei der spontanen Keimbildung in Wasserdampf," Die Naturwissenschaften, vol. 31, nos. 39/40, Sept. 24, 1943. UNIVERSITY OF FLOPin A JMENTSDER^ jT j- ^^N SCIENCE LIBRARY u 7011 --'ILE.FL 32611-7011 USA Washington November 1953 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 13 68 SUPERSATURATION IN THE SPONTANEOUS FORMATION OF NUCLEI IN WATER VAPOR*! By Adolf Sander and Gerhard Damkohler I. STATEMENT OF THE PROBLEM According to experience, a certain supersaturation is required for condensation of water vapor in the homogeneous phase; that is, for incep- tion of the condensation, at a prescribed temperature, the water- vapor partial pressure must lie above the saturation pressure. The condensation starts on so-called condensation nuclei. Solid or liquid suspended par- ticles may serve as nuclei; these particles may either a priori be present in the gas phase (dust, soot), or may spontaneously be formed from the vapor molecules to be condensed themselves. Only the second case will be considered below. Gas ions which facilitate the spontaneous formation of nuclei may be present or absent. The supersaturations necessary for spontaneous nucleus formation are in general considerably higher than those in the presence of suspended particles. The condensed, thermodynamically stable phase pertaining to water vapor below 0° is the ice. According to all experiences so far, one must nevertheless ass\mie in this temperature range the spontaneously formed primary particles to be predominantly liquid, in accordance with Ostwald's law of stages. The question is whether this is still valid at arbitrarily low temperatures, or whether not perhaps, after all, below a certain tem- perature the nuclei themselves already originate as (minute) crystals. This problem has so far been treated only theoretically^ but not experi- mentally. Thus the supersaturation of water vapor required for sponta- neous formation of nuclei was measured in a temperature range as wide as possible, namely between +35° and -75° • *"Ubersattigung bei der spontanen Keimbildung in Wasserdampf . " Die Naturwissenschaften, vol. 31, nos. 39/^+0, Sept. 2}\ , 19^+3, pp. ^60-^4-65. From the department for motor research of the Hermann Goring Institute for Aviation Research. 2r. Becker and W. Doring, Ann. Physics 2}\, pp. 719-752, 1935. - M. Volmer, Kinetik der Phasenbildung, p. 200 ff. Dresden-Leipzig 1939 • NACA TM 1368 II. METHOD OF INVESTIGATION AND APPARATUS The operating principle was as follows: Especially purified air of known pressure, known temperature, and water- vapor content (lying still below saturation) is adiabatically expanded to a certain terminal pressure. During this process cooling off and, also, for a corresponding high expan- sion ratio, supersaturation occur. If the supersattiration is sufficiently high, fog formation will be observed. The critical supersaturation / P]_ partial pressure of water vapor at the final temperature \ \ p^ saturation pressure of water vapor at the final temperature/ is attained when due to the expansion about one fog droplet per cubic centimeter and second becomes visible. The purification of the intial air and the preadjustment of the water- vapor content were carried out according to the scheme represented in figure 1. This scheme also contains the various methods of operation used. The suspended particles of the initial air could be removed optionally either by a Schott bacteria filter (G5 on 3^ fictitious pore diameter according to Bechhold = l.USp.) or by a layer of absorbent cotton of lOO-cm length (it is true that such a layer of only lO-cm length also had proved to be sufficient) . The drying was done in two exchangeable freezing traps containing a cotton wool filter of 5-cm length and cooled by liquid oxygen. The water-vapor saturator through which was sent the entire air or only a partial flow line (for the latter case the two flow manometers) con- sisted of two washing bottles connected in series with adjoining moist cotton wool filter; the entire arrangement was kept at a certain temper- ature by a Hoppler thermostat. The final adjustment of the water-vapor content was made in the sepa- rator as shown in figure 2 which was kept at the same temperature as the observation sphere b proper. The gas flow leaving the separator a, being fully saturated there, was no longer completely saturated later in the expansion sphere b since a pressure drop of about lO-mm Hg appeared in the capillary connecting tubing between a and b at the steady flow velocity of 5- cubic cm/sec used for flushing through and filling. A methanol bath was used as a cold thermostat which was disposed in a large Dewar vessel (500-mm height, 250-mm inside diameter) with observa- tion strips. The cooling agent was liquid oxygen which was from time to time injected into an immersed glass tube g. The stirring was done mechanically, by means of an electrically driven stirrer of propeller type f. The temperature measurement was performed with a Hg- thermometer cali- brated at the PTR or with a self-manufactered WH -tension thermometer con- nected to it which was moreover compared with a second model arrangement. NACA TM 1368 For expansion of the gas imder investigation, in the ohservation sphere h (O.Jl) the glass stopcock c (boring 10-mm) was opened quickly toward a large prevacuum vessel (l2Z) (not shown in figure 2) in which various pressures could be measurably adjusted. In special tests with an expansion sphere of the same size as the observation sphere which contained, however, a metal membrane manometer for mirror reading, it was possible to determine with the aid of films that the expansion time lasted about 0.1 second and the subsequent time of constant pressure more than O.3 second. No gas vibrations were observed with the connecting tubing used (about 2-m length and 20-mm inside diameter) between expansion sphere (O.^l) and prevacuum vessel (l2Z). The observation sphere was coated on the outside with a black lacquer (graphite + vinidur adhesive solution PC 20) in order to keep off scattered light. The illuminating light ray came from an arc lamp through a lens system, entered from below into the observation sphere b through the observation strips of the Dewar vessel and was lost in the expansion cock c. In the first tests, we had operated with a small film projector. However, the intensity of light of that projector was found to be too slight to recognize reliably the condensate particles which are extremely small just at low temperatures. The observation was made obliquely from above through the observation strips of the Dewar vessel. The content of ions of the expansion gas was either the natural one or it had been reduced to zero in the customary manner, by applying a field of about 50 volt/cm. For this purpose, two opposite inner segments of the observation sphere had been silver-plated and connected with four B-batteries in series (^ 500 v) by platinimi fused through the wall. The observation sphere as well as the entire remaining apparatus could be pumped out with a low-absolute-pressure aggregate, and could then, after it had been left standing for a while, be examined as to density by means of a Geissler tube. III. TEST RESULTS If figure 3^ the critical supersaturations measured p /p are rep- resented as a function of the absolute temperature T. Therein p^ sig- nifies the saturation pressure of the supercooled water as it was taken from Robitzsch's tables^. Only for the curve branch on the upper right, with a jump at the onset, reference was made to the satxiration pressure ^M. Robitzsch, Ausfuhrliche Tafeln zur Berechnung der Luftfeuchtigkeit. Leipzig 19^1. NACA TM 1368 of ice, again using Robitzsch's figures. In order to exclude systematic errors as far as possible the measuring points were obtained by very dif- ferent methods. There were three possibilities of variation: (a) The type of air purification and preadjustment of the water-vapor content according to the scheme in figure 1 (marked by capital Latin letters) . (b) The type of final water-vapor content according to the scheme in figure 3 (marked by Roman numerals) . (c) Selection of the initial temperature in the observation sphere so that for a certain expansion end temperature various temperature dif- ferences (from 2k° to 35°) could be adjusted between center and wall of the sphere . In figure 3^ "the measiiring points are distinguished only with respect to variation possibility (b) . However, none of the methods used for adjust- ment of the water- vapor content shows any systematic deviations. On the contrary, all measuring points lie so satisfactorily about the solidly drawn ciirve of mean values that one is quite justified in excluding a falsification of the measured values by insufficient purification of the air (variation possibility (a)) or by insufficiently adiabatic expansion (variation possibility (c)). Only at the lowest temperatures the meas- uring points show somewhat more scatter the cause of which is, however, in the poor visibility of the condensate particles, reduced more and more with decreasing temperature. In the temperature region investigated, the critical supersaturations measured P]_/p ( speed of nucleus formation J = 1 particle/cubic cm/sec) can be satisfactorily represented by the following interpolation formulas: In -^ = 1^ - 1.521 above -62° without ions^ (l) p T In — = ^ - 1.537 above -62° with ions^ (2) p T 00 In -J- = 1312 _ 3.7I18 below -62° with or without ions5 (3)' p ■ T p = saturation pressure of liquid water. 5p = saturation pressure of ice. NACA TM 1368 From the curves in figure 3 one can read off: 1. The influence of the gas ions favoring condensation disappears at -62°. 2. At the same temperature, a break in the supersatiiration temper- ature curve appears, in such a manner that the supersatioration pressures measured at lower temperatures may lie higher but certainly not lower than one should expect on the basis of the curve branch valid at higher tem- perat\ires. (Compare the dashed extrapolation curve.) In addition to these two quantitative findings there is a qualitative one: 3. At very low temperatures, one finds a scintillating of the con- densate particles; at -62° it is observable with certainty, at higher temperatures one sometimes imagines seeing it. A rigorous temperature limit for the start of sqj-ntillating cannot be defined. IV. DISCUSSION OF THE TEST RESULTS AND COMPARISON WITH THE THEORY USED SO FAR From the quantitative findings 1 and 2, one may conclude that at -62° there starts a more or less sudden change in the spontaneous process of nucleus formation. The disappearing of the ion influence below -62° would suggest that the nucleus fonning at lower temperatures is in a higher order state requiring more space than the nucleus type originating at higher temper- atures, for surely the ion influence favoring the condensation must be understood to mean that the water dipoles in the inhomogeneous field of the ion are attracted and tend to arrange themselves as closely as pos- sible around the latter whereby part of the surface work to be expended for nucleus fonnation is compensated by electrostatic attraction energy. This molecule grouping of maximijm density about a central ion will hardly be the molecule arrangement which must take place in ice and thus also in the crystal nucleus as is suggested by the difference in density between water and ice at 0°. It would therefore be understandable if the gas ions would favor the spontaneous formation of crystal nuclei either not at all or at least less than the formation of droplet nuclei. The scintillating of the formed condensate particles, observed with certainty at -62°, also supports the theory of a primary crystal- nucleus formation although the latter cannot be proved directly by that fact, in our opinion, for a water droplet, too, could suddenly crystallize throughout NACA TM 1368 after a certain time and be transformed into a scintillating minute crys- tal. In what time this would be possible \inder our test conditions, we are not able to tell. The break in the super saturation temperature curve found at -62° likewise points at a sudden variation in the process of nucleus formation. However, the direction of this break is strange and in contradiction to the theory used so far. According to Becker and Doring as well as to Volmer° there should always be favored that type of nucleus which requires for its formation the lesser partial pressure in the vapor phase. This conception has the advantage of representing a perfect analogy to the selection of the condensed phase thermodynamically stable in the respective case where, for a prescribed temperature, there always forms the phase which possesses the lower saturation pressure. However, the present report would indicate another process for the formation of the nucleus because of the required partial pressures, for below -62° there would have originated precisely that type of nucleus which requires for its formation a higher water vapor partial pressure than the type of nucleus stable at higher temperature, as one can recognize by comparing the extrapolation curve plotted in dashed lines with the actual measuring points. According to Becker and Doring as well as to Volmer, the break in the supersaturation temperature curve in figure 3^ seen from below, should not be convex, but concave; however, this precisely could not be observed within the comparatively high measuring accuracy. In the theoretical treatment of the spontaneous process of nucleus formation (in absence of ions), Becker and Doring as well as Volmer start out from the same fundamental physical concept: To a vapor molecule, further vapor molecules attach themselves on the basis of the natural fluctuations in successive single steps. Thus aggregates of a higher num- ber of molecules originate each of which may go over into the next highest aggregate by addition of another vapor molecule, into the next lowest aggregate by subtraction of a vapor molecule. The process of nucleus for- mation itself is interpreted as a stationary chain of reactions so that every aggregate occurs with a certain frequency. Then an expression for the speed of nucleus formation may be derived, in principle, in a simple manner. An explicit evaluation requires, of course, certain simplifying assimiptions; they were made in a somewhat different manner by Becker-Doring and by Volmer. We checked their calculations and arrived under the same physical presuppositions of theory but on the basis of a somewhat more accurate calculation at a new formula. It yields numerical values for the speed of nucleus formation which lie between those of Becker-Doring and of Volmer. In the absence of ions, we have therefore for the spontaneous formation of droplet nuclei the following theoretical relations: 6 "Compare especially the figure on p. 202 of his book (cited in footnote 2) . MCA TM 1368 Becker-Doring {~ to Volmer II ) J = Aj^ ~ ^'^ K kT 3tkT (M Volmer I J = ^l^A "K kT (5) Sander-Damkohler J = 2n. K ZiWiOk, / Ak kT 3TtkT W^ _2 n. k; (6) Wn R k M T 0. 0, K Therein signify: nianber of nuclei formed per cubic cm per second (=» to the number of fog particles observed per cubic cm) number of vapor molecules per cubic cm number of vapor molecules impinging per second on 1 square cm at Nl Pi the partial pressure p 1 W 1 V/2jtMRT Loschmidt number (= 6.022k X 10^3) * gas constant per g-mol (= 8.315 X 10' erg/deg) R/N = Boltzmann constant (= I.3807 X 10" erg/deg) molecular weight of the vapor to be condensed absolute temperature surface of the vapor molecule assumed to be spherical surface of the droplet nucleus assimied to be spherical which is in equilibrium with the external water-vapor partial pressure p. NACA TM 1368 "K ntunber of vapor molecules in the drop nucleus [n„ ^ 100) X vaporization heat per molecule ^for water X k J .k x 10" ■'-3 ergj a surface tension A^ crO^r/3 = work of nucleus formation At^ may, with the Thomson equation dOy- Pt a -^ = kT In ^ (7) (in^ Poo be traced hack to the supersaturation P-i /p wherein p and p signify: ' p^ partial pressure of the vapor to be condensed at the temperature T p saturation pressure of the vapor to be condensed at the temper- ature T With there follows Ok = CnK^'^3 (g) KT 3kT 27 \kT/ I'T^ For spherical droplets there applies with the condensate density d C3 = 36jr( 1 (10) and therewith Ak l6nNL/M\2/a\3 1 kT 3j,3 In mf^ <-> NACA TM 1368 9 According to equations (ll) and (h) to (6), there pertains to a certain super satxirat ion Pt/Poo ^ perfectly defined speed of nucleus formation J. If the latter becomes 1 particle/cubic cm/sec, we obtain the critical supersaturation observed in our measurements which is plotted in figtire 3' In figures ^(a) and ^(b), the experimental supersatiirations (in the absence of ions) of the present report are compared to the theoretically calculated curves. For the latter, the numerical values of MoserT were used for the surface tension of water above 0°. They lie highest among the known values of literature" (compare fig. 5) and are probably, for this reason, too, the most correct ones, all the more so because one can very easily lower the surface tension by slight contaminations with surface-active matter, but is hardly able to increase it. Below 0° the surface tension values had to be extrapolated. As may be recognized from figure ^4-, our new theoretical formula shows the best agreement with o\ir measuring points, at least at and above 0°9. Below 0°, however, our first extrapolation of the surface tension values performed at first arbitrarily (curve branch b in fig. 5) yields supersaturations which are too high. We employed therefore the inverse method. Under the assumption that our new formula ( 6) correctly renders the experimental data in the entire temperature range to -62°, we calculated from them backward the sxirface tension of the water and obtained thus the curve branch c in figure 5- It is pronouncedly curved; however, in view of the still more pronouncedly cambered curves d and c of Ramsay and Shields and of Weinstein-'-'-' for water, and of the glycerin curve f and g-"--^ (glycerin is also strongly associated) this would not be unthinkable. It is noteworthy that the curve branch c has a maximum for the surface tension of the water at about -50°^ that is, not far from the point where the break in the supersaturation temperature curve (compare fig. 3) was found. '''Moser, L. B. Eg. Ila, l48. "Compare also Ramsay and Shields, and Weinstein, L. B. I, 199 • 9 n •^The slight differences between experiment and theory above are most probably real and probably based on the fact that in our stationary chain of reactions for excessive water-vapor partial pressure the molecule aggregates exceeding the nucleus size are overheated because the conden- sation heat cannot be carried off with sufficient rapidity. This point, not yet taken into consideration in the theory used so far, will be discussed more thoroughly elsewhere. ^°L. B. I, 199. 11- L. B. I, 255 and L. B. Ila, I56. 10 NACA TM 1368 For the speed of crystal-nucleus formation Becker and Doring also had derived a formula which is hased on the same fundamental physical concepts as the formula for the fonnation of droplet nuclei. It is true that a considerably larger number of simplifying assumptions was necessary in the derivation of the crystal-nucleus formula because in crystal forma- tion three dimensions may grow independently of each other and an aggregate of n molecules can therefore assume very different shapes, in contrast to the sphere- shaped droplet. The Becker-Doring formula for the speed of crystal-nucleus formation in the absence of ions reads with our above symbols p^ ii J^iz^W^Oi^e ^ (12) Therein At^ = ^'^v/'i represents the work of formation of the solid crystal nucleus. It depends on the interfacial tension between solid and gaseous phase ( still unknown at present) as well as on the surface of the determinative crystal nucleus; the form of the latter must be as compact as possible, according to Becker and Doring, but is not exactly defined. For a cube-shaped nucleus (we, too, shall calculate below with such a nucleus) there results from equation (8) c - 6 l^Y" (13) idN L; and hence from equation (9) Ak 32Nl /M\2/a kT T,3 \d/ VT (iM One should not overrate the importance of single nimierical values obtained with the equations (l2) and (l^); however, a temperature vari- ation of the critical supersaturation (^ for J = 1 cm-3 8-lj NACA TM 1368 11 is significant since equation (l2) represents solely the general Arrhenius expression for a reaction velocity with the activation heat A which K appears perfectly plausible and has been assumed by Volmer for the nucleus formation even before the report of Becker and Doring appeared. In figure 6, there are plotted as fiinctions of the temperature the saturation pressure of supercooled water (ctunre a) and of ice (curve b), furthennore the critical supersaturation pressures measured in the present report (curve c) which are required for spontaneous nucleus formation in the absence of ions, and finally the supersaturation curves d, , ^^o' and d-Q for the spontaneous formation of ice nuclei calculated with the equations (l2) and (lU). The arbitrarily assumed interfacial tensions o = 60, JO, and 80 erg/square cm between solid and gaseous phase correspond to those supersaturation curves. All curves were based on the saturation pressures of the tables of Robitzsch (footnote 5)- One can see that the experimental curve for the supersaturation pressures of the droplet-nucleus fonnation can be intersected by an ice-nucleus curve in the temperature range investigated only when the interfacial tension of the ice crystals lies approximately between 68 and 72 erg/square cm. If it (the_ inter- facial tension) were independent of the temperature, a convex viewed from the abscissa axis break in the supersaturation pressure temperature curve would never occur but always only a concave one; however, such a concave break is precisely what was not found in the experiment. To explain a convex break, one would have to assimie a slight dependence on temperature of the interfacial tension approximately as it is represented in figure 5 as the curve h-'-^. 12rp^g temperature coefficient to be read from figure 5> curve h: -da/dT «« 0.062 erg/square cm degrees is, with respect to order of magnitude, completely in accord with a relation indicated by R. Fricke (Zur physikatischen Chemie, vol. 52, 19^2, pp. 28U-29^) ^ = - nk Z m ^ (15) dT ^ Va ' ^' . wherein n = number of molecules per square cm surface, v. and v , X a respectively = fund^imental frequencies of the centers of the molecules vibrating in the interior of the crystal or on the crystal surface, and the summation L is to be extended over all lattice vibrations. If one assimies that only one distinguishable lattice vibration is decisive and that the molecules situated on the surface are bound normal to it by about half the spring force as the molecules in the interior of the crystal, there applies footnote continued on following page 12 NACA TM 1368 We cannot yet state reliably at present how the break at -62°, indi- cated by our measuring points in figure 3f is to be explained. Should it be based on the transition of spontaneous droplet-nucleus formation to spontaneous ice-nucleus formation - and we have named indications for this being the case - we would have to give up the prevailing notion regarding the cause of such a transition (that always the type of nucleus forms which requires the least supersaturation pressxare) . One will give up this concept at first only reluctantly, particularly because of the above- mentioned analogy with the transition from the vapor-water to the vapor- ice equilibritmi. Nevertheless this notion entertained so far, regarding the transition of one type of nucleus to the other, does not take into consideration a point which seems to us essential: the mobility of the molecules in the nucleus surface. A droplet nucleus of almost spherical shape can form only if the molecules being newly acquired push in between the surface molecules already present, that is, if they are absorbed by the surface. In the case of a crystal nucleus, in contrast, such a pushing- in need not take place since the molecule being newly acquired, is only added on, that is, in principle, adsorbed. The first process presupposes a considerable mobility of the sTjrface particles, the latter does not. If a sort of two-dimensional melting point existed, that is, if the surface mobility of the particles would suddenly disappear at a certain temperature, no droplet nuclei could form any longer below this / In -i « In l/^ = O.3U6I+ since, furthermore, n water molecules \2/3 n = ' 6.02 X 10 ^3 V 18 r^(: 3.33 X 10 22 ,2/3 = 1.033 X 10 15 fall to the share of 1 square cm of the crystal surface, one obtains according to (l5) -dcr/dT «. (1.033 X lO^^j ^1,3807 x 10-^70.3^61+ = O.OU9U erg/square cm deg since the tangential frequencies in the crystal s\irface also will differ somewhat from the corresponding frequencies in the crystal interior, this theoretical value of 0.0U9^ erg/square cm degree would have to be increased slightly and would then come siirprisingly close to our value, inferred experimentally, of O.062 erg/square cm degree. NACA TM 1368 13 temperature and the crystal nucleus would "be left as the only primary condensation form, regardless whether the vapor partial pressure neces- saiy for the formation of this crystal nucleus is higher or lower than that of the droplet nucleus. This conception could explain the strange "break in the supersaturation temperature curve found by us. Also, this explanation does not perhaps imply an invalidation of the nucleus forma- tion theory used so far but merely limits in a special manner the temper- ature range of the droplet-nucleus and of the crystal-nucleus formulas. The two regions would not overlap,- as was assimed a priori by Becker and Doring as well as by "Volmer; rather, the two temperatiire regions would be separated by the melting point of the two-dimensional surface phase. If we denote it in the absolute temperature scale by Tg, corresponding to the "baking temperature" known from fritting processes and if we denote likewise by Tg the standard three-dimensional melting point, there would result from our measurements Tg/Tg = 211/273 = 0.77. This value probably be fitted into the sequence determined by Tammann-^J. can Tb/ts = 0.33 0.52 0.57 0.90 for metals oxides salts C-compounds Translated by Mary L. Mahler National Advisory Committee for Aeronautics 13g. Tammann, Z. angew. Chem. 39, 869, I926 - Gottinger Nachr. Math. naturwiss. Kl. 1930, 227. Ik NACA TM 1368 Compressed air 150 atmospheres absolute pressure ^ 2 Safety value Safety value Bacteria filter •4 Purification 100 cm absorbent cotton Drying ^ Op-trap I 0„-trap 2 50 Y Flow Manometer I -Quantitative measurement Flow Manometer 11 Soturator Heating coils ■Vapor content (dotted) 8 9 f^ F^^ External a , 10 1 II Low -absolute - pressure pump Separator and expansion vessel Mettiods of operation Partial flow line I Partial flow line II Position of the stop cocks 5 6 7 A Bacteria filter Not used © B 100 cm absorbent cotton Not used e e © C Bacteria filter 100 cm absorbent cotton © (0) D 100 cm absorbent cotton 100 cm absorbent cotton Q O) Figure 1.- Air purification and preadjustment of the water -vapor content. NACA TM 1368 15 Low-pressure vessel Saturator Liquid oxygen NH, -tension thermometer ■External air Arc lamp (a) Separator (b) Observation sphere (c) Expansion stop cock (d) Silver plating (e) Feeler ot the NH3- tension thermometer (f) Stirrer (g) Cooling tube (h) Mirror Figure 2.- Apparatus. Id HACA TM 1568 Method of Partial Air ahead Condensate in .,?^7''°' .^ operation flow lines of separator separator spheres * jon«; * I n m n V I or 2 Too humid Yes I or 2 Too dry Yes I or 2 Separator not used 1 Separator not used 2 Separotor not used No O • No X> \» Yes 6 i No 6 i No 9 f 300 280 260 Figure 3.- Waier-vapor s-jpersaturaiions accoriiiie '-: ",5S"-3. NACA TM 1368 17 S 2.5 2.0 a.- Becker and Doring (Vol mem) b-. Sander and Damkbhier ic:Volnner I d: Experiment Second extrapolation of surface tension 320 300280 260 240 220 200 K Figure 4.- Theoretical and experimental supersaturation curves for water vapor. 18 NACA TM 1368 80 0^ 100 Figure 5.- Surface tension of water, glycerin, and ice. 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