April, 1962
AQUEOUS SOLUBILITY AND
entropy-temperature-volume surfaces were developed, the intersections of the primitive and derived surfaces would have to be similar to those
THE
GAS-HYDIIATES
605
described in this paper because the projection of such surfaces onto the temperature-volume plane would have to be identical.
AQUEOUS SOLUBILITY AND THE GAS-HYDIiATES. THE METHANE-WATER SYSTEM' BY D. N. GLEW Contribution No. 66 f r o m the Exploratory Research Laboratory, D m Chemical of Canada, Limited, Sarnia, Ontario, Canada Received October 19, 1061
A model for aqueow solution consistent with the nature of the hydrogen-bonded liquid water solvent is formulated and developed with particular reference to the energetic and volumetrio properties of aqueous methane. The standard enthalpy functions for methane hydrate equilibria are derived and show the hydrate to be CH4.5.75H20. A correlation at 0" ia given between the heat of hydrate decomposition to ice and the heat of solute evolution from aqueous solution for methane and other hydrate-formers, which thermochemically requires that the heat of melting of these gas hydrates M-nH20 is equal to the heat of melting of n mole of ice.
Introduction The tetrahedral nearest neighbor coordination of water exhibited in ices I, 11, and 111, cubic ice and in the solid gas hydrate^,^,^ together with the small energy changes for the high pressure ice transitions, indicate that the water molecule invariably exerts a tetrahedrally directed force field in which the four hydrogen bonds with nearest neighbors do not rupture4 and are not critically sensitive t o angular distortions as large as 31 to -39' from the regular tetrahedral angle. The persistence of general three-dimensional tetrahedral fourfold coordination of nearest neighbors in liquid water postulated by Bernal and Fowler5 has received support from later X-ray scattering6J and Raman spectral measurement^,^^^ which may be interpretedlO in terms of complete tetrahedral fourfold coordination of nearest water neighbors a t lower temperatures with thermal distortion and bending of hydrogen bonds * 2 6 O without significant rupture. Eley" presented an aqueous solubility theory for the inert gases, consistent with the Bernal and Fowler water model, in which the water a t low temperatures was assumed to occupy quasi-lattice sites maintaining tetrahedral coordination and where the solute occupied interstitial sites or cavities formed a t low energy expenditure within the hydrogen-bonded solvent water structure. I n this way Eley was able to physically interpret (i) the invariably positive enthalpy of transfer AHP0(l,+g) of aqueous solute to the gas phase a t lower
temperatures, (ii) the anomalously large positive entropies of transfer AS2O(L-.g) predominantly responsible for the poor solvent properties of water for weakly-interacting solutes, and (iii) the very large solute molar heat capacities responsible for the large negative heat capacity changes ACp2O(ll-.g) characteristic of aqueous solubility. Frank and Evans12 considered the large AS20( I p g ) values in terms of generalized iceberg formation of the water next to the solute and the large aqueous heat capacities deriving from the energy requirement to break down such structures a t higher temperatures. Powell and L a t i m e P showed that the addition of weakly-interacting solutes to water decreased the free-volume of the solution which was interpreted in the sense that the presence of a solute reduces the librational freedom of the water, or that the water molecules are more restricted when adjacent to the weaklyinteracting solutes. Independent evidence regarding the nature of aqueous solvation was deduced from reaction kinetics14 and consequent solubility workl5 independently and simultaneously indicated16 that the nature of the water solvent surrounding weakly interacting aqueous solutes should be likened geometrically to those coordination polyhedra experimentally observed2v3 in the solid gas-hydrates. Such water polyhedra containing no rupture hydrogen bonds provide a natural interpretation of the low energy cavities of Eley's solution theory, l1 consistent with a three-dimensionally hydrogen-bondcd liquid water p h a ~ e . ~ . ' The ~ (1) Presented a t Professor J. H. Hildebrand's 80th Birthday Symobvious cooperative alignment of the water moleposium, Lewis Hall, University of California, Berkeley, September 13, 1961. cules within such polyhedra or hydration shells (2) (a) w. F. Claussen, J . Chem. Phya., 19, 259, 662, 1425 (19.51); likewise provides an interpretation of the larger (b) L. Pauling and R. E. Marsh. Proc. Nafl. Acad. Sci., 38, 112 (1952). entropy losses characteristic of aqueous solutes (3) M. v. Stackclberg and H. 12. Muller, Z. Eleklrochem., 68, 25 and the reduced librational frcedom13 of the water (1954). (4) R. L. McFarlan, J. Chem. P h y s . , 4, 60, 263 (1936). adjacent to the solute, in that the preferred regular (5) J. D. Bernal and R. H. Fowler, ibid., 1, 515 (1933). (6) J. Morgan and l3. E. Warren, ibid., 8 , 666 (1938).
(7) C. L. van Panthaleon van Eck, €1. Mendel, and J. Fahrenfort, Proc. Roy. Sac. (London), A24T,472 (1968). (8) M. Magat, Trans. Faraday SOC.,33, 114 (1937). (9) E. F. Gross, "Hydrogen Bonding," ed. D. Hadti, Pergamon Press. London, 1959, p. 203. (10) J. A. Pople, P r o c . Roy. Sac. (London), 8206,163 (1951). (11) D. D. Eley. Trans. Paraday Soc., 36, 1281 (1939).
(12) H. 9. Frank and M. W. Evans, J. Chem. Phys., 13, 507 (1945). (13) R. E. Powell and W. hf. L a t h e r , ibid., 19, 1139 (1951). (14) D. N. Qlew and E. A. hfoelwyn-IIughes, Proc. Roy. SOC. (London), A211, 234 (1952). (15) D. N. Glew and E. A. Moelwyn-Hughes, Diacuaaiona Faraday SOC., NO. 16, 160 (1983). (16) W. F. Claussen aud M. F. Polglase, J. A m . Chem. Soc., 74, 4817 (1952).
606
D. N. GLEW STANDARD ENTHALPY AND
I,
oc. 0 25 50 75
100
TZBLE I HEATC A P A C I T Y C H A N C E S FOR M E T H A N E EVOLUTION
WinklerMorrison and Billrt AC xo AHzo, caI./i'eg.cal./mole mole
4690 3206 1833 571 - 880
125 ... dACpo -(cal./deg.*-mole) dl'
Vol. 66
-61.6 -59.8 -55.3 -50.6 -43.8
...
+ 0.18
Cluussen and PolglaseCulberson and McKettu
FROhl AQUEOUS SOLUTION
Four seta combined
Eley model
cal./rnole
AHYO,
cal./deg.mole
Allto, cal./rnole
A C )z0, cal.fdeg.mole
449 1 3124 1872 735 - 287 -1194
-57.0 -52.4 -47.8 -48.2 -38.6 -34.0
462 1 3173 1869 707 -312 -1188
-60.7 -55.0 -49.3 -43.0 -37.9 -32.2
-62.7 -57.8 -52.1 -45.7 -38.5 -30.7
+ 0.23
f0 . 1 8 t o 0 . 3 3
ACptQ,
+ 0.18
tetrahedral hydrogen-bond forming directions of the water molecule (ice I, cubic ice) force alignment. arid increase orientation of those water molecules adjacent to the solute in maintaining their three hydrogen bonds with nearest, water neighbors also adjacent to the solute, thc fourth bonds radially directed away from the solute maintain honding of the hydration shell with the external 1)ulk water. Thc central at,tract.ive forccs betwecn the solute arid the surrounding hydration shell furt.her cooperatively stabilize the wat.cr members i i i ;i manner tending to promote intra-shell hydrogen bonding and the allied reduction of hydratioii watcr librational freedom. 111 110 sriise is it. considrrrd t'liat the water molecules adjacciit to the solute arc permanently immobilized or rigid as in solid structures, but rather as being subjept to greater orientational constraints permitting reduced hydrogen bond bending as comparrd with those water molccules within thc bulk liquid. Solubility of Methane in Water.---The implicat.ioiis of this :iqueous solution mod(2l will be examined with reference to t'hc quantitative values for the thermodynamic funothns for thc solubility arid the g:is hydiutc equi1ihi:i of the mcthniie-\\:ater system. .-lnalysis of the unit prcssurc Henry 1:iw coilstants Hifor the aqucous met'hano soluhilit,yequilibrium
ACpxO,
cal./deg.mole
heat capacity, and temperature derivative of heat (enpacity changes for reaction 1 as dcrived by standard methods20 from the solubility temperature equations for the Winkler-3Iorrison and I3illct measurements, for the Claussen and Polglase-Culberson and hlcKetta measurements, and for all data sets combined. I n the final column of 'l'able I are shown thcorctical values for ACp2O(l,+g) calculatcd by IZley's methodll using the equation
which had been derived assuming that the aqueous methane heat capacity is equal to the sum of that heat capacity CUP due to the methane inside the water cavities considered as square wells plus the heat capacity C, necessary to maintain the cavity volume Vfll against thermal collapse. at1 and 01" arc the respective coefficients of thermal expansion and compressibility of the water solvent21 and Vfll is thc aqueous methane molar volume dcrived in the next section. The measure of agrecmciit between the standard enthalpy changes and their tcmperature derivatives in Table I obtained from the two indcpendent solubility data sets allows confidence to be placed in the values for the heat capacity changes ACp2O(l14g)and in the reality of an experimentally significant change of ACp20(ll+g) with temperature, which is demonstrated here for the first time characterized by the generalized standard thermo- for aqueous solute transfer to the gas phase. The dynamic function vhangc denoted A.Y?O(ll+g), excellent agreement of the Eley model values for showed that the measurcmcnts of Winkler" be- both the absolute value for ACp2O(ll+g) and for its tween 0 aiid 50' agreed to within 0.22% with those temperature derivative gives a posteriori support of hforrison and Billet,l* while those measure- for the model which indicatcs that the anomalously mcnt,s of Culberson and Mcl(et8ta1gand of Claussen large aqueous solute heat capacities derive preand Polglasols :ig:iin agree c*losclywith each other, dominantly froin the energy requirement to mainh i t show :I 4..j% greater riiothaile so1ril)ility than tain the solute cavity against thermal breakdown. t,he former :iuthors'. The solubility d a h thus were d dl' , . ACp?(ll+g) fitted with rcapoct t,o tempcruture ixriation equa- The niodd similarly shows that tions in pairs and filially :dl d:it,a ivcrc combined. should be positive for all n.e~ikly-iriter:icting aqucIn obt'aining a t,rue reprcsrnt,ation of tho variation solutes having positive aqucous molar volumes. of aqueous methane solubility with temperature oilsMolar Volume of Aqueous Methane.- -The high it \vas ncccss:vy to usc an equation of the type pressure aqueous methano soluhility measurcme~its log H2 = A,/?' Zj log 'I' CY' I ) , \vhich implies Culberson and McKetjt,:i19were trc:it>rdby standtjhat ACp2"(11-tg) is a l i i i w r frinction of absolute of ard thermodyn:imic met,hotis2?to yicld t,hc aqueous tcmpcraturc 7'. (20) D. N. Clew and R . E, Itobcartson, .I. I'hlln. Chem., 60, 332 In Tnble I arc prestintcd the stiindard enthalpy,
+
+ +
(17) I.. u'.V'inkIer, Ber., 34, 1408 (1901). (18) T. . I . 3lorrisim and 1'. I3illat. ./. Chem. Soc., :381,L (1952). (19) 0. L. C'uIIwrsun and .I. J . ~ l c l i r t t a J. r . , .I. I ' e l r o l . T e c h n o ! . 7'mits. ;1.1..11.1:., 192, 223 ( l ! # . j l ) .
(1086). (21)
h'. E. I h r s e y , Properties of Ordinary \Vntcr-Substance," Rrinhold I'ubl. Corp., New York. N. Y . , 1940. (22) I . R . Kricllcvsky and J. S.Kessrnovsky, J. .Am.Ckeni. Soc.. 67, 2IG8 (1033). 'I
AQUEOUS SOLUBILITY AND
April, 1962
methane molar volumes, denoted Vpll, shown with standard errors in Table 11. These molar volumes fitted as a linear function of T give a methane molar volume V'$l = 34.37 f 0.54 ml. a t 0' and an aqueous methane thermal expansion coefficient deg.-l. or$ = 1180 =t190 X The third and fourth columns in Table I1 show the aqueous methane molar volumes, denoted $$, as directly determined by dilatometryz3 and by d e n s i t ~ m e t r ywhich, , ~ ~ excepting one low value, are 1.6 ml. greater than the VJ1 values, this difference being significantly larger than the experimental errors. Kritchevsky and IliinskayaZ3were first to note this difference for a number of other weakly-in teracting aqueous solutes and have provided a theoretical explanation for the effect. TABLE I1 METHANE MOLARVOLUMES
COMPARISON OF fiQUEOVS t,
"C.
V&, mL./molelg
(34 4 f 0 . 5 )
0
16.8 23.0 25 29.1 35.1 37.8 50 71.1 104.4 137.8
+all,
ml./mole2a
92'1,
ml./mole2*
36 f 0 . 5 33.2 36 3
35.6 f 0 . 4
37 f 0 . 5 38 0 38 2
35.4 f 0 . 4 38 i0 . 5 37.4 f 0 . 8 40.3 f 0 . 9 39 4 f 0 . 5
THE
GAS-HYDRATES
607
expansion arising from the orientation and reduction of bending freedom of the hydration shell water. The presence of the solute within the water structure a t lower temperatures thus is visualized as orienting the hydration shell members and, through radial hydrogen bonding, the shell's near water neighbors, in the sense of reducing the average hydrogen-bond bending from the f26' characteristicla of the bulk to some smaller value, this reduced bending moving second nearest water neighbors further away thereby increasing the water molar volumez5and reducing the water free-volume. l 3 Evidence further supporting a more ordered hydration shell in the vicinity of weakly-interacting solutes is provided by (i) the increasez5of maximum density temperature for water by addition of small quantities of ethyl alcohol, the order in the vicinity of the ethyl group being enhanced, (ii) the increased26viscosity of aqueous solutions of benzene and of diethyl ether above that for pure water, these liquid solutes themselves being less viscous than water, and (iii) the decreasedz7compressibility of aqueous diethyl ether solution below the value for pure mater, the solute decreasing the bending angles of adjacent water, compression requiring increase of the water bending angles. Methane Hydrate-Water-Gas Equilibrium.Examination of the methane equilibriurn constants for the reaction C&.nHzO(h)
nHzO(11)
+ CHdd
(4)
A simpler alternative is suggested here, consistent characterized by the thermodynamic functioiis with our solution model. Since the 4211-Vvzll AXo(h+li g) showed that no set of hydrate equilibdifference is observed in other aqueous systems it is rium measurements with water yet exists of sufreasonably assumed to derive as a general effect ficient accuracy or over a sufficiently large temperafrom the water solvent rather than from any ture range to justify evaluation of more than the special nature of the solute species. Apparently standard enthalpy change AHo(h+llg) appropriusing the pure water ate to the average data temperature. The hydrate the 4211 values were derivedz3Vz4 molar volume VI(II1for the water in the dilute meth- equilibrium measurements were treated by standane solutions, the difference between this and the ard methods developed earlier33adapted to higher water partial molar volume Vll' having been pressure systems in which corrections for gas phase neglected. When the volume of aqueous methane fugacity, gas and solution phase compositions, and solutions is expressed in terms of the partial molar condensed phase volumes have been included. The quantities, the dilatometric m e a ~ u r e m e n t sand ~~~~ ~ equilibrium constants were fitted in sets as funcsolubility pressure coefficient mea~urernents~g are tions of temperature to yield average AHo(h+llg) consistent provided that values which have been reduced to 0' using33 ACpo(h-llg) = +48.2 cal./deg.-mole. Vlll := Vlo 11 + ($211 - V&) nz/nl (3) The equilibrium measurements of VillardZ8yield a that is, the water partial molar volume V&expands standard enthalpy change AHo(h+llg) = 12,896 linearly with the solute mole ratio of the solution. Such an increase of water molar volume has been cal./mole at 0'; those of Roberts, Brownscombe, ~~ Deatoii and Frost,30 13,093; demonstratedz5 for aqueous ethyl alcohol and is and H o ~ v e ,12,818; absent for aqueous hydrogen peroxide; this locates iL'IcLeod and Campbell,3112,420; the four measureof Kobayashi and were over insufthe expansion in the vicinity of the weakly-inter- ments ficient temperature range to justify an independent acting ethyl group. Accordingly o u r model for the dissolution of (26) H. M. Chadwell, J. Am. Chem. Soe., 48, 1912 (1926). (27) T. W. Richards and H. M. Chadwell, ibid., 47, 2283 (1925). methane in water at 0' is considered as involving (i) (28) P. Villard. Compt. rend.. 107, 395 (1888). interstitial expansion of V,'l = 34.37 ml./mole (29) 0. L. Roberts, E. R . Brownsoombe, and L. S. Ilowe, Oi2 and within the water to form cavities occupied by the Gas J . , 38, No. 12, 37 (1940). (30) W.hl. Deaton and E. M. Frost, Jr., "Gas Hydrates and their methane solute, (ii) this cavity formation being associated with the concommitant 1.6 ml. water Relation to the Operation of Natural-Gas Transmisslon Lines," U.S. (23) I. Kritohessky and A. Iliinskaya, Acta Phvsieochzm. U.R.S.S , 20, 327 (24) W. L. Masterson, J. Chem. Phys., 22, 1830 (1954). ( 2 5 ) A. G. Mitchell and W. F. K. Wynne-Jones, Dzscussions Faraday Sac., No. lS, 161 (1953).
(1945).
Bureau of Xines Monograph 8, 1946. (31) H. 0. MoLeod, Jr., and J. M.Campbell, J. Petrol. Technol., 18,
590 (1961). (32) R. Kobayashi and D. L. Kats, J . Petrol. Teehnol. Trans. A.I.M.E., 186, 66 (1949). (33) D. N. Glew, Can. J. C h e m , 38, 208 (1960).
D. N. GLEW
608
Vol. 66
cnthalpy change evaluation. These values of AHo(h+llg) were weighted in direct proportion to the number of measurements to yield a mean value and standard error AHo(h-+llg) = 12,830 140 cal./mole a t 0". Methane Hvdrate-Water Eauilibrium.-The equilibrium reaction
gests that the interaction energy of aqueous methane with its hydration shell in solution is equal to that of methane with its solid coordination polyhedra in the hydrate. This equality arises from environmental similarlity in respect to coordination and spatial geometry of the solution hydration shells with those completely hydrogen-bonded hydrate polyhedra. Since the aqueous methane CHd*nHzO(h) nHzO(l1) CH4(11) (5) molar volume 34.37 ml. a t 0" corresponds closely is representative of methane hydrate melting to with that of the 20 coordinated sites of both the liquid water and aqueous methane and is character- structure I and TI hydrate lattices3 and is signifiized by t,he thermodynamic functions AXo(h+ cantly smaller than the 40-55 ml. volume rangezbva ll). Reaction 5 also may be considered the differ- for the 24 coordinated sites it is inferred that aqueence of reaction equations 4 and 1 and the func- ous methane is 20 coordinated by water a t lower tion AXO(h-4) the difference AXo(h+llg) temperatures. AXzo(li+g). To ascertain the more general validity of the Each of the hydrate equilibrium measure- solution model already applied to methane, values m e n t ~ ~ *with - ~ * water was evaluated to yield the for AHzo(Il+g) and AHo(h+slg) have been equilibrium constant for (5) and these equilibrium derived from the literature for other hydrateconstants were fitted in sets as functions of tem- formers and are presented in Table 111. The perature. The standard enthalpy changes AHo- second column under C gives the aqueous solution (h+l) from each data set were derived and aver- hydration numbers assigned from the known gas aged as previously to yield the standard enthalpy hydrate ~ t r u c t u r e s . ~All ~ ~ AH2°(11+g) values change AHo(h+ll) = 8228 & 130 cal./mole for have been obtained from the earlier work, l6 except methane hydrate melting to aqueous solution at 0". those values for ethane and propane,18 sulfur diMethane Hydrate-Ice-Gas Equilibrium.-The oxidel38 and ethyl chloride," which have been methane hydrate equilibrium r n e a s ~ r e m e n t swith ~~ derived from primary data. The errors on the ice for the reaction solubility enthalpy changes a t 0" are assessed at *200 cal., except for ethyl chloride where the CHid&O(h) nHzO(s1) CHI($) (6) error is *500 cal. The AHo(h+slg) values have characterized by the thermodynamic functions been derived from the hydrate measurements indiAXo(h+slg), were treated by standard method@ cated, hydrogen sulfides; chlorine, bromine, and to yield the equilibrium constants for (6). I n com- sulfur dioxideae; ethane and propane.* The corbination with33 ACpo(h+slg) = -2 cal./deg.- rected enthalpy change values given by von Stackelmole the standard enthalpy change for hydrate berglo have been used for methyl bromide, methyl decomposition to ice and gaseous methane is AHo- iodide, and ethyl chloride. The errors on the hy(h+slg) = 4553 102 cal./mole a t 0". drate ent,halpy changes are larger and are assessed The difference of reaction equations 4-6 repre- a t *500 cal. sents the fusion reaction of n mole of ice to yield n TABLEI11 mole of water, while correspondingly the difference AND HYDRATE ENTHALPY CHANGEFOR SOLUBILITY AHo(h+llg) AHo(h+slg) = 8277 f 172 EQUILIBRIA cal. at 0" is the standard enthalpy of fusion21of n mole of ice and is n1435.7 cal., whence the hydrate Temperature 0" formula number n = 5.765 f 0.120. This experiC AHaO (li-+g), A H 0 (h+sig), cal./mole GW solution csl./mole mental value for n agrees with the ideal crystal20 4621 4553 CHd lographic value 5.75 expected from structural 20 5140 5550 HzS work.ZaJ The cavities within the methane hydrate 5850 CzHs 24 5560 lattice at 0" thus are indicated by presently avail24 6180 6500 C1, able data to be 99.7% occupied by methane with a 24 7420 7700 so2 standard error of &2.1%, in contrast to the 83% 24 7390 8100 CHIBC occupancy predicted theoretically. 34,36 24 8750 8300 Brz Comparison of Aqueous Solution and Gas Hy28 6860 6300 CsHs drate Properties.-Comparison of the heat re28 8500 7300 CHsI quirement (8277 d.) for the melting of nmole of ice 28 8400 8700 CyHsCl to water with Alio(h+ll) = 8228 cal. at 0" for the melting of solid C€14.nH20to liquid aqueous The correlation in Table TI1 thermochemically methane reveals that the presence of methane in necessitates, as for methane, that these larger, more the hydrate melting reaction little affects the heat polarizable, and dipolar solutes do not significantly requirement for melting of n mole of solid water change the enthalpy of hydrate melting from that to the liquid state. value for the melting of the equivalent n mole of ice. The thermochemically equivalent comparison of Thus in the same way as the hydrate-former stabilAHo(h+slg) = 4553 cal. for reaction 6 with that (36) C. E. Maas and 0. M a w , J . A m . Chem. Soc., 60, 1362 (1928). AH,O(l,+g) = 4621 cal. for reaction 1 a t 0" sug-
*
__
+
-
+
*
-
(34) J. H. van der Waals and J. C. Platteeuw. "Advances in Chemical Physics," Vol. 11, ed. I. Prigogine, Interrcienco Publishers Ino., New York, N. Y.. 1959, p. 1. (35) D. N. Glew, Nature, 184, 546 (1969).
(37) N. Nioloux, Ann. phyeol. phymcochim. biol., 6, 434 (1929). (38) F. E. C. Scheffer and G. Meyer, Verslag Cewone Veroader. Afdel. Naluurk. Koninkl. N e d . Akad. Welenshap., 27,1104,1305 (1919). (39) H. W. B. Rooacboom, Rsc. traa. chim., 4, 65 (1885). (40) M. v. Staokelherg, Naturwiss., 36, 359 (1949).
April, 1962
l ~ E L h T I O NBETWEEN CRYSTALLIZATION
izes the more open hydrogen-bonded solid hydrate lattices, so these aqueous solutes stabilize and orient their hydration shells in aqueous solution. In conclusion it would appear just as liquid water a t lower temperatures has been considered a broken-down ice-like (not necessarily ice I-like) structure maintaining short range order through fourfold hydrogen-bonding of water nearest neigh-
RATEAND LIQUIDSTRUCTURE
GO9
bors, so may aqueous solutions of non-hydrogenbonding non-electrolytes be considered as consisting of individually hydrated solute species, in which the hydration shell adjacent t,o the solute contains no ruptured hydrogen bonds and can be likened to broken-down gas-hydrate type polyhedra maintaining orientational short range order through intra-shell hydrogen-bonding,
ON THE RELATION BETWEEN CRYSTALLIZATION RATE AND LIQUID STRUCTURE BYDAVID TURNBULL General Electric Research Laboratory, Schenectady, New York Received October 1.9, 1081
Most liquids, including many which crystallize to close-packed structures, exhibit, when mote-free, a high resistance to the initiation of crystallization even though the crystallization front, once formed, pro agates a t a vcry high velocity. The implications of this behavior to the validity of various models for liquids afe discusscb: Also the conditions for glass-forriiation, the knowledge on the interior stability of superheated crystals, and the relation between crystal growth rate and viscosity for various types of liquids are reviewed.
I n connection with investigations made by him and his on the crystallization rate of liquid white phosphorus, Professor Hildebrand pointed out that the crystallization behavior of simple liquids has important implications concerning the validity of some models for the structure of liquids. In view of this and the author's own interest in the subject it seems appropriate to try to review the knowledge on the relation of the crystallization rate of pure liquids to the arrangement and structure of the constituent molecules. Actually in the light of some common conceptions of liquid structure the crystallization behavior of simple liquids exhibits some striking features. Perhaps the most striking feature is the extraordinary resistance of these liquids to the initiation of crystallization even though the crystallization front once formed propagates at a very high velocity. I'or example, the rate of initiation of crystallization in mote-free metallic liquids is detectable only if the undercooling exceeds 10 to 30%, depending on the metal, of the thermodynamic crystallization temperature, T,. Further some liquids composed of structurally simple molecules, even though they are very fluid at T,, can be undercooled to the glassy state. I n marked contrast with this behavior the propagation velocity of the crystallization front in these substances is a t least as large 5s expected from the molecular mobility indicated by the liquid-state fluidity and selfdiffusion cocfficicnt. This behavior and what it seems to suggest about liquid state models are discussed more fully in what follows. To begin the mode of crystallization of a pure liquid should be recalled. Thermodynamically (1) J. H.Ilildebrand a n d G. J. Rotariu, J . Am. Chem. Soc., 78, 2524 (1951). (2) R. E. Powell, T. P Gilman, and J. 11. ITildebrand. %bid.,1 8 , 252; ( 1951).
this crystallization is a first-order transition and it occurs, even a t wide departures from equilibrium, by a process of nucleation and growth; that is, it begins at and propagates from certain centers, all reaction occurring a t crystal-liquid interfaces. To describe the rate of crystallization then two constants are needed: the frequency of appearance (nucleation), I , of crystallization centers and the rate, u,of propagation of the crystal-liquid interface from these centers. Crystal Growth Rate.-Experience indicates that in many instances the rate of growth of crystals into undercooled liquids is proportional to the liquid state fluidity, 4. However, to establish a theoretical relation4 between u and 4 some important assumptions must be made. First we suppose that the transfer of a molecule from the liquid to the crystal occurs in the following two steps: (1) reorientation of the molecule to a position favorable for the second step which is (2) the incorporntion of the molecule into the crystal. Assuming that step 2 can occur only a t a certain fraction, a, of the surface sites and that the reaction rate a t these sites is governed by the reorientation process (1) the following expression for u is obtained6 ffD,
u = __ [1
- exp(Ag/kT)]
where Ag is the molecular free energy change, a is the molecular displacement, and D, is the kinetic constant (in units of distance2/time) for the process. When the molecules are structurally complex the molecular reorientation step (1) may require motions very similar to those which are necessary for liquid state self-diffusion. If we assume that D, actually is identical with the liquid state self diffusion coefficient, D,and that D and the fluidity (4) For derivatiori and references to older literature, see references
(3) J. If. IIildebrand, (a) Discusstons Faraday Soc., No, 16, 9 11953): (b) "Growth and Perfection of Cryatals," edited by Doremus. Roberts, and Turnbull, John W h y & Sons,New York. N. Y., 1958, pp.
5 and 6 . ( 5 ) W. B. IIillig and D. Turnbull, J . Chem. l'hys.. 24,914 (1956). (6) D.Turnbull and M. H. Cohcn, "Modern Aspects of the Vitreous State," edited by J. D. Mackenzie, Butterworths, London, 1960, pp.
310-318.
3862.
