NUCLEATION THEORY, REVIEW
ucleation in Phase Transitions V I C T O R K. L A M E R COLUMBIA UNIVERSITY, N E W Y O R K , N . Y .
An
understanding of nucleation processes is fundamental in the preparation of all types of colloidal dispersions by condensation methods. For example, i n meteorology, there i s fog formation and t h e artificial production of rain by seeding with appropriate nuclei; i n metallurgy, t h e initiation and production of new phases which profoundly affect t h e physical properties; and in chemical engineering, supercooling, superheating, overcompression, and production of supersaturation and precipitation by chemical means. Controlled relief of supersaturation and continued maintenance of an unstable phase are matters of practical importance. I n this paper, the conceptual and phenomenological aspects are reviewed in light of contributions of J. W. Gibbs, Ostwald, Farkas, Frenkel, Volmer, Becker, and Doering. More recent contributions of La Mer and Pound, dealing with preparation of very monodisperse colloids, rate of nucleation in polycomponent systems, and further examples of homogeneous and heterogeneous nucleation i n gaseous, liquid, and metallic systems are reviewed i n t h i s and t h e following paper as a background for the papers t h a t follow.
N
UCLEATION is the process of generating within a nietastable mother phase the initial fragments of a new and more stable phase capable of developing spontaneously into gross fragments of the stable phase. Nucleation is consequent'ly a study of the initial stages of the kinetics of such transformations. Nucleation, like ordinary chemical kinetics, involves an activation process leading to the formation of unstable int,ermediate states known as embryos. The critical rate-determining embryo is called a nucleus. A nucleus (or germ) differs from an equal number of normal molecules in possessing an excess of surface energy sufficient to produce t'he aggregate as a new phase in the presence of the mother phase. This energy lvhich is calculable with the aid of a general formula of J. W. Gibbs ( 5 )furnishes the key- to the problem of t'he activation energy. The available experiment'al data are in much better agreement with the Trolmer-Becker-Doering theory than is fully justified by the theoretical uncertaint,ies of the elementary physical concepts. To make further progress, i t i d 1 be necessary to recognize, as Reiss discusses (Zl),that the boundary separating a liquid from a vapor phase is not a geometric surface, but a transition layer in which the properties of the liquid pass continuously into the properties of the vapor. Similar v i e w obtain for a solid-liquid interface. When dealing x i t h a large drop, the thickness of this transition layer is so small relative to the radius of t'he drop that little fractional error is incurred by locating an arbitrary dividing surface anywhere in the t,ransition layer. Under t'hese conditions, Gibbs' formula for the work of formation of a nucleus is sat>isfactory. I t can be shox-n, however, t,hat, in order to make the thermodynamic and mechanical concepts of surface tension correspond, it is necessary t o choose a special dividing surface wit,hin the transition layer, the so-called surface of tension. The radius of the drop is defined in terms of this surface of tension, but this can be located for thermodynamic purposes only by st,atistical mechanical means (averages). When the droplet is of the size of a critical nucleus, the thickness of the transition layer is comparable to the radius, so that errors of 100% in the size of the droplet can be incurred by exercising a freedom of' choice which only restricts the surface of tension to lie within the transition layer. This is the fundamental difficulty in making t,he activation energy of the present classical theory more quantitative and .accept&ble theoretically, since the uncertainties in the frequency factor are of relatively minor importance. Nevertheless, in spite of these theoretical restrictions, the theory xyorks remarltablp well. 1270
H I STOR ICAL
Early in the 18th century, Fahrenheit found that water coul(1 be successfully supercooled in some closed vessels but not in others, for no apparent reason, and also that violent concussion often induced rrystallization. Inoculation with ice crystals or even just opening the vessels would induce crystallizat.ion. Lowitz (1785) extended these observations to glacial acetic acid and demonstrated the ineffectiveness of foreign crystals as compared to the effectiveness of inoculating with crystals of the supercooled liquid. Gay Lussac (1819) found that supersaturated Glauber's salt could be crystallized by introducing various gases; mechanical shock was unsuccessful in sealed tubes, but' successful in open tubes. Gay Lussac's gases probably contained active foreign nuclei of which he >vas not aware. In an)case, by 1850 it was quite generally recognized that the relief of supersaturation could be initiated by the introduction of minute traces of certain foreign bodies which could be classified under those mysterious contact effects called catalytic by Berxelius. Lecocq de Boisbaudron (1866) noted that spontaneous precipitation occurs only in highly supersaturated solution, whereas many slightly supersat'urated solutions apparently never crystallize spontaneously. H e also found, for example, in the case of the sulfates of the bivalent metals like copper and magnesium, that different hydrates could be crystallized from the same supersaturated solution by inoculation with the appropriate crystalline form. Even viith spontaneous crystallization, the most stable phase did not always separate. Gernez (1865-75) noted that an isomorphic nucleus was inore effect'ive than a nonisomorphic nucleus, but required a higher degree of supersaturation than a nucleus of the salt itself. These observations have received their most recent veyification from investigat'ions carried out in the General Electric laboratories on the supercooling of water. To effect the crystallization of supercooled clouds for the purpose of inducing rainfall, Vonnegut (39) searched for a crystal resembling ice most closely in its crystal structure. Silver iodide differs from ice only in minor deviations in the parameters of the crystal lattice, yet these minor differences are sufficient to require that water be supercooled from 4" to 6 " C. to initiate crystallization. Schaefer ($4)and Vonnegut have found that as the crystal habit of the ice-forming nucleus deviates more and more from that of ice greater degrees of supercooling are required; for example, some atmospheric dusts and pollens require a supercooling of 20" C., others 30' C. to initiate the reaction. The General Electric investigators believe that self-nucleation of undercooled water commences at -39" to -41' C.; the latter figure being that found by Cwilong.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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NUCLEATION-Theory, Self-nucleation implies that all known foreign nuclei have been removed. This is always difficult to prove. The evidence for self-nucleation can only be inferred from the reproducibility of results which is indeed very good in Schaefer’s most recent report. De Coppet, in 1872, brought forth evidence that the average life of a supersaturated solution was related inversely to the degree of supersaturation. Later, the rate of nucleation and the size of the critical nucleus will be discussed when considering some interpretations of La Mer and Dinegar (16) on time lags of precipitation in supersaturated barium sulfate. Ostwald ( 2 0 )in a series of ingenious experiments endeavored to ascertain the size of an effective nucleating crystal. H e employed supercooled phenyl salicylate and fine hairs t o transfer his inoculating particle t o the melt. By triturating phenyl salicylate with lactose or powdered quartz and by repeated tenfold dilution with these indifferent diluents he estimated the lower limit to be about 2 X 10-6 cc. He also inoculated sodium chlorate solutions with a platinum spatula upon which measured amounts of successively diluted solutions of sodium chlorate had been evaporated. A limiting grams was indicated. It is a tribute to Ostwald’s mass of scientific judgment that he did not attach much quantitative importance to these findings since he recognized that they represented only upper limiting values. Today it is known from the work of Volmer and Flood that a nucleus of about 80 water molecules is sufficient. The recent interpretations of Christiansen ( 2 ) and of La Mer and Dinegar (16) point strongly to the view that the critical nucleus for barium sulfate is none other than the unit cell consisting of four sulfate and four barium ions, or, as is even more likely, this cell minus one or two of the ions. Ostwald’s contribution t o nucleation rests primarily upon the order and clarity to which he reduced the welter of confused data existing a t the time. H e is responsible for popularizing most of the concepts and ideas concerning supersaturated states that are taught elementary students today, and particularly for popularizing some of Gibbs’ fundamental contributions on thermodynamic stability and metastable states. The following extracts from Ostwald’s summary are of immediate interest for the present theme; they are translated and abridged from (27): “Among supersaturated solutions there are some which, if nuclei are excluded, will apparently last indefinitely under certain conditions, without ever spontaneously forming a solid phase. Such solutions will be called metastable. “There are other supersaturated solutions in which, even if nuclei are excluded, the solid phase will spontaneously appear after a limited time. Such solutions are called unstable. ‘Wetastable solutions always have a lower concentration than unstable solutions of the same substances. Through increase in concentration, therefore, a metastable solution can be converted t o the unstable condition. The concentration a t which this transition occurs may be called the metastable limit. “The metastable limit is primarily a function of the nature of the substances, of temperature and of pressure. In addition, it is affected by other factors that remain to be investigated. “If a supersaturated solution is in the neighborhood of its metastable limit, it will readily precipitate crystals spontaneously when exposed to disturbing effects such as variations in pressure, temperature, local evaporation, or the like. Since any transgression of the limit, in however small a region, may a t once produce crystallization a t that point, with ensuing propagation throughout the mass, the persistence of such a solution does not depend on the average value of the variables of condition, as indicated by our ordinary measuring devices, but upon the most minute deviations in the direction of the limit. Their persistence is therefore determined by the values that these deviations may assume. Hence, and because of the extremely small quantities in
June 1952
Review-
which nuclei are effective, there are many unknown and uncontrolled factors, as such called accidental, which can produce crystallization, often long before the metastable limit for the prevailing average temperature and pressure is reached.” The C. T. R. Wilson cloud chamber in which a fog is produced by the adiabatic expansion of saturated water vapor is a well-known and apt example of metastable limits. Here the effectiveness of foreign nuclei-such as dusts, ions, and particles from radioactive disintegrations-in lowering the limit of necessary supersaturation is particularly easy to demonstrate. When all known foreign nuclei are removed, self-nucleation takes over a t a well-defined supersaturation limit of 4.2 a t 25’ C. The theory of J. J. Thomson for the effect of electrically charged particles upon lowering the supersaturation limits, which is t o be found in almost all textbooks as the basic theory of the operation of the cloud chamber, requires modification today in certain fundamental concepts. J. J. Thomson’s theory is correct in recognizing the effect of electric charge, but it disregards self-nucleation and the fact that uncharged nuclei can be effective (52).
I
I
STABLE
UNSTABLE
Figure 1.
I
MOST STABLE
STABLE
STABLE
Stability of Brick on Table
To understand and measure quantitatively Ostwald’s generalizations regarding metastable and unstable states and also the life of metastable states, Gibbs’ criteria of stability ( 5 ) and the theory of fluctuations will be examined as a foundation for developing the now-classical theory of the kinetics of phase transitions given by Volmer (97),Becker and Doering ( I ) , and Frenkel
(4). EQUILIBRIUM, STABILITY, AND METASTABLE STATES
The usual criterion of equilibrium given in many texts on chemical thermodynamics, namely, that AF = 0, Le., that the increment in free energy in the process of interest is zero, is only a degenerate form of Gibbs’ criteria. As such, it is inadequate for the problem a t hand since not only does it not distinguijh unstable from stable states but it fails completely to give any understanding of the concept of metastability and the nature of the metastable state. In Figure 1, four possible states of a rectangular parallellepiped-for example, a brick or a book-lying upon a table are pictured, with the curve for E, t h e potential energy of the system, traced by the representative point corresponding t o the center of gravity as it passes through the successive states. If the stable state, A , is subjected t o a small finite displacement, the system returns t o its configuration of minimum potential energy when the stress producing the displacement is removed. On the other hand, the system in an unstable state, B , never returns automatically t o its original position of maximum, E. Instead, it proceeds t o a more stable state of lower E , stopping usually a t the next most adjacent state. Although A represents a state stabl’e to small finite displacements, C repre-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1271
-NUCLEATION-Theory,
Review
sents a more st,able state; in fact, C‘ is the state of absolute stability for any types of displacement compatible with the conditions of this system. The metastable state, D,can be produced by cutting oft the corner edge of the brick, which leads, in the case pictured, to a small depression in the region of the maximum position of the representative point. I t should be obvious that a progressive rounding off or beveling of the acute edge will produce types of maximum behavior which exhibit progressively higher orders of contact of the curve with its tangent a t the maximum until finally, when a sharp cutoff is produced, a definite depression appears as is indicated for D in the figure. The form of the curve as regards its orders of contact ( I O ) at the maximum m ill determine the behavior of the particular metastable state. For the immediate purpose, the essential difference between B and D as pictured resides in the st’atement that B cannot survive even infinitesimal displacements, whereas D can survive finite displacements whose magnitude is determined by the height of the surrounding energy barriers. In the absence of a depression, the order of the finiteness of the displacement is conditioned by the order of contact of the curve with the tangent at the maximum. Thus, it is seen that, even in the rudimentary process of progressively turning a brick end over end, a number of different orders of metastability are encountered which arise from relatively minor changes in acuteness of the angle forming the edge of the brick. One of the tasks in understanding the stability of metastable states in the more complex chemical systems involved in phase transitions will be to interpret their detailed molecular behavior in terms of the simple physical analogy of beveling as depicted above. When passing from simple mechanical systems to the comples systems of chemistry, it becomes necessary to consider the entropy of the system. This means that in Figure I,E must be replaced a t constant temperature and pressure by the Gibbs function, G = E - T S pi!, with the further proviso that in seeking for the equilibrium expression all possible variations, i.e., virtual displacements, which do not destroy the system consistent with the constraints imposed upon it must be investigated. I n Figure 1, state D differs from state A only in the height of the barriers, which in turn determines the rate at which systems are converted into more stable configurations. In the last analysis, the qualitative distinction between D and A is determined by the magnitude of the rate of reaction recognized (or ignored) in the description of the etate. I n most experiments, gross observations indicate a critical limit, but more refined observations disclose a blunting of the critical phenomena. In short, it will be found that minute changes in conditions (supersaturat’ion,supercooling, etc.) vi11 produce such enormous changes in rate t h a t the transition is a critical phenomenon in appearance only. Present views of chemical reaction velocity are the culmination of fundamental contributions such as those of iirrhenius (1889) on energy of activation; of Marcelin (1915), Tolman (192022), and Brdnsted (1922) on critical complexes; of Scheffer and Brandsma (1926-29), La Mer (1933), and Rodebush (1933) extending the concept of energy of activation to free energy and entropy of activation; and of Wigner, Polanyi, and Eyring (193335) on the absolute rate a t which systems cross the free energy barrier of activation. In principle a t least, the rate of reaction can be predicted if the height of the free energy barrier can be calculated for the process of interest, which means determining the height of the saddle point ( 3 ) at) B which separates states A and B of Figure 1 on a free energy diagram. I n practice it is impossible to calculate the free energy of activation without recourse to experimental rate measurements in any but the simplest and most rudimentary of chemical reactions. However, in phase transitions, the quantity which is
+
1272
comparable to free energy of activation in ordinary chemical reactions equals what Gibbs called in 1878 the work of forming a “fluid of different phase within any homogeneous fluid.” This Tvork is one third the product of the interfacial tension, U , times s, the suiface area of the pal tide of the produced new phase
w=
1/3
(us)
This simple result is the master key to the problem of nucleation, but its importance and application remained completely u11noticed until resurrected by T‘olmer a half century later. As this author states (translated from German): “It is strange that Ost’iTaId, who v a s especially interested in the subject and iTho must have thoroughly penetrated the meaning of the work of Gibbs in the course of translating it, should have failed to recognize its importance.” To be sure, Gibbs’ discussion ( 6 ) on the possibility of the formation of a fluid of different phase within any homogeneous fluid is difficult to abstract briefly, but his summary (8) is most pertinent and illuminating, and in retrospect clear and concise, as the following quotation ( 7 )illustrates: “The most simple case of a system with a surface of discontinuity is that of two coexisting phases separated by a spherical surface, the outer mass being of indefinite extent. When the interior mass and the suiface of discontinuity are formed entirely of substances which are components of the surrounding mass, the equilibrium is always unstable; in other cases, the equilibrium may be stable. Thus, the equilibrium of a drop of water in an atmosphere of vapor is unstable, but may be made stable by the addition of a little salt.” Gibbs also says (a), “The study of surfaces of discontinuity throws considerable light upon the subject of the stability of such phases of fluids as have a less pressure than other phases of the same components with the same temperature and potentials. Let the pressure of the phase of which the ctability is in question be denoted by p’, and that of the other phase of the same temperature and potentials by p“. A spherical mass of the second phase and of a radius determined by the equation 2u = (p” - p’)r
(27)
mould be in equilibrium Fith a surrounding mass of the first phase. This equilibrium, as we have just seen, is unstable when the surrounding mass is indefinitely extended. A spherical mass a little larger would tend to increase indefinitely. The work required t o form such a spherical mass, by a reversible process, in the interior of an infinite mass of the other phase, is given by the equation
w = us - (p” - 1)’)d’
(28)
The term us represents the work spent in forming the surface, and the term ( p ” - p‘)w”the work gained in forming the interior mass. The second of these quantities is always equal t o two thirds of the first. The value of W is thereforc positive, and the phase is in strictness stablr, the quantity W affording a kind of measure of its stability. We may easily express the value of W in a form which does not involve any geometrical magnitudes, viz.
where p”,p’, and u may be regarded as functions of the temperature and potentials. I t will be seen that the stability, thus measured, is infinite for an infinitesimal difference of pressures, but decreases very rapidly as the difference of pressures increases. These conclusions are all, however, practically limited to the case in which the value of r, as determined by Equation 27, is of sensible magnitude.” If (p” - p ) = 2u/r is substituted in Gibbs’ Equation 29 IT,* =
(47rrZu) =
INDUSTRIAL AND ENGINEERING CHEMISTRY
1/3
us
(1’
Vol. 44, No. 6
NUCLEATION ,Theory, These quotations may be paraphrased as follows: A phase may persist beyond the ordinary limit of stability a t which it can coexist with another under comparable pressure. However, work must be performed for the new phase to appear, the amount necessary to form the new phase being equal to that required to form a spherical droplet of the new phase within the mother phase with components having the same (chemical) potentials. This requirement determines the excess pressure t o which the new phase is subjected and hence in turn determines the spherical radius through Gibbs’ Equation 28.
. . . . . . . .. . . . . . . . . . .A . . . 0
e
-
* :
0
*
*@
.
0
0
0
0
Figure 2.
W
-
0
Work of Formation of Phase B from Phase A
= work of formation of B us (p” p ’ ) V” = ss/3 = 16rru3/3(p” p’)*
-
ceedingly rare that no cognizance need be taken of this possible mechanism. Instead, the nucleus arises by the stepwise bimolecular addition of molecules according to the scheme:
mAl = B , Bm AI = Bm+l B(i-11 A I Bi
+
+
i
Here At represents a kinetically independent unit of phase A ; B, is an embryo of phase B containing i units. Note that this process is kinetically bimolecular in mechanism but is of the reaction order i in concentration dependence since the preceding steps are in equilibrium m is usually 2, whereas i in the case of water vapor condensation is about 80-i.e., this reaction is of the 80th order. The embryos of the type Bc are continually forming and disappearing by the reverse processes of dissociation. The concentration of embryos of a given size is related to their corresponding energies by the expression
0
0
0
Review-
8
p”
n(B,) = Ce-AGJloT
(3)
where AG$ is the work of formation of BI. This means that the concentration falls off rapidly with increasing size. The energies of the embryos may be represented by points on the rising energy barrier curve (see Figure 3) and their corresponding concentration by the density of parallel lines. Embryos consequently have a transitory existence in homogeneous phases by virtue of fluctuations in local density to which corresponding stepwise fluctuations in energy are related. Such fluctuations are occurring continuously but it is only under very special, almost critical, conditions that they become of sufficient magnitude t o nucleate the phase for a transition to a more stable state.
= surface
u = tendon
-
I
$1
= 2c/r = interior
I
The a p earance of a boundary surface of tension sets th,e limits of t i e region of metastability. The lower limit is p “ = p , which corresponds to coexistence as bulk phases; the upper limit is p’ >> p ’ , but is indefinite because the great reduction in size of the droplet makes the concept of surface tension and a separate phase meaningless. Nevertheless, within the limits to which surface tension has meaning for very small droplets, the treatment gives an answer of great orienting significance to the question of the minimum size of the particle of the new phase which can be produced spontaneously (see Figure 2). How is it possible for a system to gain this work a t constant temperature and pressure and thus spontaneously undergo a phase transition? The answer lies in considering its fluctuations.
I
I
I
FLUCTUATIONS, EMBRYOS, AND NUCLEI
If one could observe with the aid of a supermicroscope the instantaneous densities and energies of an exceedingly small element of volume, he would find that these quantities are far from constant even though the volume element is immersed in a phase that is in a state of complete thermodynamic equilibrium. The energy and density of the volume elements fluctuate rapidly with time. It is only their time average values which correspond to the macroscopic value of the energy and density of the phase. Abundant direct experimental evidence now exists for accepting this basic postulate of statistical mechanics-for example, the pioneer researches of Perrin on colloidal suspensions and the perfect agreement of all light-scattering measurements with the theory of Einstein and Smoluchowski that scattering arises from such density fluctuations. There is a prevalent but mistaken notion that the reactive nucleus arises by a simultaneous collision of all of the individual molecules of which it is composed. A nucleus in water-fog formation consists of about 80 water molecules, but the probability of the simul taneous collision of these 80 water molecules is so exJune 1952
Figure 3.
AG (or
A G) as Function of r
In physical chemistry, the term nucleus (or germ) is reserved in self-nucleation processes for the embryo at the top of the energy barrier. In other words, it is the minimum-sized embryo which is capable of initiating further spontaneous growth t o produce the new phase. Fragments of the stable phase larger than the critical nucleus can of course initiate the spontaneous reaction, but the application of the term nuclei to these fragments implies an action similar to that involved in nucleation by foreign nuclei, and is strictly not a self-nucleation process. VAPOR PRESSURES AND INTERNAL PRESSURES OF SMALL DROPLETS
Equations for the thermodynamic properties of embryos and nuclei can be derived by several equivalent procedures, but the
INDUSTRIAL AND ENGINEERING CHEMISTRY
1273
Review
-NUCLEATION,Theory,
most illuminating procedure is t o follow the general method proposed by Gibbs-namely, t o determine the total free energy of the system and to apply the method of variations (virtual displacements) subject to the constraints imposed upon the system-to discover the equilibrium laws that subsist. Since detailed derivations along these lines have been given recently by Frenkel ( 4 ) only pertinent portions will he presented here The function, G, of a one-component system consisting of a phase A (say its vapor) and a phase B (one liquid fog droplet) is given a t a fixed temperature, T , and total pressure, P , by
Here AT* and N B represent the number of molecules of phase A and of phase B; N A is the number of molecules of vapor, IT^ the number in the fog droplet, and r is the radius of the droplet. The chemical potential, p ( F / S of Lewis and Randall), in the Dresent develoDment is referred in each case t o the molecule rather than to the gram or the mole as the unit of masas. The last term is the surface free energy, which is composed of the product of the surface area, 4 ~ r 2and , the interfacial tension, u. In Equation 4, G (5 of Gibbs; @ of Frenkel; F of Lewis and Randall) is the extensive property, and q = G / n , the intensive molal free energy. Gihhs ( 5 ) , in contrast to followers of the G. N, Lewis school, dismisses summarily the molal property; instead, he emphasizes and employs exclusively the chemical potential cc = ( d G / d n ) T , P , even though the two are identical in the frequently encountered case of a one-component system. KO doubt this preference was dictated by his interest in surfaces, where he undoubtedly recognized, although he does not mention i t , that the use of the molal G / n , instead of the partial molal p = d G / b n , will lead to incorrect results even in onecomponent systems when treating the surface phase. Thus Equation 4 is not homogeneous in the number of molecules, since the last term containing r2 corresponding t o the surface energy is proportional to n2/3,whereas the first two terms involve n to the first power. G / n and dG/& obviously differ by a factor of 2/a in the term for the surface contribution. The reader should he alerted for some unfortunate errors which have crept into the literature recently from failure t o recognize this point. The equilibrium state of the system is found by taking, consistent with the constraints (IYA ~ V B )= constant; 6.v~= - G L X r ~ , all possible variations, 6, which do not destroy the system and equating to zero (8G = 0), yielding
+
NB(P) =
RT In p ,
+
(7)
N%(P)
then the vapor pressure, p, of a droplet of radius, r, a t the total pressure P A P will be greater than p , in order t o conserve equilibrium. Differentiating Equation 7 and substituting ( d p ) ~= V d p (dP = dp)yields
+
T’A is the volume per molecule of the vapor and equals k T / p for an ideal gas. Keglecting V Bin respect to VA yields
(d)
kT d In p - 2 u V ~ d 01’
This result should be designated the Kelvin-Gihbs equation for the effect of radius of curvature upon the vapor pressure of a spherical droplet, since Kelvin (66) derived a simpler but essentially equivalent equation in 1869-71 (not 1881 as is stated in many texts), which Gibbs generalized and improved (1876-78). r is the radius of the droplet which is in equilibrium with the saturated vapor. If in deriving Equation 8, V B ~ had P been employed for V ~ d p , then
dp/dP = VB/T’,
(11)
when r -+ a, which is Poynting’s equation for the effect of total pressure upon vapor pressure. In fact, the substitution of the integrated form of Poynting’s equation for a perfect gas, namely
VBAP = k T l n p / p ,
(12)
into Gihbs’ Equations 2 i and 28, yields most directly the important equations for the work of nucleogenesis in terms of the degree of supersaturation, p / p , , namely
W
= as - kT In p / p ,
(13)
(5)
The bracket equals 2 r ( V ~ / 4 mwhen ~ ) it is noted that N B = 4f13/3VB,where V Bis the volume of one molecule in the liquid phase Consequently,
When the radius of curvature of the liquid droplet, !, approaches T
zero, this term vanishes and the familiar equation, P B = PA, is recovered, valid for the equilibrium of phases in bulk, where the Contribution of Furfare energy is negligible. When T is small, PA no longer equals N E , because the interior of the droplet B is now subjected to an additional pressure, AP = ( p ” - p’) = 2 0 / r (Gibbs’ Equation 2 7 ) , which arises from the compressive effect of the surface tension. [The symbol P for total pressure is introduced t o recognize the possibility that the atmosphere Burrounding the drop usually contains an inert gas (air). P is therefore equal to the partial vapor pressure of the droplet, p’, plus the partial pressure of the inert gas considered to be insoluble in the liquid droplet. d p is of course equivalent t o dP. ] The effect of M B > P A means that if the chemical potential of the liquid in bulk ( T = a ) and a t pressure P is characterized by the fixed value p , through the equation 1274
I t should be noted that when p = p m ,W becomes infinite. Equations 13 and 14 will later he derived in a more detailed way to elucidate the piocess involved. STABILITY OF POLYDISPERSE COLLOIDAL DISPERSIONS
Since the vapor pressure of a droplet is greater than the vapor pressure of the same liquid in bulk ( T . = a ), sustained equilibrium between a droplet and its mot.her phase can he achieved only by the droplet containing some nonvolatile solute-salt, sulfuric acid, et?.-which will reduce the vapor pressure of the volatile component until it is precisely the same as that of the mother phase. This means that for every droplet of a given composition, there is only one critical radius which will allow permanent coexist,ence with the pure volatile component in bulk. Droplets of radii smaller than the critical value will evaporate and condense upon droplets of larger radii. In t,he presence of droplets of large radii, even those of critical radii will in time evaporate and condense upon those of larger radii. This also means that all polydisperse aerosols are unstable thermodynamically unless each droplet contains a contaminant in just the correct proportion, in accordance with Raoult’s law to satisfy Equation 6. The fact that many polydisperse aerosols exist for relatively long periods in a nonequilihrium state without appreciable change is due to the fact that their vapor pressures are so low,
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 6
NUCLEATION-Theory for example, dioctyl phthalate, p = 10' mm. of mercury a t 25 C., that the rate of exchange of vapor molecules between different size droplets is too slow t o relieve a situation which has usually been produced by the rapid chilling of a saturated vapor. On the other hand, aerosol droplets of a volatile material like toluene (vapor pressure = 30 mm. a t room temperature) adjust their sizes almost instantaneously t o changes in the vapor pressures of the ambient atmosphere. If the time of size adjustment is 1 second for toluene, it should require approximately 30/10-7 = 3 X 106 seconds (30 days) in the case of dioctyl phthalate to achieve the same size adjustment; by that time most dioctyl phthalate aerosols will have coagulated and settled out. La Mer and Gruen (17') have investigated the effect of the Kelvin equation upon the growth of dioctyl phthalate aerosols when equilibrated with toluene atmospheres arising from master solutions of dioctyl phthalate and toluene. Their data constitute the first unambiguous direct experimental proof of the validity of Equation 10 in the range 0.08- t o 1.0-micron radius and indicate one reason for the occurrence of polydispersity in the aerosols of nature. WORK O F FORMATION O F EMBRYOS AND NUCLEI
When a liquid B is dispersed as fine droplets, the free energy, G = N B ~ Bis, increased by the energy required t o create the surface, namely, 4 ~ 1 . ~The ~ . work of creating a new phase B from a mother phase A , however, is a different matter. Here the process is: homogeneous phase A = droplets phase B dispersed in A . The increase in free energy per droplet is plotted in Figure 3 as a function of r. This equation may be derived as follows: AG = =
Gfma1
- Ginitial
+
= G - GO
, Review
for the frequency factor in the general rate expression for a chemical process rate = y e - A G / k T ( 18) where AG is the free energy of activation just calculated, and K is a transmission coefficient representing reflections of the representative point in crossing the energy barrier in generalized phase space, is of little value in the present development, since the highly convenient assumption that K = 1 cannot be made, a priori. Volmer and then Farkas ( 8 7 ) have calculated the frequency factor for vapor t o liquid by the direct methods of the kinetic theory of gases. The essential improvement introduced by Becker and Doering ( 1 ) consisted in replacing the differential rate equations corresponding to the consecutive steps involved in the activation process leading t o nucleus formation by a set of difference equations. By some clever mathematical reductions, set forth in the monographs of Volmer and of Frenkel, a value of v is obtained for what should be called the quasi-equilibrium rate of nucleation. By this, it is meant that when embryos are removed as liquid droplets, they are replaced by an equivalent number of molecules of vapor, and the whole series of reactions is in a steady state. Kantrowitz (11)in a later paper discusses nonequilibrium nucleation. The final expression is not only detailed but contains some uncertainties. These uncertainties appear t o cancel one another with the end result that Y = loz5with a final uncertainty of =tl in the exponent for supersaturated water vapor. The working equation is J = 10%e - A G / k T , in terms of the rate of droplet formation per second in 1 cc. To express the rate in terms of the supersaturation ratio, In p / p , , r = 2 u V g / ( k ~ l ~ ~is/ sub~,) stituted in the expression for AG and
+ 4?rr2u) - ( N A + N B ) P A
( N A P A NBPB
= -(PA
- / J B ) N B f 4?rr2U
is obtained for the exponential term. The term (In p/p,)Zin the exponential so completely dominates the rate that it makes very little difference whether or not Y is determined precisely, as shown in Table I.
Substituting ( M A - P B ) / V B= 2u/r,, where re is the critical radius of the droplet for (unstable) equilibrium, yields
TABLE Ia P/Pm
AG has a maximum a t
value of
T
= r,; hence
(
=
~
;r'G)
0, corresponding to the desired
AG/2.3kT
J
tb b
(AGLX.
=
1
3 (4?rurC2)
RATE O F NUCLEUS FORMATION
5000
., ..
4
3
2
37
94 10-6@
10-12
lOS* years
10s years
5
24
12
10 0 . 1 sec.
10-13 sec.
1013
Taken from ($7). Time t h a t must elapse for the appearance of the first droplet in 1 cc.
(17)
the result obtained in Equation 1. The dotted curve in Figure 2 corresponds t o the first term involving r2 and represents the increase in G with increasing surface when the system remains as phase A . The heavy curve represents the sum of the two terms of Equation 16 and corresponds to AG when a more stable phase B is created in the presence of phase A . Since for all values of r > rC, G decreases monotonically, further growth of the fragment is spontaneous. The spontaneous conversion of phase A into phase B continues until the potential .of A , as determined by its vapor pressure, has become equal to that of phase B. In the case of a supercooled one-component liquid and its immiscible solid phase, the conversion goes to completion a t constant temperature and pressure.
The direct substitution of Eyring's (9)
1.1
The prodigious increase in rate produced by increasing p / p m from 4 t o 5 makes the rate of nucleation appear to be a critical phenomenon a t p / p , = 4.2, whereas, in fact, the procesm is in reality only a rate process of such high order (80 or more) that it is exceedingly sensitive to changes in concentration. HETEROGENEOUS NUCLEATION
Nucleation which is initiated by foreign nuclei (heterogeneous nucleation) arises from the catalytic effect of their surfaces, as well as from the walls of a vessel, grain boundaries, pores, etc. The essential condition is that the surface must be wet by the phase formed in the presence of the mother phase. Volmer (88) has considered the case of a flat surface, s, upon which an embryo of phase B is forming, with a contact angle, 0, between the two phases. The interfacial energy equation UA,S
UB,s
f U B . A COS 0
then holds. When this expression is taken into account, the fundamental expression (19) for W , the work of formation of a June 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
1275
-N
UCLEATI ON-Theory,
Review
+
nucleus, is multiplied by a factor [ ( Z cos 0 ) (1 - cos 0)*/4]. When 8 is H O D , the condition for a droplet just touching the surface, the factor is unity and the limiting equation for self-nucleation is recovered.
- __r
CRlTlCLL LIMITING
SUPJ~SAT~LIATION RAPID SELF-NUCLEATION
I
P A R T I A L R E L I E F O F SUPERSATURATION
/
GROWTH BY DIFFUSION
m I TIME
I
-.
Figure 4. Schematic Representation of Concentration of Molecularly Dissolved Sulfur before and after Nucleation as Function of T i m e ( 1 4 )
As e decreases t o zero, the factor involving cos e also decreases monotonically t o the limit zero. This means that for a new phase which spreads perfectly upon the foreign nucleus, no activation energy is necessary. It appears from Turnbull's analysis (26) that this factor involving contact angle is not only sufficient to give an explanation for the lowered degrees of supersaturation needed in the presence of foreign nuclei, but the expression also furnishes an explanation of the effect of thermal history on nucleation rate. It has long been recognized that a melt or solution which has once been crystallized will recrystallize more easily when the crystals are heated and held above the melting point for some time before recooling below this point. This behavior has been assigned the vague term "memory." It now appears that memory consists in minute crystals persisting above the melting point by hiding in the cracks, crevices, and pores of the vessel by rirtue of the effects of contact angle as a means of preserving the last vestiges of these forms of nuclei. PREPARATION
OF MONODISPERSED AEROSOLS AND
HYDROSOLS
One of the more practical results of the theory of nucleogenesis is the explanation and the guidance ( 1 4 ) it has offered in preparing monodisperse colloids. La Mer and Sinclair found t h a t aerosols of very uniform particle size could be produced by the regulated slow cooling of supersaturated vapors diluted with inert gases containing appropriate numbers of foreign nuclei (18). Under such conditions, the vapors condensed uniformly on the nuclei and not upon the walls of the condenser. The appearance of such highly monodispersed preparations in which the average deviation of particles ( 1 2 ) from their mean radii did not exceed 10% is signalized by the appearance of a new and beautiful optical effect-higher order Tyndall spectra (IO, 13, 18)--which has served as a rapid means of measuring both the size and the number of the particles. These optical methods have opened up a new field of investigation in colloids Equally monodisperse sulfur hydrosols (SS),which exhibit the same optical effects as the oil aerosols, can be prepared by the reaction of dilute aqueous solutions of hydrochloric acid (0.001 t o 0.003 ill) and sodium thiosulfate (0.001 t o 0.002 M ) . I n this case, the process is one of self-nucleation, As illustrated in Figure 4 from La Mer and Dinegar ( 1 4 ) ,molecularly dispersed sulfur is continuously generated by the reaction; the concentra1276
tion of sulfur (Sz or Ss) increases steadily, passing the point of saturation and penetrating the level a t which the rate of selfnucleation becomes appreciable. It was shown that, owing to the term (In p / p , ) z , equivalent to (In S/So)z, in the nucleation rate equation, the rate increases prodigously as the transgression limit is approached. When the generating solution is dilute, corresponding t o a low rate of sulfur production, the sudden appearance of nuclei relieves the supersaturation so rapidly and effectively t h a t the region of nucleation (11) is restricted in time. KO new nuclei are produced after the first outburst. The nuclei so produced grow uniformly by a diffusion process forming a highly monodisperse preparation, for which the laws of growth have been worked out (22, 2 3 , 3 1 , 3 3 ) . When the generating solution is more concentrated and the corresponding rate of molecularly dispersed sulfur production is correspondingly larger, the supersaturation cannot be relieved sufficiently rapidly to prevent a cascading of nuclei. I n other words, region I1 extends over a wide interval of time and a polydisperse preparation results. This theory of the preparation of monodisperse colloids has been tested with other hydrosolse.g., silver chloride, sulfur by dilution of an acetone solution with water, etc. ( 1 4 ) . When a monodispersed sulfur hydrosol is produced by the addition of water to a solution of sulfur in acetone or in alcohol, both the concentration of sulfur in the organic solvent and the rate of addition of wat,er determine the monodispersity, the nature (crystalline or supercooled liquid) of the disperse phase, and the time lag in the formation of the colloid. In the case of rapid additions of water, with an initial concengram atoms of sulfur in alcohol, there is a slowly intration of creasing turbidity, while with an initial concentration of 2.5 to 2.9 X gram atoms, a sharp end point and a rapid increase in turbidity are found. The higher order Tyndall spectra (monodisperse) were observed a t 3 x 10-3 gram atoms of sulfur; crystalline precipitates. above 5 X With slow additions of water, it was observed first that the solution became increasingly turbid very slowly. When sufficient water, corresponding t o the sharp end point for rapid addition, was reached, an exceedingly rapid increase in turbidity appeared. These findings merit more careful quantitative study in the light of these studies on barium sulfate (16). SUPERSATURATION OF BARIUM SULFATE-TIME
LAGS
The kinetic behavior just not,ed qualitatively for sulfur has been demonstrated more effectively with barium sulfate. Many years ago Von Weimarn (SO) noted that the character of barium sulfate precipitates and the rate of t,heir appearances following the mixing of barium thiocyanate and manganese sulfate solutions depended upon the degree of supersaturation. Our recalculations of his data indicate a time lag of over a year when the supersaturation is 3.2-fold, whereas crystals appear in a few seconds for a supersaturation of 29-fold. T o avoid the disturbances of local supersaturation, La Mer and Dinegar (16) produced sulfate ion, in the presence of barium ion, by the reaction of thiosulfate and persulfate. Under these conditions, they found the crit,ical transgression limit, Le., the supersaturation ratio .\/Kss/Ksp corresponding to the ratio of the mean molality of the ions, to be const,ant and equal to 21.5 over a wide range of supersaturation when corrected for ion activity coefficients. Von Weimarn found values ranging from 7 to 48 primarily because he neglected corrections for activity coefficients. When his direct mixing experiments were repeated, taking care to avoid local supersaturations, the time lag of appearance increased from 97 seconds a t a supereaturation of 19.3 to 2400 seeonds a t a supersaturation of 9.5 (15). Christiansen and Nielsen ( 2 ) have performed similar experiments using a very rapid mixing technique. They observed time lags of 0.001 t o 0.1 second in the more concentrated range.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 6
'
Review
NUCLEAT1ON-Theory, When their data and those of La Mer and Dinegar (16)are plotted on a log C (ionic mean molality) versus log time plot as shown in Figure 5, the data fall on a straight line having a slope of approximately six for a time range of 2.4 x 106-fold and a concentration range of 12-fold.
T I M E LAG BARIUM SULFATE
“/3r 21 -3
0
I I
I
I
0
I
I 2
I
3
T i m e Lag, Barium Sulfate
If the slope of six is accepted, this means t h a t the rate of nucleation of barium sulfate from its ions is a seventh-order reaction, which in turn means t h a t the rate-determining step consists in the addition of a seventh ion t o a cluster of 3 B a + + and 3S0,‘ ions; thus Ba++
+ SO4--
= Bas04
+ Ba++ = Ba2SO4++ Bas04 + SOa-- = Ba(S+)z--
BaS04
I
B a + + +Ba4’(SO&++)
or (Ba0(S04)3 Sod-- -+ Ba?(SO&--
+
I n this mechanism the reaction consists entirely of bimolecular steps, but the concentration dependence of the rate is of the seventh order, since the concentration of the cluster Bas(SOJa is proportional to the sixth power (third power for B a + + and So
