382
DANIEL E. ROSNER
Lifetime of a Highly Soluble Dense Spherical Particle1 by Daniel E. Rosner AeroChem Research Laboratories, Inc., Subsidiary of Sybron Corporation, Princeton, New Jersey 086.40 (Received August 6, 1968)
The lifetime of an isolated spherical solute particle in a solvent is expressed in closed form, simultaneously incorporating the major effects of (i) limitations to solute escape by molecular diffusion, (ii) nonzero interfacial velocities for highly soluble solute-solvent systems, and (iii) kinetic limitations t o atom (-ion-molecule) detachment at the solute-solvent interface. Illustrative calculations are displayed in compact form and well-known results are recovered in the extremes of sparing solubility, infinite interfacial rate constant, or interface kinetics-controlled dissolution. Assumptions and approximations underlying the general phenomenological expression (and each of its more familiar limiting forms) are reviewed and criteria governing its domain of validity are indicated.
I. Introduction The classical problem of the lifetime of a dissolving quasi-spherical particle in an isothermal solvent plays a central role in the design of equipment fox dissolving powdered materials2 and the relevant theory is often applied in research to estimate fundamental transport properties3 (e.y., solutesolvent diffusion coefficients). Rot surprisingly, simple approximate relations for the effective particle lifetime are available only in certain limiting cases,4 perhaps the best linomn of which correspond to (i) external solute diffusion control or (ii) rate control due to interface atom (-ion-molecule) detachment kinetics. Especially for small particlesj6 siinultaneous consideration of rate limitations due to both solute escape via cliff usion and interface kinetics is necessary. However, until now this has been accomplished only for the case of sparingly soluble systems, using a method first exploited by Berthoud and Valeton.6 Unfortunately, 1his “resistance additivity” method breaks down for systems of appreciable solubility because it fails to account for the distortion of solute concentration profiles and radial convection associated with rapid recession of the solid-liquid interface.? I n the present paper it is shown that a potentially useful closed-form expression for the particle lifetime can be obtained even for highly soluble systems, We conclude with a brief discussion of the assumptions underlying this general result.
IT. Extended Quasi-Stationary Model I n keeping with our objectives, we consider the dissolution of an isolated sphere of pure solute (see Figure 1) subject to the following set of simplifying assumptions.8 A I . The interface kinetics arc well described by a reversible first-order rate law of tho Berthoud form: YS = k(p/ps)(c,,t - e,), where the rate constantg k is nominally the same at all positions on the sphere surface. Here (csat - c,) represents the prevailing The Journal of Physical Chemistry
undersaturation at the liquid-solid interface, using solute mass fraction as the concentration variable. AR. Thc physical properties of the solution (density, p , effective Fick diffusion coefficient, D , etc.) are replaced by appropriate (constant) mean values and the solute-solution system is considered to be isothermal. AS. The instantaneous concentration field surrounding the dissolving sphere is approximated by thc steady-state concentratioii field surrounding a (fictitious) spherical solid of the same size, through which solute is being artificially forced at a mass rate equal to the instantaneous rate of dissolution. This is an extended form of the nowfamiliar quasi-stationary (1) Supported by the Propulsion Division of the U. S. Air Force Office of Scientific Research under Contract AF 49(638)-1664; originally issued as AeroChem TP-178, July 1968. (2) See, e.g., the early work of A. W. Hixson and collaborators, Ind. Eng. Chena., 23, 923, 1002, 1160 (1931); 33, 478 (1941); for a recent survey, see P. H. Calderbank in “Mixing-Theory and Practice,” V. W. Uhl and J. B. Gray, Ed., Vol. 11, Academic Press, New Yorlc, N. Y., 1967, pp 1-114. (3) Analogous to the recent bubble dissolution studies of I. M. Krieger, G. W. Mulholland, and C. S. Dickey, J . Phy6. Chem., 71, 1123 (1967). (4) See, e.g., E. A. Moelwyn-Hughes, “Physical Chemistry,” 2nd rev ed, Pergamon Press, Oxford, 1964, and “The Kinetics of Reactions in Solution,” 2nd ed, Oxford University Press, London, 1947, p 374 ff. See section V of the present paper for a discussion of the
underlying assumptions. ( 6 ) Of, course, regardless of initial sine, a t some point in the life span of a dissolving particle it will become small enough so that solutesolvent molecular diffusion does not limit the rate of further dissolution. (6) A. Berthoud, J . Chim. Phys., 10, 624 (1912); J. J. P. Valeton, 2. KTkt., 59, 136, 336 (1924); 60, 1 (1924). (7) D. E. Rosner, Int. J. Heat Mass Tran8jer, 9 , 1233 (1966). (8) ,Discussion of the validity of these assumptions is postponed until aection V. (9) k is here treated as a phenomenological coefflcicnt to be determined from experiment. For a priori estimates of k based on microscopic models, see W. K. Burton, N.Cabrera, and F. C. Frank, Phil. Trans. R o y . SOC.(London), A243, 299 (1951), and F . 0. Frank, “Growth and Perfection of Crystals,” John Wiley and Sons, Inc., New York, K . Y., 1958. The present model is readily cxtended to embrace phenomenologicai dissolution rate laws which are nonlinear in the prevailing undersaturation, ( h a t - cw), at the liquidsolid interface [see section V, footnote 231.
LIFETIMEOF A HIQHLYSOLUBLE SPHERICAL PARTICLE
383 liquid side both by "convection" and by molecular diffusion. Thus, in the absence of solute mass accumulation within the control volume, the solute mass balance equation at the liquid-solid interface becomes
where use has been made of Fick's first law for the diffusion contribution. According to the extended quasistationary assumption (A3) we introducel'
Figure 1. Configuration and notstion; dissolution of an isolated spherical particle.
Combining eq 1 and 2 with the first-order interface kinetics expression (Al) provides an implicit equation for the quasi-steady solute concentration cw established a t the surface of instantaneous radius R
-
k ( ~ . . ~ e,)
(QS) approximation (the extension being necessary to account for the principal effectsof interface motion). A4. The initial particle radius Ro is very large compared to the particle radius at which the saturation concentration becomes strongly particlesize dependent." Thus, over most of the particle lifetime, c,.~ is essentially constant (independent of instantaneous particle size and, therefore, time). A6. Solute transport by convection due to (i) relative motion between the particle and hulk solvent or (ii) bouyant diffusion boundary layers is neglected. The concentration field surrounding the particle is assumed to be spherically symmetric a t all times.
III. Analysis Let us consider solute mass fluxes in and out of a (locally) flat control volume moving in such a way as to always straddle a unit area of receding solute surface (Figure 2). On the solid solute side of this moving "pillbox" control volume, solute enters at the rate m". Relative to this control volume, solute escapes on the
= -
R
[ + 1-e,
In 1
cw
e,
~
However, the dissolution mass flux m" is related to the rate of particle surface recession by ps
dR
=
(4)
dt
so that the radius-time history of the dissolving particle is completely defined by the result of combining eq 1 and 4, viz.
1
(5)
together with the c,(R) relation defincd implicity by eq 3. To obtain a general, closed-form expression for the time t ~ it .takes a particle to go from initial radius Ro to final radius R ,( .t...d, = 0. it is first convenient to introduce the following dimensionless variables parameters
R / R o (normalized particle radius)
W T 3
2 ( p l p s ) ( D t l R 0 ~In) (1
+ B) (dimensionless time)
B E 7"
Csat
- c,
1 - C8.t
(effective solubility parameter)
c, - e.
-
t . 8 C
Gat
(dimensionless interfacial undersaturation) WRE SOLUTE
Figure 2. Solute rnw balance at moving liquid-solute interface.
(10) See. e.@., J. W. Mullen. "Crystnllization," Butterworth and Co. Ltd., London. 1961, p 33 IT. (11) See. e.g., (a) D. B. Spalding. "Some Fundament& of Combustion." Butterworth and Co. Ltd., London. 1955: (b) D.B. Spalding. "Convective Mass Transfer," MeGmw-Hill Book Co.. New York. N. Y.. 1963: and (e) J. M. Lommel and B. Chalmers. T7ons. Met. Soc. A I M E . 215,499 (1969). Valum 7.9, Numb e F e h w IDES
384
DANIELE. ROSNER 1.0
QUASI-STEADY TREATMENT
08
-I O6 Tilts Ill
6
04
8-
10-1
6-
02 4
6
8
100
2
4
6
8 101
0
Za(kR/D)(I-c,,,)
02
0 CHEM CONTROL at t = O
Figure 3. Relation between dimensionless undersaturation a t interface and dimensionless first-order rate constant (cf. eq 12) for highly soluble systems.
04
06
IO DIFFUSION CONTROL
08
bo l+VO
Figure 4. Correlation of dense particle lifetimes for highly soluble systems in region of mixed control (cf. eq 14).
[LR(t)/Dl(l -
csat)
(dimensionless rate constant)
(10)
Then eq 5 becomes
dr
In (1
+
+ B)
and eq 3 becomes an implicit relation for q ( x ; B ) , viz.
(see Figure 3). At time t = 0, 9 takes on the value ?lo related to xo by eq 12. Since (from their definitions) x / x o = R,eq 12 is equivalent to the following R(7; qo, B ) relation
-
70
(””)+ BV
(1
+
B B ) In (1
+ B)
-
for
1 - 70
+
_ _ +0 1 TO
(15)
and
In 1
a=---
sive solubility (via B > 0). Consideration of the special cases of eq 14 (see below) suggests the concise representation shown in Figure 4, in which 1/rlife is shown plott,ed against (1 - ro)/(l 70) with B as a parameter (in the range 0 2 B 2 10). This abscissa has the property of mapping the entire region of mixed control (irom initial interface kinetics control to initial diffusion control) onto the interval from zero to unity. Similarly, the reciprocal lifetime l/rlife also varies between zero and unity.12 With regard t o limiting behavior, it is easy to show that the l / r l i f e curves in Figure 4 have the slopes
B
+
1E Br ) 70 In (
In (1
To obtain the dimensionless particle lifetime r l i f e from eq 11 and 13, it now proves convenient to eliminate the normalized particle size (iz from eq 11 in favor of 7 (with the help of eq 13) and then integrate between 1 the limits of qo and 1 (since eq 12 reveals that 7 when x -+ 0). The closed-form result -f
1+ B ) for __1+ .
1 TO
which can be used to express l / r l i f oin simple form for particles which (initially) are either near interface kinetics control or near diffusion control, respectively. For arbitrary 70 but in the limit of sparing solubility,13 B is small ( B > 1 (19)
indicating a lifetime which increases as the square of the initial radius (or the z//3 power of the initial mass). Conversely, if the particle dissolution is initially chemically controlled [Le., (kRo/D)(l - csat) > pcSat). Indeed this inequality not only guarantees that the initial transient but when = 0(1), will be short compared to he, it also ensures the accuracy of the extended QS approximation at later timesaZ0 Since surface free energy and charge modify csat for sufficiently small particles’O (high interface curvature), it is necessary to restrict eq 14 to cases for which csat = constant during most of the particle’s lifetime. Considering only the effect of surface free energy u (analogous to surface “tension” of a liquid) in the diffusion limit (for sparing solubility), this implies3 Ro >> 4u/(nslcT)-a condition equivalent to the statement that the initial particle radius should be very large compared to the critical radius of a solute “germ” or embryo needed to nucleate aosupersaturated solution (usually of the order of 10-100 A). I n practice, the condition that k be nominally constant over the surface of the particle (imposed in order that the particle remain spherical during the course of the dissolution) implies either that the particle be vitreous or, if crystalline, be comprised of crystallites of size negligible compared to Ro, with negligible preferential grain boundary attack. Assumption 5 implies that the Reynolds number (based on settling velocity) and the Grashof number (dimensionless bouyancy group) both be small (see, e.g., ref l l a , b for relevant correlation formulas). If the heat of solution (& per unit mass of solute) is nonzero, then even an initially is0 thermal system will develop temperature gradients as the dissolution proceeds ( L e . , the particle surface temperature will depart from T , by an amount which allows the heat generated/absorbed at the liquid interface to be conducted awaylto the surface through the adjacent solution). However, consideration of this QS energy balance reveals that a suficient condition for the neglect of this thermal effect is
where X is the thermal conductivity of the solution. This condition is usually adequately satisfied;21 e . g . , for T\’aCl(s)-HzO(l) at rr 20°, the left-hand side of the inequality (23) is about 5 X T h e Journal of Physical C‘ht?miatry
DANIEL E. ROSNER Finally, we come to the assumption of constant (mean) properties in nondilute systems (cf. A2). While necessary to achieve the simple general results presented, there is no question that breakdown of this assumption (e.g., due to the concent,ration dependence of density and diffusion coefficient) will limit the accuracy with which eq 14 can be applied in particular cases. These additional complications, which frequently accompany n o n d i l ~ t e n e s s ,will ~ ~ ultimately necessitate more exact variable property calculations for particular solute-solvent systems. However, eq 14 now provides a readily evaluated reference condition which in many cases should correctly predict the dominant effects of extensive solubility on particle lifetimes under conditions of mixed control. 2 3
Acknowledgments. I t is a pleasure to acknowledge the computational assistance of Mrs. L. Paul and the helpful comments of Drs. 11. Epstein, A. R. Cooper, Jr., R. J. Leonard Jr., R. Prober, and S. C. Kurzius.
(16) See, e.g., (a) A. R. Cooper, Jr., J. Chem. Phys., 38, 284 (1963): (b) A. R . Cooper, Jr., T r a n s . F a r a d a y SOC.,58, 2468 (1962); (c) D. W. Readey and A. R . Cooper, Jr., Chem. Eng. Sci., 2 1 , 917 (1966); (d) K. B. Bischoff, i b i d . , 18, 711 (1963). (17) J. R. Philip, J . Atmos. Sci., 22, 196 (1965); see also G. Luchak and G. 0. Langsroth, Can. J . Res., A28, 574 (1950). (18) F. A. Williams, J . Chem. Phys., 33, 133 (1960). (19) The subscript zero appearing in our previous analysis identified symbols formally evaluated a t zero time within the quasi-steady a p prozimation. I n diffusion limited systems of sparing solubility, the transient will die out a t a time large enough to ensure (rot)’/% >> no. For example, the dissolution rate falls within 10% of the QS value when (rrDt)’/Z = lORo or, in the present notation when T > 64(pc,,t/ p s ) . Accordingly, this latter quantity must be small compared mit,h riiie 2 1. When B is not small, and the system is diffusion limited, a rough estimate of the initial transient period can be made using the one-dimensional transient dissolution analysis of Cooper.16b Generally, one anticipates that the principal effect of this initial transient is equivalent to some shift in the effective starting radius of the particle or a corresponding shift in the time scale. However, a detailed treatment of this transient is beyond the scope of this paper. (20) When ps + p concentration profile distortion is due largely t o the receding boundary per se (not true radial convection relative to an observer fixed in space) and is overestimated by our extended-QS approximation a t large values of the saturation parameter B. However, for sufficiently small values of B (say, B < 2), boundary recession or equivalent true radial convection produce nearly identical profile distortion. This observation, which is the basis of A3, coupled with the CR 4 q transformation, enables the closed-form solution, eq 14, to be obtained. A3 and eq 14 can be further generalized to more accurately include arbitrary solvent jsolute density ratios, p/ps. Fortunately, a closed form QS result is still possible (whicli reproduces eq 14 in the limit p j p g --f 0) (cf. M . Epstein and D. E. Rosner, in preparation). (21) I n part because the effective Ficlr diffusivity is usually much smaller than the thermal diffusivity of the solution. I n aqueous systems this diffusivity ratio is typically of the order of 10-2. By way of contrast, thermal effects in liquid-vapor systems will often prevent direct application of the present results to the closely related problem of droplet evaporation into a background gas. (22) See, e.g., A. Emanuel and D. R. Olander, Int. J. Heat Mass Transfer, 7, 539 (1964). (23) If the interface kinetics are well described by a nonlinear power law: w g a (cast - elp)n where n # 1 (cf, e.g., R. F. StricklandConstable, “Kinetics and Mechanism of Crystallization,” Academic Press, London, 1968, p 25) then the present approach leads t o an expression of the form rlife = fct(qo, B , n),involving only a numerical quadrature to determine r1iie for any value of the exponent 72.
387
LATTICEPARAMETER YARIATIOR'SIN SODIUM FAUJASITES Glossary of Symbols
Greek Symbols
B
11
C
D k k
(1) m"
M n
Q R (R
r (s )
t
T VB
X
z
Solubility parameter; eq 8 Mass fraction of solute Effective Fick diffusion coefficient Berthoud first-order rate constant Boltzmann constant Liquid Mass flux across solid-liquid interface RIolecular weight Number density (per unit volume) Heat of solution (per gram of solute) Particle radius R/&; normalized particle radius Radial coordinate Solid Time Absolute temperature Recession velocity of interface, = m " / p s Moles of water (of hydration) per mole of solute ( k R / D ) ( l - cBat); dimensionless rate constant; eq 10
p u T
Dimensionless local undersaturation; eq 9 Thermal conductivity of solution; eq 23 Density Surface free energy Dimensionless time; eq 7
Miscellaneous O( ) ,
Order of magnitude symbol
Subscripts chem Chemically controlled (interface kinetics) Diffusion controlled diff S Pertaining to solute life Life (of dissolving particle) At liquid-solid interface R' sat Saturated 0 Evaluated at t = 0 m Far from particle surface
Variation of the Lattice Parameter with Aluminum Content in Synthetic Sodium Faujasites.
Evidence for Ordering of the Framework Ions
by E. Dempsey, G. H. Kuhl, and D. H. Olson Mobil Research and Development Corporation, Research Department, Central Research Divisioti, Princeton, N e w Jersey (Received August 8 , 1068)
OS5.$0
A plot of crystal lattice parameter against si1icon:aluminum ratio for a range of hydrated sodium X- and Y-zeolites shows two distiiict breaks a t silicoii: aluminum ratios near 1.4: 1 and 2 : 1. A similar plot for dehydrated calcium materials published earlier by Breck and Flanigeii shows similar breaks. It is suggested that the breaks are related to phase changes occurring in the materials as the rilicoii-aluminum ordering changes.
Introduction
Experimental Section
Although the framework structures of at least 25 zeolites are known, information concerning the ordering of silicon and aluminum ions in the framework has been reported for only a handful of these structures. Such information is vital for the development of any detailed theory of crystallization or reaction mechanisms. At this time, no concrete experimental data have been presented pertaining to the ordering of silicon and aluminum ions in faujasite-type zeolites. During the course of an accurate determination of the variation of the cubic lattice parameter with aluminum content in hydrated sodium faujasites, we noted distinct discontinuities in what was expected t o be a continuous linear relationship. The experimental data and their possible implications concerning silicon-aluminum ordering is discussed.
All of the sodium synthetic faujasites used in this study were of high quality, as judged by their chemical analyses, X-ray scattering power, and sorption capacity for water and cyclohexane. The X-ray measurements were made on a Siemens X-ray diffractometer equipped with a scintillation counter, pulse-height analyzer, and strip-chart recorder. A scan speed of 0,25"/min was used with a 2-cm/min chart speed. Before the measurements were made, the samples were equilibrated for 16 hr in a 75% r.h. constant-humidity cabinet. The cubic lattice parameters, ao, were measured using the double scanning diff ractometry technique,l which minimizes 0 20 errors. I n most instances, four diffraction peaks in the 50-60" 20 (Cu) range were measured and the (1) H. W. King and L. F. (1962).
S'assamillet, Advan. X - R a y Arial., 5 , 78 Volume 73, h'umbsr 2 Februaru 1960
