Comments on foam stability, Ostwald ripening, and grain growth

Comments on Foam Stability, Ostwald Ripening, and Grain Growth A discussion is presented of the applicability of well-established theories of interface-reactioncontrolled Ostwald ripening of second-phase precipitate distributions, and of the evolution of grain distributions in polycrystalline materials, to the problem of foam evolution by interbubble gas diffusion. Two limiting cases are developed in detail: evolution of a narrow size distribution of foam bubbles, and characteristics of a bubble distribution at asymptotically large times. © 1985 Academic Press, Inc.

INTRODUCTION An analysis of foam stability, in which effects of interbubble gas diffusion and gravity-induced drainage were included, has recently been presented by Monsalve and Schechter (1). The purpose of the present note is to reconsider the foam-stability problem, and in particular, to draw attention to the mathematical equivalence that exists between foam evolution by interbubble gas diffusion and a certain type of "Ostwald ripening," the theory of which has been extensively treated over past years by various investigators. This equivalence extends, in addition, to the problem of grain growth in polycrystalline materials, as is also discussed. Some well-known results of Ostwald-ripening theory are applied here to the foam-stability problem. Specifically, two stages of ripening are considered: The first corresponds to an earlier stage of the process, for which the size distribution of bubbles is taken to be "narrow" in bubble-radius space, whereas the second corresponds to asymptotically large times. EVOLUTION OF THE BUBBLE DISTRIBUTION IN FOAM To briefly review, Monsalve and Schechter (1) considered the bubble size-distribution function, F(r, t), which they defined such that F(r, t)dr is the total number of bubbles having equivalent sphere radii that lie within the range r to r + dr. They demonstrated that F satisfies the continuity equation OF 0 ~ + ~r (~pV) = 0

[ 1]

where if(r, t) is the rate at which the radius of a bubble, having radius r at time t, changes with time. Actually, the above partial differential equation, in that general form, has found widespread use in the analysis of size distributions that characterize various types of inclusions in solids, e.g., solid particles (2-7), gas-filled pores (8, 9), dislocation loops (10), as well as inclusions in liquids, e.g., crystalline precipitates (6, 7, 11).

The specific functional form for ~b(r, t) used by Monsalve and Schechter (1) had been suggested by Lemlich (12), i.e., ~p(r, t) = K ( I

- ~)

[2]

where K is a combination of various parameters that are pertinent to the mass-transport process and are independent of r and t. The quantity r2~ is given by r21 ~ M2(t)/Ml(t)

[3]

where, in general, Mj(t) is the fth algebraic moment of the size distribution, i.e., Mj(t) =--

fo

rJF(r, t)dr.

[4]

Given the functional form of the size distribution at some initial time, ti, Eqs. [1]-[4] can be used to calculate its form at times t > t i. It is interesting to observe that one can combine Eqs. [1]-[4] to show that 343, which is clearly proportional to the total foam volume, is independent of time. RELATION BETWEEN FOAM STABILITY, OSTWALD RIPENING, AND GRAIN GROWTH A crucially important observation, relative to the foam-stability problem summarized in Eqs. [1]-[4], is the following: This particular problem is mathematically equivalent to that of interface-reaction-controlled Ostwald ripening of a size distribution of second-phase particles situated within a host matrix. In other words, that process is also described by Eqs. [1]-[4], as has been shown by a number of investigators, e.g., (4, 5); only the parameters that are combined to yield the constant K are different. Briefly, Ostwald ripening consists of the net transport of matter from smaller particles to larger ones, so that the larger ones grow, the smaller ones dissolve. The driving force for the process is the accompanying reduction of net interfacial energy. Another very important observation is the following: Ostwald-ripening theory has been applied to a problem

569 0021-9797/85 $3.00 Journal of Colloid and Interface Science, Vol. 107, No. 2, October 1985

Copyright © 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

570

NOTES

that is quite similar to that of foam evolution, i.e., the growth of grains in single-phase, polycrystalline materials, e.g., by Hillert (13) and others. (In fact, both two- and three-dimensional polycrystals were considered by Hillert (13).) Indeed, grain growth amounts essentially to a solid-state analog of foam evolution due to interbubble gas diffusion. Not surprisingly, therefore, it follows from Hillert's analysis (13) that Eqs. [1]-[4] also apply to the coarsening of a system of grains in a three-dimensional polycrystal. Again, only the parameters that are combined to yield K are different. Hillert also pointed out the mathematical identity that exists between his problem and that of interface-reaction-controlled Ostwald ripening.

tended to evolve toward the unimodal asymptotic size distribution. Numerical studies of grain growth in polycrystals have been carried out by various investigators, e.g., (21-28) and by others, e.g., as cited in (26). In addition, numerical studies of the evolution of a distribution of foam bubbles have also been reported, including that of Lemlich (12), EVOLUTION OF A NARROW SIZE DISTRIBUTION OF FOAM BUBBLES One particular problem that was considered by Monsalve and Schechter (1) was the initial variation with time of the foam surface area, A(t), given by

REVIEW OF OSTWALD-RIPENING STUDIES

A(t) = bM2(t)

Detailed theoretical studies of various types of Ostwaldripening kinetics have been carried out over a period spanning more than two decades. Based on some of the reported results, the following conclusions can be drawn regarding the specific problem summarized by Eqs. [1][4]: For an arbitrary initial size distribution, F(r, ti), the solution of this set of equations is analytically intractable. However, analytic approaches can be employed for two limiting cases, namely, for a "narrow" size distribution and for asymptotically large times, as now described. At asymptotically large times, the distribution function approaches a functional form that can be expressed analytically and that is independent of the form of F(r, t~). This procedure was illustrated, for example, in the classic work of Lifshitz and Slyozov (2, 3) and Wagner (4), commonly referred to as the LSW theory. It was later shown by Markworth (14) that this type of asymptotic size distribution could be conveniently described in terms of a mathematical "similarity transformation." Some, albeit considerably fewer, analytical studies have been reported relative to Ostwald-ripening kinetics of earlier times, for which the asymptotic conditions are not yet satisfied. In these studies, F(r, ti) was taken to be relatively "narrow" in particle-radius space, that is, the standard deviation of the distribution function (which is a measure &its "width") is small compared to the mean radius. Early work in this area was conducted by Wagner (4), and more extensive studies were later carried out by Markworth, who considered distributions of both precipitate particles (15) and pores (16). A primary result of their studies was that, at these earlier times, the major change of the size distribution with time consisted of its becoming "wider" in particle-radius space, with relatively little concomitant variation of the arithmetic mean radius. It is also noted that a considerable number of numerical analyses of Ostwald-ripening kinetics have been carried out, a sample of which includes the work of Kampmann and Kahlweit (17), Robertson and Pound (18), and Thomas and coworkers (19). In a similar vein, numerical studies of the ripening of gas-filled pores were reported by Markworth and Oldfield (20). One result of particular interest was that reported by Robertson and Pound (18), who found that even an initially trimodal distribution

where b is a proportionality constant. In particular, they showed that two bubble distributions having the same values of M2(ti) and M3(Q (they actually took ti = O) are not necessarily characterized by the same A(t). It is of interest to apply some results obtained for the interfacereaction-controlled Ostwald ripening of a narrow precipitate distribution (15) to this problem. Toward this end, consider the variance, /z2(t), of the size distribution, defined as

Journal of Colloid and Interface Science. Vol. 107,No. 2. October1985

[5]

/~2(t) -= ((R - (R)) 2) = (R 2) - ( R ) 2

[6]

where, in general, ( W ) =- M;(t)/Mo(t) with M0 being independent of time, to a good approximation, for a narrow size distribution. (Clearly, from the manner in which F is defined by Monsalve and Schechter (1), M0 is the total number of bubbles in the foam.) One can use Eq. [16] of Ref. (15) to show that, for interfacereaction-controlled Ostward ripening of a narrow size distribution, P,2(t) ~ tt2(ti)exp[(T)~ ( t - t i ) ]

[7]

where the parameter K used in Eq. [7] is that used in Eq. [2], and where (R)i is the value of ( R ) at time ti, recalling that the variation of ( R ) with time is relatively slight (15). Moreover, Eq. [7] was shown to be valid regardless of the functional form of F(r, ti). It was also shown (15) that, in this approximation, ( R ) 2 ~ ( R ) 2 + 2[a2(t,) - uz(t)].

[81

Combining Eqs. [5]-[8], we obtain K

From Eq. [9], we conclude that, for a narrow size distribution of foam bubbles, the total surface area decreases with time at an exponentially increasing rate. However, since ~2(t) ~ ( R ) 2 in this case, one can readily show that

NOTES A(t3 - A(t) ~ 1 A(ti) so that the fractional change of surface area within this regime is small compared to unity. Of course, as the distribution function becomes progressively wider, the "narrowness" assumption becomes less and less applicable. ASYMPTOTIC SIZE DISTRIBUTION OF FOAM BUBBLES At large times, the size distribution of foam bubbles approaches its asymptotic form that is characterized by a similarity transformation which, for the problem described by Eqs. [1]-[4], has been shown by Markworth (14) to have the form e(r, t) = t-2g(w)

[10]

w ~ r/r21

[1 1]

r2j = (Kt/2) l/z.

[12]

where with The function g(w) has been shown (e.g., (4, 5, 13)) to have the form Bw [ -3w g(w) = (2 - w)' e x p t ~ J

[13]

for0 < w < 2 , andg(w) = 0 f o r w>t2, w h e r e B i s a normalization parameter. It is of interest to observe that the algebraic moments of the asymptotic distribution function exhibit very simple power-law-type variation with time. Combining Eqs. [4], and [10]-[12], we obtain [ K~ ~J+w2 M~(t) = mj~-~) t (j-3)/2

[141

where mj

wJg(w)dw.

[15]

We see, from Eqs. [13]-[15], that the value of the parameter B can be expressed in terms of the timeindependent moment, 1143. CONCLUSION In conclusion, it is clear that a large body of knowledge exists that is directly applicable to the problem of foam evolution due to interbubble gas diffusion. Correlation of this knowledge with experimental data should add new insights relative to foam behavior. REFERENCES 1. Monsalve, A., and Schechter, R. S., J. Colloid Interface Sci. 97, 327 (1984). 2. Lifshitz, I. M., and Slezov, V. V., Soviet Phys. JETP 35, 331 (1959).

5 71

3. Lifshitz, I. M., and Slyozov, V. V., J. Phys. Chem. Solids 19, 35 (1961). 4. Wagner, C., Z. Elektrochem. 65, 581 (1961). 5. Psarev, V. I., Phys. Metals Metallogr. (Engl. Transl.) 26, 115 (1968). 6. Hanitzsch, E., and Kahlweit, M., in "Industrial Crystallization," p. 130. The Institution of Chemical Engineers, London, 1969. 7. Kahlweit, M., in "Physical Chemistry--An Advanced Treatise" (H. Eyring, D. Henderson, and W. Jost, Eds.), Vol. X, "Solid State" (W. Jost, Ed.), p. 719. Academic Press, New York, 1970. 8. Markworth, A. J., Metall. Trans. 4, 2651 (1973). 9. Slezov, V. V., and Sagalovich, V. V., J. Phys. Chem. Solids 44, 23 (1983). 10. Kosevich, A. M., Saralidze, Z. K., and Slezov, V. V., Soy. Phys. Solid State 6, 2707 (1965). 11. Hohmann, H. H., and Kahlweit, M., Ber. Bunsenges. Phys. Chem. 76, 933 (1972). 12. Lemlich, R., Ind. Eng. Chem. Fundam. 17, 89 (1978). 13. Hillert, M., Acta MetalL 13, 227 (1965). 14. Markworth, A. J., Metallography 3, 197 (1970). 15. Markworth, A. J., Ber. Bunsenges. Phys. Chem. 75, 533 (1971). 16. Markworth, A. J., in "Defects and Transport in Oxides" (M. S. Seltzer and R. I. Jaffee, Eds.), p. 397. Plenum, New York, 1974. 17. Kampmann, L., and Kahlweit, M., Ber. Bunsenges. Phys. Chem. 74, 456 (1970). 18. Robertson, D., and Pound, G. M., J. Crystal Growth 19, 269 (1973). 19. Thomas, S. W., Brook, P. A., and Moon, J. R., Metal Sci. 13, 655 (1979). 20. Markworth, A. J., and Oldfield, W., Mater. Sci. Eng. 10, 159 (1972). 21. Hunderi, O., Ryum, N., and Westengen, H., Acta MetalL 27, 161 (1979). 22. Hunderi, O., Acta MetalL 27, 167 (1979). 23. Hunderi, O., and Ryum, N., Acta Metall. 29, 1737 (1981). 24. Hunderi, O., and Ryum, N., Acta Metall. 30, 739 (1982). 25. Srolovitz, D. J., Anderson, M. P., Grest, G. S., and Sahni, P. S., Scr. Metall. 17, 241 (1983). 26. Anderson, M. P., Srolovitz, D. J., Grest, G. S., and Sahni, P. S., Aeta MetalL 32, 783 (1984). 27. Srolovitz, D. J., Anderson, M. P., Sahni, P. S., and Grest, G. S., Acta Metall. 32, 793 (1984). 28. Anderson, M. P., Srolovitz, D. J., Grest, G. S., and Sahni, P. S., Mater. Res. Soc. Syrup. Proc. 21, 467 (1984). ALAN J. MARKWORTH Battelle, Columbus Laboratories 505 King Avenue Columbus, Ohio 43201 Received November 26, 1984; accepted January 29, 1985 Journal of Colloid and Interface Science, Vol, 107, No. 2, October 1985