Bubble Nucleation in Superheated Liquid-Liquid Emulsions C. T. AVEDISIAN 1 AND R. P. ANDRES 2 School of Engineering and Applied Science, Princeton University, Princeton, New Jersey 08540 Received April 11, 1977; accepted October 7, 1977 An investigation of the process by which bubbles form in superheated liquid-liquid emulsions is reported. A theoretical model for the rate of bubble nucleation is developed that is valid for any binary liquid system. Because of the interesting "microexplosion" phenomenon postulated to occur during combustion of fuel/water emulsions, special attention is focused on hydrocarbon/ water/surfactant mixtures. The model predicts that in these systems nucleation takes place at the interface between the two liquids but that the vapor bubbles rest entirely within the hydrocarbon phase. The maximum superheat limits of emulsions containing 83% by volume hydrocarbon (benzene, n-decane, n-dodecane, n-tetradecane, and n-hexadecane), 15% water and 2% surfactant were measured experimentally at 1 atm and found to be in good agreement with the predictions of the theoretical model. The superheat limits of these emulsions are lower than those of either pure liquid. How- ever, for microexplosion of an emulsified fuel droplet, the superheat limit of the fuel/water emulsion must be less than the boiling point of the fuel. Only the n-hexadecane/water emulsion satisfies this criterion at atmospheric pressure. 1. INTRODUCTION evaporation from the surface of an emulsi- It is almost axiomatic that improvements fied fuel drop is primarily from the con- in atomization and vaporization of high- tinuous fuel phase. Thus, when such a drop boiling-point fuels result in cleaner, more ef- is heated, its temperature approaches the ficient combustion. A promising approach in boiling point of the pure fuel. With a high- this regard is to emulsify the fuel with water boiling point fuel this means that the water prior to burning (1). The effect of adding becomes superheated. This metastable con- water is twofold. First, the water serves as a dition is maintained only as long as no vapor diluent. It reduces both liquid-droplet and bubbles form within the drop. Each emul- gas-phase temperatures in the flame, sion exhibits a critical maximum tempera- thereby moderating undesirable reactions ture, termed its limit of superheat, at which that lead to production of soot and NOx. the probability of bubble nucleation be- Secondly, as first observed by Ivanov and comes appreciable. If the drop temperature Nefedov (2), emulsified fuel droplets tend to reaches this superheat limit, it disintegrates explode when heated, thereby solving the spontaneously, shattered by the internal problem of fuel atomization. formation of vapor bubbles and the conse- This latter "microexplosion" phenome- quent rapid vaporization of superheated non has its origin in the volatility difference water. between the water and the fuel. Since the A necessary condition for microexplosion water is dispersed and isolated in the emul- of an emulsified fuel droplet is that the limit sion as relatively immobile microdroplets, of superheat of the emulsion be less than the boiling point of the fuel. In order to make 1 Department of Aerospace and Mechanical Sciences. practical use of this condition one must have 2 Department of Chemical Engineering. a theoretical expression for the limit of 438 0021-9797/78/0643 -0438502.00/0 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 439 superheat. Thus, it is necessary to under- volatile liquid are in agreement with the ex- stand the process by which vapor bubbles perimental facts. What remains is to extend nucleate in superheated emulsions. these theoretical treatments to include In what follows we will not consider cases in which both components of an emul- heterogeneous nucleation on solid motes or sion are of comparable volatility. In the impurities. We focus instead on the process present paper we formulate such a general by which vapor bubbles form homogeneously theoretical model and compare its predic- either within one of the immiscible liquids tions with experiment. or at the surface between the two liquids. Since critical size bubbles or nuclei are 2. THERMODYNAMICS OF BUBBLE generally much smaller than the dispersed FORMATION droplets present in practical emulsions, the The minimum work required to form a interface can be treated as a flat surface. vapor bubble at a flat liquid-liquid inter- The problem reduces to that of homogene- face is (11) ous bubble nucleation in a system of two immiscible liquids separated by a flat W = V"(p' - p") + ~ n~"(ix~" - txi') interface. i There have been several studies of bubble + o'aSa + o'bSb -- O'abSab. [1] nucleation in which a drop of a volatile test liquid is immersed in an immiscible In this equation V" is the volume of the bub- host fluid and is heated until it vaporizes ble; p' is the ambient pressure in the liquid; (3-10). Except in the case of water (4), p" is the vapor pressure in the bubble; n~" the superheat limits measured in these ex- is the number of molecules of component i periments do not seem to be dependent on in the bubble;/x~" and/x~' are the chemical the nonvolatile host fluid and are in excellent potentials of component i in the vapor and agreement with a kinetic model for nuclea- in the liquid, respectively; o-a, ob, and oab tion that assumes bubbles form homogene- are the surface tensions associated with the ously within the test drop. respective interfaces; and S~, Sb, and Sab Moore (3) was the first to consider the are the respective interfacial areas (see Fig. effect of the host fluid. He postulated that 1). The first term in Eq. [1] accounts for discrepancies between theory and experi- the pV work required to form the bubble. ment in the water case were due to nuclea- The second term represents the chemical tion at the surface of the water drop. Moore work required to introduce n/' molecules of developed a thermodynamic model for such species i into the bubble (the summation ex- interfacial nucleation which takes into tends over all species present in the bubble). account the surface tensions of both liquids The last three terms are the work required to and the interfacial tension between them. create or destroy the respective interfacial Apfel (9) and Jarvis et al. (10) have also surfaces. We implicitly assume in Eq. [1] studied this problem and have derived and in all that follows that the bubble is in kinetic expressions for the rate of bubble thermal and mechanical equilibrium with nucleation at the interface between a vola- the surrounding liquid. tile and a nonvolatile liquid: These treat- The detailed lenticular shape of such a ments are in agreement with the few exist- bubble is shown in Fig. 1. A force balance ing experiments but neglect the contribution yields of the less volatile liquid to bubble growth. o-~b = o-~ cos 0 + o-~ cos tO, [2] Thus, existing kinetic models for the rate of bubble nucleation in pure liquids and at and the interface between a volatile and a non- o-a sin 0 = crb sin 0. [3] Journal of Colloid and Interface Science, gol. 64, No. 3, May 1978 440 AVEDISIAN AND ANDRES I I LIQUID b ~b S\ INTERFACE Dab 2 rb LIQUID a Fro. 1. Geometry of lenticular bubble at a liquid-liquid interface. Solving these equations one obtains 16 W = -- 1ro-a(p " - p')-~ 3 O'a 2 -- Orb 2 -F O-ab 2 M. = cos0 = , [4] 20"aO'ab + E n;'(m" - t~,'), [8] i and O'b 2 -- O'a 2 Jr" Grab 2 where or is a generalized surface tension Mb = cos q~ = [5] which depends on the relative magnitudes of 20"bO'ab o-a, o'b, and Gab. In addition The lenticular shape that has been as- sumed in the derivation of Eq. [8] can exist r. = 2cra/(p" - p') [6] in mechanical equilibrium only when and > Io'o- o'.1. rb = 2o-J(p" - p'). [7] If Eq. [9] is not satisfied, the bubble is Equation [1] can be transformed through spherical in shape and rests entirely within the use of Eqs. [4]-[7] and straightforward the liquid with the lower surface tension. geometrical considerations to yield Thus, there are three distinct possibilities: LIQUID b (FUEL) Ga ~O'b +O'ab o-ob>lol)-o-o I o-b >Io'o +O"ab (A) (B) (C) LIQUID o (WATER) FIG. 2. Possible locations of bubble formation at liquid-liquid interface. Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 441 1. (Tab > I(Ta - (Tb[, the bubble is lenticular is characterized completely by its composi- (Fig. 2B), and the minimum work of forma- tion, (na",nf, . • . no"). Equation [8] repre- tion is given by Eq. [8] with sents the reversible work required to form such a bubble. However, this equation can cr = [o-aa(V2 - 3AMa + ¼Ma 3) be simplified. In many cases of practical + o-ba(1A - 3AMb + ¼M~a)]l;a; [10] interest or, and o-~ are insensitive to vapor composition, and the vapor can be treated 2. o'b -- (Ta + o'a~, the bubble is spherical as an ideal gas mixture. In this approxima- resting entirely within liquid a (Fig. 2A), tion and the minimum work of formation is given by Eq. [8] with 16 W(ni",nf , . n/') = -- ~rcr~(p '' - p,)-2 o- = o-a; and [11] 3 P i" 3. o'a >- (Tb + (Tab, the bubble is spherical + ~, n('kTln- [13] resting entirely within liquid b (Fig.