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 assumed 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 formation, (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. 2C), i Pi* and the minimum work of formation is given In this equation o- depends only on the by Eq. [8] with temperature, pressure, and composition of the liquids; k is Boltzmann's constant; T is (T = o'b. [12] the absolute temperature; p/' is the partial The criterion for bubble formation within pressure of component i in the bubble liquid a corresponds to liquid a spreading (p" = ~ p~"); and p~* is the partial pressure on liquid b; while the criterion for bubble of i when in chemical equilibrium with formation within liquid b corresponds to the surrounding liquid. liquid b spreading on liquid a (10). The bub- The volume of such a bubble is ble is lenticular only if neither liquid spreads 32 on the other. In such a situation it takes V = -- ¢ro-3(p" - p,)-3. [14] less work to create a lenticular bubble at the 3 interface than a spherical bubble in either Thus, the total number of molecules in liquid. the bubble, n", is related to the total vapor Nucleation at the liquid-liquid interface pressure in the bubble as follows: in an emulsion always results in the lowest work of formation because both liquids can contribute to the vapor pressure in the bub- n" = rr(T3(p" - p ) ]-~. [15] ble. If the bubble forms entirely in the more volatile liquid and the second liquid has a Combining Eqs. [13] and [15] one obtains negligibly small vapor pressure, however, W(nl",nf, . . . nc") the additional work required to form the bubble away from the interface is quite Yi" = W?(n r') + ~ ni"kT in [16] small. As there are generally more sites in i Y~* the bulk than at the interface, nucleation where within the volatile liquid is the dominant process in such cases. In all other cases, where the work of bubble formation at the interface is substantially less than in either liquid, the dominant process is at the × (p"-p') +p"ln P* . [17] interface. A bubble that is in thermal and mechani- In Eq. [16] Yi" is the mole fraction of com- cal equilibrium with the surrounding liquid ponent i in the bubble, and yi* is the

Journal of Colloid and Interface Science, Vol, 64, No. 3, May 1978 442 AVEDISIAN AND ANDRES equilibrium mole fraction of i(yi* = n~*/ This last case corresponds to what Jarvis n* = Pt*/P*). Equation [16] reduces to the et al. (10) term bubble blowing. Here, re- model of Jarvis et al. (10) when there is quirements of mechanical equilibrium force only one vapor species (n" = nl"). the bubble to rest entirely in liquid b while The work of formation represented by species i is assumed to be present entirely Eq. [16] exhibits a saddle point when in liquid a. A definition for St in this situa- p" = p* and y~" = Yt*. This particular bub- tion must be rather inexact. In what follows ble is termed the nucleus. Its work of forma- we assume that ha is equal to the average tion is area occupied by a molecule in the liquid a/vapor interface. Errors in estimating this W*(nl*,n2* .... nc*) quantity have only a relatively small effect 16 on the theoretical limit of superheat. _ __ zrcra(p, _ p,)-2. [18] 3 The above discussion proceeds in a simi- where lar fashion if species i is present only in P* = ~, Pt*. liquid b, giving rise to the three situations: i 1. (Tab > ](Ta-o-b], This critical embryo plays a central role in determining the nucleation rate. St = 8~-(Tb2(p" -p')-2(1 - Mb); [23] All bubbles grow by evaporation of mole- 2. (Tb >- (Ta + o~b, Si = A~; and [24] cules from the surrounding liquid. Neglect- ing for the time being the concurrent con- 3. (Ta :> O-b + (Tab, densation process, the rate at which mole- St = 16~'(Tb~(p" - p')-z. [25] cules of species i evaporate into a bubble is Now we are in a position to estimate the -f~ = fiiSt. [19] rate of bubble nucleation. If the absolute rates of evaporation and condensation where fit = Pi*(2rrmtkT) -1/2 is the maximum which produce the various bubble species rate of evaporation per unit surface area, are much faster than the net rate of nuclea- mt is the mass of an i molecule, and St is the tion, the population of these bubbles should effective surface area of the bubble as seen correspond closely to the predictions of by the i molecules. equilibrium thermodynamics. At equi- The effective surface area, S~, takes on librium (12) different values depending on the relative position of the bubble with respect to the Xeq(nl",n2 ", • • • no") liquid-liquid interface. Assuming that the = N exp(-W(nl",n2", . . . nc")/kT), [26] i species is present only in liquid a, we have three possibilities: where X eq(nl",n2 ", . . . n~") is the equilibrium 1. (Tao > ] (T, - (Tb], the bubble is lenticu- number of bubbles having the composition lar (Fig. 2B), and (na",n~", . . . n~"), N is the total number of molecules in the liquid-liquid interfacial Si = 87r(Ta2(p'' -p')-2(1 - Ma); [201 region when the dominant process is at the interface or the total number of molecules in 2. (To -> (Ta + (Tab, the bubble is spherical the volatile liquid when the dominant resting entirely within liquid a (Fig. 2A), and process is in the bulk, and W(nl",n2",... n~") Si = 16~'(Ta2(p" _p,)-2; and [21] is given by Eq. [16]. The assumption that X(n~",n2", . . . nc") 3. (Ta -> o-b + (Tab, the bubble is spherical = xeq(na",n2 ", • . . no") can never be valid resting entirely within liquid b (Fig. 2C), and for bubbles that are much larger than the Si = Aa. [221 critical nucleus because W(nl",n2", . . . no") Journal of ColZQid.and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 443

-oo when n" ~ ~ for superheated sys- let nucleation from a supersaturated binary tems. However, if this approximation is as- vapor. The multicomponent generalization sumed valid for the nucleus, if every stable of Reiss's treatment was provided by bubble that forms is assumed to pass Hirschfelder (14). Andres (15) also looked at through the nucleus state, and if any bubble this problem and extended the range of ap- that becomes larger than the nucleus is as- plicability of the Reiss-Hirschfelder formu- sumed to grow to macroscopic size with lation. In what follows we will for simplicity probability F, then an "equilibrium rate" assume a binary system, i.e., an emulsion of estimate for the rate of nucleation can be ob- two pure liquids. A general multicomponent tained as formulation may be found in Ref. (15). The net rate of a typical reaction step, J = F(~ ~ON exp(-W*/kT). [27] i (na",nb") ~ (na" + 1, no"), is The utility of this expression lies in its Ia(na t! ,no tt ) = ka(na tv ,no v!)X(na v! ,no tt) simplicity. The surprising fact is that Eq. - ka(na" + 1, nb")X(nd' + 1, no") [28] [27] with F = 1 yields reasonable estimates of superheat limits for both pure liquids and where X(na",n(') is the number of bubbles liquid-liquid emulsions. with composition (na ....,n~ ), ka(na- ....,no ) is de- In the next section we present a more fined by Eq. [19], and ka(na- ....+ 1, nb ) is de- detailed kinetic description of nucleation in fined below in terms of ka(na",nb"), emulsions of two pure liquids. This descrip- X"q(na",no"), and X~q(nd' + 1, no"). Making tion yields a" steady-state" estimate for the use of microscopic reversibility, one obtains rate of nucleation which can be cast in the ka(na vt ,no vt) xeq(nd ' + 1, no") same form as Eq. [27] and which provides [29] an explicit expression for F. ka(na + 1, no") Xeq(nd',no ")

3. RATE OF HOMOGENEOUS NUCLEATION Defining X(na",no") The actual stochastic process by which f(nd',no") - [30] Xeq(na",no ") microscopic bubbles grow to macroscopic size can be modeled as a series of chemical Eq. [28] becomes reactions in which vapor bubbles are con- Ia(na",no") = ka(na ct ,no u )X eq (na tt ,no tt ) verted from one type embryo to another by addition or loss of single molecules from the × [f(na",no") -f(na" + 1, nb")]. [31] surrounding liquid. When there is only one When na* and nb* are large, this expression chemical species involved in these reac- can be approximated as (16) tions, they take the form of a one-dimen- sional random walk or diffusion process. Ia ~- - kaX eq Of [32] The steady-state kinetic treatment of such OHa. " single component systems is well estab- lished and forms the basis of existing models The net rate of increase of bubbles of for the rate of bubble nucleation. These species (na",nb") is simply theories assume that only one of the liquids dX(nd',nb") in the emulsion exhibits an appreciable dt vapor pressure (10). Kinetic treatment of systems in which = [Ia(na" - 1, no") - la(nd',no") there are more than one active component + Ib(na", no" - 1) - Io(nd',nv")]. [33] is more complicated. The problem was first studied by Reiss (13). He derived an expres- If a pseudo steady-state is assumed and a sion for the steady-state rate of liquid drop- continuous approximation to the discrete

Journal of CoUoid and Interface Science, Vol. 64, No. 3, May 1978 444 AVEDISIAN AND ANDRES model is imposed (16), this equation be- 11 = kiN exp(- W*/kT) / comes oI~ Olb - W* -- + - 0. [34] × f_~ exp(W~/: )d(~1-~1" ). [39] Ona" Onb" In the region near (~1",~2"), W(~I,~2)/kT Equation [34],along with the definitions can be expanded to yield for I, X eq, and k, given by Eqs. [32], [26], and [19], respectively, and the boundary (W - W*)/kT ~ an (~1 -- ~1") 2 conditions f(0,0) = 1 and f(%~) = 0, com- 2 pletely define the steady-state nucleation problem. The key simplifications introduced "4- a12(~ 1 -- ~1")(~2 -- ~2") by Reiss to obtain an analytical solution are: (a) all embryos that grow to macro- + a22 (~2- ~2")2, [40] 2 scopic size pass through the region around the saddle point in the W(na",nb") surface; where all < 0 and a22 > 0. Substituting and (b) an orthogonal coordinate system Eq. [40] into Eq. [39] and integrating yields (~,~2) can be defined in this region so that the only nonvanishing component of the I1 = -kiN exp(-W*/kT)( ]an[ ]1/z nucleation flux lies in the ~ direction. \ 2Ir ] Introducing a general orthogonal transformation,

g~ = cos 6na" + sin 6nb", [35] Thus, the nucleation flux is g2 = -sin 6nd' + cos ¢knb", J = Ild~2 ~- FIN exp(- W*/kT) and assuming 12 = O, the flux in the gl direction at a fixed value of ~2 is (13-15) + lalzl . [42] I1 = _£~xeq(~l,~2 ) __0f, [36] Rewriting Eq. [42] in the form exhibited where by Eq. [27], one obtains: -kl = - kakb [37] J--- F(~ + -k~)Nexp(-W*/kT), [43] ka sin 2 4) + kb cos 2 4) where Integrating this expression along gl, one obtains

djl~Oe -f(~1,~2) -~ +1 Before the steady-state nucleation rate ~1 =0 can be calculated, we must derive explicit o expressions for all, al2, and a22 and deter- = [~lxe"(~l,g2)l-hrldgl. [38] mine the angle ~b. From Eq. [16] the minimum work of forming a bubble is The major contribution to this integral comes from the region near the nucleus. W(n. tt ,non ) = W?(n")

Assigning kl and I1 constant values in this ya tt yb n region and removing them from the integral + na"kT In + nb"kT In --, [45] yields Ya* Yb* Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 445

where Wt(n") is given by Eq. [17]. Ex- F(6). The values of 4) and F(49 obtained panding this equation around (na*,nb*) and by these two approaches are compared in introducing the coordinate transformation Sect. 5. defined by Eq. [35], we obtain (15) For most liquid systems a precise value of F is not necessary for estimating the r/*all maximum superheat. Thus, it is useful to develop and test simple approximations for ÷-- 1 / cos 24)- 2~sin 4) cos Ya* / F. The simplest model is, of course, r = 1. [51] + -~ + sin ~ 4), [46] Yb* A more accurate approximation is to choose 1~*a12 4) so that Yb* -(Y°: cos q~ tan 4) - , [52] Ya*Y~* Ya* + ~(1 - 2 cos 2 4)), [47] and evaluate F(4)) using Eq. [44]. Still another approach is to estimate F by f/*a22 neglecting the volatility of one of the 1 liquids when estimating the rate of nucle- = --OZ +-- sin2 4) + 2o~ sin (b cos 4) ation. In this limit (yo* = 0), Ya* of J = I~ = -kaN exp(-W/kT) -- [53] ÷ -a +-- ,) cos24), [48] Ona 't Yb* where Integrating as before yields Ya* ~ Yb*, and / 2~r ~ 1/2 J . [54] 3p* - [49] 2p* - p' Thus, F is equal to Thus, F is a function of p* and p' through L o~, ofy~* and Yb*(Y~* --- 1 - y~*), ofk~ and ~, ka + kb and of 4), where only the value of(b remains to be specified. Reiss (13) and later Hirsch- x (pa*-p'~*/2 felder (14) argued that 4) should be chosen _t__ / (2~'~, ,~-,/2 [55] \£Pa*+P') " "~a , " to diagonalize the Jacobian matrix This expression was obtained by Jarvis 02W/kT et al. (10). Ona"Onb" n*] " In Sect. 5 we compare these various ap- This is equivalent to picking 4) so that proximations for F with the more accurate a12(6) = 0. Making use of Eq. [47] one model given by Eq. [44] in which q~ is obtains chosen to maximize F(4)). First, however, we describe a series of experiments de- 2o/ signed to measure the nucleation temper- tan 24) = [50] (Ya* - Yb*)/(Ya*Yb*) atures or, more properly, the superheat limits of hydrocarbon/water emulsions. If Andres (15) showed that this evaluation of these measurements reflect homogeneous (5 is only valid when ka = kb- In general nucleation at the hydrocarbon/water inter- 4) should be chosen so as to maximize face, Eqs. [43] and [44] should be able to

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 446 AVEDISIAN AND ANDRES predict the superheat limits that are ob- For the experiment to be meaningful there served. must be a higher probability for nucleation within the drop than at the interface between the drop and the surrounding liquid. 4. EXPERIMENTAL METHOD AND OBSERVATIONS As discussed in Sect. 2, this means that the continuous phase in the emulsion should The experimental method employed in spread on the nonvolatile liquid used in the the present work is a modification of the column. Spreading occurred for the alkanes classical technique developed by Moore (3) used in the present experiments when they and subsequently used by Eberhart et al. contained a small amount of surfactant in (6). A small drop (-1 mm diameter) of the solution (2% by volume). Glycerine is a desired emulsion is injected into the bottom questionable host liquid for superheating of a column filled with a heavier immiscible, benzene, however. A drop of benzene con- nonvolatile liquid (glycerine). The column taining surfactant spread initially on is heated to produce a stable temperature glycerine but ultimately formed a lens. profile which is hotter at the top than at A schematic illustration of the apparatus the bottom. As the drop rises, it is heated is shown in Fig. 3. It consisted of a Pyrex until at some point bubble nucleation oc- tube 4.6 cm o.d., 4.0 cm i.d., and 60 cm curs. This method minimizes nucleation on long with rubber stoppers inserted at each solid surfaces and enables test emulsions end. The column was wrapped with 33 turns to be heated above the boiling point of their of No. 20 gage nichrome wire secured with continuous phase. Sauereisen cement. The wire spacing varied

THERMOCOUPLES~VENT ~ N2GAS NICHiOMEWIRE~ ~GLYCERINE~{~

TOVARIAC ! ~PYREX TUBE /RISINGTEST DROP NEEDLE~ ~ SEPTL~ RUBBERSTOPPER~ I "----SYRINGE

FIG. 3. Schematic of experimental apparatus.

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 447 from 2.5 cm at the bottom to 0.5 cm at the decane, were tested. They contained ap- top. Power was provided by an ac current proximately 83% by volume of the hydro- regulated by a Variac. Temperatures were carbon, 15% water, and 2% surfactant. The measured by two No. 24 gage copper- emulsions were prepared in small samples constantan thermocouples enclosed in "L" of about 30 ml by first mixing the hydro- shaped Pyrex tubes which were immersed carbon with surfactant and then adding directly into the glycerine. The thermo- water and briskly agitating the mixture for couples were connected to a Leeds & about 10 min. The hydrocarbons were ob- Northrup Model 8686 potentiometer and tained from Humphrey Chemical Co. with a calibrated to about +0. I°C. stated purity of 99.6% or better. The A nitrogen atmosphere was maintained water was singly distilled. No special at- above the glycerine to reduce oxidation tempt was made to further purify the liquids and subsequent darkening of the glycerine. other than normal filtering. The surfactants All experiments were done at atmospheric were mixtures of Span 85 and Tween 85 pressure. (alkane emulsions) and Span 80 and Tween A small solid brass cylinder with a brass 80 (benzene emulsions) available from Atlas tube soldered on the end was fitted into a Chemical Co. For the alkane/water emul- hole in the bottom rubber stopper. A 2 mm sions HLB = 5.5 was used. HLB = 7.7 diam hole was drilled through the cylinder gave a stable benzene/water emulsion. and a rubber septum was placed over the Difficulty in preparing stable emulsions hole. Droplets were injected into the system was the greatest restriction we faced in through a No. 22 gage hypodermic needle, studying a wider class of systems. The 5 cm long, which was inserted through the alkane/water emulsions stood for only a few rubber septum. minutes before visible separation occurred An experiment consisted of stabilizing the in the mixing beaker. The benzene/water temperature profile (requiring about 4 hr) emulsions lasted an average of 15 rain be- and injecting 1 drop of the test emulsion. fore visible separation. Examination of the The drop slowly rises until a vapor bubble alkane/water emulsions under a microscope forms within it. At this point the drop showed internal water droplet diameters rapidly vaporizes. ranging from 0.01 mm to 0.1 ram. Benzene/ Attention was focused on a test section of water emulsions exhibited more uniform approximately 10 cm in length near the top water droplets of approximately 0.01 mm of the column. This region is just below the diameter. maximum temperature in the column and In principle the dispersed water droplets exhibits an almost linear temperature are isolated in the hydrocarbon phase. In gradient of about l°C/cm. The temperature reality they settle slowly. This produces in this section was adjusted until out of ap- a clear region at the top of a rising emul- proximately 60 emulsion drops not one sion drop. This settling also leads to sub- made it to the top of the column without stantial loss of water through the emulsion/ vaporizing. While most of the drops vapor- glycerine interface. To minimize possible ized before reaching the test section, several effects of this water loss on the column drops in each series vaporized in this region. fluid, the glycerine was completely replaced The superheat limit of the emulsion was after injection of approximately 100 drops. taken to be the temperature at the highest Often for small drops (<0.5 mm diameter) location in this test section where the latter all the water disappeared from the emulsion drops vaporized. before the test section was reached. The Emulsions of water in benzene, n-decane, drop would then rise through the column n-dodecane, n-tetradecane, and n-hexa- usually without exploding. Therefore, data

Journal of Colloid and Interface Science, Vol. 64, No. 3. May 1978 448 AVEDISIAN AND ANDRES were taken only with drops of approxi- ing as vapor and such events are easily con- mately 1 to 2 mm in diameter. These drops fused with homogeneous nucleation. retained the majority of their water The possibility exists that water loss to phase during transit through the column. the glycerine phase gives rise to bubble They rose through the test section at a rate nucleation near the boundary between the of approximately 8 cm/sec and could be as- emulsion drop and the host fluid. Unfortu- sumed to be in thermal equilibrium with nately, homogeneous nucleation measure- their surroundings. ments when the water/surfactant phase was The water/hydrocarbon mixtures appeared sprayed directly into the hydrocarbon phase to be immiscible even at the temperatures proved impractical. The initial disturbance involved in the nucleation experiments. of the drops as they hit the liquid surface Except for the settling phenomenon dis- and the tendency of the drops to collide cussed above, the emulsions retained their with the walls of the column precluded milky appearance up to the point of explo- careful homogeneous nucleation experi- sion. We tested for miscibility by sequen- ments with this configuration. tially replacing the glycerine in the column The emulsion drops did not explode in with dodecane, tetradecane, and hexadec- the sense of producing a "pop" or "bang" ane and spraying water/surfactant solution as do pure liquids. Instead they exhibited into the top of the column through a 0.15 a sort of "breaking up" or "puffing" apart. mm capillary. We could not detect any dis- This is indicative of the slower transport solving of the water drops as they fell processes involved in rapid vaporization through the column at temperatures in ex- of an emulsion. The temperature de- cess of 200°C. In addition to these observa- pendence of the nucleation rate (Eqs. [43] tions, Dryer et al. (1) have used high speed and [44]), while still quite sharp, is also less photography to study burning n-dodecane/ abrupt than in the case of the pure liquids. water drops suspended from a quartz fiber. Both of these phenomena tend to broaden Their observations indicate immiscibility of the temperature range in which nucleation is observed. water and n-dodecane at the saturation temperature of n-dodecane (216°C). The measured superheat limits of the hydrocarbon/water emulsions, Tin, are tabu- Most of the emulsion drops vaporized be- lated in Table I. These measurements are fore reaching the test section. For example, accurate to within _+2°C, although there n-dodecane/water emulsions vaporized in a range of approximately 200-253°C and benzene/water emulsions nucleated in a TABLE I range of approximately 140-201°C. The Comparison of Normal Boiling Temperatures of major cause of this scatter is thought to be Various Pure Liquids with the Superheat Limits of the presence of microscopic air bubbles in- Their Respective Water Emulsions troduced during emulsion preparation. The Emulsion Tb T,~ Tc~ Tcb vaporization of these drops occurs in the manner described by Sideman and Taitel n-Decane/water 174 226 228 229 (17). Because of their relatively slow rate of n-Dodecane/water 216 253 250 251 n-Tetradecane/water 252 253 259 260 vaporization at the lowest temperatures in n-Hexadecane/water 287 263 267 267 the column, drops containing visible air bub- Benzene/water 80 201 211 212 bles could easily be eliminated from con- Freon E-9/water 290 228 c 231 232 sideration. Drops containing microscopic aF=l. air bubbles, however, can reach relatively b F from Eq. [44]. high levels in the column before disappear- c Jarvis et al. (10).

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 449 seems to be the chance for a systematic general for fuel/water emulsions. The effec- error in the benzene/water system. tive surface tension, o-, used in calculating Calibration runs with pure alkanes evi- W* in such cases is the surface tension denced much less experimental scatter (i.e., of the fuel, ~rb. This constitutes a great nearly all the drops vaporized at one location simplification in the theoretical treat- in the test section). These experiments yielded ment of these systems. superheat values identical to those in the The effect of surfactant on the vapor pres- literature to within _+ I°C (7). Pure benzene ex- sure and surface tension of the hydrocarbon hibited greater experimental scatter than the phase is believed to be small. This hypothe- alkanes. However, treating this data as with sis was tested by determining the limit of the emulsions (i.e., focusing on those drops superheat of drops of benzene and decane that vaporize at the highest location in the which contained surfactant. In both cases test section) yielded a superheat limit the superheat temperatures were the same identical to the accepted value (7). as measured in the absence of surfactant. The possibility exists that neglect of the sur- 5. DISCUSSION factant is no longer valid for the lower vapor pressure alkanes studied, but the superheat Calculation of the superheat tempera- limits of these liquids were too high to be ture of a binary emulsion from the formula- measured in our apparatus. tions presented in Sect. 3 requires the fol- Because the calculations are so sensitive lowing: (a) location of the nucleus relative to surface tension, we list below the ref- to the liquid-liquid interface, (b) surface erences from which we took surface tension tension and vapor pressure of both liquids data: (a) n-decane, Eberhart et al. (6); (b) as a function of temperature, and (c) deter- n-hexadecane and n-tetradecane, Dickinson mination of what nucleation rate is com- (20); (c) water, Volyak (21); and (d) benzene mensurate with the experiment. and n-dodecane, Jasper (22). The data of The nucleus position can be predicted by Jasper (22) were extrapolated to zero sur- the criteria given in Sect. 2. To make this face tension at the critical point. prediction, the interfacial tension between Equilibrium vapor pressure data as a the two liquids as well as their individual function of temperature were correlated by surface tensions must be known. Jennings curve fitting the data of Gallant (23) and (18) has measured the interracial tensions Weist et al. (24). The values ofpa* and Pb* of benzene/water and n-decane/water sys- used in Eq. [18] were obtained from the tems to 176°C and Aveyard and Haydon (19) equilibrium vapor pressures, Pae* and Pbe*, have reported dodecane/water, tetradecane/ by the relation (7) water, and hexadecane/water interfacial tensions up to 37.5°C. The effect of sur- (P~* - P') = ~i(P~e* - P'), [56] factant, however, is to lower this interfacial where 6~ is the Poynting correction factor. tension. We therefore tested the experi- We assume droplet explosions are due to mental systems to see whether the fuel a single nucleation event inside the drop so (hydrocarbon + surfactant) spread on that a nucleation rate corresponding to the water. In all cases the hydrocarbon surfac- total water/fuel surface area in a drop is tant mixture (liquid b) spread on water appropriate. Following Eberhart et al. (6), (liquid a) at 25°C indicating that o-a > o-b the critical nucleation rate can be approxi- +crao. Assuming that spreading also occurs mated as at high temperature, a spherical nucleus forms within the hydrocarbon as shown in Fig. 2C. This behavior should be quite J= dt '

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 450 AVEDISIAN AND ANDRES where J is based on the total water/fuel calculation. The various superheat tempera- interfacial area inside a drop. dT/dt (°C/sec) tures were so close, however, that only is the droplet heating rate. N/No is the frac- one temperature is listed. tion of drops which survive to a tempera- We consider the agreement between the ture T, and L reflects the temperature highest measured temperatures and the cal- dependence of the nucleation rate. From culated temperatures quite good. Moreover, Eq. [43] it can be shown that J increases by the predicted temperatures using the simpfi- a factor of about 20 per degree so that L ~ 3. fled equilibrium model with F = 1 and the re- Generally the mean nucleation temperature fined steady-state model are essentially is defined so that N/No = ½ (6). For an the same. This is due to the predominant average emulsion drop diameter of 1.5 mm, dependence of the nucleation rate on the ex- the rate of rise in the column was measured ponential term, exp[-W*/kT], which is the to be about 8 cm/sec in a temperature gra- same in both cases. dient of l°C/cm. Therefore, dT/dt = 8°C/ The differences between the various sec. Assuming 15% by volume of water dis- models for 4) and for F presented in Sect. 3 persed as individual droplets of 0.07 mm are illustrated in Table II. The temperature diameter, the total water/fuel interfacial at which the calculations were made is indi- area,A, per emulsion drop is 0.2 cm 2. There- cated in the first column. Because of its fore from Eq. [57], J/A ~- 102 nuclei/cm2 importance in differentiating between the sec. Andres (15) formulation and the Reiss- In order to estimate N, we adopt a pic- Hirschfelder (13, 14) formulation, the ratio ture of the interface as comprising two ko/ka is also given. The differences between molecular layers of thickness equal to the these models is small unless ko/ka is very diameters of a water and fuel molecule, re- different from unity. The predictions of the spectively. In this approximation two models are indistinguishable for the emulsions we studied with the single excep- N tion of benzene/water. The predictions of

- hoCta + '~0d0, [58] A Eq. [52] are also very close to those of the Reiss-Hirschfelder model (Eq. [50]). There where da and d0 are effective diameters is not much added computational effort in- (cm) of water and fuel molecules respec- volved in using the Reiss-Hirschfelder tively, and ha and h0 are the number den- value for ~b for binary systems. The gen- sities (molecules/cm3) of water and fuel. eralization of Eq. [52] to multicomponent We can now estimate emulsion superheat systems, however, is much simpler as it in- temperatures using Eq. [43]. The results are volves determination of the direction of a compared with our measured values in ray from the point (0,0,0... 0) to the point Table I. In Table I we also compare the (nl*,n2* .... , no*) while the Reiss-Hirsch- measurement of Jarvis et al. (10) for a water felder approach involves evaluation of the drop rising in a column filled with Freon E-9 eigenvalues of a c x c dimension Jacobian with our theoretical predictions. All tem- matrix. peratures are to the nearest degree. To is Finally, it is apparent that the value of F the normal boiling point of the fuel and Tm obtained by assumingy0* = 0 is too low for is the measured superheat limit of the emul- the emulsions studied. sion. Tc a is the predicted temperature using Table II shows that differences in F and F = 1 in Eq. [43] and Tc° is the temperature hence the predicted nucleation rate at a calculated using the more refined steady- given temperature can be significant. The state value for F (Eq. [44]). Several approxi- effect of such differences on the critical mations were used to estimate 4) in this last superheat limit is nevertheless small. This

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 451

TABLE II Comparison of Theoretical Estimates of 4' and F

Emulsion T (°C) /~0/ko cb~ F ~ (ob F ° q~¢ F c F a n-Decane/water 229.4 16.7 10.6 0.0168 10.6 0.0168 7.4 0.0157 0.00147 n-Dodecane/water 250.7 5.0 4.5 0.0290 4.5 0.0290 3.0 0.0282 0.00447 n-Tetradecane/water 260.1 2.3 2.4 0.0372 2.4 0.0372 1.6 0.0367 0.00814 n-Hexadecane/water 267.4 1.0 1.2 0.0426 1.2 0.0426 0.8 0.0423 0.0137 Benzene-water 212.3 132.0 71.7 0.0208 41.9 0.00924 40.5 0.00868 0.000196 Freon E-9/water 231.5 0.2 0.4 0.0405 0.4 0.0405 0.3 0.0404 0.0223

Andres model, 4, which maximizes F. 4) from Eq. [501. 4) from Eq. [521. a F from Eq. [55]. same phenomenon should carry over in the vaporization. Solid particles or surfaces can case of bubble nucleation in multicompo- also promote nucleation. An example of this nent miscible mixtures. It is no wonder that latter phenomenon is the observed disrup- single component homogeneous nucleation tive burning of dodecane/water emulsion formulations have been used with some drops suspended from a quartz fiber (1). success in estimating superheat tempera- The predictions of homogeneous nuclea- tures in systems with two active species tion theory and our measurements indicate (25). The critical modification required is to that a free burning dodecane/water drop use the proper effective surface tension and will not undergo a microexplosion at at- to sum the partial pressures of the vapor mospheric pressure. species when evaluating W*/kT. It is also important to note that the pres- A simple working formula to predict the ence of a superheated water phase within a superheat limits of fuel/water emulsions in fuel drop does not mean that the drop will flames can be based on the energy required vaporize disruptively. The driving force for to form a critical size nucleus at the fuel/ a microexplosion is present but there water interface (Fig. 2C). A best fit of our is a nucleation barrier that must be over- experimental measurements yields come before disruptive vaporization occurs. Equation [59] can be used to predict the W* 16¢ro-b3(To) maximum nucleation temperatures of fuel/ = 66, kT~ 3kT~(pa*(T~) + pb*(T~) _p,)2 water emulsions if the surface tension and [59] vapor pressure of the fuel phase are known. Systems where Tc/Tb < 1 will burn dis- with p' = 1 atm. ruptively even in the absence of air bubbles The ability of this expression to correlate or solid particles. our measurements is shown in Fig. 4. Any systems falling within the region TJTb < 1 CONCLUSIONS in Fig. 4 should burn disruptively at atmospheric pressure. Maximum superheat temperatures of It is important to remember that there emulsion drops measured using the column may be many drops that vaporize disrup- heating method can be predicted by means tively at lower temperatures than To. The of homogeneous nucleation theory. The presence of microscopic air bubbles in hydrocarbon/water emulsions studied here emulsified fuel drops can cause premature correspond to a very low interfacial tension

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978 452 AVEDISIAN AND ANDRES

1.4

1,3 W~(Tc) =66 KTc

1.2

Te (°K/°K Tb o IJ /

o n- DECANE ,~. n - DODECANE e n- TETRADECANE I.C ~ n- HEXADECANE • BENZENE [] FREON Eg(lo)

0.9 //~ D J I I I i 0"-(3.8AV 0.9 1.0 I.I 1,2 1.3 1.4 Tm/Tb (°K/eK) FIG. 4. Comparison for some fuel/water emulsions of the highest measured superheat temperatures, Tin, with temperatures calculated by using W*/kTc = 66. and to a spherical nucleus forming at the Maximum superheat temperatures of the hydrocarbon/water interface but surrounded emulsions are well correlated by W*/kTc by the hydrocarbon phase. = 66. Temperatures calculated from this There is little difference between super- correlation are the highest one should heat limits calculated using a simple equi- expect an emulsified fuel droplet to reach at librium rate model for nucleation and atmospheric pressure. calculated using a steady-state rate formulation. The differences in the absolute nuclea- ACKNOWLEDGMENTS tion rate calculated from these two models We are sincerely grateful to I. Glassman for intro- can, however, be significant. ducing us to the interesting behavior of emulsified Based on the superheat limits measured, fuels and for his support and advice given through- only the hexadecane/water emulsions out the course of the work. Conversations with F. L. Dryer, C. K. Law, D. W. Naeglie, and R. J. Santoro should burn disruptively via homogeneous have also been helpful. Financial support for R. P. nucleation at 1 atm. The n-decane, n-do- Andres was provided under NSF Grant GK-4160. decane, n-tetradecane, and benzene emul- The experimental efforts and the work of C. T. sions all have nucleation temperatures Avedisian, an NSF trainee, was supported by NSF/ higher than the boiling points of the respec- RANN Grant AER 76-08210 (I. Glassman/F. L. Dryer, Principal Investigators), which is monitored by the tive fuel. There were differences in the ex- Division of Conservation Research and Technology, plosive character of these emulsions. They ERDA. All support is gratefully acknowledged. ranged from an inaudible "puff" in the alkane/water emulsions to a sharp report in REFERENCES the benzene/water system. These differ- 1. Dryer, F. L., Report No. 1224, Department of ences are indicative of the relative rates of Aerospace and Mechanical Sciences, Princeton the transport processes involved in these University, April 1975; Dryer, F. L., Rambach, microexplosions. G. D., and Glassman, I., Report No. 1271,

Journal of Colloid and Interface Science, VoL 64, No. 3, May 1978 BUBBLE NUCLEATION IN EMULSIONS 453

Dept. of Aero. and Mech. Sci., Princeton Univ., 13. Reiss, H., J. Chem. Phys. 18, 840 (1950). Apr. 1976. 14. Hirschfelder, J. O., J. Chem. Phys. 61, 2690 (1974). 2. Ivanov, V. M., and Nefedov, P. I., NASA Tech- 15. Andres, R. P., J. Chem. Phys., submitted for publi- nical Translation NASA TT F-258, Jan. 1965. cation (1978). 3. Moore, G. R., AIChE J. 5, 458 (1959); Ph.D. 16. Cohen, E. R., J. Statistical Phys. 2, 147 (1970). thesis, University of Wisconsin (1956); (Publica- tion 17,330, Univ. Microfilms, Ann Arbor, 17. Sideman, S., and Taitel, Y., Int. J. Heat Mass Mich.). Transfer 7, 1273 (1964). 4. Blander, M., Hengstenberg, D., and Katz, J. L., 18. Jennings, H. Y., J. Colloid Interface Sci. 24, J. Phys. Chem. 75, 3613 (1971). 323 (1967). 5. Kagan, Y., Russ. J. Phys. Chem. 34, 42 (1960). 19. Aveyard, R., and Haydon, D.A., Trans. Faraday 6. Eberhart, J. G., Kremsner, W., and Blander, M., Soc. 61, 2255 (1965). J. Colloid Interface Sei. 50, 369 (1975). 20. Dickinson, E., J. Colloid Interface Sci. 53, 467 7. Blander, M., and Katz, J. L., AIChE J. 21, 833 (1975). ( 1975). 21. Volyak, L. D., Dokl. Akad. Nauk, SSSR 74, 307 8. Porteous, W., and Blander, M., AIChE J. 21, (1950). 560 (1975). 22. Jasper, J. J., J. Phys. Chem. Ref. Data 1, 841 9. Apfel, R. E., Harvard University Acoustics Re- search Laboratory, Technical Memorandum 62, (1972). Feb. 1970; J. Chem. Phys. 54, 62 (1971). 23. Gallant, R. W., "Physical Properties of Hydro- 10. Jarvis, T. J., Donohue, M. D., and Katz, J. L., carbons," Vol. 1. Gulf Publ., Houston, 1968. J. Colloid Interface Sci. 50, 359 (1975). 24. "Handbook of Chemistry and Physics" (R. C. 11. "The Collected Works of J. Willard Gibbs", Vol. Weist, S. M. Selby, and C. D. Hodgman, Eds.), 1, p. 260. Longmans, Green, New York, 1928. 46th ed., Chem. Rubber, Cleveland, 1965. 12. Frenkel, J., "Kinetic Theory of Liquids," p. 384. 25. Mori, Y., and Nagatani, T., Int. J. Heat Mass Oxford Univ. Press, London, 1946. Transfer 19, 1153 (1976).

Journal of Colloid and Interface Science, Vol. 64, No. 3, May 1978