Curvature Dependency of Surface Tension in Multicomponent Systems

Transcription

1 HERMODYNAMICS Curvature Dependeny of Surfae enson n Multomponent Systems Erk Santso Dept. of Chemal Engneerng, North Carolna State Unversty, Ralegh, NC Abbas Froozabad Reservor Engneerng Researh Insttute, Palo Alto, CA DOI /a Publshed onlne September 1, 2005 n Wley InterSene ( he effet of urvature on the surfae tenson of droplets and bubbles n both sngle and multomponent systems s modeled usng the bas equatons from lassal thermodynams. he three expressons used n our work are the Gbbs adsorpton equaton for multomponent systems, the relaton between the surfae tenson at the surfae of tenson and the dstane parameter, and the Maleod Sugden equaton for surfae tenson and ts extenson to multomponent systems. he Peng Robnson equaton of state s used to desrbe the bulk phases. We also assume that the surfae tenson expresson remans vald n terms of the propertes of the bulk phases for both flat and urved nterfaes. For a flat surfae we have developed a rgorous thermodynams approah for the alulaton of the olman dstane parameter. For urved surfaes, the results from our model reveal a derease n surfae tenson wth urvature n bubbles and a nonmonoton behavor n droplets for sngle-omponent systems. Our predtons are n good agreement wth the lterature results when the nterfae s desrbed usng the framework of the densty funtonal theory by three dfferent groups. For multomponent systems, the results show that the surfae tenson n a bubble, although monoton wth urvature, an nrease or derease n a large bubble dependng on the temperature and omposton of the mxture. In a droplet, the surfae tenson an have a nonmonoton behavor smlar to that of sngle-omponent systems Ameran Insttute of Chemal Engneers AIChE J, 52: , 2006 Keywords: urvature effet on multomponents, surfae tenson, multomponent mxtures Introduton Surfae tenson s a prme parameter n the bas formulaton of a large number of proesses nludng nuleaton and luster formaton. he work of formaton of a rtal-sze nuleus s proportonal to the surfae tenson to the power three. 1 he A. Froozabad s also afflated wth: Dept. of Chemal Engneerng, Yale Unversty, New Haven, C Correspondene onernng ths artle should be addressed to A. Froozabad at abbas.froozabad@yale.edu Ameran Insttute of Chemal Engneers solute lusterng n supersaturated solutons and onentraton gradents n a vertal olumn of a supersaturated soluton may also depend on the surfae tenson of nanopartles. 2 In addton to applatons n nuleaton and solute lusterng n supersaturated solutons, there s a wde nterest n the mportant role of surfae tenson n determnng the behavor of small droplets and bubbles nludng ol-reovery proesses. For some tme t has been reognzed that when a luster of a new phase s small (that s, has a hgh urvature), the surfae tenson s sze dependent (that s, urvature dependent). In an early paper, olman 3,4 derved the expresson for the AIChE Journal January 2006 Vol. 52, No

2 effet of droplet sze on surfae tenson n a sngle-omponent system. A key parameter n olman s work s the parameter, the dstane between the surfae of tenson and the equmoleular dvdng surfae. For a plane surfae of separaton, olman omputed for a varety of substanes, nludng water. 3 Hs results show that the dstane s postve and of the order of 1 to 3.5 Å (of the order of the ntermoleular dstanes n lquds) for pure substanes over the range of ondtons studed. Sne the early work of olman, many nvestgators have studed the urvature dependeny of surfae tenson and nuleaton theory. here s generally onsensus that the urvature dependeny of surfae tenson may ndeed result n sgnfant varaton of the work of formaton of the rtal nuleus 5 and the nuleaton rate. 6,7 However, there s wdespread onfuson and ontroversy n the lterature on the effet of urvature on the surfae tenson of a bubble and a droplet n sngle-omponent systems. Defay and Prgogne 8 provde results for the effet of urvature on the surfae tenson of water at 18 C for both bubbles and droplets. her results show an nrease of the surfae tenson wth nreasng urvature for a bubble, whereas there s a derease of surfae tenson wth nreasng urvature for a droplet (see able 15.7 of Defay and Prgogne 8 ). On the other hand, Kashhev 1 presents results for water bubbles at 583 K and droplets at 293 K, both dereasng wth nreasng urvature (see Fgure 6.1 of Kashhev 1 ). Kashhev used a postve value of of 1 Å for water droplets and bubbles. Another example s the work of Hadagapou, 9 whh provdes results from densty funtonal theory showng an nrease of the surfae tenson for a droplet wth nreasng urvature, followed by a derease at very hgh urvatures. Guermeur et al. 10 also studed the effet of urvature on the surfae tenson for ntrogen bubbles and droplets. her results show that the surfae tenson n the bubble dereases wth nreasng urvature. It has a nonmonoton behavor n the droplet. On the other hand, as was stated earler, many authors predt that the surfae tenson n, say, a droplet, should derease rapdly wth the radus (see, for example, Lee et al., 11 and the referenes theren). In a dfferent approah, Shmelzer and Badakov 12 argue that the Gbbs method for determnng the referene states for the desrpton of bulk propertes of the rtal nuleus does not gve a orret desrpton of the bulk propertes of the rtal lusters at hgh supersaturaton. hey postulate that at a nonequlbrum state, the hemal potentals of the nterfae and the ambent phase (n ther termnology, ths s, the bulk phase other than the luster bulk phase) are the same. hey make other postulatons and obtan new expressons, nludng a new expresson for the pressure dfferene between the luster phase and the other bulk phase gven by p p 2/R ( ), where p s the pressure, s the densty, s the hemal potental, s the surfae tenson, and R s the radus of the rtal luster (or rtal nuleus); s the luster phase, and s the ambent phase. All pertan to a rtal luster. (hese are nomenlature from Shmelzer and Badakov 12 ; later, we wll use our own.) Obvously, the expresson for (p p ) from Shmelzer and Badakov 12 s dfferent from the Laplae equaton gven by p p (2/R). When the work from Shmelzer and Badakov 12 s used, the surfae tenson both for a bubble and for a droplet derease wth urvature and approah zero at the spnodal. For a droplet, frst there s a slght nrease (not noteable) followed by a derease wth urvature. he work of rtal luster also approahes zero at the spnodal. We wll return to ths pont later. Most of the work on urvature dependeny of the surfae tenson s lmted to sngle omponents. Shmelzer et al. 5 developed an empral relaton for n multomponent systems for a droplet. In a more reent work, Badakov et al. 13 use an approah based on the work of Shmelzer and Badakov 12 to study the effet of urvature on surfae tenson n mxtures. he results show that smlar to a sngle-omponent system, the surfae tenson vanshes at the spnodal for a two-omponent mxture. In ths work, we present results to show that there are dfferenes n surfae tenson urvature dependeny of pure omponents and multomponent systems. he purpose of our work s to derve the expressons for the urvature dependeny of the surfae tenson for bubbles and droplets n both sngle and multomponent systems. he dervatons are based on the general expressons from lassal thermodynams usng the work of Gbbs. We use an equaton of state to desrbe the bulk-gas and lqud-phase propertes. herefore, there s no need to make assumptons regardng ompressblty and ompostonal effets. In the followng, we frst derve bas thermodynam relatons for the urvature dependeny of the surfae tenson usng a new smple approah. hen we apply the derved expressons oupled wth a surfae tenson model to obtan the bas expressons for bubbles and droplets n sngle-omponent systems. Next, the bas expressons n multomponent systems, also for both bubbles and droplets, are obtaned. We also provde numeral examples. At the end we make several onludng remarks. hermodynam Relatons for the Curvature Dependeny of Surfae enson he two fundamental expressons that form the bass of ths work an be dretly obtaned from the bas postulates of thermodynams and have been derved by Ono and Kondo. 14 he frst expresson s the Gbbs adsorpton equaton that, for a multomponent system wth a spheral nterfae, an be wrtten as 14 d s d d (1) ada 1 In ths expresson, s the surfae tenson; s s the entropy per unt area of the nterfae (the supersrpt denotes a surfae property); and are, respetvely, the number of moles per unt area and hemal potental of omponent n the nterfae; s the number of omponents; a s the radus of the nterfae; and [/a] represents the hange n the surfae tenson when the mathematal dvdng surfae between the two phases s dsplaed. he partular dvdng surfae for whh [/a] 0 s alled the surfae of tenson. hroughout ths work, all propertes referred to the surfae of tenson are dentfed wth the subsrpt s. he seond key equaton mentoned above relates the surfae tenson to the radus of the nterfae and an be wrtten as January 2006 Vol. 52, No. 1 AIChE Journal

3 ln s ln a s 2 1 /a s 1/3/a s a s 2 1 2/a s 1 /a s 1/3/a s 2 Note that n the above equaton, the dervatve s taken along a path where the temperature and omposton of the ontnuous bulk phase (that s, mole fraton of the omponents x ) are held onstant (we refer to the small bulk phase as the luster phase n the work). he parameter s the dstane between two dvdng surfaes: one s the surfae of tenson and the other s the dvdng surfae, defned by (2) ṽ 0 (3) 1 where ṽ s the partal molar volume of omponent n the ontnuous bulk phase (denoted by the supersrpt ). Parameters and propertes referred to the dvdng surfae defned by Eq. 3 wll be dentfed by the subsrpt v. he parameter s gven by a v a s (4) Note that the dstane parameter defned above s not related to geometry and that the two rad are onstant for a gven bubble or a droplet. One of the most hallengng tasks n the past has been the estmaton of. Some authors have made alulatons to show that for a flat nterfae s postve, some others suggest that t s negatve, and there s a thrd group who estmate to be zero for a flat nterfae. he mplaton for beng zero for a flat nterfae s that there s no adsorpton at the nterfae. he parameter for a flat nterfae may also have dfferent sgns n a bubble and n a droplet (see Bartell 15 and referenes theren). Our goal n ths work s to fnd a sutable approah n predtng ths parameter. o aheve ths goal, we need to fnd a learer relaton between and other propertes of the system. he total volume of the two-phase system based on the dvdng surfae defned by Eq. 3 an be expressed as V V v V v N,v 1 ṽ 1 N,v ṽ where N s the number of moles of omponent. We have ntrodued the supersrpt to dentfy the luster phase. Usng the mass balane of omponent to elmnate N,v from Eq. 5 we obtan V 1 (5) N ṽ N,v ṽ ṽ (6) 1 In Eq. 6, we have used Eq. 3 to elmnate the amounts of adsorpton at the nterfae. he total volume of the two-phase system based on the surfae of tenson leads to V 1 N ṽ N,s 1 ṽ ṽ A s 1,s ṽ In Eq. 7, A s s the surfae area of the nterfae. Equatng the rght-hand sdes of Eqs. 6 and 7 and smplfyng, we obtan V v N,v ṽ V s 1 1 N,s ṽ A s 1,s ṽ By ntrodung the molar onentratons (that s, N,v V N,s /V s ) and rearrangng we obtan (7) (8) / V v V s A s 1,s ṽ 1 1 (9) ṽ he equaton above provdes a relatonshp between the szes of the luster alulated for the two dvdng surfaes. It also reveals that when there s adsorpton the dstane parameter may not be zero for a flat nterfae. One an wrte Eq. 9 n terms of the rad as a v a s 2/3a s 3 1,s ṽ 1 1 By ombnng Eqs. 4 and 10 we obtan 1 3 a s a s 1 1 1,s ṽ ṽ ṽ 1/3 (10) 1/3 1 (11) whh provdes. o study the urvature dependeny of the surfae tenson, one may rearrange Eq. 11 to a s a s ,s ṽ ṽ (12) Equaton 12 relates to the propertes of the two bulk phases and the nterfae for a multomponent system. For a sngleomponent system, Eq. 12 smplfes to a s a s 1 s (13) where s the molar densty (n ths work we use the notaton for the molar densty of omponent n a mxture and as the molar densty of a sngle-omponent flud). olman 4 derved Eq. 13 usng an ntegral approah; we use the relatonshps from lassal thermodynams n our dervaton. Note that one may not use Eq. 13 to generalze t to multomponent mxtures wthout ertan assumptons. 5 By substtutng Eq. 12 nto Eq. 2 we fnd an equaton for the urvature dependeny of the surfae tenson that does not nlude : ln s ln a s ,s ṽ,s ṽ a s 1 1 ṽ (14) AIChE Journal January 2006 Vol. 52, No

4 Note that n the above equaton the problem has shfted from obtanng the parameter to obtanng the amounts of adsorpton of the omponents at the nterfae. o proeed further, we relate the surfae tenson to the propertes of both bulk phases to obtan the sum that nludes the amounts of adsorpton. he sum n Eq. 14 that ontans the amounts of adsorpton an be related to the surfae tenson by ombnng Gbbs adsorpton equaton and the hemal equlbrum ondtons. By wrtng Eq. 1 for the surfae of tenson and replang the surfae hemal potentals by the hemal potentals n the bulk ontnuous phase we obtan d s s s d s, d 1 (15) he dfferental of the hemal potentals of the ontnuous bulk phase an be expressed as 1 d s d ṽ dp 1 dx x P, x k, Substtutng Eq. 16 nto Eq. 15 leads to s P x,s ṽ 1 1,..., (16) (17) Equaton 17 s the sought relatonshp. Provdng a model for the surfae tenson as a funton of the propertes of the two bulk phases, Eqs. 14 and 18 an be ombned to evaluate the urvature dependeny of the surfae tenson. In the followng two setons we wll present the methodology frst for sngleomponent and then for multomponent systems. Sngle-Component Systems For a sngle-omponent system, Eq. 14 smplfes to ln s 2 s ln a s 2 s a s Also, Eq. 17 beomes (18) s s (19) P s o proeed further we desre an equaton for the equlbrum surfae tenson. For a sngle-omponent system, the wellknown Maleod Sugden equaton for the surfae tenson of a flat nterfae an be used: 1/4 L V (20) In Eq. 20, s the surfae tenson of the flat nterfae, s the parahor, and the supersrpts L and V denote lqud and vapor, respetvely. Equaton 20 provdes the surfae tenson of nonpolar fluds wth a remarkable auray over a wde range of temperature ondtons 19 and, although t was ntrodued emprally, 16,17 t was later derved theoretally. 18 In what follows we wll use two dfferent approahes: wth respet to Eq. 20, we assume that (1) the expresson s vald for large values of the nterfae radus and obtan the lmtng dependeny of for a flat nterfae (that s, ); then assume that s onstant to obtan the urvature dependeny of the surfae tenson; and (2) the expresson apples to a surfae wth a urved nterfae, whh mples that an hange wth urvature. he assumptons have to be examned for ther valdty. Let us use the terms model and model to refer to the two methods. It turns out that n ether approah t wll be neessary to obtan as a funton of urvature assumng that Eq. 20 s vald for a urved nterfae. For the model we wll take the lmt as the nterfae radus tends to nfnty. hus, we wll replae by s n Eq. 20. Let us frst onsder a bubble n a ontnuous bulk lqud phase and replae L and V n Eq. 20 by and, respetvely. he expresson for surfae tenson wll take the form s 1/4 (21) he droplet n a ontnuous bulk vapor phase wll be onsdered later. By substtutng Eq. 21 nto Eq. 19 we obtan s 4 s 3/4 P P (22) he seond dervatve n the square brakets n Eq. 22 s gven by C P (23) where C s the sothermal ompressblty of phase, whh an be readly obtaned from an equaton of state. Next we fnd an expresson for the frst dervatve n Eq. 22. From the han rule we wrte P P P P C P P (24) he dervatves n Eq. 24 are taken along an equlbrum path. herefore, we an wrte P P P P P (25) Replang the dervatves of the hemal potentals wth respet to the pressure of ther respetve phases by the nverse of the molar denstes, we obtan 314 January 2006 Vol. 52, No. 1 AIChE Journal

5 ln s 8 ln a s 3/4 s 2 C 2 C 8 3/4 s 2 C 2 C a s (29) Equaton 29 an be ntegrated numerally to ompute the urvature dependeny of the surfae tenson. In Fgure 1a we show the surfae tenson s as a funton of the radus of the surfae of tenson for a bubble of n-pentane at K usng the two models. he Peng Robnson equaton of state 20 and the standard fourth-order Runge Kutta method were used for alulatons n Fgure 1 and smlar alulatons to be presented later. he parahor of n-pentane was taken as 230. From ths graph we note that: (1) the surfae tenson s onstant down to a s 100 nm, and (2) the model predts a sharper varaton of the surfae tenson wth urvature for a s 10 nm; the dfferene between the two models nreases as the radus dereases. Fgure 2a shows the varaton of wth urvature as predted by the model. In ths fgure, t s also lear that there s almost no dfferene between the two models for szes down to a s 10 nm, but as the radus dereases further, grows sharply, whh aounts for the steeper varaton of the surfae tenson predted by ths model at larger urvatures. hs hange n an be explaned by the fat that the pressure, and thus the denstes, starts to hange muh faster at these large urvatures. Granasy 21 provdes results smlar to Fgure Fgure 1. Curvature dependeny of the surfae tenson n n-pentane at K from the and models. (a) Bubble, (b) droplet. P P (26) By ombnng Eqs. 22, 23, 24, and 26 we get s 4 s 3/4 2 C 2 C (27) Equatons 21 and 27, together wth an equaton of state, allow us to alulate the amount of adsorpton at the surfae of tenson. For the model, we wll use Eq. 27 to obtan the lmtng value of as the radus of the nterfae goes to nfnty. By substtutng Eq. 27 nto Eq. 13 and takng the lmt as a s 3 we obtan 3/4 4 2 C 2 C s (28) Usng as the onstant value of we numerally ntegrate Eq. 2 and alulate the varaton of s wth urvature. For the model, we wll substtute Eq. 27 nto Eq. 18 to get Fgure 2. Curvature dependeny of the parameter n n-pentane at K from the model. (a) Bubble, (b) droplet. AIChE Journal January 2006 Vol. 52, No

6 2a based on the semempral van der Waals/Cahn Hllard theory. Note that vs. the radus s postve n Fgure 2a. For a droplet n a ontnuous vapor phase, t s neessary to exhange the supersrpts and n Eq. 21. Repeatng the proedure we used for a bubble we obtan, for the lmtng value of for a flat nterfae, 3/4 4 2 C 2 C (30) Note that the above expresson for n a droplet s dfferent from the expresson derved n Bartell. 15 here s no need to assume that densty or volume an be expanded lnearly wth pressure. here s also no need for other assumptons exept a need for a general expresson for. Aordng to Eq. 30, even when C 0, may not be zero. Substtutng from Eq. 30 nto Eq. 2 and ntegratng, we obtan the varaton of wth urvature for the model. For the onstant model, we obtan ln s 8 ln a s 3/4 s 2 C 2 C 8 3/4 s 2 C 2 C a s (31) Fgure 1b shows the varaton of the surfae tenson wth the radus of the surfae of tenson usng the two models for a droplet of n-pentane at K. For a droplet, the dfferene between the model and the model s more pronouned: for the model, the surfae tenson nreases monotonally wth dereasng radus whereas the model predts that the surfae tenson wll go through a maxmum at an approxmate radus of 1.1 nm and then dereases rapdly wth dereasng radus. Hadagapou 9 and Guermeur et al. 10 predted a smlar nonmonoton behavor. Fgure 2b shows the varaton of wth urvature as predted by the model, where the varaton of wth urvature s smlar to that of the bubble, beng almost onstant at small urvatures and then growng fast as the radus dereases. However, s negatve for the droplet and hanges sgn and auses the surfae tenson to go through a maxmum. A negatve value for n a droplet has been presented n Bartell 15 and Granasy 21 usng dfferent approahes. In Bartell, 15 s approxmated by C /3, based on a number of assumptons. Based on another set of assumptons Laaksonen and MGraw 6 obtaned C. Granasy 21 dsusses the dfferene between the densty funton (DF) approah n predtng a negatve and the moleular dynams (MD) smulatons, whh predt a poston for a droplet. Aordng to the defnton of dstane parameter n Eq. 4, there s only onssteny when from bubbles and droplets have the same magntude but dfferent sgn. Both mply that the equmoleular dvdng surfae s loser to the bulk lqud phase than the surfae of tenson. In other words, the hange n sgn s explaned by the fat that n the bubble the dstanes are measured from the gas sde and n a droplet they are measured from the lqud sde. However, for a multomponent system, ths s not the ase as expeted (as we wll show later). he results shown above for the urvature dependeny of surfae tenson n bubbles and droplets have smlar trends to those of Hadagapou 9 and Guermeur et al. 10 Fgure 3a shows Fgure 3. Curvature dependeny of the surfae tenson n ntrogen at 77.3 K from the model and from Guermeur et al. 10 (a) Bubble, (b) droplet. a omparson between the results of Guermeur et al. for bubbles of ntrogen at 77.3 K and the results for the same system obtaned wth our model (ntrogen parahor of 52 was used n our alulatons based on surfae tenson data for ntrogen). Fgure 3b shows the same omparson for a droplet of ntrogen at the same temperature. In both fgures we observe that the trends predted by our model and the work of Guermeur et al. are the same, although the two models are very dfferent. Guermeur et al. desrbe the nhomogeneous flud n the nterfae usng a stress tensor n the framework of gradent theory, and predt a sharper varaton of the surfae tenson wth urvature for the bubbles and a smaller varaton for the droplets. Although they also obtan a maxmum n the surfae tenson plot for the droplet, the maxmum s at a smaller urvature and the orrespondng surfae tenson s also smaller. he results for our model (not shown) are farther from ther results n both ases. o ompare wth Hadagapou s results, we omputed the surfae tenson as a funton of urvature for a droplet of argon at a redued temperature of 0.8 (argon parahor of 52 was used n our alulatons). Argon was hosen for omparson beause t s a substane well suted for Hadagapou s treatment of thermodynam propertes. he results, n terms of the redued varables defned n hs work, are shown n Fgure 4. In ths ase, our model predts a smaller varaton of the surfae tenson wth urvature than Hadagapou s model, although our model predts a smaller droplet radus for the maxmum 316 January 2006 Vol. 52, No. 1 AIChE Journal

7 Multomponent Systems For multomponent systems we follow an approah smlar to that of sngle-omponent systems. Wenaugh and Katz have extended Eq. 20 to multomponent systems 23,24 : 1/4 L V (32) 1 Fgure 4. Curvature dependeny of the surfae tenson n Argon at a redued temperature of 0.8 from the model and from Hadagapou. 9 he new varables appearng n the dmensonless groups plotted are d (moleular dameter) and k (Boltzmann s onstant). surfae tenson. In vew of the smplty of our model, t s nterestng to note that ts predtons are lose to those of Hadagapou, whh s based on densty funtonal theory. Before proeedng to multomponent systems, we must pont out a defeny wth respet to the work of rtal luster formaton at the spnodal (that s, at the lmt of stablty). he work of rtal luster formaton s expeted to vansh as we approah the spnodal. In the lassal theory of nuleaton, the barrer heght approahes a fnte value at the spnodal, whh s not orret. 22 When one aounts for the effet of urvature on the surfae tenson, the work of rtal luster formaton at the spnodal redues, but may not vansh. hs s to be expeted beause of the extremely small sze of the lusters, onsttutng say some 40 atoms, at the spnodal. Instead of usng lassal thermodynams, one may use statstal or moleular thermodynams for very small lusters (that s, at very hgh supersaturaton) to aount for nonlassal effets. 22 In our approah, smlar to that of Guermeur et al., 10 the surfae tenson for a droplet s greater than the flat nterfae for large droplets but beomes smaller for smaller droplets. For bubbles, the surfae tenson remans less than. hese results are onsstent wth the work of Oxtoby and Evans, 22 that lassal and nonlassal barrer heghts ross from droplet formaton but do not for bubble formaton. Shmelzer and oworkers, n a dfferent approah, as we stated earler, use ertan postulatons that lead to the vanshng of the work of rtal luster formaton as well as the vanshng of the surfae tenson at the spnodal (Shmelzer and Badakov 12 and referenes wthn). Fgure 5 shows the plot for the work of rtal luster formaton (n dmensonless unts) vs. a s n a bubble of n-pentane at K. Note that as the spnodal s approahed, W/k dereases. Despte a small W of about 1.5k t does not vansh at the spnodal. he droplet also has a smlar behavor. he work of rtal luster formaton for the Argone bubble at the spnodal (for r 0.8) s about 2.5k whh s not far from k. he above equaton provdes good predtons n nonpolar multomponent hydroarbon mxtures. 23,24 o apply Eq. 32 to obtan the urvature dependeny of the surfae tenson, we wll follow an approah smlar to the one used n the prevous seton. Agan, we wll explore two optons: (1) assume that Eq. 32 s vald for urved nterfaes wth a very large radus, use t to obtan the lmtng value for the flat nterfae, and then use to ntegrate Eq. 2; and (2) assume that Eq. 32 s vald for a urved nterfae, thus allowng for to vary wth urvature. o keep the termnology used n prevous seton, we wll all the frst approah the model and the seond the model. Smlar to a sngle-omponent system, we need to fnd an expresson for as a funton of urvature assumng that Eq. 32 s vald for a urved nterfae. For the model we wll take the flat nterfae lmt n the resultng expresson. hus, we wll replae by s n Eq. 32. Frst we wll onsder a bubble n a ontnuous bulk lqud phase and replae L and V n Eq. 20 by and, respetvely. Wth the altered notaton, Eq. 32 takes the form 1/4 s (33) 1 Substtutng Eq. 33 nto Eq. 17 leads to 1,s ṽ 3/4 4 s 1 P P (34) Fgure 5. Work of luster formaton (rtal) vs. radus of the surfae of tenson n the n-pentane bubble at K. AIChE Journal January 2006 Vol. 52, No

8 As n Eq. 22, the seond dervatve nsde the square brakets n Eq. 34 an be readly obtaned from an equaton of state: C P 1,..., (35) he frst dervatve, however, s more omplated to evaluate. Let us wrte the dfferental of the varable n two dfferent ways: frst as a funton of the varables of phase and then as a funton of the varables of phase d d d P, x 1 1 P dp x x dx P, x k, d P, x 1 1 P dp x x dx P, x k, 1,..., (36) 1,..., (37) Consderng a proess at onstant temperature and omposton of the phase, whh are the ondtons for whh Eq. 2 holds, and equatng Eqs. 36 and 37 and wrtng the result n terms of the dervatves wth respet to P P P 1 1 P P x P x x P x, x k, 1,..., (38) Half of the dervatves n the rght-hand sde of Eq. 38 an be dretly obtaned from an equaton of state C P x ṽ ṽ P, x k, 1,..., (39) 1,..., 1,..., 1 (40) Equaton 40 s obtaned from the defnton of x and the Gbbs Duhem equaton n terms of partal molar volumes (see problem 1.5 of Chapter I n Froozabad 25 ). he symbol n Eq. 40 s the Kroneker delta. By substtutng Eqs. 39 and 40 nto Eq. 38 we obtan C P P 1 ṽ ṽ 1 And, for P x P P P P x P x x 1 1 1,..., 1 (41) x P 1 ṽ ṽ 1 However, from the sum ondton of phase 1 1 x P x P x x P x (42) (43) By substtutng Eq. 43 nto Eq. 42 we obtan Eq. 41 wth. hus, we an regard Eq. 41 as vald for all 1,...,. o use Eq. 41 we need to fnd the dervatves of the pressure and omposton of phase wth respet to the pressure of phase. We use an approah smlar to the one used for the sngleomponent system. Beause the dervatves appearng n Eq. 41 are taken n an equlbrum path, we use 1,..., (44) Dfferentatng Eq. 44 wth respet to the pressure of phase at onstant temperature and omposton of phase we obtan P P 1,..., (45) he dervatve on the left-hand sde of Eq. 45 s the partal molar volume of omponent n phase. For the dervatve n the rght hand-sde, one an wrte P 1 1 P x P x P P x x P, x k, 1,..., (46) he frst dervatve on the rght-hand sde s the partal molar volume of omponent n phase. By substtutng Eq.46 nto Eq. 45 we obtan 318 January 2006 Vol. 52, No. 1 AIChE Journal

9 ṽ ṽ P P 1 1 x P P x, x k, 1,..., (47) Upon multplyng Eq. 47 by x and takng the sum we fnd x ṽ v P P 1 x l 1 k1 x k P x l x 1 x k P, x lk, (48) where v s the molar volume. From the Gbbs Duhem equaton for phase x 1 x k P Combnng Eqs. 48 and 49 provdes 0 (49), x lk, P P 1 x ṽ ṽ 1 Substtuton of Eq. 50 nto Eq. 47 leads to 1 1 x P x x P, x k, ṽ ṽ ṽ 1 (50) 1,..., (51) Note that, although Eq. 51 s vald for 1,...,, only ( 1) of the equatons are ndependent beause we already used ther sum to obtan Eq. 50. he frst ( 1) equatons n Eq. 51 an be used to obtan the dervatves of the ompostons of phase wth respet to the pressure of phase, provded we use an equaton of state to obtan the dervatves of the hemal potentals wth respet to the ompostons n phase. Substtutng Eqs. 35, 41, and 50 nto Eq. 34 yelds Fgure 6. Curvature dependeny of the surfae tenson n a bnary mxture of propane and n-otane from the and models. 1 (a) For a bubble of equmolar lqud mxture at 300 K; (b) for a droplet of equmolar gaseous mxture at 400 K.,s ṽ 3/4 4 s C 1 C 1 ṽ 1 ṽ x P x x P (52) x where the dervatves of the omposton of phase are obtaned by solvng the system of Eq. 51. For the model we wll substtute Eq. 52 nto Eq. 12 and take the lmt as a s 3 to obtan b 4 3/4 1 C 1 ṽ x P 1 ṽ x 1 1 x P C ṽ (53) Note that the supersrpt b ndates the lmtng value for the bubble. Unlke n sngle-omponent systems, n mxtures the dvdng surfae defned by Eq. 3 s dfferent for dfferent hoes of the phase beause the two phases have dfferent partal molar volumes. hus, the lmtng value obtaned usng the equatons for a bubble may be dfferent from that AIChE Journal January 2006 Vol. 52, No

10 Fgure 7. Curvature dependeny of the parameter n a bnary mxture of propane and n-otane from the model. (a) For a bubble of equmolar lqud mxture at 300 K; (b) for a droplet of equmolar gaseous mxture at 400 K. obtaned wth the equatons for a droplet not only n sgn, but also n magntude. o avod ambguty, n the dsusson for multomponent systems we wll use the symbols b for the former and d for the latter. After substtutng Eq. 53 nto Eq. 2 and ntegratng we an alulate the urvature dependeny of the surfae tenson. For the model, Eq. 52 s substtuted nto Eq. 14 and the resultng equaton s ntegrated numerally. Fgure 6a shows the varaton of surfae tenson wth the radus of the bubble n an equmolar lqud mxture of propane and n-otane at 300 K, usng the and the models. he parahors n the surfae tenson model are: C 3, 150; nc 8, 400. he bnary nteraton parameters n the Peng Robnson equaton of state were taken as zero. he behavor shown s smlar to that obtaned for the sngle-omponent system. he varaton n surfae tenson s mportant only for bubble szes of 10 nm. he overall hange n surfae tenson n the two-omponent system onsdered s smaller than that n the sngleomponent system. he varaton of wth urvature for the model s shown n Fgure 7a, whh s onsstent wth the varaton of the surfae tenson. he equatons for a droplet n a ontnuous bulk vapor phase an be obtaned by exhangng the and supersrpts n Eq. 33 and repeatng the above proedure. In the followng equatons wll refer to the vapor phase and to the lqud phase 1,s ṽ 3/4 4 s 1 1 ṽ 1 ṽ x P x x P (54) x he dervatves of the omposton n Eq. 54 wll be found by solvng the system of Eq. 51. For the lmtng value of n a droplet d 4 3/4 1 C 1 ṽ x P 1 ṽ x 1 1 x P x C ṽ (55) A omparson of Eqs. 53 and 55 reveals the sgn dfferene as well as the phase dentty dfferenes. For the model, we wll substtute Eq. 55 nto Eq. 2 and ntegrate to alulate the urvature dependeny of the surfae tenson. For the model, Eq. 54 s substtuted nto Eq.14 and the result s ntegrated numerally. Fgure 6b shows the varaton of the surfae tenson wth the radus of the droplet n an equmolar gaseous mxture of propane and n-otane at 400 K. he same model as that of the bubble example was used. he behavor of the droplet n the multomponent system s smlar to that of the sngle-omponent droplet. Fgure 7b, showng the varaton of as a funton of urvature, ndates a stronger nrease of toward the hgh urvature regon. Fgure 8 shows the behavor of the parameter b (takng the lqud as the ontnuous bulk phase) as a funton of temperature for an equmolar lqud mxture of propane and n-otane. he behavor s somewhat dfferent from that observed n the sngleomponent system; b beomes negatve at temperatures loser to the rtal pont. hs s an mportant new result, showng that the surfae tenson for the bubble an ether nrease or derease wth urvature dependng on the temperature; ths may be the ase for a large bubble. In Fgure 9 we show the effet of omposton at dfferent temperatures on the parameter b. At low temperatures and pressures, where the system an be regarded as deal, b nreases monotonally wth omposton of propane, growng from the value for pure n-otane to the value for pure propane. At hgher temperatures, the urve frst dereases and then nreases wth omposton. As the temperature nreases toward the rtal pont, the urve beomes monotonally dereasng and b for hgher ompostons of propane beomes negatve. hs shows that the surfae tenson nrease or derease wth urvature de- 320 January 2006 Vol. 52, No. 1 AIChE Journal

11 pends not only on the temperature but also on the omposton of the mxture the nrease for a large bubble. As the system approahes the lmt of stablty (that s, the spnodal) the behavor hanges. We have arred out alulatons for ternary systems; the behavor found s smlar to that shown n Fgures 6 9. Beause the features are smlar to what we obtaned for the bnares, we have not nluded the results for the sake of brevty. Conludng Remarks We have derved the expressons for the olman dstane parameter of a flat nterfae n both pure and multomponent fluds. Based on physal nterpretaton, the dstane parameter of a flat nterfae n a sngle-omponent system for a bubble should be postve and that of a droplet should be negatve, both wth the same absolute value. he models that predt the same sgn are physally nonsstent. A thermodynam model s also presented for the effet of urvature on the surfae tenson n bubbles and droplets for both sngle and multomponent systems. he results for the nonpolar hydroarbon systems that we have studed reveal that: (1) here s generally a derease of surfae tenson n bubbles wth nreasng urvatures n sngle-omponent systems. he dstane parameter for these systems s postve. (2) For a droplet, f the dstane parameter s assumed onstant and equal to the value for the flat nterfae, there s an nrease n the surfae tenson wth nreasng urvature n sngle-omponent systems. he dstane parameter of the flat nterfae s generally negatve. Wth a varable dstane parameter model, the behavor of the surfae tenson s nonmonoton; the surfae tenson nreases wth urvature frst and then dereases. he predted results from our thermodynam model wth a varable dstane parameter are n qualtatve agreement wth the work of Hadagapou 9 and Guermeur et al. 10 based on densty funtonal theory. It s also n lne wth the work of Oxtoby and Evans. 22 (3) In multomponent systems the dstane parameter of the flat nterfae for a bubble may be ether postve or negatve dependng on the temperature and omposton. he surfae Fgure 8. Parameter b vs. temperature for a bubble n an equmolar mxture of propane and n-otane. Fgure 9. Parameter b vs. omposton for a bubble n a lqud mxture of propane and n-otane at dfferent temperatures. tenson may thus nrease or derease wth urvature at dfferent ondtons for a large bubble. (4) In both a bubble and n a droplet the surfae of tenson dereases for a small luster(of moleules) from the model presented n the work. hs s further evdene n support of the valdty of the results from our model. Aknowledgments he work was supported by DOE DE-FC26-00BC15306, and member ompanes of Reservor Engneerng Researh Insttute. Lterature Cted 1. Kashhev D. Nuleaton: Bas heory wth Applatons. Oxford, UK: Butterworth Henemann; Larson MA, Garsde J. Solute lusterng and nterfaal tenson. J Cryst Growth. 1986;76: olman RC. he superfal densty of matter at a lqud vapor boundary. J Chem Phys. 1949;17: olman RC. he effet of droplet sze on surfae tenson. J Chem Phys. 1949;17: Shmelzer JWP, Gutzow I, Shmelzer J. Curvature-dependent surfae tenson and nuleaton theory. J Collod Interfae S. 1996;178: Laaksonen A, MGraw R. hermodynams, gas lqud nuleaton, and sze-dependent surfae tenson. Europhys Lett. 1996;35: MGraw R, Laaksonen A. Interfaal urvature free energy, the Kelvn relaton, and vapor lqud nuleaton. J Chem Phys. 1997;106: Defay R, Prgogne J. Surfae enson and Adsorpton. London: Longman; Hadagapou I. Densty-funtonal theory for spheral drops. J Phys Condens Matter. 1994;6: Guermeur R, Bquard F, Jaoln C. Densty profles and surfaetenson of spheral nterfaes Numeral results for ntrogen drops and bubbles. J Chem Phys. 1985;82: Lee DJ, elo da Gama MM Gubbns KE. A mrosop theory for spheral nterfaes Lqud drops n the anonal ensemble. J Chem Phys. 1986;85: Shmelzer JWP, Badakov VG. Knets of ondensaton and bolng: Comparson of dfferent approahes. J Phys Chem B. 2001;105: Badakov VG, Boltashev GS, Shmelzer JWP. Comparson of dfferent approahes to the determnaton of the work of rtal luster formaton. J Collod Interfae S. 2000;231: Ono S, Kondo S. Enylopeda of Physs. Flügge S, ed. Vol. 10. Berln: Sprnger-Verlag; 1960: Bartell LS. olman s delta, surfae urvature, ompressblty effets, and the free energy of drops. J Phys Chem B. 2001;105: Maleod DB. Relaton between surfae tenson and densty. rans Faraday So. 1923;19: AIChE Journal January 2006 Vol. 52, No

12 17. Sugden S. he varaton of surfae tenson wth temperature and some related funtons. J Chem So. 1924;125: Fowler RH. A tentatve statstal theory of Maleod s equaton for surfae tenson and the parahors. R So Lond Ser A Phys S. 1937: Polng BE, Prausntz JM, O Connell JP. he Propertes of Gases and Lquds. 5th Edton. New York, NY: MGraw-Hll; Peng DY, Robnson DB. New two-onstant equaton of state. Ind Eng Chem Fundam. 1976;15: Granasy L. Semempral van der Waals/Cahn Hllard theory: Sze dependene of the olman length. J Chem Phys. 1998;109: Oxtoby DW, Evans R. Nonlassal nuleaton theory for the gas lqud transton. J Chem Phys. 1988;89: Wenaugh CF, Katz DL. Surfae tensons of methane propane mxtures. Ind Eng Chem. 1943;35: Froozabad A, Katz DL, Soroosh H, Saadan VA. Surfae tenson of reservor rude ol gas systems reognzng the asphalt n the heavy fraton. SPE Reservor Eng. 1988: Froozabad A. hermodynams of Hydroarbon Reservors. New York, NY: MGraw-Hll; Manusrpt reeved Aug. 16, 2004, and revson reeved Apr. 28, January 2006 Vol. 52, No. 1 AIChE Journal