Modeling Interfacial Tension in Liquid Liquid Systems Containing Electrolytes

Transcription

1 Artcle pubs.acs.org/iecr Modelng Interfacal Tenson n Lqud Lqud Systems Contanng Electrolytes Pemng Wang* and Andrzej Anderko OLI Systems Inc., 240 Cedar Knolls Road, Sute 3, Cedar Knolls, New Jersey 07927, Unted States ABSTRACT: A comprehensve model has been developed for calculatng the nterfacal tenson (σ) n lqud lqud systems wth or wthout electrolyte components. The model conssts of an equaton for computng the nterfacal tenson of two-lqudphase nonelectrolyte systems and an expresson for the effect of the electrolyte concentraton. The dependence of the nterfacal tenson on the electrolyte concentraton was derved by combnng the Gbbs equaton wth a modfed Langmur adsorpton sotherm that represents the nterfacal excess of the solute speces. The Langmur adsorpton formalsm was extended by ntroducng the effects of bnary nteractons between solute speces (ons or molecules) on the nterface. The equaton for the nterfacal tenson of nonelectrolyte lqud lqud systems was derved usng a general thermodynamc framework that was emprcally extended by ntroducng an effectve nterfacal area that s defned for each component and takes nto account the effects of other components at the nterface. The model was found to reproduce expermental data for a varety of lqud lqud systems. In partcular, the nterfacal tenson of ternary systems can be accurately predcted usng parameters determned from only bnary data. Furthermore, the nterfacal tenson model was coupled wth a prevously developed thermodynamc model to provde actvty coeffcents and equlbrum concentratons n coexstng lqud phases. Ths makes t possble to reproduce the effects of specaton and saltng out or saltng n. Because of the couplng of the thermodynamc model wth nterfacal tenson calculatons, the varaton of σ wth electrolyte concentraton can be reasonably predcted even wthout ntroducng electrolytespecfc parameters n the nterfacal tenson model. Thus, the model can be used to estmate the electrolyte effect on σ n the absence of expermental data. Wth regressed model parameters, the average devatons between the calculated results and expermental data were 0.50 mn m 1 for 30 bnary nonelectrolyte systems, 0.88 mn m 1 for 23 ternary nonelectrolyte systems, and 0.16 mn m 1 for 26 systems wth onc components. 1. INTRODUCTION Interfacal tenson s one of the most mportant propertes characterzng nhomogeneous lqud systems. Understandng the behavor of lqud lqud nterfaces s of paramount sgnfcance for the ratonal desgn of numerous processes such as those encountered n coatngs and adsorpton, ndustral separatons nvolvng extracton, enhanced ol recovery, and emulsons and suspensons n the food and pharmaceutcal ndustres. For example, nterfacal tenson s an mportant factor that nfluences the parttonng behavor of speces (molecules or ons) between two lqud phases. The degree of nterfacal adsorpton of speces s drectly assocated wth nterfacal tenson. Furthermore, the propertes of the lqud lqud nterface and, n partcular, the value of the nterfacal tenson provde nsght nto the mechansm of dstrbuton of speces n lqud lqud systems, whch s of nterest for the desgn of lqud lqud extracton n separaton processes. Because of ther practcal mportance, studes of nterfacal tenson and nterfacal chemstry have represented an area of actve research over the past century. Accurate determnaton of nterfacal tenson for varous lqud lqud systems s mportant for ganng nsght nto the nterfacal processes for the transfer of mass and energy across the phase boundary. Over the past several decades, a large amount of expermental nterfacal tenson data has been reported for varous lqud lqud systems, especally for mxtures of organc lquds and water. Interfacal tenson data have also been farly reported for systems contanng electrolytes. Thus, the expermental data that are avalable n the lterature provde a bass for a comprehensve analyss of the factors that nfluence the nterfacal tenson n two-lqud systems nvolvng nonelectrolyte (e.g., organc + water) and electrolyte components. Varous approaches have been reported n the lterature to model the lqud lqud nterfacal tenson. Most of the methods that are avalable for the predcton of nonelectrolyte lqud lqud nterfacal tenson are based on the mutual solublty as an nput varable 1 10 because the mscblty and nterfacal tenson reflect the same ntermolecular forces and are nherently related. 5,11 Interfacal tenson between two lqud phases has also been correlated wth the surface tensons of the two coexstng lqud phases by the smple Antonow s rule 12 or through an nteracton parameter that characterzes the smlarty of the ntermolecular forces between the two lquds. 13 The publshed nterfacal tenson models range from applcatons of the thermodynamc equaton of Shan and Prausntz, 1 as exemplfed by the models of Fu et al. 2 and Bahraman and Danesh, 7 to those based on soluton theores such as the quaslattce and regular soluton approxmatons, 3,9 the gradent theory, 14 and the scalng theory of crtcal phenomena. 15 Moreover, a model based on the Gbbs Langmur monolayer adsorpton sotherm has been reported. 10 There are also emprcally based models 5,12,16 that are often Receved: December 14, 22 Revsed: Aprl 24, 23 Accepted: Aprl 25, 23 Publshed: Aprl 26, Amercan Chemcal Socety 6822 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

2 Industral & Engneerng Chemstry Research convenent n practcal engneerng calculatons. Assessments and comparsons of varous methods that are avalable for estmatng nterfacal tenson n nonelectrolyte systems have also been reported. 11,17,18 In contrast to nonelectrolyte systems and also to lqud gas surface tenson, the lqud lqud nterfacal tenson behavor of systems contanng electrolytes s much less comprehensvely understood, even though the nterfaces of two mmscble electrolyte solutons have been studed for many years 19 and models for the electrolyte effect on nterfacal tenson have been reported. The most recently publshed models for such systems nclude those of Ber et al., 20 Onuk, 21 and dos Santos and Levn. 22,23 Because of the low delectrc constants of organc solvents, especally of ol components, on nteractons at ol water nterfaces have been assumed to be smlar to those at the lqud gas nterfaces n treatments usng the delectrc contnuum theory. 22 Indeed, the lqud lqud nterfacal tenson data that are reported as a functon of electrolyte concentraton seem to show a trend smlar to that of the lqud gas surface tenson, namely, the nterfacal tenson ncreases nearly lnearly wth electrolyte concentraton at large onc strengths for most of the reported systems. On the other hand, lmtng behavor derved from the theory of Onsager and Samaras 24 for lqud gas surfaces at low onc strengths cannot be expected for lqud lqud nterfaces because of on parttonng between the two phases. 20 Rather, a lmtng behavor that scales wth the square root of the onc strength has been derved for lqud lqud nterfacal tenson. 20,21 For more concentrated electrolyte solutons, an approach that combnes the thermodynamc treatment emboded n the Gbbs equaton 12 wth an adsorpton sotherm has been proposed. 25,26 In such an approach, the adsorpton sotherm defnes the nterfacal concentratons of the electrolyte components. The actvtes requred n the Gbbs equaton are then determned by ntroducng an actvty coeffcent model. The applcablty range of ths approach s generally lmted by that of the actvty coeffcent model. For example, the Messner 27 method for predctng actvty coeffcents was used by L and Lu 26 n ther nterfacal tenson model, and therefore, the applcablty range of the L Lu 26 model concdes wth that of the Messner model, whch s generally vald up to a few molal for most electrolyte systems. 27 These models have been successfully appled to represent nterfacal tenson between an organc solvent and aqueous electrolyte solutons. It should be noted that these models do not take nto account the chemcal specaton n electrolyte solutons, whch can be sgnfcant for hghly assocated systems such as sulfurc acd. For example, t s noteworthy that nterfacal tenson n the benzene water H 2 SO 4 system shows a mnmum as a functon of acd concentraton, whch would be dffcult to represent usng the exstng models wthout takng nto account the complex specaton patterns of sulfurc acd. Also, these models neglect the presence of electrolytes and water n the organc, or second lqud, phase and assume that the actvtes of the organc solvent and possble metal organc complexes are neglgble n the aqueous phase. 25,26 Such assumptons mght be justfed for systems n whch the solvent components (.e., water and an organc compound) show a wde mscblty gap and the organc solvent s hghly hydrophobc, but mght lead to unreasonable results for systems n whch both lqud phases contan sgnfcant or non-neglgble amounts of onc speces (e.g., onc lquds + water) or both phases are water-domnated (e.g., aqueous two-phase systems, or ATPSs). In a lqud lqud system contanng electrolytes, nterfacal tenson s determned not only by the concentratons of electrolytes, but also by the mutual solublty of the solvent components, whch s often a functon of temperature and pressure. Thus, lqud lqud nterfacal tenson strongly depends on the equlbrum concentratons of the mxedsolvent electrolyte system. In addton, n systems wth strong on assocaton effects, nterfacal tenson lke most thermophyscal propertes s expected to be nfluenced by the concentratons of both ons and assocated on pars. Thus, a comprehensve treatment of nterfacal tenson n mxed-solvent electrolyte systems requres one to take account not only the on solvent and on on nteractons that predomnate n aqueous solutons, but also the solvent solvent and on par solvent nteractons. Ths s possble only f a comprehensve, specaton-based framework s used as a thermodynamc foundaton for the nterfacal tenson model. The objectve of ths work s to develop a comprehensve, engneerng-orented model for predctng lqud lqud nterfacal tenson n systems contanng nonelectrolyte and electrolyte components. The model s desgned to account for specaton effects, such as complexaton or on assocaton, whch can be obtaned from a separate specaton-based thermodynamc model. The model developed n ths study conssts of two parts: computaton of the nterfacal tenson of nonelectrolyte lqud lqud systems and (2) evaluaton of the dependence of the nterfacal tenson on the electrolyte concentraton. 2. INTERFACIAL TENSION IN NONELECTROLYTE LIQUID LIQUID SYSTEMS The nterfacal tenson model for nonelectrolyte systems s derved on the bass of a rgorous thermodynamc framework that was ntroduced by Shan and Prausntz. 1 For completeness, the dervaton of ths framework s summarzed n Appendx A. In ths approach, the nterfacal tenson (σ) s related to the actvtes of a component n the bulk lqud phases (a = x γ ) and at the nterface (x nt γ nt ) nt nt σ A a = xγ = x γ exp RT where A nt s the partal molar nterfacal area of component. The exponental term n eq 1 can be nterpreted as a correcton to the product x nt nt γ and s necessary to account for the nterfacal effects determned by A nt. Combnng eq 1 wth the mole fracton constrant for the nterfacal phase, that s nt x = 1 results n the expresson x γ nt σ A nt exp = 1 γ RT nt Artcle Whereas the actvty coeffcents of the components n the bulk lqud phases can be determned drectly from the actvty coeffcent model, 28 those at the nterface, γ nt, need to be evaluated separately, together wth the partal molar nterfacal area, A nt. (2) 6823 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

3 Industral & Engneerng Chemstry Research To evaluate the actvty coeffcents of the components at the nterface, Backes et al. 6 assumed that the composton of the nterfacal regon s a pont on the te lne that lnks the bulk phases n an N-dmensonal concentraton space, where N stands for the number of components n the system. The actvty coeffcents of the components at the nterface are then calculated usng the same model as for the bulk lqud phases. An alternatve approach was proposed by Bahraman and Danesh, 7 who employed lattce theory and the regular soluton assumpton to relate the actvty coeffcents of each component at the nterface (γ nt ) to those n the two bulk lqud phases (γ α and γ β ) nt 1/2 γ = ( γγ α β ) By applyng eq 3 and the equlbrum condton x α γ α = x β γ β, eq 2 becomes nt α β σ 1/2 A ( xx ) exp = 1 RT (4) where the sum n eq 4 s over all components n the system. In ths study, we adopt the approxmaton of Bahraman and Danesh, 7 namely, eq 4. Computaton of the nterfacal tenson, σ, from eq 4 requres a pror determnaton of the partal molar nterfacal area A nt. Sprow and Prausntz 29 estmated A nt from the partal molar volume of component n the nterfacal mxture ( v nt ) nt nt 2/3 1/3 A A = ( v ) ( N ) (5) where N A s Avogadro s number. The partal molar nterfacal area can also be estmated from the van der Waals area of the molecule as descrbed by Bond. 30,31 The estmaton of the partal molar nterfacal area usng eq 5 and other methods 30,31 has been based on sphercal molecules, whch can be a crude assumpton. The partal molar volume of a component n the nterfacal phase should, n general, depend on the temperature, pressure, and composton of the two lqud phases that are adjacent to t. In vew of the absence of a rgorous method for calculatng the partal molar nterfacal area, a mxng rule s nt proposed n ths study for estmatng A usng eq 5. The mxng rule uses the pure-component molar volume of component as a reasonable reference pont for the partal molar volume. Then, t adds bnary terms that account for the effects of other components on the partal molar volume n the mxture. The bnary terms need to be weghted by the concentratons of components n the coexstng phases because the nterfacal concentraton s ntermedate between those n the bulk phases α and β. Accordngly, the mxng rule for the nterfacal molar volume of component s expressed as nt 0 α β 0 v = v + ( xj + xj ) kjvj j (6) where v 0 and v 0 j are the molar volumes of the pure lquds and j, respectvely; x α j and x β j are the equlbrum concentratons (n mole fractons) of component j n lqud phases α and β, respectvely; and k j s an adjustable parameter that can be determned from expermental nterfacal tenson data. Equaton 6 reflects the effect of the compostons of the two coexstng lqud phases on the partal molar volume of the component at the nterface. The parameter k j represents the extent of ths effect. When all k j values are zero, v nt becomes (3) equal to the pure-component lqud volume, v 0. Equaton 6 remans nvarant to dvdng any component nto two dentcal pseudocomponents, and therefore, t satsfes an mportant nternal consstency check for emprcal mxng rules (.e., t avods the so-called Mchelsen Kstenmacher syndrome 32 ). By combnng the equlbrum composton of the two lqud phases wth the partal molar nterfacal area A nt from eqs 5 and 6, the nterfacal tenson, σ, can be obtaned by solvng eq 4. It should be noted that the equlbrum concentratons n the two lqud phases, x α and x β, whch are requred for calculatng the nterfacal tenson n eq 4, are determned here from the thermodynamc actvty coeffcent model. 28 The thermodynamc model was developed prevously and has been extensvely valdated usng expermental data on phase equlbra and other thermodynamc propertes for a varety of systems ncludng those wth lqud lqud phase splttng. 33,34 3. EFFECT OF ELECTROLYTE CONCENTRATION ON INTERFACIAL TENSION 3.1. Thermodynamc Treatment: The Gbbs Equaton. For a gven system at constant temperature and pressure, the change wth composton of the nterfacal tenson σ between two lqud phases, α and β, can be expressed usng the Gbbs equaton dσ = Γ σ dμ where μ s the chemcal potental of speces and Γ σ s the nterfacal excess of component, 12 whch represents the excess (a postve dfference from the bulk value) or defcency (a negatve dfference) of any component, per unt area, at the nterface. Γ σ depends on the compostons of the two lqud phases and the nterface. Equaton 7 thus descrbes the change of nterfacal free energy between two equlbrum states of a two-phase system. Based on the theory of the Gbbs dvdng nterface, 12,35 the nterfacal excess can be defned on the bass of any arbtrarly chosen dvdng nterface n the nterfacal regon. Followng Shan and Prausntz, 1 a dvdng nterface was selected such that the nterfacal excess of any solvent component s dentcally equal to zero, or Ns σ l Γl dμ l = 0, where the sum s over all solvent components l and N s s the number of solvent components (mscble or mmscble). Such a defnton of the dvdng nterface s equvalent to a reference state n whch any devaton from the nterfacal tenson of an mmscble solvent mxture s attrbuted to the effect of the solute (.e., electrolyte) concentraton. To transton from eq 7 to a practcal equaton that can work together wth engneerng-orented Gbbs energy models for lqud lqud equlbrum (LLE) calculatons, t s necessary to consder the general LLE condtons for systems wth onc components. The lqud lqud equlbrum crteron for such systems was dscussed n a prevous study. 33 Fundamentally, the chemcal potentals of each neutral speces n the two coexstng phases, α and β, are equal α β Artcle μ = μ (8a) When the system contans electrolytes (e.g., C νc A νa ), the lqud lqud equlbrum condton can be defned as 33 (7) 6824 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

4 Industral & Engneerng Chemstry Research μ α β = μ CνAν CνA c a c νa (8b) where μ = νμ + νμ CνA c ν C a A c a (8c) Equatons 8b and 8c lead to the equalty of the mean actvtes of the electrolyte n the two lqud phases at equlbrum α β ± ± ( a ) = ( a ) (8d) where a ± =(a C ν c a A ν a ) 1/ν and ν = ν c + ν a. Such a defnton s based on the fact that only the chemcal potental of an electrcally neutral salt s expermentally accessble because of the electroneutralty condton for every phase. Consequently, ths constrant can be appled to all caton anon pars when the system contans multple ons. 33 Notwthstandng ths thermodynamc relatonshp, t has been noted n the lterature that a complete thermodynamc treatment of lqud lqud equlbra n multcomponent electrolyte systems requres calculaton of the contact potental between the two phases n equlbrum. 36 Specfcally, the electrochemcal potental of the onc speces should nclude an electrostatc contrbuton n addton to a chemcal term chem 0 μ = μ + zfϕ = μ + RT ln a + zfϕ where ϕ s the electrostatc potental. The contact potental can then be obtaned as the dfference n the electrostatc potentals n two phases. The equalty of the electrochemcal potentals of the neutral caton anon pars n both phases at equlbrum as represented by eq 8b results n the cancellaton of the terms that contan ϕ. In fact, Haynes et al. 36 noted that knowledge of the nterfacal electrostatc potental dfference between two phases n equlbrum s not necessary for the determnaton of equlbrum compostons or any other thermodynamc propertes of two-lqud-phase systems, but s useful for understandng the ntermolecular forces actng on the charged speces at the nterface. An approprate expresson for the composton dependence of the nterfacal tenson can then be derved by consderng the fact that the chemcal potental of speces, μ, should be the same at equlbrum n each of the two lqud phases and at the nterface. Thus, for neutral speces σ α β μ = μ = μ = μ N N N N (10a) μ 0 = μ + RT ln a N N N (9) (10b) dμ = RT d lna N N (10c) where a N s the actvty of speces N n the lqud phase, whch s the same n the phases α and β. For an electrolyte C νc A νa σ α β CνAν CνAν CνAν CνA c a c a c a c νa (11a) μ = μ = μ = μ μ = νμ + νμ CνA c νa c C a A 0 0 c c a A = ( νμ + νμ ) + ( νcrt lnac + νart ln aa) (11b) dμ = νrt d lna + νrt d lna CνA c C a A c νa (11c) where a C and a A are the actvtes of caton C and anon A, respectvely, whch satsfy the equlbrum crteron defned n eq 8d. In eq 11a, the electrostatc potental terms cancel. Thus, eq 7 can be expressed to nclude contrbutons from neutral speces (N) and the electrcally neutral caton anon pars (C νc A νa ) σ N σ CνA c νa CνA c νa dσ = Γ dμ Γ dμ N It can be easly shown that CνA c νa N σ CνA c νa CνA c νa CνA c νa Γ dμ = Γ I σ I dμ chem I where I denotes the consttuent ons and chem I dμ = RT d lna σ Γ = ν Γ σ I I CνA ν ( I) I (12) (13a) (13b) c a (13c) In eq 13b, the superscrpt chem n dμ chem I s used to dstngush the chemcal contrbuton to the chemcal potental from the complete electrochemcal potental, dμ I, whch ncludes the electrostatc term for on I that was canceled n dμ Cνc A νa. The sum n eq 13c ncludes all electrolytes C νc A νa that contan the on I, and ν I s the number of ons I (caton or anon) n C νc A νa. Substtutng eqs 10c and 11c nto eq 12 and applyng eqs 13a 13c yelds dσ = RT Γ σ d lna (14) Because of the selecton of the Gbbs dvdng nterface, the sum n eq 14 ncludes only solute speces, and denotes both neutral and charged speces Modfed Langmur Adsorpton Model. The most convenent way of determnng the nterfacal excess, Γ σ,sby ntroducng an adsorpton sotherm. Followng our prevous work on surface tenson 37 and the methodology of Desnoyer et al. 25 and L and Lu 26,38 for modelng the vapor lqud and lqud lqud nterfacal tensons of aqueous electrolyte solutons, an extended verson of the Langmur sotherm s used n ths study. Further, t s coupled wth a specaton-based thermodynamc model 28 wthn the framework of the Gbbs equaton so that any change of nterfacal tenson due to chemcal specaton can be explctly represented as a functon of electrolyte composton. The actvtes that are requred for the calculaton of nterfacal tenson are also determned from the thermodynamc model. The use of an extensvely valdated thermodynamc equlbrum model makes t possble for the nterfacal tenson model to be applcable over a wde range of electrolyte concentratons (.e., from nfnte dluton to the fused salt lmt) and for any composton of solvent mxtures. For modelng nterfacal tenson, the adsorpton sotherm needs to be appled to all solute speces, namely, ons, on pars, and neutral molecules, that contrbute to adsorpton. Furthermore, the dervaton of the expresson for nterfacal tenson needs to take nto account the nteractons between solute speces at the lqud lqud nterface. In the classcal Langmur model, the equlbrum condton for the compettve adsorpton of multple speces can be expressed by k (1 θ) a = k θ,a j,d j Artcle (15) 6825 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

5 Industral & Engneerng Chemstry Research where k,a and k,d are the adsorpton and desorpton rate constants, respectvely, of speces ; θ s the nterfacal coverage fracton of the adsorbed speces ; and a s the actvty of speces. Equaton 15 can be rearranged to gve θ = Ka 1 + Ka j j j (16) where K = k,a /k,d s the adsorpton equlbrum constant for speces. Assumng that the adsorpton layer has a fxed capacty for a gven speces, the nterfacal fracton θ can be related to the nterfacal excess Γ σ by ntroducng a mum nterfacal excess, Γ σ,0, 25,26,38 that s θ = σ σ,0 Γ Γ (17) Combnng eqs 16 and 17 leads to the followng expresson for Γ σ Γ σ = Γ σ,0 Ka j 1 + Ka j j (18) By substtutng eq 18 nto eq 14 and ntegratng the resultng expresson from a = 0 (for whch the lqud lqud nterfacal tenson s that of the solute-free system,.e., σ ms ) to the actual value of a, an expresson for the effect of electrolyte concentraton on nterfacal tenson s obtaned as σ,0 σ σ = Γ Ka ms RT ln Ka j j j (19) Equaton 19 was prevously obtaned by L and Lu 26 and successfully appled to calculate the nterfacal tenson between organc solvents and aqueous electrolyte solutons at mostly moderate concentratons (up to ca. 5 m) by assumng a complete dssocaton of the electrolytes. A further extenson of eq 19 s necessary, however, when onc concentratons n twolqud-phase systems reach hgher levels n ether of the two lqud phases (e.g., n some onc lqud + water mxtures or n aqueous two-phase systems) or when the effect of chemcal specaton on nterfacal tenson becomes sgnfcant. To develop an extenson of eq 19, t can be noted that nteractons between adsorbed speces n the nterfacal regon cannot be neglected, n partcular at elevated concentratons of the solutes. Thus, n a multcomponent system, the nterfacal excess of speces s lkely to be affected by the presence of other speces. Although classcal approaches such as the Frumkn and related sotherms can account for the effects of nteractons between speces n an adsorbed layer, 39 such sotherms do not lead to an analytcal expresson for the nterfacal tenson when combned wth the Gbbs equaton (eq 14). Numercal ntegraton of eq 14 n conjuncton wth a more complex sotherm s possble, but such an approach would requre calculatng the actvtes numerous tmes and would be very cumbersome, especally for process smulaton applcatons. Therefore, followng our prevous work on vapor lqud surface tenson, 37 an extended expresson for the electrolyte effect on nterfacal tenson s ntroduced to allow for parwse nteractons. The methodology used n the dervaton of ths extenson was descrbed n detal n a prevous study. 37 In ths extended expresson, bnary parwse contrbutons are ntroduced, and the mean actvtes of speces and k, a k, are used rather than those for sngle speces. Ths expresson s an emprcal extenson of eq 19 n whch the sngle-speces contrbutons K a are replaced by parwse contrbutons K k a k σ,0 σ σ = Γ Kkak ms RT k ln K a k j where a = ν ( a a ν k 1/ ) ν k k k jk jk (20) (21) ν = z k, ν k = z, and ν k = ν + ν k. For neutral speces k (or ), ν = 1 (or ν k = 1). Because of the selecton of the Gbbs dvdng nterface n the present study, eq 20 does not explctly reflect the effects of the chemcal nature of the solvent and ts composton on the electrolyte-related ncrement n the nterfacal tenson, σ σ ms. However, the parameters n eq 20 (.e., Γ σ,0 k and K k ) are unavodably dependent on the solvent envronment n whch the electrolyte components fnd themselves. If such dependence were gnored, the nterfacal tenson ncrement due to the presence of electrolytes would be the same n all solvents at the same electrolyte actvtes. Therefore, eq 20 needs to be further extended to ntroduce the specfcty of Γ σ,0 k and K k to the solvent envronment. In a prevous study of the surface tenson of mxed-solvent electrolyte systems, 37 the dependence of σ σ ms on solvent composton was ntroduced by usng a factor of (x m x n ) 1/2, where x m and x n are the salt-free mole fractons of the solvents m and n, respectvely. For lqud lqud systems, the solvent composton at the nterface s expected to be ntermedate between those n the two lqud phases. Thus, the factor (x m x n ) 1/2 needs to be redefned usng the average mole fractons of the solvent components n the two lqud phases to approxmate the composton n the nterfacal regon. Thus, the dependence of σ σ ms on the solvent composton n a lqud lqud system contanng N s solvent components and N e solutes (ons, on pars, neutral molecules) s expressed as Ns σ σms = RT ( xm xn ) where Ne m Ne k Ns n 1/2 K a σ,0 Γ, k, mn ln K j, k mn k 1 α β α β x = x + x x = x + x 2 ( ), 1 m m m n 2 ( n n ) a jk mn jk Artcle (22) (22a) and x m α, x m β, x n α, and x n β are the salt-free mole fractons of solvents m and n n phases α and β, respectvely. It should be noted here that the parameters Γ σ,0 k and K k n eq 20 are σ,0 redefned n eq 22 as Γ k,mn and K k,mn, respectvely, to reflect the specfcty of these parameters to the solvent envronment. 4. PARAMETER EVALUATION 4.1. Parameters n the σ ms Model for Nonelectrolyte Systems. The model for calculatng the nterfacal tenson (σ ms ) of partally mscble nonelectrolyte mxtures (eqs 4 6) has an adjustable parameter, k j, that can be determned usng expermental data for bnary mxtures. The model does not mpose any nherent constrants on the parameters k j and k j. However, for alkane + water systems and most polar + nonpolar pars (e.g., water + arene, alcohol + toluene, and acetone + n-hexane) that were tested n ths study, one of the k j 6826 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

6 Industral & Engneerng Chemstry Research Artcle Table 1. Parameters of Eqs 4 6 and 23 for Selected Bnary Lqud Lqud Nonelectrolyte Mxtures components parameters j (0) k j k j t ( C) no. ponts Δσ (mn m 1 ) ref(s) n-pentane a water pentane a water n-hexane a water , 43, 75, n-heptane a water , 77 80, 82, 83 n-octane a water , 77 80, 82, 83 n-nonane a water , 80, 82, 83 n-decane a water , 78 83, a n-c 11 H 24 water , 80, 82 a n-c 12 H 26 water , 78 80, 82, 83 a n-c 13 H 28 water , , 82 a n-c 14 H 30 water , 79, 82, 90 a n-c 15 H 32 water , 78, 79, 82, 83 a n-c 16 H 34 water , 78, 79, 82, 83 a n-c 22 H 46 water cyclohexane b water , 87, 91, 92 benzene b water , 45, 68, 75, 77, 82, 85, 86, 88, toluene b water , 77, 81, ethylbenzene b water , 77, 97 o-xylene b water , , 47 decaln b water phenol b water n-butanol b water , heptanoc acd b water n-butyl acetate b water , , 99 trethylamne b water b CCl 4 water , 68, 88 b CHCl 3 water , 68, 95 ntrobenzene b hexane benzene b formc acd octane b phenol a k j =0,k j =0. b k j = k j, k j = k j. or k j parameters was set equal to zero, for example, k j 0 and k j = 0, whereas for most other systems, the equalty of these parameters was assumed, namely, k j = k j. Only four polar/ nonpolar pars requred two dstnct parameters, that s, k j k j. Accordng to eqs 5 and 6, the nterfacal tenson has an nherent temperature dependence due to the temperature dependence of the lqud molar volumes of the mxture components, ν 0. However, the temperature dependence of the lqud volumes mght not provde a suffcent varaton of σ ms wth temperature because the change n nterfacal tenson wth temperature can be more or less pronounced than that mpled by the lqud molar volumes. Thus, an addtonal temperature dependence of the bnary parameter k j s ntroduced, when necessary, to represent the varaton of σ ms wth temperature (0) kj = kj exp[ kj ( T T0 )] (23) where T s n Kelvn and T 0 = K Parameters Γ k σ,0 and K k. Most nterfacal tenson data that are avalable n the lterature for lqud lqud systems contanng electrolytes are lmted to a sngle temperature (.e., 25 C) or cover only a narrow temperature range. Therefore, an explct temperature dependence of the bnary parameters Γ k σ,0 and K k s necessary only for systems for whch the data cover relatvely wde temperature ranges (e.g., C n the case of onc lqud + water systems). The dependence of Γ k σ,0 and K k on temperature can be expressed by the smple functons σ,02 σ,0 σ, Γ = Γ + Γ k k k T K k (2) Kk = Kk + T (24a) (24b) It should be noted that the parameters Γ k σ,0 and K k are solventdependent, as mpled by eqs 22 and 22a. For example, these parameters can be dfferent for a gven speces par n water + benzene and n water + hexane mxtures. 5. RESULTS AND DISCUSSION 5.1. Lqud Lqud Interfacal Tenson (σ) n Nonelectrolyte Systems and Varaton of σ wth Mutual Solublty. Expermental data for a number of bnary and ternary lqud lqud systems were used to valdate the mxng rule descrbed n secton 2. For all systems for whch the nterfacal tenson model was tested, thermodynamc model parameters 28 were frst developed to provde approprate lqud lqud equlbrum compostons as nput for nterfacal tenson modelng. Table 1 lsts the parameters k j for selected bnary systems, together wth the average error, defned by 6827 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

7 Industral & Engneerng Chemstry Research Artcle Fgure 1. Comparson of the calculated nterfacal tenson wth lterature values for bnary lqud lqud systems. References to the lterature data are gven n Table 1. n Δ σ = ( σ σ )/ n k exp, k cal, k (25) where n s the number of expermental data ponts. Lterature data sources are also ncluded n the table. These results are further vsualzed n Fgure 1, whch compares the calculated nterfacal tensons wth expermental data. Usng the parameters determned from bnary data, the nterfacal tensons of ternary systems can be predcted. For most of the ternary systems for whch nterfacal tenson data are reported, only a few systems have more than one consttuent bnary subsystem for whch nterfacal tenson data are avalable. Table 2 summarzes the predcted results for Table 2. Results for Ternary Mxtures Predcted Usng the Bnary Parameters n Table 1 solvent mxture j k t ( C) no. ponts Δσ (mn m 1 ) benzene water formc acd CCl 4 water n-heptane n-hexane water n-c 10 H 22 25, three such ternary systems usng the bnary parameters lsted n Table 1. The predcted and expermental nterfacal tensons are n good agreement, as llustrated n Fgure 2. When expermental data and, hence, bnary parameters for some of the consttuent bnary systems are not avalable, the nterfacal tenson data for ternary systems were used to determne the mssng parameters for the pars for whch bnary data are unavalable or cannot be drectly determned (e.g., n the ternary system benzene + water + ethanol, the water + ethanol bnary s fully mscble, and therefore, no lqud lqud nterfacal tenson exsts). In general, for a ternary system, only the parameters for one mssng par are necessary to model the nterfacal tenson. The par that s most prone to form two lqud phases (e.g., a polar/nonpolar par) s then selected to determne the parameters k k by fttng the σ data n ternary ref systems. Table 3 summarzes the nterfacal tenson results for such systems. Thus, the parameters for the par {, k} were determned usng data for ternary systems, whereas those for {, j} were taken from Table 1. The nterfacal tenson results that were obtaned usng the parameters n Tables 1 and 3 are further compared wth lterature data n Fgure 3 for fve ternary lqud lqud systems of the type benzene + water + polar organc (where polar organc = acetone, acetc acd, -propanol, ethanol, and methanol). Strong effects of the polar organc components on the nterfacal tenson are evdent, as these components cause a sgnfcant decrease n the nterfacal tenson upon beng added to the benzene + water mxtures. The results that were obtaned for the nonelectrolyte lqud lqud systems ndcate that the model (eqs 4 6) can accurately reproduce the expermental data and s capable of predctng nterfacal tenson n ternary mxtures usng parameters obtaned from data for bnary subsystems (cf. Fgure 2). It s noteworthy that the values of the parameter k j are usually postve, wth the excepton of that for the heptanoc acd + water mxture. The postve values of these parameters ndcate a certan augmentaton of the nterfacal molar volumes of the components and, consequently, ther nterfacal molar areas compared to those for the pure lquds. Ths can be attrbuted to systematc varatons of the ntermolecular nteractons on the nterface, whch result from the dssmlarty of the two fluds n the system. These parameters have a general trend of becomng less postve as the number of carbons (or the molecular weght) ncreases. Ths trend can be observed for alkanes and for alkylbenzenes, as seen n Table 1 for ther mxtures wth water. An ncrease n the number of carbons causes an ncrease n the nterfacal tenson, whch s accompaned by a smultaneous decrease n the solublty of the hydrocarbon n water. Ths s demonstrated n Fgure 4, where both nterfacal tenson and hydrocarbon solublty n water are plotted as functons of the number of carbons. The ncrease n the nterfacal tenson and, hence, n the nterfacal Gbbs free energy appears to be a result of an augmentaton n ntermolecular nteractons at the nterface due to an ncreased dssmlarty n the coexstng fluds as the hydrocarbon chan 6828 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

8 Industral & Engneerng Chemstry Research Artcle Fgure 2. Predcted nterfacal tensons n ternary lqud lqud systems: (a) benzene + formc acd + water, (b) n-c 7 H 16 + CCl 4 + water, (c) n- C 6 H 14 + n-c 10 H 22 + water. The symbols are from the lterature, 43,45,50 and the lnes were predcted usng eqs 4 6 and the parameters n Table 1. length ncreases. At the same tme, as the number of carbons ncreases, the lqud molar volume contrbuton (v j 0 ) to the nterfacal molar volume (v nt ), as defned by eq 6, also ncreases. The ncreases n the nterfacal molar volume (v nt ) wth number of carbons can be largely explaned by the ncrease n the lqud molar volume, leadng to a decreased, although stll postve, bnary parameter k j as the v j 0 and k j parameters partally compensate each other. The strong relatonshp between the mutual solublty and the nterfacal tenson can also be demonstrated by analyzng the three bnary systems phenol + water, trethylamne + water, and n-butanol + water. The results for these systems are shown n Fgures 5 7. These results show that the nterfacal tenson decreases n lockstep wth ncreasng mutual solublty as the temperature changes. In the phenol + water mxtures (Fgure 5), the mutual solublty of phenol and water ncreases, whereas σ decreases wth temperature. The nterfacal tenson drops to zero as the temperature approaches the upper crtcal soluton temperature (UCST), above whch the components become fully mscble. Conversely, the trethylamne + water system exhbts a lower crtcal soluton temperature (LCST), as shown n Fgure 6. In ths case, the nterfacal tenson s zero at the LCST pont at whch complete mscblty s reached and ncreases wth temperature as the mscblty gap wdens (or, equvalently, the mutual solublty decreases). For the n-butanol + water system (Fgure 7), the solublty of n-butanol n water shows a mnmum. Correspondngly, σ exhbts a mum as a functon of temperature and then decreases as the mscblty gap shrnks wth a further ncrease n temperature. The nterfacal tenson model accurately reproduces the varaton of σ wth temperature on the bass of the thermodynamc mxedsolvent electrolyte (MSE) model, whch provdes the lqud lqud equlbrum compostons. Thus, both σ and LLE can be reproduced smultaneously Effects of the Ionc Concentraton on the Interfacal Tenson. Applcatons of the new nterfacal tenson model to systems wth onc components focused on two classes of mxtures: electrolytes n base systems n whch nterfacal tenson s well-defned and expermentally known (e.g., n nonelectrolyte lqud lqud systems) and (2) onc systems for whch a base system of an mmscble nonelectrolyte mxture does not exst or cannot be determned. Systems that belong to the frst class nclude organc + water + salt mxtures where the organc + water subsystem s a partally mscble base system. Examples of the second class nclude onc lqud + water mxtures and aqueous two-phase systems such as poly(ethylene glycol) + water + salt ternares. As a frst step toward understandng and predctng the lqud lqud nterfacal tenson as a functon of onc addtves, only systems wth smple electrolytes were studed n the present work. Ionc 6829 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

9 Industral & Engneerng Chemstry Research Artcle Table 3. Results for Ternary Mxtures Predcted Usng the Bnary Parameters n Table 1 a solvent mxture parameters b j k (0) k k (0) k k t ( C) no. ponts Δσ (mn m 1 ) ref(s) benzene water acetc acd , 45 benzene water butyrc acd benzene water acetone benzene water methanol , 53 benzene water ethanol , , 54 benzene water -propanol , 53 benzene water n-propanol benzene water NH n-hexane water acetone n-hexane water ethanol n-hexane water -propanol n-hexane water n-propanol n-heptane water n-propanol toluene water ethanol toluene water -propanol toluene water n-propanol CCl 4 water n-propanol CHCl 3 water acetone cyclohexane water -propanol , , 100 n-butyl acetate water n-propanol a Interfacal tenson data for ternary systems were used to determne addtonal bnary parameters for the pars {, k} for whch no data on the correspondng bnary systems were avalable. b k k = k k = 0 for all {, k} pars. Fgure 3. Varatons of the nterfacal tenson wth the equlbrum mole fracton of the polar organc component n the aqueous phase of the ternary lqud lqud systems polar organc + benzene + water (25 C), where polar organc = acetone, acetc acd, -propanol, ethanol, and methanol. The symbols denote lterature data, 6,45,51 54 and the lnes were calculated usng eqs 4 6. surfactants were not consdered here because they have specfc structural and compostonal characterstcs and physcochemcal propertes that requre a dfferent treatment. Expermental nterfacal tenson data for systems contanng onzable components are avalable from varous lterature sources but are much less extensve than surface tenson data for vapor lqud systems. The majorty of the avalable data have been reported for systems that contan a sngle electrolyte component. Publshed data for systems wth mxed electrolytes are sparse. 25,43 Moreover, t should be noted that thermodynamc model parameters 28 must be developed pror to modelng nterfacal tenson to provde the necessary specaton, actvty coeffcents, and mscblty gap compostons. However, n some cases, the exact chemcal composton of the organc solvent s not explctly stated, 25 and relevant lqud lqud equlbrum data and thermochemcal propertes that are necessary to construct a phase equlbrum model as a thermodynamc foundaton for modelng nterfacal tenson are lackng. Thus, n the present study, the nterfacal tenson model s appled only to systems for whch the relevant thermodynamc propertes can be calculated. 28 Nonetheless, the avalable lterature data for systems wth well-defned solvents and electrolyte solutes provde a sound bass for testng the new model. Table 4 lsts the adjustable parameters n eqs 22, 24a, and 24b, namely, Γ k σ,0 and K k, for selected systems contanng onc components. If necessary, the parameters of eqs 4 6 are also ncluded (e.g., for onc lqud + water systems). The performance of the model for aqueous salt + organc solvent systems s llustrated n Fgures 8 and 9. In Fgure 8, the calculated nterfacal tenson s compared wth lterature data for the systems n-hexane + water + electrolyte, n-dodecane + water + electrolyte, and benzene + water + electrolyte as a functon of electrolyte concentraton for varous salts and acds and a base. Fgure 9 shows the results for varous combnatons of organc solvents and aqueous NaCl. These results show that the effects of electrolytes on nterfacal tenson can dffer dependng on the system. For the aqueous electrolyte + organc systems studed n ths work, nterfacal tenson ncreases wth electrolyte concentraton wth the excepton of the aqueous sodum chlorde + o-xylene system, for whch a decrease wth salt concentraton s observed (Fgure 9). The aqueous sulfurc acd + benzene system (Fgure 8d) shows a decrease n nterfacal tenson wth acd concentraton at low H 2 SO 4 molaltes. Then, the nterfacal tenson reaches a shallow mnmum and rses wth a further ncrease n the acd concentraton. Such behavor can be assocated wth changes n the specaton of sulfurc acd solutons. For ths system, t s necessary to ntroduce the Γ k σ,0 and K k parameters for both the 6830 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

10 Industral & Engneerng Chemstry Research Artcle Fgure 4. Varatons of the nterfacal tenson and solublty of hydrocarbons n water at 25 C wth the number of carbons n the hydrocarbon molecules n mxtures of (a) alkanes + water (ncludng n-alkanes, -pentane, and cyclohexane) and (b) alkylbenzenes + water (ncludng benzene, toluene, ethylbenzene, and o-xylene). The values of nterfacal tenson were calculated usng eqs 11a 11c, 12, and 13a 13c based on lterature data, and the solubltes are the averaged expermental data from the complaton of Frenkel et al. 55. Fgure 5. Varatons wth temperature of (a) the nterfacal tenson and (b) the correspondng mutual solublty n phenol + water mxtures. In panel a, the symbols denote expermental data, 56 and the lne was calculated usng eqs 4 6. In panel b, the symbols denote expermental mutual solubltes, and the lnes were calculated usng the MSE model. 28,33. H 3 O + /HSO 4 and H 3 O + /SO 2 4 pars to represent the varaton of nterfacal tenson wth acd concentraton. In our prevous work on modelng surface tenson, 37 negatve values of Γ σ,0 k were observed for systems n whch surface tenson ncreases wth electrolyte concentraton, thus ndcatng a negatve adsorpton (or defcency) of the electrolyte on the soluton surface. Conversely, a postve Γ σ,0 k value led to a decrease n surface tenson wth electrolyte concentraton. However, such trends were not observed for all systems n modelng nterfacal tenson. Ths s due to the fact that the value of σ ms, namely, the baselne nterfacal tenson n the absence of electrolytes, s calculated based on the equlbrum concentratons of nonelectrolyte components on a salt-free bass. Such salt-free nonelectrolyte compostons n electrolytecontanng systems can be dfferent from the equlbrum compostons n the absence of salts, because of the electrolyte effects on lqud lqud equlbra that are assocated wth saltng-n (ncreased solublty) or saltng-out (reduced solublty) effects. For systems wth a saltng-out effect, the reduced mutual solublty n the presence of an electrolyte leads to a greater value of σ ms compared to that obtaned for the equlbrum composton n the absence of the electrolyte. Thus, when the onc bnary parameters (Γ σ,0 k and K k ) are regressed to reproduce the varaton of nterfacal tenson wth electrolyte concentraton, the elevated σ ms value can result n ether a postve or a negatve Γ σ,0 k value, dependng on how strong the saltng-out effect s. Ths s demonstrated n Fgure 10, where the nterfacal tensons n the systems benzene + aqueous HCl and benzene + aqueous NaCl are plotted as functons of the salt concentraton. To elucdate the effect of saltng n or saltng out on the nterfacal tenson, predctons obtaned wth zero values of Γ σ,0 k and K k (whch are equvalent to σ ms n eq 22) are shown as dashed lnes. They are compared n Fgure 10 wth the results obtaned usng regressed Γ σ,0 k and K k values n Table 4 (σ, sold lnes). For the system benzene + aqueous HCl, the calculated σ ms value, based on the equlbrum salt-free concentratons of the solvent components (benzene and water), s greater than σ (cf. Fgure 10a), and therefore, a 6831 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,

11 Industral & Engneerng Chemstry Research Artcle Fgure 6. Varatons wth temperature of (a) the nterfacal tenson and (b) the correspondng mutual solublty n trethylamne + water mxtures. In panel a, the symbols denote expermental data 56 and the lne was calculated usng eqs 4 6. In panel b, the symbols denote expermental mutual solubltes, and the lnes were calculated usng the MSE model. 28,33. Fgure 7. Varatons wth temperature of (a) the nterfacal tenson and (b) the correspondng mutual solublty n n-butanol + water mxtures. In panel a, the symbols denote expermental data, 6,66 69 and the lne was calculated usng eqs 4 6. In panel b, the symbols denote expermental mutual solubltes, 58,70 74 and the lnes were calculated usng the MSE model. 28,33. postve Γ k σ,0 value s obtaned for the {H 3 O +,Cl } par. In contrast, for the system benzene + aqueous NaCl, the σ ms values that ncorporate the saltng-out effect are smaller than the total σ values (Fgure 10b), and therefore, a negatve Γ k σ,0 s obtaned for the {Na +,Cl } par. Thus, Γ k σ,0 n the present model represents the excess (postve ncrement) or defcency (negatve ncrement) of the electrolyte at the nterface usng, as a reference state, a salt-free lqud lqud system whose composton vares wth the electrolyte concentraton. As shown n Fgure 10, the σ values that are predcted wthout the Γ k σ,0 and K k parameters (.e., the dashed lnes) capture the trend of nterfacal tenson wth electrolyte concentraton based solely on the salt effects on lqud lqud equlbra. Thus, for systems for whch no data are avalable, eqs 4 6 and 22 can predct a reasonable nterfacal tenson trend as long as accurate lqud lqud equlbra can be obtaned from thermodynamcs. 28 The nternal lnkage wth phase equlbra, as quantfed by the saltng-n or saltng-out effects, provdes the present model wth a dstnctve capablty to predct the varaton of σ wth electrolyte concentraton. Ths s an advantage of the new model compared to other exstng nterfacal tenson models for electrolyte systems. For systems contanng onc lquds and water, both water and the undssocated onc lqud molecules [e.g., BMIM N- (CF 3 SO 2 ) 0 2 and EMIM N(CF 3 SO 2 ) 0 2 ] need to be treated as solvent components because of the strong on assocaton effects that are evdent n the treatment of thermodynamc propertes. 34 At the same tme, the dssocated caton and anon are treated as solute speces because they can exst n sgnfcant amounts n both lqud phases. In such systems, specaton can change dramatcally wth the concentraton of the onc lqud. 34 The bnary parameters that are used n the model for ths type of systems nclude the Γ σ,0 k and K k parameters between the caton and the anon n eq 22 and the k j parameter between the solvent components [ e.g., H 2 O and BMIM N(CF 3 SO 2 ) 0 2 ]n eqs 4 6. For example, the best ft for the system BMIM N- (CF 3 SO 2 ) 2 + H 2 O was obtaned when the parameters σ,0 Γ BMIM,NTf2 /H 2 O,IL 1, K BMIM,NTf2 /H 2 O,IL 1, and k H2 O,IL 1 were ntroduced [where NTf 2 = N(CF 3 SO 2 ) 2,IL 1 = BMIM N(CF 3 SO 2 ) 0 2, and k IL1,H 2 O = k H2 O,IL1]. These parameters are lsted n Table 4. As 6832 dx.do.org/ /e303460c Ind. Eng. Chem. Res. 23, 52,