Thermodynamics of Solubility of Surfactants in Water

＊ Mitsuru TANAKA Emeritus Professor, Department of Chemistry, Fukuoka University (Fukuoka, 814-0180, JAPAN)

Edited by M. Iwahashi, Kitasato Univ., and accepted January 18, 2005 (received for review November 8, 2004)

Abstract: The thermodynamic equations to quantitatively well explain the temperature and pressure effects on the solubility of surfactants in water have been obtained by the application of the small system thermodynamics to a multi-component micellar solution coexisting at equilibrium with the crystalline solid phases of the component surfactants. The equation theoretically predicts the existence of the critical micellization temperature, the Krafft temperature, in the temperature effect on solubility at given pressure, and that of the critical micellization pressure; the Tanaka pressure, in the pressure effect on solubility at given temperature. Key words: CMC, critical micellization pressure, Krafft temperature, micelle, small system thermodynamics, solubility, surfactant, Tanaka pressure

micelles to stably form in the temperature and pressure 1 Introduction region where T ≥ TK and P ≤ PC. Solubility of ionic surfactant in water has been found It can be concluded from these that the principle of to increase with temperature at given pressure, in which antagonistic effects of temperature and pressure holds the increasing rate of solubility with increased tempera- for the solubility of surfactants in water. ture changes from small value to large value over a nar- It is suitable to apply“Thermodynamics of Small row range of temperature (critical micellization temper- Systems”developed by Hill (9) to surfactant micelles. ature range) around a certain temperature TK called Hall (10)-(15) has intensively applied the small system Krafft temperature (1). thermodynamics mainly to non-interacting micelles of Solubility of ionic surfactants (1) (2) (3) (4) and that nonionic surfactants in solution. of nonionic surfactants (5) have been found to decrease The present author has made some reinterpretation with increasing pressure, in which the decreasing rate about the application of small system thermodynamics of solubility with increased pressure rapidly changes to non-interacting (16) and interacting micelles (17) in from large negative value to small negative value over a solution, and has removed some ambiguity of the refer- narrow range of pressure (critical micellization pressure ence Gibbs free energy and activity of micelles etc. range) around a certain pressureat PC at given tempera- remained in the references (9), (10) and (12) by the use ture. PC has been defined as critical solution (or micel- of Gibbs-Duhem type equations of micellar solution lization) pressure, the Tanaka pressure (6) (7) (8). obtained in the small system thermodynamics. It has been found that solubility-temperature curve In the present paper, the thermodynamic equations to intersects with CMC-temperature curve at TK at given quantitatively well explain the temperature and pressure pressure (1) (2), and solubility~pressure curve inter- dependency of solubility of surfactants in water have sects with CMC-pressure curve at given temperature (1) been derived by the application of small system thermo- (2). It is seen from these that it is just possible for dynamics to a multi-phase system, in which a solution

＊ Correspondence to: Mitsuru TANAKA, 2-27-2, Chihara-dai minami, Ichihara 290-0142, JAPAN E-mail: [email protected]

Journal of Oleo Science ISSN 1345-8957 print / ISSN 1347-3352 online 259 http://jos.jstage.jst.go.jp/en/ M. Tanaka

m of interacting multi-component micelles coexists with mi is the chemical potential of monomeric i in micelles the crystalline solid phases of the component surfac- (m), e is the subdivision potential of micelles, and xr tants at equilibrium. The multi-phase system can rea- and xM are defined as sonably be treated as a completely open (T, P, mi) sys- c tem (9) at complete equilibrium (11). The present theo- xNÊ NNs ˆ yx [2c] rr=+Ë Â iM ¯ = r M ry of solubility is developed on the basis of the theory i=0 s of solution of interacting micellar small systems devel- where Ni is the number of molecules of monomeric i in oped in the previous paper (17). the bulk solution. It should be stressed that the present theory owes It should be noted in eq. [2b] that the reference free much to Hall’s theory of solution of non-interacting energy Gr(T, P) has the physical meaning of the Gibbs micelles of nonionic surfactants (13). free energy of completely hydrated Mr at infinite dilu- tion at given T and P.

It is convenient here to transform Gr(T, P) to new ref- 2 Theoretical N erence free energy mr (T, P, yr) defined as

2･1 Basic Equations of Micelles in Solution N m r ()TPy,,rr= G() TP ,+ kTy ln r [3] In order to constitute the small system thermodynamics of solubility of surfactants in water, it is suitable to Eq. [2b] can be rewritten as the followings. reexamine from another point of view the basic equa- N mmrr= ()TPy,,rMMMM+= kTa ln a x g [4a, b] tions derived in the previous paper (17). Notations used here are all the same as in (10-14) and the previous where rM is the mean activity coefficient defined as fol- paper (17) excepting that 0 in stead of 1 refers to the lows, solvent water. lnggMrr= Â y ln [5] Consider a solution system of interacting micelles, r t the (T, P, mi) system, in which the total numbers Ni of From eq. [2a] and [4b], we obtain t solute species i(=1 → c) are dissolved in N0 of solvent c N r m water 0, and are formed NM micelles with polydispersi- mmeeer =+=-Â NkTai i MM; ln M [6a, b] i=1 ty characterized by the number Nr(=yrNM) of the rth r micellar species Mr including Ni molecules of the vari- It has been used in eq. [4a, b] and [6a, b] that gr = gM ous species i(=1 → c). Here the distribution function of for all r if the mean reference free energy`G = (T, P,`Ni) N Mr is defined as yNNrrM= / where MNMr= Â r . = Â r yTPyrrm (),,r is taken to be controlled by eq. It should be noted here that for solutions sufficiently [15] bellow, as will be shown in appendix 3･3. s s s s dilute in micelles as is usual, the presence of solvent Regarding mi = mi (T, P, ai ) in eq. [1], dmi can be water 0 in the micelles may to a good approximation be written as follows ignored in the thermodynamic equations of micelle for- s ˜ s s s dSdTVdm i =-i +˜i P +kTdln ai [7a] mation (12, 14), (17) (see appendix 3･1). s The chemical potential mi of monomeric species i in where we have the bulk solution (s) and the chemical potential mr of s ffs s SSTPka˜i =-ii(), ln VVTP˜i =i (), [7b, c] interacting Mr can reasonably be obtained in the following forms (see appendix 3･2). N s f s s s s Taking the exact differential of mr (T, P, yr) in eq. [3] mmi =+ii()TP,ln kT aaxi =i gi [1b] and [6a], we obtain sssss mm=+0 ()TP,ln kT aax = g N N N 00 0000 [1a] dSdTVdm r =-r +r P +kTdln yr [8a]

c r m mmeri=+N i c Â [2a] r m i=1 =+Â Ndi mei d M [8b] i=1 GTP kTa a x = rrrrr(),ln+=g [2b] where we have

260 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) Thermodynamics of Solubility of Surfactants in Water

c N r m ∂e SSTPky= , -=ln NS -Ê M ˆ [9a] (see appendix. 3･2). r rri() Â i Ë ¯ i=1 ∂T Py, r c It should be noted that equations [12]-[15] have the N r m Ê ∂e M ˆ VVTPNVr = ri(), =-Â i Ë ¯ [9b] same mathematical forms as the ideal equations of non- i=1 ∂ P Ty, r interacting micelles, in which gr = gM = 1 for all r (16). m m `Si and`Vi are respectively the partial molar entropy It is generally reasonable to take the reference free and volume of monomeric i in micelles defined by energyG so as to be controlled by eq. [15], the equation

m m of the same form as in the case of ideal micelle (see m Ê ∂ mmi ˆ m Ê ∂ i ˆ S =- V =- [10a, b] appendix 3･3). i Ë ∂T ¯ i Ë ∂ P ¯ Py,,rrTy Here,G,S andV can be interpreted as the mean It should be noted in the derivatives of eq. [9a, b] and intrinsic free energy, entropy and volume of micellar

[10a, b] that constant yr is equivalent to constant`Ni small systems in solution. r according to NyNiri= Â . Eq. [15] is just the expression of the first and second From eq. [8a] and [8b] we obtain the basic equation laws of thermodynamics with respect to`G(T, P,Ni) of

of Gibbs-Duhem type for a micellar species Mr as the micellar small systems. following. Eq. [13b] and [15] show that unlike macroscopic sys- c  N N r m tems, G of small systems is not a linear homogeneous dSdTVdPNemMr=- +r -Â i di +kTdln yr [11]    i=1 function of Ni. Eq. [16] and [17] show that S and V are

The mean chemical potential of micelles mM is also not the linear homogeneous functions ofNi.

obtained by multiplying eq. [4a] by yr, summing over It should be noted in eq. [14] that the Gibbs-Duhem

all r and using eq. [5], the definition of gM, as follows. equation for macroscopic phase c m mmMrr==Â r yGTPNkTa(),, i+ ln M[12a, b] 0 =-SdT + VdP -Â Nii dm [14]* i=1 c m does not hold for the micellar small systems because of =+Â Niime [12c]   i=1 the nonlinear homogeneity of G(T, P, Ni). from whichG the mean reference free energy is derived

as 2･2 Distribution Function yr of Interacting c N m Micellar Species Mr in Solution GT(),, P Nirr==+ÂÂ ymme Nii M [13a, b] r i=1 Here our micellar solution is reasonably assumed to

r be at complete equilibrium (11), in which hold e = 0 NyNiri= Â r [13c] s m and mi = mi = mi (see appendix 3･1).

Multiplying eq. [11] by yr and summing over all r, For a quasi-static process of micelle formation at s m we obtain complete equilibrium, de = 0 and dmi = dmi = dmi hold. c m Eliminating deM from eq. [11] and [14], and rearrang- dSdTVdemMii=- +PNd -Â [14] i=1 ing the result, we obtain From eq. [14] and the exact differential of eq. [13b], we obtain kTdln yr = c c m N N r dG=++ SdT VdPÂ m i dNi [15] ()SSdTVVr - --()r dP+-Â() Ni Ni dm i [18] i=1 i=1 s m where`S and`V are the mean reference entropy and vol- where has been used dmi = dmi = dmi. N ume of micelles obtained by Suitably substituting eq. [9a] and [9b] for Sr , and N c Vr and eq. [16] and [17] forS andV in eq. [18], we N m Ê ∂e M ˆ SySNS==Â rr Â ii - [16] obtain a useful equation r i=2 Ë ∂T ¯ Py, r c r m m kTdln yri=-Â() N Ni () Si dT-+ Vi dP dm i [19] c i=1 N m ∂e VyVNSÊ M ˆ [17] ==-Â rr Â ii Eq. [18] and [19] show that yr = yr(T, P, mi) for com- r Ë P ¯ i=2 ∂ Ty, r pletely open (T, P, mi) system. The basic equations [12] to [15] can directly be derived from the Gibbs-Duhem type equation [A11]

261 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) M. Tanaka

2･3 Development of d`Nk (k = 1 → c) in monomeric k in micellar form, and

Terms of dT, dP and dmi r Multiplying eq. [18] by yrNk , and summing over all m m Nk Ê ∂lnaM ˆ SSˆk =+k kT [24] 2 Ë ∂T ¯ r, we obtain the following equation for d`Nk (k = 1 → Nk Py, r c). ˆ m m Nk Ê ∂lnaM ˆ VVk =+k kT [25] kTdNk =-() SNk SNk dTVNVNdP--()k k 2 Ë ∂T ¯ Nk Ty, r c +-Â()NNi k NNik dm i [20] 2 i=1 Since`Nk /`Nk ≪ 1 for sufficiently large`Nk, the fol- where we have lowings hold to good approximation. N r N r ˆ m m ˆ m m SNkrr==ÂÂr y S Nk ,, VNkrrr y V Nk SSk ªªk VVk k [26a, b] Eq. [23] can be rewritten to NN yNr Nr ik= Â r ri k [21] c cM È m 2 ˆ m ˘ dNx()kM =+Â NNSiki Nkk S dT Eq. [20] is essentially the same form of equation as kT ÎÍikπ ˚˙ c derived by Hall (12) for non-interacting micelles of cM È m 2 m ˘ -+Â NNViki NVkkˆ dT [27] nonionic surfactants in solution. kT ÎÍikπ ˚˙ r c Multiplying eq. [19] by yrNk and summing over all r, cM +-Â NNdikmgikMM Nc dln we obtain kT i=1 c m m kTdNkkiki=-Â() N1 N N N() S dT-+ Vi dP dm i [22] t i=1 2･5 The Total Concentration Ck of Compo- Eq. [22] is also obtained from substitutions of eq. nent Surfactant k in Micellar Solution

[9a], [9b], [16] and [17] in eq. [20] and [21]. We define for simplicity the concentrations ck and cM

Eq. [20] and [22] show that`Nk = `Nk (T, P, mi) for of monomeric k and micelle M, and the total concentra- t t micelle with polydispersity. Since mi = mi (T, P, C i ) as tion Ck of monomeric k as the followings. t will be shown later, we have`Nk =`Nk (T, P, C i ), where s t t t ck = Nk / N0 cM = NM / N0 [28a, b] Ci is the total concentration of monomeric i in micellar solution. t t t Ck = Nk / N0 [29] Eq. [22] shows that if micelles are in monodisper- t sion,`Nk (k = 1 → c) have no dependency on T, P and where N0 is the total number of molecules of solvent t t s m m Ci . It is evident from this that the mass action law is water 0 and Nk = Nk + Nk , where Nk =`NkNM is the only applicable to the equilibrium of formation of number of monomeric k in micelles monodispersed micelles. The equations of mass balance hold in micellar solutions as follows,.

2･4 The Concentration of Monomeric t CccNk =+kMk [30a] Species i in Micellar Form

Putting de = 0 and deM = － d (kt ln aM) in eq. [11] at t dCk =+ dckMk d() c N [30b] complete equilibrium of micellar solution (11), multi- r s plying the result by Nk yr, and summing over all r, we where ck ≈ xk and cM ≈ xM are good approximations for

obtain the equation for d(`NkaM) on suitable arrange- solutions dilute in micelles.

ment, Substituting eq. [27] in eq. [30b], and using dck = c s s aM È m 2 ˆ m ˘ ckdlnak －ckdlngk in the result, we generally obtain the dNa()kM =+Â NNSikk Nkk S dT kT ÎÍ ikπ ˚˙ following equation on arrangement. c aM È m 2 m ˘ c -+NNVikk NVkkˆ dP [23] c ÍÂ ˙ Cdt At cd as M È NNSm N2 Sˆ m ˘ dT kT Î ikπ ˚ k lnk =+kk ln ÍÂ iki +kk˙ C kT Îikπ ˚ aM + Â NNdikm i kT i=1

NkaM is regarded as the effective concentration of

262 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) Thermodynamics of Solubility of Surfactants in Water

c c M È m 2 ˆ m ˘ t -+Â NNViki NVkk dP [31] dP and dCi (i = 1 → c). The equations obtained can be kT ÎÍikπ ˚˙ applied to discuss the temperature and pressure depend-

c ency of CMC (10) (17). cM Ê 2 ˆ ++NdkkmmÂ NNd iki kT Ë ikπ ¯ 2･6 The Solubility of Surfactants in Water t t where Ak and gk are assumed to be When c component micellar solution coexists at t t t equilibrium with l crystalline solid phases (c) of the Ak = Ckgk [32] component surfactants i(= 1 → l), the following l

t t s equilibrium conditions should hold for the components Cdk lnggk =++ cdkk ln() cMk N d ln g M [33] i(= 1 → l). t t It is apparent that Ak = Ck in the case of non-interact- s ˜ s s s ddSdTVmmii==i +˜i dP + kTdln ai [37a] ing micellar solution. WhenNk is sufficiently large, ck and cM are usually small, even in the saturated solution c c c s ==-+dSdTm i i VdPi [37b] of surfactant k, so that gk and gM are not so far from t unity. It is most probable that Ak should have the same from which we obtain t tendency of dependence on T, P and mi(i = 1 → c) as Ck s ˜ s c ˜ s c t t kTdln ai =-() Si Si dT--() Vi Vi dP [38] in solution of interacting micelles. To replace Ak by Ck c in eq. [31] will not bring about so large error, and will where the superscript c as in mi refers to crystalline make things easy to be understood. solid phase (c). Here the following approximate equations may con- Now we treat a mixed micellar solution being at veniently be used in stead of eq. [27] and [31]. equilibrium with l coexisting crystalline solid (c) phases i, k(= 1 → l) of surfactants where the mixed c cM ÈÊ 2 ˆ m m ˆ dc()Mk N =+NSkk Â NNSiki dT micelles solubilize the components j(= l + l → c) not kT ÍË ¯ Î ikπ having the crystalline solids coexisted. Component j c -+Ê NV2 ˆ m NNVm ˆ dP˘ [34] may be hydrophobic substances or amphiphiles like Ë kk Â iki¯ ˙ ikπ ˚ alcohols. c cM Ê 2 ˆ ++NdkkmmÂ NNd iki Suitably applying the equilibrium conditions of the kT Ë ikπ ¯ s s form of eq. [38] for d lnak and d lnai (i = 1 → l, i ≠ k) t in eq. [36], we obtain the equation for dCk of a micellar c c dCt =+ c dln as M ÈÊ NS2 ˆ m + NNSm ˆ dT solution saturated with surfactants k and i(≠ k) as fol- k kk ÍË kk Â iki¯ kT Î ikπ lows. c Ê NV2 ˆ m NNVm ˆ dP˘ [35] -+kk Â iki ˙ Ë ikπ ¯ ˚ t 1 Ï ˜ s c dCk =-ÌcSkk() Sk c kT cM Ê 2 ˆ Ó ++NdkkmmÂ NNd iki l kT Ë ikπ ¯ È m c +-cNNSSMikiÍÂ ()i s s Î i=1 Substituting eq. [7a] for dmk and dmi in eq. [35], we c m ˜ c ˘¸ obtain the following equation for micellar solution at +-Â NNjk() S j Sj ˙˝ dT complete equilibrium. j=+l 1 ˚˛ 1 Ï ˜ s c --ÌcVkk() Vk [39] c kT t cM È 2 ˆ m ˜˜s m s ˘ Ó dCk =-NSkk() Sk +-Â NNSik() i Si dT l kT ÎÍ ikπ ˚˙ È m c +-cNNVVMikiÍÂ ( i ) c c Î i=1 T --M ÈNV2 ˆ m V˜˜s +- NNVm Vs ˘ dP[36] Í kk()k Â ik() i i ˙ c kT Î ikπ ˚ m ˜ s ˘¸ + Â NNjk V j- VdPj ˝ c ()˙ 2 s s j=+l 1 ˚˛ ++()ccNdakMkln k+ Â cM NNNdikln a i c ikπ s + Â cNNdaM jkln j Solving simultaneously c equations of the form of eq. j=+l 1 s s [36] (k =1→ c) with respect to dlnak , dlnak can be m m expressed in terms of (c+2) experimental variables, dT, where Si and Vi for i = k should strictly be taken to

263 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) M. Tanaka

m m Sˆk and Vˆk in eq. [24] and [25]. mation here, the present (c + 1) component system t dCj of solute component j is written as follows. should have (2 + l) phases; micelle phase, free solution phase and l crystalline solid phases. Gibbs phase rule l t cM È m c dC j =-Â NNij() S i Si should give (c + 1 － l) independent variables to the kT ÎÍ i=1 t c present system, T or P being independent at given Cj . m ˜ s ˘ +-Â NNlj() S l Sl dT Thus, the phase separation model can not explain the l=+l 1 ˚˙ change of solubility with T at given P and C t and the l j cM È m c --Â NNij() V i Vi [40] change of solubility with P at given T in the region at T kT ÎÍ i=1 c > TK and P < PC where micelles are present. m ˜ s ˘ +-Â NNlj() V l Vl dP The pseudo-phase separation model has been pro- i=+l 1 ˚˙ posed in order to temporize in such fatal inconven- 2 s ++()ccNdjMjln a j iences of the micelle phase model. We have never

c heard that the thermodynamics of pseudo-phase has s + Â cNNdaMljln l ever been developed and applied to micelles. It is seri- l=+()l 1 ljπ ously problematic that the ordinary thermodynamics of macroscopic phase is still applied to micelles in spite of m m where Sl and Vl for l = j should strictly be taken to micelle being defined as pseudo-phase. It is unreason- m m Sˆ j and Vˆj in the same form as eq. [24] and [25]. able to apply the concept of phase to micelles, and also t The total concentration Ck in eq. [39] is just the solu- it is most unreasonable to stretch the phase concept to bility of surfactant k in an aqueous solution including other colloidal and interfacial systems without any (l － 1) component surfactants i(≠ k) and (c － l) solu- experimental and theoretical examinations. bilizates. 2 It can intuitively be seen from eq. [39] that whenNk 2･7 Solubility of Single Component Non- and NNikare large, once cM has reached a small value, ionic Surfactant 1 in Water t the changing rate of solubility Ck with temperature For a micellar solution of single component surfac- t ()∂∂CTk should rapidly become very large over the tant (k = 1), eq. [36] is simplified to narrow range of temperature with increasing tempera- cN2 ture, and the changing rate of C t with pressure dC t =-M 1 SSdTˆ ms˜ k 1 kT ()11 ∂∂CPt should rapidly change over the narrow range ()k cN2 of pressure with increasing pressure. --M 1 VVdPˆ ms˜ [41] kT ()11 In the present (c + 1) component system, we have (1 2 s + l) phases; a micellar solution phase and l crystalline ++()ccNda11M ln 1 t solid phases. In eq. [39], dCk is expressed in terms of (c s + 2 － l) independent variables, dT, dP and d ln aj ( j = Eq. [41] is available to define CMC of single compo- l + 1 → c). nent nonionic surfactant (11) (17) and to discuss its Simultaneously solving (c － l) equations of the dependency on temperature and pressure. s s form of eq. [40] with respect to dlnaj , dlnaj can be Eq. [41] can be used to analyze the light scattering t expressed in terms of dT, dP and dCj ( j = l + 1 → c). data of nonionic micellar solutions of single component s t Substituting dlnaj thus obtained in eq. [39], dCk can be by the use of a suitable light scattering equation (13), 2 expressed in terms of (c + 2 － l) independent experi- from which c1, cM,N1 andN1 are all obtained (20). t mental variables, dT, dP and dCj , T and P being both When a micellar solution coexists at equilibrium with t independent at experimentally given Cj . the crystalline solid of the solute surfactant 1, eq. [41] Number of independent variable (c + 2 － l) shows is rewritten by the use of the equilibrium condition [38] that Gibbs phase rule should hold for the multi-phase for i = 1 as follows. system including l crystalline solid phases and a micel- tsc1 2 mc dC =-cS˜ S+- cN Sˆ S dT- lar solution phase at complete equilibrium, at which 11111kT []()M ()11 s m holds e = 0 in addition to mi = mi = mi. 1 ˜ sc 2 mc cV11- V 1+- cNM 1 Vˆ11 V dP [42] Applying the phase separation model for micelle for- kT []()()

264 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) Thermodynamics of Solubility of Surfactants in Water

tsc1 ˜ dC1111=-cS() S + from which we obtain the temperature and pressure kT { dependencies of the solubility of one component non- cNS2 ˆmc- S+- NNS ms S˜ dT- ionic surfactant in water as the followings. M []}1 ()11 1222()

t 1 sc Ê ∂C1 ˆ 1 sc 2 mc ˜ [47] ˜ ˆ [43] cV11()- V 1+ Á ˜ =-[]cS11() S 1+- cNM 1() S11 S kT { Ë ∂TkT¯ P 2 mc ms˜ cNVM 1 ()ˆ11- V+- NNV 1222() V dP+ t []} Ê ∂C1 ˆ 1 sc 2 mc Á ˜ =-cV11˜ - V 1+- cNM 1 Vˆ11 V [44] Ë ¯ []()() s ∂ PkTT cNNdaM 12ln 2 It is most probably reasonable to assume that in aqueous solutions of nonionic surfactant t cM mc2 ˆ ms˜ dC212112=-[]NN() S S+- N() S22 S dT- ˜ sc mc kT SS11> SSˆ11> [45a, b] cM mc2 ˆ ms˜ [48] []NN12() V 1- V 1+- N 2() V22 V dP+ ˜ sc mc kT VV11> VVˆ11> [46a, b] 2 s t ()ccNda22+ M ln 2 Eq. [43] and [45] predict that ()∂∂CT1 > 0 , and 2 sinceN1 is large, once cM has increased to reach only a s small value with increased temperature (see appendix Eliminating dlna2 from eq. [47] and [48], and using the t 3･4), ()∂∂CT1 should increase from small positive approximation [26a, b], we obtain an approximate t t value to large positive value over a narrow temperature equation for dC1 written in terms of dT, dP and dC2 as range around a certain temperature TK. The narrow follows, range of temperature should be called critical micelliza- t 1 Ï ˜ sc tion temperature range (CMTR). dC111=-ÌcS()i S + t kT Ó Eq. [44] and [46] predict that ()∂∂CP1 < 0 , and that once cM has decreased to a small value with È ˘¸ t 2 mcNN12 ms˜ cNM 1 qq11 S- S 1+- 2 SSdT22 increased pressure (see app. 3･4), ()∂∂CP1 will fall Í () 2 ()˙˝ Î N1 ˚˛ from large negative value to small negative value over a narrow pressure range around a certain pressure PC. The Ï 1 ˜ sc narrow range of pressure may be defined as critical --ÌcV11()i V + [49] kT Ó micellization pressure range (CMPR), and PC is defined as critical solution pressure (2) (6) or critical micelliza- 2 È mcNN12 ms ˘¸ cNM 1 qq11 V- V 1+- 2 VV22˜ dP tion pressure. Í ()2 ()˙˝ Î N1 ˚˛ It can be seen from these that micelles will be stably formed in the temperature and pressure range where T cNNM 12 t > T and P< P (2). + dC2 K C ccN+ 2 The prediction is consistent with the experimental 22M behavior of the pressure dependence of the solubility of where we have 2 nonionic surfactant C12E6 in water (5). 2 cNNM 12 N1 q1 =-10> [50a] ccN+ 2 2･8 Solubility of Two Component Nonionic 22M Surfactant 2 cNM 2 q2 =-10> Firstly consider a mixed micellar solution of nonion- 2 [50b] ccN22+ M ic surfactant D1 and D2, coexisting only with the crys- c t talline solid D1. D2 is not always a surfactant but may Eq. [49] shows that once micelles are present, C1 be a solubilizates like alcohol. rapidly increases with the increase of T at given P and t Here eq. [39] and [40] are simplified to C2, and so on. Secondly consider a two component micellar solu-

265 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) M. Tanaka

tion of nonionic surfactant D1 and D2 both coexisting at ponent micelle of nonionic surfactant i(= 1, 2) at the c equilibrium with their respective crystalline solids, D1 same cM (20,21). c and D2. Thirdly we treat a case where nonionic surfactants

Using the equilibrium condition [38] here, eq. [35] D1 and D2 form mixed crystalline solid of constant can be rewritten for the present solution system as fol- composition. Here for simplicity we treat the solubility (c) lows. of one to one mixed crystals D12 . It is evident that the following conditions should be applied to the equations ts dC11=+ c dln a 1 [51] and [52].

2 tt tt cNM ()11 (cm ) () ( cm ) CC== dCi dC [55a, b] 1 DDSdTVdP- [51a] 12 2 kT [] This is applicable to the solubility of 1-1 ionic surfac- 1 sc 21() (cm ) =-cS˜ S+ cND S dT- kT []11() 1M 1 tant as will be mentioned next.

1 sc 21() (cm ) cV˜ - V+ cND V dP [51b] kT []11() 1M 1 2･9 Solubility of an Ionic Surfactant in Water Here we take a micellar solution of 1-1 ionic surfac- ts dC22=+ c dln a 2 tant D12 with an electrolyte X23 added which coexists at equilibrium with the ionic crystalline solid D (c). Here cNN 12 + M 12DD()22SdTVdP (cm )- () ( cm ) [52a] subscripts 1, 2 and 3 refer respectively to the surface kT [] active ion, counterion and the similion. 1 sc ()2 (cm ) =-cS˜ S+ cNND S dT- kT []22() 2M 12 It can be assumed that electro-neutrality should hold in the micellar solution and also in the coexisting ionic 1 sc ()1 (cm ) cV˜ - V+ cNNM D V dP [52b] kT []21() 1 12 crystalline solid. We have the equation of mass balance to show electro-neutrality

In these equations we have tt dC21=+ dC dCX [56]

()1 (cm ) Ê mNN12 mˆ Ê cNN12 c ˆ D SS=+ˆ1 SS21-+ S2 [53a] wher CX is the concentration of added electrolyte X23. Á 2 ˜ Á 2 ˜ Ë N1 ¯ Ë N1 ¯ The chemical potential mi of individual ion i and the equilibrium condition [38] is hypothetical but has dis- Ê 2 ˆ Ê 2 ˆ ()2 (cm ) mN2 ˆ m cN2 c D SS=+Á 1 SS21˜ -+Á S2 ˜ [53b] tinct theoretical meaning. Seeing from purely theoreti- Ë ¯ Ë ¯ NN12 NN12 cal view point, the same equations as eq. [51] and [52] are applicable to explain the behavior of solubility of D()1 V (cm )and D()2 V (cm )are given by S being replaced ionic surfactant. by V respectively in eq.[53a] and [53b]. Here it is most probably reasonable (17) (18) (19) to Very rough approximation obtains the following rela- assume in eq. [51b] that tions (17). ˜ sc ()1 (cm ) SS11->0 D S > 0 [57a, b] 2 2 NN12 N 1ªª= N2 NN12 N 2 N 1 b [54] sc ()1 (cm ) VV˜11->0 D V > 0 [58a, b] Then, we haveDD()VcmcmSS ( )ª () , DD()VcmcmVV ( )ª () (n Under the assumptions [57] and [58], eq. [51b] can = 1, 2) (see appendix 3･4). These may to rough approx- well explain the experimental behavior (1) (2) of the imation be interpreted as the mean entropy and volume temperature and pressure dependences of the solubility c changes on the dissolution of crystalline solids D1 and of the surface active ion 1 and of the ionic surfactant. c c D2 to form mixed micelles of composition b. The chemical potential m12, the partial molar entropy c c (c) If mixed micelles of nonionic surfactants 1 and 2 can and volume,`S12 and`V12, of the ionic crystal D12 which 2 2 be treated as perfect mixtures, c1, c2, NNN121,,, N2 are experimentally determinable can reasonably be and N1N2 of mixed micelles at given cM can be calculat- expressed as follows 00 0 2 ed from the values of cNii, and (Ni ) of single com-

266 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) Thermodynamics of Solubility of Surfactants in Water

()c cc mmm12 =+12 [59a] From eq. [65] we obtain at given CX

cccccc t SSSVVV12=+ 1 2 12 =+ 1 2 [59b, c] Ê ∂C112ˆ cc ˜ sc Á ˜ = ()SS12- 12 [66] Ë ∂T ¯ ()cckT12+ The Gibbs-Duhem equation for the macroscopic PC, X ionic crystalline phase with electro-neutrality should be t Ê ∂C112ˆ cc ˜ sc cc c Á ˜ =- ()VV12- 12 [67] 0 =-SdTV12 + 12dP - dm 12 [60] Ë ∂ P ¯ ()cckT+ TC, X 12 t t The equilibrium condition of experimental meaning In this region, we should have c1 = C1 and c2 = C2. should be From eq. [64] we obtain the equations at given CX

sssc t ddddmmmm12=+ 1 2= 12 [61] Ê ∂C ˆ cc ss () 112È SS˜ Á ˜ = Í()12- 12 + s Ë ∂T ¯ PC, ()cckT12+ Î Using eq. [6] for dmi (i = 1, 2) and eq. [60], the equilib- X rium condition [61] is rewritten as c c M NS21() (cm ) M NN()2 S (cm )˘ [68] 1 DD+ 12 ˙ ss˜ s c˜ s c c1 c2 ˚ kTdln a12 a=-() S 12 S12 dT--() V 12 V 12 dP [62] t Ê ∂C112ˆ cc È ss where have been used Á ˜ =- VV˜ - + Ë ∂ P ¯ ()cckT+ Í()12 12 TC, X 12 Î SSS˜˜˜sss=+ VVV˜˜˜ sss =+ 12 1 2 12 1 2 [63a, b] c c M NV21() (cm ) M NN()2 V (cm )˘ 1 DD+ 12 ˙ [69] From experimental view point, the equilibrium con- c1 c2 ˚ dition [62] and the equation of mass balance [56] should be applied to eq. [51a] and [52a]. The following and the equation of Clausius-Clapeyron type equation is resultantly obtained for the solubility of 1-1 P ionic surfactant in water. Ê ∂ ˆ = Ë ¯ t ∂ T CC1 , X

t dC1 = ˜ sccM 2 ()1 (cm ) cM ()2 (cm ) ()SS12- 12 ++NS1 DDNN12 S c c 1 2 [70] cc sccM 21() (cm ) 12 È ˜ c cm c cm SS12- 12 ++NS1 D ˜ sc M 2 ()1 ( ) M ()2 ( ) Í() VV12- 12 ++NV1 DDNN12 V ()cckT12+ Î c1 () c1 c2 c M NN()2 S (cm )˘ dT 12D ˙ c2 ˚ It is most probably reasonable to assume here that ˜ sc()n ( cm ) cc c SS12-> 12 0012D S >(,)n = [71a, b] 12 È VS˜ sc M NV21() (cm ) - Í()12- 12 + 1 D [64] ()cckT12+ Î c1 VS˜ sc->0012D()n V ( cm ) >(,)n = [72a, b] c 12 12 M NN (2 ))()VdPcm ˘ + 12D ˙ c2 ˚ Under the assumptions [71a, b], eq. [66] and [68] c well explain the experimental facts of the temperature - 2 dC t cc+ X effect on the solubility C1 of 1-1 ionic surfactants in 12 2 water (1), which show that since`N1 is large, once cM In the temperature and pressure region where has reached a small value with increase of temperature t micelles are absent (cM = 0) in solution, [64] should be (appendix. 3･4), ()∂∂CT1 rapidly increases from a rewritten to small positive value to large positive value over a nar- t row range of temperature. dC1 = Under the assumptions [72a, b], eq. [67] and [69] cc 12 SSdT˜ sc- --VVdP˜ sc [65] well explain the experimental behavior of the pressure ()cckT+ []()12 12() 12 12 12 effect on the solubility of ionic surfactants (1-4), which

c2 show that once cM decreases to a small value with - dCX t cc12+ increased pressure (appendix 3･4), ()∂∂CP1 falls

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rapidly from large negative value to small negative type relating TK to PC as follows. value over a narrow range of pressure. t Ê ∂PC ˆ It is seen from eq. [64] that the solubility C1 should = Ë T ¯ ∂ K C X decrease with increase of CX, the concentration of added electrolyte. È ˘ ()1 (cm ) NN12 ()2 ( cm ) N1 ˆsc ÍDDS + S ˙ --()SS12 12 It is experimentally known that CMC-Temperature b N2 N2 Î 1 ˚ 1 [77] and CMC-Pressure curves at given CX intersect respec- È ()1 (cm ) NN12 ()2 ( cm )˘ N1 sc tively with solubility-temperature and solubility-pres- DDV + V --VVˆ12 12 Í 2 ˙ 2 () sure curves at respective certain points. These points of Î b N1 ˚ N1 intersection have respectively been defined as the Krafft temperature TK and the critical solution (or micelliza- where b =`N2`/N1 is the mean degree of counterion bind- tion) pressure PC, the Tanaka pressure (2) (6) (7) (8). ing on micelles (see appendix 3･4). CMC has conveniently been defined as a total surfac- Assuming eq. [71] and [72], eq. [77] suggests that t tant concentration Ck of main micelle forming surfac- ()∂∂PTCK> 0, because the first terms have sufficient- t tant k at which the degree of micellization `(Nkcm/Ck ) ly large positive values compared with the second terms has a certain fractional value ak within the critical on the denominator and numerator. Experiments show micellization range (CMR) (4) (10) (17). that ()∂∂PTCK> 0 , in which PC increases almost lin-

Then it is reasonable to define TK and PC respectively early with increase of TK (2) (19). as a certain temperature and pressure at which Nc Ct = a within the critical micellization tem- ()km k k 3 Appendixes perature and pressure ranges. From eq. [51a] we obtain for the surface active ion 1 3･1Basic Equations of Small System Ther- modynamics of Micellar Solutions C t 1 dCln t = Small system thermodynamics (9) (10) should gener- c 1 1 ally give the following equations to a multi-component s cM 21() (cm ) () 1 ( cm ) daln 1 +-NSdT1 []DDVdP [73] micellar solution of surfactants, where solvent water 0 ckT1 t is now taken to a component of micelles. There have C1 dNcln 1 M = been used the same notations as in the reference (10). c1 t cM C1 21() (cm ) () 1 ( cm ) NSdT1 []DD- VdP[74] dGtt=- S dT + V t dP + ckT1 Nc1 M c c s s m m Subtracting [73] from [74], we obtain ÂÂmmei dNi ++i dNi dNM [A1] i==00i t c c C1 s s s m m ddalna11=- ln + GNti=+ÂÂmmei Ni i + NM [A2] c1 i==00i c c 1 21() (cm ) () 1 ( cm ) s s m m NSdT1 []DD- VdP [75] NdMttemm=- SdTVdP + -ÂÂ Nd ii - Ndi i [A3] NkT2 i==00i

t s m In the same way from eq. [52a], we obtain for the coun- Experimental variables are T, P and Ni (= Ni + Ni ). terion 2 at given CX The equilibrium condition should be

t c C2 s s m s ddalna 22=- ln + []dGt t =-Â()mmi i dNi += e dN 0 [A4] TPN,,i c2 i=0

1 ()22 (cm ) () ( cm ) NN12[]DD S dT- V dP [76] which should hold only if NkT2 s m mi = mi = mi e =0 [A5] Summing eq. [75] and [77], using eq. [62] for s s dlna1 a2 in the result and putting dlnak = 0 for given It should be noted here that our micellar solution is at s m fractional values of ak(k = 1, 2) at a point (TK , PC , complete equilibrium (11), in which not only mi = mi

CMC), we obtain an equation of Clausius-Clapeyron = mi but also the condition of subdivision equilibrium e

268 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) Thermodynamics of Solubility of Surfactants in Water

c m r = 0 holds. mmri=+Â Ni e [A13b] It has been known that solvent water 0 is contained in i=1 considerable amount in micelles (22). However, for c s s solutions sufficiently dilute in micelles, the presence of 0 =-SdTtt + VdP -Â N i dmmi -Â r Ndrr[A14] water 0 can to a good approximation be ignored in the i=0 thermodynamic equations of micelle formation (12) Here we have (14), as is shown bellow. s m t s mmMrr= Â r y [A15] Since N0 ≫ N0 =`N0NM and N0 ≈ N0 for solutions sufficiently dilute in micelles, we can have the following approximations at equilibrium of micelle formation. 3･2 Derivations of Eq. [2b] for mr, [12b] for

m mM and [15] for`G ss mm sÊ N ˆ ss NNmm+=+ NÁ1 0 ˜ mmª N [A6] Eq. [1a] and [1b] are reasonable expression of the 00 0 0 0Ë N s ¯ 000 0 chemical potentials of solvent water 0 and monomeric ss mm ss Nd00mmm+ª N 0 d 0 Nd 00 [A7] solute i in free solution to show respectively the devia- tions from Raoult’s law and Henry’s law (16) (17). Using [A6] and [A7], [A1] can approximately be Substituting eq. [1a] and [1b] in [A14] at given T and rewritten as follows, P, and assuming the following relations in the result

c s s dGtt=- S dT + VdP + 0 =+kTÂÂ xi dln ai r xr d ln ar [A16a] i=0 c c c ss s s m m s s mmme00dN++ÂÂi dNi i dNi + dNM [A8] 0 =+ÂÂxdi lnggi r xdr ln r [A16b] i==11i i=0 It should be noted here that the interactions of we obtain monomeric i of micelles with solvent water 0 may rea- xd kT a m Â r rr[]()m - ln rTP, = 0 [A17] sonably be treated as being implicitly included in mi (i

= 1 → c) Since xr are all independent, [A17] requires the follow-

The following equations can be derived from [A8]. ing equation for mr

m rr= GTP(),ln+ kTa r [2b] dGtt=- S dT + V t dP +

c c Substituting eq. [1a] and [1b] in [A11] at given T and s s m ÂÂmmi dNi ++MM dN m i NMi dN [A9] P, we obtain i==01i c c c s s m s s 0 =-kTÂÂ xi dln ai - xMM dmm + i xMi dN [A18] GNNti=+Â mmi MM [A10a] i==01i i=0 c The following equation can be assumed here. mme=+m N MiÂ i [A10b] c i=1 s s 0 =+Â xdi ln ai xMM d ln a [A19a] SdT VdP i=0 0 =-tt + - c xds s x d c c 0 =+Â i lnggi MM ln [A19b] s s m i=0 Â Ndi mmmi -+ NMM d Â iNdNMi [A11] i=0 i=1 from [A18] and [A19a], we obtain

The following equations are available to solution of c dkTam dN poly-dispersed micelles in good approximation (11) []()mmMM- ln TP, = Â i i [A20] (17). i=1

c which shows that (mM － kT ln aM) should be a function s s dGtt=- S dT + V t dP +Â mm i dNi +Â r rr dN [A12] of T, P and`Ni, say`G(T, P,`Ni). Then we resultantly i=0 c obtain from [A20] the following equations. s s GNti=+Â mmi Â r rr N [A13a] i=0 m MiM= GTPN(),,+ kT ln a [12b]

269 J. Oleo Sci., Vol. 54, No. 5, 259-271(2005) M. Tanaka

3 4Dependence of c on T and P in Satu- c ･ M m dG=++ SdT VdPÂ m i dNi [15] rated Micellar Solution i=1 m s Putting de = 0 and dmi = dmi = dmi in eq. [14] at the where we have defined the thermodynamic functions`S complete equilibrium of micelle formation, we obtain and`V as follows to good approximation

G ∂G Ê ∂ ˆ =-S Ê ˆ =-V [A21] -=kTdln a Ë ¯ Ë ¯ M ∂T PN,,ii∂ P TN c --()SkadTVdPNdln Mii+-Â m [A27] i=1 3･3 That gr = gM for all Mr under the Mean Reference Gibbs Free Energy`G(T, P, Using eq. [16] and [17], [A27] is rewritten to

`Ni) obeying Eq. [15]

In general, activity coefficient is determined by what -=kTdln aM to take the reference free energy`G(T, P,`Ni). It is rea- c c c ˜ m ˜ m sonable to take the mean reference free energy so as to -+-ÂÂNSii dT NViidP Â Ni dm i [A28] satisfy eq. [15], the equation of the same form as`G of i==11i i = 1 ideal non interacting micelles. where we have Subtracting [A14] from [A11], we obtain m m kT ∂ln a SS˜ =+ Ê M ˆ [A29] c i i Ë ¯ m N ∂T Py, r dyddNmmmMrr=+ÂÂr i i [A22] i=1

m m kT ∂ln a Substitution of eq. [2b] for m and [12b] for m in [A22] VV˜ =+Ê M ˆ [A30] r M i i N Ë ∂ P ¯ at given T and P results in Ty, r

c c dG kT dym dN NN []TP, =-Â r lngmrr +Â i i [A23] = Â i [A31] i=1 i=1 where have been used eq. [5] and dxlnMrr= Â r ydx ln . For simplicity we consider a micellar solution of sin- Taking reasonably eq. [15] for d`G, we obtain result- gle component 1-1 ionic surfactant coexisting with the antly from eq. [A23] crystalline solid at equilibrium. We have the following equilibrium conditions. Â r lng rrdy = 0 [A24] s c c c dddmmmii===-+i SdTVdPii i ;, =12 [A32]

Using dypr=-Â rpπ dy for arbitrary p, [A24] can be Suitably substituting [A32] for dmi in [A28], we rewritten as resultantly obtain

dy N11()cmN () cm Â rpπ ()lnggrpr- ln = 0 [A25] daln =-DDSdT VdP [A33] M kT kT

Since dyr(r ≠ p) are all independent, [A25] holds only if where we have ≠ gr = gp for all r( p). Using the definition of gM eq. [5] ()cm˜˜ m m c c DSSSSS=+12bb-+ 12 [A34] here, we obtain ()() ()cm˜ m m c c gr = gM, for all r [A26] DVVVVV=+()12bb-+() 12 [A35]

It should be noted here that [A26] does not always and b =`N2 `/N1 is the mean degree of counterion binding mean monodispersity of micelles in solution as Hall of micelles made from`N1 surface active ions 1. (12) suggested. We obtain from [A33]

Ê ∂ln aM ˆ N1 ()cm = DS [A36] Ë ∂T ¯ P kT

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canoate in Aqueous Solution, J. Solution Chem., Vol. 17, 125- Ê ∂ln aM ˆ N1 ()cm 137 (1988). =- DV [A37] Ë ∂ P ¯ T kT 7. S. KANESHINA and G. SUGIHARA, Progress Chem. Ind. (Korean Chem. Soc.), Vol. 15, 74-90 (1975). It can most probably be assumed that 8. E. KISSA, Surfactant Science Series, Vol. 50. Fluorinated Sur- ˜ m c ()cm ˜ m c ()cm SSii >=i (,),12 DS > 0 and VVi > i , DV > 0. factans, Synthesis ･ Properties ･ Application, Chapter 6 (1994). Then [A36] and [A37] show that when a micellar solu- 9. T.L. HILL, Thermodynamics of Small Systems, Part 1, 2, Ben- tion of ionic surfactants coexists with the ionic crys- jamin, New York (1963). 10. D.G. HALL and B.A. PETHICA, Nonionic Surfactant, Chapt. talline solid at equilibrium, aM and so cM should increase with increase of temperature at given pressure, 16 (M.J. SHICK, ed.), New York (1967). 11. D.G. HALL, Nonionic Surfactant, Chapt. 5 (M.J. SHICK, ed.), and decrease with increase of pressure at given temper- Dekker, New York and Basel (1987). ature. 12. D.G. HALL, Thermodynamics of Solutions of Ideal Multi-com- It can be predicted from eq. [22] that when an ionic ponent Micelles, Part 1, Trans. Faraday Soc., Vol. 66, 1351- micellar solution coexists with the ionic crystalline 1358 (1970). solid,`N1 and`N2 increase with temperature and 13. D.G. HALL, Thermodynamics of Solutions of Ideal Multi-com- decrease with pressure. ponent Micelles, Part 2, Trans. Faraday Soc., Vol. 66, 1359- 1368 (1970). 14. D.G. HALL, The Application of the Thermodynamic Theory of Acknowledgement Ideal Multi-Component Micelles to Ionic Micelles, Kolloid- Zeitschrift und Zeitschrift fur Polymere, Vol. 250, 895 (1972). The author expresses his sincere thanks to Dr. Prof. 15. D.G. HALL, Thermodynamics of Solutions of Polyelectrolytes, T. Tomida and G. Sugihara for their kind suggestions, Ionic Surfactants and Other Colloidal Systems, J. Chem. Soc., intensive discussions and strong encouragements for Faraday Trans. 1, Vol. 77, 1121 (1981). him to write the present paper. 16. M. TANAKA, Y. MURATA, Y. HOMMA and G. YANOS, Ther- It should be stressed here that the author could not modynamics of Solutions of Multi-component Micelles, Fuku- develop the theory in the present paper without making oka University Science Reports, Vol. 22, 175 (1992). 17. M. TANAKA, New Interpretation of Small System Thermody- reference to the excellent works of the late Dr. Denver namics Applied to Ionic Micelles in Solution and Corrin- G. Hall about the application of the small system ther- Harkins Equation, J. Oleo Sci., Vol. 53, 183-196 (2004). modynamics to micelles. 18. M. TANAKA, S. KANESHINA, K. SHIN-NO, T. OKAJIMA and T. TOMIDA, Partial Molal Volumes of Surfactants and its Homologous Salts Under High Pressure, J. Colloid & Interface References Sci., Vol. 46, 132-138 (1974). 1. Y. MOROI, Micelles: Theoretical and Applied Aspects, Plenum 19. M. TANAKA, S. KANESHINA, G. SUGIHARA, N. NISHIKI- Press (1992), New York and London. DO and Y. MURATA, Pressure Study of Surfactant Solutions, 2. M. TANAKA, S. KANESHINA, T. TOMIDA, K. NODA and K. Solution Behavior of Surfactants (K.L. MITTAL, ed), Vol. 1, p. AOKI, The Effect of Pressure on Solubility of Ionic Surfactants 41, Plenum Press, New York (1961). in Water, J. Colloid & Interface Sci., Vol. 44, 525-531 (1973). 20. M. OKAWAUCHI, M. SHINOSAKI, Y. IKAWA and M. TANA- 3. Y. MOROI, R. MATUURA, T. KUWAMURA and S. INOKU- KA, A Light Scattering Study on Micelle Formation of Nonionic MA, Anionic Surfactants with Divalent Gegenions of Separate Surfactants as a Function of Concentration and Pressure by Electric Charge: Solubility and Micelle Formation, J. Colloid & Applying Small System Thermodynamics, J. Phys. Chem., Vol. Interface Sci., Vol. 113(1), 225-231 (1986). 91, 109-112 (1987). 4. Y. MOROI, Y. MURATA, Y. FUKUDA, Y. KIDO, W. SETO and 21. M. TANAKA, M. SHIGEMATSU and K. GONDO, Light Scat- M. TANAKA, Solubility and Micelle Formation of Bolaform- tering of Two Component Nonionic Surfactant Solution, Yamada Type Surfactants: Hydrophobic Effect of Counterion, J. Phys. Conference XIX, Ordering and Organization in Ionic Solutions, Chem., Vol. 96(21), 8610-8613 (1992). 345-353 (N. ISE and I. SOGAMI, ed.), World Scientific, Singa- 5. N. NISHIKIDO, N. YOSHIMURA, M. TANAKA and S. pore, New Jersey, Hong Kong (1987). KANESHINA, Effect of Pressure on the Solution Behavior of 22. K. NAKASHIMA and K. TAKEUCHI, Water Content in Nonionic Surfactants in Water, J. Colloid & Interface Sci., Vol. Micelles of Poly (ethylene oxide)Poly (propylene oxide)Poly 78, 338 (1980). (ethylene oxide) Triblock Copolymers in Aqueous Solutions as 6. Y. IKAWA, S. TSURU, Y. MURATA, M. OKAWAUCHI, M. Studied by Fluorescence Spectroscopy, Applied Spectroscopy, SHIGEMATSU and G. SUGIHARA, A Pressure and Tempera- Vol. 55, 1237-1244 (2001). ture Study on Solubility and Micelle Formation of Perfluorode-

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