The 1987 study of Shinoda, Kobayashi, and Yamaguchi measured the critical micelle concentration (cmc) of lithium perfluorooctanesulfonate (LiFOS) between 30.5 and 183 °C by 19F NMR, extracted the enthalpy and entropy of transferring a monomer from a micelle into water with a correction for temperature-dependent counterion binding, and decomposed them into hydrogen-bond (mixing), electrostatic, and 'iceberg' (structured hydration) contributions, inferring that iceberg formation raises the cmc by roughly 3-7 times near 30 °C and persists well above 0 °C. The open problem is to establish a molecular-level, quantitative account of structured hydrophobic hydration - its microscopic structure, the temperature range over which it contributes to solution thermodynamics, and its transferability across solute classes (hydrocarbon chains, fluorinated chains, rare gases, ethylene oxide units, simple inorganic solutes) - sufficient to predict the temperature dependence of aqueous solubility and critical micelle concentrations, including the 30.5-183 °C LiFOS data. Later literature replaced the iceberg framework with quantitative statistical-mechanical theories validated for hydrocarbon- and noble-gas-like solutes and began resolving fluorinated hydration shells spectroscopically, but the microscopic structure and mechanism of the hydration shell, the temperature range over which structured hydration contributes, and the cross-class universality of any single quantitative balance remain unresolved.

Background

Hydrophobic hydration. When a nonpolar molecule - a hydrocarbon, a fluorocarbon, or a noble gas - dissolves in water, the process is strongly disfavored in free energy and anomalously temperature-dependent: dissolution carries a large positive heat capacity, so both its enthalpic and entropic components change strongly with temperature. To explain this, Frank and Evans (1945) proposed that water forms an 'iceberg' around nonpolar solutes: a locally more ordered, icelike hydration shell whose formation lowers the enthalpy but costs entropy. The word has survived as shorthand for hydration-shell structuring even though its literal meaning has been debated ever since.

Surfactants and the critical micelle concentration. Amphiphilic molecules such as detergents dissolve as single molecules (monomers) only up to a concentration called the critical micelle concentration (cmc), above which they self-assemble into micelles, aggregates that bury their oily tails away from water. The cmc therefore measures the free-energy penalty of exposing an apolar chain to water, and because hydrophobic hydration anomalies feed into that penalty, the cmc of ionic surfactants shows a peculiar nonmonotonic temperature dependence - typically a shallow minimum near room temperature followed by a rise at higher temperature. Counterions (here Li+) partially bind to the charged micelle surface; the degree of binding is captured by an empirical parameter KgK_g, defined by ln⁡(cmc)=−Kgln⁡Cg+const\ln(\mathrm{cmc})=-K_g\ln C_g+\mathrm{const} for added counterion concentration CgC_g, and it itself varies with temperature, which must be corrected for when converting the slope of cmc versus temperature into an enthalpy.

The source paper. Shinoda and co-workers had argued since the 1970s that the aqueous solubility and cmc of paraffin-chain compounds could be read as a thermodynamic balance between (i) a large positive enthalpy of mixing the apolar chain with water, (ii) electrostatic (headgroup and counterion) contributions, and (iii) 'iceberg' hydration, whose enthalpic gain and entropic cost nearly cancel near room temperature. In the 1987 paper, Shinoda, Kobayashi, and Yamaguchi chose lithium perfluorooctanesulfonate (C8F17SO3Li\mathrm{C}_8\mathrm{F}_{17}\mathrm{SO}_3\mathrm{Li}, LiFOS), a fluorinated surfactant stable enough to be studied to 183 °C, and measured its cmc from the break in the 19F NMR chemical shift of the CF3 (terminal trifluoromethyl) group versus molality: 0.0078 molal at 30.5 °C, rising to 0.082 molal at 183 °C. From the temperature dependence of the mole-fraction cmc, with the measured temperature-dependent KgK_g, they obtained the partial molal enthalpy and entropy for the transfer micelle to singly dispersed monomer, ΔHˉ2=1.9\Delta\bar{H}_2=1.9 kcal/mol and ΔSˉ2=6.27\Delta\bar{S}_2=6.27 cal/(mol K) at 30.5 °C, rising to 14.5 kcal/mol and 31.9 cal/(mol K) at 183 °C, and decomposed them as

ΔHˉ2=ΔHˉ2(H bonds)+ΔHˉ2(elec)+nΔHˉi\Delta\bar{H}_2=\Delta\bar{H}_2(\mathrm{H\ bonds})+\Delta\bar{H}_2(\mathrm{elec})+n\Delta\bar{H}_i ΔSˉ2=−R(1+Kg)ln⁡Xcmc+ΔSˉ2(H bonds)+nΔSˉi\Delta\bar{S}_2=-R(1+K_g)\ln X_{\mathrm{cmc}}+\Delta\bar{S}_2(\mathrm{H\ bonds})+n\Delta\bar{S}_i

where nn is the number of structured water molecules per solute. At 30.5 °C the iceberg residual amounted to nΔHˉi=12.4n\Delta\bar{H}_i=12.4 kcal/mol and nΔSˉi=34.9n\Delta\bar{S}_i=34.9 cal/(mol K); the ratio Tˉi=ΔHˉi/ΔSˉi≈355\bar{T}_i=\Delta\bar{H}_i/\Delta\bar{S}_i\approx 355 K suggested that the enthalpic and entropic consequences of iceberg formation nearly cancel. Omitting the iceberg term was estimated to lower the cmc by a factor of about 3-7 near 30 °C. The authors proposed that iceberg formation - attributed to dense, low-free-energy local packing of water around the solute, connected to the relatively high free energy of ordinary ice I (the familiar crystalline ice phase, ice I, that is stable at 0 °C) - can occur well above 0 °C for many solutes, including hydrocarbons, fluorocarbons, rare gases, ethylene oxide, and some inorganic compounds.

What the 1987 treatment is not. It is a macroscopic, residual-term analysis: 'iceberg' appears as a fitted contribution inferred by difference, not as a molecular structure; the temperature at which iceberg formation disappears is located by extrapolation rather than measurement; and the quantitative balance was calibrated on a single surfactant.

Modern context. Later statistical mechanics replaced much of this scaffolding. The information theory model of Hummer, Garde, García, Pohorille, and Pratt (1996) predicts cavity-formation and inert-gas hydration thermodynamics from just the density and the oxygen-oxygen radial distribution of water; Lum, Chandler, and Weeks (1999) made hydrophobicity explicitly length-scale dependent (small solutes are wrapped by straddling, structure-enhancing water, while extended surfaces drive dewetting); scaled particle theory - the classic framework computing the work of inserting a hard cavity of given radius into a solvent - was revived and calibrated against simulation by Ashbaugh and Pratt (2006); reviews by Southall, Dill, and Haymet (2002) and Dill, Truskett, Vlachy, and Hribar-Lee (2005) survey this shift. Within these frameworks the hydration entropy and enthalpy converge at characteristic 'entropy-convergence temperatures', the modern, molecularly grounded descendant of the 'iceberg disappearance temperature'. On the experimental side, Raman multivariate curve resolution (Raman-MCR) spectroscopy isolates the vibrational spectrum of water in the hydration shell of a solute and can directly test hydration-shell structure, including around micelles and around fluorinated groups. High-temperature solubility measurements of inert gases and methane in water (300-600 K) map the regime where the hydration anomalies fade, and direct molecular information about fluorinated hydration shells has only begun to appear (2019). The open question is whether these modern tools can deliver what the 1987 decomposition could only parametrize.

Problem Statement

The open question comprises three linked parts that jointly serve one objective - a molecular-level, quantitative account of structured ('iceberg'-type) hydrophobic hydration and its thermodynamic consequences:

(1) Microscopic structure and mechanism. Determine, at the molecular level, the structure, energetics, and dynamics of the hydration shell that forms around solutes in water - hydrocarbon chains, fluorinated chains (as in lithium perfluorooctanesulfonate, C8F17SO3Li\mathrm{C}_8\mathrm{F}_{17}\mathrm{SO}_3\mathrm{Li}), rare gases, ethylene oxide units, and simple inorganic solutes (small neutral molecules and simple electrolytes) - and establish or refute the mechanism proposed for it: dense, low-free-energy local packing of water around the solute that lowers both enthalpy and entropy. What observable signatures (for example, hydration-shell spectra, tetrahedral order - the degree to which the four neighbors of a water molecule occupy near-tetrahedral hydrogen-bond positions - local density, hydrogen-bond populations) distinguish structured hydration from ordinary solvation, and what do they show for each solute class?

(2) Temperature regime and its continuous evolution. Determine, from measurement or molecular modeling rather than long extrapolation, the temperature range over which structured hydration contributes to solution thermodynamics - that is, the range over which its signature remains resolvable above the uncertainty of the measurement or model used, with the temperature at which 'iceberg formation' effectively disappears understood as the point beyond which it is no longer resolvable - and how the hydration shell evolves continuously with temperature and with solution composition (for example, added salt, cosolutes, or concentration) for systems beyond the single surfactant studied in the source, at least one system from each solute class named in part (1). This includes the regime up to the 183 °C bound of the source data and the supercooled regime below it.

(3) Universality of the quantitative balance. Establish whether a single quantitative decomposition of solution and micellization thermodynamics into mixing (hydrogen-bond), electrostatic (including temperature-dependent counterion binding KgK_g), and hydration-structure terms - the balance calibrated on lithium perfluorooctanesulfonate between 30.5 and 183 °C - transfers across the other solute classes, and whether it predicts the measured temperature dependence of aqueous solubility and critical micelle concentrations. Where it fails, identify what replaces it and why (for example, class-specific water-fluorine interactions).

References

  1. Kozo Shinoda, Makoto Kobayashi, Nobuyoshi Yamaguchi. 'Effect of "Iceberg" formation of water on the enthalpy and entropy of solution of paraffin chain compounds: the effect of temperature on the critical micelle concentration of lithium perfluorooctane sulfonate.' The Journal of Physical Chemistry 1987, 91(20), 5292-5294. DOI 10.1021/j100304a031. https://doi.org/10.1021/j100304a031
  2. Henry S. Frank, Marjorie W. Evans. 'Free Volume and Entropy in Condensed Systems III. Entropy in Binary Liquid Mixtures; Partial Molal Entropy in Dilute Solutions; Structure and Thermodynamics in Aqueous Electrolytes.' The Journal of Chemical Physics 1945, 13, 507-532. DOI 10.1063/1.1723985. https://doi.org/10.1063/1.1723985
  3. Kozo Shinoda. '"Iceberg" formation and solubility.' The Journal of Physical Chemistry 1977, 81, 1300-1302. DOI 10.1021/j100528a016. https://doi.org/10.1021/j100528a016
  4. Kozo Shinoda. 'Characteristic property in aqueous solutions: effect of iceberg formation of water surrounding solute on the solubility (or cmc) and its peculiar temperature dependence.' Advances in Colloid and Interface Science 1992, 41, 81-100. DOI 10.1016/0001-8686(92)80008-L. https://doi.org/10.1016/0001-8686(92)80008-L
  5. Rosa Crovetto, Roberto Fernández-Prini, María Laura Japas. 'Solubilities of inert gases and methane in H2O and in D2O in the temperature range of 300 to 600 K.' The Journal of Chemical Physics 1982, 76(2), 1077-1086. DOI 10.1063/1.443074. https://doi.org/10.1063/1.443074
  6. Roberto Fernández-Prini, Rosa Crovetto, Maria L. Japas, Daniel Laria. 'Thermodynamics of dissolution of simple gases in water.' Accounts of Chemical Research 1985, 18(7), 207-212. DOI 10.1021/ar00115a003. https://doi.org/10.1021/ar00115a003
  7. G. Hummer, S. Garde, A. E. García, A. Pohorille, L. R. Pratt. 'An information theory model of hydrophobic interactions.' Proceedings of the National Academy of Sciences USA 1996, 93(17), 8951-8955. DOI 10.1073/pnas.93.17.8951. https://doi.org/10.1073/pnas.93.17.8951
  8. Ka Lum, David Chandler, John D. Weeks. 'Hydrophobicity at small and large length scales.' The Journal of Physical Chemistry B 1999, 103, 4570-4577. DOI 10.1021/jp984327m. https://doi.org/10.1021/jp984327m
  9. David Chandler. 'Interfaces and the driving force of hydrophobic assembly.' Nature 2005, 437, 640-647. DOI 10.1038/nature04162. https://doi.org/10.1038/nature04162
  10. Henry S. Ashbaugh, Lawrence R. Pratt. 'Colloquium: Scaled particle theory and the length scales of hydrophobicity.' Reviews of Modern Physics 2006, 78, 159-178. DOI 10.1103/RevModPhys.78.159. https://doi.org/10.1103/RevModPhys.78.159
  11. Noel T. Southall, Ken A. Dill, A. D. J. Haymet. 'A view of the hydrophobic effect.' The Journal of Physical Chemistry B 2002, 106, 521-533. DOI 10.1021/jp015514e. https://doi.org/10.1021/jp015514e
  12. Ken A. Dill, Thomas M. Truskett, Vojko Vlachy, Barbara Hribar-Lee. 'Modeling water, the hydrophobic effect, and ion solvation.' Annual Review of Biophysics and Biomolecular Structure 2005, 34, 173-199. DOI 10.1146/annurev.biophys.34.040204.144517. https://doi.org/10.1146/annurev.biophys.34.040204.144517
  13. E. Fisicaro, E. Pelizzetti, R. Bongiovanni, E. Borgarello. 'Perfluorinated surfactants: properties and applications I. Thermodynamic properties.' Colloids and Surfaces 1990, 48, 259-275. DOI 10.1016/0166-6622(90)80233-T. https://doi.org/10.1016/0166-6622(90)80233-T
  14. V. Tomašić, A. Chittofrati, N. Kallay. 'Thermodynamic properties of aqueous solutions of perfluorinated ionic surfactants.' Colloids and Surfaces A 1995, 104(1), 95-99. DOI 10.1016/0927-7757(95)03260-K. https://doi.org/10.1016/0927-7757(95)03260-K
  15. Alfredo González-Pérez, Juan M. Ruso, María J. Romero, Elena Blanco, Gerardo Prieto, Felix Sarmiento. 'Application of thermodynamic models to study micellar properties of sodium perfluoroalkyl carboxylates in aqueous solutions.' Chemical Physics 2005, 313(1-3), 245-259. DOI 10.1016/j.chemphys.2005.01.010. https://doi.org/10.1016/j.chemphys.2005.01.010
  16. Hong-Un Kim, Kyung-Hee Lim. 'A model on the temperature dependence of critical micelle concentration.' Colloids and Surfaces A 2004, 235(1-3), 121-128. DOI 10.1016/j.colsurfa.2003.12.019. https://doi.org/10.1016/j.colsurfa.2003.12.019
  17. Joshua A. Long, Blake M. Rankin, Dor Ben-Amotz. 'Micelle structure and hydrophobic hydration.' Journal of the American Chemical Society 2015, 137(33), 10809-10815. DOI 10.1021/jacs.5b06655. https://doi.org/10.1021/jacs.5b06655
  18. João R. Robalo, Louis M. Streacker, Denilson Mendes de Oliveira, Petra Imhof, Dor Ben-Amotz, Ana Vila Verde. 'Hydrophobic but water-friendly: favorable water-perfluoromethyl interactions promote hydration shell defects.' Journal of the American Chemical Society 2019, 141(40), 15856-15868. DOI 10.1021/jacs.9b06862. https://doi.org/10.1021/jacs.9b06862
  19. João R. Robalo, Denilson Mendes de Oliveira, Dor Ben-Amotz, Ana Vila Verde. 'Influence of Methylene Fluorination and Chain Length on the Hydration Shell Structure and Thermodynamics of Linear Diols.' The Journal of Physical Chemistry B 2021, 125, 13552-13564. DOI 10.1021/acs.jpcb.1c08601. https://doi.org/10.1021/acs.jpcb.1c08601
  20. Henry S. Ashbaugh. 'Reversal of the temperature dependence of hydrophobic hydration in supercooled water.' The Journal of Physical Chemistry Letters 2021, 12(34), 8370-8375. DOI 10.1021/acs.jpclett.1c02399. https://doi.org/10.1021/acs.jpclett.1c02399
  21. Claudio A. Cerdeiriña, Pablo G. Debenedetti. 'Water anomalous thermodynamics, attraction, repulsion, and hydrophobic hydration.' The Journal of Chemical Physics 2016, 144(16), 164501. DOI 10.1063/1.4947062. https://doi.org/10.1063/1.4947062