The hydrophobic effect

Current Opinion in Colloid & Interface Science 22 (2016) 14–22

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The hydrophobic effect Bengt Kronberg SP Technical Research Institute of Sweden, Dept. of Chemistry, Materials and Surfaces, P.O. Box 5607, SE-11486 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 18 January 2016 Accepted 7 February 2016 Available online 16 February 2016 Keywords: Hydrophobic effect Water structuring Cavity formation Entropy of transfer Enthalpy of transfer Free energy of transfer Enthalpy–Entropy compensation Solubility Micellization Protein unfolding

a b s t r a c t This review is a brief discussion on the development of the understanding of hydrophobicity, or the hydrophobic effect. The hydrophobic effect is primarily discussed in terms of partitioning of hydrocarbons between a hydrophobic environment and water as well as solubility of hydrocarbons in water. Micellization of surfactants is only briefly reviewed. It is emphasized that (i) the cause of the hydrophobic effect, e.g. the low solubility of a hydrocarbon in water, is to be found in the high internal energy of water resulting in a high energy to create a cavity in order to accommodate the hydrophobe, (ii) the “structuring” of water molecules around a hydrophobic compound increases the solubility of the hydrophobe. The “structuring” of water molecules around hydrophobic compounds is discussed in terms of recent spectroscopic findings. It is also emphasized that (iii) the lowering of entropy due to a structuring process must be accompanied by an enthalpy that is of the same order of magnitude as the TΔS for the process. Hence, there is an entropy–enthalpy compensation leading to a low free energy change for the structuring process. The assumption of a rapid decay of the entropy with temperature provides an explanation of the enthalpy–entropy compensation so often found in aqueous systems. It is also emphasized (iv) that the free energy obtained from partitioning, or solubility limits, needs to be corrected for molecular size differences between the solute and the solvent. The Flory–Huggins expression is a good first approximation for obtaining this correction. If the effect of different molecular sizes is not corrected for, this leads to erroneous conclusions regarding the thermodynamics of the hydrophobic effect. Finally, (v) micellization and adsorption of surfactants, as well as protein unfolding, are briefly discussed in terms of the hydrophobic effect. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The hydrophobic effect is manifested in a variety of situations involving amphiphilic compounds, such as the self-assembling into micelles in bulk and into monolayers at hydrophobic surfaces. It is also manifested in the unfolding of proteins and there are many other examples. The driving force is the poor interaction of hydrocarbons, or hydrocarbon moieties, with water, hence forcing the hydrocarbon moiety into a water-free environment. The cause of this poor interaction is the subject of this presentation. 2. Frank and Evans versus Shinoda and Hvidt Back in 1945 Frank and Evans suggested that the low solubility of hydrocarbons in water is due to “iceberg formation” (or ordering) of water molecules around the hydrocarbon [1]. Experimentally it has been found that the entropy change upon transferring a hydrocarbon from a nonpolar environment into water, Δw o S, is large and negative. (Here the index o represents “oil” i.e. a hydrophobic environment. This could be the liquid phase of the hydrocarbon itself, the interior of

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http://dx.doi.org/10.1016/j.cocis.2016.02.001 1359-0294/© 2016 Elsevier Ltd. All rights reserved.

a micelle or some other hydrophobic environment, see below.) This large and negative entropy of transfer was attributed to structuring of water molecules around the hydrophobic moiety. Since the enthalpy change, Δw o H, for the same process, is negligible at room temperature there is a large and positive free energy change, Δwo G, when a hydrocarbon moiety is introduced into water: w Δw o Gðlarge and positiveÞ ¼ Δo Hðsmall or zeroÞ

ð1Þ

−T Δw o Sðlarge and negativeÞ This large and positive free energy change implies a poor interaction of the hydrocarbon moiety with water. Frank and Evans therefore concluded that the poor interaction, or low solubility, of hydrocarbons with water is due to structuring of water (“iceberg formation”) around the hydrocarbon. When interpreting the temperature dependence of micelle formation a similar picture evolves (but now with opposite signs), i.e. there is a large and positive entropy of micellization accompanied with a small, or zero, enthalpy of micellization, resulting in a large and negative free energy of micellization. Here the small enthalpy change on micelle formation is often interpreted in terms of interaction of the headgroups of the amphiphile [2].

B. Kronberg / Current Opinion in Colloid & Interface Science 22 (2016) 14–22

In the late 70'ies Shinoda [3•,4], and later Hvidt [5], presented an alternative and astounding explanation of the low solubility of hydrocarbons in water. They showed that the formation of “icebergs” around a hydrocarbon moiety, i.e. water structuring, would increase the solubility in water and hence the low solubility needed an alternative explanation. Their analysis suggested that it is the high cohesive energy in liquid water that is the cause for the low solubility of hydrocarbons in water. The arguments are based on the graph shown in Fig. 1a, showing the solubility in water as a function of the inverse of temperature for three hydrocarbons. According to the van't Hoff equation the slope of the solubility versus 1/T gives us the enthalpy, ΔH, accompanied with the mixing of the two liquids: d ln x2 ΔH ¼− R dð1=TÞ

ð2Þ

where x2 is the solubility limit expressed in mole fraction. This slope is normally a straight line (Fig. 1b) since the heat of solution does not change appreciably with temperature [6]. Indeed the solubility of hydrocarbons in water shows such a pattern, but only at high temperatures. The deviation from the straight line increases dramatically at lower temperatures and this deviation is attributed to the structuring of water around the hydrocarbon molecule (Fig. 1a). A conclusion from the figure is then that the structuring of water (iceberg formation) increases the solubility, as indicated with the arrows in the figure. Hence the cause of the poor solubility of hydrocarbons in water needs an alternative explanation. 3. On the terminology “water structure” or “ordering of water” When Frank and Evans coined the term “iceberg”, they referred to a microscopic region around a hydrophobic molecule, in which “water molecules are tied together in a quasi-solid structure”. Later, the word “iceberg” began to be taken literally, i.e. water molecules were assumed to form ice-like (tetrahedrally ordered) structures around hydrophobic molecules. However, this picture is not supported by experiments: Neutron scattering shows that there is hardly any difference in the state of water molecules when a hydrophobe is introduced [7]. On the other hand molecular dynamics calculation indicate that the structural changes of water in the vicinity of small nonpolar solutes cannot be deduced from the water radial distribution functions, explaining why this increased ordering is not observed through neutron diffraction experiments. The molecular dynamics study shows a slower translational and re-orientational dynamics of water near hydrophobic groups, resembling water at low temperatures [8]. However, this issue is still under debate [9].

15

Femtosecond mid-infrared spectroscopy shows that hydrophobic groups are surrounded by water molecules that display much slower orientational dynamics than the bulk liquid and that are therefore effectively immobilized [10]. Bakulin [11••] showed, using two-dimensional IR spectroscopy combined with molecular dynamics simulations, that water molecules in the hydrophobic solvation shell do not exhibit an increased tetrahedral ordering compared with the bulk. These water molecules are not ice-like frozen since they have librational degrees of freedom and can even rotate. However, the rate of the reorientation is dramatically decreased near the hydrophobic groups due to a substantial decrease in the water translational mobility. Hence, the hydrogenbond network around hydrophobic groups is not more rigid, or icelike, compared with the bulk but that the hydrogen bond dynamics is different. This slowdown has further been investigated through a combination of molecular dynamics simulations with mid-infrared pump–probe spectroscopy [12]. The authors show that H-bond breaks through two competing mechanisms: The first is switching through an associative partner exchange and the second though a dissociative breaking, characterized by an unbound state. The first mechanism is shown to occur less often near a hydrophobic surface, thus creating OH groups that do not switch, causing a distinct slower timescale in the reorientational dynamics. As will be discussed further below there are two issues that are fundamental for the hydrophobicity or the hydrophobic effect. The first is that the hydrophobic effect is proportional to the surface area of the hydrophobic moiety (see Fig 5c) and the second is that the entropy due to this “structuring” rapidly decreases with temperature (see Fig. 2a). We will here for the sake of simplicity use the terminology “water structuring” for the state of water with low entropy in the vicinity of a hydrophobic molecule, even though this is not defined here on a molecular level. 4. Ordering of the water leads to an enthalpy–entropy compensation Patterson and Barbe [13••] showed that a change in entropy due to any ordering, or structuring, process must be accompanied by a change in enthalpy. This is easily realized considering that a change in structuring should also be reflected by a change in the heat capacity. The relation between these three entities is: −T

ws ws dΔw dΔw ws oS oH ¼ Δw o Cp ¼ dT dT

ð3Þ

Here the index ws stands for water structuring. If we make the plausible assumption that the structuring of water around a hydrophobic moiety decreases rapidly with temperature, and eventually vanishes

Fig. 1. (a) The solubility of three hydrocarbons, expressed as log x2, in water as a function of 1/T, illustrating that only at high temperatures we obtain the expected straight line according to Eq. (1). At lower temperatures the structuring of water increases the solubility i.e. going from the dashed lines to the full drawn lines (redrawn from Ref. [3•]). (b) Solubility of iodine in some solvents, expressed as log x2, versus the inverse of the temperature showing the expected straight line (redrawn from Ref. [6]).

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Fig. 2. (a) The entropy for a process where structuring is involved plotted versus the absolute temperature (redrawn from Ref. [13••]). Experimental points represent the entropy of transfer of toluene from a hydrophobic environment (the liquid polymer polydimethylsiloxane) into water [14]. The full drawn curve through the points is a fit using Eq. (6) with a fitted value of τ = 60 K. (b) Temperature dependence of the thermodynamic parameters for the same process.

The shaded area, i.e. the rectangle in Fig. 2a, is equal to T Δwo Sws. Hence, the enthalpy of the ordering process is equal to the total area, i.e.:

seen in Fig. 2b the enthalpy and entropy are similar in magnitude resulting in a small free energy. This is realized from the Eqs. (6) and (7), since the constant τ is a small number (ca. 60–70 K). Thus, plotting Δwo Hws versus T Δwo Sws gives a slope slightly larger than unity. Hence, the assumption of an exponential decay of the entropy with temperature provides an explanation of the enthalpy–entropy compensation so often found in aqueous systems. From the equations above we also have an understanding of the heat capacity Δwo Cws p associated with the hydrophobic effect. It is of opposite sign of the entropy and has only a weak temperature dependence.

ws ws ws ¼ T Δw þ Δw Δw oH oS oG

5. Comparing with experimental results

at high temperatures, we will have the situation depicted in Fig. 2a showing the entropy of transfer, Δwo Sws, as a function of temperature. The hatched area under the Δwo Sws curve in Fig. 2a is equal to Δwo Gws according to: ws ¼ Δw oG

Z

∞ T

ws Δw dT oS

ð4Þ

ð5Þ

From the sizes of the areas in Fig. 2a we have for an ordering process, or structure formation, the following picture: T Δwo Sws is large and negws ative, Δw is large and negative and Δw oH o G is small and negative. All three quantities go rapidly towards zero with increasing temperature (Fig. 2b). Hence, we realize that the original assumption that the enthalpy of water structuring around a hydrophobe can be ignored is wrong. Indeed the enthalpy is of the same order of magnitude as T Δwo Sws for the water structuring process. As is realized from an inspection of Fig. 2b, the structuring of water around a hydrophobic moiety is associated with a negative free energy change. This implies that the structuring of water increases the solubility of the hydrophobe, as was pointed out by Shinoda and Hvidt [3•–5]. Barbe and Patterson [15,16•] suggest that the structuring process might decrease in magnitude with T in an exponential fashion, i.e., ws TΔw ∝ ‐ T e−T=τ oS

ð6Þ

ws Δw ∝ ‐ ðT þ τÞ e−T=τ oH

ð7Þ

ws Δw ∝ ‐ τ e−T=τ oG

ð8Þ

ws Δw o Cp ∝

T −T=τ e τ

ð9Þ

The curve in Fig. 2a is a fit using Eq. (6) and the points represent the entropy of transfer of toluene from a hydrophobic environment (the liquid polymer polydimethylsiloxane) into water [14]. The good fit suggests that the assumption of an exponential decay of the entropy, Eq. (6), is reasonable. The relations above give an understanding of the enthalpy–entropy ws ws compensation phenomenon since Δw = (1 + τ/T) T Δw oH o S . As

When we compare the above thermodynamic functions with experimental results we find that, independent of temperature, the experimental enthalpy of transferring a hydrophobic moiety into water, Δwo Hexp, differs by a constant value compared with that calculated for the structuring of water, Δwo Hws (Fig. 3). This discrepancy is attributed to the energy required to form a cavity in the water, Δwo Hcavity, in order to accommodate the hydrocarbon. Transferring a hydrophobic moiety into water requires the formation of a cavity in the water in order to accommodate the hydrophobe (Fig. 4). This involves breaking, or bending, of hydrogen bonds and this in turn is associated with a large and positive enthalpy. The entropy contribution to this process is negligible and hence the free energy of cavity formation is also large and positive: exp ws cavity Δw ¼ Δw þ Δw oH oH oH

ð10aÞ

exp ws cavity ¼ Δw þ Δw Δw oG oG oG

ð10bÞ

exp ws ¼ Δw Δw oS oS

ð10cÞ

exp ws Δw ¼ Δw o Cp o Cp

ð10dÞ

The enthalpy of cavity formation can be split up in two contributions. The first contribution is only dependent on the size of the molecule being transferred into water, irrespective of whether or not the molecule is a hydrophobe (first arrow in Fig. 4). The second contribution (second arrow in Fig. 4) involves insertion of the hydrophobe into the cavity and hence encompasses dispersion forces between the hydrocarbon and the water molecules. For hydrophilic molecules energy is regained because there are possibilities for the inserted molecule to interact with the water, by for example hydrogen bonding or cavity dipole–dipole interaction, hence lowering Δw . This ability to oH

B. Kronberg / Current Opinion in Colloid & Interface Science 22 (2016) 14–22

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w w Fig. 3. Schematic drawing of the thermodynamic parameters T Δw o S, Δo H and Δo G for the transfer of a hydrophobic molecule from a nonpolar environment into water; (a) Contribution due to water structuring. (b) Contribution due to cavity formation in water and (c) the sum of the two contributions, which corresponds to experimental results. Note that in (c) the enthalpy contribution is around zero at room temperature, which is the situation described by Eq. (1) (redrawn from Ref. [17]).

lower the energy is absent for hydrophobic molecules, giving rise to the poor interaction with water, with low solubility as a consequence. This is the mechanism behind the hydrophobic effect. The creation of a cavity requires a large and positive enthalpy (Fig. 3b). The existence of a cavity contribution is only an interpretation of the thermodynamic analysis. This idea can be extended in models where the water molecules are involved in two stages, relaxed (low energy) and un-relaxed (normal water) [18] or three stages, clustering water, free water and structured water [19]. However, it is beyond the scope of this presentation to review various models leading to the hydrophobic effect. Comparing Eqs. (10c) and (10d) with Eqs. (6) and (9) we draw the following conclusions: First, these equations explain that if Δwo Sexp for a series of solutes is plotted against Δwo Cexp p at one experimental temperature, a straight line is obtained with a slope giving the constant τ. This was first observed by Sturtevant [20], who found that the ratio, Δwo Sexp/ Δwo Cexp p , for transfer of a series of hydrocarbons at 25 °C was nearly the

exp exp same for all hydrocarbons. Second, we note that Δw and Δw o Cp oS have different signs and, remembering that τ is a small number (60–70) compared to T, Δwo Cexp is large in magnitude. Finally, we recogp nize that in a small temperature interval the Δwo Cexp term does not vary p exp considerably compared to Δw . This is confirmed experimentally oS and has led many researchers to wrongly assume that Δwo Cexp is indep pendent on temperature.

6. How to extract thermodynamic information from experimental data We now consider the simplest case, which is the transferring of a hydrophobic molecule, or moiety, from a hydrophobic environment into water at infinite dilution. The hydrophobic environment could be the pure liquid of the hydrophobe, a dilute solution in another hydrophobic liquid, the interior of a micelle, or the interior of a monolayer (or bilayer) of an adsorbed surfactant. It could also be the interior of

Fig. 4. Illustration of the two contributions to the transfer of a hydrocarbon into water. The cavity formation contribution is the creation of a cavity in the water and insertion of the hydrocarbon molecule into the cavity. The water structuring contribution is the relaxation of water molecules around the hydrocarbon into a more ordered state. The same principles apply to any hydrocarbon moiety, for instance the hydrocarbon tail of a surfactant (redrawn from Ref. [17]).

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a protein, having conformations such that the hydrophobic parts are hidden from the water. Hydrophobicity is here defined as the free energy change upon this transfer process.

The standard molar Gibbs free energy of transfer between the oil and the water phases is defined in terms of the difference between the interaction parameters:

6.1. Thermodynamic background

exp Δw oG ¼ χ1;w −χ1:o RT

The partitioning of a molecule (Component 1) between an aqueous phase (w) and a hydrophobic, or oil, phase (o) is related to the standard molar Gibbs free energy of transfer from phase (o) to phase (w), Δwo G, through: exp Δw o o w oG ¼ ln γw 1 − ln γ1 ¼ ln c1 − ln c1 RT

ð11Þ

Here c is the concentration and γ the activity coefficient of the molecule in the respective phase. The use of Eq. (11) deserves some discussion. The activity coefficient, and hence Δwo Gexp, does not only reflect the difference in molecular interaction but also a difference in molecular size of the components. Using mole fractions as concentration variables in Eq. (11) neglects the molecular size difference and hence the effect of molecular size difference shows up in the interaction part and obscures the interpretation of the hydrophobic effect. There is therefore a need for an alternative expression. The Flory–Huggins theory offers an expression of the activity coefficient taking into account the effect of molecular size differences. The Flory–Huggins expression for the free energy has previously been used for the transfer of hydrocarbons into water giving a free energy of transfer that is only dependent on the surface area of the hydrophobe [18,21] (see below). Hence the Gibbs free energy of transfer can be split up into a combinatorial and an interaction contribution. It is only the interaction contribution that is a measure of the strength of the hydrophobic interaction. The use of the Flory–Huggins expression of the combinatorial entropy has been scrutinized since it is believed to overestimate the effect of the relative size of the molecules giving a too large entropy contribution (by ca. 10%) [22]. However, it has been found that the use of the Flory– Huggins expression is a reasonable first approximation to express the effect of mixing molecules of different size and that it is a better approach than using the mole fractions in Eq. (11), which neglects the differences in molecular sizes. It might be argued that the Flory–Huggins expression is based on a lattice theory and is therefore only valid for chain like polymers. However, Tompa [23] has demonstrated that the Flory–Huggins expression can be deduced without the assumption of a lattice model and hence the Flory–Huggins expression can be considered as an acceptable first approximation for a correction for the molecular size difference. Hence, using the Flory–Huggins expression creates a free energy that is only dependent on the solute/solvent contact area. In the Flory–Huggins theory the chemical potential of a molecule (Component 1) in solution with Component i as the solvent (could be either w or o) has the following expression:   μ i1 − μ •1 V1 ¼ ln a1 ¼ ln ϕ1 þ 1− ð1−ϕ1 Þ þ χ1;i ð1−ϕ1 Þ2 RT Vi

ð12Þ

Here ϕ1 is the volume fraction of the hydrocarbon moiety, V1 and Vi the molar volumes of the hydrocarbon moiety and the solvent, respectively, χ1,i the interaction parameter between the two components and μ•1 the chemical potential of the pure Component 1. Equating the chemical potential of Component 1 in the water and oil phases gives at infinite dilution:

w

0 ¼ ln xo1 − lnx1 − ln

  Vo V1 V1 − − þ χ1;o −χ1;w Vw Vo Vw

ð13Þ

ð14Þ

and hence (still at infinite dilution):   exp xo Δw Vo V1 V1 oG ¼ ln w1 − ln − − RT x1 Vw Vo Vw

ð15Þ

Using Eq. (15) gives a free energy of transfer that is free from any combinatorial contributions originating from molecular size differences. We are now in a position to apply these equations to the processes associated with the hydrophobic effect. Below we will present analyses of partitioning of a hydrophobic solute at infinite dilution and solubility of hydrocarbons in water. We emphasize that the entropy (non-combinatorial) of transfer, obtained from the temperature dependence of the free energy calculated from Eq. (15), asymptotically approaches zero at higher temperatures (Fig. 2a). If an appropriate correction to entropy of transfer is not performed, then the entropy becomes zero at a certain temperature, Ts, and positive above this temperature. The significance of Ts has in literature wrongly been discussed in terms of the temperature at which structuring of water around a hydrophobe ceases to exist [24,25•]. It has been shown that this is an artifact due to not correcting for the combinatorial entropy [26]. It was shown that Ts is the temperature at which the (positive) combinatorial contribution to the entropy of transfer and the (negative) contribution from the water structuring balance each other. The temperature at which the structuring of water ceases (or at least is not detectable) is when the heat capacity of transfer is zero and this occurs around 150 °C [26]. 6.2. Application to the partitioning of hydrophobic molecules between an organic phase and water The relation between the partitioning of a hydrophobic molecule between water and a hydrophobic environment (oil) is given by Eq. (15). Using liquid–liquid chromatography very accurate measurements of the partitioning of hydrocarbons between a hydrophobic stationary phase and water can been obtained [27•]. The relation between the partition coefficient and the retention volumes and the non-combinatorial free energy of transfer into water is: exp Δw oG RT

p

¼ ¼

w

Cp1 V1 V1 þ ln w þ Vw Cw V 1 1 p w R MK V −V V1 V1 ln þ ln w þ P V V1 Vw

ln

ð16Þ

Here Cp1 and Cw 1 are the molar concentrations of the probe in the stationary and mobile phases, respectively. VR, VMK and VP are the retention volume of the hydrophobic molecule, the retention volume of a non-interacting marker and the retention volume of the stationary phase, respectively. The latter is a nonpolar liquid polymer (polydip

w

methylsiloxane). V1 and V 1 are the partial molar volumes of the hydrophobic molecule at infinite dilution in the stationary phase (polymer) and water phase, respectively. Fig. 5a shows the specific net retention volume, i.e. (VR − VMK)/m, where m is the total mass of the stationary polymer phase (polydimethylsiloxane) versus temperature for toluene and ethylbenzene. Significant is the maximum occurring around room temperature. This corresponds to a minimum in solubility in water of the probes. Poor interaction of the probe with water “pushes” the probe into the stationary phase, hence increasing the retention volume.

B. Kronberg / Current Opinion in Colloid & Interface Science 22 (2016) 14–22

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Fig. 5. (a) The specific retention volume of toluene and ethylbenzene as a function of temperature (redrawn from Ref. [14]). (b) Non-combinatorial free energy of transfer from a nonpolar liquid to water expressed per unit surface area of four different alkylbenzenes (benzene to propylbenzene) as a function of temperature (redrawn from Ref. [18]). (c) The extracted transfer functions for the transfer of toluene from PDMS into water as a function of temperature (redrawn from Ref. [14]).

exp From the temperature dependence of Δw , both the enthalpy, oG exp 2 exp w exp Δw H = − T d (Δ G /T)/d (1/T), and the entropy, Δw = o o oS exp − d (Δw ) / dT, of transfer can be obtained. If the measurements oG exp are very accurate also the heat capacity of transfer, Δw = o Cp exp d (Δw H )/dT, can be extracted. o Using Eq. (16) and expressing the free energy of transfer per unit surface area of the hydrophobic molecule produces a master curve for a series of alkylbenzenes (Fig. 5b). This unique curve is the result of having subtracted the combinatorial part from the experimentally obtained Gibbs free energy of transfer. Hence the entropy of transfer derived from this curve is also non-combinatorial and solely a function of the structuring of water around the hydrophobe. The same results have been obtained from direct measurements of the partitioning of benzene between water and alkanes [21]. The resulting thermodynamic transfer functions are shown in Fig. 5c for the transfer of toluene, displaying a large and negative entropy of transfer (as expected from the above discussion). The results also show that the enthalpy of transfer is negative at low temperatures, zero at around 20 °C and positive at higher temperatures. This is in line with the expected results outlined above (Fig. 3c).

from a van't Hoff plot is highly correlated [28,29]. Enthalpy–entropy compensation plots should therefore never be used when the entropy and enthalpy are obtained from the same set of data. Several methods for finding a real compensation effect have been suggested in the literature [30]. One of these methods, adopted here, is plotting the enthalpy versus the free energy. 7. Solubility data At the solubility limit of a hydrocarbon in water the hydrocarbon is partitioned between the aqueous phase, which is assumed to be at infinite dilution in water, and its own liquid, which is considered to be a pure liquid, free from water. Eq. (13) is thus written as: 0 ¼ ln xw 1 þ ln

  V1 V1 þ 1‐ þ χ1;w Vw Vw

ð17Þ

From Eq. (14) we hence find:    V1 V1 exp ¼ RT χ1;w ¼ − RT ln xw þ 1− Δw 1 þ ln oG Vw Vw

ð18Þ

6.3. Finding the cavity contribution The next step is to find the two contributions to the transfer functions, viz. the cavity formation and the water structuring contribution. This can be accomplished by assuming that (i) the water structuring is negligible at high temperatures and that (ii) the Gibbs free energy of cavity formation is temperature independent. Thus, extrapolation of exp exp the points in a graph where Δw is plotted versus Δw to the oH oG w exp w exp w exp Δo H = Δo G line, i.e. when Δo S = 0, gives us the Gibbs cavity cavity free energy of cavity formation: Δw = Δw . Such a plot is oG oH shown in Fig. 6a for the transfer of toluene and ethylbenzene. The resulting transfer functions for toluene are shown in Fig. 6b where the contributions from the water structuring, as was schematically illustrated in Fig. 3a, are plotted. Notice the enthalpy (Δwo Hws)–entropy ws (TΔw o S ) compensation resulting in a small free energy of transfer w ws (Δo G ) (cf. Fig. 2b). It is tempting to plot the enthalpy versus the entropy of transfer in order to find the cavity contribution when the entropy of transfer is zero. However such plots are commonly linear due to compensation of errors, i.e. the uncertainty in the estimates of enthalpy and entropy

Eq. (18) describes the Gibbs free energy of transfer from the pure liquid to water and from the temperature dependence the entropy and enthalpy of transfer are obtained as described above for the partitioning process. We are now in a position to be able to understand the temperature dependence of the solubility of hydrocarbons in water: First we exp recognize that Δw is the sum of the cavity contribution and the oG water structuring contribution, Eq. (10b). Second we write Eq. (8) as ws Δw (= − A τ e− T/τ), where A is a constant. Then we can write oG the following expression for the solubility of a hydrophobic compound in water: ln xw 1 ¼−

   1  w cavity V1 V1 Δo H −A τ e−T=τ − ln − 1− RT Vw Vw

ð19Þ

cavity and τ are constants. where A, Δw oH Fig. 7a shows the solubility of toluene as obtained from Ref. [31]. The minimum in solubility at about room temperature is a result of the two terms which contribute oppositely to the hydrophobic effect, i.e. (i) the

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Fig. 6. (a) Enthalpy of transfer versus the Gibbs free energy of transfer for toluene and ethylbenzene as obtained from the data in Fig. 5a. The crossing with the Δwo Hexp = Δwo Gexp line gives cavity the cavity contribution, Δw . (b) The different contributions to the transfer functions of toluene for the transfer from a nonpolar environment to water as a function of temperature oH (both figures are redrawn from Ref. [14]).

formation of a cavity in the water and (ii) the structuring of water molecules in the vicinity of the hydrocarbon surface, Eq. (19). Such a minimum in solubility is general for all nonpolar hydrocarbons in water. The full drawn line in Fig. 7a is the result of a fitting using Eq. (19), 7.1. Why is it that the minimum in solubility occurs at the same temperature for different alkanes? In analyzing the solubility of hydrophobic compounds it turns out exp that the minimum in solubility, i.e. when Δw = 0 occurs at the oH same temperature, Tmin, for all compounds, viz. at 20–25 °C. At a first glance this might be surprising, but taking a closer look at the consequences from what has been outlined above it is to be expected: at exp the point of minimum solubility the experimentally obtained Δw oH is zero and hence we have:

cavity ws Δw ¼ − Δw oH oH

ð20Þ

If we assume that Δwo Hws obeys Eq. (7) and that Δwo Hcavity is independent of temperature we have: cavity Δw ¼ A ðTmin þ τÞe−Tmin=τ oH

ð21Þ

Here A is a constant that is proportional to the surface area of the cavity hydrophobe. Since the enthalpy of cavity formation, Δw , also oH should be proportional to the surface area, we realize that the expression in Eq. (21) is independent of the size of the hydrophobic moiety and hence Tmin is the same for all molecules having the same chemistry, i.e. the same interaction with water, such as aliphatic hydrocarbons. Hence, the temperature at which the solubility is at minimum, Tmin, is expected to be a universal temperature for all aliphatic hydrocarbon moieties. Note that the entropy of transfer only asymptotically approaches zero as the temperature is raised. Hence, there is no temperature at which the entropy crosses the zero entropy line. This is a result of correcting for the combinatorial entropy due to different sizes of the molecules as discussed above.

Fig. 7. (a) Temperature dependence of the solubility of toluene in water (data from Ref. [31]). The full drawn line is a fitting using Eq. (19). (b) The temperature dependence of the CMC values of three alkyl (C10, C12 and C14) trimethylammonium chlorides. (Redrawn from Ref. [32].) The upswing at lower temperatures in both graphs is due to the structuring of water around the hydrophobe.

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8. Micellization Formation of micelles, as well as of liquid crystalline phases, is typical consequences of the hydrophobic effect. Realizing that water behaves as a normal liquid only at high temperatures we can understand the temperature dependence of micellization of ionic surfactants. Non-ionic surfactants based on a polyoxyethylene chain have a strong temperature dependence emanating from this chain and hence are more complex in their analysis. We shall therefore restrain this short discussion to ionic surfactants. Thermodynamically, the micellization process can simply be viewed as a phase separation. However, in applying Eq. (18) to micellization of ionic surfactants we also need to consider the counterions. Hence, using the phase separation model and taking the molecular size difference into account the free energy of transfer from water to the micelle becomes [33]: Δw o G ¼ ð2−αÞRT χ1;w   V1 V1 ¼ − ð2−αÞRT ln xCMC þ ln þ 1− Vw Vw

ð22Þ

Here Component 1 is the surfactant, xCMC is the critical micelle concentration expressed in mole fraction and α is the degree of dissociation. In the derivation of Eq. (22) it is assumed that the micelle interior is free from water. Fig. 7b shows the temperature dependence of the critical micelle concentration, CMC, of three alkyltrimethylammonium chlorides (C10, C12 and C14) [32]. Note the striking similarity in the temperature dependence with the data obtained from the solubility data of toluene. We note with interest the surfactants do not display a minimum at the same temperature as is the case with aliphatic hydrocarbons. The reason for this is most likely due to that not all the hydrocarbon moiety is completely transferred into a non-polar environment [32]. A part of the surfactant hydrophobe in the micelle is exposed to water. This is especially so for more hydrophilic surfactants and for surfactants where the polar part is not attached at the end of the alkyl chain [34]. Also it is found that for surfactants with very low CMC the interior of the micelle is more hydrophobic compared to the micelles of similar surfactants with higher CMC [35]. These factors most likely alter the temperature at which there is a minimum in the CMC as a function of temperature. We note that the existence of a minimum of the CMC with temperature is general for ionic surfactants. This has recently been reported for anionics [36], cationics [32] and Gemini surfactants [37]. Also alkyl glycosides display a minimum in CMC with temperature [38,39]. The general thermodynamic analysis of micelle formation then follows the same thread of thought as outlined above for the dissolution of hydrocarbons in water (but with a reversed sign). 9. Concluding remarks We have discussed the hydrophobicity, or the hydrophobic effect, in terms of transferring hydrophobic moieties from a hydrophobic environment into water. Hydrophobicity is here defined as the free energy change upon this transfer process. The hydrophobic effect is also manifested through adsorption of surfactants at surfaces or the folding of proteins. The analysis of surfactant adsorption requires a molecular model in order to obtain the free energy of adsorption and from the temperature dependence the enthalpy and entropy of adsorption can be obtained. The adsorption of ionic surfactants is only weakly dependent on the temperature and for ionic surfactants there is a maximum adsorption at the temperature where the CMC is at a minimum [17]. The adsorption of non-ionic surfactants, based on a polyoxyethylene chain, has a strong temperature dependence emanating from this chain and requires an even more elaborate analysis [40,41].

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Analyses of the protein unfolding and ligand binding reveal that the thermodynamics is similar to the transfer of hydrophobic molecules into water [24,25•,42–44,45••,46] In models the protein interior can be treated as an organic liquid [47], but other modes such as the “packingdesolvation” model have also been proposed [25•,48]. Thus, the force responsible for the compactness and integrity of protein structures is most likely due to the hydrophobic effect. Unfortunately, the analysis cannot be as exact as in the dissolution of hydrophobic compounds, since the entropy of the unfolding process is not corrected for the combinatorial contribution. Interestingly, in analyzing the enthalpy versus the heat capacity of protein unfolding we can circumvent this problem since in these two quantities there is no combinatorial contribution involved. Murphy et al. showed that there is a linear correlation between the enthalpy and the heat capacity change on protein unfolding for 11 different proteins at 25 °C [42]. If we assume that both quantities reflect the structuring of water around the hydrophobic moieties of the proteins, then Eqs. (7) and (9) should be valid. Thus, the slope of the curve in the figure should give a value of τ close to that obtained for the dissolution of hydrophobic molecules in water. Indeed, the obtained value is 77 K, which should be compared to a value of 65 ± 5 K for the dissolution of hydrocarbons. 10. Conclusions It is clear that the hydrophobic effect can be understood in terms of the two contributions originating from the cavity formation and from the water structuring as has been illustrated by the examples discussed here. The origin of the large energy involved in the cavity formation is due to the facts that the water molecules are small and the hydrogen bonds between the water molecules are strong; thus, the cohesive energy density of water is high. It was shown that, whereas the cavity contribution is dominating, the temperature dependence is entirely determined by the water structuring, or rearrangement, in the vicinity of a hydrophobe. We concluded that the lowering of entropy due to structuring must be accompanied by an enthalpy that is of the same order of magnitude as the TΔS for the process. Hence, there must be an entropy–enthalpy compensation leading to a low free energy change for the structuring process. As a result of the analysis it was shown that the temperature, Tmin, at which the solubility of alkanes in water is at minimum, is universal. Finally, it was discussed that the free energy obtained from partitioning, or solubility limits, needs to be corrected for molecular size differences between the solute and the solvent. It was concluded that the Flory–Huggins expression is a good first approximation for obtaining this correction. Erroneous conclusions regarding the thermodynamics of the hydrophobic effect will be the result if the effect of different molecular sizes is not corrected for. Also micelle formation, surfactant adsorption and protein unfolding were briefly discussed in terms of the hydrophobic effect. Acknowledgment The author wishes to especially thank Professor Krister Holmberg for reading many drafts of the earlier version of this paper, and for making many detailed and helpful comments. References and recommended readings•,•• [1] Frank H, Evans M. 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. J Chem Phys 1945;13:507–32. [2] Berg J. An introduction to interfaces and colloids — a bridge to nanoscience. London World Scientific; 2010 139.

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of special interest. of outstanding interest.

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