Chemical Physics Letters 497 (2010) 33–36

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Chemical Physics Letters

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Significance of the Tolman length at a molecular level

Giuseppe Graziano *

Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 June 2010In final form 28 July 2010Available online 30 July 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.07.083

* Fax: +39 0824 23013.E-mail address: graziano@unisannio.it

It is shown that, by assuming valid a relationship between the work of cavity creation in a liquid and thebulk liquid–vapour surface tension, corrected for curvature effects, and by reproducing calculated DGc

values to create in water and n-hexane a cavity suitable to host a methane molecule, the obtained esti-mates of the Tolman length prove to markedly depend on temperature, losing a precise physical meaning.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The occurrence of an empirical correlation between the solubil-ity of gases in liquids and the corresponding bulk liquid–vapoursurface tension was noted by Battino and colleagues long timeago [1,2]. It was surmised that the bulk liquid–vapour surface ten-sion, c1 (where the subscript1 indicates that a plane, flat surfacehas an infinite radius of curvature), could play a role in the processof cavity creation [3], an unavoidable step to insert a solute mole-cule in a liquid [4]. Specifically, the Gibbs energy cost of cavity cre-ation should be proportional to the bulk liquid–vapour surfacetension:

DGcðRcÞ / c1 ð1Þ

where DGc(Rc) is the reversible work to create a cavity of radius Rc

in the liquid of interest. Note that there are two size measures of aspherical cavity: (a) the radius of the spherical region from whichall parts of the liquid molecules are excluded, indicated by rc and(b) the radius of the spherical region from which the centres of li-quid molecules are excluded, indicated by Rc [5]. It results thatRc = rc + r1, where r1 is the radius of the liquid molecules, and thesolvent accessible surface area SASAc ¼ 4pR2

c [6]. Actually, it wasconsidered that, since the cavities relevant for solubility have amolecular size, the c1 quantity should be corrected for curvatureeffects [7], according to the formula devised by Tolman [8]:

cðRcÞ ¼ c1½1� ð2 � dTolman=RcÞ� ð2Þ

where c(Rc) is the liquid–vapour surface tension for a bubble of Rc

radius, and dTolman is the first-order curvature correction to c1and is known as the Tolman length. From a theoretical point ofview, the Tolman length has received a lot of attention in recentyears, even though some points are still not settled down [9–11].These theoretical approaches are not of real interest in the presentcase because they are devoted to liquid–vapour interface formation

ll rights reserved.

and not to cavity creation in a liquid. On the basis of Eq. (2), it ispossible to write down:

DGcðRcÞ ¼ 4p � R2c � c1½1� ð2 � dTolman=RcÞ� ð3Þ

This relationship has to be considered macroscopic in origin andshould be right for a small bubble. However, over the years, it hasbeen applied also to molecular-sized cavities [12–16]. Note, in thisrespect, that a cavity is conceptually different from a bubble be-cause the former has to be empty by definition, whereas the lattercontains molecules in the vapour phase.

Ashbaugh and Pratt, A&P, have recently claimed that the loca-tion of the maximum in the cavity contact correlation function,G(Rc), identifies the minimum length for which a macroscopicdescription of cavity thermodynamics should be valid in a given li-quid [17]; i.e., Eq. (3) should hold. Their molecular dynamics, MD,results showed that the G(Rc) maximum, at room temperature, oc-curs at Rc � 2.8–3.0 Å in water [17], and at Rc � 3.0 Å in n-hexane[18], suggesting that the macroscopic description of DGc shouldhold also for very small (i.e., molecular-sized) cavities. In addition,the temperature dependence of Rc,max proved to be small in bothliquids and, in any case, Rc,max decreased on increasing tempera-ture [18]. Therefore, according to A&P results, Eq. (3) should holdfor molecular-sized cavities over a large temperature range.

In this Letter I would like to emphasize that, by considering Eq.(3) right to reproduce the DGc values emerging from computersimulations, the Tolman length becomes a purely fitting parame-ter, losing its physical meaning. This is strongly evident when thetemperature dependence of dTolman is determined for water andn-hexane. The conclusion is that Eq. (3) has no theoretical ground,and cannot shed light on the molecular mechanism of hydropho-bicity because c1 and its curvature correction are macroscopicthermodynamic quantities.

2. Results

I have decided to apply Eq. (3) to the DGc values calculated byA&P, via MD simulations [17,18], to create a cavity suitable to host

34 G. Graziano / Chemical Physics Letters 497 (2010) 33–36

a methane molecule in both SPC/E water and a CHARMM model ofn-hexane (i.e., a chain of six CH3/CH2 units, each having a radius of1.9 Å), over the 270–370 K temperature range. For such a cavity theradius rc = 1.9 Å in both liquids, whereas the radius Rc = rc +r(H2O) = 1.9 + 1.4 = 3.3 Å in water, and Rc = rc + r(CH2) = 1.9 + 1.9 =3.8 Å in n-hexane. The DGc values calculated by A&P are listed inthe fourth column of Table 1: they are large and positive in bothliquids, but larger in water than in n-hexane because the watermolecules have an effective size, r(H2O) = 1.4 Å, smaller than thatof the CH3/CH2 units constituting the n-hexane molecules,r(CH2) = 1.9 Å [19,20].

In addition, the DGc values show a qualitatively different tem-perature dependence in the two liquids: they increase with tem-perature in water, but decrease with temperature in n-hexane.The latter point is very important in order to apply Eq. (3) andtry to connect the DGc values in a given liquid with its bulk li-quid–vapour surface tension c1. In fact, the experimental c1 val-ues decrease with temperature in both liquids [21], even thoughthe values of water are markedly larger than those of n-hexaneand the rate of decrease is larger in the latter liquid (see the secondand third columns of Table 1). The percentage decrease over the270–370 K temperature range amounts to 50% of the c1 value at270 K in n-hexane, and to 23% of the c1 value at 270 K in water.Therefore, the temperature dependence of c1 is qualitatively sim-ilar to that of DGc in n-hexane, but contrasts with that of DGc inwater.

Application of Eq. (3) leads to the values of the Tolman length,dTolman, listed in the last column of Table 1. The latter are alwayspositive and decrease with temperature in water (i.e., from1.03 Å at 270 K to 0.67 Å at 370 K), whereas they start small posi-tive and then become negative in n-hexane (i.e., from 0.27 Å at270 K to zero at 310 K and to �0.79 Å at 370 K). According to a geo-metric derivation of Eq. (2) [7], the quantity 2�dTolman is expected tobe close to the size of solvent molecules or units. This does not holdover the whole temperature range investigated in the case of n-hexane, and is true only at low temperature in the case of water.The change of sign for dTolman in n-hexane is a simple consequenceof the fact that, on increasing temperature, the DGc/SASAc numbersbecome larger than the c1 values; a situation that does not happenin the case of water (see the third and fifth columns of Table 1). Inother words, the change of sign for dTolman in n-hexane would bethe result of a purely numerical exercise, unless Eq. (3) would havea robust theoretical ground. But this does not seem to be the case(for more, see below).

Such a situation is similar to the finding that the ratio DGc/SA-SAc, evaluated for very large cavities in common organic solvents atroom temperature, by several authors performing MD simulations

Table 1For water and for n-hexane, values of the experimental bulk liquid–vapour surface tension [accessible surface area of the cavity, SASAc, and of the so-called Tolman length over the 2

TK

c1dyne cm�1

c1J mol�1 �

Water 270 76.0 457.9290 73.2 440.8310 70.1 422.2330 66.8 402.3350 63.3 381.2370 59.6 358.8

n-Hexane 270 20.9 125.6290 18.8 113.1310 16.7 100.7330 14.6 88.2350 12.6 75.7370 10.5 63.3

[22,23], or both classic and modified SPT calculations [5,24],proved to be larger than the experimental value of the bulk li-quid–vapour surface tension c1 of such liquids. Similarly, Huangand Chandler obtained that the DGc/SASAc ratio for a Lennard–Jones fluid was larger than the corresponding bulk liquid–vapoursurface tension at the selected temperature and pressure [25];analysis of their data by means of Eq. (3) clearly produced negativevalues for dTolman [26]. It is worth noting that the dTolman values cal-culated in this Letter are specific for a cavity with Rc = 3.3 Å inwater and Rc = 3.8 Å in n-hexane; the dTolman values calculated byfitting, at a single temperature, the DGc/SASAc trend should beindependent of Rc. In general, however, the idea of a curvature cor-rection becomes ambiguous when the cavity shape is far fromspherical.

There is a final non-trivial point. In applying Eq. (3), I have usedthe experimental values of c1 for both water and n-hexane [21].However, the DGc values are calculated in a computer model ofthe selected liquid, and so one should take into account the c1 va-lue of such model. This could produce strange situations. In thecase of the SPC/E water model there are different c1 estimates:(a) Huang et al. obtained c1 = 73.6 dyne cm�1, at 298 K and1 atm [26]; (b) Vega and de Miguel obtained c1 = 63.6 dyne cm�1,at 300 K and 1 atm [27]; (c) Godawat et al. obtainedc1 = 56 dyne cm�1, at 300 K and 1 atm [28]; note that the experi-mental value is c1 = 71.7 dyne cm�1, at 300 K and 1 atm [21].When the above estimates are contrasted against the DGc/SASAc

trend determined by Huang et al. [26,29] in SPC/E water at 298 Kand 1 atm (see Fig. 1), it should be clear that, by assuming validEq. (3) to establish a link between DGc and c1, the Tolman lengthbecomes a purely fitting parameter.

It is worth noting that the DGc/SASAc trend versus Rc shows theoccurrence of a crossover from small-to-large length scale cavities(see Fig. 1). This led Lum et al. to claim that the crossover should bean indication of a change in the physics of cavity creation [29,30]. Ido not think that such a claim is right because the crossover isqualitatively well reproduced by means of classic scaled particletheory, SPT, calculations in both water and common organic sol-vents, suggesting that the physics of cavity creation does notchange [5]. This is because a complete enthalpy–entropy compen-sation characterizes the structural reorganization of solvent mole-cules upon cavity creation, as shown by statistical mechanicalanalyses [4,5,31]. In water this means that the creation of a largecavity (i.e., Rc � 10 Å) would surely cause the loss of some H-bondsthat, in turn, would produce a gain of translational/configurationaldegrees of freedom of the corresponding water molecules: the en-thalpy increase is balanced by the entropy increase, and DGc is so-lely due to the solvent-excluded volume effect [5,31].

21], of the work to create a cavity with rc = 1.9 Å [18], of the ratio of DGc to the solvent70–370 K temperature range. See text for further details.

2DGc

kJ mol�1DGc/SASAc

J mol�1 �2dTolman

Å

23.5 171.7 1.03125.3 184.9 0.95826.8 195.8 0.88527.8 203.1 0.81728.7 209.7 0.74229.3 214.1 0.665

19.5 107.5 0.27418.7 103.3 0.16418.3 100.6 0.00217.5 96.4 �0.17717.0 93.7 �0.45116.3 89.6 �0.788

0 3 6 9 12 15 180

100

200

300

400

(J m

ol-1

angs

trom

-2)

Rc (angstrom)

Fig. 1. Trend of DGc/SASAc versus Rc obtained by Huang et al. [26,29] in SPC/E waterat 298 K and 1 atm (black curve); values of the bulk liquid–vapour surface tensionc1 for the SPC/E water model calculated by Huang et al. [26] at 298 K and 1 atm(solid line), by Vega and de Miguel [27] at 300 K and 1 atm (dashed and dotted line),and by Godawat et al. [28] at 300 K and 1 atm (dashed line).

G. Graziano / Chemical Physics Letters 497 (2010) 33–36 35

3. Discussion

The use of Eq. (3) implies that: (a) the temperature dependenceof c1 is the experimental one for each liquid and (b) the need to fitthe DGc values over a large temperature range renders dTolman atemperature dependent quantity. However, there is no physicalexplanation for the finding that dTolman is always positive for water,whereas it starts positive and becomes negative just above 310 Kfor n-hexane. This is a simple consequence of the mathematical fit-ting and there is no significance to attach to the Tolman length at amolecular level. Moreover, Eq. (3) cannot provide a rationalizationof the datum that DGc is a quantity increasing with temperature inwater, but decreasing with temperature in n-hexane and in a Len-nard–Jones liquid [17,18,32].

For a small convex bubble, the geometric and physical expecta-tion is that the surface tension should be smaller than the bulk c1value. Tolman wrote [8]: ‘It is concluded that surface tension canbe expected to decrease with decrease in droplet size over a widerange of circumstances’. According to Eq. (2) and its general deriva-tion, this implies that dTolman should be a positive quantity, notstrongly dependent on temperature, for all liquids. The real prob-lem is that a cavity does not correspond to a very small bubble be-cause the cavity region has to be empty/void by definition (i.e.,there is no vapour in equilibrium with the liquid [23]). In fact, cav-ity creation does not produce an interface but a boundary layer: thereare molecules on both sides of an interface, instead there are mol-ecules only on one side of a boundary layer. All the componentsshould have the same chemical potential on each side of an inter-face, but this thermodynamic restriction does not hold for a bound-ary layer such that existing at the surface dividing a cavity from theliquid. As a consequence, the reversible work necessary to create acavity in a liquid should not be assimilated to the reversible worknecessary to maintain the liquid–vapour interface of a bubble,where vapour molecules bombard the interface. A strictly quanti-tative correlation between these two Gibbs energy quantitiesshould not be expected.

Ashbaugh [32], by means of Monte Carlo simulations, calculatedthe cavity contact correlation function and the work of cavity cre-ation for a Lennard–Jones liquid, over a large temperature range,from the triple point up to the critical point. By fitting the simula-tion results with a polynomial Rc expression for DGc, containing theliquid–vapour surface tension, the Tolman length and two otherparameters (see Eq. (7) in Ref. [32]), he found that the obtained val-

ues of c1 were systematically larger than the true ones, and thoseof dTolman were always negative over the entire temperature rangeinvestigated. The other two parameters in the DGc expression donot seem to have a clear physical meaning and their values werenot reported. In addition, Ashbaugh exactly wrote [32]: ‘The sol-vent profile in contact with the cavity possesses two interfaces: aliquid–vapour interface where the bulk solvent density drops tothat of a vapour film surrounding the cavity and a vapour–wallinterface where the vapour film contacts the cavity. The interfacialfree energies of both surfaces can have different curvaturecorrections.’

The latter sentences clarify the difficulties of equating the crea-tion of a cavity in a liquid to the formation of a liquid–vapour inter-face. The main question is that there is no theoretical connectionbetween the probability of finding zero molecules in a given regionof a liquid (so to have a cavity) and the bulk liquid–vapour surfacetension of the liquid itself. This fundamental point has been ne-glected with the idea that, for very large, macroscopic cavities,the bulk liquid–vapour surface tension should necessarily play arole [17,29]. A basic reply is that also cavities suitable to host glob-ular proteins cannot be considered macroscopic by any means, andcan be produced by means of molecular-scale density fluctuationsat equilibrium (i.e., there is no need to consider the formation of aninterface) so that DGc is solely due to the solvent-excluded volumeeffect [5,31].

The finding that DGc largely depends on SASAc should not be as-sumed as a clear evidence that c1 plays a role. In several contin-uum solvent models, nonpolar hydration Gibbs energies areassumed to depend linearly on solute’s SASA (i.e., Poisson–Boltz-mann/surface area, PBSA; generalized Born/surface area, GBSA;conductor-like screening model, COSMO; the SMx models; for acomprehensive review see the article by Cramer and Truhlar[33]). The proportionality constant, however, assumes differentvalues in the various computational approaches and no one ofthese values corresponds to c1 of water. This finding has led sev-eral authors to raise concerns about the widespread and oftenuncontrolled use of SASA based formulas [22,23,34,35]. Actually,the SASAc dependence of DGc originates from the solvent-excludedvolume effect associated with cavity creation, as it has been shownby the geometric derivation of classic DGc(SPT) formulas [36–38]as well as free energy calculations [22]. This is further confirmedby the finding that the DGc magnitude increases with SASAc onchanging the cavity shape but keeping the cavity van der Waalsvolume fixed [38,39].

In conclusion, I have tried to show that, since cavity creationdoes not produce an interface but a boundary layer, the relationshipbetween DGc and c1 has no theoretical ground and its use rendersthe Tolman length a purely fitting parameter.

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