Surface tensions and surface potentials of acid solutions
aa r X i v : . [ c ond - m a t . s o f t ] N ov Surface tensions and surface potentials of acid solutions
Alexandre P. dos Santos and Yan Levin Instituto de F´ısica, Universidade Federal do Rio Grande do Sul,Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS,Brazil
A theory is presented which allows us to quantitatively calculate the excess sur-face tension of acid solutions. The H + , in the form of hydronium ion, is found to bestrongly adsorbed to the solution-air interface. To account for the electrostatic poten-tial difference measured experimentally, it is necessary to assume that the hydroniumion is oriented with its hydrogens pointing into the bulk water. The theory is quan-titatively accurate for surface tensions and is qualitative for electrostatic potentialdifference across the air-water interface.1 . INTRODUCTION Electrolyte solutions are of fundamental interest for a variety of disciplines. Over a hun-dred years ago Hofmeister observed a strong dependence of the stability of protein solutionson the specific nature of electrolyte. While some ions tend to stabilize protein solutions,often denaturing them in the process, others destabilize them favoring protein precipitation.A few years after Hofmeister, Heydweiller observed that salt increases the surface tensionof the air-water interface. Furthermore, Heydweiller noticed that the relative effect that ionshave on the surface tension follows closely the Hofmeister series, suggesting that the twophenomena are related.Over the last hundred years there has been a great effort to understand how ionic speci-ficity influences stability of protein solutions and how it affects the surface tension of theair-water interface. Langmuir was probably the first to attempt to construct a quantita-tive theory of surface tensions of electrolyte solutions. Appealing to the Gibbs adsorptionisotherm, Langmuir concluded that the excess surface tension of electrolyte solution was aconsequence of ionic depletion from the interfacial region. However, no clear explanationfor this depletion was provided. A few years after, Wagner argued that this depletion wasthe result of interaction between the ions and their electrostatic images across the air-waterinterface. Onsager and Samaras simplified Wagner’s theory and obtained a limiting lawwhich they argued was universally valid for all electrolytes at sufficiently small concentra-tion. More recently, Levin and Flores-Mena used a direct free energy calculation to obtainsurface tension of strong electrolytes. To have a quantitative agreement with experiments,these authors stressed the fundamental importance of ionic hydration. Nevertheless, thetheory of Levin and Flores-Mena was not able to predict correctly surface tensions of allelectrolyte solutions. Bostr¨om et al. suggested that Hofmeister effect and the ionic speci-ficity are a consequence of dispersion forces arising from finite frequency electromagneticfluctuations. This theory predicted that weakly polarizable cations should be adsorbed atthe air-water interface. This, however, was contradicted by the experimental measurementsof the electrostatic potential difference and by the simulations on small water clusters ,as well as by the subsequent large scale polarizable force fields simulations and the pho-toelectron emission experiments . These experiments and simulations showed that someanions — and not cations — are present at the solution-air interface . To explain this,2evin extended the traditional Born theory of ionic solvation to account for ionic polar-izability. The new theory predicted that highly polarizable anions can actually prefer aninterfacial solvation. In a followup work, Levin et al. used this theory to quantitativelycalculate the surface tensions and the surface potentials of 10 different electrolyte solutionsand to reproduce the Lyotropic, Hofmeister, series.While almost all salts lead to increase the air-water surface tension, acids tend to lowerit . The only explanation for this is a strong proton adsorption at the water-air inter-face . It is well known that H + ion forms various complexes with water molecules .It has also been suggested that the high surface adsorption of H + is related to the hydronium(H O + ) geometry . This ion, has a trigonal pyramidal structure with the hydrogens locatedat the base of the pyramid . In this form hydrogens are good hydrogen-bond donors, whileoxygen is a bad hydrogen-bond receptor . This favors hydronium ion to be preferentiallylocated at the interface, with the hydrogens pointing towards the bulk water and the oxygenpointing into the gas phase . Explicit solvation energy calculations confirm this picture .The sign of electrostatic surface potential difference is related with the relative popula-tion of cations and anions at the interface. Because of high adsorption of hydronium ions,one would naturally expect that the electrostatic potential difference across the air-waterinterface for acid solutions should be positive. The experiments, however, show that thesurface potential difference for acids has the same sign as for halide salts, i.e. is predom-inantly negative . Frumkin suggested that this apparently strange behavior might bea consequence of the incomplete dissociation of acid molecules. A different explanationwas advanced by Randles who argued that presence of hydroniums at the interface leadsto a preferential orientation of water molecules resulting in a dipole layer with a negativeelectrostatic potential difference across it. This conclusion is in agreement with the theoryproposed in the present paper, as well as with the recent molecular dynamics simulations .In this paper we present a theory that allows us to quantitatively calculate surface ten-sions of acid solutions using only one adjustable parameter related to the strength of thehydronium adsorption to the interface. Predictions of the theory are compared with theexperimental measurements. The theory is then used to estimate the electrostatic potentialdifference across the water-air interface for various acid solutions.3 I. MODEL AND THEORY
We consider an acid solution in a form of a drop of radius R , where r = R is the positionof the Gibbs dividing surface (GDS) . The water and air will be modeled as uniformdielectrics of permittivities ǫ w = 80 and ǫ o = 1, respectively. The surface tension can beobtained by integrating the Gibbs adsorption isotherm equation:d γ = − Γ + d µ + − Γ − d µ − , (1)where µ ± = k B T ln( c b Λ ± ) are the chemical potentials and Λ ± are the de Broglie thermalwavelengths. In this equation + sign corresponds to the hydronium ion, and − sign to theanion. The bulk ion concentration is c b = ρ + (0) = ρ − (0), where ρ ± ( r ) are the ionic densityprofiles. The ion excess per unit area due to existence of the interface isΓ ± = 14 πR (cid:20) N − πR c b (cid:21) , (2)where N is the total number of acid “molecules”. The ionic density profiles, ρ ± ( r ), will becalculated using a modified Poisson-Boltzmann (mPB) equation, as discussed later in thepaper.Anions are divided into two categories: kosmotropes and chaotropes. The theory of elec-trolyte solutions showed that chaotropes, Br − , I − , NO − , and ClO − , lose their hydrationsheath near the GDS and are partially adsorbed to the interface. On the other hand kos-motropes, F − , Cl − , and SO − remain hydrated in the interfacial region and are repelledfrom the GDS.To bring an ion of radius a h to distance z > a h from the GDS requires : W ( z ; a h ) = q ǫ w Z ∞ dke − s ( z − a h ) k [ s cosh( ka h ) − k sinh( ka h )] s [ s cosh( ka h ) + k sinh( ka h )] , (3)of work. In this equation s = p ( κ + k ) and κ = p πq c b /ǫ w k B T is the inverse De-bye length. The kosmotropic ions remain strongly hydrated in the interfacial region andencounter a hardcore-like repulsions from the GDS at a distance of one hydrated ionic ra-dius. On the other hand, strongly polarizable chaotropic anions (Br − , I − , NO − and ClO − )loose their hydration sheath and can move cross the water-air interface. However, to avoidthe large electrostatic energy penalty of exposing the charge to a low-dielectric (air) en-vironment, the electronic charge density of a chaotropic anion redistributes itself so as to4emain largely hydrated . The fraction of ionic charge which remains inside the aqueousenvironment, x ( z ), can be calculated by minimization the polarization energy U p ( z, x ) = q a ǫ w (cid:20) πx ( z ) θ ( z ) + π [1 − x ( z )] ǫ w [ π − θ ( z )] ǫ o (cid:21) + (1 − α ) αβ (cid:20) x ( z ) − − cos [ θ ( z )]2 (cid:21) . (4)In the above equation α is the relative polarizability defined as α = γ i /a , where γ i is the ionicpolarizability, a is the unhydrated (bare) radius, and θ ( z ) = arccos[ − z/a ]. Performing theminimization we obtain x ( z ) = (cid:20) λ B πǫ w a ǫ o [ π − θ ( z )] + (1 − α ) α [1 − cos [ θ ( z )]] (cid:21) / (cid:20) λ B πa θ ( z ) + λ B πǫ w a ǫ o [ π − θ ( z )] + 2 (1 − α ) α (cid:21) , (5)where λ B = q /ǫ w k B T is the Bjerrum length.The force that drives chaotropic ions towards the interface results from water cavitation.To introduce an ion into an aqueous environment requires creating a cavity, which perturbesthe hydrogen bond network of water molecules. For small ions, the free energy cost of forminga cavity is entropic and is proportional to the volume of the void formed . As the ionmoves across the GDS, its cavitational free energy decreases. This results in a short-rangeattractive potential between the anion and the GDS : U cav ( z ) = νa for z ≥ a , νa (cid:16) za + 1 (cid:17) (cid:16) − za (cid:17) for − a < z < a , (6)where ν ≈ . k B T / ˚A is obtained from bulk simulations . For hard (weakly polarizableions) the cavitational free energy gain is completely overwhelmed by the electrostatic free en-ergy penalty of moving ionic charge into the low dielectric environment. For soft polarizableions, however, the electrostatic penalty is small, since most of the ionic charge remain insidethe aqueous environment. The total potential of a soft anion, therefore, has a minimum inthe vicinity of the GDS, see Fig. 1.The H + ions (protons) do not exist as a separate specie in water. Instead they formcomplexes with water molecules, H O + and H O +5 29–31 . Because of its favorable geometry(trigonal pyramidal), the hydronium ion (H O + ) adsorbs to the water-air interface with apreferential orientation of oxygen towards the air. We model this attraction by a squarewell potential with a range of a hydrogen bond 1 .
97 ˚A, U hyd ( z ) = z ≥ .
97 ˚A , − . k B T for 0 ≤ z < .
97 ˚A . (7)5 z [Å] -202468 po t e n ti a l s [ K B T ] ClO I - Br - NO SO F - Cl - H + FIG. 1. Potentials for all ions at 1M. The GDS is at z = 0 ˚A. The bulk cavitational potential isnot considered for kosmotropes, since it does not change along the drop. The depth of the potential is then adjusted to obtain the experimentally measured surfacetension of HCl. The same potential is then used to calculate the surface tensions of allother acids. We should stress, however, that one should not attach too much meaning tothe specific value of the potential depth. The real proton transfer is a quantum mechanicalprocess, therefore there is bound to be some arbitrariness in how one models it at a classicallevel. Here we have chosen the range of the square well potential to be one hydrogen bond.If one changes this distance, the depth of the potential will have to be modified to obtainan optimal fit of the surface tension of HCl solution. However once this is done, the valuesof the surface tension of other acids will not be significantly affected. Thus, the strength ofH + potential is the only free parameter of the theory. The total potential felt by H + is then, U H ( z ) = U hyd ( z ) + W ( z ; 0), see Fig. 1.While the kosmotropic anions feel only the potential W ( z ; a h ) and the hardcore repulsionfrom the GDS, the chaotropic anions are influenced by the total potential : U tot ( z ) = W ( z ; a ) + νa + q ǫ w a for z ≥ a ,W ( a ; a ) z/a + U p ( z ) + U cav ( z ) for 0 < z < a ,U p ( z ) + U cav ( z ) for − a < z ≤ . (8)In Fig. 1, we plot the potentials felt by various ions at 1M concentration, as a function ofthe distance from the GDS. 6 ABLE I. Ion classification into chaotropes (c) and kosmotropes (k). Effective radii (hydrated orpartially hydrated) for kosmotropes and (bare) for chaotropes, for which we have also include thepolarizabilities from Ref. .Ions chao/kosmo radius (˚A) polarizability (˚A )F − k 3.54 *Cl − k 2 *Br − c 2.05 5.07I − c 2.26 7.4NO − c 1.98 4.48ClO − c 2.83 5.45SO − k 3.79 * The ionic density profiles can now be obtained by integrating the mPB equation: ∇ φ ( r ) = − πqǫ w [ ρ + ( r ) − ρ − ( r )] , (9) ρ + ( r ) = N e − βqφ ( r ) − βU H ( z ) R R πr dr e − βqφ ( r ) − βU H ( z ) ,ρ chao − ( r ) = N e βqφ ( r ) − βU tot ( r ) R R + a πr dr e βqφ ( r ) − βU tot ( r ) ,ρ kos − ( r ) = N Θ( R − a h − r ) e βqφ ( r ) − βW ( z ; a h ) R R − a h πr dr e βqφ ( r ) − βW ( z ; a h ) , where Θ is the Heaviside step function, ρ chao − ( r ) is the density profile for chaotropic anionsand ρ kos − ( r ) for kosmotropic ones.Once the ionic density profiles are calculated, the surface tensions can be obtained byintegrating the Gibbs adsorption isotherm (eq 1). The ionic radii and polarizabilities arethe same as were used in our previous work on surface tension of electrolyte solutions .In Table I we summarize this data.The depth of the potential U hyd ( z ) (eq 7) is adjusted to fit the HCl experimental data (see Fig. 2), this is the only adjustable parameter of the theory. We find that a squarewell potential of depth − . k B T results in an excellent fit of the experimental data forHCl in the range of concentrations from 0 to 1 M. The excess surface tension of all otheracids is then calculated using the same potential U hyd ( z ). The predictions for the surface7 concentration [M] -2-1.5-1-0.500.5 s u rf ace t e n s i on [ m N / m ] HFHClHBrHI
FIG. 2. Surface tensions for HF, HCl, HBr, and HI. The symbols are the experimental data forHCl and the lines are the results of the present theory. tensions of HF, HBr, and HI are plotted in Fig. 2. Unfortunately, we have no experimentaldata to compare for these halogen acids. For H SO and HNO (see Fig. 3), we find agood agreement between the theory and experiment. For HClO the theory overestimatesthe surface tension. This is similar to what was found for sodium perchlorate salt . Thedifficulty is that ClO − is a large weakly hydrated ion. Since the cavitational energy growswith the cube of ionic radius, a small error in radius leads to a big error in surface tension. concentration [M] -3-2-10 s u rf ace t e n s i on [ m N / m ] H SO HNO HClO FIG. 3. Surface tensions for H SO , HNO , and HClO . The symbols are the experimental data and the lines are the results of the present theory. Finally, we use the theory to calculate the electrostatic potential difference across thesolution-air interface. The surface potential difference ∆ χ = φ ( R + a ) − φ (0) predictedby the present theory has a wrong sign compared to Frumkin’s experimental measurements8 ABLE II. Surface potential differences for various acids at 1M concentration. Contributions fromelectrolyte and aligned water dipoles.Acids calculated [mV] Frumkin [mV]HF 85 . − . − − − − . − − . − − − — positive instead of negative . Positive sign reflects a strong adsorption of hydroniumions to the GDS. The simple dielectric continuum theory presented here, however, does notaccount for the structure of the interfacial water layer. Since the hydronium ion at theGDS has a preferential orientation with the hydrogens pointing towards the bulk, presenceof many such ions will result in a dipole layer. Note that in the absence of hydroniums,the water dipoles predominantly point along the interface . The hydronium layer producesan electric field E = 4 πpN h /ǫ o dA , where N h is the number of hydroniums at the interface, p is the water dipole moment, d is the dipole length, and A is the interfacial area. If wesuppose that all the hydroniums are perfectly aligned, the potential difference across thedipolar layer will be ∆ χ w = − πp Γ + /ǫ o . Using the dipole moment of a water molecule, p = 1 .
85 D, we obtain the dipole layer contribution to the overall electrostatic potentialdifference. Adding this to ∆ χ , we obtain the total electrostatic surface potential differenceacross the solution-air interface. In Table II, we list the surface potentials of various acidsat 1M concentration. Clearly these values are an exaggeration of the total electrostaticpotential difference across the interface, since at finite temperature there will not be perfectalignment of interfacial hydronium ions. Nevertheless, the theory should provide us an orderof magnitude estimate of the electrostatic potential difference. In fact, for most acids we finda reasonable agreement between the predictions of the theory and Frumkin’s experimentalmeasurements . A noticeable exception is the HF. Experimental potential for hydrogenfluoride measured by Frumkin is negative, while we find a large positive value. Frumkin’svalue for HF is clearly outside the general trend for halogen acids. In his classical review9f electrolyte solutions, Randles did not mention Frumkin’s result for HF acid, whilediscussing his other measurements. We can only suppose that Randles also did not have acomplete confidence in this particular value. Experimental measurements of excess surfacepotentials are very difficult. This is probably the reason why Frumkin’s measurements ofsurface potentials of acids have not been repeated in over 90 years. III. CONCLUSIONS
In this paper we have developed a theory for surface tensions of acid solutions. Thehydronium adsorption to the interface was modeled by a square well potential, the depthof which is the only adjustable parameter of the theory. The agreement between the theoryand experiments is very reasonable for different acid solutions at concentrations from 0to 1M. In order to account for the experimental values of the excess electrostatic surfacepotential, we must require a preferential orientation of hydronium ion at the interface, withthe hydrogens pointing into the bulk. With this assumption we get a qualitative agreementwith the experimental measurements of the excess electrostatic potentials of various acidsolutions. At the moment this is the only theory that can account (quantitatively) for thesurface tensions and (qualitatively) for the surface potentials of acid solutions.
IV. ACKNOWLEDGMENTS
This work was partially supported by the CNPq, INCT-FCx, and by the US-AFOSRunder the grant FA9550-09-1-0283.
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