Surface tensions and surface potentials of acid solutions
A theory is presented which allows us to quantitatively calculate the excess surface tension of acid solutions. The \ceH+, in the form of hydronium ion, is found to be strongly adsorbed to the solution-air interface. To account for the electrostatic potential difference measured experimentally, it is necessary to assume that the hydronium ion is oriented with its hydrogens pointing into the bulk water. The theory is quantitatively accurate for surface tensions and is qualitative for electrostatic potential difference across the air-water interface.
Electrolyte solutions are of fundamental interest for a variety of disciplines. Over a hundred years ago Hofmeister observed a strong dependence of the stability of protein solutions on 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 Heydweiller (1910) observed that salt increases the surface tension of the air-water interface. Furthermore, Heydweiller noticed that the relative effect that ions have on the surface tension follows closely the Hofmeister series, suggesting that the two phenomena are related.
Over the last hundred years there has been a great effort to understand how ionic specificity influences stability of protein solutions and how it affects the surface tension of the air-water interface. Langmuir Langmuir (1917) was probably the first to attempt to construct a quantitative theory of surface tensions of electrolyte solutions. Appealing to the Gibbs adsorption isotherm, Langmuir concluded that the excess surface tension of electrolyte solution was a consequence of ionic depletion from the interfacial region. However, no clear explanation for this depletion was provided. A few years after, Wagner Wagner (1924) argued that this depletion was the result of interaction between the ions and their electrostatic images across the air-water interface. Onsager and Samaras Onsager and Samaras (1934) simplified Wagner’s theory and obtained a limiting law which they argued was universally valid for all electrolytes at sufficiently small concentration. More recently, Levin and Flores-Mena Levin and Flores-Mena (2001) used a direct free energy calculation to obtain surface tension of strong electrolytes. To have a quantitative agreement with experiments, these authors stressed the fundamental importance of ionic hydration. Nevertheless, the theory of Levin and Flores-Mena was not able to predict correctly surface tensions of all electrolyte solutions. Boström et al. Boström et al. (2001) suggested that Hofmeister effect and the ionic specificity are a consequence of dispersion forces arising from finite frequency electromagnetic fluctuations. This theory predicted that weakly polarizable cations should be adsorbed at the air-water interface. This, however, was contradicted by the experimental measurements of the electrostatic potential difference Frumkin (1924); Jarvis and Scheiman (1968) and by the simulations on small water clusters Perera and Berkowitz (1991); Dang and Smith (1993); Stuart and Berne (1999), as well as by the subsequent large scale polarizable force fields simulations Jungwirth and Tobias (2006, 2002); Horinek et al. (2009); Brown et al. (2008) and the photoelectron emission experiments Markovich et al. (1991); Ghosal et al. (2005); Garrett (2004). These experiments and simulations showed that some anions — and not cations — are present at the solution-air interface Perera and Berkowitz (1991); Dang and Smith (1993); Stuart and Berne (1999); Jungwirth and Tobias (2002); Horinek et al. (2009); Brown et al. (2008); Jungwirth and Tobias (2006); Markovich et al. (1991); Ghosal et al. (2005); Garrett (2004). To explain this, Levin Levin (2009) extended the traditional Born theory of ionic solvation to account for ionic polarizability. The new theory predicted that highly polarizable anions can actually prefer an interfacial solvation. In a followup work, Levin et al. Levin et al. (2009); dos Santos et al. (2010) used this theory to quantitatively calculate the surface tensions and the surface potentials of 10 different electrolyte solutions and to reproduce the Lyotropic, Hofmeister, series.
While almost all salts lead to increase the air-water surface tension, acids tend to lower it Weissenborn and Pugh (1996); Randles and Schiffrin (1966). The only explanation for this is a strong proton adsorption at the water-air interface Levering et al. (2007); Mucha et al. (2005); Jungwirth and Tobias (2006); Petersen et al. (2004); Petersen and Saykally (2005); Iyengar et al. (2005). It is well known that \ceH+ ion forms various complexes with water molecules Eigen (1964); Zundel (2000); Marx et al. (1999). It has also been suggested that the high surface adsorption of \ceH+ is related to the hydronium (\ceH3O+) geometry Petersen et al. (2004). This ion, has a trigonal pyramidal structure with the hydrogens located at the base of the pyramid Mucha et al. (2005). In this form hydrogens are good hydrogen-bond donors, while oxygen is a bad hydrogen-bond receptor Petersen et al. (2004). This favors hydronium ion to be preferentially located at the interface, with the hydrogens pointing towards the bulk water and the oxygen pointing into the gas phase Mucha et al. (2005). Explicit solvation energy calculations confirm this picture Dang (2003).
The sign of electrostatic surface potential difference is related with the relative population 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-water interface for acid solutions should be positive. The experiments, however, show that the surface potential difference for acids has the same sign as for halide salts, i.e. is predominantly negative Randles and Schiffrin (1966); Frumkin (1924). Frumkin Frumkin (1924) suggested that this apparently strange behavior might be a consequence of the incomplete dissociation of acid molecules. A different explanation was advanced by Randles Randles, J. E. B. (1963) who argued that presence of hydroniums at the interface leads to a preferential orientation of water molecules resulting in a dipole layer with a negative electrostatic potential difference across it. This conclusion is in agreement with the theory proposed in the present paper, as well as with the recent molecular dynamics simulations Mucha et al. (2005).
In this paper we present a theory that allows us to quantitatively calculate surface tensions of acid solutions using only one adjustable parameter related to the strength of the hydronium adsorption to the interface. Predictions of the theory are compared with the experimental measurements. The theory is then used to estimate the electrostatic potential difference across the water-air interface for various acid solutions.
Ii Model and theory
We consider an acid solution in a form of a drop of radius , where is the position of the Gibbs dividing surface (GDS) Ho et al. (2003); Levin et al. (2009). The water and air will be modeled as uniform dielectrics of permittivities and , respectively. The surface tension can be obtained by integrating the Gibbs adsorption isotherm equation:
where are the chemical potentials and are the de Broglie thermal wavelengths. In this equation sign corresponds to the hydronium ion, and sign to the anion. The bulk ion concentration is , where are the ionic density profiles. The ion excess per unit area due to existence of the interface is
where is the total number of acid “molecules”. The ionic density profiles, , will be calculated using a modified Poisson-Boltzmann (mPB) equation, as discussed later in the paper.
Anions are divided into two categories: kosmotropes and chaotropes. The theory of electrolyte solutions dos Santos et al. (2010) showed that chaotropes, Br, I, NO, and ClO, lose their hydration sheath near the GDS and are partially adsorbed to the interface. On the other hand kosmotropes, F, Cl, and SO remain hydrated in the interfacial region and are repelled from the GDS.
To bring an ion of radius to distance from the GDS requires Levin and Flores-Mena (2001):
of work. In this equation and is the inverse Debye length. The kosmotropic ions remain strongly hydrated in the interfacial region and encounter a hardcore-like repulsions from the GDS at a distance of one hydrated ionic radius. 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 avoid the large electrostatic energy penalty of exposing the charge to a low-dielectric (air) environment, the electronic charge density of a chaotropic anion redistributes itself so as to remain largely hydrated Levin (2009). The fraction of ionic charge which remains inside the aqueous environment, , can be calculated by minimization the polarization energy Levin (2009)
In the above equation is the relative polarizability defined as , where is the ionic polarizability, is the unhydrated (bare) radius, and . Performing the minimization we obtain
where 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 perturbes the hydrogen bond network of water molecules. For small ions, the free energy cost of forming a cavity is entropic and is proportional to the volume of the void formed Lum et al. (1999); Chandler (2005). As the ion moves across the GDS, its cavitational free energy decreases. This results in a short-range attractive potential between the anion and the GDS Levin et al. (2009):
where Å is obtained from bulk simulations Rajamani et al. (2005). For hard (weakly polarizable ions) the cavitational free energy gain is completely overwhelmed by the electrostatic free energy penalty of moving ionic charge into the low dielectric environment. For soft polarizable ions, however, the electrostatic penalty is small, since most of the ionic charge remain inside the aqueous environment. The total potential of a soft anion, therefore, has a minimum in the vicinity of the GDS, see Fig. 1.
The H ions (protons) do not exist as a separate specie in water. Instead they form complexes with water molecules, HO and HO Eigen (1964); Zundel (2000); Marx et al. (1999). Because of its favorable geometry (trigonal pyramidal), the hydronium ion (HO) adsorbs to the water-air interface with a preferential orientation Mucha et al. (2005) of oxygen towards the air. We model this attraction by a square well potential with a range of a hydrogen bond Å,
The depth of the potential is then adjusted to obtain the experimentally measured surface tension of HCl. The same potential is then used to calculate the surface tensions of all other acids. We should stress, however, that one should not attach too much meaning to the specific value of the potential depth. The real proton transfer is a quantum mechanical process, therefore there is bound to be some arbitrariness in how one models it at a classical level. 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 obtain an optimal fit of the surface tension of HCl solution. However once this is done, the values of the surface tension of other acids will not be significantly affected. Thus, the strength of H potential is the only free parameter of the theory. The total potential felt by H is then, , see Fig. 1.
While the kosmotropic anions feel only the potential and the hardcore repulsion from the GDS, the chaotropic anions are influenced by the total potential Levin et al. (2009):
In Fig. 1, we plot the potentials felt by various ions at M concentration, as a function of the distance from the GDS.
|Ions||chao/kosmo||radius (Å)||polarizability (Å)|
The ionic density profiles can now be obtained by integrating the mPB equation:
where is the Heaviside step function, is the density profile for chaotropic anions and for kosmotropic ones.
Once the ionic density profiles are calculated, the surface tensions can be obtained by integrating the Gibbs adsorption isotherm (eq 1). The ionic radii and polarizabilities are the same as were used in our previous work on surface tension of electrolyte solutions Levin et al. (2009); dos Santos et al. (2010). In Table 1 we summarize this data.
The depth of the potential (eq 7) is adjusted to fit the HCl experimental data Weissenborn and Pugh (1996) (see Fig. 2), this is the only adjustable parameter of the theory. We find that a square well potential of depth results in an excellent fit of the experimental data for HCl in the range of concentrations from to M. The excess surface tension of all other acids is then calculated using the same potential . The predictions for the surface tensions of HF, HBr, and HI are plotted in Fig. 2. Unfortunately, we have no experimental data to compare for these halogen acids. For HSO and HNO (see Fig. 3), we find a good agreement between the theory and experiment. For HClO the theory overestimates the surface tension. This is similar to what was found for sodium perchlorate salt dos Santos et al. (2010). The difficulty is that ClO is a large weakly hydrated ion. Since the cavitational energy grows with the cube of ionic radius, a small error in radius leads to a big error in surface tension.
Finally, we use the theory to calculate the electrostatic potential difference across the solution-air interface. The surface potential difference predicted by the present theory has a wrong sign compared to Frumkin’s experimental measurements — positive instead of negative Randles and Schiffrin (1966); Frumkin (1924). Positive sign reflects a strong adsorption of hydronium ions to the GDS. The simple dielectric continuum theory presented here, however, does not account for the structure of the interfacial water layer. Since the hydronium ion at the GDS has a preferential orientation with the hydrogens pointing towards the bulk, presence of many such ions will result in a dipole layer. Note that in the absence of hydroniums, the water dipoles predominantly point along the interface Kathmann et al. (2008). The hydronium layer produces an electric field , where is the number of hydroniums at the interface, is the water dipole moment, is the dipole length, and is the interfacial area. If we suppose that all the hydroniums are perfectly aligned, the potential difference across the dipolar layer will be . Using the dipole moment of a water molecule, D, we obtain the dipole layer contribution to the overall electrostatic potential difference. Adding this to , we obtain the total electrostatic surface potential difference across the solution-air interface. In Table 2, we list the surface potentials of various acids at 1M concentration.
|Acids||calculated [mV]||Frumkin Frumkin (1924) [mV]|
Clearly these values are an exaggeration of the total electrostatic potential difference across the interface, since at finite temperature there will not be perfect alignment of interfacial hydronium ions. Nevertheless, the theory should provide us an order of magnitude estimate of the electrostatic potential difference. In fact, for most acids we find a reasonable agreement between the predictions of the theory and Frumkin’s experimental measurements Frumkin (1924). A noticeable exception is the HF. Experimental potential for hydrogen fluoride measured by Frumkin is negative, while we find a large positive value. Frumkin’s value for HF is clearly outside the general trend for halogen acids. In his classical review of electrolyte solutions, Randles Randles, J. E. B. (1963) did not mention Frumkin’s result for HF acid, while discussing his other measurements. We can only suppose that Randles also did not have a complete confidence in this particular value. Experimental measurements of excess surface potentials are very difficult. This is probably the reason why Frumkin’s measurements of surface potentials of acids have not been repeated in over 90 years.
In this paper we have developed a theory for surface tensions of acid solutions. The hydronium adsorption to the interface was modeled by a square well potential, the depth of which is the only adjustable parameter of the theory. The agreement between the theory and experiments is very reasonable for different acid solutions at concentrations from 0 to 1M. In order to account for the experimental values of the excess electrostatic surface potential, we must require a preferential orientation of hydronium ion at the interface, with the hydrogens pointing into the bulk. With this assumption we get a qualitative agreement with the experimental measurements of the excess electrostatic potentials of various acid solutions. At the moment this is the only theory that can account (quantitatively) for the surface tensions and (qualitatively) for the surface potentials of acid solutions.
This work was partially supported by the CNPq, INCT-FCx, and by the US-AFOSR under the grant FA9550-09-1-0283.
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