Interactions between Small Inorganic Ions and Uncharged Monolayers on the Water/Air Interface

The interaction of several simple electrolytes with uncharged insoluble monolayers is studied on the basis of tensiometric and potentiometric data for the surface electrolyte solution|air. The induced adsorption of electrolyte on the monolayer is determined via a combination of data for equilibrium spreading pressure and surface pressure versus area isotherms. We show that the monolayer-induced adsorption of electrolyte is not only strongly ion-specific but also surfactant-specific. The comparison between the ion-specific effects on a carboxylic acid monolayer at low pH and an ester monolayer shows that the anion series follows the same order while the cation series reverses. The effect of the electrolyte on the chemical potential of the monolayer shows attraction between the surfactant and the ions at low monolayer densities, but at high surface densities, repulsion seems to come into play. In nearly all investigated cases, a maximum of monolayer-induced electrolyte adsorption is observed at intermediate monolayer densities. This suggests specific interactions between the surfactant headgroup and the ions. The Volta potential data for the monolayers are analyzed on the basis of the equations of quadrupolar electrostatics. The analysis suggests that the ion-specific effect on the Volta potential is due to the ion-specific decrement of the bulk dielectric constant of the electrolyte solution. Moreover, we present evidence that in most cases the effect of the electrolyte on the orientation of the adsorbed dipoles cannot be neglected. Instead, the change in the ion distribution in the electric double layer seems to have a small effect on the Volta potential.


S1 Experimental data
: References to the original sources of the experimental data used in the paper.

S2.1 Activity coefficients
The conversion between molal activity a el,m and molality C el,m is done according to the relation a el,m = γ el,m C el,m The molal based activity coefficients of the electrolytes were calculated with an empirically corrected Debye-Hückel law where z i are the ion's charges and I m is the molal ionic strength of the solution The coefficient A is calculated according to S15 where e is the elementary charge, N a is Avogadro's constant, ϵ 0 is the dielectric permittivity of vacuum and ρ w is the density of water. The electrolyte specific coefficients B and β i were collected from the literature (refer to table S1). The values of the parameters are presented on table S2.

S2.2 Molarities
The conversion between molarities C el,M and molalities of the solutions was done using where V el is the partial molar volume of the electrolyte and S ρ is an empirical parameter compensating for solute-solvent interactions. Both can be found on table S2.

S2.3 Osmotic pressure
The osmotic pressure is by definition where Using equation S2, γ w,x can be found analytically where k is = I m /C el,m . Figure Figure S1: Comparison of monolayer-induced adsorption of electrolyte ∆Γ el calculated by differentiating an explicitly interpolated function (solid lines) and differentiating numerically with two/three point finite difference (points).

S2.4 Differentiation of ∆π sp
To make the results easier for comparison, the approach, opted for here, was to interpolate the ∆π sp (p osm ) data with a polynomial before differentiating analytically. The choice of explicit interpolation over finite point difference is motivated by the smoothing of the experimental noise, characteristic for the method. The interpolation was done with a 1 -st or 2 -nd degree polynomial, with a fixed intercept ∆π sp (0) = 0. A higher than 2 -nd degree polynomial was excluded from the consideration. A binome was chosen only if: 1. There are more than three experimental points. That is to impose at least one degree of freedom for the fit.
2. The standard deviation s of the linear fit is larger than 0.1 mN/m. Deviations of 0.1 mN/m or less are comparable to the experimental uncertainty. In that case raising the degree of the polynomial risks overfitting the experimental noise.
3. The standard deviation of the quadratic fit is 20% or more lower than the standard deviation of the linear fit. This is to ensure that there is a reasonable improvement to be gained at the cost of more free parameters.
The fits are compared to the experimental data graphically on figure S2. The fitting parameters and standard deviations are presented in table S3. The relevant fitting function is: From the comparison of the calculated ∆Γ el using the interpolated lines and with finite point difference, the error of the method is evaluated (see S1a). The vertical distance between the points and lines was averaged over all electrolytes and points to give an error of 0.026 nm −2 . That is quite small compared to the characteristic monolayer-induced adsorptions ∆Γ el calculated. However, it should be kept in mind that the average error is weighted heavily by well resolved data, e.g. CaCl 2 on OA. The actual error is much larger for systems with only a few data points, e.g. KCl on ES.

S2.5 Calculating ∆µ s
To integrate the isotherms, they were first interpolated explicitly. The OA isotherms were described as a polylog function comparison of the fits it presented in figure S3.
The EP data are harder to interpolate as there is a phase transition around 10 mN/m.
At equilibrium the phase transition should be described by horizontal line connecting the LE and LC phases. However, the interactions between the domains in the heterogeneous region (Het) determine slower kinetics of compression. S16,S17 As such, the measured points between LE and LC are rarely close enough to equilibrium. Here the Het region is fitted with a 4 th degree polynomial S Het (π) = c 4 π 4 + c 5 π 3 + c 6 π 2 + c 7 π + c 8 (S11) In the final reconstruction of the isotherms the phase transition region is replaced by a horizontal line π = π pt . Furthermore, the dynamics of the Het region also effects the first S-9   points in the LC region. S16,S17 Thus, only the last two points of the isotherm are used to describe the LC region. They were fitted with a straight line The line is then extended to intersect with π pt . The function is once again forced to go thought the point (S sp , π sp ). The general procedure is as follows: 1. The points visually determined bellow the phase transition region are fitted to Equation S10 with π ref = π sp and S ref as a free parameter.
2. The highest two points are fitted to a straight line (equation S12).
S-10 3. The data points left in-between are fitted to a polynomial of the fourth degree (equation S11).
4. The intersection point between the fit for the LE region and the Het region is determined as the phase transition pressure π pt and area S pt .
5. The data point division for the fitting procedures are checked with regards to π pt and if inconsistencies arise items 1 to 5 are redone.
The final fitting parameters are tabulated in table S5. Figure S4 shows a graphical comparison of the final fits with the experimental data.  Formulas S10, S11 and S12 can be analytically integrated to calculate the chemical potentials. When we account for the different phases equation 23 becomes for OA ans EP respectfully. The resulting potentials as a function of the surface pressure can be seen in figure S5.

S2.6 Differentiation of ∆µ s
Unfortunately, formulae S13 and S14 can not be differentiated analytically with respect to the electrolyte concentration. In order to do that the parameters c i 's functional dependence on the concentration/osmotic pressure needs to be determined. The current data is too sparse for that. Thus, ∆µ s was calculated from each isotherm and then differentiated numerically using second order (three point) central finite difference. The two phases of EP further complicate the differentiation. To calculate (∂∆π/∂p osm ) µs or (∂∆µ s /∂p osm ) π the surfactant should be in the same phase at all three concentrations. This is illustrated on S6.

S3.1 Adsorption of H + at oleic acid monolayers
Oleic acid (HOa) at W|A dissociates leading to charging of the monolayer.
The equilibrium condition is where C H + = 10 −2 M is the bulk concentration of the protons, and ϕ S is the surface charge.
We will assume that the surface acid dissociation constant of oleic acid K HOa is equal to the bulk one (approx. 10 −5 M). Γ Oa -and Γ HOa are the surface concentrations of the dissociated and undissociated oleic acid, respectively. They are related trough the mass balance: where Γ HOa,t is the total adsorption of oleic acid. Furthermore, the surface charge eΓ Oa -and potential are related as N a e 2 Γ 2 Oa -2εkT C H + = e −eϕ S /kT + e eϕ S /kT − 2 (S17) Equations S15, S16 and S17 can be solved numerically to find the three unknowns Γ Oa -, Γ HOa and ϕ S . At monolayer density Γ HOa,t = 1/25 Å −2 (dense monolayer) and 25 • C, only about 0.1% of the monolayer is dissociated. This gives rise to −0.5 mV surface potential and 0.004 nm −2 adsorption of H + . For context, for most electrolytes, in the concentration range we are interested in, the electrolyte adsorptions on W|OA is in the order of e.g. −0.5 nm −2 . S-14

S3.2 Excess size of the depletion layer
Provided the depletion and diffuse layer do not overlap, one can divide the ion surface excess Γ i into two parts: non-electrostatic contribution Γ sp i , due to the image potential u im,i and an ion-specific interaction potential u i acting on the i-th ion, and diffuse ion layer contribution Γ diff i , due to the charging of the interface as a result of Γ sp i . Within the linear Gouy theory it can be shown that the diffusive contributions cancel out from the electrolyte excess, i.e.
Γ el is determined from the non-electrostatic interactions only, Γ el = i Γ sp i / i ν i . S18 Moreover, at concentrations above 0.3 M, the image forces can be neglected due to electrostatic screening (Aveyard's approximation). If we finally assume that the interaction potential u i is concentration independent, the negative slope R * el ≡ −∂∆Γ ϵ el /∂C el,M (see figure 7) can be found as: The integrals under the sum in equation S18 have dimension of length and can be thought of as negative excess size of the ion depletion layer R * i in the presence of monolayer. Therefore, the quantity R * el is the mean excess size of the electrolyte depletion layer, related to the ion characteristic R * i as νR * el = i ν i R * i . The results in figure 7 are in agreement with the prediction of a linear slope up to concentration 2 mol/kg. At higher concentrations the interaction potential u i seems to be concentration dependent. The calculated R * el , with V s = 32 mL/mol, are presented in table S6. In the MS and similar models the W|A surface is defined as the plane of discontinuity of the medium properties. When it comes to the image forces the relevant property is the dielectric permittivity ε of the medium. It is represented as a stepwise function from ε 0 in the air phase to ε w ≈ 78ϵ 0 in the water phase (see figure S7b). It has been experimentally found that the distance between the plane of ε discontinuity and the average position of the outside edge of the uppermost layer of water molecules, i.e. the hydrophobic gap, is approximately one effective water molecule radius. Thus, to ensure agreement between the theory and the experiment, the plane of ε discontinuity is positioned in the middle of the uppermost layer of water molecules S18,S19 (see figure S7a). The position of the plane of ε discontinuity is relevant not only for the image forces acting on the dissolved ions, but also tightly related to the immersion of a surfactant molecule. An amphiphile molecule can be divided into polar and apolar segments (head and tail). The dividing point between them we call the hydrophilic-lyophilic centre (HL centre). Here we make the assumption that, when introducing a surfactant molecule in the system, the HL centre positions itself on the plane of ε discontinuity. However, the addition of the surfactant changes the ε profile and offsets the plane of ε discontinuity, thus indirectly effecting the electrolyte. In this section we are interested in roughly estimating this shift.
Fist of all, we neglect any correction on the position of the surfactant molecules themselves. We assume, the average position of the HL centre is the same for diluted and con- (d) Figure S7: (a and c) schematic representation of the structure of the interface W|A and W|M, respectively. (b and d) approximated stepwise ε profile through W|A and W|M, respectively.
S-17 thermore, we assume that each segment has the same dielectric response as a bulk phase of the closest substance, e.g. the head of an aliphatic acid behaves as formic acid ε ≈ 58ε 0 .
Thus, defining the ε profile boils down to finding the the dielectric permittivities of the two intermediary layers. That can be done with the Clausius-Mossotti relation. However, in the interest of simplicity and brevity, we will instead assume a linear relationship between ε and the concentrations: where α i is the molecular polarizability of the i -th component and the summation is over all constituents. Using equation S19 one can express the polarizabilities α i through the dielectric permittivity ε ∞ i and molar volume V ∞ i of the pure substance, e.g. for alkane Thus, we can find the dielectric permittivities of the head and tail layers as: where we used the condition i C i (z)V i = 1 to remove the concentration of water in the head layer. Here C tail and C head are the concentrations of tails and heads in their respective layers.
The plane of ε discontinuity is located at some distance d from the origin (see figure S8).
Since the plane of ε discontinuity corresponds to the plane of ε zero excess, the following condition holds: Within the current model, the last equation can be restated as: Combining that with equations S20 and S21 gives an explicit solution for the shift of the plane of ε discontinuity: As it can be seen within these rough approximations the shift of the plane of ε discontinuity is a linear function of the surfactant adsorption Γ s . The closer the segments are to the two respective mediums in terms of dielectric permittivities, the smaller d. The maximum possible shift is at maximum monolayer density, which we assume is the equilibrium spread monolayer Γ s,sp . The product Γ s,sp V ∞ i is simply the characteristic length of the segment, e.g. Γ s,sp V ∞ formic acid ≈ 3.5 Å. Furthermore, the fraction (ϵ ∞ i − ϵ j ) / (ϵ w − ϵ 0 ) is generally small. The resulting shift d sp is about an order of magnitude lower than the effective water radius R w , e.g. for OA d sp ≈ 0.3 Å (see figure S7d). Thus we can conclude, that any effect coming from the offset of the plane of ε discontinuity is quite small.

S3.3.2 Depletion layer thickness
Within the MS model the work required for an ion to penetrate into the topmost layer of molecules is assumed infinite. Thus, for a monovalent ion i on W|A the thickness of the depletion layer R i is simply (see figure S7a) This simple assumption works incredibly well and allows for the qualitative prediction of the surface tension of a large set of systems. When a dense monolayer is spread on the surface, the topmost layer is now of surfactant molecules, that have a different size. Taking into account that R i is defined in relation to the plane of ε discontinuity, it can be found as (see figure S7c) The correction d is small. On the other hand, L 0 head could be quite larger than R w , e.g. for OA L 0 head is approximately 3.5 Å compared to R w = 1.4 Å. The difference ∆R i ≡ L 0 head −R w ∼ 2 Å results in monolayer-induced adsorption ∆Γ i ∼ −∆R i C i ∼ −0.12 nm −2 , which is comparable with the ∆Γ el calculated above.
For a monolayer of intermediate densities, some ions will be underneath water molecules and some underneath surfactant molecules. Therefore, R i is a weighted averaged value. If we assume there is no specific interaction between the ions and surfactant molecules, the thicknesses are simply wighted by the surface coverage θ = Γ s /Γ s,sp Note that R i,av is also a linear function of Γ s . Thus, the larger size of the surfactant molecule on its own can not explain the complicated relationship between ∆Γ el and Γ s presented in figure 11. S-20

S3.3.3 Partial molar volume
Lets find the volume V M d between an arbitrary plane M (see figure S9) and the plane of ε discontinuity. Only a part of the surfactant molecule is inside the slice M d. That part we characterise with a volume V s (d). Then, the volume the surfactant molecules occupy in the slice M d is V s (d)n s , where n s is the number of molecules on the surface. Applying the same logic for the water molecules, we can express V M d as This is equivalent to the condition 14. Dividing both sides by V M d , we get where C M d . This allows us to write Here V w (0) is a constant, characteristic of water. We assume that V w (0) is equal to the bulk molecular volume of water molecules. For the latter we know that V w C w = 1. If we subtract S-21 that from equation S30 and integrate over the depth z, we get This is a version of equation 15 in the absence of electrolyte. The surfactant molar volume V s that appears in equation 15 and all that follow it is defined by this derivation as the volume of a surfactant molecule penetrating beneath the plane of ε discontinuity at maximum surface density. It can be found as the depth of penetration L 0 head − d sp times the cross-sectional area of the head group. If we assume the cross-sectional area of the carboxylic group is 16.5 Å 2 , the molar volume of OA is approximately 32 mL/mol. That is just below the molar volume of formic acid (35 mL/mol).