Nanomolar surface-active charged impurities
account for the zeta potential of hydrophobic
surfaces
Yuki Uematsu,∗,†,‡ Douwe Jan Bonthuis,¶ and Roland R. Netz§†Department of Physics, Kyushu University, 819-0395 Fukuoka, Japan‡Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS,
Sorbonne Université, Université de Paris, F-75005 Paris, France¶Institute of Theoretical and Computational Physics, Graz University of Technology, 8010
Graz, Austria§Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
E-mail: uematsu.yuki@phys.kyushu-u.ac.jp

Abstract
The electrification of hydrophobic surfaces is an intensely debated subject in physical chemistry. We theoretically study the ζ potential of hydrophobic surfaces for varying pH and salt concentration by solving the Poisson-Boltzmann and Stokes equations
with individual ionic adsorption affinities. Using the ionic surface affinities extracted
from the experimentally measured surface tension of the air-electrolyte interface, wefirst show that the interfacial adsorption and repulsion of small inorganic ions such2−
as H3 O+ , OH− , HCO−
3 , and CO3 are irrelevant for the ζ potential observed in ex-

periments because the surface affinities of these ions are too small. Even if we takehydrodynamic slip into account, the characteristic dependence of the ζ potential on1

pH and salt concentration cannot be reproduced. Instead, to explain the sizable experimentally measured ζ potential of hydrophobic surfaces, we assume minute amountsof impurities in the water and include the impurities’ acidic and basic reactions withwater. We find good agreement between our predictions and the reported experimental
ζ potential data of various hydrophobic surfaces. Our theory suggests that the impurities consist of a mixture of weak acids (pKa = 5 to 7) and weak bases (pKb = 12) at
a concentration of the order of 10−7 M.

Introduction
The ζ potential is derived from experimentally measured electro-osmotic or electrophoretic
mobilities using the Helmholtz-Smoluchowski equation and within this simple theoreticalframework describes the electrostatic potential at the shear plane. 1,2 Because the ζ potential
is an important and useful quantity for characterizing the properties of electriﬁed surfaces,
theoretical and experimental studies of the ζ potential have been performed for over a hundred years. For most metal oxides, such as silica, 3 aluminium oxide, 4 and titanium oxides, 4
the surface charge originates from the protonation and deprotonation of the oxide surface,
and the pH determines the surface charge density by charge regulation. 5 Whereas a quantitative description of the surface charge and the ζ potential of these metal oxides is welldeveloped, the surface charge of other surfaces is still poorly understood.
In particular, solid surfaces such as Teﬂon AF (amorphous ﬂuoropolymer), 6 carbons 7–10
and boron-nitride, 11 sulﬁde minerals, 12–15 and silver halides 16,17 exhibit a negative ζ potential
at neutral pH and an isoelectric point located around pH = 1 to 4. 18 The contact angle ofwater droplets at these surfaces approaches, or exceeds, 90◦ , 19,20 meaning that these surfaces
are hydrophobic. The ζ potentials of gas bubbles 21–35 and oil droplets 36–45 show behavior
very similar to these hydrophobic solid surfaces, suggesting that the electriﬁcation of solid,
liquid and gaseous hydrophobic surfaces is governed by a universal mechanism. The natureof this mechanism, however, is still an open question in physics and chemistry 46–49 because2these interfaces are chemically inert, i.e. they do not have dissociable groups.
A widely used explanation for the puzzling observation of negative ζ potentials at hydrophobic surfaces is hydroxide ion (OH− ) adsorption, 42,50–54 because ideal puriﬁed waterat neutral pH includes only hydroxide ions as anions. This explanation is supported by
the potential of mean force of OH− ions on air, graphene, and boron nitride surfaces calculated by ab-initio molecular dynamics simulations 53–56 and by surface-sensitive molecular
spectroscopy. 57 However, it contradicts experimental measurements of the surface tensionof NaOH solutions at the air interface, because the surface tension of a NaOH solution ishigher than the surface tension of a NaCl solution of the same concentration, whereas thechloride ion (Cl− ) is usually considered to be surface-inactive. 58 This experimental ﬁnding
implies that OH− is more strongly repelled from the air-water interface than Cl− . The same
experiments show that H3 O+ weakly absorbs onto the air interface because most acid solutions decrease the surface tension. 49,58 H3 O+ adsorption is supported by MD simulations
that predict the potential of mean force on the air surface 58–60 and surface-sensitive molecular spectroscopy. 61 Therefore, based on experimental surface tension data, if only H3 O+
and OH− are present in water, the ζ potential should be positive, which contradicts theexperimental observations. Other explanations for the negative ζ potential suggested in
the literature are polarization of the interface, 62 bicarbonate adsorption, 48 charge transfer
between water molecules, 47,63 and impurity eﬀects. 46,49,64
Assuming a charged sphere (or capillary) with the surface potential ψ0 immersed inan electrolyte solution, the linearized electro-hydrodynamic equations determine the electrophoretic (or electro-osmotic) mobility µ as µ = (εε0 ψ0 /η)f (ψ0 , κR, {λi }, . . . ), where ε isthe dielectric constant, ε0 is the vacuum permittivity, η is the solution viscosity, κ is theinverse of the Debye length, R is the radius of the sphere (or capillary) and λi is the molar
conductivity of ions of type i. The function f is model-dependent, and analytical expressions
are known only for limiting cases. For solid particles, f = 1 for the planar limit κR → ∞,corresponding to the Helmholtz-Smoluchowski formula, 65 while f = 2/3 for the point limit3κR → 0, which is the Hückel formula. 66 For arbitrary κR and ψ0 , f can be calculated numerically. 67,68 Note that the electrophoretic and electro-osmotic mobilities depend on theviscous properties of the interfacial layer, which is of particular importance for gas bubbles
and oil droplets, where f depends on the gas or oil viscosity. 69–71
150(a) metal oxides(b) hydrophobes

(a)

100

ζ [ mV ]50(b)alumina

Teflon AF

titania

octadecane

silicahydrogen

0-50

0.1 mM

10 mM

1 mM

100 mM

-100
-150
2

4

6

8

10

12

2

4

6

8

10

12

100

(c) air, 10mM(d) hexadecane, 1mM(c)ζ [ mV ]

50

0

(d)

air-1

hexadecane-1

air-2

hexadecane-2

air-3

hexadecane-3

air-4

hexadecane-4

-50

-100
2

4

6

8

10

12

2

4

6

pH

8

10

12

pH

Figure 1: Collection of experimental ζ potentials of (a) metal oxides: alumina, 4 titania, 4 andsilica, 3 (b) hydrophobic systems: Teﬂon AF surfaces, 6 octadecane droplets, 43 and hydrogen 31
and (c) air bubbles in 10 mM NaCl solution: air-1, 35 air-2, 35 air-3, 33 and air-4, 26 and (d)hexadecane droplets in 1 mM NaCl solution: hexadecane-1 and -2, 46 hexadecane-3, 45 andhexadecane-4. 41 In a and b, the colors represent diﬀerent bulk salt concentrations, whereas
the shapes of the symbols represent the diﬀerent surface types. The broken lines are guides
to the eye. More details are given in Appendix A.
In experiments, the electrophoretic mobility is measured and converted into the surface
potential using a model-dependent mobility formula. Typically, the Helmholtz-Smoluchowski
formula is used for this. Since the Helmholtz-Smoluchowski formula neglects ion adsorption
at the surface as well as interfacial dielectric anomalies, the converted surface potential is
not identical to the real surface potential. Furthermore, surface slip and the inhomogeneity4

of the surface viscosity complicate the deﬁnition of the shear plane. Therefore, the converted
quantity is called the ζ potential instead of surface potential. In Fig. 1ab we show a collectionof experimental ζ potentials of metal oxides and diﬀerent hydrophobic surfaces for diﬀerent
salt concentrations as a function of pH. For metal oxides, the ζ potentials decrease as the
pH increases and exhibit isoelectric points where the ζ potential vanishes. The isoelectric
points depend on the type of metal oxide, but are almost independent of the salt concentration. The ζ potentials of the hydrophobic systems corresponding to Teﬂon AF surfaces,
octadecane droplets and hydrogen bubbles are plotted in Fig. 1b. They all exhibit isoelectric points around pH = 4 18 and show a similar dependence on pH and salt concentration.
Naive comparison of the data in Figs. 1a and 1b would suggest that the surface charge onhydrophobic surfaces is similar to the one on silica, which is puzzling since silica has dissociable surface groups, in contrast to the hydrophobic surfaces. In Figs. 1c and 1d, we show
a collection of experimental ζ potentials of air bubbles in a 10 mM NaCl solution 26,33–35 andhexadecane droplets in a 1 mM NaCl solution. 41,45,46 Even though the materials are the same,the experimental ζ potentials show a large spread, which implies that the electriﬁcation of
the hydrophobic surfaces is governed by an uncontrolled factor in the experiments such asmeasuring techniques and protocols, methods of synthesis or puriﬁcation of chemicals.
The ζ potentials of hydrophobic surfaces have been quantitatively explained by assuming
OH− adsorption. 50 The ﬁt of that model to the experimental data yields an interfacial ionicproduct of the H3 O+ and OH− concentrations around 6, 50 which is substantially lower thanthe bulk value of 14. A simple calculation demonstrates that the surface adsorption energy of
OH− in this case equals −(14−6) ln 10×kB T = −18.4kB T (the minus sign means adsorption),
which is even larger than the adsorption energy of typical ionic surfactants and, as mentioned
above, contradicts the experimental surface tension data of bases. In fact, using experimental
surface tension data and assuming an adsorption layer thickness of 0.5 nm, we have previously
estimated the surface adsorption energy to be −15.6kB T for the surfactant dodecylsulfate,+1.6kB T for OH− and −0.9kB T for H3 O+ , which means that OH− ions are repelled from5

and H3 O+ ions are slightly attracted to the air-water interface. 72
In this paper, we propose the presence of minute quantities of basic and acidic impurities
to explain the experimental ζ potentials of hydrophobic surfaces in a way that is consistent
with the ion aﬃnities extracted from surface tension measurements. In a previous calculation,
we have proposed acidic impurity eﬀects, 49 which reproduces the correct trend of the ζpotential as a function of pH and salt concentration, as well as the Jones-Ray eﬀect, 72,73 thedisjoining pressure of the water wetting ﬁlm on metal oxide surfaces, 49,74 and other featuresof the air-water interface. 75–78 Here, we ﬁrst demonstrate that for surface aﬃnities that areconsistent with experimental electrolyte surface tension data, interfacial eﬀects that involve
water ions and dissolved CO2 cannot explain the experimentally measured ζ potentials ofhydrophobic systems even if hydrodynamic slip eﬀects are taken into account. In a second
step we introduce surface-active charged impurities in the solution and perform a detailed
analysis of their eﬀect on the ζ potential. Finally, we study the eﬀects of the impurity
concentration and the impurity pKa and pKb values and propose a consistent model for theexperimentally measured negative ζ potential of hydrophobic surfaces.

Model
In this section, we ﬁrst introduce the governing electrostatic and hydrodynamic equations fora planar surface. We construct a model to calculate the ζ potential of hydrophobic surfaces
including the adsorption of water ions, salt ions and charged impurities. All surface aﬃnities
of ions are extracted from experimental surface tension data or from molecular dynamics
simulations. Finally, we compare the calculated ζ potentials to experimental data on planar
Teﬂon AF surfaces, 6 which have the advantage that it can be assumed that ions do notpartition into the hydrophobic material. For oils, in contrast, both inorganic and organic
ions can penetrate from the aqueous phase into the oil phase, 79 and ions naturally exist inambient air at a concentration of ∼ 103 /cm3 (∼ 10−17 M). 80

6

Electrostatic and hydrodynamic equations
We consider an interface located at z = 0 between a hydrophobic material (z < 0) andan electrolyte solution (z > 0). Diﬀerent types of mobile ions are present in the electrolyte
solution, whereas we assume that no ions are present in the hydrophobic material. To obtain
the ion distribution and the electrostatic potential proﬁle, we solve the Poisson-Boltzmann
equation given by 81ddε⊥ (z)ε0 ψ(z) = −ρ(z) for z > 0,
dz
dz

(1)

where ε⊥ (z) is the solution dielectric proﬁle near the interface, ε0 is the electric permittivityof vacuum, ψ(z) is the electrostatic potential. The right hand side of eq. 1 is the charge
density, which we assume to obey the Boltzmann distribution including ionic adsorption
potentials Ui (z),ρ(z) = eXqi cbi e−eqi ψ(z)/kB T −Ui (z) ,(2)iwhere e is the elementary charge, i refers to the ion type, qi is the charge in units of e, cbi isthe ion bulk concentration and kB T is the thermal energy. When we apply an electric ﬁeld
Ex in tangential direction to the surface, the Stokes equation for the tangential coordinate
in the absence of a pressure gradient is given byddη⊥ (z) ux (z) + ρ(z)Ex = 0,
dz
dz

(3)

where η⊥ (z) is the viscosity proﬁle for the tangential shear ﬂow, and ux (z) is the solution
velocity in the tangential direction.
In order to simplify the solution of eqs. 1-3, we use box proﬁles for the interfacial proﬁlesε⊥ (z), Ui (z), and η⊥ (z), the parameters of which have been previously obtained by molecular7

dynamics simulation, 82

ε⊥ (z) = ε + (εint − ε)θ(z − z ∗ )(4)

Ui (z) = αi θ(z − z ∗ )(5)η⊥ (z) = η + (ηint − η)θ(z − z ∗ ),(6)

where ε is the bulk dielectric constant of the solution, εint is the interfacial dielectric constant,
αi is the adsorption energy of an ion of type i, η is the bulk viscosity of the solution, ηint
is the interfacial viscosity. For simplicity, the width of the interfacial layer z ∗ is assumed tobe identical for the dielectric proﬁle, the ionic potentials of mean force, and the viscosity
proﬁle, which is a quite realistic assumption when compared with simulation results. 82 Whenelectrostatic interactions are neglected, the integrated surface excess of type i ions is given byz ∗ (e−αi −1)cbi , which is equivalent to the linear version of the Langmuir adsorption isotherm, 83
the term in front of the bulk concentration is equivalent to the adsorption coeﬃcient.
Eq. 1 is solved with the boundary conditions dψ/dz|z=0 = 0 and ψ|z→∞ = 0, which reﬂect
the zero-charge condition at the surface and the absence of an electrostatic potential in bulk.
Eq. 3 is solved with the boundary conditions ux |z=0 = 0 and du/dz|z→∞ = 0, which correspond to the no-slip boundary condition at the surface and the zero-shear condition in bulk.
A boundary condition with ﬁnite surface slip length is regularly used for the hydrodynamic
boundary condition at hydrophobic surfaces. 84 We incorporate surface slip eﬀects at solid
surfaces by using the no-slip boundary condition at z = 0 together with a decreased viscosity
in the interfacial layer.
In planar geometry, the electro-osmotic mobility is deﬁned by the bulk velocity divided
by the applied electric ﬁeld, µ = ux |z→∞ /Ex , and the ζ potential is deﬁned by ζ = −ηµ/εε0 .
From the solution of eqs. 1 and 3, the ζ potential is given by, 81
εint η
εint ηζ=ψ∗,
ψ0 + 1 −
εηint
εηint
8

(7)

where ψ ∗ = ψ|z=z∗ and ψ0 = ψ|z=0 . For the case ε⊥ (z) = ε and η⊥ (z) = η, i.e. when there
are no interfacial dielectric and viscosity eﬀects, the ζ potential is identical to the surface
potential, ψ0 .

Parametrization of the experimental system
KCl-HCl-KOH solutions and pH effects
In the experiments on Teﬂon AF surfaces, 6 KCl was used as the electrolyte and HCl and
KOH were used for the adjustment of the pH. Water dissociates into H3 O+ and OH− ionsaccording to the reaction
+
−
b
b
◦2
2H2 O −↽−−⇀
− H3 O + OH , cH3 O cOH = Kw c ,(8)with the ionic product Kw = 10−14 and the standard concentration c◦ = 1 M. In eq. 8, as
well as in the following eqs. 11, 12, and 15-18, we use the ideal-gas approximation for theion activities. Within this approximation, the pH is deﬁned by pH = − log10 (cbH3 O /c◦ ).
The control parameters in the experiments are the bulk hydronium ion concentration,
cbH3 O , and the bulk salt concentration, cbsalt . The hydroxide ion concentration is determinedby eq. 8, but the potassium and chloride ion concentrations are normally not equal to cbsalt
because HCl and KOH added for pH adjustment contain potassium and chloride ions. Solving
the charge balance equation, we can determine the concentrations of potassium and chloride
ions as follows. We deﬁne the charge-weighted sum of concentrations (excluding KCl) byP
C = i6=K,Cl qi cbi , by which we obtain
cbK = cbsalt , cbCl = cbsalt + C for C > 0,(9)cbK = cbsalt − C, cbCl = cbsalt for C < 0.

(10)

and otherwise

9Eqs. 9 and 10 hold even when impurities and (bi)carbonate ions are present in the solution.
Acidic and basic impurities
Although ultrapure water was used in the experiments, 6 we assume that trace amounts ofsurface-active charged impurities are likely to be present in the water. We consider two kindsof impurity, AH and B. AH is an acidic impurity which is deprotonated according to−+
AH + H2 O −↽−−⇀− A + H3 O ,cbA= 10pH−pKa ,
b
cAH

(11)

where Ka is the acid reaction constant. B is a basic impurity which is protonated accordingto+−
B + H2 O −↽−−⇀− BH + OH ,cbBH
= 1014−pH−pKb ,
cbB

(12)

where Kb is the basic reaction constant; alternatively, the acid reaction constant pKa of the
protonated base equals 14 − pKb . The impurities are assumed to be surface-active and thetotal concentrations of acids and bases, deﬁned by
cbA,tot = cbA + cbAH ,(13)

cbB,tot = cbB + cbBH ,

(14)

and

respectively, are used as the adjustable model parameters.
Dissolution of carbon dioxides
Because the Teﬂon AF experiments were performed using degassed ultrapure water under
nitrogen atmosphere, 6 we do not need to consider the dissolution of carbon dioxide. However,other experiments have been done in ambient air conditions, 31 and therefore we also treat
the eﬀect of CO2 dissolution. When water is in equilibrium with the ambient atmosphere,10

carbon dioxide dissolves into the water according to the reaction

CO2(gas) −↽−−⇀− CO2(aq) , Hcc =

cbCO2
,
cair
CO2

(15)

where the Henry constant is given by Hcc = 0.86 and the ambient concentration of CO2 is−5given by cair
M. 85 The dissolved CO2 reacts with the water according to theCO2 = 1.6 × 10

following reactions−−
CO2(aq) + H2 O ↽−⇀− H2 CO3(aq) ,cbH2 CO3= KCO2 ,
cbCO2

(16)

−
+
H2 CO3(aq) + H2 O −↽−−⇀− HCO3 + H3 O ,cbHCO3 cbH3 O= KH2 CO3 c◦ ,cbH2 CO3

(17)

2−
+
−−⇀
HCO−
3 + H2 O ↽−− CO3 + H3 O ,cbCO3 cbH3 O= KHCO3 c◦ ,cbHCO3(18)

where the equilibrium constants are KCO2 = 2.6 × 10−3 , KH2 CO3 = 1.7 × 10−4 , and KHCO3 =4.7 × 10−11 . 85 The pH of water with dissolved CO2 follows as pH = 5.6 using the chemical
equilibrium eqs. 8 and 15-18, which agrees well with the experimental pH of aqueous solutions
that are in contact with ambient air. 74,85
Interfacial properties
For a hydrophobic diamond surface, we have previously determined εint /z ∗ = 8.3 /nm 86 anda slip length of b = 2.1 nm (with ηint = η) 87 from molecular dynamics simulations. Fixingz ∗ = 0.5 nm, we obtain from these values εint = 4.2 and ηint /η = 0.2 (assuming vanishing
surface slip, b = 0). For Cl− and Na+ , we have determined αCl = 1.0 and αNa = 1.2 byintegrating the potential of mean force obtained by molecular dynamics simulation. 88 Allother αi values are determined by ﬁtting the experimental air-water surface tension using
11the present model with εint = ε (see Appendix B where we determine the surface aﬃnities
of carbonate ions). 49,72 Regarding the interfacial viscosity, we consider ηint 6= η only insection III.A.1 and Appendix C, and otherwise ηint = η. Since the eﬀect of a modiﬁed
interfacial dielectric constant is not signiﬁcant at small surface charge density, as shown inAppendix D, we use εint = ε in the main text. The surface aﬃnities for A− and BH+ are
for concreteness taken from dodecylsulfate and dodecyldimethylammonium data, without
implying any speciﬁc identities of the impurities. The surface aﬃnities of all ions are shownin Table 1.
Table 1: Ion-speciﬁc surface aﬃnities to the air-water interface.
Ion
H3 O +
OH –
HCO−
3
CO2−
3
Na+
K+
Cl –
A–
BH+

Ref.

αi
−0.9
1.6
−0.4
1.4
1.2
1.2
1.0
−15.6
−14.5

72
72
72
Appendix B
72, 8849
72, 8872
72

Results and discussion
In this section, we solve the model described in the last section and present the ζ potential forvarying pH and bulk salt concentration. We also investigate the eﬀects of CO2 dissolution,
slip, and the presence of impurities. First, we solve the chemical equilibrium equations for
all ions, eqs. 8 to 18, and determine the bulk concentration of each ion type. Then, we
solve the Poisson-Boltzmann equation (eq. 1) using the analytic solutions for monovalention species. 81 When we consider divalent CO2−
3 originating from CO2 dissolution, we employ
a numerical solution of eq. 1. To calculate the ζ potential, we do not need to solve the Stokes
equation (eq. 3), but substitute ψ0 and ψ ∗ into eq. 7.

12

10

0.1

10

0

(b) without CO2+

K(c) with CO2

Cl-

0

H3O+
OH-

-0.1

ci [ M ]ζ [ mV ]

5

b

-0.2
1

0

5

7

9 11

-2

10

-4

HCO-3
CO23

without / with CO2
0.1 mM
1 mM
10 mM

-5

(a)

3

10

1

3

5

7

9

11

1

pH

3

5

7
pH

9

11

1

3

5

7

9

11

pH

Figure 2: (a) ζ potential of the hydrophobe-KCl solution interface as a function of pH
calculated with (broken lines) and without (solid lines) CO2 -dissolution eﬀects. The inset
shows a magniﬁcation. We use T = 298 K, z ∗ = 0.5 nm, εint = ε = 78, ηint = η, and all αiare summarized in table. 1. (b) Plot of the ion bulk concentrations as a function of pH for
the 0.1 mM KCl solution without CO2 . (c) Plot of the ion bulk concentrations as a functionof pH for the 0.1 mM KCl solution with CO2 .

In the absence of impurities
First, we calculate the ζ potential of the hydrophobe-electrolyte interface without impurities.
We demonstrate that even if we include CO2 dissolution and surface slip, the theory cannot reproduce the experimentally measured ζ potentials. Note that we extract the surface
aﬃnities of the ions from experimentally measured electrolyte surface tensions and thus use
a model for electriﬁed electrolyte interfaces that is consistent with all available experimental
data.
Fig. 2a shows the ζ potential of a hydrophobe-KCl solution interface without impurities
as a function of pH. The solid lines include only the water dissociation according to eq. 1,
whereas the broken lines include CO2 dissolution according to eqs. 15-18 as well as water
dissociation. In Fig. 2bc the bulk ion concentrations are plotted as a function of pH for ﬁxed
cbsalt = 0.1 mM with/without CO2 , which follow from solving the chemical reaction eqs. 8-10
and 15-18.
Both with and without CO2 dissolution, the ζ potentials are in the range of −5 mV <
ψ0 < 10 mV, which is much smaller than the magnitude of the ζ potentials measured inexperiments. For a 0.1 mM KCl solution without CO2 , the main bulk ionic components for13

acidic conditions (pH < 4) are H3 O+ and Cl− , as shown in Fig. 2b. Because the interface
aﬃnity of H3 O+ is more pronounced than the interface aﬃnity of Cl− , αH3 O < αCl , the ζpotential for acidic conditions is positive. At neutral pH (4 < pH < 10), the main bulk ioniccomponents are K+ and Cl− , and thus, the ζ potential is slightly negative because αK > αCl .
For basic conditions (pH > 10), the main bulk ionic components are K+ and OH− . Because
αK < αOH , the ζ potential is again positive.
For a 0.1 mM KCl solution in the presence of CO2 , the main ionic components vary with
pH. For pH < 4 and 4 < pH < 7, the main ionic components in bulk are the same as those
without CO2 , whereas for 7 < pH < 10, K+ and HCO−
3 are the main ionic components
in bulk, and thus, the ζ potential becomes negative because αK > αHCO3 . For pH > 10,
the contribution of CO2−
becomes important, and the ζ potential is mostly negative but
3increases rapidly for very large pH.
These results suggest that in order to reproduce ζ potentials of magnitude similar to theexperimental values, which are around −50 mV, it is necessary to assume the presence of
charged components with very large surface aﬃnities.
Effect of hydrodynamic slip
Interfacial slip, which in our model is equivalent to the presence of an interfacial layer witha low viscosity, enhances the ζ potential. Fig. 3 shows the ζ potential of KCl solutions in
the presence of slip. We use an interfacial viscosity ηint = 0.2η in Fig. 3a and ηint = 0.05η in
Fig. 3b. The corresponding slip lengths are given by, respectively, b = z ∗ (η/ηint − 1) = 2 nm,
which is identical to the slip length found for hydrophobic diamond surfaces in molecular
dynamics simulations, 87 and b = 9.5 nm. 6 The results in Fig. 3 show that slip enhances the ζpotential but does not change the pH dependence of the curves as long as ηint is independentof pH. Comparing the calculated curves with the experimental data, both with and without
CO2 , one sees that the pH dependence does not agree with the experimental data. In
addition, the slip length b = 9.5 nm used in Fig. 3b is very large compared to typical slip

14lengths obtained in molecular dynamics simulations of hydrophobic surfaces. 89 In Appendix
D we furthermore demonstrate that interfacial dielectric eﬀects do not signiﬁcantly modify
the ζ potential of hydrophobic surfaces.
5050

ζ [ mV ]ηint / η = 0.2

ηint / η = 0.05

0

0

-50

-50

-100

without / with CO2

-150

(a)

-100

0.1 mM
1 mM
10 mM

1

3

5

7

9

without / with CO2

-150
11
1

(b)

pH

0.1 mM
1 mM
10 mM

3

5

7

9

11

pH

Figure 3: Eﬀect of slip on the ζ potential of the hydrophobe-KCl solution interface as afunction of pH, calculated with (broken lines) and without (solid lines) CO2 . The interfacial
viscosity is set to (a) ηint /η = 0.2, corresponding to a slip length of b = z ∗ (η/ηint − 1) = 2 nm,
and to (b) ηint /η = 0.05, corresponding to a slip length of b = z ∗ (η/ηint − 1) = 9.5 nm. Other
parameters are the same as those used in Fig. 2. The symbols represent the experimental
data for Teﬂon AF surfaces. 6

In the presence of impurities
Because the eﬀect of slip is not suﬃcient to explain the pH dependence of the experimental
ζ potential, we consider the presence of impurities. 46,72 Speciﬁcally, we consider the eﬀect ofacidic and basic surface-active impurities that are present in water as well as the eﬀect of
surface-active anionic impurities that are present in the KOH solution added for adjustmentof the pH. For simplicity we do not consider the presence of CO2 nor the eﬀects of slip in
the remainder of the paper.
Acidic surface-active impurities in water
We use the ionic aﬃnity for acidic impurities αA = −15.6 which is extracted from theexperimental surface tension of the interface between sodium dodecylsulfate solution and15

air. 72 Fig. 4 shows the comparison between the experimental and theoretical ζ potentials foracidic impurity concentrations cbA,tot = 50 nM and 500 nM. We use pKa = 5 (broken lines),
which is the value typical for the carboxyl acid group, and pKa = 7 (solid lines), which
describes the experimental data better. In fact, the deprotonated impurity species dominates
the surface charge of the hydrophobic interface. For impurity concentration cbA,tot = 500 nM
with pKa = 7 the experimental Teﬂon AF ζ potential data are described quite well.
Figure 5 displays the pKa and pKb values of typical acidic and basic groups that arepresent in surfactant molecules. 90 The most ubiquitous acids in nature are carboxylic acids
with a bulk pKa around 5, 90 which is in fact equal to the pKa found by ﬁtting a similar
impurity model to hexadecane data. 46 The fact that our best ﬁt pKa = 7 is slightly higher
may be speciﬁc to the experimental conditions or reﬂect a shift of the pKa , which in fact is
expected for acidic groups that are close to low-dielectric surfaces. 91
The slip eﬀect on the impurity-induced ζ potential is considered in Appendix C. There
we show that using b = z ∗ (η/ηint − 1) = 2 nm, which is identical to the slip length foundfor hydrophobic diamond surfaces in molecular dynamics simulations, 87 we cannot obtain avalue for cbA,tot that reproduces the experimental data. Interfacial dielectric eﬀects reduce
the ﬁtted impurity concentration by ∼ 40%, as shown in Appendix D, which is insigniﬁcantin the context of the present study.
Basic surface-active impurities in water
In strongly acidic conditions the experimental data exhibit positive ζ potentials which cannot
be reproduced with acidic impurities and which suggests the additional presence of basic impurities. For the surface aﬃnity of the basic impurity species we use the value αBH = −14.5,
which was previously extracted from the experimental surface tension of dodecyldimethylammonium chloride solutions. 72 Since the parameters cbA,tot = 500 nM and pKa = 7 for theacidic impurity species accurately describe the experimental ζ potential data in the pH rangeof 4 < pH < 8, we do not change these parameters in the following analysis.

16

50

ζ [ mV ]

0
-50
-100
b

b

(a) cA,tot = 50 nM

-150
1

3

5

7

(b) cA,tot = 500 nM
9

11 1

3

5

7

9

11

pH

pH
50

Experimentbcsalt = 0.1 mM
1 mM
10 mM

ζ [ mV ]0
Theory-50

pKa = 5, cbsalt = 0.1 mM
1 mM
10 mM

-100

b
pKa = 7, csalt = 0.1 mM(c) pH = 7
-150
-9
-8
-7
-6
10
10
10
10
b
cA,tot [ M ]

1 mM
10 mM

-5

10

Figure 4: (a,b) Eﬀects of acidic impurities on the ζ potential as a function of pH for pKa = 7(solid lines) and for pKa = 5 (broken lines). (c) ζ potential as a function of cbA,tot . The
symbols in (a,b) represent the experimental data. 6 The parameters are the same as thoseused in Fig. 2 and all αi are summarized in table 1.

strongweak
4.76

acetic acid

3.34

methylamine

7.15

10.0

15.5

4-nitrophenol

phenolethanol

4.20

9.13

13.0

trimethylamine

aniline4-nitroaniline

Figure 5: Collection of a few pKa and pKb values for typical acidic and basic groups. 90

17

Fig. 6 shows the eﬀect of basic impurities on the ζ potential for pKb = 12 (solid lines)
and pKb = 4 (broken lines). Typical bases such as fatty primary amines have pKb values
around 4, 90 but such a value leads to neutralization of deprotonated acidic impurities at theinterface and bad agreement with the experimental data, as seen in Fig. 6. We ﬁnd good
agreement between experiment and theory for a basic impurity concentration cbB,tot = 600 nM
and pKb = 12. The value pKb = 12 indicates a very weak base which is only half protonatedeven at a low pH of pH = 2. Figure 5 shows that bases that have such a low pKb do in fact
exist, but their presence will depend on the speciﬁc experimental conditions.
100

Experimentcbsalt = 0.1 mM

ζ [ mV ]

50

1 mM
10 mM

0

Theory, pKa = 7
cbA,tot = 500 nM, cbB,tot = 600 nM-50bpKb = 4, csalt = 0.1 mM
1 mM

-100

10 mM
pKb = 12, cbsalt = 0.1 mM

-150

1 mM

1

3

5

7

9

11

10 mM

pH

Figure 6: The eﬀect of additional basic impurities on the ζ potential as a function of pH. The
symbols represent the experimental ζ potential of Teﬂon AF, 6 whereas the lines depict thetheoretical ζ potential. Impurities with pKb = 4 (strong base, broken lines) and 12 (weakbase, solid lines) are examined. The parameters are the same as those used in Fig. 2 and all
αi are summarized in table 1.

Negatively charged surface-active impurities in KOH
Although the solid lines in Fig. 6 are in good agreement with the experimental data formost values of the pH, there is still a discrepancy for strongly basic conditions (pH > 8). We
demonstrate next that this small discrepancy can be eliminated by additionally assuming thepresence of an acidic surface-active impurity in KOH. The rationale behind this assumption
is that alkali hydroxides are strongly hygroscopic and cannot be puriﬁed by roasting, whichmakes it diﬃcult to keep them pure. We implement the presence of impurities in KOH bymodifying the total acid impurity concentration according to c̃bA,tot = cbA,tot +νcbOH , where ν isthe ratio of OH− to A− in KOH salts and therefore is a measure of the fraction of impurities18

in KOH. For simplicity, we use the same surface aﬃnity and the same pKa for the impurity
that is present in KOH as for the acidic impurity that is present in water.
Fig. 7 shows the eﬀect of an impurity present in KOH. The broken lines are the sameas the solid lines in Fig. 6, and thus, they are calculated without an impurity in KOH. The
solid lines are calculated in the presence of an impurity in KOH using the impurity fraction
ν = 0.05. From the comparison with the experimental data we see that in the presenceof an impurity in KOH good agreement between theory and experiment is achieved also instrongly basic conditions.
50

Experimentcbsalt = 0.1 mM
1 mM
10 mM

ζ [ mV ]0

Theory, pKa = 7, pKb = 12-50

cbA,tot = 500 nM, cbB,tot = 600 nMbν = 0.00, csalt = 0.1 mM
1 mM

-100

10 mM
ν = 0.05, cbsalt = 0.1 mM

-150

1 mM

1

3

5

7

9

11

10 mM

pH

Figure 7: Eﬀect of an additional impurity present in KOH on the ζ potential. The brokenlines are calculated without an impurity in KOH, whereas the solid lines account for an
impurity in KOH. We use the impurity fraction ν = 0.05 for the ratio of A− to OH− in KOH
salts. The impurity species in KOH has the same pKa value and the same surface aﬃnity
as the acidic impurity present in water. Other parameters are the same as those used inFig. 2 and all αi are summarized in table 1. The symbols represent the experimental data
for Teﬂon AF surfaces. 6

Two kinds of acidic impurities in the water
Accurate ﬁts to the experimental data can also be achieved without assuming impurities
in the added KOH, by taking more types of water impurities into account instead. We
consider the eﬀect of a second acidic impurity in water with a diﬀerent pKa value. Keeping
a basic impurity with cbB,tot = 600 nM and pKb = 12 and the ﬁrst acidic impurity A−
1= 500 nM and pKa1 = 7, we introduce a second acidic impurity A−
parameters at cb,tot
2
A1
which is characterized by a bulk concentration cbA2 ,tot and a value pKa2 . The broken lines inFig. 8 include only one kind of acidic impurity, whereas the solid lines include two diﬀerent
1950

Experimentbcsalt = 0.1 mM
1 mM
10 mM

ζ [ mV ]0

Theory, pKa = 7, pKa = 9, pKb = 12

-50

1

b

b

2

cA ,tot = 500 nM, cB,tot = 600 nM
1bbcA ,tot = 0 µM, csalt = 0.1 mM
2

1 mM

-100

10 mM
b
b
cA ,tot = 2 µM, csalt = 0.1 mM
2

-150

1 mM

1

3

5

7

9

11

10 mM

pH

Figure 8: Eﬀect of the presence of a second acidic impurity on the ζ potential. The brokenlines are predictions without a second acidic impurity species, whereas the solid lines are
predictions in the presence of a second acidic impurity species. Keeping cbA1 ,tot = 500 nM, pKa1 = 7, αA1 = −15.6, as well as cbB,tot = 600 nM, pKb = 12, αBH = −14.5, we use
cbA2 ,tot = 2 µM, αA2 = −15.6, and pKa2 = 9. Other parameters are the same as those usedin Fig. 2. The symbols represent the experimental data for Teﬂon AF surfaces. 6
acidic impurities with pKa1 = 7 and pKa2 = 9. A ﬁt to the experimental cbsalt = 0.1 mM data
(red circles) in the basic pH range yields cbA2 ,tot = 2 µM. We see that with a second impurity
we obtain good agreement between theory and experimental data even in basic conditions
and without assuming the presence of impurities in the added KOH.

What is responsible for the negative ζ potential of hydrophobic
surfaces?
In the previous sections, we have demonstrated that water ions and dissolved CO2 cannot
explain the experimentally measured negative ζ potential of hydrophobic surfaces. We thenproceeded to discuss the eﬀects of various impurities on the ζ potential of hydrophobic
surfaces. Our results suggest that the ζ potential of hydrophobic surfaces originates fromsurface-active charged impurities that are present in the water and possibly also in the salts
used for adjusting the pH. We show that the experimental data for Teﬂon AF surfaces
can be reproduced quite well if we assume the presence of acidic impurities (pKa = 7)
with surface aﬃnity αA = −15.6 at a concentration of cbA,tot = 500 nM. The agreementwith the experimental data at low pH is improved if we additionally assume the presenceof basic impurities (pKb = 12) with surface aﬃnity αBH = −14.5 at a concentration of2050(a) hydrogen(a)(b) octadecane

ζ [ mV ]

0

(b)
b

b
cA,tot = 80 nM

cA,tot = 300 nM

pKa = 6

pKa = 5bcB,tot = 0 nM

cB,tot = 300 nMb

pKb = 12-50
0.1 mM

10 mM

1 mM

100 mM

-100
1

3

5

7

9

11

13 1

3

5

50
(c) hexadecane, 1mM

7
pH

9

11

13

(c)
b

pKa = 7, cA,tot = 30 nM

ζ [ mV ]0bcA,tot = 200 nM
pKa = 5, cbA,tot = 30 nM
bcA,tot = 200 nM-50

hexadecane-1
hexadecane-2

-100
1

3

5

7

9

11

13

pH

Figure 9: Comparison between theory and experimental data for (a) hydrogen bubbles, 31(b) octadecane droplets, 43 and (c) hexadecane droplets 46 in 1mM NaCl solution. Acidic
impurities cbA,tot = 80 nM with pKa = 6 are used for the theoretical modeling in (a) andcbA,tot = 300 nM with pKa = 5 are used in (b). In (c) acidic impurities at concentration
cbA,tot = 30 nM (broken lines) and 200 nM (solid lines) are used with pKa = 5 (blue) and 7(black). Basic impurities cbB,tot = 300 nM with pKb = 12 are used in (a). Since the data
do not extend to low pH values we set cbB,tot = 0 nM in (b) and (c). Other parameters are
the same as those used in Fig. 4. Slip eﬀects, interfacial dielectric eﬀects and dissolution ofcarbon dioxide are not considered.

21

cbB,tot = 600 nM. Since the carbon number of dodecylsulfate is 12, a simple estimate of theadsorption energy per number of carbons gives −15.6kB T /12 = −1.3kB T . If the carbon
number of the impurity’s alkyl chains increases by 2, the impurity concentration needed to
produce a certain ζ potential reduces approximately by a factor of 10, because e2.6 ≈ 13.
Conversely, if the impurities have shorter alkyl chains the concentration needed to produce acertain ζ potential becomes larger. Therefore, our estimate for the impurity concentration of
the order of 100 nM is only an example, and it is possible that signiﬁcantly smaller impurity
concentrations cause the ζ potential of hydrophobic surfaces because we do not know thechemical nature of the impurity and therefore cannot characterize precisely its surface aﬃnity.
The ultrapure water standard, for example, Milli-Q ultrapure water, prescribes a resistivity of at least 18.2 MΩ·cm and a total organic carbon (TOC) concentration of at most
5 µg/L. In terms of concentrations, this conductivity corresponds to approximately 100 nM
of charged impurities while this TOC equals 5 µg/L= 420 nM in carbon, and 35 nM in dodecylsulfate. That means that ultrapure water already includes impurities at concentrations ofa few tens of nano-molar. In addition, when electrophoretic or electro-osmotic mobilities aremeasured, there are several possibilities of how water can be contaminated further, including contamination due to storage of water in plastic or glass containers 92 or contamination
due to contact with the ambient atmosphere. 93 Charged impurities in ultrapure water haveindeed been detected by mass spectrometry and liquid chromatography. 92–95
Regarding our extracted pKa and pKb values, the ﬁtted value pKa = 7 is substantiallylarger than the literature value of carboxylic acids pKa = 5, which is expected for acidic
groups that are close to low-dielectric surfaces. 91 We also note that the experimental isoelectric points extracted from ζ potential measurements vary substantially in the range pH = 1
to 4, depending on the data set. 18 In Fig. 9a and b we show a comparison of our model predictions with hydrogen bubble 31 and octadecane 43 data, respectively. For the hydrogen bubble
data, 80 nM acidic impurities with pKa = 6 and 300 nM basic impurities with pKb = 12give good agreement. For the octadecane data, 300 nM acidic impurities with pKa = 5 ﬁt

22

the data quite well, here basic impurities are not needed to describe the data because theoctadecane data do not extend to small pH values. In Fig. 9c two diﬀerent hexadecane
droplet data sets 46 are compared with theoretical predictions using 30 nM and 200 nM acidic
impurities with pKa = 5 (blue broken and solid lines). The predictions using pKa = 7 (black
broken and solid lines) clearly disagree with the experimental data. In fact, a similar impurity model using pKa = 5 was previously shown to describe the experimental data, 46 andit was suggested that acidic impurities originate from hexadecane because diﬀerent gradesof purity were used: 99.8 % (hexadecane-1, open squares) and 99 % (hexadecane-2, open
circles). In this scenario, the impurities (presumably fatty acids) dissolve in both the oil andthe water phase. Alternatively, it is possible that hydrophobic surfaces exhibit immobile
ionic impurities. For example, the ζ potential of polymethylmethacrylate (PMMA) particles
is believed to be governed by ionic end-groups of the polymers. 96 Summarizing, because it
is likely that diﬀerent experiments contain diﬀerent combinations of impurities, the spreadof pKa values is rather plausible.
For the basic impurity, the ﬁtted pKb = 12 corresponds to a very weak base. Since
the most ubiquitous base, ammonium, has a strong basic reaction constant, pKb = 4, thissuggests that the basic impurity is not a derivative of ammonium. Instead, the basic impurity
could be a fatty alcohol, 97 an ester, 98 an ether, 98 or an acid, 99 because the oxygen atom inthese molecules is only protonated in strongly acidic conditions. It is known that a dilute
alcohol in water is more basic than water, 97,98 for example, isopropanol in 1M sulfuric acid has
a basic reaction constant of pKb = 13.7, which is close to our estimate for the basic impurity
reaction constant. 97 In fact, Teﬂon AF is a copolymer of tetraﬂuoroethylene (C2 F4 ) and 2,2bis(triﬂuoromethyl)-4,5-diﬂuoro-1,3-dioxole (C5 F8 O2 ), and the ratio of the two monomers is
C2 F4 : C5 F8 O2 = 1 : 2. 6 Therefore, many ether bonds (−COC−) exist on the surface, which
in principle could act as a weak base. 98 On the other hand, the hydrogen data also suggest
the presence of basic impurities. In fact, hydrogen bubbles are produced by electrolysis ofwater, 31 which provides a possible source of contamination with basic impurities.

23

For the experimentally observed enhancement of the ζ potential at large pH, we propose
two scenarios: a negatively charged impurity present in the salt used to adjust the pH, or asecond acidic impurity with a diﬀerent pKa value present in the water. Both scenarios arelikely to occur in typical experimental conditions. In practice, impurities that are present
in experiments will presumably consist of many diﬀerent types of chemicals, and they will
exhibit a spectrum of surface aﬃnities, diﬀerent bulk concentrations, and diﬀerent pKa andpKb values. Even the mass spectrum of ultrapure water shows many peaks, 93,94 the origin of
which are not very clear at present. Monitoring impurity distributions of pure water whilemeasuring the ζ potential of hydrophobic surfaces at the same time would be highly desirable
in future experiments.

Conclusions
We calculate the ζ potential of hydrophobic surfaces as a function of pH and salt concentration by solving the Poisson-Boltzmann and hydrodynamic equations including ion adsorption, where the ionic surface aﬃnities have been extracted independently from surface-tension
data at the air-electrolyte interface. The interface adsorption and repulsion of small inor2−
ganic ions such as H3 O+ , OH− , HCO−
are not suﬃcient to reproduce the ζ
3 , and CO3

potential observed in experiments because their aﬃnities for hydrophobic surfaces are small– of the order of |αi | ≈ 1. The presence of hydrodynamic surface slip also is not suﬃcient toproduce agreement with experimental ζ potential data. However, when we introduce a small
amount of charged surface-active impurities in the water, we ﬁnd good agreement between
our calculations and the literature experimental data for Teﬂon AF surfaces 6 if the impurity
acid and basic reaction constants are suitably chosen. Our comparison with experimental
data suggests that the dominant impurity type is a weak acid with pKa = 5 to 7, in orderto reproduce the experimental ζ potentials at very low pH additionally a weak base with
pKb = 12 at concentrations of a few hundred nano-molar is needed.

24

We do not know the identity of the impurities, but we presume that at hydrophobic
surfaces, many diﬀerent types of surface active impurities can be found in addition to waterions, bicarbonate and carbonate ions. Even though the assumed bulk concentration of the
impurities is only nano-molar, they accumulate at a hydrophobic surface because of theiramphiphilic nature. The experimentally measured ζ potential at hydrophobic surfaces will
reﬂect the distribution of pKa and pKb values of the impurities. Regarding the origin of
the impurities, we consider diﬀerent possibilities such as impurities in the water, contaminations from experimental processes, as well as contaminations present in added salts, acids,
bases, gaseous phases, oil phases, 46 and on hydrophobic surfaces. 96 The variety of possible
contamination sources can easily explain that even in nominally identical systems the ζ potentials exhibit a signiﬁcant spread as shown in Fig. 1cd. Given the almost undetectably
small amounts of impurities necessary to reproduce the experimentally observed ζ potential
eﬀects, it will be diﬃcult to produce a positive proof that impurities are in fact responsible
for the nonzero ζ potentials of hydrophobic solutes. Our ﬁndings thus merely suggest one
possible scenario that explains the ζ potentials of hydrophobic surfaces over a large range ofpH values and salt concentrations. It is crucial to realize, though, that the present theory
is consistent with experimental and theoretical work on electrolyte surface tensions, 58 theJones-Ray eﬀect, 72,73 the disjoining pressure of wetting water ﬁlms on silica 49,74 and experimental investigations of the hydrodynamic boundary conditions at air-water interfaces. 75–78

Appendix A: Detailed description of Fig. 1
In Fig. 1 a and b, the colors represent diﬀerent bulk salt concentrations, whereas the shapes ofthe symbols represent diﬀerent surface types. Filled squares correspond to alumina (γ-Al2 O3 )powders with radius R = 45 to 50 nm in KNO3 -HNO3 -KOH solution and the conversion intothe ζ potential follows Ref. 100. 4 The ﬁlled circles correspond to titania (TiO2 , rutile) powders with radius R = 45 nm in KNO3 -HNO3 -KOH solution and the conversion into the ζ25

potential follows Ref. 100. 4 The ﬁlled triangles correspond to planar fused silica (SiO2 ) in
KCl solution where acid and base are not speciﬁed and streaming current measurements areused for determining the electro-osmotic mobility (i.e. f = −1). 3 The open squares correspond to planar Teﬂon AF (poly(tetraﬂuoroethylene-co-2,2-bis(triﬂuoromethyl)-4,5-diﬂuoro1,3-dioxole) ((C2 F4 )n (C5 F8 O2 )m ) in KCl-HCl-KOH solution where streaming current measurements are used for determining the electro-osmotic mobility. 6 The open circles correspond to octadecane (C18 H38 ) droplets with radius R = 150 nm in NaCl solution where acidand base are not speciﬁed and f = 1 is used for conversion. 43 The open triangles correspondto hydrogen bubbles with radius R = 10 to 70 µm in NaCl-HCl-NaOH solution and f = 1
for conversion. 31
In Fig. 1 c and d, the shapes of the symbols represent the diﬀerent experimental resultson the same materials. The broken lines are guides to the eye. All data in c and d aremeasured in NaCl-HCl-NaOH solution. The ﬁlled squares correspond to air bubbles with
sub-micrometer diameters in a 10 mM solution and f = 1 for conversion. 35 The ﬁlled circles
correspond to air bubbles with diameters of a few tens of micrometers and f = 1 for conversion. 34 The ﬁlled up-pointing triangles correspond to R = 290 nm, where Henry’s function 67
is used for conversion. 33 The ﬁlled down-pointing triangles correspond to R = 5 µm andf for conversion is not speciﬁed. 26 The open squares and circles correspond to hexadecane
droplets in a 1 mM solution with the diameter 2R = 150 − 200 nm and f = 1 for conversion,
whereas the purity of hexadecane is diﬀerent between squares (99.8%) and circles (99%). 46
The open up-pointing triangles correspond to diameters of about 300 nm and f = 1 forconversion. 33 The open down-pointing triangles correspond to diameters 3 − 8 µm and f = 1
for conversion. 41

26

100

Washburn, Na2CO3

4

Matubayasi, Na2CO3

3

Ozdemir, NaHCO3

2

10-9

0

10
0

0.5
1
cbsalt [ M ]Na+(c) NaHCO3

H3O+
OH-

1.5

HCO-3
CO23

10-6

1

(a)

(b) Na2CO3

10-3

Ozdemir, Na2CO3

cbi[ M ]∆γ [ mN / m ]

5

CO2
H2CO3

-12

10-8

10-6

10-4
cbsalt [ M ]10-2

100 10-8

10-6

10-4

10-2

100

cbsalt [ M ]Figure 10: (a) Comparison of experimental and theoretical surface tensions of NaHCO3
and Na2 CO3 solutions. The symbols represent the experimental data, 101–103 whereas thelines are theoretical predictions. The blue and red solid lines are the calculations including
the reactions eqs. 8 and 16-18. The orange and light blue broken lines are the calculationconsidering only the reactions Na2 CO3 → 2Na+ + CO2−
or NaHCO3 → Na+ + HCO−
3
3,∗
without the reactions eqs. 8 and 16-18. We use T = 298 K, z = 0.5 nm, εint = ε = 78, andall αi are summarized in table 1. (b) Plot of the reactant concentrations as a function of
cbNa2 CO3 . (c) Plot of the reactant concentrations as a function of cbNaHCO3 .

Appendix B: Estimate of the surface affinity of carbonate ions to the air interface
In this section we estimate the aﬃnity of carbonate ions to the air-water interface, αCO3 ,using experimental surface tension data of Na2 CO3 solutions. Because the carbonate ion,
CO2−
3 , is divalent, the exact solution of eq. 2 is not available, therefore eq. 2 needs to besolved numerically. Since the solution of Na2 CO3 is basic, we take the dissociation of waterand the reaction of carbonates into account.
Fig. 10a shows the surface tension of Na2 CO3 and NaHCO3 solutions as a function of
their bulk concentration. We do not use the experimental data of Na2 CO3 taken from Ref.
103 for ﬁtting because they deviate from the other experimental data. 101,102 To obtain thesurface aﬃnity of CO2−
3 , we calculate the surface tension of Na2 CO3 solutions by solving thechemical equilibrium equations 8 and 16-18 together with conservation of the total carbon
concentration. Here we do not use the reaction eq. 15, because the dissolution of CO2
from the ambient air into the solution is very slow. If we wait for equilibration of the27reaction described by eq. 15, NaHCO3 and Na2 CO3 solutions relax and thereby release CO2 .
We also neglect the surface excess of neutral molecules CO2 and H2 CO3 , even though thesurface excess of these neutral species enter the surface tension diﬀerence, because theirbulk concentrations are much smaller than the main cation and anion concentrations in thebulk. We use ε = 78, T = 298 K, z ∗ = 0.5 nm, αNa = 1.2, αH3 O = −0.9, αOH = 1.6,
αHCO3 = −0.4. 72 Note that αHCO3 has been obtained by assuming only the dissociation
NaHCO3 → Na+ + HCO−
3 . We obtain αCO3 = 1.4 for the best ﬁt (blue line in Fig. 10a).
When we assume only the reaction Na2 CO3 → 2Na+ + CO2−
3 , the result is almost thesame (orange broken line in Fig. 10a), showing that the reactions, eqs. 8 and 16-18 are notimportant for the surface tension of Na2 CO3 solutions in this concentration range.
The red line in Fig. 10a depicts the theoretical surface tension of a NaHCO3 solution
considering the chemical equilibrium described by equations 8 and 16-18. Here we also
neglect the surface excess of neutral molecules, and we use the same parameters as for the
calculation of the Na2 CO3 surface tension. The light blue broken line in Fig. 10a is thetheoretical surface tension of a NaHCO3 solution assuming only NaHCO3 → Na+ + HCO−
3,
which is almost the same as the red solid line. The agreement between the two theoreticallines shows that the surface tension of Na2 CO3 and NaHCO3 solution can be predicted bytreating the solutions as strong electrolytes, neglecting the reactions, eqs. 8 and 16-18.
Figs. 10b and c show the bulk ion concentrations due to the chemical equilibrium eqs. 8
and 16-18 for Na2 CO3 and NaHCO3 solutions. We ﬁnd that the dominant ions in Na2 CO3solutions at cbsalt > 10 mM are Na+ and CO2−
3 ions, whereas the dominant ions in NaHCO3
solutions at cbsalt > 10 mM are Na+ and HCO−
3 . These results show that carbonate equilibria
in electrolyte are rather complex.

28

Appendix C: Slip effect on the ζ potential caused byacidic impurities
In this section we examine how slip eﬀects modify the impurity-induced ζ potential. Fig. 11
shows the ζ potential including slip eﬀects in the presence of acidic impurities. The slip
length b = z ∗ (η/ηint − 1) = 2 nm is identical to the slip length found for hydrophobic
diamond surfaces in molecular dynamics simulations. 87 Except for ηint , the parameters are
the same as those used in Fig. 4.
In Fig. 11a (cbA,tot = 50 nM), the eﬀect of the salt concentration on the ζ potential issuppressed compared to the case without slip, shown in Fig. 4a, meaning that the three
curves for diﬀerent salt concentration are very similar to each other. When the impurity
concentration is higher (cbA,tot = 500 nM, Fig. 11b), the slip eﬀect is even more drastic
and the order of the absolute ζ potential with respect to the salt concentration is reversed
at basic conditions and thus diﬀerent from the experimental data and also diﬀerent fromthe prediction without slip in Fig. 4b. To linear order in κz ∗ , the slip eﬀect modiﬁes theζ potential by the factor (1 + κb), 81,104 which explains the trends seen in Fig. 11a and b.
Fig. 11c shows the ζ potential at pH = 7 as a function of cbA,tot and demonstrates the reversalof the salinity dependence for cbA,tot > 100 nM for both pKa = 5 and 7.
Fig. 11 demonstrates that the predicted ζ potential diﬀers from the experimental results for ﬁnite slip, even though hydrophobic surfaces exhibit a ﬁnite slip length. 77 It is atpresent not clear where this discrepancy comes from, possibly the adsorption of impurities
onto the interface modiﬁes the interfacial eﬀective viscosity, as suggested earlier in related
works. 77,105–107

29

50
ηint / η = 0.2

ηint / η = 0.2

ζ [ mV ]

0
-50
-100
-150
b

b

(a) cA,tot = 50 nM

-200
1

3

5

(b) cA,tot = 500 nM

7

9

11 1

3

5

7

50

11

Experimentηint / η = 0.2bcsalt = 0.1 mM0ζ [ mV ]

9

pH

pH

1 mM
10 mM

-50

TheorypKa = 5, cbsalt = 0.1 mM

-100

1 mM
10 mM
b
pKa = 7, csalt = 0.1 mM

-150

1 mM

(c) pH = 7
-200
-9
-8
-7
-6
10
10
10
10
b
cA,tot [ M ]

10 mM

-5

10

Figure 11: Slip eﬀect on the ζ potential caused by acidic impurities for ηint /η = 0.2. Other
parameters are the same as those used in Fig. 4. (a,b) ζ potential as a function of pH for
pKa = 7 (solid lines) and for pKa = 5 (broken lines). (c) ζ potential as a function of cbA,tot .
The symbols in (a,b) represent the experimental data. 6

10100
0.1 mM0ζ [ mV ]

5

50

1 mM
10 mM

-0.2

0
-0.4
1

3

5

7

-50

9 11

0

εint = 4.2

εint = 4.2, 78.0

-5

(a)

-100

0.1 mM
1 mM
10 mM

1

3

5

7

cbA,tot = 300 nM
pKa = 7

-150
9

11

pH

(b)

1

3

5

7

9

11

pH

Figure 12: (a) Diﬀerence between the ζ potential with and without interfacial dielectriceﬀects. The presence of impurities and carbon dioxide is not considered, and the parameters
except for εint are the same as those used in Fig. 2. The broken lines are calculated using
εint = ε = 78, and the solid lines are calculated using εint = 4.2. The inset shows amagniﬁcation. (b) ζ potential predicted for an acidic impurity concentration of cbA,tot =300 nM and pKa = 7, where εint = 4.2 is used and all other parameters are the same as thoseused in Fig. 4b. The symbols represent the experimental data for Teﬂon surfaces AF. 630

Appendix D: Effect of interfacial dielectric constant on
the ζ potential
In this section, we demonstrate that interfacial dielectric eﬀects are not relevant for the ζpotential of hydrophobic surfaces. Fig. 12a shows the diﬀerence between the ζ potential withand without a modiﬁed interfacial dielectric constant. Here we do not consider impurities orcarbon dioxide dissolution, and the parameters except for εint are the same as those used inFig. 2. The broken lines are calculated using εint = ε = 78, and the solid lines are calculatedusing εint = 4.2, which is an estimate based on the interfacial capacitance of a hydrophobewater interface from molecular dynamics simulation. 86 The diﬀerence between the results
for the two scenarios is very small, we thus conclude that dielectric interfacial eﬀects cannot
explain the experimental ζ potential without the presence of impurities.
Fig. 12b shows the ζ potential predicted in the presence of acidic impurities for cbA,tot =300 nM and pKa = 7, where εint = 4.2 is used and other parameters are the same as thoseused in Fig. 4b. The ﬁtted impurity concentration 300 nM is smaller than 500 nM used inFig. 4b, which is caused by the presence of the modiﬁed interfacial dielectric constant, but
this diﬀerence is not rather marginal.

Acknowledgement
Funding by the Deutsche Forschungsgemeinschaft (DFG) via grant NE810/11 is gratefully
acknowledged. YU was supported by JSPS Overseas Research Fellowships No. 201860001.

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Graphical TOC EntrypHζ

44