Chapmono Template

Chapmono Template

Author
July 7, 2019
Abstract

In this chapter we review the electrostatic properties of charged membranes in aqueous solutions, with or without added salt, employing simple physical models. The equilibrium ionic profiles close to the membrane are governed by the well-known Poisson-Boltzmann (PB) equation. We analyze the effect of different boundary conditions, imposed by the membrane, on the ionic profiles and the corresponding osmotic pressure. The discussion is separated into the single membrane case and that of two interacting membranes. For the one membrane setup, we show the different solutions of the PB equation and discuss the interplay between constant-charge and constant-potential boundary conditions. A modification of the Poisson-Boltzmann theory is presented to treat the extremely high counter-ion concentration in the vicinity of a charge membrane. The two membranes setup is reviewed extensively. For two equally-charged membranes, we analyze the different pressure regimes for the constant-charge boundary condition, and discuss the difference in the osmotic pressure for various boundary conditions. The non-equal charged membranes is reviewed as well, and the crossover from repulsion to attraction is calculated analytically for two limiting salinity regimes (Debye-Hückle and counter-ions only), as well as for general salinity. We then examine the charge-regulation boundary condition and discuss its effects on the ionic profiles and the osmotic pressure for two equally-charged membranes. In the last section, we briefly review the van der Waals interactions and their effect on the free energy between two planar membranes. We explain the simple Hamaker pair-wise summation procedure, and introduce the more rigorous Lifshitz theory. The latter is a key ingredient in the DLVO theory, which combines repulsive electrostatic with attractive van der Waals interactions, and offers a simple explanation for colloidal or membrane stability. Finally, the chapter ends by a short account of the limitations of the approximations inherent in the PB theory.

Chapter 1 Charged Membranes: Poisson-Boltzmann theory, DLVO paradigm and beyond

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1.1 Introduction

It is of great importance to understand electrostatic interactions and their key role in soft and biological matter. These systems typically consist of aqueous environment in which charges tend to dissociate and affect a wide variety of functional, structural and dynamical properties. Among the numerous effects of electrostatic interactions, it is instructive to mention their effect on elasticity of flexible charged polymers (polyelectrolytes) and cell membranes, formation of self-assembled charged micelles, and stabilization of charged colloidal suspensions that results from the competition between repulsive electrostatic interactions and attractive van der Waals interactions (Verwey and Overbeek 1948, Andelman 1995, 2005, Holm, Kekicheff and Podgornik 2000, Dean el al., 2014, Churaev, Derjaguin and Muller 2014).

In this chapter, we focus on charged membranes. Biological membranes are complex heterogeneous two-dimensional interfaces separating the living cell from its extra-cellular surrounding. Other membranes surround inter-cellular organelles such as the cell nucleus, golgi apparatus, mitochondria, endoplasmic reticulum and ribosomes. Electrostatic interactions control many of the membrane structural properties and functions, e.g., rigidity, structural stability, lateral phase transitions, and dynamics. Moreover, electric charges are a key player in processes involving more than one membrane such as membrane adhesion and cell-cell interaction, as well as the overall interactions of membranes with other intra- and extra-cellular proteins, bio-polymers and DNA.

How do membranes interact with their surrounding ionic solution? Charged membranes attract a cloud of oppositely charged mobile ions that forms a diffusive electric double layer (Gouy 1910, 1917, Chapman 1913, Debye and Hückel 1923, Verwey and Overbeek 1948, Israelachvili 2011). The system favors local electro-neutrality, but while achieving it, entropy is lost. The competition between electrostatic interactions and entropy of ions in solution determines the exact distribution of mobile ions close to charged membranes. This last point shows the significance of temperature in determining the equilibrium properties, because temperature controls the strength of entropic effects as compared to electrostatic interactions. For soft materials, the thermal energy is also comparable to other characteristic energy scales associated with elastic deformations and structural degrees of freedom.

It is convenient to introduce a length scale for which the thermal energy is equal to the Coulombic energy between two unit charges. This is called the Bjerrum length, defined as:

(1.0)

where is the elementary charge, is the vacuum permittivity111Throughout this chapter we use the SI unit system. and the dimensionless dielectric constant of water is . The Bjerrum length is equal to about at room temperatures, .

A related length is the Gouy-Chapman length defined as

(1.0)

At this length scale, the thermal energy is equal to the Coulombic energy between a unit charge and a planar surface with a constant surface-charge density, . The Gouy-Chapman length, , is inversely proportional to . For strongly charged membranes, is rather small, on the order of a tenth of nanometer.

In their pioneering work of almost a century ago, Debye and Hückel introduced the important concept of screening of the electrostatic interactions between two charges in presence of all other cations and anions of the solution (Debye and Hückel 1923). This effectively limits the range of electrostatic interactions as will be further discussed below. The characteristic length for which the electrostatic interactions are screened is called the Debye length, , defined for monovalent 1:1 electrolyte, as

(1.0)

with being the salt concentration (in molar), and is the inverse Debye length. The Debye screening length for 1:1 monovalent salts varies from about in strong ionic solutions of M to about in pure water, where the concentration of the dissociated OH and H ions is M.

The aim of this chapter is to review some of the basic considerations underlying the behavior of charged membranes in aqueous solutions using the three important length-scales introduced above. We will not account for the detailed structure of real biological membranes, which can add considerable complexity, but restrict ourselves to simple model systems, relying on several assumptions and simplifications. The membrane is treated as a flat interface with a continuum surface charge distribution or constant surface potential. The mobile charge distributions are continuous and we disregard the discreteness of surface charges that can lead to multipolar charge distributions.

This chapter is focused only on static properties in thermodynamic equilibrium, excluding the interesting phenomena of dynamical fluctuations and dynamical responses to external fields (such as in electrochemistry systems). We mainly treat the mean-field approximation of the electric double-layer problem and the solutions of the classical Poisson Boltzmann (PB) equation. Nevertheless, some effects of fluctuations and correlations will be briefly discussed in section 1.10. We will also discuss the ion finite-size in section 1.4, where the ‘Modified PB equation’ is introduced.

The classical reference for the electric double layer is the book of Verwey and Overbeek (1948), which explains the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory for stabilization of charged colloidal systems. More recent treatments can be found in many textbooks and monographs on colloidal science and interfacial phenomena, such as Evans and Wennerström (1999), Israelachvili (2011), and in two reviews by one of the present authors, Andelman (1995, 2005).

1.2 Poisson-Boltzmann Theory

In Fig. 1.1, a schematic view of a charged amphiphilic (phospholipid) membrane is presented. A membrane of thickness is composed of two monomolecular leaflets packed in a back-to-back configuration. The constituting molecules are amphiphiles having a charge ‘head’ and a hydrocarbon hydrophobic ‘tail’. For phospholipids, the amphiphiles have a double tail. We model the membrane as a medium of thickness having a dielectric constant, , coming essentially from the closely packed hydrocarbon (‘oily’) tails. The molecular heads contribute to the surface charges and the entire membrane is immersed in an aqueous solution characterized by another dielectric constant, , assumed to be the water dielectric constant throughout the fluid. The membrane charge can have two origins: either a charge group (e.g., ) dissociates from the polar head-group into the aqueous solution, leaving behind an oppositely charged group in the membrane; or, an ion from the solution (e.g., ) binds to a neutral site on the membrane and charges it (Borkovec, Jönsson and Koper 2001). These association/dissociation processes are highly sensitive to the ionic strength and pH of the aqueous solution.

When the ionic association/dissociation is slow as compared to the system experimental times, the charges on the membrane can be considered as fixed and time independent, while for rapid association/dissociation, the surface charge can vary and is determined self-consistently from the thermodynamical equilibrium equations. We will further discuss the two processes of association/dissociation in section 1.8. In many situations, the finite thickness of the membrane can be safely taken to be zero, with the membrane modeled as a planar surface displayed in Fig. 1.2. We will see later under what conditions this simplifying limit is valid.

Figure 1.1: A bilayer membrane of thickness composed of two monolayers (leaflets), each having a negative charge density, . The core membrane region (hydrocarbon tails) is modeled as a continuum medium with a dielectric constant , while the embedding medium (top and bottom) is water and has a dielectric constant, .

Let us consider such an ideal membrane represented by a sharp boundary (located at ) that limits the ionic solution to the positive half space. The ionic solution contains, in general, the two species of mobile ions (anions and cations), and is modeled as a continuum dielectric medium as explained above. Thus, the boundary at marks the discontinuous jump of the dielectric constant between the ionic solution () and the membrane (), which the ions cannot penetrate.

The PB equation can be obtained using two different approaches. The first is the one we present below combining the Poisson equation with the Boltzmann distribution, while the second one (presented later) is done through a minimization of the system free-energy functional. The PB equation is a mean-field (MF) equation, which can be derived from a field theoretical approach as the zeroth-order in a systematic expansion of the grand-partition function (Podgornik and Žekš 1988, Borukhov, Andelman and Orland 1998, 2000, Netz and Orland 2000, Markovich, Andelman and Podgornik 2014, 2015).

Consider ionic species, each of them with charge , where and is the valency of the ionic species. It is negative () for anions and positive () for cations. The mobile charge density (per unit volume) is defined as with being the number density (per unit volume), and both and are continuous functions of .

In MF approximation, each of the ions sees a local environment constituting of all other ions, which dictates a local electrostatic potential . The potential is a continuous function that depends on the total charge density through the Poisson equation:

(1.0)

where is the total charge density and is a fixed external charge contribution. As stated above, the aqueous solution (water) is modeled as a continuum featureless medium. This by itself represents an approximation because the ions themselves can change the local dielectric response of the medium (Ben-Yaakov, Andelman and Podgornik 2011, Levy, Andelman and Orland 2012) by inducing strong localized electric field. However, we will not include such refined local effects in this review.

The ions dispersed in solution are mobile and are allowed to adjust their positions. As each ionic species is in thermodynamic equilibrium, its density obeys the Boltzmann distribution:

(1.0)

where , and is the bulk density of species taken at zero reference potential, .

Figure 1.2: Schematic illustration of a charged membrane, located at , with charge density . Without lost of generality, we take . For the counter-ion only case, the surface charge is neutralized by the positive counter-ions. When monovalent (1:1) electrolyte is added to the reservoir, its bulk ionic density is .

  Boltzmann distribution via electrochemical potential A simple derivation of the Boltzmann distribution is obtained through the requirement that the electrochemical potential (total chemical potential) , for each ionic species is constant throughout the system (1.0) where is the intrinsic chemical potential. For dilute ionic solutions, the ionic species entropy is taken as an ideal gas one, . By substituting into Eq. (1.2), the Boltzmann distribution of Eq. (1.2) follows. This relation between the bulk ionic density and chemical potential is obtained by setting in the bulk, and shows that one can consider the chemical potential, , as a Lagrange multiplier setting the bulk densities to be . Note that we have introduced a microscopic length scale, , defining a reference close-packing density, . Equation (1.2) assumes that the ions are point-like and have no other interactions in addition to their electrostatic one.

We now substitute Eq. (1.2) into Eq. (1.2) to obtain the Poisson-Boltzmann Equation,

(1.0)

For binary monovalent electrolytes (denoted as 1:1 electrolyte), , the PB equation reads,

(1.0)

Generally speaking, the PB theory is a very useful analytical approximation with many applications. It is a good approximation at physiological conditions (electrolyte strength of about ), and for other dilute monovalent electrolytes and moderate surface potentials and surface charge. Although the PB theory produces good results in these situations, it misses some important features associated with charge correlations and fluctuations of multivalent counter-ions. Moreover, close to a charged membrane, the finite size of the surface ionic groups and that of the counter-ions lead to deviations from the PB results (see sections 1.4 and 1.8 for further details).

As the PB equation is a non-linear equation, it can be solved analytically only for a limited number of simple boundary conditions. On the other hand, by solving it numerically or within further approximations or limits, one can obtain ionic profiles and free energies of complex structures. For example, the free energy change for a charged globular protein that binds onto an oppositely charged lipid membrane.

In an alternative approach, the PB equation can also be obtained by a minimization of the system free-energy functional. One can assume that the internal energy, , is purely electrostatic, and that the Helmholtz free-energy, , is composed of an internal energy and an ideal mixing entropy, , of a dilute solution of mobile ions.

The electrostatic energy, , is expressed in terms of the potential :

(1.0)

while the mixing entropy of ions is written in the dilute solution limit as,

(1.0)

Using Eqs. (1.2) and (1.2), the Helmholtz free-energy can be written as

where the sum of the first two terms is equal to and the third one is . The variation of this free energy with respect to , , gives the Poisson equation, Eq. (1.2), while from the variation with respect to , , we obtain the electrochemical potential of Eq. (1.2). As before, substituting the Boltzmann distribution obtained from Eq. (1.2), into the Poisson equation, Eq. (1.2), gives the PB equation, Eq. (1.2).

1.2.1 Debye-Hückel Approximation

A useful and quite tractable approximation to the non-linear PB equation is its linearized version. For electrostatic potentials smaller than at room temperature (or equivalently ), this approximation can be justified and the well-known Debye-Hückel (DH) theory is recovered. Linearization of Eq. (1.2) is obtained by expanding its right-hand side to first order in ,

(1.0)

where is the ionic strength of the solution. The first term on the right-hand side of Eq. (1.0) vanishes because of electro-neutrality of the bulk reservoir,

(1.0)

recovering the Debye-Hückel equation:

(1.0)

with the inverse Debye length, , defined as,

(1.0)

For monovalent electrolytes, , with , and Eq. (1.1) is recovered. Note that the Debye length, , is a decreasing function of the salt concentration.

The DH treatment gives a simple tractable description of the pair interactions between ions. It is related to the Green function associated with the electrostatic potential around a point-like ion, and can be calculated by using Eq. (1.0) for a point-like charge, , placed at the origin, , ,

(1.0)

where is the Dirac -function. The solution to the above equation can be written in spherical coordinates as,

(1.0)

It manifests the exponential decay of the electrostatic potential with a characteristic length scale, . In a crude approximation, this exponential decay is replaced by a Coulombic interaction, which is only slightly screened for and, thus, varies as , while for , is strongly screened and can sometimes be completely neglected.

1.3 One Planar Membrane

We consider the PB equation for a single membrane assumed to be planar and charged, and discuss separately two cases: (i) a charged membrane in contact with a solution containing only counter-ions, and (ii) a membrane in contact with a monovalent electrolyte reservoir.

As the membrane is taken to have an infinite extent in the lateral directions, the PB equation is reduced to an effective one-dimensional equation, where all local quantities, such as the electrostatic potential, , and ionic densities, , depend only on the -coordinate perpendicular to the planar membrane.

For a binary monovalent electrolyte (1:1 electrolyte, ), the PB equation from Eq. (1.2), reduces in its effective one-dimensional form to an ordinary differential equation depending only on the -coordinate:

(1.0)

where is the rescaled dimensionless potential and we have assumed that the external charge, , is restricted to the system boundaries and will only affect the boundary conditions.

We will consider two boundary conditions in this section. A fixed surface potential (Dirichlet boundary condition), , and constant surface charge (Neumann boundary condition), . A third and more specialized boundary condition of charge regulation will be treated in detail in section 1.8. In the constant charge case, the membrane charge is modeled via a fixed surface charge density, in Eq. (1.2). A variation of the Helmholtz free energy, , of Eq. (1.2) with respect to the surface potential, , , is equivalent to constant surface charge boundary:

(1.0)

Although we focus in the rest of the chapter on monovalent electrolytes, the extension to multivalent electrolytes is straightforward.

The boundary condition of Eq. (1.0) is valid if the electric field does not penetrate the ‘oily’ part of the membrane. This assumption can be justified (Kiometzis and Kleinert 1989, Winterhalter and Helfrich 1992), as long as , where is the membrane thickness (see Fig. 1.1). All our results for one or two flat membranes, sections 1.3-1.4 and 1.5-1.7, respectively, rely on this decoupled limit where the two sides (monolayers) of the membrane are completely decoupled and the electric field inside the membrane is negligible.

1.3.1 Counter-ions Only

A single charged membrane in contact with a cloud of counter-ions in solution is one of the simplest problems that has an analytical solution. It has been formulated and solved in the beginning of the 20th century by Gouy (1910, 1917) and Chapman (1913). The aim is to find the profile of a counter-ion cloud forming a diffusive electric double-layer close to a planar membrane (placed at ) with a fixed surface charge density (per unit area), , as in Fig. 1.2.

Without loss of generality, the single-membrane problem is treated here for negative (anionic) surface charges () and positive monovalent counter-ions (cations) in the solution, and , such that the charge neutrality condition,

(1.0)

is fulfilled.

The PB equation for monovalent counter-ions is written as

(1.0)

where is the reference density, taken at zero potential in the absence of a salt reservoir. The PB equation, Eq. (1.0), with the boundary condition for one charged membrane, Eq. (1.0), and vanishing electric field at infinity, can be integrated analytically twice, yielding

(1.0)

so that the density is

(1.0)

where is a reference potential and is the Gouy-Chapman length defined in Eq. (1.1). For example, for a choice of , the potential at vanishes and Eq. (1.0) reads

(1.0)

Although the entire counter-ion profile is diffusive as it decays algebraically, half of the counter-ions ( per unit area) accumulates in a layer of thickness close to the membrane,

(1.0)

As an example, we present in Fig. 1.3 the potential (in mV) and ionic profile (in M) for a surface density of , leading to a Gouy-Chapman length, . The figure clearly shows the build-up of the diffusive layer of counter-ions attracted by the negatively charged membrane, reaching a limiting value of . Note that the potential has a weak logarithmic divergence as . This divergency is a consequence of the vanishing ionic reservoir (counter-ions only) with counter-ion density obeying the Boltzmann distribution. However, the physically measured electric field, , properly decays to zero as , at .

Figure 1.3: The electric double layer for a single charged membrane in contact with an aqueous solution of neutralizing monovalent counter-ions. In (a) the electrostatic potential (in mV) is plotted as function of the distance from the membrane, , Eq. (1.0). The charged membrane is placed at with . The zero of the potential is chosen to be at the membrane, . In (b) the density profile of the counter-ions, (in M), is plotted as function of the distance . Its value at the membrane is and the Gouy-Chapman length, , is marked by an arrow.

1.3.2 Added Electrolyte

Another case of experimental interest is that of a single charged membrane at in contact with an electrolyte reservoir. For a symmetric electrolyte, , and the same boundary condition of constant surface charge , Eq. (1.0), holds at the surface. The negatively charged membrane attracts the counter-ions and repels the co-ions. As will be shown below, the potential decays to zero from below at large ; hence, it is always negative. Since the potential is a monotonic function, this also implies that is always positive. At large , where the potential decays to zero, the ionic profiles tend to their bulk (reservoir) densities, .

The PB equation for monovalent electrolyte, Eq. (1.0), with the boundary conditions as explained above can be solved analytically. The first integration of the PB equation for 1:1 electrolyte yields

(1.0)

where we have used that is implied by the Gauss law and electro-neutrality, and chose the bulk potential, , as the reference potential. A further integration yields

(1.0)

where is an integration constant, . Its value is determined by the boundary condition at .

The two ionic profiles, , are calculated from the Boltzmann distribution, Eq. (1.2), and from Eq. (1.0), yielding:

(1.0)

For constant surface charge, the parameter is obtained by substituting the potential from Eq. (1.0) into the boundary condition at , Eq. (1.0). This yields a quadratic equation, , with as its positive root:

(1.0)

For constant surface potential, the parameter can be obtained by setting in Eq. (1.0),

(1.0)

We use the fact that the surface potential is uniquely determined by the two lengths, and , and write the electrostatic potential as

(1.0)

where , in accord with our choice of . In Fig 1.4 we show typical profiles for the electrostatic potential and ionic densities, for (). Note that this surface charge density is ten times larger than of Fig 1.3. For electrolyte bulk density of M, the Debye screening length is .

The DH (linearized) limit of the PB equation, Eq. (1.0), is obtained for small surface charge and/or high electrolyte strength, . This limit yields and the potential can be approximated as

(1.0)

As expected for the DH limit, the solution is exponentially screened and falls off to zero for .

The opposite counter-ion only case, considered earlier in section 1.3.1 is obtained by formally taking the limit in Eqs. (1.0)-(1.0) or, equivalently, . This means that and from Eq. (1.0) we recover Eq. (1.0) for the counter-ion density, , while the co-ion density, , vanishes.

Figure 1.4: The electric double layer for a single charged membrane in contact with a 1:1 monovalent electrolyte reservoir of concentration , corresponding to . The membrane located at is negatively charged with , yielding . Note that the value of is ten times larger than the value used in Fig. 1.3. In (a) we plot the electrostatic potential, as function of , the distance from the membrane. The value of the surface potential is . In (b) the density profiles of counter-ions (solid line) and co-ions (dashed line), (in ), are plotted as function of the distance from the membrane, . The positive counter-ion density at the membrane is (not shown in the figure).

For a system in contact with an electrolyte reservoir, the potential always has an exponentially screened form in the distal region (far from the membrane). This can be seen by taking while keeping finite in Eq. (1.0)

(1.0)

Moreover, it is possible to extract from the distal form an effective surface charge density, , by comparing the coefficient of Eq. (1.0) with an effective coefficient from the DH form, Eq. (1.0),

(1.0)

Note that is calculated for the nominal parameter values in Eq. (1.0). The same concept of an effective is useful in several situations other than the simple planar geometry considered here.

1.3.3 The Grahame Equation

In the planar geometry, for any amount of salt, the non-linear PB equation can be integrated analytically, resulting in a useful relation known as the Grahame equation (Grahame 1947). This equation is a relation between the surface charge density, , and the limiting value of the ionic density profile at the membrane, . The first integration of the PB equation for a 1:1 electrolyte yields, Eq. (1.0), . Using the boundary condition, Eq. (1.0), and simple hyperbolic function identities gives a relation between and

(1.0)

and via the Boltzmann distribution of , the Grahame equation is obtained

(1.0)

This equation implies a balance of stresses on the surface, with the Maxwell stress of the electric field compensating the van ’t Hoff ideal pressure of the ions.

For large and negative surface potential, , the co-ion density, , can be neglected and Eq. (1.0) becomes

(1.0)

For example, for a surface charge density of (as in Fig. 1.4) and an ionic strength of , the limiting value of the counter-ion density at the membrane is  M, and that of the co-ions is  M. The very high and unphysical value of should be understood as an artifact of the continuum PB theory. In physical situations, the ions accumulate in the membrane vicinity till their concentration saturates due to the finite ionic size and other ion-surface interactions. We will further explore this point in sections 1.4 and 1.8.

The differential capacitance is another useful quantity to calculate and it gives a physical measurable surface property. By using Eq. (1.0), we obtain

(1.0)

As shown in Fig. 1.6, the PB differential capacitance, has a minimum at the potential of zero charge, , and increases exponentially for .

1.4 Modified Poisson-Boltzmann (mPB) Theory

The density of accumulated counter-ions at the membrane might reach unphysical high values (see Fig. 1.4). This unphysical situation is avoided by accounting for the solvent entropy. Including this additional term yields a modified free-energy and PB equation (mPB). Taking this entropy into account yields a modified free-energy, written here for monovalent electrolyte:

(1.0)

This is the free energy of a Coulomb lattice-gas (Borukhov, Andelman and Orland 1997, 2000, Kilic, Bazant and Ajdari 2007). Taking the variation of the above free energy with respect to , , gives the ionic profiles

(1.0)

with being the bulk volume fraction of the ions. For simplicity, is taken to be the same molecular size of all ionic species and the solvent.

  Entropy derivation of the mPB Let us start with a homogenous system containing an ionic solution inside a volume , with cations, anions and water molecules, such that . The number of different combinations of cations, anions and water molecules is . Therefore, the entropy is (1.0) where we have used Stirling’s formula for . We now consider a system of volume . The entropy of such system can be written in the continuum limit as (1.0) where and , are the densities of the cations, anions and water molecules, respectively, and is the total number of molecules in the volume . In this last equation we have used the lattice-gas formulation, in which the solution is modeled as a cubic lattice with unit cell of size . Each unit cell contains only one molecule, .

In the above equation we have also used the equilibrium relation

(1.0)

valid in the bulk where . Variation with respect to , , yields the mPB equation for 1:1 electrolyte:

(1.0)

For small electrostatic potentials, , the ionic distribution, Eq. (1.0), reduces to the usual Boltzmann distribution, but for large electrostatic potentials, , this model gives very different results with respect to the PB theory. In particular, the ionic concentration is unbound in the standard PB theory, whereas it is bound for the mPB by the close-packing density, . This effect is important close to strongly charged membranes immersed in an electrolyte solution, while the regular PB equation is recovered in the dilute bulk limit, , for which the solvent entropy can be neglected.

For large electrostatic potentials, the contribution of the co-ions is negligible and the counter-ion concentration follows a distribution reminiscent of the FermiDirac distribution

(1.0)

where electro-neutrality dictates . In Fig 1.5 we show for comparison the modified and regular PB profiles for a 1:1 electrolyte. To emphasize the saturation effect of the mPB theory, we chose in the figure a large ion size, .

Figure 1.5: Comparison of the modified PB (mPB) profiles (black solid lines) with the regular PB one (dashed blue lines). In (a) we show the counter-ion profile, and in (b) the co-ion profile. The parameters used are: ion size , surface charge density and 1:1 electrolyte ionic strength . Note that while the PB value at the membrane is , the mPB density saturates at .

The mPB theory also implies a modified Grahame equation that relates the surface charge density to the ion surface density, . First, we find the relation between and the surface potential, ,

(1.0)

This equation represents a balance of stresses on the surface, where the Maxwell stress of the electric field is equal to the lattice-gas pressure of the ions. The surface potential can also be calculated

(1.0)

with the dimensionless parameter .

For large surface charge or large surface potential, the co-ions concentration at the membrane is negligible, , and the surface potential, Eq. (1.0) is approximated by

(1.0)

and from Eq. (1.0) we obtain the Grahame equation,

(1.0)

Note that in the dilute limit , the Grahame equation reduces to the regular PB case, Eq. (1.0).

Figure 1.6: Comparison of the differential capacitance, , calculated from the regular PB theory (dashed red line), Eq. (1.0), (chosen so that it corresponds to and ), and from the mPB theory, Eq. (1.0). The mPB differential capacitance is calculated for . For low , it shows a camel shape (black solid line), while for high , it shows a unimodal (dash-dotted blue line).

It is also straightforward but more cumbersome to calculate the differential capacitance, , for the mPB theory. From Eq. (1.0) we obtain,

(1.0)

Although it can be shown that for the mPB differential capacitance reduces to the standard PB result, the resulting is quite different for any finite value of . The main difference is that instead of an exponential divergence of at large potentials, decreases for high-biased . For rather small bulk densities, , the shows a behavior called camel-shape or double-hump. This behavior is also observed in experiments at relatively low salt concentrations. As shown in Fig. 1.6, the double-hump has a minimum at and two maxima. The peak positions can roughly be estimated by substituting the closed-packing concentration, , into the Boltzmann distribution, Eq. (1.2), yielding . Using parameter values as in Fig. 1.6, is estimated as as compare to the exact values, .

Furthermore, it can be shown that for high salt densities, , exhibits (see also Fig. 1.6) a unimodal maximum close to the potential of zero charge, rather than a minimum as does . Such results that take into account finite ion size for the differential capacitance are of importance in the theory of confined ionic liquids (Kornyshev 2007, Nakayama and Andelman 2015).

1.5 Two Membrane System: Osmotic Pressure

Figure 1.7: Schematic drawing of two asymmetric membranes. The planar membrane located at carries a charge density , while the membrane at has a charge density of . The antisymmetric membrane setup is a special case with , while in the symmetric case, .

We consider now the PB theory of two charged membranes as shown in Fig. 1.7. The two membranes can, in general, have different surface charge densities: at and at . The boundary conditions of the two-membrane system are written as , and using the variation of the free energy, :

(1.0)

It is of interest to calculate the force (or the osmotic pressure) between two membranes interacting across the ionic solution. The osmotic pressure is defined as , where is the inner pressure and is the pressure exerted by the reservoir that is in contact with the two-membrane system. Sometimes the osmotic pressure is referred to as the disjoining pressure, introduced first by Derjaguin (Churaev, Derjaguin and Muller 2014).

Let us start by calculating the inner and outer pressures from the Helmholtz free energy. The pressure ( or ) is the variation of the free-energy with the volume:

(1.0)

with , being the system volume, the lateral membrane area, and is the inter-membrane distance. As the interaction between the two membranes can be either attractive () or repulsive (), we will analyze the criterion for the crossover () between these two regimes as function of the surface charge asymmetry and inter-membrane distance.

  General derivation of the pressure The Helmholtz free-energy obtained from Eq. (1.2) can be written in a general form as, , where we use the Poisson equation to obtain the relation, . As the integrand depends only implicitly on the coordinate through , one can obtain from the Euler-Lagrange equations the following relation (Ben-Yaakov et al. 2009). (1.0) where the sum is over ionic species. Let us understand the meaning of the constant on the right-hand side of the above equation. For uncharged solutions, the Helmholtz free-energy per unit volume contains only the entropy term, , and from Eq. (1.5), we obtain . A known thermodynamic relation is , with the total chemical potential defined as before, , implying that the right-hand side constant is . However, even for charged liquid mixtures, the electrostatic potential vanishes in the bulk, away from the boundaries, and reduces to the same value as for uncharged solutions. Therefore, we conclude that the right-hand side constant is , yielding (1.0) If the electric field and ionic densities are calculated right at the surface, we obtain the contact theorem that gives the osmotic pressure acting on the surface. Another and more straightforward way to calculate the pressure, is to calculate the incremental difference in free energy, , for an inter-membrane separation , i.e. . The calculation of can be done by including an additional slab of width in the space between the two membranes at an arbitrary position. We remark that the validity of the contact theorem itself is not limited to the PB theory, but is an exact theorem of statistical mechanics (Henderson and Blum 1981, Evans and Wennerström 1999, Dean and Horgan 2003).

We are interested in the osmotic pressure, . For an ionic reservoir in the dilute limit, Eq. (1.0) gives , where is the ionic species bulk density. Thus, the osmotic pressure can be written as

(1.0)

and for monovalent 1:1 ions:

(1.0)

At any position between the membranes, the osmotic pressure has two contributions. The first is a negative Maxwell electrostatic pressure proportional to . The second is due to the entropy of mobile ions and measures the local entropy change (at an arbitrary position, ) with respect to the ion entropy in the reservoir.

1.6 Two Symmetric Membranes,

For two symmetric charged membranes, at , the electrostatic potential is symmetric about the mid-plane yielding a zero electric field, at . It is then sufficient to consider the interval with the boundary conditions,

(1.0)

As is constant (independent of ) between the membranes, one can calculate the disjoining pressure, , from Eq. (1.0), at any position , between the membranes. A simple choice will be to evaluate it at (the mid-plane), where the electric field vanishes for the symmetric case,

(1.0)

and for monovalent ions, , we get

(1.0)

where is the mid-plane concentration of the species. It can be shown that the electro-neutrality condition implies that the osmotic pressure is always repulsive for any shape of boundaries (Sader and Chan 1999, Neu 1999) as long as we have two symmetric membranes .

Note that the Grahame equation can be derived also for the two-membrane case with added electrolyte. One way of doing it is by comparing the pressure of Eq. (1.0) evaluated at one of the membranes, , and at the mid-plane, . The pressure is constant between the two membranes, thus, by equating these two pressure expressions, the Grahame equation emerges

(1.0)

By taking the limit of infinite separation between the two-membranes and , the Grahame equation for a single membrane, Eq. (1.0), is recovered.

1.6.1 Counter-ions Only

In the absence of an external salt reservoir, the only ions in the solution for a symmetric two-membrane system, are positive monovalent counter-ions with density that neutralizes the surface charge,

(1.0)

The PB equation has an analytical solution for this case. Integrating twice the PB equation, Eq. (1.0), with the appropriate boundary conditions, Eq. (1.6), yields an analytical expression for the electrostatic potential:

(1.0)

and consequently the counter-ion density is

(1.0)

In the above we have defined and chose arbitrarily . We also introduced a new length scale,