Atoms, molecules and ions constitute the building blocks of most materials. Even very small objects, such as, for example, colloidal particles (i.e., particles with a few nm or µm in size) contain at least several thousands and often millions/billions of such building units. The constituting molecules (or atoms) in one object interact with all molecules (or atoms) in another object via van der Waals interactions. Thus, as will be shown below, the interaction between surfaces is to a great extent defined by the van der Waals interaction between their respective components.

For surfaces interacting in a liquid medium, an entropic repulsive force (the electric double-layer repulsion) arising from electrostatic interactions between ions in the medium will also be observed. The van der Waals and electric double-layer interactions form the core of the DLVO theory, named after the scientist that developed it: Derjaguin and Landau in Russia in 1941 [2] and Verwey and Overbeek [3] in 1948 in The Netherlands.

The DLVO theory is particularly useful to describe and predict the stability of colloidal dispersions. A colloidal system is by definition composed of microscopic particles dispersed throughout a medium. How long the particles will remain dispersed is a measure of the colloidal stability of the system. The colloidal stability depends on the interplay between the attractive and repulsive forces acting between the particles. Thus, in the frame of the DLVO theory, the colloidal stability of a given system depends on the relative strengths of the attractive van der Waals and the repulsive electric double-layer interactions, which will be described in detail in the following sections.

Van der Waals forces between surfaces

Let us imagine having an infinite planar wall. If we place a molecule at a given distance from the wall, van der Waals interactions will exist between this molecule and all the molecules forming the wall. The total energy of interaction between the molecule and the wall can be approximated as the sum of all these interactions, as proposed in the pioneering work by Hamaker [4] and Bradley [5].

To find an expression for this interaction, consider the situation depicted in Fig. 1. Here, a molecule is placed at a distance d from an infinite planar wall. The position of the molecule is defined as the origin point of a Cartesian coordinate system, with the wall plane found at a distance d in the y-axis. Consider now the interaction of the molecule with a volume differential within the wall at the coordinates (x, y, z). The volume differential will be given by dV = dx*dy*dz, and its separation r from the molecule will be given by \(r=\sqrt^+^+^}\). The van der Waals interaction between the molecule and all molecules within the volume differential will be given by:

$$W}_}\left(r\right)=-\rho \frac^}\mathrmV=-\rho \frac^+^+^\right)}^}\mathrmx\mathrmy\mathrmz,$$

(3)

where \(\rho\) is the density (in molecules per volume unit) of the wall.

Fig. 1

A single molecule interacting with an infinite wall

Integrating over the space occupied by the wall (i.e., taking the sum of the interaction between the molecule and all possible volume differentials), an expression for the total interaction is obtained:

$$_}\left(r\right)=_^ _^ _^-\frac^+^+^\right)}^}\mathrmx\mathrmy\mathrmz=-\frac^}.$$

(4)

An interesting consequence of Eq. (4) is the fact that the strength of the interaction is observed to decrease with d−3, in contrast with the r−6 dependence observed between molecules. This implies that the van der Waals interaction between a molecule and infinite wall has much longer range than between single molecules.

A similar analysis can be used to estimate the total van der Waals interaction between two infinite planar walls (Fig. 2). In this case, the interaction between two volume differentials is calculated and integrated over the volume of the two walls (i.e., one calculates the sum of all possible interactions between molecules in one wall and molecules in the other one). To obtain a finite value, the interaction is calculated per unit area. The separation r between volume differentials would be given by \(r=\sqrt_-_\right)}^+_-_\right)}^+_-_\right)}^}\) and thus:

$$_}\left(d\right)=_^_^ _^ _^-\frac__}_-_\right)}^+_-_\right)}^+_-_\right)}^\right)}^}\mathrm_x}_y}_z}_=-\frac__C}^}=-\frac_}^},$$

(5)

where \(_= ^__C\) is the Hamaker constant describing the interaction between the two walls. Equation (5) implies that the van der Waals interaction between planar surfaces has much longer range than between molecules since its strength decreases with the square of the separation. This means that the short-range van der Waals forces acting between molecules can give rise to long-range forces between macroscopical objects.

Fig. 2

Two infinite planar walls interacting with each other

The pairwise additivity assumptions made in the derivations above ignore, among other phenomena, the effect of atoms and molecules in the vicinity of two molecules interacting. More rigid approaches to estimate the interaction between two planar walls result, however, in an expression identical as given by Eq. (5). The only difference is the way the Hamaker constant is calculated (see, e.g., the derivation by Israelachvili [6]). Furthermore, it can be shown that the same expression can be used to describe the interactions between two objects in a medium. Again, the only difference is the value of the Hamaker constant, which will be dependent not only on the materials interacting, but also on the medium in which the interaction takes place. Thus, the interaction per unit area between Wall 1 and Wall 2 in Medium 3 (Fig. 3) is described by the equation:

Fig. 3

Two infinite planar walls interacting in a medium

$$_}\left(d\right)=-\frac_}^}.$$

(6)

The Hamaker constant for the interaction in a medium \(_\) is related to the Hamaker constant of the different materials in vacuum as given by the combination rule: \(_\approx \left(\sqrt_}-\sqrt_}\right)\left(\sqrt_}-\sqrt_}\right)\). An interesting consequence of the combination rule is that the Hamaker constant for the interaction in a medium \(_\) can be negative if \(_<_<_\) (or vice versa), i.e., if the value of the Hamaker constant of the medium lies between the values of the Hamaker constants of the two materials interacting. In such cases, the van der Waals interaction between the walls will be repulsive. Notice that these repulsive van der Waals interactions are possible only in a medium and only if the two materials interacting are different. Between two similar materials or between any materials in vacuum, the interaction is always attractive.

In the frame of the DLVO theory, a final step to be able to estimate the strength of van der Waals interaction between colloidal particles is to find expressions to describe the interaction between particles with finite geometries. In particular, the interaction between spherical objects is of relevance within the field of colloids. A very useful relationship to easily determine the force F(d) acting between two spheres is given by the Derjaguin approximation [7], which states that this force will be proportional to the interaction energy between two planar surfaces W(d) as long as the separation d between the spheres is shorter than their radii:

$$F\left(d\right)=\frac_\right)+\left(1/_\right)},$$

(7)

where R1 and R2 are the radii of the two spheres interacting. In the case of spheres with the same radius R, Eq. 7 simplifies to \(F\left(d\right)=\pi RW(d)\). The Derjaguin approximation applies to all kinds of interactions and is therefore very useful to extrapolate the results obtained when considering two infinite surfaces to what would be expected for two finite spheres. The energy of the interaction between two spheres Ws(d) can then be calculated from:

$$_}\left(d\right)=-_^F\left(d\right)\mathrmd.$$

(8)

In the case of van der Waals interactions between two identical spheres with radius R, combining Eqs. 6, 7 and 8 gives us the expression:

$$_(\mathrm)}\left(d\right)=-\frac_R}.$$

(9)

Remarkably, the van der Waals interaction between spherical particles is thus shown to decay linearly with the inverse of the separation and thus behaves similarly to the Coulomb interaction between ions (Eq. 1).

Comparing Eqs. 6 and 9 reveals that the distance dependence of van der Waals interactions varies with the geometry of the objects interacting. Thus, the range of the interaction between spherical particles is much longer than for planar surfaces and other geometries, as summarized and exemplified in Fig. 4.

Fig. 4

Distance dependence of the van der Waals interactions for selected geometries

Electric double-layer repulsion between surfaces in a liquid

The second force described by the DLVO theory is the repulsion between surfaces in a liquid arising due to their respective electric double layers. As will be discussed below, this repulsion, although commonly referred to as “electrostatic,” is in fact entropy-driven. Indeed, the “pure” electrostatic, energy-driven, interaction between two equally charged surfaces in a liquid is, counterintuitively, attractive. Why this is the case and what then drives the net repulsion described by the DLVO theory will be discussed in the following sections.

Surfaces in a liquid: the electric double-layer and the Poisson-Boltzmann equation

A surface submerged in a polar liquid will probably acquire a certain electrical charge. There are several mechanisms by which surface charges appear. For example, in water and other hydrogen-bonding liquids, one can expect the dissociation of acidic or basic groups at the surface, resulting thus on the formation of, respectively, negative or positive surface charges. The surface can also acquire a charge if ions from the solution adsorb or bind onto it. What is important to keep in mind is that, regardless of how the surface acquires a charge, the solution will acquire the same amount of charges but with opposite sign, usually in the form of counterions such that the system in its totality remains neutral. These counterions are the complementary product of the reaction by which charges at the surface are generated. For example, in the case of dissociation of an acid at the surface:

$$}_)}\underset}_)}^+}_)}^.$$

Thus, for every negative charge at the surface there is a positive ion in solution. Similarly, in the case of ion adsorption:

$$}_)}^+}_)}^+}_)}\underset}_)}^+}_)}^.$$

In this case, Cl− in solution adsorbs onto a binding site on the surface, rendering the surface negatively charged. The sodium counterion remains in solution. Thus, for every negative charge that is transferred to the surface, there is a positive charge remaining in the solution.

The spatial distribution of the counterions in the liquid medium close to the surface is not random but results from a compromise between their mixing entropy (which favors a homogeneous distribution of the ions in the available volume) and the electrostatic attraction between the surface and the counterions, which favors the accumulation of counterions at the interface. Some of the ions will form a compact layer at the surface, forming a so-called Stern layer. The rest will be distributed in a “cloud” expanding from the surface, with the counterion concentration decreasing as the separation from the surface increases [8]. This forms the so-called diffuse layer (Fig. 5).

Fig. 5

Depiction of a negatively charged wall with surface charge density σ immersed in a liquid. A compact layer of counterions form close to the surface (Stern layer). The rest of the counterions are distributed in a diffuse layer

To better describe the distribution of ions in the liquid medium, it is necessary to consider that the total chemical potential µ of a single ion in the solution is given by the sum of the electrostatic and entropic contributions mentioned above:

$$\mu =ze\psi \left(x\right)+kT\mathrmn\left(x\right),$$

(10)

where z is the valence of the ion, e is the elementary charge, \(\psi \left(x\right)\) is the electrical potential at a given position x in the solution, \(k\) is the Boltzmann constant (k = 1.38065 × 10–23 J K−1), T is the absolute temperature, and \(n(x)\) is the ion concentration at position x. At equilibrium, the chemical potential µ of the ion must be the same at all values of x. Given that only electrical potential differences (and not absolute values of the electrical potential) are ever meaningful, we can define an arbitrary position x where the potential is set to \(\psi\) = 0 and the ion concentration is thus \(_\). From Eq. 10, it can then be shown that \(n\left(x\right)\) is given by the Boltzmann distribution:

$$n\left(x\right)=_}^}.$$

(11)

A second needed relationship between the ion concentration and the potential can be derived from the Poisson equation, which states that:

$$\rho \left(x\right)=-\varepsilon _\frac}^\psi \left(x\right)}x}^},$$

(12)

where \(\rho \left(x\right)\) is the bulk charge density, roughly defined as the “charge concentration,” at position x and given by:

$$\rho \left(x\right)=e\sum __(x).$$

(13)

Combining Eqs. 11, 12 and 13, one obtains the Poisson-Boltzmann (PB) equation:

$$\frac}^\psi (x)}x}^}=-\frac_}\sum __}^_e\psi \left(x\right)}}.$$

(14)

By applying appropriate limiting conditions, Eq. 14 can be used to determine the potential \(\psi (x)\), ion concentration \(n\left(x\right)\) and electric field \(E(x)=\left|\mathrm\psi (x)/\mathrmx\right|\) at any point x near a charged surface, or, more importantly for the purpose of this text, at any point x in between two surfaces. The equation thus lies at the heart of the DLVO theory.

Electric double-layer interaction between a charged and a neutral planar wall in the absence of electrolyte

Consider the situation depicted in Fig. 6, where a non-charged planar wall is placed at x = 0 and a charged planar wall with a surface charge density σ is placed at x = d. Counterions accumulate in the gap between the walls, ensuring thus electroneutrality. Imagine now that outside of the gap there is a reservoir of solvent in which no ions are present. The mixing entropy of the trapped counterions would increase if the solvent from the reservoir would flow in the gap, effectively pushing the walls apart. Thus, a repulsive osmotic pressure acts at all points within the gap. This pressure is highest at the vicinity of the charged surface (at x = d, where the concentration of ions is highest) and lowest right next to the uncharged surface (at x = 0). An opposing, attractive pressure is also found in the gap, and it has an electrostatic (enthalpic) origin: counterions in solution are attracted to the charged wall (and vice versa) and thus they effectively pull the walls together. This pressure is also highest at x = d, and it is zero right at the interface between the medium and the uncharged wall (i.e., at x = 0) since the latter experiences no electrostatic pull. The total pressure PDL(d) is the sum of both these contributions, and it should be uniform across the gap. Since the electrostatic contribution at x = 0 is zero, the total pressure acting in the system is given by the osmotic pressure at this point, i.e.:

$$_}\left(d\right)=kT_,$$

(15)

where \(_\) is the concentration of counterions at x = 0. This concentration will be dependent on the separation d between the walls.

Fig. 6

Representation of a neutral surface placed at a separation d from a negatively charged surface with surface charge density σ

To be able to calculate the value of the pressure acting between the walls, \(_(d)\) needs to be determined. This can be done with help of the PB equation (Eq. 14), which, for the system depicted in Fig. 6, becomes:

$$\frac}^\psi (x)}x}^}=-\frac_}z_\mathrm}^},$$

(16)

where the potential at x = 0 has been set to zero (i.e., \(\psi \left(0\right)=0)\). A further limiting condition needed to solve Eq. (16) is given by the electric field at x = 0, which is zero, i.e.,\(E\left(0\right)=\psi \left(x\right)/\mathrmx\right|}_=0\)

The solution to Eq. 16 is thus (see Engström et al. [9]):

$$\psi \left(x\right)=\frac\mathrm\left(\mathrm\left(Kx\right)\right),$$

(17)

where

As stated above, the electroneutrality condition must be fulfilled, i.e., that the surface charge density must be compensated by the sum of the bulk charge densities at all positions in the gap, i.e.:

$$\sigma =-_^\rho \left(x\right)\mathrmx=\varepsilon __^\left(\frac}^\Psi \left(x\right)}x}^}\right)\mathrmx=\varepsilon _\left(\Psi \left(x\right)}x}\right)}_-\Psi \left(x\right)}x}\right)}_\right)=\varepsilon _\left(E\left(d\right)-E\left(0\right)\right)=\varepsilon __},$$

(19)

where \(_}= E\left(d\right)\) is the electric field directly at the surface of the charged wall. Combining Eqs. 17 and 19, one obtains:

$$_}=\Psi \left(x\right)}x}\right|}_=-\frac\mathrm\left(Kd\right)=\left|\frac_}\right|.$$

(20)

For a given system where σ is known, \(_\) can be numerically determined for a given d using Eq. 20, thus allowing calculating the pressure acting between the surfaces.

Electric double-layer interaction between two identical charged infinite walls in the absence of electrolyte

Consider now two identical charged walls interacting in a liquid. In this case, the electrostatic contribution to the total pressure will be zero right at the midplane, and the pressure acting between the walls is thus given by the concentration of ions at this point [9]. It is thus convenient to set x = 0 and \(\psi =0\) at the midplane. The charged surfaces are thus located at x =  ± (d/2), as depicted in Fig. 7. Each half of the system depicted in Fig. 7 is identical to the system illustrated in Fig. 6, with the neutral surface being replaced by the midplane. The solution for the PB equation is thus similar to the case described above. The only difference is that the separation d in Eq. 20 must be replaced by d/2 [9]:

Fig. 7

Representation of two identical charged walls placed at a separation d from each other

$$_}=-\frac\mathrm\left(K\frac\right)=\left|\frac_}\right|.$$

(21)

Thus, two charged surfaces separated by a distance d will interact with the same strength as a charged and a neutral surface separated by d/2 [10].

Isolated charged surfaces in an electrolyte solution

The cases described in the previous sections are unlikely to be found in real systems, particularly if water is the medium. In such case, most surfaces will acquire a charge, and there will always be other ions in solution besides the surface counterions. All these ions need to be considered when solving the PB equation. For two surfaces interacting in an electrolyte, the approach used in previous cases does not provide an analytical solution concerning the concentration of ions at any point in the gap. Instead, it becomes necessary to treat each wall as an isolated surface and to define \(\psi =0\) at a point where the concentration of the ions is known. Thus, x = 0 is defined right at the surface of the wall, and \(\psi =0\) is set at x = \(\infty\) (see Fig. 8). Thus, \(_=_\), i.e., the concentration of ions at an infinite separation from the surface. This is equal to the bulk concentration of ions.

Fig. 8

Representation of an isolated charged wall in an electrolyte solution

The solution to the PB equation (Eq. 14) becomes much more cumbersome when an electrolyte solution is present. Table 1 summarizes the expressions for \(\psi \left(x\right)\) obtained for symmetrical electrolytes (i.e., electrolytes where the anion and the cation have the same absolute valence z) and for a mixture of 2:1 (e.g., CaCl2) or 1:2 (e.g., Na2SO4) with a 1:1 (e.g., NaCl) electrolyte. The equations are expressed in terms of the parameters \(_\) and \(_\left(\infty \right)}\), which represent, respectively, the surface potential (i.e., the potential at x = 0) and the bulk concentration of the ion i. In both equations, the decay of the potential as the distance from the surface increases is described by a parameter \(\kappa\) dependent only on the composition and temperature of the electrolyte and not on the properties of the surface. The inverse of this parameter (i.e., \(^\)) is known as the Debye length and is a measure of the thickness of the electric double layer. Increasing the ionic strength of the solution decreases the magnitude of the Debye length, meaning that the electric double layer becomes thinner. Adding an electrolyte will indeed cause an increase of the counterion concentration near the surface, screening the surface charges already at small values of x.

Table 1 Solutions of the PB equation for a symmetrical electrolyte (known as the Gouy-Chapman theory) and for a mixture of 2:1 (or 1:2) and 1:1 electrolytes mixed at a ratio a = \(_\)/\(_\) according to unpublished work by the author

The second equation in Table 1 is reduced to the first one when a = ∞ (i.e., when only the monovalent electrolyte is present). Setting a = 0 (i.e., when only the asymmetrical electrolyte is present) results in the equation proposed by Andrietti et al. [11] for the case of 2:1 (or 1:2) electrolytes, which is in turn similar to the solutions given by Grahame [12].

A relationship between the surface potential \(_\) and the surface charge density \(\sigma\) can be obtained from the PB equation by considering that electroneutrality is required:

$$\sigma =-_^\rho \left(x\right)\mathrmx=\varepsilon __^\left(\frac}^\psi \left(x\right)}x}^}\right)\mathrmx=-\varepsilon _\psi \left(x\right)}x}\right)}_=\sqrt_kT\sum _\left(}^_e_}}}-1\right)}.$$

(22)

Equation 22 and the equations in Table 1 can be dramatically simplified when the value of the surface potential is low. For absolute values of \(_\ll kT/(ze)\) (the so-called “Debye approximation”), the equations in Table 1 are both reduced to:

$$\psi \left(x\right)=_}^.$$

(23)

Equation 23 is called the Debye-Hückel equation and is a very useful approximation valid for all kinds of electrolytes. The equation results in a very exact description of the potential decay near the surface when the surface potential is low (< 25 mV at room temperature) as long as the surface potential and the Debye length are determined accurately; see Fig. 9 (left). Even at large surface potentials, the differences between the predictions from Eq. 23 and from the equations show in Table 1 differ only slightly (see Fig. 9, right). For 1:1 electrolytes, the deviation is actually negligible. To simplify the description of the electric double-layer interaction between surfaces, in the following sections the Debye-Hückel equation will be used instead of the exact solutions in Table 1.

Fig. 9

Comparison of the predictions of exact solutions to the PB equation for different electrolytes and the Debye-Hückel approximation. The Debye length is set equal in all systems (1.34 nm). Left: Surface potential: − 15 mV. Right: Surface potential − 50 mV. The legend is valid for both figures

From Eq. 23, a simple relationship between the surface potential and the surface charge density is obtained:

$$\sigma =-\varepsilon _\psi \left(x\right)}x}\right)}_=\varepsilon _\kappa _.$$

(24)

Electric double-layer interaction between two identical charged surfaces in an electrolyte solution

When considering the interaction between two charged walls in an electrolyte, it is necessary to determine how the ions distribute in the gap. The Boltzmann distribution can be used to determine the concentration at any point in between the surfaces. The potential at any position can be estimated by the weak overlap approximation which considers that the potential at any point in the gap between the walls will be given by the sum of the potentials of each wall treated separately. From the Debye approximation, the potential at any point x between two similar walls (see Fig. 10) with a surface potential \(_\) at a separation d is:

Fig. 10

Representation of two identical charged walls placed at a separation d from each other in an electrolyte solution

$$\psi \left(x\right)=_\left(}^+}^\right)=\frac_\kappa }\left(}^+}^\right).$$

(25)

From Eqs. 11 and 25, the ion distribution in the gap can be calculated. Figure 11 shows the distribution of co- and counterions in the gap between two surfaces, as well as the total concentration of ions at different points in the gap:

Fig. 11

Distribution of ions in the gap between two charged surfaces (\(\sigma\) = 5 mC/m.2) placed at a separation d = 10 nm in an electrolyte (NaCl 10 mM)

As can be observed, the total concentration of ions is higher near the walls and lowest at the midplane. The midplane concentration is, however, higher than the bulk concentration (20 mM in the case illustrated in Fig. 11). The total pressure acting between the walls will be given by the difference between the osmotic pressure at the midplane, where the electrostatic contribution is zero, and at the bulk, i.e.:

$$_}\left(d\right)=kT\left(\sum _\left(d/2\right)-\sum _\right),$$

(26)

Substituting the Boltzmann distribution (Eq. 11) in Eq. 26, we obtain:

$$_}\left(d\right)=kT\sum \left[_\left(}^_e_}}-1\right)\right],$$

(27)

where \(_\) is the potential at the midplane. Expanding the exponents into series, and assuming that the value of \(_\) is small, Eq. 30 simplifies into:

$$_}\left(d\right)=\frac_^}_^.$$

(28)

The midplane potential \(_\) calculated from the Debye and weak overlap approximations is:

$$_=2 \psi \left(d/2\right)\approx _}^},$$

(29)

The pressure is thus given by:

$$_}\left(d\right)=2\varepsilon _^_^}^=\frac^}_}}^.$$

(30)

The energy of interaction per unit area W(d) can be obtained by integration of Eq. 30. In terms of the surface charge density, one obtains:

$$_}\left(d\right)=-_^_}\left(d\right)\mathrmd=\frac^}_}}^.$$

(31)

Using the Derjaguin approximation, the force Fs(DL) and the energy of interaction \(}_(\mathrm)}\) between spherical particles can thus be expressed as:

$$_(\mathrm)}\left(d\right)=\frac^}_}}^,$$

(32)

and

$$_(\mathrm)}\left(d\right)=\frac^}^\varepsilon _}}^,$$

(33)

respectively. The relevance of Eqs. 30–33 is that they are valid in all kinds of electrolytes, including mixtures, as long as the Debye approximation holds. Indeed, at absolute values of \(_\ll kT/(ze)\) (around 25 mV for a 1:1 electrolyte at room temperature), the equations give a very exact description of the experimental observations as long as the Debye length is determined with precision. For larger potentials, the approximation is still rather good qualitatively, although it should be used with caution.

For more accurate results with symmetrical electrolytes, the weak overlap approximation and the corresponding equation in Table 1 can be used to get an expression for \(_\). The resulting expression for the pressure acting between the surfaces is slightly more complicated than Eq. 30, but can still be integrated to obtain the energy of interaction.

For asymmetrical electrolytes or for mixtures, an equation for the pressure between walls can be obtained. However, this expression cannot always be integrated analytically to estimate the energy of interaction. Numerical approximations are thus often needed.

Summarizing the discussion above, the repulsion acting between two charged surfaces in a liquid is driven by entropy and not by a direct electrostatic repulsion between them. Indeed, purely electrostatic interactions would result in an attractive force, since the counterions attract the surface towards them. The entropy contribution as predicted by the PB equation is, however, always dominant. In Sect. “Shortcomings of the PB equation” below, examples where the electrostatic contribution actually dominates (leading to an attractive force) will be discussed.

The total interaction between surfaces in an electrolyte

For two surfaces interacting in a liquid, the total interac