DLVO theory

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The DLVO theory (named after Boris Derjaguin and Lev Landau, Evert Verwey and Theodoor Overbeek) explains the aggregation and kinetic stability of aqueous dispersions quantitatively and describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so-called double layer of counterions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, . For two spheres of radius each having a charge (expressed in units of the elementary charge) separated by a center-to-center distance in a fluid of dielectric constant containing a concentration of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa potential,

Contents

where


Overview

DLVO theory is a theory of colloidal dispersion stability in which zeta potential is used to explain that as two particles approach one another their ionic atmospheres begin to overlap and a repulsion force is developed. [1] In this theory, two forces are considered to impact on colloidal stability: Van der Waals forces and electrical double layer forces.

The total potential energy is described as the sum of the attraction potential and the repulsion potential. When two particles approach each other, electrostatic repulsion increases and the interference between their electrical double layers increases. However, the Van der Waals attraction also increases as they get closer. At each distance, the net potential energy of the smaller value is subtracted from the larger value. [2]

At very close distances, the combination of these forces results in a deep attractive well, which is referred to as the primary minimum. At larger distances, the energy profile goes through a maximum, or energy barrier, and subsequently passes through a shallow minimum, which is referred to as the secondary minimum. [3]

At the maximum of the energy barrier, repulsion is greater than attraction. Particles rebound after interparticle contact, and remain dispersed throughout the medium. The maximum energy needs to be greater than the thermal energy. Otherwise, particles will aggregate due to the attraction potential. [3] The height of the barrier indicates how stable the system is. Since particles have to overcome this barrier in order to aggregate, two particles on a collision course must have sufficient kinetic energy due to their velocity and mass. [2] If the barrier is cleared, then the net interaction is all attractive, and as a result the particles aggregate. This inner region is often referred to as an energy trap since the colloids can be considered to be trapped together by Van der Waals forces. [2]

For a colloidal system, the thermodynamic equilibrium state may be reached when the particles are in deep primary minimum. At primary minimum, attractive forces overpower the repulsive forces at low molecular distances. Particles coagulate and this process is not reversible. [4] However, when the maximum energy barrier is too high to overcome, the colloid particles may stay in the secondary minimum, where particles are held together but more weakly than in the primary minimum. [5] Particles form weak attractions but are easily redispersed. Thus, the adhesion at secondary minimum can be reversible. [6]

History

In 1923, Debye and Hückel reported the first successful theory for the distribution of charges in ionic solutions. [7] The framework of linearized Debye–Hückel theory subsequently was applied to colloidal dispersions by Levine and Dube [8] [9] who found that charged colloidal particles should experience a strong medium-range repulsion and a weaker long-range attraction. This theory did not explain the observed instability of colloidal dispersions against irreversible aggregation in solutions of high ionic strength. In 1941, Derjaguin and Landau introduced a theory for the stability of colloidal dispersions that invoked a fundamental instability driven by strong but short-ranged van der Waals attractions countered by the stabilizing influence of electrostatic repulsions. [10] In 1948, Verwey and Overbeek independently arrived at the same result. [11] This so-called DLVO theory resolved the failure of the Levine–Dube theory to account for the dependence of colloidal dispersions' stability on the ionic strength of the electrolyte. [12]

Derivation

DLVO theory is the combined effect of van der Waals and double layer force. For the derivation, different conditions must be taken into account and different equations can be obtained. [13] But some useful assumptions can effectively simplify the process, which are suitable for ordinary conditions. The simplified way to derive it is to add the two parts together.

van der Waals attraction

van der Waals force is actually the total name of dipole-dipole force, dipole-induced dipole force and dispersion forces, [14] in which dispersion forces are the most important part because they are always present. Assume that the pair potential between two atoms or small molecules is purely attractive and of the form w = −C/rn, where C is a constant for interaction energy, decided by the molecule's property and n = 6 for van der Waals attraction. [15] With another assumption of additivity, the net interaction energy between a molecule and planar surface made up of like molecules will be the sum of the interaction energy between the molecule and every molecule in the surface body. [14] So the net interaction energy for a molecule at a distance D away from the surface will therefore be

where

Then the interaction energy of a large sphere of radius R and a flat surface can be calculated as

where

For convenience, Hamaker constant A is given as and the equation becomes

With a similar method and according to Derjaguin approximation, [16] the van der Waals interaction energy between particles with different shapes can be calculated, such as energy between

Double layer force

A surface in a liquid may be charged by dissociation of surface groups (e.g. silanol groups for glass or silica surfaces [17] ) or by adsorption of charged molecules such as polyelectrolyte from the surrounding solution. This results in the development of a wall surface potential which will attract counterions from the surrounding solution and repel co-ions. In equilibrium, the surface charge is balanced by oppositely charged counterions in solution. The region near the surface of enhanced counterion concentration is called the electrical double layer (EDL). The EDL can be approximated by a sub-division into two regions. Ions in the region closest to the charged wall surface are strongly bound to the surface. This immobile layer is called the Stern or Helmholtz layer. The region adjacent to the Stern layer is called the diffuse layer and contains loosely associated ions that are comparatively mobile. The total electrical double layer due to the formation of the counterion layers results in electrostatic screening of the wall charge and minimizes the Gibbs free energy of EDL formation.

The thickness of the diffuse electric double layer is known as the Debye screening length . At a distance of two Debye screening lengths the electrical potential energy is reduced to 2 percent of the value at the surface wall.

with unit of m1, where

The repulsive free energy per unit area between two planar surfaces is shown as where

The interaction free energy between two spheres of radius R is [18]

Combining the van der Waals interaction energy and the double layer interaction energy, the interaction between two particles or two surfaces in a liquid can be expressed as where W(D)R is the repulsive interaction energy due to electric repulsion, and W(D)A is the attractive interaction energy due to van der Waals interaction.

Effect of shear flows

Alessio Zaccone and collaborators investigated the effects of shear-flow on particle aggregation [19] which can play an important role in applications e.g. microfluidics, chemical reactors, atmospheric and environmental flows. Their work showed a characteristic lag-time in the shear-induced aggregation of the particles, which decreases exponentially with the shear rate. [20]

Application

Since the 1940s, the DLVO theory has been used to explain phenomena found in colloidal science, adsorption and many other fields. Due to the more recent popularity of nanoparticle research, DLVO theory has become even more popular because it can be used to explain behavior of both material nanoparticles such as fullerene particles and microorganisms.

Shortcomings

Additional forces beyond the DLVO construct have been reported to also play a major role in determining colloid stability. [21] [22] DLVO theory is not effective in describing ordering processes such as the evolution of colloidal crystals in dilute dispersions with low salt concentrations. It also cannot explain the relation between the formation of colloidal crystals and salt concentrations. [23]

Related Research Articles

<span class="mw-page-title-main">Colloid</span> Mixture of an insoluble substance microscopically dispersed throughout another substance

A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend the definition to include substances like aerosols and gels. The term colloidal suspension refers unambiguously to the overall mixture. A colloid has a dispersed phase and a continuous phase. The dispersed phase particles have a diameter of approximately 1 nanometre to 1 micrometre.

In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as where the vector r is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as 1/r2, the average number of particles at each distance r is proportional to r2, assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.

<span class="mw-page-title-main">Van der Waals equation</span> Gas equation of state which accounts for non-ideal gas behavior

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change, liquid vapor. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is: where is pressure, is temperature, and is molar volume, is the Avogadro constant, is the volume, and is the number of molecules. In addition is the universal gas constant, is the Boltzmann constant, and and are experimentally determinable, substance-specific constants.

<span class="mw-page-title-main">Electrophoresis</span> Motion of charged particles in electric field

Electrophoresis is the motion of charged dispersed particles or dissolved charged molecules relative to a fluid under the influence of a spatially uniform electric field. As a rule, these are zwitterions.

<span class="mw-page-title-main">Dilatant</span> Material in which viscosity increases with the rate of shear strain

A dilatant material is one in which viscosity increases with the rate of shear strain. Such a shear thickening fluid, also known by the initialism STF, is an example of a non-Newtonian fluid. This behaviour is usually not observed in pure materials, but can occur in suspensions.

In plasmas and electrolytes, the Debye length, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.

Electroacoustic phenomena arise when ultrasound propagates through a fluid containing ions. The associated particle motion generates electric signals because ions have electric charge. This coupling between ultrasound and electric field is called electroacoustic phenomena. The fluid might be a simple Newtonian liquid, or complex heterogeneous dispersion, emulsion or even a porous body. There are several different electroacoustic effects depending on the nature of the fluid.

<span class="mw-page-title-main">Double layer (surface science)</span> Molecular interface between a surface and a fluid

In surface science, a double layer is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The DL refers to two parallel layers of charge surrounding the object. The first layer, the surface charge, consists of ions which are adsorbed onto the object due to chemical interactions. The second layer is composed of ions attracted to the surface charge via the Coulomb force, electrically screening the first layer. This second layer is loosely associated with the object. It is made of free ions that move in the fluid under the influence of electric attraction and thermal motion rather than being firmly anchored. It is thus called the "diffuse layer".

<span class="mw-page-title-main">Surface conductivity</span>

Surface conductivity is an additional conductivity of an electrolyte in the vicinity of the charged interfaces. Surface and volume conductivity of liquids correspond to the electrically driven motion of ions in an electric field. A layer of counter ions of the opposite polarity to the surface charge exists close to the interface. It is formed due to attraction of counter-ions by the surface charges. This layer of higher ionic concentration is a part of the interfacial double layer. The concentration of the ions in this layer is higher as compared to the ionic strength of the liquid bulk. This leads to the higher electric conductivity of this layer.

Sedimentation potential occurs when dispersed particles move under the influence of either gravity or centrifugation or electricity in a medium. This motion disrupts the equilibrium symmetry of the particle's double layer. While the particle moves, the ions in the electric double layer lag behind due to the liquid flow. This causes a slight displacement between the surface charge and the electric charge of the diffuse layer. As a result, the moving particle creates a dipole moment. The sum of all of the dipoles generates an electric field which is called sedimentation potential. It can be measured with an open electrical circuit, which is also called sedimentation current.

In molecular physics, the Hamaker constant is a physical constant that can be defined for a van der Waals (vdW) body–body interaction:

Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.

Bacterial adhesion involves the attachment of bacteria on the surface. This interaction plays an important role in natural system as well as in environmental engineering. The attachment of biomass on the membrane surface will result in membrane fouling, which can significantly reduce the efficiency of the treatment system using membrane filtration process in wastewater treatment plants. The low adhesion of bacteria to soil is essential key for the success of in-situ bioremediation in groundwater treatment. However, the contamination of pathogens in drinking water could be linked to the transportation of microorganisms in groundwater and other water sources. Controlling and preventing the adverse impact of the bacterial deposition on the aquatic environment need a deeply understanding about the mechanisms of this process. DLVO theory has been used extensively to describe the deposition of bacteria in many current researches.

In surface chemistry, disjoining pressure according to an IUPAC definition arises from an attractive interaction between two surfaces. For two flat and parallel surfaces, the value of the disjoining pressure can be calculated as the derivative of the Gibbs energy of interaction per unit area in respect to distance. There is also a related concept of disjoining force, which can be viewed as disjoining pressure times the surface area of the interacting surfaces.

Adsorption of polyelectrolytes on solid substrates is a surface phenomenon where long-chained polymer molecules with charged groups bind to a surface that is charged in the opposite polarity. On the molecular level, the polymers do not actually bond to the surface, but tend to "stick" to the surface via intermolecular forces and the charges created by the dissociation of various side groups of the polymer. Because the polymer molecules are so long, they have a large amount of surface area with which to contact the surface and thus do not desorb as small molecules are likely to do. This means that adsorbed layers of polyelectrolytes form a very durable coating. Due to this important characteristic of polyelectrolyte layers they are used extensively in industry as flocculants, for solubilization, as supersorbers, antistatic agents, as oil recovery aids, as gelling aids in nutrition, additives in concrete, or for blood compatibility enhancement to name a few.

<span class="mw-page-title-main">Derjaguin approximation</span>

The Derjaguin approximation (or sometimes also referred to as the proximity approximation), named after the Russian scientist Boris Derjaguin, expresses the force profile acting between finite size bodies in terms of the force profile between two planar semi-infinite walls. This approximation is widely used to estimate forces between colloidal particles, as forces between two planar bodies are often much easier to calculate. The Derjaguin approximation expresses the force F(h) between two bodies as a function of the surface separation as

<span class="mw-page-title-main">Particle deposition</span>

Particle deposition is the spontaneous attachment of particles to surfaces. The particles in question are normally colloidal particles, while the surfaces involved may be planar, curved, or may represent particles much larger in size than the depositing ones. Deposition processes may be triggered by appropriate hydrodynamic flow conditions and favorable particle-surface interactions. Depositing particles may just form a monolayer which further inhibits additional particle deposition, and thereby one refers to surface blocking. Initially attached particles may also serve as seeds for further particle deposition, which leads to the formation of thicker particle deposits, and this process is termed as surface ripening or fouling. While deposition processes are normally irreversible, initially deposited particles may also detach. The latter process is known as particle release and is often triggered by the addition of appropriate chemicals or a modification in flow conditions.

<span class="mw-page-title-main">Double layer forces</span>

Double layer forces occur between charged objects across liquids, typically water. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The strength of these forces increases with the magnitude of the surface charge density. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The theory due to Derjaguin, Landau, Verwey, and Overbeek (DLVO) combines such double layer forces together with Van der Waals forces in order to estimate the actual interaction potential between colloidal particles.

A depletion force is an effective attractive force that arises between large colloidal particles that are suspended in a dilute solution of depletants, which are smaller solutes that are preferentially excluded from the vicinity of the large particles. One of the earliest reports of depletion forces that lead to particle coagulation is that of Bondy, who observed the separation or "creaming" of rubber latex upon addition of polymer depletant molecules to solution. More generally, depletants can include polymers, micelles, osmolytes, ink, mud, or paint dispersed in a continuous phase.

In condensed matter physics and physical chemistry, the Lifshitz theory of van der Waals forces, sometimes called the macroscopic theory of van der Waals forces, is a method proposed by Evgeny Mikhailovich Lifshitz in 1954 for treating van der Waals forces between bodies which does not assume pairwise additivity of the individual intermolecular forces; that is to say, the theory takes into account the influence of neighboring molecules on the interaction between every pair of molecules located in the two bodies, rather than treating each pair independently.

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