# Properties of colloidal solutions

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## Chapter: Pharmaceutical Drugs and Dosage: Colloidal dispersions

Properties of colloidal solutions : Kinetic properties, Electrical properties, Colligative properties, Optical properties

Properties of colloidal solutions

## Kinetic properties

Properties of colloidal systems that arise from the motion of particles with respect to the dispersion medium are known as kinetic properties. These include Brownian motion, diffusion, sedimentation, and osmosis.

### 1. Brownian movement

Brownian motion results from asymmetry in the force of collisions of mol-ecules of the dispersion medium on the dispersed phase. This results in ran-dom dispersed-phase particle motion, called Brownian movement. Since the speed of motion of the molecules of the dispersion medium increases with temperature, Brownian motion is a function of temperature. Increase in temperature generally increases Brownian motion of dispersed-phase par-ticles. The velocity of the particles also increases with decreasing particle size, which can be attributed to lower inertia of smaller particles. Similarly, increasing the viscosity of the medium decreases Brownian movement due to greater resistance to movement of the dispersed-phase particles.

### 2. Diffusion

Colloidal particles are subject to random collisions with other dispersed-phase particles, usually with a greater force, in addition to the molecules of the dispersion medium. This leads to the overall movement, called diffu-sion, of the dispersed-phase particles from a region of high concentration to a region of low concentration. The rate of diffusion of the dispersed-phase particles is given by Fick’s first-law equation:

where:

dM is the mass of substance diffusing in time dt across a cross-sectional area S

dC over the diffusion distance dx

D is the diffusion coefficient

The diffusion coefficient of the dispersed phase, D, is related to the fric-tional coefficient, f, of the particles, which quantitates the resistance to the movement of particles in the dispersion medium. The diffusion coefficient, D, and the frictional coefficient, f, are inversely related to each other and are linearly dependent on temperature, T, as explained by the Einstein’s law of diffusion:

Df = kT                  (9.2)

where:

k is the Boltzmann constant

T is the absolute temperature

f is the frictional coefficient

The Boltzmann constant is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas and is obtained by dividing the gas constant, R, by the Avogadro’s number, N, that is, the number of molecules per mole of a substance. The Boltzmann constant has the dimensions of energy over temperature, same as entropy, and is quantitatively 1.38064852(79) × 10−23 J/K. Thus,

k = R/N

The frictional coefficient is dependent on the size of particles and the viscosity of the dispersion medium by the equation:

f = 6πηr                         (9.3)

where:

η is the viscosity of the medium

r the radius of the particle

Thus, diffusion coefficient depends on the viscosity and temperature of the dispersion medium and the size of the dispersed phase by the equation:

D = kT/6πr                      (9.4)

This equation indicates that the diffusion coefficient is inversely propor-tional to the viscosity of the medium and the radius of the diffusing par-ticles, while it is directly proportional to the temperature.

Expressing the Boltzmann constant in terms of the gas constant and the

Avogadro’s number yields the Stokes–Einstein equation:

D = RT/6πηrN                      (9.4)

However, this equation assumes spherical particles and does not take into account particle shape effects, which can be important in the case of com-plex molecules, such as proteins, and linear polymers that can entangle during movement. In addition, greater the asymmetry or deviation from sphericity, greater the resistance to flow.

### 3. Sedimentation

When stored undisturbed, the dispersed phase tends to separate out from the dispersion medium and concentrate in one region of the dispersion. When the dispersed-phase density is higher than that of the dispersion medium, the dispersed phase accumulates at the bottom, or sediments, and this process is called sedimentation. This is the case for most aqueous suspensions. When the dispersed-phase density is lower than that of the dispersion medium, such as in the case of aqueous emulsions, the dispersed phase accumulates toward the top of the container, or creams, and this pro-cess is called creaming. Both these phenomena are governed by the same physics, and for simplicity; this section will focus on sedimentation.

The rate of settling of particles, that is, the velocity (v) of sedimentation, is given by the Stokes’ law equation:

where:

ρ is the density of the particles

ρ0 is the density of the dispersion medium

η0 is the viscosity of the dispersion medium

g is the acceleration due to gravity

In a centrifugation experiment, g is replaced by angular acceleration ω2x, where ω is the angular velocity and x is the distance of the particle from the center.

Stokes’ law was derived for dilute dispersions of spherical particles. It does not take into consideration deviation of particle shape from sphericity and interparticulate interactions, especially at high dispersed-phase con-centration. Thus, Stokes’ law may not be quantitatively exactly applicable to the concentrated dispersions. However, the qualitative, or rank-order, effects of the factors indicated by the Stokes equation still hold true. For example, an increase in the mean particle size or in the difference between the densities of the solid and liquid phases increases the rate of sedimenta-tion. Using the Stokes equation, creaming of an emulsion or sedimentation of a given suspension can be reduced by forming smaller particles, increas-ing the viscosity of continuous phase and/or decreasing the density differ-ence between two phases.

## Electrical properties

Electrical properties of the dispersed phase refer to the electrostatic charge on the surface of the particles and its impact on the interaction of the dis-persed-phase particles with each other and with the dispersion medium.

### 1. Surface charge

Surface charge on the dispersed phase plays an important role in the following:

·           Physical stability of colloids. Greater the electrostatic repulsion among the dispersed-phase particles, greater the physical separation and uni-form appearance of the colloidal dispersion. However, when settled, the cake formed may not be easily redispersible. Therefore, a balance of electrostatic charge on the particles is sought that promotes forma-tion of uniform dispersion and also allows easy redispersibility on settling.

·           Filtration efficiency of submicron particles, which can be diminished considerably by particle aggregation or particle affinity for the filtra-tion membrane.

·           Determining the conformation of macromolecules such as polymers, polyelectrolytes, and proteins by influencing macromolecule–solvent interactions and intramolecular interactions within the polymers and macromolecules.

Most substances acquire a surface electric charge when brought in con-tact with an aqueous medium by ionization, ion adsorption, and/or ion dissolution.

### 1.1 Ionization

Surface charge arising from ionization on the particles is the function of the pH of the environment and the pKa of the particle’s surface functional groups. For example, proteins and peptides acquire charge through the ionization of surface carboxyl and amino groups to COO and NH3+ ions, respectively. The state and extent of ionization of these groups and the net molecular charge depend on the pH of the medium and the pKa of the functional groups, as determined by the Henderson–Hasselbalch equation.

Macromolecules such as proteins have many ionizable groups. Thus, at pH below its isoelectric point (PI), the protein molecule bears an overall positively charge, and at pH above its PI, the protein molecule bears an overall negative charge—even though there may be domains within the protein structure that would be uncharged or bear the opposite charge. At the PI of a protein, the total number of positive charges equals the total number of negative charges in the protein, resulting in the net charge being zero.

As an illustration for the amino acid alanine, which has one amino and one carboxylate group, this phenomenon may be represented as:

Ionized molecule has stronger electrostatic, dipole, and hydrogen bond interactions than unionized functional group or molecule. Thus, ioniza-tion increases dispersed-phase–aqueous-solvent interactions and generally stabilizes the dispersion. Addition of salt to solutions of ionized proteins can reduce protein–solvent interactions and protein solubility. Thus, addi-tion of salt to precipitate the protein of interest is a common procedure in experimental sciences. In solutions of multiple proteins, increasing salt concentration can sequentially precipitate proteins in the increasing order of their aqueous solubility.

However, at the isolectric point, proteins with multiple functional groups can self-associate through interactions of oppositely charged functional groups. Thus, often, a protein is least soluble at its isoelectric point due to the attractive interactions between different protein molecules. At the isolectric point, water-soluble salts such as ammonium sulfate, which par-tially neutralize surface charges and reduce interparticle attractions, may increase protein solubility.

Surfaces that are already charged usually show a tendency to adsorb coun-terions from solution. For example, a positively charged surface selectively adsorbs chloride (Cl) ions from a salt (NaCl) solution. This results in an excess of the countercharge (i.e., negative charge in the example of Cl ions) on the surface of the dispersed phase, compared with the bulk solution. A second layer of charged coions concentrate above the countercharged surface. In the aforementioned example, the free Na+ ions in solution form a second layer over the Cl ions on the surface. These two layers of electri-cal charge on a charged surface are together called electrical double layer. Figure 9.1 shows the electrical double layer on a surface, with the first layer of negatively charged counterions and the second layer of positively charged coions.

Figure 9.1 Electrical double layer and zeta potential of colloidal particles. (a) schematic of electrical double layer at the separation between two phases, showing distri-bution of ions and (b) changes in potential with distance from particle surface.

A lower net surface charge results on charged surfaces that form electrical double layer due to unequal adsorption of oppositely charged ions, resulting in only partial neutralization of particle surface charge.

This phenomenon can also enable nonpolar surfaces to develop charge by adsorption of charged solutes from solution. For example, surfactants strongly adsorb by the hydrophobic effect and determine the surface charge when adsorbed.

### 1.3 Ion dissolution

Ionic substances can acquire a surface charge by unequal dissociation of the oppositely charged ions. For example, in a dispersion of silver iodide particles with excess [I] in solution, the dispersed particles carry a negative charge. This is due to the suppression of dissociation of the I ions on the surface of particles by the common-ion effect.

AgI  ↔ Ag + + I

Similarly, the net charge on AgI particles is positive if excess Ag+ ions are present in the solution. In this case, therefore, the silver and iodide ions are referred to as potential-determining ions, since their concentrations deter-mine the electric potential at the particle surface.

### 2. Electrical double layer

As explained in Section 9.4.2.1.2, the surface charges of dispersed-phase particles influence the distribution of the nearby ions in the polar dis-persion medium. Ions with opposite charge (known as counterions) are attracted toward the surface, and ions with like charges (known as coions) form a second layer on the concentrated layer of counterions. This leads to the formation of an electric double layer made up of a neutralizing excess of counterions close to the charged surface and coions. The electri-cal double-layer theory explains the distribution of ions with the chang-ing magnitude of the electric potentials, which occur in the vicinity of the charged surface.

At a particular distance from the surface, the concentration of anions and cations is equal; that is, conditions of electrical neutrality prevail in bulk solution. The system as a whole is electrically neutral, even though there are regions of unequal distribution of anions and cations. This is illustrated in Figure 9.1. The first layer extends from aa’ to bb’ and is tightly bound to the surface. This rigid layer attached to the particle surface is called the stern layer. The second layer extends from bb’ to cc’ and is more diffuse. Hypothetical planes are defined as boundaries around the surface of par-ticles that define true hydrated particle size of electrically double-layered dispersed-phase particles (called stern plane) and the plane that defines the movement of these particles in solution (called shear plane). The stern plane is at the center of the first layer of hydrated ions from the surface. The shear plane is the boundary of the first layer of hydrated ions from the surface.

The thickness of the electrical double layer is defined by the Debye–Huckel radius or length parameter, which characterizes the distance from surface at which the particle charge is completely screened by other charges in solu-tion. This parameter is dependent on the electrolyte concentration of the aqueous media. The thickness of the electrical double layer shrinks with increase in electrolyte concentration in solution.

### 2.1 Nerst and zeta potentials

Electrothermodynamic or Nerst potential (E) is defined as the difference in potential between the actual surface and the electroneutral region of the solution. This is the potential at the particle surface (aa’ in Figure 9.1). However, when the particles are set in motion by electrical forces, such as electrophoresis, a small layer of solvent with oppositely charged ions moves concurrently with the particles. The boundary of this layer is termed the shear plane (bb’), since this distinguishes the moving from the stationary part of the solvent. The electrical potential at the shear plane bb’ is known as the electrokinetic or zeta potential, ζ. The ζ potential is defined as the difference in potential between the surface of the tightly bound layer (shear plane) and the electroneutral region of the solution.

The ζ potential, rather than the Nerst potential, truly governs the degree of repulsion between the adjacent, similarly charged, dispersed particles. Therefore, measurement and optimization of ζ potential is needed for the stability of dispersed systems. The ζ potential can be impacted by all three mechanisms discussed in the previous section, viz., ionization, ion dissolu-tion, and ion adsorption. In addition, surfactant molecules that adsorb by the hydrophobic effect on the surface of the dispersed phase can affect the ζ potential.

### 2.2 DLVO theory

DLVO theory is named in honor of Russian physicists B. Derjaguin and L. Landau and Dutch pioneers in colloid chemistry, E. Verwey and J. Overbreek. These scientists independently formulated the theories of interaction forces between colloidal particles in the 1940s to help predict colloidal stability of charged particles in dispersion. This theory explains the stability of dispersed colloids in aqueous suspensions on the basis of the balance of two opposite forces between the dispersed-phase particles: electrostatic force of repulsion and van der Waals force of attraction.

DLVO theory of colloidal stability states that the only interactions involved in determining the stability of colloidal dispersed particles are electric repulsion (VR) and van der Waals attraction (VA) and that these interactions are additive. Therefore, the total potential energy of interaction (VT) is given by:

VT =VA +VR                           (9.7)

Thus, a stable dispersion is obtained when the repulsive forces dominate, while a physically unstable dispersion is obtained when attractive forces dominate.

### 3. Electrophoresis

Electrophoresis is the movement of charged particles (with the attached ions and the solvent in the tightly attached first electrical layer) relative to the stationary liquid dispersion medium, under the influence of an applied electric field. Migration of particles in an electric field occurs due to the motion of the particle and its counterion cloud away from the electrode of the same charge and toward the electrode of the opposite charge.

Electrophoretic mobility (μ) of a molecule is a function of its net charge (Q) and size (radius, r). Thus,

µ = Q/r

Experimentally, the electrophoretic mobility is determined as the particle velocity (v) per unit electrical field (E). Thus,

µ = v/E                         (9.8)

Hence, electrophoresis experiments can be used to determine the net sur-face charge on the particles.

## Colligative properties

Colligative properties are the properties that depend only on the number of nonvolatile molecules in solution, without regard to their size or molecular weight, or the solute–solute or solute–solvent interactions.

### 1. Lowering of vapor pressure

Addition of a nonvolatile solute to a solvent lowers its vapor pressure, since solute occupies some of the surface of the solvent. This, therefore, reduces the rate of evaporation of the solvent.

The extent of decrease in the vapor pressure with the addition of solute to a solvent is given by Raoult’s law, which states that the vapor pressure of an ideal solution is dependent on the vapor pressure of each individual component, weighted by the mole fraction of that component in solution. Thus,

PA = XAPA0                  (9.9)

where:

PA is the vapor pressure of the colloidal solution

XA is the mole fraction of solute in the solvent

PA0 is the vapor pressure of the pure solvent

Thus, the reduction of vapor pressure of a solvent is directly proportional to the concentration of solute in that solvent.

### 2. Elevation of boiling point

Addition of a nonvolatile solute leads to the elevation of boiling point due to the nonvolatile solute displacing the corresponding number of solvent mol-ecules from the surface and, consequently, reducing the number of solvent molecules that are able to escape into the vapor phase from the solution. The extent of increase in the boiling point by the addition of a nonvolatile solute is given by:

ΔTb = Kb m                    (9.10)

where:

Kb is the molal boiling point elevation constant

ΔTb is the elevation of boiling point

m is the molal amount of solute in solution

Thus, the extent of elevation of boiling point is directly proportional to the concentration of nonvolatile solute in solution. The extent of this effect is different for each solvent. The value of Kb for different solvents is available in literature.

### 3. Depression of freezing point

Addition of a nonvolatile solute results in reduction in solvent–solvent interactions; This leads to depression of freezing point of the solvent.

The extent of decrease in the freezing point is given by:

ΔTf = Kfm                        (9.11)

where:

Kf is the molal freezing point depression constant

ΔTf is the depression of freezing point

m is the molal amount of solute in solution

Similar to the case with the elevation of boiling point, the extent of depression of freezing point is directly proportional to the concentration of nonvolatile solute in solution. The extent of this effect is different for each solvent. The value of Kf for different solvents is available in literature.

### 4. Osmotic pressure

Osmosis involves flow of solvent molecules through a membrane toward its concentration gradient, which is opposite of the concentration gradient of the solute in solution. The use of a membrane with a well-defined pore size leads to its semipermeable nature; that is, only molecules below a certain size or molecular weight are able to pass through the membrane. Thus, the use of a membrane through which the colloidal solutes are not able to diffuse, with solutes at different solution concentration on either side of the membrane, promotes solvent flow from a solution of low solute concen-tration to a solution of high solute concentration. This process of solvent flow is called osmosis, and the relative difference in the pressure of solvent generated by the concentration gradient on either side of the membrane is called osmotic pressure, π. The osmotic pressure is given by:

where:

R is the gas constant

T is temperature in Kelvin

M is the difference in the molar concentration of solute in solution, which is defined as the number of moles of solute, n, per unit volume of solution, v

Thus, osmotic pressure of a solution is directly proportional to its solute concentration and temperature, through their impact on the motion of sol-vent molecules. The higher the solute concentration and the temperature, the higher the osmotic pressure.

## Optical properties

Colloidal solutions scatter light, since their particle diameter is within the range of wavelength of visible light. This phenomenon is known as Tyndall effect. Thus, light passing through a colloidal solution with particle diameter of ~200 nm leads to scattering, resulting in turbid or milky appearance. This property is utilized in quantifying the number of suspended par-ticulates in a liquid or gas colloidal solution by using a turbidimeter or nephelometer by calibrating the amount of turbidity at different concentra-tions of a colloidal solution.