MES 712

Interfacial Phenomena

Colloidal Stability Supplement

As was discussed during class, nearly all surfaces in water are charged. Therefore, as many aspects of industrial science utilize particles in water, the interaction of these particles is of great importance. Colloidal stability is the term used to describe the ability/inability of small particles, one linear dimension between 1 nm and 1000 nm, to aggregate in solution and then to precipitate out of solution. To determine whether particles aggregate in solution, the interaction of the particle double layers must be considered, along with various other non-electrostatic forces. The first derivation of this situation was done separately around the time of World War 2, by Derjaguin and Landau in the Soviet Union and Verwey and Overbeek in the Netherlands. Because these were done nearly simultaneously and separately, the theory developed is usually called DLVO theory. In addition to DLVO forces, non-DLVO forces are also now recognized but will not be considered in this class. The forces can be divided into attractive and repulsive forces. The sum of the attractive and repulsive forces is the total force between two particles. To begin, we will examine the repulsive forces, then the attractive forces.

Repulsive forces Repulsive forces develop between particles primarily due to the overlap of the diffuse double layers around the particles. First, we will consider the case of two parallel plates of identical material some distance, H, apart, as shown in Figure 1. When H is a very large distance as compared to the Debye length (1/), the osmotic pressure between the plates is given by ; where n is the total bulk concentration of ions. As the plates are brought together from H=, to some distance H=2l apart, the osmotic pressure changes to , where n+ is the concentration of positive ions and n- is the concentration of negative ions, both given by

the Boltzmann equation –> . Substituting in the individual nion

terms into the P2 equation, , where n is the

bulk ion concentration,zi is the absolute value of the valence, e is the electronic charge constant, and l is the overlapped double layer potential at the midpoint of separation, l. The potential difference between H=2l and H= is then P2-P1 and is called the effective pressure, Peff. P2 can be simplified to P2=2nkboltzT coshU; where coshU is the hyperbolic

cosine of . Thus, .

However, Peff is also the negative of the differential of the repulsive for between surfaces,

Vr, so that and . The 2 comes from there being 2 surfaces

going from infinite separation to 2l separation.

Therefore, . To integrate this, the value of U must be remembered to be a function of x based on the l term in U. The value of l/( at any value of separation x=H for H between ∞ and 2l can be found from Gouy-Chapman

H

Double layer potential

No overlapP1=2nkboltzT

H = 2l

Overlapped double layer

ll

H

Double layer potential

No overlapP1=2nkboltzT

H = 2l

Overlapped double layer

ll

Figure 1. Double layer repulsion between parallel plates.

theory. Another way to write the GC equation is

where

tanh is the hyperbolic tangent and 0(mV) is the surface potential in milliVolts. With

this equation for (x), the integration Vr gives , for two

identical surfaces at any separation H0. If the surfaces are not identical, , where 1, and 2, are the s of the two different interacting surfaces. This equation gives Vr in units of interaction energy per unit surface area, so that the surface area of the particles must be included to get the total repulsive interaction energy.

That was two plates, what about 2 spheres, or a sphere and a plate. For two spheres, of arbitrary radii a1 and a2, the starting point is the same as two plates except that the interaction area is now defined by the shape of the spheres, rather than the total face area. In general, the Derjaguin approximation is used (see Figure 2), which results in the

interaction area being . Vr for the sphere-sphere case then becomes

. Two important cases with respect to the radii

are 1. a1=a2=a for which , and 2. when a1>>a2 for

which a1a2/(a1+a2) ≈ a2 and . Case 2 is the case in

which a sphere-interacts with a small plate.

Sphere radius a1 Sphere radius a2

Sum interactions between surface points is

2a1a2/(a1+a2)

Sphere radius a1 Sphere radius a2

Sum interactions between surface points is

2a1a2/(a1+a2)Figure 2. Derjaguin approximation.

The equations given above are valid only under a small set of conditions. For instance, the plate-plate Vr is only strictly correct for small surface potentials and symmetrical electrolytes. The sphere-sphere Vr is correct when the separation is much less than either particle radius (H0<<ai), ai > 5 and symmetrical electrolytes. Many other equations for

Vr have been derived for various other conditions and these are shown in the tables at the end of this supplement. These derivations and much other important surface chemistry material can be found in the references given at the end of this supplement. One final derivation of Vr I will include is that by Hogg, Healey and Fuerstenau (HHF). This inclusion is primarily because Fuerstenau is a graduate of SDSM&T. HHF assumed small surface potentials for two spheres and ai >> 1, and found that

,

where Φ1 and Φ2 and i is 1 or 2 respectively. This is the end of the repulsive

force section, now onto attractive forces.

Attractive forces Attractive forces primarily arise from what are termed London-van der Waals or sometimes just van der Waals forces. London-van der Waals (vdW) forces arise from the interaction of like materials. Three main types of vdW forces are generally recognized: 1. Debye – which is the name for interactions between permanent and induced dipoles; 2. Keesom – between 2 permanent dipoles; and 3. London between 2 induced dipoles. Va will be negative (and Vr positive) by convention. Two approaches have been developed to evaluate these forces. Hamaker developed a microscopic approach based upon pairwise summation of all intermolecular interactions, while Lifshitz has developed a macroscopic approach using the electromagnetic properties of the media. Usually, the Hamaker formulation is used due to convenience and retardation effects are included when necessary. Figure 3 shows the Va equations for a variety of geometries. One important factor in the equations in Figure 3 is the constant A, called the Hamaker constant. In Hamaker’s work Va is calculated by summing the interactions of all molecules in one particle, with all the molecules in the other particle. Hamaker used a double integration procedure rather than the summation to derive the first expression in Figure 3, for the case when H0 << ai. This equation is only good to about

, but often Va is zero at this distance so the equation works well quite frequently.

Another important point about the Hamaker constant A is that subscripts denoting the materials involved and the surrounding medium are necessary to ensure correct calculation of A. Thus, A11 represents the Hamaker constant between material 1 and itself in a vacuum. A12 is the Hamaker constant between materials 1 and 2 in a vacuum, and A12=(A11A22)0.5. A132 is the Hamaker constant between materials 1 and 2 in medium 3. A132=A12+A33-A13-A23≈ (A12

0.5 -A33

0.5)(A220.5

–A330.5), so that A131, the Hamaker constant

between material 1 and itself in medium 3, = (A11 0.5

-A330.5)2. Hamaker constants of the

types A11 and A12 can be calculated if the characteristic frequencies and the dielectric constants for the media are known. Typical Hamaker constant values (A131) lie between about 0.5 and 15 x10-20 Joules when medium 3 is water. Mineral particles are towards the high end, while biological materials are at the low end of this range.

Non-DLVO forces Five main types of non-DLV forces have been identified. These are Born repulsion due to overlap of electron clouds which is very large at very short (< 1 nm) separations, hydration effects as surface ions hydrate which acts over at most about 1.5 nm, hydrophobic interactions between non-polar molecules, which are often long range > 60 nm, but relatively weak, steric interactions between material with adsorbed polymers in which the polymer chain extend into solution and can not be penetrated by other chains. Steric interactions generally stabilize the colloid. Polymer bridging, in which polymers adsorb to more than one particle causing aggregation, is the final non-DLVO force. Both steric interactions and polymer bridging take place at about the radius of gyration of the polymer. These forces can be taken into account as part of the total interaction energy, but only Born repulsion has a well-defined equation governing it.

Two spheres of radii a1, a2separated by distance H0

One sphere of radius a1, and a flat plateseparated by distance H0

Va = -Aa1a2/(6H0(a1+a2))

Va = -Aa1/(6H0)

Two infinite flat platesseparated by distance H0

Va = -A/()Per unit area

Two spheres of radii a1, a2separated by distance H0

One sphere of radius a1, and a flat plateseparated by distance H0

Va = -Aa1a2/(6H0(a1+a2))

Va = -Aa1/(6H0)

Two infinite flat platesseparated by distance H0

Va = -A/()

Two spheres of radii a1, a2separated by distance H0

One sphere of radius a1, and a flat plateseparated by distance H0

Va = -Aa1a2/(6H0(a1+a2))

Va = -Aa1/(6H0)

Two infinite flat platesseparated by distance H0

Va = -A/()Per unit area

Two cylinders of radii a1, a2And both length L separated by distance H0

Va = -[AL/(12(2)0.5H0)](a1a2/(a1+a2))0.5

Two crossed cylinders of radii a1, a2separated by distance H0

Va = -A(a1a2)0.5/6H0

Two non-infinite flat plates of thickness separated by distance 2d=H0

Va = -[A/(48)][1/d2 + 1/(d+)2 – 2/(d+/2)2

Per unit area

Two cylinders of radii a1, a2And both length L separated by distance H0

Va = -[AL/(12(2)0.5H0)](a1a2/(a1+a2))0.5

Two crossed cylinders of radii a1, a2separated by distance H0

Va = -A(a1a2)0.5/6H0

Two non-infinite flat plates of thickness separated by distance 2d=H0

Va = -[A/(48)][1/d2 + 1/(d+)2 – 2/(d+/2)2

Per unit area

Figure 3. Attractive van der Waals forces for various geometries.

DLVO Theory In DLVO theory the main objective is to calculate the total interaction energy of the system. This is done by adding Vr and Va (Vt=Vr+Va) addition can be done because the repulsive (Vr) and attractive (Va) energies were defined with opposite signs. Figure 4 shows the three types of interaction energy curves that can be obtained. Figure 4a has the case in which the interaction energy starts out initially attractive. A secondary minimum at about 14 nm occurs when the repulsive energy begins to increase in magnitude more strongly than the attractive energy magnitude. This causes a cross-over to repulsion as the distance between surfaces decreases. Eventually, the energy barrier is reached in which the maximum repulsive total interaction energy occurs. The interaction energy then drops as the attractive forces begin to steeply increase, but at no time does the total energy become attractive. This type of interaction is the most stable type. Particles will stay apart by the distance at which the secondary minimum occurs. Providing energy to the system, by stirring for instance, will never result in aggregation, because the total energy never becomes attractive. In Figure 4b, a meta-stable total interaction energy curve is shown. This curve is similar to that in 4a, except that the total energy between contact and the energy barrier does become attractive so that a primary minimum occurs. In this case, particles will behave as in 4a except, that if more energy than the energy barrier is provided, the interparticle distance will shrink to the primary minimum distance and the dispersed particles will become aggregated. If the energy barrier is less than 5kboltzT (~2x10-20 J at 298 K), Brownian motion of the particles will cause aggregation of the particles (this is often called fast flocculation), and is equivalent to Figure 4c. In 4c, the total interaction energy does not become repulsive until distances less than the primary minimum, thus the particles will go to the primary minimum distance upon interaction.

-4E-18

-3E-18

-2E-18

-1E-18

0

1E-18

2E-18

3E-18

4E-18

0 2 4 6 8 10 12 14 16 18 20

Distance (nm)

Inte

ract

ion

Ener

gy (J

)

VaVrVt

Secondary Minimum

Energy Barrier

(a)

-4E-18

-3E-18

-2E-18

-1E-18

0

1E-18

2E-18

0 1 2 3 4 5 6 7 8 9 10

Distance (nm)

Inte

ract

ion

Ener

gy (J

)

VaVrVt

Secondary Minimum

Energy Barrier

Primary Minimum

(b)

-4E-18

-3.5E-18

-3E-18

-2.5E-18

-2E-18

-1.5E-18

-1E-18

-5E-19

0

5E-19

1E-18

0 1 2 3 4 5 6 7 8 9 10

Distance (nm)

Inte

ract

ion

Ener

gy (J

)

VaVrVtPrimary Minimum

(c)Figure 4. Three stability curve types.

As the total interaction energy is the sum of the repulsive and attractive energies, the stability of the colloidal solution can be altered by changing the conditions. In general, the attractive energy is difficult to change other than by changing the temperature. As the repulsive energy is due to the double layer interactions, changing the double layer can change the Vt value. The primary controllers of Vr are the surface potential of the colloidal particle and the salt concentration in the solution. The magnitude of Vr decreases as the surface potential decreases or the solution ionic strength increases. Flocculation can be initiated by sufficiently decreasing Vr. Usually, this is done by increasing the ionic strength. This is called salting out. Also, the valence of the salt added is important. The Schulze-Hardy rule indicates that the salting out ability of an ion increases as z6. This comes from the critical coagulation concentration (CCC), the concentration at which no barrier exists to coagulation/flocculation/aggregation. K

. At large , the hyperbolic tangent term goes to one, so that

CCCz-6. For lower magniutude , CCC z-2 is a closer approximation to reality.