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Structure and Properties of Non-Metallic Materials Lecture 4: Colloids Professor Darran Cairns firstname.lastname@example.org Colloids Shapes of Colloid Particles
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Professor Darran Cairns
Typical shapes of colloid particles: (a) spherical particles of polystyrene latex, (b) fibres of chrysotile asbestos, (c) thin plates of kaolininite.
Monodisperse inorganic colloids. (a) Zinc sulphide (spherulite); (b) cadmium carbonate
If you look at a dilute suspension of colloidal particles (for example plant pollen in water) under a microscope each particle moves in a random jiggling motion. This motion is known as Brownian Motion after Botanist Robert Brown who reported the phenomenon in 1827.
In addition to his Nobel Prize winning work on the photoelectric effect and his celebrated work on special and general relativity Albert Einstein found time to characterize Brownian motion.
The movement of a colloidal particle in suspension has the characteristics of a random walk. In a random walk the mean of the total displacement is always zero, but the mean value of the square of the displacement is proportional to the number of steps and thus is proportional to time.
If the displacement vector after time t is R then
Where ais related to the diffusion constant and can be determined following Einstein and Smoluchowski’s argument
Equation of motion of particle
Drag coefficient for spherical particle of radius a
If the motion of the particle is truly random then <x2>=<y2>=<z2> and therefore <R2>=3<x2>
Multiplying equation of motion by x and rearranging and useing the identity d(x2)/dt=2x(dx/dt)
Rewriting again using the above identity and taking averages
But from the equipartition of energy for any object in thermal equilibrium at temperature T we can write (m(vx)2)/2=kBT/2
Giving us a total mean squared displacement
The motion of the particle is diffusive, with a diffusion coefficient D given by the Einstein formula
For a sphere diffusing in a liquid and therefore
This relationship is often used to determine the size of unknown colloidal particles using dynamic light scattering. Dynamic light scattering can be used to measure the diffucion coefficients and the radius of the particles can be calculated from this.
Form of potential due to double layer can be found by solving
If the potential is small sinh(x) ~ x (the Debye-Huckel approximation) the form of the potential is
Where κ-1 is the Debye screening length
Models for the electric double layer around a charged colloid particle: (a) diffuse double layer model, (b) Stern model
Distribution of ions near a charged surface, according to Debye-Huckel theory. The dotted line illustrates the form of the potential near the surface.
Stabilisation of colloids with grafted polymers. When the particles come close enough for the grafted polymers to overlap, a local increase in polymer concentration leads to a repulsive force of osmotic origin.
The depletion interaction. Polymer coils are excluded from a depletion zone near the surface of the colloidal particles; when the depletion zones of two particles overlap there is a net attractive force between the particles arising from unbalanced osmotic pressures.
A water-borne latex suspension imaged by environmental scanning electron microscopy, showing the formation of ordered regions.
A single close-packed layer, illustrating that there are two sites on which a second close-packed layer can be placed: b and c.
Phase diagram for charged spheres in a polyelectrolyte solution as a function of the volume fraction of spheres Φand the concentration of salt as calculated for spheres of radius 0.1 mm with surface charge 5000e.
Calculated phase diagram for a colloid of hard spheres with non-adsorbing polymer added to the solution. The ratio of the sizes of the colloidal spheres to the radii of the polymer molecules is 0.57
Aggregation with and without rearrangement. In (a) the attraction is weak enough to allow the particles to rearrange following aggregation – this produces relatively compact aggregates. In (b) the attractive energy is so strong that once particles make contact, they remain stuck in this position. Particles arriving later tend to stick on the outside of the cluster, as access to its interior is blocked, resulting in much more open aggregates with a fractal structure.
Four stages in the construction of a simple deterministic fractal model for particle aggregates. In (a) five particles are formed in the shape of a cross. In (b) five of these crosses are joined together to form a larger cross. The process is extended in (c) and (d) in two dimensions. Each time the mass is increased by a factor of 5, the lateral extension is increased by a factor of 3. The fractal dimension of this pattern is D=log 5/ log 3=1.465
Relative viscosity as a function of volume fraction for model hard sphere lattices, in the limit of low shear rates (filled symbols) and high shear rates (open symbols). The solid lines are fitting functions and the dashed line is the Einstein prediction for the dilute limit. Squares are 76 nm silica spheres in cyclohexane, triangles are polystyrene spheres of radii between 54 and 90 nm in water.
Relative viscosity as a function of shear rate for model hard-sphere lattices. The shear rate γ is plotted as the dimensionless combination, the Peclet number Pe=6πη0a3γ/kBT. The solid line is for polystyrene lattices of radii between 54 and 90 nm in water; the circles are 38 nm polystyrene lattices in benzyl alcohol, and the diamonds 55nmpolystyrene spheres in a meta-cresol.
Schematic of a thermosetting gel. The system consists of a mixture of short chains with reactive groups at each end, and cross-linker molecules, each with four functional groups capable of reacting with the ends of the chains. As the reaction proceeds the chains are linked together by the cross-linker to form an infinite network.
Schematic of a vulcanization reaction. The system consists of a mixture of long chains. Initially, the chains are entangled but not covalently linked. The reaction proceeds by chemically linking adjacent chains, leading to the formation of an infinite network.
Thermoreversible gelation by the formation of microcrystals. At low temperatures (bottom) adjacent chains form small crystalline regions which act as cross-links. Above the melting temperature, the crosslinks disappear (top).
A triblock copolymer (SBS) can form a thermoplastic elastomer. The end blocks microphase separate to form small, spherical domains. When these domains are glassy they act as cross-links for the rubbery centre blocks; the rubber can be returned to the melt state by heating above the glass transition of the end blocks.
The percolation model. We start with array of points, to which bonds are added at random (left). As more bonds are added clusters of points are formed (middle), which ultimately join to form a cluster which spans the entire system (right).Percolation model
Three generations of a Cayley tree
Definitions of branches and neighbours on a Cayley tree
Number of bonds N out to nth generation of a Cayley tree with a coordination number z and a probability of bond formation f is given by
Below some critical value of f finite clusters (sols) are formed. Above fc an infinite cluster (gel) forms.
The gel fraction in the classical model of gelation for a coordination number z=3.
The effect of dangling ends on the shear modulus of a gel near the gel point. The bonds shown with dashed lines are part of the infinite network, but do not contribute to the elastic modulus.
Card-house structure of kaolinite due to +ve charge on edge and –ve charge on surface. Occurs at low pH.
(a) A sketch of the ideal kaolinite layer structure (Al(OH)2)2O(SiO2)2. One hydoxyl ion is situated within the hexagonal ring of apical, tetrahedral oxygens and there are three others in the uppermost plane of the octahedral sheet. The two sheets combined make up the kaolinite layer. (b) simplified schematic diagram of a kaolinite crystal. Note that the upper and lower cleavage faces of the perfect crystal would be different. A typical crystal would contain a 100 or so such layers. (c) A typical kaolinite crystal of aspect ratio (a/c) about 10. Note the negative charges on the cleavage faces or basal planes and positive charges around the edges (these are eliminated at pHs above about 7).
Simplified model of ink impression and absorption. (1) The ink is hydraulically pressed into the pores. As the roller pulls away, the ink film splits by cavitation and rupture and the surface is thoroughly wet. (2) the paper relaxes and draws ink into the larger pores. (3) The capillary forces take over and draw liquid into the smaller pores, leaving most of the pigment behind. (4) The ink continues to spread slowly over all the surfaces
Scanning electron microscope studies of bread dough morphology. (a) Dried dough showing the starch granules embedded in the gluten matrix. (b) Environmental SEM (ESEM) of fresh dough showing hydrated starch granules and the thick gluten matrix that holds the dough together
Milk is a classic food colloid. It is basically an oil-in-water emulsion, stabilized by protein particles. Fresh unskimmed cows milk contains 86% water, 5% lactose, 4% fat, 4% proteins and 1% salt. The milky appearance is due to large colloidal particles called casein micelles. Casein micelles are polydisperse associates of proteins bound together with colloidal calcium phosphate. The fat globules are dispersed in water and are stabilized by a membrane of proteins and phospholipids at the oil/water interface.
Model for the structure of a casein micelle. Casein protein subunits are linked by colloidal calcium phosphate to produce a raspberry-like structure
Typical structure of ice-cream revealed in an electron micrograph. (a) Ice crystals, average size ~50 μm, (b) air cells, average size ~100-200 μm, (c) unfrozen material. [From W.S. Arbuckle, Ice Cream, 2nd Edition, Avi Publishing Company (1972)]