Soft2007 Colloid

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Introduction to colloidal

dispersions

R.A.L. Jones, 4



Colloid

A colloid or colloidal dispersion is a system of two or

more components; a type of mixture intermediate between

homogeneous solution and heterogeneous mixtures with

properties also intermediate between a solution and a

mixture. Examples: butter, milk, cream, aerosols (fog, smog,

smoke), asphalt, inks, paints, glues and sea foam, are

colloids.

This field of study was introduced in 1861 by Scottish

scientist Thomas Graham. The size of dispersed phase particles in a colloid range

from 0.001 to 1 micrometers.



Milk

Cow's milk contains, on average, 3.4% protein, 3.6%

fat, and 4.6% lactose

Milk supplies 66 kcal of energy per 100 grams.



Butter

Commercial butter is about 80% butterfat and 15% water;

traditionally-made butter may have as little as 65% fat and

30% water.

Butterfat consists of many moderate-sized, saturated

hydrocarbon chain fatty acids. It is a triglyceride, an esterderived from glycerol and three fatty acid groups.

http://www.scientificpsychic.com/fitness/fattyacids.html





Properties of colloids

High area of interface

Dispersion, aggregation, sedimentation

Stabilation: charge, cover by polymer etc.

Shear thickening or thinning

+

+

+

+-

-

--

-

-

-

- -

--

-

-

-



A single colloidal particle in a liquid

Fluid mechanics point of view

Gravity

Grag is the force that resists the movement of a solid

object through a fluid (a liquid or gas).

Grag

Gravity



Randomly moving sphere in a liquid

Expression for the frictional

force exerted on small

spherical objects in a viscous

fluid. The force (Stokes law

1851) can be written asF = 6 a v ,

where v is the velocity, a the

radius of the particle,

viscosity of a liquid.

Reynolds numberRe = va/

where is the density.

The gravitation force for

one sphere is

F g = 4/3 a 3 g,

where is the density

difference between thesphere and the liquid.

In balance F = F g . From this

the terminal velocity isv t = 2a 2 g/(9 ).



Suspension of colloidal spheres

Particles are in random motion

How this is characterized: random walk

The directions of the successive steps are not correlated.

The mean value of the square of the displacements isproportional to the number of steps and thus the time t .

Denote the displacement vector by R(t). Then

<R(t)2 > = a t.

The factor a is related to the diffusion coefficient D.



Randomly moving spheres in a liquid

Diffusion coefficient for the particles?

Equation of motion for a particle:

m d 2 R/dt 2 + b dR/dt = F,

where F is random force resulting from collisions of the

spheres with the liquid molecules, m the mass and b thedrag coefficient.

Drag force proportional to the velocity (Stokes law):

b = 6 a.




Use Cartesian coordinates x, y, z.

The sphere can move in random directions, all

coordinates behave the same way <x 2 > =<y 2 > =<z 2 > :

<R 2 > = 3 <x 2 >

Langevin equation of motion (x -direction)m d 2 x/dt 2 + b dx/dt = F,

The applied force is the random force F resulting from

collisions of the solute molecules with the sphere.

Inserting the relationsd(x 2 )/dt = 2x dx/dt and xd 2 x/dt 2 = d/dt(x dx/dt) – (dx/dt)2




Equation of motion

m/2 (d 2 x 2 /dt 2 ) -m(dx/dt)2 = -b/2(dx 2 /dt) + Fx

Take the average

b/2 d<x 2 >/dt = <xF> - m d/dt<xdx/dt> + 1/2m<dx/dt> 2 .

Because the direction of the random force is uncorrelated

with the position of the particle, <xF> = 0 .

The position and the velocity of the particle are not

correlated, <x dx/dt> = 0. From the equipartition theorem <1/2 m(dx/dt)2 > = 1/2 kT



Equipartition theorem

The theorem of classical statistical mechanics and

thermodynamics states: the internal energy of a system

composed of a large number of particles at thermal

equilibrium will distribute itself evenly among each of the

quadratic degrees of freedom allowed to the particles of the system.

For example, the equipartition theorem says that the

mean internal energy associated with each degree of

freedom of a monatomic ideal gas is the same.

For a molecule of gas, each component of velocity has anassociated kinetic energy. This kinetic energy is, on

average, kT/2 , where k is Boltzmann constant and T

temperature.




Denoting X = <dx 2 /dt>

one obtains a first order equation

(dX/dt) + b/mX = 2kT/m

The general solution is X = 2kT/b + C exp(-b/mt)

Assume that t >> b/m , one obtains

<x 2 > = 2kT/bt or <x 2 > = 2Dt.




The motion is diffusive.

Einstein formula for the diffusion coefficient D = kT/b.

Sphere diffusing in a liquid D = kT/(6 a)

(Stokes-Einstein equation)



Light scattering and diffusion coefficient

Let one consider solution of colloidal particles.

The random motion of the particles causes intensity

fluctuations of the scattered light.

The autocorrelation function of the scattering intensity

g(q,t) can be presented asg(q,T) = <I(q,t) I(q, t+T)>/<I(q,t)> 2 .

Magnitude of the scattering vector q = 4 n sin / where n

is the refractive index of the solution, the wavelength

and the scattering angle.



Light scattering and diffusion coefficient

The correlation function g

can be fitted by a sum of

exponential functions giving

the decay time t which is

related to the diffusioncoefficient of the particles D

as

1/t = q 2 D .

D is affected by the

interaction between the

particles as

D = D 0 (1+a )

where is the concentrationof the particles and a an

interaction parameter.

The hydrodynamic radius R h

of the particles in associated

with D 0 asD 0 = kT/(6 R h )



Example. SAXS and DLS

For a solution of fluorinated colloidal particles about the

same particle size distribution was obtained by using

SAXS and DLS.

DLS: He/Ne laser SAXS: 1.54 Å

J. Wagner et al. Langmuir 2000, 16, 4080-4085.



Diffusion Coefficients for Four Hydrocarbons

in Water

Price and Söderman. Self-diffusion coefficients of somehydrocarbons in water: Measurements and scaling relations.

J. Phys.Chem. A 104, 5892-5894, 2000.

0.70 ± 0.06201.3/3.61.450.764o -xylene

0.77 ± 0.06180.5/3.51.420.95c -hexane

0.85 ± 0.33218.5/3.74.260.294n -hexane

0.86 ± 0.10192.8/3.65.720.225n -pentane

D we (10-9 m2 s-1)V (Å3)/R (Å)d D bulk

c (10-9 m2 s-1)b (10-3 kg (ms)-1 )molecule



Forces between colloidal particles

System with a large amount of surface

Interfacial energy is large.

Jones book estimate: may be much bigger than kT.

Are dispersions then unstable? Other forces between particles of electrostatic origin.



Interactions in colloidal solution

Excluded Volume Repulsion: This refers to the

impossibility of any overlap between hard particles.

Electrostatic interaction: Colloidal particles often carry an

electrical charge and therefore attract or repel each other.The charge of both the continuous and the dispersed

phase, as well as the mobility of the phases are factors

affecting this interaction.




van der Waals forces: This is due to interaction between

two dipoles that are either permanent or induced. Even if

the particles do not have a permanent dipole, fluctuations

of the electron density gives rise to a temporary dipole in

a particle. This temporary dipole induces a dipole inparticles nearby. The temporary dipole and the induced

dipoles are then attracted to each other. This is known as

van der Waals force, and is always present, is short-

range, and is attractive.




Entropic: According to the second law of thermodynamics,

a system progresses to a state in which entropy is

maximized. This can result in effective forces even

between hard spheres.

Steric: between polymer-covered surfaces or in solutionscontaining non-adsorbing polymer can modulate

interparticle forces, producing an additional repulsive

steric repulsion force (which is predominantly entropic in

origin) or an attractive depletion force between them.



Example. Concentrated Silica ColloidalDispersions

The structure factors of colloidal silica dispersions at

rather high volume fractions (from 0.055 to 0.22) were

determined by small-angle X-ray scattering and fitted with

both the equivalent hard-sphere potential model (EHS)

and the Hayter-Penfold/Yukawa potential model (HPY). Both of these models described the interactions in these

dispersions successfully, and the results were in

reasonable agreement.

D. Qiu et a. A Small-Angle X-ray Scattering Study of the Interactionsin Concentrated Silica Colloidal Dispersions.

. Langmuir, 22 (2), 546 -552, 2006.



SAXS of Concentrated Silica ColloidalDispersions

The strength and range of the interaction potentials

decreased with increasing particle volume fractions, which

suggests shrinkage of the electrical double layer arising

from an increase in the counterion concentration in the

bulk solution. However, the interactions at the average interparticle

separation increased as the volume fraction increased.

D. Qiu et a. A Small-Angle X-ray Scattering Study of the Interactions inConcentrated Silica Colloidal Dispersions.. Langmuir, 22 (2), 546 -552, 2006.



Forces between colloidal particles: van der

Waals

Interaction between two atoms (particles)

Potential between two particles is assumed to be of

form -C/r 6

For two macroscopic bodies with separation h :

U(h) = -C/r 6 1 2 dV 1 dV 2

where 1 and 2 are the number densities of the

volume elements dV 1 and dV 2 .

.



Example.

An atom at the distance D from

medium of density .

The interaction between the molecule

and a ring of radius x whose centre

is z away from the molecule is

-2 x dx dz C/(x 2 +z 2 )3 .

The total interaction energy is

w(D) = -2 C D dz 0

x dx/(x2+z3)3

= -2 C/(12 D 3 ).D

x

x

z

y

r dxdz



Example.

Assume that instead of one atom there is a sheet of

atoms.

The unit area and thickness of this atomic sheet is dz at

the distance z from the semi-infinite sheet.

The energy per unit area is simply

-22C/12z3,

where is the density.

The constant A = 2C is called the Hamaker constant.

Then the interaction energy is

U(h) = -A/12h2 h z-3 dz.



van der Waals and Hamaker constant

Surface-surface interaction

The total interaction energy per unit area between two

semi-infinite sheets of the same material was

U = 2 /12D2

The constant A = 2 C is the Hamaker constant. Magnitude

around 10-19 J



vdW interaction energies between particles of

different geometries

Atoms w ~ -1/r6

Two spheres w ~ -1/r

Atom-surface w ~ -1/r3

Sphere-surface w ~ -1/r

Two parallel cylindersw ~ -1/r3/2

Two surfaces w ~ -1/r2

Israelachvili p 177

;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

12

-1010

-108

-106

-104

-102

-100

r

w ( r )

atoms

spherescylinderssurfaces



Problems

Pairwise additivity of the forces. In practice all other atoms

in the system have to be taken into account.

For large separations, over 10 nm, the effects of the finite

speed of the propagation of fields arising from the

fluctuating dipoles become significant.

More powerful approach – Lifshitz theory.



Charged surfaces

Electrostatic interactions are important.

In water: counterions of the colloids float freely: Screening

Screened Coulomb interaction: exponential decay in

strength with distance

Consider two ionized parallel surfaces in the water.

Counter ions are in the solution, but provide charge

neutrality. The counter ions are attracted to the surfaces.

Their concentration profile is diffuse.



Charged surfaces in water

Temperature T

Let be the electrostatic potential and the number density of

ions of valency z at any point between the surfaces.

Chemical potential = ze + kT ln

Boltzmann distribution for the density of ions at any point = 0 exp(-ze /kT) (0 is assumed 0)

Poisson-Bolzmann equation gives the

concentration profile for the counter ions.-

-

-

-

-

-

-

-D

+



Poisson-Bolzmann equation

The charge of the ions is ze.

Poisson equation for the net charge density at point x:

ze = - 0 d 2 /dx 2

Combine with the Boltzmann distribution

d 2

/dx 2

= - ze 0 = -(ze 0 0 ) exp(-ze /kT)

PB: d 2 /dx 2 = -(ze 0 0 ) exp(-ze /kT)

Poisson-Bolzmann equation is a non-linear second-order

differential equation.



Surface charge

PB equation

d 2 /dx 2 = -(ze 0 0 ) exp(-ze /kT)

Boundary conditions

Symmetry requirement: the field must vanish at the midplane /dx = 0

Overall electroneutrality: total charge in the gap must be

equal (opposite sign) to that at surfaces.

Let be the charge density at surfaces. Then

= - 0 D/2

ze dx = 0 0 D/2

d 2

/dx 2

dx = 0 (d /dx)D/2

= 0 (d /dx)s

The field E s at the surface is (d /dx)s = / 0 .

-D/2 D/2





Counterion concentration profile away from a

surface

PB equation d 2 /dx 2 = -(ze 0 0 ) exp(-ze /kT)

Solution (x,T) = (kT/ze) ln( cos2 Kx),

where K(T)2 = (ze)2 0 0 kT

At x=0, =0 and /dx = 0 for all K At any point x

E x = d /dx = 2kTK/ze tanKx

The counter ion distribution profile

(x,T) = 0 exp(-ze /kT)

= 0 / cos2 Kx

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

x

1 / c o s

2 K

x

K=0.5, 0.3333, 0.25, 0.2, 0.1667



Limitations of PB equation

Continuum, mean field

Breaks down at small distances

Ion-correlation effects (polarization, vdW)

Finite-ion size

Discreteness of surface charges Solvation forces



Electrostatic double layer forces

Consider pair of parallel ionized surfaces

Overall charge neutralization achieved by counter ions.

Electrostatic potential (x) at a distance x from the

surface.

Boltzmann equation gives the density of ions: n = n(0)exp (-ze (x)/(kT)) where k Boltzmann constant and T

temperature, ze ion charge and the potential (x).

The potential is determined by the distribution of net

charge (z) by the Poisson equation = d2 /dx2.

Counter ions balance the surface charge = ze.

Poisson-Boltzmann equation

d2 /dx2 = -(ze n(0)/ ) exp(-ze /kT)



Electrostatic double layer forces

Example. Surface is in contact with a solution of an

electrolyte which is a solution of a univalent salt (e.g.

sodium chloride).

Concentration of positive ions n + = n(0) exp(-ze /kT)

Concentration of negative ions n - = n(0) exp(ze /kT) The net charge density = ze(n + +n - )

d 2 /dx 2 = -(ze n(0)/ ) sinh (-ze /kT)



Debye-Hückel approximation

Boundary conditions for solution:

Isolated plate: and /dx approach 0 as x approaches

infinity.

If the potential is small one can approximate sinh x ~ x.

Debye-Hückel approximation for the potential(x) = (0) exp(- x)

where

= (2e 2 n(0)z 2 /( 0 kT) )1/2 .



Debye screening length

Electric fields are screened in an electrolyte.

Screening is the damping of electric fields caused by the

presence of mobile charge carriers.

The length which characterizes the screening 1/ is calledthe Debye screening length.



Stabilizing colloids

Coating with a polymer layer.

When two particles approach each other, the

concentration of the polymer inside the gap increases.

This increases the osmotic pressure and causes a

repulsive force.



Spatial Correlation of Spherical Polyelectrolyte

Brushes in Salt-Free Solution As Observed by

Small-Angle X-ray Scattering

Linear chains of poly(acrylic acid) (PAA) are chemically

grafted onto the surface of a colloidal poly(styrene)

particle.

Robillard et al. Macromolecules33, 9109-9114, 2000



Depleting interactions

A solution contains both particles and e.g. dissolvedpolymer.

Polymer coils are excluded from a depletion zone bear the

surface of the colloid particles. When the depletion zones

of two particles overlap there is a net attractive force

between the particles arising from unbalanced osmoticpressure.



Depleting interactions

Dilute solution of particles

Osmotic pressure P osm = N/V kT

N number of polymers in volume V of solution.

The net potential F dep = -P osm V dep

V dep is the total volume of particles from which thepolymers are excluded.



Model potential describing the depletion effect

First theoretical consideration Asakura and Oosawa J.

Chem Phys. 22, 1255, 1954

Suspension of colloidal particles, radius a , and non-

absorbing polymer

Depletion potential U = infinity for r 2a

U = A for 2a < r < 2a+rg

U = 0 for r > 2a+rg

Here rg is the radius of the polymer molecule

The constant A is related to the polymer osmotic pressureand the volume of the overlapping depletion zones

between two particles.



Results

With the model potential the phase behaviour of the

mixture could be predicted.

At low polymer concentrations fluidlike arrangement of

colloidal particles

At high polymer concentrations colloidal fluid-crystalcoexistence

Also metastable gel phases

S.MiIlett et al. Phys. Rev. E, 1995, 51, 1344-1352.



Stability and phase behaviour of colloids

Interaction from repulsive to attractive

Adding salt

Add poor solvent

Remove grafted chains from the surface

Add non-adsorbing polymer to increase the size of thedepletion zone.



Crystallization of hard sphere colloids

Stable suspension: the forces between colloidal particles

are repulsive at all distances.

Spherical particles: The systems crystallizes as the

concentration of particles reaches high enough level.

Colloidal crystals: true long range order

Simplest model: hard sphere model

Good model for colloidal particles stabilized by polymer

coating. The coating need to be thin compared to the

radius of the particles.



Origin of the phase transition

A way of thinking: excluded volume

Two spheres cannot overlap. This leads to repulsive force

between the spheres.

Ideal gas of N atoms in a volume V: S i = k ln(aV/N)

Gas atoms have a finite volume b . The volume accessibleto any given atom is V-Nb.

Now the entropy S = k ln(a(V-nb)/N) = S i – k ln(1-bN/V)

If the free volume of atoms is low S = S i – k b N/V

The corresponding free energy F = F i + k T b N/V

There is effective repulsion between particles whichcauses the crystallization.



Structures

Close packed: HCP or FCC



Colloids with weakly attractive interactions

Grafted colloids

Weak interaction: liquid-solid transition



Colloids with strongly attractive interactions

Aggregation



Crystal-nucleation rate (Auer and Frenkel

Nature 2001, 1020-1023

Probability (per particle) that a spontaneous fluctuation

will result in formation of a critical nucleus depends on the

free energy

G c as P c = exp(- G c /kT)

Free energy needed to form the nucleus G(r) = 4/3 r 3 + 4 r 2 ,

where is the number density of the solid and the solid-

liquid interfacial energy density.

The maximum of G at r = 2 /( |) where | is the

difference in chemical potential of the solid and the liquid. The nucleation rate f per unit volume is proportional to the

probability P :

f = c P c = c exp(- G c /kT) = c exp(-16 /3 3 /( |)2 )



Complementary Use of Simulations and Molecular-

Thermodynamic Theory to Model Micellization

Brian C. Stephenson, Kenneth Beers, and Daniel

Blankschtein. LANGMUIR Volume: 22 Issue: 4 Pages:

1500-1513 Published: 2006

Computersimulation

Molecular-thermodynamic

model

ThermodynamicDescription of

Micelle aggregation

Property

predictiongmic

Soft2007 Colloid

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