1. Last Lecture:
The thermodynamics of polymer phase separation is similar to that of simple liquids, with consideration given to the number of repeat units, N.
For polymers, the critical point occurs at cN=2, with the result that most polymers are immiscible.
As cN decreases toward 2, the interfacial width of polymers becomes broader.
The Stokes’ drag force on a colloidal particle is Fs=6phav.
Colloids undergo Brownian motion, which can be described by random walk statistics: <R2>1/2 = n1/2 , where is the step-size and n is the number of steps.
The Stokes-Einstein diffusion coefficient of a colloidal particle is given by D = kT (6pha)-1.

2. Colloids under Shear and van der Waals’ Forces
3SM
Colloids under Shear and van der Waals’ Forces
5 March, 2009
Lecture 7
See Jones’ Soft Condensed Matter, Chapt. 4 and Israelachvili, Ch. 10 &11

3. Flow of Dilute Colloidal Dispersions
The flow of a dilute colloidal dispersion is Newtonian (i.e. shear strain rate and shear stress are related by a constant h).
In a dispersion with a volume fraction of particles of f in a continuous liquid with viscosity ho, the dispersion’s h is given by a series expression proposed by Einstein:
A typical value for the constant b is 2.5; the series can usually be truncated after the first two or three terms, since f must be << 1 for the equation to hold.

4. Viscosity of Soft Matter Often Depends on the Shear Ratess
Newtonian:
(simple liquids like water) h ss h
or thickening:
Shear thinning h

5. Flow of Concentrated Colloidal Dispersions
At higher f, h is a function of the shear strain rate, , and the flow is non-Newtonian. Why?
Shear stress influences the arrangement of colloidal particles.
At high shear-strain rates, particles re-arrange under the applied shear stress. They form layers or strings along the the shear plane to minimise dissipated energy. Viscosity is lower.
At low shear-strain rates, Brownian diffusive motion is able to randomise particle arrangement and destroy any ordering imposed by the shear stress. Viscosity is higher.

6. Effects of Shear Stress on Colloidal Dispersions
Under a shear stress
With no shear
Confocal Microscope Images
MRS Bulletin, Feb 2004, p. 88

7. The Characteristic Time for Shear Ordering, tS
Both the shear strain rate and the Brownian diffusion are associated with a characteristic time, t. A y F Dx v g
The characteristic time for the shear strain, ts, is simply:
Slower shear strain rates thus have longer characteristic shear times. One can think of ts as the time over which the particles are re-distributed under the shear stress.

8. Characteristic Time for Brownian Diffusion, tD
The characteristic time for Brownian diffusion, tD, can be defined as the time required for a particle to diffuse the distance of its radius, a. a
So,
Substituting in an expression for the Stokes-Einstein diffusion coefficient, DSE:

9. Competition between Shear Ordering and Brownian Diffusion: Peclet Number, Pe
To determine the relative importance of diffusion and shear strain in influencing the structure of colloidal dispersions, we can compare their characteristic times through a Peclet number:
(a unitless parameter)
Substituting in values for each characteristic time:
Thus, when Pe >1, diffusion is slow (tD is long) relative to the time of shear strain (tS). Hence, the shear stress can order the particles and lower the h. Shear thinning is observed!

10. A “Universal” Dependence of h on Pe
Data for different colloids of differing size and type
Shear thinning region
Small a; Low
Large a; High
When Pe <1, tD is short in comparison to tS, and the particles are not ordered because Brownian diffusion randomises them.

11. van der Waals’ Energies between Particles
The van der Waals’ attraction between isolated molecules is quite weak.
However, because of the additivity of forces, there can be significant forces between colloidal particles.
Recall the London result for the interaction energy between pairs of non-polar molecules:
The total interaction energy between colloidal particles is found by summing up w(r) for the number of pairs at each distance r.
Remember that all types of van der Waals interaction energies will follow this general form (r -6) but the value of C will vary depending on the particular type of molecules.

12. Interaction Energy between a Molecule and a Ring of the Same Substancex
 
r is the molecular density in the condensed state: number/volume
Israelachvili, p. 156

13. Interaction Energy between a Molecule and a Ring of the Same Substance
The cross-sectional area of the ring is dxdz.
The volume of the ring is thus V = 2pxdxdz.
If the substance contains r molecules per unit volume in the condensed phase, then the number of molecules in the ring is N = rV = 2prxdxdz.
The distance, r, from the molecule to the ring is:
The total interaction energy between the molecule and N molecules in the ring can be written as:

14. Interaction Energy between a Molecule and a Slab of the Same Substancex
Semi- slab
 

15. Interaction Energy between a Molecule and a Slab of the Same Substance
For a ring of radius, x:
Let the molecule be a distance z = D from a semi- slab.
The total interaction energy between the molecule and slab is found by integrating over all depths into the surface. A slab can be described by a series of rings of increasing size.

16. Attractive Force between a Molecule and a Slab of the Same Substance • D
Force is obtained from the derivative of energy with respect to distance:

17. Interaction Energy between a Particle and a Slab of the Same Substance
  x dz
Slice Thickness = dz
z =0 z
z =2R
2R-z R x D z
D+z
R = particle radius

18. Interaction Energy between a Colloidal Particle and a Slab of the Same Substance
For a slice of thickness dz and radius x, the volume is px2dz.
Each slice contains N = rV = rpx2dz molecules.
For a single molecule in the particle at a distance z+D, the interaction energy with the slab is:
To calculate the total interaction energy between a particle and the slab, we need to add up the interactions between every slice (with N molecules) and the slab.

19. Interaction Energy between a Colloidal Particle and a Slab of the Same Substance
For a sphere with a radius of R, the chord theorem tells us that x2 = (2R - z)z. Substituting in for x2:
But if D<<R, which is the case for close approach when vdW forces are active, only small values of z contribute significantly to the integral, and so integrating up to z = will not introduce much error. We can also neglect z in the numerator as z <<R when energies are large.

20. Attractive Force between a Colloidal Particle and a Slab of the Same Substance
Integrating:
It is conventional to write a Hamaker constant as A = r2p2C. Then,
Note that although van der Waals interactions vary with molecular separation as r -6, particle/surface interaction energy varies as D -1.
The force between the particle and a slab is found from the derivative of W(D):
Units of A:

21. Hamaker Constants for Identical Substances Acting Across a Vacuum
A = p2Cr2
Substance C (10-79 Jm6) r (1028 m-3) A (10-19 J)
Hydrocarbon
CCl
H2O
“A” tends to be about J for many substances. Why?
For non-polar, uncharged molecules, recall the definition of the London constant: C  o2
If v = molecular volume, we know that r  1/v and ao r3  v
So, roughly we see: A Cr2 ao2r2  v2/v2 = a constant!

22. Surface-Surface Interaction Energies
The attractive energy between two semi- planar slabs is !
Can consider the energy between a unit area (A) of surface and a semi- slab. dz z D
z=0
Unit area
In a slice of thickness dz, there are N =rAdz molecules. In a unit area, A = 1, and N = rdz.
We recall that for a single molecule:

23. Surface-Surface Interaction Energiesz D
z=0 dz
Unit area z
To find the total interaction energy per unit area, we integrate over all distances for all molecules:
z=D
z =  

24. Summary of Molecular and Macroscopic Interaction Energies
Colloidal particles 
If R1 > R2:
Israelachvili, p. 177

25. What Makes Adhesives Stick to a Variety of Surfaces?
Soft polymers can obtain close contact with any surface - D is very small.
Then van der Waals interactions are significant.

26. Significance of W(D) for Planar Surfaces
Per unit area:
Typically for hydrocarbons, A = J. Typical intermolecular distances at “contact” are D = 0.2 nm = 0.2 x 10-9 m.
To create a new surface by slicing an  slab in half would therefore require -1/2 W(D) of energy per unit area of new surface.
Hence, a typical surface energy, g, for a hydrocarbon is 30 mJ m-2.

27. Adhesion Force for Planar Surfaces
As we’ve seen before, the force between two objects is F = dW/dD, so for two planar surfaces we find:
As W is per unit area, the force is likewise per unit area. Thus, it is a pressure, P = F/A.
Using typical values for A and assuming molecular contact:
This pressure corresponds to nearly 7000 atmospheres! But it requires very close contact.

28. Gecko’s stick to nearly any surface – even under water – because of van der Waals attractive forces
A Gecko
Pads on feet
Setae
Images from
Spatulae

29. Synthetic “Gecko” Tape
But van der Waals’ forces also cause attraction between the fibers!
When polymer fibers make close contact to surfaces, they adhere strongly.

30. Ordering of Colloidal Particles
Numerous types of interactions can operate on colloids: electrostatic, steric, van der Waals, etc.
Control of these forces during drying a colloidal dispersion can create “colloidal crystals” in which the particles are highly ordered.
MRS Bulletin,
Feb 2004, p. 86

31. Electrostatic Double Layer Forces
Colloidal particles are often charged. But, colloidal liquids don’t have a net charge, because counter-ions in the liquid balance the particle charge. - + + + + + + +
-ve particle surface + + + + + +
The charge on the particles is “screened”.

32. The Boltzmann Equation
+++++ y
Charged surface x
Ions (both + and -) have a concentration at a distance x from a surface that is determined by the electrostatic potential y(x) there, as given by the Boltzmann Equation:
Here e is the charge on an electron, and z is an integer value. - n x
“Bulk” concentration no +

33. The Poisson Equation
In turn, the spatial distribution of the electrostatic potential is described by the Poisson equation:
The net charge density, r, (in the simple case in which there are only ions to counter-balance the surface charge) is
But n+ and n- can be given by the Boltzmann equation, and then the Poisson-Boltzmann equation is obtained:

34. Solutions of the Poisson-Boltzmann Equation
We recall that
So,
The P-B Equation then becomes:
But when x is small, sinh(x)  x, and so for small y:
In this limit, a solution of the P-B equation is
where -1 is called the Debye screening length.

35. The Debye Screening Length, k-1x
k-1 + -
It can be shown that  depends on the ionic (salt) concentration, no, and on the valency, z, as well as e for the liquid (e = 85 for water).
For one mole/L of salt in water, k-1 = 0.3 nm. As the salt concentration increases, the distance over which the particle charge acts decreases.

36. Interaction Potential
Colloidal particles with the same charge will repel each other. + x
But the repulsion is not significant at a separation distance of x > ~ 4k -1
But there is still an attractive van der Waals’ force between particles.
The sum of these contributions makes the DLVO interaction potential:
Van der Waals
Electrostatic

37. Effect of Ionic (Salt) Concentration
Low salt
High salt D R D

38. Packing of Colloidal Particles
When mono-sized, spherical particles are packed into an FCC arrangement, they fill a volume fraction f of 0.74 of free space. e 4r r e3 =
f =
Sphere V
Occupied V
Can you prove to yourself that this is true?
When randomly-packed, f is typically 0.6 for spherical particles. Interestingly, oblique spheroid particles (e.g. peanut M&Ms) fill a greater fraction of space when randomly packed.
The Debye screening length can contribute to the effective particle radius and prevent dense particle packing of colloidal particle dispersed in a liquid (e.g. water).

39. Effect of Salt on Packing of Charged Particles
salt concentration f
Disordered
Ordered: FCC packing
0.7
Long screening length
Short screening length

40. Electrokinetic Effects
If particles have a charge, q, they can be moved by an electric field. E a q FS FE
At equilibrium, the force from the applied electric field, FE, will equal the Stokes’ drag force, FS.
The mobility, m, of a particle is then obtained as:
Mobility measurements can be used to determine colloidal particle charge.

41. What’s wrong with this picture?
An artist’s conception of a “nano-robot” landing on a red blood cell in flowing blood and injecting a “medicine”.

42. Need to consider:
Brownian motion – in both the nano-robot and the cells
Drag force – acts on nano-robot and makes it difficult to propel
Attractive surface forces – the nano-robot will be attracted from all sides to cells and other objects. How to control the robot orientation?

43. Problem Set 4
1. The glass transition temperature of poly(styrene) is 100 C. At a temperature of 140 C, the zero-shear-rate viscosity of a poly(styrene) melt is measured to be 7 x 109 Pas. Using a reasonable value for To in the Vogel-Fulcher equation, and an estimate for the viscosity at Tg, predict the viscosity of the melt at 120 C.
2. A polymer particle with a diameter of 300 nm is dispersed in water at a temperature of 20 C. The density of the polymer is 1050 kg m-3, and the density of water is 1000
kgm-3. The viscosity of water is 1.00 x 10-3 Pa s.
Calculate (a) the terminal velocity of the particle under gravity, (b) the Stokes-Einstein diffusion coefficient (DSE), (c) the time for the particle to diffuse 10 particle diameters, and (d) the time for the particle to diffuse one meter.
Explain why DSE will be affected by the presence of an adsorbed layer on the particles. Explain the ways in which the temperature of the dispersion will also affect DSE.
3. A water-based dispersion of the particle described in Question 2 can be used to deposit a clear coating on a surface. A 200 mm thick layer is cast on a wall using a brush. Estimate how fast the brush must move in spreading the layer in order to have a significant amount of shear thinning. (Note that with a low shear rate, such as under gravity, there is less flow, which is desired in this application.)
4. For charged colloidal particles, with a radius of 0.1 mm, dispersed in a solution of NaCl in water, calculate the Debye screening length when the salt concentration is (a) 1 mole per litre; (b) 0.01 moles per litre; and (c) 10-5 moles per litre. In the absence of salt, the particles pack into a random, close-packed arrangement at a volume fraction f of For each of the three salt concentrations, calculate the volume fraction f of particles when the particles are randomly close-packed. You should assume that the radius of the particles is equal to the true particle radius plus the Debye screening length.

44. Problem Set 4
1. The glass transition temperature of poly(styrene) is 100 C. At a temperature of 140 C, the zero-shear-rate viscosity of a poly(styrene) melt is measured to be 7 x 109 Pas. Using a reasonable value for To in the Vogel-Fulcher equation, and an estimate for the viscosity at Tg, predict the viscosity of the melt at 120 C.
2. A polymer particle with a diameter of 300 nm is dispersed in water at a temperature of 20 C. The density of the polymer is 1050 kg m-3, and the density of water is 1000
kgm-3. The viscosity of water is 1.00 x 10-3 Pa s.
Calculate (a) the terminal velocity of the particle under gravity, (b) the Stokes-Einstein diffusion coefficient (DSE), (c) the time for the particle to diffuse 10 particle diameters, and (d) the time for the particle to diffuse one meter.
Explain why DSE will be affected by the presence of an adsorbed layer on the particles. Explain the ways in which the temperature of the dispersion will also affect DSE.
3. A water-based dispersion of the particle described in Question 2 can be used to deposit a clear coating on a surface. A 200 mm thick layer is cast on a wall using a brush. Estimate how fast the brush must move in spreading the layer in order to have a significant amount of shear thinning. (Note that with a low shear rate, such as under gravity, there is less flow, which is desired in this application.)