1. 3 Dilute Solution Thermodynamics, Molecular Weights, and Sizes 71 3
3 Dilute Solution Thermodynamics, Molecular Weights, and Sizes Introduction / The Solubility Parameter / Thermodynamics of Mixing / Molecular Weight Averages / Determination of the Number-Average Molecular Weight / Weight-Average Molecular Weights and Radii of Gyration / Molecular Weights of Polymers / Intrinsic Viscosity / Gel Permeation Chromatography / Mass Spectrometry / Instrumentation for Molecular Weight Determination / Solution Thermodynamics and Molecular Weights / Concentrated Solutions, Phase Separation Behavior, and Diffusion Phase Separation and Fractionation / Regions of the Polymer–Solvent Phase Diagram / Polymer–Polymer Phase Separation / Diffusion and Permeability in Polymers / Latexes and Suspensions / Multicomponent and Multiphase Materials / 186 References /

2. Molecular weight Small molecules have exact molecular weights
hexane
M = 84 g/mol
heptane
M = 100 g/mol
paraffin wax carbons, M = 350 –700 g/mol
but polymers …
Polyethylene (n=2200)
M = g/mol
Polyethylene (n=2205)
M = g/mol
Many polymer properties depend on the molecular weight
For example: Tm, melt viscosity, mechanical properties…

4. Molecular weight distribution
ideal narrow distribution
In practice often very broad
The Schultz distripution is typical for
Chain-growth radical polymerizatio
The Schults-Flory distripution is typical for
step-growth polymerization

5. Molecular weight averages
Number average
Weight average
z-average
Viscosity average

6. Example 1 Demographics Average population 723,500/4= 180 875
A good way to understand the difference between the number average molecular weight and the weight average molecular weight is to compare some American cities.
Let's take four cities, say, Memphis, Tennessee; Montrose, Colorado; Effingham, Illinois; and Freeman, South Dakota. Now we'll take a look at their populations.
Average population
723,500/4=
Now we see that of these four cities, that average population is 180,875.
But we could look at it a different way. Until now we've been worried about "the average city". What is the population of "the average city"? But let's forget about cities for a moment, and think about people. What size city does the average person among the populations of these four towns live in?
If you look at the numbers you can see that the average person doesn't live in a town of a population of 180,000. Take a look there. most of the people in the combined populations of the four towns live in Memphis, a town with a lot more than 180,000 people. So how do we calculate the size of town that the average person lives in, if the simple average doesn't work?
What we need is a weighted average. This is an average that would account for the fact that a large city like Memphis holds a larger percentage of the total population of the four cities than Montrose, Colorado. Doing this involves a little bit of math that looks scary but really isn't. All we do is take the total number of people in each city, then multiply that number by that city's fraction of the total population. Take all the answers we get for each city and add them up, and we get an answer that we'll call the weight average population of the four cities.

7. Let's walk through this to show what I mean. Take Memphis
Let's walk through this to show what I mean. Take Memphis. It has a population of 700,000. The total population of our four cities is 723,500. So the fraction of people who live in Memphis is...
, or we might say, 96.75% of the people live in Memphis. Now let's take our fraction, , and multiply that by the population of Memphis:
So our weight average population of the four cities is about 677,600. We can say from this figure that the average person lives in a city of about 677,600. That is more believable than saying that the average citizen lives in a city of 180,000

8. Example 2 polymers

9. The number average molecular weight
The number average molecular weight is the total weight of the sample divided by the number of molecules in the sample.

10. The weight average molecular weight
Number average = g/mol
Weight average = g/mol  Polydispersity index PDI = Mw/Mn = 1.05

11. Example 3 Blend: 1 g Monomer, M1 = 100 g/mol
9 g Polymer, M1 = g/mol
What are Mn, Mw, and Mz ?
Number of molecules? Number of moles ni=mi/Mi , Number of molecules Ni=ni*NA = (mi*NA)/Mi
n1=1g/100g/mol=10-2 mol, n2=9g/ g/mol =9*10-5 mol
Mn is sensitive to the admixture of low molecular mass
Mw is sensitive to the admixture of high molecular mass
Mw always exeeds Mn (or is equal)
Ratio Mw/Mn measures the range of molecular sizes (PDI)

12. Polydispersity index PDI
If all chains are equal in length
In general

13. How to measure Mn

14. Osmotic Pressure and Mn
Osmotic pressure () is a thermodynamic colligative property that measures the free energy difference between a polymer solution and a pure solvent.
There is a free energy gain in mixing polymer with solvent that makes more solvent to flow into the polymer solution.
The osmotic pressure is determined from the height difference h as
where  is the solvent density and g is the gravitational acceleration.
In equilibrium state, for dilute polymer solutions (similar to ideal gas)  is the thermal energy kT times the number density of chains cNA/M.
The van’t Hoff law :
Osmotic pressure is a colligative property, it is simply proportional to the number density and gives number average Mn in case of polydisperse sample.

15. Osmotic pressure, Mn Notice the analogy with ideal gas law:
Osmotic pressure p depends on the molecular weight as follows:
Notice the analogy with ideal gas law:
Ideal gas law:
n is in moles. n/V is equal to c/M 
Setting the gas pressure equal to the osmotic pressure P = p 

16. Measuring of Mn by Osmotic Pressure
To obtain Mn, osmotic coefficients (/c) data, measured at various low concentrations, must be extrapolated to the zero concentration.
Concentration dependence of osmotic coefficient for three poly(a-methylstyrene) samples in toluene at 25°C. The data corresponding to dilute solutions for these three samples are shown, with lines fit to the lowest concentration data.
(Source: I. Noda, N. Kato, T. Kitano and M. Nagasawa, Macromolecules 1981, 16, 668].
Polymer-polymer interactions must be taken into account.
This is the ideal gas contribution.
Two body interactions are represented by the second virial coefficient A2.
At higher concentrations, the higher-order terms have to be taken into account.

18. Viscosity average molecular weight

19. Viscosity average molecular weight
Relative viscosity
h0 is the viscosity of solvent and h is the viscosity of the polymer solution
Specific viscosity
Specific viscosity, divided by the consentration and extrapolated to zero consentration, yields the intrinsic viscosity

20. Mark-Houwink relationship
where K and a are the unique constants for each solvent-polymer pair at a particular temperature.

22. Gel permeation chromatography (GPC)
Gel permeation chromatography (GPC) or size exlusion chromatography (SEC) makes use of the size exlusion principle. Depending on the size of the molecule, defined by its hydrodynamic radius, they can or cannot enter the small pores in a bed of cross-linked particles. The smaller molecules diffuse into the pores via Brownian motion and are dealyed.
GPC measures the molar mass distripution

25. Mass spectrometry: MALDI, TOF
Molecular weight distripution, absolute method
MAtrix-assisted Laser Desorption Ionization (MALDI), a soft ionization technique for transfering large molecular ions into a mass spectrometer with minimum fragmentation.
Time of flight (TOF)technique
Given equal force, the more mass, the slower the acceleration. For us this means that the big heavy polymer molecules will take a lot longer to get to the detector at the end of the chamber. So the polymers will hit the detector, the small ones first, then the big ones. They hit completely in order by mass.

27. Light Scattering

28. How a Light Scattering Setup Looks Like?
Laser
Goniometer
Sample
Detector

29. What is Light Scattering?
The phenomenon occurs because
- the molecules are polarized by the electric field of the passing light
- fluctuation of density and concentration of particles
- Static Light Scattering, SLS
The intensity is averaged over a fairly long time (1-2 s)
- Dynamic Light Scattering, DLS
Fast fluctuations of intensity of scattered light ( s)

30. Static and Dynamic Light Scattering
Static Light Scattering Dynamic Light Scattering
time scale
> 1 sec
Molar mass Mw
Radius of gyration <Rg2>½
Second virial coefficient A2
time scale
0.1 sec < t < 1 sec
Hydrodynamic radius Rh
Diffusion coefficient D
relax  Rh 1 / Dtrans

31. Sperling book..

32. (see PDF file)
The intensity of the scattered light depends on the polarizability (to be defined later) and the polarizability depends on the molecular weight.
Besides molecular weight dependence, light scattering also has a direct dependence on particle size.  radius of gyration of the polymer molecule
As with osmotic pressure, we expect all light scattering experiments to be done in non-ideal solutions. Nonideality complicates the data analysis, but, like osmotic pressure, allows you to determining a virial coefficient, A2
Rayleigh theory - applies to small particles
Many polymers will violate this criterion and the light scattering results will have to be corrected for large particle size effects.
The correction method involves extrapolation techniques that extrapolate light scattering intensity to zero scattering angle. Theother is an extrapolation to zero concentration to remove the effect of non-ideal solutions

33. We begin by describing the theory for light scattering off a small particle in an ideal solution. At the origin the field is time dependent and described by
If the particle at the origin is polarizable, the incident electric field will induce a dipole moment in that particle. The magnitude of the dipole moment is proportional to the field. The proportionality constant is called the polarizability

34. Equipment that measures scattered light is typically only sensitive to the intensity of light. Thus, squaring the amplitude of Es gives the scattered light intensity
The above results are for incident light polarized in the z direction. Experiments, however, are usually done with unpolarized light.

35. We now have the scattered light intensity for scattering off a single particle. For scattering off n moles of particles or nL particles (L is Avagadro’s number) in a dilute solution of volume V ,
As a function of , the scattered intensity is proportional 1/l4. This strong wavelength dependence makes short wavelength light scatter more than long wavelength light. This effect explains why the sky is blue and sunsets appear red.

36. 7.3 Ideal Polymer Solutions with Small Particles
First, the polarizability can be thought of as a difference in the index of refraction between the polymer and the solvent. In other words light scattering only occurs in mediums that have an inhomogeneous index of refraction.
Writing c as nM/V (in units of g/ml) yields

39. Sperling

40. To correct for large particles, we merely need to do the light scattering experiments at zero scattering angle. Unfortunately, these experiments cannot be done.
We thus do a second extrapolation, an extrapolation to zero scattering angle.

41. Sperling book..

43. Sperling book..

48. Polymer processing injection moulding, extrusion, blow moulding...
Extrusion & injection moulding
blown film extrusion

49. Nobel prize winners in polymer science