3 Dilute Solution Thermodynamics, Molecular Weights,

and Sizes 71

3.1 Introduction / 71

3.2 The Solubility Parameter / 73

3.3 Thermodynamics of Mixing / 79

3.4 Molecular Weight Averages / 85

3.5 Determination of the Number-Average Molecular Weight / 87

3.6 Weight-Average Molecular Weights and Radii of Gyration / 91

3.7 Molecular Weights of Polymers / 103

3.8 Intrinsic Viscosity / 110

3.9 Gel Permeation Chromatography / 117

3.10 Mass Spectrometry / 130

3.11 Instrumentation for Molecular Weight Determination / 134

3.12 Solution Thermodynamics and Molecular Weights / 135

4 Concentrated Solutions, Phase Separation Behavior,

and Diffusion 145

4.1 Phase Separation and Fractionation / 145

4.2 Regions of the Polymer–Solvent Phase Diagram / 150

4.3 Polymer–Polymer Phase Separation / 153

4.4 Diffusion and Permeability in Polymers / 172

4.5 Latexes and Suspensions / 184

4.6 Multicomponent and Multiphase Materials / 186

References /

Molecular weight

hexane M = 84 g/mol

Small molecules have exact molecular weights

heptane M = 100 g/mol

Polyethylene (n=2200)M = 61 600 g/mol

but polymers …

Polyethylene (n=2205)M = 61 740 g/mol

Many polymer properties depend on the molecular weightFor example: Tm, melt viscosity, mechanical properties…

paraffin wax 25 - 50 carbons, M = 350 –700 g/mol

2200CH2 CH2

CH2 CH2 2205

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

Molecular weight distribution

Molecular weight(or degree of poloymerization)

Molecular weight(or degree of poloymerization)

Molecular weight averages

Number average

Weight average

z-average

Viscosity average

i

ii

ii

iii

n MnN

MNM

i

ii

ii

iii

iii

iii

w MwW

MW

MN

MNM

2

iii

iii

iii

iii

z MW

MW

MN

MNM

2

2

3

a

iii

i

aii

v MN

MNM

/11

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.

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.

Average population

723,500/4= 180 875

Example 1 Demographics

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...

...0.9675, or we might say, 96.75% of the people live in Memphis. Now let's take our fraction, 0.9675, and multiply that by the population of Memphis:

9675.0500,723

000,700

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

http://www.pslc.ws/macrog.htm,

Example 2 polymers

1000 10000 100000 10000000

5

10

15

20

25

Num

ber

of m

olec

ules

Molecular weight g/mol

Number of Molecules

Mass of each Molecule

1 120000

2 100000

3 95000

4 90000

5 85000

8 80000

10 75000

17 70000

25 65000

17 60000

10 55000

8 50000

6 45000

4 40000

2 35000

1 30000

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

The number average molecular weight

Number of Molecules, Ni

Mass of Each Molecule, Mi

Total Mass of Each Type of Molecule, NiMi

1 120000 120000

2 100000 200000

3 95000 285000

4 90000 360000

5 85000 425000

8 80000 640000

10 75000 750000

17 70000 1.19E6

25 65000 1.625E6

17 60000 1.02E6

10 55000 550000

8 50000 400000

6 45000 270000

4 40000 160000

2 35000 70000

1 30000 30000

Total number of molecules Ni

Total weight of the sample NiMi

123 8095000

g/mol81365123

8095000nM

Number average = 65 813 g/mol

Weight average = 69 145 g/mol Polydispersity index PDI = Mw/Mn = 1.05

The weight average molecular weightNumber of Molecules,

Mass of Each Molecule,

Total Mass of Each Type of Molecule,

Weight Fraction Type of Molecule

Ni Mi NiMi wi=(NiMi/NiMi) (wiMi)

1 120000 120000 0.015 1779

2 100000 200000 0.025 2471

3 95000 285000 0.035 3345

4 90000 360000 0.044 4002

5 85000 425000 0.053 4463

8 80000 640000 0.079 6325

10 75000 750000 0.093 6949

17 70000 1.19E6 0.147 10290

25 65000 1.625E6 0.201 13048

17 60000 1.02E6 0.126 7560

10 55000 550000 0.068 3737

8 50000 400000 0.049 2471

6 45000 270000 0.033 1501

4 40000 160000 0.020 791

2 35000 70000 0.009 303

1 30000 30000 0.004 111

Total number of molecules

Ni

Total weight of the sample NiMi

Weight average molecular weight

wiMi

123 8095000 69145

Blend: 1 g Monomer, M1 = 100 g/mol9 g Polymer, M1 = 100 000 g/mol

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/100 000g/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)

Example 3

What are Mn, Mw, and Mz ?

ii

ii

iAi

iiA

i

i

ii

iii

n n

m

Nn

MNMm

N

MNM

molgmolmol

gM n /991

10*910

1052

ii

iii

iii

iii

w W

MW

MN

MNM

2

molgg

molg

molg

M w /9001010

100000*9100*1

iii

iii

iii

iii

z MW

MW

MN

MNM

2

2

3

molg

molg

gmol

gg

molg

gmol

gg

M z /99989100000*9100*1

)100000(*9)100(*1 22

If all chains are equal in length

In general

Polydispersity index PDI

0.1n

w

M

M

1002 n

w

M

M

How to measure Mn

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.

The osmotic pressure is determined from the height difference h as

where is the solvent density and g is the gravitational acceleration.

gh

There is a free energy gain in mixing polymer with solvent that makes more solvent to flow into the polymer solution.

In equilibrium state, for dilute polymer solutions (similar to ideal gas) is the thermal energy kT times the number density of chains cNA/M.

M

RT

M

NkT

cA

c

0

limThe 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.

nc M

RT

c

0

lim

Osmotic pressure p depends on the molecular weight as follows:

Osmotic pressure, Mn

nc M

RT

c

0

lim

nRTPV Ideal gas law:

n is in moles. n/V is equal to c/M RTM

cP

Setting the gas pressure equal to the osmotic pressure P = p

nM

RT

c

Notice the analogy with ideal gas law:

Measuring of Mn by Osmotic PressureTo 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].

nc M

RT

c

0

lim

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.

...

...

22cA

M

cRT

ccAM

cRT

n

i jjiij

n

i j

jiiji j

jiij wwAccAc

A22

1

...1

2 cAMcRT n

At higher concentrations, the higher-order terms have to be taken into account.

Viscosity average molecular weight

0 relRelative viscosity

Specific viscosity 1 relsp

Specific viscosity, divided by the consentration and extrapolated to zero consentration, yields the intrinsic viscosity

0c

sp

c

h0 is the viscosity of solvent and h is the viscosity of the polymer solution

Viscosity average molecular weight

Mark-Houwink relationship aVKM

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

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

Gel permeation chromatography (GPC)

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.

maF

Mass spectrometry: MALDI, TOFMolecular weight distripution, absolute method

1. MAtrix-assisted Laser Desorption Ionization (MALDI), a soft ionization technique for transfering large molecular ions into a mass spectrometer with minimum fragmentation.

2. Time of flight (TOF)technique

Light Scattering

How a Light Scattering Setup Looks Like? 

Laser

Goniometer

Sample

Detector

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 (10-6 - 10-7 s)

I

E0 incidentE

scattered

Lightsource

Sample

Detector

G2(t)=<I(t) I(t+n)>

<I>

I(t)

time

I(t+)I(t+4)

I(t+5)I(t+3)

I(t+2)I(t)

43

2

time scale> 1 sec

 Molar mass Mw

Radius of gyration <Rg2>½

Second virial coefficient A2

Static Light Scattering Dynamic Light Scattering

time scale0.1 sec < t < 1 sec

 Hydrodynamic radius Rh

Diffusion coefficient Drelax Rh 1 / Dtrans

Static and Dynamic Light Scattering

Sperling book..

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 moleculeAs 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

(see PDF file)

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

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.

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.

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

Sperling

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.

Sperling book..

Sperling book..

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