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**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 /

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**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…

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**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

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**Molecular weight averages**

Number average Weight average z-average Viscosity average

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**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.

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**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

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Example 2 polymers

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**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.

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**The weight average molecular weight**

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

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**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)

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**Polydispersity index PDI**

If all chains are equal in length In general

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How to measure Mn

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**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.

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**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

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**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.

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**Viscosity average molecular weight**

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**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

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**Mark-Houwink relationship**

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

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**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

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**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.

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Light Scattering

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**How a Light Scattering Setup Looks Like? **

Laser Goniometer Sample Detector

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**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)

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**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

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Sperling book..

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(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

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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

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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.

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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.

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**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

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Sperling

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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.

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Sperling book..

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Sperling book..

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**Polymer processing injection moulding, extrusion, blow moulding...**

Extrusion & injection moulding blown film extrusion

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**Nobel prize winners in polymer science**

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