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Questions

1. Derive the Poiseuille equation and discuss if the placement of the viscometer (parallel orperpendicular) affects the accuracy of the measurements

To derive the Poiseuille equation, consider a solid cylinder of fluid or radius r inside a

hollow cylindrical pipe of radius R

The force driving fluid flow is Fp(results from pressure):

= (2)The viscous drag also exerts a force Fv:

=(2) At equilibrium and constant speed, the net force should be 0:

+ = 02=(2)

The second equation above can be rearranged to get

=

2(2)=

2

Because of friction from the walls of the cylindrical tube, we know that fluid flow should

be fastest at the center and slowest at the edges. Empirically, this is the case:

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Using these boundary conditions, we can integrate the differential equation to get:

0

=2

()= ( 4)(2 2)=> parabolicUsing the equation of continuity,

==

4 (2 2)(2)= (

2

0

) (2 3) 0

=2

42 4

4= 4

8 Integrating and rearranging, we obtain the Poiseuille equation:

= 4t

8

As can be seen on the equation above, viscosity depends on the pressure difference P

between the two ends of the tube, and that this pressure difference is the gravitational

force acting on the chunk of liquid with height h and density . Therefore, placing the

viscometer in any position other than perpendicular to the lab bench will affect the value

of by making it smaller (P on the fluid segment is largest when the viscometer isperpendicular to the lab bench and 0 when the viscometer is parallel to the lab bench)

Reference: Fundamentals of Physics, Halliday, 11e

= 0 => = 0 => = (center)

==> v=0 (edge)

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2. Discuss the methods used to determine the molar masses of polymers and the operationalprinciples behind them.

The molecular weight of polymers can be characterized using GPC (gel permeation

chromatography), LALS (low angle light scattering), vapor phase osmometry, membrane

osmometry, and by measuring the intrinsic viscosity.

a) GPC is a type of size exclusion chromatography that separates analytes on the basisof size or hydrodynamic radius by using porous beads packed in a column; smaller

analytes enter the pores more easily and therefore have larger retention times while

larger analytes are eluted more quickly. It can be used to determine the relative

molecular weight of polymer samples as well as the distribution of molecular weights

by measuring the molecular volume and shape function defined by its intrinsic

viscosity => Mn, Mw,Mz

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To relate the elution volume Vel to M, a calibration curve for a given column is prepared

by measuring the elution volume for a series of monodisperse polymer standards of

known molecular mass

polymers having the same hydrodynamic volume elute at the same time and have thesame elution volume

The signal intensity hiis proportional to the polymer concentration ciin an aliquot of

volume Vel

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Reference: Fundamentals of Polymer Science, Painter, Coleman

b) LALS is used in conjuction with SEC (size exclusion chromatography) to study themolecular mass distribution of a polymer. The eluent of the SEC column is allowed

to pass through both a refractive index detector, which measures the concentration of

the solution as a function of time, and through a laser scattering cell, which measures

the scattered intensity as a function of time under a small angle with respect to the

laser beam. The laser signal and the concentration are then translated into a curve that

yields both Mnand Mw

Reference: The application of size exclusion chromatography equipped with RI and

LALLS detectors to study network formation, Polymer Bulletin, 27, 281-287,

Petrovic, MacKnight.

c) Vapor phase osmometry is a relative technique used to determine Mnof a polymer ina dilute solution that is based on the principle that the vapor pressure of a solution is

lower than that of the pure solvent at the same temperature and pressure. It uses the

fact that the magnitude of the vapor pressure decrease is directly proportional to the

= 1/(

)

= = (+1 )1/

=

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molar concentration of the solute and measures the change in voltage (which is

proportional to the change in temperature)

A calibration curve is used to determine the calibration factor K:

= 22

The molecular weight is then calculated using:

2= 2

the wick provides a vapor-saturated atmosphere

syringes are used to inject thestandard and the sample

thermistors detect changes intemperature

voltage changes are recorded system at an equilibrated

temperature

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Reference: Characterization And Analysis of Polymers, Wiley&Sons, 2008

d) Membrane osmometry is an absolute technique used to determine Mnby measuringthe osmotic pressure of a solution separated from pure solvent by a semipermeable

membrane

0 =1=( 212+ 22 12 1 22)

=chemical potentialn2= partial specific volume of polymer

c= interaction parameter

c2= solute concentration

M2= molecular mass of the polymer

Therefore,

2

=

2

+22

1

(1

2 1)2

The first term is the vant Hoff equation for osmotic pressure at infinite dilution while the

second term is the deviation from ideal behavior of the polymer solution and is related to

the second virial coefficient. The osmotic behavior of the polymer solution is govered by

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the ideal solution law when the interaction parameter is and when the second virial

coefficient is 0.

Reference: Modern Methods of Polymer Characterization, Wiley & Sons, 1991, pg.208-

222

3. Does the temperature need to be constant during the density and viscosity measurements?Why or why not?

The temperature needs to be constant during the viscosity measurements because viscosity

generally decreases with increasing temperature and increases with decreasing temperature:

As can be seen in the graph above, the viscosity of water changes dramatically with temperature.

The density of liquids (g/mL) is much less sensitive to variations in temperatures because liquids

are incompressible and their volumes do not change significantly with temperature (unlike gases).

It Therefore, it is crucial to maintain constant temperature during the viscosity measurements.

Reference: Dortmund Data Bank

4. How does solvent affect polymers? Point out the effect of solvent to a in the Mark-Houwink equation.

The properties of polymers are characterized by the interaction between the solvent and the

polymer (in dilute solution). In a good solvent, intermolecular forces between the solvent and

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the monomer subunits dominate over the intramolecular interactions, causing the polymer

appears swollen and occupies a large volume. In a bad solvent, intramolecular forces

dominate and the chain contracts. The theta solvent is a solvent in which the second virial

coefficient is 0, allowing the intermolecular polymer-solvent repulsion to exactly balance the

intramolecular monomer-monomer attraction. In theta solvents, the polymer behaves like an

ideal random coil. The Mark-Houwink equation relates the intrinsic viscosity [] and the

molecular weight M:

[]= KMa Both a and K depend on the particular polymer-solvent system. The value of a indicates the

rigidity of the polymer in a given solvent. For a theta solvent, a=0.5. Most good solvents

yield a=0.8 and 0.5

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Possible Causes of Negative Error

1. Not controlling the temperature: keeping the temperature below 30 C causes positiveerror in the viscosity measurements since viscosity increases with increasing temperature

2. Not completely cooling down the polymer solution: since high temperatures decrease theviscosity values, this will cause negative error in the viscosity measurements.