1

VISCOSIMETRY - APPLICATION

Laboratory excersise 1: Determination of percentage of 1,2-diol bonds in

polyvinyl alcohol from limiting viscosity number1

Introduction

While the basic chemical structure of a synthetic polymer is usually well understood, many physical

properties depend on such characteristics as chain length, degree of chain branching, and molar

mass, which are not easy to specify exactly in terms of a molecular formula. Moreover the

macromolecules in a given sample are seldom uniform in chain length or molar mass (which for a

linear polymer is proportional to chain length); thus the nature of the distribution of molar masses is

another important characteristic. A polymer whose molecules are all of the same molar mass is said

to be monodisperse; a polymer in which the molar masses vary from molecule to molecule is said to

be polydisperse. Specimens that are approximately monodisperse can be prepared in some cases by

fractionating a polydisperse polymer; this fractionation is frequently done on the basis of solubility in

various solvent mixtures.

This experiment is concerned with the linear polymer polyvinyl alcohol 2CH CHOHn

(PVOH), which is prepared by methanolysis of the polyvinyl acetate (PVAc) obtained from the direct

polymerization of the monomer vinyl acetate 3 2CH COO CH=CH . As ordinarily prepared,

polyvinyl alcohol shows a negligible amount of branching of the chains. It is somewhat unusual

among synthetic polymers in that it is soluble in water. This makes polyvinyl alcohol commercially

important as a thickener, as a component in gums, and as a foaming agent in detergents.

A characteristic of interest in connection with PVOH and PVAc is the consistency of orientation of

monomer units along the chain. It is assumed in the formula 2CH CHXn

, where X denotes –

OH nebo –OOCCH3 group that all monomer units go together "head to tail", it means the groups

2CH a CHX are bonded. skupina. However, occasionally a monomer unit will join onto the

chain in a "head-to-head" fashion, yielding a chain of the form

2 2 2 2CH CHX CHX CH CHX CH CHX CHm m

The frequency of head-to-head linkage depends on the relative rates of the normal growth-step reaction α

k

R M R M (1.1)

and the abnormal reaction ,

k

R M R M (1.2)

where R is the growing polymer radical, the arrow representing the predominant monometer

orientation). The rates in tum must depend on the activation energies:

1 Shoemaker, Garland, Nibler, "Experiments in Physical Chemistry," McGraw-Hill Publishing Company, Toronto

using different experimental equipment; The book is at disposal in the Department of physiacal chemistry library

2

exp

exp

exp

a

a

a

EA

k RT ES

Ek RTA

RT

(1.3)

where a a aE E E is the additional thermal activation energy needed to produce abnormal

addition and S is the steric factor representing the ratio of the probabilities that the abnormally and

normally approaching monomer will not be prevented by steric or geometric obstruction from being

in a position to form a bond. If the (1.1) reaction is faster than the reaction (1.2), it is probable that

the activation energy of the first reaction aE is lower than the activation energy of the second

reaction aE . This hypothesis was confirmed by Flory and Leutner, when they found out that the

frequency of head-to-head linkages CHX CHX in PVAc increase with increasing

polymerization temperature. The values of steric factor S and difference of activation energies

a a aE E E determine wheter the formation of the “head-to-head linkages“ CHX CHX is

reduced by the steric effect or by thermal activation barrier.

In this experiment, the method of Flory and Leutner will be used to determine the fraction of head-

to-head attachments in a single sample of PVOH. The method depends on the fact that in PVOH a

head-to-head linkage is a 1,2-glycol structure, and 1,2-glycols can bespecifically and quantitatively

cleaved by periodic acid or periodate ion. Treatment of PVOH with periodate should therefore break

the chain into a number of fragments, bringing about a corresponding decrease in the effective

molar mass. All that is required is a measurement of the molar mass of a specimen of PVOH before

and after treatment with periodate.

Experimental determination of molar mass of polymers

For the determination of very high molar masses, freezing-point depressions, boilingpoint elevations,

and vapor-pressure lowerings are too small for accurate measurement. Osmotic pressures are of a

convenient order of magnitude, but measurements are timeconsuming. The technique to be used in

this experiment depends on the determination of the intrinsic viscosity of the polymer. However,

molar-mass determinations from osmotic pressures are valuable in calibrating the viscosity method.

Two other methods, light scattering and gel permeation chromatography (GPC), could also be used

as alternatives or in addition to the viscosity technique. Light scattering is a classical method for

determining molar masses, and if scattering as a function of angle is measured, it is also possible to

learn something of the polymer shape, i.e., to distinguish rodlike from spherical forms. In the case of

gel permeation chromatography, molecules of different sizes are carried by a liquid solvent over a

substrate gel network of uniformly sized pores, which can be chosen in a range from 10 to 100,000

nm. Smaller solute molecules diffuse into and out of the pores and hence exhibit a longer retention

time than larger solute molecules, which are not retained and travel through the column with the

solvent. This method is simple, but to obtain accurate molar masses of a given polymer type from the

retention times it is necessary to reproduce the flow rates carefully and to calibrate the column with

polymers of known molar mass and similar chemical composition.

As it was mentioned above, molar mass will be determined by viscosity measurement in this work.

Viscosity will be mesured by the Stabinger SVM 3000 instrument from Anton Paar company, which is

3

a special rotation viscometer. The rotation viscometer is, generally, based on the fact that there is a

function dependence of torque, which develops at constant rotation of a body in liquid, on viscosity

of this liquid. Stabinger SVM3000 differs from standard rotational viscometers by a patented feature:

hollow cylinder freely floats in liquid and is centred in the cell during rotation of outer jacket by

centrifugal forces. This arrangment eliminates unwanted influence of mechanical friction between

mechanical parts of normal rotational viscometers, where rotating body is connected directly to the

motor.

Theory of evaluation of molar mass from viscosity measurement

Einstein showed that viscosity of a fluid in which small rigid spheres are present in dilute and

uniform suspension is related to the viscosity0 of the pure fluid (solvent) by the expression

0

51

2

v

V

(1.4)

where v is the volume occupied by all the spheres and V is the total volume. The left side of the

equation (1.4) is denoted as a increment of relative viscosity i (specific viscosity sp ). For

nonspherical particles the numerical coefficient on the right side of equation (1.4) is higher than 5 2 ,

but should be still a constant for any given shape provided the rates of shear are sufficiently low to

avoid preferential orientation of the particles.

Limiting viscosity number (formerly intrinsic viscosity) is defined as the ratio of the specific

viscosity to the weight concentration of solute, in the limit of zero concentration

2 20 0

2 2 0

1lim lim lni

w ww w

(1.5)

where 2w is conventionally defined as the concentration in grams of solute per 100 mL of

solution/suspension. Both 2i w and 2 01 lnw show a reasonably linear concentration dependence at

low concentrations 2w . The dependence 2 01 lnw vs.

2w usually has small negative slope, while

2i w vs. 2w plot has larger positive slope. Therefore both plots can be used for the extrapolation to

obtain , although 2 01 lnw for a single dilute solution will give a fair approximation to .

If the internal density of the spherical particles (polymer molecules) is independent of their size (i.e.,

the volume of the molecule is proportional to its molar mass), the intrinsic viscosity should be

independent of the size of the particles and hence of no value in indicating the molar mass. This,

however, is not the case; to see why we must look into the nature of a polymer macromolecule as it

exists in solution.

Gauss (statistical) polymer coil

A polymer such as PVOH contains many single bonds, around which rotation is possible. If the

configurations around successive carbon atoms are independent and unrelated, it will be seen that

two parts of the polymer chain more than a few carbon atoms apart are essentially uncorrelated in

regard to direction in space. The molecule is then "statistically coiled" and resembles a loose tangle

of yarn (see Fig.1):

4

Figure 1. – a scheme of Gauss (statistical) polymer coil

Simple statistical treatments show that the mean distance between the two ends of the chain, and

indeed also the effective mean diameter d of the coiled molecule regarded as a rough sphere, should

be proportional to the square root of the chain length and thus to the square root of the molar mass:

1 2d M (1.6)

The volume mv occupied by a molecule should then vary as 3 2M . The number of molecules in a

given weight of polymer varies inversely with the molar mass M; hence the total volumeof the spheres is

3 2 1 2cVv M cVM

M (1.7)

Two equations (1.5) and (1.7) give

1 2KM (1.8)

where K is a constant. This treatment is much simplified; it ignores among other things the problem

of "excluded volume"; that is, the fact that the chain cannot coil altogether randomly because it is

subject to the restriction that no two parts of the chain may be at the same point in space at the

same time. This restriction becomes more and more important the higher the molar mass. Even

more serious is the effect of solvent; the above treatment tacitly assumes a "poor solvent," which

would barely get the polymer into solution. A "good solvent," by solvating the polymer, makes the

size of the statistical coil increase faster with chain length than it otherwise would (equation (1.7)),

owing to enhancement of the excluded volume effect. Accordingly, instead of equation (1.8) we

might write

aKM (1.9)

where K and a are empirical parameters characteristic both of the polymer itself and of the

solvent. The exponent a varies from about 0.5, for well-coiled polymer molecules in a poor solvent,

to as much as 1.7 for a rigidly extended "rodlike" polymer molecule.

Flory and Leutner,l working with monodisperse specimens of PVOH differing from one another in

molar mass over a wide range (obtained by fractionating polydisperse commercial PVOH),

established a correlation between the molar mass, as determined from osmotic pressure

measurements, and the increment of relative viscosity i . They found that for PVOH in aqueous

solution at 25°C the values of K a a parameters from equation (1.9) are

4 0,762,0 10 M (1.10)

and if molar mass is expressed then

5

1,3247,6 10M (1.11)

where molar mass M is in the g/mol unit. molární hmotnost je v g/mol. The equation (1.11) also

holds for a polydisperse sample of PVOH, but the'molar mass in this case is M , the viscosity-

average molar mass defined below.

Number-Average Molar MassNM and Viscosity-Average Molar Mass M

For a polydisperse polymer, any determination of molar mass must yield an average of some sort.

When a colligative property such as osmotic pressure is used, the average is a number average NM 2

0

0

N

MP M dM

M

P M dM

(1.12)

where P M is the molar-mass distribution function; that is, P M dM is proportional to the

number of molecules with molar masses M and M dM . Viscosity-Average Molar Mass M

obtained from the value of limiting viscosity number is given by the equation3

1

0

0

aM P M dM

M

MP M dM

(1.13)

and its value is different from the NM value in the case of polydisperse polymer (certainly except for

the case of monodisperse polymer, where values of molar mass determined by osmotic pressure and

viscosity measurement, respectivally, are the same ). The ratio NM M is a constant given by a

form of the distribution function P M and by value of the parameter a . A commonly encountered

distribution function, and one that is likely to be valid for PVOH, is one that arises if the probability of

a chain-termination reaction during the polymerization is constant with time and independent of the

chain length already achieved. It is also the most likely function for the product resulting from

cleavage with periodate if the head-to-head structures CHX CHX can be assumed to be

randomly distributed along the PVOH chains. This distribution function is

2 If there is a countable number of monodisperse fractions, then N i i

i

M x M , where ix je molar fraction

of i-th fraction and iM is its molar mass

3 If there is a countable number of monodisperse fractions, then

1 a

a

i i

i

M W M

, where iW is mass

fraction of i-th fraction and iM is its molar mass

6

1

NM M

N

P M eM

(1.14)

If this distribution function is used in the equations (1.12) a (1.13), the folowing relation can be

derived:

1

1 1a

NM M a a (1.15)

where is the gamma function (Euler integrál of the second kind). For 0,76a (the value given

above for polyvinyl alcohol) we obtain

1,89NM M (1.16)

For the sake of completeness it can be added that so called mass-average molar mass WM can be

defined WM 4

2

0

0

W

M P M dM

M

MP M dM

(1.17)

Series of different molar mass values for one sample is as follows:

pro 0,5 1

pro 1

N W

N W

M M M a

M M M a

(1.18)

Determination of Frequency of Head-to-Head CHX CHX Bonds in a PVOH Sample

This determination is based on the assumption that degradation arises exclusively from cleavage of

1,2-glycol structures and that all such structures are cleaved. The relative frequency of head-to-

head CHX CHX bonds in a PVOH sample is equal to the increase in the number of molecules

present in the system, divided by the total number of monomer units represented by all molecules in

the system:

0

0 0

M C C

CM CM

N N N

N N

(1.19)

MN is a number of reversly head-to-head oriented monomer units, 0

CMN is a total number of all

monomer units in the whole system, 0

CN is a total number of all polymer particles in an original

4 If there is a countable number of monodisperse fractions, then W i i

i

M W M , where iW is mass fraction

of i-th fraction and iM is its molar mass

7

sample and CN is a total number of all polymer particles in the sample after cleaving of the 1,2-

glycol structures. Because numbers of particles in the relation (1.19) are in inverse proportion to the

respective molar masses, it is possible to write

0

1 1

1

N NM M

M

(1.20)

where aN NM M jsou number average molar masses before and after degradation (cleaving of the

1,2-glycol bonds), respectivally, and 0M is molar mass of the monomer unit, so 44 g/mol in the case

of PVOH. Therefore

44 1 1N NM M (1.21)

and using the equation (1.16) we get the relation with viscosity-average molar masses

83 1 1M M (1.22)

Experimental setup

1. Preparation of the stock solutions. The preparation of PVOH solutions is time-consuming,

because it is necessary to dissolve PVOH in hot water and to stir gently to avoid a formation of

foam. Therefore it is better to prepare one stock solution with the highest concentration and

other solutions with lower concentrations will be prepared by a dilution of this stock solution.

The stock solution of the original sample of PVOH is prepared by dissolution of ca. 4.2 g of the

polymer (the mass is determined using an analytical balance with accuracy 0.1 mg) in ca. 200 ml

of hot distilled water using gentle stiring to minimize amount of foam. After dissolving and cooling

down the solution is poured quantitatively into 250 ml volumetric flask (it is necessary to wash

out the rest of the solution from the original bottle into the volumetric flask by pure distilled

water). Then pure distilled water is added up to the mark of the volumetric flask and the solution

is slowly mixed. The stock solution of the degradated PVOH is prepared in a similar way. Again ca.

4.2 g of the original polymer (the mass is determined using an analytical balance with accuracy

0.1 mg) is dissolved in ca. 200 ml of hot distilled water using gentle stiring to minimize amount of

foam. After dissolving and cooling down, 1.25 g of solid KIO4 is added into the solution and it is

warmed up to 70°C and stirred until the periodate is dissolved. Then the solution is cooled down

and is poured quantitatively into 250 ml volumetric flask (again, it is necessary to wash out the

rest of the solution from the original bottle into the volumetric flask by pure distilled water). Then

pure distilled water is added up to the mark of the volumetric flask and the solution is slowly

mixed. The prepared stock solutions of the original and degradated PVOH, resp. have mass

concentration around 16 g/l (the accurate value of concentration is calculated).

2. Preparation of diluted solutions. 50 and 25 ml of the stock solutions are pipetted into 100 ml

volumetric flasks and refilled by pure distiled water up to the mark. Particular mass

concentrations 2w (unit g/l) are to be calculated.

3. Measurement of viscosity. Measurement of dynamic viscosity is performed on Stabinger SVM

3000 at temperature 25°C according to following procedure

8

a) First, the button MODE on the front panel of the apparatus (see fig. 2) is pressed and the

procedure M6-REPEAT S is chosen by arrows buttons (see fig. 2 SET/MODE). The choice is

confirmed by ENTER. The chosen procedure is based on a performance of three subsequent

measurements, when an operator is asked automatically to refill the cells by new amount of

the sample. The final result is an arithmetic average of these three values. The apparatus

automatically checks, if a value of standard deviation is within a defined range. If the value of

standard deviation is out of the range, the operator is asked to repeat measurement. This

procedure (MODE) minimizes problems with insifficiently washed cells.

Fig.2: Keyboard on the front panel of the SVM3000. MENU – main menu; PRINT-

saving or printing of results; SET- set of temperature value and sample info; MODE- choice of the apparatus mode – it determines experimental procedure; START – button ti start a measurement (repeated, too), or button to switch on/off a motor durinf filling of the cells; PUMP – air pump for drying of the cell to be switched on/off; INFO- opens further screans with further information

b) The menu for writing of sample name and defining temperature of measurement is brought

about by pressing of the button SET (fig.2). The external keybord or the keybord from the

front panel (see fig. 2) is used for listing of menu items. The chosen item is confirmed by key

ENTER.

c) Make sure that the motor of the outer rotor is switched off (if it is not true, it is necessary to

switch the rotor off by pressing of the button START (see fig. 2). The liquid sample is slowly

sucked into 5 ml syrringe without bubbles (if there are bubbles in the syrringe, they should

be removed by turning the surringe over and pressing the piston). After that the syrringe is

put into the port D1 (see fig.3) and at least 2 ml of the liquid sample is inserted into the

apparatus by gentle pressing5 of the syrringe piston. Then the motor of the rotor is switched

5 The power for pushing should not excess ca. 20 N. It corresponds to weight of the 2 kg body.

9

on for ca. 10 seconds by pressing of the button START on the front panel (see fig. 2) to

ensure optimal filling of the cells. After ca. 10 seconds the motor is switched off by pressing

START button and further 1 ml of the sample is inserted into the apparatus (liquid should

appeared in the waste tube – see fig. 3). Then measurement can be started.

Fig.3: Scheme of the inject part of the Stabinger SVM3000.

d) The real measurement is started by further pressing of the START button and it is waited

until the apparatus obtains the first value of viscosity, switches off the motor and asks the

operator to push further 1 ml of liquid. After pushing of the sample the START button is

pressed to start the second measurement and it is. Then the third measurement is

performed in the same way as previous measurements. If there is no problem, the apparatus

calculates the final value as an arithmetic average from the three values and the operator

saves the result by the button PRINT (fig.2). If the standard deviation is out of the defined

range, measurement has to be repeated after pushing further amount of the sample into the

apparatus.

4. Data treatment. The values of limiting viscosity numbers both for original and degradated

polymer are obtained by extrapolation of the 2i w vs. 2w dependencies. After that the viscosity

average molar masses aM M

are obtained

from

values according to the equation (1.11).

Finally the value of the relative frequency of head-to-head CHX CHX bonds in a PVOH

sample is calculated from the equation (1.22) and the relation this value to the rate constants

values k a k in the equation (1.3) is discussed.

10

Laboratory excersise 2: Finding of numerical values of parameters in equation describing correlation between molar mass and limiting viscosity number (intrisnic viscosity) in the case of polyvinyl alcohol Introduction

While the basic chemical structure of a synthetic polymer is usually well understood, many physical

properties depend on such characteristics as chain length, degree of chain branching, and molar

mass, which are not easy to specify exactly in terms of a molecular formula. Moreover the

macromolecules in a given sample are seldom uniform in chain length or molar mass (which for a

linear polymer is proportional to chain length); thus the nature of the distribution of molar masses is

another important characteristic. A polymer whose molecules are all of the same molar mass is said

to be monodisperse; a polymer in which the molar masses vary from molecule to molecule is said to

be polydisperse. Specimens that are approximately monodisperse can be prepared in some cases by

fractionating a polydisperse polymer; this fractionation is frequently done on the basis of solubility in

various solvent mixtures.

This experiment is concerned with the linear polymer polyvinyl alcohol 2CH CHOHn

(PVOH), which is prepared by methanolysis of the polyvinyl acetate (PVAc) obtained from the direct

polymerization of the monomer vinyl acetate 3 2CH COO CH=CH . As ordinarily prepared,

polyvinyl alcohol shows a negligible amount of branching of the chains. It is somewhat unusual

among synthetic polymers in that it is soluble in water. This makes polyvinyl alcohol commercially

important as a thickener, as a component in gums, and as a foaming agent in detergents.

Experimental determination of molar mass of polymers

For the determination of very high molar masses, freezing-point depressions, boilingpoint elevations,

and vapor-pressure lowerings are too small for accurate measurement. Osmotic pressures are of a

convenient order of magnitude, but measurements are timeconsuming. The technique to be used in

this experiment depends on the determination of the intrinsic viscosity of the polymer. However,

molar-mass determinations from osmotic pressures are valuable in calibrating the viscosity method.

Two other methods, light scattering and gel permeation chromatography (GPC), could also be used

as alternatives or in addition to the viscosity technique. Light scattering is a classical method for

determining molar masses, and if scattering as a function of angle is measured, it is also possible to

learn something of the polymer shape, i.e., to distinguish rodlike from spherical forms. In the case of

gel permeation chromatography, molecules of different sizes are carried by a liquid solvent over a

substrate gel network of uniformly sized pores, which can be chosen in a range from 10 to 100,000

nm. Smaller solute molecules diffuse into and out of the pores and hence exhibit a longer retention

time than larger solute molecules, which are not retained and travel through the column with the

solvent. This method is simple, but to obtain accurate molar masses of a given polymer type from the

retention times it is necessary to reproduce the flow rates carefully and to calibrate the column with

polymers of known molar mass and similar chemical composition.

As it was mentioned above, molar mass will be determined by viscosity measurement in this work.

Viscosity will be mesured by the Stabinger SVM 3000 instrument from Anton Paar company, which is

11

a special rotation viscometer. The rotation viscometer is, generally, based on the fact that there is a

function dependence of torque, which develops at constant rotation of a body in liquid, on viscosity

of this liquid. Stabinger SVM3000 differs from standard rotational viscometers by a patented feature:

hollow cylinder freely floats in liquid and is centred in the cell during rotation of outer jacket by

centrifugal forces. This arrangment eliminates unwanted influence of mechanical friction between

mechanical parts of normal rotational viscometers, where rotating body is connected directly to the

motor.

Theory of evaluation of molar mass from viscosity measurement

Einstein showed that viscosity of a fluid in which small rigid spheres are present in dilute and

uniform suspension is related to the viscosity0 of the pure fluid (solvent) by the expression

0

51

2

v

V

(2.1)

where v is the volume occupied by all the spheres and V is the total volume. The left side of the

equation (2.1) is denoted as a increment of relative viscosity i (specific viscosity sp ). For

nonspherical particles the numerical coefficient on the right side of equation (2.1) is higher than 5 2 ,

but should be still a constant for any given shape provided the rates of shear are sufficiently low to

avoid preferential orientation of the particles.

Limiting viscosity number (formerly intrinsic viscosity) is defined as the ratio of the specific

viscosity to the weight concentration of solute, in the limit of zero concentration

2 20 0

2 2 0

1lim lim lni

w ww w

(2.2)

where 2w is conventionally defined as the concentration in grams of solute per 100 mL of

solution/suspension. Both 2i w and 2 01 lnw show a reasonably linear concentration dependence at

low concentrations 2w . The dependence 2 01 lnw vs.

2w usually has small negative slope, while

2i w vs. 2w plot has larger positive slope. Therefore both plots can be used for the extrapolation to

obtain , although 2 01 lnw for a single dilute solution will give a fair approximation to .

If the internal density of the spherical particles (polymer molecules) is independent of their size (i.e.,

the volume of the molecule is proportional to its molar mass), the intrinsic viscosity should be

independent of the size of the particles and hence of no value in indicating the molar mass. This,

however, is not the case; to see why we must look into the nature of a polymer macromolecule as it

exists in solution.

Gauss (statistical) polymer coil

A polymer such as PVOH contains many single bonds, around which rotation is possible. If the

configurations around successive carbon atoms are independent and unrelated, it will be seen that

two parts of the polymer chain more than a few carbon atoms apart are essentially uncorrelated in

regard to direction in space. The molecule is then "statistically coiled" and resembles a loose tangle

of yarn (see Fig.1):

12

Figure 1. – a scheme of Gauss (statistical) polymer coil

Simple statistical treatments2 show that the mean distance between the two ends of the chain, and

indeed also the effective mean diameter d of the coiled molecule regarded as a rough sphere, should

be proportional to the square root of the chain length and thus to the square root of the molar mass:

1 2d M (2.3)

The volume mv occupied by a molecule should then vary as 3 2M . The number of molecules in a

given weight of polymer varies inversely with the molar mass M; hence the total volumeof the spheres is

3 2 1 2cVv M cVM

M (2.4)

Two equations (2.2) and (2.4) give

1 2KM (2.5)

where K is a constant. This treatment is much simplified; it ignores among other things the problem

of "excluded volume"; that is, the fact that the chain cannot coil altogether randomly because it is

subject to the restriction that no two parts of the chain may be at the same point in space at the

same time. This restriction becomes more and more important the higher the molar mass. Even

more serious is the effect of solvent; the above treatment tacitly assumes a "poor solvent," which

would barely get the polymer into solution. A "good solvent," by solvating the polymer, makes the

size of the statistical coil increase faster with chain length than it otherwise would (equation (2.4)),

owing to enhancement of the excluded volume effect. Accordingly, instead of equation (2.5) we

might write

aKM (2.6)

where K and a are empirical parameters characteristic both of the polymer itself and of the

solvent. The exponent a varies from about 0.5, for well-coiled polymer molecules in a poor solvent,

to as much as 1.7 for a rigidly extended "rodlike" polymer molecule.

The equaton (2.6) is valid for polydispersed polymers, too. In this case, viscosity-average molar mass

M , whis is defined below, is placed in the equation (2.6).

13

Number-Average Molar MassNM and Viscosity-Average Molar Mass M

For a polydisperse polymer, any determination of molar mass must yield an average of some sort.

When a colligative property such as osmotic pressure is used, the average is a number average NM 6

0

0

N

MP M dM

M

P M dM

(2.7)

where P M is the molar-mass distribution function; that is, P M dM is proportional to the

number of molecules with molar masses M and M dM . Viscosity-Average Molar Mass M

obtained from the value of limiting viscosity number is given by the equation7

1

0

0

aM P M dM

M

MP M dM

(2.8)

and its value is different from the NM value in the case of polydisperse polymer (certainly except for

the case of monodisperse polymer, where values of molar mass determined by osmotic pressure and

viscosity measurement, respectivally, are the same ). The ratio NM M is a constant given by a

form of the distribution function P M and by value of the parameter a . A commonly encountered

distribution function, and one that is likely to be valid for PVOH, is one that arises if the probability of

a chain-termination reaction during the polymerization is constant with time and independent of the

chain length already achieved. It is also the most likely function for the product resulting from

cleavage with periodate if the head-to-head structures CHX CHX can be assumed to be

randomly distributed along the PVOH chains. This distribution function is

1

NM M

N

P M eM

(2.9)

6 If there is a countable number of monodisperse fractions, then N i i

i

M x M , where ix je molar fraction

of i-th fraction and iM is its molar mass

7 If there is a countable number of monodisperse fractions, then

1 a

a

i i

i

M W M

, where iW is mass

fraction of i-th fraction and iM is its molar mass

14

If this distribution function is used in the equations (2.7) and (2.8), the folowing relation can be

derived:

1

1 1a

NM M a a (2.10)

where is the gamma function (Euler integrál of the second kind).

For the sake of completeness it can be added that so called mass-average molar mass WM can be

defined WM 8

2

0

0

W

M P M dM

M

MP M dM

(2.11)

Series of different molar mass values for one sample is as follows:

pro 0,5 1

pro 1

N W

N W

M M M a

M M M a

(2.12)

Experimental setup

1. Preparation of the stock solutions. The preparation of PVOH solutions is time-consuming,

because it is necessary to dissolve PVOH in hot water and to stir gently to avoid a formation of

foam. Therefore it is better to prepare one stock solution with the highest concentration and

other solutions with lower concentrations will be prepared by a dilution of this stock solution.

The stock solution of the each sample of PVOH with known molar mass is prepared by dissolution

of ca. 4.2 g of the polymer (the mass is determined using an analytical balance with accuracy

0.1 mg) in ca. 200 ml of hot distilled water using gentle stiring to minimize amount of foam. After

dissolving and cooling down the solution is poured quantitatively into 250 ml volumetric flask (it is

necessary to wash out the rest of the solution from the original bottle into the volumetric flask by

pure distilled water). Then pure distilled water is added up to the mark of the volumetric flask and

the solution is slowly mixed. The prepared stock solutions of the PVOH have mass concentration

around 16 g/l (the accurate value of concentration is calculated).

2. Preparation of diluted solutions. 50 and 25 ml of the stock solutions are pipetted into 100 ml

volumetric flasks and refilled by pure distiled water up to the mark. Particular mass

concentrations 2w (unit g/l) are to be calculated.

8 If there is a countable number of monodisperse fractions, then W i i

i

M W M , where iW is mass fraction

of i-th fraction and iM is its molar mass

15

3. Measurement of viscosity. Measurement of dynamic viscosity is performed on Stabinger SVM

3000 at temperature 25°C according to following procedure

a) First, the button MODE on the front panel of the apparatus (see fig. 2) is pressed and the

procedure M6-REPEAT S is chosen by arrows buttons (see fig. 2 SET/MODE). The choice is

confirmed by ENTER. The chosen procedure is based on a performance of three subsequent

measurements, when an operator is asked automatically to refill the cells by new amount of

the sample. The final result is an arithmetic average of these three values. The apparatus

automatically checks, if a value of standard deviation is within a defined range. If the value of

standard deviation is out of the range, the operator is asked to repeat measurement. This

procedure (MODE) minimizes problems with insifficiently washed cells.

Fig.2: Keyboard on the front panel of the SVM3000. MENU – main menu; PRINT-

saving or printing of results; SET- set of temperature value and sample info; MODE- choice of the apparatus mode – it determines experimental procedure; START – button ti start a measurement (repeated, too), or button to switch on/off a motor durinf filling of the cells; PUMP – air pump for drying of the cell to be switched on/off; INFO- opens further screans with further information

b) The menu for writing of sample name and defining temperature of measurement is brought

about by pressing of the button SET (fig.2). The external keybord or the keybord from the

front panel (see fig. 2) is used for listing of menu items. The chosen item is confirmed by key

ENTER.

c) Make sure that the motor of the outer rotor is switched off (if it is not true, it is necessary to

switch the rotor off by pressing of the button START (see fig. 2). The liquid sample is slowly

sucked into 5 ml syrringe without bubbles (if there are bubbles in the syrringe, they should

be removed by turning the surringe over and pressing the piston). After that the syrringe is

put into the port D1 (see fig.3) and at least 2 ml of the liquid sample is inserted into the

16

apparatus by gentle pressing9 of the syrringe piston. Then the motor of the rotor is switched

on for ca. 10 seconds by pressing of the button START on the front panel (see fig. 2) to

ensure optimal filling of the cells. After ca. 10 seconds the motor is switched off by pressing

START button and further 1 ml of the sample is inserted into the apparatus (liquid should

appeared in the waste tube – see fig. 3). Then measurement can be started.

Fig.3: Scheme of the inject part of the Stabinger SVM3000.

d) The real measurement is started by further pressing of the START button and it is waited

until the apparatus obtains the first value of viscosity, switches off the motor and asks the

operator to push further 1 ml of liquid. After pushing of the sample the START button is

pressed to start the second measurement and it is. Then the third measurement is

performed in the same way as previous measurements. If there is no problem, the apparatus

calculates the final value as an arithmetic average from the three values and the operator

saves the result by the button PRINT (fig.2). If the standard deviation is out of the defined

range, measurement has to be repeated after pushing further amount of the sample into the

apparatus.

4. Data treatment. The values of limiting viscosity numbers are obtained by extrapolation of the

2i w vs. 2w dependencies for particular samples with known molar mass. The obtained

experimental data vs. molar mass is obtained for determination of parameter values of the

equation (2.6). E.g.the equation (2.6) can be rearranged to obtain linear dependence ( ln vs.

ln M ) and then the least squares method can be applied. Finaly the obtained results will be

9 The power for pushing should not excess ca. 20 N. It corresponds to weight of the 2 kg body.

17

compared to literature values of K and a parameters (a=0,76; K = 2,010-4), especially with regard

to prospective polydispersity of samples and the way, in which known values of molar masses

were obtained.

Literature

Shoemaker, Garland, Nibler, "Experiments in Physical Chemistry," McGraw-Hill Publishing Company, Toronto (1989).