G RUNDMODUL IN MAKROMOLEKULARER C HEMIE 2:C HARAKTERISIERUNG VON P OLYMEREN

Skript zum Praktikum

(Lab course „macromolecular chemistry: physicalchemistry“)

Experiment 1:Experiment 2:Experiment 3:Experiment 4:Experiment 5:Experiment 6:Experiment 7:

Light ScatteringSize-exclusion Chromatography (SEC)OsmometryCD-SpectroscopyViscosity of Polymer SolutionsDynamic Mechanical CharacterizationX-ray Diffraction

1 - 1

EXPERIMENT 1: LIGHT SCATTERING

1 Introduction Light scattering occurs when polarizable particles in a sample are placed in the

oscillating electric field of a beam of light. The varying field induces oscillating

dipoles in the particles and these radiate light in all directions. This important and

universal phenomenon represents the basis to answer the following questions why is

the sky blue and why are fog and emulsions opaque. Light scattering has been

utilized in many areas of science to determine particle size, molecular weight, shape,

thermodynamic properties, diffusion coefficients etc.

2 Light Scattering

2.1 Static Light Scattering

In static light scattering the time average value of the scattered intensity is measured

as function of the scattering angle. This allows to determine the weight average of

the molar mass Mw , the z-average of the squared radius of gyration <R 2g >z and the

second virial coefficient of the osmotic pressure A2. Therefore historically, light

scattering is one of the most effective methods to determine molar mass and to

obtain size information of polymers or biopolymers – without reference to standards.

As mentioned above the oscillating electric field of light induces oscillating dipoles

within molecules and these therefore radiate light in all directions. The wavelength of

the scattered light is identical with the wavelength of the incident beam. Therefore

the trace of light within a strong scattering medium can be observed as a weak

shining beam (Tyndall effect). Tyndall explored this phenomena systematically in

1871 and he observed that this effect is much more pronounced using blue light as

compared to to red light.

Some years later Rayleigh started to explore light scattering. With the assumption of

disorderly distributed molecules in space he found – applying the Maxwell theory of

Light Scattering

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electrodynamics – that the so called Rayleigh ratio of scattered intensity and primary

beam intensity is given by:

(2-1) R( ) = 0

²IIr =

1

240

)²cos1(²8k

kkN

with R( ): Rayleigh ratio as a function of scattering angle I: Intensity of the scattered light I0: Intensity of the primary beam r: Distance of detector and scattering volume 0: Wavelength of the primary beam in vaccum Nk: Number of scattering centers k: Polarizability of the scattering center k : Angle between primary and scattered beam Using vertically polarized light of a laser the scattering intensity (of small particles) is

indepent from the scattering angle and the so called polarization term (1 + cos2 )

equals 2.

The measured scattering intensity can only be determined in relative units. Therefore

the absolute scattering intensity of several pure liquids were measured using special

experimental setups. These liquids (e.g. toluene or benzene) are utilized as

calibration standards and with their help the absolute scattering intensity of other

liquids and solutions are determined. For the measurement of the Rayleigh ratio of

any solution the following formula has to be applied:

(2-2) R( ) = dardsdards

solventsolution RRI

IItan

tan

With RRstandard : Absolute scattering intensity of the standard.

All specific parameters of the scattering apparatus (e.g. distance r of the detector,

size of the scattering volume, primary beam intensity of the laser) are therefore

eliminated.

Einstein and Smoluchowski developed the fluctuation theory. Here the fluctuation of

the polarizability in a liquid or a solution is described as a function of fluctuations of

the density and the fluctuation of the concentration due to thermal movements of the

molecules. Scattering can only occur if there are differences of the refractive index of

a small volume compared to its neighborhood.

Light Scattering

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For small particles we can write:

(2-3) R( ) =

dcdcMRT

dcdnnRT

ddnn

N L 0

02

0

2204

0

24

With NL : Avogardro constant , 0 : Density of the solution and solvent n, n0 : Refractive index of solution and solvent R: Universal gas constant T : Absolute temperature in Kelvin K : Isothermal compressibilty M0 : Molar mass of the solvent c : Concentration of the solute material : Difference of the chemical potential of solution and solvent

:dcdn Refractive index increment due to concentration changes

:ddn Refractive index increment due to density changes

In equation (2-3) there are two contributions for the Rayleigh ratio. The first term

describes the contribution of density fluctuations, the second term the contribution of

concentration fluctuations in the solution. For diluted solutions it can be assumed

that the contribution of the density fluctuations of solution and solvent are the same.

Therefore the scattering of the dissolved substance is given by:

(2-4) R( ) = R( )solution – R( )solvent =

dcd

cRTMdcdn

Nn

L0

02

40

20

24

The change of the chemical potential with the concentration can be described as a

change of the osmotic pressure with concentration:

(2-4) - dcdM

dcd

0

0

with : Osmotic pressure

Use of a series development of the osmotic pressure with respect to concentration

yields:

Light Scattering

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(2-6) .......321 232 cAcA

MRT

dcd

with A2, A3 : Virial coefficients of the osmotic pressure

Introducing equ. (2-5) and (2-6) in equ. (2-4) yields:

(2-7) ......321)(

232 cAcA

MRKc

with :4 2

40

20

2

dcdn

NnK

L

Optical constant

M : Molar mass of the dissoluted material

Equ. (2-7) is only valid for small particles which are randomly distributed in space

and therefore behave like a point dipole. For polymers with dimensions in the range

of the wavelength of the light applied (particles larger than /20) interference of the

scattered light occurs (fig. 2-1).

Fig 2-1: Interference of two primary beams scattered at the centers i and j of a large particle (larger than /20).

Light Scattering

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The occurrence of interference leeds to a weakening of the scattered intensity with

increasing scattering angle. For molecules of dimensions less than /2 therefore a

continuous decrease with increasing scattering angle is expected. With larger

dimensions minima of the scattering intensity as a function of the scattering angle

are observed. At the angle = 0° the path difference of the scattered elemental

waves is zero, therefore the scattered intensity of large particles is not influenced by

interference effects. Therefore extrapolating the scattered intensity to zero angle =0

allows us to interprete the result in terms of the Rayleigh theory. Measurements at

=0 are not possible because the primary beam intensity is much larger than the

scattered beam intensity (factor of 106).

Note that the effective scattering volume is also a function of the scattering angle. If

a sin( ) correction is conducted (Fig. 2-3) for small isotropic scattering molecules the

scattering intensity is constant.

Fig. 2-3: Changing of the effective scattering volume as a function of the scattering

angle .

For large particles the dependence of the scattered intensity is expressed by the

form factor P(q). For the calculation of P(q) it has to be considered that due to

thermal (Brownian) motion a particle adopts all possible orientations in space in a

very short time (less than 1 msec). Therefore an average value of all possible

orientations and distances are measured. Theoretically, the following expressions

are derived:

(2-8) N N

ijrqiN

qP1 1

2 exp1)(

with 2

sin4

0

nqq : Norm of the scattering vector q

Light Scattering

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N: Number of scattering centers within a particle

jiij rrr : Distance of scattering centers i and j

n: Refractive index of the solvent

Integration over all possible orientations from = 0 to 2 and = 0 to yields:

(2-9) N

i

N

j ij

ij

qrqr

NqP

sin1)( 2 with i < j

Formula (2-9) depends only on the distances of the scattering centers within a single

particle. For small values of the scattering vector q, the form factor P(q), which only

depends on shape and size of the particle, can be rewritten as a polynomial series:

(2-10) N

i

N

jijr

NqqP ....!3

1)( 22

2

(break off after the 2nd term)

The mean squared radius of gyration is defined by:

(2-11) N

i

N

i

N

jijig r

Nr

NR 2

222

211

with ir : Distance vector of scattering center i from the center of mass of the

particle

Application of equ. (2-11) and (2-10) yields for monodisperse particles:

(2-12) .....311)( 22

gRqqP

Polydispersity does not only influence the form factor but also the mean squared

radius of gyration:

(2-13)

iii

iigii

zg Mm

RMmR

2

2

with zgR 2 : z-average of the square of the radius of gyration

igR 2 : Square of the radius of gyration of particle i

mi : Mass of the particle i

Mi : Molar mass of the particle i

Therefore we can rewrite eq. (2-7):

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(2-14) ......32)(

1)(

232 cAcA

qPMRKc

zw

Thus the molar mass is a mass average value and the mean square of the radius of

gyration is a z-average.

For the most particle shapes the form factor decreases continously with increasing

scattering vector; only spheres and ellipsoidal particles show sharp and distinct

minima at larger angles. The form factor can be calculated for particles of different

shapes. By comparison of the experimental function with the theoretical prediction or

direct fitting of the experimental values the form factor of the particle and its

dimension can be determined. This especially holds for small angle x-ray scattering

(SAXS) and small angle neutron scattering (SANS). Fig. 2-4 is a sketch of the form

factor of a gaussian coil as a function of the scattering angle.

Fig. 2-4: Form factor P( ) of a Gaussian coil as a function of the scattering angle

Applying eq. (2-13) and (2-12) and using the relationships

,11

1 xx

or xx

11

1

we get the famous so called Zimm Equation (B.H. Zimm, J. Chem. Phys., 16, 1099

(1948)):

Light Scattering

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(2-15) .....23111

)( 222 cARq

MRKc

zgw

By plotting the data )(R

Kc as a function of kcq2 we can extrapolate for c 0 and

q2 0. From the intercept of both the c-dependence as well as the q² dependence we

can calculate the mass average of the molar mass Mw and from the slopes we get

the z-average of the mean square of the radius of gyration zgR 2 (from q2

dependence) and the second virial coefficient of the osmotic pressure A2 (from c

dependence).

Fig. 2-5: Extrapolation of light scattering data according to Zimm; : experimental values of Kc/R( ), : extrapolated values.

2.2 Dynamic Light Scattering

Whereas in static light scattering the time average of the scattering intensity is

measured in dynamic light scattering the fluctuations of the scattering intensity due

to Brownian motion of the particles are correlated by means of an intensity-time

autocorrelator. The correlator monitors the scattering intensities in small time

intervals over a total observation time t=n with n the number of time intervals .

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Typically n=100-500 and =1-1000 sec. The autocorrelation function g2(t) is than

calculated as

(2-16) )nt(I)t(I)t(g 02

where the brackets <….> denote an average over typically 106-108 single

correlations.

From g2(t) the correlation function of the electric field g1(t) is derived by

(2-17) 2

1

21 A

A)t(g)t(g

with A an experimentally determined baseline, i.e.

(2-18) 22 I)t(glimA

t

In this limit the intensities I(t=0) and I(t ) are not correlated, i.e.

(2-19) 200 I)t(I)t(I)t(I)t(I

Since

(2-20) 220

I)t(glimt

the intensity correlation function decays from <I2> to <I>2.

For scattering centers undergoing Brownian motion it can be shown that g1(t) is the

Fourier transformation of the van Hove space time correlation function

)t,R(G which expresses the probability that one particle moves a distance R

within a time interval t. For particles undergoing Brownian motion )t,R(G is given

by a Gaussian distribution of mean square displacements R2 :

(2-21) )t(R)t(Rexp)t(R)t,R(G

2

22

32

23

32

(2-22) Rd)Rqiexp()t,R(G)t,q(g1

(2-23) )tDqexp()tRexp()t(g 221 6 with )t(RD 26

The diffusion coefficient we measure is a z-average value and applying the Stokes-

Einstein equation we can calculate the hydrodynamic radius of a corresponding

sphere:

(2-24) zzh

h DkT

RR

061

Light Scattering

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with k Boltzmann factor, T temperature in K, 0 viscosity of the solvent.

2.3 Theta Temperature The virial coefficients are closely connected to the chemical potential and therefore

contain contributions of the enthalpy of dilution and the entropy of dilution as well.

According to thermodynamics both quantities can be determined from the

temperature dependency of the second virial coefficient A2. There exists a special

case, where the contribution of enthalpy and entropy compensate each other; then

A2 equals zero (but not A3 = 0 at the same time!) and the solution behaves pseudo ideal. In contrast a real ideal solution is characterized by the fact that both the

enthalpy and entropy of dilution get zero and not by the fact that these relatively

large quantities compensate each other. In some solvents we can realize this

pseudo ideal condition with A2 = 0 simply by changing the temperature. According to

Flory this temperature is called the theta temperature and the corresponding solvent

is called a theta solvent. The theta condition is of great importance in polymer

science due to the fact that the macromolecules adopt their unperturbed dimensions.

This not only holds in solution but also for polymers in the melt as shown by Kirste

using neutron scattering.

2.4 Molar Mass Dependence of the Radius of Gyration zgR 2

The value of the radius of gyration can be determined by scattering without

knowledge of the shape of the molecule. But for different molecular shapes there are

characteristic dependencies of the radius of gyration on the molar mass. For a wide

range of molar masses this dependency can be described by a scaling law:

(2-25) a

zg MKR 2

(2-26) MaKRzg logloglog 2

The exponent a only depends on the shape of the scattering particle:

a = 2 for rods,

a = 1.2 for a Gaussian coil in a good solvent,

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a = 1 for Gaussian coils with unperturbed dimensions (theta condition)

a = 2/3 for compact spheres.

The behaviour of Gaussian coils demonstrates that thermodynamic interactions are

changing their dimensions. The coils expand in a better solvent.

Light Scattering

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3 Experiment 3.1 Apparatus

For the measurement of the scattered light intensity as a function of the scattering

angle special experimental setups are applied. Due to their high light intensity, the

very low divergence of their beam and their high stability with respect to the spatial

position typically lasers are used in modern light scattering apparatus. This allows to

measure at low concentrations and to determine low molar masses (lower than

1000).

Fig. 3-1: Sketch of a light scattering apparatus (goniometer type) with incident beam I0 , 0 , scattered beam i , scattering angle and the distance r of the scattering

detector (photomultiplier tube) from the scattering volume . The detection of the scattered intensity is performed by a photomultiplier tube or an

Avalanche photodiode. In a goniometer setup the complete secondary detection

optics or fiber optics is rotated around the center of the apparatus. In a multi angle

light scattering apparatus (MALLS) the scattered light is measured simultaneously at

several angles (typically 10 – 20) with one detector per detection angle. In both,

goniometer and MALLS set up, the sample cuvette is placed in a toluene bath, which

minimizes reflections (refractive index matching to glass) and serves as a

temperature bath.

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3.2 Experiment and Data Evaluation

The weight average molar mass Mw, the second virial coefficient A2 and the radius of

gyration (z-average) 2/12

zgR of a polystyrene sample dissolved in toluene (or

cyclohexane) will be determined by laser light scattering. Therefore a solution of

polystyrene in toluene (or cyclohexane) of known concentration has to be prepared

using a measuring flask of 100ml at 20°C. From this solution 5, 10, 15, 20 and 25ml

are taken and diluted in measuring flasks of 25ml volume. To remove dust, which

considerably disturbs every light scattering measurement, these solutions of different

polymer concentration are then filtered through a 0.2 µm filter suitable for organic

solvents. Prior to use the sample cells (cylindrical quartz cuvette of 1 cm diameter)

are cleaned from dust using a special glass apparatus for condensing acetone.

Furthermore a cuvette of pure toluene is prepared also by filtration (0.2µm organic

filter). All filtration operations are conducted in a laminar flow cabinet to exclude dust

contamination. The outer side of every cuvette has to be cleaned with methyl ethyl

ketone before placing it into the toluene bath of the measuring cell.

The following measurements have to be performed: 1) Calibration of the primary

intensity and correction for the scattering volume by measuring pure toluene 2) The

pure solvent, i.e. toluene or cyclohexane, respectively 3) The different concentrations

of the sample.

The absolute scattering intensities with the correction of the scattering volume are

calculated by

(2-27) 2

2

Solvent

Toluene

Toluene

SolventSolutionTolueneSolventSolution n

nI

IIRR)(R)(R)(R

with RRToluene=1.27 10-5 cm-1 the Rayleigh ratio (absolute scattering intensity) of

toluene at 632.8 nm and 20°C, nToluene=1.4960 refractive index of toluene,

nCyclohexane=1.4260 respectively.

The data are evaluated by the application of the Zimm equation (equ. 2-7, 2-15)

caclulating Kc/R( ) using a refractive index increment of polystyrene in toluene of

0.109 ml/g (0.187 ml/g in cyclohexane) at 632.8nm. All the data Kc/R( ) are plotted

against q² + kc with an appropriate constant k using millimeter paper sheets and the

value Mw determined from the intercepts and <Rg²>z1/2 and A2 from the slopes. All

the experimental results should be given with units and also discussed with respect

to the errors. The experimental notes which have to be written do not need to include

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details of the theory. A description of the experiment, the parameters and the

discussion of the result is sufficient.

4. Literature

a) H.G. Elias, Makromoleküle, 1981

b) B. Vollmert, Grundriß der Makromolekularen Chemie, 1988

c) J. Springer, Einführung in die Theorie der Lichtstreuung verdünnter Lösungen

großer Moleküle, 1970

d) R. J. Young, P.A. Lovell, Introduction to Polymers, 1991

e) S. F. Sun, Physical Chemistry of Macromolecules, John Wiley, 1994

f) K. A. Stacey, Light Scattering in Physical Chemistry, 1956

g) Ch. Tanford, Physical Chemistry of Macromolecules, John Wiley, 1961

h) P. Kratochvil, in “Classical Light Scattering from Polymer Solutions”,

ed. A. D. Jenkins, 1987

i) M. Schmidt, “Simultaneous Static and Dynamic Light Scattering”, in

Dynamic Light Scattering, ed. H. Brown, Oxford, 1993, p. 372

j) B. Chu, Laser Light Scattering, 2nd edition, Academic Press, 1991

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EXPERIMENT 2: SIZE EXCLUSION CHROMATOGRAPHY (SEC) 1 Introduction Mechanical, rheological and thermodynamic properties of polymers are mainly

influenced by their molar mass (for example melt viscosities) but also depend on their

molar mass distribution functions. Molar mass distributions are mainly influenced by the

polymerization mechanism involved in polymer synthesis and their knowledge is also

important with respect to kinetics. The knowledge of the molar mass distribution

therefore allows conclusions on the polymerization mechanism.

For polymer processing a broader molar mass distribution is desirable due to the

softening properties of the oligomer content. For other purposes a narrower molar mass

distribution is necessary, for instance for engine-oil additives (viscosity index

improvement). For physico-chemical investigations, narrowly distributed polymers are

needed, because most of the properties depend on molar mass.

Nowadays the most important indirect method for the determination of the molar mass

and its complete distribution is size exclusion chromatography (SEC) also known as gel

permeation chromatography (GPC) or gel filtration chromatography (GFC, especially for

proteins in water). Therefore this experiment will demonstrate the practical use of

SEC/GPC.

2 Basic Principles of SEC/GPC SEC is a special application of the High Performance Liquid Chromatography (HPLC)

but in an ideal case without interaction due to absorption or partition. A polymer solution

(typical concentration 0.1% by weight) passes through a column of a porous gel/material

at pressures of 30-100 bar and a flow rate of typically 1ml/min. In contrast to HPLC the

separation is based on a size exclusion mechanism and not on absorption. The pores of

volume Vx contain the same solvent than the interstitial volume Vo (dead volume of the

packing). Molecules which are larger than the size of the pores can only pass through

the interstitial volume and can not penetrate into the pores of the packing material and

therefore leave the column first at the same volume Vo called the upper exclusion limit (‘total exclusion’) above of which the sizes of the larger particles can not be resolved.

Molecules which are smaller than the pore size enter all pores and leave them without a

SEC

2 - 2

separation. They all elute with the same maximum elution volume (total permeation

volume) Vtot=Vo+Vx (see Fig. 2-1), which exceeds the interstitial volume by the total pore

volume Vx . This is the lower limit of the range of separation, which is called the

separation threshold (‘total permeation’).

Vx V0 Vtot Fig.2-1: Scheme of a volume element of a SEC-separation column with total Volume Vtot

, pore volume of the gel phase Vx and the interstitial volume V0 .

The physical property which determines the elution volume is the hydrodynamic volume

of the polymer coil. If the elution volume of two different polymers is the same they

exhibit identical hydrodynamic volumes although their topology or chemical structure

might be different. Conversely it is possible that polymers of the same molar mass (but

different chemical structure or topology) have a different hydrodynamic volume and

therefore elute with a different elution volume. The driving force for the penetration of

the polymer coils into the pores is the gradient of concentration between the pore

volume Vx and the interstitial volume Vo. There is a concentration gradient into the pores

if it is only filled with solvent. If the coil has entered the pore the concentration gradient

will be in direction of the interstitial volume and the coil diffuses out of the pore.

It is usual and practical to indicate the steric exclusion limit and the separation threshold

by molar mass values which enable the user immediately to determine the problems for

which the various gel types are suitable. The solvent and the polymer should be stated

in addition due to the fact that the dimensions of the polymer and the gel as well depend

on the solvent. For samples having molecular sizes above the upper exclusion limit, the

distribution constant, K , is zero. As the molecular sizes decreases, K increases,

reaching its maximum value K=1 at the separation threshold. While in the absorption

mechanism also higher values of K may occur (corresponding to an enrichment of the

substance in the stationary phase) in the steric exclusion mechanism the highest

possible concentration in the stationary phase is equal to that in the solution. From c’’=c’

it follows that K=1. Particles whose sizes lie between the steric exclusion limit and the

SEC

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separation threshold are more or less strongly retained. Their elution volume Ve range

between the dead volume Vo and the maximum retention volume Vtot:

Ve = Vo + K Vx with 0 K 1

The distribution constant K is therefore a measure of the volume fraction of the pore

volume for a given size.

2.1 Calibration Procedure In contrast to light scattering or membrane osmosis SEC is a relative method for the

determination of molar masses and therefore needs calibration. SEC is nothing more

than a separating technique which, operated on its own, does not provide either the

molar mass distribution or the mean values of molar mass. As a direct result, elution

curves are obtained which, at best, show which amounts of the sample leaves the

column at a certain volume. The values Mi and its relative amounts Hi(Mi) for calculation

of the quantities being of actual interest must be determined by means of calibration

procedures.

The latter are established using, if possible, several well defined polymers as calibration

standards. Calibration relationships therefore are only valid for polymers of the same

chemical structure and topology as the calibration standards applied. The calibration

standards have to be characterized by means of an absolute method of molar mass

determination (light scattering, osmosis). The elution volume of such samples decrease

with the logarithm of molar mass (Moore, 1964):

(2-1) Ve = C1 – C2 logM

The constants C1 and C2 can be taken from the linear part of the graphical

representation of logM vs. Ve, which at the same time shows the limitations of the

separating range. Both, the upper exclusion limit, and the separation threshold, do not

represent sharp points but rather a broad transition regime of poor resolution. Usually a

sigmoidal function is observed as the experimental calibration curve (Fig. 2-2) which can

be interpolated by a polynomial function:

(2-2) logM = A + B Ve + C Ve2 + D Ve

3 + …

SEC

2 - 4

Eqn. (2-1) describes the relationship as it is established in the calibration: the elution

volume is the dependent variable determined as a function of the molar mass. The slope

factor can be calculated from the positions of two points on the linear part of the

calibration curve:

(2-3) SMM

VVC

III

IeIIe

)log(,,

2

The selectivity factor S defined in this way is also used in the characterization of

columns.

Fig. 2-2: Typical sigmoidal calibration curve obtained from polystyrene calibration

standards of different molar mass in THF at 35°C with columns of pore size 106, 105,

104 and 103A.

Because it is possible that due to a change of external conditions (temperature, solvent

quality) the gel swells stronger or shrinks and therefore a change ( 5%) of the elution

volume occurs. Therefore every sample contains 20 ppm of toluene (if the eluent is

SEC

2 - 5

THF) as an internal standard, which allows to correct the experimental elution volumes

by:

(2-4) exp

tanexp

toluene

dardstoluene

ecorr

e VVVV

The standard peak of toluene dardstolueneV tan is defined by one single measurement. With this

method it is also possible to correct fluctuations of the solvent flow by the pump.

SEC

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2.2 Calibration Functions by Universal Calibration Molar masses and molar mass distributions can only be determined quantitatively, if the

calibration standards and the unknown sample are chemically identical, i.e. consist of

identical monomers and exhibit the same chain architecture. The reason for this

restrictive condition is that the hydrodynamic volume is different for different polymers

with the same molar mass.

Benoit found that when plotting Vh vs. logM even the curves for linear and branched

polystyrenes did not coincide. For equal molar masses, linear samples showed smaller

elution volumes than star-shaped ones, which in turn had smaller volumes than samples

of a comb-like structure. The respective intrinsic viscosities (for equal M) had been

found to decrease in the following order:

combstarchain ][][][

These observations could be explained by Einstein’s viscosity law originally derived for

spheres

(2-5) M

VN.. hA5252

which yields

(2-6) M.constVh

Although the constant depends slightly on the chain architecture and on the solvent

quality, log([ ] M) vs. Vh indeed yielded a common curve, not only for the polystyrene

samples of different topology, but also for polymethylmethacrylate, polyvinylchloride,

polyphenylsiloxane, polybutadiene and graft copolymers. Thus log([ ] M) vs. Ve

represents a universal calibration (Fig. 2-3) which, however, requires [ ] to be known.

SEC

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Fig. 2-3: Universal calibration according to Grubisic, Rempp and Benoit (1967)

As mentioned above the hydrodynamic volume is related to the Staudinger-Index and

the molar mass by:

Vh = const. [ ] M

The Staudinger-Index is related to the molar mass according the Mark-Houwink relation:

MK][

Therefore we get: 1MKVh

For two different polymers with the same hydrodynamic volume it helds:

2,1, hh VV 222

111

21 MKMK

and

11

11

2

12

21 )( MKKM or in logarithmic representation:

12

1

2

1

22 log

11log

11log M

KKM

With the help of the Mark-Houwink coefficients the hydrodynamic volume of a known

standard is attached to the molar mass of a different polymer. This procedure has to be

done for every point of the calibration curve which results in a complete calibration curve

of the new polymer.

SEC

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Tab. 2-1: Some Mark-Houwink-Constants at 25°C in THF

Polymer K/10-3 ml g-1 Polystyrene 13.60 0.714 Polyisobutylene 36.04 0.629 Polymethylmethacrylate 12.98 0.688 Polybenzylmethacrylate 59.00 0.868 Poly(tert.-butylmethacrylate) 4.92 0.750 Poly(tert.-butylacrylate) 3.30 0.800 Poly(tert.-butylvinylketone) 4.14 0.795 2.3 Band Broadening due to Axial Diffusion

If a monodisperse sample (e.g a protein or low molar mass sample like toluene) enters a

SEC-apparatus the original rectangular concentration profile changes into a Gaussian-

like profile (Fig. 2-4). This is due to axial diffusion. The signal exhibits a characteristic

width and shape, the so called kernel. Usually polymers are broadly distributed and the

width of the kernel is small as compared to the distribution itself. Characterizing very

narrowly distributed samples requires the correction of axial dispersion.

Fig. 2-4: Concentration profile of a sample before entering and after leaving a SEC-column The extent of the broadening of a low molar mass liquid can be used to charaterize the

separation efficiency of a SEC-system. The number of theoretical plates N is the square

of the ratio of the peak maximum to the standard deviation . Another definition uses

the broadness of the peak at which the amplitude has decayed to 1/e. A newer

definition utilizes the width b defined as the distance between the intercepts of the

tangent at the points of inflection with the baseline (Fig. 2-5).

SEC

2 - 9

Fig. 2-5: Detector signal as a function of elution volume Ve : Gaussian peak shape of standard deviation . For true Gaussian shapes all three definitions are equivalent:

2max,

2max,

2max, 168

bVVV

N eee

N can be normalized to the length of the column L simply by multiplication with its

reciprocal value. The reciprocal value of the number of theoretical plates 1/N is known

as the height equivalent theoretical plates HETP=L/N which can also be used for the

characterization of columns.

3 Experiment 3.1 Apparatus A SEC-system consists of the following components: solvent reservoir, solvent

degasser, high pressure injection valve (optional with an autosampler for automatic

injections), inline filter, column oven (to reduce solvent viscosity and therefore pressure)

with guard column (100A) and separation columns and detectors usually a UV-detector

with variable wavelength (or a photodiode array detector PDA allowing to detect the

whole UV/VIS-spectra as a function of the elution volume) and a refractive index

detector (differential refractometer). All high pressure components are connected by

stainless steel or PEEK (for corrosive media) capillaries. Recording of the

chromatograms and data evaluation is performed by a computer program.

SEC

2 - 10

Trouble may occur due to:

change of the solvent quality due to moisture may change the elution volume.

Furthermore the formation of peroxides is possible (THF, dioxane) which could

damage the column support.

dust or non-soluble solids damage the high pressure injection valve and block the

column. Therefore the sample solution is filtered by passing through a 0.45µm filter.

To protect the columns a platinum inline filter and a guard column are used.

pressure fluctuations cause baseline instabilities (especially for RI-detection) which

are mainly due to damaged high pressure seals in the pump or an insufficient

degassing of the solvent.

temperature fluctuations also cause baseline instabilities, especially for RI-detection;

therefore a temperature control unit is highly recommended for both, the columns

and the RI-detector (better than 0.05°C).

3.2 Column Materials The stationary phase of SEC for use with organic solvents consists mainly of highly

crosslinked spherical poly(styrene-co-divinylbenzene) of 3, 5, 7 or 10µm diameter.

Other supports are based on polymethacrylates, porous silica or porous glass. The

resolution capability (with respect to molar mass) depends mainly on the size and

uniformity, as well as the packing density of the spheres (without channels or large

cavities). Because the denser package of the smaller particles the interstitial volume

decreases with decreasing particle size and therefore the resolution increases. The

solvent applied should be a solvent for the support polymer. Otherwise no swelling of

the support material occurs and the pore volume is not accessible and separation due to

size exclusion is not possible. By application of columns with different pore sizes the

range of molar mass separation can be extended from 10 M 107. The use of ‘mixed

gel’ columns (‘linear columns’) which apply a mixture of gels of different pore sizes is

also possible. Due to historical reasons the pore size is given as the contour length of a

polystyrene chain which just fits into the pore.

SEC

2 - 11

Tab. 3-1: Pore size and range of separation

Pore size Range of separation

102 A 10 M 103

104 A 3 102 M 3 105

106 A 104 M 107

3.3 Detectors

In routine SEC-measurements detection by the index of refraction and UV/VIS

absorption are common. The amount of polymer which is necessary (1mg dissolved in

1ml is sufficient) to get a good signal to noise ratio is quite small.

RI: The difference between the index of refraction from polymer solution to the pure

solvent is detected. The difference of the index of refraction of a solution and the solvent

is proportional to the mass of the solute. Therefore a polymer chain of molar mass M

gives the same signal intensity like two chains of molar mass M/2. For oligomers it

should be taken into account that the index of refraction can depend on molar mass

which has to be considered for calibration.

UV/VIS: There are two different modes of operation: if the wavelength of the absorption

maximum of the repeat unit is used for detection the signal is proportional to the mass

concentration of the solution ( =260 nm for polystyrene, =230 nm for PMMA and

PtBMA). For oligomers the signal may also depend on molar mass. In special cases the

wavelength corresponding to the absorption maximum of an endgroup (or terminating

agent) could be selected as a signal proportional to the number of chains.

Other detectors used are light scattering detectors (in combination with RI or UV/VIS-

detectors for the determination of concentration) which allow the measurement of the

molar mass distribution without the use of calibration standards and at the same time

allow the determination of Rg-M-relationships (from the angular dependency of the

scattered light).

Online viscosity detection allows a simultaneous determination of the Staudinger-Index

and to establish an universal calibration (see above).

SEC

2 - 12

4. Theory 4.1 Molar mass distributions

For the calculation of molar mass distribution from kinetic assumptions it is useful to

apply the degree of polymerization Pn instead of the molar mass. The degree of

polymerization is the number of repeat units connected to a polymer chain. The molar

mass Mi is related to the degree of polymerization by:

Mi=Mo Pn with Mo the molar mass per repeat unit

With size exclusion chromatography size distributions are determined. Two

representations are common.

The number distribution h(P) gives the relative number of polymer chains of a given

degree of polymerization P. With n(P) as absolute number (or number fraction) of

polymer chains of degree of polymerization we define:

PPn

PnPh)(

)()( and 1)(P

Ph

The weight distribution gives the relative mass (weight fraction) of polymer chains of a

given degree of polymerization with m(P) the absolute mass (or weight fraction) of

polymer chains of degree of polymerization P:

PPm

PmPm)(

)()(

Both distribution functions are related by m(P)=P M0 n(P) with M0 the molar mass per

repeat unit.

4.2 Average Values of the Molar Mass

Polymers are usually characterized by average values of the molar mass distribution.

The degree of polymerization is defined as the average value of the number of

monomers per polymer chain.

chainsofnumbermonomersofnumber

Pn

The number average degree of polymerization corresponds to the mean value (1st

moment) of the number distribution. The i-th moment <xi> is defined by:

)()(

xfxfx

xi

i

SEC

2 - 13

The zeroth-moment of a distribution corresponds to the area defined by the distribution

function (usually 1), the 1st moment <x> corresponds to the mean value of the

distribution. The standard deviation is given by =<x²> - <x>². The indices h (number)

or w (weight) indicate to which type of distribution the values are related.

Therefore we get for the degree of polymerization:

1)(

)()(

0

1 PPhP

PPh

PPhPn

For the weight average value of the degree of polymerization we obtain:

ww PPPwPw

PPwP )(

)()(

w(P) equals h(p) P and therefore:

h

hw P

P

PPhPPh

P22

)()(

The polydispersity D and the nonuniformity U are defined by:

n

w

PPD and 11 D

PPU

n

w

Introducing the average values leads to:

2

2

2

22

2

2

1nn

hh

h

h

PP

PP

P

PU

The polydispersity index is identical with the square of the normalized (with respect to

Pn) standard deviation of the number average distribution. For a monodisperse sample

the nonuniformity U equals zero and the polydispersity D equals 1. For Gaussian

distribution (symetric) the full width at half maximum is given as:

UPP n22

SEC

2 - 14

4.3 Typical Molar Mass Distribution Functions of Polymers 4.3.1 Living Polymerization

For a living polymerization the degree of polymerization is given by:

][][][ 0

IMMP t

n = ][][ 0

IM

with [M]0 : monomer concentration at time zero [M]t : monomer concentration at time t

[I]: initiator concentration : degree of coupling ( =1 for monofunctional initiators, =2 for Na/naphthalene) : extent of reaction

It leeds to the extremely narrow Poisson distribution:

!1exp)(

1

PPh

P

!11exp)(

1

PPPw

P

Pn is given by: 1nP

Pw is given by: n

nn

nnw P

PPP

PP 1112

Therefore we get : nnn PPP

U 1112 with Pn >> 1

4.3.2 Polycondensation

For the polycondensation reaction the so called most propable or Flory distribution is

found:

)1()( 1PPh The most frequent particles are always monomers! 12)1()( PPPw

The average values are:

11

nP and 11

wP and therefore 1n

w

PPU

4.3.3 Free Radical Polymerization In radical polymerizations the degree of polymerization is given by the kinetic chain

length which is the ratio of the rate of propargation to the sum of the rates of chain

termination and chain transfer:

IM

RRR

trt

P

SEC

2 - 15

Therefore the degree of polymerization is given by

2nP for termination by recombination

and

nP for termination by disproportionation.

For an ideal radical polymerization the Schulz-Zimm distribution function is obtained:

nn PPP

PPh exp)( 1 with the degree of coupling

For chain termination by recombination: = 2

For chain termination by disproportionation: = 1

If = 1 then

nn

PPP

Ph /exp1)( which is a simply decaying function.

If = 2 then

nn PPP

PPh 2exp2)(

2

this is a function with a maximum which is slowly

decaying.

The polydispersity D is given by:

1D which means that for recombination D = 1.5 and for disproportionation D = 2.

Fig. 4-1: Comparison of Poisson-distributions (---) and Schulz-Zimm-distributions with Pn=100 and =1 (…..) and =2 (- - -).

SEC

2 - 16

4.4 Determination of the Molar Mass Distribution from Experimental Data An elugram registers a detector signal as function of the elution volume (time). By use of

a calibration function the elugram is transformed into a number distribution or a weight

distribution. The area of the detector signal is proportional to the concentration.

Fig. 4-2: Molar mass distribution function from SEC-chromatograms and corresponding calibration curves

With a mass proportional detection (RI-detector) the signal intensity corresponds to a

mass of the polymer.

An easy procedure for the evaluation of the data is the following: In the elugram a

certain elution volume Vei corresponds to a molar mass Mi (see calibration curve). The

corresponding height of the detector signal Hi is proportional to a mass concentration ci

of molar mass Mi. If the frequency Hi is normalized it corresponds to a mass fraction. For

the average values it becomes:

ii

i

ii

i

ii

i

i

iin MH

HMc

cMm

mnMn

M///

i

ii

i

ii

ii

iiw H

MHcMc

MnMn

M2

Therefore a table has to be established consisting of the following columns:

No. Ve(ml) Hi (mm) Hi (%) Hi (%) Mi Hi/Mi HiMi

This allows also the installation of the integral and differential mass distribution by

plotting Hi vs Mi and Hi vs Mi respectively.

Nowadays these calculations are done with the help of computer systems which allow a

fast and precise evaluation of the experimental data.

SEC

2 - 17

4.4 Composition of a Block-Copolymer (PMMA/PS) It would be ideal if there are two detectors which yield only one signal of each

component of the copolymer. Both components would be detectable independently. It is

sufficient if the two detectors respond in a different manner as for instance a UV/VIS-

and a RI-detector. The two signal intensities S1 and S2 correspond to:

2121111 cfcfS

2221212 cfcfS

with the sensitivity factors fij of detector i and component j which have to be determined

by calibration measurements of the homopolymers. The concentrations cj have to be

normalized according to:

21

11 cc

cw and 21

22 cc

cw

The concentration determination would be better the higher the fii and the lower the fij

values are. For instance in UV-detection an absorption maximum is selected where the

other monomer does not absorb. Because mostly there is also a difference in refractive

index increment usually a combination of UV- ( = 260 nm for polystyrene) and RI-

detection is chosen.

In a zero order approximation all chains in a given interval of the elution volume are of

the same length and composition. This is not exactly because the detectors yield an

average composition and the block length in a certain elution volume element are not

the same but exist with an unknown distribution.

If we are using the zero order approximation then the two calibration curves of PMMA

and Polystyrene can be weighted and we get an approximate distribution of the molar

mass. The weighing procedure is supposed to be linear:

2211 PwPw)V(P e

At the corresponding elution volume Ve the values P1 and P2 are taken from the

calibration curves and the values are weighted by the experimental mass fractions.

The analytics of copolymers done by this method is helpful but not highly precise. But a

range of molar mass is obtainable and the most important result is qualitative

dependency of comonomer composition as a function of molar mass. The block

efficiency as well as the amount of termination can be calculated directly.

SEC

2 - 18

5 Experimental Task

1. For the polystyrenes obtained by radical and anionic polymerization the following

quantities should be determined by using the easier slice method (20 slices for the

radical sample, 10 slices for the anionic sample):

- number and mass distribution

(mass distribution both differential and integral)

- the average values of Mn , Mw and Pn , Pw

- the nonuniformity U

- discuss the termination reaction with respect to the molar mass distribution

The calibration curve and its polynomial representation will be given by the

assistant.

2. Compare the elugrams of the sample obtained by free radical polymerisation, the

sample obtained by controlled radical polymerization and the sample by anionic

polymerization.

3. Measure and discuss the molar mass distribution of the block copolymer (PMMA/PS)

obtained by anionic polymerization. Discuss the extent of chain termination after the

dosage of the second monomer (at the computer).

4. Calculate a universal calibration curve of a polystyrene calibration obtained from a

PMMA-calibration. The following calibration data of PMMA have to be applied:

No. Ve / ml Molar mass / g/mole

1 26.60 480,000 2 28.49 195,000 3 29.66 96,000 4 31.18 32,000 5 32.37 11,000 6 33.62 5,200 7 38.14 1,002

5. From the elugram of 1,2-dichlorobenzene the number of theoretical plates N and the

HETP of the applied column combination have to be calculated. Ask the assistant for the

length of the columns applied. Calculate the selectivity factor S of the columns applied.

SEC

2 - 19

6 Additional Remarks For taking the SEC-measurements the following samples are used:

1. conventional free radical polymerization

2. controlled radical polymerization

3. anionic polymerization/macromonomer synthesis

4. block-copolymer

Please bring these samples to the assistant of the SEC-experiment.

7 Questions

1. Which are the contributions to the SEC-column volumes?

2. Which methods do you know for determination of absolute values of molar mass?

3. How can you determine molar mass distributions?

4. How can you proceed for a continuous degassing of SEC/HPLC-solvents?

5. Which parameters influence the selectivity factor S and the number of theoretical

plates N?

6. Which are the major demands concerning pumps and detectors?

7. How does a differential refractometer work?

8 Literature 1 W.W. Yau, J.J. Kirkland, D.D. Bly, Modern Size-Exclusion Liquid

Chromatography,

John Wiley and Sons, New York (1979)

2 G. Glöckner, Polymer Characterization by Liquid Chromatography, Elsevier

(1987)

3 J. F. Johnson, R.S. Porter, Analytical Gel Permeation Chromatography, Wiley

N.Y.

(1968)

4 M.J.R. Cantow, Polymer Fractionation, Academic Press N.Y. (1971)

3 - 1

EXPERIMENT 3: OSMOTIC PRESSURE 1. Theory

1.1. Introduction

Due to the fact that many properties of polymers are influenced by their molar

mass, the determination of the molecular weight is very important.

Not all molecules of a polymer sample have the same molar mass, as a result

of the polymerization mechanismen. That means a polymer sample has a

molar mass distribution which is defined by diffent averages, like the number

average Mn or the weight average Mw of the molar mass.

1.2. Membrane Osmometry

One of the methods yielding the number average molecular weight Mn is the

determination of the osmomtic pressure of a polymer solution via membrane

osmometry.

1.2.1. Osmotic Pressure

The osmotic pressure of a solution of a component 2 in a solvent

(component 1) is defined as:

V1 : molar volume of the solvent

µ1 = µ1 - µ1pure : difference between the chemical potential of the solvent in

the solution and the pure solvent

a1 : activity of the solvent in the solution

)1(ln 111

1 aVRT

V

Osmotic Pressure

3 - 2

Setting a1 = x1 = 1 – x2 (x : molar fraction) and using the Taylor-expansion for

the logarithm leads to the osmotic pressure of a solution:

The higher terms can be omitted for infinite dilution:

With V1 (n1 + n2) V and 21

22 nn

nx follows :

c2 : concentration of component 2

M2 : Molar mass of component 2

The deviations from the ideal behavior are treated in analogy to the gas

pressure by a virial expansion:

with

for infinite dilution.

A2, A3 : second, third virial coefficients

)2(...)32

()1ln(32

22

21

21

xxxVRTx

VRT

)3(21

xVRT

id

)4(22

2

MRT

c

RTnVid

)5(...)1( 22322

22

cAcAM

RTc

)6(2 2

2

1,2 M

VAid

Osmotic Pressure

3 - 3

With

ni

ii

ges

igesii

MnMn

ncVM

nnVMnc ,

(7)

follows for a polymer in solution:

c: concentration of the polymer in solution

Mn: number average of the molar mass

B = RTA2

)9(

)8(...)(

lim2

nc

n

MRT

c

BcMRT

c

Osmotic Pressure

3 - 4

2. Experimental Setup

Measurements of the osmotic pressure of a polymer solution can be carried

out in the type of cell represented schematically in figure 1. The polymer

solution is separated from the pure solvent by a membrane, permeable only to

solvent molecules.

Initially, the chemical potential µ1 of the solvent in the solution is lower than that

of the pure solvent, and solvent molecules tend to pass through the membrane

into the solution in order to attain equilibrium. This causes a build up of

pressure in the solution compartment until, at equilibrium, the pressure exactly

counteracts further net solvent flow. This pressure is the osmotic pressure.

Figure 1: Basic setup of a membrane osmometer

Osmotic Pressure

3 - 5

In this case, the osmotic pressure can be expressed by the following equation,

with the density of the solution and the gravitational constant g:

g h = (9)

Modern instruments make use of a pressure sensor to detect the osmotic

pressure. A scheme of an instrument is shown in figure 2.

Figure 2: Scheme of a membrane osmometer

The important parts of this instrument are the membrane (21), the pressure

sensor (9) which has to be calibrated before measurement, and the solution

compartment (15).

The compartment between the membrane and the pressure sensor is filled

with the solvent, the upper part of the main chamber is filled with the solution.

Osmotic Pressure

3 - 6

3. a) Determination of the number-average molar mass of a polyethyleneoxide sample with high

polydispersity In order to determine the Mn of the polyethyleneoxide sample, 4 solutions in

degassed water with 4, 6, 7 and 8 mg/ml have to be prepared.

b) Determination of the number-average molar mass of a polyethyleneoxide sample with low polydispersity In order to determine the Mn of the polyethyleneoxide sample, 4 solutions in

degassed water with 4, 5, 6 and 8 mg/ml have to be prepared.

4.Questions

1. Calculate Mn and A2 in SI-Units of the measured samples. Use the intercept

and the slope of the /c versus c-Diagramm.

2. Describe and compare the results obtained from the osmosis experiment

with the GPC result of the same sample.

3. Which other methods yield the number average molar mass (minimum 5).

Discuss the advantages and disadvantages of these methods (including

membrane osmometry) and subdivide them into absolute and relative

methods.

5.Literature

J.M.G. Cowie, Polymers: Chemistry & Physics of modern Materials, 2nd Ed.,

Blackie Academic & Professional, London 1991.

Skript zum Versuch 4:

Ermittlung von Überstrukturen durch CD-Spektroskopie

Makromolekulares Praktikum SS2010

Kapitel 1: Einleitung

Die Konstitution eines Moleküls erlaubt häufig die Variation der räumlichen Gestalt durch Annahme verschiedener Konformationen. (Beispiel: Cyclohexan kann als Sessel, Wanne oder Twist-Form vorliegen).Im Bereich der Polymere, Proteine, Polysacharide oder Oligonukleotide bedingt die Primärstruktur in vielen Fällen, dass in Lösung unter bestimmten Voraussetzungen nicht ein statistisches Knäuel vorliegt, sondern eine definierte Überstruktur, d.h. eine feste räumliche Anordnung der einzelnen Kettensegmente bezogen aufeinander. Da jede Abweichung der Konformation eines Makromoleküls von der Gestalt des statistischen Knäuls mit einem Entropieverlust verbunden ist, müssen enthalpische Wechselwirkungen die Ursache für das Auftreten von geordneten Strukturen sein. Solche enthalpischen Effekte können, z.B. aus der Wechselwirkung zwischen den Basenpaaren der Polynucleotide („Stapelenergien“) oder durch intramolekulare Wasserstoffbrückenbindungen in Polypeptiden („α-Helix“) entstehen. Es können aber auch abstoßende Kräfte, z.B. durch Coulomb-Wechselwirkungen gleichnamiger Ladungen auftreten, die in Polyelektrolyten für eine Aufweitung des Knäuels verantwortlich sein können. Methoden zur Bestimmung der Überstruktur von Biomolekülen sind neben der CD- bzw. ORD-Spektroskopie z.B. die NMR- , IR-Spektroskopie bzw. die Röntgendiffraktometrie. Letztere Methode besitzt den Nachteil, dass von der zu analysierenden Substanz ein Einkristall gezüchtet werden muss, damit die Sekundärstruktur analysiert werden kann. Dies ist in vielen Fällen nicht möglich, wodurch diese Methode auf eine limitierte Anzahl von biologisch relevanten Systemen beschränkt ist. Bezüglich der NMR- und IR-Spektroskopie hat die CD-Spektroskopie den Vorteil, dass die benötigte Probenmenge wesentlich geringer ist. Ein weiteres Argument für die CD- bzw. ORD-Spektroskopie ist die relative einfache Interpretation der Spektren bezüglich der Sekundärstruktur, was sich in der NMR- bzw. IR-Spektroskopie teilweise als äußerst schwierig erweist. Insbesondere die NMR-Spektroskopie versagt bei der Ermittlung der Sekundärstruktur, wenn die Molekülstruktur kompliziert ist oder das Molekulargewicht die Grenze von 500000 g/mol überschreitet. Beide Methoden können aus diesen Gründen nicht zur Ermittlung der Sekundärstruktur von großen Proteinen bzw. Polynucleotiden (DNA) verwendet werden. Die CD-Spektroskopie bzw. ORD-Spektroskopie bietet in diesem Fall eine verlässliche Alternative um die oben genannte Problemstellung zu lösen. Neben diesen Vorteilen gibt es allerdings auch Nachteile die zu diskutieren sind. Die CD- bzw. ORD-Spektroskopie stellen spezielle Formen der UV-Absorptionsspektroskopie dar, mit der man alleine in der Regel nicht unterscheiden kann, ob man die Überstruktur von einer Komponente oder eines Mehrkomponentengemischs bestimmt. Somit ist es im Vorhinein sehr wichtig sicher zu stellen, dass es sich bei der zu untersuchenden Probe um ein Einkomponentensystem handelt. Die NMR- bzw. IR-Spektroskopie besitzt gegenüber diesem Problem einen entscheidenen Vorteil, da mit diesen beiden Methoden zwischen der Anzahl der verschiedenen Komponenten und deren Sekundärstrukturen in vielen Fällen unterschieden werden kann, weil das beide Methoden zusätzlich Informationen über den molekularen Aufbau der zu untersuchenden Komponente(n) liefern. Des weiteren können in der CD- bzw. ORD-Spektroskopie nur Lösungsmittel bzw. Salze verwendet werden, die im zu untersuchenden Absorptionsbereich selbst nicht absorbieren. Da dieser in der Regel zwischen 180 und 260nm liegt, sind eine Vielzahl von Lösungsmitteln nicht einsetzbar (siehe Anhang).

Kapitel 2: Polarisiertes Licht 2.1 Polarisationsarten

Auf Grund der Tatsache das in der CD-Spektroskopie bzw. ORD-Spektroskopie die Verwendung von zirkular- bzw. linear-polarisiertem Licht unerlässlich ist, werden in diesem Kapitel die Grundlagen über die verschiedenen Polarisationsarten noch einmal angesprochen.

Elektromagnetische Strahlung (Radiowellen, UV-Licht usw.) ist meist eine Transversalwelle mit rechten Winkeln zwischen dem Wellenvektor k , der in Ausbreitungsrichtung zeigt, und den Vektoren des elektrischen und magnetischen Feldes E bzw. B (siehe Abb.1). Aus der Zeit, als dies noch nicht bekannt war und die Polarisation des Lichtes als transversale mechanische Schwingung des hypothetischen Äthers erklärt wurde, stammt eine Festlegung der Polarisationsrichtung, die sich später als die Schwingungsrichtung des magnetischen Feldvektors herausstellte. Da die meisten Wechselwirkungen elektromagnetischer Strahlung mit Materie elektrischer Natur sind, wird die Polarisationsrichtung heute meist auf den elektrischen Feldvektor bezogen.

Abb.1: Schematische Darstellung einer elektromagnetischen Welle

Man unterscheidet drei Arten von Polarisation, die man durch Richtung und Betrag des elektrischen Feldvektors in einem festen Raumpunkt beschreiben kann:

• Lineare Polarisation: Der elektrische Feldvektor zeigt immer in eine feste Richtung und die Auslenkung ändert bei Voranschreiten der Welle ihren Betrag und ihr Vorzeichen periodisch (mit fester Amplitude).

• Zirkulare Polarisation (auch als drehende Polarisation bezeichnet): Der elektrische Feldvektor dreht sich bei Voranschreiten der Welle mit konstanter Winkelgeschwindigkeit um den Wellenvektor und ändert seinen Betrag dabei nicht.

• Elliptische Polarisation: Der elektrische Feldvektor rotiert um den Wellenvektor und ändert dabei periodisch den Betrag. Die Spitze des Feldvektors beschreibt dabei eine Ellipse.

Die zirkulare und elliptische Polarisationen entstehen hierbei durch Überlagerung zweier linear polarisierter Wellen.

Jede beliebige Polarisation kann als Überlagerung zweier Basispolarisationen dargestellt werden. Daher können die lineare und die zirkulare Polarisation als Grenzfälle der elliptischen Polarisation aufgefasst werden, umgekehrt lässt sich aber auch jede elliptische Polarisation als eine Überlagerung einer linear- und einer zirkular-polarisierten Welle beschreiben.

Für die mathematische Beschreibung von Polarisationen, beispielsweise bei der Beschreibung der Reflexion an einer Grenzfläche, werden häufig die folgenden beiden Varianten verwendet:

1. Zwei linear polarisierte Wellen, deren Polarisationsrichtungen senkrecht aufeinander stehen, werden überlagert. Abhängig von der Phasenbeziehung und dem Amplitudenverhältnis der beiden Wellen ergeben sich folgende Ausgangspolarisationen:

• Bei verschwindender Phasendifferenz (oder einer Phasendifferenz, die einem Vielfachen von π entspricht) und unterschiedlicher Amplitude ist die Ausgangspolarisation linear und die Richtung hängt vom Amplitudenverhältnis ab.

• Bei einem Phasenunterschied von π/2 und gleichen Amplituden ist die Ausgangspolarisation zirkular.

• In jedem anderen Fall ist die Ausgangspolarisation elliptisch.

2. Zwei zirkular polarisierte Wellen, eine rechts- und eine linksdrehend, werden überlagert. Abhängig von der Phasenbeziehung und dem Amplitudenverhältnis der beiden Wellen ergeben sich folgende Ausgangspolarisationen:

• Bei gleichen Intensitäten und variabler Phasendifferenz ist die Ausgangspolarisation linear und die Richtung hängt von der Phasenlage der Basispolarisationen ab.

• Wenn eine der Basispolarisationen eine verschwindende Amplitude hat, ist die Ausgangspolarisation die jeweils andere zirkulare Polarisation.

• In jedem anderen Fall ist die Ausgangspolarisation elliptisch.

2.2 Erzeugung von linear polarisiertem Licht

Die gängigste Methode zur Erzeugung von linear polarisiertem Licht, besteht in der Verwendung von Polarisationsfiltern (Polaroidfolien). Diese bestehen aus speziellen Kunststofffolien in denen die einzelnen Molekülketten parallel zueinander ausgerichtet sind (z.B. durch mechanisches Strecken). Zusätzlich wird die Folie noch mit einer Iodverbindung dotiert. Dadurch werden in den Molekülketten Elektronen eingelagert, die sich aber nur längs der Ketten bewegen können. Senkrecht zu den Kettenmolekülen sind die Elektronen unbeweglich. Trifft nun Licht dessen elektrischer Feldvektor parallel zu den Molekülketten orientiert ist auf die Folie, so werden die eingelagerten Elektronen durch das elektrische Feld entlang der Molekülketten beschleunigt. Die dazu notwendige Energie muss von dem einfallenden Licht aufgebracht werden, wodurch dieses absorbiert wird. Ein Polarisationsfilter ist demnach für Licht, dass parallel zu den Ketten orientiert ist undurchlässig. Fällt dagegen Licht, dessen elektrischen Feldvektor senkrecht zu den Molekülketten orientiert ist auf den Filter, so werden die Elektronen nicht beschleunigt und das einfallende Licht kann den Filter passieren (Abb.2).

Abb.2: Erzeugung von linear polarisiertem Licht mittels einer Polarisationsfolie

Unpolarisiertes Licht

Polarisiertes Licht

Polarisator

2.3 Erzeugung von zirkular polarisiertem Licht

Zirkular polarisiertes Licht kann man erzeugen, indem man eine linear polarisierte Lichtwelle durch einen Kristall sendet, in dem die Fortpflanzungsgeschwindigkeit von der Schwingungsachse des elektrischen Feldes abhängt (doppelbrechende Materialien z.B. Quarz). Abbildung 3 zeigt eine linear polarisierte Welle, deren elektrischer Feldvektor im 45 Grad Winkel zur optischen Achse in den Kristall eintritt. Auf Grund des Superpositionsprinzips kann man die linear polarisierte Welle in ihre Vektoren aufspalten, welche in Abbildung 3 senkrecht bzw. waagrecht zur optischen Achse verlaufen.

Abb.3: Erzeugung einer zirkular polarisierten Welle

Die doppelbrechenden Eigenschaften des Materials sorgen dafür, dass sich die waagrechte Komponente des elektrischen Feldvektors langsamer durch den Kristall fortpflanzt, verglichen mit der senkrechten Komponente. Die Länge des Kristalls (λ/4 Plättchen) ist so gewählt, dass beim Austritt eine Phasenverschiebung von 90 Grad zwischen den beiden Komponenten auftritt. Anschließende Addition der beiden Komponenten ergibt eine polarisierte Welle, deren elektrischer Feldvektor mit konstanter Amplitude im Uhrzeigersinn um die Ausbreitungsachse der elektromagnetischen Welle zirkuliert (rechts-zirkular polarisiertes Licht; Abbildung 4).

Abb.4: Darstellung einer zirkular polarisierten Welle

Eine Verdopplung der Länge des Kristalls würde wieder zu polarisiertem Licht (orthogonal zur ursprünglichen Ausbreitungsrichtung) führen, während dessen eine Verdreifachung der Länge links- zirkular polarisiertes Licht erzeugen würde.

Moderne CD-Spektrometer verwenden KERR- oder POCKELS-Zellen zur Erzeugung von zirkular polarisiertem Licht. Bestimmte Kristalle (z.B. Ammonium- oder Kaliumdihydrogenphosphat) ändern ihre doppelbrechenden Eigenschaften unter dem Einfluss elektrischer Felder. Diesen Effekt nutzt man in der POCKELS-Zelle aus, mit der sowohl links- als auch rechts-zirkular polarisiertes Licht mit einer Wechselfrequenz im Bereich von einigen kHz erhalten wird, wenn man ein elektrisches Wechselfeld an den Kristall anlegt.

Kapitel 3: Grundlagen der CD- bzw. ORD-Spektroskopie

3.1 ORD-Spektroskopie

Optische Rotation und Zirkulardichroismus sind zwei stets miteinander gekoppelte Erscheinungsformen der optischen Aktivität und resultieren aus der Wechselwirkung eines Mediums mit linear polarisiertem Licht. Ihr Verständnis wird erleichtert, wenn linear polarisiertes Licht nach Fresnel als eine Überlagerung zweier gleicher Anteile von rechts- und links-zirkular polarisierten Lichtes auffasst (Abb. 5).

Abb.5: Links. Zusammensetzung des linear polarisierten Lichtes aus zwei gleich großen Komponenten links- und rechts-zirkular polarisierten Lichtes. Projektion der elektrischen Feldstärkevektoren E entgegen der Propagationsrichtung. Rechts Situation nach Austritt des Lichtes nicht absorbierter Wellenlänge aus einem optisch aktiven Medium

Optisch aktive Medien besitzen für diese beiden Komponenten verschieden Brechungsindizes, n l≠nr sowie, da n=c/ν (c=Lichtgeschwindigkeit im Vakuum), verschiedene

Propagationsgeschwindigkeiten l≠r und damit verschiedene Winkelgeschwindigkeiten der Projektionen der elektrischen Feldstärken (siehe Abb. 5).

Während des Durchtritts durch eine Schicht der Dicke d´ gewinnt die eine gegenüber der anderen Komponente einen zeitlichen Vorsprung (∆ t)

(Gl. 3-1)

Dieser entspricht einer Phasenverschiebung (φ)

(Gl. 3-2)

Die Phasendifferenz bleibt mit Austritt aus dem optisch aktiven Medium konstant. Das ausgetretene Licht ist also weiterhin linear polarisiert, jedoch ist die Polarisationsebene gegenüber derjenigen vor dem Eintritt in das Medium um den Winkel α gedreht. (λvak. ist die Wellenlänge des verwendeten

Lichts im Vakuum).

(Gl. 3-3)

Um den Drehwinkel als Stoffkonstante angeben zu können, bezieht man auf eine Schichtdicke d in Dezimetern und die Konzentration in g∙cm3. Bei der Angabe der sogenannten spezifischen Drehung[]

T müssen zugleich Wellenlänge und Temperatur spezifiziert werden.

(Gl.3-4)

Für den Vergleich ähnlicher Stoffe empfiehlt sich eine molare Basis. Ist M die Molmasse, so beträgt die molare Drehung []

T

(Gl. 3-5)

Bei Polymeren ist der Bezug auf die Molmasse des Gesamtmoleküls aufgrund der Polydispersität nicht sinnvoll. Hier bezieht man auf die (bei Heteropolymeren mittlere) Molmasse der Polymerisationseinheit, so z.B. bei Proteinen auf die mittlere Aminosäurerestmasse (MRW mean

t=nl−nr

c⋅d ´

=2⋅ t=2c

nl−nr ⋅d ´= 2vak.

nl−nr⋅d ´ [rad ]

[]T=

M100

[ ]T=

M100

⋅

c⋅d [grad⋅cm2

dezimol ]

=/2=180vak.

nl−nr⋅d ´ [grad ]

[]T=

c⋅d [grad⋅cm3

g⋅dm ]

residue weight). Die molare Drehung pro mittleren Aminosäurerest [m]T ist.

(Gl.3-6)

Die optische Drehung ist ferner abhängig vom Brechungsindex nλ des Lösungsmittels bei der

betreffenden Wellenlänge. Durch den Lorentz-Faktor werden alle Messungen auf ein einheitliches Medium reduziert. Die reduzierte molare Drehung [m´ ]

T lautet

(Gl. 3-7)

3.2 Zirkulardichroismus

Der andere Aspekt der Wechselwirkung von linear polarisiertem Licht mit einem optisch aktiven Medium besteht darin, dass dieses für die links- und rechts-zirkular polarisierte Komponente einen unterschiedlichen Extinktionskoeffizienten besitzt l≠r , d.h. es zeigt Circulardichroismus. Die Projektion der elektrischen Vektoren haben also nicht allein eine unterschiedliche Winkelgeschwindigkeit, sondern ferner eine unterschiedliche Länge (Abb.6).

Abb.6: Elliptische Polarisation bei Austritt des Lichtes aus einem Medium im Absorptionsbereich eines optisch aktiven Elektronenüberganges: Zirkulardichriosmus. (Projektionsrichtung wie in Abb. 5)

Das Licht verlässt verläßt das Medium mit elliptischer Polarisation. Die Elliptizität θ ist definiert als der

[m ]T=

MRW100

⋅

c⋅d [grad⋅cm2

dezimol ]

[m ´ ]T=

3

n22

⋅MRW100

⋅

c⋅d [grad⋅cm2

dezimol ]

Arcustangens des Verhältnisses von kleiner zu großer Halbachse (siehe Abb. 6)

(Gl. 3-8)

Da die Transmission dem Quadrat der Länge elektrischen Feldvektors proportional ist, gilt

sowie (Gl. 3-9 a und b)

Daraus folgt für den Tangens der Elliptizität

(Gl. 3-10)

Durch weitere geschickte Umformungen gefolgt von einer Taylorentwicklung kann man einen Zusammenhang zwischen der Elliptizität und der Differenz zwischen den Extinktionskoeffizienten von recht- und links-zirkular polarisiertem Licht herleiten.

bzw. (Gl. 3-11 und 3-12)

Die Elliptiziät ist somit dem Zirkulardichroismus proportional. Für die Elliptizität ist ein analoger Formalismus eingeführt worden, wie für die Rotation. Dementsprechend gilt für die spezifische und molare Elliptizität []

T bzw. [M ]T sowie bei Polymeren für die auf die mittlere Molmasse der

Bausteine bezogene Elliptizität [MRW ]T

(Gl. 3-13)

(Gl. 3-14)

(Gl. 3-15)

=arctanbc

E l=k⋅T l E r=k⋅T r

tan=ba=T r−T l

T rT l

=2,303⋅ E4

[ rad ]

[ ]T=

c⋅d

[M ]T= M

100⋅

c⋅d

[MRW ]T=MRW

100⋅

c⋅d

=360⋅2,303⋅E2⋅4

[grad ]

3.3 Vergleich zwischen optischer Rotation und Zirkulardichroismus

Wie aus der λ-Induzierung ersichtlich, sind opische Rotation und Zirkulardichroismus bzw. Elliptizität von der Wellenlänge anhängig. Die Auftragung des optischen Drehwinkels gegen die Wellenlänge bezeichnet man als ORD-Spektrum, die des Zirkulardichroismus oder der Elliptizität als CD-Spektrum. Je nach dem ob die Brechzahl oder der Extinktionskoeffizient der links-zirkular polarisierten größer oder kleiner sind als die der rechts-zirkular polarisierten Komponente, haben optische Drehung oder Elliptizität ein positives oder negatives Vorzeichen. Abbildung 7 zeigt schematisch das ORD- und das CD-Spektrum im Bereich einer isolierten Absorptionsbande. Beide spektralen Erscheinungen werden nach ihrem Entdecker als Cotton-Effekt bezeichnet. Nach dem Vorzeichen der CD-Bande unterscheidet man zwischen negativem und positivem Cotton-Effekt. Bei einem positiven Cotton-Effekt tritt im ORD-Spektrum der positive Teil stets auf der langwelligen Seite des Nulldurchgangs auf. Es fällt auf, dass die ORD-Kurve der ersten Ableitung der CD-Kurve ähnelt. Wie viele andere dispersive und absorptive Phänomene sind optische Rotation und Zirkulardichroismus ineinander überführbar. Aus einem ORD-Spektrum lässt sich insofern nicht zu viele Cottoneffekte überlagern durch Anwendung der Kronigs-Kramer-Transformationsgleichungen das zugehörige CD-Spektrum berechnen und umgekehrt.

Abb.7a,b: Cotton-Effekte von Absorptionsbanden bei λ0 . a [α] als Funktion von λ: ORD-Spektrum, b [θ] als Funtion von λ: CD-Spektrum. Links positiver Cotton-Effekt. Rechts negativer Cotton-Effekt

Wie in Abbildung 7 angedeutet, wird optische Drehung auf Grund Ihrer dispertiven Natur noch in größerer Entfernung von der Lage des optisch aktiven Übergangs wahrgenommen, wohingegen der Zirkulardichroismus natürgemäß auf den Bereich der Absorptionsbande begrenzt ist. Hierin liegen auch die Vor- und Nachteile der einen gegenüber der anderen Methode begründet. Heute steht die CD-Spektroskopie ganz im Vordergrund. Ihre Überlegenheit beruht hauptsächlich auf der Auflösung sich überlagernder Banden. Dennoch greift man wegen spezifischer Vorteile auf die ORD-Spektroskopie zurück. Sie stellt eine Absolutmethode dar, während die CD-Skala mit Standartsubstanzen (Camphersulfonsäure) kalibriert werden muss. Ferner erlaubt die ORD-Spektroskopie Schlüsse auf Existenz und Vorzeichen von Cotton-Effekten in Wellenlängenbereichen, die der direkten Messung aus technischen Gründen oder wegen stark absorbierender Lösungsmittel unzugänglich sind. Bei Denaturierungsstudien von Proteinen mit hohen Konzentrationen an Harnstoff

oder Guanidinhydrochlorid reicht z.B. bei kleinen Schichtdicken die Transparenz gerade noch aus um das Helix-bedingte ORD-Minimum bei 233nm noch zu erfassen, während das Minimum bei 222nm im CD-Spektrum auf Grund des schlechten Signal zu Rausch Verhältnisses nicht mehr detektierbar ist.

Kapitel 4: Molekulare Ursachen der Chiralität

Die Quantentheorie der optischen Aktivität geht auf Rosenfeld (1928) zurück. Nach ihrer Kernaussage ist die optische Aktivität eines Moleküls daran gebunden, dass ein bestimmter Elektronenübergang 0→i sowohl elektrisch als auch magnetisch erlaubt ist und das die beiden Übergangsmomente nicht senkrecht aufeinander stehen.

(Gl. 4-1)

Die Rotationsstärke R0i ist der Dipolstärke eines isotropen Übergangs analog. Sie ist für einen isolierten Cotton-Effekt sowohl über das CD- als auch über das ORD-Spektrum zugänglich. Entspricht die CD-Bande einer Gauss-Kurve, so ist

(Gl. 4-2)

Dabei ist [θ ]0i die molare Elliptizität im Maximum der i-ten Bande bei der Wellenlänge λ0i und ∆0i ist

das Wellenlängenintervall, über das [θ ]0i auf [θ ]0i/e abfällt, beide in nm. Aus einer ORD-Kurve ist die Rotationsstärke nach folgender Formel zu erhalten:

(Gl. 4-3)

Hier steht [A´]0i für die Amplitude der Kurve, Lorentz-korrigiert, ∆0i für den halben

Wellenlängenabstand der beiden Extrema, den sogenannten Dämpfungsabstand in nm.

Seit erstmal van´t Hoff die optische Drehung mit dem Molekülaufbau verknüpft hat, ist die bekannteste Quelle optischer Aktivität das asymmetrische Kohlenstoffatom (siehe Abb.8).

R0i= Jm0i⋅mi0

R0i=4,118⋅10−57⋅[]0i⋅0i

0i

[Am3 s ]

R0i=7,366⋅10−57 [A´ ]0i⋅0i

3

0i⋅0i° 20,0030i3[Am3 s ]

Abb.8: L- und D-Enatiomer der Milchsäure (Bild und Spiegelbild)

Beide Enantiomere der Milchsäure besitzen in einer achiralen Umgebung die gleiche chemische Reaktivität, verhalten sich aber aufgrund ihrer strukturellen Asymmetrie wie Bild und Spiegelbild zueinander.

Für die CD- bzw. ORD-Spektroskopie ist von unabdinglich, dass die zu untersuchenden Strukturen optische Aktivität zeigen. Zunächst kann dies jedes absorbierende Molekül sein, das keine Symmetrieebene oder Symmetriezentrum aufweist. Dazu gehören z.B. alle in der Natur vorkommenden Zucker, Aminosäuren oder Polynucleotide. Optische Aktivität kann jedoch auch hervorgerufen werden, wenn ein Molekül sich in einer asymmetrischen Mikroumgebung befindet. So kann z.B. Helicen (Abbildung 9) nicht eben gebaut sein, sondern muss die Gestalt einer Schraubenfläche haben. Damit ist aber auch das π-System chiral geworden, was wiederum eine optische Aktivität induziert.

Abb.9: Rechts und linksdrehendes Helicen (inhärent dissymetrisches Chromophor)

Die meisten chromophoren Gruppen von chiralen Strukturen besitzen meistens eine lokale Symmetrie. Ihr Elektronensystem ist somit achiral und sollte mit einem links bzw. rechts zirkularpolarisiertem Lichtstrahl gleich stark interferieren (Abbildung) und die Richtung des elektrischen Feldvektor von linear polarsiertem Licht nicht beeinflussen. Sind derartige Chromophore aber in eine chirale Umgebung eingebaut, wie z.B. die Ketogruppe in Campher oder die Amidgruppe in Lysin, dann ist zumindest noch eine dissymmetrische Störung der ansonsten symmetrischen Orbitale festzustellen.

Abbildung 10: Beispiel für inhärent symmetrische, aber dissymmetrisch gestörte Chromophore. Die Zahlen unter den Formeln geben die das Absorptionsmaximum an

Die optische Aktivität vieler Biopolymere ergibt sich allerdings nicht nur durch die Bindung ihrer chromophoren Gruppen an asymmetrische Kohlenstoffatome (symmetrisch gestörte Chromophore), sondern auch durch die Ausbildung von Überstrukturen wie z.B. α-Helices oder β-Faltblätter. Die Kopplung von Chromophoren führt dazu, dass die optische Aktivität ein empfindlicher Monitor der Sekundärstruktur von Biomolekülen ist und entsprechend gerne zur Strukturanalyse genutzt wird.

NO2 COORR

RH2

H2 N3

290nm 340nm 500nm 260nm 340nm

290nm, 340nm 210nm

500nm 290nm

250nm 330nm265nm

315nm, 450nm

200nm

OC C C C O SC S S Se Se

C N N N C C

SS S

C

S

C C

N

OCu

O

NH

O

O

H

C

Kapitel 5: Optische Übergänge bei Proteinen oder Polypeptiden in der CD-Spektroskopie

Im Fall von Proteinen oder Polypeptiden unterscheidet man im Fall der CD-Spektroskopie zwischen zwei Spektralbereichen, den Nah UV-Bereich (250-300nm), welcher Informationen über die Tertiärstruktur beinhaltet und den Fern UV-Bereich (170-250nm), mit dem Aussagen über die Sekundärstruktur des Proteins getroffen werden können. Bezüglich des Fern UV-Bereichs dienen als Chromophore bei Proteinen die Amidgruppen (dissymmetrisch gestörtes Chromophor). Ihre π-Elektronen sind über die C-, N- und O-Atome delokalisiert (partieller Doppelbindungscharakter). Im Grundzustand ist das bindende π-Orbital mit zwei Elektronen besetzt, ebenso wie das nichtbindende n-Orbital. Das energetische höher liegende π*-Orbital ist unbesetzt. Durch Absorption eines Photons kann ein Elektron aus dem π- oder dem n-Orbital in das π*-Orbital angehoben werden. Diese Prozesse bezeichnet man als π→π* bzw. n→π* Übergang.

Abbildung 11: π→π* und n→π* Übergang der Amidgruppe

Der n→π*-Übergang ist verboten, daher ist sein molarer Extinktionskoeffizient sehr klein (εmax=100M-1cm-1). Das Maximum des Übergangs liegt zwischen 210 und 230nm. Die Frequenz des

Übergangs ist sensitiv auf H-Brückenbindungen. Der n→π*-Übergang hingegen ist intensiv (εmax=7000M-1cm-1). Er liegt im Bereich 180-200nm. Die Lage des Absorptionsmaximum hängt empfindlich von der Umgebung der Bindung und damit auch von der Konformation des Peptids ab. Aus absorptionsspektroskopischen Untersuchungen lassen sich daher Aussagen über die Konformation bzw. über die Konformationsänderung von Polypeptiden gewinnen. Liegt das Polypeptid in der Random coil Konformation vor, ist im CD-Spektrum ein stark negatives Minimum bei etwa 195nm zu beobachten (π→π*-Übergang) und ein schwach positives Maximum bei 212nm (n→π*-Übergang). Ist die α-Helix die dominierende Überstruktur des Proteins, ist der π-π*-Übergang entartet. Er besteht aus zwei Komponenten, einer bei 191nm und einer bei 208nm. Sie sind einem senkrecht bzw. parallel zur Helixachse liegendem Betrag zuzuschreiben. Der n→π*-Übergang liegt hier bei etwa 222nm.

Besteht die Überstruktur des Proteins hauptsächlich aus einem β-Faltblatt wird ein Maximum bei 196nm (π→π*-Übergang) und ein Minimum bei 212nm (n→π*-Übergang) im CD-Spektrum detektiert.

Abbildung 12: Basisspektren von α-Helix, β-Faltblatt und einer Struktur in Zufallsknäuelkonformation

Kapitel 6: Aufbau eines CD-Spektrometers

Der schematische Aufbau eines CD-Spektrometers ist in Abbildung 1 gezeigt.

Abb. 13: Schematischer Aufbau eines CD-Spektrometers

Unpolarisiertes monochromatisches Licht, erzeugt durch eine Xenonlampe und einen Monochromator (optisches Gitter oder Prisma), wird mittels eines Polarisationsfilters linear polarisiert und anschließend durch eine POCKELS-Zelle ( siehe Kapitel 2) in rechts bzw. links zirkular polarisiertes Licht umgewandelt. Das rechts bzw. links zirkular polarisierte Licht wird wellenlängenabhängig abwechselnd auf die Probe eingestrahlt und die Absorption mit Hilfe eines Photomultipliers bzw. einer

Halbleiterdiode detektiert. Anschließend wird mittels eines Computers die Differenz zwischen rechts und links polarisiertem Licht bestimmt und die Elliptizität in Abhängigkeit der Wellenlänge im Spektrum aufgetragen.

Kapitel 7: Qualitative Bestimmung der Sekundärstrukturanteile

Neben der qualitativen Betrachtung der CD-Spektren aus dem man meistens mit bloßem Auge den dominierenden Anteil der Sekundärstruktur bestimmen kann, ist man natürlich auch daran interessiert, aus den Spektren von Proteinen die genauen Anteile der jeweiligen Sekundärstruktur zu ermitteln. Dazu macht man einen halbempirischen Ansatz, bei dem man davon ausgeht, dass sich ein Protein als Anneinanderreihung von Regionen mit α-helikaler Struktur, β-Faltblatt und Zufallsknäuelstruktur beschreiben lässt, und sich das CD-Spektrum aus den Beiträgen der einzelnen Strukturen additiv zusammensetzt, wobei Seitenketteneffekte vernachlässigt werden. Als Basis für die Rechnungen benutzt man die Elliptizität, die sich aus den CD-Spektren von Modellsubstanzen für die Strukturmerkmale ergibt. Die Sekundärstrukturanteile der Modellsubstanzen wurden durch röntgendiffraktrometrische Messungen bestimmt. Für die Berechnung der Anteile wird folgender Ansatz gewählt (Gleichung 6-1).

(Gl: 6-1)

[MRWi ]T sind die Elliptizitäten für die reinen Strukturmerkmale i, fi der Anteil des

Sekundärstrukturmerkmals im Protein (fα+fβ+frc=1). Zum einen lassen sich nach dieser Gleichung die

CD-Spektren für verschiedene Gehalte an α-Helix, β-Faltbaltt und Zufallsknäuel berechnen zum anderen aber auch gemessene Spektren anpassen. Mit Hilfe eines Computerprogramms können die Parameter fi solange variiert werden, bis die berechneten Punkte auf dem gemessenen Spektren liegen. Gängige Computerprogramme um die Sekundäranteile zu berechnen, sind z.B. CONTIN von Provencher und Glöckner, SELCON von Sreerama und Woody oder CDSSTR von Johnson. Programmpakete welche die oben genannten Programme verwenden um die Sekundärstrukturanteile zu berechnen sind z.B. CD-Pro oder Dichroweb. Beide Pakete sind kostenlos im Internet erhältlich.

[MRW ]T= f ⋅[MRW ]

T f ⋅[MRW ]T f ⋅[MRW ]

T

Anhang: Geeignete Lösungsmittel, Puffer und Kuvetten für die CD-Spektroskopie

Aufgrund der Tatsache, dass der Wellenlängenbereich in der CD-Spektroskopie zwischen 180 und 350nm liegt, sind viele Lösungsmittel nicht geeignet, da sie meistens unterhalb von 230nm eine starke Absorption aufweisen. Dasselbe gilt für viele Puffer und Salze die zur Herstellung von physiologisch wässrigen Medien verwendet werden. Die nachfolgenden Tabellen geben einen Überblick welche Lösungsmittel und welche Puffer für die CD-Spektroskopie verwendet werden können.

Tabelle 1: Geeignete Lösungsmittel in der CD-Spektroskopie für eine Kuvette mit der Schichtdicke von 1mm

Tabelle 2: Geeignete Puffer bzw. Salze in der CD-Spektroskopie

Puffer 210nm 200nm 190nm 180nm

170nm 0 0 0 0NaF 170nm 0 0 0 0Borat 180nm 0 0 0 0NaCl 205nm 0 0.02 >0.5 >0.5

210nm 0 0.05 0.3 >0.5

195nm 0 0 0.01 0.15Natriumacetat 220nm 0.03 0.17 >0.5 >0.5Glycin 220nm 0.03 0.1 >0.5 >0.5Diethylamin 240nm 0.4 >0.5 >0.5 >0.5NaOH, pH=12 230nm >0.5 2 >2 >2Borat/NaOH, pH=9,1 200nm 0 0 0.09 0.3Tricine, pH=8.0 230nm 0.22 0.44 >0.5 >0.5Tris, pH=8.0 220nm 0.02 0.13 0.24 >0.5Hepes, pH=7.5 230nm 0.37 0.5 >0.5 >0.5Pipes, pH=7.0 230nm 0.2 0.49 0.29 >0.5Mops, pH=7.0 230nm 0.1 0.34 0.28 >0.5Mes, pH=6.0 230nm 0.07 0.29 0.29 >0.5

    Absorption in einer 10mM Lösung in einer 0.1cm Zelle bei...Kein Absorbtion überhalb von…

NaClO4

Na2HPO4

NaH2PO4

Lösungsmittel

182HFIP 174.5TFE 179.5MeOH 195.5EtOH 196MeCN 1851,4-Dioxan 231Cyclohexan 180n-Pentane 172

λCut off/nm

H2O

Gleiche Ansprüche muss man auch bezüglich des Materials der Kuvetten stellen. Besteht die Kuvette aus einem Glas, welches in dem oben angegebenen Wellenlängenbereich stark absorbiert, kann diese nicht in der CD-Spektroskopie eingesetzt werden. Aus diesem Grund finden bei dieser Spektroskopiemethode nur Kuvetten aus reinem Quarzglas (Suprasil) Verwendung.

Abbildung 9: Transmission von leeren Kuvetten aus verschiedenen Gläsern

5 - 1

EXPERIMENT 5: VISCOSITY OF POLYMER SOLUTIONS

Please note: The solutions have to be prepared one day in advance! Therefore you have to

appear in room 01 157 building K (9-11 a.m., 14-16 p.m., tel: 39-24639, time required: ca. 1 h).

Otherwise the experiment cannot be performed. To make an appointment please call me or

send an email (andreas.eich@wee-solve.de).

Topics of these experiments are the viscometric characterization of dilute polymer solutions (part

1) and an introduction to the rheology of concentrated polymer solutions (part 2). The expression

“viscometry” is used for rather simple measurements, when the viscosity only depends on

composition and temperature. The study of complex flow and deformation behaviour is subject of

“rheological” measurements.

The viscometry of dilute solutions deals with isolated polymer molecules. The interest of these

studies is to determine molecular parameter (e.g. molecular mass, hydrodynamic volume, coil

expansion in good solvents). This kind of viscosity measurements are performed in capillary or

rolling ball viscometers.

The rheology of concentrated polymer solutions and polymer melts deals with interacting polymer

molecules. In this course we focus on the dependency of the viscosity on shear rate. The

measurements are performed on a rotational rheometer.

For convenience some important expressions are translated to German in the appendix. In the text

the words are written in italics.

5 - Viscosity of polymer solutions

5 - 2

Table of content

1) Introduction ..................................................................................................................................... 3

2) Molecular models ............................................................................................................................ 5

2.1) Polymer solution: Polymers regarded as rigid particles without draining ............................. 5

2.2) Polymer solutions: Polymers regarded as drained coils (without entanglements) ............... 6

2.3) Concentrated polymer solutions and polymer melts: Influence of entanglements .............. 8

3) Dilute polymer solutions ................................................................................................................ 10

3.1) Definitions ............................................................................................................................. 10

3.2) Dependence of the viscosity on molecular weight at infinite dilution................................. 10

3.3) Influences on [η] ................................................................................................................... 12

3.4) Dependence of viscosity on polymer concentration ............................................................ 12

3.5) From dilute to concentrated solutions: Coil overlap ............................................................ 13

4) Concentrated polymer solutions ................................................................................................... 14

4.1) Shear rate dependence of viscosity ...................................................................................... 14

4.2) Shear thinning of polymer melts and concentrated polymer solutions............................... 14

5) Experimental Methods .................................................................................................................. 16

5.1) Capillary viscometer.............................................................................................................. 16

5.2) Rotational viscometer / rheometer ...................................................................................... 17

5.3) Falling ball viscometer .......................................................................................................... 17

6) Experimental Part .......................................................................................................................... 18

6.1) Rotational viscometry ........................................................................................................... 18

6.2) Capillary viscometry .............................................................................................................. 18

7) Appendix: Translation of important expressions .......................................................................... 19

8) Literature ....................................................................................................................................... 19

5 - Viscosity of polymer solutions

5 - 3

1) Introduction

Fig. 1 sketches a shear flow of a liquid between two parallel plates:

Fig. 1: Parallel plate model

Moving the plate on top in x1-direction and keeping the lower plate stationary, the liquid phase

shows some resistance to the motion in form of friction. Molecules of the liquid in contact with

the moving plate show the same velocity as the plate itself. The next liquid layer is moving slightly

slower than the top layer. (Here, we assume laminar flow of the liquid which means the absence

of turbulences.) This velocity gradient continues until, at the bottom layer, the velocity v = 0 is

reached. The velocity gradient between the two plates is called shear rate γɺ .

1 2d /dxv xγ =ɺ (1)

In fig. 1 exists a constant velocity gradient between the two plates. In this case γɺ is the ratio of the

velocity of the top plate and the gap width ∆x2.

The force 1xF acting tangential on a plate of surface area A (see fig. 1) is called shear stress τ21.

1

121

xF

Aτ = , [τ 21] = N/m

2 = Pa (2)

The indices denote the directions of the surface normal (here: x2) and the force (here: x1),

respectively. In practice, the indices are often omitted.

For ideal viscous liquids, there is a linear relation between shear stress and shear rate, Newton’s

law (eq. (3)):

21τ η γ= ɺ (3)

η : shear viscosity, (dynamic viscosity), unit: Pa s (SI), 1 Pa s = 10 Poise (CGS)

1 It is also common to use σ instead of τ as abbreviation for the shear stress.

5 - Viscosity of polymer solutions

5 - 4

Samples obeying this relation are called Newtonian. An important value to keep in mind is the

viscosity of water at 20 °C: 2

20 C

H Oη ° = 1 mPa s (exact 1.003 mPa s).

The viscosity provides a quantitative measure for the internal friction forces of a liquid sample.

Multiplication of both sides of eq. (3) with γ = ∆ ∆ɺ1 2/xv x - considering ∆ = ∆ ∆

1 1 /xv x t - results in

an expression for the amount of energy dissipated per volume and time:

2E

V tη γ

∆=

∆ ⋅∆ɺ (4)

5 - Viscosity of polymer solutions

5 - 5

2) Molecular models

Several different molecular models have been developed to describe the frictional properties –

and therefore the viscosity - of polymer solutions and melts. In a short overview, the basic

concepts are described qualitatively in the following text. Some quantitative details are derived in

the next sections.

2.1) Polymer solution: Polymers regarded as rigid particles without draining

To understand how particles influence the viscosity of a suspension, we have a look on a shear

flow as shown in fig. 2. We regard. First we assume that the suspended particles (non-drained) do

not rotate but only perform a translational motion as depicted in fig. 2b.

Fig. 2: Influence of a particle on shear flow for non-rotating and rotating particles. (Taken

from Hiemenz, Principles of Colloid and Surface ChemistryA)

The non-rotating particles – capturing a volume fraction ϕ - do not contribute to the velocity

gradient. As far as the gradient is concerned, the particles might as well be allowed to settle to the

stationary wall (fig.3):

Fig. 3: Scheme of a suspension of particles with volume fraction ϕ a) homogenous distributed particles, b) sedimented particles.

By this process the remaining liquid layer which develops the gradient is reduced to the fraction

(1-ϕ) of the original gap width (fig. 3(b)); therefore the shear rate in this layer is increased by the

factor 1/(1-ϕ). We now consider the shear stress τ – which should be equal in fig. 3(a) and 3(b) –

starting from τ = ηSolv v/d for the pure solvent:

5 - Viscosity of polymer solutions

5 - 6

η τ ηϕ ϕ

= =− −

3(b) 3(a)

Solv

1

(1 ) 1

v v

d d (5)

ηsolv , η : Viscosity of the solvent and suspension, respective

Since ϕ < 1, η > ηsolv. Further considerations lead to the famous Einstein viscosity equation of

suspended spheres which is given in section 3.2 (for details confer Hiemenz, Polymer ChemistryB).

If we assume particle rotation (cf. fig. 2c), the shear field is not as much disturbed as in case of

non-rotating particles. But then the energy has to be taken into account which is necessary for the

particle rotation. Due to friction the rotating energy is continuously dissipated which leads to an

additional contribution to the viscosity.

2.2) Polymer solutions: Polymers regarded as drained coils (without entanglements)

Debye model

If the polymer is drained by solvent, the viscosity is determined by the frictional forces that the

polymer chain experiences by the solvent molecules. Debye calculated the flow dynamics of a

drained polymer in 1946. A picture of this model is shown in fig. 4. In this model the polymer chain

is divided into segments that can be set equal to the repeating unit.

Fig. 4: a) The velocity gradient in a flowing liquid. b) Velocities relative to the center of mass

of a polymer molecule. (Taken from Hiemenz, Polymer ChemistryB)

As described for particles, the shear induces rotation of the polymer coils. This can be easily

understood by shifting the origin of the coordinate system to the center of mass of a single

polymer molecule. Each segment experiences frictional forces to other polymer segments or

solvent molecules. These forces are taken into account by a segmental friction factor. With this

model, a quantitative evaluation leads to proportionality between viscosity and the molecular

weight of the polymer.

5 - Viscosity of polymer solutions

5 - 7

Bead-spring model

While the Debye model is designated to flow behaviour, another similar model explains

viscoelastic behaviour: The bead-spring-model (Rouse 1953, Zimm 1956) takes into account the

entropic properties of the polymer chain.

Fig. 5: Scheme of the bead-spring model. (Taken from Ferry, Viscoelastic Properties of

PolymersC)

In this model the polymer chain is divided into subsections that are sufficiently large to display

rubber like elasticity (springy behaviour at small elongations due to entropy elasticity of Gaussian

chains). The subsections of the chain have a roughly spherical shape and present a mechanical

drag with respect to the remainder of the melt or solution that is quantified with a subsection

friction factor. The two elements of a single subsection can be represented in series as a spring of

no volume and a rigid bead.

The calculation for the bead-spring model according to Rouse doesn’t account for hydrodynamic

interactions between the segments (limit of free draining). The calculation results in an equation

which correlates different modes of the relaxation time τRouse with the zero shear viscosity2 η0,

molar mass M2 and concentration c2 of the polymer3:

ητ

π= 0 2

Rouse, p 2 22

6 1M

c RT p, with mode p = 1,2,3… (6)

(Don’t be confused: The relaxation time is abbreviated with the same letter τ as the shear stress.)

The longest relaxation time (p=1) is the terminal relaxation time for the total rearrangement of the

molecule.

In the more complex evaluation due to Zimm the hydrodynamic interaction of beads in the same

chain is taken into account.

A disadvantage of the bead-spring model is the lack of interactions between different polymer

molecules. Besides, the model explains linear viscoelastic behaviour but not the shear rate

dependence of viscosity.

2 Viscosity at shear rate zero: 0

0limγ

η η→

=ɺ

3 The index 2 denotes the polymer in a binary mixture, whereas index 1 denotes the solvent. The indices can be

omitted if there are no ambiguities.

5 - Viscosity of polymer solutions

5 - 8

2.3) Concentrated polymer solutions and polymer melts: Influence of entanglements

The Debye and the bead-spring model lead to a proportionality η0 ~ M2 which is found

experimentally for dilute polymer solutions and melts of low molecular weight polymers. For

polymers with molar masses above a critical mass Mc, a power law η0 ~ M23.4

is observed (fig. 6).

Fig. 6: Dependence of melt zero shear viscosity on molecular weight

The critical molecular mass Mc, is roughly two times the molar mass of the polymer chain length

between two entanglements Me: Mc ≈ 2 Me. The entanglements of different polymer molecules

are the origin for the strongly increased sensitivity of the viscosity on molar mass above Mc.

Concentrated polymer solutions give similar results. Here the entanglement density is described

by the product c2M which replaces ρ M (ρ : density) in case of polymer melts.

Bueche model

To describe the motion of entangled polymer chains, Bueche developed in 1952 a theory which is

sketched in Fig. 7.

Fig. 7: Variation of melt zero shear viscosity with molecular weight. (Taken from Hiemenz,

Polymer ChemistryB)

5 - Viscosity of polymer solutions

5 - 9

In this model the motion of a polymer chain is coupled with the motion of other chains, whereas

different orders of coupling are distinguished. The quantitative calculation leads to a η0 ~ M3.5

relationship close to the experimental results. However, the drawback of this model is that

independent chain mobility vanishes at the onset on entanglement.

Reptation model

The reptation model (Doi, Edwards 1978, based on a concept of de Gennes 1971) describes the

movement of a polymer chain in a tube which is formed by other polymer molecules.

Fig. 8: Reptation model for entanglements. (Taken from Hiemenz, Polymer ChemistryB)

The reptation model is based on the diffusion rate of the chain segments in the tube. The

relaxation time τ0 of one segment can be expressed as 20 /2l kTξ (with segmental friction

parameter ξ, segment length l0, Boltzmann constant k); the relaxation time of the whole molecule

with n segments equals τ = τ0 n3 ~ M

3. The expression for the zero shear viscosity shows the same

molecular weight dependence η ~ M3. Even though the experimental 3.4 power law dependence

is not derived, the reptation model has become the favoured model for entangled polymer

systems due to other successful predictions of experimental findings, e.g. the effects of branching.

5 - Viscosity of polymer solutions

5 - 10

3) Dilute polymer solutions

3.1) Definitions

For dilute polymer solutions it is common to use special variables derived from the viscosity of the

polymer solution and of the pure solvent, respectively:

Viscosity of the solution: η

Viscosity of the solvent: ηsolv

Relative viscosity: η

ηη

=rel

solv

(7)

Specific viscosity: η η

η ηη

−= = −solv

spec rel

solv

1 (8)

Reduced viscosity: spec

redc

ηη = (9)

These variables are used to determine the intrinsic viscosity that is defined as:

[ ]2

spec

020

limc cγ

ηη

→→

=ɺ

, unit: ml/g (10)

3.2) Dependence of the viscosity on molecular weight at infinite dilution

In 1906 A. Einstein published an expression for the viscosity of dilute dispersions of solid spherical

particles in a liquid, known as Einstein’s law of viscosity:

η η ϕ η= +solv 2 solv2.5 (11)

2ϕ : volume fraction of the solid particles.

We assume that a polymer coil in solution behaves like a rigid sphere, which means no draining of

the coils. Then the volume fraction of the sphere can be replaced by the volume fraction of the

polymer coils, leading to

η ηϕ

η

−=solv

coil

solv

2.5

or using the intrinsic viscosity [η]:

[ ] coil

2

2.5c

ϕη = (12)

The volume fraction of all polymer coils coilϕ is related to the sphere-equivalent volume of one

polymer coil Vη and to the number of polymer coils N2 in the sample.

5 - Viscosity of polymer solutions

5 - 11

2

coil

N V

V

ηϕ = (13)

Notice that the polymer coil volume Vη contains not only polymer but also solvent molecules.

Using the relations between the number of polymer molecules, N2, the molar amount, n2, the

Avogadro number, NA, the polymer mass, m2, the molar mass, M2, and the mass concentration, c2,

=2 A 2N N n , =2 2 2/n m M , =2 2m c V ,

equation (13) can be rewritten by

ηϕ = 2coil

2

A

cN V

M (14)

Combination of eq. (12) and (14) leads to

[ ] ηη = ⋅ A

2

2.5V

NM

(15)

Eq. (15) relates the intrinsic viscosity with the molecular mass and the hydrodynamic volume of

the polymer molecules. Now we discuss two special cases:

(1) In case of rigid spheres their hydrodynamic volume ⟨Vη⟩ is proportional to their mass: The

fraction on the right side of eq. (15) and therefore [η] becomes independent of the particle

size or mass, respectively. E.g., a dilute dispersion of 1000 particles with ⟨Vη⟩ = 1µL exhibits

the same viscosity as a dispersion of 500 particles with ⟨Vη⟩ = 2µL. So it is not possible to

determine the size/mass of rigid spheres from the viscosity of dilute dispersions.

(2) In case of polymer solutions, the hydrodynamic volume ⟨Vη⟩ is not proportional to the

molar mass. If we regard the special case of θ-conditions4, by definition the squared radius

of gyration ⟨rg2⟩ is proportional to the molecular weight. Then the relation between

hydrodynamic volume and molecular weight is given by ⟨Vη⟩ ~ ⟨rg2⟩3/2

~ M23/2

. Inserting

this relation in eq. (15) yields the molecular weight dependence of [η] at θ-conditions:

[ ] 0.5

θθK Mη =

, Kuhn-Mark-Houwink equation (16)

with M = M2: molecular weight of the polymer and

K constant of proportionality

Universally, the KMH-equation is formulated:

[ ] aK Mη = (17)

Which value attains the exponent a in the case of rigid spheres (case (1))?

4 The dimension of a polymer coil can be calculated by random walk statistics. Therefore it is assumed that every

segment of a polymer chain can be placed freely around connected segments. This assumption is somewhat

erroneous: Actual polymer repeat units occupy finite volumes and therefore exclude other segments from occupying

the same space. This excluded volume effect leads to an expansion of the polymer coil. On the other hand the

preference of polymer-polymer and solvent-solvent contacts as compared with polymer-solvent contacts in poor

solvents leads to a shrinkage of the coil. At θ-conditions, both effects neutralise each other.

5 - Viscosity of polymer solutions

5 - 12

3.3) Influences on [ηηηη]

For polymer solutions the Kuhn-Mark-Houwink exponent normally exhibit values between 0.5 and

1. The exponent increases with solvent quality: Good solvents expand the polymer coils, and

therefore the intrinsic viscosity is larger for a polymer dissolved in a good solvent than in a bad

solvent. The expansion of a polymer dissolved in a good solvent compared to a θ - solvent is

expressed by the expansion coefficient αη:

[ ][ ]

3

θ θ

V

V

ηη

η

ηα

η= = (18)

As temperature influences the solvent quality K and a are also temperature dependent.

Additionally, the intrinsic viscosity changes with the architecture and the stiffness of the

molecules:

The values for K and a of numerous polymer-solvent systems are listed in the literature. For an

unknown system, these parameters are determined by measuring monodisperse polymers of

different molecular weights.

If K and a are known, the molecular weight of a polymer sample with unknown chain length can be

determined by simply measuring the viscosity of a dilute solution.

For polydisperse samples, this approach yields the viscosity average of the molecular weight:

[ ] ηa

K Mη = (19)

with

1

η

aa

i i

i

M w M

= ∑ ,

iw being the weight fraction of polymer species i .

Please note that Mη is measured in dimensionless mass units (Dalton, corresponding to g/mol) to

avoid fractal units.

3.4) Dependence of viscosity on polymer concentration

Intrinsic viscosities are determined by measuring the reduced viscosity of dilute polymer solutions

as function of concentration and extrapolating to infinite dilution.

The extrapolation method of Huggins, for instance, bases on a plot of the reduced viscosity vs.

polymer concentration:

[ ] [ ]2specred H 2

2

k cc

ηη η η= = + (20)

with kH: Huggins coefficient

According to Schulz-Blaschke the independent variable is the specific viscosity instead of the

concentration c2. Therefore here ηred is plotted vs. ηspec:

[ ] [ ]red SB speckη η η η= + (21)

with kSB: Schulz-Blaschke coefficient

5 - Viscosity of polymer solutions

5 - 13

The Huggins coefficient kH and the Schulz-Blaschke coefficient kSB describe the concentration

dependence of dilute solution viscosity. They depend on the polymer-solvent system and

temperature as well as on molecular weight of the polymer.

In case of aqueous solutions of polyelectrolytes in the absence of salts, equations (20) and (21) do

not show a linear dependency even for diluted solutions: The electrostatic repulsion of the electric

charges of the polymer chain leads to an expansion of the polymer coil and therefore an increase

of ηred with increasing dilution. In this case the extrapolation c2 → 0 do not yield reliable results.

0,0 0,5 1,0 1,5 2,0 2,5

6

8

10

12

14

16

water/CMG 860

25 °C; no salt

ηsp

ec c

-1 /

L g

-1

c / gL-1

Fig. 9: Non-linear concentration dependence of a polyelectrolyte solutionD.

CMG: Carboxymethyl guar

3.5) From dilute to concentrated solutions: Coil overlap

The intrinsic viscosity is used to define a reduced (dimensionless) concentration c*:

[ ] 2*c cη= (22)

*c classifies polymer solutions in respect to their effective polymer concentration :

*c < 1: dilute solution, polymer coils and embedded solvent molecules moving in phase

*c > 6: concentrated solution (also called network solution), strongly interpenetrating polymer

coils are forming a macroscopic polymer network drained by the independently moving

solvent molecules.

c* is traditionally designated “degree of coil overlap”. This term is misleading, because [η] only

describes the volume of polymer coils in very dilute solutions.

5 - Viscosity of polymer solutions

5 - 14

4) Concentrated polymer solutions

4.1) Shear rate dependence of viscosity

Newtons law of viscosity (eq. 3) - which means a viscosity independent of shear rate - can be

applied only for few samples, e.g. homogeneous low molecular liquids. Most samples of practical

interest like dispersions or entangled polymer solutions and melts, show a strong non-linear

relation between shear stress τ and shear rate γɺ , corresponding to a shear rate dependent

viscosity η. In most cases the viscosity decreases with increasing shear rate; this rheological

behaviour is called shear thinning. The opposite behaviour – viscosity increase with shear – is

named shear thickening (fig. 10).

Fig. 10: Scheme of flow and viscosity curves with different flow behaviour

The experimental results are depicted in a plot shear stress vs. shear rate (called flow curve) or

viscosity vs. shear rate (called viscosity curve or also flow curve). Often logarithmic scaling is used

for both axes.

Flow curves are determined by rotational viscometers which are able to measure the viscosity at

controlled shear rate or shear stress within a range of several decades.

4.2) Shear thinning of polymer melts and concentrated polymer solutions

Polymer solutions show at low shear rates a constant viscosity, the plateau value η0 is called zero-

shear viscosity (fig. 11). Reaching the critical shear rate γɺ =1/τ0 the sample shows shear thinning.

This regime is called “power-law regime” and can extend over several decades in shear rate.

5 - Viscosity of polymer solutions

5 - 15

Fig. 11: Shear thinning of polymer melts and concentrated polymer solutions

According to GraessleyE, the shear thinning is caused by the following mechanism: in polymer

solutions and polymer melts polymer chains with molecular weights exceeding a critical limit Mc

are entangled with each other. In the unperturbed state (no shear), Brownian motion of the

polymer segments causes release of some entanglements and formation of new ones, until a

thermal equilibrium state with a constant density of entanglements is reached. At low shear rate,

the shear motion causes release of the entanglements. Since in this regime the motion due to

shear is small compared to the thermal Brownian motion, however, there is sufficient time to

allow for reformation of the released entanglements, and the overall entanglement density

therefore remains constant. At higher shear rate, the time becomes insufficient for reformation of

all released entanglements, consequently the number density of entanglements decreases with

increasing shear rate and the sample shows shear thinning. The critical shear rate c

γɺ where shear

thinning starts is the reciprocal of the characteristic relaxation time0

τ , which is the time needed to

form a new entanglement at thermal equilibrium.

It is observed that the onset of shear thinning for low to moderately concentrated polymer

solutions is governed approximately by the longest bead-spring relaxation time, eq. (6) with p=1.

Therefore it is convenient to introduce reduced variables: The viscosity is replaced by η/η0 and the

shear rate by η γɺ0 2 2/M c RT , which is roughly the product of shear rate and relaxation time of the

bead-spring model. Using these reduced variables, most of the observed variation among different

samples and systems can be removed.

5 - Viscosity of polymer solutions

5 - 16

5) Experimental Methods

The shear stress which is necessary to obtain a shear flow can generally be realised by a

mechanical drive or external pressure. Whenever gravitation is the driving force, we take in

account the hydrostatic pressure and/or buoyancy. In these cases the kinematic viscosity ν is

measured, including the density ρ of the sample:

ην

ρ= , [ν ]

2 mm

s= (SI) ,

2 m m1 1 cSt

s= (Centistokes, CGS) (23)

The kinematic viscosity is frequently used for samples which are measured by capillary

viscometers (e.g. oil).

5.1) Capillary viscometer

There exist different kinds of capillary viscometers. Nowadays in most cases Ubbelohde capillary

viscometers are used.

Fig. 12: Different types of capillary viscometers

Using capillary viscometers, the viscosity of Newtonian samples can be determined using the law

of Hagen-Poiseuille:

( ) 4

8

g h p RV

t h

ρ π

η

∆ + ∆∆=

∆ ∆ (24)

with V∆ the sample volume flowing through the capillary within the time interval t∆ , ρ the

density of the liquid, g the gravitational constant, h∆ the length of the capillary and p∆ the

difference in external pressure at the beginning and at the end of the capillary. R is the radius of

the capillary.

For the determination of intrinsic viscosities, only relative viscosities and no absolute values have

to be measured. Assuming identical densities of pure solvent and polymer solution, the viscosity is

5 - Viscosity of polymer solutions

5 - 17

proportional to the time which a defined sample volume needs to flow through the capillary.

Therefore the relative and the specific viscosity can be expressed by:

ηη

η= =rel

solv solv

t

t , η

−= solv

spec

solv

t t

t

An advantage of capillary viscometers is that the sample temperature can be easily controlled

(using conventional thermostats). This is very important since the viscosity depends strongly on

sample temperature.

5.2) Rotational viscometer / rheometer

Rotational viscometers are equipped with various measuring geometries; the most important are

cone/plate, plate/plate and concentric cylinder (frig. 13). In all cases, they consist of two axially

symmetric parts separated by the sample liquid. One part is kept stationary; the other is rotated

by a motor.

Fig. 13: Measuring geometries: Cone/plate, plate/plate and concentric cylinder

In the lab course we use a cone/plate geometry as shown in fig. 13. For this geometry, the

relations between the primary measured variables angular velocity φɺ respective torque M and the

rheological measures shear rate γɺ respective shear stress τ are given by:

/γ φ α= ɺɺ , τ = 33 /2πM R (25)

The advantage of cone and plate in contrast to two parallel plates is the constant shear rate in the

gap.

There exist two different measurement principles for rotational rheometers: 1) control of the

shear rate and measurement of the shear stress, and 2) controlling the shear stress and detecting

the resulting sample deformation. The choice between both methods depends on the rheometer

technology and the measuring procedure.

5.3) Falling ball viscometer

In the simplest case, this type of viscometer is a cylinder filled with the sample liquid and

containing a falling bead of radius R. After a short period of acceleration the ball reaches a

constant velocity v, which is defined by the equilibrium of gravitational force and viscous friction F.

The quantity which is measured here is the time it takes the ball to fall a defined distance.

5 - Viscosity of polymer solutions

5 - 18

6) Experimental Part

6.1) Rotational viscometry

The measurements with the rotational rheometer are performed together with the supervisor to

avoid damage of the instrument!

1. Measure the viscosity of three stock solutions of a high viscous poly-dimethylsiloxane sample

(M = 650000 g/mol) in an oligodimethylsiloxane at T = 20, 40 and 60 °C as a function of shear

rate. The concentrations are 0.14, 0.28 and 0.42 g/mL.

2. Plot log(η) vs. log(γɺ ) for all measurements. In a second graph, plot the reduced variables

log(η/η0) vs. log 0 2 2( / )M c RTη γɺ and discuss the result. Why coincide the measurements at

different temperatures better than for different concentrations?

6.2) Capillary viscometry

1. For the experiment, prepare 5 solutions of polystyrene in cyclohexane (concentrations ca. 3; 6;

9; 12 and 14 mg/ml) and 1 solution of PS in toluene (conc. ca. 6 mg/ml) one day in advance. All

samples are stirred overnight. For the calculation of exact concentrations (mass/volume at

measuring temperature), take the thermal expansion between room temperature and 34.5 °C

into account. The thermal expansion coefficients α=1/V (∆V/∆T) are 1.23·10-3

K-1

for

cyclohexane and 1.09·10-3

K-1

for toluene in the temperature range between room temperature

and 35 °C.

2. Measure the viscosity at TΘ = 34.5 °C for each of these solutions, using the Ubbelohde capillary

viscometers. Use filters to fill the solution into the capillary. Start by determining the flow time

for the pure solvent (for each capillary!).

3. Use the Schulz-Blaschke approach to determine kSB and [ ]η of PS in cyclohexane.

4. Use kSB = 0.31 (from literature, valid for PS in toluene at 34.5 °C) and the specific viscosity you

obtained for the PS/toluene system to determine [ ]η of this system.

5. Because of T = 34.5 °C corresponds to the θ-temperature of the PS/cyclohexane system, use the

determined intrinsic viscosities to calculate the coil expansion coefficient of PS for the

transition from the θ-solvent cyclohexane to the good solvent toluene.

6. Use the KMH-equation to calculate the molecular mass of the polymer coil for the

PS/cyclohexane system at θ-conditions (Kθ = 0.082 ml/g) and determine the mean sphere-

equivalent coil radius <Rη2>

0.5 for this system by combining eq. (15) with the definition of a

sphere: 30.5

24

3V Rη ηθ

θ

π = ⋅

5 - Viscosity of polymer solutions

5 - 19

7) Appendix: Translation of important expressions

Buoyancy – Auftrieb

Draining – Durchspülen

Flow curve – Fließkurve

Friction – Reibung

Shear rate – Scherrate

Shear stress – Schubspannung

Shear thickening – Scherverdickung bzw. scherverdickend (auch: dilatant)

Shear thinning – Scherverdünnung bzw. scherverdünnend (auch: pseudoplastisch, strukturviskos)

Torque – Drehmoment

Velocity – Geschwindigkeit

Viscosity curve – Viskositätskurve

Zero shear viscosity – Nullscherviskosität

8) Literature

A P. C. Hiemenz, Principles of colloid and surface chemistry, 2

nd ed. (1986), Marcel Dekker Inc.

B P. C. Hiemenz, Polymer Chemistry - The Basic Concepts (1984), Marcel Dekker Inc., New York

C J. D. Ferry, Viscoelastic Properties of Polymers, 3

rd Ed (1980), John Wiley, New York

D Badiger, Gupta, Eckelt, Wolf, Macromol. Chem. Phys. 209 (2008), p. 2087-2093

E W. W. Graessley, The Entanglement Concept in Polymer Rheology, Adv. Polym. Scie. 16 (1974)

6 - 1

EXPERIMENT 6: DYNAMIC MECHANICAL CHARACTERIZATION OF POLYMERS IN THE GLASS TRANSITION RANGE

1. Introduction

Investigations of the mechanical properties of polymers can be carried out from two very different points of view:

On one hand a prerequisite of the practical use of a material is knowledge about its mechanical properties. The relationship between deformation, stress and material property is described by a “constitutive equation”. An example for such a constitutive equation is Hooke’s law for an elastic spring:

force = force constant elongation

Such constitutive equations are usually derived empirically from experience.

On the other hand the macroscopic mechanical response of a specimen is closely related to its microscopic structure. When a force is applied to a material, its atoms change position in response to the force, this change is named strain. Measurements of the mechanical properties of polymers aim at the interpretation and understanding of macroscopic behavior on a molecular level.

The connecting element between these two views is statistical mechanics, where one tries to derive constitutive equations from a molecular model (as an example see Experiment Rubber Elasticity).

The ideal cases of mechanical properties of condensed matter are the ideal elastic body (Hooke’s body) and the ideal liquid (Newton’s liquid). The elastic body has a defined form and, by the application of external forces, is deformed to a new equilibrium form. In the ideal case of small extensions the energy needed for the deformation is stored elastically and after removal of the external force the elastic body returns to its initial form. A viscous liquid has, however, no defined form and flows irreversibly under the influence of an external force. Real materials show mechanical properties between these two extremes. While the stress-strain-behavior of metals and molecular crystals or inorganic glasses is in good agreement with Hooke’s law (small strains, no plastic deformation !), and for low molecular weight liquids in laminar flow Newton’s law is valid (low shear rates !), polymers –due to their structure– can show viscous and elastic behavior, depending on the temperature or time-/frequency-scale of the experiment. The viscous component causes a strong time dependence of the mechanical behavior. Unlike for metals, for the modulus of polymers the time- or frequency-dependence has to be taken into account.

In this lab course the relation between the temperature-dependence and time-dependence in the range around the glass transition is investigated. This script starts with a phenomenological description of viscoelastic behavior, introduces the measurement technique and discusses the Williams-Landel-Ferry equation for the description of the temperature-dependence. The derivation of the fundamental equations is limited to simple

Dynamic Mechanical Characterization

6 - 2

types of deformation (stretching, shear). The complete description using tensor notation can be found in the textbooks.

1.1 Experiments for the measurements of mechanical behavior

In the investigation of mechanical properties either the deformation is given and the tensile force is measured; the proportional constant is called modulus (Young’s modulus E for uniaxial tension, shear modulus G for shear tension). Or the mechanical stress is given and the deformation resulting in the material is measured. The proportional constant (deformation per stress unit) is called compliance (D for stretching, J for shear). For materials whose mechanical properties are not time-dependent applies E = 1/D, G = 1/J. This simple relationship is not valid for time-dependent mechanical properties ( E(t) 1/D(t); G(t) 1/J(t) ). Other experiments for the measurement of mechanical properties we want to mention are: biaxial tension, compression, elongational flow.

1.2 Linear viscoelasticity, Boltzmann superposition principle

In a linearly viscoelastic material the observed mechanical property varies linearly with its trigger, i.e. stress and strain are proportional. Polymers obey this condition at small strains. In this case the modulus is a material property, which depends only on temperature and the time or frequency, but not on the amount of strain.

An important assumption in the theory of linear viscoelasticity is the Boltzmann superposition principle. It states that strains of all deformations can be added linearly. Thus it allows to describe the state of stress or strain of a specimen from its entire deformation history.

Figure 1: Schematic curve of the storage modulus E', the loss modulus E", and the loss factor tan as

a function of temperature, time or reciprocal frequency

Figure 1 schematically shows the modulus of an amorphous polymer as a function of temperature or time/frequency. At low temperatures or high frequency amorphous polymers

Dynamic Mechanical Characterization

6 - 3

are glassy solids with a high modulus of 109 – 1010 Pa. With increasing temperature the modulus decreases in a relatively narrow temperature range by 3 – 4 orders of magnitude (glass transition range, see Experiment DSC). For high molecular weight flexible chain molecules follows a range of almost constant modulus (“rubber plateau”, 105 – 107 Pa). At sufficiently high temperature the modulus of an un-crosslinked linear polymer drops to zero and the polymer behaves like a liquid. Covalently crosslinked polymers do not flow and their modulus is almost temperature-independent till the decomposition temperature.

1 How does the modulus of a crosslinked rubber change with increasing temperature? Draw a diagram !

So in linear viscoelasticity measurements the relationship of stress and strain in dependence of time or frequency and of temperature are of interest. The following experiments come into question:

1. Stress relaxation measurement: The specimen is strained in a defined way and the stress is measured as a function of time or temperature.

2. Retardation or creep experiment: A defined force is applied and the deformation of the specimen is observed.

3. Dynamic mechanical measurements: Stress or strain is varied periodically. Usually sinusoidal oscillations are used at a frequency in cycles/s ( in rad/s). The reciprocal of is the oscillation cycle and defines the time scale of the experiment.

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1.3 Hooke’s body, Newton’s liquid

For the ideal specimens Hooke’s body and Newton’s liquid, the stress resulting from a strain is linear proportional

a) to the deformation (Hooke)

b) to the rate of deformation (Newton)

At sufficiently small strains these linear dependencies are also valid for real specimens, wherein the proportional constants (modulus, viscosity) are important material properties.

a) uniaxial tension b) shear

Figure 2: Definition of directions, deformations, and stresses by projection of a volume element on

one of the base areas.

Indexing:

1st index = area-normal index of the area, where the stress or strain acts

2nd index = direction of the stress or strain

Hooke’s law for tension

The strain 11 in direction x (see Figure 2a) is defined as

10

111 r

r (1)

Young’s modulus E is defined with Hooke’s law for uniaxial tension:

1111 E (2)

where 11 is the stress related to the initial cross-sectional area.

Hooke’s law for shear

The shifts u1 increase linearly with x2 (see Figure 2b). The shear strain 21 is therefore defined as

tandxd

2

121

u (3)

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6 - 5

The shear modulus G results from the (required) linear dependence of the shear stress 21 from the shear process

2121 G (4)

Young’s modulus represents the response of the material to changes in form and volume, while the shear modulus describes only the reaction to changes in form. E and G can be converted using a third material constant, Poisson’s ratio :

GE )1(2 (5)

For incompressible materials is = 0.5.

Newton’s liquid

A liquid which obeys Newton’s law is characterized by a linear dependence of the shear stress on the shear strain rate, where the proportional constant is the viscosity :

td

d 2121 (6)

In contrast to Hooke’s body the mechanical deformation work is completely dissipated. The energy dissipated per time and volume unit is proportional to the viscosity and the squared shear rate.

1.4 Description of viscoelastic properties by mechanical models

The basic viscoelastic phenomena can be described by simple macroscopic mechanical models. In the simplest form viscoelastic properties are described as a combination of an elastic spring (elastic element) and a viscous dashpot (damping element). The two components can be combined in series (Maxwell model) or in parallel (Voigt-Kelvin model). Other models involving more complex combinations of the two elements have also been discussed. Here we want to discuss the Maxwell model.

Figure 3: Maxwell element

The elastic spring of modulus E obeys Hooke’s law. The Newtonian dashpot contains a liquid of viscosity which obeys Newton’s law.

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6 - 6

The elements are in series and therefore the overall strain is given by the sum of the strains of the spring and the damping element

viseltot (7)

while the stress is identical in each element

viseltot (8)

With equations (2) and (6) results for the time-dependence of the overall strain

tEt d

d1dd (9)

This differential equation yields for a stress-relaxation experiment (d /dt = 0, = o)

tE

t

tEtE

d-

o e)( n integratio

d -d 0dd1

(10)

i.e. an exponential decline of the stress from the initial value o = E o with the “rate constant” E/ . The quotient E/ has the dimension of a reciprocal time, it is characteristic for a certain combination of elastic and viscous component. Its reciprocal /E is called relaxation time of the relaxation process.

After division by o results the time-dependent elasticity modulus

t

EttE-

oo

e)()( (11)

Eo is called relaxation strength.

2 Can a creep experiment in first approximation be described by the Maxwell model as well?

The Maxwell model can also be used as a simple example to demonstrate the effect of a dynamic load on the mechanical response of a viscoelastic body.

In a dynamic experiment with forced oscillation the strain is given by

)sin(o t (12)

If the specimen only showed elastic behavior (validity of Hooke’s law), for the stress would result

)sin()sin( oo ttEE (13)

with oo E .

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6 - 7

For a liquid, which obeys Newton’s law, applies

)cos(dd

o tt

(14)

i.e. stress and strain are 90° out of phase. Equation (14) also shows that the stress increases linearly with frequency.

The differential equation, which describes the Maxwell element at dynamic load, is given by

tE

tdd1)cos(o (15)

or

Et

tEdd)cos(o (15a)

The universal solution for this differential equation is

)sin(C)cos(B tt (16)

After differentiation and insertion into the differential equation B and C can be determined by comparison of the coefficients. With /E = results

)]cos(1

)sin(1

[ 2222

22

o tEtE (17)

After division by o the dynamic modulus is

)cos(1

)sin(1

)( 2222

22

tEtEE (18)

The first term of this equation is in phase with the rotation, while the second is 90° out of phase. The following parameters are introduced as a convention

22

22

1 ' EE (19)

221 " EE (20)

)cos(")sin('* tEtEE

The storage modulus E' describes the term which is in phase with the deformation and therefore a measure for the elastic response of the system. The loss modulus E" is the term which is 90° out of phase and proportional to the energy loss per oscillation period. E' and E" depend on the relaxation strength (Eo), the frequency ( ), and the characteristic relaxation time .

Alternatively the dynamic stress can be defined via the stress amplitude and the phase angle :

))cos()sin()cos()(sin()sin( oo (21)

Dynamic Mechanical Characterization

6 - 8

By comparison of the coefficients results

)cos( 'o

oE (22)

)sin("o

oE (23)

and )tan('"

EE

tan( ) is called loss factor. It characterizes the ratio of dissipated and elastically stored energy.

For the mechanical response of a simple viscoelastic body (Maxwell element) –according to equations (19), (20) – the dependence on the product is crucial. If the product 1, then E' and E" have the value ½ E. If > 1, then applies E' E ; E" 0, if < 1, then applies E' 0 ; E" 0. E has a maximum at = 1, when the excitation frequency is identical with the characteristic frequency (reciprocal relaxation time) of the viscoelastic process. The loss factor decreases in the Maxwell model with increasing , tan( ) =1/( ), i.e. at a given characteristic relaxation time the elastic fraction increases with increasing frequency.

Real viscoelastic materials can not be described by the Maxwell model. An empirical description can be achieved by combination of many Maxwell elements in parallel. Each of these Maxwell elements is characterized by its modulus Ei and its relaxation time i. Furthermore an additional time-independent element (spring with Ee) can be introduced to describe the behavior of crosslinked polymers.

The storage modulus and loss modulus then become

e2i

2

2i

2

i 1)(' EEE (24)

2i

2i

i 1)(" EE (25)

If a very large number of relaxation elements is considered, the discrete relaxation processes Ei, i can be described by a continuous relaxation function E( ) and equations (24) and (25) become

0

22

22

e d1

)( )(' EEE (26)

0

22 d1

)()(" EE (27)

Due to the wide characteristic time or frequency range in real viscoelastic materials viscoelastic functions are usually plotted logarithmically. In order to display the distribution

Dynamic Mechanical Characterization

6 - 9

function on a logarithmic scale, the integral term is extended with . With d / = d(ln ) and H( ) = E( ) results for E' and E"

)(lnd1

)()(' 22

22

e HEE (28)

)(lnd1

)()(" 22HE (29)

Figure 4: run of the response functions

H( ) is the relaxation time spectrum. This function describes the distribution of the relaxation processes that characterize a specimen, independent of the type of experiment. The functions 2 2/(1+ 2 2) and /(1+ 2 2) indicate how much a certain relaxation process ( ) at the frequency contributes to the mechanical response of the system (Figure 4). H( ) still has the dimension of a modulus (Pa). Alternatively H( ) can be written as a product RT/Me h( ) , where h( ) now is a dimension-less distribution function, which describes the

contribution of the single relaxation processes in units of RT/Me. Basis of that is the description of polymer melts analogously to covalently crosslinked rubbers (see Script Rubber Elasticity). Me is a characteristic molecular weight of high molecular weight polymers, which describes the density of the network of entangled chain molecules.

3 In zero approximation E" is directly proportional to H( ). Why?

When not the frequency, but the temperature varies in the experiment, a direct description by mechanical analogies is no longer possible. Here the temperature dependence of the characteristic relaxation processes must be known.

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6 - 10

2. Time-temperature-superposition

The complete characterization of the viscoelastic properties of polymers requires measurements in a very wide time- or frequency-range (ca. 10-20 decades). The experimentally accessible range is much smaller. In the case of dynamic mechanical measurements the frequency can usually be varied over a maximum of 3-4 decades.

Figure 1 has already shown schematically that the total range of mechanical properties can be scanned both by variation of the frequency and by variation of the temperature.

The effect of a temperature change on the modulus can be derived from the generalized Maxwell model. A temperature change influences the fore factor RT/Me , and also the molecular relaxation times i (with Ei = gi RT/Me)

)(1

)(g/R)('

12

i2

12

i2

ie111 TT

MTE T (30)

)(1

)(g/R)('

22

i2

22

i2

ie222 TT

MTE T (31)

In order to compare the relaxation functions of the two temperatures, they are first converted to the same fore factor. For this purpose one of the temperatures is chosen as reference temperature (here T1):

)(1

)(g/)(')('

22

i2

22

i2

i22112 TT

TTEE Tred (32)

The shift of the relaxation time of a certain molecular relaxation process with temperature is given by

)(/)( 12 TTaT (33)

where aT is called shift factor. Because of the form of equation (32) the same result is received when is multiplied with aT instead of , i.e. the modulus E'( ) has after correction to E'red at the frequency and the temperature T2 the same value as at a frequency aT and the temperature T1. The function aT (T) describes the change of the characteristic frequency of a relaxation process with the temperature.

If all relaxation processes have the same temperature dependence, i.e. if aT (T) f( i), it is possible to use the time(frequency)-temperature-superposition principle. To this end each isotherm is first reduced to a reference temperature. The resulting reduced values at different temperatures can be shifted onto a single curve by multiplying all frequencies measured at a temperature with a factor aT. With a logarithmic time(frequency) axis this equals an addition of log(aT). Figure 5 shows schematically how such a shift is performed along the frequency axis.

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6 - 11

Figure 5: Schematic procedure for the construction of master curves

Figure 6: Family of isotherms of the shear relaxation modulus(G(t)) of poly(isobutylene)

Figure 7: Master curve constructed from the data in Figure 6

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6 - 12

Instead of a series of curves at a limited frequency range for different temperatures results by this reduction an image of the viscoelastic properties over a wide frequency range at a reference temperature. Such a curve is called isothermal master curve. Additionally the shift factors log(aT) (T) in terms of the reference temperature To are obtained.

Figure 6 shows a set of isotherms, which are shifted to form a viscoelastic master curve in Figure 7. These are the results of shear creep experiments.

The applicability of the time-temperature-superposition principle implies that the same shift factors can be used for E', E" and tan .

4 When does the time-temperature-superposition principle fail?

2.1 Temperature dependence of viscoelastic behavior

The temperature dependence of the shift factors log(aT) can in many cases be described by the Williams-Landel-Ferry(WLF) equation

ref

refT TT

TT

2

1

c)(c

)alog( (34)

where Tref is the reference temperature and c1, c2 are constants which can be determined from the experimental data by linearization of the WLF equation. The form of the WLF equation is independent of the reference temperature, the constants c1 and c2 depend on the choice of Tref, but the product c1c2 is independent of Tref. In the theory of free volume a molecular meaning can be assigned to the coefficients c1 and c2.

When the glass transition temperature of a polymer is chosen as reference temperature, similar values of the constants of the WLF equation result for different polymers.

The WLF equation describes the temperature dependence of the viscoelastic behavior in a temperature range approximately till Tg + 70 K. At higher temperatures the behavior is better described by a constant activation energy (Arrhenius behavior). In the range where the system obeys the WLF equation an apparent temperature-dependent flow activation energy can be calculated from the temperature dependence of ln(aT):

T

H T

1d

)a(lndR (35)

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6 - 13

3. Assignments

In this lab course the dynamic mechanical behavior of a sample of weakly crosslinked natural rubber is investigated in a range of –100°C to room temperature (Rheometrics Solid Analyzer). Both a measurement at constant frequency and variable temperature ("temperature sweep") and a series of frequency-dependent measurements at various temperatures are performed.

The temperature sweep experiment is supposed to give a general impression of the mechanical behavior. In particular it is used to determine suitable temperatures for the frequency-dependent measurements.

The measurements are performed in a film geometry. The specimen is clamped in the apparatus at both ends. The dimensions are determined before the measurements. (Details Messaufbau?)

As the stress varies from the glass transition temperature to the rubber plateau by a factor 1000, the preload is determined from the last force value for each measuring point. The measurements at constant temperature are performed at a frequency range of 0.05 to 100 rad/s.

1.) Perform the measurements under the guidance of the lab technician.

2.) Discuss possible error sources a) in the isochronous measurement, b) in the isothermal measurements.

3.) Discuss the progress of E', E", and tan as a function of temperature.

4.) Construct a viscoelastic master curve from the frequency-dependent measurements at different temperatures.

5.) Analyze the temperature dependence: Can the behavior be described using the WLF equation? Calculate the constants of the WLF equation!

6.) Determine the temperature dependence of the activation energy.

4. Literature

J.D. Ferry “Viscoelastic properties of polymers”, 3rd Ed. Wiley New York 1980

P.C. Hiemenz “Polymer Chemistry – The Basic Concepts”, Marcel Dekker 1984

R.J. Young, P.A. Lovell “Introduction to polymers”, 2nd Ed. Chapman&Hall London 1991

7 - 1

EXPERIMENT 7: X-RAY DIFFRACTION

1. Theory

1.1 Morphology and diffraction behavior of semi-crystalline polymers

The arrangement of chains and the morphology are vitally important for the physical properties of solid polymers. Regularly built macromolecules are basically able to crystallize. However, the absolute equilibrium state, where fully stretched macromolecules are packed parallel analogously to the crystal structure of the corresponding oligomers, cannot be realized for kinetic reasons during freezing of the melt or precipitation from solution. Meta stable lamellar structures are formed instead during crystallization. Beside the crystalline areas exist amorphous areas, in which the chains have a coil conformation similar to the melt. The volume fractions of the crystalline and amorphous phase, vc and va = 1 vc (two-phase model), determine the degree of crystallization of the specimen. They depend on the thermal history and can also be varied in the solid state by annealing.

A two-phase structure (amorphous and crystalline areas) clearly appears in the X-ray diffraction diagram. In the fully amorphous state, the diffuse halo characteristic for liquid structures is observed. It is caused by short-range ordered states which also exist in unordered systems between neighboring molecules or chain segments, wherein the maximum represents the most probable distance. At increasing crystallinity this halo is superposed by sharp reflexes, which are caused by the crystalline areas. Basically it is possible to separate the diffraction curves into fractions of intensity for amorphous and crystalline areas. When a two-phase system without transition areas is assumed, for the overall intensity results

I( ) = (1 vc) Ia( ) + vcIc( ) (1)

where Ia represents the intensity of the purely amorphous case, Ic the intensity of the purely crystalline case. is half the scattering angle.

Vice versa results from (1) for vc

)()()()(

ac

ac II

IIv (2)

wherein any angle can be chosen.

Even low concentrations of defects in the regular chemical structure of the polymer chains, e.g., by copolymerization, branching or entanglements, result in a drastic broadening of the melting range with a shift to lower temperatures (cf. Experiment DSC). This “partial melting” is also reflected in the X-ray diffraction diagram. Figure 1 shows for a branched polyethylene (Lupolen 1800S) how the intensity of the crystalline reflexes decreases with increasing temperature while the amorphous halo increases.

X-ray Diffraction

7 - 2

Figure 1: X-ray diffraction diagrams of LDPE

1.2 The counter goniometer

The counter goniometer is particularly suitable for the registration of a diffraction diagram. Figure 2 shows its ray path.

F focus of the X-ray tube

B aperture unit

D detector

DK detector circle

FK focussing circle

A puncture of the rotation axis perpendicular to FK

Figure 2: Ray path of a counter goniometer

The primary radiation emitted by the X-ray tube is diffracted at a lamellar specimen and registered at the counter tube which moves along the detector circle. Crystalline reflexes appear at the angles which obey Bragg’s equation

2 dhkl sin = n (3)

X-ray Diffraction

7 - 3

where dhkl is the lattice distance for the lattice family with the Miller indices (hkl), n is the reflection order, and is the X-ray wavelength (when a copper anode is used = 1.542 Å). From the angular position of the halo the intermolecular distance of neighboring groups can be estimated, again by using Bragg’s equation (3).

1.3 Morphology and diffraction behavior of semi-crystalline polymers after stretching

When a polymer foil or fiber, in which the chains are at first arranged isotropically (i.e. without any preferential direction), is stretched, then an orientation results. In amorphous polymers an elongation of the previously coiled molecules takes place, which can be followed by a crystallization; in semi-crystalline polymers crystallites are destroyed followed by directional recrystallization. Usually this transition to the oriented state does not take place homogeneously and equally for the whole sample, but after formation of a constriction (“necking”). After this neck has appeared at an elongation of several percent at one particular point of the sample, during additional stretching it extends along the sample and thus converts it from the isotropic to the oriented state at a constant applied force. The changes in morphology during the orientation of the chains leads to a dramatically changed diffraction behavior. Usually so-called “fiber diagrams” result after stretching.

The diffraction pattern of unstretched foils or fibers in a plane film camera in the unoriented state consists of Debye-rings. After stretching, when all crystallites have oriented along a specific axis, the Debye-rings transform to sickle-shaped single reflexes, which are arranged on series of layer lines. The sickle-shaped broadening of the reflexes along the original Debye-ring can be interpreted as an incomplete orientation along the fiber direction.

The formation of the layer lines can be explained by means of the Ewald construction (Figure 3). The diffraction curves are identical with those of rotating single crystals. In both cases the reciprocal lattice has the form of concentric circles. Reflexes on the equator are from the lattice planes (hk0). The reflexes of the (hk1) planes are on the first-order layer lines. On the meridian are all (00l) reflexes.

The length of the designated c-axis can be determined from the distance of the layer lines. Bragg’s equation applies:

2c sin = n

where results from Ah2tan (see Figure 4).

h is the height of the 1st layer measured along the meridian, and A is the distance between the plane of the film and the sample.

Literature

L.E. Alexander, “X-ray diffraction Methods in Polymer Science”, Malabar 1985, especially Chapter 3.

X-ray Diffraction

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Figure 3: Formation of a “fiber diagram” of a uniaxially oriented sample

(a) distribution of reciprocal lattice points relative to the fiber axis

(b) Geometric relation between the distribution of reciprocal lattice points and the

diffraction pattern (fiber diagram)

S , So unit vectors in the direction of the incident and scattered x-ray, respectively

(from: M. Kakadu, N. Kusai, “X-ray diffraction by Polymers”, Elsevier Publ. Camp.

Amsterdam, 1972)

Figure 4: Illustration for the interpretation of 00l reflexes in a fiber diagram

X-ray Diffraction

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2. Assignments

Task 1 Investigation of the partial melting of low density polyethylene (LDPE) by X-ray diffraction measurements with a counter goniometer

a) Interpretation of diffraction diagrams at 26°C, 50°C, 75°C, 90°C, 100°C and 115°C (melt) in a range of angles between 2 = 10° and 2 = 40°

b) Under the assumption of the 2-phase model, the course of partial melting can be analyzed in the following way: After subtraction of the background the measured scattering curves are planimetered. According to a universal law of scattering theory the overall intensity should be unchanged. If the calculated values deviate anyway, the reason may be geometric effects like a change of the position of the sample in the ray, roughness of the sample surface, etc.

totalcc IwI

totalca IwI )1(

total

a

ac

c

ac

cc I

IFF

FII

Iw 1

First only the sum of amorphous and crystalline intensity is known from the experimental data. However, the amorphous fraction Ia of the semi-crystalline diffraction diagram can be determined from the intensity Ia0 in the melt, when we assume that the amorphous areas of the semi-crystalline sample have the same structure as the totally amorphous polymer in the melt. Therefore we can expect that the diffraction curves of the totally amorphous sample in the melt and the amorphous fraction of the semi-crystalline diffraction diagram are congruent. For all diffraction angles applies the following relationship between the integral intensity of the melt Ia0 and of the amorphous fraction of the semi-crystalline sample Ia (T):

T

aT

a Ihh

TI 1)()(

)( 00

T = measurement temperature

hT ( ) = intensity at temperature T at a fixed angle

h0 ( ) = intensity in the melt at a fixed angle

T = standardization factor Itotal (T)/Ia0

c) Plot the obtained degrees of crystallinity as a function of temperature. Please discuss the result !

X-ray Diffraction

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Task 2 Determine the volume of the unit cell of LDPE at room temperature and the temperature dependence of the available cell parameters

Polyethylene crystallizes in the orthorhombic crystal system. Here applies for the lattice distances dhkl as a function of the lattice parameters:

21

2221cl

bk

ah

dhkl

The c-value of the unit cell is 2.54 Å.

Task 3 Measurement of the fiber diagram of a nylon-6 fiber in a flat-plate chamber

A nylon-6 fiber is oriented by stretching, fixed in the stretched state and tempered for 20 hours. Then the structure shown in Figure 5 results. The identity period corresponds to the distance c. Determine from the flat-film measurement (distance sample-film 4.46 cm) the identity period in c-direction.

(Lit. J.L. White, J.E. Spruiell, J. Appl. Polym. Sci.: Appl. Polym. Symp. 1978, 33, 91-127)

Figure 5: Arrangement of the chains in nylon-6 crystals (a-c-plane)

X-ray Diffraction

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3. Suggestions for your homework

After this Experiment you should also be able to talk about the following topics:

a) Information obtained by X-ray diffraction diagrams

b) Definition and properties of the reciprocal lattice

c) Construction and properties of the Ewald sphere

d) Position of the lattice plane families, where the 200- and 110-reflexes come from, in the real PE lattice