Skript zum Praktikum. (Lab course macromolecular chemistry: physical chemistry )

Transcription

1 GRUNDMODUL IN MAKROMOLEKULARER CHEMIE : CHARAKTERISIERUNG VON POLYMEREN Skript zum Praktikum (Lab course macromolecular chemistry: physical chemistry ) Experiment : Experiment : Experiment 3: Experiment 4: Experiment 5: Experiment 6: Experiment 7: Experiment 8: Light Scattering Size-exclusion Chromatography (SEC) Osmometry Thermodynamics of Polymer Solutions Viscosity of Polymer Solutions Dynamic Mechanical Characterization X-ray Diffraction Asymmetrical Flow Field-Flow Fractionation

2

3 EXPERIMENT : LIGHT SCATTERING 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. Light Scattering. 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 M w, the z-average of the squared radius of gyration <R g > z and the second virial coefficient of the osmotic pressure A. 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 87 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 -

4 Light Scattering electrodynamics that the so called Rayleigh ratio of scattered intensity and primary beam intensity is given by: (-) R(θ) = Ir ² = I 0 8π ² λ 4 0 k= N α ( + cos ² θ ) k k with R(θ): Rayleigh ratio as a function of scattering angle θ I: Intensity of the scattered light I 0 : Intensity of the primary beam r: Distance of detector and scattering volume λ 0 : Wavelength of the primary beam in vaccum N k : Number of scattering centers α k : Polarizability of the scattering center k θ: 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 ( + cos θ) equals. 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: I solution I solvent (-) R(θ) = RRs tandard I s tandard With RR standard : 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 α 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. -

5 Light Scattering For small particles we can write: (-3) R(θ) = 4π dn dn M 0 c ρ n RTβ + n RT λ N L dρ dc ρ d µ 0 0 dc With N L : Avogardro constant ρ, ρ 0 : Density of the solution and solvent n, n 0 : Refractive index of solution and solvent R: Universal gas constant T : Absolute temperature in Kelvin K β : Isothermal compressibilty M 0 : Molar mass of the solvent c : Concentration of the solute material µ : Difference of the chemical potential of solution and solvent dn : Refractive index increment due to concentration changes dc dn : Refractive index increment due to density changes dρ In equation (-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: (-4) R(θ) = R(θ) solution R(θ) solvent = 4π n 4 λ N 0 0 dn dc RTM 0c d ρ0 dc L µ The change of the chemical potential with the concentration can be described as a change of the osmotic pressure with concentration: d µ M 0 dπ (-4) - = dc ρ dc 0 with : Osmotic pressure Use of a series development of the osmotic pressure with respect to concentration yields: - 3

6 Light Scattering dπ (-6) = RT + A + 3 c A3c +... dc M with A, A 3 : Virial coefficients of the osmotic pressure Introducing equ. (-5) and (-6) in equ. (-4) yields: Kc (-7) = + A c + 3A3c +... R( θ ) M 4π n0 dn with K = : 4 λ N dc 0 L Optical constant M : Molar mass of the dissoluted material Equ. (-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 λ/0) interference of the scattered light occurs (fig. -). Fig -: Interference of two primary beams scattered at the centers i and j of a large particle (larger than λ/0). - 4

7 Light Scattering The occurrence of interference leeds to a weakening of the scattered intensity with increasing scattering angle. For molecules of dimensions less than λ/ 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 θ = 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 θ=0 allows us to interprete the result in terms of the Rayleigh theory. Measurements at θ=0 are not possible because the primary beam intensity is much larger than the scattered beam intensity (factor of 0 6 ). Note that the effective scattering volume is also a function of the scattering angle. If a sin(θ) correction is conducted (Fig. -3) for small isotropic scattering molecules the scattering intensity is constant. Fig. -3: Changing of the effective scattering volume as a function of the scattering angle θ. 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 msec). Therefore an average value of all possible orientations and distances are measured. Theoretically, the following expressions are derived: N N r rr (-8) P( q) = exp( iqr ij ) N with q = r q 4π n θ = sin λ 0 : Norm of the scattering vector q r - 5

8 Light Scattering N: Number of scattering centers within a particle r r r = : Distance of scattering centers i and j ij i j n: Refractive index of the solvent Integration over all possible orientations from ϕ = 0 to π and θ = 0 to π yields: (-9) ( qr ) N N sin ij P( q) = with i < j N qr i j ij Formula (-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: N N q P ( q) r ij... 3! N (-0) = + i j (break off after the nd term) The mean squared radius of gyration is defined by: N r (-) Rg = ri = N i N N i N j r ij with r i : Distance vector of scattering center i from the center of mass of the particle Application of equ. (-) and (-0) yields for monodisperse particles: (-) P ( q) = q Rg Polydispersity does not only influence the form factor but also the mean squared radius of gyration: (-3) R g z = i i m M i i i m M R g i i with R : z-average of the square of the radius of gyration g z R : Square of the radius of gyration of particle i g i m i : Mass of the particle i M i : Molar mass of the particle i Therefore we can rewrite eq. (-7): - 6

9 Light Scattering Kc (-4) = + Ac + 3A3c +... R( θ ) M wpz ( q) 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. -4 is a sketch of the form factor of a gaussian coil as a function of the scattering angle. Fig. -4: Form factor P(θ) of a Gaussian coil as a function of the scattering angle θ Applying eq. (-3) and (-) and using the relationships + x x, or x + x we get the famous so called Zimm Equation (B.H. Zimm, J. Chem. Phys., 6, 099 (948)): - 7

10 Light Scattering Kc (-5) = + q R g + A c +... R( θ ) M 3 z w By plotting the data Kc R(θ ) as a function of q + kc we can extrapolate for c 0 and q 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 M w and from the slopes we get the z-average of the mean square of the radius of gyration R (from q dependence) and the second virial coefficient of the osmotic pressure A (from c dependence). g z Fig. -5: Extrapolation of light scattering data according to Zimm; : experimental values of Kc/R(θ), ο : extrapolated values.. 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 τ over a total observation time t=n τ with n the number of time intervals τ. - 8

11 Light Scattering Typically n= and τ=-000 µsec. The autocorrelation function g (t) is than calculated as (-6) g ( t ) = I( t = 0 ) I( t = n τ ) where the brackets <.> denote an average over typically correlations. From g (t) the correlation function of the electric field g (t) is derived by single (-7) g g( t ) A ( t ) = A with A an experimentally determined baseline, i.e. (-8) A = lim g t ( t ) = I In this limit the intensities I(t=0) and I(t ) are not correlated, i.e. (-9) Since I ( t = 0 )I( t = ) = I( t = 0 ) I( t = ) = I (-0) lim g ( t ) = t 0 I the intensity correlation function decays from <I > to <I>. For scattering centers undergoing Brownian motion it can be shown that g (t) is the Fourier transformation of the van Hove space time correlation function G( R,t r ) which expresses the probability that one particle moves a distance R within a time interval t. For particles undergoing Brownian motion G( R,t r ) is given by a Gaussian distribution of mean square displacements R : r π 3 (-) ( ) 3 R G( R,t ) = R ( t ) exp 3 R r r r r (-) g ( q,t ) G( R,t ) exp( i q R ) dr ( t ) ( t ) (-3) g( t ) = exp( 6 R t ) = exp( q D t ) with D = 6 R ( t ) 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: (-4) R h = R h z = kt 6πη D 0 z - 9

12 Light Scattering with k Boltzmann factor, T temperature in K, η 0 viscosity of the solvent..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 A. There exists a special case, where the contribution of enthalpy and entropy compensate each other; then A equals zero (but not A 3 = 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 A = 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..4 Molar Mass Dependence of the Radius of Gyration 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: R g z (-5) R g z = K M a (-6) ( R ) K a M log g z = log + log The exponent a only depends on the shape of the scattering particle: a = for rods, a =. for a Gaussian coil in a good solvent, - 0

13 Light Scattering a = for Gaussian coils with unperturbed dimensions (theta condition) a = /3 for compact spheres. The behaviour of Gaussian coils demonstrates that thermodynamic interactions are changing their dimensions. The coils expand in a better solvent. -

14 Light Scattering 3 Experiment 3. 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 000). Fig. 3-: Sketch of a light scattering apparatus (goniometer type) with incident beam I 0, λ 0, scattered beam i θ, scattering angle θ 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 0 0) 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. -

15 Light Scattering 3. Experiment and Data Evaluation The weight average molar mass M w, the second virial coefficient A and the radius of gyration (z-average) ( ) / R of a polystyrene sample dissolved in toluene (or g z 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 00ml at 0 C. From this solution 5, 0, 5, 0 and 5ml are taken and diluted in measuring flasks of 5ml volume. To remove dust, which considerably disturbs every light scattering measurement, these solutions of different polymer concentration are then filtered through a 0. µm filter suitable for organic solvents. Prior to use the sample cells (cylindrical quartz cuvette of 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.µm 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: ) Calibration of the primary intensity and correction for the scattering volume by measuring pure toluene ) 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 I Solution I Solvent (-7) ntoluene R( θ ) = R( θ ) Solution R( θ ) Solvent = RRToluene IToluene nsolvent with RR Toluene = cm - the Rayleigh ratio (absolute scattering intensity) of toluene at 63.8 nm and 0 C, n Toluene =.4960 refractive index of toluene, n Cyclohexane =.460 respectively. The data are evaluated by the application of the Zimm equation (equ. -7, -5) caclulating Kc/R(θ) using a refractive index increment of polystyrene in toluene of 0.09 ml/g (0.87 ml/g in cyclohexane) at 63.8nm. All the data Kc/R(θ) are plotted against q² + kc with an appropriate constant k using millimeter paper sheets and the value M w determined from the intercepts and <R g ²> / z and A 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 - 3

16 Light Scattering 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, 98 b) B. Vollmert, Grundriß der Makromolekularen Chemie, 988 c) J. Springer, Einführung in die Theorie der Lichtstreuung verdünnter Lösungen großer Moleküle, 970 d) R. J. Young, P.A. Lovell, Introduction to Polymers, 99 e) S. F. Sun, Physical Chemistry of Macromolecules, John Wiley, 994 f) K. A. Stacey, Light Scattering in Physical Chemistry, 956 g) Ch. Tanford, Physical Chemistry of Macromolecules, John Wiley, 96 h) P. Kratochvil, in Classical Light Scattering from Polymer Solutions, ed. A. D. Jenkins, 987 i) M. Schmidt, Simultaneous Static and Dynamic Light Scattering, in Dynamic Light Scattering, ed. H. Brown, Oxford, 993, p. 37 j) B. Chu, Laser Light Scattering, nd edition, Academic Press, 99-4

17 EXPERIMENT : SIZE EXCLUSION CHROMATOGRAPHY (SEC) 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. 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.% by weight) passes through a column of a porous gel/material at pressures of bar and a flow rate of typically ml/min. In contrast to HPLC the separation is based on a size exclusion mechanism and not on absorption. The pores of volume V x contain the same solvent than the interstitial volume V o (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 V o 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 -

18 SEC separation. They all elute with the same maximum elution volume (total permeation volume) V tot =V o +V x (see Fig. -), which exceeds the interstitial volume by the total pore volume V x. This is the lower limit of the range of separation, which is called the separation threshold ( total permeation ). V x V 0 V tot Fig.-: Scheme of a volume element of a SEC-separation column with total Volume V tot, pore volume of the gel phase V x and the interstitial volume V 0. 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 V x and the interstitial volume V o. 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= 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=. Particles whose sizes lie between the steric exclusion limit and the -

19 SEC separation threshold are more or less strongly retained. Their elution volume V e range between the dead volume V o and the maximum retention volume V tot : V e = V o + K V x with 0 K The distribution constant K is therefore a measure of the volume fraction of the pore volume for a given size.. 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 M i and its relative amounts H i (M i ) 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, 964): (-) V e = C C logm The constants C and C can be taken from the linear part of the graphical representation of logm vs. V e, 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. -) which can be interpolated by a polynomial function: (-) logm = A + B V e + C V e + D V e

20 SEC Eqn. (-) 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: Ve, II Ve, I (-3) C = S log( M M ) I II The selectivity factor S defined in this way is also used in the characterization of columns. Fig. -: Typical sigmoidal calibration curve obtained from polystyrene calibration standards of different molar mass in THF at 35 C with columns of pore size 0 6, 0 5, 0 4 and 0 3 A. 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 0 ppm of toluene (if the eluent is - 4

21 SEC THF) as an internal standard, which allows to correct the experimental elution volumes by: s tan corr exp V (-4) Ve = Ve exp V dard toluene toluene The standard peak of toluene s dard V tan toluene is defined by one single measurement. With this method it is also possible to correct fluctuations of the solvent flow by the pump. - 5

22 SEC. 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 V h 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: [ η ] > η chain > [ η] star [ ] comb These observations could be explained by Einstein s viscosity law originally derived for spheres (-5) [ η ] = which yields. 5 N = M V. A h 5 ρ (-6) = const. [ η ] M V h Although the constant depends slightly on the chain architecture and on the solvent quality, log([η] M) vs. V h 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([η] M) vs. V e represents a universal calibration (Fig. -3) which, however, requires [η] to be known. - 6

23 SEC Fig. -3: Universal calibration according to Grubisic, Rempp and Benoit (967) As mentioned above the hydrodynamic volume is related to the Staudinger-Index and the molar mass by: V h = const. [η] M The Staudinger-Index is related to the molar mass according the Mark-Houwink relation: [ η] = K M α Therefore we get: V h = K M α + For two different polymers with the same hydrodynamic volume it helds: V = V h, h, K M = K M α + α + and K α + α + M = ( M ) or in logarithmic representation: K log M = K α + + log log α + K α + M 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. - 7

24 SEC Tab. -: Some Mark-Houwink-Constants at 5 C in THF Polymer K/0-3 ml g - α Polystyrene Polyisobutylene Polymethylmethacrylate Polybenzylmethacrylate Poly(tert.-butylmethacrylate) Poly(tert.-butylacrylate) Poly(tert.-butylvinylketone) 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 Gaussianlike profile (Fig. -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. -4: Concentration profile of a sample before entering and after leaving a SECcolumn 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 σ. Another definition uses the broadness of the peak β at which the amplitude has decayed to /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. -5). - 8

25 SEC Fig. -5: Detector signal as a function of elution volume V e : Gaussian peak shape of standard deviation σ. For true Gaussian shapes all three definitions are equivalent: N = V e,max σ = Ve,max Ve,max 8 = 6 β b 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 /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. 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 (00A) 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. - 9

26 SEC 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. 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 0µm 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 0 M 0 7. 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. - 0

27 SEC Tab. 3-: Pore size and range of separation Pore size Range of separation 0 A 0 M A 3 0 M A 0 4 M Detectors In routine SEC-measurements detection by the index of refraction and UV/VIS absorption are common. The amount of polymer which is necessary (mg dissolved in ml 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/. 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 (λ=60 nm for polystyrene, λ=30 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/VISdetectors 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 R g -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). -

28 SEC 4. Theory 4. Molar mass distributions For the calculation of molar mass distribution from kinetic assumptions it is useful to apply the degree of polymerization P n instead of the molar mass. The degree of polymerization is the number of repeat units connected to a polymer chain. The molar mass M i is related to the degree of polymerization by: M i =M o P n with M o 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: n( P) h( P) = and h( P) = n( P) P P 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: m( P) = m( P) m( P) P Both distribution functions are related by m(p)=p M 0 n(p) with M 0 the molar mass per repeat unit. 4. 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. number of monomers P n = number of chains The number average degree of polymerization corresponds to the mean value ( st moment) of the number distribution. The i-th moment <x i > is defined by: x i = i x f ( x) f ( x) -

29 SEC The zeroth-moment of a distribution corresponds to the area defined by the distribution function (usually ), the st moment <x> corresponds to the mean value of the distribution. The standard deviation σ is given by σ=<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: ) ( ) ( ) ( 0 = = = P P h P P P h P P h P n For the weight average value of the degree of polymerization we obtain: = = = w w P P P w P w P P w P ) ( ) ( ) ( w(p) equals h(p) P and therefore: h h w P P P P h P P h P ) ( ) ( = = The polydispersity D and the nonuniformity U are defined by: n w P P D = and = = D P P U n w Introducing the average values leads to: n n h h h h P P P P P P U σ = = = The polydispersity index is identical with the square of the normalized (with respect to P n ) standard deviation of the number average distribution. For a monodisperse sample the nonuniformity U equals zero and the polydispersity D equals. For Gaussian distribution (symetric) the full width at half maximum is given as: U P P n = = σ - 3

30 SEC 4.3 Typical Molar Mass Distribution Functions of Polymers 4.3. Living Polymerization For a living polymerization the degree of polymerization is given by: with P n [ M ] 0 [ M ] t [ M ] 0 = κ = κ α [ I ] [ I ] [M] 0 : monomer concentration at time zero [M] t : monomer concentration at time t [I]: initiator concentration κ : degree of coupling (κ= for monofunctional initiators, κ= for Na/naphthalene) α: extent of reaction It leeds to the extremely narrow Poisson distribution: ( ν ) exp ν h( P) = ( P )! P w( P) = P P exp( ν ) ν ( ν + ) ( P )! P n is given by: P = ν + n P w is given by: P w = P n + P n = P n + Pn P n + Therefore we get : 4.3. Polycondensation U = P + P P with P n >> n n n For the polycondensation reaction the so called most propable or Flory distribution is found: P h ( P) = α ( α) The most frequent particles are always monomers! w( P) = P ( α ) α P The average values are: P n = and α Free Radical Polymerization + α Pw P w = and therefore U = = α α P In radical polymerizations the degree of polymerization is given by the kinetic chain length ν which is the ratio of the rate of propargation to the sum of the rates of chain termination and chain transfer: ν = RP R + R t tr [ M ] [] I n - 4

31 SEC Therefore the degree of polymerization is given by P = ν for termination by recombination n and P = ν for termination by disproportionation. n For an ideal radical polymerization the Schulz-Zimm distribution function is obtained: κ h( P) = P κ P κ κ exp P n P n with the degree of coupling κ For chain termination by recombination: κ = For chain termination by disproportionation: κ = If κ = then h ( P) = exp( P / Pn ) which is a simply decaying function. P n If κ = then h( P) = P P exp P n P n this is a function with a maximum which is slowly decaying. The polydispersity D is given by: κ + D = which means that for recombination D =.5 and for disproportionation D =. κ Fig. 4-: Comparison of Poisson-distributions (---) and Schulz-Zimm-distributions with P n =00 and κ= (..) and κ= (- - -). - 5

32 SEC 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-: 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 V ei corresponds to a molar mass M i (see calibration curve). The corresponding height of the detector signal H i is proportional to a mass concentration c i of molar mass M i. If the frequency H i is normalized it corresponds to a mass fraction. For the average values it becomes: M M n w nim i mi ci = = = n / M / M = i mi nim i M i = = = n M c i i ci i i ci H M i H i i i H Therefore a table has to be established consisting of the following columns: No. V e (ml) H i (mm) H i (%) ΣH i (%) M i H i /M i H i M i This allows also the installation of the integral and differential mass distribution by plotting ΣH i vs M i and H i vs M i respectively. Nowadays these calculations are done with the help of computer systems which allow a fast and precise evaluation of the experimental data. i H i / M i - 6

33 SEC 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/VISand a RI-detector. The two signal intensities S and S correspond to: S S = f c + f c = f c + f c with the sensitivity factors f ij of detector i and component j which have to be determined by calibration measurements of the homopolymers. The concentrations c j have to be normalized according to: w c = and c + c w c = c + c The concentration determination would be better the higher the f ii and the lower the f ij 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- (λ = 60 nm for polystyrene) and RIdetection 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: P(V e ) w P + w P At the corresponding elution volume V e the values P and P 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. - 7

34 SEC 5 Experimental Task. For the polystyrenes obtained by radical and anionic polymerization the following quantities should be determined by using the easier slice method (0 slices for the radical sample, 0 slices for the anionic sample): - number and mass distribution (mass distribution both differential and integral) - the average values of M n, M w and P n, P w - 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.. 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. V e / ml Molar mass / g/mole , , , , , , ,00 5. From the elugram of,-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. - 8

35 SEC 6 Additional Remarks For taking the SEC-measurements the following samples are used:. conventional free radical polymerization. controlled radical polymerization 3. anionic polymerization/macromonomer synthesis 4. block-copolymer Please bring these samples to the assistant of the SEC-experiment. 7 Questions. Which are the contributions to the SEC-column volumes?. 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 W.W. Yau, J.J. Kirkland, D.D. Bly, Modern Size-Exclusion Liquid Chromatography, John Wiley and Sons, New York (979) G. Glöckner, Polymer Characterization by Liquid Chromatography, Elsevier (987) 3 J. F. Johnson, R.S. Porter, Analytical Gel Permeation Chromatography, Wiley N.Y. (968) 4 M.J.R. Cantow, Polymer Fractionation, Academic Press N.Y. (97) - 9

36 EXPERIMENT 3: OSMOTIC PRESSURE. Theory.. 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 M n or the weight average M w of the molar mass... Membrane Osmometry One of the methods yielding the number average molecular weight M n is the determination of the osmomtic pressure of a polymer solution via membrane osmometry.... Osmotic Pressure The osmotic pressure π of a solution of a component in a solvent (component ) is defined as: µ RT Π = = ln a V V () V : molar volume of the solvent µ = µ - µ pure : difference between the chemical potential of the solvent in the solution and the pure solvent a : activity of the solvent in the solution 3 -

37 Osmotic Pressure Setting a = x = x (x : molar fraction) and using the Taylor-expansion for the logarithm leads to the osmotic pressure π of a solution: Π = RT V ln( x ) = RT V ( x x 3 x ) () The higher terms can be omitted for infinite dilution: Π id = RT V x (3) With V (n + n ) V and x n n + n = follows : Π idv Π = c = n RT RT M (4) c : concentration of component M : Molar mass of component The deviations from the ideal behavior are treated in analogy to the gas pressure by a virial expansion: Π c = RT( M + A c + A c ) (5) with V A, id = M (6) for infinite dilution. A, A 3 : second, third virial coefficients 3 -

38 Osmotic Pressure With ΣniM i c =, n V cv ΣniM i M = = n Σn ges i ges = Σn = M n i (7) follows for a polymer in solution: Π c lim c RT = ( M n Π = c + Bc +...) RT M n (8) (9) c: concentration of the polymer in solution M n : number average of the molar mass B = RTA 3-3

39 Osmotic Pressure. Experimental Setup Measurements of the osmotic pressure π of a polymer solution can be carried out in the type of cell represented schematically in figure. The polymer solution is separated from the pure solvent by a membrane, permeable only to solvent molecules. Initially, the chemical potential µ 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 : Basic setup of a membrane osmometer 3-4

40 Osmotic Pressure In this case, the osmotic pressure can be expressed by the following equation, with the density of the solution ρ and the gravitational constant g: g h ρ = π (9) Modern instruments make use of a pressure sensor to detect the osmotic pressure. A scheme of an instrument is shown in figure. Figure : Scheme of a membrane osmometer The important parts of this instrument are the membrane (), the pressure sensor (9) which has to be calibrated before measurement, and the solution compartment (5). 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. 3-5

41 Osmotic Pressure 3. a) Determination of the number-average molar mass of a polyethyleneoxide sample with high polydispersity In order to determine the M n 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 M n of the polyethyleneoxide sample, 4 solutions in degassed water with 4, 5, 6 and 8 mg/ml have to be prepared. 4.Questions. Calculate M n and A in SI-Units of the measured samples. Use the intercept and the slope of the π/c versus c-diagramm.. 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, nd Ed., Blackie Academic & Professional, London

42 EXPERIMENT 4: THERMODYNAMICS OF POLYMER SOLUTIONS All participants are requested to register the day before the hand-on training starts in the laboratory 0 3 building K to prepare the solutions (time required: approx. h). Otherwise the experiment cannot be carried out within one day. Introduction The practical importance of polymers is beyond doubt as becomes obvious in every-day life. The significance of these products is not restricted to the area of materials, macromolecules are also of great pharmaceutical importance and as essential modifying agents in many applications. Most of the synthetic compounds are prepared and processed in the liquid state, i.e. in solution or in the molten state. Detailed knowledge on this state is therefore indispensable. In particular it is essential to know the limits of complete miscibility with a low molecular weight solvent as a function of temperature, pressure and composition. Furthermore it is often mandatory to be acquainted with shear induced changes in the segregation of a second phase, which may either be liquid or solid. The present experiments are meant to provide some insight into the physicochemical features of polymer containing mixtures. Truly binary systems (non-uniformity U=0) For the present consideration we assume that the polymer consists (like typical low molecular weight compounds) of one kind of molecules only. In other words we assume that all polymer chain have the same length (molar mass). For synthetic polymers this assumption is never true. In this case the number average molar mass M n - obtained by counting the molecules (osmosis) is always less than the weight average M w - resulting from weighting (light scattering). Concerning the definition of M n and M w please consult the literature. The width of the molecular weight distribution can be quantified by the molecular non-uniformity U defined as M w U = () M n In the limit of a uniform material U 0. With some polymerization or fractionation techniques it is possible to realize very small U values. In these cases the present consideration become approximately true. 4 -

43 Thermodynamics of poymer solutions Phenomenology An example for a typical phase diagram obtained for small U values is shown in Fig. for the system@ cyclohexane/polystyrene. In this case a second liquid phase is segregated from the homogenous solutions upon cooling as well as upon heating. Only within a certain limited temperature range the components are completely miscible. For some systems the miscibility gap at low T and that at high T overlap. In this case it is impossible to observe complete miscibility at any temperature (constant pressure). There only exists a characteristic T where the polymer can take up the largest amount of solvent (swelling of the polymer) and the solvent is able to take up a limited amount of solute. With many systems one does not observe phase separation upon cooling, because the solvent solidifies before it becomes sufficiently poor to induce demixing. Analogously the solvent often boils off at atmospheric pressure before the two-phase state is reached. Fig. : Phase separation upon cooling and upon heating for the system cyclohexane/polystyrene and the indicated molar masses (M w in Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4 -

44 Thermodynamics of poymer solutions kg/mole); w is the weight fraction of polystyrene. Saeki, S, et al. Macromolecules 6(), The measuring data of Fig. were obtained by cooling or heating a given homogeneous solution until it becomes turbid at the so called cloud point temperature T cp because of the segregation of a second phase. The reason for this milky appearance lies in the normally pronouncedly different refractive indices of the components. The dependence of T cp on the composition of the mixture is called cloud point curve. For small U the two cloud points belonging to a given molar mass and constant temperature (cf. Fig. ) constitute the compositions of the coexisting phases. If one adds successively polymer to a certain amount of solvent one moves along a line parallel to the abscissa until the cloud point at the lower polymer concentration is reached. Up to that point the mixtures is homogeneous. As further polymer is added, the first droplet of a second phase (gel phase) is segregated from this mixture (sol phase). The composition of the gel is in the present case given by the second cloud point at the given temperature. Adding more polymer does not change the compositions of the coexisting phases but only increases the volume of the polymer rich phase until the last droplet of the sol phase disappears. The mixture remains homogeneous up to the pure polymer. The line connecting the points representing the sol and the gel phase, respectively, for T=const. is called tie line. A more detailed analysis of the phase diagram reveals that the two-phase regime can be subdivided into two areas, within one the mixture unstable within the other it is metastable. The line separating these regions is called spinodal line. Fig. shows the situation schematically for a system exhibiting a so called upper critical solution temperature (UCST, phase separation upon cooling). In this case the tie lines degenerate into a single point (at the critical temperature T c and at w c, the critical weight fraction of the polymer; w c, is for U=0 given by the maxima of the cloud point curves) as T is raised. For the opposite case (phase separation upon heating) we speak of a system exhibiting a lower critical solution temperature (LCST). Subject to the condition U = 0 the ends of the tie lines (the so called coexistence curve) coincides with the cloud point curve. For mixtures of low molecular weight compounds (U = 0 is automatically fulfilled) the critical composition (extrema of the coexistence curves) are normally close to : mixture. With polymer/solvent systems w c is the more shifted towards lower values, the higher the molar mass of the polymer becomes (cf. Fig. ). In the limit of infinitely long chains w c 0. The temperature at which this situation is reached, is normally Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-3

45 Thermodynamics of poymer solutions called Θ (theta temperature, cf. viscometric experiments). As can be seen from Fig., there exist two theta temperatures for the system cyclohexane/polystyrene, one for endothermal conditions, corresponding to the UCSTs, and another one for exothermal conditions, corresponding to the LCSTs. Fig. : Schematic phase diagram (after Derham and Goldsbrough and Gordon 974) for solutions of a molecularly uniform polymer. Polymer lean phase (sol): A stabile; B metastable; C unstable, segregation of a gel phase. Polymer rich phase (gel): D stabile; E metastable; F unstable, segregation of a sol phase. Binodal curve and spinodal curve touch each other at the critical point. Within the metastable regime a solution may remain homogeneous upon standing for a very long time, despite the possibility to reduce the Gibbs energy upon phase separation. Under these conditions the demixing process takes place via nucleation and growth. For values of temperature and composition located inside the spinodal line, on the Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-4

46 Thermodynamics of poymer solutions other hand, phase separation takes place spontaneously, because any fluctuation in concentration will inevitably right away lead to a reduction in the Gibbs energy. Consistent with the different demixing processes, the morphology of the two phase systems looks markedly different as demonstrated in Fig. 3. Fig. 3: Micrographs of the phase separated system phenetol/polyisobuten 87. Upper picture: A solution was slowly cooled from the homogeneous region (75 C) to 5 C into the metastable region ( K/h; mechanism: nucleation and growth). Lower picture: This time a solution was cooled rap- Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-5

47 Thermodynamics of poymer solutions idly into the unstable region ( K/s; mechanism: spinodal decomposition); M. Heinrich thesis, Mainz 99 In the case of a nucleation and growth mechanism the individual droplets of the minor phase formed in the early state of the process grow slowly. They are dispersed in the matrix of the corresponding coexisting phase and can become rather large. For spinodal decomposition, on the other hand, the size of the coexisting phases is usually at least one order of magnitude less and the morphology is co-continuous, i.e. for each phase it is possible to find paths through the entire system without the necessity of penetrating into the other coexisting phase. With mixtures of low molecular weight liquids these structures are quickly lost upon standing. The driving force for this process is the minimization of the interface (contributing to high values of the Gibbs energy). Eventually the coexisting phases are separated macroscopically and divided by a meniscus. With polymer mixtures the morphologies prevailing at the early stages of phase separation are often frozen in (e.g. because of the glassy solidification of one phase upon cooling) and constitute the basis of some special properties of such blends. Fig. 4 shows an example for the spatial distribution of the phases in a commercial product. Fig. 4: Scanning electron micrograph of a 70/30 blend of EPM and polypropylene; the EPM phase was extracted with heptane, leaving the PP. Encyclopedia of Polymer Sci. Vol. 9, p. 779 Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-6

48 Thermodynamics of poymer solutions Binodals and spinodals The following discussion in terms of phenomenological thermodynamics is based on the Gibbs energy, G, of the system. Quantities referring to one mole of mixture are characterized by a stroke above the symbol ( X ), those referring to one mole of segments (where the segment can be defined arbitrarily and is normally defined by the volume of the solvent or set 00 ml/segment) by a double stroke ( X ). The latter option is considerable more suitable for polymer containing systems, because of the fact that one mole of a truly high molecular material has a mass of approximately one ton. It is, however, essential to keep in mind that mole fractions are still the basis for all thermodynamic consideration due to the fact that segments are bound together and do not constitute independent units. Once the size of a segment is defined (e.g. in terms of volumes), one can calculate the number N i of segments of a polymer species i as N V i i = () V seg In many cases the molar volume of the solvent is set equal to the molar volume of the segment. The Gibbs energy of n moles of component and n moles of component is calculated from the segment molar or molar quantities as ( ) ( ) G = G n N + n N = G n + n (3) The coexistence of different phases under equilibrium is bound to the condition that the chemical potential µ must be identical in all phases. We are presently only interested in liquid/liquid phase equilibria (i.e. the two phases have the same state of aggregation); this means that we need only account for differences in the Gibbs energy of mixing and can write µ = µ (4) ' i '' i For many purposes volume fractions ϕ are employed as composition variables. For a binary mixture containing components that are made up of more than one segment, ϕ is given as nn i i ϕ i = nn+ nn With the definition of the chemical potential of component (5) Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-7

49 Thermodynamics of poymer solutions µ G = n pt,, n (6) and analogously of component we obtain the following relations (cf. eq()) ( nn + nn ) G µ = = n ( ) pt,, n G ϕ = N G+ nn+ nn ϕ n (7) where ϕ = n ϕϕ n (8) so that eq (7) becomes nn + nn G µ = N G+ N ϕϕ nn ϕ (9) After some rearrangement we obtain = N G+ ( ) G µ ϕ ϕ (0) In terms of molar quantities this equation reads G µ = G+ ( x) () x Because of the above relations one can obtain the chemical potential of component by means of the tangent to the curves describing the composition dependence of the Gibbs energy of mixing from the intercept with the ordinate (ϕ =), as demonstrated in the lower part of Fig. 5. Analogously the chemical potential of component results from the intercept at ϕ =. The chemical potentials of a given component must be identical in the coexisting phases as formulated in eq (4). In case the system exhibits limited mutual solubility it is therefore possible to determine the composition of the coexisting phases by means of a common tangent (double tangents, cf. upper curves in the lower part of Fig. 5). Repeating this construction for different tempera- Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-8

50 Thermodynamics of poymer solutions tures and plotting T on the ordinate and the corresponding compositions on the abscissa yields the binodal curve shown in the upper part of Fig. 5. T = 60 K N = N = 3 80 T c = 300 K 300 g = g c K - ( T - T c ) 30 ϕ c = Binodale T c T / K 80 Spinodale 60 0 G / (J/mol) µ / N µ N ,0 0, 0,4 0,6 0,8,0 ϕ c ϕ Fig. 5: How to construct a phase diagram knowing the composition dependence of segment molar Gibbs energy of mixing Another totally equivalent possibility to determine the composition of the coexisting phases makes use of the condition that the Gibbs energy of any equilibrium system must become minimum. Out of any conceivable combination of coexisting phases the one with the lowest Gibbs energy of the entire system will under these Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-9

51 Thermodynamics of poymer solutions conditions be realized. To find that minimum for a given over-all (brutto) composition b of the mixture ϕ, one calculates the Gibbs energy G b of the entire system for all ' b " b possible pairs ϕ < ϕ and ϕ > ϕ. Fig. 6 gives an example for this procedure, for the thermodynamic conditions used to calculate the uppermost curve of the lower b part of the previous diagram and setting ϕ equal to 0.. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-0

52 Thermodynamics of poymer solutions Fig. 6: Segment molar Gibbs energy of mixing for a phase separated b system (constant over-all composition ϕ = 0.) as a function of the composition of the coexisting phases. The exact location of the minimum is hard to read from this representation. For " ' this reason we reduce the number of variables by one, introducing ϕ = ϕ - ϕ and keeping (for purely heuristic reasons) the phase volume ratio constant at the equilibrium value. The result of this evaluation is shown in the following graphs for various b ϕ values. It is self-evident that the tie lines calculated from the minima in G must not depend on the over-all starting composition (lying inside the two phase regime). Another interesting feature consists in the fact that G may initially rise as ϕ becomes larger (Fig. 7) before the minimum is reached. This behavior is indicative for the passage of metastable states. Fig. 7: Segment molar Gibbs energy of mixing for a phase separated system (at the indicated over-all compositions) as a function of the difference ϕ in the composition of the Fig. 8: As. Fig. 7 but for different over-all composition Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4 -

53 Thermodynamics of poymer solutions coexisting phases. Fig. 9: As. Fig. 7 but for different over-all composition b The upper curve of Fig. 8, corresponding to ϕ = 0,588, is the first one, which does no longer exhibit the initial ascend upon rising the over-all polymer concentration. This implies that we have chosen an over-all composition located on the spi- b nodal line. As this ϕ metastable. b In case one selects a ϕ value is surpassed the mixtures become unstable instead of value located inside the homogeneous region of the phase diagram the Gibbs energy of the hypothetically phase separated mixture increases steadily as ϕ rises. This situation is depicted in Fig. 9. The points of inflection of the curves of the lower part of Fig. 5, representing the spinodal conditions in terms of Gibbs energy, are mathematically given by the condition Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4 -

54 Thermodynamics of poymer solutions G = 0 () ϕ In the critical point of the system, where the binodal line and the spinodal line touch, the minima and the points of inflection coincide and the third derivative also becomes zero 3 G = 0 (3) ϕ 3 In the vicinity of the critical composition of the system and close to the critical temperatures the curve G ( ϕ) is almost linear as demonstrated in Fig. 5. Flory-Huggins theory Processes taking place at constant temperature and constant pressure are normally dealt with in terms of changes in the Gibbs energy G, which are made up of an enthalpy contribution where T is the absolute temperature. H and an entropy contribution S according to G = H T S (4) Perfect mixing takes place athermally ( H = 0) and the volume of the mixture does not differ from the sum of the volumes of its constituents (volume of mixing V = 0). In this case the driving force for the formation of a molecularly disperse mixture consists exclusively of the changes in entropy associated with the mixing process, i.e. in the higher number of arrangements of the molecules in the mixed state. The just described limiting situation is usually called perfect mixing (by approximation sometimes realized with mixtures of gases or mixed crystals) and the following relation holds true S R perf = + (5) x ln x x ln x where R is the universal gas constant and x i are mole fractions. For the Gibbs energy of mixing we thus obtain perf perf G = T S (6) Real mixture normally deviate considerably from the behavior described above. In order to maintain a well defined reference state one introduces so called excess Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-3

55 Thermodynamics of poymer solutions quantities, measuring the deviation from perfect mixing, as formulated in the following equations. perf E G = G + G (7) where E E G = H T S (8) This procedure is very useful for mixtures of low molecular weight compound. For polymer solutions and polymer blends the deviation from perfect conduct is, however, so pronounced that another reference behavior is advantageous. For linear macromolecules Flory and Huggins have therefore developed the concept of combinatorial mixing. To this end each molecule is subdivided into individual segments, which are in their size usually fixed by the volume of the solvent or (by definition) by a volume of 00 ml/segment (cf. page 7). This approach uses a lattice onto which the different segments of the individual molecules can be placed, as shown by the two-dimensional sketches of Fig. 0. Fig. 0a: Lattice model for a mixture of low molecular weight compounds Fig. 0b: Lattice model for a mixture of chain molecules The situation for a mixture of low molecular weight compounds (N = N = ) is depicted in part a of this graph for an equal number of black and white entities. Let us assume that this sketch stands for one mole of mixture. The combinatorial entropy can then be easily calculated from eq (9). Part b of this Figure differs from part a only by the fact that we invariably connect 5 of the white molecules and 0 of the black molecules by a chemical bond to form a white penta-mer (N = 5) and a black Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-4

56 Thermodynamics of poymer solutions deca-mer (N = 0). As a consequence of this action we have reduced the number of moles from to 0.5, without changing the mass of the system. From the manifold of possibilities to place the segments of the chain molecules on the lattice, the authors have come to the following expression for the so-called combinatorial entropy of mixing for one mole of segments (instead of molecules), which is again an idealization like the corresponding expression for the perfect entropy of mixing comb S = ϕ ln ϕ + ϕ ln ϕ (9) R N N By analogy to mixtures of low molecular weight components we quantify the deviation from this limiting behavior. To this end we introduce residual contribution according to Initially R G comb R G = G + G (0) was considered to be exclusively of enthalpic nature and a composition independent interaction parameter, here called g, was introduced by means of the following relation H = g ' ϕϕ () RT g was meant to measure ½ of the change in enthalpy associated with the destruction of a contact between two segments of component and two segments of component to yield two contacts between a segment of and a segment of. Despite the fact that experiments have very early demonstrated convincingly that g is neither independent of composition nor necessarily of enthalpic nature, this formalism is still widespread and helpful for the understanding of some central features of polymer containing mixtures. For the integral Gibbs energy of mixing per mole of segments the Flory-Huggins equation reads G = ϕ ln ϕ + ϕ ln ϕ + g ϕ ϕ () RT N N where g is redefined as R G g = (3) RTϕ ϕ and contains enthalpic as well as entropic contributions. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-5

57 Thermodynamics of poymer solutions The integral Flory-Huggins interaction parameter g is experimentally inaccessible. The only information that is available stems from the measurement of chemical potentials, normally that of the solvent (e.g. via vapor pressure measurements or via osmosis). For crystalline polymers the chemical potential of the polymer in the mixture becomes accessible form liquid/solid equilibria. In view of this situation and because of the already mentioned concentration dependence of g we must differentiate the integral equation () and end up with the following expressions G RT g = + ln ϕ + ln ϕ + g ( ϕ ϕ ) + ϕ ϕ ϕ N N N N ϕ (4) G RT g g = + g + ( ϕ ϕ ) + ϕϕ ϕ ϕ ϕ ϕ ϕ N N (5) 3 G RT = + + ϕ ϕ ϕ ϕ ϕ ϕ ϕ 3 g g g 6 3 ( ϕ 3 ϕ) ϕ 3 N N (6) By means of the above relations one obtains the following expression for the chemical potential of component µ = ln ϕ+ ϕ + χ ϕ RT N N N N (7) where χ is given by g µ χ = g + ϕ = (8) R ϕ RT Nϕ and for the chemical potential of component µ = ln ϕ + ϕ+ ξ ϕ RT N N N N (9) where ξ is given by g µ ξ = g + ϕ = (30) R ϕ RT Nϕ For the integral interaction parameter the following equations hold true Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-6

58 Thermodynamics of poymer solutions ϕ g = χ dϕ = ξ d ϕ ϕ 0 0 ϕ ϕ (3) g = ϕ χ + ϕ ξ (3) Demixing into two liquid phases is bound to the existence of a hump in the function ( ) G ϕ as discussed earlier. The contribution G ( ϕ ) inevitably runs above R its tangents and does consequently exclude demixing; it is only the residual contribution G ( ϕ ), which may induce phase separation as demonstrated in Fig.. Only if the interaction parameter g exceeds a certain critical value, depending on the chain lengths of the components, the deviation from combinatorial behavior becomes large enough to produce the required hump. Under the (unreasonable) assumption that g does not depend on composition, all interaction parameter become identical and one can calculate the critical interaction parameter g c and the critical volume fractions c from the condition that the binodal curve and the spinodal curve touch each other as the conditions become critical. By means of the eqs () and (5) one can calculate the spinodal if g is known and with the eqs (3) and (6) the critical point becomes accessible. From (3) and (6) one obtains N c c comb N ϕ = N ϕ (33) ( ϕ ) ϕ = N (34) c c N ϕ + N ϕ = N (35) c c ϕ = c N N + N (36) and from the eqs () and (5) g c = + Nϕc Nϕc (37) Insertion of eq (36) yields Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-7

59 Thermodynamics of poymer solutions g c N + N N + N ( N ) + N = + = N N N N N N (38) Despite the deficiencies of the Flory-Huggins theory this approach is very helpful in understanding some basic features. For example the fact that the mutual miscibility associated with a certain unfavorable interaction between the components (positive g values) decreases rapidly as the number of segments N i becomes larger. Similarly it explains that critical volume fractions around 0.5 can only be expected if the chain length of the components is not too different. Otherwise the critical composition is shifted to the side of the component containing fewer segments. Fig. : Segment molar Gibbs energy of mixing and its combinatorial and non-combinatorial (residual) contributions as a function of composition. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-8

60 Thermodynamics of poymer solutions Quasi binary systems (non-uniformity U>0) Synthetic polymers are seldom molecularly uniform. This implies that the number of species of their solution in a single solvent is typically on the order of several thousands, despite the fact that chemically speaking we have only two components. To indicate this feature we are in this case talking about quasi-binary systems. This short chapter describes some additional effects observed with such solutions. The example shown in Fig. presents a phase diagram measured for solutions of polystyrene (most probable molecular weight distribution U = ) in cyclohexane. Fig. : Cloud point curve (full line) and coexistence curves for different over-all concentrations (broken lines) measured for the system cyclohexane/polystyrene The most striking feature is the discrepancy between the cloud point curve (full line) and the binodal curves (connection of the end-points of the tie lines). Because of an uneven distribution of polymer species differing in molar mass upon the coexisting phases one obtains an individual binodal curve for each starting composition. Normally the binodal curves are interrupted and only for critical composition one obtains a closed curve passing through the critical point, which is shifted out of the maximum towards higher polymer concentration. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-9

61 Thermodynamics of poymer solutions Upon phase separation the original polymer is fractionated. This means that the shorter chains accumulate in the polymer lean phase (sol) for entropic reasons (larger number of possible arrangements), whereas the longer chains prefer the polymer reach phase (gel) for enthalpic reasons (fewer unfavorable contacts between the polymer segments and solvent molecules). The distribution coefficient of the different polymeric species of a given sample varies considerably with chain length as can be seen from the GPC diagram (differential molecular mass distribution) shown in Fig H O/-POH/PAA T = 40 C MS - Lauf C 0.6 w lgm w 3BP < w 3c G = 0.45 Feed Sol Gel log (M / g/mol) Fig. 3: Differential molar mass distribution of the starting polymer (feed) and of the polymer fractions contained in the coexisting phases (sol and gel) as determined by GPC experiments for the system water/- propanol/poly(acrylic acid). G (not to be confused with the Gibbs energy) is the mass ratio of the polymer contained in the sol and in the gel, respectively. K. Meißner thesis Mainz 994 In this graph the sum of sol and gel must yield the value of the starting material (feed). At the M value at which the curves for sol and gel intersect, 50% of the species that are present in the feed reside in each phase; below that characteristic M value the percentage is higher in the sol and above it in the gel phase. Liquid/liquid phase equilibria of the present kind are used for preparative fractionation on a technical scale. It is obvious that a sharp cut through the molecular weight distribution Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-0

62 Thermodynamics of poymer solutions would be best. In reality fractionation is much less efficient. In order to quantify the success, one uses the so called Breitenbach-Wolf plot (Fig. 4). To that end the logarithm of the ratio of polymer with a given molar mass M that is found in the sol phase and in the gel phase, respectively, is mapped out as a function of M. In such graphs the ordinate value becomes zero at the M value at which the molecular weight distributions for sol and gel intersect. The steepness of the curves increases with rising quality of fractionation. In the unrealizable case of sharp cuts through the molecular weight distribution the curve would run parallel to the ordinate and its position on the M axis determines where this section takes place (i.e. fixes the G value, cf. Fig. 3). / ((-G) wgel lgm )] log [G w Sol lgm Auftragung nach Breitenbach - Wolf H O/-POH/PAA 5.6w G = 0.45 MS - Lauf C 40 C w BP 3 = M / kg/mol Fig. 4: Breitenbach-Wolf plot for the fractionation displayed in Fig. 3. K. Meißner thesis Mainz 994 Ternary systems The description of three component systems requires three independent variables in the case of constant pressure: T and two composition variables. Because of the additional variable it is according to the Gibbs phase law possible that three phases coexist within a certain range of composition, in contrast to binary systems, for which only three phase lines are feasible. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4 -

63 Thermodynamics of poymer solutions The Gibbs phase triangle In order to avoid three-dimensional representations one normally depicts the isothermal situation and uses the so-called Gibbs phase triangle for that purpose as demonstrated in Fig. 5. Fig. 5: How to read the composition of a ternary mixture in a Gibbs phase triangle The corners of the triangle represent the pure components, the three edges (of unit length) the binary subsystems and the interior of the triangle stands for ternary mixtures. There are no restrictions concerning the particular nature of the composition variable, as long as the sum of all components yields unity. The most common method (out of several) to read the concentrations is demonstrated in Fig. 5 Gibbs energy of mixing The extension of the integral Flory-Huggins equation to K components yields the following expression K K K G = ϕ ln ϕ + g ϕ RT i i ij i ϕ j (39) i= Ni i= j= i+ For its derivation it was tacitly assumed that interactions between two types of segments (ij) suffice to describe the mixture and that no ternary interaction parameters g ijk are required. For K = 3 we obtain the relation for mixtures of three components. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4 -

64 Thermodynamics of poymer solutions For the construction of the phase diagram in terms of phenomenological thermodynamics (by analogy to that described for binary systems) we must now use a three dimensional representation as demonstrated in Fig. 6. The hump of the binary case becomes a fold in the ternary and the tangent turns into a tangential plane. Fig. 6: Segment molar Gibbs energy of mixing for a ternary system as a function of its composition Cosolvency and co-nonsolvency Bound to special thermodynamic conditions it is possible that a mixture of two low molecular weight liquids can dissolve any amount of a given polymer, whereas each of these liquids alone exhibits a miscibility gap with the polymer. How this phenomenon, termed cosolvency, looks like in a Gibbs phase triangle is shown in Fig. 7a. Similarly an area of immiscibility may show up for ternary mixtures, despite the fact that phase separation is absent for all three binary subsystems. This particular behavior, called co-nonsolvency by analogy to cosolvency is shown in Fig. 7b. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-3

65 Thermodynamics of poymer solutions Fig. 7: Schemes describing the phenomena of cosolvency and co-nonsolvency The easiest way to rationalize cosolvency is offered by the so-called single liquid approximation of Scott. It treats the mixture of the low molecular weight liquids and as one component (index <>) and obtains for its interaction with the polymer (index 3) the following relation * * * * g = 3 ϕ g + 3 ϕ g < > 3 ϕ ϕ g (40) where the asterisks of the volume fractions indicate that these variables refer to the low molecular mixture only, according to * ϕi ϕi = ϕ + ϕ (4) with i = or According to this approach cosolvency is due to a very unfavorable interactions between the components of the mixed solvent (large g ), which do not yet suffice to induce their demixing but which are large enough to reduce g <>3 below its critical value. In other words the formation a homogeneous mixture may lower the Gibbs energy of the ternary system more (because of the avoidance of - contacts via the insertion of polymer segments) than the prevention of the less unfavorable -3 and - 3 contacts (associated with phase separation). Co-nonsolvency can be explained by very favorable, normally negative g values. Under these conditions the third term of eq (40) may become dominant and g <>3 Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-4

66 Thermodynamics of poymer solutions can exceed its critical value in spite of the fact that g 3 and g 3 are well below. Here the reason for demixing lies in the formation of many favorable - contacts in one of the coexisting phases. Exercises. Establishment of a phase diagram for the ternary system acetone/diethyl ether/polystyrene a. Determination of cloud points at 0 C by means of titration b. Swelling experiments with the binary subsystems solvent/polymer. Swelling experiments with the system cyclohexane/polystyrene at room temperature 3. Interpretation of a plot of light transmittance as a function of temperature for a solution of polystyrene in cyclohexane of known composition with respect to its cloud point. 4. Draw a schematic phase diagram from the info of experiments and 3, keeping in mind that the theta temperature of the system is 34 C. 5. Determination of the molecular weight distribution of the polystyrene sample used and of the fractions obtained with the system acetone/-butanone/polystyrene by means of GPC and evaluation of the fractionation efficiency by means of a Breitenbach-Wolf plot. 6. Calculation of g = g c (.9 + group number * 0.005) by means of the critical interaction parameter g c for N = and N =. Also calculate the critical composition ϕ c. Plot the combinatorial part and the residual part (for the calculated g) of G and G itself as a function of composition and determine the tie lines and spinodal composition graphically. 7. Discuss the slopes of G ( ϕ ) in the limit of ϕ 0 and ϕ (analytically by differentiating the Flory-Huggins relation). Note: Please do not copy the script when describing your experiments. The idea is that you make clear how the measurements were performed and evaluated. To this end it is recommended that the derivation of the relevant equations is presented or at least commented. Please collect the data in tables. Furthermore a reasonable estimate of the experimental uncertainties, differentiating between systematic and random errors, must be given. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-5

67 Thermodynamics of poymer solutions Experimental details All participants are requested to show up in the lab 0 3 of building K (Welder- Weg 3, st story) to prepare the solutions (required time ca. - hours). Otherwise it is impossible to perform the experiments in one day. Preparation of the solutions Note the weight of the flask (including a magnetic stirrer) and weigh in the required amounts of the components; write down the individual data. ad item 5): Separate 50 mg of the gel phase and approximately.5 g of the sol phase formed by the system acetone/-butanone/polystyrene that has formed upon standing at room temperature and deposit these solutions in a small glass flask. The volatiles are removed and the polymer is dried over night in the oven. ad item a): Prepare a mixed solvent containing 3 parts (weight) of diethyl ether and parts acetone. Prepare two sets of polystyrene solutions in this mixed solvent of the following concentrations: 5, 0, 5, 0 and 5 wt%. ad items b and ): Fill g of polymer into each of three flasks (note the precise weight) and add one of the solvents acetone, diethyl ether or cyclohexane to prepare approximately 5 ml of the solutions. The solutions in acetone or diethyl ether are placed in the refrigerator over night, whereas that in cyclohexane is kept at room temperature. Titrations a. Switch on the thermostat and the temperature control unit. b. Control the weight of the solutions prepared the previous day (to control loss of solvent) c. Cool the solution in an ice bath. d. Fill a burette that can be held at constant temperature with diethyl ether. e. Titrate the prepared homogeneous polymer solutions in the mixed solvent with diethyl ether until they become cloudy. f. The composition of the mixture at the cloud point is determined by weighting the flask. g. Empty the burette and rinse it with acetone. Fill it with acetone and repeat items e and f. h. Switch off all apparatus and clean all containers thoroughly. Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-6

68 Thermodynamics of poymer solutions Swelling experiments Decant the supernatant solvent and determine the solvent content of the swollen remaining polymer. Cloud point curve The participants will be briefed on that in the lab. GPC measurements Dissolve the three different polymer samples (staring material, polymer contained in the gel and in the sol phase, respectively) in THF such that the concentration amounts to approximately mg/ml. Toluene is used as an internal standard for the calibration of the GPC curve. Additional information is supplied when the experiments are performed. Literature ) Comprehensive Polymer Science, Polymer Characterization Vol.,.Auflage, Pergamon Press, 989 ) R. Koningsveld, W.H. Stockmayer, N. Nies: Polymer Phase Diagrams, Oxford Univ. Press 00 3) H.-G. Elias, Makromoleküle, 5. Auflage, 990, Hüthig & Wepf, Basel 4) G. Glöckner, Polymercharakterisierung durch Flüssigkeitschromatographie, Hüthig & Wepf, Heidelberg, 98 5) P.J.Flory, Principles of Polymer Chemistry,.Auflage, 953, Cornell University Press, Ithaca,N.Y. 6) R.L. Scott, J.Chem.Phys. 7 (949), 68 7) J.M. Prausnitz, S. v. Tapavicza, Thermodynamik von Polymerlösungen. Eine Einführung, Chemie-Ing.-Techn. 47 (975), 55 8) G. Rehage, D.Möller, O. Ernst, Entmischungserscheinungen in Lösungen von molekular uneinheitlichen Hochpolymeren, Makromol. Chemie 88 (96-5), 3 9) Encyclopedia of Polymer Science and Technology, Vol. (985), Wiley Interscience Publication, N.Y. 0) G. Wedler, Lehrbuch der physikalischen Chemie,.Auflage (985), Verlag Chemie, Weinheim ) B.A. Wolf, Zur Thermodynamik der enthalpisch und der entropisch bedingten Entmischung von Polymerlösungen, Fortschritte i. d. Hochpolymerenforschung 0(97), 09 Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-7

69 Thermodynamics of poymer solutions ) B.A. Wolf, R.J. Molinari, True Cosolvency, Makromol.Chem. 73 (973) 4 3) B.A. Wolf, G. Blaum, Measured and Calculated Solubility of Polymer in Mixed Solvents, J.Polym.Sci., Phys.Ed., (975), 5 Thermodynamics of polymer solutions (R. Horst and B.A. Wolf) 4-8

70 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 0 57 building K (9- a.m., 4-6 p.m., tel: , time required: ca. h). Otherwise the experiment cannot be performed. To make an appointment please call me or send an (andreas.eich@wee-solve.de). Topics of these experiments are the viscometric characterization of dilute polymer solutions (part ) and an introduction to the rheology of concentrated polymer solutions (part ). 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 -

71 Viscometry Table of content ) Introduction... 3 ) Molecular models... 5.) Polymer solution: Polymers regarded as rigid particles without draining... 5.) Polymer solutions: Polymers regarded as drained coils (without entanglements) ) Polymer solutions: Influence of entanglements ) Dilute polymer solutions... 3.) Definitions... 3.) Dependence of the viscosity on molecular weight at infinite dilution ) Influences on [η] ) Dependence of viscosity on polymer concentration ) From dilute to concentrated solutions: Coil overlap ) Concentrated polymer solutions ) Shear rate dependence of viscosity ) Shear thinning of polymer melts and concentrated polymer solutions ) Experimental Methods ) Capillary viscometer ) Rotational viscometer / rheometer ) Falling ball viscometer ) Experimental Part ) Rotational viscometry ) Capillary viscometry ) Appendix: Translation of important expressions ) Literature

72 Viscometry ) Introduction Fig. sketches a shear flow of a liquid between two parallel plates: Fig. : Parallel plate model Moving the plate on top in x -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. ɺ γ = d v /dx () x In fig. 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 x. The force F x acting tangential on a plate of surface area A (see fig. ) is called shear stress τ. τ =, [τ ] = N/m = Pa () F x A The indices denote the directions of the surface normal (here: x ) and the force (here: x ), respectively. In practice, the indices are often omitted. Ideally, there is a linear relation between shear stress and shear rate, Newton s law (eq. (3)): τ = ηɺ γ (3) η : shear viscosity, (dynamic viscosity), [η] = Pa s (SI), Pa s = 0 Poise (CGS) 5-3

73 Viscometry Samples obeying this relation are called Newtonian. An important value to keep in mind is the 0 C viscosity of water at 0 C: η HO = mpa s (exact.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 ɺ γ = v / x - considering v = x t - results in x an expression for the amount of energy dissipated per volume and time: x / E V t = η γ ɺ (4) 5-4

74 Viscometry ) 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..) Polymer solution: Polymers regarded as rigid particles without draining To understand how particles influence the viscosity of a dispersion, we have a look on fig. and 3 where we regard non-drained particles. First we assume that the suspended particles do not rotate but only perform a translational movement as depicted in fig. b. Fig. : Influence of a particle on shear flow for non-rotating and rotating particles. (Taken from Hiemenz, Principles of Colloid and Surface Chemistry ) The non-rotating particles capturing a volume fraction ϕ - do not contribute to the velocity gradient. Therefore the spheres might as well be allowed to settle to the stationary wall. Fig. 3: Scheme of a suspension of particles with volume fraction ϕ a) homogenous distributed particles, b) sedimented particles. (Taken from Hiemenz, Polymer Chemistry ) Therefore the remaining liquid layer is reduced to (-ϕ) of the original gap width, and the shear rate is increased to /(-ϕ) compared to the shear rate in a pure solvent. The increase of shear rate leads to an increase of the shear stress and therefore of the viscosity. 5-5

75 Viscometry If we assume particle rotation (cf. fig. c), 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 (cf. eq. 4)..) 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 946. 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 Chemistry ) 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 derivation leads to proportionality between viscosity and the molecular weight of the polymer. Rouse model While the Debye model is designated to flow behaviour, another similar model explains viscoelastic behaviour: The spring-bead- or Rouse model (Rouse 953, Zimm 956) takes into account the entropic properties of the polymer chain. 5-6

76 Viscometry Fig. 5: Scheme of the Rouse model. (Taken from Ferry, Viscoelastic Properties of Polymers 3 ) 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 Gaussian entropy elasticity). 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 Rouse model results in an equation which correlates the (Rouse) relaxation time τ R of viscoelastic properties with the zero shear viscosity η 0, molecular weight M and concentration c of the polymer: 6 η0m τr = (5) π c RT The index denotes the polymer in a binary mixture, whereas index denotes the solvent. The indices can be omitted if there are no ambiguities. A disadvantage of the Rouse model is the lack of interactions between different polymer molecules. For this reason it is mainly applicable for dilute solutions. Besides, the model explains linear viscoelastic behaviour but not the shear rate dependence of viscosity. Viscosity to shear rate zero: η0 = lim ɺ γ 0 η 5-7

77 Viscometry.3) Polymer solutions: Influence of entanglements To obtain the influence of entangling on the relaxation time in the Rouse model, each entanglement is treated as a separate bead-spring interaction between a molecule as a whole and the medium (cf. fig. 6). Fig. 6: Incorporation of entanglements in the Rouse model. (Taken from Graessley, The Entanglement Concept in Polymer Rheology 4 ) The beads represent entanglement sites which are distributed uniformly along the chain contour. For a large number of entanglements per macromolecule, E, the following proportionality can be derived: η0m τr (6) Ec RT Of course, E depends on the polymer concentration. In highly concentrated systems E ~ c is found in many cases. The Debye and the Rouse model lead to a proportionality η ~ M which is found experimentally for melts of low molecular weight polymers. For polymers with molar mass above a critical mass M c, the power law η ~ M 3.4 is observed (cf. fig. 7). 5-8

78 Viscometry Fig. 7: Dependence of melt zero shear viscosity on molecular weight Entanglements of different polymer molecules are the origin for the strongly increased sensitivity of the viscosity on molar mass above M c. Concentrated polymer solutions give similar results. Here the entanglement density is described by the product c M which replaces ρ M in case of polymer melts. Bueche model To describe the motion of entangled polymer chains, Bueche developed in 95 a theory which is sketched in fig. 8. Fig. 8: Variation of melt zero shear viscosity with molecular weight. (Taken from Hiemenz, Polymer Chemistry ) 5-9

79 Viscometry 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 η ~ M 3.5 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 978, based on a concept of de Gennes 97) describes the movement of a polymer chain in a tube which is formed by other polymer molecules. Fig. 9: Reptation model for entanglements. (Taken from Hiemenz, Polymer Chemistry ) The reptation model is based on the diffusion rate of the chain segments in the tube. The relaxation time τ 0 of one segment can be expressed as ξ l0/kt (with segmental friction parameter ζ, segment length l 0 ); the relaxation time of the whole molecule with n segments equals τ = τ 0 n 3 ~ M 3. The expression for the zero shear viscosity shows the same molecular weight dependence η ~ M 3. 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. 5-0

80 Viscometry 3) Dilute polymer solutions 3.) 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: η 0 Relative viscosity: η ηrel = (7) η 0 η η0 Specific viscosity: ηspec = = ηrel (8) η Reduced viscosity: η 0 spec η red = (9) c These variables are used to determine the intrinsic viscosity that is defined as: [ η] η = lim c c 0 ɺ 0 γ spec with [η] = ml/g (0) 3.) Dependence of the viscosity on molecular weight at infinite dilution In 906 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 +.5ϕ η () 0 0 ϕ : 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 η0 =.5ϕcoil η 0 or using the intrinsic viscosity [η]: ϕ η =.5 () c [ ] coil The volume fraction of all polymer coils ϕ coil is related to the sphere-equivalent volume of one polymer coil V η and to the number of polymer coils N in the sample. 5 -

81 Viscometry N Vη ϕ coil = (3) V Notice that the polymer coil volume V η contains not only polymer but also solvent molecules. The number of polymer coils can be substituted by the mass concentration c and the molecular weight M of the polymer c ϕ = N V (4) coil L M η Combination of eq. () and (4) leads to Vη η = N (5) M [ ].5 L Eq. (5) relates the intrinsic viscosity with the molecular weight and the hydrodynamic volume of the polymer molecules. Now we discuss two special cases: () In case of rigid spheres their hydrodynamic volume V η is proportional to their mass: The fraction on the right side of eq. (5) and therefore [η] becomes independent of the particle size or mass, respectively. E.g., a dilute dispersion of 000 particles with V η = µl exhibits the same viscosity as a dispersion of 500 particles with V η = µl. So it is not possible to determine the size/mass of rigid spheres from the viscosity of dilute dispersions. () In case of polymer solutions, the hydrodynamic volume V η is not proportional to the molar mass. If we regard the special case of θ-conditions, by definition the squared radius of gyration r g is proportional to the molecular weight. Then the relation between hydrodynamic volume and molecular weight is given by V η ~ r g 3/ ~ M 3/. Inserting this relation in eq. (5) yields the molecular weight dependence of [η] at θ-conditions: 0.5 [ η ] = K M θ θ Kuhn-Mark-Houwink equation (6), with M = M : molecular weight of the polymer and K constant of proportionality Universally, the KMH-equation is formulated: a [ η ] = KM (7) Which value the exponent a attains in the case of rigid spheres (case ())? 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 -

82 Viscometry 3.3) Influences on [η] For polymer solutions the Kuhn-Mark-Houwink exponent normally exhibit values between 0.5 and. 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 α η : 3 [ η] Vη αη = = (8) η V [ ] θ η θ 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: [ η ] = KMη a (9) with M η a a wimi, i i = w being the weight fraction of polymer species i. Please note that M η 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: η η spec [ ] [ ] red = = η + khc η (0) c with k H : Huggins coefficient According to Schulz-Blaschke the independent variable is the specific viscosity instead of the concentration c. Therefore here η red is plotted vs. η spec : [ ] k [ ] η = η + η η () red SB spec with k SB : Schulz-Blaschke coefficient 5-3

83 Viscometry The Huggins coefficient k H and the Schulz-Blaschke coefficient k SB 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 (0) and () 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 η red with increasing dilution. In this case the extrapolation c 0 do not yield reliable results. 6 η spec c - / L g water/cmg C; no salt 0,0 0,5,0,5,0,5 c / gl - Fig. 0: Non-linear concentration dependence of a polyelectrolyte solution. CMG: Carboxymethyl guar 3.5) From dilute to concentrated solutions: Coil overlap The intrinsic viscosity is used to define a reduced (dimensionless) concentration c*: c* [ η] c = () c * classifies polymer solutions in respect to their effective polymer concentration : c * < : 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. 5-4

84 Viscometry 4) Concentrated polymer solutions 4.) 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. ). Fig. : 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.) Shear thinning of polymer melts and concentrated polymer solutions Polymer solutions show at low shear rates a constant viscosity, the plateau value η 0 is called zeroshear viscosity (fig. ). Reaching the critical shear rate γɺ=/τ 0 the sample shows shear thinning. This regime is called power-law regime and can extend over several decades in shear rate. 5-5

85 Viscometry Fig. : Shear thinning of polymer melts and concentrated polymer solutions According to Graessley 4, the shear thinning is caused by the following mechanism: in polymer solutions and polymer melts polymer chains with molecular weights exceeding a critical limit M c 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 γɺ c where shear thinning starts is the reciprocal of the characteristic relaxation time τ 0, which is the time needed to form a new entanglement at thermal equilibrium. It is observed that the onset of shear thinning is governed approximately by the Rouse relaxation time, eq. (5) and (6). Therefore it is convenient to introduce reduced variables: The viscosity is replaced by η/η 0 and the shear rate by τ ɺ Rγ = η0m ɺ γ / crt. Using these reduced variables, most of the observed variation among different samples and systems can be removed. 5-6

86 Viscometry 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: η mm ν =, [ν] = (SI), ρ s mm = cst(centistokes, CGS) (3) s The kinematic viscosity is frequently used for samples which are measured by capillary viscometers (e.g. oil). 5.) Capillary viscometer There exist different kinds of capillary viscometers. Nowadays in most cases Ubbelohde capillary viscometers are used. Fig. 3: Different types of capillary viscometers Using capillary viscometers, the viscosity of Newtonian samples can be determined using the law of Hagen-Poiseuille: V = t ( ρg h + ) 8η h p πr 4 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 5-7 (4)

87 Viscometry 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: η t t t0 ηrel = =, ηspec = η 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.) Rotational viscometer / rheometer Rotational viscometers are equipped with various measuring geometries; the most important are cone/plate, plate/plate and concentric cylinder (frig. 4). 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. 4: Measuring geometries: Cone/plate, plate/plate and concentric cylinder In the lab course we use a cone/plate geometry as shown in fig. 4. For this geometry, the relations between the primary measured variables angular velocity φ ɺ respective torque M and the rheological measures shear rate γɺ respective shear stress τ are given by: ɺ γ = ɺ φ/ α, τ = 3 M/πR (5) 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: ) control of the shear rate and measurement of the shear stress, and ) 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. 5-8

88 Viscometry 6) Experimental Part 6.) Rotational viscometry The measurements with the rotational rheometer are performed together with the supervisor to avoid damage of the instrument!. Measure the viscosity of three stock solutions of a high viscous poly-dimethylsiloxane sample (M = g/mol) in an oligodimethylsiloxane at T = 0, 40 and 60 C as a function of shear rate. The concentrations are 4, 8 and 4 g/ml.. Plot log(η) vs. log(γɺ) for all measurements. In a second graph, plot the reduced variables log(η/η 0 ) vs. log( η 0 M γɺ / c RT) according to the Rouse theory, eq. (5), and discuss the result. Plot log (η/η 0 ) vs. log( η 0 M γɺ / c x RT) and adjust the exponent x between and by trial and error to obtain the best coincidence of the different curves. 6.) Capillary viscometry. For the experiment, prepare 5 solutions of polystyrene in cyclohexane (concentrations ca. 3; 6; 9; and 4 mg/ml) and solution of PS in toluene (conc. ca. 6 mg/ml) one day in advance. All samples are stirred overnight.. Measure the viscosity at T Θ = 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 k SB and [ η ] of PS in cyclohexane. 4. Use k SB = 0.3 (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 weight of the polymer coil for the PS/cyclohexane system at θ-conditions (K θ = 0.08 ml/g) and determine the mean sphereequivalent coil radius <R η > 0.5 for this system by combining eq. (5) with the definition of a sphere: π Vη = R θ η 3 θ 5-9

89 Viscometry 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 P. C. Hiemenz, Principles of colloid and surface chemistry, nd ed. (986), Marcel Dekker Inc., New York) P. C. Hiemenz, Polymer Chemistry - The Basic Concepts (984), Marcel Dekker Inc., New York) 3 J. D. Ferry, Viscoelastic Properties of Polymers, 3 rd Ed (980)., John Wiley, New York 4 W. W. Graessley, The Entanglement Concept in Polymer Rheology, Advances in Polymer Science 6 (974) 5-0

90 EXPERIMENT 6: DYNAMIC MECHANICAL CHARACTERIZATION OF POLYMERS IN THE GLASS TRANSITION RANGE. 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-strainbehavior 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 6 -

91 Dynamic Mechanical Characterization types of deformation (stretching, shear). The complete description using tensor notation can be found in the textbooks.. 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 = /D, G = /J. This simple relationship is not valid for time-dependent mechanical properties ( E(t) /D(t); G(t) /J(t) ). Other experiments for the measurement of mechanical properties we want to mention are: biaxial tension, compression, elongational flow.. 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 : Schematic curve of the storage modulus E', the loss modulus E", and the loss factor tan δ as a function of temperature, time or reciprocal frequency Figure schematically shows the modulus of an amorphous polymer as a function of temperature or time/frequency. At low temperatures or high frequency amorphous polymers 6 -

92 Dynamic Mechanical Characterization are glassy solids with a high modulus of 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, 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. 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:. Stress relaxation measurement: The specimen is strained in a defined way and the stress is measured as a function of time or temperature.. 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 (ω in rad/s). The reciprocal of ω is the oscillation cycle and defines the time scale of the experiment. 6-3

93 Dynamic Mechanical Characterization.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 : Definition of directions, deformations, and stresses by projection of a volume element on one of the base areas. Indexing: st index = area-normal index of the area, where the stress or strain acts nd index = direction of the stress or strain Hooke s law for tension The strain γ in direction x (see Figure a) is defined as γ r = r 0 () Young s modulus E is defined with Hooke s law for uniaxial tension: σ = E γ () where σ is the stress related to the initial cross-sectional area. Hooke s law for shear The shifts u increase linearly with x (see Figure b). The shear strain γ is therefore defined as γ du = dx = tan α (3) 6-4

94 Dynamic Mechanical Characterization The shear modulus G results from the (required) linear dependence of the shear stress σ from the shear process σ = G γ (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 ν: E = ( + ν)g (5) For incompressible materials is ν = 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 η: σ = η dγ dt (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..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 η which obeys Newton s law. 6-5

95 Dynamic Mechanical Characterization 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 γ tot = γ el + γ vis (7) while the stress is identical in each element σ tot = σ el = σ vis (8) With equations () and (6) results for the time-dependence of the overall strain dγ dσ = dt E dt σ + η (9) This differential equation yields for a stress-relaxation experiment (dγ/dt = 0, γ = γ o ) dσ E dt σ + η = 0 integration dσ E = - dt σ η σ( t) = σ o e E - dt η (0) i.e. an exponential decline of the stress from the initial value σ o = E γ o with the rate constant E/η. The quotient E/η has the dimension of a reciprocal time, it is characteristic for a certain combination of elastic and viscous component. Its reciprocal η/e is called relaxation time τ of the relaxation process. After division by γ o results the time-dependent elasticity modulus t - τ σ( t) E( t) = = Eo e () γ E o is called relaxation strength. o 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 γ = γ o sin( ωt) () If the specimen only showed elastic behavior (validity of Hooke s law), for the stress σ would result with σ = E. o γ o σ = E γ = E γ sin( ωt) = σ sin( ωt) (3) o o 6-6

96 Dynamic Mechanical Characterization For a liquid, which obeys Newton s law, applies dγ σ = η = η ω γ o cos( ωt) (4) dt i.e. stress and strain are 90 out of phase. Equation (4) also shows that the stress increases linearly with frequency. The differential equation, which describes the Maxwell element at dynamic load, is given by or dσ σ ω γ o cos( ωt) = + (5) E dt η dσ Eσ ω E γ o cos( ωt) = + (5a) dt η The universal solution for this differential equation is σ = B cos( ωt) + C sin( ωt) (6) After differentiation and insertion into the differential equation B and C can be determined by comparison of the coefficients. With η/e = τ results ω τ ωτ σ = γ o [ E sin( ωt) + E cos( ωt)] (7) + ω τ + ω τ After division by γ o the dynamic modulus is ω τ ωτ E( ω ) = E sin( ωt) + E cos( ωt) (8) + ω τ + ω τ 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 ω τ E ' = E (9) + ω τ ωτ E " = E (0) + ω τ E* = E' sin( ωt) + E" cos( ωt) 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 (E o ), the frequency (ω), and the characteristic relaxation time τ. Alternatively the dynamic stress can be defined via the stress amplitude σ and the phase angle δ: σ = σ sin( ωτ + δ) = σ (sin( ωτ) cos( δ) + sin( δ) cos( ωτ)) () o o 6-7

97 Dynamic Mechanical Characterization By comparison of the coefficients results σ E ' = γ o o σ E " = γ E" and = tan( δ) E' o o cos( δ) sin( δ) () (3) tan(δ) 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 (9), (0) the dependence on the product ωτ is crucial. If the product ωτ, then E' and E" have the value ½ E. If ωτ >, then applies E' E ; E" 0, if ωτ <, then applies E' 0 ; E" 0. E has a maximum at ωτ =, 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 ωτ, tan(δ) =/(ωτ), i.e. at a given characteristic relaxation time τ 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 E i and its relaxation time τ i. Furthermore an additional time-independent element (spring with E e ) can be introduced to describe the behavior of crosslinked polymers. The storage modulus and loss modulus then become τi τi ω E '( ω) = Ei + Ee (4) + ω ωτi E "( ω) = Ei (5) + ω τ i If a very large number of relaxation elements is considered, the discrete relaxation processes E i, τ i can be described by a continuous relaxation function E(τ) and equations (4) and (5) become 0 ω τ E '( ω) = Ee + E( τ) dτ (6) + ω τ 0 ωτ E "( ω) = E( τ) dτ (7) + ω τ 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 6-8

98 Dynamic Mechanical Characterization function on a logarithmic scale, the integral term is extended with τ. With dτ/τ = d(lnτ) and H(τ) = τ E(τ) results for E' and E" + ω τ E '( ω ) = E e + H ( τ) d(ln τ) (8) + ω τ + ωτ E "( ω ) = H ( τ) d(ln τ) (9) + ω τ Figure 4: run of the response functions H(τ) 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 ω τ /(+ω τ ) and ωτ/(+ω τ ) indicate how much a certain relaxation process (τ) at the frequency ω contributes to the mechanical response of the system (Figure 4). H(τ) still has the dimension of a modulus (Pa). Alternatively H(τ) can be written as a product ρrt/m e h(τ), where h(τ) now is a dimension-less distribution function, which describes the contribution of the single relaxation processes in units of ρrt/m e. Basis of that is the description of polymer melts analogously to covalently crosslinked rubbers (see Script Rubber Elasticity). M e 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(τ). 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. 6-9

99 Dynamic Mechanical Characterization. Time-temperature-superposition The complete characterization of the viscoelastic properties of polymers requires measurements in a very wide time- or frequency-range (ca. 0-0 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 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 ρrt/m e, and also the molecular relaxation times τ i (with E i = g i ρrt/m e ) ω τi ( T ) E' ( ω) T = ρrt / M e g i (30) + ω τ ( T ) i ω τi ( T ) E' ( ω) T = ρrt / M e g i (3) + ω τ ( T ) 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 T ): i ω τi ( T ) E' ( ω) red = E' ( ω) T ρt / ρt g i (3) + ω τ ( T ) The shift of the relaxation time of a certain molecular relaxation process with temperature is given by i a T = τ T ) / τ( T ) (33) ( where a T is called shift factor. Because of the form of equation (3) the same result is received when ω is multiplied with a T instead of τ, i.e. the modulus E'(ω) has after correction to E' red at the frequency ω and the temperature T the same value as at a frequency ω a T and the temperature T. The function a T (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 a T (T) f(τ 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 a T. With a logarithmic time(frequency) axis this equals an addition of log(a T ). Figure 5 shows schematically how such a shift is performed along the frequency axis. 6-0

100 Dynamic Mechanical Characterization 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 6 -

101 Dynamic Mechanical Characterization 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(a T ) (T) in terms of the reference temperature T o 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 δ. 4 When does the time-temperature-superposition principle fail?. Temperature dependence of viscoelastic behavior The temperature dependence of the shift factors log(a T ) can in many cases be described by the Williams-Landel-Ferry(WLF) equation c ( T Tref ) log( a T ) = (34) c + T T ref where T ref is the reference temperature and c, c 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 c and c depend on the choice of T ref, but the product c c is independent of T ref. In the theory of free volume a molecular meaning can be assigned to the coefficients c and c. 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 T g + 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(a T ): H = d(ln a T ) R (35) d T 6 -

102 Dynamic Mechanical Characterization 3. Assignments In this lab course the dynamic mechanical behavior of a sample of weakly crosslinked natural rubber is investigated in a range of 00 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 000, 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 00 rad/s..) Perform the measurements under the guidance of the lab technician..) Discuss possible error sources a) in the isochronous measurement, b) in the isothermal measurements. 3.) Discuss the progress of E', E", and tan δ 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, 3 rd Ed. Wiley New York 980 P.C. Hiemenz Polymer Chemistry The Basic Concepts, Marcel Dekker 984 R.J. Young, P.A. Lovell Introduction to polymers, nd Ed. Chapman&Hall London

103 EXPERIMENT 7: X-RAY DIFFRACTION. Theory. 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, v c and v a = v c (twophase 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(Θ) = ( v c ) I a (Θ) + v c I c (Θ) () where I a represents the intensity of the purely amorphous case, I c the intensity of the purely crystalline case. Θ is half the scattering angle. Vice versa results from () for v c I( Θ) I a ( Θ) v c = () I ( Θ) I ( Θ) wherein any angle Θ can be chosen. c a 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 shows for a branched polyethylene (Lupolen 800S) how the intensity of the crystalline reflexes decreases with increasing temperature while the amorphous halo increases. 7 -

104 X-ray Diffraction Figure : X-ray diffraction diagrams of LDPE. The counter goniometer The counter goniometer is particularly suitable for the registration of a diffraction diagram. Figure shows its ray path. F B D DK FK A focus of the X-ray tube aperture unit detector detector circle focussing circle puncture of the rotation axis perpendicular to FK Figure : 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 Θ which obey Bragg s equation d hkl sin Θ = nλ (3) 7 -

105 X-ray Diffraction where d hkl is the lattice distance for the lattice family with the Miller indices (hkl), n is the reflection order, and λ is the X-ray wavelength (when a copper anode is used λ =.54 Å). From the angular position of the halo the intermolecular distance of neighboring groups can be estimated, again by using Bragg s equation (3)..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 (hk) 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: c sin Θ = n λ where Θ results from h tan Θ = (see Figure 4). A h is the height of the st 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 985, especially Chapter

106 X-ray Diffraction 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) v v, S unit vectors in the direction of the incident and scattered x-ray, respectively S o (from: M. Kakadu, N. Kusai, X-ray diffraction by Polymers, Elsevier Publ. Camp. Amsterdam, 97) Figure 4: Illustration for the interpretation of 00l reflexes in a fiber diagram 7-4

107 X-ray Diffraction. Assignments Task 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 6 C, 50 C, 75 C, 90 C, 00 C and 5 C (melt) in a range of angles between Θ = 0 and Θ = 40 b) Under the assumption of the -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. a I = w c c I total I = ( w ) I c total w c c c = = = I c I + I a F c F + F a I I a total First only the sum of amorphous and crystalline intensity is known from the experimental data. However, the amorphous fraction I a of the semi-crystalline diffraction diagram can be determined from the intensity I a0 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 Θ applies the following relationship between the integral intensity of the melt I a0 and of the amorphous fraction of the semi-crystalline sample I a (T): I T h ( Θ) = I T a ( ) a0 h0 ( Θ) β T T h T (Θ) h 0 (Θ) = measurement temperature = intensity at temperature T at a fixed angle Θ = intensity in the melt at a fixed angle Θ β T = standardization factor I total (T)/I a0 c) Plot the obtained degrees of crystallinity as a function of temperature. Please discuss the result! 7-5

108 X-ray Diffraction Task 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 d hkl as a function of the lattice parameters: d hkl h = a k + b + l c The c-value of the unit cell is.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 0 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. 978, 33, 9-7) Figure 5: Arrangement of the chains in nylon-6 crystals (a-c-plane) 7-6

109 X-ray Diffraction 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 00- and 0-reflexes come from, in the real PE lattice 7-7

110 Experiment 8: Asymmetrical Flow Field Flow Fractionation (AF-FFF) of Polymers and Colloids Asymmetrical Flow Filed Flow Fractionation (AF-FFF) is a method for the separation and size characterization of polymers and colloids in solution. AF-FFF belongs to the family of Field Flow Fractionation techniques (FFF), first introduced by Giddings and coworkers in 970. The basic principle of operation of any FFF technique is based on a laminar flow between two parallel plates in a thin channel resulting in the formation of a parabolic flow profile, as shown in Figure and Figure. A force field acts perpendicular to this flow onto particles or molecules that have been introduced into the channel. The kind of force field defines the individual FFF-technique. Figure : Scheme of a FFF channel Figure : Mechanism of an FFF separation of two components X and Y across the parabolic flow profile resulting in different flow velocities of X and Y.

111 Field-Flow-Fractionation FFF Theory of FFF: dc ( x) Flow density J x = D + Uc ( x), D : diffusion coefficient,u : field induced velocity dx at equilibrium: J = 0 concentration profile: height direction x fromthe wall tothe wall U c( x) = c0 exp x, c 0 : conc. at the wall, x : coordinate in channel D D x effective layer thickness: l = c( x) = c0 exp, l: characteristic average of distance U l (depending on sample and field) kt F D = (Stokes Einstein), U =, F : force, f : friction coefficient f f kt => l =, typically < l < 0 µ m F Normal Retention Mode ( Brownian ) L Retention time t R =, L : channel length, v : particle velocity ( v of an exponential v distribution in a parabolic flow profile) Parabolic flow profile: Retention (as in Chromatography) described by dimensionless retention parameter λ l kt λ = =, w : channel height (typically µm) w F w average zone velocity v in z-direction Retention ratio Integral expression: v ( x) v( x) c( x) c v = (averaging over channel height) R =, v : average v of sample component, v ( x ) v( x) R w c( x) v( x) dx 0 = w w c( x) dx 0 0 v( x) dx : average v of solvent Flow profile of an isoviscous liquid between two parallel plates (Hagen-Poiseuille): p v( x) = x( w x), η : viscosity of solvent η L

112 => average velocity v (x) : v( x) pw = ηl Integration yields: R = 6λ coth λ λ if w >> l (requirement for efficient operation): λ -> 0, thus lim ( ) limit λ 0 lim R = 6λ λ 0 - Term and so Alternative description: V t0 R = 0 =, V 0 : dead volume (channel volume), V R : retention volume, t 0 : dead-(void-) time V R R t R t : retention time t w F w R = = t 6l 6 kt 0 t R ~ F seperation if F is high enough (typically: 0-6 N) size selectivity: S d = d logtr d log R Flow FFF viscous force on particle due to cross flow: F = f U kt U = D = 6πη R U t ~ f, D, R, thus S (GPC typically S ~ 0. ) R d Band broadening: average plate height H H = H neg + H long H i, H neg : velocity gradient mass transfer (non-equilibrium), H long : + longitudinal diffusion: can often be neglected, neglected d H i : instrumental effects: can often also be Theory of separation systems with non-uniform flow profile leads to: χw L H neg = v, v : average Carrier-velocity =, χ = f ( λ ) 0 D t 3 for small λ ( λ 0.06) : χ 4λ 0 L Dt plate height N = = 3 H w 4λ typical example of channel parameters yields: H of 0.8 mm N 550 possible

113