1 MOLECULAR WEIGHTS AND THEIR CHARACTERIZATION I. Molecular weight dispersion A. Most bulk polymer samples have a variety of different sized molecules. B. Ambiguity must be resolved to create a measure to compare samples. II. Number-average molecular weight – Mn W Ni Mi Mn A. Definition N Ni 1. Mi – molecular weight for a specific total length of polymer molecules. 2. Ni – number of molecules with a specific molecular weight. B. Measured with colligative properties, osmosis III. Weight-average molecular weight – Mw Mi Wi Ni Mi2 Mw A. Definition W Ni Mi B. Measured with light scattering IV. Z-average molecular weight – Mz Ni M3i Mz A. Definition Ni Mi2 B. Measured with ultracentrifugation V. Viscosity-average molecular weight – Mv A. Definition Mv 1 a N M N M 1 a i i i i 1. The value of “a” comes from Mark-Houwink equation: KMav a. Theoretically, a = 0.5 for random coil in solvent. b. a = 2.0 for a rigid rod. 2. “a” is an experimentally determined parameter 3. Therefore, Mv is not definition from first principles B. Measured with viscometry 2 VI. Polydispersity A. The different molecular weights can be ordered according to their values. Mn < M v < Mw < Mz B. Often polydispersity is measured with the polydispersity index. M PDI w Mn C. PDI can very from 1 to approximately 20. D. Consequences of polydispersity 1. Increased solubility 2. Decreased strength 3. Degradation of polymer first occurs in low M molecules Stolen from: http://openlearn.open.ac.uk/mod/resource/view.php?id=196637 3 VII. Molecular Weight Separations A. Fractionation 1. The solubility of polymer is inversely proportional to its molecular weight. 2. Adding a nonsolvent to a polymer solution cause higher molecular weight molecules to precipitate. 3. Relatively monodisperse fractions can be collected by carefully adding increasing amounts of nonsolvent. B. Gel Permeation Chromatography (GPC) 1. Also known as Size-Exclusion Chromatography (SEC) 2. Crosslinked polystyrene beads (Sephadex) have pore sizes that can be controlled through processing conditions. 3. High pressure pumps force eluent through column. 4. Smaller molecules become entangled in pores and proceed slowly through the column. Larger molecules move quicker through the column. 5. Fractionation is possible with a series of columns with Sephadex beads of differing porosity. VIII. Molecular Weight Characterizations A. Osmometry 1. van’t Hoff equation cRT Bc2 Mn RT Bc c Mn a. plot of vs. c yields RT/Mn as the intercept - find number average molecular weight at infinite dilution (ideal solution) cRT RT Bc 2 lim c 0 Mn c Mn - ideal solution also exists at theta temperature where B(T)= 0 - at θ-temperature, configuration of coil at random-walk. b. c is concentration is g/L 2. very accurate and sensitive technique 3. classical methods are slow (several weeks); however modern methods are relatively quick (15 minutes) 4 B. End group analysis 1. Use titrimetry to measure number of end groups m Mn n 2. End groups than can be titrated include carboxylate, hydroxyl, amino, ethylene oxide, etc… C. Ebulliometry, Cryometry, Vapor pressure lowering 1. Boiling point elevation, freezing point depression, vapor pressure lowering are not very concentration-sensitive colligative properties 2. Rarely used in molecular weight determination. D. Refractometry 1. Refractive index varies with concentration 2. Convenient for quick measurements 3. Need monodisperse standards to create calibration curve. E. Light Scattering 1. Measure intensity of scattered light from a polymer solution over very large angles. 2. Debye and Zimm formulated a theory of light scattering in polymer solutions that relates to the osmotic pressure for polymer molecules assumed to be spheres. Hc 1 R RT c 3. The optical constant, H, is related to refractive index, n0, the refractive index gradient with concentration, n c , and the wavelength of the incident radiation, λ. NA – Avogadro’s number, 1 = 3.1416… 22 n 2 n H 1 40 N A c 2 - the refractive index gradient depends on the polymer/solvent pair - quality of the molecular weight determination depends on the quality of refractive index gradient 5 4. The Debye/Zimm theory becomes more sophisticated when the light scattering intensities depend on the shape as well as the size of the polymer molecule. cRT Bc 2 Mn RT 2Bc c M n Hc 1 2Bc R R solvent M w P - R(solvent) is a correction factor for scattering of the solvent molecules 5. Rayleigh ratio, R(), is the fraction of light scattered at angle per unit length. i r 2 R (for dilute gases) I0 1 cos i() – intensity of scattered light per unit volume r – distance away from scattering center 6. When the scattering is limited to small angle scattering the Rayleigh ratio reduces to 3 R 161 7. is the turbidity from Lambert’s law (Beer’s law with turbidity) I e l I0 - turbidity, l – length of cell 8. P() is called the scattering form-factor and is related to the radius of gyration of the polymer particles. P 2 2 2 R 2 K2 R K 1 e g 4 g R K 4 g K 41 sin 2 9. For small angles, P(), expands as a series to P 1 R g2 K 2 3 1 R g2 1612 sin 2 2 3 2 6 10. The basic light scattering equation for polymer solutions has two limits. Hc 1 2Bc R R solvent M w P a. The zero concentration limit 1 Hc lim c 0 R Mw 1 41 2 2 2 1 R g sin 2 3 i. for a specific angle, measure Hc/R for several concentrations of polymer. ii. use the Hc/R vs. concentration data to find the intercept (c = 0) iii. make several other plots for different angles and find intercepts. iv. plot the intercepts as a Hc/R vs. plot to find weight-average molecular weight and radius of gyration. b. The zero scattering angle limit 1 Hc lim 2Bc 0 R M w i. for a specific concentration, measure Hc/R for several small angles of scattering of polymer. ii. use the Hc/R vs. angle data to find the intercept ( = 0) iii. make several other plots for different concentrations and find intercepts. iv. plot the intercepts as a Hc/R vs. c plot to find weight-average molecular weight and radius of gyration. 7 = 45 = 60 = 75 = 90 = 105 = 120 = 135 = 150 Kc/R() 106 = 0 11. Zimm Plot a. The two separate extrapolation processes above (c = 0, = 0) can be simplified into a single double-extrapolation plot called a Zimm plot. c = 0.0006 g/cm3 c = 0.0005 g/cm3 c = 0.0004 g/cm3 c = 0.0003 g/cm3 c = 0.0002 g/cm3 c = 0.0001 g/cm3 c = 0 g/cm3 100c + sin2(/2) b. The plot is made by plotting Hc/R vs. the sum of concentration and angle, (sin2(/2)). An appropriate scale factor for the concentration must be used to ensure that all data plotted can be seen. (Otherwise data points may overlap. Thus the need for an “appropriate” scale factor.) c. With the Zimm plot, the intercept yields the number-average molecular weight, whereas, two slopes can be found from the double extrapolation to yield the radius of gyration and the second virial coefficient. 8 F. Dynamic Light Scattering 1. Scattering off of polymer particles causes an interference pattern. a. Scattering function G 1 e 2Dq 2 . D – diffusion constant, q – wavevector of incident light 2. Diffusion of particles causes interference pattern to change. a. Stokes-Einstein relationship D kT (assuming hard sphere model) 6R b. Delay time Dq 2 1 3. Molecular weight is related to the size of the polymer particle, R. 4. Technique can be used to explain morphological changes of polymer particles. G. Viscometry 1. Definitions a. Relative viscosity, r . 0 0 – viscosity of pure solvent. r b. Specific viscosity, sp sp 0 . 0 c. Reduced viscosity, red red sp c . d. Inherent viscosity, inh inh ln r c e. . Intrinsic viscosity, [] sp . c lim c 0 9 2. Viscosity of dilute polymer solutions with constant concentration increase with increasing MW. K M w a – Mark – Houwink equation a. Because K constants can’t be determined theoretically, viscometry is useful for relative measurements or measurements against standards. i. Often MW standards (measured with other methods such as light scattering) are used to create calibration curves. ii. Viscometry yields very quick determinations, thus the bother of creating standard solutions may have a high benefit. b. The shape of the molecule significantly impacts the viscosity of the polymer solution by affecting the exponent in the Mark – Houwink equation. i. a = 0.5 ii. a = 0 iii. a = 2 random coil hard spheres rigid rods 3. Volume of molecule is considered in the Flory equation. 3 r02 2 M r0 – end-to-end distance in solvent 3 r02 2 – hydrodynamic volume – Flory expansion factor – empirical fitting parameter 10 H. Ultracentrifugation 1. Sedimentation a) Forces i) Gravitational Fg mg m – mass of polymer particle ii) Buoyancy Fb mv2g v 2 – specific volume of polymer particle - density of solvent ii) Friction force Ff fu u – velocity of particle b) At terminal velocity of particle, sum of the forces equals zero. Fg Fb Ff mg mv2g fu 0 u m 1 v2 g f i) Define sedimentation constant. s 1 v2 u ter f g ii) Unit of the sedimentation constant is the Svedberg. iii) s has dimensions of time. 1 S = 10-13 s 2. Centrifugation a) Method I (with sedimentation and diffusion coefficients) i.) Centrifugation is similar to sedimentation replacing gravitational acceleration with centrifugal acceleration. g 2 x – angular velocity x – distance from rotation axis 1 v2 u ter f 2 x m 1 v2 2 x u f s 11 ii) The Stokes-Einstein equations, which examine viscosity theoretically from a microscopic perspective yield an expression for the frictional force on a smooth sphere in terms of the thermal energy and the diffusion coefficient. Ff kTu kT fu f D D iii.) Substitute for f. m 1 v2 2 x f u f u m 2 s f 1 v2 x 1 v2 kTs RTs m Mz D 1 v2 D 1 v2 iv.) Spinning causes a relatively distinct boundary to occur between the high concentration portion and a low concentration portion. v.) This boundary moves toward the cell bottom over time. dx xb t x u ter dx 2 dt s 2 2 s dt 2st ln b x x x x 0 x b ,i b,i vi.) Centrifugation is similar to sedimentation replacing gravitational acceleration with centrifugal acceleration. g 2 x vii.) Thus a plot of ln(xb) versus time yields the sedimentation constant. 1 d ln x b s 2 dt viii.) From the sedimentation constant and a known diffusion constant, the z-average molecular weight can be determined. 12 b) Method II (with sedimentation coefficient only) i) Determination of the diffusion coefficient can be difficult; thus a method without making a diffusion constant determination can be preferred. ii.) The flux of polymer particles away from the rotation axis due to centrifugal forces is in quasi-equilibrium with the flux of particles moving toward the rotation axis due to diffusion. (Quasiequilibrium implies that the boundary is changing very slowly in comparison of the time scale of the fluxes.) diff cent. cent. uc dc diff D dx dc D uc dx u dc 1 D dx c iii.) Substitute for the diffusion coefficient and terminal velocity into the equation from the balance of forces above. u m 1 v2 2 x f m m f u kT u 2 2 1 v2 x 1 v2 D x kT u kT dc 1 2 2 1 v2 x D 1 v2 x dx c Mz RT dc 1 2 1 v2 x dx c 2 dc M 1 v2 x dx c RT 13 iv.) Integration of the equation yields a usable expression. 2 2 2 2 c 2 M z 1 v2 x 2 x1 dc M 1 v2 x dx ln c RT 2RT c1 v.) Clever substitution leads to a linear plot where the molecular weight is found by measuring how the boundary changes with concentration with a plot of ln c vs. x2. c dc d ln c ln c2 ln c1 ln 2 c c1 x 22 x12 1 2 x dx 2 d x 2 2 d ln c M z 1 v2 RT d x2 Mz d ln c RT 2 2 d x 1 v2 I. MALDI-MS (Matrix – assisted laser deposition/ionization mass spectroscopy) 1. Polymer and “matrix substance” are dissolved in solvent. 2. Solution is deposited on probe and dried. 3. Matrix molecules absorb UV light. a. Vibrational relaxation causes polymer to desorb from surface. b. Desorbed polymer molecules are carried into mass spectrometer.