Part I. POLYMER STRUCTURE AND PROPERTIES Chapter 2. Molecular Weight & Polymer Solutions. 2.1. Number Ave. Molecular weight 2.2. Polymer Solutions 2.3. Measurement of Number Ave. Molecular Weight 2.4. Measurement of Weight Ave. Molecular Weight 2.5. Viscometry 2.6. Molecular Weight Distribution Chapter 2. MW & Polymer solutions. 2.1 Number Ave. & wt Ave. Molecular weight MW Physical properties ex) Higher MW polymer : Tougher ???? Too high MW “What do we mean by high molecular weight ?” Low MW : High MW Factors ? Where the boundary ? CH2 CH2 * ex) Polyethylene vs Polyamide ex) Low MW polymer Initial processing H H ex) Vinyl polymer MW 105 ~ 106 N (CH2)5 N H * O C (CH2)4 C OH n O O Polyamide w/ polar group MW 15000 ~ 20000 H N H (CH2)5 C OH What we have calculated is something called a number average, which is defined mathematically below. If you’re not used to dealing with summations, this looks Horrible. To give you a feel for how it works, and also introduce a different average- the weight average let’s consider a ridiculous example. Molecular weight Number of end group Free volume Viscoelastic property etc (Example) “Number average molecular weight” (Example) “Number average molecular weight” “Weight average molecular weight” Now, let’s say we had a sample with 5 (moles of) chains of “length” (degree of polymerization or DP) 100, 5 (moles of) chains of length 150 and 5 (moles of ) chains of 200. Nx is simply the number ( of moles) of chains of the “x-species”. We have three species in our sample; Chains of DP 100, chains of DP 150 and chains of DP 200, whose weights M, are therefore 10,000, 15,000, and 20,000, respectively. Before proceeding, see if you can substitute correctly into the equation opposite. What about the weight average molecular weight ? Now note that the total weight of species x present is just the molecular weight of each chain of type x multiplied by the number of chains of this type (e.g. 5 chains, each of weight 10,000 means that Wx is 50,000): WE can now substitute in the equation at the top to obtain a different form of the equation for weight average. “Molecular Weight Distribution” Number and weight average molecular weight [Text P37] 9 moles having 30,000 molecular weight and 5 moles having 50,000 molecular weight Substituting grams for moles: In each instance, we see that Mw is greater than Mn. n Please determine Mn and Mw now !! Mn Mw PDI 4 moles having 1 molecular weight and 2 moles having 5 molecular weight 2.33 3.86 1.65 4 moles having 4 molecular weight and 2 moles having 5 molecular weight 4.33 4.38 1.01 What difference can we see after comparison ??? We have seen that ave. molecular weight is not unique. It turnes out that there are more than two ways to define an average. Look at the definitions of number and weight average again. You can see that we can go from number to weight average by multiplying each of the terms inside the summations by Mx. Higher order averages can be constructed in the same way; e.g. the z-average The ratios of these averages can be related to the moments of the distribution and tell us about its breadth and “skewedness”. Effect of MW on the Physical properties. Mechanical Strength C B A: * H2 C H2 C * B: * H2 C H2 C * 1000 2000 (Mn)A= 28000 (g/mole) (Mn)A= 56000 (g/mole) MW A Useful MW Optimum MW 20000-40000 (Min) MW 5000-10000 Ni A: * H2 C H2 C * N: Number of repeating unit 1000 Distribution ? I (chain length) ** MW Distribution ⓐ Tensile strength Modulus Thermal expansion coefficient Free Volume Refractive index Dielectric constant Etc. wt% ⓑ DP What properties ? (a) , (b) * Molecular Weight ? wt% A ⓐ ⓑ wt% DP B ⓐ ⓑ DP Molecular Weight Distribution (MWD) (a) Wx Nn (b) Mn Mw Mw Mz Mx Mx Mn Polydisperse Polymer : Practical Polymer Mn Mw Mz Mn Mw Mz Monodisperse polymer ? Polydispersity Index (PDI) = Mw Mn 2.2 Polymer Solution Dissolution ① Solvent molecule diffuse through the Polymer matrix Swollen gel ② Gel breaks up and the molecules are dispersed into a true solution Slow Process ∝ T Network polymer ? Choice of Solvent Polymer Handbook Thermodynamic Principles Semiempirical relationship - When a polymer dissolves spontaneously, ∆G is Negative !! The ∆S invariably has a (+) value arising from increased conformational mobility of the polymer chains ∴ ∆H determine Sign of ∆G Heat of mixing ∆Hmax for binary system Concentration & Energy Parameters Cohesive Energy Vm: total volume of the mixture V1, V2: molar volume (molecular weight/density) F1, F2 : volume fraction DE1, DE2: Energy of vaporization If (∆E/V)1/2 is replaced by the symbol d, the equation is written more simply Solubility Parameter For dissolution (negative ∆G), ∆Hmix must be small ∴ (d1- d2)2 must be small! d1 ≈ d2 similar solubility parameters <Solvent 1, Polymer 2> Cohesive E : Energy needed to remove a molecule from its nearest neighbors. ≈ heat of vaporization per Volume for volatile Compound For solvent, d1 From the latent heat of vaporization (∆Hvap ) D vap Since Polymers have negligible vapor pressure !! d2 “Group “Group molar attraction constants” Contribution” expect many physical properties !! “Group Contribution” G values are additive for a given structure Ex) PS * * n r=d=1.05 Unit mass = 104 i) Small’s ii) Hoy’s ?? Trouble Molecular Interaction ? Strong Dipolar interactions (e.g. Hydrogen Bonding) Polymer-Solvent System How the polymer molecules behave in that solvent ? Resultant size Hydrodynamic volume in solution Depend on ① interaction btw solvent & polymer molecule ② Chain branching ③ Conformational effects (arising from the polarity & steric bulk of the substituent) ④ restricted rotation Resonance ex) Type of Polyamide O C H N O H C N Mean Square Average Distance for Linear Polymer Mean Square Average Radius of Gyration about COG for a branched Polymer Average shape of the Coiled molecule “Spherical” “The greater the affinity of the solvent for polymer, the larger will be the Sphere” “Hydrodynamic Volume” Solvent Interaction 3/22/2005 r 2 = r 02 a 2 s 2= s 02 a 2 Expansion factor *Unperturbed dimension : No solvent Effect r0, s0 Combination of free rotation and intramolecular steric and polar interaction *Expansion factor For a linear polymer a a a=1 interaction btw Solvent and Polymer r2= 6 s2 a >1 in a good solvent ! better solvent “ Ideal” statistical Coil Solubility ∝ f(temp) in a given solvent a ∝ f(temp) “No solvent interaction” For a given polymer in a given solvent Lowest T (a=1) Theta (θ) temperature “Theta State of Polymer” in solution. From the stand point of MW determinations, significance of the Parameters Dilute Solution Viscosity “Flory-Fox Equation” Where [ŋ] is the Intrinsic Viscosity M is average MW. Φ proportionality constant ~ 3X1024 mol-1 Substituting r02 a2 for r2 Since r0 & M are Constant set k= F(ro2 M-1)3/2 at θ Temp, a=1 a ∝ f (polymer, solvent, T) “Mark-Houwink-Sakurada” eqn. In q solvent “Mark-Houwink-Sakurada” equation 2.3. Measurement of Molecular Weight No relation with the kind of molecule!! 2.3. Measurement of Number Average Molecular Weight 2.3.1. End-group analysis : No average MW of any Linear Polymer having End-groups. very low concentration. Limit of MW ~ 50,000 “If high MW or low MW polymer ???” 1) Titration 2) Elemental Analysis 3) Measurement of activity of a Radioactive -tagged end group 4) UV spectroscopic determination of an end group w/ a characterizable chromophore Points 1. Not valid for branched polymer unless the number of branches is known. 2. In a linear polymer, twice as many end groups as polymer molecules. 3. If the polymer contains different groups at each end of the chain and only one characterizable End group is being measured, the number of this type is equal to the number of polymer molecules. 4. Measurement of molecular weight by end-group analysis is only meaningful when the mechanisms of initiation and termination are well understood. Typical Example H3COOC COOCH3 + O O * O HOCH2CH2OH + 2CH3OH O n * to determine No average MW of the linear Polyester Titrate the carboxyl and hydroxyl end groups by standard methods. *Carboxyl : Titration *Hydroxyl : Titration A weighed sample of polymer is dissolved in an appropriate solvent (acetone) A sample is acetylated w/ excess acetic anhydride Titrated w/ standard base to a phenolphthalein end point. Liberated acetic acid <together w/ carboxyl end groups> is similarly titrated “Two steps” From the two titrations # of milliequivalents of carboxyl and hydroxyl 2 in the numerator : two end groups counted per molecule • Shortly : Difficulty !! 5000 ~10000 “Valid” OSMOSIS 2.3.2. Membrane Osmometry : Number ave. MW most useful!! Definition : Osmotic Pressure The pressure at equilibrium (no further passage of the solvent) Dynamic Equilibrium Method Apply a counter pressure to the measuring tube connected to the solution compartment Prevents flow of Solvent and maintains equal liquid levels in the two measuring tubes. Static Equilibrium Method : Long period of time for Equilibrium ∴Dynamic method Encompass horizontal membrane Separating solution and solvent Cells :Measures osmotic pressure directly via a strain guage transducer attached to a flexible diaphragm in the Solvent Cell Semipermeable Membranes Cell acetate, Cell nitrate, rubber, PVA Osmotic Pressure is related to MW by the van’t Hoff equation P : osmotic pressure. P = pgΔh R : 0.082 L atm mol-1K-1 T ; Kelvin. C: Conc in g/L ρ : solvent density in g/cm3 g : acceleration due to gravity 9.81 m/s2 Δh: the difference in heights of solvent and solution in an A2 : 2nd virial coefficient (measure of the interaction btw solvent & polymer) Intrinsic parameter Osmotic pressure : allowing the system to reach equilibrium and measuring the hydrostatic head that develops. WAIT and WAIT for equl. !!! Thin is referred to as the “Static Equilibrium Method” P/C : dyne L g-1 cm-1, J kg-1 A plot of Reduced osmotic pressure (π/c) vs concentration is linear with the intercept equal to RT/Mn and slope equal to A2 What does A2 mean ? A2 : Measure of Solvent-Polymer interaction ∴ Slope = 0 at General Mn ~ 50000 ~ 2x106 Tθ Theta Condition! Versatile Method Preferred !! Error Source : Low-molecular-wt species diffuse through the Membrane! B A 2.3.3. Cryoscopy & Ebulliometry Cryoscopy : freezing-point depression Ebulliometry : boiling point elevation C : cont of cm3 T : freezing of boiling T of the solvent (K) R : Gas constant r: solvent density ∆Hv ∆Hf : latent heats of fusion & vaporization per gram of solvent A2 : 2nd virial coeff. Mn : No Ave. MW !! Error source : sensitivity of the Methods of measuring Mn < 20000 Preferred (not useful) ∆Hv ∆Hf 2.4. Measurement of weight Average MW 2.4.1. Light scattering : useful, popular method (~osmometry) Scattering for a pure liquid finite nonhomogenieties in the distribution of molecules within adjacent area that give rise to differences in density + solvent molecular scattering The amplitude of intensity of the scattered light Concentration, size, polarizability • Refractive index depends on conc. and amplitude of vibration Turbidity n0 : refractive index of the solvent l : wavelength of the incident light N0 : Avogadro # dn : specific refractive increment dc n vs c slope !! General : constant (polymer, solvent, Temp) Who is this man ? As molecular size approaches in magnitude the l of light Corrections must be made for interference btw scattered light coming from different parts of the molecules. James Clerk Maxwell !!! To determine molecular wt P(θ) : function of the angle θ A2 : 2nd-virial coeff. Depends on the shape of the molecule in solution Experimental data : extrapolation to both c=0, θ=0 where p(θ) is equal to 1 Intercept corresponds to 1 Mw Error source : Dust! Availability : Mw 10,000 ~ 10,000,000 2.4.2. Ultracentrifugation : Intricate and expensive instrument : Not widely used : applicable to natural polymer : useful to determine Mz of synthetic P Stimulus Strong centrifugal field Distribute them according to size perpendicularly to the axis of rotation “Sedimentation” 2.5. Viscometry Dilute solution viscosity : simplest popular Not absolute method Stand polymer solution : calibration • Conc. : 0.5g/100ml Folw time : sec Temp : 30.0 ± 0.01 oc •Viscosity Expression Rel. viscosity (ηrel) Specific viscosity (ηsp) : fractional increase in η Name by IUPAC Dimensionless ! •Intrinsic viscosity : eliminate conc. effect • Inherent viscosity : approximate indication of MW C : g/100ml g/cm3 Reduced, inherent, intrinsic η dL/g cm3/g (less) •Intrinsic viscosity Mark-Houwink-Sakurada eqn. • For most common polymers “a” varies 0.5~0.8 (random coil polymer in θ solvent) For more rodlike extended-chain polymers, where hydrodynamic vol is relatively large. A •K 1.0 10-3 ~ 0.5 “solvent” “temperature” • Error source : chain branching too broad MWD solvation of polymer molecules alternating or block 2.6. MW Distribution 2.6.1. Gel permeation chromatography (GPC) SEC : size exclusion chromatography highly porous materials Separates the polymer molecules according to size, “Molecular sieving” Principle : i) small molecules : diffuse into the pore Slowly travel ii) higher MW fractions are thus eluted first • Detection of polymer fraction i) ii) Refractive index UV or IR detector • Typical GPC Detector response vs vol. of dilute polymer solvent (retention vol.) elution vol. • Major problem w/ calibrating a GPC column : Few standard samples (narrow MWD) * Ex) * n MW 600~2.5 million Universal calibration method [η]•M independent of polymer type “Universal calibration parameter” Plot of log [η]•M vs elution vol in THF Single curve (~ linear) • log([η]•M ) : constant for all polymers for a given column, Temp, elution vol If assuming reference polymer(PS) is P1 sample polymer P2 From Mark-Houwink-Sakurada To determine the M2 at a given retention t, column must be calibrated w/ standard PS fraction (same solvent, same temp) Semilogarithmic calibration plot : linear Over a broad range of MW • K, a polymer handbook M1 from calibration of col. K, a M2 can be readily calculated ! • Problem ? 2.6.2. Fractional Solution - extract a polymer in a soxhlet apparatus - Solvent (+Nonsolvent) low Dissolution of high MW polymer