Chap.. Molecular Weight and Polymer Solutions. Number Average and Weight Average Molecular Weight A) Importance of MW and MW Distribution M.W. physical properties As M.W., toughness, viscosity ) Optimum MW, MW Distribution depends upon application via processing and performance tradeoffs ) Typical MW values for commercial polymers a) Vinyl polymers in the 0 5 and 0 6 range b) Strongly H-bonding polymers in the 0 4 range e.g., 5,000-0,000 for Nylon
Intermolecular Interactions Increasing Interaction Strength Type of Interaction Characteristics Approximate Strength Examples Dispersion Forces Short Range Varies as -/r 6 0. - 0.5 kcal/mole Poly(ethylene) Polystyrene Dipole/dipole Interactions Short Range Varies as -/r 4 0.5 - kcal/mole Poly(acrylonitrile) PVC Strong Polar Interactions and Hydrogen Bonds Complex Form but also Short Range - 0 kcal/mole Nylons Poly(urethanes) Coulombic Interactions Long Range Varies as /r 0-0 kcal/mole Surlyn (Ionomers) Surlyn: copolymers such as ethylene/methacrylic acid, DuPont adds zinc, sodium, lithium or other metal salts.
3 B) Number Average Molecular Weight, M n ) Very sensitive to the total number of molecules in solution and sensitive to the low molecular weight monomers and oligomers - Determined by End Group Analysis and Colligative Properties (freezing point depression, boiling point elevation, osmotic pressure) ) M n N M i N i i where N i # of molecules (or # of moles) having MW M i 3) Example 9 moles of MW 30,000 and 5 moles of MW 50,000 M n 37,000
4 C) Weight Average Molecular Weight, M w ) Sensitive to the mass of the molecules in solution sensitive to the very highest MW species - Determined by Light Scattering and Ultracentrifugation ) WM i i NiMi Mw Wi NiMi 3) Example 9 moles of MW 30,000 and 5 moles of MW 50,000 M w 40,000 4) Note: Mw a) M w M b) n polydispersity index (PDI) Mn c) For a sample having a single MW (Monodisperse) M w M n M M d) Polydisperse M w M w > M > n Mn w n
5 D) General Molecular Weight Expression & M z and M v a+ NiMi a NiMi ) M a 0 for M n 0 < a < for M v a for M w a for M z For polydisperse sample M z > M w > M v > M n ) Z average, is closely related to processing characteristics a M z 3 NiMi NiMi 3) Viscosity average MW, M v, has 0 a and closer to (i.e., to M w ) M v N M i N M i a + i i a.8 NiM i Mv NiM i in a typical case 0.8
. Polymer Solutions 6 A) Steps Dissolving a Discrete Molecule and a Polymer ) Discrete Molecule Dissolution Steps for a Crystalline Sample ) Polymer Dissolution Steps a) Solvent diffusion i) Solvation & swelling ii) Gel formation iii) Network polymers stop at this stage Degree of swelling correlated with crosslink density b) True dissolution i) Untangling of chains ii) Very slow process and may not occur on timescale of real world
7 B) Thermodynamics of Polymer Dissolution ) Choosing a Solvent for Polymers a) Polymer Handbook! lists solvents and nonsolvents for common polymers b) Rule of Thumb: Like dissolves Like ) G H - T S a) G must be negative for spontaneous dissolution b) S will be positive because of greater mobility in solution c) need H to be negative or at least not too positive
8 3) H mix (δ - δ ) a) H mix Enthalpy of mixing (dissolution) b) δ Solubility Parameter of one component c) δ Solubility Parameter of the other component 4) In practice, H is seldom negative and we simply try to keep it from getting too positive 5) We see that we want the polymer and the solvent to have as similar of Solubility Parameters as possible
9 C) Solubility Parameters (δ) ) δ is related to the heat of vaporization of the sample H mix V mix E V Where V mix total volume of mixture E V V, V molar volumes E V φ, φ volume fractions E, E energies of vaporization E, V V E cohesive energy desities (CED) δ solubility parameter CED energy needed to remove a molecule from its nearest neighbors heat of vapoization per volume for a volatile compound φ φ
0 ) For small molecules these can be measured experimentally E H vap RT where H vap latent heat of vaporization δ H R gas constant vap 3) δ s of solvents are tabulated V RT
4) For conventional polymers these can be estimated using tables a) Group Molar Attraction Constants b) Table. Group molar attraction constants G[(cal cm 3 ) / mol - ] Group small Hoy CH 3 CH CH C CH CH C 6 H 5 CH C O CO 4 47.3 33 3.5 8 85.99-93 3.03 90 6.5 9 84.5 (phenyl) 735 - (aromatic) - 7. (ketone) 75 6.7 (ester) 30 3.6
c) δ d G M i) G the individual Group Molar Attraction Constants of each structural fragment ii) d density iii) M molecular weight d) For polystyrene CH CH C 6 H d.05, repeating unit mass 04 5 Small s G values δ Hoy s G values δ.05.05 ( 33 + 8 + 735) 04 ( 3.5 + 85.99 + 6( 7.) ) 04 9.0 9.3 e) Major problem with solubility parameters: They do not take into account strong dipolar forces such as hydrogen bonding.
D) Hydrodynamic Volume (V h ) in Solution 3 ) The apparent size of the polymer in solution ) Reflects both the polymer chain itself and the solvating molecules in inner and outer spheres r end-to-end distance s radius of gyration r mean-square end-to-end distance s mean-square radius of gyration V ( ) For a linear polymer: r 6 s h r 3
Root-Mean-Square End-to-End Distance 4 r o N l (r o ) 0.5 N 0.5 l If N 0,000, l ; (r o )0.5 00!!! l r
3) Hydrodynamic Volume is related to an Expansion Factor, α 5 a) The greater the affinity of solvent for polymer, the larger will be the sphere. r r 0 α s s 0 α r 0, s 0 unperturbed dimension size of macromolecule exclusive of solvent effects combination of free rotation and intramolecular steric and polar interactions α expansion factor r α r 0 s s interactions between solvent & polymer b) α for the non-expanded polymer in the ideal statistical coil having the smallest possible size c) as α increases, so does the Hydrodynamic Volume of the sample 0
E) Theta (θ) State ) Solubility varies with temperature and the nature of the solvent 6 ) There will be a minimal dissolution temperature call the Theta Temperature and at that point the solvent is said to be the Theta Solvent 3) The Theta State at this point is the one in which the last of the polymer is about to precipitate 4) Compilations of Theta Temperatures & Solvents are available in the literature
F) Intrinsic Viscosity & Molecular Weight 7 ) [η] Intrinsic Viscosity η rel η η 0 η sp η η η 0 0 η red η sp c η lnη rel sp η inh [ η] ( ηred ) ( ηinh ) c 0 c 0 c c ) Flory-Fox equation ( ) 3 ( ) φ r φ r [ ] 0 α η M M where φ proportionality constant Flory constant 3 x 0 4 mol - Vh 3 c 0 ( ) 3 r [ η]m
Rearranged to 3 3 0 0 M [ ] ( ) ( ) 3 3 η φ r α M φ r M M α K α 8 where K At T θ, α ( ) 3 r 0 M φ Q 0 M [ η] KM θ r θ solvent At T θ, α α(m) M 0~0. [ η] 0.8 KM good solvent α M 0. 3) Mark- Houwink-Sakurada Equation a) [ ] a η KMv b) K and a are characteristic of the particular solvent/polymer combination c) a 0.5 (θ solvent) ~ 0.8 (good solvent)
9.3 Measurement of Number Average Molecular Weight M n General Considerations ) Most methods give only averages Exceptions are: GPC, Light Scattering, MS ) Most methods results vary depending on the structure of the sample need to calibrate each sample and/or know some structural information such as branching 3) Most methods have limited sensitivities and/or linear ranges 4) Most methods require expensive instrumentation 5) There can be substantial disagreements between the results of different techniques
.3. End- group Analysis ) Basic principles a) The structures of the end groups must be different from that of the bulk repeating units (e.g., CH 3 vs. CH in an ideal polyethylene) 0 b) If you detect the concentration of the end group and know the total amount of sample present you can calculate the average MW, M n. i) need to have either a perfectly linear polymer (i.e., two end groups per chain) or need to know information about the amount of branching ii) the M n values that come out for linear polymers must typically be considered an upper bound since there may be some branching c) Detection of concentrations of end groups i) Titration, using either indicators or potentiometric techniques ii) Spectroscopy - IR, NMR, UV-Vis iii) Elemental Analysis iv) Radioactive or Isotopic labels Mn Linear polyerster HOOC~~~~~OH sample weight sample weight [ COOH] + [ OH] [ COOH] + [ OH]
) Strengths a) The requisite instruments are in any department b) can be quite quick c) Sometimes this information comes out free during polymer structural studies 3) Weaknesses a) does not give MW distribution information b) need to know information about the structure - identity and number of end groups in each polymer molecule c) limited to relatively low MW for sensitivity reasons i) Practical upper limit ; 50,000 ii) 5,000-0,000 is typical MW range iii) Can be high with some detections types radioactive labeling of end groups fluorescent labeling of end groups
.3. Membrane Osmometry Static equilibrium method: No counterpressure, Long time Fig.. Membrane Osmometry
Dynamic equilibrium method: Counterpressure, short time 3 Fig..3 Automatic Membrane Osmometry
Van't Hoff equation π c RT M n + A c where π osmotic pressure ρg h R gas constant 0.08 L atm mol - K - 8.34 J mol - K - c concentration [g L - ] ρ solvent density [g cm -3 ] g acceleration due to gravity 9.8 m s - h difference in heights of solvent and solution [cm] A second virial coefficient measure of interaction between solvent and polymer A 0 at T θ, A > 0 at T > θ 4 Major source of error: low-m.w.-species diffuse through membrane M n (obtained) > M n (actual) 50,000 < M n <,000,000
5 π c RT M n + A c
6 µ s G n s T,P,n p
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30 de TdS -PdV dh d(e + PV) TdS PdV+ PdV VdP TdS + VdP dg d(h - TS) TdS + VdP - TdS SdT - SdT + VdP dg - SdT + VdP ( + µ s dn s + µ p dn p ) G P T,n s, n p V
3 n s n s n p n s
3 n s n s n s
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35 n p np n s n p n p n s n s n s
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39.3.3 Cryscopy and Ebulliometry Freezing-point depression ( T f ) Boiling-point elevation ( T b ) T c RT f + c 0 ρ Hf Mn A c T c RT b + c 0 ρ HvMn A c where T freezing point or boiling point of the solvent ρ solvent density H f latent heat of fusion H v latent heat of vaporization A second virial coefficient The most sensitive thermister ; 0-4 C Upper limit 40,000
.3.4 Vapor Pressure Osmometry - Polymer solution and pure solvent are placed on thermistor beads. - Solvent vapor condenses onto polymer solution. - Temp rise of solution due to heat of evaporization. - Vapor pressure of solution is increased to that of pure solvent. - Temperature difference between the solution and the solvent droplet is measured as the resistance difference R between the thermistor beads. 40
In a dilute solution, the vapor pressure of a solvent is given by Raoult s Law where P P 0 x P partial pressure of solvent in solution P 0 vapor pressure of pure solvent x mole fraction of solvent 4 x -x x mole fraction of solute P P 0 (-x ) vapor pressure lowering P P 0 -P P 0 x It is assumed that T, H v, and P constant. Clausius-Clapeyron equation dp dt P H RT integrated to yield T v RT P P H v where P vapor pressure T absolute temperature H v enthalpy of vaporization R gas constant
Substitution of the equations For small pressure changes, P 0 P x For very small n where n x n + n n n T T o RT P x P Hv RT x H n number of moles of solvent n number of moles of solute v 4 w M w M RT n RT RT w RT m T M Hv n Hv Hv w M Hv M where w weight of solvent w weight of solute m w w 000 M 000 g molality ( kg )
43 n v v M m 000 RT 000 M m M H RT 000 M M m H RT T λ Where λ heat of vaporization per gram of solvent n M m 000 RT T λ
The energy of laser beam is transferred to the matrix which is partially vaporized, carrying intact polymer into the vapor phase and charging the polymer chains..3.5 Matrix-Assisted Laser Desorption Ionization Mass Spectrometry (MALDI-MS or MALDI-TOF) 44 N Laser (337 nm) Analyte: Polymer as large as 0 6 Da Matrix: UV absorbing 0 4 x molar excess
All ions are rapidly accelerated to ideally the same high-kinetic energy by an electrostatic field and expelled into a field-free region (flight-tube) where they physically separate from each other based on their mass-to-charge (m/z) ratios. 45
Linear time-of-flight matrix-assisted laser desorption ionization mass spectrometer 46
Figure.5 MALDI mass spectrum of low-molecular-weight PMMA 47
48.4 Measurement of M w.4. Light Scattering ) Laser light-scattering photometer (Figure.7)
) Polymer molecule in solution (and its associated solvent molecules) has a different refractive index than neat solvent 49 they behave as tiny lenses and scatter light a) scan detector over a range of angles or use multiple detectors b) measure scattered intensity as a function of angle and concentration c) Use Zimm plot to extrapolate to infinite dilution and to zero degrees
50 Kc ( + cos θ) R θ M w + 6π 3λ s sin θ + A c where K π no 4 λ N dn dc o n o refractive index of the solvent λ wavelength of the incident light N o Avogadro s number dn dc specific refractive increment I r R θ θ Rayleigh Ratio Io r distance from scatterer to detector I o θ I θ
k scaling factor Figure.6 Zimm plot of light-scattering data Kc ( + cos θ) R θ M w + 6π 3λ s sin θ + A c 5 Kc ( + cos θ) R θ θ 0 M w + A c Kc ( + cos θ) R θ ( + cos θ) Kc 6π θ + s sin Rθ Mw 3λMw c 0
.5 Viscometry Not an absolute method Measured at concentrations of about 0.5g/00mL of solvent A) Viscosity Measurement ) Table. Dilute Solution Viscosity Designations 5 Common Name IUPAC Name Definition Relative viscosity Specific viscosity Reduced viscosity Inherent viscosity Intrinsic viscosity Viscosity ratio - Viscosity number Logarithmic viscosity number Limiting viscosity number η t η rel ηo to η η o t t o η sp η rel η t η η red inh η C o η C ln η C sp rel o η rel C sp [ η ] ( η ) C 0 C 0 inh
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55 ) Basic Principles a) Mark-Houwink-Sakurada Equation [ ] a η KMv log[η] logk + a logm v i) K and a are characteristic of the particular solvent/polymer combination ii) M v Viscosity Average Molecular Weight M v N M i N M i a + i iii) a 0.5 (θ solvent) ~ 0.8 (good solvent) i a
56 b) Measurement of [η] i) Make up 5-6 solutions at different concentrations of the same sample and of pure solvent ii) measure the time it takes each of them to flow through the viscometer iii) extrapolate to viscosity at zero concentration which gives the intrinsic viscosity Capillary viscometers: Ubbelohde Cannon-Fenske
57 c) Measurement of [η], K and a Plot the [η] values against the MW values from another technique and get K and a from the intercept and slope log[η] logk + a logm v d) Determination for a polymer of known structure i) Look up K and a in the Polymer Handbook (Table.3) ii) Use the [η] values to calculate M v directly
Table.3 Representative Viscosity-Molecular Weight Constants a Polymer Polystyrene (atactic) c Polyethylene (low pressure) Poly(vinyl chloride) Polybutadiene 98% cis-,4, %, 97% trans-,4, 3%, Polyacrylonitrile Poly(MMA-co-St) Solvent Cyclohexane Cyclohexane Benzene Temperature 35 d 50 5 Molecular Weight Range X0-4 8-4 e 4-37 e 3-6 f K b X 0 3 a b 80 6.9 9.5 58 0.50 0.599 0.74 Decalin 35 3-00 e 67.7 0.67 Benzyl alcohol cyclohexanone Toluene Toluene DMF g DMF 55.4 d 0 30 30 5 5 4-35 e 56 7-3 f 3.7 5-50 f 30.5 5-6 f 9.4 5-7 e 6.6 3-00 f 39. 0.50.0 0.75 0.753 0.8 0.75 30-7- mol % 7-9 mol % -Chlorobutane -Chlorobutane 30 30 5-55 e 7.6 4.8-8 e 4.9 0.67 0.63 PET m-cresol 5 0.04-. f 0.77 0.95 Nylon 66 m-cresol 5.4-5 f 40 0.6
.6 M.W. Distribution.6. Gel Permeation Chromatography (GPC) Size Exclusion Chromatography (SEC) 59
60 ) Basic Principles 3~0 µm Pore size: 0.5~0 5 nm M.W.: 00~4x0 7 microporous gel particle styrene and divinyl benzene copolymer
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Injection Exclusion Partial permeation Total permeation 6 0 V o V o + kv i V o + V i Retention volume Retention volume (V r ) elution volume V r V 0 + kv i Where V 0 Interstitial (or void) volume between porous gel particles V i Pore volume within the porous gel particles k Partition coefficient between V i and the portion accessible to a given solute 0 ~ very large molecule which cannot penetrate any available pore volume very small molecule which can penetrate all the available pore volume
Gives Polystyrene (or Poly(vinyl alcohol)) Equivalent MWs Figure.0 Typical gel permeation chromatogram 63
Reference polymer: monodisperse polystyrene or poly(vinyl alcohol) Figure. Typical semilogarithmic calibration plot 64 Exclusion Total permeation
) Universal calibration 65 [η] M.5 N A V h Einstein viscosity relation universal calibration parameter constant for all polymers for a given column, temp, and elution volume Polymer reference polymer (e.g. polystyrene) Polymer polymer to be fractionated For equal elution volumes of two different polymers [η] M [η] M Mark-Houwink-Sakurada relationship a a [ η ] K [ η] K + a + KM K M M a M ( + a) M logk + ( a ) log K + logm + M K + a log + log M + a K + a
Figure. Universal calibration for GPC log([η]m) 66 0 9 log([η]m) is plotted with V r All polymers fit on the same curve 0 8 0 7 0 6 0 5 8 0 4 6 8 30 Retention volume