Chapter 2. Molecular Weight and Polymer Solutions 2.1 Number average and weight average molecular weight 2.2 Polymer solutions 2.3 Measurement of number average molecular weight 2.4 Measurement of weight average molecular weight 2.5 Viscometry 2.6 Molecular weight distribution POLYMER CHEMISTRY

2.1 Number Average and Weight Average Molecular Weight A. The molecular weight of polymers   a. Some natural polymer (monodisperse) : All polymer molecules have same molecular weights.  b. Synthetic polymers (polydisperse) : The molecular weights of  polymers are distributed         c. Mechanical properties are influenced by molecular weight    much lower molecular weight ; poor mechanical property    much higher molecular weight ; too tough to process    optimum molecular weight ; 105 -106 for vinyl polymer     15,000 - 20,000 for polar functional group containing polymer (polyamide) POLYMER CHEMISTRY

B. Determination of molecular weight Absolute method : mass spectrometry colligative property end group analysis light scattering ultracentrifugation.   b. Relative method : solution viscosity c. Fractionation method : GPC POLYMER CHEMISTRY

C. Definition of average molecular weight   a. number average molecular weight ( Mn )                        Mn= (colligative property and  end group analysis) b. weight average molecular weight ( Mw)                   Mw= (light scattering)  i i Ni N M WiMi Wi POLYMER CHEMISTRY

c. z average molecular weight ( MZ )                          MZ= (ultracentrifugation) d. general equation of average molecular weight : M =                                ( a=0 , Mn            a=1 , Mw            a=2 , Mz         )   e. Mz > Mw > Mn    C. Definition of average molecular weight NiMi3 NiMi2 NiMia+1 NiMia POLYMER CHEMISTRY

D. Polydispersity index : width of distribution   polydispersity index (PI) = Mw / Mn ≥ 1 POLYMER CHEMISTRY

E. Example of molecular weight calculation   a. 9 moles, molecular weight (Mw) = 30,000     5 moles, molecular weight ( Mw) = 50,000 (9 mol x 30,000 g/mol) + (5 mol x 50,000 g/mol) Mn= = 37,000 g/mol 9 mol + 5 mol 9 mol(30,000 g/mol)2 + 5 mol(50,000 g/mol)2 Mw = = 40,000 g/mol 9 mol(30,000 g/mol) + 5 mol(50,000 g/mol) POLYMER CHEMISTRY

E. Example of molecular weight calculation   b. 9 grams, molecular weight ( Mw ) = 30,000     5 grams, molecular weight ( Mw ) = 50,000 = 35,000 g/mol Mn = 9 g + 5 g (9 g/30,000 g/mol) + (5 g/50,000 g/mol) Mw = = 37,000 g/mol POLYMER CHEMISTRY

2.2 Polymer Solutions A. Process of polymer dissolution : two step   first step : the solvent diffuses into polymer masses to make               a swollen polymer gel    second step : swollen polymer gel breaks up to solution POLYMER CHEMISTRY

2.2 Polymer Solutions B. Thermodynamics of solubility : Gibb's free energy relationship                             G =H - TS    ΔG < 0 : spontaneously dissolve    T and ΔS are always positive for dissolving process.    Conditions to be negative ΔG,    ΔH must be negative or smaller than TΔS. POLYMER CHEMISTRY

C. Solubility parameter : δ                  Hmix=Vmix[( )1/2-( ) 1/2]212       ψ1, ψ2 = volume fraction    ΔE1/V1, ΔE2/V2 = cohesive energy densities    δ1, δ2 = solubility parameter                                          δ1, δ2   = (                  )1/2 Hmix= Vmix(δ1 – δ2)212 E = Hvap- RT δ1 = ( )1/2                      if δ1= δ2, then Hmix= 0        V1 E1 V2 E2 V E V H vap - RT POLYMER CHEMISTRY

D. Small's and Hoy's G parameter   a. Small(designated G derived from Heat of vaporization, Table 2.1) δ =                               ( d : density , M :  molecular weight of unit ) ex) polystyrene           δ = = 9.0 b. Hoy(designated G based on vapor pressure measurement, Table 2.1) ex) polystyrene :           dG M M 104 1.05(133+28+735) dG M 104 1.05[131.5+85.99+6(117.1)] = 9.3 POLYMER CHEMISTRY

E. Hydrodynamic volume of polymer molecules in solution.   to be depended on followings polymer-polymer interaction b. solvent-solvent interaction c. polymer-solvent interaction d. polymer structure ( branched or not ) e. brownian motion r = end-to-end distance s = radius of gyration Figure 2.1 Coil molecular shape r 2 = ro22 s2= so22  = (r2)1/2 (ro2)1/2 The greater the value of α, the ‘better’ the solvent  α = 1, 'ideal' statistical coil.

2.2 Polymer Solutions F. theta(θ) temperature and theta(θ) solvent        The lowest temperature at which α=1 : theta(θ) temperature  blink The solvent satisfied this condition : theta(θ) solvent  point G. Flory-Fox equation : The relationship among hydrodynamic volumes,     intrinsic viscosity and molecular weight         [η] : intrinsic viscosity                      M : average molecular weight                       ψ : Flory constant (3×1024/mol)                       r : end-to-end distance [η] = (r2)3/2 M POLYMER CHEMISTRY

2.2 Polymer Solutions H. Mark-Howink-Sakurada equation : The relationship between intrinsic viscosity and molecular weight                                  [η] : intrinsic viscosity                         K , a : constant for specific polymer and solvent                           M : average molecular weight I. Important properties of polymer solution : solution viscosity    a. paint spraying and brushing b. fiber spinning [η] = KMa POLYMER CHEMISTRY

2.3 Measurement of Number Average Molecular Weight 2.3.1 End-group Analysis A. Molecular weight limitation up to 50,000 B. End-group must have detectable species     a. vinyl polymer : -CH=CH2    b. ester polymer : -COOH, -OH    c. amide and urethane polymer : -NH2, -NCO    d. radioactive isotopes or UV, IR, NMR detectable functional group POLYMER CHEMISTRY

2.3 Measurement of Number Average Molecular Weight 2 x 1000 x sample wt Mn = C. meq COOH + meq OH D. Requirement for end group analysis    1. The method cannot be applied to branched polymers.   2. In a linear polymer there are twice as many end of the chain and groups as polymer molecules. 3. If having different end group, the number of detected end group       is average molecular weight. 4. End group analysis could be applied for polymerization mechanism identified E. High solution viscosity and low solubility : Mn = 5,000 ～ 10,000 POLYMER CHEMISTRY

FIGURE 2.2 Schematic representation of a membrane osmometer.

2.3.2 Membrane Osmometry  RT ( )C=0 = + A2C c Mn   A. According to van't Hoff equation                        limitation of : 50,000～2,000,000     The major error arises from low-molecular-weight species diffusing     through the membrane. ( c  )C=0 = Mn RT + A2C FIGURE 2.3 Automatic membrane osmometer [Courtesy of Wescan Instruments, Inc.]

FIGURE 2.4. Plot of reduced osmotic pressure (/c) versus concentration (c). Mn RT C Slope = A2 POLYMER CHEMISTRY

2.3.3 Cryoscopy and Ebulliometry   A. Freezing-point depression (Cryoscopy)              Tf  : freezing-point depression,         C :  the concentration in grams per cubic centimeter         R :  gas constant         T : freezing point       Hf: the latent heats of fusion         A2 : second virial coefficient ( C Tf )C=0 = Hf Mn RT2 + A2C POLYMER CHEMISTRY

2.3.3 Cryoscopy and Ebulliometry   B. Boiling-point elevation (Ebulliometry)             Tb   : boiling point elevation      H v  : the latent heats of vaporization     We use thermistor to major temperature. (1×10-4℃)     limitation of Mn : below 20,000 ( C Tb )C=0 = HvMn RT2 + A2C POLYMER CHEMISTRY

2.3.4 Vapor Pressure Osmometry   The measuring vapor pressure difference of solvent and solution  drops.            λ : the heat of vaporization per gram of solvent                         m : molality     limitation of Mn : below 25,000   Calibration curve is needed to obtain molecular weight of polymer sample    Standard material : Benzil T = ( 100 RT2 )m POLYMER CHEMISTRY

2.3.5 Mass spectrometry   A. Conventional mass spectrometer for low molecular-weight compound   energy of electron beam : 8 -13 electron volts (eV) POLYMER CHEMISTRY

B. Modified mass spectrometer for synthetic polymer    a. matrix-assisted laser desorption ionization mass spectrometry       (MALDI-MS)    b. matrix-assisted laser desorption ionization time-of-flight       (MALDI-TOF) c. soft ionization      sampling : polymers are imbedded by UV laser absorbable organic                compound containing Na and K. d. are calculated by using mass spectra. e. The price of this mass is much more than conventional mass. f. Up to = 400,000 for monodisperse polymers. POLYMER CHEMISTRY

FIGURE 2.5. MALDI mass spectrum of low-molecular-weight poly(methyl methacrylate). POLYMER CHEMISTRY

2.3.6 Refractive Index Measurement   A. The linear relationship between refractive index and 1/Mn . B. The measurement of solution refractive index by refractometer. C. This method is for low molecular weight polymers. D. The advantage of the method is simplicity. POLYMER CHEMISTRY

 2.4 Measurement of Weight Average Molecular Weight 2.4.1 Light Scattering   A. The intensity of scattered light or turbidity(τ) is depend on following factors   a. size   b. concentration   c. polarizability   d. refractive index   e. angle   f. solvent and solute interaction g. wavelength of the incident light POLYMER CHEMISTRY

g. wavelength of the incident light    C  : concentration                          no: refractive index of the solvent         λ : wavelength of the incident light     No : Avogadro's number     dn/dc : specific refractive increment      P() : function of the angle,θ     A2 : second virial coefficient      Zimm plot (after Bruno Zimm) : double extrapolation of concentration  and angle to zero (Fig 2.6)  = HcMW 32 3 H = 4No No2(dn/dc)2  Hc = MP() 1 + 2A2C POLYMER CHEMISTRY

FIGURE 2.6. Zimm plot of light-scattering data. sin2/2 + kc  Hc Mw 1 C=0 Experimental Extrapolated POLYMER CHEMISTRY

2.4.1 Light Scattering B. Light source    High pressure mercury lamp and laser light. C. Limitation of molecular weight( ) : 104～107 FIGURE 2.7. Schematic of a laser light-scattering photometer. POLYMER CHEMISTRY

2.4.2 Ultracentrifugation A. This technique is used   a. for protein rather than synthetic polymers. b. for determination of Mz B. Principles : under the centrifugal field, size of molecules are distributed perpendicularly axis of rotation. Distribution process is called sedimentation. POLYMER CHEMISTRY

2.5 Viscometry A. IUPAC suggested the terminology of solution viscosities as following.   Relative viscosity :    : solution viscosity                                         o: solvent viscosity                                          t : flow time of solution                                         t o: flow time of solvent    Specific viscosity : Reduced viscosity : Inherent viscosity : Intrinsic viscosity :  rel = o  = to t rel - 1 sp = o  - o = to t - to c rel = sp inh = In rel [] = ( )c=o=(ηinh)C = 0 POLYMER CHEMISTRY

FIGURE 2.8. Capillary viscometers : (A) Ubbelohde, and (B) Cannon-Fenske. POLYMER CHEMISTRY

B. Mark-Houwink-Sakurada equation [η] = KMa log[η] = logK + alogMv                      [η] = KMa log[η] = logK + alogMv (K, a : viscosity-Molecular weight constant, table2.3)                  Mv        is closer to Mw       than Mn        Mw > Mv > Mn POLYMER CHEMISTRY

TABLE 2.3. Representative Viscosity-Molecular Weight Constantsa Polymer Polystyrene (atactic)c Polyethylene (low pressure) Poly(vinyl chloride) Polybutadiene 98% cis-1,4, 2% 1,2 97% trans-1,4, 3% 1,2 Polyacrylonitrile Poly(methyl methacrylate-co-styrene) 30-70 mol% 71-29 mol% Poly(ethylene terephthalate) Nylon 66 Solvent Cyclohexane Cyclihexane Benzene Decalin Benzyl alcohol Cyclohexanone Toluene DMFg DMF 1-Chlorobutane M-Cresol Temperature, oC 35 d 50 25 135 155.4d 20 30 Molecular Weight Range  10-4 8-42e 4-137e 3-61f 3-100e 4-35e 7-13f 5-50f 5-16f 5-27e 3-100f 5-55e 4.18-81e 0.04-1.2f 1.4-5f ab 0.50 0.599 0.74 0.67 1.0 0.725 0.753 0.81 0.75 0.63 0.95 0.61 Kb 103 80 26.9 9.52 67.7 156 13.7 30.5 29.4 16.6 39.2 17.6 24.9 0.77 240 aValue taken from Ref. 4e. bSee text for explanation of these constants. cAtactic defined in Chapter 3. d temperature. eWeight average. fNumber average. gN,N-dimethylformamide. POLYMER CHEMISTRY

2.6 Molecular Weight Distribution 2.6.1 Gel Permeation Chromatography (GPC) A. GPC or SEC (size exclusion chromatography)   a. GPC method is modified column chromatography.   b. Packing material: Poly(styrene-co-divinylbezene), glass or silica bead swollen and porous surface. c. Detector : RI, UV, IR detector, light scattering detector   d. Pumping and fraction collector system for elution.  e. By using standard (monodisperse polystyrene), we can obtain Mn , Mw . POLYMER CHEMISTRY

FIGURE 2.9. Schematic representation of a gel permeation chromatograph. POLYMER CHEMISTRY

Elution volume (Vr) (counts) Baseline Detector response FIGURE 2.10. Typical gel permeation chromatogram. Dotted lines represent volume “counts.” Elution volume (Vr) (counts) Baseline Detector response POLYMER CHEMISTRY

Heterograft copolyner Poly (methyl methacrylate) Poly (vinyl chloride) FIGURE 2.11. Universal calibration for gel permeation chromatography. THF, tetrahydrofuran. Log([η]M) 109 108 107 106 105           Polystyrene (linear) Polystyrene (comb) Polystyrene (star) Heterograft copolyner Poly (methyl methacrylate) Poly (vinyl chloride) Styrene-methyl methacrylate graft copolymer Poly (phenyl siloxane) (ladder) Polybutadiene            18 20 22 24 26 28 30 POLYMER CHEMISTRY Elution volume ()5 ml counts, THF solvent)

FIGURE 2.12. Typical semilogarithmic calibration plot of molecular weight versus retention volume. Retention volume (Vr) (counts) 106 105 104 103 Molecular weight (M) POLYMER CHEMISTRY

)log( ) + ( )logM1 B. Universal calibration method          to be combined Mark-Houwink-Sakurada equation [η]1M1 = [η]2M2 logM2 = ( 1 + a2 1 )log( K2 K1 ) + ( 1 + a1 )logM1 POLYMER CHEMISTRY

2.6.2 Fractional Solution   Soxhlet-type extraction by using mixed solvent. Reverse GPC : from low molecular weight fraction                  to high molecular weight fraction Inert beads are coated by polymer sample. POLYMER CHEMISTRY

2.6.3 Fractional Precipitation Dilute polymer solution is precipitated by variable non-solvent mixture. Precipitate is decanted or filtered Reverse fractional solution : from high molecular weight fraction to    low molecular fraction POLYMER CHEMISTRY

2.6.4. Thin-layer Chromatography (TLC)   Alumina- or silica gel coated plate. Low cost and simplicity. Preliminary screening of polymer samples or monitoring polymerization processes. POLYMER CHEMISTRY