18 2 Polymer Size and Polymer Solutions separated from a solution by a semi-permeable membrane that allows passage of solvent but not solute molecules,solvent will flow through the membrane into the solution.As the liquid level rises in the solution compartment,the hydrostatic pressure increases until it prevents further passage of solvent or,more exactly,until solvent flow is equal in both directions.The pressure at equilibrium is the osmotic pressure.A schematic representation of an osmometer is given in Fig.2.4 Osmotic pressure is related to molecular weight by the van't Hoff equation extrapolated to zero concentration: 月.+Ac RT (2.24) whereπ，the osmotic pressure,is given by π=Pg△h (2.25) where R is the gas constant,0.082 L atm mol-K-(CGS)or 8.314 J mol-K-1 (SD);T is the temperature in kelvins;C is the concentration in grams per liter;p is the solvent density in grams per cubic centimeter,g is the acceleration due to gravity,9.81 m/s2;Ah is the difference in heights of solvent and solution in centimeters;and A2 is the second virial coefficient (a measure of the interaction between solvent and polymer).A plot of reduced osmotic pressure,n/C,versus concentration(Fig.2.5)is linear with the intercept equal to RT/M and the slope equal to A2,units for /C are dyn Lgcm(CGS)or Jkg(SD).Because A2 is a measure of solvent-polymer interaction,the slope is zero at the theta temperature. Thus osmotic pressure measurements may be used to determine theta conditions. Matrix assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF MS)is developed recently to determine the absolute molecular weight of large molecule.The polymer sample is imbedded in a low molecular weight organic compound that absorbs strongly at the wavelength of a UV laser. Upon UV radiation,organic compound absorbs energy,then energy transfer to polymer to form ions.Finally,the ions are detected.At higher molecular weight, the signal to noise ratio is reduced.From the integrated peak areas,reflecting the number of ions (N)and the average molecular weight(Mi),both M and Mw can be calculated.Figure 2.6 shows that a low molecular weight(M3,000)poly(3- Fig.2.5 Plot of reduced osmotic pressure versus n/C concentration [2] RI Slope=A2
separated from a solution by a semi-permeable membrane that allows passage of solvent but not solute molecules, solvent will flow through the membrane into the solution. As the liquid level rises in the solution compartment, the hydrostatic pressure increases until it prevents further passage of solvent or, more exactly, until solvent flow is equal in both directions. The pressure at equilibrium is the osmotic pressure. A schematic representation of an osmometer is given in Fig. 2.4. Osmotic pressure is related to molecular weight by the van’t Hoff equation extrapolated to zero concentration: p C c¼0 ¼ RT M n þ A2C ð2:24Þ where p, the osmotic pressure, is given by p ¼ qgDh ð2:25Þ where R is the gas constant, 0.082 L atm mol-1 K-1 (CGS) or 8.314 J mol-1 K-1 (SI); T is the temperature in kelvins; C is the concentration in grams per liter; q is the solvent density in grams per cubic centimeter, g is the acceleration due to gravity, 9.81 m/s2 ; Dh is the difference in heights of solvent and solution in centimeters; and A2 is the second virial coefficient (a measure of the interaction between solvent and polymer). A plot of reduced osmotic pressure, p=C, versus concentration (Fig. 2.5) is linear with the intercept equal to RT=M n and the slope equal to A2, units for p=C are dyn Lg-1 cm-1 (CGS) or Jkg-1 (SI). Because A2 is a measure of solvent–polymer interaction, the slope is zero at the theta temperature. Thus osmotic pressure measurements may be used to determine theta conditions. Matrix assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF MS) is developed recently to determine the absolute molecular weight of large molecule. The polymer sample is imbedded in a low molecular weight organic compound that absorbs strongly at the wavelength of a UV laser. Upon UV radiation, organic compound absorbs energy, then energy transfer to polymer to form ions. Finally, the ions are detected. At higher molecular weight, the signal to noise ratio is reduced. From the integrated peak areas, reflecting the number of ions Nj and the average molecular weight ð Þ Mi , both M n and M w can be calculated. Figure 2.6 shows that a low molecular weight Mð Þ w 3;000 poly(3- Fig. 2.5 Plot of reduced osmotic pressure versus concentration [2] 18 2 Polymer Size and Polymer Solutions
2.3 Measurement of Molecular Weight 19 (a) 4双11 (b) 4E014 21 8000 8000 6000 6000 [n'e] 4000 4000 2000 2000 0 15002000250030003500400045005000 00260028003000320034003600 m/z m/z Fig.2.6 MALDI mass spectrum of low molecular weight poly(3-hexyl thiophene)(a)whole spectrum,(b)magnified area between m/z 2,400 and 3,800 hexyl thiophene)was measured by MALDI-TOF MS.The spectrum shows the molecular weight distribution and the difference between every peak is equal to the repeating unit 3-hexyl thiophene of 167. The absolute weight average molecular weight(M)can also be measured by light scattering method.The light passes through the solution,loses energy by absorption,conversion to heat,and scattering.The intensity of scattered light depends on concentration,size,polarizability of the scattering molecules.To evaluate the turbidity arising from scattering,one combines equations derived from scattering and index of refraction measurements.Turbidity,t,is related to concentration,c,by the expression t=HcMw (2.26) where His H=32n(dn/de) 34N0 (2.27) and no is the refractive index of the solvent,A is the wavelength of the incident light,and No is Avogadro's number.The expression dn/dc,referred to as the specific refractive increment,is obtained by measuring the slope of the refractive index as a function of concentration,and it is constant for a given polymer, solvent,and temperature.As molecular size approaches the magnitude of light wavelength,corrections must be made for interference between scattered light coming from different parts of the molecules.To determine molecular weight,the expression for turbidity is rewritten as Hc 1 tM.P(0+2A2C (2.28)
hexyl thiophene) was measured by MALDI-TOF MS. The spectrum shows the molecular weight distribution and the difference between every peak is equal to the repeating unit 3-hexyl thiophene of 167. The absolute weight average molecular weight Mð Þ w can also be measured by light scattering method. The light passes through the solution, loses energy by absorption, conversion to heat, and scattering. The intensity of scattered light depends on concentration, size, polarizability of the scattering molecules. To evaluate the turbidity arising from scattering, one combines equations derived from scattering and index of refraction measurements. Turbidity, s, is related to concentration, c, by the expression s ¼ HcM w ð2:26Þ where H is H ¼ 32p3 3 n2 0ð Þ dn=dc 2 k4 N0 ð2:27Þ and n0 is the refractive index of the solvent, k is the wavelength of the incident light, and N0 is Avogadro’s number. The expression dn=dc, referred to as the specific refractive increment, is obtained by measuring the slope of the refractive index as a function of concentration, and it is constant for a given polymer, solvent, and temperature. As molecular size approaches the magnitude of light wavelength, corrections must be made for interference between scattered light coming from different parts of the molecules. To determine molecular weight, the expression for turbidity is rewritten as Hc s ¼ 1 M wPð Þh þ 2A2C ð2:28Þ 2448.771 2281.781 2616.048 2114.975 2949.110 1948.380 3450.147 3616.721 3782.791 3949.088 4117.025 6000 8000 Intens. [a.u.] 2616.048 2949.110 2783.182 3116.5263283.505 3450.147 3616.721 2480.130 6000 8000 Intens. [a.u.] 1781.261 1613.847 1445.919 1367.571 4614.864 4781.324 4947.428 5114.582 2000 4000 2703.324 2869.867 3037.597 2537.347 3205.285 3373.805 3538.679 3704.432 2000 4000 0 1500 2000 2500 3000 3500 4000 4500 5000 m/z 0 00 2600 2800 3000 3200 3400 3600 m/z (a) (b) Fig. 2.6 MALDI mass spectrum of low molecular weight poly(3-hexyl thiophene) (a) whole spectrum, (b) magnified area between m/z 2,400 and 3,800 2.3 Measurement of Molecular Weight 19
20 2 Polymer Size and Polymer Solutions Fig.2.7 Zimm plot of light- scattering data of polymer [2] 8=0 Hc c=0 Experimental M. .Extrapolated sin2 0/2+kc where P(0)is a function of the angle,0,at which t is measured,a function that depends on the shape of the molecules in solution.A2 is the second virial coeffi- cient.Turbidity is then measured at different concentrations as well as at different angles,the latter to compensate for variations in molecular shape.The experi- mental data are then extrapolated to both zero concentration and zero angle,where P(0)is equal to 1.Such double extrapolations,shown in Fig.2.7,are called Zimm plots.The factor k on the abscissa is an arbitrary constant.The intercept corre- sponds to 1/Mw. A major problem in light scattering is to obtain perfectly clear,dust-free solutions.This is usually accomplished by ultra centrifugation or careful filtration. Despite such difficulties,the light-scattering method is widely used for obtaining weight average molecular weights between 10,000 and 10,000,000.A schematic of a laser light-scattering photometer is given in Fig.2.8. Intrinsic viscosity is the most useful of the various viscosity designations because it can be related to molecular weight by the Mark-Houwink-Sakurada equation: n]=KM (2.29) where My,is the viscosity average molecular weight,defined as ΣNM+a 1/a M,= ΣNM: (2.30) Log K and a are the intercept and slope,respectively,of a plot of log n versus log Mw or log Mn of a series of fractionated polymer samples.Such plots are linear (except at low molecular weights)for linear polymers,thus log[n]log K+alogM (2.31)
where Pð Þh is a function of the angle, h, at which s is measured, a function that depends on the shape of the molecules in solution. A2 is the second virial coeffi- cient. Turbidity is then measured at different concentrations as well as at different angles, the latter to compensate for variations in molecular shape. The experimental data are then extrapolated to both zero concentration and zero angle, where Pð Þh is equal to 1. Such double extrapolations, shown in Fig. 2.7, are called Zimm plots. The factor k on the abscissa is an arbitrary constant. The intercept corresponds to 1=M w. A major problem in light scattering is to obtain perfectly clear, dust-free solutions. This is usually accomplished by ultra centrifugation or careful filtration. Despite such difficulties, the light-scattering method is widely used for obtaining weight average molecular weights between 10,000 and 10,000,000. A schematic of a laser light-scattering photometer is given in Fig. 2.8. Intrinsic viscosity is the most useful of the various viscosity designations because it can be related to molecular weight by the Mark-Houwink-Sakurada equation: ½ ¼ g KM a v ð2:29Þ where M v, is the viscosity average molecular weight, defined as M v ¼ RNiM1þa i R NiMi 1=a ð2:30Þ Log K and a are the intercept and slope, respectively, of a plot of log ½ g versus log M w or log M n of a series of fractionated polymer samples. Such plots are linear (except at low molecular weights) for linear polymers, thus log½ ¼ g log K þ alogM ð2:31Þ Fig. 2.7 Zimm plot of lightscattering data of polymer [2] 20 2 Polymer Size and Polymer Solutions
2.3 Measurement of Molecular Weight 21 Fig.2.8 Schematic of a laser Partially transmitting mirror light scattering photometer [2] Polarizer Fully reflecting mirror Laser Light- scattering Thermostat cell for sample Analyzer Photomultiplier Photomultiplier Amplifier Imbalance Correlator amplifier Data acquisition system Fig.2.9 A modified Ubbelohde viscometer with improved dilution characteristics Factors that may complicate the application of the Mark-Houwink-Sakurada relationship are chain branching,too broad of molecular weight distribution in the samples used to determine K and a,solvation of polymer molecules,and the presence of alternating or block sequences in the polymer backbone.Chain entanglement is not usually a problem at high dilution except for extremely high molecular weights polymer.Ubbelohde type viscometer is more convenient to use
Factors that may complicate the application of the Mark-Houwink-Sakurada relationship are chain branching, too broad of molecular weight distribution in the samples used to determine K and a, solvation of polymer molecules, and the presence of alternating or block sequences in the polymer backbone. Chain entanglement is not usually a problem at high dilution except for extremely high molecular weights polymer. Ubbelohde type viscometer is more convenient to use Laser Correlator l0 Imbalance amplifier Data acquisition system Photomultiplier Polarizer Partially transmitting mirror Fully reflecting mirror Lightscattering cell for sample Amplifier Thermostat Analyzer Photomultiplier Fig. 2.8 Schematic of a laser light scattering photometer [2] Fig. 2.9 A modified Ubbelohde viscometer with improved dilution characteristics 2.3 Measurement of Molecular Weight 21
22 2 Polymer Size and Polymer Solutions Cone Polymer 'N R Plate Fig.2.10 Schematic of cone-plate rotational viscometer [2] for the measurement of polymer viscosity,because it is not necessary to have exact volumes of solution to obtain reproducible results.Furthermore,additional solvent can be added (assuming the reservoir is large enough);thus concentration can be reduced without having to empty and refill the viscometer.A schematic of the Ubbelohde type viscometer is given in Fig.2.9. The viscosity of polymer can also be measured by the cone-plate rotational viscometer as shown in Fig.2.10.The molten polymer or polymer solution is contained between the bottom plate and the cone,which is rotated at a constant velocity (2).Shear stress ()is defined as 3M t=27R3 (2.32) where M is the torque in dynes per centimeter(CGS)or in newtons per meter(SD), and R is the cone radius in centimeters.Shear rate (r)is given by i- (2.33) where 2 is the angular velocity in degrees per second (CGS)or in radians per second (SI)and a is the cone angle in degrees or radians.Viscosity is then t 3aM kM 1=；=2πR3Q (2.34) where k is 31 k= 2πR3 (2.35) Gel permeation chromatography (GPC)involves the permeation of a polymer solution through a column packed with microporous beads of cross-linked poly- styrene.The column is packed from beads of different sized pore diameters,as shown in Fig.2.11.The large size molecules go through the column faster than the small size molecule.Therefore,the largest molecules will be detected first.The
for the measurement of polymer viscosity, because it is not necessary to have exact volumes of solution to obtain reproducible results. Furthermore, additional solvent can be added (assuming the reservoir is large enough); thus concentration can be reduced without having to empty and refill the viscometer. A schematic of the Ubbelohde type viscometer is given in Fig. 2.9. The viscosity of polymer can also be measured by the cone-plate rotational viscometer as shown in Fig. 2.10. The molten polymer or polymer solution is contained between the bottom plate and the cone, which is rotated at a constant velocity ð Þ X . Shear stress ð Þs is defined as s ¼ 3M 2pR3 ð2:32Þ where M is the torque in dynes per centimeter (CGS) or in newtons per meter (SI), and R is the cone radius in centimeters. Shear rate ð Þr_ is given by r_ ¼ X a ð2:33Þ where X is the angular velocity in degrees per second (CGS) or in radians per second (SI) and a is the cone angle in degrees or radians. Viscosity is then g ¼ s r_ ¼ 3aM 2pR3X ¼ kM X ð2:34Þ where k is k ¼ 3a 2pR3 ð2:35Þ Gel permeation chromatography (GPC) involves the permeation of a polymer solution through a column packed with microporous beads of cross-linked polystyrene. The column is packed from beads of different sized pore diameters, as shown in Fig. 2.11. The large size molecules go through the column faster than the small size molecule. Therefore, the largest molecules will be detected first. The Fig. 2.10 Schematic of cone-plate rotational viscometer [2] 22 2 Polymer Size and Polymer Solutions