Caution: Contents of the lecture notes are copyrighted ( 有版權 Textbook of Physical Polymer Science). (Do not use outside the class room purposes without permission from the publisher.) Physical Polymer Science Lecture notes by Prof. E. M. Woo adapting from: Textbook of Physical Polymer Science (Ed: L. H. Sperling, 3rd Ed) Chap 3: Molecular Weights Of Polymers CONTENTS: Effects of molecular weight on properties - MW distribution - Determination/measurements of MW - MW and intrinsic viscosity - GPC 1
3.7 MOLECULAR WEIGHTS OF POLYMERS 3.7.1 Molecular Weight of Commercial Polymers The molecular weight of polymers used in commerce varies from about 30,000 to over 1,000,000 g/mol. Sometimes conflicting requirements include the use of high enough molecular weights to obtain good physical properties, and low enough molecular weights to permit reasonable processing conditions, such as melt viscosity. Poly(vinyl chloride) - Commercial poly(vinyl chloride) vinyl polymers range from 60,000 to about 90,000 g/mol. The restrictions above hold in this case. Poly(methyl methacrylate) - Those polymers that are used in such products as Plexiglas have high molecular weights with broad distributions. PMMA molding from low-mw oligomers A viscous syrup containing low-molecular-weight polymer and monomer is poured into a mold and allowed to polymerize. Late in the polymerization, the phenomenon known as auto-acceleration takes place, where the molecular weight increases dramatically owing to a suppression of the termination step. This high molecular weight (PMMA) produced at the end may be over 1 * 10 6 g/mol, contributing strength and toughness to the final sheet. 2
Molecular Weight of Polyethylene Ultra-high-molecular-weight polyethylene (UHMWPE) is a subset of the polyethylene. Also known as high-modulus polyethylene, (HMPE), or highperformance polyethylene (HPPE). It has extremely long chains, with a molecular mass usually between 3.5 and 7.5 million [1] g/mol (or amu, Da). The longer chain serves to transfer load more effectively to the polymer backbone by strengthening intermolecular interactions. This results in a very tough material, with the highest impact strength of any thermoplastic presently made. [2] Note: Molecular weight unit sometime is expressed as; amu or Dalton (Da). Amu: atomic mass unit (symbol: u or amu) or dalton (symbol: Da) is the standard unit that is used for indicating mass on an atomic or molecular scale (atomic mass) Da (Dalton) is numerically equivalent to 1 g/mol. UHMWPE is produce by synthesis process based on metallocene catalysts, resulting in UHMWPE molecules typically having 100,000 to 250,000 monomer units per 3 molecule each, compared to HDPE's 700 to 1,800 monomers.
Mw of cellulose Cellulose: This natural polymer occurs with extremely high molecular weights, sometimes in the several millions range, and with molecular weight distributions of M w /M n in the range of 10 to 50 [quite wide distribution]. For commercial applications such as rayon (which is re-processed cellulose), the polymer is deliberately degraded down to the 50,000 to 80,000 g/mol range to increase processibility. The better products often utilize the higher end of this range. 4
Many properties of cellulose depend on its chain length or degree of polymerization, the number of glucose units that make up one polymer molecule. Cellulose from wood pulp has typical chain lengths between 300 and 1700 units; cotton and other plant fibers as well as bacterial cellulose have chain lengths ranging from 800 to 10,000 units [MW up to millions]. [10] Molecules with very small chain length resulting from the breakdown of cellulose are known as cellodextrins; in contrast to long-chain cellulose, cellodextrins are typically soluble in water and organic solvents. 5
3.4 MOLECULAR WEIGHT AVERAGES There are four molecular weight averages in common use; the number-average molecular weight, Mn; the weight-average molecular weight, Mw; the z-average molecular weight, Mz; and the viscosity-average molecular weight, Mv. M v =[Σ i N i M i 1+a /Σ i NiMi] 1/a a=0.5=0.8 for most polymers These are defined below in terms of the numbers of molecules Ni having molecular weights Mi, or in terms of wi, the weight of species with molecular weights Mi. See textbook p. 76. 6
3.7.2 Thermodynamics and Kinetics of Polymerization effects on Mw and PDI of polymers The synthesis of polymers, with attendant aspects of the thermodynamics and kinetics of polymerization that occupy entire textbooks (62,63), is to a very significant extent beyond the coverage of this text. Indeed, organic polymer science is often taught as a mate course to physical polymer science. However, since the thermodynamics and kinetics of polymerization affect both the molecular weights and the polydispersity index obtained, the most salient features of these areas will be explored. Polymer synthesis itself was briefly discussed in Section 1.4. 7
3.7.2.1 Thermodynamics of Chain Polymerization Under standard conditions, the Gibbs free energy, G 0, is related to the equilibrium constant of the polymerization, K, by (3.64) Consider a chain polymerization of monomer M: (3.65) where the rate constant of the forward reaction, polymerization, is k p and the rate constant of the reverse reaction, depolymerization, is given by k dp. Then (3.66) 8
Cont d When the forward and reverse reactions have equal rates, namely when polymerization-depolymerization propagation rates are equal, the concept of ceiling and floor temperatures arises. Most polymerizations have ceiling temperatures, temperatures above which the monomer cannot be polymerized, but the polymer will spontaneously depolymerize back to the monomer. Commercially this fact leads to an important method of polymer recycling whereby scrapped polymer is heated under anaerobic (no air) conditions to allow distilling off the resultant monomers. 9
3.7.2.2 Kinetics of Chain Polymerization p. 92 The kinetic chain length, v, of a radical chain polymerization is the average number of monomer molecules consumed for each radical initiating a chain. Thus, at steady state, where Ri, Rp, and Rt represent the rates of initiation, propagation, and termination, respectively. - The quantity f is the initiator efficiency factor, the fraction of initiator molecules that actually initiate a polymerization on decomposition. Frequently f is about 0.8. - The number-average degree of polymerization, DPn in reaction (3.72), is equal to 2v for termination by combination. - Termination by combination yields a polydispersity index of PDI=1.5, 10 while termination by chain transfer yields a PDI of 2.0, ideally.
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3.7.2.3 Thermodynamics of Step Polymerization (3.74) (3.75) For step-growth polymerizations, the fractional conversion p is given by [COO] = p[m] 0, where [M] 0 is the concentration of ester groups. Then DP n is given by: (3.76) and the corresponding weight-average degree of polymerization is given by This leads directly to the polydispersity index, DP w /DP n, PDI = 1 + p (3.78) (3.77) PDI thus depends on the value of p (fractional conversion). 12
3.7.3 Molecular Weight Distributions -If the termination reaction in chain polymerization is by disproportionation, then the polydispersity index, Mw/Mn, is 2.0. -Termination by combination yields a polydispersity index of 1.5. -Stepwise polymerizations, such as polyester formation, yield a value of PDI=2 (because p=1.0). -Anionic polymerizations yield surprisingly narrow distribution, with values sometimes less than 1.05. Of course, polymerization need not be ideal in its kinetics. Branching may occur, which broadens the molecular weight distribution. There may even be two or more peaks in the molecular weight distribution. A powerful method for directly observing the shape of the distribution curve is gel permeation chromatography (see Section 3.9). In general, the polydispersity index can be determined from an analysis of the kinetics of the reaction; in practice, various phenomena cause 13 the products to be much broader in molecular weight distributions (67).
Note: Proteins are almost the only source of truly monodisperse polymers (PDI=1.0), with perfect stereoregularity. Nature makes all these molecules exactly alike [All molecules have same Mn no MW distribution]. NOTE: -By contrast, other natural polymers like cellulose or amylose have very broad distributions, as mentioned previously. 14
3.8 INTRINSIC VISCOSITY Intrinsic viscosity measurements are carried out in dilute solution and result in the viscosity-average molecular weight; see Figure 3.4 and equation (3.34). Consider such a dilute solution flowing down a capillary tube (Figure 3.12). 3.8.1 Definition of Terms Several terms need defining. The solvent viscosity is η 0, usually expressed in poises, Stokes, or, more recently, Pascal. seconds, Pa s. [Note: 1 P = 1 g s -1 cm -1 Fo r UI, Pa s = kg m -1 s -1,therefore: 1 Pa s = 10 P = 1000 cp. The viscosity of the polymer solution is η. The relative viscosity is the ratio of the two, 15
The specific viscosity is the relative viscosity minus one: Usually η sp is a quantity between 0.2 and 0.6 for the best results. The specific viscosity, divided by the concentration and extrapolated to zero concentration, yields the intrinsic viscosity: For dilute solutions, where the relative viscosity is just over unity, the following algebraic expansion is useful: Then, dividing ln η rel by c and extrapolating to zero concentration also yields the intrinsic viscosity: Note that the natural logarithm of η rel is divided by c in equation (3.88), not itself. The term (ln η rel )/c is called the inherent viscosity. η rel 16
3.8.2 The Equivalent Sphere Model In assuming a dilute dispersion of uniform, rigid, noninteracting spheres, Einstein (69,70) derived an equation expressing the increase in viscosity of the dispersion,.., and derivation by Flory, etc. [Procedures skipped. See text book] 17
3.8.3 The Mark Houwink Sakurada Relationship In the late 1930s and 1940s, Mark, Houwink, and Sakurada arrived at an empirical relationship between the molecular weight and the intrinsic viscosity (71): where K and a are constants for a particular polymer solvent pair at a particular temperature. Equation (3.97) is known today as the Mark Houwink Sakurada equation. This equation is in wide use today, being one of the most important relationships in polymer science and probably the single most important equation in the field. More generally, it should be pointed out that a varies from 0 to 2.0, as shown in Table 3.11 18
The quantity K is often given in terms of the universal constant Φ, where r 02 represents the mean square end-to-end distance of the unperturbed coil. If the number-average molecular weights are used, then Φ equals 2.5x10 21 dl/mol cm 3. A theoretical value of 3.6x10 21 dl/mol cm 3 can be calculated from a study of the chain frictional coefficients (17). For many theoretical purposes, it is convenient to express the Mark Houwink Sakurada equation in the form: Same as the theoretical derivation according to Flory (17), Eq. 3-96. 19
3.8.4 Intrinsic Viscosity Experiments - p. 101 In most experiments, dilute solutions of about 1% polymer are made up. The quantity η rel should be about 1.6 for the highest concentration used. The most frequently used instrument is the Ubbelhode viscometer, which equalizes the pressure above and below the capillary. Several concentrations are run and plotted according to Figure 3.14. Two practical points must be noted: Figure 3.15 Figure 3.14 Double logarithnmic plots of [h] versus MW for anionically synthesized polystyrenes, which were then fractionated leading to values of MW /Mn of less than 1.06. Filled circles in benzene, half-filled circles in toluene, 20 and open circles in dichloroethylene, all at 30 C (75). The arrows indicate the axes to be used. Units for [h] in 100 ml/g.
GEL PERMEATION CHROMATOGRAPHY (GPC) for determination of Mw s of polymers 21
3.9 GEL PERMEATION CHROMATOGRAPHY (GPC) - p. 103 Note: MW measurements, etc.. [] For details, see Instrumental Analysis Noting that GPC is a relative molecular weight method, such instrumentation needs to be calibrated. Narrow molecular weight distribution, anionically synthesized polystyrenes are used most often for the purpose. Other polymers used for calibration include poly(methyl methacrylate), polyisoprene, polybutadiene, poly(ethylene oxide), and the sodium salt of poly(methacrylic acid). Molecular weight ranges available start at low oligomers of only a few hundred g/mol, up to 20,000,000 g/mol. In all cases, use of narrow molecular weight distribution standards is preferred. Table 3.14 22
3.9 Gel permeation chromatography (GPC) p.103 Or Size Exclusion Chromatography 資料來源 : http://ethesys.lib.ccu.edu.tw/etd-db/etd- search/getfile?urn=etd-1102105-145413&filename=etd- 23 1102105-145413.pdf
Eluent [from Wiki] The eluent (mobile phase) should be a good solvent for the polymer, should permit high detector response from the polymer and should wet the packing surface. The most common eluents in for polymers that dissolve at room temperature GPC are tetrahydrofuran (THF), o-dichlorobenzene and trichlorobenzene at 130 150 C for crystalline polyalkynes and m-cresol and o-chlorophenol at 90 C for crystalline condensation polymers such as polyamides and polyesters. Gel - [packing materials] Gels are used as stationary phase for GPC. The pore size of a gel must be carefully controlled in order to be able to apply the gel to a given separation. Other desirable properties of the gel forming agent are the absence of ionizing groups and, in a given solvent, low affinity for the substances to be separated. Commercial gels like PLgel, Sephadex, Bio-Gel (cross-linked polyacrylamide), agarose gel and Styragel are often used based on different separation requirements. [5] Note: Styragel (commercial name) = 聚苯乙烯型交联共聚物 24
3.9 Gel permeation chromatography (GPC) The type of column packing depends on whether the polymer is water-soluble or organic soluble. For water soluble polymers, column packings consist of a range of materials, including silicas, hydroxyethyl methacrylate copolymers, chitosan, and highly cross-linked poly(vinyl alcohol). Organic soluble polymer-based columns most often contain porous, densely cross-linked polystyrene [Styragel], but porous silicas and highly cross-linked poly(vinyl alcohol) are also used. The size of the pores determines the size of the molecule that can diffuse in and out by Brownian motion. The larger molecules are restricted to entering only the larger pores. Since the motion in and out of the pores is random, the residence time in the pores of the short chains is longer. Hence the larger, high molecular weight polymer chains elute first from the column. 25
Packing Material -Silica based SEC packing materials generally exhibit higher resolving power than polymer based materials. -However, polymer based materials show greater stability for use with high ph eluents. Polymeric packing materials are generally available in larger particle sizes, which may be more practical for large scale preparative separations. Glycerol Bonded Silica Silica Poly(hydroxymethacrylate) Divinylbenzene Styrene Poly(styrene-co-Divinylbenzene) Poly(vinyl alcohol) - crosslinked Note: Styrene and divinylbenzene react to form the copolymer styrenedivinylbenzene, S-DVB or Sty-DVB. [crosslinked into gels ] The resulting cross-linked polymer is mainly used for the production of ion exchange resins or column packings. [3] 26
detectors There are several types of detectors; see Figure 3.17 (79). These are classified as either (I) concentration-sensitive detectors, or (II) molar mass (molecular weight) sensitive detectors. The refractive index (RI) detector is most popular concentration-sensitive detector, measuring the change in refractive index as the concentration of polymer in the solution changes. When the polymer chains arrive at the detector, then the refractive index of the solution changes, providing a measure of the polymer concentration. While most polymers have a different refractive index than the solvent (usually higher), if the refractive indices of both the polymer and solvent are substantially the same, the method cannot be used. Another detector group of methods involves the input of ultraviolet light (UV), with the output being fluorescence or absorption by the polymer. Polymers, like polystyrene, absorb strongly in UV light very powerful detector. Molecular-weight sensitive detectors: viscometry or light scattering less often 27 used.
The most sensitive detector is the differential UV photometer and the most common detector is the differential refractometer (DRI). When characterizing copolymer, it is necessary to have two detectors in series. [4] For accurate determinations of copolymer composition at least two of those detectors should be concentration detectors. [6] The determination of most copolymer compositions is done using UV and RI detectors, although other combinations can be used. Standard for GPC: Polystyrene standards with PDI of less than 1.2 are typically used to calibrate the GPC. [4] Unfortunately, polystyrene tends to be a very linear polymer and therefore as a standard it is only useful to compare it to other polymers that are known to be linear and of relatively the same size. 28
3.9.5 Calibration Noting GPC is a relative molecular weight method, such instrumentation needs to be calibrated. Narrow molecular weight distribution, anionically synthesized polystyrenes are used most often for the purpose. Other polymers used for calibration include poly(methyl methacrylate), polyisoprene, polybutadiene, poly(ethylene oxide), and the sodium salt of poly(methacrylic acid). Molecular weight ranges available start at low oligomers of only a few hundred g/mol, up to 20,000,000 g/mol. In all cases, use of narrow molecular weight distribution standards is preferred. GPC Separation of Anionically Synthesized Polystyrene; M n =3,000 g/mol, PDI=1.32 GPC Separation of Free-Radical Synthesized Polystyrene; M n =24,000 g/mol, PDI=4.96. 29
By determining the retention volumes (or times) of monodisperse polymer standards (e.g. solutions of monodispersed polystyrene in THF), a calibration curve can be obtained by plotting the logarithm of the molecular weight versus the retention time or volume. Once the calibration curve is obtained, the gel permeation chromatogram of any other polymer can be obtained in the same solvent and the molecular weights (usually M n and M w ) and the complete molecular weight distribution for the polymer can be determined. A typical calibration curve is shown to the right and the molecular weight from an unknown sample can be obtained from the calibration curve. [From Wikipedia] End of Chap. 3 - Molecular Weights of Polymers. 30