Chapter 2
`Polymer Size and Polymer Solutions
`
`The size of single polymer chain is dependent on its molecular weight and mor-
`phology. The morphology of a single polymer chain is determined by its chemical
`structure and its environment. The polymer chain can be fully extended in a very
`dilute solution when a good solvent is used to dissolve the polymer. However, the
`single polymer chain is usually in coil form in solution due to the balanced
`interactions with solvent and polymer itself. We will discuss the size of polymer
`first, and then go to the coil formation in the polymer solution.
`
`2.1 The Molecular Weight of Polymer
`
`The molecular weight of polymer determines the mechanical properties of poly-
`mers. To have strong durable mechanical properties, the polymer has to have
`molecular weight much larger than 10,000 for structural applications. However,
`for thin film or other special application, low molecular weight polymer or oli-
`gomer sometime is adequate. As shown in Fig. 2.1, above (A), strength increases
`rapidly with molecular weight until a critical point (B) is reached. Mechanical
`strength increases more slowly above (B) and eventually reaches a limiting value
`(C). High molecular weight polymer has high viscosity and poor processability.
`The control of molecular weight and molecular weight distribution (MWD) is
`often used to obtain and improve certain desired physical properties in a polymer
`product.
`Polymers, in their purest form, are mixtures of molecules of different molecular
`weights. The reason for the polydispersity of polymers lies in the statistical variations
`present in the polymerization processes. The above statement is true for common
`polymerization reaction such as free radical chain polymerization, step polymeri-
`zation, etc. However, cationic or anionic chain polymerization as so called living
`polymerization has small MWD. Low dispersity can also be obtained from emulsion
`polymerization, and new polymerization techniques such as living free radical
`polymerization including nitroxide-mediated polymerization (NMP), atom transfer
`radical polymerization (ATRP), and reversible addition–fragmentation chain
`
`W.-F. Su, Principles of Polymer Design and Synthesis,
`Lecture Notes in Chemistry 82, DOI: 10.1007/978-3-642-38730-2_2,
`Ó Springer-Verlag Berlin Heidelberg 2013
`
`9
`
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`

`10
`
`2 Polymer Size and Polymer Solutions
`
`Fig. 2.1 Dependence of
`mechanical strength on
`polymer molecular weight [1]
`
`transfer polymerization (RAFT). The chemistry of different polymerization reac-
`tions will be discussed in detail in the subsequent chapters.
`ð
`Þ is total weight (W) of all the molecules
`Number-average molecular weight Mn
`in a polymer sample divided by the total number of molecule present, as shown in
`Eq. 2.1, where Nx is the number of molecules of size Mx, Nx is number (mole)
`fraction of size Mx
`
`ð2:1Þ
`Mn ¼ W=RNx ¼ R NxMx=RNx ¼ R NxMx
`Analytical methods used to determine Mn include (1) Mn \ 25,000 by vapor
`pressure osmometry, (2) Mn 50,000–2 million by membrane osmometry, and (3)
`Mn \ 50,000 by end group analysis, such as NMR for –C=C; titration for car-
`boxylic acid ending group of polyester. They measure the colligative properties of
`polymer solutions. The colligative properties are the same for small and large
`molecules when comparing solutions at the same mole fraction concentration.
`Therefore, the Mn is biased toward smaller sized molecules. The detailed mea-
`surement methods of molecular weight will be discussed in Sect. 2.3. Weight-
`average molecular weight is defined as Eq. 2.2, where Wx is the weight fraction of
`Mx molecules, Cx is the weight concentration of Mx molecules, and C is the total
`weight concentration of all of the polymer molecules, and defined by Eqs. 2.3–2.5.
`Mw ¼ R WxMx ¼ R CxMx=RCx
`ð2:2Þ
`Wx ¼ Cx=C
`ð2:3Þ
`Cx ¼ NxMx
`ð2:4Þ
`C ¼ R Cx ¼ R NxMx
`ð2:5Þ
`Light scattering is an analytical method to determine the Mw in the range of
`10,000–10,000,000. It unlike colligative properties shows a greater number for
`larger sized molecules than for small-sized molecules. Viscosity-average molec-
`ð
`Þ is defined as Eq. 2.6, where a is a constant. The viscosity and
`ular weight Mv
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`

`2.1 The Molecular Weight of Polymer
`
`11
`
` 1=a
`
`weight average molecular weights are equal when a is unity. Mv is like Mw, it is
`
`
`
`greater for the larger sized polymer molecules than for smaller ones.
`ð2:6Þ
`Mv ¼ R Ma
`1=a¼ R NxMaþ1
`=RNxMx
`x Wx
`x
`A measure of the polydispersity in a polymer is defined as Mw divided over Mn
`ð
`Þ. For a polydispersed polymer, Mw [ Mv [ Mn with the differences
`Mw= Mn
`between the various average molecular weights increasing as the molecular-weight
`distribution (MWD) broadens, as shown in Fig. 2.2.
`For example, consider a hypothetical mixture containing 95 % by weight of
`molecules of molecular weight 10,000, and 5 % of molecules of molecular weight
`100. The Mn and Mw are calculated from Eqs. 2.1 and 2.2 as 1,680 and 9,505,
`respectively. The use of the Mn value of 1,680 gives an inaccurate indication of the
`properties of this polymer. The properties of the polymer are determined primarily
`by the molecules with a molecular weight of 10,000 that makes up 95 % of the
`weight of the mixture. The highest % fraction of molecular weight of molecule
`will contribute the most toward the bulk property. It is desirable to know the
`molecular weight distribution, then to predict the polymer properties. At present,
`the gel permeation chromatography (GPC) technique has been advanced to be able
`to easily measure Mn; Mv; Mw; simultaneously and calculate PDI using only one
`sample. All the measurements of molecular weight of polymers are carried out
`using polymer solutions. Therefore, the accuracy of molecular weight measure-
`ment is dependent on the behavior of polymer solution. Usually, a calibration
`curve is established first using a specific polymer dissolving in a specific solvent.
`Polystyrene standard dissolved in tetrahydrofuran (THF) is the most popular cal-
`ibration curve used in GPC. If the measured polymer exhibits different behavior in
`THF from that of polystyrene, then a deviation from the actual molecular weight is
`occurred. For example, a conducting polymer, poly (phenylene vinylene), con-
`taining rigid rod molecular structure shows a higher molecular weight when the
`standard of coil structured polystyrene is used. More detailed discussion of GPC is
`in Sect. 2.3.
`
`Fig. 2.2 Distribution of
`molecular weights in a typical
`polymer sample [1]
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`12
`
`2 Polymer Size and Polymer Solutions
`
`2.2 Polymer Solutions
`
`Polymer solutions occur in two stages. Initially, the solvent molecules diffuse
`through the polymer matrix to form a swollen, solvated mass called a gel. In the
`second stage, the gel breaks up and the molecules are dispersed into a true solu-
`tion. Not all polymers can form true solution in solvent.
`Detailed studies of polymer solubility using thermodynamic principles have led
`to semi-empirical relationships for predicting the solubility [2]. Any solution
`process is governed by the free-energy relationship of Eq. 2.7:
`ð2:7Þ
`DG ¼ DH TDS
`When a polymer dissolves spontaneously, the free energy of solution, DG, is
`negative. The entropy of solution, DS, has a positive value arising from increased
`conformational mobility of the polymer chains. Therefore, the magnitude of the
`enthalpy of solution, DH, determines the sign of DG. It has been proposed that the
`heat of mixing, DHmix, for a binary system is related to concentration and energy
`parameters by Eq. 2.8:
`
`"
`
`DHmix ¼ Vmix
`
`
`
`DE1
`V1
`
`
`
`
`1=2 DE2
`V2
`
`
`
`1=2
`
`#
`2;1;2
`
`ð2:8Þ
`
`where Vmix is the total volume of the mixture, V1 and V2 are molar volumes
`(molecular weight/density) of the two components, ;1 and ;2 are their volume
`fractions, and DE1 and DE2 are the energies of vaporization. The terms DE1=V1
`and DE2=V2 are called the cohesive energy densities. If ðDE=VÞ1=2 is replaced by
`the symbol d, the equation can be simplified into Eq. 2.9:
`ð2:9Þ
`DHmix ¼ Vmixðd1 d2Þ2;1;2
`The interaction parameter between polymer and solvent can be estimated from
`DHmix as:
`
`ð2:10Þ
`
`ðd1 d2Þ2
`
`v12 ¼ V1
`RT
`The symbol d is called the solubility parameter. Clearly, for the polymer to
`dissolve (negative DG), DHmix must be small; therefore, ðd1 d2Þ2 must also be
`small. In other words, d1 and d2 should be of about equal magnitude where
`d1 ¼ d2, solubility is governed solely by entropy effects. Predictions of solubility
`are therefore based on finding solvents and polymers with similar solubility
`parameters, which requires a means of determining cohesive energy densities.
`Cohesive energy density is the energy needed to remove a molecule from its
`nearest neighbors, thus is analogous to the heat of vaporization per volume for a
`
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`

`volatile compound. For the solvent, d1 can be calculated directly from the latent
`heat of vaporization DHvap
`using the relationship of Eq. 2.11:
`ð2:11Þ
`DE ¼ DHvap RT
`R is the gas constant, and T is the temperature in kelvins. Thus, the cohesive
`
`
`energy of solvent is shown in Eq. 2.12:
`d1 ¼ DHvap RT
`
`13
`
`ð2:12Þ
`
`2.2 Polymer Solutions
`
`
`
`
`
`1=2
`
`V
`
`Since polymers have negligible vapor pressure, the most convenient method of
`determining d2 is to use group molar attraction constants. These are constants
`derived from studies of low-molecular-weight compounds that lead to numerical
`values for various molecular groupings on the basis of intermolecular forces. Two
`sets of values (designated G) have been suggested, one by Small [3], derived from
`heats of vaporization and the other by Hoy [4], based on vapor pressure mea-
`surements. Typical G values are given in Table 2.1. Clearly there are significant
`differences between the Small and Hoy values. The use of which set is normally
`determined by the method used to determine d1 for the solvent.
`G values are additive for a given structure, and are related to d by
`d ¼ d R G
`M
`where d is density and M is molecular weight. For polystyrene –[CH2–
`CH(C6H5)]n–, for example, which has a density of 1.05, a repeating unit mass of
`104, and d is calculated, using Small’s G values, as
`
`ð2:13Þ
`
`Table 2.1 Representative group molar attraction constants [3, 4]
`G[(cal cm3)1/2mol-1]
`Chemical group
`Small
`214
`133
`28
`
`H 3C
`
`CH 2
`CH
`
`C
`
`CH2
`CH
`C6H5 (phenyl)
`(aromatic)
`CH
`
`C
`
`O (ketone)
`
`CO2
`
`(ester)
`
`-93
`
`190
`19
`735
`–
`275
`
`310
`
`Hoy
`147.3
`131.5
`86.0
`
`32.0
`
`126.0
`84.5
`–
`117.1
`262.7
`
`326.6
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`

`14
`
`2 Polymer Size and Polymer Solutions
`
`d ¼ 1:05ð133 þ 28 þ 735Þ
`
`104
`
`¼ 9:0
`
`or using Hoy’s data,
`
`d ¼ 1:05 131:5 þ 85:99 þ 6ð117:1Þ½ Š
`
`
`
`¼ 9:3
`
`
`
`104
`
`Both data give similar solubility parameter. However, there is limitation of
`solubility parameter. They do not consider the strong dipolar forces such as
`hydrogen bonding, dipole–dipole attraction, etc. Modifications have been done by
`many researchers and available in literature [5, 6].
`Once a polymer–solvent system has been selected, another consideration is how
`the polymer molecules behave in that solvent. Particularly important from the
`standpoint of molecular weight determinations is the resultant size, or hydrody-
`namic volume, of the polymer molecules in solution. Assuming polymer molecules
`of a given molecular weight are fully separated from one another by solvent, the
`hydrodynamic volume will depend on a variety of factors, including interactions
`between solvent and polymer molecules, chain branching, conformational effects
`arising from the polarity, and steric bulkiness of the substituent groups, and
`restricted rotation caused by resonance, for example, polyamide can exhibit res-
`onance structure between neutral molecule and ionic molecule.
`
`NH+
`
`-
`
`CO
`
`NH
`
`CO
`
`Because of Brownian motion, molecules are changing shape continuously.
`Hence, the prediction of molecular size must base on statistical considerations and
`average dimensions. If a molecule was fully extended, its size could easily be
`computed from the knowledge of bond lengths and bond angles. Such is not the
`case, however, with most common polymers; therefore, size is generally expressed
`
`Fig. 2.3 Schematic
`representation of a molecular
`coil, r = end to end distance,
`s = radius of gyration [2]
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`

`2.2 Polymer Solutions
`
`15
`
`in terms of the mean-square average distance between chain ends, r2, for a linear
`polymer, or the square average radius of gyration about the center of gravity, s2,
`for a branched polymer. Figure 2.3 illustrates the meaning of r and s from the
`perspective of a coiled structure of an individual polymer molecule having its
`center of gravity at the origin.
`The average shape of the coiled molecule is spherical. The greater the affinity of
`solvent for polymer, the larger will be the sphere, that is, the hydrodynamic
`volume. As the solvent–polymer interaction decreases, intramolecular interactions
`become more important, leading to a contraction of the hydrodynamic volume. It
`is convenient to express r and s in terms of two factors: an unperturbed dimension
`(r0 or s0) and an expansion factor að Þ. Thus,
`r2 ¼ r2
`0a2
`s2 ¼ s2
`0a2
`
`Þ1=2
`a ¼ r2ð
`
`1=2
`
`r2
`0
`
`ð2:14Þ
`ð2:15Þ
`
`ð2:16Þ
`
`The unperturbed dimension refers to the size of the macromolecule exclusive of
`solvent effects. It arises from a combination of free rotation and intramolecular
`interactions such as steric and polar interactions. The expansion factor, on the
`other hand, arises from interactions between solvent and polymer. For a linear
`polymer, r2 ¼ 6s2. The a will be greater than unity in a ‘‘good’’ solvent, thus the
`actual (perturbed) dimensions will exceed the unperturbed dimensions. The greater
`the value of a is, the ‘‘better’’ the solvent is. For the special case where a ¼ 1, the
`polymer assumes its unperturbed dimensions and behaves as an ‘‘ideal’’ statistical
`coil.
`Because solubility properties vary with temperature in a given solvent, a is
`temperature dependent. For a given polymer in a given solvent, the lowest tem-
`perature at which a ¼ 1 is called the theta hð Þ temperature (or Flory temperature),
`and the solvent is then called a theta solvent. Additionally, the polymer is said to
`be in a theta state. In the theta state, the polymer is on the brink of becoming
`insoluble; in other words, the solvent is having a minimal solvation effect on the
`dissolved molecules. Any further diminish of this effect causes the attractive forces
`among polymer molecules to predominate, and the polymer precipitates.
`From the standpoint of molecular weight determinations, the significance of
`solution viscosity is expressed according to the Flory-Fox equation [7],
`Þ3=2
`g½ Š ¼ U r2ð
`ð2:17Þ
`M
`where g½ Š is the intrinsic viscosity (to be defined later), M is the average molecular
`weight, and U is a proportionality constant (called the Flory constant) equal to
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`
`g½ Š ¼ U r2
`0a2
`M
`
`
`Equation 2.18 can be rearranged to
`g½ Š ¼ U r2
`
`1
`
`M
`
`0
`
`16
`
`2 Polymer Size and Polymer Solutions
`
`approximately 3 9 1024. Substituting r2
`0‘a2 for r2, we obtain Mark-Houwink-
`Sakurada equation:
`
`
`
`3=2
`
`ð2:18Þ
`
`
`
`1
`
`M
`
`0
`
`3=2
`
`, then
`
`ð2:19Þ
`
`ð2:20Þ
`
`3=2 M1=2a3
`
`Since r0 and M are constants, we can set K ¼ U r2
`g½ Š ¼ K M1=2a3
`At the theta temperature, a ¼ 1 and
`ð2:21Þ
`g½ Š ¼ K M1=2
`For conditions other than the theta temperature, the equation is expressed by
`ð2:22Þ
`g½ Š ¼ K Ma
`Apart from molecular weight determinations, many important practical con-
`siderations are arisen from solubility effects. For instance, one moves in the
`direction of ‘‘good’’ solvent to ‘‘poor’’, and intramolecular forces become more
`important, the polymer molecules shrink in volume. This increasing compactness
`leads to reduced ‘‘drag’’ and hence a lower viscosity which has been used to
`control the viscosity of polymer for ease of processing.
`
`2.3 Measurement of Molecular Weight
`
`Many techniques have been developed to determine the molecular weight of
`polymer [8]. Which technique to use is dependent on many factors such as the size
`of the polymer, the ease of access and operation of the equipment, the cost of the
`analysis, and so on.
`For polymer molecular weight is less than 50,000, its molecular weight can be
`determined by the end group analysis. The methods for end group analysis include
`titration, elemental analysis, radio active tagging, and spectroscopy. Infrared
`spectroscopy (IR), nuclear magnetic resonance spectroscopy (NMR), and mass
`spectroscopy (MS) are commonly used spectroscopic technique. The IR and NMR
`are usually less sensitive than that of MS due to the detection limit.
`Rules of end group analysis for Mn are: (1) the method cannot be applied to
`branched polymers unless the number of branches is known with certainty; thus it
`is practically limited to linear polymers, (2) in a linear polymer there are twice as
`
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`

`2.3 Measurement of Molecular Weight
`
`17
`
`many end groups as polymer molecules, (3) if the polymer contains different
`groups at each end of the chain and only one characteristic 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. To
`determine the number average molecular weight of the linear polyester before
`cross-linking, one can titrate the carboxyl and hydroxyl end groups by standard
`acid–base titration methods. In the case of carboxyl, a weighed sample of polymer
`is dissolved in an appropriate solvent such as acetone and titrated with standard
`base to a phenolphthalein end point. For hydroxyl, a sample is acetylated with
`excess acetic anhydride, and liberated acetic acid, together with carboxyl end
`groups, is similarly titrated. From the two titrations, one obtains the number of
`mini-equivalents of carboxyl and hydroxyl in the sample. The number average
`molecular weight (i.e., the number of grams per mole) is then given by Eq. 2.23:
`ð2:23Þ
`
`Mn ¼ 2  1000  sample wt:
`meqCOOH þ meqOH
`The 2 in the numerator takes into account that two end groups are being counted
`per molecule. The acid number is defined as the number of milligrams of base
`required to neutralize 1 g of polyester which is used to monitor the progress of
`polyester synthesis in industry.
`Of the various methods of number average molecular weight determination based
`on colligative properties, membrane osmometry is most useful. When pure solvent is
`
`Fig. 2.4 Schematic
`representation of a membrane
`osmometer [2]
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`

`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:
`
` 
`
`p C
`
`þ A2C
`
`c¼0
`
`ð2:24Þ
`
`¼ RT
`Mn
`where p, the osmotic pressure, is given by
`ð2:25Þ
`p ¼ qgDh
`where R is the gas constant, 0.082 L atm mol-1K-1 (CGS) or 8.314 J mol-1K-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= Mn and the slope
`equal to A2, units for p=C are dyn Lg-1cm-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
`ð
`Þ, both Mn and Mw can
`and the average molecular weight Mi
`number of ions Nj
`
`be calculated. Figure 2.6 shows that a low molecular weight Mw  3;000ð
`Þ poly(3-
`
`Fig. 2.5 Plot of reduced
`osmotic pressure versus
`concentration [2]
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`

`2.3 Measurement of Molecular Weight
`
`19
`
`2949.110
`2783.182
`
`3116.526
`
`3283.505
`3450.147
`
`3616.721
`
`2616.048
`2480.130
`
`2537.347
`
`2703.324 2869.867
`
`3037.597
`
`3205.285
`
`3373.805
`
`3538.679 3704.432
`
`(b)
`
`8000
`
`6000
`
`4000
`
`2000
`
`Intens. [a.u.]
`
`0
`
`00
`
`2600 2800 3000 3200 3400 3600
`m/z
`
`4614.864
`
`4781.324
`
`4947.428
`
`5114.582
`
`(a)
`8000
`
`6000
`
`4000
`
`2949.110
`2616.048
`
`3450.147
`3616.721
`
`3782.791
`
`3949.088
`
`4117.025
`
`2448.771
`2281.781
`
`2114.975
`
`1948.380
`
`1781.261
`
`1613.847
`
`2000
`
`1445.919
`
`1367.571
`
`0
`
`Intens. [a.u.]
`
`1500 2000 2500 3000 3500 4000 4500 5000
`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 Mwð
`Þ 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 Hc Mw
`
`ð2:26Þ
`
`where H is
`
`H ¼ 32p3
`3
`
`ð2:27Þ
`
`ð2:28Þ
`
`ð
`Þ2
`0 dn=dc
`n2
`k4N0
`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
`¼
`
`Hc
`s
`
`1
`
`MwP hð Þ þ 2A2C
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`

`20
`
`2 Polymer Size and Polymer Solutions
`
`Fig. 2.7 Zimm plot of light-
`scattering data of polymer [2]
`
`where P hð Þ 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 experi-
`mental data are then extrapolated to both zero concentration and zero angle, where
`P hð Þ 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:
`
`g½ Š ¼ K Ma
`v
`
`where Mv, is the viscosity average molecular weight, defined as
`Mv ¼ R NiM1þa
`ð2:30Þ
`i
`R NiMi
`Log K and a are the intercept and slope, respectively, of a plot of log g½ Š 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 g½ Š ¼ log K þ alog M
`
`1=a
`
`ð2:29Þ
`
`ð2:31Þ
`
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`

`2.3 Measurement of Molecular Weight
`
`21
`
`Fig. 2.8 Schematic of a laser
`light scattering photometer [2]
`
`Polarizer
`
`Laser
`
`Partially transmitting mirror
`
`Fully reflecting mirror
`
`Photomultiplier
`
`Light-
`scattering
`cell for
`sample
`l0
`
`Thermostat
`
`Analyzer
`
`Photomultiplier
`
`Amplifier
`
`Correlator
`
`Imbalance
`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
`
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`

`22
`
`2 Polymer Size and Polymer Solutions
`
`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 Xð Þ. Shear stress sð Þ is defined as
`s ¼ 3M
`2pR3
`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 _rð Þ is given by
`_r ¼ X
`a
`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
`¼ 3aM
`¼ kM
`ð2:34Þ
`g ¼ s
`_r
`2pR3X
`X
`
`ð2:32Þ
`
`ð2:33Þ
`
`where k is
`
`k ¼ 3a
`2pR3
`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
`
`ð2:35Þ
`
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`

`2.3 Measurement of Molecular Weight
`
`23
`
`Fig. 2.11 Simple illustrations of the principle of gel permeation chromatography (GPC) [9].
`(Adapted from I.M. Campbell, Introduction to Synthetic Polymers, Oxford, 1994, p. 26 with
`permission)
`
`smallest size molecules will be detected last. From the elution time of different
`size molecule, the molecular weight of the polymer can be calculated through the
`calibration curves obtained from polystyrene standard.
`For example,
`the synthesis of diblock copolymer: poly(styrene)-b-poly(2-
`vinylpyridine) (PS-b-P2VP) can be monitored by the GPC. Styrene is initiated by
`sec-butyl lithium first and then the polystyrene anion formed until the styrene
`monomer is completely consumed. Followed by introducing the 2-vinyl pyridine, a
`PS-b-P2VP block copolymer is finally prepared (Fig. 2.12). As shown in Fig. 2.13,
`the GPC results show that the PS anion was prepared first with low PDI (1.08).
`After adding the 2-VP, the PS-b-P2VP block copolymer was analyzed by GPC
`again, the PDI remained low, but the molecular weight has been doubled. More
`detailed discussions of anionic polymerizations will be present in Chap. 8.
`
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`

`24
`
`2 Polymer Size and Polymer Solutions
`
`sec-Butyl Lithium +
`
`PS-Li+
`
`+
`
`N
`
`PS-PVP
`
`Fig. 2.12 Synthesis of PS-b-P2VP via anionic polymerization
`
`PDI~1.08 Mn=30,000
`PS
`PS-PVP PDI~1.1 Mn=60,000
`
`200
`
`150
`
`100
`
`50
`
`0
`
`Intensity
`
`Fig. 2.13 GPC traces of PS
`homo polymer and PS-b-
`P2VP block copolymer. The
`right peak is PS. The peak of
`copolymer is shifted to the
`left due to the addition of
`P2VP
`
`-50
`
`0
`
`5
`
`20
`15
`10
`Elusion Time (mins)
`
`25
`
`30
`
`35
`
`2.4 Problems
`
`1. A ‘‘model’’ of a linear polyethylene having a molecular weight of about
`200,000 is being made by using a paper clip to represent one repeating unit.
`How many paper clips does one need to string together?
`2. In general, the viscosity of polymer is reduced by increasing temperature.
`How might the magnitude of this effect compare for the polymer in a ‘‘poor’’
`solvent or in a ‘‘good’’ solvent? (This is the basis for all weather multi vis-
`cosity motor oils.)
`3. From the practical standpoint, is it better to use a ‘‘good’’ solvent or a ‘‘poor’’
`solvent when measuring polymer molecular weight? Explain.
`4. Discuss the value of knowledge of the molecular weight and distribution of a
`polymer to the polymer scientist and engineer. Which method would you use
`to obtain this information on a routine basis in the laboratory and in the
`production respectively? Why? Which method would you use to obtain this
`information for a new polymer type which is not previously known? Why?
`5. What would be the number average, weight average molecular weight and
`polydispersity of a sample of polypropylene that consists of 5 mol of 1000
`unit propylene and 10 mol of 10,000 unit propylene?
`6. A 0.5000-g sample of an unsaturated polyester resin was reacted with excess
`acetic anhydride. Titration of the reaction mixture with 0.0102 M KOH
`required 8.17 mL to reach the end point. What
`is the number average
`
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`

`2.4 Problems
`
`25
`
`Elution time (min.)
`13.0
`13.5
`14.0
`14.5
`15.0
`15.5
`16.0
`16.5
`
`Intensity
`0.5
`6.0
`25.7
`44.5
`42.5
`25.6
`8.9
`2.2
`
`0
`
`molecular weight of the polyester? Would this method be suitable for deter-
`mining any polyester? Explain.
`7. Explain how one might experimentally determine the Mark-Houwink-Saku-
`rada constants K and a for a given polymer. Under what conditions can you
`1? How can r2
`1 be used to measure chain
`use g½ Š
`M
`M
`to measure r2
`0
`branching?
`8. The molecular weight of poly(methyl methacrylate) was measured by gel
`permeation chromatography in tetrahydrofuran at 25°C and obtained the
`above data:
`The polystyrene standard (PDI * 1.0) under the same conditions gave a
`linear calibration curve with M = 98,000 eluting at 13.0 min. and M = 1,800
`eluting at 16.5 min.
`a. Calculate Mw and Mn using the polystyrene calibration curve.
`b. If the PDI of polystyrene is larger than 1.0, what errors you will see in the
`Mn and Mw of poly(methyl methacrylate).
`c. Derive the equation that defines the type of molecular weight obtained in
`the ‘‘universal’’ calibration method in gel permeation chromatography.
`
`9. Please explain why the gel permeation chromat