ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS phần 4 pptx

MACROMOLECULAR MASSES AND SIZES BY SCATTERING TECHNIQUES 181 to be scattered in a solid angle  per unit volume of the sample. The total scattering section can be split into two contributions, one coherent and the other noncoherent: (dσ/d) = (dσ/d) c +(dσ/d) I (6.37) In an experiment of light or X-ray scattering, both scattered and incident radia- tions are always out of phase, which is also true for the neutron scattering by most of the isotopes but not for all; certain isotopes, like protons, scatter neutrons in phase. In contrast, deuterium scatters neutrons out of phase and in a coherent manner. The concept of coherent scattering length was introduced to account for the phase of the scattered neutrons: (dσ/d) c =   i b i  2 b takes positive values for isotopes scattering out of phase and negative values for in-phase scattering centers. Thus, for hydrogen we have b H =−0.374 ×10 −12 cm and, for deuterium, b D =+0.667 ×10 −12 cm Under these conditions, the scattered intensity can be written as I(θ) ∼ = K n M 0 cP(θ) where K n is the contrast factor, which is expressed as K n =   i b i −  b j  2 N a M 2 u (6.38) M u is the molar mass of a repetitive unit. The first sum refers to repetitive units, and the second sum refers to solvent molecules. For neutron scattering, as in the case of light scattering, the total scattered intensity corresponds to the difference between scattering by the repetitive units and by the solvent molecules. Recalling that in light scattering, the “contrast factor” K is written as 2π 2 ˜n 2 λ 4 0 N a ( d ˜n dc ) 2 In X-ray scattering, the scattering cross section corresponds to dσ d = r 2 e (Z) 2 182 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES where r e is the radius of an electron (2.81 ×10 −15 m) and Z is the difference between the number of electrons of constitutive polymer atoms and those of solvent. Z can be deduced from the electron density of the various components, ρ e,u (repetitive unit) and ρ e,s (solvent): (Z) 2 = (ρ e,u −ρ e,s ) 2 V 2 u where V u is the volume of a repetitive unit. As indicated earlier, the intensity can be easily deduced, I(θ) ∼ = K x M u cP(θ) (6.39) where K x is the contrast factor, which is equal to K x = γ 2 e (ρ e,u −ρ e,s ) 2 V 2 u N a M 2 u (6.40) For the three scattering techniques a same overall expression is obtained for the scattered intensity Kc I = 1 M w  1 + s 2 q 2 3 +···  (6.41) that can be used to determine M w , A 2 , s 2 . Due to their very short wavelength, neutrons and X-rays serve to measure small radii of gyration. In order to get the same effect, the quantity corresponding to P(θ) must be the same for light scattering (λ =450 nm) or X-ray (λ =0.1 nm) scattering to avoid the effects due to interferences because they are detectable for sizes smaller than those measured in light scattering, that is, when s 2  1/2 λ 0 /20. This imposes that experiments be carried out at very small angles (θ 2 ◦ ); thus in the two latter cases the detector has to be placed sufficiently away (1 m) from the sam- ple so that there is enough separation between the incident and the scattered beams. On the other hand, these scattering techniques give information at a scale much smaller than that resulting from light scattering by photons; thus one can measure radii of gyration s 2  1/2 of [1 −9 2 (<s 2 )/3] much smaller macromolecules. At these small angles, P(θ) can be approximated by the Guinier function [1 −q 2 (s 2 /3)], which allows the determination of the radius of gyration of macromolecules independently of their shape; the higher limit for measurements by SAXS, SANS, and light scattering is 5 nm, 20 nm, and 200 nm, respectively. Synchrotron sources of radiation generate X-rays, whose wavelengths vary from 0.06 to 0.3 nm. With laboratory equipment, K α radiation of Cu is used with a wavelength of 0.154 nm. Neutrons are produced by nuclear reactors and have to be slowed down in order to obtain “cold” neutrons of about 1 nm wavelength. MASS SPECTROMETRY APPLIED TO POLYMERS 183 6.3. MASS SPECTROMETRY APPLIED TO POLYMERS Mass spectrometry is useful for the determination of molar masses of simple molecules. It is based on the vaporization and the ionization of the entities to be studied. Sent into an electric (or magnetic) field, each of these charged species undergoes a deflection proportional to its m/z ratio, where m represents the mass of the particle and z represents the number of charges carried. Because of their thermal instability and their lack of volatility, polymers cannot be characterized by this technique. The use of “MALDI” (Matrix-Assisted Laser Desorption Ionization) equipped with a “time of flight” (TOF) spectrometer offers an elegant and effective method for the precise and absolute determination of polymer molar masses. Initially designed for the characterization of biomolecules, MALDI-TOF spectrometry is now exten- sively applied to synthetic polymers; in this technique the polymer to be studied is dispersed in an organic matrix that is volatilized under the effect of a laser radiation whose wavelength is in the absorption range of the matrix. Owing to the specific molecular interactions between polymer and matrix—which plays an important role in the chain desorption—each polymer requires a suitable matrix. For example, 1,8,9-trihydroxyanthracene is well suited to poly(methyl methacrylate). In order to avoid interchain entanglements, it is essential experimentally to keep the polymer concentration in the matrix very low. Upon volatilization of the matrix, the polymer species are desorbed and ionized (by a mechanism still ill-known), which makes them sensitive to the accelerating effect of an electric field. For polymer analysis, the positive mode—the one which causes the formation of a positive charge on the chain by interaction with a cationic species (proton or metal cation)—is generally used. In order to obtain simple and easy to analyze spectrograms, chains should be monocationized. The separation of polymers (pol −H + or pol −Met + ) according to their molar mass is obtained through the time of flight necessary for each of the species to travel from the target to the detector. Although molar masses up to 1.5 ×10 6 g·mol −1 could be measured, most of the published studies pertain to polymers of molar mass ranging between 1000 and 2 ×10 4 g·mol −1 ; this technique is an extremely sensitive one since quantities of matter in the range of femtomoles can be detected. Problems encountered in MALDI-TOF mass spectrometry are due to: • The low resolution beyond molar masses of about 2.5 ×10 4 g·mol −1 • The difficulty to “cationize” certain nonpolar polymers • The difficulty to desorb macromolecules of high molar masses Remark. The upper limit of molar masses for well-resolved signals depends on the molar mass of the monomer unit. Beyond that limit, an envelope of the constituting signals is observed. 184 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES The example chosen to illustrate this technique of characterization is a sample of a macrocyclic polystyrene cyclized through an acetal function ( M n =6900 and M w /M n =1.5) and represented here: - O O O O ~~ ~ ~ PS (n−5) In the MALDI-TOF spectrogram shown in Figure 6.12, the exact degrees of polymerization (n) of the chains present in the samples can be seen without ambi- guity through their mass and the various molecular groups carried by the chains identified; each signal corresponds to chains whose molar mass is given by the relation M signal = (104n +2344 +23) where 23 represents the molar mass of Na + added into the matrix in order to cationize the chains. The accuracy of this technique is of great interest for the study of the mechanisms of polymerizations. 5471.1 5782.8 6407.4 6719.1 7031.8 7447.7 7655.3 7863.2 8072.0 8279.8 8487.0 8694.7 6095.5 Figure 6.12. MALDI-TOF mass spectrogram of a macrocyclic polystyrene whose formula is given above. VISCOSITY OF DILUTE SOLUTIONS—MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 185 6.4. VISCOSITY OF DILUTE SOLUTIONS—MEASUREMENT OF MOLAR MASSES BY VISCOMETRY Molar masses and molecular dimensions of polymers are accessible not only through scattering techniques but also from viscosity measurements. Indeed, the response of macromolecules to the application of hydrodynamic forces can give information about their volumes and their dimensions and thus indirectly about their molar masses. By definition, the viscosity of a liquid is proportional to the product of the flow time of a characteristic volume times its density: η ÷t × ρ (6.42) The simplest case corresponds to a flow behavior described by the Newton law, in which the viscosity (η) is the ratio of the shear stress (σ) to the shear rate (˙γ) and more specifically the slope of the straight line drawn from the variation of the shear stress versus the shear rate (η = σ/˙γ). Liquids exhibiting such behavior are called Newtonian, and their viscosity is independent of the shear rate. Remark. Non-Newtonian liquids do not have this linear variation of σ versus ˙γ, in the entire range of shear rates. Except for very high molar mass samples, viscosities of dilute polymer solutions are Newtonian, and the relations between the polymer molar mass and its viscosity are established in this context. More than the proportionality constant, which relates the viscosity to the flow time, experimenters are interested in comparing the viscosity of a polymer solution with that of pure solvent (η 1 ). This has given rise to the notion of relative viscosity (η r =η/η 1 ), specific viscosity [η sp = (η −η 1 )/η 1 = η γ −1], or reduced viscosity (η red =η sp /c 2 ). The specific viscosity is an indicator of the increase in viscosity due to the addition of a polymer, whereas the reduced viscosity characterizes the propensity of a given polymer to increase the relative viscosity and is also called intrinsic viscosity [η] in the limit of infinite dilutions: [η] =  η sp c 2  c 2 →0 Thus, intrinsic viscosity has the dimension of a specific volume. 6.4.1. Variation of Viscosity with Concentration The viscosity of a dispersion of sufficiently diluted rigid particles—so that their effect is thus simply additive—can be described by the Einstein viscosity relation: η = η 1 [1 +B 1  2 +B 2  2 2 ···] (6.43) 186 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES c 2 (g/dL) h sp /c 2 or ln (h r /c 2 ) ln (h r /c 2 ) h sp /c 2 0.7 0.9 1.0 0.2 0.4 0.8 1.0 [h] Figure 6.13. Example of extrapolation to zero concentration of the variation of η sp /c 2 and ln(η r /c 2 ) against concentration. with B 1 =2.5 when particles are nonsolvated rigid spheres and B 2 =14.1; in this equation, B 1 and B 2 can take different values, depending on the shape and the size of the particles under consideration. In the previous expression the volume fraction of the particles  2 =V 2 /V 1 can be replaced by their mass concentration c 2 , after observing that V 2 is equal to N 2 V H (i.e., the product of the number of particles times their hydrodynamic volume) and that the concentration (c 2 ) and the number (N 2 ) of these particles in the volume (V ) are related by N 2 /V = c 2 ·N a /M Thus, the reduced viscosity can be described by η red = 5/2(V H N a /M) +B 2 (V H N a /M) 2 c 2 +··· corresponding to η red ≡ η sp /c = [η] +k H [η] 2 c 2 +··· where [η] is equal to 5/2(V H N a /M) and k H , which is called the Huggins coefficient, is equivalent to 4B 2 /25. k H can be obtained by plotting the linear variation of η sp /c 2 against c 2 . k H takes values close to 1/3 when the polymer is in a good solvent and can grow up to 0.5–1 if a bad solvent is used. k H is thus a criterion of the quality of solvent. 6.4.2. Relation Between Viscosity and Molar Mass of a Polymer Long ago, Staudinger had the intuition that the molar mass of a polymer and its viscosity must be related and thus postulated that [η] must be proportional to the VISCOSITY OF DILUTE SOLUTIONS—MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 187 0.74 0.50 [h] (ml/g) 10 10 2 10 3 10 3 10 4 10 5 10 6 10 7 M w (g/mol) polystyrene in θ conditions (cyclohexane, T = 34.5°C) polystyrene in a good solven t Figure 6.14. Variation of [η] against molar mass in the case of polystyrene solutions in a good solvent and under θ conditions. molar mass, which was subsequently proved true for rigid polymers (Figure 6.14). Actually, molar mass and viscosity could be related in an empirical manner through the Mark–Houwink–Sakurada (M–H–S) equation: [η] = KM α (6.44) where K and α are constants varying with the polymer, solvent, and temperature under consideration. The value taken by the exponent α gives information about the conformation of the polymer in a given solvent and even its shape. Thus α is equal to 0 for spheres, 0.5 for statistical coils in a nonperturbed state (θ conditions), ∼0.8 for chains in solution in a good solvent, and 2 for rigid rods. For polymers with a wormlike shape, the coefficient α is intermediate between that of perturbed chains and rods—that is, between 0.8 and 2. Due to the non-Gaussian character of small chains—which are not statistical coils—the “M–H–S” equation applies strictly only to chains whose molar masses are higher than 2 ×10 4 g·mol −1 . The constants K and α for small chains in a given solvent are therefore different from those determined for Gaussian chains with the same repetitive units. To know the type of average molar mass accessible by viscosity measurements, the following reasoning can be used; because η sp is equal to [η]c 2 within the limit of low concentrations, η sp can also be written as η sp =  i η sp,i =  i [η i ]c i By dividing this equation by c 2 ≡  i c i and by considering that w i = c i /c, one obtains η sp /c 2 = [η] =  i [η i ]w i 188 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES The viscosity of a sample is thus the mass average of viscosities of the collection of chains present in the medium. Average molar masses obtained from viscosity experiments are called viscosity average molar masses. M v = ([η]/K) 1/α =   i w i [η i ]/K  1/α =   i ω i M α i  1/α (6.45) 6.4.3. Determination of Molecular Dimensions from Intrinsic Viscosity Measurements 6.4.3.1. Case of Rigid Spheres. As previously shown, the intrinsic viscosity corresponds to a specific volume—that is, to the hydrodynamic volume of 1 g of the polymer analyzed within the limit of infinite dilutions. Insofar as relations can be established between the hydrodynamic volume of a polymer of a given conformation and its molar mass, use can be made of viscosity measurements to determine the molar mass of a sample. If the polymer analyzed is of spherical shape, its hydrodynamic volume corresponds to the volume of an equivalent sphere of radius R sph : [η] = 5 2 N a M V H = 10π 3 N a R 3 sph M (6.46) Knowing the relation between the square of the radius of this sphere and its radius of gyration, we obtain s = (3/5)(R 5 e −R 5 i )/(R 3 e −R 3 i ) where R e and R i are the outer and inner radii of a partially hollow sphere; in our case related to solid spheres, R i =0andR e ≡ R sph ; the conversion factor Q (R sph = Q sph,s ) is thus equal to (5/3) 1/2 and then one obtains: [η] = 10π 3 N a (5/3) 3/2 s 3 M = φ sph,s s 3 M (6.47) with φ sph,s equal to 13.57 ×10 24 mol −1 . Knowing that the relation between the molar mass of a spherical object and its radius of gyration is written as s = (3/5)(4πρ)N 1/3 a M 1/3 (6.48) it is easily shown that the intrinsic viscosity of a sphere is independent of its molar mass and depends only on its density. 6.4.3.2. Case of Statistical Coils. The same reasoning—that is, identifying a polymer (here a statistical coil) to an equivalent sphere of radius R —can be VISCOSITY OF DILUTE SOLUTIONS—MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 189 applied here: [η] = 5 2 N a V H M = 10π 3 N a R 3 H M = 10π 3 N a Q 3 s 3 M = φ s 3 M (6.49) with φ = 10 π 3 N a Q 3 . The problem is to establish the relation between the hydrodynamic radius and the radius of gyration of such equivalent sphere and thus to calculate Q,whichis supposed to depend on the distribution of segments in the statistical coil. Free Draining Model. In this model, known as the “Rouse model,” the polymer is represented as beads interconnected by massless springs free of hydrodynamic interactions. The solvent passes freely through the statistical coil and exerts fric- tional forces on the segments, or the beads: each center or friction point moves independently as if the other points do not exist. The viscosity (η) is thus the prod- uct of the frictional coefficient ξ of a segment and a global factor F, which takes into account not only the friction undergone by each of the X segments but also the conformation effects: F = (ρN a /6)(s 2  0 /M)X In dilute solutions, the density (ρ ≡ m coil /V coil ) can be considered as identical to the concentration: C 2 =m 2 /V ,wherem 2 is the mass of polymer and V its volume. Viscosity can be written as η = η 1 (η/η 1 ) ∼ = η 1 (η −η 1 )/η 1 = η 1 η sp and η/ρ ∼ = η 1 η sp /c = η 1 [η] This leads to the following equation: [η] = N a ξX 6η 1 s 2  0 M = N a ξ 6η 1 M seg M = K η M (6.50) With s 2  0 /M being constant, it is included in K η . Thus, the free draining model predicts that the exponent α of molar mass is equal to 1 for chains without excluded volume as in the Staudinger empirical formula. Except for a few cases, this model is not very realistic because it neglects hydrodynamic interactions between elements of chains. Unperturbed Statistical Coils. In the Kirkewood and Riseman model, the poly- mer is represented as a collection of beads interconnected by bonds of length L 190 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES and interacting with each other. This method involves the calculation of the pertur- bations due to the interactions between repeating units and to the long-range ones induced by the chains. Using the Oseen formula, these authors obtained [η] = π 2/3 N a (Q ·f(Q) s 2  3/2 0 M = φ s 2  3/2 0 M = φ  s 2  0 M  3/2 M 1/2 (6.51) Through the function Q · f (Q), this model describes both the case of the free flow and that of impermeable chains. Depending on the degree of friction between the solvent and the chain, Q · f (Q) varies from values close to 0, in the case of free draining chains, to 1.26 for impermeable chains. In the latter case, φ is equal to 4.22 ×10 24 mol −1 . Because the exponent α is not supposed to vary with the molar mass in this approach, it is simply assumed that the degree of permeability of the chains is related to their molar mass. The product ( s 2  0 /M ) 3/2 then varies with M . Statistical Coils in Perturbed Mode. This model is also called the Flory–Fox model. In the previous model, the effect of excluded volume was not taken into consideration. According to the Flory and Fox analysis, the latter model applies only to the case of unperturbed chains under θ conditions; the case of chains in a good solvent requires a separate treatment. The Flory–Fox model is based on the assumption that long-range interactions and the perturbations that they cause do not modify the flow of a solution: [η] θ = φ θ s 2  3/2 0 M = φ θ  s 2  0 M  3/2 M 1/2 = K θ M 1/2 (6.52) where φ θ is a constant that is not supposed to vary with the molar mass and is thus said to be “universal.” Indeed, the conversion factor, which relates the unperturbed radius of gyration to the hydrodynamic radius, is independent of the structure of the polymer, with the distribution of segments in an unperturbed polymer being independent of its structure. For the same chains under θ conditions,s 2  0 /M is also independent of the molar mass of the sample. Since Q θ =R H,θ /s 0 =0.87 for such nonperturbed chains (in the case of spheres, Q sph =(5/3) 1/2 as it was shown previously), φ θ is equal to 4.22 ×10 24 mol −1 , which corresponds to the value deter- mined by Kirkewood and Riseman for impermeable chains. Figure 6.15 shows how φ varies in reality. To take into consideration the continuous increase of the exponent α (in [η] = KM α ) with the molar mass in a good solvent, Flory and Fox conditioned the degree of permeability of the chains to their mass, and they did it through the excluded volume effect. Considering that for perturbed chains s varies with M 2 as s 2  1/2 = KM 3/5 (6.53) [...]... deconvolution intensity of the crystalline 3 band at 10.7 mm (arbitrairy unit) 0.3 intensity of the amorphous band at 8.8 mm crystalline band amorphous band 2 1 0.2 0.1 1.06 0 1.10 20 1. 14 40 1.18 60 80 1.22 density 100 degree of crystallinity Figure 6.26 Variation of intensity of two different absorption bands for a series of polyhexamethyleneadipamide (PA-6,6) as a function of their density 6.7.5... deconvolution of the signals and a comparison of the resulting surfaces, an absolute value of the degree of crystallinity (Figure 6.25) can be determined 6.7 .4. 2 Infrared Spectrometry Theoretical studies of the effect of the degree of crystallinity on vibration spectra account only partially for the appearance or the disappearance of certain absorption bands when comparing amorphous polymers with semicrystalline... phases instead of additivity of volumes leads to densities (ρc , ρa , ρ) and volumic degree of crystallinity (Xv ) There is a simple relation between Xm and Xv : Xm = V Vc Xv = ρc Xv ρa The volumetric method requires the knowledge of the densities of a totally amorphous sample and that of a completely crystalline one To obtain ρa , the values of the density (or of the specific volume) of a given polymer... interplanar spacing for a family of crystal plane, θ is the angle of the incident X-ray beam with the set of crystal planes, and λ is the wavelength of the incident radiation fiber axis X-rays Figure 6.22 Schematic representation of the orientation of crystallites in a fiber resulting from the stretching of a semicrystalline polymer (see Figure 5 .40 ) DETERMINATION OF THE DEGREE OF CRYSTALLINITY (X) 201 The... (6.70) Figure 6.23 Debye–Scherrer pattern of a semicrystalline isotactic polypropylene sample Each peak of the diffracted intensity corresponds to a family of crystal planes DETERMINATION OF THE DEGREE OF CRYSTALLINITY (X) 203 and θ is the angle of transmission of either the diffracted or the scattered beam and λ the wavelength of incident radiation The applicability of the above relation strictly implies... (6.55) αH and αs are different and can be obtained from scattering measurements for the radius of gyration and viscosity measurements for the hydrodynamic radius For a sphere, one finds αH 3 = αs 2 .43 ; for an ellipse, one finds αH 3 = αs 2.18 ; and so on 6 .4. 3.3 Cases of Rods Examples of rigid rods among synthetic polymers are those which adopt a helical conformation either due to the size of their substituents... with gradient of density The latter technique is currently used to measure the density of polyolefins r ra Ta Tg TF Figure 6. 24 Variation of density of a semicrystalline polymer with the temperature DETERMINATION OF THE DEGREE OF CRYSTALLINITY (X) 205 Table 6.1 Density of some synthetic polymers in the amorphous state and in the crystalline state (values at 20◦ C in g·cm−1 ) ρa ρc 0.8 54 0.852 0.856... (f + 1) (f + 2) For a comb polymer including x points of branching of functionality f and p grafts, we obtain 6p2 + (f − 1)x(x 2 − 1) g= (6. 64) 6p(p + 1)(p + 2) with p = (f −1)x 6 .4. 4 Techniques of Measurement It is imperative to work with solutions of precisely known concentration and free of any dust The solutions should not be too dilute (lack of precision) or too concentrated (interchain interactions)... absorption bands is observed as a function of the density and hence degree of crystallinity (Figure 6.26) One band is due to the crystalline state, and the other one is due to the amorphous state 208 DETERMINATION OF MOLAR MASSES AND STUDY OF CONFORMATIONS & MORPHOLOGIES H Figure 6.25 Low-resolution NMR signal of the 1 H nuclei of a polymer with high degree of crystallinity Dashed lines represent two... correspondence between the elution volume (or elution time) of polymer chains and their molar mass This operation requires the prior establishment of a calibration curve from elution volumes of standard samples as a function of their molar mass Standard samples of polystyrene for polymers soluble in organic solvents or poly(ethylene oxide) for water-soluble polymers are used for such calibration curves (see . mass of Na + added into the matrix in order to cationize the chains. The accuracy of this technique is of great interest for the study of the mechanisms of polymerizations. 547 1.1 5782.8 640 7 .4 6719.1 7031.8 744 7.7 7655.3 7863.2 8072.0 8279.8 848 7.0 86 94. 7 6095.5 Figure. prior establishment of a calibra- tion curve from elution volumes of standard samples as a function of their molar mass. Standard samples of polystyrene for polymers soluble in organic solvents or. of “MALDI” (Matrix-Assisted Laser Desorption Ionization) equipped with a “time of flight” (TOF) spectrometer offers an elegant and effective method for the precise and absolute determination of