COURSE MATERIAL: Unit 3 (Part 1) Polymer Science LT8501 (Click the link Detail to download) Dr. Debasis Samanta Senior Scientist & AcSIR Assistant Professor Polymer Science & Technology Department., CSIR-CLRI, India Honorory Faculty, Anna University 1) 1
Syllabus: Unit 1 Molecular weight and its distribution by: End group analysis, osmometry, light scattering. Ultra centrifugation, gel permeation chromatography, intrinsic viscosity Books & References: 1) 2
1.1. Polymer Molecular weight and its distribution : The chain length can be expressed as degree of polymerization (number of repeating units), or, in most cases, as molar mass (molecular weight). In a specific polymer, molecules of different chain lengths may exist providing different molecular weight. Therefore, every statement about molar mass indicates an average. Hence the molecular weight may be expressed as number average molecular weight (M n bar) or weight average molecular weight (M w bar ). 1.1.1. Number average molecular weight (Mn bar ): The number molecular weight is the ordinary arithmetic mean or average of the molar masses of the individual molecule. It is determined by measuring the molecular weight of the n polymer molecules, summing the masses and dividing by n. It can be expressed as the following formula. Where n i, is defined as the number of molecules, each with molecular weight M i 1.1.2. Weight average molecular weight (Mw bar): If we had recorded the weight of each species in the sample rather than the number of molecules, the array of the data will be the weight distribution and the average value will indicate the weight average molecular weight. The differential weight distribution curve looks like a BELL shape curve as mentioned below 1) 3 1.1.3. Polydispersity index (PDI): It is expressed as the formula: Mw(bar)/Mn(bar). It is always greater than 1. The more the value of PDI, the more the polymer is polydisperse. Polymer with more PDI value indicates the wider distribution of the molecular weight of the polymer. 1.1.4. Methods of measuring the molecular weight: A number of methods are employed for measuring the molecular weight as listed below: 1.1.4.1.1. Osmometry (Absolute method): (a) Membrane osmometry: This is an absolute method of determining the molecular weight of the polymer based on the measurement of colligative
properties In this case, osmotic pressure (Π os ) is first determined using a membrane osmometer. The membrane osmometer contains two chambers, separated by semipermeable membrane (only solvent can pass, not the polymer molecules). One chamber contains the pure solvent and the second one contains the polymer solution in the same solvent. In this case, solvent molecules migrate through the membrane from the solvent chamber into that of polymer chamber and dilute it. Therefore, the volume of the polymer solution increases until an equilibration is reached between the osmotic pressure and the hydrostatic pressure generated by the diluted polymer solution. After reaching the equilibria, considering the van t Hoff s law, the following equation van be derived Hence, the osmotic pressure should be first measured at different polymer concentration. Then Π os /c is plotted vs c so that the values are linearly extrapolated to c tends to 0 and the number average molecular weight M n is determined from the y axis intercept. A 2 is second virial coefficient and for ideal solvent, A 2 is zero. For the measurement of molecular weight by membrane osmometry, it is very important that the samples should be very pure and free from oligomers. Otherwise, the small molecules or the oligomers may migrate through the membrane leading to the overestimation of molecular weight. Usually, the lower limit of M for application of membrane osmometry is approximately 10000, depending on the available membrane pore size. Further, for reliable prediction of molecular weight, aggregation phenomenon should not happen. (b) Vapour Pressure Osmometry It is based on Raoult s law, which states that for the dilute solution of a compound 2 in a solvent 1, the vapour pressure of the solvent decreases proportionally with the mole fraction of the solvent. The following equation may be deduced for the measurement of molecular weight using the vapour pressure osmometry method Where p 1,0 is the vapour pressure of the pure solvent and p1 = p 1,0 p 1, n 1 and n 2 is number of moles of the corresponding species. So, the measurement of vapour pressure of the dilute polymer solution should lead to M n. However, precise determination of vapour pressure is not easy. Hence, the effect of vapour pressure lowering is measured indirectly by determining the increase of solution temperature (due to the heat of 1) 4
condensation) when the solution is in contact with a saturated atmosphere of solvent vapour. 1.1.4.1.2. Static light scattering method (Absolute method for weight average molecular weight) An electromagnetic radiation, whose energy is insufficient for bound electrons to excite from ground state to excited state transition, may cause the emitting of absorbed radiation in all directions in space. This light is observed as scattered light. This is called Rayleigh scattering which is ideally coherent and elastic. Rayleigh and Debye proposed that only sub volume elements in a sample (whose size is determined by the wavelength of the incident radiation) contribute to the scattering, which are different in polarizability and thus refractive indices with respect to their surrounding: the scattering intensity is proportional to the square of the refractive index difference. In a pure solvent, scattering is very week for visible light, however, for solution; there is an appreciable contribution due to the dissolved materials which causes concentration fluctuations. For a much diluted solution, the scattering intensity caused by the dissolved molecule, R Θ is given by: R Θ = K.c.M K can be calculated from dn/dc (refractive index increment), refractive index of the solvent and other parameter. Hence the molecular weight of the dissolved monodisperse material can be determined by measuring R Θ at concentration c. This equation is valid for monodisperse samples. For the polydisperse compounds, the following equation is valid considering that all components i having different molecular weight M i and concentration C i, scattered independently from each other: From the above equation, knowing all the parameters, weight average molecular weight of the polymer can be calculated. 1.1.4.1.3. Intrinsic viscosity (Relative method for viscosity average molecular weight) After dissolving a polymer in a solvent, it becomes viscous. The higher the molecular weight, the more viscous the polymer solution will be. This is particularly because, the higher the molecular weight, the more strongly the solvent molecules will be bound to the polymer, reducing the mobility of the solvent molecules. 1) 5
For most polymers, there is a definite relationship between the molecular weight and solution viscosity. For the determination of the molecular weight by solution viscosity measurement, a viscometer is used where polymer solution and the pure solvent is passed separately through the capillary. Then, the time t, which a given volume of solution needs to flow through the capillary; is compared with time t 0, which is needed by the pure solvent when flowed through the capillary. If the measurement is made in a capillary viscometer of specified dimensions and at low polymer concentration, then one can obtain the value of specific viscosity (η sp ) from the following equation If this value is divided by the concentration c of the polymer in solution, one can obtain the reduced specific viscosity ((η red ). Since the dissolved macromolecules influence each other more in a concentrated solution, viscosity of the polymer solution should be determined at infinite dilution. However, such measurements are impossible in practice. Hence, it can be determined at low polymer concentration and extrapolate the values at zero concentration. This limiting value of reduced specific viscosity at infinite dilution is called intrinsic viscosity and expressed as the following equation: The Mark-Houwink-Kuhn equation gives a general description of how the molecular weight can be calculated from the intrinsic viscosity as the following equation: 1) 6 M is the viscosity average molecular weight, and K and a are the Mark- Houwink constants. There is a specific set of Mark-Houwink constants for every polymer-solvent combination. So, to measure the exact value of viscosity average molecular weight, one has to know these values for the a known polymer-solvent combination. 1.1.4.1.4. End-group Analysis (Absolute method for number average molecular weight) If a polymer has an easily detectable end functional group, their number average molecular weight (M n ) can be determined by precise analytical method. However, only polymer with molecular weight of less than 50,000 can be determined and the specific analytical method must be very accurate since the end group constitutes only a small fraction. To determine the molecular weight of the polymer by end group analysis, first, the quantitative determination of end functional group is
necessary. One of the most common methods of quantitative determination of end group is potentiometric ph titration for the determination of end functional carboxylic acid groups. Elemental analysis of halogen is also an effective technique when bromo compound is used as initiator for the radical polymerization. Spectroscopic techniques such as IR, UV-vis (for initiator with azo functional groups) and NMR spectroscopy (for step-growth polymers) are very useful for the rapid detection of end group. In certain cases, radiochemical analysis can also be taken after the introduction of radioactive nuclei such as 3 H, 14 C etc. Particularly, the amino, hydroxyl and carboxyl end groups in polyesters and polyamides can be estimated very precisely both by potentiometric ph titration and by colourimetry. Finally, the number-average molecular weight can be calculated by the analytically determined end group content according to the following formula Where E is the molecular weight of the end group and z is their number per macromolecules. e is the experimentally determined end groups in weight percentage. 1.1.4.1.5. Gel permeation chromatography or Size-Exclusion chromatography (Relative method to determine both M n, M w and PDI) Gel permeation chromatography is one of the very few methods, which can be used to determine both M n, M w and PDI simultaneously. In a GPC experiment, the polymer is separated in a column which is filled with a swollen, uniformly packed resin or gel. The gel beads are made of crosslinked polymers with little pores of different sizes in the same dimension of solvated polymer coils. A solution of the polydisperse polymer is charged on top of the swollen gel packed in a column. Only solvent molecules and those macromolecules whose size is less than the pore size can defuse into the pores of swollen gel. In this case, smaller polymer molecules are trapped inside the pores first, then come out because of solvent and caught again into the another pore and so on. Since there are fewer pores to trap the bigger molecules, those molecules pass through the column more quickly. Accordingly, the elution time increases with decreasing the molecular size. In order to detect the macromolecules that elute from the column, detectors are needed to measure the amount of macromolecules coming out of the column at a given time interval. The commercially available GPC equipment is usually fitted with Refractive Index (RI) detector, which compares the change of refractive index of the eluting solution to find out the amount of polymer. Alternatively, the polymer concentration is 1) 7
determined by UV-vis spectroscopy provided the macromolecules posses relevant absorption band. Using those collected data, a plot is made using time as X-axis and the number of polymer molecules (from the intensity of the eluting polymer solution) coming out at a given time as y axis. Since the GPC is relative method of determining the molecular weight, a calibration curve is needed to fit with the known polymer. However, when a light scattering detector is used, the absolute molar mass can also be obtained. In recent time, GPC technique became very polular, particularly in the RND labs due to its broad applicability, easy sample preparation and the large amount of resulting from full distribution curve. Importantly, the commercially available GPC equipment works automatically with small sample amounts and at different temperatures. 1.1.4.1.6. Ultracentrifugation (Absolute method) In the ultracentrifugation field, dissolved molecules or suspended particles can be either sedimented or flotate depending on the density. Under identical experimental condition, the velocity of the molecules or particles depends on the mass and shape of the dissolved/dispersed polymers/particles. Following methods can be employed for the determination of molecular weight of the polymers using the ultracentrifugation techniques. (i) Svedberg method In this case, the speed of the rotor is very high compared to their diffusion velocity. Thus, diffusion can be disregarded and a zone is formed where a concentration gradient will be created. In an ideal condition, this concentration gradient migrates from the meniscus to the bottom of the cell during the centrifugation process. The following Svedberg equation of sedimentation can be derived considering all the factors: (ii) Thus, the molecular weight M s,d can be determined by knowing the diffusion coefficient (D) of the polymer in the used solvent, the specific volume ν (bar), the density of the solvent, Q solvent, and sedimentation coefficient s (by measuring the maximum of concentration gradient at regular time interval. Sedimentation equilibrium At low rotor revolution numbers, an equilibrium state can be reached between sedimentation and diffusion. In this case, a time dependent concentration gradient is established and from the 1) 8
Svedberg equation, the molecular weight of the polymer can be determined. 2. Problem Suppose a polymer has the following distribution of molecular weight (not the real case): Number of molecules 1 20000 2 25000 5 30000 8 40000 6 50000 4 60000 1 80000 Mass of each molecules Calculation of Number average molecular weight (Unit for molecular weight is g/mol) Hence the mass of each molecule and total mass of each type of molecule can be tabulated as below. No of molecules, N i Mass of each molecule, M i Total mass of each type of molecule, N i M i 1 20000 20000 2 25000 50000 5 30000 150000 8 40000 320000 6 50000 300000 4 60000 240000 1 80000 80000 Hence Total mass NiMi=860000 Total no. Of molecules Ni=27 Hence No average molecular weight= NiMi/ Ni=31851.8 Calculation of Weight Average Molecular Weight In this case, weight fraction of each type of molecule is important. Hence, the following table summarises the weight fraction value for individual type of molecules Number molecules (N i ) of Mass each molecule of Total mass of each type of molecules Weight fraction type of molecules (nimi/ NiMi) WiMi (NiMi 2 / NiMi) 1) 9
(Mi) (NiMi) (nimi/860000) 1 20000 20000 0.023250 465 2 25000 50000 0.058139 1453.4 5 30000 150000 0.174418 5232.5 8 40000 320000 0.372093 14883.7 6 50000 300000 0.348837 17441.8 4 60000 240000 0.279069 16241.4 1 80000 80000 0.093023 7441.8 Hence Weight average molecular weight = WiMi = NiMi 2 / NiMi=465+1453.4+5232.5+14883.7+17441.8+16241.4+7441.8 Mw bar= 63159.6 g/mol 1) 10