EQ, Q, DEQuim, DQuim nd semester 017/018, IST-UL Science and Technology of Polymers ( nd cycle) Polymers Reactions and Polymers Production (3 rd cycle) Lecture 5
Viscosity easurements of the viscosity of polymer solutions (even at very low concentrations, e.g. 1wt%) provide information about viscosity average molecular weight, macromolecules dimensions and shape, and polymer-solvent interactions. Capillary viscometers: - Ostwald; - Hubbelohde (t t 0 ) t 0. c = ( 0 )/ 0. c The time taken for the solution to flow through the capillary is measured and then compared with a standard sample Höppler viscometer (queda de esfera) K visc esf L t Rotational viscometer
Shear stress Shear rate The flow of fluids through a tube of uniform cross-section under an applied pressure is given by the Hagen Poiseuille s law: 4. R. t.( P ) 8. V. L P. g. h 4. R.. g. h. t 8. V. L Kinematic viscosity: 4. R. g. h. 8. V. L t V = volume flow rate ΔP is the pressure drop across the tube of length L and radius R Upper plate is set in motion in the x-direction at a constant velocity According to the Newton s law, in the neighborhood of the surface of the moving plate, the fluid acquires a certain amount of x-momentum. This fluid, in turn, transmits some of its momentum to the adjacent layer of fluid causing it to remain in motion in the x-direction; in effect, the x-momentum is transmitted in the y-direction. The velocity gradient is a measure of the speed at which intermediate layers move with respect to each other. For a given stress, fluid viscosity determines the magnitude of the local velocity gradient. Fluid viscosity is due to molecular interaction; it is a measure of a fluid s tendency to resist flow, and hence it is usually referred to as the internal friction of a fluid. Polymers are non-newtonian! viscosity depends on the shear rate and time (at a certain temperature and pressure) Assumptions: 1. The flow is laminar, which means that the dimensionless quantity called Reynold s number, Re, < 100.. The fluid is incompressible, i.e., its density is constant. 3. The flow is independent of time, i.e., steady-state conditions prevail. 4. The fluid is Newtonian.
Effect of shear rate on polymer chain rotation: Hydrodynamic work is converted into heat, which leads to an increase in viscosity. The measurement of the flow rate of the liquid can be made through a capillary tube (part of the viscometer). By measuring the flow time of the solution, t, and that of the pure solvent, t 0, the relative viscosity can be determined. Viscosity of the polymer solutions is always greater than that of the pure solvent. This fractional increase in the viscosity resulting from the dissolved polymer in the solvent is referred to as the specific viscosity η sp. Various viscosity terms: increase in the viscosity resulting from the dissolved polymer in the solvent fractional increase in the viscosity resulting from the dissolved polymer in the solvent measures the capacity with which a given polymer enhances the specific viscosity limiting value of the reduced viscosity at infinite dilution
OLECULAR SIZE AND INTRINSIC VISCOSITY For high molecular weight flexible chain molecules, the correct model is for non-draining, inpenetrable units Einstein (1905): viscosity relation for rigid spherical particles in solution (dilute solution of suspended spheres with negligible interaction) = 0. 1 5.... 0 0 5 number of spherical particles = N A. (polymer mass) n V = c N A or 5 n.. V 4 3 R 3 Number of polymer molecules per unit volume Volume fraction of equivalente spheres R is the radius of an equivalent hydrodynamic sphere that would enhance the viscosity of the solvent medium to the same extent as would the actual polymer molecule. Viscosity of an assembly of spheres is independent of the size of the spheres, depending only on their volume fraction (reduced viscosity) 0 0 1 5 NA 4 3 5 4... R.. NA. c 3 3 3 R 5. V V = molar hydrodynamic volume (N A.R 3 ) Vol ( const.). (.). const R 3 5. V V.. 5
OLECULAR SIZE AND INTRINSIC VISCOSITY (cont.) However, real polymer molecules are neither rigid nor spherical (permeable chains). Instead the spatial form of the polymer molecule in solution is regarded as a random coil. Vol ( const.). (.). const R V 5. 3 3 3. RG 0 3 <R G > = <R G > 0. R G 3 0 N A G. 3. R G 3 0 (universal constant) G 3.6710 4 mol 1 R 3/ 3/ N 0 3 0 N 1/ is Flory s constant, which ranges from.8710 3 (for poor solvents) and.110 3 ml.mol -1.cm -3 (for good solvents), when instrinsic viscosity is in ml.g -1 and <R N > in cm. For a linear polymer of a given structural unit, the quantity <R N > 0 / is independent of. So: R where K = Φ(< R N >/) 3/ solvent quality, but decreases with the increase of the temperature: when T increases, more freedom to rotations, K is independent of the polymer molecular weight and of the less hindrances to free rotation, larger < R N >.
OLECULAR SIZE from INTRINSIC VISCOSITY ark-houwink-sakurada: Viscosity depends on molecular weight, solvent quality and polymer concentration.
ark-houwink equation: K. a Good solvent Solvent Flexible chain Good solvent R 3 5 1 10 K. or Vol R 3 Solvent R 1 Rigid chain Rod Vol R ( bastonete ) R 4 5 0. 8 K. 1 0. 5 1 K. K. or K.
from L.H. Sperling (pag. 117)
OLECULAR SIZE from INTRINSIC VISCOSITY One has to extrapolate to zero concentration to determine the intrinsic effect of the addition of the polymer to the increase in viscosity of the solution. from IT OCW
OLECULAR SIZE from INTRINSIC VISCOSITY Above the limit of infinite dilutions, there are already interactions between polymer molecules: The relative viscosity varies with concentration as a power series C pol = c HUGGINS EQUATION k = 0.7 ( solvents) k = 0.-0.4 (good solvents) KRAEER EQUATION 1 st approximation: diluted solutions with sp /c <<1% nd approximation: diluted solutions with sp <1%
embrane Osmometry ( n, A, ) The semipermeable membrane, represented by the dashed line, allows the passage of solvent but not the solute. Since it cannot pass through the semipermeable membrane, it must remain on right-hand side. The chemical potential of the solvent on the right-hand side (solution) is now less than that of the solvent on the left-hand side (pure solvent). If the external pressure on the right-hand side is maintained at PA, the liquid level on the righthand side will rise as the solvent passes from the left (higher chemical potential) to the right (lower chemical potential) to equalize the chemical potential on both sides. However, this flow of solvent can be prevented if the external pressure on the solution is increased so as to keep the liquid levels the same on both sides. The additional pressure is the osmotic pressure, π, of the solution. It arises as the driving force for solvent flow in response to the reduction of the chemical potential of the solvent due to the addition of a solute.
embrane Osmometry (cont.) from IT OCW
Osmometry
Osmometry Note: A >0 good solvent A =0 theta solvent A <0 poor solvent >1/ poor solvent =1/ theta solvent <1/ good solvent from IT OCW
Gel Permeation Chromatography (GPC) GPC is a type of size-exclusion chromatography (SEC) and it employs porous non-ionic gel beads (made of glass or cross-linked PS, with pores of various sizes and distributions), packed into a column, to separate polymers in solution. A solvent is pumped through the column and then a polymer solution in the same solvent is injected into the column. Fractionation of the polymer samples results from elution of different-sized molecules at different times. Fractionation of molecules is governed by hydrodynamic volume rather than by molecular weight. Viscometer, IR, UV, laser diffraction Solvent: should be a good solvent for the polymer, have a different refractive index from the polymer and to not have light absorption in the range for which the polymer is absorptive Fraction collector The largest polymer chains in solution cannot penetrate the pores within the cross-linked gel beads, so they will elute first as they are excluded and their retention volume is smaller. The smallest polymer chains in solution require more time to elute, because their retention volume is larger.
Gel Permeation Chromatography (GPC) The larger molecules flow straight through, while the smaller ones are temporarily held up.
Gel Permeation Chromatography (GPC) where Kse is the size exclusion equilibrium constant A s is the pore S/V ratio, and depends on the pore/column and <r > 1/ depends on solvent, T and the W of that polymer chain entering the pore.
Gel Permeation Chromatography (GPC) GPC is not an absolute method so it is necessary to calibrate the W vs. elution volume curve using known narrow fraction samples of the same polymer in the same solvent at the same temperature. Normally, researchers employ PS in THF at 3 C for PS-based calibration. Thus samples are referred to as e.g. 60000 g/mol on a PS-basis, meaning that the particular sample exited the column at an elution volume corresponding to a 60000 g/mol PS sample going through the same column using THF at 3 C (a good solvent) as the carrier medium. polymers eluted at identical elution volumes: (V h ) 1 = (V h ) Note: 5. V V.. 5 η 1 1 = η η 1 = K 1 1 a 1 η = K a log = 1 1+a log K 1 K + 1+a 1 1+a log 1
Questions A mixture of three polymer samples of similar molecular weights and chemical structures, but with different physical properties, are injected together into a sizeexclusion chromatograph. One polymer is linear, another is branched and the third has a star-shaped structure. In what order will the polymers be eluted? Chain branching will affect the V h. Linear polymer: smaller, permeates into the pores of the beads more readily than the larger branched and star molecules Star-shaped polymer is a larger molecule which will not permeate the pores and thus will be excluded. Order of elution: (1st) Star shaped (nd) branched (3rd) linear.
Resolution of SAQ16 w n = 1/ σ w i i = 9570