Viscometry. - neglect Brownian motion. CHEM 305

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1 Viscometry When a macromolecule moves in solution (e.g. of water), it induces net motions of the individual solvent molecules, i.e. the solvent molecules will feel a force. - neglect Brownian motion.

2 To describe this force, let us consider two sheets of fluid, of area A A h dv dx F F = Aηdv dx η = viscosity

3 To measure viscosity, the capillary effect is used. The force for the movement of solvent depends on the hydrostatic pressure, i.e. dx x a F up = Pπa 2 For a small cylindrical sheet, at a radial distance x, the differential force will be l df up = 2Pπx dx If the fluid is flowing through the capillary at a steady state, this force must be balanced by a frictional force, i.e. F down = -A ηdv dx = - 2πxlηdv dx where the negative sign indicates that it is in the direction opposite to the applied force.

4 The net force on the sheet due to fluid motion is the differential force felt by the two sides of the sheet, i.e. df down = - 2πlη d[x(dv/dx)] dx dx These two differential forces (up and down) are equal, therefore Integrating this equation once gives 2Pπx dx = - 2πlη d[x(dv/dx)] dx dx Px = - lη d[x(dv/dx)] dx ½ Px 2 + c 1 = -ηlx (dv/dx) and again ¼ Px 2 + c 1 ln x + c 2 = -ηlv where c 1 and c 2 are integration constants.

5 The integration constants can be obtained by looking at the boundary conditions. 1) At x=0 ¼ Px 2 + c 1 ln x + c 2 = -ηlv - cannot be infinite!!! Therefore, c 1 = 0. 2) At x=a ¼ Pa 2 + c 2 = -ηlv Therefore, c 2 = - ¼ Pa 2. = 0 Thus we can write, v = P (a 2 x 2 ) 4ηl (flow velocity).

6 Unfortunately, flow velocity is not easily measured better to use the volume rate of flow, which is defined as a dv = 2πxv dx 0 dt a = πp (a 2 -x 2 ) x dx 0 2ηl dv = πpa 4 dt 8ηl Poiseuille s law

7 Measuring viscosity Ostwald viscometer Wilhelm Ostwald ( ) Nobel Laureate 1909 (for his work in catalysis, chemical balance, and Reaction rates)

8 A fluid of density ρ is allowed to fall from height h 1 to h 2, in a determined time t. The hydrostatic pressure felt by the solution is given by ρgh. Using the equation for the volume rate of flow, h 1 a dv = πp (a 2 -x 2 ) x dx dt 2ηl 0 h 2 we can determine the time required for the total volume V to flow by integrating. The result is a l h h 2 t = 8ηl dv/h. πgρa 4 h 1 The integral is a constant for a given apparatus, which is determined by measuring the time it takes for a solution of known density to fall from h 1 to h 2. Typically one uses the pure solvent in which the macromolecule will be studied subsequently.

9 Disadvantages 1) Large volume of solution is required. 2) Shearing forces generated by the flow gradient are large. Shear stress S = F/A = η (dv/dx) -can cause distortions in the coil distribution of flexible molecules, which in turn means that the viscosity can be altered. The average shear stress in a capillary viscometer can be determined by using the equation: where we know that c 1 = 0. ½ Px 2 + c 1 = -ηlx (dv/dx)

10 This allows us to write S x = η (dv/dx) = - Px 2l for a cylindrical sheet of fluid with radius x. To obtain the average shear stress, we need to integrate the expression over all sheets, i.e. a <S> = 2πxl dx Sx 0 a 2πxl dx 0 = 2πl (-P) x 2 dx = -P a 3 2l (2l)(3) 2πl x dx a 2 /2 <S> = - Pa 3l shear stress depends on the height of the capillary Assumption: that the pressure remains constant during capillary viscosity measurement not the case!

11 To minimize shear stress, we can use a different type of viscometer, namely The relative viscosities of any two solutions is given by ω η 2 = ω 2 η 1 ω 1 The shear can be altered by changing the strength of the applied magnetic field. The shear stress is 10 4 less that in an Ostwald viscometer.

12 Effect of solute on viscosity The equations presented up to this stage all relate to the solvent. If we now include a solute, we have the complicated task of computing how a particle distorts the flow lines of a solution containing a velocity gradient. We start by calculating the energy per unit time needed to maintain the shear in the parallel plate system A F = Aηdv dx Energy = F v b = Ahη dv 2 t dx h dv dx F This allows us to define the viscosity of the rate of energy dissipation per unit volume (Ah) at unit shear (dv/dx = 1), de α η dt

13 Einstein showed that the rate of energy dissipation in a dilute macromolecular solution is defined by de dt solution = de (1 + νφ) dt solvent where φ is the fraction of the solution volume occupied by macromolecules and ν is a numerical factor related to the shape (like the Perrin factor, but not the same value ν = 2.5 for a sphere). Given that de/dt is proportional to the viscosity, we can write η r = η solution = 1 + νφ η 0 where η r is the relative viscosity and η 0 refers to the pure solvent. We can now define a specific viscosity as: η sp = η r 1 = νφ What does η sp mean physically?

14 We can further define an intrinsic viscosity as [η] = lim η sp = lim νφ c2 0 c 2 c2 0 c 2 Let us now rewrite φ in terms of a hydrated volume of the solute, V h. Recall φ is the volume fraction occupied by the solute molecules, i.e. Therefore, for a spherical solute, φ = V h N A c 2 M 2 η sp = 2.5 (V h N A c 2 ) and [η] = 2.5 (V h N A ) in cm3.g-1 M 2 M 2 or putting in the definition for the hydrated volume [η] = 2.5 (V 2 + δ 1 V 1 ) independent of molecular weight!!!!!!!!!!

15 Bovine serum albumin Hemoglobin Bushy stunt virus Lysozyme Ribonuclease A -all near spherical macromolecules - range of possible values for [η]: for DNA [η]=5000; for tropomyosin [η]=52

16 Effect of solute shape on viscosity As alluded to on the previous slide, the shape of the macromolecule has a large effect on the measured viscosity (DNA and tropomyosin which are rod-like particles vs. spherical particles). If the shape can be modelled as a rigid ellipsoid, then the intrinsic viscosity is defined as: [η] = ν (V h N A ) = ν (V 2 + δ 1 V 1 ) M 2 The factor ν is a Simha factor and is defined as: ν = (a/b) 2 + (a/b) for prolate ellipsoids 5[ln(a/b) 0.5] 15[ln(2a/b) 1.5] 15 b a ν = 16 a 15 b tan -1 (a/b) for oblate ellipsoids b a

17 Using viscosity to estimate molecular weight By combining viscosity with sedimentation or diffusion measurements, it should be possible to obtain a good estimate of molecular weight of a biomolecule, while eliminating most shape effects. Starting from the friction coefficient: f = 6πη 3V h 4π ⅓ F And combining it with the sedimentation coefficient: s = M 2 [1 - ρv 2 ] N A f ⅓ 6π 3 F = 4π M 2 [1 - ρv 2 ] V h 1/3 ηn A s

18 Using the definition for intrinsic viscosity: [η] = ν (V h N A ) M 2 We can divide the equation above (after taking the cube root) by the equation on the previous slide to yield: N A 1/3 ν 1/3 = [η] M 2 ⅓ ⅓ 6π 3 F 4π V h M 2 [1 - ρv 2 ] V h 1/3 ηn A s N 1/3 A ν 1/3 = [η] 1/3 ηn A s (162π 2 ) 1/3 F M 2/3 2 (1 V 2 ρ) Scheraga-Mandelkern equation β shape factor

19 Not very sensitive to shape Ref: Cantor and Schimmel, p. 652

20 The parameter β is not very sensitive to shape but since M 2 α (β ) -3/2, the molecular weight derived from the Scheraga-Mandelkern equation should be accurate to within 10%. For most practical applications, the intrinsic viscosity is used to determine the molecular weight by solving the equation: [η] = k M a where k and a are constants specific to the system. E.g. DNA (rod-like macromolecules) [η] α M 1.8 whereas for coils [η] α M 0.5 to M 1.0

21 Summary: Transport processes - Diffusion - Electrophoresis -Sedimentation: - sedimentation velocity - equilibrium ultracentrifugation - Viscosities All of these methods can be used to yield information on the molecular weight of a biomolecule. E.g. Lysozyme Method Chemical structure Sedimentation and diffusion Sedimentation equilibrium Viscosity (Scheraga-Mandelkern) Molecular weight g. mol

22 Sedimentation and diffusion s = M [1 - ρv 2 ] D = kt = RT N A f f N A f M = srt D [1 - ρv 2 ] Svedberg equation f /f min = [ (V 2 + δ 1 V 1 ) V 2 ]1/3 F

23 Sedimentation equilibrium c 2 (x) = c 2 (x 0 ) exp {[M 2 (1-V 2 ρ) ω 2 /2RT] (x 2 -x 02 )} ln c 2 slope α M 2 x

24 Viscosity [η] = ν (V h N A ) M 2 N 1/3 A ν 1/3 = [η] 1/3 ηn A s (162π 2 ) 1/3 F M 2/3 2 (1 V 2 ρ) Scheraga-Mandelkern equation β shape factor