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.
To describe this force, let us consider two sheets of fluid, of area A A h dv dx F F = Aηdv dx η = viscosity
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.
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.
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).
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
Measuring viscosity Ostwald viscometer Wilhelm Ostwald (1853-1932) Nobel Laureate 1909 (for his work in catalysis, chemical balance, and Reaction rates)
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.
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)
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!
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.
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
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?
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!!!!!!!!!!
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
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) 2 + 14 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
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
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
Not very sensitive to shape Ref: Cantor and Schimmel, p. 652
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
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 14 211 g. mol -1 14 100 14 500 12 400
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
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
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