Viscometry - University of British ColumbiaMeasuring viscosity Ostwald viscometer Wilhelm Ostwald...
Transcript of Viscometry - University of British ColumbiaMeasuring viscosity Ostwald viscometer Wilhelm Ostwald...
CHEM 305
Viscometry
When a macromolecule moves in solution (e.g. of water), it induces netmotions of the individual solvent molecules, i.e. the solvent moleculeswill feel a force.
- neglect Brownian motion.
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To describe this force, let us consider two sheets of fluid, of area A
F
A
dvdx
h
F = Aηdvdx
η = viscosity
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To measure viscosity, the capillary effect is used. The force for the movementof solvent depends on the hydrostatic pressure, i.e.
Fup = Pπa2
For a small cylindrical sheet, at a radialdistance x, the differential force will be
dFup = 2Pπx dx
If the fluid is flowing through the capillaryat a steady state, this force must bebalanced by a frictional force, i.e.
Fdown = -A η
= - 2πxlη
where the negative sign indicates that it is in the direction opposite to the applied force.
ax
dvdx
dx
l
dvdx
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The net force on the sheet due to fluid motion is the differential force felt by thetwo sides of the sheet, i.e.
dFdown = - 2πlη d[x(dv/dx)] dxdx
These two differential forces (up and down) are equal, therefore
2Pπx dx = - 2πlη d[x(dv/dx)] dxdx
Px = - lη d[x(dv/dx)] dx
Integrating this equation once gives
½ Px2 + c1 = -ηlx (dv/dx)
and again ¼ Px2 + c1 ln x + c2 = -ηlv
where c1 and c2 are integration constants.
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The integration constants can be obtained by looking at the boundary conditions.
1) At x=0
¼ Px2 + c1 ln x + c2 = -ηlv
Therefore, c1 = 0.
2) At x=a
¼ Pa2 + c2 = -ηlv
Therefore, c2 = - ¼ Pa2.
Thus we can write,
v = P (a2 – x2) (flow velocity).4ηl
-∞ cannot be infinite!!!
= 0
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Unfortunately, flow velocity is not easily measured – better to use the volume rateof flow, which is defined as
dV = ∫ 2πxv dxdt
= πP ∫ (a2-x2) x dx2ηl
dV = πPa4 Poiseuille’s lawdt 8ηl
0
a
0
a
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Measuring viscosity
Ostwald viscometer
Wilhelm Ostwald(1853-1932)
Nobel Laureate 1909(for his work in catalysis,chemical balance, and
Reaction rates)
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h1
h2
l
a
A fluid of density ρ is allowed to fall from heighth1 to h2, in a determined time t. The hydrostaticpressure felt by the solution is given by ρgh.Using the equation for the volume rate of flow,
we can determine the time required for the totalvolume V to flow by integrating. The result is
t = 8ηl ∫ dV/h.πgρa4
The integral is a constant for a given apparatus,which is determined by measuring the time ittakes for a solution of known density to fall fromh1 to h2. Typically one uses the pure solvent in which the macromolecule will be studiedsubsequently.
h
dV = πP ∫ (a2-x2) x dxdt 2ηl 0
a
h1
h2
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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 coildistribution of flexible molecules, which in turn means that the viscositycan be altered.
The average shear stress in a capillary viscometer can be determined byusing the equation:
where we know that c1 = 0.
½ Px2 + c1 = -ηlx (dv/dx)
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This allows us to write
Sx = η (dv/dx) = - Px2l
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.
<S> = ∫ 2πxl dx Sx∫ 2πxl dx
= 2πl (-P) ∫ x2 dx = -P a3
2l (2l)(3)2πl ∫ x dx a2/2
<S> = - Pa3l
0
a
0
a
shear stress depends onthe height of the capillary
Assumption: that the pressure remains constant during capillary viscosity measurement – not the case!
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To minimize shear stress, we can use a different type of viscometer, namely
ω
The relative viscosities of anytwo solutions is given by
η2 = ω2η1 ω1
The shear can be altered by changing thestrength of the applied magnetic field. Theshear stress is 104 less that in an Ostwaldviscometer.
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Effect of solute on viscosity
The equations presented up to this stage all relate to the solvent. Ifwe now include a solute, we have the complicated task of computinghow a particle distorts the flow lines of a solution containing avelocity gradient. We start by calculating the energy per unit time neededto maintain the shear in the parallel plate system
F
A
dvdx
h
F = Aηdvdx
Energy = F vb = Ahη dv 2
t dx
This allows us to definethe viscosity of the rateof energy dissipation perunit volume (Ah) at unitshear (dv/dx = 1),
dE α ηdt
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Einstein showed that the rate of energy dissipation in a dilute macromolecularsolution is defined by
dE = dE (1 + νφ)dt dt
where φ is the fraction of the solution volume occupied by macromolecules andν is a numerical factor related to the shape (like the Perrin factor, but not thesame value – ν = 2.5 for a sphere). Given that dE/dt is proportional to theviscosity, we can write
ηr = ηsolution = 1 + νφη0
where ηr is the relative viscosity and η0 refers to the pure solvent. We can nowdefine a specific viscosity as:
ηsp = ηr – 1 = νφ
What does ηsp mean physically?
solution solvent
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We can further define an intrinsic viscosity as
[η] = lim ηsp = lim νφc2 0 c2 c2 0 c2
Let us now rewrite φ in terms of a hydrated volume of the solute, Vh. Recall φis the volume fraction occupied by the solute molecules, i.e.
φ = Vh NA c2M2
Therefore, for a spherical solute,
ηsp = 2.5 (Vh NA c2) and [η] = 2.5 (Vh NA) in cm3.g-1M2 M2
or putting in the definition for the hydrated volume
[η] = 2.5 (V2 + δ1V1)
independent of molecularweight!!!!!!!!!!
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Ribonuclease A
Lysozyme
Bovine serum albumin
Hemoglobin
Bushy stunt virus
-all near spherical macromolecules- range of possible values for [η]: for DNA [η]=5000;
for tropomyosin [η]=52
ab
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Effect of solute shape on viscosity
As alluded to on the previous slide, the shape of the macromolecule has alarge effect on the measured viscosity (DNA and tropomyosin – which arerod-like particles – vs. spherical particles). If the shape can be modelled asa rigid ellipsoid, then the intrinsic viscosity is defined as:
The factor ν is a Simha factor and is defined as:
ν = (a/b)2 + (a/b)2 + 14 for prolate ellipsoids5[ln(a/b) – 0.5] 15[ln(2a/b) – 1.5] 15
ν = 16 a for oblate ellipsoids15 b tan-1(a/b)
[η] = ν (Vh NA) = ν (V2 + δ1V1) M2
ab
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Using viscosity to estimate molecular weight
By combining viscosity with sedimentation or diffusion measurements, itshould 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 3Vh F4π
And combining it with the sedimentation coefficient:
6π 3 F = 4π
⅓
s = M2 [1 - ρV2]NA f
⅓ M2 [1 - ρV2]Vh
1/3ηNA s
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[η] = ν (Vh NA) M2
Using the definition for intrinsic viscosity:
We can divide the equation above (after taking the cube root) by the equationon the previous slide to yield:
NA1/3 ν1/3 = [η] M2
Vh6π 3 F
4π
NA1/3 ν1/3 = [η]1/3ηNAs
(162π2)1/3F M22/3(1 – V2ρ)
⅓
⅓
M2 [1 - ρV2]Vh
1/3ηNA s
β’
Scheraga-Mandelkernequation
shape factor
CHEM 305 Ref: Cantor and Schimmel, p. 652
Not very sensitiveto shape
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The parameter β’ is not very sensitive to shape but since M2 α (β’)-3/2, themolecular weight derived from the Scheraga-Mandelkern equation should beaccurate to within 10%.
For most practical applications, the intrinsic viscosity is used to determinethe molecular weight by solving the equation:
[η] = k Ma
where k and a are constants specific to the system.
E.g. DNA (rod-like macromolecules)
[η] α M1.8
whereas for coils
[η] α M0.5 to M1.0
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Summary: Transport processes
- Diffusion
- Electrophoresis
-Sedimentation: - sedimentation velocity- equilibrium ultracentrifugation
- Viscosities
All of these methods can be used to yield information on the molecular weightof a biomolecule.
E.g. Lysozyme Method
Chemical structureSedimentation and diffusionSedimentation equilibriumViscosity (Scheraga-Mandelkern)
Molecular weight
14 211 g. mol -114 10014 50012 400
Sedimentation and diffusion
s = M [1 - ρV2] D = kT = RTNA f f NAf
M = sRTD [1 - ρV2]
Svedberg equation
f /fmin = [ ]1/3 F(V2 + δ1V1)V2
Sedimentation equilibrium
c2(x) = c2(x0) exp {[M2(1-V2ρ) ω2/2RT] (x2-x02)}
lnc 2
x
slope α M2
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[η] = ν (Vh NA) M2
Viscosity
NA1/3 ν1/3 = [η]1/3ηNAs
(162π2)1/3F M22/3(1 – V2ρ)
β’
Scheraga-Mandelkernequation
shape factor