Ch 24pages 643-650

Lecture 8 – Viscosity of Macromolecular Solutions

We have expanded the concept of diffusion coefficient from its simple relationship to size and viscosity provided by Stoke’s law

Summary of lecture 7

f R 6

fTkD b /

Summary of lecture 7

Hydration affects diffusion so that the frictional coefficient f that we measure is related to the frictional coefficient of an unhydrated molecule by the relationship:

2

112

3

0 V

VV

f

f

The denominator is the volume of an un-hydrated molecule, and the term above is the total hydrodynamic volume, including hydration. If a molecule is approximately spherical:

3

0

3

0

r

r

f

f

The diffusional properties of non-spherical objects can be calculated (for simple shapes analytically) and used to provide frictional coefficients

For a rod-like particle of length 2a and radius b, for example

Summary of lecture 7

30.0)2ln(

3/2 3/23/1

0

P

Pff

For a prolate ellipsoid where the lengths a and b are called the major and minor semi-axis lengths

1ln

12

23/1

0

PP

PPff

Viscosity

We have introduced the concept of friction by expressing the counterforce acting on a particle moving in a viscous medium, F=fu. Although Newtons’ law is exercised on each particle (e.g. electrophoresys or sedimentation), and therefore one would expect the particle to accelerate under the influence of a certain force F (an electric or centrifugal field, for example), the viscous drag defines a certain velocity as the steady state speed at which the particles move under the influence of an acting external force and of the viscosity of the medium. Friction increases with speed, so that the speed of the particle will only increase up to a point, until it will reach a steady state value u

Consider a molecule in solution. If an external force F is applied to the particle, the particle obviously accelerates according to F=ma

The particle will not accelerate for long. After a brief period, the velocity becomes constant as a result of resistance from the surrounding fluid. This velocity is called the steady state velocity vT and fulfills the condition:

Friction

fv FT

fv is the frictional force exerted by the surrounding fluid on the particle and f is the frictional coefficient of the particle

The fictional coefficient depends on the size and shape of the particle but not on its mass. For a spherical particle with radius R

Friction

fv FT

f R 6

Where is the viscosity of the fluid (Stoke’s Law)

Viscosity

Viscosity measures the resistance of fluids to flow

Consider a flowing liquid constrained between two plates (see Figure). The plates are large and the lower plate stationary. The upper plate moves in a plane parallel to the lower plate

Viscosity

As the upper plane moves, an infinitesimally thin layer of liquid sticks to each plate: a layer of liquid adheres to the lower plate and thus does not move, while a layer of liquid translates with the upper plate. Intervening layers move with velocities that vary as a function of the distance from the lower stationary plate

The velocity gradient in the direction perpendicular to the planes may be thought of as a deformation of the liquid x/y (after all:

dtdxu x /

The velocity gradient is:dydu x /

Viscosity

The velocity gradient

dydu x /

can be related to the ‘deformation’ in the fluid) and is called shear.

Viscosity

A way to look at viscosity is by considering that there will be a momentum change in different layers in the liquid; as the velocity increases creating a velocity gradient, there is a net momentum transfer in the opposite direction of the gradient

The change in velocity will induce a change in momentum p perpendicular to the motion of the planes (p=mu). We can express the change in velocity in terms of a flow of momentum perpendicular to the plane, and therefore we can introduce a ’current’ describing the flux of momentum

Viscosity

We can relate this flux to the momentum gradient by an equation analogous to Fick’s diffusion equation:

dy

duJ p

is a phenomenological proportionality constant called viscosity

The viscosity has units of gm cm-1 sec-1=1 poise. For example, the viscosity of pure water at room temperature is approximately 0.01 poise (1 centipoise)

Viscosity

Another way to look at viscosity is by introducing the shear strain, which is defined as:

dy

dx

y

xy

0lim

The shear stress is defined as the force F pushing the upper plate in the x direction divided by the area A of the upper plate:

shear stressF

A

Viscosity

The shear stress is defined as the force F pushing the upper plate in the x direction divided by the area A of the upper plate:

shear stressF

A

For many liquids, called Newtonian liquids, the shear stress and strain are related by a simple proportionality. The constant of proportionality that relates the shear stress to the velocity gradient is called the viscosity

dy

dx

dt

d

dt

dx

dy

d

Viscosity and friction are measures of the rate of energy dissipation in a flowing fluid. The addition of a macromolecule to a solution greatly increases the viscosity and thus increases the rate of energy dissipation

The rate of energy dissipation in a macromolecular solution is greater than the rate of dissipation in pure solvent:

Viscosity of Macromolecular Solutions

Einstein showed that the rate of energy dissipation in a dilute macromolecular solution is defined by the equation:

solventso t

E

t

E

ln

vt

E

t

E

solventso

1ln

represent the fraction of the solution volume occupied by macromolecules and is a numerical factor related to the shape of the macromolecule. For spherical particles =5/2, while for non-spherical molecules >5/2. For ellipsoids of rotation, for example, the asymmetry factor is dependent on the axial ratio a/b:

Viscosity of Macromolecular Solutions

Oblate Ellipsoid of Rotation:

vt

E

t

E

solventso

1ln

15

14

)2/1)/2ln(5

/

2/3)/2ln(15

/ 22

ba

ba

ba

bav

baba

v/tan

/

15

161

Because the viscosity is proportional to the rate of energy dissipation in macromolecular solutions, i.e:

Intrinsic Viscosity

we may write the ratio of the viscosity of a dilute macromolecular solution solution to the viscosity of pure solvent 0 as :

dE

dt

solution

0

1

for more concentrated solutions, higher order terms must be considered, but this approximation is valid at relatively low concentration of solute

Intrinsic Viscosity

solution

0

1

The ratio:

solutionr

0

is called relative viscosity

r

solution V C 1 10

The quantity:

is called specific viscosity

C is the concentration of the solute in gm/mL and V is the partial specific volume of the hydrated macromolecule

Intrinsic Viscosity

The intrinsic viscosity is defined as

It directly reflects the shape of the solute

Experimentally, the intrinsic viscosity is measured by measuring the specific viscosity divided by C, i.e. sp/C, and

extrapolating to C=0:

Lim

CV

C

sp

0

Intrinsic Viscosity

The difficulty with interpreting intrinsic viscosity data is that both the hydration and shape (asymmetry parameter) contribute to it. That is, if two intrinsic viscosities differ, it may be a result of variations in the asymmetry factor and/or the specific volume, which reflects hydration. In the case of viscosity calculations, the specific volume corresponds to the hydrated molecule (e.g. protein or DNA). As stated in the previous lecture, the specific volume of a hydrated molecule can be expressed from its partial specific volume and from the volume associated with hydration molecules:

V V V 2 1 1

Intrinsic Viscosity

The difficulty with interpreting intrinsic viscosity data is that both the hydration and shape (asymmetry parameter) contribute to it. That is, if two intrinsic viscosities differ, it may be a result of variations in the asymmetry factor and/or the specific volume, which reflects hydration. In the case of viscosity calculations, the specific volume corresponds to the hydrated molecule (e.g. protein or DNA). As stated in the previous lecture, the specific volume of a hydrated molecule can be expressed from its partial specific volume and from the volume associated with hydration molecules:

V V V 2 1 1V2 is the partial specific volume of the unhydrated molecule, V1 is the partial specific volume of water (essentially the

inverse of the density of water), 1 is the number of grams of

water hydrating a gram of protein