Origins of viscosity
Fluids try to achieve uniform
flow across any region (i.e. a
constant speed independent of
position)
Fluids resist a gradient of
speeds
Physical Chemistry
Lecture 3
Viscosity and sedimentation
Described by a drag force that
slows fast molecules and
speeds up slow molecules
Produces a flux of linear
momentum


Proportionality defines the
coefficient of viscosity, 
Measured in poise (10-1 kg m-1
s-1)


Gases

Increase with
temperature
50
100
150
200
250
300
Viscosity of Aniline
Liquid: aniline

Short-range attractive intermolecular forces dominate interactions
Decrease with
temperature
0.16
0.12
0.08
0.04
0
250
Viscosity usually decreases with increasing T
300
350
400
T (K)
Measurement of viscosity
Ostwald viscometry
Need a small-diameter tube
(capillary)
Measure time of flow of a
specific volume through the
capillary
May use Poiseuille’s equation
to calculate viscosity
Measure flow in the
presence of a gradient
of speed
Determine the viscosity
by measuring flow of
the material through a
tube

100
T (K)
Considers only repulsive interactions during collisions
Poiseuille’s formula for
flow through a cylindrical
tube subject to a
pressure drop that forces
material through the
tube
May also find result for
gravity as the force
causing motion
150
50
 Hard to model exactly
 Described empirically

200
0
Liquids

dv
dx
250
Importance of prediction is
 Hard-sphere approximation

Gas: oxygen
 (micropoise)
~
5 Nvave m
32 2
 Increase with square root of temperature
 Increase with square root of mass

 
Viscosity of Oxygen
Momentum transfer between “fast” and “slow” molecules
High-level kinetic theory predicts (slightly different from what is
quoted in your text)
 

dv
dx
J momentum
 (poise)

  A
Comparison of temperaturedependent viscosities
Viscosity

Fviscous
The viscous force is proportional
to the gradient of the speed

V
t
V
t

r
P
8 l
4

gravity

 r gh
8 l
4
Often do not know the
radius well
Height changes somewhat
over the experiment
Generally calibrate with a
known material to give the
viscometer coefficient, A

Viscosity determined this
way gives accurate results
t


A
1
Including excluded-volume
effects: Enskøg’s relation
Falling-ball viscometer
 
vt

R
t
Simplest kinetic theory





g
2 2
r    0  t
R
9



Example: viscosity coefficient
oxygen viscosity
Positive deviation from simple
kinetic theory at high T shows
effect of attractive forces
1
S
1
T

200
150
100
50

0
8
10
12
14
16

Nature of materials in
solution
Amount of each material in
solution
Shape of molecule
The shape modifies the
proportionality factor



Einstein’s relation for
viscosity coefficient gives a
minimum value of 2.5
Simha factor, , accounts
for shape
For ellipsoidal molecule, 
is always larger than for a
sphere
300
200
100
0
0
0.005
0.01
0.015
0.02
Number Density (mole cm -3)

2
 b0 
 b0  
  0.865 V  
 Vm 
 m  
  0 1  0175

Nature of materials in
solution
Amount of each material
in solution
Einstein’s relation for
viscosity coefficient of a
solution of solid spheres
in a continuous medium
Viscosity is a measure of
V , the volume fraction
of the large molecules


   0 1   V 
V
1   0 



   0 
   0 1  2.5V 
  0 

 0.40
 0 
V
Viscosity of polymer solutions
Specific viscosity, sp, from
Einstein’s equation linear in
concentration
General case: solution specific
viscosity not linear in
concentration
Intrinsic viscosity, []
Solution viscosity depends
on

18
Square Root of T
Viscosity of polymer solutions

Gives a density dependence
for the viscosity of a gas
A virial expansion in the
density (i.e. inverse of
molar volume)
Predicts an increase of gas
viscosity with density, as
observed experimentally
Large molecules in
solution with a smallmolecule solvent
Viscosity of Oxygen
250
 (micropoise)
 (T )  0 (T )
400
Solution viscosity
depends on
Incorporates attractive forces
qualitatively
Gives a correction factor

500
Viscosity of polymer solutions
Intermolecular interactions are
also attractive
Approximate interaction
potential by Sutherland

Viscosity of Nitrogen at 50 C
600
Inclusion of excluded-volume
interactions by Enskøg
Including attractive forces:
Sutherland’s equation

Considers molecules as
point particles with only
collisional effects
Predicts no particular
density dependence of gas
viscosity
In agreement with lowdensity measurements
 (micropoise)
Measure the
terminal velocity of
a ball falling in a
fluid
Use Stokes Law for
the viscous drag to
determine viscosity
Defined in the infinite-dilution
limit
Avoids problems such as
“entanglement”
 sp , Einstein

 sp
 k1cm
 

  0
0
 k 2 cm2
 sp
lim c
c m 0

10 r 3
cm
3m
 k3cm3
 
 k1
m
Mark-Houwink equation




Intrinsic viscosity depends on
molar mass of a polymer
Intrinsic viscosity depends on
shape
Molar mass determined from
this equation is called the
viscosity-average molar mass
 is predicted to be 0.500 for
ideal polymer
 
 K M
2
Viscosity of polymer solutions
Intrinsic solution viscosities
Example: polystyrene in benzene
Intercept gives intrinsic viscosity
One may calculate the viscosity-average molecular
weight from [] with the Mark-Houwink equation
Shape
Protein
Globular
Ribonuclease
13.7
Serum albumin
67.5
Rod
Molar mass
(kg mol-1)
Random Coil
0.06
 sp /cm
0.05
3.410-3
3.710-3
Fibrinogen
330.
2.710-2
Myosin
493.
2.210-1
39,000.
3.710-2
Tobacco mosaic virus
Polystyrene in Benzene
Intrinsic viscosity
(m3 kg-1)
Poly--benzyl-L-glutamate
340.
7.210-1
Poly--benzyl-L-glutamate
340.
1.810-1
Ribonuclease
13.6
1.610-2
Serum albumin
67.5
5.210-2
0.04
0.03
0.02
0
5
10
15
20
cm (gm cm -3)
Viscosity-determined molecular
dimensions
Mark-Houwink a constants
•Dependence of
intrinsic viscosity on
molar mass
•Coefficient depends
on
•Polymer
Polymer
Solvent
Polystyrene
Benzene, 25C
Polystyrene
a
0.74
Ideal spherical radius
(nm)
Maximum-asymmetry prolate
ellipsoid ratio
2.66
3.9
Cyclohexane, 34C
0.50
Natural rubber
Toluene, 25C
0.67
Ribonuclease
1.93
3.9
Cellulose acetate
Acetone, 25C
0.90
Serum albumin
3.37
4.4
Amylose
0.33 N KCl (aq.), 25C
0.50
Hemoglobin
3.4
4.1
Amylose
Dimethylsulfoxide, 25C
0.64
Poly--benzyl-L-glutamate
Dichloroacetic acid, 25C
0.87
Poly--bezyl-L-glutamate
Dimethylformamide, 25C
1.75
Taken from C. Tanford, Physical Chemistry of Macromolecules,
John Wiley, New York: 1961.
•Solvent
Protein
-Lactoglobulin
Collagen
40
Fibrinogen
11.2
20
Myosin
25.7
68
9.1
29
Tropomyosin
•Temperature
175
•Recognizes different
structures and shapes
Sedimentation
Particles settle in a
force field based on
mass

DNA data from Tinoco
et al., Physical
Chemistry
Gravitational settling
generally too slow
Use centrifugation
Measure the settling
speed to determine
mass
Centrifugal motion as
the force
Unit of s : svedberg =
10-13 sec
Time (min)
Plot according to
equation
r
ln    2 s t
 r0 
r
ln    2 s t
 r0 
Slope gives s , if the
frequency is known
Frequency in rad/sec
r (cm)
ln r
16
6.2687
1.836
32
6.3507
1.849
48
6.438
1.862
64
6.5174
1.874
80
6.6047
1.888
96
6.6814
1.899
Sedimentation of a DNA Sample
1.92
y = 0.0008x + 1.8231
R2 = 0.9995
1.9
1.88
ln(r)

Sedimentation coefficient
1.86
1.84
1.82
1.8
0
20
40
60
80
100
120
Time (minutes)
3
Molar mass and sedimentation
Sedimentation and
diffusion are related
Summary
Movement of molecules under forces

Equation gives the
ability to find the molar
mass from
measurements of the
diffusion coefficient and
the sedimentation
coefficient of a solution
of macromolecules

Viscous flow
Sedimentation
Useful in characterizing the material

M

RTs
D(1  V2 1 )

Viscosity coefficient is a material property
Sedimentation coefficient, S, depends on
molar mass
4