2.
MOLECULAR MASS DISTRIBUTION (MMDs)
FOR LINEAR CHAINS
2.1
2.2
2.3
The importance of MMDs
Experimental measurement of MM and MMDs
Mathematical description of MMD
2.3.1 Number distribution
2.3.2 Weight distribution
2.3.3 Distributions in Chemical Engineering
2.3.4 Moments of distribution
2.3.5 Continuous distributions
2.3.6 Analytic expression for MMDs
2.3.7 Useful mathematical tips
2.
Molecular Mass Distributions MMDs for linear chains
This section briefly describes why polymer MMDs are important.
It then describes how MMDs can be measured and finally
develops the mathematics used to describe both discreet and
continuous MMDs.
2.1
The importance of MMDs
Nearly all, but not recent Metallecene catalysed, commercial
polymers have abroad MMD and many physical properties
are sensitive to the molecular mass (or equivalently molecular
length) of chain.
a repeat unit of mm= m 0
r repeat units.
r repeat units each repeat unit with molecular mass mo.
Molecular mass of chain m = m0 r.
mrm1@cheng.cam.ac.uk
1
Chain needs to have r > ~ 100 before you can safely call it a polymer.
You often need to have r > 100 before useful different properties
develop.
“ toughness”
Polymer
Polyol
20
100 repeat uni ts r
What r do you choose? A classic Chemical Engineering
compromise.
Product
Product
quali ty
increases with
increasing r
repeat uni ts r
Process
Ease of
processing
decreases with
increasing r
repeat uni ts r
So you usually end up with a compromise.
mrm1@cheng.cam.ac.uk
2
M ~ 103 - 105 fast processing. Fibres, injection moulding.
M ~ 104 - 106 slow processing. Extrusion.
The fact that you have a MMD means that you can often tailor
a particular MMD for a particular process and product function.
A major manufacturer of bulk polymers such as BP Amoco might
have ~ 100 different grades of polyethylene, each one having a
different MMD.
2 . 2 The experimental measurement of molecular mass and
molecular mass distribution
There are a number of absolute methods of determining MMs
(see Flory, Principles of Polymer Chemistry, if you are really
interested). These methods include:a)
Osmotic Vapour Pressure Depression
b)
Light Scattering
c)
Intrinsic Viscosity
d)
Electrophoresis
The most common method used by the major commodity
chemical manufacturers is Gel Permeation Chromatography
(GPC).
The principle of operation of GPC
inject polymer soln at t=0
reference
gel column
Di fferential detector.
IR,UV,Opti cal
differential
detector
signal
The Gel
Gel column
Short molecules
pass slowly
through gel due
to Brownian
motion
Mass
fraction
with
certain m
long molecules pass
quickly through gel
polymer mmd
calibration
Elution volume
mrm1@cheng.cam.ac.uk
molecular mass m
3
2.3
Mathematical description
Molecular mass of r mer m= mo r, where r = number of
repeat units = degree of polymerisation of chain. Initially
let us consider a discreet contribution of chain lengths.
Let Nm = number of chains with a molecular mass of m
(or equivalent Let Nr = number of chain with r repeat units).
There are two (essentially) equivalent forms of presenting
data.
2.3.1 Number distribution
Plot Nm as ftn of m (or equivalently Plot Nr as ftn of r)
monodisperse
Nm
addition
Stepwise
Molecular mass m
Define number fraction
xm =
Nm
∑ Nm

 xr =

Nr 

∑ Nr 
If distribution is continuous.
Nm = Nos fraction between m and m + dm
Nmdm
xm =
∞
∫ Nmdm
o
strictly this should be mo, but integration from
o easier.
mrm1@cheng.cam.ac.uk
4
We can also define a cumulative number fraction.
m
X=
∑ xm
mo
≈
m
∫0
x m dm
2 . 3 . 2 Weight (mass) distribution
Plot molecular mass, m Nm as a ftn of m
(or equivalently, r Nr as a ftn of m)
monodisperse
M = Nm m
m
addition
Stepwise
Molecular mass m
define weight (mass) fraction
wm =
Nm m
∑ Nm m

 wr =

Nr r 

∑ Nr r
Weight fraction curve will be same form as above.
Note Neither the number fraction or weight fraction curves
are necessarily symmetric about a mean.
Second Note
We can present data in a number of ways.
Number fraction xm as a function of mol mass m
"
"
xr " "
" of degree of polymerisation
Weight fraction wm as a function of mol mass m
"
" wr as a " of degree of polymerisation
mrm1@cheng.cam.ac.uk
5
All are essentially equivalent!
We can define a cumulative weight fraction W
m
W=
∑ wm
≈
mo
m
∫0
w m dm
2 . 3 . 3 Distributions in Chemical Engineering.
A slight digression
Example 1. Exam results
x
Number of
students
σ
Ni
Marks
We usually characterise the distribution by the mean and standard
deviation.
x =
∑ N i xi
∑ Ni
mrm1@cheng.cam.ac.uk
,
σ =
(
∑
(x i
- x)
2
)
1
2
6
Example 2. Residence time distributions
q
E (t)
V
t =q / V
q
q
q
V
V
distribution
not symmetric
E (t)
V
t =q / V
Example 3 Particle size distribution
(PSD)
Number of particles
with size D - D + dD
N (D)
D
Example 4
Polymers
Number
fraction
weight
fraction
Xm
Wm
m
m
Distribution not always symmetric; so define moments µ i .
µo
µ1
µ2
=
=
=
∑ Nm
∑ Nm
m
2
∑ Nm m
mrm1@cheng.cam.ac.uk
DPo =
DP1 =
DP2 =
∑ Nr
∑ Nr
r
2
∑ Nrr
7
each moment has a different dimension
2.3.3 so define normalised moments
Molecular mass averages
Degree of polymerisation
averages
∑ Nm m j
∑ Nm m j-1
∑ Nr r j
∑ Nr r j-1
Mj =
kg kmol
Note simple linking
DP j =
M j = M o DP j
The 1st moment j = 1.
M1 = M n =
Mn
DP n
∑ Nm m
∑ Nm
DP1 = DP n =
∑ Nr r
∑ Nr
= Number average molecular mass
= Nnumber average degree of polymerisation
The 2nd Moment j = 2
N m m2
∑
M2 = Mw =
∑ Nm m
Nr r 2
∑
DP 2 = DP w =
∑ Nr r
Mw
DP w
= Weight average molecular mass
= Weight average degree of polymerisation
The z moment z > 2
Mz =
∑ N m mz
∑ Nm mz-1
So instead of "talking", about the whole distribution we often
"talk about" M n and M w as a two parameter description of the
distribution.
mrm1@cheng.cam.ac.uk
8
M
M
n
M
w
Number
fraction
n
M
w
weight
fraction
Xm
Wm
m
m
For example a commercial Polyethylene
=
=
=
Mn
Mw
M3
150,000
600,000
1,500,000
kg/mol
2 . 3 . 5 Analytic descriptions of MMDs
Strictly MMD is discreet, but often assume continuous.
∞
∫ Nm
Then ex
o
Mn =
m dm
∞
∫ Nmdm
o
∞
∫ wm
Mw =
m dm
see Ex sheet.
o
where wm is wt frac from m → m + dm.
2.3.6 Two "popular" distributions
(a) "Most probable"
w( m ) =
 m
exp

2
 mn 
mn
m
Many Stepwise Polymers model using this.
(b)
"Log normal"
 ( ln ( m / m' ))2 
w( m ) =
exp 
1
β 2
'


2

βπ m
1
mrm1@cheng.cam.ac.uk
9
m’
β
=
=
Mw
=
location of maximum
Breadth
β 2 
'

m exp
 4 


(
Mw
M z+1
=
= exp β 2 2
Mz
Mn
)
Many additional polymerisations model using this.
Stepwise
Most probable is often good fit
M
M
n
M
w
Number
fraction
n
M
w
weight
fraction
Xm
Wm
m
m
Addition
log normal is often good fit
M
M
n
M
w
Number
fraction
n
M
w
weight
fraction
Xm
Wm
m
m
M 3 > Mw > Mv > Mn
Viscosity Ave
Ratio
Mw
=
Mn
mrm1@cheng.cam.ac.uk
Polydispersity index
10
if
Mw
= 1
Mn
Mw
≈ 2
Mn
Mw
~ 5
Mn
Monodisperse
typical for stepwise
typical for addition
2.3.7 Useful notes for Tripos manipulation
Worked example. Express M n in terms of the number fraction
xm .
Mn =
∑Nm m
∑Nm
xm =
Nm
∑ Nm
Definition
Definition
manipulate xm to get xm in form of M n define.
So
Nm m
∑ Nm
xm m =
Take sum of both sides
∑ xm m
=
∑
Nm m
∑ Nm
This is a number and can be taken outside 1st ∑
∑Nm m
∑ xm m =
= M n QED
∑Nm
Mn =
∑ xm m
Now you show,
Mn =
1
w
∑ mm
mrm1@cheng.cam.ac.uk
11
Summary.
Why are MMD important?
Can you define the number and weight average molecular mass without
looking at notes?
Can you express the above averages in terms of weight fractions?
Can you define cumulative number and weight fraction in terms of r?
Why do we use normalised moments?
What other areas of Chem Eng, other than the ones already given,
utilise/ need to be described by moments?
mrm1@cheng.cam.ac.uk
12