Polymer Physics2.1. Molecular Mass and Molecular Mass Distribution
2.1.1. Polymer size and shape
2.1.2. Molecular weight average
2.2. Determination of Molecular Weight Averages and Sizes
2.2.1. Number-average molecular weight
2.2.3. Intrinsic viscosity and gel permeation chromatography
2.2.4. Solution thermodynamics and molecular weights
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Characteristics
1) High molecular weight, ranging from 25,000~106g/mol or higher
2) Distributed molecular weight
in the bulk : determined by small-angle neutron scattering(SANS)
- function of temperature, solvent, structure, crystallization, extension,
and the presence of other polymer
Determination depends on dissolving the polymer in an appropriate solvent
and measuring the required properties in dilute solvent
⇒ Solution property must be understood. → Chapter 3.
2.1. Molecular Mass and Molecular Mass Distribution
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Mi = molecular weight of molecules i
ni = number fraction of molecules i
wi = weight fraction of molecules i
N = total number of molecules
W = total weight of molecules
Number-average
MW distribution.
- absolute Mn : by osmometry, end-group analysis, colligative methods
- absolute MW : by light scattering
2) Relative Method
2.1. Molecular Mass and Molecular Mass Distribution
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~30,000 to 1,000,000g/mol
High Molecular Weight
(for increasing processibility)
down to degreaded
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- Monodisperse samples Mw/Mn = 1
- Condensation polymerization Mw/Mn = 2
- Anionic polymerization Mw/Mn < 1.05
e.g. Proteins : truly monodisperse
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: Assumes that all the chains are initiated simultaneously.
: Growth continues at approximately the same rate in each chain, until the
monomer runs out.
2
+A +A +A +A +A
)1exp( )!1(
)1( 1
N : degree of polymerization
Weight fraction distribution function (wN)
PDI for the Poisson distribution
; PDI is quite narrow Appendix 2.1-1
where Nw : weight average degree of polymerization
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The Poisson Distribution
: Many addition polymerization rxn (ex. anionic polymerization) have propagation
rates much faster than initiation rates and have essentially no termination.
: Such rxn produce narrow MW distributions that can be approximated by
“the Poisson distribution”.
The curve is the Poisson distribution
prediction
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The Schultz Distribution
: Many addition polymerizations that involve free radicals at chain ends have
termination rxn when two growing chain ends meet and when a growing chain
end meets an impurity.







Where Γ(a+1) = gamma function ()
Note that for a=1 the most-probable distribution
is recovered.
Appendix 2.1-2
Appendix 2.1-3
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For stepwise polymerization the kinetics are considered in terms of the extent of rxn, P, defined as the fraction of the functional groups reacted at time, t
If only bifunctional reactants are present (AB)
2.1. Molecular Mass and Molecular Mass Distribution
The probability of N-mer 2)1(),( 1 ppNpn N
The number fraction of N-mer
p
PDI approaches 2
unreacted (A-B) : 1-p
Functionality = 2 Linear chain, in general
> 2 Branched or cross-linked
Gelation : when a single molecules, connected by ordinary covalent bonds,
extends throughout the polymerization vessel or the point where
a three-dimensional network is formed.
⇒the viscosity of the reacting mass → ∞ at the gelation point
The critical extent of rxn. Pc at the gel point
2
1
2
1
)1(
1
)2(
1
r = vA/vB : stoichiometric imbalance
: Flory and Stockmayer equation
multifunctional molecules
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2.2.1.1. End-Group Analysis
Functional groups such as -OH and -COOH can be titrated or analyzed
instrumentally by IR etc.
2.2.1.2. Colligative Properties
Depend on the No. of molecules in a solution, and not their chemical
constitution
melting point (mp) depression
vapor pressure lowering
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c.f. osmosis
: a process in which certain kinds of molecules in a liquid are preferentially transmitted by a
semi-permeable membrane. The hydrostatic pressure then balancing osmosis is called
the osmotic pressure, symbol π.
For bp elevation and mp depression
where ΔTb, ΔTf : the boiling point elevation and freezing point depression
ρ : solvent density
ΔHυ, ΔHf : the latent heats of vaporization and fusion per gram of solvent
c : the solute concentration in grams per cm3





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For small vapour pressure of the solute,
the solvent follows Rault's vapour pressure law
ρ1 0, ρ1 : the vapor pressure of the pure solvent and the solution respectively
X2 : the mole fraction of the solute
The osmotic pressure π depends on the MW
; van’t Hoff Law See Appendix 2.2-1
20
1
1
0
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2.2.1.3. Osmotic Pressure
1) Thermodynamic Basis
For polymer solutions, the chemical potential of the solvent in the solution ≠ that
of the pure solvent.
⇒ a net flow of solvent from the pure solvent side to the solution side
2.2. Determination of Molecular Weight Averages and Sizes
2) Instrumentation (ref. to Fig)
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membranes : regenerated cellulose
or other microporous
⇒ modernized by adopting automatic
activity of the two sides is equal
: a week → a few hours.
Fig. 3.5 Design of the Stabin osmometer. The cell holds 10cm3 of solution. The
plunger is used to preset the pressure near that expected, to reduce the time to
equilibrium. The difference in heights of the solution and reference capillaraies is
read via a cathetometer. The temperature must be maintained constant, ±0.01;
otherwise the solution capillary meniscus may rise and fall like in a thermometer
tube
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---(3.41)
n c M
At finite concentrations, Interactions between one polymer
molecule and the solvent result in the second virial coefficient A2.
Multiple polymer-solvent interactions produce higher virial coeff. A3, A4.
A3 = gMnA2 2 : approximate relation
→ 0.25 for polymer solution : depends on p-s interactions.
If
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Fig. 3.6. useful plot for determining higher molecular weights or working in more concentrated solution.
A & T, solvent for a given polymer
Flory θ-temperature : the temp. at which A2=0
- π/c is independent of concentration - only one concentration need be studied to determine Mn.
- an infinite molecular weight polymer just precipitates.
2.2. Determination of Molecular Weight Averages and Sizes
3) Experimental Treatment of Data
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π (cm solvent) × g/cm3 (solvent density) = π (g/cm2)
R = 8.48×104 T in K (kelvin)
∴For a polymer c = 2×10-3g/cm3
π = 0.3cm in a solvent of 1.0g/cm3
density at 30
⇒ Light Scattering is more sensitive at higher MW.
nM
RT
c
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Light scattering
useful methods of determining the MW
- If the size of the object > the wavelength of the radiation
⇒ Radiation is to be reflected.
- If the size of the object the wavelength of the radiation
(down to atomic dimensions)
2.2. Determination of Molecular Weight Averages and Sizes
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Fig.3.6 The effect of an electromagnetic wave on a free electron. The forced
oscillations of the electron involve accelerations, which cause the
electromagnetic energy to be re-radiated.
e- absorbs the ε and begins to oscillate
e-*
- This re-radiation of ε : scattering
2.2. Determination of Molecular Weight Averages and Sizes
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1) For regular arrays of organized atoms, molecules, or particles
⇒ the radiation will be diffracted
→ scattering can be observed at special angles
( at all other angles these is total destructive interference between the scattered radiation arising from different parts of the array )
e.g. X-ray diffraction
⇒ Scattering will be observed at all angler
→ angular variation of the scattering intensity provides a measure of the size of the structures
Light Scattering : caused by fluctuations in the refractive index of the medium on the molecular or supermolecular scale.
e.g. blue of large bodies of water - due to slight fluctuations in the spacing of the water molecules.
2.2. Determination of Molecular Weight Averages and Sizes
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Theory of light-scattering of polymer solutions by Debye and Zimm
→ replaced fluctuations in the refractive index of the solvent itself
by the changes caused by the polymer molecules.
---(3.42)
TcRTR
Hc
R(θ) : Rayleigh's ratio
Iθ : light intensity observed at angle θ scattered from a volume Vs
w : distance from the source
I0 : the intensity of the incident light
2.2.2.1. Scattering Theory and Formulations
Where π : osmotic pressure
H : Optical constant for a particular polymer and solvent, determined
theoretically.
NA : Avogadro's No.
asused in the light scattering :
n, n0 : refractive indices of the solution and the solvent
π1 : 3.1416
For three modes of scattering i.e.
Light Scattering : theory formulated by Debye
X-ray Scattering : theory formulated by Guinier and Fournet
Small angle Neutron Scattering : by Kirste, Ballard, and Ibel.
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---(3.44) cA PMsolventRR
R(θ) → 3τ/16π where τ is the turbidity in Beer's Law
P(θ) : scattering form factor
= 1 ; if R(θ)=3τ/16π
P(θ) ≠ 1
In the region of very small angles.
i.e. K2Rg 2 < 1 the Guinier region
P(θ) : a measure of the radius of gyration, Rg
2.2. Determination of Molecular Weight Averages and Sizes
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(3.47)
(3.48)
θ : the angle of scatter
K (or q or Q) : the wave vector or the range of momentum transfer
(for SANS)
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Random Coil
: is a polymer conformation where the monomer subunits are oriented
randomly while still being bonded to adjacent units.
A one-dimensional random
showing all possible
The number of trajectories
position x.
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Random Coil
defined structures.
structure, are often assumed to exhibit a random coil conformation
Illustration of a 3-dimensional
(3.50)
(3.51)
= λ0 / n0















2.2. Determination of Molecular Weight Averages and Sizes
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Mw, Z-average radius of gyration, and A2 can be determined.
2.2. Determination of Molecular Weight Averages and Sizes
Fig.3.10 Illustration of a light-scattering calculation
From a plot (Hc/R(θ)) vs. sin2(θ/2)
(3.52) )intercept(16
gR
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H (=optical constant for a particular pair of polymer-solvent) depends on the kind of radiation)
∴ For X-ray scattering
where Mp : the monomeric unit molecular weight
ap, as : coherent neutron scattering lengths of the polymer mer units and solvent
ie : the Thomson scattering factor for a single e-
ρe : the electron density of the solution
p, s : polymer, solvent
From the Zimm plot
K2Rg 2 < 1 (3.55)
⇒ there must be only partial destructive interference between two
waves striking the same particle.
i.e. the waves should not be out of phase by more than 180°.
2.2. Determination of Molecular Weight Averages and Sizes
2.2.2.2. The Appropriate Angular Range
After scattering : the waves become out of phase
if 2θ=180°, one wave lags behind the other by 1/2λ
∴ radiation intensity min.
difference is attained.
θ 45∼135° for light scattering (λ=5000)
θ ≤ 1 for X-ray scattering (λ1~2)
for thermal neutrons (λ5)
c.f. Radius of Gyration (Rg)
Rg 2 : the mean square distance away from the center of gravity
For N scattering points of distance ri
the radius of a thin ring that has the same mass and same



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L : length of Rod
ks = K
r = r.m.s. of the distance between ends of the random coil
6/22 rRg
2 )cos(sin
c.f. Radius of Gyration (Rg)
See Appendix 2.2-5
scattering calculation
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Fig.3.14 A Zimm plot for cellulose tricaproate in imethylformamide.
Note that A2 is zero, indicating that at the temperature of measurement, 41, this is a Flory θ-system.
λ=5460; refractive index of dimethyl formamide is 1.43
2.2. Determination of Molecular Weight Averages and Sizes
2.2.2.3. The Zimm Plot
r2 = c M (3.60)
→ The relation equating the total distance traveled for a particle
undergoing Brownian motion as a function of time
Because of , , and , C increases
Brownian motion of Particle Polymer chain
Traveled Distance variable mer bond length fixed
Turning Angle any angle C-C bond angle fixed
Path may cross itself may not
2.2. Determination of Molecular Weight Averages and Sizes
2.2.2.4. Polymer Chain Dimensions
MW
For proper comparison MWD (or Mz/Mw) must be known
In general, the Z-average Rg values are corrected back to the Mw and the
weight averages of both quantities are reported.
: a measure of stiffness
melt and solution viscosities depend directly on
the radius of gyration of the polymer and
the chain's capability of being deformed.
2
2.2.2.5. Scattering Data
(Hint : Hydrodynamic volume)
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; should be carried out in dilute solution result in the viscosity - average MW.
Fig.3.12 The effect of shear rates on polymer chain rotation.
Hydrodynamic work is converted into heat, resulting in an
increased solution viscosity.
of the capillary.
of finite size "sees" a different shear rate in
different parts of its rotational forces on the
molecule, yielding the mechanism of
"viscosity" increase by the polymer in the
solution.
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Let η0 : solvent viscosity in poise, Stokes or Pa·s
η : polymer viscosity in poise, Stokes or Pa·s
00 t
t rel
Relative Viscosity
Intrinsic Viscosity







In shear flow, a coiled polymer exhibits
a frictional coefficient to f0.
Then Stoke's Law
2.2.3.2. The Equivalent Sphere Model
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(3.91)
where n2/ V : the No. of molecules per unit volume
Ve :
= the viscosity of an assembly of spheres is independent of
the size of the spheres, depending only on their volume
fraction
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c : concentration
N : Avogadro's Number
, and Re = Reoα
α : the expansion of the coil in a good solvent over that of a Flory θ-solvent
Re0 : roughly constant
: expresses the distance traveled by the chain in a random walk as a
function of MW.
Intrinsic
(3.97)
K, a : constants for a particular polymer-solvent pair at a particular temperature
N.B. Since Mv is difficult to obtain directly, the Mw of sharp fractions or
narrow MWD is usually substituted to determine K and a.
a = 0.5 For a Flory θ-solvent ←α=α°
= 0.8 For a thermodynamically good solvent ← α=α0.1
a = 0 for hard spheres
= 1 for semicoils
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If [η] is determined in both a Flory-θ solvent and a "good solvent"
= 1 in Flory θ-solvents
2
3
2
0
: the mean square end-go-end distance of the unperturbed coil. 2
0r
32
1
32
12
3
2
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- ηrel should be about 1.6 for the highest concentration used
Fig.3.14 Schematic of a plot of ηsp/c and
lnηrel/c versus c, and extrapolation
to zero concentration to determine [η].
2.2. Determination of Molecular Weight Averages and Sizes
1) Both lines must extrapolate to the same
intercept at zero concentration
2) The sum of the slopes of the two curves is
related through the Huggins equation
and Kraemer equation
molecular aggregation
other problems
(ref. to Fig 2.15 p.116)
As A2→0 Rg & [η] ↓
plot of log[η] vs. log Mw can give Kaud a values.
(Mw determined with Light Scattering)
2.2. Determination of Molecular Weight Averages and Sizes
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= utilize size exclusion principle
size of the molecule (defined by its hydrodynamic radius) can or
cannot enter small pores in a bed ofcross-linked polymer.
1) The Experimental Method
Stationary phase : small, porous particle (cross-linked PS)
Retention time : The length of time that a particular fraction remains
in the columns (30~60cm in length)
(ref. to Fig 3.18 p119)
2.2. Determination of Molecular Weight Averages and Sizes
2.2.3.5. Gel Permeation Chromatography
packed bed.
are temporarily held up.
2.2.3.5. Gel Permeation Chromatography
MW : determined by the retention time of the particular fraction.
N.B. the motion in and out of the gel particles by the polymer
: strictly governed by the size of the chains
Brownian motion
PMMA
PEG
PEO
PE
2.2.3.5. Gel Permeation Chromatography
efficiency of columns.
(a) Poor resolution.
2.2. Determination of Molecular Weight Averages and Sizes
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→ proportional to the polymer's hydrodynamic volume.
Since GPC depends on the hydrodynamic volume only rather than its molecular weight itself
"universal calibration", which calls for a plot of [η]M vs. elution volume can be suggested.
The universal calibration is valid for a range of topologies and chemical compositions.
However, it can't be used for highly branched materials or polyelectrolytes, which have different or varying hydrodynamic volume relationships.
32
3
2
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Fig. 3.23. The universal calibration curves for polystyrene and poly(vinyl acetate).
The number 5 in the x-axis means that the scale is in siphon "counts" of
5cm3, so that the x-ordinate 30 corresponds to an elution volume of
150cm3.(R. Dietz, private communication, November 1984.)
is the “peak” GPC molecular weight, usually the unknown.
values are close to the geometric mean of and . ˆ rM ˆ
rM nM wM
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1) the temperature where A2 = 0 for dilute solution and .
2) the temperature where Rg in the solution Rg of the bulk polymer.
3) the temperature where an infinite MW fraction would just precipitate.
2
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