Transcript Chapter 3
Exam 1
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Exam due Feb 6 th at 11:59 PM.
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Two Homework assignments due Feb 7
th
.
•
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•
D2L problems due next week
Chapter 3 Polymer Solutions How do you know if something is soluble???
Industrial Relevance of Polymer Solubility
Polymer Diblock copolymers Solvent Motor Oil Effect Colloidal suspensions dissolve at high T, raising viscosity Reduces turbulent flow Application Multiviscosity motor oil (10W40) Poly(ethylene oxide) Water Polyurethanes, cellulose esters Poly(vinyl chloride) Polystyrene Polystyrene Esters, alcohols, various Dibutyl phthalate Solvent vehicle evaporates, leaving film for glues Plasticizes polymer Poly(phenylene oxide) Triglyceride oils Mutual solution, toughens polystyrene Phase separates upon oil polymerization Heat exchange systems, lowers pumping costs Varnishes, shellac and adhesives Lower polymer Tg, making “vinyl” Impact resistant objects, appliances Oil-based paints, tough, hard coatings
Polymer Solubility
• When two hydrocarbons such as dodecane and 2,4,6,8,10 pentamethyldodecane are combined, we (not surprisingly) generate a homogeneous solution: • It is therefore interesting that polymeric analogues of these compounds, poly(ethylene) and poly(propylene) do not mix, but when combined produce a dispersion of one material in the other.
n n
•
Polymer solutions: Definitions
Configuration
: arrangement of atoms through bonds – Inter-change requires bond breaking and making (>200 kJ/mol) – Different compositions of polymers – Different isomers or stereochemistries •
Conformation
: within the constraints of a configuration, the possible arrangement(s) of atoms in space – Interchange requires bond rotations, but no bonds are broken or made (can have non-bonding , ie. hydrogen bonding)
What are the dimensions of a polymer?
• In a solution, in the solid state, in its melt, in vacuum?
Polyethylene coil Fully stretched Polyethylene
Factors Affecting Macromolecule Dimensions
Flexible coil
Polymer Conformations
Rigid rod
And everything in between
How to describe conformations with models: The freely jointed chain
• Simplest measure of a chain is the length along the backbone – For
n
monomers each of length
l
, the contour length is
nl l
1 2 3 . . .
n
First, Most Primative Model:
Freely Jointed Chain
Any bond angles & orientations are possible
A useful measure of the size of macromolecules: end-to-end distance
r
• For an isolated polymer in a solvent the end-to end distance will change continuously due to molecular motion – But many conformation give rise to the same value of
r
, and some values of
r
are more likely than others e.g., • Only one conformation with
r = nl
extended chain - a fully • Many conformation have
r =
0, (cyclic polymers) – Define the root mean square end-to-end distance – Permits statistical treatments
r
2 1 2 But we still don’t know the shape; need something more Free rotation model: Infinite number of conformations
Random Walk Model for linear polymers
Example of a “low resolution” polymer coil RW. Contour length = l ct , unit vectors to describe the length = e(l), a = vectors that connect the junction points on the chain, r = end-to-end distance of the chain A freely jointed chain in 2D from a random walk of 50 steps.
# of steps based on number atoms in chain Complications: excluded volume & steric limitations
What does this all mean??
An ideal polymer chain with 10 6 repeat units (not unusual), each unit about 6Å will have: • a rms end-to-end distance R of 600 nm • a contour length of 600 μm
A better measure of Polymer dimensions: Radius of Gyration of a Polymer Coil
The radius of gyration R g is defined as the RMS distance of the collection of atoms from their common centre of gravity.
R For a solid sphere of radius R; R g 2 R 5 0 .
632 R R g For a polymer coil with rms end-to-end distance R ; R g 1 6 R 2 1 / 2 a 6 N 1 / 2
Can be related directly to statistical distributions from polymer characterizations
Restrict the bond angles to what we see in the polymer:
Valence Angle Model
Valence angle model
• • Simplest modification to the freely jointed chain model – Introduce bond angle restrictions – Allow free rotation about bonds –
Neglecting
steric effects (for now) If all bond angles are equal to q ,
r
2
fa
nl
2 (1 q q indicates that the result is for the valence angle model • E.g. for polyethylene q = 109.5
° and cos q ~ -1/3, hence,
r
2
fa
2
nl
2
Finite Number of Conformations due to torsional & steric interactions:
Restricted Rotation Angle
The energy barrier between gauche and trans is about 2.5 kJ/mol RT~8.31*300 J/mol~2.5 kJ/mol
Equivalent Freely Jointed Chain Model
C ∞ C ∞ is a function of the stiffness of the chain. Higher is stiffer
Steric parameter and the characteristic ratio
• In general
r
2 s 2
nl
2 (1 – where s is the
steric parameter
for each polymer experimentally , which is usually determined – A measure of the stiffness of a chain is given by the
characteristic ratio C
r
2 0 /
nl
2 –
C
typically ranges from 5 - 12
Polymer Conformations
Flexible coil
The shape of the polymers can therefore only be usefully described statistically
.
Calculate size of macromolecules!
Excluded volume • Freely jointed chain, valence angle and rotational isomeric states models all ignore – long range intramolecular interactions (e.g. ionic polymers) – polymer-solvent interactions • Such interactions will affect
r
2 – Define a
r
2 a
r
2
r
2 0
Space filling?
The random walk and the steric limitations makes the polymer coils in a polymer melt or in a polymer glass “expanded”.
However, the overlap between molecules ensure space filling
Expansion Factors: Measure of solvent -Polymer Interactions
s r 2 2 : mean-square average distance between chain ends for a linear polymer : square average radius of gyration about the center of gravity for a branched polymer Better solvent
stronger interaction between solvent and polymer
larger hydrodynamic volume
r
2
r o
2 a 2
s
2
s o
2 a 2
r 0 , s 0 : unperturbed dimension
(i.e. the size of the macromolecule exclusive of solvent effects) a
: an expansion factor
r
2 6
s
2
(for a linear polymer)
a (
r
2 ) 1 2 (
r o
2 ) 2 1
greater
a a
= 1
better solvent ideal statistical coil Solubility vary with temperature
a
is temperature dependent
in a given solvent
Theta (θ) temperature (Flory temperature)
● ● For a given polymer in a given solvent,
the lowest temperature at which
a
= 1
q state / q solvent
Polymer in a
q
state
• having a
minimal solvation effect
• on the brink of becoming insoluble • further diminution of solvation effect polymer precipitation
The expansion parameter
a r
Solvent Effects on the Macromolecules
• a r depends on balance between i) polymer-solvent and ii) polymer-polymer interactions – If
polymer-polymer
are
more
• a r < 1 • Chains contract • Solvent is poor favourable than
polymer-solvent
– If
polymer-polymer
are
less
• a r > 1 • Chains expand • Solvent is good favourable than
polymer-solvent
– If these interactions are equivalent, we have theta condition • a r = 1 • Same as in amorphous melt
The theta temperature
• For most polymer solutions a r depends on temperature, and increases with increasing temperature • At temperatures above some
theta temperature
, the solvent is good, whereas below the solvent is poor, i.e.,
T
> q
T
= q
T
< q a
r > 1
a
r = 1
a
r < 1
Often polymers will precipitate out of solution, rather than contracting What determines whether or not a polymer is soluble?
Solubility of Polymers
Encyclopedia of Polymer Science, Vol 15, pg 401 says it best...
A polymer is often soluble in a low molecular weight liquid if: •the two components are similar chemically or are so constituted that specific attractive interactions such as hydrogen bonding take place between them; •the molecular weight of the polymer is low; •the bulk polymer is not crystalline; •the temperature is elevated (except in systems with LCST). The method of solubility parameters can be useful for identifying potential solvents for a polymer. Some polymers that are not soluble in pure liquids can be dissolved in a multi component solvent mixture. Binary polymer-polymer mixtures are usually immiscible except when they possess a complementary dissimilarity that leads to negative heats of mixing.
Another way of looking at Solubility: Thermodynamics of Mixing
m A moles m B moles material A material B + D G mix < 0 A-B solution D G mix > 0 immiscible blend D G mix (Joules/gram) is defined by: D G mix = D H mix -T D S mix where D H mix D S mix = H AB = S AB - (x A H A - (x A S A + x B H B + x B S B ) ) and x A , x B are the mole fractions of each material.
Polymer-solvent: volume fraction of polymers
Thermodynamics of Mixing: Small Molecules
Ethanol(1) / n heptane(2) at 50ºC Ethanol(1) / chloroform(2) at 50ºC Ethanol(1) / water(2) at 50ºC
Solvent-Solvent Solutions vs Polymer-Solvent Solutions Solvents can easily replace one another.
Polymers are thousands of solvent sized monomers connected together. Monomers can not be placed randomly!
• Entropy of mixing is generally positive • Enthalpy is also often positive: smaller the better Ideal = 0
ENTROPY OF MIXING The total configurational entropy of mixing (J/K) created in forming a solution from n 1 is: moles of solvent and n 2 D S mix ( 1 ln f 1
n
2 ln moles of solute (polymer) f 2 ) .
where f i is the component volume fraction in the mixture: f 1
x n
1 1
x n
1 1
x n
2 2 .
and f 2
x n
1 1
x n
2 2
x n
2 2 .
x i represents the number of segments in the species for a usual monomeric solvent, x i = 1 x i for a polymer corresponds roughly (but not exactly) to the repeat unit On the previous slide, f 1 , f 2 representations, but n 2 and n 1 are equivalent in the two lattice = 20 for the monomeric solute, while n 2 = 1 for the polymeric solute.
Enthalpy of Mixing
D H mix can be a positive or negative quantity If A-A and B-B interactions are stronger than A-B interactions, then D H mix > 0 (unmixed state is lower in energy) If A-B interactions are stronger than pure component interactions, then D H mix < 0 (solution state is lower in energy) An ideal solution is defined as one in which the interactions between all components are equivalent. As a result, D H mix = H AB - (w A H A + w B H B ) = 0 for an ideal mixture In general, most polymer-solvent interactions produce D H mix > 0, the exceptional cases being those in which significant hydrogen bonding between components is possible.
Predicting solubility in polymer systems often amounts to considering the magnitude of D H mix > 0.
If the enthalpy of mixing is greater than T D S mix , then we know that the lower Gibbs energy condition is the unmixed state.
Gibbs Energy of Mixing: Flory-Huggins Theory
Combining expressions for the enthalpy and entropy of mixing generates the free energy of mixing: D G mix D H mix
RT
( 1 D S mix cf
n x
2 1 1 ln f 1 ( 1
n
2 ln f 2 ln f 1 cf
n
2 ln
n x
2 1 1 ) f 2 ) The two contributions to the Gibbs energy are configurational entropy as well as an interaction entropy and enthalpy (characterized by c ; “chi” pronounced “Kigh”).
Note that for complete miscibility over all concentrations, c for the solute-solvent pair at the T of interest must be less than 0.5.
• If • If c c > 0.5, then D G mix < 0.5, then D G mix > 0 and phase separation occurs < 0 over the whole composition range.
• The temperature at which c = 0.5 is the theta temperature.
Flory-Huggins Theory
• • Flory-Huggins theory, originally derived for small molecule systems, was expanded to model polymer systems by assuming the polymer consisted of a series of connected segments, each of which occupied one lattice site.
Assuming segments are randomly distributed and that all lattice sites are occupied, the free energy of mixing per mole of lattice sites is D G mix =RT[( f A /N A )ln f A +( f B /N B )ln f B + c FH f A f B ] (3) f i is the volume fraction of polymer i N i is the number of segments in polymer i c HF is the Flory-Huggins interaction parameter
Characterization of polymers:
Length
and
time
scales are important
[Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer gels”, Prog. Polym. Sci.
23
, 533 (1998).]
Polymer solubility
• In solvent, other polymer(s) or plasticizer • Non-Newtonian properties: Viscosity, rheology • Need to understand the configuration, conformation, and dynamics of macromolecules – Statistical mechanics • Series of mathematical models • Tied to experiment (viscosity, light scattering)
Thermodynamics of Mixing: (Free energy) For solution D G < 0 D G = D H -T D S -T D S Entropy > 0, so –T D S < 0 D H can be negative or positive Polymers are more sensitive to D H
Spinoidal decomposition of a solution into two phases • Quenching (rapid cooling) of solution • Polymer rich & polymer poor phases Careful removal of polymer poor phase “microporous” Foams.
Any solution process is governed by the
free energy relationship
D G = D H － T D S D G ＜ 0 D S ＞ 0 polymer dissolves spontaneously arising from
increased conformational mobility
of the polymer chain
For a binary system,
D
mix
V mix
[( D
E
1
V
1 ) 1 2 D H mix : heat of mixing V 1 ,V 2 : molar volumes (
M
.
W
.
)
d
f 1 , f 2 : volume fractions D E 1 , D E 2 : energies of vaporization D
E 1 /V 1 ,
D
E 2 /V 2 : cohesive energy densities
( D
E
2
V
2 1 ) 2 ] 2 f 1 f 2
Enthalpy of Mixing
D H mix can be a positive or negative quantity – If A-A and B-B interactions are stronger than A-B interactions, then D H mix then D > 0 (unmixed state is lower in energy) – If A-B interactions are stronger than pure component interactions, H mix < 0 (solution state is lower in energy) • An ideal solution is defined as one in which the interactions between all components are equivalent. As a result, D H mix = H AB - (w A H A + w B H B ) = 0 for an ideal mixture • In general, most polymer-solvent interactions produce
exceptional cases being those in which significant hydrogen bonding between components is possible
.
– Predicting solubility in polymer systems often amounts to considering the magnitude of D H mix > 0.
– If the enthalpy of mixing is greater than T D S mix D H the lower Gibbs energy condition is the unmixed state.
mix > 0, the , then we know that
1 ( D
E
1
V
1 ) 2 1 2 ( D
E
2
V
2 ) 2 1
1 ,
2 : solubility parameters
D H mix = V mix ( 1 － 2 ) 2 f 1 f 2 ∴ ( 1 － 2 ) 0; D H mix small
if (
1
－
2 ) = 0
D
H mix
＝ then, D G = － T D S D G ＜ 0
0
Under this condition,
solubility is governed solely by entropy effects i.e., For an ideal solvent
s
p Cohesive energy density (
D
E/V;
2 )
●
energy needed to remove a molecule
from its nearest neighbors.
●
analogous to heat of vaporization
per volume for a volatile compound.
For
1 (solvent)
calculated
1 directly ( D
E
)
V
1 2
from
the latent heat of vaporization ( D
H vap
) ( D
H vap V
RT
) 1 2
For
2 (polymer) estimated from
group molar attraction constants
G
(due to their negligible vapor pressure)
G
● ● ● ●
Consider intermolecular forces
between molecular groups
Derived from low M.W. compounds
by
Small
by
Hoy
heat of vaporization vapor pressure measurements
Table 2.1. Representative group molar attraction constants a
G[(cal/cm 3 ) 1/2 mol -1 ] Group Small
CH 3 CH 2 CH C CH 2 CH C 6 H 5 (phenyl) CH (aromatic) C O (ketone) CO 2 (ester) a Values taken from Refs. 5 and 6.
214 133 28 -93 190 19 735 275 310
Hoy
147.3
131.5
85.99
32.03
126.5
84.51
117.1
262.7
326.6
Which set one uses is normally
determined by the method
for the solvent.
used
for
determining
1
G V
d
G M
(V: molar volume, d: density, M: MW of repeat unit)
CH 2 CH C 6 H 5
Small’s system Hoy’s system
1 .
05 ( 133 28 735 ) 104 9 .
0 1 .
05 [ 131 .
5 85 .
99 6 ( 117 .
1 )] 104 9 .
3
For taking into account the strong intermolecular dipolar forces
,
solubility parameters
d p H (dispersion forces) (dipole-dipole attraction) (hydrogen bonding)
Hydrodynamic volume
polymer size in solution
●
Interaction
between solvent and polymer molecules ●
Chain branching
●
Conformational effects
(arising from the polarity and steric bulkiness of the substituent groups)
Restricted rotation
caused by resonance O C NH O C NH
D
H
mix
and the Solubility Parameter
• The most popular predictor of polymer solubility is the solubility parameter, i . Originally developed to guide solvent selection in the paint and coatings industry, it is widely used in spite of its limitations.
• For regular solutions in which intermolecular attractions are minimal, D H mix can be estimated through: D H 1 , 2 D U 1 , 2 f 1 f 2 ( 1 2 ) 2 cal / cm 3 • • where D U 1,2 = internal energy change of mixing per unit volume, f i = volume fraction of component i in the proposed mixture, • i = solubility parameter of component i: (cal/cm • Note that this formula always predicts D H mix 3 ) 1/2 > 0, which holds only for regular solutions.
Determining the Solubility Parameter
The conditions of greatest polymer solubility exist when the solubility parameters of polymer and solvent match.
If the polymer is crosslinked, it cannot dissolve but only swell as solvent penetrates the material.
The solubility parameter of a polymer is therefore determined by exposing it to different solvents, and observing the at which swelling is maximized.
Solubility (Hildebrant) Parameters of
Material Acetone Benzene Tetrahydrofuran Carbon tetrachloride n-Decane Dibutyl amine Mineral spirits Methanol Toluene Water Xylene
Select Materials
(cal/cm 3 ) 1/2 9.9
9.2
Material Poly(butadiene) Poly(ethylene) 9.5
8.6
6.6
8.1
6.9
14.5
8.9
23.4
8.8
Poly(methylmethacrylate) Poly(tetrafluoroethylene) Poly(isobutylene) Poly(styrene) Cellulose triacetate Nylon 6,6 Poly(vinyl chloride) Poly(acrylonitrile) (cal/cm 3 ) 1/2 8.4
7.9
9.45
6.2
7.85
9.10
13.6
13.6
10.5
12.4
If
D
< 1, polymer will dissolve
Solubility Parameters of Select Materials
Partial Miscibility of Polymers in Solvents
•Idealized representation of three generalized possibilities for the dependence of the Gibbs free energy of mixing, D G m , of a binary mixture on composition (volume fraction of polymer, f 2 ) at constant P and T. •I. Total immiscibility; •II. Partial miscibility; •III. Total miscibility. •Curve II represents the intermediate case of partial miscibility whereby the mixture will separate into two phases whose compositions ( ) are marked by the volume fraction coordinates, f 2 A and f 2 B , corresponding to points of common tangent to the free-energy curve.
Factors Influencing Polymer-Solvent Miscibility
Based on the Flory-Huggins treatment of polymer solubility, we can explain the influence of the following variables on miscibility: 1. Temperature: The sign of D G mix is determined by the Flory Huggins interaction parameter, c .
As temperature rises, c decreases thereby improving solubility.
Upper solution critical temperature (UCST) behaviour is explained by Flory-Huggins theory, but LCST is not.
2. Molecular Weight: Increasing molecular weight reduces the configurational entropy of mixing, thereby reducing solubility.
3. Crystallinity: A semi-crystalline polymer has a more positive D H mix = H AB - x A H A - x B H B due to the heat of fusion that is lost upon mixing.
Theta Temperature
Partial Miscibility of Polymers in Solvents
• Phase diagrams for the polystyrene-acetone system showing both UCSTs and LCSTs. • Molecular weights of the polystyrene fractions are indicated.
Polymer-Solvent Miscibility Phase diagrams for four samples of polystyrene mixed with cyclohexane plotted against the volume fraction of polystyrene. The molecular weight of each fraction is given. The dashed lines show the predictions of the Flory-Huggins theory for two of the fractions. f 2
Practical Use of Polymer TDs Fractionation
• • • • • • Consider solution in poor solvent of two polymers, p1 and p2.
Flory-Huggins tells us that if p2 has higher molecular weight it should precipitate more readily than p1 add non-solvent until solution becomes turbid heat, cool slowly and separate precipitate
finite drop in temperature always renders finite range of molecular weight insoluble some p2 will also remain soluble!
T 1 phase clear solution 1
N
f 1 1 f 2 c 2 phase cloudy p1 p2 f 2 volume fraction polymer
Fractionation of Polymers using Precipitation
Blends are not easy to make:
• Blends are not easy to discover-often copolymers are needed to make blend with another homopolymer • Phase separation gets out of hand in immiscible systems without compatibilizers.
• Some blends must be made with solvent.
Polymer Mixtures
Immiscible
Some miscible blends
Polymer blend
poly(ethylene terephthalate) with poly(butylene terephthalate)
Some miscible blends
PMMA & poly(vinylidene fluoride)
Some miscible blends
Mini Cooper / Cooper S, radiator grille & PETE
with
Immiscible Blends
Immiscible Blends
1:2 (PTV:PCBM) 1:3 (PTV:PCBM) 1:4 (PTV:PCBM) S C 12 H 25 n 1:6 (PTV:PCBM) 1:10 (PTV:PCBM)
1. Linear polymer: copolymers
Morphology of diblock Copolymers: Random: AABABBAAABABABB Alternating: ABABABABABABABA Block: AAAAAAAABBBBBBB
PMMA-PS-PB ternary blend
•
Dilute Solution Viscosity
The “strength” of a solvent for a given polymer not only effects solubility, but the conformation of chains in solution.
– A polymer dissolved in a “poor” solvent tends to aggregate while a “good” solvent interacts with the polymer chain to create an expanded conformation.
– Increasing temperature has a similar effect to solvent strength.
• • • The viscosity of a polymer solution is therefore dependent on solvent strength.
• • • Consider Einstein’s equation: hh s (1+2.5
f ) where h is the viscosity • h s is the solvent viscosity and f is the volume fraction of dispersed spheres.
Molecular Weight and Polymer Solutions
Number Average and Weight Average Molecular Weight
M n number average molecular weight M w weight average molecular weight
Determine M. W. of small molecules Mass spectrometry Cryoscopy (freezing-point depression) Ebulliometry (boiling-point elevation) Titration Determine M. W. of polymers Osmometry
M n Light scattering
Ultracentrifugation End-group analysis M w
M w M n
Measurement of Number Average Molecular Weight End-group analysis Titration
polyester (-COOH, -OH), polyamide (-C(O)NH-), polyurethanes( isocyanate), epoxy polymer (epoxide), acetyl-terminated polyamide (acetyl)
Elemental analysis Radioactive labeling UV / NMR / (IR) Upper limit M. W.
50,000
(due to the low concentraction of end groups) preferred range: 5,000-10,000
Not applicable to branched polymers Analysis is meaningful only when
the mechanisms of initiation and termination are well understood
Membrane Osmometry
● based on colligative properties ●
most useful method for M n
(range:
50,000 to 2,000,000)
● major error source arising from the loss of low MW species ● obtained M n
values are generally higher
than other colligative measurement method
Static equilibrium method
measure the hydrostatic head
( D
h
) after equilibrium
Dynamic equilibrium method
measure the counter pressure
needed to maintain equal liquid levels
h
Osmotic pressure (
)
related to by the
n
van’t Hoff equation
extrapolated to zero concentration (C; g/L) to obtain the intercept (
C
)
C
0
RT M n
A
2
C PV
V
n RT W t M n RT
A 2 = v 2 V -1 (0.5 – c )
V W t
RT M n
1
C
C
RT M n
RT M n
RT M n C
RT M n
(
C
)
C
0
C
RT M n A
2
C
2
A
2
C A
3
C
3 ...
Cryoscopy and Ebulliometry
Thermodynamic relationships
D
T C f
C
0
RT
2 D
H f M n
A
2
C
T b C
C
0
RT
2 D
H v M n
A
2
C
for freezing-point depression for boiling-point elevation T D H
f
D H v : : A 2 : : : freezing point or boiling point (solvent) latent heat of fusion (per gram) latent heat of vaporization (per gram) second virial coefficient solvent density • •
Limited by the sensitivity of measuring
D
T f ,
D
T b
• As MW higher; D T f , D T b smaller
Upper limit
－
Mn = 40,000
• Preferred － M n ＜ 20,000
Vapor Pressure Osmometry
●
For M n
＜
25,000
●
No membrane needed
• Thermodynamic principle similar to membrane osmometry
Measurement method
→
Add a drop of solvent and solution,
on a pair of matched themistors, in an insulated chamber saturated with solvent vapor then, →
Condensation heats the solution thermistor
(until the vapor pressure of the solution equal to that of the pure solvent)
Temperature change is measured
(by resistance change of the thermistor)
and related to solution molality
D
T
RT
2 100
m
: heat of vaporization per gram (solvent)
m
=
n
g = W t / M g
n
m : molarity
Mass Spectrometry
MALDI-MS (MALDI-TOF)
MALDI-MS (matrix-assisted laser desorption ionization mass spectrometry) (TOF － time-of-flight) ●
Imbeds polymer in a matrix of low MW organic compound
● Irradiates the matrix with UV laser ● Matrix transfer the absorbed energy to polymer and vaporize the polymer ● Integrated peak areas the number of ions ●
M n , M w can be calculated Soft ionization method
field desorption (FD-MS) laser desorption (LD-MS) electrospray ionization (ESI-MS)
Refractive Index Measurement
Most
suitable for low MW
n 1
M n
polymers
Measurement of Weight Average Molecular Weight (Mw) Light Scattering
•
Most widely used method for measuring absolute Light loses energy by
absorption, conversion to heat, and scattering •
Light scattering is caused by
molecules due to the inhomogeneous distribution of
irregular change in density and refractive index
(because of the fluctuation in composition) •
Scattering intensity depends on concentration, size and polarizability
of the molecules •
Refractive index also depends on concentration
amplitude of vibration and
LS adds optical effects
Size
q
= 0 in phase
I
s maximum
q
> 0 out of phase,
I
s goes down
I s
1
q
2
R g
2 3
Scattering causes turbidity (
)
K
Kc M w
32 3 3 =
R
q
n
0 2 (
dn
/
dc
) 2 4
N
0
n 0
: wavelength of light N 0 : Avogadro’s number n : refractive index of solution
• •
dn/dc (specific refractive increment) is a constant
for a given polymer, solvent, and temperature
measured from the slope of n vs. c
Debye equation
Kc R
q 1
M w
2
A
2
c
(
A
2 : the second virial coefficient)
I
e
l I
0 I 0 : intensity of incident light I : intensity of scattered light l : length of sample solution : turbidity
i Kc
K
c i M i
c i M i
c i
Kc
Kc
c i M i
W i
W i
/
V c
M i
/
V
Kc
W i M i
W i
Kc M w
As molecular size approaches the wavelength of the light
•
Interference between scattered light
coming from different parts occurs •
Correction is needed
•
Turbidity (
or
D
R
q
) associated with large particles measured at different
concentrations (c) and
angles
( q )
Kc
D
R
q 1
M w P
( q ) 2
A
2
c
P(
q
): Particle scattering factor P(
q
)
is a function of q ;
depending on the shape
of molecule in solution
For a monodisperse of randomly coiling polymers
P
( q ) 2 (
v
2 )[
e
v
( 1
v
)]
v
( 16 2
s
2 )(
s
2 ) sin 2 ( q 2 ) s 2 : mean-square radius of gyration of the molecue s : the wavelength in solution
s
n
Zimm Plot Plot
Kc R
q
vs.
[ sin 2 ( q / 2 )+
kc
]
k is an arbitrary constant Extrapolated to both zero concentration and zero angle
intercept = 1
M w
(P( q ) = 1 at c = 0 and q = 0)
Kc R
q
•
Light scattering photometer
• Suitable range of
M w : 10,000
－
10,000,000
• Dust-free solution is required
Light source Old:
high-pressure Hg lamps plus filter
New:
laser
50 40 30 20 10 0 0 Rg = 0,0074.( M Linear PS w ) 0,64 Rg=0,0126.( R 2 = 0,99 200000 400000 600000 M w ddl (g/mol) 800000 M w )
Ultracentrifugation
• • Most
expensive
• Extensively
used with proteins Determining
M z
•
Sedimentation rate
is proportional to molecular mass •
Distributed
according to size
along the perpendicular
•
direction Concentration gradients
within the polymer solution are
observed by refractive index measurements and interferometry
Viscometry
• •
Simplest
and most widely used • Not an absolute number
Can be calibrated with the absolute MW
(by light scattering) • of fractionated polymer samples
Typical conditions:
0.5 g/100 mL; 30.0
± 0.01
℃ Viscometers: (a) Ubbelohde (b) Cannon-Fenske •
Ubbelohde type
• more convenient
not necessary to have exact volumes of solution
• additional solvent can be added (for dilution) • dust particles can greatly alter the flow time (via the capillary tube)
● ratio of ● dimensionless
Viscosity
Relative viscosity (viscosity ratio),
h
rel solution viscosity to solvent viscosity
● viscosity units (poises) or flow times cancel out (→ dimensionless)
Specific viscosity (
h
sp )
●
fractional increase in viscosity
h
sp
h h 0 h 0 h
rel
t
t
0 h h 0
t
0
t
h
t
0
rel
1
Inherent viscosity (
h
inh ; dL/g)
●
determined only by a single solution
h
inh
ln h
rel
at a specific concentration
C
●
an approximate indication of MW Intrinsic viscosity ([
h
]; dL/g)
● ● ● most useful [ h ] h
C sp
C
0
eliminating the concentration effect
by dividing h sp with C and then extrapolating to zero C
inh C
0
Table 2.2. Dilute Solution Viscosity Designations a
Common Name
Relative viscosity Specific viscosity Reduced viscosity Inherent viscosity Intrinsic viscosity
IUPAC Name
Viscosity ratio h
rel
h h 0
t t
0 － Viscosity number h
sp
h
red
h h h h 0 0
sp C
t
t
0 h
rel t
0 1
C
h
rel
Logarithmic viscosity h
inh
ln h
rel C
number 1 Limiting viscosity number [ h ] h
sp C
C
0 ( h
inh
)
C
0 a Concentrations (most commonly expressed in grams per 100 mL of solvent) of about 0.5 g/dL
[ h ]
M v
viscosity average molecular weight
K M v
log[ h ]
a
log (Mark-Houwink-Sakurada)
K
a
log
M M v
N i M N i M i
1
a i
1
a
•
plot log [
h
] vs log
M w (or log
M n
) measure a and K •
in general,
M n < M v < M w
(closer to
M w ) • ● ●
better results are obtained if the MW of fractionated samples are used (here,
M n M v M w
)
typically, a 0.5
－ 0.8, K 10 -3 － 0.5
for randomly coiled polymer
in a q solvent
a = 0.5;
M v
N i M i
1.5
N i M i
1 0.5
N i M i
1.5
N i M i
2 •
for a rodlike extended chain polymer
a = 1.0;
<
M v
M w
N
N i i M M i i
2
M w
In polymers a values have solvent dependence
[η] = KM a
Linear polymers: a Rigid rod: a = 0.5-1.0
= 1.0-2.0
Branched polymers: a = 0.21-0.28
Rigid spherical particles: a = 0
Complication in applying Mark-Houwink-Sakurada relationship
• chain branching • broad MWD sample • solvation of polymer • backbone sequences (alternating or block) • chain entanglement (when MW is extremely large)
Viscosity measurements based on mechanical shearing
•
for concentrated polymer
solutions or undiluted polymer system • more
applicable to the flow properties
of polymers
Dynamic Viscosity
Informally: resistance to flow Formally: Ratio of Shearing stress (F/A) to the velocity gradient (
dv x /d z
) Force: Pa = N m -2 = kg m s -2 Force/Area = Pa m -1 = kg m -1 s -2 Viscosity = Pa s = kg m s -1 Other units (Poise = dyne s cm -2 ) 1 Pa s = 10 Poise 1 centipoise = 1 millipascal second (mPa s)
EXAMPLES
• Water 1 mPa s • Honey 10,000 mPa s • Peanut Butter 200,000 mPa s
Flow Regimes of Typical Polymer Processing
10 3 10 10 1 10 3 Sedimentation Extrusion Chewing Pipe flow Injection molding Spraying Rolling High-speed coating Lubrication 10 5 10 3 10 1 10 1 10 3 10 5 10 7 ( s -1 ) Typical viscosity curve of a polyproprene (PP), melt flow rate (230 C/2.16 Kg) of 8 g/10 min at 230 C with indication of the shear rate regions of different conversion techniques. [Reproduced from M. Gahleitner, “Melt rheology of polyolefins”, Prog. Polym. Sci.
26
, 895 (2001).]