There’s a lot of free volume!
Density of carbon (as diamond) = 3 g/mL
Density of 12C nucleus =
12
23
mass
6
.
02

10

volume 4 R 3
nucleus
3
Rnucleus = [(protons + neutrons)1/3 ][1.2  10-13 cm]
So…carbon could be compressed to about ~ 1  1014 g/mL
WOW! I guess those electrons do a great
job of repelling each other to fill the volume.
Wiggle Room: as important to
polymers as to Hitler.
Crystal of a small alkane
Volume increases with temperature
V
T
“Segmental motion” – many C-atoms move together - ~50 in polyvinyls
Transitions can be followed through
thermodynamic state variables.
V
Of course,
water is
different!
V
Tm
T
Tm
T
Melting temperature implies a transition from left to right.
We could just as well call it the freezing point
for the transition going from right to left.
Equilibrium is highly overrated!
• Slow-cooled SiO2 = Quartz
• Fast-cooled SiO2 = window glass
• You can also glass water! Just cool it really, really fast.
Practical application of water glass:
Freeze-fracture TEM image of aqueous gel.
Water in the gel is just “stopped” dead in
its tracks without forming ice crystals
that would distort the structure.
Stopping polymers dead in their tracks
Amorphous Polymers
•Polymers that just can’t crystallize, ever.
•Polymers that could crystallize,
but weren’t given enough time
at right T.
Semicrystalline
Polymers that partially crystallized,
but contain amorphous regions.
E.G. – ethylene/octene copolymer –
hexyl branch gives amorphous region,
higher impact strength at given modulus
(E). SCB  -1 (e.g., Dow “Elite” PE)
“Toughness” induced by SCBs (S. Chum)
Red – traditional PE; blue - Elite
Impact
Strength,
Izod method B
Less crystalline, so less soluble.
Also use some crosslinks to get high E.
Modulus, MPa
(E)
Do polymer glasses or
crystals shatter?
Does it hurt to dive into water?
From how high up?
Do bookcases sag?
How long do we wait?
Do glaciers flow?
How long do we
wait?
How steep?
In all cases, the answer is….it depends.
Still, it is easy to identify water as a liquid.
Wooden bookshelves and glaciers are clearly
solid for most practical purposes.
Near Chamonix, France, is a flowing ice tunnel.
Polymer Volume Transitions
Totally crystalline
V
Totally glassy
V
T
Tm
Tg
Semi-crystalline
V
Tg
Tm T
T
This zone
makes ALL
the difference!
TOUGH ZONE
Remember! T is for the downgoing transition, but we really care
about the stuff above T .
g
g
That stuff can be melt or tough stuff,
depending on crystallinity.
Even “melty”, non-crystallizable polymers
can acquire toughness if covalent
crosslinks substitute for the crystalline
zones.
Above Tg….
Completely amorphous polymer  Viscous fluid
Frustrated, crystallizable polymer 
let’s return to that later.
Semicrystalline polymer  Tough solid
Very crystalline  Often made into fiber.
Practical Guide to Polymer Behavior
From Rudin
A molecular level view shows more local
volume at temperatures exceeding Tg
V
Greater local
motion
Restricted
local motion
Free
volume
Tg
Brittle glass
T
Melt, tough polymer
or “other”
Below Tg …….
Polymer is certainly more brittle.
Polymer might not be completely brittle, because
some motions remain that permit the polymer to
dissipate energy. These correspond to “other”
transitions that may or may not produce much of a
volume change. Transitions usually called a, b, g
V
Tother
Tg
T
Example:
Nylon is always
used below its
Tg, yet is not
brittle
Classifying Transitions Thermodynamically
This isn’t a thermo class, but you must recall
this golden oldie from PCHEM:
dG = VdP - SdT +  i dni =
 G 
 G 
 G 

dn j

 dP  
 dT   

 P T ,n
 T  P ,n
 n j  n ,T , P
i
 G 
V 

 P  T
 G 
S  

 T  P
i
i
Volume is related to a
first derivative of G.
So is entropy.
Melting Crystals vs. Librating Glass
Second order transition
First order transition
V
V
Tm
Tg
T
dV
dT
T
dV
dT
Tm
T
Discontinuity in volume,
i.e., discontinuity in a
1st derivative of G
Tg
T
Discontinuity in derivative
of volume, i.e., discontinuity
in a 2nd derivative of G
Measuring Volume Stinks!
Remember that Work = -pdV
System would have to gain some energy,
as heat, to perform that work.
It might be easier to measure heat instead.
Order of Magnitude of Transition -


V
1
V0
g ~ 0.5 r
V
(1 / V )(
)p
T

( T )
Entropy trends parallel volume
S
S
Tm
Tg
T
T
H = T S
H = 0
1st order transition
with “latent heat”
At transition, you have
to suddenly put in more
heat.
2nd order transition
no latent heat.
After transition, the
rate at which heat must
be supplied changes
Differential Scanning Calorimetry
Primitive Power Supplies
Thermometers
Sample
Reference
Suppose we keep track of RPM’s needed to maintain sample
and inert reference at same temperature as both are
heated…. Or…we could keep track of current.
1st & 2nd Order DSC Transisions
Differential heat:
the extra heat it takes to get
sample through transitions
that the inert reference does
not have.
Sample -- Reference
i
i
Tm
T
Tg
T
Real transitions depend on rate of scanning, quality
of thermal contact between sample & container, etc.
H* = Tm Q dt
From Campbell
Optimal mobility range – Tm – 10 to Tg + 30 (K)
Rate of Cryst. Highly Nonlinear
• Avrami eq. - fc = 1 - exp(-k tn) [fraction]
• N ~ 2-4. Why? Nucleation triggers rapid growth
at optimal conditions. Then it slows down as
advancing fronts meet – diffusional limits.
• Secondary nucleation best – crystals beget
crystals.
• Easy way to follow – measure  - higher c
(e.g., 1.51 vs. 1.33 g/mL for PET)
Rate and Ultimate Amount of Cryst.
Dependent on:
•
•
•
•
•
•
Conformational regularity – iso, syndio etc.
Polarity, H-bonding (intermolec. forces)
Nucleation conditions
T and P (stress)
Cooling (heating) rate
Side groups – some (-CH2- -CHOH- -CF2-C(O)- ) always fit, some don’t
At submicros. level, structures usually either planar zigzag
(PE, PVA, nylons) or helical (PP, PMMA, PTFE,
poly(peptides))
Jargon – H, 151 = helical, 15 monomers per complete turn
Other Tg Methods
• NMR T1, T2, 2H etc.
• All transitions have
characteristic
• Dielectric spectroscopy
frequencies
• Viscoelastic methods,
• Tg  as frequency 
which can directly
probe the entire
you really have to
mechanical spectrum
chill something before
as function of
it cannot slowly
frequency.
deform.
Why is “loss” high at Tg (visco., dielectric)
• T < Tg - rotation restricted, stress or
potential stored by vibrational modes
(“elastic”).
• T > Tg – stresses stored by uncoiling
• T ~ Tg = chains won’t uncoil, bonds inelastic
Some Typical Tg’s + Tm’s
Tg (oC)
-75
-20
-67
-6
-47
108
172
8
121
81
105
67
84
Poly[2,2’-(m-phenylene)-5,5’-bibenzimidazole
Tm (°C)
180
137-146
(PE)
176-200
(PP)
280 (PET)
265 (N6,6)
700-773
(Tg, PBI)
From Campbell
Tg Trends
Tg  as stiffness  (rings, double bonds)
Tg  as steric bulk  (but not side chain length – e.g.,
PMMA (105), PEMA (65), PPMA (35 °C))
Tg  as M 
i.e., Tg = Tg, - (K/Mn) ; K ~ 8 x 104 - 4 x 105
With crosslinks: Tg = Tg, - (Ks/Mn) ; Ks ~ 3.9 x 104
Tg  as intermolecular forces 
Useful thermo. correlation: 2 ~ 0.5 m R Tg - 25 m ,
m = # DOF’s of a link
1.4 < Tm/Tg < 2.0
From Billmeyer
Caveats
•A lot about this lecture is schematic; the real picture
is more complex.
Slow
•A lot depends on rate!
V
Fast
Tg
Tg
T
T
time
Modern DSC’s (like ours!) use sophisticated temperature
ramping sequences to sort out reversible (fast) from
irreversible (slow) transitions.
It really, really matters!
Challenger space shuttle.
Feynmann: http://www.feynman.org/
Plasticizers can change polymer bricks
into polymer pillows by modifying Tg.
O
O
O
O
Di-sec-octylphthalate (DOP)
Other uses:
lubricant for textiles
rocket propellant
insect repellant
perfume solvent
nail polish to prevent chipping
http://www.chemicalland21.com/industrialchem/plasticizer/DOP.htm
Tg behavior or plasticizers
Rough eq. –
Tg-1 = 1 Tg1-1 + 2Tg2-1 (Fox-Flory eq.)
More exact – based on thermo. ln(Tg/Tg1) =
[2 ln(Tg2/Tg1)] /[1 (Tg2/Tg1) + 2 ]
- Works for copolymers too!
Heat Deflection T (HDT)
• Widely reported
• T at which sample bar deflects by 0.25 mm under
center load of 455 kPa, at 2 K/min ramp.
• Amorphous – 10-20 K less than Tg
• Crystalline – closer to Tm
Effects of Additive on Tm
•
•
•
•
3 types of additives –
Isomorphous – Additive doesn’t disrupt lattice.
One-crystallizable – shows min.
Plasticizer – amorphous additive
320
Tm, ºC
Two different blends
of poly(amides)
260
200
0
2
60
Mechanical Behavior of Crystalline Polymers
• For already crystallized polymer – many polymers go
amorphous  cryst. on drawing.
• Lamellae distort to “shish-kebabs” – slip, tilt, twist to
fibrils. Annealing helps.
y
Stress

Necking
Strain softening
Break - b
Strain =  - 1 = (L/L0) - 1
PVT Behavior of Amorphous Polymers
• Rheo. Behavior – follows WLF theory.
Above Tg:
V ~ V(Tg) + [d(V0 + Vf)/dT] (T – Tg)
The dV0 accounts for polymer, the dVf for the
FV. Knowing that:
 = r - g = (1/V0) (dVf/dT), we obtain:
Williams-Landel-Ferry (WLF) Eqs.
f = fg +  (T – Tg) ; f = Vf/V0
Can subs. any ref. T0, f0 to >Tg - 20 and it
should still work. WLF proposed:
ln(/0) = f-1 – f0-1 ; subs. previous eq. to get:
log(/0) = -C1(T – T0) / [C2 + (T – T0)]
Where: C1 = (2.303 f0)-1 and C2 = (f0/).
dTg/dP ~ 0.16-0.43 K/MPa , so P, 
WLF Theory
(/0) called the “shift factor”, aT. WLF postulated
that:
aT is universal for ANY mechanical or rheological
property related to segmental motion (relax.
times, moduli).
aT ‘s use depends on type of property – up or down
WRT T? The “shifting” described by aT is known
as time-T superposition.
“Universal” WLF constants are:
0 = 1012 Pa*s ; C1 = 17.44; C2 = 51.6 K
WLF Theory –”Universal??”
For Polymer Liquids (Melts, Conc. Solutions)
Slope = 3.4 – zero shear
Log 
High shear
Slope = 1.7
Xw ~ 600
Log(Xw)
The critical Xw is where “critical entanglement” happens.
(Xw)c ~ 2 Xe , where the entanglement chain length can be
found from “overlap criterion” – where:
(# coils/vol)*(vol/coil) = 1 in dilute solution.
Entangled Melts – Reptation Theory
Characteristic t
to exit ~ Maxwellian time constant,
= /E
Then, using Einstein eq.,
 ~ L2/Dc
Where L is the path length and Dc is
a diffusivity along the path.
Constraints imposed
by nearby chains –
path is the
“primitive path”;
constraint surface
is the “tube”.
Leave tube -you’re free (like a
corn maze).