there’s a lot of free volume! density of carbon (as diamond) = 3 g/ml density of 12 c nucleus = r...
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There’s a lot of free volume!Density of carbon (as diamond) = 3 g/mL
Density of 12C nucleus =
Rnucleus = [(protons + neutrons)1/3 ][1.2 10-13 cm]
So…carbon could be compressed to about ~ 1 1014 g/mL
3
23
34
1002.612
nucleusRvolume
mass
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
T
V
“Segmental motion” – many C-atoms move together - ~50 in polyvinyls
Transitions can be followed through thermodynamic
state variables.
T
V
Tm
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.
T
V
Tm
Of course,water isdifferent!
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 timeat right T.
SemicrystallinePolymers 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
ImpactStrength,
Izod method B
Modulus, MPa (E)
Less crystalline, so less soluble.Also use some crosslinks to get high E.
Do polymer glasses or crystals shatter?
Does it hurt to dive into water?
Do bookcases sag?
Do glaciers flow?
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.
From how high up?
How long do we wait?
How long do we wait?How steep?
Near Chamonix, France, is a flowing ice tunnel.
Polymer Volume Transitions
T
V
Tm
Totally crystalline
T
V
Tg
Totally glassy
T
V
TmTg
Semi-crystallineThis zone
makes ALLthe difference!TOUGH ZONE
Remember! Tg is for the down-going transition, but
we really care about the stuff above Tg.
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
Semicrystalline polymer Tough solid
Very crystalline Often made into fiber.
Frustrated, crystallizable polymer let’s return to that later.
From Rudin
Practical Guide to Polymer Behavior
A molecular level view shows more local volume at
temperatures exceeding Tg
T
V
Tg
Restrictedlocal motion
Greater localmotion
Freevolume
Brittle glass Melt, tough polymeror “other”
Below Tg …….
Polymer is certainly more brittle.
Polymer might not be completely brittle, becausesome 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
T
V
TgTother
Example:Nylon is alwaysused below itsTg, yet is not brittle
Classifying Transitions Thermodynamically
This isn’t a thermo class, but you must recallthis golden oldie from PCHEM:
dG = VdP - SdT + i dni =
j
PTnjnPnT
dnn
GdT
T
GdP
P
G
iii ,,,,
P
T
T
GS
P
GV
Volume is related to a first derivative of G.
So is entropy.
Melting Crystals vs. Librating Glass
T
V
Tm T
V
Tg
Discontinuity in volume,i.e., discontinuity in a 1st derivative of G
T
dVdT Tm T
dVdT Tg
Discontinuity in derivativeof volume, i.e., discontinuity in a 2nd derivative of G
First order transition Second order transition
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 - g ~ 0.5 r
)(1
))(/1(
0
TV
V
T
VV p
Entropy trends parallel volume
H = T S
1st order transitionwith “latent heat”At transition, you haveto suddenly put in moreheat.
T
S
Tm T
S
Tg
H = 0
2nd order transitionno latent heat. After transition, therate at which heat mustbe supplied changes
Differential Scanning Calorimetry
Suppose we keep track of RPM’s needed to maintain sample and inert reference at same temperature as both are heated….
Primitive Power Supplies
Thermometers
Sample Reference
Or…we could keep track of current.
1st & 2nd Order DSC Transisions
Sample --Reference
Differential heat:the extra heat it takes to get sample through transitions that the inert reference does not have.
T
i
Tm T
i
Tg
Real transitions depend on rate of scanning, qualityof thermal contact between sample & container, etc.
From Campbell Optimal mobility range – Tm – 10 to Tg + 30 (K)
H* = Tm Q dt
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.• Dielectric
spectroscopy• Viscoelastic
methods, which can directly probe the entire mechanical spectrum as function of frequency.
• All transitions have characteristic frequencies
• Tg as frequency you really have to
chill something before it cannot slowly 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
From Campbell
Tg (oC)
-75-20-67-6-471081728121811056784
Some Typical Tg’s + Tm’s
Tm (°C)
180137-146
(PE)
176-200 (PP)
280 (PET)265 (N6,6)
700-773 (Tg, PBI)
Poly[2,2’-(m-phenylene)-5,5’-bibenzimidazole
Tg TrendsTg 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
From Billmeyer
1.4 < Tm/Tg < 2.0
Caveats•A lot about this lecture is schematic; the real picture is more complex.•A lot depends on rate!
T
V
Tg Tg
Slow
Fast
Modern DSC’s (like ours!) use sophisticated temperature ramping sequences to sort out reversible (fast) fromirreversible (slow) transitions.
T
time
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 plasticizersRough 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
0 60
Tm, ºC
200
320
260
Two different blends of poly(amides)
2
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.
Strain = - 1 = (L/L0) - 1
Stress
y
NeckingBreak - b
Strain softening
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)
Log(Xw)
Log
Slope = 1.7
Slope = 3.4 – zero shear
Xw ~ 600
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.
High shear
Entangled Melts – Reptation Theory
Constraints imposed by nearby chains – path is the “primitive path”; constraint surface is the “tube”. Leave tube -- you’re free (like a corn maze).
Characteristic tto exit ~ Maxwellian time constant,
= /EThen, using Einstein eq.,
~ L2/Dc
Where L is the path length and Dc is a diffusivity along the path.