Material of the Day – Metallic Clays
Methylcellulose Binder
Atomized Metallic Powders
MAT_SCI 201/301 | S23 | 1
Lecture W5-2
Diffusion in Crystalline Solids and
Polymers
Pr. Jonathan Emery
ANNOUNCEMENTS | April. 27th, 2023
Lecture Topic
-
Diffusion (Plug-in examples in supplemental video lecture.)
Logistics
-
P-set W6/Quiz W6.
Exams returned after class. Rewrites due May 4th.
Morfli: Videos now embedded into Morfli
Reading
Morfli Ch. 7
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Outline: Diffusion in Materials
•
Diffusion
•
Motivation and Basics
• Brownian Motion to Random Walk
• Discovering Fick’s Laws
• Analysis with Fick’s Laws (2nd Law Analysis in Supplemental Video)
• Diffusion Mechanisms
• The Diffusion Coefficient and Structure/Temperature
•
Polymer Structure
•
Polymers and their applications
• Polymer synthesis
• Polymers and their characteristics
•
•
•
•
Size [Video lecture]
Shape
Structure/Configuration
Crystallinity
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Steady-state Diffusion – Across Plate
The slope is the concentration gradient:
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Steady-state Diffusion
•
Flux independent of time:
(if in steady-state, J does not change with time)
•
Flux is proportional to concentration gradient
•
The two are related by a diffusion constant (𝐷𝐷):
(a material property, units m2/s)
C1
Fick’s first law of diffusion
If the slope is linear:
C2
The concentration gradient is called
the driving force for diffusion.
Why negative? Ensures flux is positive when describing mass transfer from high to low concentration.
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Example: Chemical Protective Clothing (CPC)
Methylene chloride is a common ingredient in paint removers. Besides
being an irritant, it also may be absorbed through the skin. When
using this paint remover, protective gloves should be worn.
1
2
If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive
flux of methylene chloride through the glove?
Assume steady-state diffusion.
Data:
• Diffusion coefficient of methyl chloride in butyl rubber is:
• Surface concentrations are:
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Example: Chemical Protective Clothing (CPC)
Fick’s 1st Law
Linear concentration profile.
LD50 dose = 1.25 g/kg
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Non-Steady-State Diffusion
•
In most practical case non-steady-state diffusion is occurring:
1.
2.
Flux and concentration gradient change with time at a position in the material
Implies net accumulation or depletion of diffusing species.
•
The concentration of diffusing species changes with both time and position C = C (x,t)
•
In this case Fick’s Second Law is used:
(Assumes D is independent of composition. This is a powerful approximation, but it is not always a
good assumption.)
•
Second order partial differential equations (PDE) requires boundary conditions for
physical solution. Known as a governing equation, because it governs the behavior of the
system.
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Non-Steady-State Diffusion: Testing a Solution
•
There are only some mathematical functions that satisfies Fick’s Second Law. How
can we test them?
•
How to test whether a proposed concentration profile C(x, t) satisfies Fick’s Second
Law?
•
Example:
Nope.
Not a function that can describe diffusion.
You are not solving differential equations. You are testing solutions.
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Example:
•
Cu Diffusion into a Semi-infinite Bulk of Al from a Constant Surface Concentration
Copper diffuses into a long bar of Al and the surface concentration is held constant.
Surface concentration:
Cs of Cu atoms →
bar
pre-existing conc., C0 of copper atoms
CS
C(x,t)
C0
x
Boundary conditions (BCs)
(at t = 0, constant concentration in bar)
(constant surface concentration at all time)
(semi-infinite solid)
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A Solution to the PDE
•
C(x,t) for a constant surface concentration into a semi-infinite solid can be expressed as:
•
C(x,t) is the concentration at point x at time t.
erf(z) is the error function:
•
You do not need to solve this PDE, but you should be able to use/evaluate solutions I provide.
After some time (at t = t1)
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A Solution to the PDE (VL)
•
C(x,t) for a constant surface concentration into a semi-infinite solid can be expressed as:
•
C(x,t) is the concentration at point x at time t.
erf(z) is the error function:
•
I will tabulate erf values for you. You don’t have to evaluate it to get values.
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Example: Non-steady State Diffusion (VL)
•
An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated
temperature in an atmosphere that provides a constant surface carbon concentration at 1.0 wt%.
•
If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface,
determine the diffusion coefficient of the material.
•
Use
•
Boundary conditions
1.) Find D
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Example: Non-steady State Diffusion (VL)
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Outline: Diffusion in Materials
•
Diffusion
•
Motivation and Basics
• Brownian Motion to Random Walk
• Diffusion Mechanisms
• Discovering Fick’s Laws
• Analysis with Fick’s Laws
• The Diffusion Coefficient and Structure/Temperature
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Diffusion in Solids
•
What’s odd about the picture below? How is diffusion proceeding?
• How could those silver and orange atoms possibly exchange positions?
Initially (at t = t0)
After some time (at t = t1)
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Diffusion Mechanism #1: Vacancy-mediated Diffusion
•
•
•
Atoms “exchange” with vacancies, hopping into a vacant lattice position
Vacancy-mediated diffusion is the mechanism for the motion of substitutional
impurities and self-diffusion.
Rate of diffusion depends on:
•
Number of vacancies present
• Activation energy to “hop”
• The energy in the system
• What do these three independent variables suggest?
General form – Arrhenius equation
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Diffusion Mechanism #2: Interstitial-mediated Diffusion
•
•
•
Diffusion proceeds through interstitial sites (triangular, tetragonal, octahedral…)
Preferential pathways have minimum distortion from point A to point B.
Generally faster or slower than vacancy diffusion? Why?
Position of interstitial atom before
diffusion (an atomic jump)
Position of interstitial atom after
diffusion (an atomic jump)
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The Diffusion Coefficient
But wait – what is the diffusion coefficient, exactly?
But… D0?
→ number that ties together material-dependent properties:
e.g., lattice vibrational frequencies, entropic contributions, jump distance…
It is a measured (or calculated), tabulated material property.
Deeper investigation into the mature of D0 in upper-level MSE courses.
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Diffusion and Temperature
•
D has an exponential dependence on 1/T
•
Plotting the natural log of the diffusion coefficient vs. 1/T presents the data linearly:
How to read this graph?
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Example: Diffusion Coefficient at Temperature
What is the diffusion D coefficient of Cu in Si at 350 °C if:
D(300 °C) is 7.8 x 10-11 m2/s and Qd = 41.5 kJ/mol?
transform data
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Example: Diffusion Coefficient at Temperature
What is the diffusion D coefficient of Cu in Si at 350 °C if:
D (300 °C) is 7.8 x 10-11 m2/s and Qd = 41.5 kJ/mol?
Sanity check → larger diffusion coefficient at higher temp… good.
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Factors Affecting the Diffusion Coefficient
•
Mechanism: in general, DInt. > DSub.
•
Temperature
•
Higher temperature, faster diffusion
•
Diffusion species: smaller → faster
Crystal structure: more “open” (small APF) → faster
Polymer structure more “open” (looser chain packing) → faster (more permeable)
Bonding: stronger → slower (higher Qd)
•
Lattice imperfections: “short circuit” diffusion
•
•
•
•
Much faster along dislocations, (grain boundaries), and external surfaces.
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Summary
•
Mass can be transported through solids by diffusion.
•
There are two diffusion atomistic mechanisms* that dominate in metallic and ceramic
crystals – interstitial and vacancy (aka substitutional-mediated) diffusion.
•
Fick’s first law is a differential equation that describes time-independent diffusion flux across
an (typically thin) interface.
•
Fick’s second law is a differential equation that describes time-dependent diffusion and is the
case in most practical diffusion situations.
•
Diffusion can be influences by numerous factors – host lattice, density, diffusing species,
bond type, temperature, etc.
*Neglecting so-called short-circuit paths.
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Outline: Diffusion in Materials
•
Diffusion
•
Motivation and Basics
• Brownian Motion to Random Walk
• Discovering Fick’s Laws
• Analysis with Fick’s Laws (2nd Law Analysis in Supplemental Video)
• Diffusion Mechanisms
• The Diffusion Coefficient and Structure/Temperature
•
Polymer Structure
•
Polymers and their applications
• Polymer synthesis
• Polymers and their characteristics
•
•
•
•
Size [Video lecture]
Shape
Structure/Configuration
Crystallinity
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Module Outcomes
•
Recognize the features within polymer structure hierarchy (chemistry, size, shape, architecture) and
compare simple properties (melting temperature, strength, density) of polymers with varying
structures.
•
Compute important values for polymer structures such as molecular weight, end-to-end distance,
dispersity, etc,
•
Explain the roles of primary and secondary bonding in polymeric structures.
•
Discuss how polymers occupy space due to their intermolecular bonding, size, and conformation.
•
Understand how simple random walk models can lead to polymer conformation and conjecture about
limitations and extensions of the model.
•
Differentiate between various types of polymer synthesis (chain-growth/step growth) and evaluate
the affects these differences may have on polymer structure, namely molecular weight and dispersity.
•
Correlate polymer architecture with post-consumer processability (recycling).
*Practice
this.
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Structure in Polymers
Chemistry
Conformation
Size
Architecture
H H H H H H
C C C C C C
H H H H H H
Polyethylene (PE)
H H H H H H
C C C C C C
H Cl H Cl H Cl
Poly(vinyl chloride) (PVC)
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Polymers: Name That Polymer!
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Polymers: Natural
The structure of wheat protein and response to hydration and shear.
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Ancient Human Uses of Polymers
•
Natural polymers used by ancient
peoples:
•
Wood, Rubber, Wool
• Cotton, Silk, Leather
•
Oldest known uses
•
Skins for clothing
• Textiles (felts, woven cloth)
• Wooden structures and
weapons
• Rubber ball
*Pretty subjective.
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Polymers since ~1940
•
Many plastics, rubbers, fibers used today are synthetic polymers…
•
Since WWII, synthetic polymers have revolutionized industry:
•
Cheap to produce
• Highly tunable properties
• Often superior to natural materials
• Displaced metals and wood in many applications
• Lower cost and superior/good enough properties
•
Environmental concerns…
•
Recycling
• Resource usage and conservation
• Sustainability
• Toxicity
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What is a Polymer?
Poly
mer
many
repeat unit
repeat
unit
repeat
unit
H H H H H H
C C C C C C
H H H H H H
H H H H H H
C C C C C C
H Cl H Cl H Cl
Polyethylene (PE)
Adapted from Fig. 14.2, Callister & Rethwisch 8e.
repeat
unit
H
C
H
Poly(vinyl chloride) (PVC)
H H
C C
CH3 H
H H
C C
CH3 H
H
C
CH3
Polypropylene (PP)
MAT_SCI 201/301 | S23 | 37
Polymer Composition
Most polymers are hydrocarbons*
•
Made up of H and C
Strong covalent intramolecular bonds
4 valence e- per C atom
1 valence e- per H atom
Weaker hydrogen and secondary intermolecular bonds
Polymers therefore have relatively low melting/boiling points
[or mer (C2H4)]
*Others:
polysiloxanes (silicones), polyphosphates (biological processes), S-based, B-based…
n
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Physical Properties and Chain Length
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Bulk or Commodity Synthetic Polymers
“Vinyl”
“Teflon”
“Styrofoam”
Do not memorize repeat units.
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Bulk or Commodity Synthetic Polymers (cont’d)
“Acrylic”
“Phenolics”
“Nylon”
“Polyester”
Do not memorize repeat units.
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Unsaturated Hydrocarbons
•
Each carbon not bonded to 4 other atoms
•
Other atoms or groups can bond to the original molecule
•
Double & triple bonds somewhat chemically unstable
→ can form new bonds.
•
Double bond found in ethylene or ethene (C2H4)
•
•
4-bonds, but only 3 atoms bound to C atoms
Triple bond found in acetylene or ethyne (C2H2)
Strength (dissociation energy) and chemical stability is not the same: https://pubs.acs.org/doi/full/10.1021/acs.jpca.6b03631
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Polymerization and Polymer Chemistry
•
Chain-growth/addition polymerization
•
Monomer unit daisy-chained one at a time to form linear macromolecule
• 3 stages: initiation, propagation, and termination:
•
Ethylene (monomer) reacted to form polyethylene
Initiation
Free Radical
Monomer
(ethylene)
Propagation
•
Chain forms via sequential addition of monomer units
• Active site (unpaired e-) transfers in each reaction step.
Radicals for polyethylene: TiCl3 complexes. Watch it happen.
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Polymerization and Polymer Chemistry
•
Step-growth/Condensation polymerization
Intermolecular reactions involving >1 monomer species giving off
a low MW condensate (e.g., HCl)
• Example: Nylon 6,10
•
in
Sebacoyl chloride
cyclohexane
in
6-diaminohexane
water
+
Note – there is a distinction between step-growth and condensation reactions, but the vast majority of
step-growth produce condensate.
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Let’s Make Some Nylon (Back-up Video Below)
See nylon being pulled from solution:
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Exercise 8.1.1
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Molecular Conformation of Polymers
•
Conformation is the shape of the polymer chain
•
Implies the shape that results due to the rotation about bonds.
• Chain bending and twisting are possible by rotation of carbon atoms around their chain single bonds
•
It is not necessary to break the chain bonds to alter the molecular shape
Adapted from Fig. 14.5, Callister & Rethwisch 8e.
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Chain End-to-End Distance, r
•
Rotation about bond:
•
Results in bending, coiling, or kinking of chain
• Chain can entangle with itself or other chains.
• Affects mechanical properties (elasticity and strength)
• How to model the conformation of a polymer chain?
• Why not random walk?
• (Consider limitations in homework)
• l = length between bonded atoms in chain
• n = number of bonds in the molecule
r << total chain length
For polymer chains of typical lengths
Derivation
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Polymer Molecule Classification [VL]
(Architecture)
Be able to identify/explain tfhe structural classifications above.
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Size - Molecular Weight [VL]
•
•
Molecular weight, M: Mass of a mole of chains.
MW can be extremely large.
Low M
Not all chains in a polymer
are the same length!
— i.e., there is a distribution
of molecular weights
High M
•
MW can be defined in several ways:
•
�𝑛𝑛 )
Number averaged, (𝑀𝑀
• Bin the chains into a set of ranges and determine the number fraction in each range
•
�𝑤𝑤 )
Weight averaged, (𝑀𝑀
• Based on total weight fraction of molecules within a given range.
•
Degree of polymerization, DP
• Number of monomers per polymer chain
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Molecular Weight Distribution [VL]
�𝑛𝑛 : number-average molecular weight:
𝑀𝑀
�𝑤𝑤 : weight-average molecular weight
𝑀𝑀
Break up molecular distribution into ranges:
Mi = mean molecular weight in range i
xi = number fraction of chains in size range i
wi = weight fraction of chains in size range i
Adapted from Fig. 14.4, Callister & Rethwisch 8e.
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Molecular Weight Distribution [VL]
Number Fraction
Weight Fraction
Weight fraction is always* higher than number faction
→ Shifted towards large molecular weight molecules
Adapted from Fig. 14.3, Callister & Rethwisch 8e.
*There is one case this is not true… can you think of it?
MAT_SCI 201/301 | S23 | 54
Determine the Average Mass of the Class [VL]
•
What is the average weight of the students in this class?
•
(a) Based on the number fraction of students in each mass range?
• (b) Based on the weight fraction of students in each mass range?
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Determine the Average Mass of The Class: Solution [VL]
Sort the students into weight ranges.
1.
1.
• 40 lb ranges gives the following table:
total number
total weight
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Determine the Average Mass of The Class: Solution [VL]
1. Calculate the number fraction and weight fraction of students in each range
:
For example: for the 81-120 lb range
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Determine the Average Mass of Class, Solution [VL]
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�𝑛𝑛 vs 𝑀𝑀
�𝑤𝑤 [VL]
The Use of 𝑀𝑀
• Number-average molecular weight is used when a property or interaction is dependent on number distribution.
• Thermodynamic properties depending on number of molecules in a system.
• Colligative properties (often in solution):
• Vapor pressure
• Glass transition temperature
• Important for processing
• Weight-average molecular weight is used when a property
or interaction is dependent on the weight distribution.
• Mechanical properties depend more highly on the
number of large molecules
• Tensile strength
• Rigidity
• Characterization:
• Visible light scattering
• X-ray scattering
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Degree of Polymerization (DP), Dispersity (Đ)
•
•
DP is the average number of repeat units (monomers) per chain
Dispersity Đ is a measure of the heterogeniety of the chains in the polymer
H H H H H H H H H H H H
H C C (C C ) C C C C C C C C H
H H H H H H H H H H H H
Where m = molecular weight of the repeat unit.
Example,
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Why Do We Care About Polymer Size Distribution?
• Molecular weight influences a broad spectrum of materials properties
• Mechanical performance generally improves with molecular weight
• Long chains interweave with each other → more interactions, higher thermal stability
• Large molecular weights influences melt temperature and viscosity (higher)
• Large molecular weights can increase entanglement, increasing tensile strength.
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Exercise 8.1.2
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Exercise 8.1.2
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Polymers: Chemistry, Size, Architecture, Conformation
Size
Chemistry
H H H H H H
C C C C C C
H H H H H H
Polyethylene (PE)
H H H H H H
C C C C C C
H Cl H Cl H Cl
Poly(vinyl chloride) (PVC)
Shape/Conformation
Architecture
Adapted from Figs. 14.4, 14.6 Callister & Rethwisch 9e.
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Polymer Architectures
•
Linear
•
Repeat units joined end-to-end
• Long flexible chains (“spaghetti”)
• Extensive Van der Waals and hydrogen bonding between chains
• Examples: HDPE, PVC, PS, Nylon, fluorocarbons
•
secondary
bonding
Linear
Branched
•
Side-branch chains connected to main chain
• Chain-packing efficiency reduced
• Thermodynamically unavoidable (processing)
• Lower density
• Often weaker interchain interactions
• Examples: LDPE
Branched
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Molecular Structures
•
Cross-linked
•
Adjacent chains joined by covalent bonds
• Occurs during synthesis
• Non-reversible processing
• Examples: Rubbers (via vulcanization using sulfur)
•
Cross-Linked
Network
•
Monomers forming 3 or more covalent bonds
• Form 3-D networks
• Examples: Epoxies, polyurethanes, phenol-formaldehyde
Network
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Molecular Structures for Polymers
•
Generalized chain configurations and strength
Increased Strength
secondary
Branched
bonding
Linear
Cross-Linked
Network
• Thermoset Polymers
• Thermoplastic Polymers
• Covalent bonds between chains or
• Secondary bonds between chains
formation of network polymers
*Branched
and linear may be interchanged, depending on specifics.
and network may be interchanged, depending on specifics.
*Cross-linked
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Polymer types
•
Thermoplastics
•
Thermosets
•
Linear polymers with 2 links/mer
• Branched with flexible chains
• Weak secondary bonds between chains
• Strong bonds within chains
•
•
•
They can be softened (and melted)
repeatedly by raising the temperature
• Relatively soft
• Can be recycled
• Examples: PE, PVC
*This isn’t strictly true, but it is true in practice.
However, clever people are making developments.
Crosslinked with 3 links/mer or network
polymers
• Rigid character to strong, immovable covalent
bonds
After a thermoset is formed, it cannot* be
reshaped or remelted
• Difficult to recycle*
• Examples: vulcanized rubber, epoxies, polyester
resins
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Exercise 8.1.3
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Exercise 8.3.1
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Summary
•
Most polymeric materials are composed of large molecular chains with
some size distribution.
•
Molecular:
•
Chemistry
• Size (molecular weight/degree of polymerization)
• Shape (conformation),
• Architecture (linear…network)
• Crystallinity
•
Polymers are highly engineerable materials – a key principle of this
engineering is influencing the interchain interaction.
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