3.063 Polymer Physics Spring 2007

• Viewpoint: somewhat more general than just polymers:

“Soft Matter”Polymers, Colloids, Liquid Crystals, Nanoparticles and Hybrid Organic-

Inorganic Materials Systems.Reference: Young and Lovell, Introduction to Polymer Science

(recommended)

Demos and Lab Experiences: Molecular Structural Characterization (DSC, GPC, TGA, Xray, AFM, TEM, SEM…) and Mechanical & Optical Characterization

Course Info

eltweb.mit.edu/3.063 - course website: notes, hmwks…Eric VerploegenMs. Juliette Braun Office hours - How are Wednesdays at 230-330pm?

Mini AssignmentSend Prof. Thomas– contact info/ name/year/major/email/credit/listener– and interests relevant to topics in this course– Any expertise you have on lab demos, characterization tools, cool

samples…

Hard vs. Soft Solids(for T ~ 300K)

• Hard matter: metals, ceramics and semiconductors: typically highly crystalline. U >> kT

• Soft matter: polymers, organics, liquid crystals, gels, foods, life(!). U ~ kT

• Soft matter is therefore sometimes called “delicate material” in that the forces holding the solid together and causing the particular atomic, molecular and mesoscopic arrangements are relatively weak and these forces are easily overcome by thermal or mechanical or other outside influences. Thus, the interplay of several approximately equal types of forces affords the ability to access many approximately equalenergy, metastable states. One can also “tune” a structure by application of a relatively weak stimulus. Such strong sensitivity means that soft matter is inherently good for sensing applications.

In 3.063 we aim to understand:• The Key Origins of Soft Solid Behavior:

relatively weak forces between moleculesmany types of bonding, strong anisotropy of bonding (intra/inter)

wide range of molecular shapes and sizes, distributionslarge variety of chemistries/functionalitiesfluctuating molecular conformations/positionspresence of solvent, diluententanglementsmany types of entropyarchitectures/structural hierarchies over several length scales

• Characterization of the molecular structures and the properties of the soft solids comprised of these molecules

1

0.8

0.6

0.4

0.2

010 6 10 5 10 4 10 3 10 2 10 1 10 0

Carbon nanotube

Photoresist (150 nm)

Intel 4004 Patterned atoms

HumanHair

Red blood cells

Dimensions (nm)

Virus

DuPont microstructured fibers.

2-photon polymerized photonic crystal Nature 1999

DNA molecule AFM

Scale in Soft Matter

Interactions in Soft Solids

• Molecules are predominantly held together by strong covalent bonds.

• Intramolecular - Rotational isomeric states• Intermolecular potential

– Hard sphere potential– Coulombic interaction– Lennard Jones potential (induced dipoles)– Hydrogen bonding (net dipoles)

• “hydrophobic effect” (organics in water)

Polyelectrolyte Domain Spacing Change by e-Field

nPS - nP2VP ~ 1.6nH2O = 1.33

Fluid reservoir

PS

QP2VP

Apply field

Photo of tunable gels removed dueto copyright restrictions.

Initial film thickness: ~ 3 μm

Total # of layers: ~ 30 layers

Structure of Soft Solids• SRO - (always present in condensed phases)

– intra- and inter-• LRO - (sometimes present)

– Spatial: 1D, 2D, 3D periodicities– Orientational

• Order parameters (translational, orientational…)• Defects -

– influence on properties; at present defects are largely under-appreciated; – e.g. transport across membranes…self assembly-nucleation, mutations, diseases

• Manipulation of Orientation and Defects: Develop methods to process polymers/soft solids to create controlled structures/hierarchies and to eliminate undesirable defects

Polymers/Macromolecules

H. Staudinger (1920’s) colloidal ass’y vs. molecule

“MACROMOLECULAR HYPOTHESIS”

single repeating unit(MER)

covalent bonds

A SINGLE REPEAT UNIT → MONOMER (M)MANY REPEAT UNITS → POLYMER (M)n

“Macromolecule” - more general term than polymer

POLYMERIZATIONCommon reactions to build polymers:1. Addition: Mn + M → Mn+1

2. Condensation: Mx + My → Mx+y + H2O or HCl / etc.

Think of a polymer as the endpoint of a HOMOLOGOUS SERIES:

N = 1 2 3 nDegree of

Polymerization

CH2=CH2 → – CH2 – CH2 – → – CH2 – CH2 – n≥100

Example: Polyethylene (n ≈ 104)

n<100monomer oligomer polymer

How do the properties change with n?

N

H

O

N

H

O

N

H

O

amide linkage• Members of the nylon family are named by counting thenumber of carbon atoms in the backbone betweennitrogen atoms.

O

N

H

O

N CH2

H

O

N

H

CH2

nylon 6O

N

H

O

N CH2

H

O

N

H

CH25 5

6 carbonatoms

Tm = 215°C

note nylon n=∞ = polyethylene (!) Tm ~ 140°C

nylon 2 Tm > 300°C

Proteins are “decorated nylon 2”

O

N

R

H

H

n

R = various side groups(20 possible amino acids)

R = H → glycineR = CH3 → alanine, etc.

Proteins and Polyamides (Nylons)

N

H

O

N

H

O

N

H

O

N

H

O

N

H

O

N

H

O

Hydrogen bonds

Characteristic Features of Macromolecules• Huge Range of Structures & Physical Properties• Some examples:

• Insulating --------> Conducting• Light emitting• Photovoltaic, Piezoelectric• Soft elastic ---------> Very stiff plastic

– Ultra large reversible deformation– Highly T, t dependent mechanical properties

• Zero/few crystals ------> High Crystallinity

• Readily Tunable Properties:

Weak interactions between molecules, so chains can be readily reorganized by an outside stimulus

Chain Conformations of Polymers: 2 Extremes

• 1. Rigid rod: Fully Extended Conformation – < r2>1/2 ≈ n l = L

• 2. Flexible coil: Random Walk – < r2>1/2 ≈ n1/2 l - note this is much smaller than L

n is the number of links in the chain with each link a step size l so the contour length of the chain L is nl;

As a measure of size of the chain, we can have two situations:

Flexible Coil

Flexible Coils

Rigid Rod

Rigid Rods

Extreme Conformations of (Linear) Polymers

Entanglements

Isolated molecules

.357 Magnum, 22 layers of Kevlar

Polymer Architectures

x

x

Flexible Coil

Rigid Rod

Cyclic (Closed Linear)

PolyrotaxaneInterpenetrating

Networks

Densely Cross-Linked

Lightly Cross-Linked

Regular Star-Branched

Regular Comb-Branched

Random Long Branches

Random Short Branches

Regular Dendrons

Controlled Hyperbranched(Comb-burstTM)

Random Hyperbranched

Dendrimers(StarburstR)

LINEAR CROSS-LINKED BRANCHED DENDRITIC(I) (II) (III) (IV)1930's- 1940's- 1960's- 1980's-

Figure by MIT OCW.

1-D Random Walk

R2 = N b2

-4 4x = 0

(a)

(a) Initial distribution of atoms, one to a line.

-4 4

(b)

0

2 R2

2 R2

(b) Final distribution after each atom took 16 random jumps. is the calculated root mean square for the points shown.

N steps of size b

Figure by MIT OCW.

Gaussian Distribution

P(r,n) =2π nl2

3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−3/2

exp −3r2

2nl2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ Note: units are [volume-1]

P(r,n) 4π r2 dr0∞

∫ = 1 normalized probability

r2 = r2 P(r ,n) 4π r2 dr∫

Probability of distance rbetween 1st and nth

monomer units for an assembly of polymers

is a Gaussian distribution, hereshown for 3D

1.00

0.75

0.50

0.25

0.000.0 0.5 1.0 1.5 2.0

p(R

)/p(0

)

R/<R2>

Figure by MIT OCW.

Flexible Coil Chain Dimensions

• Mathematician’s Ideal Random Walk

• Chemist’s Chain in Solution and Melt

• Physicist’s Universal Chain

3 Related Models

22 bNr =

r2 = n l2 C∞ α 2

22 nlr =

Mathematician’s Ideal Chain

21

21

2, :law Scaling nr nl ∝

n monomers of length l li = l

r l ,n = l 1 + l 2 + ......+ l n = l i

i=1

n

∑

r1,n = 0 since r l ,n{ }are randomly oriented

Consider r1,n2

12 ≠ 0

r l ,n ⋅ r l ,n = li ⋅ l j∑∑= l 1 ⋅ l 1 + l 2 + l 3 + ...( )+ l 2 ⋅ l 1 + l 2 + l 3 + ...( )+ ...( )

l1 ⋅ l1 l1 ⋅ l2 l1 ⋅ l3 ... l1 ⋅ ln

l2 ⋅ l1 l2 ⋅ l2

..

ln ⋅ l1 ln ⋅ ln

→

l2 0 0 ... 0. l2 .. . .. . .0 l2

r l ,n

2 = nl2 r l ,n

212 = n

12 l

No restrictions on chain passing through itself (excluded volume), nopreferred bond angles

Chemists’ Chain: Fixed Bond Angle (and Steric Hinderance)

Fixed θ:

• projections of bond onto immediate neighbor

li ⋅ li +1 = l2 −cosθ( )li ⋅ li +2 = l2 −cosθ( )2

In general: li ⋅ li +m = l2 −cosθ( )m

l2 , l2 −cosθ( ) , l2 cos2 θ , l2 −cos3 θ( )....

Factor out l2:

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−−

−−− −

1.1

coscos1coscos.coscos1cos

)cos(...coscoscos1

22

2

132

θθθθθθθ

θθθθ n

⎟⎠⎞

⎜⎝⎛

+−

≈∑ θθ

cos1cos12nl

Simpliest case:

Chain comprised ofcarbon-carbon single

bonds

Figure by MIT OCW.

C4

C4

C4

C2

C1

C3

(1)

(2) (3)

D

2

Special case: Rotational conformations of n-butane

CH3CH2CH2CH3

Views along the C2-C3 bond

Planar trans conformer is lowest energyV(φ)

V(φ)

High energy states

Low energy states

5

00- 180o - 120o - 60o 60o 120o 180o

10

15

20

gauche gaucheplanar trans

φ

φPotential energy of a n-butane molecule as a function of the angle of bond rotation.

Pote

ntia

l ene

rgy/

kJ m

ol-1

H

H H

H

H

H

H

H

H

HHH

H

HH

H

HH HH

H

H

HHCH3

CH3

CH3 CH3 CH3

CH3

CH3

H3C

H3C

CH3 CH3

CH3

eclipsed conformations

Gauche GauchePlanar

eclipse

eclipse eclipse

eclipse

Figure by MIT OCW.

Conformers: Rotational Isomeric State Model

• Rotational Potential V(φ) – Probability of rotation angle phi P(φ) ~ exp(-V(φ)/kT)

• Rotational Isomeric State (RIS) Model– e.g. Typical 3 state model : g-, t, g+ with weighted probabilities– Again need to evaluate taking into account probability of a φ

rotation between adjacent bonds

• This results in a bond angle rotation factor of

where

with P(φ) = exp(-V(φ)/kT)

1+⋅ ji ll

r2 = nl2 1− cosθ1+ cosθ

⎞ ⎠ ⎟

1+ cosφ1− cosφ

⎛

⎝ ⎜

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = nl2C∞

φφ

cos1cos1

−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎜⎜⎝

⎛

−+

⎟⎠⎞

+−

=∞ φφ

θθ

cos1cos1

cos1cos1C

cosφ =cos(φ)P(φ)dφ∫

P(φ)dφ∫

combining

The so called “Characteristic ratio”and is compiled for various polymers

The Chemist’s Real Chain

• Preferred bond angles and rotation angles:– θ, φ. Specific bond angle θ between mainchain atoms (e.g. C-C bonds) with rotation

angle chosen to avoid short range intra-chain interferences. In general, this is called “steric hinderance” and depends strongly on size/shape of set of pendant atoms to the main backbone (F, CH3, phenyl etc).

• Excluded volume: self-crossing of chain is prohibited (unlike in diffusion or in the mathematician’s chain model): Such contacts tend to occur between more remotesegments of the chain. The set of allowed conformations thus excludes those where the path crosses and this forces <rl,n

2 >1/2 to increase.

• Solvent quality: competition between the interactions of chain segments (monomers) with each other vs. solvent-solvent interactions vs. the interaction between the chain segments with solvent. Chain can expand or contract.

• monomer - monomer

• solvent - solvent εM-M vs. εS-S vs. εM-S• monomer - solvent

Excluded Volume

• The excluded volume of a particle is that volume for which the center of mass of a 2nd particle is excluded from entering.

• Example: interacting hard spheres of radius a– volume of region denied to sphere A due to presence of

sphere B– V= 4/3π(2a)3 = 8 Vsphere

but the excluded volume is shared by 2 spheres so Vexcluded = 4 Vsphere

Solvent Quality and Chain Dimensions

• Solvent quality: – M-M, S-S, M-S interactions: εM-M, εS-S, εM-S

Solvent quality factor αTheta θ Solvent

20

20

20

20

10

10

10 10

10

10

10 10Poor Solvent

Good Solvent

Solvent In

Solvent Out

Local Picture Global PictureSolvent quality factor

α2 = < r2 >θ

< r2 >

θ − Solvent

Good solvent

Poor solvent

Figure by MIT OCW.

Solvent Quality (α)

• good solvent: favorable εSM interaction so chain expandsto avoid monomer-monomer contacts and to maximize the number of solvent-monomer contacts.

• poor solvent: unfavorable εSM, therefore monomer-monomer, solvent-solvent interactions preferred so chain contracts.

r2 = n l2 C∞ α 2

0-εSS

-εMM-εSM

α > 1

0-εSM-εSS

-εMM

α < 1poorgood

Theta Condition:Choose a solvent and temperature such that…

• Excluded volume chain expansion is just offset by somewhat poor solvent quality and consequent slight chain contraction:

called ⇒ Θ condition (α =1)

r2

θ = n l2 ⋅local steric influence

bond angles, rotation angles⎛

⎝ ⎜

⎞

⎠ ⎟ = n l2 C∞

θ=α

2

22

r

r“solvent quality factor”

Actually expect a radial dependence to alpha, since segment density is largest at center of “chain segment cloud”.

Random Walks

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

RW

SARW

Figure by MIT OCW.

Figure by MIT OCW.