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Unformatted text preview: Lecture 4: Polymer
Conformation
Real Chains 1
Monday, October 4, 2010 Recall from last time …
A polymer can adopt many conformations
Polymers with no interactions between monomers separated by many bonds along
the chain are called IDEAL CHAINS
The mean square end to end distance for an ideal (long, linear) chain is:
R 2 ≅ C∞ nl 2 C∞ is a characteristic ratio that describes the stiffness of the chain
We have defined the Kuhn segment, b, for a polymer chain, which is related to the
number of monomers N. From the Kuhn segment, we obtain an expression for the
mean square end to end distance for ideal chains
Several models of ideal chain conformation exist: freely rotating chain, freely jointed
chain, wormlike chain, hindered rotation and rotational isomeric state
The mean square radius of gyration, <Rg2>, is defined as the average square
distance from all monomers to the center of mass of the polymer
General scaling for IDEAL CHAIN:
RN
Monday, October 4, 2010 1
2 2 Polymer Conformation
• The conformation of a polymer chain will be strongly affect by the intraand intermolecular interactions of the chain with itself and its environment
• In order to better understand and predict the 3D shape of a polymer
chain based on these interactions, we need a starting point (we did this last
time)
• Models of conformation often treat polymers as ideal chains, but the real
world (and our experiments) is more complicated
DEFINITIONS
Ideal chain: no interactions between repeat units far apart along the chain,
even if they approach each other in space (do not interact with solvent or
itself)
Real chain: attractive and/or repulsive interactions between/within repeat
units as well as with solvent (interacts with solvent and itself) 3
Monday, October 4, 2010 Today’s Concepts
Real chains do not exist in a vacuum, but rather in a solution with either solvent,
or other chains surrounding them
Real chains have interactions with the things surrounding them, and between
monomers along the chain
These interactions can be described in terms of excluded volume, which
describes the net twobody interaction between monomers
Temperature can affect excluded volume
When concentration is high, net three body interactions become important
A polymer “solution” can be described as good, theta, and poor; these can change
with concentration 4
Monday, October 4, 2010 Potential Energy versus Distance
Too crowded repulsion Potential
Energy Potential due to
repulsive forces Potential due to
attractive forces Perfect balance of
attraction and repulsion
Monday, October 4, 2010 r Too far away 5 Possible interactions of beads on the chain
•
•
•
• Dispersion forces (van der Waals)
Dipole/dipole interactions
Hydrogen bonding
Coulombic Interactions 6
Monday, October 4, 2010 van der Waals Forces, dipole dipole interactions
•
• weak interactions between dipoles (δ+ and δ )
alignment of liquid crystals, gecko adhesion ⎡⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎤
φ LJ ( r) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥
⎣⎝ r ⎠ ⎝ r ⎠ ⎦
ϕ = LennardJones Potential
(energy)
σ = hard shell radius
r = nuclear separation repulsive, exchange energy ε attractive, dispersion energy 7
Monday, October 4, 2010 Hydrogen Bonding
•
hydrogen is δ+ from bonding to O, N, F, (electronegative atoms,
forms polar covalent bond)
•
δ+ H associates with lone pairs of electrons on heteroatom,
typically O, N, sometimes (weaker) S, P, B, Cl, Br, I H
O H O N N O O O N O R O H O O N
H O R O N
H H
R O N
O R O O O O H N N
H N N H O O N
O H H O R O O
N O R H 8
Monday, October 4, 2010 Ionic Bonds
• usually metallic and nonmetallic atoms involved • metal gives up valence electrons to nonmetallic
atom(s) • two oppositely charged IONS come together to form an ionic bond + Ca2+
 9
Monday, October 4, 2010 Summary of Interaction Strengths
Characteristics Approx.
Strength
(kcal/mole) Examples Dispersion forces Short Range,
varies as 1/r6 0.2  0.5 PE, PSty Dipole/Dipole
Interactions Short range,
varies as 1/r6 0.5  2 PAN, PVC Strong Polar Interactions Complex form,
and Hydrogen Bonds
but also short
range 1 10 Nylons,
Polyurethanes Coulombic Interactions
(Ionomers) Increasing interaction strength Type of Interaction 10  20 alginate Long range,
varies as 1/r 10
Monday, October 4, 2010 Excluded Volume
•
•
•
• Excluded volume describes the volume around a monomer from which
another monomer is excluded
A positive excluded volume means that repulsion occurs between
monomers
A negative excluded volume means that the monomers are attracted to
each other
When excluded volume equals zero, this is called the theta state and it
describes an ideal chain Monomers have some repulsion,
resulting in a stretched chain ν>0
R~N3/5
Monday, October 4, 2010 Remember, this is our
starting point
Monomermonomer repulsion is
perfectly balanced by attraction ν=0
R~N1/2 Monomermonomer attraction
results in smaller coil v<0
R~N1/3 Excluded Volume
•
•
• U(r) (effective interaction potential) is the energy cost of bringing two
monomers from infinite separation to within distance r of each other.
Hardcore repulsion when r is small
Often, attractive at intermediate distances U(r) includes all interactions, including those mediated by solvent 12
Monday, October 4, 2010 Excluded Volume
• • • Boltzmann factor describes the probability of finding two
monomers separated by a distance r in a solvent at
temperature T
The Mayer ffunction is defined as the difference
between the Boltzmann factor for two monomers at
distance r and that for the case of no interaction (or
infinite distance)
The excluded volume v summarizes the net twobody
interaction between monomers Hardcore repulsion e − U (r )
kB T f (r ) = e −U ( r )
kB T −1 What is the area
under this function?
The area describes
the probability of
finding a monomer
close to another
monomer Attractive interactions
13 Monday, October 4, 2010 Excluded Volume
• The EXCLUDED VOLUME v is minus the integral of the Mayer ffunction
over the whole space: f (r ) = e
• •
• −U ( r )
kB T −1 U (r )
−
⎛
⎞
v = − ∫ f (r )d 3r = ∫ ⎜ 1 − e kB T ⎟ d 3r
⎝
⎠ EXCLUDED VOLUME is the net twobody interaction between
monomers. Hard core repulsion (r<1) makes a negative contribution to
the integration of the Mayer ffunction, and a positive contribution to
excluded volume. Attractive forces between monomer (r>1) makes a
positive contribution to the Mayer ffunction and a negative contribution
to excluded volume
A net attraction has excluded volume less than zero, and a net repulsion
has excluded volume greater than 0.
In practical terms, excluded volume is the volume around a segment
(monomer) that another segment (monomer) cannot occupy, determined
by energetics of the interaction between monomers. 14
Monday, October 4, 2010 Flory theory, cont.
•
• •
•
•
• Flory theory overestimates repulsion energy (correlations between
monomers along the chain are omitted)
The number of contacts per chain, in Flory theory, scales with N1/5, but in
computer simulations of random walks , the number of contacts
between monomers that are far apart does not scale with N. The result
is that the attractive forces are also overestimated
The two overestimations cancel each other out.
The elastic energy of the chain is also overestimated, because the ideal
chain conformational entropy is assumed
Still, Flory theory is simple, and the predictions are in good agreement
with experiment, and more sophisticated experiments.
Flory theory leads to a UNIVERSAL POWER LAW dependence of the
size of the polymer on the number of monomers N R~N υ 15
Monday, October 4, 2010 R~N υ • The Flory approximation of the scaling exponent is ν =3/5 •
• For an ideal chain, the exponent is ν= 0.5
If we again think of a chain as a fractal object, the fractal dimension of
an ideal chain is D=1/ν=2 • For a swollen chain, the fractal dimension is D=1/ν=5/3 • More sophisticated theories lead to a better estimate of the scaling
exponent ν for a swollen linear chain in 3D: υ ≅ 0.588 16
Monday, October 4, 2010 Flory Theory of a polymer in a poor solvent
• The Flory free energy for polymer chain is given by: ⎛ R2
N2 ⎞
F ≈ kT⎜ 2 + v 3 ⎟
R⎠
⎝ Nb
•
•
•
• • In a poor solvent, the excluded volume is negative, indicating a net
attraction
Both entropic and enthalpic contributions decrease with decreasing R
Strong collapse of a polymer into a point is not physically possible; need
stabilizing term in the free energy
There is an entropic cost to confining a chain, so we can add in a term to
account for that: ⎛ R 2 Nb 2
N2 ⎞
F ≈ kT⎜ 2 + 2 + v 3 ⎟
R
R⎠
⎝ Nb The free energy still has a minimum at R=0; the confinement entropy
term is still not enough
17 Monday, October 4, 2010 What now?
•
•
• Stabilization of the coil comes from other terms of the interaction part of
free energy
The interaction energy per unit volume is expressed a virial expansion in
powers of the number density of monomers cn
In a pervaded coil volume R3, the excluded volume term is the first term
in the virial series and counts twobody interactions as vcn2 ( ) Fint
2
3
≈ kT vcn + wcn + ...
3
R • The next term in the expansion counts threebody interactions as wcn3,
where w is the threebody interaction coefficient
⎛ N2
⎞
N3
Fint ≈ kT ⎜ v 3 + w 6 + ...⎟
R
⎝R
⎠ • At low concentration, the twobody term dominates the interaction, but
the threebody term becomes important at higher concentrations and
can stabilize the collapse of the globule
18 Monday, October 4, 2010 Excluded Volume Allows us to understand Chain
Behavior
• • • If the attraction between monomers just balances the effect of the hard
core repulsion the net excluded volume is zero and the chain will adopt
a nearly ideal conformation (θcondition)
If the attraction between monomers is weaker than the hard core
repulsion, the excluded volume is positive and the chain swells; the coil
size is larger than the ideal size
If there are attractive forces between monomers stronger than the hard
core repulsion, excluded volume is negative and the chain collapses; the
size of the globule is smaller than ideal 19
Monday, October 4, 2010 What happens when we deform the chains? tension compression 20
Monday, October 4, 2010 Temperature Dependence of Excluded Volume
T − Tθ 3
v≈
b
T
• For T<Tθ, the excluded volume is negative (coil collapse, poor solvent) • For T> Tθ, the excluded volume is positive and the chain is swollen • For T>> Tθ, excluded volume becomes independent of temperature • At T= Tθ, the net excluded volume is zero and the chain adopts an ideal
conformation 21
Monday, October 4, 2010 How does concentration affect chain conformation? 22
Monday, October 4, 2010 Polymer Chains in Solution
•
•
• We were treating (ideal) polymer chains as if they existed in space; in
reality, they are surrounded by something
Since polymers are never in the gas phase, they have to be surrounded
by something – solvent or other polymers
Polymer solution behavior is affected by concentration c cv mon N Av
φ= =
ρ
M mon
Volume fraction, φ: the ratio of the occupied volume of polymer in the
solution and the total volume of the solution
c: polymer mass concentration
M mon
vmon:occupied volume of a single monomer
ρ=
v mon N Av
ρ: polymer density
Mmon: molar mass of the monomer
23
Monday, October 4, 2010 Pervaded Volume, V
• The pervaded volume V is the volume of solution spanned by the
polymer chain: V ≈ R3
This volume is typically orders of magnitude larger than that occupied by
the polymer chain.
The fractal nature of the polymer chain N~RD with typical fractal dimensions
of D<3 means that most of V is occupied by solvent or other chains. 24
Monday, October 4, 2010 Solution Regimes for Flexible Polymers
The volume fraction φ, of a single molecule inside its pervaded volume V is
called the overlap volume fraction, φ* or the corresponding overlap
concentration c*:
Nv
φ* = mon
V Nv mon
M
c* = ρ
=
V
VN Av Polymer coils in dilute solutions (where the average distance between chains
is larger than their size) are far away from each other. In dilute solutions, the
solution properties are similar to that of the solvent, with slight modifications
due to the presence of the polymer. 25
Monday, October 4, 2010 Solution Regimes for Flexible Polymers: Semidilute
Solutions are called semidilute at polymer volume fractions above overlap.
While most of the polymer volume fractions in semidilute solutions is still
very low (φ<<1), the polymer coils overlap and dominate most of the physical
properties of the solution.
In semidilute solutions both solvent and other chains are found in the
pervaded volume of a given coil. The overlap parameter P is the average
number of chains in a pervaded volume randomly placed in solution: φV
P=
Nv mon 26
Monday, October 4, 2010 Solution Regimes for Flexible Polymers: Bulk
In the absence of solvent, polymers can form a bulk liquid state, called a
polymer melt.
Polymer melts are neat polymeric liquids above their glass transition
temperature and melting temperatures
A macroscopic piece of a polymer melt remembers its shape and has
elasticity on short time scales, but exhibits liquid flow (with high viscosity) at
long times.
In the polymer melt the overlap parameter is large (P>>1) and the strong
overlap with neighboring chains leads to entanglement that greatly slows the
motion of polymers. However, individual chains in a polymer melt do move
over large distances on long time scales, a property characteristic of fluids. 27
Monday, October 4, 2010 Solution Regimes for Flexible Polymers 28
Monday, October 4, 2010 Summary of Real Chains
•
• • • Real chains have interactions with their environment
If the attraction between monomers just balances the effect of the hard
core repulsion the net excluded volume is zero and the chain will adopt
a nearly ideal conformation (θcondition)
If the attraction between monomers is weaker than the hard core
repulsion the excluded volume is positive and the chain swells; the coil
size is larger than the ideal size; the chain is a selfavoiding walk
If there are attractive forces between monomers stronger than the hard
core repulsion, excluded volume is negative and the chain collapses; the
size of the globule is smaller than ideal (below Tθ) • Most chains in a poor solvent collapse into a globule, agglomerate with
other chains and precipitate from solution; this occurs far below Tθ • This is the nonsolvent limit and an individual chain in that solvent has a
fully collapsed chain; most chains therefore precipitate from solution
into a melt and then adopt ideal conformation to maximize entropy
Stretching a real linear chain in a good solvent is easier than an ideal
chain
Compressing an ideal chain is easier than compressing a real chain
Excluded volume changes with temperature in the vicinity of Tθ
29 •
•
• Monday, October 4, 2010 References
• Rubenstein and Colby Chpt 3 • Strobl Chpt 23 (Chpt 3 has more info than we’ve covered so far) • Painter and Coleman Fundamentals Chpt 9.D. • Topics in Polymer Physics by Richard Stein and Joseph Powers Chpt 2 30
Monday, October 4, 2010 ...
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This note was uploaded on 11/29/2010 for the course BME 104 taught by Professor Kasko during the Fall '10 term at UCLA.
 Fall '10
 KASKO

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