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Unformatted text preview: Lecture 3: Models for
Polymer Conformation
Ideal Chains 1
Monday, September 27, 2010 Recall from last time...
• Isomers of polymers exist
– Structural Isomerism
– Cis/trans (geometric) Isomerism
– Stereoisomerism •
•
•
• Isomerism can be characterized using techniques such as NMR spectroscopy
Different isomers of polymers behave differently (i.e. crystallization)
Isomerism is important in both synthetic polymers and natural polymers
Isomerism affects chain conformation • Depending on the side groups on a polymer chain, their arrangement, and the
backbone in general, polymers can adopt different conformations • Configuration: permanent geometry that results from the spatial arrangement
of the bonds (cis or trans, R or S)
Conformation: geometry adopted by a polymer chain as a result of bond
rotation • 2
Monday, September 27, 2010 Course Overview
1A 1010 1 nm
9 107 108 10 1m
6 10 105 104 1 mm
3 10 1 cm 102 101 organ
collagen tropohelix
red blood cell
small blood vessel
atoms,
e. coli
capillaries
critical size bone defect
bond lengths DNA, virus
eyeball
tooth 1m
0 10 person Lectures 3 and 4: enthalpic and entropic
effects on chain conformation based on
interactions of repeat units (not atoms)
O
HO O H
N
O N
H
O HO
H
N
O
OH HN
H 2N NH Monday, September 27, 2010 NH2 =
=
3 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
• 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) 4
Monday, September 27, 2010 Ideal Chains
• Polymer backbones are flexible due to rotation about sigma bonds • The large number of possible rotations (due to the large number of
bonds) means polymers can adopt many conformations (entropically
favorable) • Ideal polymer chain conformation can be described as a random walk • The basic relationship between the meansquare end to end distance of
a polymer chain and the number of repeat units in the chain is: R 2 = Nb 2
• Polymer size can alternatively (and preferably) be described by a radius
of gyration, which is related to the endtoend distance • Various models of ideal chain behavior take into account geometrical
constraints of chemical bonds and atoms 5
Monday, September 27, 2010 Flexibility Mechanisms in Polymers
• Consider a single repeat unit of PE; the C atoms adopt a tetrahedral
geometry • The bond length between carbon atoms is almost constant at l = 1.54 Å.
The angle between neighboring bonds is 68°. • The main source of polymer flexibility is the variation of torsion angles. 6
Monday, September 27, 2010 Flexibility Mechanisms in Polymers
The trans state of the torsion angle ϕ is the lowest energy conformation
of the consecutive CH2 groups. The changes in the torsion angle lead
to energy variations • The gauche (+ and ) interactions are the next lowest energy. The
energy difference between gauche and trans Δε determines the relative
probability of a torsion angle being in the gauche state in thermal
equilibrium; in general, it is also influenced by the values of the torsion
angles of neighboring monomers. • The energy barrier ΔΕ determines the dynamics of conformational
rearrangements. energy • ΔΕ
Δε 7
Monday, September 27, 2010 Flexibility Mechanisms in Polymer Chains
• Any section of the chain with consecutive trans
states is in a rodlike zigzag conformation. If all
torsion angles of the chain are in the trans state,
the chain has the largest possible value of its endtoend distance Rmax. • Rmax is proportional to the number of bonds in the
chain skeleton, n, and their projected length (lcos
(θ/2) along the contour of the chain. This is
referred to as the contour length of the chain. Rmax = nl cos
•
• • θ
2 Gauche states lead to flexibility in the chain
conformation since each gauche state alters the
conformation from the alltrans zig zag.
Typically, chains are not all trans, but broken up
by gauche states; the chain is rodlike on scales
smaller than the all translength, and flexible on
larger length scales
Typically alltrans sections comprise fewer than
ten main chain bonds Monday, September 27, 2010 8 Possible Chain Conformations
• Assume that a chain segment has three local minima (e.g. trans, gauche,
gauche seen in PE), and that the chain segment can only exist in one of these
three minima (not in the higher energy eclipsing states, for example) • To simplify, we’ll also assume all three minima have the same energy, so we
are exactly as likely to find the segment in one conformation as the next • Let’s think about the first two bonds in this segment – how many different
possibly conformations do we have, ignoring redundancies?
G’
G T’
G’ 3 2! G’
T T’
G’
G’ G T’
G’ Now, if this is just for 2 bonds,
imagine a polymer chain!
That would be an enormous number
of possible conformations.
9 Monday, September 27, 2010 Polymer Chain Conformations
• The enormous number of possible conformations allows us to use a statistical
approach to gain insight into the nature of polymeric materials and their
properties • In order to do this we need a parameter that tells us something about the
shape • We use the end to end distance R
R 10
Monday, September 27, 2010 Robert Brown (17731858)
Around 1827 Brown made a systematic study of the “swarming motion” of
microscopic particles of pollen.
This motion is now referred to as Brownian movement. (Brownian motion).
At first, “…I was disposed to believe that the minute spherical particles were in
reality elementary units of organic bodies.”
Brown then tested plants that had been dead for over a century. He remarks on
the “vitality retained by these molecules so long after the death of the plant”
Later he tested: “rocks of all ages … including a fragment of the Sphinx”
Conclusion: origin of this motion was physical, not biological.
His careful experiments showed that motion was not caused
by water currents, light, evaporation or vibration.
He could not explain the origin of this motion.
Many later experiments by others were inconclusive. But by
the late 1800’s the idea Brownian movement was caused by
collisions with invisible molecules gained some acceptance.
11
Monday, September 27, 2010 Random Walks
• For simplicity’s sake, let’s first consider a one dimensional random walk • We can repeat this random walk several times, and end up with a
distribution 12
Monday, September 27, 2010 Conformation of Polymer Chain
• Imagine a chain of 200 segments • Imagine the polymer chain has a carbon backbone, and that the bond
rotational angles limit it to trans and gauche • The occurrence of trans and gauche is weighted according to potential
energy J, Mark and B, Erman, Rubberlike Elasticity: A Molecular Primer. John Wiley and Sons, New York (1988)
Monday, September 27, 2010 13 Random Walks
Now, let’s generalize. Instead of each step having a length of 1, let the length
be l : R 2 = Nl 2 R 2 0.5 Freely jointed chain model = N 0.5 l This means that if we have a chain of 10000 bonds each with a length of l, the
average end to end distance is 100 bonds!
We’d have to stretch it to 100 times its length before the bonds would
experience stress
There are many many conformations of the chain available, and only one of
these is the fully extended chain; thermal motion would cause an extended
chain to rearrange to one of these coiled up chains
This is the origin of rubber elasticity, and we’ll revisit it later!
Monday, September 27, 2010 14 Random Walks
• For polymers, we consider steps of equal length, defined by chemical
bonds, rather than observing the movement of a particle over an
interval of time • Polymer chains have connectivity and volume constraints – they can’t
reoccupy an already occupied step, and the connectivity to the
previous monomer limits the next step • Steric issues and bond rotations geometrically limit the steps restricted unrestricted 15
Monday, September 27, 2010 Random Walks
• In the previous example, we left a few important details out • Bonds angles need to be restricted to reflect reality Freely rotating chain R 1 + cos θ
= Nl
1 − cos θ 2 2 C
C • We can also account for hindered rotation: Hindered rotation model R 2 θ C ⎛1 + cos θ ⎞⎛1 + ϕ ⎞
= Nl ⎜
⎟⎜
⎟
⎝ 1 − cos θ ⎠⎝ 1 − ϕ ⎠
2 16
Monday, September 27, 2010 Characteristic Ratio
• While the scaling relationship between <R2> and N remains the same,
we can add in a prefactor that helps to describe chain conformation
better: R 2 = C∞ Nl 2 C∞ = •
• R2
Nl 2 This characteristic ratio is the ratio of the actual end to end distance
over the random flight model
It describes chain stiffness or bond rotational freedom 17
Monday, September 27, 2010 Characteristic Ratio
• Depends on the local stiffness of the polymer chain; 79 typical for
flexible chains • Bulkier side groups tend to lead toward higher C∞, but there are many
exceptions • Flexible polymers have many universal properties independent of
repeat unit structure 18
Monday, September 27, 2010 Characteristic Ratio
Polymer C∞ b(Å) ρ(g/mL) M0(g/mol) 1,4polyisoprene 4.6 8.2 0.830 113 1,4polybutadiene 5.3 9.6 0.826 105 polypropylene 5.9 11 0.791 180 PEO 6.7 11 1.064 137 PDMS 6.8 13 0.895 381 PE 7.4 14 0.784 150 PMMA 9.0 17 1.13 655 Atactic polystyrene 9.5 18 0.969 720 19
Monday, September 27, 2010 Kuhn Length
• We’ve (somewhat arbitrarily) defined each step as a bond length, but it
doesn’t need to be defined this way • Instead of using individual monomers or bonds, we can use segments,
that, on average, behave as a freely jointed unit when taken collectively • This is called a Kuhn segment or Kuhn length (b), and it follows the
same scaling as the freely jointed chain model
2
N K bK = R 2 = C∞ Nl 2 • Kuhn segments represent a group of monomers; each monomer is not
completely freely rotating, but the larger segment is
Equivalent freely jointed chain model 20
Monday, September 27, 2010 Wormlike Chain Model (KratskyPorod Model)
•
•
•
•
• A special case of the freely rotating chain model for very small values of
the bond angle
A good model for very stiff polymers
Since the bonds angle is very low, it takes a large number of monomers
to make the equivalent freely jointed segment
The Kuhn length, b, is therefore very large
The characteristic ratio is also very large C∞ = • 1 + cosθ 4
≅2
1 − cosθ θ Worm like chains are stiff on scales shorter than b, but are not
completely rigid and can fluctuate and bend 21
Monday, September 27, 2010 Rotational Isomeric State Model
• Most successful model used to calculate the details of conformations of
different polymers • Bond lengths l and bond angles θ are fixed • Each molecule is assumed to exist only in discrete torsional states
corresponding to the potential energy minima; fluctuations about the
minima are ignored • Each of the n2 torsion angles can be in one of three states  trans,
gauche + or gauche  (t, g+, g) • The whole chain has 3n2 rotational isomeric states • In this model, the states are NOT equally probable; correlations
between neighboring torsional states are included • The relative probabilities of the states of neighboring torsional angles
are used to calculate the meansquare end to end distance and
characteristic ratio. 22
Monday, September 27, 2010 Summary: Ideal Chain Models
FJC FRC HR RIS l fixed fixed fixed fixed θ Free fixed fixed fixed ϕ Free Free Controlled by U
(ϕ ) t, g+, g Next ϕ
independent? Yes Yes Yes No C∞ 1 ⎛1 + cos θ ⎞
⎜
⎟
⎝ 1 − cos θ ⎠ ⎛1 + cos θ ⎞⎛1 + cos ϕ ⎞
⎜
⎟⎜
⎟
⎝ 1 − cos θ ⎠⎝ 1 − cos ϕ ⎠ specific 23
Monday, September 27, 2010 Radius of Gyration
• While the endtoend distance is useful for these models, it’s not
something that we can readily measure in the lab • Additionally, endtoend distance doesn’t work well for branched or
cyclic chains • We can measure the radius of gyration of our polymers using a number
of techniques (we’ll get specific about techniques in a later lecture) • 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 • <Rg2> is related to the mean square end to end distance of a polymer
chain, and varies with the architecture:
Ideal
Chains Linear Ring farm star <Rg2> = Nb2/6 Nb2/12 [(N/f)b2/6](32/f) H polymer
(Nb2/6)89/125
24 Monday, September 27, 2010 Radius of Gyration of Rodlike chains
• Relationships also exist for rigid polymers • The relationship between Rg2 and endto end distance is different than
that derived for an ideal chain: N 2b 2 L2
R=
=
12
12
2
g Rigid
Objects Disk Sphere Rod Cylinder R g2 = R2/2 3R2/5 L2/12 (R2/2)+L2/12 25
Monday, September 27, 2010 SelfAvoiding Walk
• So, we’ve accounted for the bond angle issue, and in some ways, the
hindered rotation • What about the volume issue? Two segments cannot occupy the same
space • This is a more difficult problem • The bottom line: R2 A more generalized version: ν tells us specific
information about the
polymer conformation
(shape)
Monday, September 27, 2010 0.5 = N 0.6 l R2 0.5 = constant × N ν R 2 0.5 = KM ν R = KM ν Random, selfavoiding walk in 3D Really important
Equation
26 A Brief Summary
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 ration 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 27
Monday, September 27, 2010 Summary Cont.
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
It is related to the mean square end to end distance REFERENCES:
Essentials, Chpt 8
Fundamentals Chpt 7
Strobl Chpt 2
Rubenstein/Colby Chpt 2 28
Monday, September 27, 2010 Next Time
Real chains … review the types of interactions
References for next lecture:
Strobl Chpt 23
Fundamentals Chpt 9
Essentials Chpt 11
Rubenstein/Colby Chpt 3 29
Monday, September 27, 2010 “Structure and chain conformation of a (1>6)aDglucan from the
root of Pueraria lobata (WiIId.) Ohwi and the antioxidant activity of
its sulfated derivative” H. Ciu, Q. Liu, Y. Tao, H. Zhang, L. Zhang,
K. Ding. Carbohydrate Polymers 2008, 74, 771778
• Investigating the physical properties and structure of a polysaccharide from a
medicinal herb, P. lobata • The herb is used in Eastern medicine to manage diabetes, hypertension, heart
disease
Rg 2 1 2 = 0.0112 M 0.56 ± 0.008 30
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