Transcript Slide 1

Top 10 Reasons to Understand Macromolecular Dimensions**
10. Proteins—the darn things change size before acting on a substrate.
9. Fast way to follow polymerization of monomers, aggregation of
polymers, binding of proteins or small molecules to vesicles, etc.
8. $1,000,000,000/year rests on getting size vs. mass relations to prove
branching of polyolefins.
7. Multi-viscosity motor oils use polymers that form structures whose
size depends on temperature.
6. Contractile polymers—simulating muscle.
5. How do I buy the best GPC column?
4. Assessment of early stages of crystal formation for protein
crystallography.
3. Do dendrimers change size when they bind metals?
2. Is it Alzheimer’s yet?
And the number one reason to measure and understand polymer
dimensions….
1. Will that condom stop HIV virus?
Skipping Many Pages of Written
Lecture Now
In F2003 basic info on scaling and freely jointed was done
through the device of a quiz that asked students to compute the
# of configurations and estimate the size through Stockmayer’s
mnemonic about 106 PS = 1000A.
We also know fractal dimension and that it’s ½ for FJ, that it can
differ from this for real chains.
The remaining slides in this lecture are to introduce long and
short range nonideality.
 r  nl C
2
2
The yardstick from
the freely jointed
chain model
Characteristic ratio
chemical term
short-range structure
Expansion factor
Physics
long-range nonideality
2
Characteristic Ratio
Define: Cn = characteristic ratio. It allows for expansion of
the chain (compared to FJ) due to chemical, short-term
effects such as a fixed bond angle or “crashes”. It does
not actually depend strongly on n. In the high-n (large
chain) limit, it’s called C and it is just a constant.
Actual values:
C = 6.7 for polyethylene
C = 10 for polystyrene
It should make sense that C is bigger for PS than for PE:
PS has those bulky sidechains that will require a
preponderance of extended trans configurations.
Why we need C
Three things missed in the FJ model:
•Bond angle is fixed (here, the bond angle supplement is shown).
•The monomer units have a finite size.
•Correlations between bond choices can lead to crashes.
q
FJ model
Slightly more realistic with bond angles
Why we need 2
It is hard to predict from chemical bond considerations, but
eventually the chain can come back and crash into itself. Since
two beads (monomer units) cannot occupy the same space (as in
the freely jointed chain of thin bonds) this leads to expansion.
This expansion effect gets
worse as the chain gets
longer.
Ouch!
Gets better or worse if
polymer is restricted to two
dimensions?
This expansion effect can
be opposed by attractions
between chain segments
(in a bad solvent).
Theories for the short-range,
chemical effect, C
Name
Properties
Abbreviation
Freely-rotating
model
Won’t work, but
pedagogically
useful (ironically)
for semi-stiff rods.
FR = freely
rotating model
Bond correlation
model
Oh, yes you can
conquer all those
zillions of
conformations
RIS = rotational
isomeric state
FR model
Holy cow! Now all those
off-diagonal terms don’t
fade away anymore.
In computing the term
l1  l3
First you project l3 onto l2
and then l2 onto l1. This
introduces cos(q)…twice!
The rest is just math and
tricks…see if you can do
it by following the notes
on the website.
The answer is….
1 
2
<r2> = n 
1 
where:
n 


2   (1   ) 
  2
2 

(
1


)



  cosq
The lead term is most important (it contains the large number, n, so…
 1  cosq 
 r  n 

 1  cosq 
2
2
So…..
 1  cosq 
C  
2
 1  cosq 
Not even close to experimental values (from light scattering measurements
of M and Rg) but nevertheless this freely rotating model is a useful
pedagogical tool. The full equation (atop previous slide) is used to get a
theory for weakly bending free rotators—e.g., semiflexible rods.
Also, we learned in this about the importance of projection! One bond onto
another onto another. What if we could do this better, with proper
statistics? That is what Flory did.
Hindered rotation
Cl
H
Cl
Cl
Cl
H
H
Cl
L

Energy
H
-180
H
H
H
H
H
Cl
H
H
H
-120 -60
0
60 120
180

 1  cosq  1  cos  

C , HR  

 1  cosq  1  cos  
180
 cos 

180
V ( )
kt
e
180
cosd
 0.213
Assuming a realistic potential.
V ( ) d
e

180
Bottom line: a nice example of a statistical mechanical
average of some quantity, the cosine of , but….this does
not raise C by nearly enough.
Pentane Effect
What’s missing?
Correlation! When bond configurations
repeat, it leads to a crash. We need a
properly weighted average of the bond
configurations, along with the resultant
geometrical effects.
In other words, weighted projections.
This can only be done for a limited subset of
the  angles.
Rotational Isomeric State Model

Energy
-180 -120 -60
0
60
120
180

V (60)  V (180)  
V (120)  V (120)  V (0)
The RIS model is less accurate for
simple things like <cos> but the
simplicity lets us perform the
projections….and actually
enumerate all those zillions of
conformations. That is, we can
tame that W~3n problem. For this,
and other amazing achievements,
one gets a Nobel prize.
To see how it works, consider some
simple average…
W120 cos(120)  W0 cos(0)  W120 cos(120)
cos   
W120  W0  W120
Note that a statistical weight (e.g., W120 is the probability of that angle) is
always multiplying into the thing whose average we want, cos(120), and
these terms are added.
Then you have to sum, and “normalize” by the sum of weights.
A kind of math operation that multiplies as it adds is matrix algebra.
Matrix Multiplication
 a b  e


 c d  g
f   ae  bg af  bh
  

h   ce  dg cf  gh
If you need to sum together all these terms, other
matrix multiplications can do that.
The computational miracle
Each bond gets a matrix, whose elements
represent the probabilities that it and a
nearest neighbor are in a set of states.
i
i-1
t
g_
g
+
 utt

 u g t

u
 g t
t
g+
utg 
ug  g 
ug  g 
g_
utg  
ug  g  

ug  g  

This matrix is ui. Each element
represents the probabilitiy that bond
i is in a given state and that the
preceding bond is in a given state.
This weight matrix will prevent
crashes.
1   


u  1  0 
1 0  


In trans, out trans =1
In g-, out trans = 1
In g+ , out trans = 1
In trans, out g+=
In g-, out g+ = 
In g+, out g+ = 0
Miracle, continued
The probability of the whole chain – its total
weighted number of states—is related to:
u1 x u2 x u3 x u4….
This thing, a partition function in statistical
thermodynamics, functions like the
denominator when we were computing the
average age of students: you divide it into
the numerator to get whatever average
you need.
So what? Suppose we have n = 1024
bonds, producing W~31024 conformations.
u
1024
   u    u 
 u
512 2
2
256 2
128 2  2  2
 u 64  2 2 2 2  u 32  2 2 2 2 2etc....  u 2 2 2 2 2 2 2 2 2 2
It just multiplies right up, thanks to efficient exponent laws!
If you can do that, you can also do weighted bond
projections and get <r2>.
Instead of 3 x 3 matrix multiplication, it becomes 15 x 15,
but this is still easy and fast.
Bottom line: RIS predicts C accurately.
Long range effects.
Switch back to notes on web.
The probability of all the chains is
not necessarily normalized