Subject: Polymer Chain Rigidity

From: [email-masked] (Palavo Subramanian)
Date: Thu, 8 Dec 94 07:25:59 CST

Dear Netters,

 I had posted a query on polymer rigidity a while back.  Here is a summary
of all the replies that I received.  My sincere thanks to everyone who t
took time to reply.
:
Date: Thu, 01 Dec 1994 16:21:47 -0500 (EST)
From: [email-masked]
Subject: Re.: Quantifying polymer rigidity.

Dear P. Subramanian,

Regarding the measurement of rigidity in polymers, yes, as far as
I know, the persistence length is the standard one-parameter descriptor
used to that effect. Any other measure of angular autocorrelation along 
the chain will work similarly.
I have developed an alternative measure of flexibility (or rigidity)
that conveys also the shape of the polymer.  This aspect is somewhat lost 
in the persistence length.  The basic ideas and some applications of this 
approach are in: Phys.Rev. E 49 (1994) 2417 and J.Phys.Chem. 97 (1993) 13831.  
Other authors have proposed alternative characterizations of entanglements 
which could also be used to describe rigidity. Perhaps you may find these 
descriptors useful.

Best regards,

G. Arteca
Dept. de Chimie et Biochimie,
Laurentian University, Sudbury, Ontario,
Canada P3E 2C6.
----------------------------------------------------------------------------
Date: Thu, 1 Dec 94 17:43:44 -0500
From: S. Vijayakumar <[email-masked]>

Mani:

Here is my 2 cents worth!

	Ideally, the lower the internal degrees of freedom the more  
rigid a molecule is. The predominant contributor to this would of  
course be rotations along the torsional angles.  Crudely, one can  
assume that the flexibility of a polymer is directly proportional to  
the number of accessible states and inversely proportional to the  
conformational barrier between these states.  For the former, you can  
perform a global conformational search and look at the number of  
accessible states (minimum energy conformations) within a fixed  
energy (usually about 5 Kcal) window. But the latter problem of  
estimating the barrier for conformational transition can be a bit  
more tricky and involved. In any case, you cannot consider the  
effects of temperature on the flexibility using this approach.

	A more easier approach, would be run an MC simulation of the  
polymer in vacuo or appropriate conditions at a series of  
temperatures and generate an ensemble of structure at each  
temperature. One can then think of ways to cluster these into groups  
of related conformations and figure out a composite index to estimate  
the flexibility based on the number of distinct states and the  
probability density at each state. 

	On a general basis, persistence length is really only one of  
many criteria for estimating the flexibility. However, if the  
polymers that you are interested in is unbranched then it might  
suffice and could serve as an useful practical measure. 

Regards,

Vijay

*************************************************************
S. Vijayakumar, Ph. D., DCA
Department of Chemistry,  Wesleyan University, Middletown, CT 06459
Voice:	(203) 685-2777			Fax:	(203) 685-2211
email:	[email-masked]                                          
*************************************************************
Date: 	Fri, 2 Dec 1994 05:55:12 PST
From: [email-masked]
Subject: Chain rigidity

Dear subramanian,

 concerning your question on calculating the chain rigidity : I have been using
the conformational entropy as a numeric measure. I have related the chain
rigidity, via calculated entropy, to functional properties such as the glass
transition temperature (see Macromolecules, v. 26, 1993, p. 344. Sometimes,
partition fucntions can be sued as well (Polym. Mater. Sci. Eng., v 63, 1990, p
392.)
Regards
Sundararajan
[email-masked]
----------------------------------------------------------------------------
From: [email-masked] (Dr. Tom Pierce)

A good way to quantify rigidity depends on what you will experimentally
measure as 'rigid'.  The most compelling measures correlate with experiments.

Interesting measures are:
     Radius of Gyration
     Variance of Radius of Gyration
     End-to-end distances
     Variance of end-to-end distances
     Characteristic ratio (Cinfinity)
     Kuhn segment (or Persistence Length)
     
These are single chain estimates, returning to what does rigid mean,
if you are interested in packing effects then volume calculations 
might be useful.  

Note that the relationship of Radius of Gyration to End-to_end
distance will tell you "how" gaussian your chains are. 

-- 
Sincerely, Thomas Pierce - [email-masked]  -  Computational Chemist
"These opinions are those of the writer and not the Rohm and Haas Company."
---------------------------------------------------------------------------
Return-Path: <[email-masked]>
From: [email-masked] (Craig W. Burkhart)
Subject: Chain Stiffness

Mani,

One of the most general ways to infer chain rigidity is calculation of the
characteristic ratio. It is defined, for a simple linear chain, as

C(n) = <s^2>/nl^2

where s = chain end-to-end distance
      n = number of bonds along backbone
      l = bond distance

The use of <> brackets implies an ensemble average. For a freely jointed chain,
C(n) = 1. For a jointed chain with a bond angle of 120 degrees, C(n) ~ 2.5 or
so. For polyethylene, C(n) = 6.6-6.9. This implies that polyethylene is fairly
stiff compared to ideal chains. Main-chain liquid crystal polymers have C(n) >
10 (usually). So now you have an idea of how chain stiffness can be measured in
this fashion.

If you choose to do this via simulation, you BETTER BE SURE that your
forcefield reproduces the conformational states of the chain extremely well. An
RIS dyad analysis of the chain torsional states is probably the best route.
Most of this information can be found in Flory's classic "Chain Statistics of
Macromolecules". It is MUST reading for anyone who deals with polymer chain
conformations and configuration statistics...

You have mentioned persistence length. In order to obtain the persistence
length, you have to do the same gyrations (pardon the pun) as with
characteristic ratio. Both methods are complementary...

Have fun!

Craig
-- 

[][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][]
                                        |
Papernet:  Craig W. Burkhart            | "One man's analytical is
           Goodyear Research            | another man's numerical..."
           142 Goodyear Blvd            |
           Akron, OH   44305            | - old hacker's maxim
Mouthnet:  216.796.3163                 |
Faxnet:           .3304                 |
Internet:  [email-masked]       |
[][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][]
Return-Path: <[email-masked]>
From: "Bruce E. Wilson" <[email-masked]>
Subject: Re: CCL:Polymer Chain Rigidity

If you have a set of decent probabilities (and they can be calculated)
a Rotational Isomeric State model to calculate end to end distances
for an ensemble of molecules can be done very quickly.  I've tended
to divide by the end to end distance for a fully extended chain 
(which is not the same thing as topological length) or by the mass
to get some sense of normalization.  If the polymers are "reasonably"
similar, you can just use the end to end distances for the same Mn and
get reasonable results.  If you don't have an RIS model, then the average
end to end distance for a collection of three or four starting points 
during molecular dynamics (I prefer MM2, MM3) can work, although you'll
probably want fairly long run times (hundreds of picoseconds).

The RIS calcs can be done in a day easily, if you have decent probabilities.
The MD calcs on most RS/6000 workstations may take a couple of weeks
to get decent convergence.

                        Bruce E. Wilson ([email-masked])
                        Eastman Chemical Company
--------------------------------------------------------------------------
Return-Path: <[email-masked]>
From: [email-masked] (Lawrence R. Dodd)
Subject: Re: CCL:Polymer Chain Rigidity

::: "TP" == Tom Pierce <[email-masked]> writes:

  TP> Previously, Palavo Subramanian wrote:

  >> What is a good way to 'quantify' polymer chain rigidity ?. I am
  >> looking at comparing the rigidity of Polymer A as a function of
  >> temperature and also as a function of composition(monomer ratio
  >> in the case of a copolymer).  Is there a good way to calculate
  >> this? Is Persistence Length a good measure of the rigidity?

  TP> These are single chain estimates, returning to what does rigid
  TP> mean, if you are interested in packing effects then volume
  TP> calculations might be useful.

I don't know what volume type of calculations to which Tom refers,
but we (Doros Theodorou and I) found that the molecular volume of
(united atom) polymer chains was relatively independent of chain
conformation and that the volume of an "all-trans" chain was as good
as any other (Mol. Phys., 72(6), 1313-1345).

--Mani--