How do you calculate the Kuhn length of a polymer?

No, just chemistry.In summary, the conversation discusses the derivation of a more accurate equation for the freely jointed polymer chain, which takes into account the presence of bonds between adjacent steps. The resulting equation involves the persistence length and uses the law of large numbers. The conversation also briefly touches on the use of the central limit theorem in understanding this equation. One participant mentions being in the field of materials science, while the other is in chemistry.
  • #1
etotheipi
My lecturer said that as a first approximation, we can take the polymer chain to consist of ##n## freely jointed rods of length ##l##. That's just going to give$$\langle \vec{R}_n^2 \rangle = \langle \sum_{i=1}^n \vec{r}_i \cdot \sum_{j=1}^n \vec{r}_j \rangle = \langle \sum_{i=1}^n \sum_{j=1}^n \vec{r}_i \cdot \vec{r}_j \rangle = l^2 \sum_{i=1}^n \sum_{j=1}^n \langle \cos{\theta_{ij}}\rangle$$The freely jointed condition gives ##\langle \cos{\theta}_{ij} \rangle = \delta_{ij}##, so the ##\sum_{i=1}^n \sum_{j=1}^n \langle \cos{\theta_{ij}}\rangle = \sum_{i=1}^{n} \delta_{ii} = n## and thus ##\sqrt{\langle \vec{R}_n^2 \rangle} = l\sqrt{n}##. A more accurate equation, where the rods are no longer freely jointed, is actually $$\sqrt{\langle \vec{R}_n^2 \rangle}= l\sqrt{n}\sqrt{\frac{1+\cos{\theta}}{1-\cos{\theta}}} $$I looked around but couldn't find a derivation of this, so I wondered if someone could give me a hint as to proceed with the ##\sum_{i=1}^n \sum_{j=1}^n \langle \cos{\theta_{ij}}\rangle## term in order to obtain the more accurate result? Thank you!
 
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  • #2
I think it is easier to see geometrical situation, if you write ##\sqrt{\dfrac{1+cos \theta}{1-\cos \theta}}=\left|\cot \left(\dfrac{\theta}{2}\right)\right|##.

The closest I came was: "It can be shown..." (sic!) "... that ##E(\cos \theta)=e^{-l/l_p}##, where ##l_p## is the persistence length, defined as ...".

In any case, it will need a stochastic argument, why the formula "... where the rods are no longer freely jointed, is actually ..." holds. Wiki unfortunately says "clarification needed". It feels like "assume friction". As far as I could see, the deduction uses the law of large numbers. Maybe you get to the desired result if you use the central limit theorem instead. Or you understand this on page 21 (29):
https://www.fz-juelich.de/SharedDocs/Downloads/IBI/IBI-4/EN/polymerKhkolov_pdf.pdf?__blob=publicationFile
and again without details on page 10 (25):
https://www.theorie.physik.uni-muen...06/softmatter/talks/Peter_Jensen-Polymers.pdf
 
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  • #3
Awesome, that's exactly it! Assume that bonds between adjacent steps have mean ##\langle \theta \rangle = \gamma##, and thus ##\langle \cos{\theta} \rangle## for steps separated by ##j-i## is ##(\cos{\gamma)}^{j-i}##, so$$\begin{align*}

\langle \vec{R}_n^2 \rangle = l^2 \sum_{i=1}^n \sum_{j=1}^n \langle \cos{\theta_{ij}}\rangle &= l^2 \left (\sum_{i=j} \langle \cos{\theta_{ij}} \rangle + 2\sum_{i < j} \langle \cos{\theta_{ij}} \rangle \right) \\

&= l^2(n + 2\sum_{i < j}(\cos{\gamma})^{j-i}) = nl^2 \left(1+ 2\frac{\cos{\gamma}}{1-\cos{\gamma}} \right)

\end{align*}$$and simplify for the result. Thanks!
 
  • #4
Are you doing materials science?
 
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  • #5
Mayhem said:
Are you doing materials science?

Yes! Are you?
 
  • #6
etotheipi said:
Yes! Are you?
No, just chemistry.
 

Related to How do you calculate the Kuhn length of a polymer?

1. What is the Kuhn length of a polymer?

The Kuhn length of a polymer is a characteristic length scale that represents the average distance between adjacent bonds in a polymer chain. It is a measure of the flexibility and conformational properties of the polymer.

2. How is the Kuhn length calculated?

The Kuhn length is calculated by dividing the persistence length of the polymer by the number of Kuhn segments in the polymer chain. The persistence length is a measure of the stiffness of the polymer, while the Kuhn segment is a theoretical unit of length that represents the average distance between adjacent bonds.

3. What is the significance of the Kuhn length in polymer physics?

The Kuhn length is an important parameter in polymer physics as it helps to determine the conformational properties of a polymer chain. It is also used to calculate other important properties such as the radius of gyration and the end-to-end distance of the polymer.

4. How does the Kuhn length affect the behavior of a polymer?

The Kuhn length affects the flexibility and conformational properties of a polymer chain. A longer Kuhn length indicates a more flexible polymer, while a shorter Kuhn length indicates a more rigid polymer. This can impact the behavior of the polymer in various applications, such as in the formation of gels or in determining the mechanical properties of a polymer material.

5. Can the Kuhn length be experimentally measured?

Yes, the Kuhn length can be experimentally measured using various techniques such as light scattering, neutron scattering, and atomic force microscopy. These techniques allow for the determination of the persistence length and the number of Kuhn segments in a polymer chain, which can then be used to calculate the Kuhn length.

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