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| The '''persistence length''' is a basic mechanical property quantifying the [[stiffness]] of a [[polymer]]. | | The writer is recognized by the title of Numbers Wunder. I am a meter reader. What I love doing is to gather badges but I've been using on new things lately. North Dakota is exactly where me and my husband reside.<br><br>Here is my web site: home std test kit ([http://www.escuelavirtual.registraduria.gov.co/user/view.php?id=140944&course=1 take a look at the site here]) |
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| Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional [[random walk]].
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| Formally, the persistence length, ''P'', is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain.<ref>{{cite book |title=Statistical Mechanics of Chain Molecules |first=Paul J. |last=Flory |publisher=Interscience Publishers |location=New York |year=1969 |isbn=0-470-26495-0 |url=http://openlibrary.org/b/OL5613440M/Statistical_mechanics_of_chain_molecules}}</ref>
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| Let us define the angle ''θ'' between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance ''L'' away from position 0, along the contour of the chain. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,<ref>{{cite book |title=The Theory of Polymer Dynamics |first1=M. |last1=Doi |first2=S.F. |last2=Edwards |publisher=Clarendon, Oxford |year=1986 |page=317}}</ref>
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| :<math> \langle\cos{\theta}\rangle = e^{-(L/P)} \, </math> | |
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| where ''P'' is the persistence length and the angled brackets denote the average over all starting positions.
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| In polymer science jargon, the persistence length is considered to be one half of the [[Kuhn length]], the length of hypothetical segments that the chain can be considered as freely joined. The persistence length equals the average projection of the [[end-to-end vector]] on the tangent to the chain contour at a chain end in the limit of infinite chain length.<ref>{{cite web |title=Compendium of Chemical Terminology |publisher=IUPAC |url=http://www.iupac.org/goldbook/P04515.pdf#search=%22persistence%20length%22}}</ref>
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| The persistence length can be also expressed using the bending stiffness <math> B_s </math>, the [[Young's modulus]] ''E'' and knowing the section of the polymer
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| chain.<ref>{{cite book |title=Cytoskeletal mechanics: models and measurements
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| |first=M.R.K. |last=Mofrad |publisher=Cambridge Univ Pr |url=http://books.google.ch/books?hl=en&lr=&id=HI8xTBUhnuYC&oi=fnd&pg=PA18&dq=Cytoskeletal+mechanics:+models+and+measurements&ots=EQKON4nMwi&sig=lvdjSlj8vTNcPp-70AQJLVscHhU}}</ref>
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| :<math> P=\frac{B_s}{k_BT}\, </math>
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| :<math> B_s=EI\, </math>
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| In the case of a rigid and uniform rod ''I'' can be expressed as:
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| :<math> I=\frac{\pi a^4}{4}\, </math>
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| where ''a'' is the radius.
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| For example a piece of uncooked [[spaghetti]] has a persistence length on the order of <math> 10^{18} </math> m (taking in consideration a Young modulus of
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| 5 GPa and a radius of 1 mm).<ref>{{cite web |title=Brittle failure of dry spaghetti
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| |first=G. V. |last=Guinea |publisher=Engineering Failure Analysis |url=http://linkinghub.elsevier.com/retrieve/pii/S1350630704000123}}</ref> Double-helical [[DNA]] has a persistence length of about 500 [[Angstrom]]s.
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| Another example:<ref>http://www.physicsforums.com/showthread.php?t=510330</ref> <br />
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| Imagine a long cord that is slightly flexible. At short distance scales, the cord will basically be rigid. If you look at the direction the cord is pointing at two points that are very close together, the cord will likely be pointing in the same direction at those two points (i.e. the angles of the tangent vectors are highly correlated). If you choose two points on this flexible cord (imagine a piece of cooked spaghetti that you've just tossed on your plate) that are very far apart, however, the tangent to the cords at those locations will likely be pointing in different directions (i.e. the angles will be uncorrelated). If you plot out how correlated the tangent angles at two different points are as a function of the distance between the two points, you'll get a plot that starts out at 1 (perfect correlation) at a distance of zero and drops exponentially as distance increases. The persistence length is the characteristic length scale of that exponential decay.
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| ==See also==
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| * [[Polymer]]
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| * [[Worm-like chain]]
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| * [[Freely Jointed Chain]]
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| * [[Kuhn length]]
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| * [[Paul Flory]]
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| ==References==
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| {{Reflist}}
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| [[Category:Physical quantities]]
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| [[Category:Polymer physics]]
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The writer is recognized by the title of Numbers Wunder. I am a meter reader. What I love doing is to gather badges but I've been using on new things lately. North Dakota is exactly where me and my husband reside.
Here is my web site: home std test kit (take a look at the site here)