The conformations of linear polymers

Transcription

1 The coformatos of lear polymers Marc R. Roussel Departmet of Chemstry ad Bochemstry Uversty of Lethbrdge February 19, 9 Polymer scece s a rch source of problems appled statstcs ad statstcal mechacs. I these otes, we wll look at the coformatos of polymers from ths perspectve. 1 Mathematcal prelmares 1.1 Sold agles Suppose that you wated to defe the word agle to someoe who had ever heard ths word. A agle measures how ope the space s betwee two les a plae. Oe way you could defe a agle more precsely would be based o the geometry of a crcle: Image drawg two les from a pot, ad a crcle of radus r cetered o that pot. The agle betwee the two les, θ, s the rato of the arclegth of the crcle cotaed betwee the two les to the radus of the crcle (fgure 1): θ = L r. Ths gves a agle the usual rada measure, betwee ad π. Whe we go to three-dmesoal geometry, we may wat to quatfy how ope a coe s. A coe, ths cotext, s a surface whch has a pot at oe ed ad exteds away from ths pot. We ca mage very arrow coes ad very wde coes. The sze of the opeg s, roughly, what the sold agle descrbes. Specfcally, we measure the area of tersecto betwee our coe ad a sphere of radus r (fgure ). The sold agle Ω s the defed as the rato Ω = S r. Ths formula gves the sold agle uts of steradas. The sold agle ca take ay value betwee ad 4π steradas. 1

2 r θ L Fgure 1: The agle θ ca be defed as the rato of L to r. r Fgure : A coe tersects a sphere of radus r a area S (shaded). The sold agle s the rato of S to r.

3 1. Taylor seres Suppose that f(x) s a fucto we wat to approxmate by a smpler form ear some pot x = x. For ormal fuctos (cotuous, wth cotuous dervatves), we ca use a Taylor seres, whch fts a polyomal to our fucto, dervatve-by-dervatve: f(x) f(x ) + df dx (x x ) + 1 d f x=x dx (x x ) d f x=x! dx (x x ). x=x Ofte, we use Taylor seres whe x x s very small. I that case, we ofte ust keep the frst o-zero term the Taylor expaso. The freely oted polymer We are gog to study a very crude model of a polymer soluto, amely the freely oted polymer. I ths model, we treat a polymer as beg made up of pot-lke moomers coected by rgd bods of legth l. Both the bod ad dhedral agles are urestrcted,.e. they ca assume ay value betwee ad π. Ths last assumpto s t very realstc, but t turs out that we get reasoably accurate predctos of some polymer propertes from ths model ayway. Fgure 3 shows a example of a polymer geerated ( two dmesos) by ths model. We wll cocetrate o ubrached polymers,.e. polymers whch cosst of a smple strg of moomers..1 The extet of the polymer alog a coordate axs We wll frst calculate the statstcal propertes of the ed-to-ed dstace alog a partcular coordate axs, say the x axs. I other words, we wat the probablty dstrbuto of x = x x 1, where x s the x coordate of the th moomer, ad s the total umber of moomers the cha. Defe ψ as the agle a bod makes to the x axs. If we thk of the bod as a vector, the x compoet of the bod s l x = l cos ψ. Therefore, f we kew p(ψ), the probablty desty for the agle ψ, the we could calculate the statstcal propertes of l x, ad from there the statstcal propertes of x. The probablty desty s defed such that p(ψ) dψ s the probablty that some partcular measuremet Ψ of ths agle satsfes Ψ (ψ, ψ + dψ). The rego defed by ths relatoshp s a coe whose er ad outer walls are separated by a agle ψ (fgure 4). Now mage the tersecto of ths coe wth a sphere of radus r. Sce the vectors are radomly oreted, the probablty p(ψ) dψ s ust the fracto of the sphere occuped by the tersecto,.e. the sold agle dvded by 4π, the total sold agle of a sphere. Specfcally, we wat the area of the aulus at dstace r from the org betwee agles ψ ad ψ + dψ. Because dψ s very small, we ca calculate ths area as the crcumferece of the crcle of radus r s ψ, whch s πr s ψ, multpled by the wdth of the aulus, r dψ. Thus the sold agle s πr s ψ dψ/r = π s ψ dψ. The probablty that Ψ s betwee ψ ad ψ+dψ s therefore p(ψ) dψ = 1 4π π s ψ dψ = 1 s ψ dψ. 3

4 3 y x Fgure 3: A typcal, radomly geerated 3-moomer freely oted polymer wth bod legth l = 1. The polymer s colored blue at oe ed ad yellow at the other to help you follow the lks. 4

5 dψ r r s ψ ψ Fgure 4: The aular rego at dstace r from the org betwee agles ψ ad ψ + dψ s the oe whose area s used to calculate the sold agle, ad thece the probablty desty for agle ψ. 5

6 We ca ow calculate statstcal propertes of l x. Let s start wth the average: l x = = π π l x (ψ)p(ψ) dψ l cos ψ 1 s ψ dψ =. Ths should t be a great surprse: Ay gve bod s ust as lkely to go to the left as to the rght. We ca also calculate the root-mea-squared dsplacemet alog the x axs: l x = = π π = l 3. l x = l 3. [l x (ψ)] p(ψ) dψ (l cos ψ) 1 s ψ dψ Ths s a measure of the typcal dstace alog the x axs betwee oe ed of a bod ad the other. Now suppose that we wat to kow the dstrbuto of dstaces x betwee oe ed of the polymer ad the other. If we have a large polymer, the we ca assume that, o average, each bod adds to or subtracts from the x coordate of the ed moomer a amout l x. Let + be the umber of moomers that make a postve cotrbute to x, ad be the umber of moomers that make a egatve cotrbuto to x. The x = ( + ) l x = ( + ) l 3. Sce the sg of the x dsplacemet s radomly chose from the two possbltes, ths s ust a co-tossg process. The probablty of obtag exactly + heads out of tosses s Defe so that p( + ) =! +!!. m = + x = m l 3. (1) We ca rephrase our probablty dstrbuto terms of ad m: + = 1 ( + m), ad = 1 ( m).! p(m ) = [ 1 ( + m)]! [ 1 ( m)]!. 6

7 Now we take a logarthm of p(m ) ad apply Strlg s approxmato. We are terested the case where s large. We expect that the overwhelmgly largest umber of cofguratos wll have m, so all the factorals appearg our probablty are factorals of large umbers. [ ] [ ] 1 1 l p(m ) = l! l l ( + m)! l ( m). l l 1 [ ] 1 ( + m) l ( + m) + 1 ( + m) 1 [ ] 1 ( m) l ( m) + 1 ( m) = l l 1 [ ] 1 ( + m) l ( + m) 1 [ ] 1 ( m) l ( m) = l l l [ (1 ) ( + m)( m) = l l( m ) + m ( ) m l + m = {l 1 ( [ l 1 { m = l ] ( m ) )] + m ( 1 m l 1 + m ( 1 m ) 1 + m 1 [ ( m ) ]} l 1. + m ( ) m l + m Recall that m/ s usually small. We ca apply Taylor seres to the two logarthms ths expresso. Keepg oly the frst o-zero terms, we get ( 1 m ) l 1 + m m, [ ( m ) ] ( m ) ad l 1. { ( m ) 1 ( m ) } l p(m ) + = m. Strlg s approxmato, the form we use ths course, 1 leaves out some costats whch are eglgble at large. As a result, we have lost the ormalzato of the dstrbuto p(m ). The best we ca say at ths tme s that p(m ) s proportoal to e m /(). However, that s eough for our purposes. We ca use equato 1 to elmate m favor of x. Ths gves us the uormalzed probablty dstrbuto f(x) = e 3x /(l ). 1 There are other versos of Strlg s approxmato. They are all equvalet for very large values of, but some are more accurate at smaller. )} 7

8 1.8.6 u() Fgure 5: Graph of the fucto u() = erf 3/. If we always took steps of exactly l/ 3 alog the x axs, that would be the ed of the story, gve or take ormalzg ths dstrbuto. However, ths s oly a approxmato vald for averagg over a large umber of steps. I fact, x s a cotuous varable, ot a dscrete oe lke m. We should therefore terpret f(x) as a probablty desty. The ormalzato costat C s foud as follows: C = = ( l f(x) dx l 1 ( 3/ ) l erf ) 1 3 π. Note that the lmts of tegrato are set by the maxmum possble extet of the polymer, whch we obta oly f every sgle bod pots the same drecto, ether alog the +x or x sem-axs. The fucto erf( ) s the error fucto. It ofte comes up whe computg tegrals volvg e kx. We do t eed to kow much about ths fucto sce programs lke Maple ad Excel kow how to calculate ts value. Fgure 5 shows the fucto u() = erf 3/. Notce how fast ths fucto approaches u = 1. Sce we are mostly terested loger polymer chas, we ca therefore set u() = 1 our ormalzato factor. The probablty dstrbuto becomes p(x) dx = 1 3 /(l) dx. () l π e 3x 8

9 r dr Fgure 6: Th sphercal shell used to determe the radal probablty desty. Note that ths s a Gaussa dstrbuto. It s ormalzed provded we exted the rage of tegrato to be (, ), whch of course s cosstet wth assumg that s large.. Ed-to-ed dstace three dmesos Equato apples to ay drecto space. We therefore statly kow the probablty desty for the three depedet Cartesa axes: p(x, y, z) dx dy dz = [p(x) dx] [p(y) dy] [p(z) dz] ( ) = e 3r /(l ) dx dy dz, l π where r = x + y + z. p(x, y, z) dx dy dz s the probablty that the relatve posto of the last moomer the polymer relatve to the frst s the small box of edge legths dx, dy ad dz wth oe corer at (x, y, z). Of course, we do t usually care how the polymer s oreted. What we usually wat to kow s how far the two eds are from each other. I other words, we wat p(r) dr. Ths would be the probablty that the relatve posto vector r = (x x 1, y y 1, z z 1 ) has a legth betwee r ad r+dr, rrespectve of the agle ths vector makes to the Cartesa axes. The vectors whch satsfy ths codto defe a sphercal shell of er radus r ad thckess dr. See fgure 6. What we eed to do s to tegrate (add up) p(x, y, z) dx dy dz over the teror of the shell. Sce p(x, y, z) oly depeds o r, t s essetally costat our th shell, ad we ca pull t out of the tegral. All that s left to do s to tegrate dx dy dz over our shell, but ths s ust the volume of the shell,.e. 4πr dr (area tmes thckess). The desred probablty dstrbuto s therefore p(r) dr = ( 1 l 3 π ) 3 e 3r /(l ) 4πr dr. 9

10 r Z s Fgure 7: Vectors used to calculate the radus of gyrato. The polymer s show blue. The heavy dot represets the cetre of mass. Note that there really s t aythg specal about the frst ad last moomers a cha. Provded s t too small, ths equato therefore gves the dstrbuto of dstaces betwee ay two moomers separated by bods the cha. Exercses 1. Calculate the mea dstace betwee the two eds of a polymer assumg the freely oted cha model.. Calculate the stadard devato of the dstace betwee the two eds of a freely oted polymer..3 Radus of gyrato May polymer propertes deped o the radus of gyrato, whch s the root-mea-squared dstace of the moomers from the cetre of mass of the polymer. The radus of gyrato turs out to be related to the root-mea-squared dstace betwee two moomers, so we start by computg ths quatty: r = r = l. r p(r) dr = l. (3) Aga ote that ths s the rms dstace betwee ay two moomers separated by bods. We eed to defe a umber of vectors: r s the vector from oe ed of the cha (.e. moomer 1) to moomer. s s the vector from the cetre of mass to moomer. Z s the vector from moomer 1 to the cetre of mass. (Z = s 1.) Fgure 7 llustrates these vectors. From these deftos, we have s = r Z. (4) 1

11 The cetre of mass s defed by s =. Summg equato 4 over all moomers, we get = r Z, or Z = 1 r. (5) Z = Z Z = 1 r r. (6) The radus of gyrato s, as metoed above, the rms dstace from the cetre of mass or, terms of the vectors we have ust defed, the rms value of s. We start by calculatg the squared sum of the dstaces from the cetre of mass: s = s = s s = (r Z) (r Z). Ths last relatoshp was obtaed usg equato 4. Thus we have s = r + Z Z r. (7) Let s work o the last term ths equato: Z r = Z r = Z Z = Z, where we used equato 5 to elmate the sum. Gog back ow to equato 7, we have s = r Z. Now we use equato 6: The dot product s defed by but the cose rule says that s = r 1 r r. (8) r r = r r cos θ, r + r r r cos θ = r, where r s the dstace betwee moomers ad. Combg these two equatos to elmate r r cos θ, we get r r = 1 ( ) r + r r. 11

12 Substtutg ths relatoshp to equato 8, we get s = = = r 1 r 1 r 1 = r 1 = 1 = 1 < ( r + r r) ( r + r ( r + r ) r ( r ) r r r. The last trasformato uses the fact that each of the parwse dstaces occurs twce the prevous sum. To get the radus of gyrato, we frst eed to get the average of s : s = 1 s = 1 r. For a freely oted polymer, r s gve by equato 3 wth =. We therefore have < < s = l ( ) = l 1. The last equalty s ust a reorderg of the er sum, as you ca see for yourself by wrtg dow the terms the two forms of the er sum. The sums we eed to calculate are well kow. I ust looked them up a referece book. We wll also make a few approxmatos usg the fact that we are terested large polymers,.e. polymers wth a large. Frst, 1 =1 = ( 1)/. We wll the sum ths quatty over values of from 1 to. Of course, most of these values wll be large, so we ca approxmate ths sum as /. Ths leaves us wth the followg sum: =1 = ( + 1)( + 1)/6 3 /3. Puttg t all together, we get s = l 1 3 Now we use equato 3 aga, obtag the result 3 = l 6. =1 r ) or s = r /6, s = r /6. 1

13 Ths quatty s the radus of gyrato we wated to calculate. Note that the radus of gyrato for a lear polymer ths s t true for brached polymers s drectly proportoal to the average ed-to-ed dstace. What good s the crude theory developed these pages? It turs out that several predctos of ths model the Gaussa dstrbuto of dsplacemets alog a coordate axs, the form of the probablty desty for the ed-to-ed dstace, ad the relatoshp betwee the radus of gyrato ad root mea squared ed-to-ed dstace are all roughly depedet of the model detals. Eve f we added realstc bod ad dhedral agle costrats, we would get about the same results. Exercses 1. A lear polymer has moomers whch are about 15 Å legth. Plot the radus of gyrato as a fucto of, for up to 1. 13