Statistical Mechanics of Worm-Like Polymers from a New Generating Function

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1 The University f Akrn IdeaExchange@UAkrn Cllege f Plymer Science and Plymer Engineering 9--4 Statistical Mechanics f Wrm-Like Plymers frm a New Generating Functin Gustav A. Carri University f Akrn Main Camus, gac@uakrn.edu Marcel Maruch Please take a mment t share hw this wrk hels yu thrugh this survey. Yur feedback will be imrtant as we lan further develment f ur resitry. Fllw this and additinal wrks at: htt://ideaexchange.uakrn.edu/lymer_ideas Part f the Plymer Science Cmmns Recmmended Citatin Carri, Gustav A. and Maruch, Marcel, "Statistical Mechanics f Wrm-Like Plymers frm a New Generating Functin" (4). Cllege f Plymer Science and Plymer Engineering. 6. htt://ideaexchange.uakrn.edu/lymer_ideas/6 This Article is brught t yu fr free and en access by IdeaExchange@UAkrn, the institutinal resitry f The University f Akrn in Akrn, Ohi, USA. It has been acceted fr inclusin in Cllege f Plymer Science and Plymer Engineering by an authrized administratr f IdeaExchange@UAkrn. Fr mre infrmatin, lease cntact mjn@uakrn.edu, uaress@uakrn.edu.

2 JOURNAL OF CHEMICAL PHYSICS VOLUME 11, NUMBER 1 SEPTEMBER 4 Statistical mechanics f wrm-like lymers frm a new generating functin Gustav A. Carri a) and Marcel Maruch The Maurice Mrtn Institute f Plymer Science, The University f Akrn, Akrn, Ohi Received 16 Aril 4; acceted 8 June 4 We resent a mathematical arach t the wrm-like chain mdel f semiflexible lymers. Our methd is built n a nvel generating functin frm which all the rerties f the mdel can be derived. Mrever, this arach satisfies the lcal inextensibility cnstraint exactly. In this aer, we fcus n the lwest rder cntributin t the generating functin and derive exlicit analytical exressins fr the characteristic functin, lymer ragatr, single chain structure factr, and mean square end-t-end distance. These analytical exressins are valid fr lymers with any degree f stiffness and cntur length. We find that ur calculatins are able t cature the fully flexible and infinitely stiff limits f the afrementined quantities exactly while rviding a smth and arximate crssver behavir fr intermediate values f the stiffness f the lymer backbne. In additin, ur results are in very gd quantitative agreement with the exact and arximate results f five ther treatments f semiflexible lymers. 4 American Institute f Physics. DOI: 1.163/ I. INTRODUCTION Since the grund-breaking ideas intrduced by Kratky and Prd in 1949, 1 theretical studies f semiflexible lymers based n the Kratky Prd KP mdel, called wrmlike lymers hencefrth, have been abundant. In this mdel, the lymer chain dislays resistance t bending defrmatins. The degree f resistance is determined by a free energy that enalizes the different cnfiguratins f the chain based n the extent f bending f the lymer backbne. This free energy deends n arameters elastic cnstants that are the result f many shrt-range mnmer-mnmer interactins. Exlicitly, fr the cntinuus versin f the KP mdel, called the wrm-like chain WLC mdel which is the nly mdel fr semiflexible lymers studied in this aer, the free energy is H L ds R s, 1 where R(s) is the vectrial field in three dimensins that reresents the lymer chain, s is the arc f length arameter, L is the cntur length f the lymer, and is the rati between the bending mdulus and the thermal energy. In additin, the lcal inextensibility cnstraint dr(s)/ds1 must be satisfied fr all values f the arc f length arameter. As a cnsequence f the bending rigidity, a wrm-like chain is characterized by a ersistence length that is rrtinal t the bending mdulus such that fr length scales shrter than the ersistence length the chain behaves like a rd while, fr length scales larger than the ersistence length the chain is gverned by the cnfiguratinal entry that favrs the randm-walk cnfrmatins. a Authr t whm any crresndence shuld be addressed. Electrnic mail: gac@uakrn.edu The lcal inextensibility cnstraint has nt allwed researchers t find an exact slutin t the mdel when the effects f an external field are added t the thery see. 44 in Ref. 3 and, Secs and in Ref. 4. Indeed, the cnstraint dr(s)/ds1 is written using a Dirac distributin in infinite dimensins. Deending n hw the cnstraint is written, dr(s)/ds1 r dr(s)/ds 1, we btain ath integral reresentatins fr the different statistical quantities e.g., characteristic functin with actins that are nnanalytic r nnlinear, resectively. Cnsequently, there is n exact slutin fr the WLC mdel with an external field at resent. Hwever, the ath integral reresentatin f the WLC mdel withut an external field is the same ne that describes the Brwnian mtin f a int article f mass mving n the unit shere. Since this rblem is exactly slvable many rerties f the WLC mdel such as sme crrelatin functins and the first few mments f the distributin f the end-t-end distance are knwn exactly;,3 higher rder mments can als be cmuted exactly using a recursive slutin t Schrödinger s equatin. 4 Hwever, exact exressins fr the different distributin functins are nt available at resent. Indeed, many researchers have addressed the WLC mdel and ther mdels f semiflexible lymers like the mdel f Dirac chains develed by Khldenk 5 with the urse f understanding the statistical behavir f this kind f lymers. Amng the many theretical treatments f wrm-like lymers, let us start with tw very imrtant cntributins made in 195s: first, the wrk by Daniels 6 wh develed exansins f the lymer ragatr fr a wrmlike lymer in inverse wers f the number f segments and, secnd, the classic aer by Benit and Dty 7 where the exact exressins f the mean square end-t-end distance and radius f gyratin were btained. During the fllwing tw decades, the field f statistical mechanics f wrm-like 1-966/4/11(1)/664/14/$ American Institute f Physics This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

3 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers 665 lymers saw a substantial grwth, thanks t the seminal cntributins f many researchers. Fr examle, Fixman and Kvacs develed a mdified Gaussian mdel fr stiff lymer chains under an external field external frce. 8 In this arach, they cmuted an arximate distributin fr the bnd vectrs frm which they were able t cmute the artitin functin and average end-t-end vectr. An alternative arach was rsed by Harris and Hearst 9 wh develed a distributin fr the cntinuus mdel frm which they were able t cmute the tw-int crrelatin functin and, cnsequently, the mean square end-t-end distance and radius f gyratin. A refrmulatin f the KP mdel using field-theretic methds was develed by Saitô and c-wrkers, wh cmuted exactly different averages f the end-t-end distance and tangent-tangent crrelatin functin. In additin, Gbush and c-wrkers 1 develed an asymttic exressin fr the lymer ragatr in inverse wers f the number f segments and, Yamakawa and Stckmayer 11 addressed the first rder crrectin t the mean square end-t-end distance and secnd virial cefficient due t excluded vlume interactins. Other seminal cntributins t the understanding f the WLC mdel made by Yamakawa and cllabratrs have been summarized in Ref. 3 recently. Wrm-like lymers were als studied using field-theretic methds by Freed 1 wh develed a mdified Gaussian distributin arximatin fr the WLC mdel. 13 This distributin has been rederived using different mathematical methds by Lagwski and c-wrkers, 14 and Winkler et al. 15 Similar results were btained by Zha and cllabratrs 16 wh als studied the effect f an external field. Fieldtheretic methds have als been used by Bhattacharjee and Muthukumar 17 wh emlyed the Edwards-Singh selfcnsistent arach t btain an effective Gaussian Hamiltnian which they used t cmute the mean square end-tend distance. In recent years, the advent f new exerimental methds that have allwed researchers t maniulate single mlecules has generated new mmentum in the area f statistical mechanics f wrm-like lymers. 18 Indeed, new analytical and numerical araches t wrm-like lymers have been develed by Mark and Siggia, Kry and Frey, and Samuel and Sinha 19 wh derived the frce-elngatin relatinshi redicted by the WLC mdel, Hansen and Pdgrnik wh develed a nnerturbative 1/d exansin (d being the dimensin f the embedding sace, Wilhelm and Frey 1 wh cmuted the lymer ragatr fr lymers with large bending rigidities, Winkler wh cmuted the same quantity fr any value f the stiffness f the lymer backbne using the maximum entry rincile, and Kleinert 4 wh develed a recursive slutin t Schrödinger s equatin frm where the all the mments f the lymer ragatr can be calculated exactly. Kleinert has als rsed a articular mathematical exressin fr the lymer ragatr. Mre recently, Steanw and Schütz 3 cmuted the Furier- Lalace transfrm f the lymer ragatr systematically by maing the WLC mdel nt the rblem f randm walks with cnstraints, which is related t the reresentatin thery f the Temerley-Lieb algebra. The WLC mdel is nt the nly descritin f semiflexible lymers available in the literature. Indeed, ther mdels fr semiflexible lymers have been develed. Amng these ther araches it is imrtant fr the urses f this aer t mentin the mdel f Dirac chains develed by Khldenk. 5 The advantage f this mdel is that statistical quantities such as the frce-elngatin relatinshi and single chain structure factr can be calculated exactly and, mrever, have been used t describe exerimental data and cmare well with arximate results f the WLC mdel. The rigin f the quantitative agreement between the different bservables f bth mdels can be fund in the arximate equivalence between the WLC mdel and the mdel f Dirac chains. This was rved by Khldenk using the Blch-Nrdsieck arximatin. The arximate equivalence between bth mdels and the availability f an exact slutin fr the mdel f Dirac chains make this mdel very useful fr the urse f estimating the quality f ur arximatins fr the WLC mdel. 5 In this aer we revisit the WLC mdel f semiflexible lymers with the urse f develing a new arximate methd caable f rviding clsed frm exressins fr the mst cmmnly studied statistical rerties f the mdel. The main mtivatin fr this study is t btain these exressins frm a unique arximatin t the mdel such that these exressins are accurate fr any value f the ersistence length. Since all these results are btained frm a single arximate treatment f the mdel, they are n equal fting. In ther wrds, we d nt have t devel different arximatins fr different statistical quantities r regimes with different degrees f stiffness. This feature is ne imrtant advantage f this wrk. We shw in this article that sme statistical quantities e.g., the lymer ragatr in real sace are described by very simle and accurate mathematical frmulas when ur methd is emlyed. Hwever, the cnsequences f the arximate nature f ur methd aear in tw arameters that ur calculatin cannt redict at resent therefre, they are determined a steriri by cmaring ur results with the nes btained by ther arximate treatments f the WLC mdel. Althugh this lack f self-cnsistency is the mst imrtant drawback f ur calculatin, it is als ne f its majr strengths since it allws us t cnnect ur methd with the results btained by ther research grus. Let us be mre exlicit abut the bjectives f this wrk. Our main bjective is t find a slutin t the WLC mdel caable f rerducing the statistical rerties f the mdel arximately. Furthermre, we require that ur slutin catures all the statistical rerties f the rd-like and flexible limits exactly. Mrever, ur slutin must resect the lcal, nt glbal, inextensibility cnstraint. In the crssver regin, the slutin is nt exact but, we require that it dislays the crrect hysical features as described by ther arximate and exact descritins f semiflexible lymers. Finally, ur slutin shuld be amenable t systematic and cntrlled crrectins calculated in a erturbative manner. In this aer, we nly rvide the guidelines fr the develment f such erturbatin exansin but leave the detailed mathematical calculatins fr a future article where we will als address the excluded vlume rblem. This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

4 666 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch This article is rganized as fllws. In Sec. II, we start with the KP mdel and rse a generating functin that we evaluate arximately. This sectin als cntains the cntinuus limit WLC mdel and the tw arximatins required by ur calculatin tgether with their justificatins. Afterward, we use the generating functin t calculate the characteristic functin, mean square end-t-end distance, lymer ragatr, and single chain structure factr f the mdel. In Sec. III we discuss the results f ur calculatins which are valid fr any value f the stiffness f the lymer backbne and length f the lymer chain. Mrever, fr the urse f making ur resentatin mre balanced and bjective, we resent a quantitative cmarisn f ur results with thse btained by ther researchers using different descritins f semiflexible lymers Dirac chains and different arximatins t the WLC mdel. Sec. IV cntains the cnclusins f ur wrk. The details f sme mathematical calculatins are resented in the Aendix. II. THEORY A. Review f the Kratky Prd mdel and the generating functin Let us cnsider a lymer chain as a set f n bnd vectrs (u,u 1,...,u n1 ) cnnected in a sequential manner. Furthermre, let us assume that the length f each bnd vectr is l k (Kuhn length and that airs f cnsecutive bnd vectrs try t be arallel t each ther. This rientatinal interactin is mdeled with a Bltzmann weight given by the fllwing mathematical exressin 8,17 ex n l u k k B T k1 u k, k where and k B T are the bending and thermal energies, resectively. In additin, we take int accunt the lcal inextensibility cnstraint with the fllwing term: n1 u i l k 1. 3 i The exressins given by Eqs. and 3 define the KP mdel cmletely and all the statistical rerties f the mdel such as the characteristic functin, single chain structure factr and ther crrelatin functins, rbability distributins like the lymer ragatr and their mments can be calculated. The evaluatin f these statistical rerties can be easily dne using the fllwing generating functin: n1 C kj ex k u k kj, 4 where kj is defined as fllws: kj J k iq k jk. This definitin shws that kj is an auxiliary tensr. It reresents a standard external surce cnsisting f tw terms, the first ne, J k, reresents a dilar culing with the kth bnd vectr and, the secnd ne, q, is als a dilar culing 5 that is nnzer nly when the kth bnd vectr is lcated between the jth and th nes alng the lymer chain. (z) is the Heaviside ste functin. 4 If the generating functin defined in Eq. 4 is knwn, then all the afrementined statistical rerties can be btained frm it by differentiatin with resect t J k and/r by assigning secific values t J k. Fr examle, the characteristic functin is btained frm Eq. 4 by setting all the J k variables equal t zer, Ciq (k j)(k); the lymer ragatr is the inverse Furier transfrm f the characteristic functin C(iq) and the tangent-tangent crrelatin functin, u j u k, is cmuted as the secnd-rder derivative f the generating functin with resect t J j and J k, afterward, this result shuld be evaluated fr values f J k equal t zer. Other statistical quantities can als be cmuted easily. Cnsequently, ur first ste is t evaluate C( kj ) whse exlicit mathematical exressin is C kj 1 n1 z i du i u i l k 1 ex n l u k k B T k1 u k k n1 u k k kj, where z is defined in such a way that C( kj )1. We nw rewrite the distributins in Eq. 6 as fllws: n1 u j n1 3 l j l k 1 k cs j j l k d j j l k Ô 1 j ex il k j u j 1 l j, 7 k where Ô i ( j ) is an eratr whse functin is t take the first i terms f the Taylr series in wers f j f the argument and evaluate this result fr j 1. After relacing the distributins in Eq. 6 with the integral reresentatins described in Eq. 7 the mathematical exressin fr the generating functin becmes C kj 1 n1 cs z q q l k du q d q q l k Ô 1 q ex n l u k k B T k1 u k k n1 u k kj i k 1 k k. The integrals ver the variables u k are evaluated exactly using the saddle int methd. The mathematical exressin fr the saddle int is the fllwing u k u l k k B T k1 m mj i m 1 m km. In additin, the fllwing cnstraint must be fulfilled: This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:

5 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers 667 n1 m i m 1 m mj. 1 After we relace the exressin fr the saddle int given by Eq. 9 int the exressin fr the generating functin given by Eq. 8 and writing the cnstraint, Eq. 1, using the exnential reresentatin f the delta distributin, the exressin f the generating functin becmes n1 Ô 1 q C kj 1 z du q d q csl k j l k j n1 ex i u s 1 s i sj s n n1 n1 s i k 1 k k kj K k,s i s 1 s sj, 11 where the kernel K k,s and the arameter are defined as fllws: K k,s 1 k sk n n sk and l k k B T. 1 We refer the reader t Aendix fr sme mathematical details f this art f the calculatin. Anther mathematical exressin fr the kernel K k,s is 1 ks n sk n. Nte that the cntributin t duble integral arising frm the term 1 (ks)/n in the kernel vanishes due t the cnstraint deicted by Eq. 1. Therefre, we remve these cntributins frm the definitin f the kernel K k,s hencefrth. Then, the generating functin can be written as fllws: n1 C kj 1 z du q d q csl k j l k j n1 ex i u s 1 s i sj s n n1 n1 4 s k 1 k k Ô 1 q. i kj sk n s 1 s i sj 13 The next ste invlves the integratin ver the variables q. These integrals cannt be evaluated exactly because the kernel generates nnlcal crrelatins between the variables. In ther wrds, the variable s is crrelated t j where j s. These nnlcal crrelatins make the exact evaluatin f the integrals in Eq. 13 imssible and frce us t search fr an arximate methd t evaluate them. First, we bserve that the term cntaining the csine functin inside the rduct cannt be arximated and must be ket intact. FIG. 1. Illustratin f the definitin f the effective self-crrelatin, cntinuus line, in terms f the crrelatins ragated by the kernel K k,s dashed line. Only tw aths are shwed. Bth vertical lines reresent the same lymer. Numbers indicate the variables alng the lymer chain. Otherwise, the lcal inextensibility cnstraint wuld be vilated and the mdel wuld lead t the Gaussian chain mdel. A similar argument can be made fr the first term in the exnential f Eq. 13. If this term is arximated, then the exact results f the flexible and rigid limits are lst. Since ur arach must cature these tw limits exactly and reserve the lcal inextensibility cnstraint intact then, the nly term that we are allwed t arximate is the secnd term in the exnential. Mrever, the nly quantity that we can arximate in the secnd term such that we can evaluate the q integrals is the nnlcal kernel. Let us study the secnd term in the exnential f Eq. 13. Observe that the nnlcal kernel crrelates the behavir f the jth variable with the behavirs f all the ther variables. In ther wrds, there is n self-crrelatin crrelatin f q with itself. Thus, we can reresent these crrelatins with a diagram shwn in Fig. 1. The vertical lines reresent the same lymer chain. We shw tw lines t clarify hw the different variables n the same lymer crrelate with each ther. The dashed lines reresent the crrelatins between airs f variables that are ragated by the kernel K k,s. Since K k,s des nt allw self-crrelatins, variables with the same value f the subindex are nt crrelated via the kernel K k,s. Hwever, there are aths that lead t effective self-crrelatins. One examle f this is the case where the first variable is crrelated with the secnd ne, the secnd ne with the third ne and this ne with the first ne. Anther ssible ath fr the self-crrelatin wuld be the first variable crrelating with the third ne, the third ne with the fifth ne and this ne with the first ne. Clearly, ther aths can be cnstructed. The additin f all these aths defines an effective self-crrelatin fr the first variable. We call this self-crrelatin. Similarly, this self-crrelatin can be defined fr all ther variables alng the lymer chain. We assume that this self-crrelatin is indeendent f the sitin f the variable alng the lymer chain but deends nly n the length f the lymer and its stiffness. In additin, if we culd sum the cntributins frm all the aths, then we wuld be able t cmute directly frm the mdel. Unfrtunately, this is equivalent t slve the mdel exactly which is nt ssible at resent. Thus, is left as an adjustable arameter t be determined a steriri by an This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

6 668 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch arriate criterin. Using the cncet f effective selfcrrelatin, we arximate the kernel as fllws: K k,s 1 /l k,nl k ks, 14 where ks is Krnecker s delta. At this int it is relevant t analyze the cnsequences f this first arximatin. Observe that the relacement f K k,s by a diagnal term imlies that the internal crrelatins are taken int accunt n an average sense. Thus, this lwest level f arximatin fr ur generating functin cannt describe crrelatin functins like u j u k accurately. Anther imrtant quantity that is affected by this first arximatin is the single chain structure factr. Hwever, we will shw that the cnsequences f this arximatin n the structure factr are negligible. It is wrth mentining that the results btained using the arximatin mentined in the revius aragrah are amenable t crrectins calculated in a cntrlled manner. One way f imrving the accuracy f the results is t take int accunt the resence f the nearest neighbrs exlicitly in Eq. 13. This arach is similar t the Bethe arximatin fr the Ising mdel and wuld lead t crrelatins between nearest neighbr variables, s and s1, via the kernel K s,s1. Cnsequently, anther effective crrelatin, 1 (/l k,nl k ), wuld aear in the thery. Further imrvements culd be made by accunting exlicitly fr the crrelatins with nearest neighbrs in higher shells. Anther methd is the erturbative ne. In this case we have t add and subtract t the kernel in Eq. 13 and assume that the cntributins arising frm the difference between and the kernel are small. A simle Taylr exansin in wers f the difference between and the kernel leads t a standard erturbative analysis. In this aer, we fcus n the lwest rder cntributin t the generating functin and stne the analysis f higher rder crrectins fr a future ublicatin. Using the arximatin described befre the exressin f the generating functin given in Eq. 13 can be written as fllws: n1 C kj 1 z du q csl k j d q l k j Ô 1 q n1 ex i u s 1 s i sj s n n1 4 k k 1 q i kj. 15 The evaluatin f the integrals ver the variables q is simlified by the use f sherical crdinates. The resulting mathematical exressin fr the generating functin is C kj (n) z du q n1 Ô 1 q ex n 4 qj u qj 1 q n l k qj u l k erf 1 1 q n l k qj u l k l k n 1 erf 1 q n l qj u l k l k k n. 16 Afterward, we aly the eratr Ô 1 ( q ) t the integrand in Eq. 16. As a result, we btain the fllwing exressin fr the generating functin: C kj 1 n1 z du q ex u n l sinh k n n qj u l k n n, 17 qj u where shuld be understd as ( ). Finally, we take the cntinuus limit f the exressin given in Eq. 17 t btain an exressin valid fr the WLC mdel. Fr this urse, we rewrite the arameters f the discrete mdel in terms f the cntur length Lnl k and ersistence length l 3 k / f the chain i.e., (L, )]. Afterward, we slit the rduct ver q segments in Eq. 17 int tw rducts. The first ne is the rduct ver all the ersistence lengths that can fit in a lymer with cntur length L. The secnd ne is the rduct ver all the segments that can fit in ne ersistence length. Clearly, this factrizatin f the rduct has tw imlicatins. First, it assumes that an integer number f segments frms a ersistence length and, secnd, that the lymer has an integer number f ersistence lengths. We will exlain hw t reslve this limitatin after resenting the secnd arximatin f ur arach. After factrizing the rduct and rewriting all the arameters in terms f the nes f the cntinuus mdel, Eq. 17 becmes This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

7 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers C kj 1 z du ex u sinh l k L L l k L L L/ /l k q,q j u. 18,q j u C kj 1 z du ex u sinh L L qj u L L qj u L/. 669 At this int it is relevant t say that the WLC mdel des nt deend n the Kuhn length, l k, exlicitly. Indeed, the mdel deends n tw length scales determined by which is the ultravilet cutff f the mdel and L which reresents the infrared cutff f the thery. If the hysical henmenn f interest has a characteristic length scale shrter than, then the WLC mdel cannt describe it. Fr these length scales, the WLC mdel redicts the behavir f a structureless rigid rd. Cnsequently, ne can naturally assume that the rduct ver the segments inside a ersistence length in Eq. 18 can be arximated by the argument f the rduct where l k is relaced by, Observe that the wer L/ was riginally btained under the assumtin that it was an integer number. Hwever, since L/ is renrmalized by the fluctuatins, as discussed befre, it can be a real number. Althugh this des nt change the underlying cncetual hysics f the arximatins, it des have imrtant imlicatins frm the cmutatinal int f view. If L/ is real, then whenever the imaginary art f the rati sinh L L qj u L L, 1 qj u /l k q sinh l k L L,q j u l k L L,q j u sinh L L j u L L. 19 j u This is the secnd arximatin f ur arach and has tw imrtant cnsequences. On the ne hand, this arximatin remves l k frm the mdel in agreement with the fact that the WLC mdel des nt deend n l k. On the ther hand, it assumes a secific functinal frm fr the rduct ver q. This functin is exact if ne ersistence length were t behave like a rigid rd. Hwever, even in the case f very shrt lymers (L ), sme fluctuatins with resect t the rigid rd cnfiguratin are ssible. These fluctuatins can be accunted fr if we define the rati L/ used in ur calculatins as a functin f the true rati L/. In ther wrds, the fluctuatins renrmalize the rati L/ f ur arach. The functinal relatinshi between the rati L/ used in ur calculatins and the true ne is unknwn. Thus, we will cnstruct this relatinshi later by cmaring ur results with the nes btained frm ther treatments f the WLC mdel. Finally, we relace the result btained in Eq. 19 int the exressin f the generating functin given by Eq. 18 t btain the mst general exressin fr the generating functin C( kj ), changes sign, there will be a cut n the cmlex lane. Crssing these cuts generates discntinuus behavir fr the different functins. These discntinuities are artifacts f the extensin f ur arximatins frm integer values f L/ t real nes. Thus, when using any f the mathematical exressins derived hereafter, we must make sure that they are evaluated n nly ne Riemann sheet s that cuts are avided. B. Characteristic functin Let us start with the evaluatin f the characteristic functin. This functin is btained frm the mathematical exressin f the generating functin given in Eq. when the external field qj is relaced with iq where q is the scattering wave vectr. Then, the mathematical exressin fr the characteristic functin is Cq 1 i qz du ex u u /L (uiq L / ) v /L (uiq L / sinhv L/ ) v dv. This exressin can be simlified further by an integratin by arts ver the variable u and with the hel f the fllwing definitins: h(z)h(z)i Imh(z) and h(z )h(z), where hz sinhz L/l /z L/l 1. As a result, the exressin fr the characteristic functin becmes This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

8 67 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch Cq qzl 1 du ex u Im sinh L uiq L L uiq L L/ 1 L/. 3 Observe that in the limit L/ the integral reresentatin f the characteristic functin given in Eq. 3 araches the asymttic limit given by the frmula C L/l q sinq L/ q, 4 which, if the rati L/ is an integer, is the characteristic functin f the randm flight mdel. 5 On the ther hand, in the limit L/ the characteristic functin araches the rdlike limit 6 C L/l q sinql. 5 ql In additin, the exressin f the characteristic functin given in Eq. 3 has an analytical slutin fr the entire range f values f L/, which is 1 Cq qzl L/l L/ Im du ex u L L uiq L L L/ 1 uiq L, 6 where we have exanded the functin sinh x (z) in wers f z. 4 This exressin fr the characteristic functin avids all the cuts intrduced by the extensin f L/ t real values. C. Mean square end-t-end distance We nw rceed t evaluate the mean square end-t-end distance R and the nrm z frm the exansin f the characteristic functin in wers f the wave vectr. First, we change variables as fllws: vvx L u i u L qx q, Cq 1 z L ex t 4 dt L t L q sinh L/ t 6t L/ sinht t L t L 1 L 1 t cthtcsch tt L/, 9 where, in rder t satisfy the cnditin C()1, we define z as fllws: uut L t, 7 then, the exressin f the characteristic functin given by Eq. becmes Cq 1 z L dt ex t 4 L t 1 dx 1 sinht it qx q t it qx q L/. 8 In the limit f small wave vectrs, the integral reresentatin f the characteristic functin given in Eq. 8 has the fllwing asymttic behavir: z L L/ A L/ L,. A a b (L/, / ) is defined by the fllwing frmula: A a b L, dt ex t 4 L sinhb t t a The secnd-rder derivative f the characteristic functin with resect t the wave vectr evaluated at q determines the exressin f the mean square end-t-end distance. The final exressin fr R is This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

9 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers 671 R 6 Cq q q L z L dt ex t 4 L sinhl/ t t L/ L t L 1 L 1 t cthtcsch tt, 3 r, in terms f A a b (L/, / ), R L/ A L/ L L, L/ A L/ L, L 1 L 1 L L/ 1 A L/ L, L/ A L/ D. Plymer ragatr L, L 1 1 L. 33 Let us nw rceed t evaluate the lymer ragatr. Fr this urse, the starting int is the Furier transfrm f the characteristic functin which is PR 1 3 dq CqexiqR. 34 After integrating ut the angular variables and using the arity rerties f the characteristic functin, we can rewrite the integral in Eq. 34 as fllws: PR 1 dq sinqrqcq R The change f variables t ( /L ) u transfrms the exressin f the characteristic functin given in Eq. 3 in the fllwing way: Cq qz dt ex t 4 Im sinhtiq L/ tiq L/ L Relacing this integral exressin fr the characteristic functin int the integral reresentatin f the lymer ragatr shwed in Eq. 35 gives the fllwing result: PR 8 Rz dt ex t 4 L dq sinqrim sinhtiq L/ tiq L/ 1 L/ 3 L/ 1 Rz, 37 which can be rewritten using the Taylr exansin f the functin sinh x (z). 4 The result is PR L/ dt ex t 4 L L t Im dq sinqr ex iq L. t/ iq L/ 1 38 The integral ver q is slvable exactly. Thus, the resulting exressin fr the ragatr is the fllwing: PR L/ 3 L/ 1 RzL/ dt ex 1 t 4 L L/ R L/ ex t 4 L Rt/. [LR/ ] L Rt/ [LR/ ] L L/ R dt L/ The tw integrals resent in the exressin f the lymer ragatr shwed in Eq. 39 are exactly slvable. The results are the fllwing: 39 dt ex t 4b at b exa b1erfab b. 4 Cnsequently, the final exressin fr the lymer ragatr is given by the fllwing frmula: This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

10 67 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch PR ex R L L/ 3 LzRL/ 1erf R 1 L 1erf R L [LR/ ] L/ LR [LR/ ] L/ LR L/ L/, 41 which is valid fr any value f the ersistence length and cntur length f the chain. E. Structure factr Finally, we cmute the single chain structure factr frm the well-knwn frmula SQ L L/ dvl/ vcq,v. 43 Fr the urse f clarity, let us define the rati L/ in Eq. 43, then the exressin f the structure factr becmes Sq L L dslscq,s. 4 First, we define tw dimensinless variables sv and Qq where is the true ersistence length f the lymer chain. The resulting exressin fr the structure factr is where SQ dvvcq,v, 44 CQ,v t(v) Qvzv4 v t(v)1 tv Im du ex u v tv tv tv v tv uiqv tv v uiqv tv v 45 t(v)1 and tv zzv v t(v) A t(v) tv tv,. v 46 We have intrduced tw new functins f v, namely t(v) and (v). The frmer is equal t the rati L/ used in ur slutin t the mdel. We remind the reader that this rati can differ frm the true ne, as discussed in Sec. II f this aer. Thus, it shuld be a functin, called t(v), f the true value f the rati L/. The latter functin is the rati L/ used in ur calculatins. Mre discussin abut these tw functins and their exlicit mathematical exressins are resented in the fllwing sectin. The final exressin fr the structure factr is btained by relacing the exressins f C(Q,v) and z(v) given in Eqs. 45 and 46 int the integral reresentatin f the structure factr shwed in Eq. 44. mdel, namely and L/. Althugh the hysical cncets behind the arximatins used t define these arameters were justified in the receeding sectin, we nw face the rblem f determining these arameters quantitatively. Fr this urse, we have t define a criterin that shuld build TABLE I. List f values used t cnstruct Eqs. 48 and 49. t III. RESULTS AND DISCUSSION We begin the discussin f ur results by cnsidering the tw arameters f ur arximate treatment f the WLC This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

11 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers 673 TABLE II. Cnstants fr Eq. 48. Cnstant Value a a a a a E3 FIG.. Plt f () vs. Cntinuus line least squares fit frm Eq. 48, circles numerical values frm Table I. the functinal relatinshis between these tw arameters and the arameters used in ther araches t the WLC mdel. At this int, ne very relevant bservatin is that ur arach targets a generating functin that deends n recircal sace variables q exlicitly. Cnsequently, if the arximate mathematical exressin fr the generating functin is accurate then, all the results that deend n recircal sace variables, such as the single chain structure factr and the characteristic functin, will be accurate because they are determined frm the generating functin directly r thrugh integral eratrs. This imlies that the magnitude f any errr made in the evaluatin f the generating functin will remain the same r be reduced in all statistical rerties cmuted frm the generating functin that deend n recircal sace variables. On the cntrary, the magnitude f the errrs will be magnified by the inverse Furier transfrm required t cmute real sace rerties. This leads us t the cnclusin that in rder t have accurate results fr recircal and real sace statistical quantities, the best way t determine the arameters and L/ is t require that ur result fr ne real sace rerty agrees n a quantitative level with the same real sace rerty redicted by ther treatments f the WLC mdel. Thus, ur criterin fr the determinatin f and L/ is that the height and lcatin f the maximum f the radial distributin functin fr the end-t-end distance agree with the nes btained frm ther treatments f the WLC mdel. Fr this urse, we chse tw classic results: the wrk by Gbush et al. 1 wh calculated the lymer ragatr as an exansin arund the fully flexible limit and the results btained by Wilhelm and Frey 1 wh studied the rdlike limit. We start by rewriting ur arameters in dimensinless frm. We call these arameters t and which are defined as fllws: t L/ and L/. 47 In additin, we define the variable r as R/L where R is the end-t-end distance. Then, we extract the functinal relatinshi between ur new arameters, t and, and the arameter, defined as the true rati L/, by matching the sitin and height f the eak in the radial distributin functin f the end-t-end distance 4r P(r) btained by using ur methd with the nes btained frm the studies f Gbush et al. 1 and Wilhelm and Frey. 1 Table I shws the values btained frm the fitting rcedure and, Eqs. 48 and 49 are the least square fits t these data ints, a a 1a a 3 a 4 3, 48 tb b 1 b b 3 3 b 4 4, 49 where the values f the cefficients are given in Tables II and III. Equatins 48 and 49 are valid fr values f in the interval.1,4. Hwever, these fits can be extended t any value f using the rcedure described befre. Fr the urse f illustratin we ltted Eqs. 48 and 49 in Figs. and 3, resectively. Bth figures dislay very gd quantitative agreement between the data and the least square fits. Cnsequently, Eqs. 48 and 49 can be used t make redictins fr the lymer ragatr fr intermediate values f. The deendence f () n is shwed in Fig.. Observe that () decreases with increasing fr small values f. This imlies that as the chain becmes shrter, the self-crrelatins becme weaker. This is a clear cnsequence f the kernel K k,s which araches zer fr shrt chains because it is rrtinal t sk. On the ther hand, () increases with increasing fr large values f lng chains which imlies that the self-crrelatins becme weaker. This is the natural cnsequence f the fact that, in TABLE III. Cnstants fr Eq. 49. Cnstant Value b b E1 b E b E3 b E5 FIG. 3. Plt f t() vs. Cntinuus line least squares fit frm Eq. 49, circles numerical values frm Table I. This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

12 674 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch FIG. 4. Nrmalized radial distributin functin, 4r P(r), vs r. Cmarisn between ur results cntinuus lines, Eq.41, and Gbush s results symbls fr the fllwing values f : 35circles, 5squares, 18diamnds, 15triangles u, 1triangles dwn, and 1 stars. FIG. 6. Mean square end-t-end distance divided by L ) r vs the arameter. Our result dashed line, exact result slid line, Ref. 1 circle and Ref. 1 diamnds. this limit, the chain is in the flexible regime. Thus, all the effects that are cnsequences f the semiflexible nature f the lymer backbne becme negligible. Therefre, () dislays a minimum fr values f clse t fur. The deendence f t() n is shwed in Fig. 3. This lt shws the exected mntnic behavir. Furthermre, fr large enugh values f, the relatinshi is linear. This result can be ratinalized as fllws: the lnger the chain is, the less imrtant the fluctuatins arund the rdlike cnfiguratin fr ne ersistence length are. Thus, ur L/ shuld arach the true value f L/ as the chain becmes lnger. Let us nw turn the attentin t the lymer ragatr P(R) whse exressin is shwed in Eq. 41. It is wrth bserving that aart frm sme refactrs, the deendence f the ragatr n R is mathematically very simle. It is just the sum f tw terms each f which is the rduct f an exnential, a term invlving an errr functin and a finite sum f wers. Cnsidering that this exressin was derived frm the characteristic functin, Eq. 36, via an inverse Furier transfrm, we find this exressin t be remarkably simle. Figure 4 shws the radial distributin functin 4r P(r) redicted by ur calculatins using Eqs. 48 and 49, and the ne redicted using the exressin derived by Gbush et al. 1 Bth results are in excellent quantitative agreement. Figure 5 shws the same quantity near the rdlike limit. In this case, we cmare ur results with the nes btained in Ref. 1. As in the revius case, the quantitative agreement is very gd fr all the values f studied. The small deviatins bserved between bth results can be crrected in a erturbative way as described in the receeding sectin. It is imrtant t bserve that the results fr the radial distributin functin arising frm Refs. 1 and 1 disagree fr 1. But, indeendently f which ne is the mst accurate ne, ur methd can be made t agree with it by the arriate selectin f the values f and t. Figures 4 and 5 als shw hw bth asymttic behavirs are cnnected. The lcatin f the eak in ur radial distributin functin fr chains with a fixed cntur length, L, mves tward larger values f r when the ersistence length f the lymer backbne increases. This is the crrect result because the stiffer the lymer backbne, the higher the energetic enalty t bend the chain. Cnsequently, thse cnfiguratins f the macrmlecule with small endt-end distance will be mre and mre hindered as the ersistence length increases, while thse cnfiguratins with large end-t-end distance shuld be mre favred. Therefre, the eak shifts tward larger values f r when the ersistence length increases. Mrever, the height f the eak must increase t reserve the nrmalizatin f the radial distributin functin. Figure 6 shws fur different results fr the mean square end-t-end distance divided by L ). Our result which is reresented by the dashed line was btained by relacing the exressins f () and t() shwed in Eqs. 48 and 49 int the mathematical frmula fr R given by Eq. 33. The slid line crresnds t the exact slutin. The circles FIG. 5. Nrmalized radial distributin functin, 4r P(r), vs r. Cmarisn between ur results cntinuus lines, Eq. 41, and Frey s results symbls fr the fllwing values f : 1triangles dwn, 5circles, squares, 1diamnds, and.5 triangles u. FIG. 7. Same data as in Fig. 6 ltted differently nly tw data sets are shwed. Our result dashed line and exact result slid line. This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

13 J. Chem. Phys., Vl. 11, N. 1, Setember 4 Statistical mechanics f wrm-like lymers 675 FIG. 8. Characteristic functin C(q), Eq. 6, vs wave vectr q. Cntinuus lines this wrk, symbls Steanw and Schütz (L 1 G 1 (k,)(k,) in their ntatin. Circles (35), squares (3), diamnds (), triangles u (1), and triangles dwn (8). FIG. 1. Structure factr S(q) vs wave vectr q. Cmarisn f ur results cntinuus lines, Eq.44, with Khldenk s results symbls. Stars ( 35), squares (), circles (1), (8), and triangles (4). and diamnds crresnd t the mean square end-t-end distance btained frm Refs. 1 and 1, resectively. This figure shws that ur result is in very gd quantitative agreement with the exact result ver the whle range f values f. In articular, it catures the fractal dimensins f the flexible and rigid limits crrectly, as shwed in Fig. 7. Let us nw cnsider recircal sace quantities. Fr this urse, we start with the characteristic functin C(q). Figures 8 and 9 shw C(q) as a functin f q fr ten different values f.1,.5,1,,4,8,1,,3, and 35. The symbls are the results btained using the methd f Steanw and Schütz fr n1 in their ntatin. 3 These figures clearly shw that ur results cature the cntinuus change in the behavir f this functin which changes frm a mntnically decreasing functin in the flexible limit large values f t an scillating functin in the rdlike regime small values f. In additin, ur cmutatins redict that, fr a fixed, the decrease f the characteristic functin fr small values f q shuld be faster fr flexible lymers than fr rigid nes. This is a cnsequence f the fact that fr a fixed ersistence length the flexible limit is achieved with large cntur lengths thus, the mean square end-t-end distance is larger and the decrease in the characteristic functin is mre rnunced than in the rigid limit which is realized by shrt cntur lengths thus, small mean square end-t-end distances. In additin, Figs. 8 and 9 shw that the results btained using ur methd can describe recircal sace rerties accurately. Based n the afrementined exressin fr the characteristic functin, we have als cmuted the single chain structure factr S(q) whse final exressin is shwed in Eq. 44. Furthermre, we have cmared it with the redictin arising frm the mdel fr semiflexible Dirac chains develed by Khldenk 5 which was successfully tested against exerimental data 7 and Mnte Carl simulatins based n a discrete frm f the Kratky Prd mdel fr wrm-like chains. 8 We resent this cmarisn in Figs. 1 and 11 where we ltted S(q) as a functin f q fr nine different values f the ersistence length. The symbls crresnd t Khldenk s mdel and the lines are the results f ur calculatins. These figures deict an excellent quantitative agreement between bth results fr the whle range f values f studied. This agreement is t be exected due t the arximate equivalence, via the Blch Nrdsieck arximatin, between the WLC mdel and the mdel f Dirac chains. The bserved quantitative agreement between ur arximate results and the exact nes fr Dirac chains imlies that ur arximatin t the WLC mdel is accurate and the use f an effective self-crrelatin des nt affect the redicted S(q) significantly. IV. CONCLUSIONS In this article we have develed a new arach t the WLC mdel f semiflexible lymers based n a new generating functin. The advantage f ur arach is fivefld. First, the evaluatin f the mst relevant statistical rerties f the mdel is straight frward since they can be btained as derivatives r integrals f the generating functin. This FIG. 9. Characteristic functin C(q), Eq. 6, vs wave vectr q. Cntinuus lines this wrk, symbls Steanw and Schütz (L 1 G 1 (k,)(k,) in their ntatin. Circles (4), squares (), diamnds (1), triangles u (.5), and triangles dwn (.1). FIG. 11. Structure factr S(q) vs wave vectr q. Cmarisn f ur results cntinuus lines, Eq.44, with Khldenk s results symbls. Stars ( ), diamnds (1), squares (.5), and circles (.1). This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3

14 676 J. Chem. Phys., Vl. 11, N. 1, Setember 4 G. A. Carri and M. Maruch makes the evaluatin f the generating functin the fundamental ste fr the slutin f the WLC mdel. In ur treatment f the mdel, we devised tw arximatins that were able t cature the flexible and rigid limits exactly and, mrever, rvided a smth crssver behavir between the tw afrementined limiting regimes. This crssver behavir has all the crrect qualitative features as shwed in the receeding sectin and, furthermre, agrees n a quantitative level with the results btained by five ther research grus. Secnd, all the results btained using ur methd are based n the same arximatins. In rder wrds, there is n need t devel different arximatins fr different statistical rerties r regimes with different degrees f stiffness. This uts all ur results n equal fting. Third, ur methd rvides clsed frm mathematical exressins fr all the rerties studied in this aer. Sme f these exressins, like the ne f the lymer ragatr in real sace, turned ut t be very simle. Furth, anther imrtant advantage f this wrk is that it resects the lcal, nt glbal, inextensibility cnstraint. Cnsequently, this methd culd be used t study the frce-elngatin relatinshi f wrmlike lymers. We will study this tic in a future nte. Finally, ur treatment f the WLC mdel was able t rvide a slutin t this mdel arund which a standard erturbative treatment can be develed. In this aer we did nt ursue the develment f such erturbative treatment. Hwever, we rvided the guidelines fr it. The detailed calculatins will be the tic f a future ublicatin. Perhas, ne f the mst imrtant cntributins f ur wrk is the derived lymer ragatr. The mathematical exressin btained fr the ragatr is very simle. It cntains tw terms each f which is the rduct f an exnential, a term which invlves an errr functin and a finite sum f wers which is very similar t the exressin f the lymer ragatr f the randm flight mdel. ACKNOWLEDGMENTS This aer is based un wrk surted by the Natinal Science Fundatin under Grant N. CHE Als, acknwledgment is made t The Ohi Bard f Regents, Actin Fund Grant N. R566, and The University f Akrn fr financial surt. APPENDIX: SOME MATHEMATICAL ASPECTS OF THE EVALUATION OF THE GENERATING FUNCTION In this Aendix we exlain hw the final exressin f the generating functin, Eq. 11, was btained. Fr the urse f clarity we define the variables k as fllws: k i k 1 q kj, A1 then, the integrals ver the variables u k in Eq. 8 can be evaluated exactly using the saddle int methd which is the slutin t the fllwing equatins: u 1 u, A where was defined in Eq. 1. The slutins t these equatins are given in Eq. 9. Mrever, the cnstraint stated in Eq. 1 is a cnsequence f these equatins. Therefre, if this cnstraint is nt satisfied, then Eq. 9 is nt the saddle int slutin. The terms aearing in Eq. 8 can be rewritten as fllws: and als 1 n u k1 u k n k k n1 k n1 u k k k n1 m k m m km k. m, A5 A6 We can simlify the sum f these tw terms further if we nte that n1 k n1 m n1 k n m km k k n1 m n k n1 k n1 m k m n m km k k m k m n1 m k n1m mk1 n m km k k n n1 m k n1k n1 m mk k mk1 n1 m m k, A7 k where the last duble sum vanishes because f the cnstraint given by Eq. 1. After changing the rder f summatin we btain the fllwing exressin: n1 n1 n1 k1 m m km k m m km k. A8 k k1 Aart frm the term k the first duble sum can be searated int tw terms: ne fr mk1 and the ther ne fr mk such that the frmer cancels the secnd duble sum in the receeding exressin. As a result we btain n1 k k k k1 n1 k n1 m n1 n1 mk m km k m mkmk k. A9 If we nw use the cnstraint fr the k s, Eq. 1, we btain the quadratic term in Eq G. Prd, Mnatsch. Chem. 8, ; O. Kratky and G. Prd, Recl. Trav. Chim. Pays-Bas 68, N. Saitô, K. Takahashi, and Y. Yunki, J. Phys. Sc. Jn., H. Yamakawa, Helical Wrm-like Chains in Plymer Slutins Sringer, Berlin, u n1 u n n1, A3 4 H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Plymer Physics, and Financial Markets Wrld Scientific, Singare, 3. u k1 u k u k1 k, k1...n, A4 5 A. L. Khldenk, Phys. Lett. A 141, ; Ann. Phys., 186 This article is cyrighted as indicated in the article. Reuse f AIP cntent is subject t the terms at: htt://scitatin.ai.rg/termscnditins. Dwnladed t IP: On: Thu, 3 Ar 14 16:1:3