Thermally Induced Local Failures in Quasi-One-Dimensional Systems: Collapse in Carbon Nanotubes, Necking in Nanowires and Opening of Bubbles in DNA
Cristiano Nisoli, Douglas Abraham, Turab Lookman, Avadh Saxena
TThermally Induced Local Failures in Quasi-One-Dimensional Systems: Collapse inCarbon Nanotubes, Necking in Nanowires and Opening of Bubbles in DNA.
Cristiano Nisoli , Douglas Abraham , , Turab Lookman and Avadh Saxena Theoretical Division and Center for Nonlinear Studies, Los Alamos National Lab, Los Alamos NM 87545 USA Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road Oxford, OX1 3NP England (Dated: November 4, 2018)We present a general framework to explore thermally activated failures in quasi one dimensionalsystems. We apply it to the collapse of carbon nanotubes, the formation of bottlenecks in nanowires,both of which limit conductance, and the opening of local regions or ”bubbles” of base pairs instrands of DNA that are relevant for transcription and danaturation. We predict an exponentialbehavior for the probability of the opening of bubbles in DNA, the average distance between flat-tened regions of a nanotube or necking in a nanowire as a monotonically decreasing function oftemperature, and compute a temperature below which these events become extremely rare. Thesefindings are difficult to obtain numerically, however, they could be accessible experimentally.
PACS numbers: 61.46.-w, 65.80.-g, 68.35.Rh, 87.14.gk
Nanowires [1], carbon nanotubes [2] and DNA fila-ments [3] are examples of Quasi One Dimensional (Q1D)mesoscopic systems characterized by one very long di-mension, on the molecular length scale, compared to theothers that are of the order of nanometers. Their theoret-ical and technological interest is very high and motivatesan analysis of the mechanisms governing their failure,yet numerical computations become impractical in thedilute regime. It has been experimentally observed andtheoretically predicted that nanotubes of radii above acritical value of about 3 nm collapse globally along theirlength with a pancake-dumbell cross-section [4, 5, 6, 7].However, for carbon nanotubes of radii below the criticalvalue, and therefore globally stable, we expect that ther-mal fluctuations will also cause collapse in local regionsto impact their structural and transport properties. Thenecking of nanowires under thermal fluctuations has sim-ilar consequences. DNA filaments at physiological tem-perature are known to locally open to form the so calledDNA bubbles [8, 9], relevant to transcription and den-tauration processes. The size of these bubbles has beeninvestigated by computationally intensive studies basedon the Peyrard-Bishop-Dauxois model (PBD) [10, 11] atnon-zero temperature [12], however, a theoretical under-standing of their behavior is still lacking.We present here a very general statistical mechan-ics framework to study such thermally activated eventswhich is applicable to a variety of Q1D systems, and weshow how to implement it practically to gain some insighton the phenomena described above. We predict an expo-nential behavior for bubbles in DNA (which has recentlyemerged from numerical model simulations but has sofar not been accessible experimentally) and the depen-dence of the average distance between collapsed regionsin a nanotube or necking in a nanowire, which effectivelyinfinite below a critical temperature and monotonicallydecreases to a non-zero minimum at very high tempera-tures. Following previous work [13], we consider a Q1D sys-tem like the one depicted in Fig. 1, and, for simplicity,restrict ourselves to one degree of freedom, x ( l ), where l is the coordinate along the length of the wire. Theboundaries are allowed to oscillate under an elastic en-ergy k x (cid:48) ( l ) and they interact via a local potential V ( x ).For instance, for a collapsing carbon nanotube, x ( l ) rep-resents the shorter axis of the collapse, as we describedthe collapsed posrtion as a “pancake” of heigh x . Inmodels for solitons in DNA, e.g. the celebrated Peyrard-Bishop model [10], x represents the distance betweencomplementary bases, whereas V ( x ) is a Morse poten-tial, chosen because exactly solvable, although a simplesquare well would suffice [14]. In strained, 2-dimensionalnano-islands, V ( x ) represents the elastic interaction be-tween the edges. The locality of the potential is often anapproximation which is valid when the typical length oc-cupied by the collapse failure or instability is longer thanthe lateral size of the wire. The problems we describethus posses a characteristic length, l c below which col-lapse cannot happen, simply because there is not enoughavailable space. In a carbon nanotube l c is the lengthneeded to go from a cylindrical to a collapsed configura-tion and back. Different is the case of DNA: we will be ld x c x(l) FIG. 1: A schematic depiction of localized failure in a Q1Dsystem, where x c defines the collapse width, d is the lateralsize at equilibrium, and x ( l ) is the relevant degree of freedomfor our problem, allowed to fluctuate in the coordinate l , 0 ≤ l ≤ L . a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t interested in studying the length of bubbles, and thus wewill consider “defect” the joined double filament configu-ration: because of that l c = 3 . x reppresent theirlateral size.This approach provides us with a natural discretiza-tion length for our problem: that, as we shall see later,considerably simplifies matters. For specificity we define“failure” as x dropping below some critical value x c , avalue that is suggested by the problem itself and mighthave nothing to do with the particular shape of V ( x ). Fora nanowire it will be the threshold below which band gapsopen and conductivity drops. For a carbon nanotube, itmight be the distance corresponding to the metastableflattened state. For bubble in DNA it will be the dis-tance at which bases are linked by hydrogen bonds.With these assumptions, the partition function for apiece of the wire of lenght nl c , starting and ending in afailure is given by a path integral over the allowed fluc-tuations Z n = (cid:90) D x ( l ) e − β R nlc d l [ k x (cid:48) ( l )+ V ( x ( l )) ] (1)with x (0) = x ( n l c ) ≤ x c , and n integer. The normal-ization in D x ( t ) is chosen so that standard transforma-tions [16, 17] allow us to write Z n as density matrix Z n = x − c (cid:90) x c d x (cid:90) x c d y (cid:104) x | e − n ˆ H β | y (cid:105) (2)of the one-particle, temperature dependent hamiltoniangiven by ˆ H β = − l c βk d d x + βl c V ( x ) . (3)Now, the probability that a system of length nl c will notfail anywhere in the middle is given by P n = Ω n Z n (4)where Ω n is the sum over fluctuations such that x ( l ) >x c : that is, the sum over the fluctuations that only leadto failure at the boundaries, orΩ n = (cid:90) x ( l ) >x c D x ( l ) e − β R nlc d l [ k x (cid:48) ( l )+ V ( x ( l )) ] , (5)again with x (0) = x ( nl c ) ≤ x c .We divide all the fluctuations in the definition of Z n into n groups: those that never lead to failure, those thatlead to failure after the first n − n −
2, . . . , 2, 1 steps.In this way Ω n enters the convolution equation Z n = Ω n + v n − (cid:88) m =1 Ω m Z n − m , (6) reminiscent of the work of M. E. Fisher et al. [15], Thevertex term v , a length, is introduced to account for thepath integration between 0 and x c around each m l c . Forsmall x c one expects v = λx c .We can solve (6) for Ω n by deconvoluting via the gener-ating functions ˜ Z ( u ) = (cid:80) ∞ n =1 Z n u n , Ω( u ) = (cid:80) ∞ n =1 Ω n u n thus obtaining ˜Ω( u ) = ˜ Z ( u )1 + v ˜ Z ( u ) , (7)from which Ω n can be derived by power expansion of˜Ω( u ). For instance, by direct differentiation of (7), andusing ˜ Z (0) = 0 one immediately finds P = 1 . (8)As expected by definition, a piece of length l c cannot fail.For simplicity, let us assume (justifications follow later)that only the temperature dependent effective Hamilto-nian in equation (3) has a bound ground state ψ ( x ) ofenergy (cid:15) o (functions of temperature) and that Z n reducesto a projector on that ground state, Then we can write Z n = (1 − p ) u − n v − where1 − p = vx − c (cid:90) x c ψ ( x )d x , (9)and u = exp( (cid:15) ). Summing simple series and followingthe procedures described above one finds Ω n = v − (1 − p ) p n − u − n and immediately P n = p n − (10)where, correctly, P = 1 and thus again, a piece of length l c does not fail. This exponential behavior of (10) hasbeen found in numerical studies of bubbles in DNA, asexplained later [12]. From the average ¯ n one finds theaverage distance ¯ l = l c ¯ n between failures for a long wire¯ l = l c − p (11)which shows that the lower is 1 − p (related to the proba-bility of penetration below x c of the eigenstate) the longeris the distance between failures. Also the probability P n ,and hence ¯ l or ¯ n are independent of u and thus of theactual eigenvalue of the state, as expected, since the prob-lem is invariant under uniform shift of the spectrum of(3): of course when more than one eigenvalue is relevant,then only the differences among them plays a role.When can we apply the outmost bound eigenvalue ap-proximation? We leave to the reader to prove that from(7), when the entire spectrum of (3) is retained, ˜Ω( u ) isa meromorphic function with real positive poles of or-der one. In the approximation of infrequent failures, orlarge bubbles in DNA (acceptable at physiological tem-perature) where the average distance ¯ n is large, only the d ∆ ! b x c c c c ! ! p " max T FIG. 2: Top: Plot of V ( x ) in (3). The system is in equilibriumbetween d and d − ∆, with a hard core repulsion at d and zero,and a square potential barrier at d − ∆ of strength (cid:15) o . Dashedlines are eigenstate of eigenvalue larger and smaller than β(cid:15) b .Bottom: the behavior of 1 − p ( T ) ∝ ¯ l − , for the potentialabove: notice saturation at high temperature, whereas defectsare virtually inexistent at for T /T c < . smallest pole is relevant. Then, if the spectrum of (3) issuch that (cid:15) s − (cid:15) (cid:29)
1, one only considers the boundgroundstate, as the correction to p is proportional toexp( (cid:15) − (cid:15) ). We will see that in the cases we studyboth approximations are fulfilled.For specificity we consider now the steplike potentialdepicted in Fig. 2, designed to capture in a simplifiedpicture the essential features of carbon nanotube col-lapse and the rich phenomenology of nanowire necking(whereas a different approach will be described later forDNA bubbles). We assume that the system is in equi-librium between distances d, d − ∆, with ∆ (cid:28) d as theinaccessible region is much wider than the size of theQ1D system. The infinite repulsion at d ensures stabilityof the wire toward large fluctuations, and, in the case ofcarbon nanotubes, it accounts for the very large energyneeded to break sp bonds, whereas the finite height (cid:15) b of the square repulsion allows for some penetration andthus a non zero probability of localized failure 1 − p in (9).Hard core repulsion at zero prevents negative values of x , which would lead to an artificial additional extensiveentropy.Now, a look at (3) shows that for very small T , thepotential barrier β(cid:15) b is very large in the Schr¨odinger op- erator, and thus the probability 1 − p from (9) is verysmall, because the ground state provides little tunneling.As the temperature increases, the barrier decreases, theground state has larger probability for penetration andthus 1 − p also increases. There is a definite tempera-ture T c such that when T > T c , there are no more boundstates of eigenvalue less than β(cid:15) b and the behavior of theground state in the forbidden zone becomes sinusoidalrather than exponential, and thus 1 − p becomes muchlarger. Yet 1 − p does not increase indefinitely with tem-perature, but saturates to a value easy to compute: atlarge T , β(cid:15) p goes to zero, the potential tends to a flatinfinite well of size d , and thus the ground state reducesto ψ ( x ) = (cid:112) /d sin ( xπ/d ) from which one finds imme-diately, for small x c , the saturation probability(1 − p ) max = π vx c d . (12)There is therefore a minimum distance between neckingof nanowires or localized collapses of nanotubes, given by¯ l min = 2 π l c d x c λ , (13)and attained at very large temperature. This is a generalfeature of localized failures in Q1D systems whose lateralstability is protected by an infinite potential barrier. Thisis clearly not the case for DNA, where instead denatu-ration, or complete separation of the two filaments, isknown to occur, as we will see later. No doubt this sat-uration effect could disappear should the defects comeclose enough to interact attractively–an interaction wecompletely neglect. Our estimates below show that thisis not the case for carbon nanotubes. T c can be found by solving graphically for the eigen-value equation of the potential in Fig. 2. If ∆ (cid:28) d , stateswith energy less than β(cid:15) p disappear at T c = ∆ (cid:112) k(cid:15) b /π. (14)Numerical solution of the eigenvalue equation (Fig. 2)shows that until about 0 . T c no defects are present, yet1 − p grows steeply around T c [where it is roughly (1 − p ( T c )) (cid:39) (1 − p ) max / (cid:39) (cid:39) k ∼ (cid:15) b ∼ − eV ˚A − , then we obtain T c ∼ − eV,or of the order of 10 –10 K. At very low temperaturesthe eigenstates reduce to plane waves in an infinite wellof width ∆, and their eigenvalues are proportional to T ,which is small: nevertheless, with the numbers above onefinds (cid:15) − (cid:15) (cid:29) T (cid:29) x c = 0 .
34 nm (the equilibrium distance in the vander Waals interaction between graphene sheets), d = 2 . < d c ( d c is the critical value above which the glob-ally collapsed configuration is stable, between 3 and 3.5nm [5, 6, 7]) λ ∼ l c ∼ l min ∼
100 nm, or less. A few considerations: as d comesclose to d c , the metastability of V ( x ) around x c becomessignificant and the potential in Fig. 2 does not prop-erly describe the situation anymore. Conversely, as d gets smaller, ¯ l min also decreases, as smaller amplitude isneeded in thermal fluctuations to induce failure, yet thepotential barrier (cid:15) b increases, and with it the tempera-ture T c . Finally a reversed case exist for large ( d > d c ),fully collapse nanotubes, which, conversely, we expect toshow local thermally inflated regions. From (13) one cansee that in this case, as d is larger, so is ¯ l min , of the orderof many hundreds of nanometers.Many theoretical and computational studies have beendevoted to the study of bubbles in DNA [3, 8, 9, 10, 11,12], because of their importance for genetic transcrip-tion. In an inverted approach, now it is the joined doublestrand that we considered a “defect“, and we ask our-selves what is the probability of existence of a bubble, orfilament separation, of a certain length. We can use theformalism above by considering a square potential [14],of width x c = 0 . (cid:15) b = 0 .
33 eV, [10], roughlyrepresenting the chemical bonding between complemen-tary bases, on a semi-infinite line. As for the elastic con-stant we use k = 3 10 − eV/˚A [10]: that is consistentwith the disappearance of a bond state and thus denatu-ration at a certain critical temperature T c = x c √ k(cid:15) b /π .With the numbers above, we can see that our single eigen-value approximation is justified for any reasonable tem-perature; therefore we find that the probability of a bub-ble decays exponentially with its length as P n = p n , aresult confirmed by computationally intensive numericalstudies [12]. Also, since p increases when the localizationof the ground state in the potential well decreases, theaverage length of a bubble will be larger in the regionwhere the bond between complementary bases is weaker,as expected for instance in regions rich of softer AT bases,a phenomenon demonstrated by recent work [18]. Also¯ l increases with temperature, to reach the value p = 1,thus ¯ l = ∞ at the critical temperature for denatura-tion. For temperatures very close to denaturation, thesingle eigenvalue approximation might break down, andthe probability might not then behave exponentially.In conclusion we have calculated the statistics ofthermally-induced localized structural failures, defects orinstabilities in Q1D systems. For the collapse of carbonnanotubes, and necking of nanowires, we predict a crit-ical temperature below which the occurrence of failurebecomes extremely rare. We also show that the behavior of the average distance between defects is monotonicallydecreasing in temperature and approaches a minimumavearage distance for very high temperatures. These re-sults may be verified by experimental measurements oftransport at different temperatures. For DNA, we predictthat the probability of bubble openings decreases expo-nentially with the length of the bubble. It increases withtemperature and is likelier to occur where the strength ofbonding of complementary bases is weaker. These resultsmay be verified by simulations using the PBD model.Discussions on DNA modeling with Boian Alexandrov(Los Alamos National Laboratory) was as useful as pleas-ant. This work was carried out under the auspices of theNational Nuclear Security Administration of the U.S. De-partment of Energy at Los Alamos National Laboratoryunder Contract No. DE-AC52-06NA25396. [1] P.A. Serena and N. Garcia (Editors), Nanowires
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