Non-monotonic dependence of the rupture force in polymer chains on their lengths
We consider the rupture dynamics of a homopolymer chain pulled at one end at a constant loading rate. Our model of the breakable polymer is related to the Rouse chain, with the only difference that the interaction between the monomers is described by the Morse potential instead of the harmonic one, and thus allows for mechanical failure. We show that in the experimentally relevant domain of parameters the dependence of the most probable rupture force on the chain length may be non-monotonic, so that the medium-length chains break easier than the short and the long ones. The qualitative theory of the effect is presented.
Probing the mechanical response of single molecules to an external load has attracted considerable interest in recent years. In molecular failure experiments Rief et al. (1997); Mehta et al. (1999); Grandbois et al. (1999); Cui et al. (2006); Friedsam et al. (2003); Neuert et al. (2007) and experiments on protein unfolding Strunz et al. (1999) a force grows linearly in time until a molecule breaks or changes its structure. The dynamic force spectroscopy Evans (2001); Strunz et al. (1999) delivering the spectrum of the rupture forces at different loading rates gives deep insights into the internal dynamics of molecules Garg (1995); Dudko et al. (2003, 2006); Dias et al. (2005). In all these experiments polymers play an outstanding role, either as elastic couplers or as a subject of study Embrechts et al. (2008); Kühner and Gaub (2006); Friedsam et al. (2003). It was shown, that even the mechanical properties of passive polymer spacers can affect the outcome of pulling experiments Friedsam et al. (2003); Kühner and Gaub (2006); Neuert et al. (2007). Recently a strong impact of the polymer size on its rupture dynamics has been reported Embrechts et al. (2008); Fugmann and Sokolov (2009).
The overall picture of the homopolymer mechanical failure is very similar to the one of the failure of macroscopic fibers, as represented as a sequence of links. Under very slow ramping of the external load each link of the chain is subjected to the same force, and it is mostly the microscopic mechanism of a failure of a single link, which discriminates between the two situations. In a macroscopic case the single link failure is due to the preexisting defects, so that the survival probability of a link under force typically follows a power law. The probability that all the links are intact is then given by
and tends to a Weibull law for fibers long enough. Moreover, the longer fibers typically break at smaller forces, since the probability to find a weak link grows with the fibers’ length, see e.g. Padgett et al. (1995). In a microscopic homogeneous polymer chain, the single link breakdown is thermally activated; the breakdown probability at a given force at a given time follows an exponential law (as obtained from the Kramers theory of thermally activated barrier crossing), and therefore tends to a Gumbel distribution. However, the fact that the longer chains break more easily still holds: the larger is the number of links, the higher is the probability that one of them breaks Embrechts et al. (2008). At higher loading rates the situation with the polymer chain gets more involved. As we proceed to show, a complex interplay between the temporal aspects of thermally activated single link breakdown and the force redistribution along the chain leads to new features of the polymer failure problem. Thus, the behavior of the most probable force at breakdown as a function of the chain’s length gets non-monotonic: the medium-length chains break more easily than the short and the long ones. The effect is not small and therefore is pertinent to experimental observation. The breakdown force is reduced by compared to the extremal cases of a single bond and of an infinitely long chain at the chosen values of the parameters, and the effect is expected to be the more pronounced the softer the bonds are since then the thermally activated breakdown happens earlier while the Rouse time, being a measure for the timescale of force propagation, becomes large.
In what follows we first discuss our theoretical model and present the results of its numerical simulations. We then turn to the analytical description of the behavior observed, giving a full qualitative picture of the effect and then give its simple explanation.
Like in Fugmann and Sokolov (2009) our model corresponds to a chain of monomers interacting via the Morse potential
which parameterizes the interaction energy in terms of dissociation energy and stiffness . This is a simple prototype of an intramolecular interaction potential which offers fragmentation. Otherwise, the model is identical to the Rouse one Doi and Edwards (1986): we disregard hydrodynamical interactions and describe the interaction of the monomers with the heat bath via independent white noises. The constant loading enters through an additional time dependent potential of the form
with loading rate . The load is denoted by and is applied at one end of the chain while the other end is fixed.
At smaller loads the overall potential has two extrema, , a minimum corresponding to a metastable state of the pulled bond, and a maximum providing the activation barrier. There exists a critical load for which the extrema merge at and disappear. In the purely deterministic dynamics the Morse bond breaks exactly at . Since the system is in contact to a heat bath at temperature , its overdamped dynamics is described by a set of coupled Langevin equations
with , Gaussian white noise , the Boltzman constant , the friction coefficient and . We introduce (), () and (in the following is referred to as force). The diffusion coefficient is denoted by ().
The set of coupled equations (4) is integrated by use of a Heun integration scheme. A chain is considered as broken as soon as one of the reaction coordinates overcomes the activation barrier. Statistics stem from an ensemble of at least simulation runs.
In Fig. 1 we present the numerically obtained most probable rupture force as a function of the chain length for a fixed value of the loading rate . For small chain lengths the numerically obtained most probable rupture force (symbols) follows the scaling relation given in Eq. (7) reaching a minimal value for an intermediate value of . A further increase in the chain length results in an increase of the most probable rupture force and eventually in a saturation of the latter. A theoretical description following the ansatz in Fugmann and Sokolov (2009) (dashed line) gives in a good agreement the breakdown force for shorter chains as well as in the saturation regime, i.e. for very long ones. The curve obtained from Eq. (8) of the present contribution (solid line) gives the correct qualitative behavior in the whole domain of the chains’ lengths.
The distribution of the position of the breakdown in a chain of links is shown in Fig. 2. For a small loading rate, i.e., (solid line), there is only a very slight decrease of the rupture probability density along the chain. Virtually all bonds contribute equally to the rupture process. The situation changes drastically when passing to higher loading rates (dashed and dashed-dotted lines). The rupture probability density decreases fast along the chain. For only half of the chain contributes to the rupture process.
In Fig. 3 we present the distribution of the position of breakdown in the chain for a fixed value of the loading rate and two different values of the length . One readily infers that the probability of a bond breakdown at a given site decays for a longer chain (solid line) faster with the distance from the pulled end than it does for a shorter one (dashed line). Thus, although the longer chains offers a larger number of possible breakdown sites, the number of bonds which contribute to the rupture process becomes smaller reaching a constant—loading rate dependent— value in the limit of a semiinfinite system.
In order to derive an analytical description of the chain rupture process we first recall the model describing the single bond rupture. The probability that a single breakable bond remains intact can be expressed through the following kinetic equation Tshiprut et al. (2006); Dudko et al. (2003, 2006); Raible et al. (2004) with being the Kramers rate Kramers (1940); Hänggi et al. (1990). Taking we can rewrite the kinetic equation in the form . The measured probability density function (PDF) for the rupture forces then is .
Under the assumption that is close to when bond rupture occurs, it is usual to expand the potential around the inflection point up to the third order in deviations from Garg (1995); Dudko et al. (2003, 2006). We note that the breakdown properties of the chain only depend on the behavior of the potential close to the point of critical load, which are universal Dudko et al. (2006); Lin et al. (2007); Friddle (2008). The Morse potential gives a convenient parametrization of the situation in terms of dissociation energy and stiffness. In the limit of small loading rates the most probable rupture force follows the scaling relation Dudko et al. (2003); Fugmann and Sokolov (2009)
with and .
Passing to a chain of breakable bonds we consider the rupture dynamics of different bonds to be independent. The probability that a bond is intact is denoted by . The pulling force at the chain end is . The equation for the survival probability of the chain reads:
If the typical rupture time is much larger than the Rouse time of the system, , (which is the case either for short chains or for small loading rates) each bond in the chain experiences the same force . The probability that a bond breaks in an interval is and it is given by the same expression as with changed for . This gives the scaling relation for the most probable rupture force Fugmann and Sokolov (2009)
The opposite situation is more involved. First, we have to calculate . To do this, we note that the barrier crossing events are very rare: Most of the time the motion of the monomers takes place close to the quadratic potential minima. Therefore, like in Ref. Fugmann and Sokolov (2009), the dynamics of the chain can be approximated by a Rouse one. At difference to Fugmann and Sokolov (2009) the boundary condition at the grafted end is explicitly taken into account. This leads to the dependence of the force profile on the chain’s length as shown in Fig. 4.
Linearizing the force profile close to the pulled end and inserting it into Eq. (6) we eventually, derive the PDF
with the probability function
, and . is the upper incomplete Gamma function.
The analytical description given by Eq. (8) (solid line in Fig. 1)) agrees well with the outcome of the numerical simulations. The nonmonotonous behavior of the rupture forces is nicely reproduced and in the limit of short and long chains the theory agrees also qualitatively. Furthermore the chain lengths which minimizes the most probable rupture force coincide with the ones derived from the numerical simulations. Deviations can result from the harmonic approximation in the derivation of the force profile: the decay of the force profile along the chain for a soft Morse potential is expected to be more pronounced than in the Rouse chain, so that less bonds contribute to the rupture process. This might explain the shift of the theoretical curve to lower rupture forces compared to numerical data points.
The overall behavior can therefore be explained as follows. A short chain has only a few bonds that can break. Each of them feels practically the same force , since the forces acting on the bonds decrease only slightly with their distance from the pulled end, see Fig. 4. The longer is the chain, the more breakable bonds are present, however each of them is subject to the tension which is smaller than and decays with the chain’s length. The interplay between the thermally activated rupture of a single bond and the force distribution along the chain generates a non-monotonous behavior of the typical rupture forces.
Let us summarize our findings. Compared to the single bond breaking, the existence of the chain introduces new aspects into rupture dynamics, the most important being the delayed stress propagation along the chain. We show that the most probable rupture force decreases with the length of the chain as for short chains and saturates at the value depending on the loading rate for very long ones. In between it can exhibit a non monotonous behavior: the most probable rupture force attains its minimum for a certain intermediate chain length. These results are obtained via direct numerical simulations of a Rouse-like model (however with anharmonic Morse interaction potential between the monomers instead of a harmonic one for a genuine Rouse chain) and are well reproduced by a generalization of an analytical approach put forward in our previous study Fugmann and Sokolov (2009). The qualitative explanation of the effect involves a complex interplay between the force propagation into the chain and the extreme-value statistic underlying rupture. Since the effect is rather dependent on the different timescales in the system under study than on the specific parameter values it is pertinent to experimental observation.
Acknowledgements.The authors thankfully acknowledge valuable discussions with W. Ebeling. This research has been supported by DFG within the SFB 555 research collaboration program.
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