Polymer desorption under pulling: a novel dichotomic phase transition
We show that the structural properties and phase behavior of a self-avoiding polymer chain on adhesive substrate, subject to pulling at the chain end, can be obtained by means of a Grand Canonical Ensemble (GCE) approach. We derive analytical expressions for the mean length of the basic structural units of adsorbed polymer, such as loops and tails, in terms of the adhesive potential and detachment force, and determine values of the universal exponents which govern their probability distributions. Most notably, the hitherto controversial value of the critical adsorption exponent is found to depend essentially on the interaction between different loops. The chain detachment transition turns out to be of the first order, albeit dichotomic, i.e., no coexistence of different phase states exists. These novel theoretical predictions and the suggested phase diagram of the adsorption-desorption transformation under external pulling force are verified by means of extensive Monte Carlo simulations.
pacs:05.50.+q, 68.43.Mn, 64.60.Ak, 82.35.Gh, 62.25.+g
Introduction - The manipulation of single polymer chains has turned recently into an important method for understanding their mechanical properties and characterization of the intermolecular interactions Strick (); Celestini (). Such manipulation is mainly triggered by the progress in atomic force microscopy (AFM) Rief () as well as by the development of optical/magnetic tweezers technique Bustamante (). This rapid development has been followed by theoretical considerations, based on the mean - field approximation Sevick (), which provide important insight into the mechanism of polymer detachment from adhesive surfaces under the external force pulling. A comprehensive study by Skvortsov et al. SKB () examines the case of a Gaussian polymer chain. We also note here the close analogy between the force detachment of adsorbed chain and the unzipping of a double - stranded DNA. Recently, DNA denaturation and unzipping have been treated by Kafri et al. Kafri () using the Grand Canonical Ensemble (GCE) approach Poland () as well as Duplantier’s analysis of polymer networks of arbitrary topology Duplantier (). An important result for the properties of adsorbed macromolecule under pulling turns to be the observation Kafri () that the universal exponents (which govern polymer statistics) undergo renormalization due to excluded volume effects, leading thus to a change of the order of DNA melting transition from second to first order. In this Letter we use similar methods to describe the structure and detachment of a single chain from a sticky substrate when the chain end is pulled by external force.
Single chain adsorption - Starting with the conventional (i.e., force-free) adsorption, we recall that an adsorbed chain is build up from loops, trains, and a free tail. One can treat statistically these basic structural units by means of the GCE approach where the lengths of the buildings blocks are not fixed but may rather fluctuate. The GCE-partition function is then given by
where is the fugacity and , , and denote the GCE partition functions of loops, trains and tails, respectively. The building block adjacent to the tethered chain end is allowed for by . The partition function of the loops is defined as , where is the connective constant and is the exponent which governs surface loops statistics. It is well known that for an isolated loop Vanderzande (). We will argue below that changes value due to the excluded volume interaction between a loop and the rest of the chain. The train GCE-partition function reads whereby one assumes that each adsorbed segment gains an additional statistical weight with the dimensionless adsorption energy . Eventually, the GCE partition function for the chain tail is defined by . For an isolated tail Vanderzande () but again the excluded volume interactions of a tail with the rest of the chain increase the value of . Using the generating function method Rudnick (), is obtained as where the pole is given by the condition so that the free energy is and the fraction of adsorbed monomers . In terms of the so called polylog function, which is defined as Erdelyi (), the equation for reads
A nontrivial solution for in terms of (or the adsorption energy ) appears at the critical adsorption point (CAP) where is determined from as and is the Riemann function. In the vicinity of the CAP the solution attains the form
where is a constant. Then the average fraction of adsorbed monomers is . A comparison with the well known scaling relationship where is the so called adsorption (or, crossover) exponent Vanderzande () suggests that
This is a result of principal importance. It shows that the crossover exponent , describing polymer adsorption at criticality, is determined by the exponent which governs polymer loop statistics! If loops are treated as isolated objects, then so that . In contrast, excluded volume interactions between a loop and the rest of the chain lead to an increase of and , as we show below.
Probability distributions of loops and tails - How does the length distribution of polymer loops and tails close to the CAP look like? From the expression for , given above, and eq.(3) we have . This is valid only for since a solution for eq.(2) for subcritical values of the adhesive potential does not exist. Nontheless, even in the subcritical region, , there are still monomers which occasionally touch the substrate, creating thus single loops at the expense of the tail length. The partition function of such a loop-tail configuration is . On the other side, the partition function of a tail conformation with no loops whatsoever (i.e., of a nonadsorbed tethered chain) is . Thus the probability to find a loop of length next to a tail of length can be estimated as , which is valid at . In the vicinity of the CAP, , the distribution will be given by an interpolation between the expressions above. Hence, the overall loop distribution becomes
The same reasonings for a tail leads to the distribution
In eqs.(5) - (6) are constants. Close to CAP these distributions are expected to attain a U - shaped form (with two maxima at and ), as predicted for a Gaussian chain by Gorbunov et al. Gorbunov (). For the average loop length the GCE-partition function for loops yields . Close to the CAP, diverges as . The average tail length can be obtain as . Again, using the properties of the polylog function, one can show that close to the average tail length diverges as . Note that this behavior corresponds to a length of adsorption blob .
Role of interacting loops and tails - Consider the number of configurations of a tethered chain in the vicinity of the CAP as an array of loops which end up with a tail. Using the approach of Kafri et al. Kafri () along with Duplantier’s Duplantier () graph theory of polymer networks, one may write the partition function for a chain with building blocks: loops and a tail. Consider a loop of length while the length of the rest of the chain is , that is, . In the limit of (but with ) one can show SBVRAMTV () that where the surface exponent and are critical bulk and surface exponents Duplantier (). The last result indicates that the effective loop exponent becomes
Thus, , in agreement with earlier Monte Carlo findings Eisenriegler (). One should emphasize, however, that the foregoing derivation is Mean-Field-like ( appears as a product of loop- and rest-of-the-chain contributions) which overestimates the interactions and increases significantly the value of , serving as an upper bound. The value of , therefore, is found to satisfy the inequality , i.e., depending on loop interactions, .
Adsorption under detaching force - Using the GCE approach now we treat the case of self-avoiding polymer chain adsorption in the presence of pulling force, thus extending the consideration of Gaussian chains by Gorbunov et al. Skvortsov (). Under external detaching force , the tail GCE-partition function in eq. (1) has to be replaced by where is the end-to-end distance distribution function for a self-avoiding chain DesCloizeaux (). After some straightforward calculations the tail GCE-partition function can be written as
Here the dimensionless force , the exponents , and with and . The function has a branch point at , i.e., . One may, therefore, conclude that the total GCE-partition function has two singularities on the real axis: the pole , and the branch point . It is known (see, e.g., Sec. 2.4.3. in Rudnick ()) that for the main contributions to come from the pole and the branch singular points, i.e.,
Thus, for large only the smallest of these points matters. On the other hand, depends on the dimensionless adsorption energy only (or, on ) whereas is controlled by the external force . Therefore, in terms of the two control parameters, and , the equation determines the critical line of transition between the adsorbed phase and the force-induced desorbed phase . In the following this line will be called detachment line. Below it, , or above, , either or , respectively, contribute to . The controll parameters, and , which satisfy this equation, denote detachment energy and detachment force, respectively. On the detachment line the system undergoes a first-order phase transition. The detachment line itself terminates for in the CAP, ,
where the transition becomes of second order. In the vicinity of the CAP the detachment force vanishes as . This first order adsorption-desorption phase transition under pulling force has a clear dichotomic nature (i.e., it follows an “either - or” scenario): in the thermodynamic limit there is no phase coexistence! The configurations are divided into adsorbed and detached (or stretched) dichotomic classes. Metastable states are completely absent. Moreover, the mean loop length remains finite upon detachment line crossing. The average tail length , on the contrary, diverges close to the detachment line. Indeed, at the average tail length is given by . At the detachment line, , it diverges as .
Reentrant phase behavior - Recently, it has been realized Mishra () that the detachment line, when represented in terms of dimensional variables, force versus temperature , goes (at a relatively low temperature) through a maximum, that is, the desorption transition shows reentrant behavior! Below we demonstrate that this result follows directly from our theory. It can be seen that the solution of eq.(2) at large values of (or, at low temperature) can be written as so that the detachment line, , in terms of dimensionless parameters is monotonous, . Note, however, that the same detachment line, if represented in terms of the dimensional control parameters, force versus temperature (with a fixed dimensional energy ), shows a nonmonotonic behavior . This curve has a maximum at a temperature given by .
Monte Carlo Simulation - We have investigated the force induced desorption of a polymer by means of extensive Monte Carlo simulations using a coarse grained off-lattice bead-spring model MC_Milchev () of a polymer. Fig. 1a shows the variation of the order parameter (average fraction of adsorbed monomers) with changing adhesive potential at fixed pulling force whereas Fig.1b depicts vs. force for various . The abrupt change of the order parameter is in close agreement with our theoretical prediction. Using the threshold values of and for critical adsorption/desorption in the thermodynamic limit , one can construct the adsorption - desorption phase diagram for a polymer chain under pulling shown in Fig.2 which is among the central results of this work. The detachment lines, obtained from MC data and the numerical solution of almost coincide, and the slope of vs is close to unity, according to the prediction . Also indicated by the shaded area in Fig.2 is the reentrant image of the same phase diagram, obtained when the numerical solution of is plotted in dimensional units versus .
In Fig. 3a we show the PDF of tail length at different strength of adsorption in the absence of pulling. This confirms the U - shape of predicted by eq.(6). While for the agreement with eq. 6 is perfect, for long tails are slightly overestimated by eq.(6). This small discrepancy reflects the dominance of our “single loop & tail” approximation - multiple loops would effectively reduce the tail size. Fig. 3b shows the divergency of close to the critical point . For chain of finite length , the tail length divergence at is replaced by a rounding into a plateau since but away from the measured slope extrapolates to the theoretical prediction . In the presence of pulling force one observes a remarkable feature of the order parameter probability distribution
- an absence of two peaks in the vicinity of the critical strength of adsorption, , which still keeps the polymer adsorbed at pulling force - Fig. 4. At the distribution is flat, indicating huge fluctuations so that any value of is equally probable. Close to , one observes a clear maximum in the distribution , indicating a desorbed chain with for , or a completely adsorbed chain with for . This lack of bimodality in the manifests the dichotomic nature of the desorption transition which rules out phase coexistence.
In conclusion, we have demonstrated that a full description of the force induced polymer chain desorption transition can be derived by means of the GCE approach, yielding the average size and probability distribution functions of all basic structural units as well as their variation with changing force or strength of adhesion. The detachment transition is proved to be of first order albeit dichotomic in nature thus ruling out phase coexistence. The critical line of desorption, while monotonous when plotted in dimensionless units of detachment force against surface potential, becomes non-monotonous in units of force against temperature, thus outlining a reentrant phase diagram. In addition, we show that the crossover exponent, , governing polymer behavior at criticality, depends essentially on interactions between different loops so that . All these predictions appear in very good agreement with our Monte Carlo computer simulation results.
Acknowledgments We are indebted to A. Skvortsov, L. Klushin, J.-U. Sommer, and K. Binder for useful discussions during the preparation of this work. A. Milchev thanks the Max-Planck Institute for Polymer Research in Mainz, Germany, for hospitality during his visit in the institute. A. Milchev and V. Rostiashvili acknowledge support from the Deutsche Forschungsgemeinschaft (DFG), grant No. SFB 625/B4.
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