The Longitudinal Superdiffusive Motion of Block Copolymer in A

Article The Longitudinal Superdiﬀusive Motion of Block Copolymer in a Tight Nanopore

Waldemar Nowicki

Faculty of Chemistry, Adam Mickiewicz University in Poznań,ul. Uniwersytetu Poznańskiego8, 61-614 Poznań, Poland; [email protected]

Received: 7 November 2020; Accepted: 3 December 2020; Published: 8 December 2020

Abstract: The structure and dynamic properties of polymer chains in a confined environment were studied by means of the Monte Carlo method. The studied chains were represented by coarse-grained models and embedded into a simple 3D cubic lattice. The chains stood for two-block linear copolymers of different energy of bead–bead interactions. Their behavior was studied in a nanotube formed by four impenetrable surfaces. The long-time unidirectional motion of the chain in the tight nanopore was found to be correlated with the orientation of both parts of the copolymer along the length of the nanopore. A possible mechanism of the anomalous diffusion was proposed on the basis of thermodynamics of the system, more precisely on the free energy barrier of the swapping of positions of both parts of the chain and the impulse of temporary forces induced by variation of the chain conformation. The mean bead and the mass center autocorrelation functions were examined. While the former function behaves classically, the latter indicates the period of time of superdiffusive motion similar to the ballistic motion with the autocorrelation function scaling with the exponent t5/3. A distribution of periods of time of chain diffusion between swapping events was found and discussed. The influence of the nanotube width and the chain length on the polymer diffusivity was studied.

Keywords: diﬀusion; lattice model; Monte Carlo method

1. Introduction In most cases, the self-diffusion process can be described as a relation between the mean square displacement (MSD) and time, expressed as a scaling law with the exponent equal to 1. However, in some systems, this simple relationship should be formulated in a more general way, as follows [1]: D E r(t)2 tv (1) ∼ Depending on the value of the anomalous diffusion exponent, v, a subdiffusion (0 < v < 1) or superdiffusion (v > 1) can be distinguished [2–4]. The origin of the discrepancy between the properties of the system, and the scaling law with the exponent 1 is that its derivation requires that the Brownian particle should move in an infinite structureless medium. This assumption is generally incorrect when the Brownian motion takes place in a complex medium or when the diffusively migrating objects should be treated as structured species whose structural elements’ motions are only partly independent, but mutually correlated in general. Moreover, Equation (1) describes self-diffusion in a long period of time only. The subdiffusion deviations from the simple scaling exponent v = 1 have been predicted even for simple stochastic chains. According to the Rouse theory [5,6], the MSD of the beads forming the polymer chain increases with time in three time regimes: in the first and the last regime, the MSD scales with v = 1 (but with two different diffusion coefficients), whereas, between these two diffusional asymptotes, the MSD increases as the power of t1/2. The diffusion of the theta chain (Zimm model [6,7])

Polymers 2020, 12, 2931; doi:10.3390/polym12122931 www.mdpi.com/journal/polymers Polymers 2020, 12, 2931 2 of 19 behaves similarly: The short, intermediate, and longtime MSD vs. time dependencies are power laws with exponents v = 1, 2/3, and 1, respectively. The deviations of v values predicted by the reptation theory are similar. At very short time of self-diffusion, Brownian motion occurs in three basic regimes: (i) a ballistic-like motion before any molecular collisions [8]; (ii) a motion affected by hydrodynamic forces of the fluid [9], which begins when the moving object interacts with fluid molecules; and (iii) the actual random Brownian motion fractal in nature. The motion in the last regime—the random Brownian motion—causes the MSD values to increase linearly with time. The motion in the first regime—the free unidirectional ballistic motion scaling with t2—cannot last for long and occurs at timescales once deemed immeasurably small by Einstein [10]. The ballistic motion can be met not only in the transport of electrons in conductors, phonons in solids, and photons in opaque media but also in disordered liquids, colloids, bubbles, grains, and so forth that are on the verge of jamming [11]. This particular motion was observed for extended periods of time for particles in the air as a result of low viscosity of dispersion medium [5,12] and in short times in liquid water as a higher-density medium [10]. The transition between ballistic and diffusive motion is highly dependent on the properties and structure of a particular liquid [13]. Unusually long ballistic motions have been reported by research groups studying particles in polymer melts [14–16], polymer-grafted gold nanoparticles [17], polymer rings [18], DNA [19], enzymes [20], and even bacteria [21]. Dissolved polymer molecules located in the confined environment show unique behavior different from those of free macromolecules in the bulk phase [22]. The nanoconfinement alters not only the statics [23,24] but also the Brownian dynamics of a polymer chain [25,26]. The specific behavior of macromolecules results from a decrease in conformational entropy of macromolecules due to the excluded volume effects [27]. The constraints of conformation can induce the entropy-driven processes such as segregation of macromolecules [28,29], translocation through narrow pores [30,31] and translocation through connected chamber-pore system [32], prestretching before threading through a nanopore [33,34], etc. Investigation of Brownian polymer motions in nanoconfinement permitted development of methods for sequencing, manipulations, sorting, and separation of DNA molecules [35,36]. The shift of the ballistic/diffusive transition point towards longer times can be induced by the introduction of some geometrical constraints to space in which diffusion takes place such as slits, nanochannels, or by the limitation of diffusion to two-dimensional square lattice [37]. Anomalous diffusion has been repeatedly observed in the systems containing branched polymers [38,39]. As reported by Romiszowski and Sikorski, the longitudinal unidirectional motion of the star-branched chains can be observed in a very narrow nanochannel [40]. This motion happens when one arm points towards a particular direction, while the other two arms are left behind the branching point. The authors of References [40] have postulated a mechanism of the ballistic motion based on the fact that the two arms, located opposite to the direction of motion push the whole structure towards the direction of motion. As observed, since the nanochannel is still large enough to enable the arm’s ends to change their position passing from one side of the molecule to the other, the direction of the longitudinal motion of the chain changes chaotically. A similar anomalous diffusion governed by the active noise amplitude has been observed by Saito et al. [41]. The explanation of the longitudinal unidirectional motion of star macromolecules in tight confinement suggests that similar behavior of a wider class of asymmetric chains can be expected, e.g., even in the simple case of chains built of two parts of different properties. Therefore, the aim of this study is the examination of diffusion of a simple asymmetric chain—a block copolymer—composed of two parts of the same number of beads but of different energies of interaction between the segments. The energies were chosen so that one half of the chain could be treated as immersed in the athermal solution, and the other one was characterized by a set of interactions preferring bead–bead interaction (at the Flory coefficient χ = 2). The difference in the energies produces differences in the local conformational entropies and radii of gyration of both parts of the chain. This asymmetric structure is Polymers 2020, 12, 2931 3 of 19 supposed to be subjected to the Brownian motion that is disturbed by a longitudinal ballistic motion similar to that described in References [40], since the reversal of the chain parts in a nanochannel is unlikely. The paper also focuses on investigation of the effect of the width of the nanochannel and the length of the chain on its diffusive properties. The paper is organized as follows: In Section2, the details of the simulation technique applied to the model representation of the polymer chain and geometry of the systems are presented. The results of a single simulation and the analysis of the evolution of geometric and thermodynamic properties of the migrating chain are gathered in Sections 3.1 and 3.2, respectively. Sections 3.3 and 3.4 contain the analysis of the bead and the chain mass center autocorrelation functions. The distribution of the periods between position swapping of chain parts is analyzed in Section 3.5. The geometrical aspects and thermodynamics of a single swapping event in very short time steps are presented in Section 3.6. Then, Section 3.7 presents a hypothetical explanation of the long-lasting asymmetric diffusion of the studied copolymers. A summary of the results is given in Section 3.8.

2. Model and Simulation Method The study was performed by employing the simple and relatively fast Metropolis Monte Carlo (MC) method [42], chosen because of the ease of its implementation. As a result of the model assumptions, the studied system represents a damping dynamic model, holding only the activation aspect of diffusion and neglecting the inertia effects [43]. The simulations were performed on the three-dimensional cubic lattice of the lattice constant b. The confined space was a rectangular nanochannel. The length of the nanochannel was 600b and the width D was varied from 3b to 8b (depending on the simulation, in most simulations D = 3b). At both ends of the channel, the periodic boundary conditions were applied. The polymer chains are represented by SAWs (self-avoiding walks) embedded in the lattice. Each monomer is identified by the site on the lattice (B bead), so each chain is a sequence of N consecutive sites occupied by B beads. The beads succeeding along the chain are located in adjacent lattice sites at the distance b. The interaction energies of the type bead—bead and bead—solvent molecule and solvent molecule—solvent molecule were incorporated into the system as εPP, εPS, and εSS, respectively. For simplicity, εPS and εSS were assumed to be equal to zero, whereas εPP was dependent on the assumed type of bead. Two different types of beads were considered: one characterized by εPP = 0 (i.e., immersed in the athermal solution) and the other with εPP not equal to zero. In most cases, εPP was assumed as -kBT corresponding to the solution of the Flory coefficient χ = 2 [6]. The study neglects all other intermolecular interactions, as well as interaction with nanochannel walls, except for the excluded volume effect. The studied chain consisted of N beads built of two equal parts: the half-chain P1 of bead type ε /k T = 0 and the half-chain P2 of type ε /k T = 1. P1 and P2 PP B PP B − half-chains can be interpreted as immersed in the poor and good solvent with the tendency to form globules and loose conformation, respectively [6]. Fluctuations in the conformational entropy of the chains were examined by using the modified SCM (statistical counting method) [44] from the following equation:

N 1 S X− = ln(ω ) (2) k i B i=1 where N denotes the number of beads in the chain, kB is the Boltzmann constant, and ωi is the eﬀective coordination number of each succeeding bead in the chain, which is equal to the number of lattice sites occupied by the solvent molecules. The modiﬁcation consists in the application of the “phantom chain”: Initially, the whole chain was removed from the simulation space, and then—following the chain path—the ωi value was calculated in a step-by-step manner. After each step, one succeeding bead was put back to the chain structure. Polymers 2020, 12, 2931 4 of 19

The temporary internal energy of the system was calculated as the sum of interaction energies of all current bead pairs m occupying the adjacent lattice sites εPP , 0

m U X = ε (3) k T PP B i=1

The Helmholtz free energy, A [45], of the system was calculated as the sum of i/energetic contributions, U, and ii/entropic contributions, S, associated with the chain:

A = U TS (4) − The chain translocation was studied by means of the Metropolis MC method [46]. The elementary micromodification applied in the simulation consisted of (i) random selection of the bead to be moved; (ii) a shift of the bead to a new position (using elementary Verdier–Stockmayer algorithms which include the kink-jump, crankshaft (two segments), reptation, or end movements [47]; (iii) checking whether the new position does not violate the topological constraints or excluded volume condition; and, finally, (iv) the Metropolis criterion. The applied reptation algorithm includes a simultaneous removal of a monomer from one end of the chain, addition of a monomer to the other end, and shift of all monomers along the chain to ensure continuity of the parts P1 and P2. The system energy defined by Equation (2) was used to calculate the Boltzmann factor in the standard Metropolis algorithm at a constant reduced temperature T = 1. All types of elementary micromodifications were employed with the frequency proportional to the number of objects to which they can be applied, because such a procedure provided the correct timescale of the simulation [43,48]. The number of beads in the chains tested was N = 20 200 (in most simulations N = 100). ÷ Each single simulation was a result up to 108/N MC steps (one MC step corresponds to the number of shifts needed to give each of the beads the possibility to move once, i.e., 100 elementary macromolecule micromodifications of a macromolecule of N = 100). The unit of the corresponding time period, t, was defined as one MC cycle. Part of the results was presented as functions of elementary time tE = t/N. Each simulation started with 105 steps of the equilibration of conformation, using the annealing with a hyperbolic cooling schedule [49]. The data collected from the simulation contain positions of mass centers of the whole coil and its parts P1 and P2 and all components of free energy of the system, including internal energy and conformational entropy of the macromolecule. Moreover, on the basis of the coordinates of all beads, as well as those of the center of the coil, the autocorrelation function of the whole chain, g, is defined as follows: D E g(∆t) = (x(t) x(t + ∆t))2 (5) − and the mean autocorrelation function of beads belonging to part P1 or P2, gB, is defined as follows:

N/2 2 XD E g (∆t) = (x (t) x (t + ∆t))2 (6) B N B,i − B,i i=1 where x is the position of the center of mass and xB,i is the position of bead i taken from the simulation trajectory, respectively. Hence, the autocorrelation function was calculated as an average of values obtained for the whole directory and all assumed time steps, ∆t, which is the actual parameter of the function. The gB function was additionally averaged over the positions of the monomers belonging to part P1 or P2. Polymers 2020, 12, 2931 5 of 19

In the course of simulations, the mutual position of parts P1 and P2 with respect to the chain mass center (the mutual orientation parameter) was detected and stored in the variable M. The parameter M was defined as follows:   1 if rP1 rP2 > δ/2  | | − | | M =  1 if rP1 rP2 < δ/2 (7)  − | | − | | −  0 if r r < δ/2 || P1| − | P2|| where rP1 and rP2 are vectors identifying the Cartesian coordinates of mass centers of P1 and P2 relative to the mass center of the whole chain, respectively, and δ = 0.01b. This definition is equivalent to the following expression applicable for the chain in a tight nanopore:   1 if xP1 xP2 > δ  − M =  1 if xP1 xP2 < δ (8)  − − −  0 if x x < δ | P1 − P2| The M parameter took a value equal to +1 when the center of mass of P1 part was to the right of the mass center of P2 (its x coordinate was higher than that of P2), 1 in the opposite situation, and 0 − when the difference in positions was small. In the bulk phase, the parameter behaved similarly, but the criterion of its value was based on the distance of the mass centers of parts P1 and P2 from the mass center of the whole chain. The parameter was introduced to check if the direction of chain motion was correlated with the mutual positions of parts P1 and P2. Forces exerted by parts of the chain, resulting from changes in their free energy, were calculated as partial derivatives: ! ∂A F = (9) ∂x T,V where T and V denote the temperature and the volume of the system, respectively, whereas their averaged impulses in the time period ∆t = tE were computed as the force integrals over the time of movement: acting on P1, P2, and at the whole chain.

tZ+∆t tZ+∆t ! 1 1 ∂A I = Fdt = dt (10) t t ∂x T,V t t

The averaged impulses were computed for forces acting on part P1, part P2, and on the whole chain.

3. Results

3.1. The Chain in a Tight Nanopore Visualization of most often observed conﬁgurations of the studied chain in the course of evolution in the nanopore is schematically shown in Figure1. In the course of the simulation, both parts of the chain (P1 and P2) occupied diﬀerent positions along the nanopore, except during short periods when they overlapped each other during swapping of their mutual locations. The probability of both orientations of the chain parts was the same. The conformation of P1 was usually more compact than that of P2 as a result of the assumed stronger bead-to-bead interactions. Polymers 2020, 12, x FOR PEER REVIEW 5 of 19 where rP1 and rP2 are vectors identifying the Cartesian coordinates of mass centers of P1 and P2 relative to the mass center of the whole chain, respectively, and δ = 0.01b. This definition is equivalent to the following expression applicable for the chain in a tight nanopore:

 −>δ 1ifxxP1 P2 =− − <−δ Mxx 1if P1 P2 (8)  −<δ 0ifxxP1 P2

The M parameter took a value equal to +1 when the center of mass of P1 part was to the right of the mass center of P2 (its x coordinate was higher than that of P2), −1 in the opposite situation, and 0 when the difference in positions was small. In the bulk phase, the parameter behaved similarly, but the criterion of its value was based on the distance of the mass centers of parts P1 and P2 from the mass center of the whole chain. The parameter was introduced to check if the direction of chain motion was correlated with the mutual positions of parts P1 and P2. Forces exerted by parts of the chain, resulting from changes in their free energy, were calculated as partial derivatives:

∂A F = ∂ (9) x T,V where T and V denote the temperature and the volume of the system, respectively, whereas their averaged impulses in the time period Δt = tE were computed as the force integrals over the time of movement: acting on P1, P2, and at the whole chain.

tt+Δ tt +Δ 11 ∂A  I== Fdt dt ∂ (10) ttxT,V tt

The averaged impulses were computed for forces acting on part P1, part P2, and on the whole chain.

3. Results

3.1. The Chain in a Tight Nanopore Visualization of most often observed configurations of the studied chain in the course of evolution in the nanopore is schematically shown in Figure 1. In the course of the simulation, both parts of the chain (P1 and P2) occupied different positions along the nanopore, except during short periods when they overlapped each other during swapping of their mutual locations. The probability of both orientations of the chain parts was the same. The conformation of P1 was usually more compactPolymers 2020 than, 12 ,that 2931 of P2 as a result of the assumed stronger bead-to-bead interactions. 6 of 19

Figure 1. The visualization of the conformation of the macromolecule in the nanochannel. FigureThe self-attracting 1. The visualization part of macromoleculeof the conformation P1 is of marked the macromolecule in blue, whereas in the the nanochannel. athermal part The P2 self- in Polymersattractingred (2020N =, 200,12 part, x εFOR of /macromoleculek PEERT = REVIEW1 and 0 forP1 is P1 marked and P2, in respectively, blue, whereasD = the4b). athe Arrowsrmal indicatepart P2 in the red direction (N = 200, of 6 of 19 PP B − εmovementPP/kBT = −1 ofand the 0 wholefor P1 chain.and P2, respectively, D = 4b). Arrows indicate the direction of movement of 3.2. theLong-Time whole chain. Trajectories 3.2. Long-Time Trajectories As result from the analysis of the chain trajectory recorded for a relatively long time (107tE), the 7 studiedAs result chain fromshowed the a analysis chaotic diffusive of the chain motion trajectory that generally recorded was for not a relativelyspatially oriented. long time However, (10 tE), the studied chain showed a chaotic diﬀusive motion that generally was not spatially oriented. However, for shorter time periods (105tE, Figure 2), the situation changed and the observed movements proved 5 forto shorterbe directed time toward periods one (10 ortE the, Figure other2), end the of situation the nanopore changed over and relatively the observed long times. movements Figure proved 2 shows tothe be position directed of toward the mass one orcenter the other of the end whole of the chain nanopore along overthe nano relativelypore longand the times. mutual Figure orientation2 shows theparameter, position M, of theas a mass function center of oft. As the seen, whole the chain direction along of the the nanopore temporary and unidirectional the mutual orientationmotion was parameter,correlated M,withas the a function mutual ofpositionst. As seen, of P1 the and direction P2 of the temporary unidirectional motion was correlated with the mutual positions of P1 and P2

Figure 2. The dependence of the position of the center of mass of copolymer along the nanopore on the numberFigure of2. The MC dependence steps (N = 100, of theε /positionk T = 1of and the 0center for P1 of and mass P2, of respectively, copolymerD along= 3b ,the the nanopore coordinate on PP B − xthe= 0 numberwas assumed of MCas steps a starting (N = 100, point εPP/ ofkBT the = − fragment1 and 0 for of P1 trajectory and P2, shownrespectively, in the D Figure). = 3b, the The coordinate mutual orientation,x = 0 was assumedM, along as the a xstarting-axis is point marked of inthe green. fragment The of chart trajectory is based shown on the in data the recordedFigure). The in periods mutual 3 oforientation,tE = 10 elementary M, along the steps. x-axis is marked in green. The chart is based on the data recorded in periods of tE = 103 elementary steps. The swapping of P1 and P2 positions (illustrated by the change in the sign of parameter M) forcedThe the swapping change in of the P1 direction and P2 ofposi motiontions (illustrated in such a way by the that change the leading in the part sign of of the parameter chain was M) P1,forced whereas the change athermal in the P2 wasdirection the following of motion one. in such The resultsa way that presented the leading in Figure part2 of well the illustrate chain was the P1, unusualwhereas behavior athermal of P2 the was chain the in follow the nanopore.ing one. However,The results the presented results do in not Figure explain 2 allwell detailed illustrate chain the motionsunusual and behavior are rather of the smoothed, chain in sincethe nanopore. they are collected However, for the long results periods. do not In consequence,explain all detailed one cannot chain expectmotions a direct and correlationare rather smoothed, of the chain since motion they with are its co thermodynamicllected for long properties.periods. In Indeed, consequence, the values one ofcannot conformational expect a direct entropy, correlation as well asof internalthe chain energy motion change with its vs. thermodynami the diﬀusion time,c properties. shown in Indeed, Figure the2, seemvalues to beof conformational completely chaotic entropy, (see Figure as well3). as internal energy change vs. the diffusion time, shown in Figure 2, seem to be completely chaotic (see Figure 3).

Polymers 2020, 12, x FOR PEER REVIEW 6 of 19

3.2. Long-Time Trajectories

As result from the analysis of the chain trajectory recorded for a relatively long time (107tE), the studied chain showed a chaotic diffusive motion that generally was not spatially oriented. However, for shorter time periods (105tE, Figure 2), the situation changed and the observed movements proved to be directed toward one or the other end of the nanopore over relatively long times. Figure 2 shows the position of the mass center of the whole chain along the nanopore and the mutual orientation parameter, M, as a function of t. As seen, the direction of the temporary unidirectional motion was correlated with the mutual positions of P1 and P2

Figure 2. The dependence of the position of the center of mass of copolymer along the nanopore on

the number of MC steps (N = 100, εPP/kBT = −1 and 0 for P1 and P2, respectively, D = 3b, the coordinate x = 0 was assumed as a starting point of the fragment of trajectory shown in the Figure). The mutual orientation, M, along the x-axis is marked in green. The chart is based on the data recorded in periods of tE = 103 elementary steps.

The swapping of P1 and P2 positions (illustrated by the change in the sign of parameter M) forced the change in the direction of motion in such a way that the leading part of the chain was P1, whereas athermal P2 was the following one. The results presented in Figure 2 well illustrate the unusual behavior of the chain in the nanopore. However, the results do not explain all detailed chain motions and are rather smoothed, since they are collected for long periods. In consequence, one cannot expect a direct correlation of the chain motion with its thermodynamic properties. Indeed, the values of conformational entropy, as well as internal energy change vs. the diffusion time, shown in PolymersFigure 20202, seem, 12, 2931 to be completely chaotic (see Figure 3). 7 of 19

Polymers 2020, 12, x FOR PEER REVIEW 7 of 19

Figure 3. The dependence of the entropy, internal energy, and Helmholtz free energy of the chain on Figurethe number 3. The dependenceof MC steps of(N the = 100, entropy, εPP/k internalBT = −1 and energy, 0 for and P1 Helmholtzand P2, respectively, free energy D of = the 3b). chain The ontime changes in the mutual orientation parameter, M, along the x direction are marked in green. The chart the number of MC steps (N = 100, εPP/kBT = 1 and 0 for P1 and P2, respectively, D = 3b). The time − 3 changesis based in on the the mutual data orientationrecorded in parameter, periods of M,tE =along 10 elementary the x direction steps. are marked in green. The chart is 3 based on the data recorded in periods of tE = 10 elementary steps. The dependence of the force, F, calculated from Equation (9), acting on the whole chain on the motionThe dependencetime calculated of thefor force,the sameF, calculated data, is also from chaotic Equation (see (9),Figure acting 4), and on the the whole correlation chain between on the motionthe force time and calculated the mutual for theorientation same data, parameter is also chaotic is elusive. (see Figure 4), and the correlation between the force and the mutual orientation parameter is elusive.

Figure 4. The dependence of the net force acting of the chain on the number of MC steps (fragment ofFigure the trajectory 4. The dependence from Figure of2 ,theN net= 100,forceε acting/k T of= the1 chain and 0 on for the P1 number and P2, of respectively, MC steps (fragmentD = 3b). of PP B − Thethe mutual trajectory orientation from Figure parameter, 2, N = 100,M, alongεPP/kBT the = −x1 directionand 0 for P1 is markedand P2, inrespectively, green. The D chart = 3b). is The based mutual on 3 theorientation data recorded parameter, in periods M, ofalongtE = the10 xelementary direction is steps. marked in green. The chart is based on the data recorded in periods of tE = 103 elementary steps. 3.3. Bead Autocorrelation Function 3.3.The Bead results Autocorrelation gathered Function above point to a distinct deviation of the copolymer motion from the expectationsThe results based gathered on the simple above di pointffusion to behavior.a distinct However,deviation theyof the do copolymer not provide motion a quantitative from the characterizationexpectations based of motion. on the Therefore,simple diffusion for a deeper behavi analysisor. However, of the they asymmetrical do not provide chain a di quantitativeffusion in thecharacterization nanopore, the of autocorrelation motion. Therefore, function for a for deeper individual analysis beads of the and asymmetrical the whole chainchain wasdiffusion recorded. in the Thenanopore, autocorrelation the autocorrelation functions for func beadstion located for individual in P1 and P2beads parts and of thethechain, wholeg Bchain, are shownwas recorded. in Figure The5. The functions were recorded for a large range of steps starting at ∆t = t . The bead autocorrelation autocorrelation functions for beads located in P1 and P2 parts of the chain,E gB, are shown in Figure 5. functions obtained are in line with expectations, when compared to the autocorrelation functions The functions were recorded for a large range of steps starting at Δt = tE. The bead autocorrelation predictedfunctions by obtained the Rouse are and in line Zimm with models expectations, [5–7]. The when plot hascompared three distinct to the autocorrelation regions: for a short functions and 1 longpredicted time, the by motion the Rouse dynamics and Zimm is diff usionalmodels (the[5–7].gB Thefunction plot scaleshas three with distinct∆t ). In regions: the intermediate for a short time, and g Blongincreases time, tothe the motion power dynamics of 2/3. This is exponentdiffusional corresponding (the gB function to the scales subdi withffusion Δt1 region). In the agrees intermediate exactly withtime, the gB value increases obtained to the from power the of Zimm 2/3. modelThis exponent for both corresponding theta and athermal to the solvents subdiffusion [6,7] and region is similar agrees exactly with the value obtained from the Zimm model for both theta and athermal solvents [6,7] and is similar to the exponents predicted for the theta solvent in the Rouse model (v = 1/2) or the one- dimensional reptation dynamics (v = 1/4 and 1/2) [5,6]. Polymers 2020, 12, 2931 8 of 19 to the exponents predicted for the theta solvent in the Rouse model (v = 1/2) or the one-dimensional Polymers 2020, 12, x FOR PEER REVIEW 8 of 19 reptation dynamics (v = 1/4 and 1/2) [5,6].

Figure 5. The log–log plot of the bead autocorrelation function, gB, and the chain autocorrelation Figure 5. The log–log plot of the bead autocorrelation function, gB, and the chain autocorrelation function, g, in the nanochannel (D = 3b, N = 100, εPP/kBT = 0, and 1 for P1 and P2, respectively). function, g, in the nanochannel (D = 3b, N = 100, εPP/kBT = 0, and −1 for− P1 and P2, respectively). The The slopes of diﬀusion, subdiﬀusion, and quasi-ballistic modes (equal 1, 2/3, and 5/3) are marked slopes of diffusion, subdiffusion, and quasi-ballistic modes (equal 1, 2/3, and 5/3) are marked in gray. in gray.

The part of the whole chainchain autocorrelation function, g, forfor large Δ∆t overlaps the bead chain autocorrelation function gB. In the region of short times, the whole chain dynamics (represented by autocorrelation function gB. In the region of short times, the whole chain dynamics (represented by g) gshows) shows also also asymptotically asymptotically diff usionaldiffusional behavior, behavior, but,as but, expected, as expected, with a smallerwith a valuesmaller of value the di ffofusion the diffusioncoefficient. coefficient. In the range In the of rang intermediatee of intermediate times, the times, whole the chainwhole undergoes chain undergoes superdi superdiffusionffusion with a withg vs. ∆a tgrelationship vs. Δt relationship scaling withscaling the with exponent the exponent 5/3. Such 5/3. an unexpectedSuch an unexpected result seems result to beseems caused to bybe causedthe asymmetry by the asymmetry of the chain of and the its chain long-time and its quasi-ballistic long-time quasi-ballistic motion in tight motion confinement, in tight confinement, impeding the impedinginversion ofthe chain inversion parts. Theof observedchain parts. effect The is similarobserv toed the effect “ballistic is similar motion” to observedthe “ballistic in a systemmotion” of observedthree-arm in star a system chains of in three-arm the narrow star nanochannel chains in the of D narrow= 3b [40 nanochannel]. However, of the D exponent= 3b [40]. observedHowever, here the exponentwas smaller observed than v =here2 corresponding was smaller than to the v = Newtonian 2 corresponding uniform to motion.the Newtonian uniform motion.

3.4. The The Mass Mass Center Center Autocorrelation Autocorrelation Function The results collectedcollected inin FiguresFigures6 –6–88 show show the the whole whole chain chain autocorrelation autocorrelation function, function,g, obtained g, obtained for fordiff differenterent chain chain lengths lengths and and nanopore nanopore widths. widths. The Th calculatede calculated autocorrelation autocorrelation functions functions based based on theon theshift shift of the of chainthe chain mass mass center center in three in dimensionsthree dimensions and along and the alongx coordinate the x coordinate parallelto parallel the nanopore to the nanoporeaxis, are identical axis, are with identical the accuracy with ofthe the accuracy standard of deviation the standard of presented deviation results. of Therefore,presented onlyresults. the Therefore,results obtained only the for results 3D shifts obtained are presented for 3D shifts below. are presented below. As shown in Figure6, all chain motions in short and long times are di ffusive with the g vs. ∆t relationship described by scaling exponent 1. In the intermediate time, the slope of the log–log plot decreases with increasing width of the nanopore and goes down starting from 5/3 for D/b = 3 to almost 1 for the chain located in 3D space without any geometrical constraints (denoted as D = inf). The slope of the g vs. ∆t dependence equal to 5/3 for D = 3b practically does not depend on the chain length (evident in the range N = 100–200). However, the range of the intermediate superdiffusive part of the curve shifts with the chain length increasing towards higher ∆t values. To check whether the observed phenomenon is not an artificial effect of the assumed method or model, some additional simulations for the symmetric chain wholly immersed in the athermal solvent (εPP/kBT = 0 for P1 and P2) were performed. The results collected in Figure8 show practically no superdiffusive bias in the g vs. ∆t dependence.

Figure 6. The log–log plot of the autocorrelation function, g, in nanochannels of different widths (N =

100, εPP/kBT = 0 and −1 for P1 and P2, respectively). The slopes of diﬀusion and quasi-ballistic modes (equal 1 and 5/3) are marked in gray. Polymers 2020, 12, x FOR PEER REVIEW 8 of 19

Figure 5. The log–log plot of the bead autocorrelation function, gB, and the chain autocorrelation

function, g, in the nanochannel (D = 3b, N = 100, εPP/kBT = 0, and −1 for P1 and P2, respectively). The slopes of diffusion, subdiffusion, and quasi-ballistic modes (equal 1, 2/3, and 5/3) are marked in gray.

The part of the whole chain autocorrelation function, g, for large Δt overlaps the bead chain autocorrelation function gB. In the region of short times, the whole chain dynamics (represented by g) shows also asymptotically diffusional behavior, but, as expected, with a smaller value of the diffusion coefficient. In the range of intermediate times, the whole chain undergoes superdiffusion with a g vs. Δt relationship scaling with the exponent 5/3. Such an unexpected result seems to be caused by the asymmetry of the chain and its long-time quasi-ballistic motion in tight confinement, impeding the inversion of chain parts. The observed effect is similar to the “ballistic motion” observed in a system of three-arm star chains in the narrow nanochannel of D = 3b [40]. However, the exponent observed here was smaller than v = 2 corresponding to the Newtonian uniform motion.

3.4. The Mass Center Autocorrelation Function The results collected in Figures 6–8 show the whole chain autocorrelation function, g, obtained for different chain lengths and nanopore widths. The calculated autocorrelation functions based on the shift of the chain mass center in three dimensions and along the x coordinate parallel to the nanopore axis, are identical with the accuracy of the standard deviation of presented results. Polymers 2020, 12, 2931 9 of 19 Therefore, only the results obtained for 3D shifts are presented below.

Polymers 2020, 12, x FOR PEER REVIEW 9 of 19 Figure 6. The log–log plot of the autocorrelation function, g, in nanochannels of different widths Figure 6. The log–log plot of the autocorrelation function, g, in nanochannels of different widths (N = (N = 100, ε /k T = 0 and 1 for P1 and P2, respectively). The slopes of diffusion and quasi-ballistic Polymers 2020, 12, PPx FORB PEER REVIEW− 9 of 19 100,modes εPP (equal/kBT = 0 1 and and − 51/3) for are P1 marked and P2, in respectively). gray. The slopes of diffusion and quasi-ballistic modes (equal 1 and 5/3) are marked in gray.

Figure 7. The log–log plot of the autocorrelation function, g, in the nanochannel for different chain

lengths (D = 3b, εPP/kBT = 0 and −1 for P1 and P2, respectively). The slopes of diff usion and quasi- Figure 7. The log–log plot of the autocorrelation function, g, in the nanochannel for different chain ballisticFigure 7.modes The log–log(equal 1 plot and of5/3) the are autocorrelation marked in gray. function, g, in the nanochannel for different chain lengths (D = 3b, ε /k T = 0 and 1 for P1 and P2, respectively). The slopes of diffusion and PP B − quasi-ballisticlengths (D = 3 modesb, εPP/kB (equalT = 0 1and and −1 5 /for3) are P1 markedand P2, inrespectively). gray. The slopes of diffusion and quasi- ballistic modes (equal 1 and 5/3) are marked in gray.

Figure 8. The log–log plot of the autocorrelation function, g, in the nanochannel and 3D space (N = 100, Figure 8. The log–log plot of the autocorrelation function, g, in the nanochannel and 3D space (N = εPP/kBT = 0 for P1 and P2—athermal solution). The slopes of diffusion and quasi-ballistic modes (equal 100,1 and εPP 5//k3)BT are = 0 marked for P1 and in gray. P2—athermal solution). The slopes of diffusion and quasi-ballistic modes (equalFigure 1 8.and The 5/3) log–log are marked plot of in the gray. autocorrelation function, g, in the nanochannel and 3D space (N = Results presented in Figure7 indicate that there are three scaling time regions of the mean 100, εPP/kBT = 0 for P1 and P2—athermal solution). The slopes of diffusion and quasi-ballistic modes autocorrelationAs(equal shown 1 and in function.5/3) Figure are marked 6, The all firstchain in gray. and motions the third in short regions and are long similar times to are the diffusive monomer with autocorrelation the g vs. Δt relationshipfunction: in thedescribed initial regionby scaling where exponentν = 1 and 1. theIn th dieff usionintermediate coefficient time, is smallerthe slope than of forthe thelog–log monomer, plot decreasesAs shown with increasing in Figure width6, all chain of the motions nanopore in and short goes and down long starting times are from diffusive 5/3 for Dwith/b = the3 to galmost vs. Δt 1relationship for the chain described located in by 3D scaling space exponentwithout any 1. In geometrical the intermediate constraints time, (denoted the slope as of D the = inf). log–log plot decreasesThe slope with of increasing the g vs. widthΔt dependence of the nano equalpore toand 5/3 goes for downD = 3b starting practically from does 5/3 fornot D depend/b = 3 to onalmost the chain1 for thelength chain (evident located inin 3D the space range without N = any 100–200). geometrical However, constraints the (denotedrange of asthe D =intermediate inf). superdiffusiveThe slope partof the of gthe vs. curve Δt dependence shifts with equalthe chain to 5/3 length for D increasing = 3b practically towards does higher not depend Δt values. on the chainTo lengthcheck whether(evident the in observed the range phenomenon N = 100–200). is not anHowever, artificial effectthe range of the ofassumed the intermediate method or model,superdiffusive some additional part of the simulations curve shifts forwith the the symmetric chain length chain increasing wholly towards immersed higher in theΔt values.athermal solventTo ( checkεPP/kBT whether = 0 for P1 the and observed P2) were phenomenon performed. isThe not resu an artificiallts collected effect in ofFigure the assumed 8 show practically method or nomodel, superdiffusive some additional bias in thesimulations g vs. Δt dependence. for the symmetric chain wholly immersed in the athermal solventResults (εPP/ kpresentedBT = 0 for P1in andFigure P2) 7were indicate performed. that th Theere areresu threelts collected scaling in time Figure regions 8 show of practicallythe mean autocorrelationno superdiffusive function. bias in Thethe gfirst vs. andΔt dependence. the third re gions are similar to the monomer autocorrelation function:Results in the presented initial region in Figure where 7 ν indicate= 1 and the that diffusion there are coefficient three scaling is smaller time than regions for the of monomer, the mean andautocorrelation in the final region function. where The ν first is still and equal the thirdto 1, butregions the diffusio are similarn coefficient to the monomer takes the autocorrelation same value as forfunction: the monomers, in the initial because region all where monomers ν = 1 and move the diffusion with the coefficient same velocity is smaller as the than whole for thechain. monomer, In the intermediateand in the final region, region as wherea result ν isof stillthe equalcollective to 1, motion but the causeddiffusio byn coefficientthe asymmetry takes ofthe the same monomers value as infor both the partsmonomers, of the chain, because which all monomersforces the tempor move arywith unidirectional the same velocity motion as of the the whole chain masschain. center, In the intermediate region, as a result of the collective motion caused by the asymmetry of the monomers in both parts of the chain, which forces the temporary unidirectional motion of the chain mass center, Polymers 2020 12 Polymers 2020, 12, , x, FOR 2931 PEER REVIEW 10 10of of19 19 the g = f(Δt) dependence scales with higher exponent than ν = 1. The g = f(Δt) relationship for the Polymersand in 2020 the, 12 final, x FOR region PEER REVIEW where ν is still equal to 1, but the diffusion coefficient takes the same10 of value 19 intermediate region significantly depends on the nanopore width—the narrower the nanopore the as for the monomers, because all monomers move with the same velocity as the whole chain. In the thehigher g = thef(Δ tscaling) dependence exponent scales tending with tohigher ν = 5/3. exponent The anomalous than ν = diffusion1. The g =practically f(Δt) relationship vanishes for in the intermediate region, as a result of the collective motion caused by the asymmetry of the monomers in intermediatebulk phase. The region increase significantly in the monomer depends numberon the nanopore in the chain width—the influences narrower the analyzed the nanopore dependence the both parts of the chain, which forces the temporary unidirectional motion of the chain mass center, higheronly through the scaling the increase exponent in tendingthe initial to ordinate ν = 5/3. Thebut atanomalous the same totaldiffusion diffusion practically coefficient. vanishes in the the g = f(∆t) dependence scales with higher exponent than ν = 1. The g = f(∆t) relationship for the bulk phase. The increase in the monomer number in the chain influences the analyzed dependence intermediate region significantly depends on the nanopore width—the narrower the nanopore the only3.5. Time through Periods the between increase Swaps in the initial ordinate but at the same total diffusion coefficient. higher the scaling exponent tending to ν = 5/3. The anomalous diffusion practically vanishes in the bulkProbability phase. The density increase functions in the monomerin the time number periods in between the chain the influences swapping the of P1 analyzed and P2 dependencepositions, 3.5. Time Periods between Swaps resultingonly through in the theinversion increase of in the the direction initial ordinate of chain but movement, at the same are total presented diffusion in coe Figuresfficient. 9–11 in the logarithmicProbability scale. density As seen, functions the functions in the time are veryperiods wide between and behave the swapping as the three-modal of P1 and P2 log-normal positions, resultingdistributions.3.5. Time in Periods the Three inversion between maxima of Swaps arethe evdirectionident in ofthe chain case movement,of diffusion arein the presented narrow nanochannel.in Figures 9–11 The in first the logarithmicmaximumProbability can scale. be interpreted densityAs seen, functions the as functionsa result in the of are the time very small periods wide fluctuations betweenand behave of the P1 swappingas and the P2 three-modal centers’ of P1 and positions log-normal P2 positions, over distributions.theresulting center of in theThree the whole inversion maxima chain ofare thewhen ev directionident both in partsthe of case chain prac oftically movement,diffusion overlap in arethe each presentednarrow other nanochannel. in Figuresthe course9 –The11 of in first the the maximumswappinglogarithmic eventcan scale.be at interpreted M As = 0 seen, (see as theFigure a functionsresult 2). ofThe the are other small very two fluctuations wide maxima and behave seem of P1 toand as be the P2 produc three-modalcenters’ed bypositions the log-normal mutual over themigrationdistributions. center ofof athe number Three whole maxima of chain segments arewhen evident causingboth inparts the prac casemovementstically of diff usionoverlap of parts in theeach of narrow the other chain nanochannel.in theand courseof the Thewholeof the first swappingchain.maximum event can beat interpretedM = 0 (see Figure as a result 2). The of the other small two fluctuations maxima seem of P1 andto be P2 produc centers’ed positions by the mutual over the migrationcenter of of the a wholenumber chain of segments when both causing parts practically the movements overlap of each parts other of the in the chain course and of of the the swapping whole chain.event at M = 0 (see Figure2). The other two maxima seem to be produced by the mutual migration of a number of segments causing the movements of parts of the chain and of the whole chain.

Figure 9. The probability density functions of periods, Δt, between swapping of P1 and P2 mass

centers in the logarithmic scale for different chain lengths (tE = 5 × 107, D = 3b, εPP/kBT = 0 and −1 for P1 Figure 9. The probability density functions of periods, ∆t, between swapping of P1 and P2 mass centers Figureand P2, 9.respectively). The probability density functions of periods, Δt, between7 swapping of P1 and P2 mass in the logarithmic scale for diﬀerent chain lengths (tE = 5 10 , D = 3b, εPP/kBT = 0 and 1 for P1 and × 7 − centersP2, respectively). in the logarithmic scale for different chain lengths (tE = 5 × 10 , D = 3b, εPP/kBT = 0 and −1 for P1 and P2, respectively).

Figure 10. The probability density functions of periods, ∆t, between swapping of P1 and P2 mass Figurecenters 10. in The the probability logarithmic density scale for functions diﬀerent of nanopore periods, width Δt, between (N = 100, swappingε /k T =of0 P1 and and1 P2 for mass P1 and PP B − centersP2, respectively). in the logarithmic scale for different nanopore width (N = 100, εPP/kBT = 0 and −1 for P1 and FigureP2, respectively). 10. The probability density functions of periods, Δt, between swapping of P1 and P2 mass

centers in the logarithmic scale for different nanopore width (N = 100, εPP/kBT = 0 and −1 for P1 and P2, respectively). Polymers 2020, 12, x FOR PEER REVIEW 11 of 19 Polymers 2020, 12, 2931 11 of 19

Figure 11. The probability density functions of time periods, ∆t, between swapping of P1 and P2 mass Figurecenters 11. inThe the probability nanochannel density and functions in the 3D of space time in periods, the logarithmic Δt, between scale swapping at different of P1 nanopore and P2 mass width centers in the nanochannel and in the 3D space in the logarithmic scale at different nanopore width (N = 100, εPP/kBT = 0 for P1 and P2). (N = 100, εPP/kBT = 0 for P1 and P2). The second maximum is practically independent of the chain length (Figure9) but slightly depends onThe the nanoporesecond maximum width (Figure is practically 10). Since independent the longitudinal of the chain chain extension length is (Figure larger than9) butD andslightlyD > b, dependsthe chain on can the benanopore represented width by (Figure the de Gennes10). Since blob the model, longitudinal e.g., a string chain of extension self–avoiding is larger compression than D andblobs D >of b, diameter,the chain Dcan[25 be,50 represented–53]. However, by the in contrastde Gennes to Referenceblob model, [25 e.g.,], reporting a string theof self–avoiding relaxation time compressiondecreasing withblobsD ofto diameter, the power D [25,50–53].1.3, the time However, between swapsin contrast decreases to Reference with the [25], nanopore reporting width the to a − relaxationpower of time about decreasing0.2 0.2. with D to the power −1.3, the time between swaps decreases with the − ± nanoporeThe width third maximumto a power depends of about on −0.2 the ± nanopore0.2. width in the way similar to that of the second one, whileThe its third position maximum and heightdepends strongly on the varynanopore with thewidth chain in the length. way similar The smaller to that the of chainthe second length, one,the while higher its the position probability and height of the swapstrongly of P1 vary and with P2 positions, the chain whichlength. results The smaller in a smaller the chain time length, between theswappings higher the and probability the higher of numberthe swap of of swap P1 and events. P2 positions, The increase which in results∆t caused in a bysmaller increasing time betweenN remains swappingsin a quantitative and the agreementhigher number with of the swap Zimm events. model The [6]. increase in Δt caused by increasing N remains in a quantitativeIn summary, agreement the three with successive the Zimm modes model of distribution [6]. are associated with the swapping of larger andIn larger summary, elements the ofthree the chainsuccessive starting modes with of the distri swappingbution caused are associated by the move with of the a single swapping monomer of larger(the relaxationand larger time elements is independent of the chain of bothstartingN and withD ),the by swapping the fragment caused of the by chain the move of the of size a single of blob monomer(the relaxation (the relaxation time depends time is on independentD) and both of relativelyboth N and large D), partsby the P1 fragment and P2 of (the the relaxation chain of the time sizedepends of blob on (theN). relaxation time depends on D) and both relatively large parts P1 and P2 (the relaxationThe time distributions depends collected on N). in Figure 11 show that swapping of identical parts of the chain described byThe the thirddistributions maximum collected is a slightly in Figure rarer 11 event, show as that compared swapping to the of swappingidentical parts of parts of the of di chainfferent describedinteraction by energiesthe third (Figure maximum 10), butis a theslightly maximum rarer event, still remains. as compared However, to the since swapping the symmetry of parts of of the differentchain does interaction not force energies the directed (Figure motion, 10), the but autocorrelation the maximum function still remains. does not However, detect any since anomalous the symmetrydiffusion of behavior the chain of does the chainnot force with the identical directed P1 motion, and P2 the parts autocorrelation (see Figure8). function does not detect any anomalous diffusion behavior of the chain with identical P1 and P2 parts (see Figure 8). 3.6. A Single Swapping Episode 3.6. A Single Swapping Episode Since the prolonged movement of the chain in one direction seems to be related to the difficulty encounteredSince the prolonged by the swapping movement process, of the the chain trajectory, in one conformation, direction seems and to thermodynamics be related to the ofdifficulty the chain encounteredwere examined by the in swapping the immediate process, surroundings the trajecto ofry, the conformation, swapping point. and thermodynamics of the chain were examinedThe details in the of theimmediate trajectory surroundings of P1 and P2of the in theswapping course point. of a single swap episode are shown in FigureThe details 12. The of datathe trajectory collected of here P1 were and recordedP2 in the withcourse the of time a single step 10swap5 (1000 episodeN) times are smaller shown thanin Figurethat of12. the The data data shown collected in Figure here2 were. In the recorded course ofwith di fftheusive time motion, step 10 after5 (1000 quickN) times approach smaller of centersthan thatof of P2 the to P1data (P2 shown is more in mobile),Figure 2. bothIn the parts course overlap of diffusive each other motion, and after irregularly quick approach fluctuate of around centers the ofcommon P2 to P1 center(P2 is more of mass. mobile), Finally, both P1 andparts P2 overlap segregate each and other relax. and The irregularly direction fluctuate of motion around of the massthe commoncenter of center the whole of mass. chain Finally, occurring P1 and in the P2 same segregat timee is and determined relax. The by directio the mutualn of orientationmotion of the parameter mass centerM: Initially, of the thewhole chain chain moves occurring to the right,in the then same fluctuates time is arounddetermined a certain by thex coordinate, mutual orientation and finally parameterreturns back, M: Initially, since the the swapping chain moves positions to the causes right, the then inversion fluctuates of direction around ofa certain motion. x coordinate, and finally returns back, since the swapping positions causes the inversion of direction of motion. Polymers 2020, 12, x FOR PEER REVIEW 12 of 19

Polymers 2020, 12, x FOR PEER REVIEW 12 of 19 Polymers 2020, 12, 2931 12 of 19

Figure 12. Short parts of trajectories of s P1 and P2 and their radii of gyration (N = 100, εPP/kBT = −1 and 0 for P1 and P2, respectively, D = 3b). The chart is based on the data recorded at each elementary

step,Figure tE. 12. Short parts of trajectories of s P1 and P2 and their radii of gyration (N = 100, εPP/kBT = 1 and Figure 12. Short parts of trajectories of s P1 and P2 and their radii of gyration (N = 100, εPP/kBT− = −1 0 for P1 and P2, respectively, D = 3b). The chart is based on the data recorded at each elementary step, tE. Theand 0swapping for P1 and ofP2, P1 respectively, and P2 positions D = 3b). The is accompchart is basedanied on by the the data change recorded in conformationat each elementary of both partsstep, ofThe the t swappingE. chain. As of seen, P1 and the P2size positions variations is accompanied concern mainly by theP2. changeThe approach in conformation of the centers of both of both parts partsof the to chain. each Asother seen, results the size in variationsa decrease concern in Rg of mainly P2, whereas P2. The approachthe overcoming of the centers of P2 ofthrough both parts P1 The swapping of P1 and P2 positions is accompanied by the change in conformation of both increasesto each other its Rg results. At the in end a decreaseof the translocation, in Rg of P2, all whereas beads of the P2 overcoming gather in a ofcompact P2 through coil at P1 one increases end of parts of the chain. As seen, the size variations concern mainly P2. The approach of the centers of both P1its (bothRg. At parts the endcollapse). of the translocation,Finally, P2 expands all beads tend ofing P2 gatherto the free in a compactconformation coil at unperturbed one end of P1 by (both the otherpartsparts partto collapse). each of the other Finally,chain. results P2 expandsin a decrease tending in to R theg of free P2, conformation whereas the unperturbedovercoming byof theP2 otherthrough part P1 of increasesthe chain.Figure its 13 R gdescribes. At the end the of chain the translocation, conformation allat abeads swapping of P2 gatherevent in in terms a compact of two coil parameters: at one end the of distanceP1 (bothFigure betweenparts 13 collapse).describes mass centers Finally, the chain of P1P2 conformationandexpands P2 and tend theing at sum a to swapping of the their free radii conformation event of gyration. in terms unperturbed As of twoshown, parameters: initially by the other part of the chain. thethe conformation distance between is relatively mass centers expanded—the of P1 and distance P2 and between the sum mass of their centers radii of of P1 gyration. and P2 | AsxP1- shown,xP2| is largerinitiallyFigure than the 13the conformation describes sum of their the ischain radii. relatively conformation Just before expanded—the swapping, at a swapping distance the conformation event between in terms mass becomes of centerstwo parameters: more of P1 compact and the P2 distance between mass centers of P1 and P2 and the sum of their radii of gyration. As shown, initially (decrease|xP1-xP2| isin larger RgP1 + than RgP2 the). Then, sum of two their parts radii. of Just the beforechain swapping,move in anti-parallel the conformation directions—during becomes more the conformation is relatively expanded—the distance between mass centers of P1 and P2 |xP1-xP2| is swapping,compact (decrease the sum in ofR gP1their+ RradiigP2). of Then, gyration two parts slightly of the increases chain move as a in result anti-parallel of P1 and directions—during P2 elongation larger than the sum of their radii. Just before swapping, the conformation becomes more compact alongswapping, the nanochannel. the sum of their When radii the of swapping gyration ends, slightly the increases sum of R asg values a result evidently of P1 and de P2creases elongation as a result along (decrease in RgP1 + RgP2). Then, two parts of the chain move in anti-parallel directions—during ofthe formation nanochannel. of compact When dense the swapping coils located ends, close the to sum each of other.Rg values Finally, evidently both parts decreases relax, owing as a result to the of entropicswapping,formation elasticity. the of compact sum of their dense radii coils of located gyration close slightly to each increases other. Finally, as a result both of parts P1 and relax, P2 owing elongation to the alongentropic the elasticity.nanochannel. When the swapping ends, the sum of Rg values evidently decreases as a result of formation of compact dense coils located close to each other. Finally, both parts relax, owing to the entropic elasticity.

Figure 13. The distance between mass centers of P1 and P2 and the sum of their radii of gyration for Figurethe fragment 13. The ofdistance trajectory between from mass Figure centers 12( N of= P1100, andε P2/k andT = the1 andsum 0of for their P1 radii and P2,of gyration respectively, for PP B − the fragment of trajectory from Figure 12 (N = 100, εPP/kBT = −1 and 0 for P1 and P2, respectively, D = D = 3b). The chart is based on the data recorded at each elementary step. 3b). The chart is based on the data recorded at each elementary step. FigureThe evolution 13. The distance of thermodynamic between mass properties centers of P1 of bothand P2 chain and partsthe sum (see of Figuretheir radii 14) of is gyration disturbed for by the randomThethe fragment evolution noise similar of trajectoryof thermodynamic to the from large Figure scale pr 12 dependenceoperties (N = 100, of ε PPboth/ shownkBT =chain −1 inand parts Figure 0 for (s P13ee. and However,Figure P2, respectively, 14) one is disturbed can D notice = by a thesmall random3b positive). The noisechart ﬂuctuation is similar based onto of thethe the data large Helmholtz recorded scale dependen at energy each elementary accompanyingce shown step. in Figure the start 3. However, of the swapping one can process. notice The mechanism of the appearance of this small Helmholtz energy barrier is entropic in nature. The evolution of thermodynamic properties of both chain parts (see Figure 14) is disturbed by the random noise similar to the large scale dependence shown in Figure 3. However, one can notice Polymers 2020, 12, x FOR PEER REVIEW 13 of 19 a Polymerssmall positive 2020, 12, xfluctuation FOR PEER REVIEW of the Helmholtz energy accompanying the start of the swapping process.13 of 19 The mechanism of the appearance of this small Helmholtz energy barrier is entropic in nature. a small positive fluctuation of the Helmholtz energy accompanying the start of the swapping process. The mechanism of the appearance of this small Helmholtz energy barrier is entropic in nature. Polymers 2020, 12, 2931 13 of 19

Figure 14. The dependence of entropy, internal energy, and Helmholtz energy of the chain on the number of elementary steps, tE, for the fragment of trajectory from Figure 12 (N = 100, εPP/kBT = −1 and Figure 14. The dependence of entropy, internal energy, and Helmholtz energy of the chain on the 0 for P1 and P2, respectively, D = 3b). The chart is based on the data recorded at each elementary step. Figurenumber 14. of elementaryThe dependence steps, tof, forentropy, the fragment internal of energy, trajectory and from Helmholtz Figure 12 energy( N = 100, of εthe /kchainT = on1 andthe E PP B − number0 for P1 andof elementary P2, respectively, steps, tDE, =for3b the). The fragment chart is of based trajectory on the from data Figure recorded 12 (N at = each 100, elementary εPP/kBT = −1 step. and The0 for relationship P1 and P2, respectively, A = f(x) (not D presented= 3b). The chart here, is butbased shown on the for data a shorterrecorded part at each of trajectory elementary in step. Figure 17 in SectionThe relationship 3.7) allowsA the= f( xcalculation) (not presented of force here, caused but shown by the for free a shorterenergy partfluctuations. of trajectory The in resulting Figure 17 plotin SectionshownThe relationship in3.7 Figure) allows 15 A the is = stillf( calculationx) (nothighly presented irregular, of force here, causedbut butit shows byshown the a freestatisticallyfor a energy shorter significant fluctuations. part of trajectory trend: The Initially, in resulting Figure theplot17 forcein shownSection takes in 3.7) positive Figure allows 15 values; theis still calculation at highly the moment irregular, of force just ca but usedbefore it shows by the the afirst free statistically swapping energy fluctuations. significant (when coils trend: The P1 resultingand Initially, P2 startplotthe forcetoshown overlap), takes in Figure positive the force 15 values;is is still negligibly highly at the irregular, momentsmall, whereas justbut beforeit shows the thereleasing a statistically first swapping chain significantproduces (when coilsatrend: force P1 Initially,highly and P2 fluctuatingthestart force to overlap), takes but of positive significantly the force values; is negligiblynega at thetive moment values. small, Since whereasjust before the theforce the releasing isfirst calc swappingulated chain as produces (whenthe partial coils a forcederivative P1 and highly P2 ofstartfluctuating the Helmholtzto overlap), but ofenergy the significantly force (Equation is negligibly negative (9)), its values.small, positive whereas Since values the the mean force releasing isproviding calculated chain energy produces as the to partial the a forcecoil derivative and highly its compression,fluctuatingof the Helmholtz but whereas of energysignificantly the (Equationnegative nega forces (9)),tive its values.correspond positive Since values to the the force meanchain is providingstretching.calculated energyas the partial to the coilderivative and its ofcompression, the Helmholtz whereas energy the (Equation negative forces(9)), its correspond positive values to the mean chain providing stretching. energy to the coil and its compression, whereas the negative forces correspond to the chain stretching.

Figure 15. The net force caused by changes in the free energy exerted by both parts of the chain (N = 100, Figureε /k 15.T = The1 andnet force 0 for P1caused and P2,by changes respectively, in theD =free3b ).energy The chart exerted is based by both on theparts data of recordedthe chain at (N each = PP B − 100,elementary εPP/kBT = step.−1 and 0 for P1 and P2, respectively, D = 3b). The chart is based on the data recorded at eachFigure elementary 15. The netstep. force caused by changes in the free energy exerted by both parts of the chain (N =

100,The ε mechanicalPP/kBT = −1 and asymmetry 0 for P1 and of theP2, respectively, properties of D both= 3b). parts The chart of the is chainbased ison clearly the data illustrated recorded at by the impulsesTheeach mechanical elementary of forces [54 asymmetrystep.] exerted by of both the partsproperties of the of chain. both Theparts force of the impulse, chain asis clearly an integral illustrated of the force by theover impulses time, gives of forces the information [54] exerted about by both the parts force of cumulative the chain. contributionThe force impulse, to the variationas an integral of the of chain the forcemomentum, overThe mechanicaltime, exerted gives theasymmetry by information a certain of part the about ofpr theoperties the chain. force of cumulative Theboth results parts contributionof are the only chain qualitative, isto clearly the variation since illustrated the of forcethe by chaintheimpulse impulses momentum, is not of equivalent forces exerted [54] toby exerted the a certain momentum by both part parts of of the di ofﬀ usionch theain. chain. transfer The Theresults as force the are systemimpulse, only qualitative, under as an consideration integral since of the the is forceforcenot Newtonian.impulse over time, is givesnot equivalentthe information to the about momentum the force cumulativeof diffusion contribution transfer as to the the systemvariation under of the considerationchainFigure momentum, 16 is shows not Newtonian.exerted the impulses by a certain exerted part by bothof the parts chain. of the The chain results vs. timeare only of motion. qualitative, The forces since were the forcecalculated impulse on the is basisnot equivalent of ﬂuctuation to ofthe the momentum Helmholtz energyof diffusion of P1 andtransfer P2 separately as the system (Equation under (9)) considerationand then integrated is not Newtonian. (Equation (10)) starting at a certain time before swapping. As shown, except for the initial steps in which the mass centers of both parts of the chain approach each other, P2 pushes P1 all the time. Polymers 2020, 12, x FOR PEER REVIEW 14 of 19

Figure 16 shows the impulses exerted by both parts of the chain vs. time of motion. The forces were calculated on the basis of fluctuation of the Helmholtz energy of P1 and P2 separately (Equation (9)) and then integrated (Equation (10)) starting at a certain time before swapping. As shown, except for the initial steps in which the mass centers of both parts of the chain approach each other, P2 pushes P1 all the time. Polymers 2020, 12, 2931 14 of 19

Figure 16. Impulses of forces caused by changes in the free energy exerted by both part P1 and P2 of Figurethe chain 16. Impulses (N = 100, ofε forces/k T =caused1 and by 0 changes for P1 and in the P2, free respectively, energy exertedD = 3b ).by The both chart part is P1 based and P2 on of the PP B − thedata chain recorded (N = 100, at each εPP/k elementaryBT = −1 and step. 0 for P1 and P2, respectively, D = 3b). The chart is based on the data recorded at each elementary step. 3.7. Mechanism of Unidirectional Motion 3.7. Mechanism of Unidirectional Motion The results presented above allow the following description of the phenomena occurring in theThe studied results system. presented The chaoticabove allow and stochasticallythe following symmetricaldescription of di theffusion phenomena of beads occurring constituting in the the studiedasymmetrical system. chain The ischaotic a collective and st propertyochastically which symmetrical produces temporaldiffusion unidirectionalof beads constituting motion of the the asymmetricalasymmetrical chain chain is located a collective in a tight property confinement, which hinderingproduces thetemporal swapping unidirectional of chain parts. motion The of motion the asymmetricalviolates the second chain located law of thermodynamicsin a tight confinement, [45] but hindering only for the a relatively swapping short of chain period parts. of time.The motion violatesThe the phenomenon second law of of thermodynamics the longitudinal [45] motion but only of the for asymmetricala relatively short chain period represented of time. by two sub-chainsThe phenomenon of different of Flory the interactionlongitudinal coe motionfficients of shouldthe asymmetrical be related to chain higher represented conformational by two entropy sub- chainsand higher of different diffusional Flory mobility interactio ofn the coefficients weaker interacting should be sub-chainrelated to P2.higher The conformational P1 fragment of theentropy chain andis less higher mobile, diffusional as compared mobility to P2,of the since weaker instead interacting of quasi-free sub-chain bead motionsP2. The P1 in fragment P2 limited of only the chain by the isrequirement less mobile, ofas thecompared chain continuity, to P2, since its instead beads formof quasi-free aggregates bead which motions can in move P2 limited only in only a collective by the requirementway or their of motion the chain is associated continuity, with its breakingbeads form of attractiveaggregates interaction which can with move one only bead in anda collective creating waya new or their one with motion another is associated one. Changes with breaking in the radius of attractive of gyration intera ofction P2, as with well one as changesbead and in creating entropy, a arenew asymmetric one with another in time: one. The Changes decrease in in the both radius parameters of gyration is faster of P2, than as well their as increase changes because in entropy, some arelattice asymmetric nodes are in time: blocked The by decrease P1 beads in both which parame releaseters them is faster relatively than their slowly. increase As a because result, the some fast latticemovement nodes is are inhibited blocked more by stronglyP1 beads than which the slowrelease one them as in relatively the scallop slowly. effect [ 55As]. a There result, is nothe inertia fast movementeffect incorporated is inhibited in more the model, strongly but than there the are slow some on geometricale as in the scallop constraints effect instead.[55]. There In consequence,is no inertia effectthe P2 incorporated sub-chain exerts in the amodel, pushing but pressure there are on some allobjects geometrical around constraints it, among in othersstead. onIn consequence, the stopper in thethe P2 nanochannel sub-chain exerts formed a pushing by the morepressure compact on all andobjects stronger around interacting it, among sub-chain others on P1.the stopper As a result, in theit producesnanochannel the net formed force by whose the fluctuationsmore compact averaged and stronger over a relativelyinteracting long sub-chain period ofP1. time As area result, directed it producestowards the P1. net Swapping force whose is a rare fluctuations event and averaged requires ov overcominger a relatively of the long entropic period barrier. of time are directed towardsThe P1. trajectory Swapping mapping is a rare of event the free and energy requires changes overcoming along the of the diff usionentropic trajectory, barrier. for a very small partThe of trajectorytrajectory wheremapping only of unidirectional the free energy motion changes occurs along (M the= diffusion1), is shown trajectory, in Figure for 17 a .very The small figure partdemonstrates of trajectory changes where inonly entropy, unidirectional internal energy,motion andoccurs finally (M = free 1), is Helmholtz shown in energyFigure in17. the The course figure of demonstrateschain diffusion. changes As is in clearly entropy, visible, internal the changesenergy, an ind the finally Helmholtz free Helmholtz free energy energy are caused in the course mainly of by chainentropic diffusion. effects. As The is clearly Figure visi demonstratesble, the changes that thein the energy Helmholtz vs. position free energy of migrating are caused species mainly shows by entropicasymmetry effects. of the The Helmholtz Figure demonstrates free energy versusthat the the energy position vs. ofposition the chain. of migrating The energy species profile shows shown asymmetryhere reminds of the Helmholtz figure of energy free energy profile versus characteristic the position of behavior of the ofchain. the systems The energy in which profile the shown ratchet heremechanism reminds ofthe motion figure takesof energy place profile [56]. characteristic of behavior of the systems in which the ratchet mechanism of motion takes place [56]. Polymers 2020, 12, x FOR PEER REVIEW 15 of 19

PolymersPolymers2020 2020, 12, 12, 2931, x FOR PEER REVIEW 1515 of of 19 19

Figure 17. The entropy, S, internal energy, U, and Helmholtz free energy, A, profiles obtained from Figurethe simulation 17. The entropy, (N = 100,S, internal εPP/kBT energy,= −1 andU 0, andfor P1 Helmholtz and P2, respectively, free energy, A D, profiles= 3b). The obtained chart is from based the on the data recorded at each elementary step, tE. simulationFigure 17. ( NThe= 100,entropy,ε /k ST, internal= 1 and energy, 0 for P1 U and, and P2, Helmholtz respectively, freeD energy,= 3b). TheA, profiles chart is obtained based on from the PP B − datathe recordedsimulation at ( eachN = 100, elementary εPP/kBT step,= −1 andtE. 0 for P1 and P2, respectively, D = 3b). The chart is based on The necessary condition of the ratchet mechanism of asymmetric diffusion is the asymmetric the data recorded at each elementary step, tE. dependenceThe necessary of energy condition on position of the ratchetof migrating mechanism species, of as asymmetric schematically diff shownusion isin the Figure asymmetric 18. Such a dependenceprofileThe induces necessary of energy asymmetry condition on position ofof the of migrating ratchetdriving mechanismforce. species, In asconsequence, of schematically asymmetric the showndiffusion slow inmotion Figureis the causedasymmetric18. Such by a a profiledependencerelatively induces small of asymmetry energy but long-lasting on of position the driving force of migrating force.is more In effect consequence,species,ive asthan schematically thethe slowfast one. motion shown The caused effect in Figure byis similar a relatively18. Such to the a smallprofileslow-open, but induces long-lasting fast-close asymmetry force actuation is moreof the mechanism e ffdrivingective than force. of thethe In fast scallopconsequence, one. Themotion, eff ectthe the is slowsimilar molecular motion to the motors-drivencaused slow-open, by a fast-closerelativelyinternal actuation sliding small butof mechanism polymericlong-lasting offilaments theforce scallop is inmore singly motion, effect flagellated theive molecularthan theeukaryotes fast motors-driven one. [57,58]; The effect however, internal is similar sliding instead to ofthe of polymericslow-open,the inertia filaments fast-closeor viscous in singlyactuation effects flagellated being mechanism the eukaryotes cause of ofthe [ 57scallop scallop,58]; however, or motion, eukaryote instead the motimolecular of theon, inertiathe motors-drivenmotion or viscous of the effinternalanalyzedects being sliding copolymer the cause of polymeric of is scallopdue to filaments its or eukaryoteenergetic in singly asymmetry. motion, flagellated the motion eukaryotes of the analyzed [57,58]; copolymer however, instead is due to of itsthe energetic inertia asymmetry.or viscous effects being the cause of scallop or eukaryote motion, the motion of the analyzed copolymer is due to its energetic asymmetry.

Figure 18. Schematic presentation of the energy and the force dependence on the position of an object movingFigure as 18. a resultSchematic of the presentation ratchet mechanism. of the energy and the force dependence on the position of an object moving as a result of the ratchet mechanism. TheFigureforces 18. exerted Schematic by presentation part P2 on P1 of of the the energy chain and computed the forc ase dependence partial derivatives on the position of the free of an Helmholtz object The forces exerted by part P2 on P1 of the chain computed as partial derivatives of the free energymoving (Figure as 17 a) result and averagedof the ratchet for positionsmechanism. of the chain mass center are collected in Figure 19. HelmholtzAlthough energy this force (Figure ﬂuctuates, 17) and it is averaged generally directedfor positions towards of the increasing chain mass values center of the arex-coordinate, collected in thatFigure is,The in 19. the forces current exerted direction by part of the P2 chain on P1 motion. of the This chain seems computed to support as thepartial hypothesis derivatives of a collapsingof the free mechanismHelmholtz ofenergy the chain (Figure diﬀusion 17) and that averaged occurs between for posi swappingtions of the moments. chain mass center are collected in Figure 19. PolymersPolymers2020 2020, 12, 12, 2931, x FOR PEER REVIEW 1616 of of 19 19

Figure 19. The average force, F, exerted by part P2 on P1 of the chain observed for a very short time periodFigure vs. 19. theThex averagecoordinate force, of F the, exerted chain by mass part center P2 on ( NP1= of100, the chainε /k observedT = 1 and for 0a forvery P1 short and time P2, PP B − respectively,period vs. theD = x3 bcoordinate). The chart of is the based chain on themass data center recorded (N = at100, each εPP elementary/kBT = −1 and step, 0t Efor. P1 and P2, respectively, D = 3b). The chart is based on the data recorded at each elementary step, tE. 3.8. Conclusions InAlthough solutions this of the force two-block fluctuates, copolymers, it is generally whose directed monomer towards properties increasing differ in values the interaction of the x- energiescoordinate, with that the solventis, in the molecules, current direction placed in of the the constrained chain motion. environment, This seems namely to support in a tight the hypothesis nanopore, theof anomalousa collapsing di mechanismffusion of polymer of the chain is observed. diffusion The that surprising occurs between behavior swapping of the system moments. consists in the unexpected longitudinal diffusion of molecule along the nanopore. The mechanism of the specific di3.8.ffusion Conclusions behavior involves two phenomena: In solutions of the two-block copolymers, whose monomer properties differ in the interaction 1. The unidirectional motion of the chain between swapping episodes. The mechanism of the energies with the solvent molecules, placed in the constrained environment, namely in a tight unidirectional motion relies on the overcoming the asymmetric landscape of the Helmholtz free nanopore, the anomalous diffusion of polymer is observed. The surprising behavior of the system energy versus the actual chain position. This asymmetry produces the temporal impulse of consists in the unexpected longitudinal diffusion of molecule along the nanopore. The mechanism of force acting between active P2 and passive P1 subchains. Changes in the force impulse produce the specific diffusion behavior involves two phenomena: asymmetrical changes in the momentum of the chain and finally the temporal asymmetric motion 1. ofThe the unidirectional chain. motion of the chain between swapping episodes. The mechanism of the 2. Theunidirectional change in motion the direction relies on of the the overcoming diffusion motion the asymmetric requires overcominglandscape of ofthe the Helmholtz Helmholtz free freeenergy energy versus barrier the actual accompanying chain position. the exchange This asymmetry of positions produces of both the temporal chain parts impulse in confined of force environments.acting between Such active events P2 areand rather passive rare. P1 subchains. Changes in the force impulse produce asymmetrical changes in the momentum of the chain and finally the temporal asymmetric Themotion anomalous of the chain. diffusion of two-block copolymers depends strongly on the width of the nanopore and2. onThe the change chain length. in the Thedirection observed of the abnormal diffusion diff motiusionon is requires well illustrated overcoming by the of anomalous the Helmholtz behavior free of the autocorrelationenergy barrier functionaccompanying of motion the of theexchange mass center of positions of the whole of chain.both chain This particularparts in behaviorconfined deals withenvironments. the specific Such diffusion events of are the rather chain rare. in the intermediate range of the autocorrelation function, where the scaling exponent takes values tending to 5/3 for very tight confinement. The probability densityThe function anomalous of time diffusion required toof swap two-block parts of copolymers the chain is three-modaldepends strongly and consists on the of thewidth swapping of the causednanopore by theand move on the of achain single length. monomer, The aobserved fragment abnormal of the chain diffusion of the size is well of a blob,illustrated and by by both the partsanomalous of the two-block behavior of copolymer. the autocorrelation function of motion of the mass center of the whole chain. ThisThe particular observed behavior dependencies deals with can the be explained,specific diffusion in detail, of onthe thechain basis in the of the intermediate blob theory. range of the autocorrelation function, where the scaling exponent takes values tending to 5/3 for very tight Funding:confinement.This research The probability received no densit externaly function funding. of time required to swap parts of the chain is three- Conflictsmodal and of Interest: consistsThe of the author swapping declares caused no conflict by the of interest. move of a single monomer, a fragment of the chain of the size of a blob, and by both parts of the two-block copolymer. ReferencesThe observed dependencies can be explained, in detail, on the basis of the blob theory.

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