Irreversible Adsorption of Wormlike Chains - American Chemical Society

Aug 7, 2017 - Supercomputing Center, Korea Institute of Science and Technology Information, Daejeon 34141, Korea. ABSTRACT: .... bending over and alig...
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Irreversible Adsorption of Wormlike Chains: Alignment Effects Yunha Kim,† Min-Kyung Chae,† Nam-Kyung Lee,*,†,‡ Youngkyun Jung,§ and Albert Johner*,‡ †

Department of Physics, Sejong University, Seoul 05006, South Korea Institute Charles Sadron, CNRS 23 Rue du Loess, 67034 Cedex 2, Strasbourg, France § Supercomputing Center, Korea Institute of Science and Technology Information, Daejeon 34141, Korea ‡

ABSTRACT: Following the recent scaling theory developed for irreversible adsorption of wormlike chains (WLCs) from a dilute solution, we address several subtle issues regarding the buildup of irreversible adsorption layers. By performing Brownian dynamics simulations for WLCs, we study zipping dynamics of an adsorbing chain and alignment effects due to the interaction with preadsorbed WLCs. The irreversible chemisorption proceeds by zipping on the surface with a typical loop size s0. The measured loop-size distribution comprises a slowly decreasing small loop regime and a large loop regime decreasing ∝ s−5/2, separated by the typical size s0. From the computational study of an incoming chain interacting with a preadsorbed one lying flat on the surface, we provide the crossing probability. A strong bias toward crossing (vs alignment and reflection) is observed, and no strong cooperativity of chain alignment is found.

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INTRODUCTION Polymer coatings are relevant to many industrial processes aiming at the control of surface properties.1 Coatings can be achieved by attaching each polymer either by one (strong) specific anchoring group, which is called grafting, or by potentially any of its monomers which is called (diffuse) adsorption. A polymer interacting with the substrate by any monomer can flatten down and adsorb even for a free energy gain per monomer below the thermal energy kBT due to cooperativity. This is known as weak adsorption and is the focus of most theoretical studies.2−5 Weak adsorption formally leads to equilibrium layers although some of the kinetic processes involved can be very slow. (Main processes ranked by increasing relaxation times are adsorption, polymer exchange with the solution, and washing.6,7) For example, layers can usually not reach equilibrium when compressed against each other or compressed against a wall. This situation is often handled theoretically by constraining slow variables8 such as the adsorbance measuring the monomers pertaining to an adsorbed chain per unit area or the surface concentration measuring the monomers directly in contact with the substrate. In practice, when the free energy gain per monomer exceeds kBT, the adsorption process appearsa irreversible as it cannot reach equilibrium within experimental times.9,10 Truly irreversible adsorption, where a monomer once adsorbed on the substrate cannot move anymore and particularly does not go off the surface again, was considered more recently both experimentally11,12 and theoretically.13−18 If adsorption further proceeds from dilute enough solution,19 polymers adsorb irreversibly one after another without simultaneously competing for the area. This is a variant of random sequential adsorption (RSA): in © XXXX American Chemical Society

contrast to original RSA where the binding is entirely rejected in the case of any overlap, here partial binding of a polymer is allowed. RSA was originally designed for rigid colloids20 (mainly hard spheres21,22 sometimes rods23,24) and applied to (globular) protein adsorption.25 There are several classes regarding the monomer adsorption mechanism. The two extreme cases are physisorption where a monomer sticks upon first contact and chemisorption where it sticks after many attempts, and the polymer samples the available configuration space (phase space) prior to every binding. The idea behind chemisorption is that there is a high enough barrier against entering the reaction between a monomer and the surface for the initial configuration to relax before any reaction event. Chemisorption hence incorporates some aspects of equilibrium statistics while physisorption is ruled by first passage statistics. Consider a polymer at some stage of its chemisorption process. The reaction rate of a given monomer is the product of the probability for the monomer to be within the reaction distance zc from the wall, loosely “at contact”, by the reaction rate at contact Q (Figure 1). Both these quantities zc and Q are time-independent. The probability for a given monomer to be at contact with the wall is conditioned by earlier bindings in the same polymer and interactions with the wall or with previously adsorbed polymers. As most other studies we will consider purely repulsive monomer−wall interactions (apart from the chemical binding reaction). Received: April 13, 2017 Revised: July 23, 2017

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according to the loop distribution on a impenetrable wall. In contrast to flexible chains, zipping of the filament onto the wall was predicted to proceed by small hopping loops s0 ∼ l1/3b2/3 larger than b (see also Figure 3). Multiple zipping proceeding Figure 1. (a) Schematics of semiflexible chain adsorbing on the surface of reaction zone size zc. Monomers within the reaction zone are indicated as dark gray. (b) Zipping of a single stiff chain on the surface where coarse-grained surface monomers are not explicitly shown. Black colored monomers represent adsorbed monomers.

The simplest polymer model considers ideal (noninteracting) completely flexible chains without any orientational correlations along their path. Realistic polymers are subject to monomer−monomer interactions, the consequences of which were worked out extensively26,27 also near surfaces,28 and are not completely flexible. Rigidity is especially important for biofilaments and attracted more attention recently.29−31 Rigidity reduces the occurrence of self-contacts. In many instances excluded volume is not relevant to rigid chains in 3D dilute solutions. In contrast, at high enough concentration, excluded volume between locally rodlike filaments triggers alignments and, eventually, some nematic ordering. Some orientational order was for instance observed in simulations of 2D dense semiflexible chains (see Figure 2a)30 and dense solutions near a wall,32 similarly to liquid crystalline polymers.33

Figure 3. (a) Adsorption process. New adsorbing monomers are indicated as red. A secondary loop can be formed by adsorption of an internal monomer of an existing loop. A hopping loop forms by adsorption of a monomer belonging to a tail. (b) Distribution of the hopping-loop size with zc = 1.3σ for chains of three different values of the persistence lengths l = 120σ (●), 240σ (△), and 360σ (◇) (N = 100, Q = 0.001). The average hopping loop size as a function of zc for realistic values of zc is shown in the inset. (c) Distribution of hopping loop sizes for various reaction zone sizes with l = 240σ. Open circles (○) indicate data for zc = z*c , and filled symbols are for various reaction zone width zc ranging from 1.2 to 2.0σ with an interval of 0.1σ. The dashed line indicates the asymptotic behavior of loop size ∝ s−5/2.

from distant nucleation prevails only deeper in the flexible regime and will not be addressed here.19 Focusing on (very) rigid filaments, we suggested that the ensemble of almost straight adsorbed filaments can be described by a random Poisson line mosaic (Figure 2b) where an incoming filament has to find free available polygons for adsorption. Larger free polygons are exponentially rare but energetically favored over smaller ones where the filament has to buckle in order to reach the surface. As we pointed out earlier, two lines drawn at random in the surface generically cut (i.e., with probability 1). The relative angle α at crossings is not distributed uniformly but according to the law P(α) = (1/2)sinα, favoring orthogonal crossings (no crossings under zero angle are expected). In addition, for short-range interfilament repulsion, crossing orthogonally is indeed energetically more favorable than bending over and aligning, when the two filaments start to interact. Indeed, crossing needs a hopping loop only marginally larger than the typical one. In order to gain further insight into the ordering problem, we explicitly take into account interactions between the incoming filament and preadsorbed ones in the early stages when alignment is possible. Testing the theory for chemisorption including the final jammed regime still appears very computationally demanding. In contrast, extensive simulation data36,37 is published for chains with various rigidities and surface affinities adsorbing at equilibrium with the solution.38 In the present work we numerically study multichain problems focusing on the nucleation of an alignment defect.

Figure 2. (a) Local nematic order in a 2D liquid of semiflexible polymers at density ρ = 0.35 with persistence length l = 16σ and N = 25630 (picture adapted from ref 30) (b) Poisson lines with the line density τ = 0.2, which illustrates the stiff chain adsorption at early stage of irreversible chemisorption. When the mesh of the array becomes tighter than s0, an incoming chain jumps from favorable polygonal hole to hole. The line density τ = Nl/D counts the total number of lines (Nl) cutting the circle of diameter D circumscribed about the square (only the square is shown).

Often rigid polymers are described theoretically by the continuous wormlike chain model (WLC) which penalizes curvature κ according to the Hamiltonian H = ∫ S0(Bκ2/2) ds, where the integral runs along the filament of length S.34 In the WLC model the orientation correlation along the filament decreases exponentially with the curvilinear distance. The decay length is simply linked to the flexural modulus B. In 3D, it coincides with the persistence length l, l = B/kBT. The WLC model proved very useful to describe the elasticity of doublestranded DNA35 and was more generally applied to biofilaments.30 Below we will use the WLC model for filaments of length S, diameter b, and persistence length l. In our previous work19 essentially based on scaling theory, we showed that chemisorption of a single filament proceeds B

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by forming small loops of typical size s0. We call this zipping, albeit zipping usually refers to adsorption proceeding by small loops of size ∼b. The simple zipping mechanism is expected to fail deeper in the flexible regime for chain lengths S > S* ∼ l5/3b−2/3 > l where it is preempted by distant adsorption along the chain creating new nucleation sites for multiple zipping.19 From the simulation of the single chain adsorption process, we record external loops resulting from the adsorption of a tail monomer, called hopping loops below, and discard secondary loops which form subsequently upon readsorption of an existing loop Figure 3a. Formally, the size distribution is similar for both types of loops up to finite size effects (see Figure 2 in ref 19). Figure 3b shows the size distribution of hopping loops for zc = 1.3σ. The measured hopping-loop distribution clearly shows the predicted ∝ s−5/2 decay for large loops (Figure 3b). The probability distribution of small loops shows a much weaker size dependence. An important point is that the total reaction rate integrated over the hopping-loop size is dominated by the crossover size. We are now eligible to define the typical loop size more quantitatively than in the scaling description.19 For example, the large s asymptotics P(s) = (s/ s0)−5/2 defines the typical size s0 as the x-intercept. We obtain 1.85, 2.52, and 3.47 as s0 for l = 120σ, 240σ, and 360σ with zc = zc*. Alternatively, the associated average hopping loop size ⟨s⟩ could be considered, which is also determined by the crossover loop size and is found to increase with the rigidity (see Figure 3a, inset). This increase is compatible with the dependence s0 ∝ l1/3 of the scaling prediction for the few persistence lengths tested. The average hopping size further shows a sublinear increase with zc − z*c for the retained range of zc (1.1−1.4) as shown in the inset of Figure 3b. The simple picture developed here and in ref 19 breaks down qualitatively for large zc, as seen from Figure 3c. For the system investigated, the −5/2 power law is lost for the highest values of zc shown. The large zc regime, which is not realistic, rather follows the softer s−3/2 power law reflecting the probability for the monomer to be in the reaction zone albeit fluctuations typically drive it further away from the surface. More precisely, at large zc, the small section does not have to loop back in a strict sense as the adsorption points are distributed all over the reaction zone width.b To recover the expected loop distribution (s/s0)−5/2, the section of length s has to go out of the reaction zone by at least ∼zc. Conversely, the s−3/2 distribution is cut exponentially by the confinement free energy FW in a slit of width W several zc large. According to refs 39 and 40 for strong confinement FW = 1.1s/(W2/3l1/3), in thermal units. As a consequence, the s−3/2 power law is cut and the (s/s0)−5/2 takes over at large s. Note that it is a coincidence that the same slope obtains for an ideal flexible chain as there are no flexible loops here. Because of the softer power law decay, the average hopping loop size is rather dominated by larger “loops”. The s−3/2 regime is dominant for unphysically large zc. It is however of interest that its signature appears in the crossover region s ≳ s0 for the smaller, retained, values of zc.

We perform Brownian dynamics (BD) simulations for WLC where we take into account excluded volume interactions between all monomers, which allows us to investigate their relevance for multichain problems at finite local concentration. Simulations prove especially useful for intermediate stiffness where scaling predictions are weaker. The article is organized as follows. First, we present zipping dynamics and show the corresponding zipping loop distributions during chemisorption. We then consider the problem of one incoming chain adsorbing in the presence of a preadsorbed one and discuss the crossing vs alignment and reflection probability. Next, we study the behavior of an adsorbing chain in the presence of wider obstacles of height b, mimicking a nematic region. Finally, we discuss the impact of our findings on the general chemisorption of WLCs.

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MODEL We model the polymer using a bead−spring chain consisting of N beads with diameter b ≈ σ. The bead−bead interactions were modeled by the fully repulsive Weeks−Chandler−Andersen (WCA) potential: UWCA(r) = 4ϵ[(σ/r)12 − (σ/r)6 + 1/4] for r < 21/6σ and 0 elsewhere. Here ϵ and σ represent the strength and range of the WCA potential, respectively; r denotes the center-to-center distance between two beads or the distance of the bead center from the wall. The chain connectivity is ensured by the finite extension nonlinear elastic (FENE) potential between two consecutive beads. The chain stiffness is taken into account by an angular potential, Ubend = l(1 − cos θ), in which θ is the angle between two successive bond vectors of a chain. The surface is represented by an array of Lennard-Jones (LJ) beads of diameter b similar to monomer beads, and the bead− wall interactions were also modeled by the same fully repulsive WCA potential. We run Brownian dynamics (BD) simulations. A monomer is considered to be inside the reaction zone if its center is within the distance zc from the plane defined by the centers of the surface beads as shown in Figure 1a. Thus, if strict contact is required for reaction, zc = zc* = 21/6σ. In practice, we choose z c = 1.1−1.5σ. In the spirit of chemisorption, a new binding is accepted with a small probability Q per unit BD time, to let configurations equilibrate. We choose Q = 0.001 and checked that further decrease of Q does not affect the adsorption results.

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ZIPPING DYNAMICS We first consider the zipping dynamics of a single WLC of length S. A (long) chain arrives at the surface from the dilute solution and typically attaches by a central monomer rather than by an end.19 We assume for simplicity that the surface is not yet covered with polymers, which is the case at early stages of adsorption, and then the chain subsequently adsorbs freely and zips on the surface as illustrated in Figure 1b. Globally the subsequent reaction occurs according to the loop distribution p(s) ∝ s−5/2 as shown in our previous study.19 Nonetheless, due to chain stiffness, a small stretch close to the bound monomer is almost surely within the reaction zone. In a simplified view the width of the reaction zone is simply zc ∼ b which defines the strand length s0 ∼ l1/3b2/3, beyond which thermal fluctuations take monomers out of the reaction zone. In our previous work19 this led us to the conjecture ∼(s/s0)−5/2 for the probability to find a monomer at a distance s > s0 within the reaction zone. This suggests that chemisorption proceeds

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MULTICHAIN EFFECTS: CROSSING VERSUS ALIGNMENT Crossing Probability. For dilute enough solution, a given adsorbing chain has time to zip and complete its adsorption before any new chain comes in and competes for the surface.19 An incoming chain adsorbs in the presence of already adsorbed chains with “frozen-in” configurations. A central question is whether, or to what extent, the final jammed layer does exhibit C

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for zc = 21/6σ = z*c , and similar results are obtained for somewhat larger zc. Chains cross almost surely when they start orthogonal to the straight obstacle. In such a starting geometry a costly sharp bend is required to avoid crossing. In contrast, noncrossing is more likely for chains starting with small angles where negligible bending is sufficient to reflect the configuration, which is more advantageous than hopping over the straight chain. This equilibrium argument applies to each individual hopping step. The free energy penalty for bending can be avoided by forming a large loop, which is statistically rare. For stiffer chains, this picture is more clear-cut, and also the criterion for almost surely crossing is more stringent. Furthermore, the starting angles allowing almost surely crossing shifts to the smaller values with increasing stiffness. We expect this trend to persist for even stiffer chains, which is not easy to investigate numerically. Because the relative orientation between the two chains is not random, the distribution Pc(α) has to be further averaged with weight sin α (normalized between 0 and π/2) which gives a bias in favor of crossing. The resulting global crossing probability is shown in Figure 5b. It increases with rigidity and surpasses 80%. The crossing probability is central in this contribution. Obviously, the quenched distribution of the incidence angle α plays an important role. At this stage it is worthwhile to try and contrast the chemisorption case with an annealed model. For simplicity, we choose an annealed 2D WLC model with crossings. The chains are 2D everywhere but allowed to cross with a finite penalty E. Some aspects of this model can be solved analytically (see Appendix). Let us consider the very same crossing problem of a chain facing a straight configuration (Figure 4, top left) for this annealed model. As shown in the Appendix, the statistics for a given angle of incidence α are not so drastically different. What makes a huge difference is the statistical weight of incidence angle α. For chemisorption this weight is quenched and biased toward orthogonal incidence (weight sin α). For the 2D annealed model the distribution of α is thermalized according to the total partition function (summing up both crossing and noncrossing partition function); the statistics of the incidence angle is unbiased for E = 0 and biased toward small angles for the more realistic cases E > 1 kBT. It is hence expected that alignment prevails for the 2D annealed model and, likely, for reversible adsorption. It is worthwhile to attempt a direct comparison of alignment effects between reversible adsorption and chemisorption in the future. Next, we study the events where the chain crosses and discuss the distribution of the crossing-loop size. In Figure 6, we show crossing loop size distributions for chains N = 100 with y = 10σ and N = 200 with y = 20σ. According to the data set for N = 200 (Figure 6d), the average crossing loop size distribution does not dramatically depend on the starting orientation. Strikingly the (relative) importance of end binding is enhanced at small α; the scenario seems then to be that the adsorbing chain initially aligns and finally crosses with a small end loop. At large α crossing via end-binding occurs very rarely and by a large loop, likely in early stages. The crossing loop distribution resembles the hopping loop distribution, and the large loops show apparent power law decay ∝ s−2.1. For N = 100 (Figure 6c), the distribution at smaller angles are considerably more noisy as crossing events are rare. The chain is so short that crossing loop size is almost equivalent to the total chain length, which results in more endbinding than for long chain. As shown in Figure 6b, the

local orientational order as observed in the 2D equilibrium liquid30 (Figure 2a). In our qualitative scaling description we argued that bending and alignment is the rule for a chain coming in almost parallel to a preadsorbed one but not the general rule for scarcely covered surfaces. To gain an insight for this problem, we investigate the adsorption of a chain starting out from a binding point located a distance y from a (preadsorbed) straight LJ chain under an angle α. The geometry is also defined in Figure 4 (top left). As

Figure 4. One adsorbing chain in the presence of an adsorbed straight chain: (top left) definition of the starting geometry; (other panels) some of the obtained (not always typical) configurations.

in the previous section, we consider chains consisting of N = 100 and N = 200 monomers with l = 60−360σ. These chains are locally rigid at scales much smaller than l but less so at the scale of the total chain ∼Nb. The starting distance y away from the preadsorbed straight polymer is chosen to be y = 10σ and 20σ for N = 100 and 200, respectively. Chain configurations are equilibrated under this constraint prior to any adsorption. The starting position y is not crucial provided it is large enough for the chain to sample the zipping statistics in front of the obstacle. The set of values chosen for y and Q were determined from exploratory simulation runs. A few, not always typical configurations obtained from the simulation are shown in Figure 4 for various incident angle α. In some cases the configuration crosses the preadsorbed chain. In other cases it is reflected. Rarely it crosses but also shows an aligned section. To be quantitative, we measure the crossing probability for a given starting angle α. Figure 5 shows results

Figure 5. (a) Crossing statistics for a chain (N = 100) starting out a distance y = 10σ from a straight adsorbed configuration under the angle α as a function of α. Four persistence lengths are investigated: l = 60σ (□), 120σ (●), 240σ (△), and 360σ (◇). Here strict contact is required for binding (zc = z*c = 21/6σ). (b) Global crossing probability for various l for chains N = 100 with y = 10σ (●) and for N = 200 with y = 20 (△). The global crossing probability is obtained by integral of Pc(α) with weight sin α. D

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Figure 6. Statistics of the crossing loop size. (a) Illustration of the crossing loop and average crossing-loop size for various α. (b) Incident angle dependence of the average crossing loop size for N = 100 and y = 10σ with zc = z*c for chains with various persistence lengths l = 60 (□), 120 (●), 240 (△), and 360 (◇). (c) Statistics of the crossing loop size for different starting orientations α for N = 100, l = 120σ, y = 10σ, and zc = zc*. (d) Statistics of the crossing loop size for N = 200, l = 120σ, y = 20σ, and zc = 1.2σ.

crossing loop size is almost not sensitive to the incident angle α. We thus compute the crossing loop size sc by taking average over all α for different stiffness. We obtain sc = 34.6, 41.1, and 47.5 with zc = 1.2σ and N = 200 for l = 120σ, 240σ, and 360σ. These values increase with stiffness, regardless of the incident angle, roughly s0 ∝ l1/3. The proportional constant sc/(lb2)1/3 is about 7 ± 0.2. Scaling is less working for large zc (and short chains), as expected.c Multichain Effects: Nucleation of Nematic Patches. Assuming there is local alignment of the two chains, how does the adsorption pattern develop further? To shed some light on this question, we directly simulate the sequential adsorption of a few chains. A set of configurations are shown in Figure 7a for N = 50 and l = 20σ. The snapshots of first raw show sequential adsorption of three chains near the straight preadsorbed chain starting with small incident angle α = 15°, where chains are reflected from the preadsorbed ones and are aligning to some extent. In the second raw, a set of configurations (snapshots) in

the process of fourth chain adsorption crossing the frozen-in adsorbed chains are shown. It is clear that many different situations can occur here. We address the simpler question whether a wider obstacle composed of a number of adjacent straight LJ chains significantly affects the crossing probability. Results are displayed in Figure 7b for an adsorbing chain

Figure 8. For a 2D single stiff chain, the probability to be confined in the half-plane (noncrossing probability) is shown under scaling parameter x = α(2l/9 y)1/3 when chains of various l start from a distance of y = 10σ with angle α. The dot-dashed line represents the theoretically obtained noncrossing probability P0 = Znc/Z0 of reversibly adsorbing chain with E = 0 (Z0 = 0.5). The red, black, blue, and green lines are equilibrium probability for the infinitely long chains for several values of crossing energy penalty E = 5, 4, 3, and 2. For each value of E, three lines correspond to Z0 = 0.5, 0.75, and 1.0 from top to bottom.

starting under an angle of π/4. The recorded data show very weak dependency of the crossing probability on the width of obstacles in the explored range (at least up to width 10σ). We do not find strong cooperatively of alignment and hence do not expect frequent nucleation of nematic patches.

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CONCLUSIONS We show that chemisorption of a WLC in the stiff regime (S ≤ l) proceeds by simple zipping with a average hopping-loop size ⟨s⟩ larger than the monomer size (of the order of five monomers in the simulation) weakly increasing with rigidity. Larger hopping loops are scarce and distributed according to (s/s0)−5/2, with a characteristic size s0 slightly smaller than ⟨s⟩. (In scaling theory these both quantities share a common scale.)

Figure 7. (a) Configurations for the sequential adsorption of three chains (l = 20σ, N = 50) in the presence of a straight chain. Each chain is injected with angle α = 15° and from additional distance 5σ away. In the second row, we show a series of snapshots for a fourth adsorbing chain (shown as red) and the end configuration. Statistics for crossing a wide obstacle composed of a number of adjacent straight chains with α = π/4 for (b) N = 100, y = 10σ, and (c) N = 200, y = 20σ. E

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rescale the angle α. Let us start with the partition Z1/2 of a WLC restricted to the half-plane y > 0. The partition function Z1/2 can be obtained by the general Grosberg equation:43−45 a master equation describing the variation of the partition function Z for a WLC of length s ending at coordinate y with the tangent t, after infinitesimal growth

Our numerical findings match the prediction of our earlier scaling theory.19 Lines randomly thrown on a surface, if allowed to cut each other, form a Poisson mosaic as shown in Figure 2a. In this structure the angle under which two lines cut is not distributed uniformly but according to the density (1/2) sin α normalized between 0 and π. Taking this extra weight into account for the WLCs investigated, we evaluate the global crossing probability for several persistence lengths. For stiff chains we find that most encounters of two chains result in a crossing. However, in larger samples of the adsorption layer there will be some encounters without crossing, where chains locally align. Note that for reversible adsorption, not investigated in all details, the polar angle of the incoming chain is annealed and its thermally weighted distribution biased toward small-angles and alignment. For chemisorption, it is then important whether local alignments can be nucleation points for large regions with orientational order. These regions would, at the very least, appear as defects in the Poisson mosaic. Our data, for widths of preoriented regions up to ten chain diameters (Figure 7), suggest that the existence of a wider aligned region, more than one chain broad, does not significantly promote the alignment of the subsequently adsorbing chain. Our findings globally support our previous scaling description19 where chains first build a 2D Poisson array of crossing chains making most of the adsorbance. At a later stage, when the Poisson array becomes tight, incoming chains must jump from hole to hole and build up a more diffuse outer layer as discussed in ref 19. We considered WLC with purely repulsive short-range excluded volume interchain interactions and short-range repulsive interactions with the substrate besides the chemical reaction. Moderate increase of the repulsion range does not change our results, which are still dominated by almost orthogonal crossings, much. There are however cases where the interactions are more complex, especially for charged filaments. More complex, attractive, interchain interactions were considered recently for a single WLC sticking to an interface.41 We also considered reversible plasmid adsorption on an oppositely charged surface at low monovalent salt where the filament/ substrate interaction is attractive and longer-ranged and where the crossing penalty involves longer-ranged electrostatic repulsions42 which energetically favor orthogonal crossings. Besides the WLC model is a simple and useful continuous model, but at the scale of a few monomers, chemical details, which are out of the scope of this paper, may matter.

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∂Z 1 ∂ 2Z ∂Z = − ty 2 ∂s 2l ∂θ ∂y

(A1)

with θ the polar angle of the tangent at the chain end. The boundary condition for the half-plane partition function Z1/2 is Z1/2(y = 0) = 0 with ty > 0, which states that a chain ending at the wall cannot be grown out of the forbidden half-plane y < 0.39,46 A common way to solve eq A1 is an eigenfunction expansion47 Znc = ∑ψke−ϵks. We may restrict ourselves to the ground state approximation, only retaining the lowest eigenvalue ϵ, which is valid for asymptotically long chains. Further, for the chain restricted to a half-plane without any long-range surface potential, the lowest eigenvalue is 0 and the ground state eigenfunction ψ satisfies46 0=

∂ψ 1 ∂ 2ψ + sin θ 2l ∂θ 2 ∂y

(A2)

The polar angle θ is chosen such that ty = −sin θ, turning the boundary condition into Znc(y = 0) for sin θ < 0. It is possible to solve eq A2 for small angles θ < 1 where sin θ ≈ θ by a scaling ansatz ψ = yβf(x) where the scaling variable x = θ(2l/ 9y)1/3 has been introduced above. After linearization of sin θ, the Grosberg equation (eq A2) turns into the Kummer equation: 0=x

d2f dx

2

+

⎛2 ⎞ df ⎜ − x⎟ + βf ⎝3 ⎠ dx

(A3) 48

which is solved by the confluent hypergeometric function U, f(x) = U(−β, 2/3, x). The boundary f(−∞) = 0 quantizes the admissible values of β, further asking that the ground state eigenfunction has no node selects the value β = −1/6 for depletion.46 Equipped with the partition function ⎛ 5 2 2lα 3 ⎞ −2lα 3/9y Z1/2(y , α) = y1/6 U ⎜ , , ⎟e ⎝ 6 3 9y ⎠

(A4)

where α is positive for a chain growing out from the distance y towards the boundary (α = −θ) and Kummer’s transform was used. We can now proceed to the crossing and noncrossing probabilities. For an ideal WLC, where the difference between a crossing and noncrossing is purely geometric (E = 0), the total partition function Z0 counting all configuration is independent of y and α. In this case the noncrossing probability is simply P0(y,θ) = Z1/2(y,θ)/Z0; the shape of this function is shown in Figure 8. Let us now turn to the more realistic case where chains pay some free energy penalty E for crossing, the partition function Znc is unaffected, but the partition function for crossing is reduced. The reduction of statistical weight applies to each crossing configuration individually. For the sake of simplicity, we apply the global weight e−E to Zc, Zc = (Z0 − Znc)e−E; it is assumed that E is almost independent of α but may depend on l.d (We get some support from the weak dependence of the crossing loop distribution on α found for chemisorption.) The noncrossing probability with penalty E is linked to the one without penalty (E = 0) through

APPENDIX

Noncrossing Probability for the Annealed 2D Model with Crossings

It is instructive to try and compare with what is expected for 2D equilibrium configurations of WLCs with crossings, although this is intrinsically different from the irreversible adsorption process studied here which does not obey detailed balance. Let us start with the partition function of a WLC restricted to a half-plane. The relevant dimensionless parameter at equilibrium is the bending energy involved in the reflection of the configuration, expressed in thermal units, Fb ∼ lα3/y, where α < 1 is assumed. We choose the equivalent parameter x = α(2l/ 9y)1/3, suggested by the analytical theory, developed below, to F

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Macromolecules Pnc =

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P0 −E

P0 + e (1 − P0)

REFERENCES

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The decay of the noncrossing probability at small α is more sluggish when E increases. In Figure 8, we plot the noncrossing probabilities computed with various values of crossing penalty E, together with chemisorption simulation data, under the same scaling parameter x = α(l/9y)1/3. We show that for small α the noncrossing probability Pnc for the annealed model almost matches with the simulation results of chemisorption with a choice of appropriate E (for x < 0.5). The total probability of noncrossing (or crossing) is obtained from the weight sin θ for irreversible chemisorption, as shown earlier, while for the annealed model the angle α is thermalized and subjected to thermal weighting by the total partition function Znc(α) + Zc(α) under the given incident angle α. In the irreversible case the contribution to the crossing/noncrossing probability with the weight of sin θ is biased toward α = π/2, while in the annealed case it is not biased when formally vanishing crossing penalty applies and biased toward small angles when higher crossing penalty applies. As a consequence, crossing is favored for chemisorption while alignment is favored in the 2d annealed model with crossings.

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Article

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (N.-K.L.). *E-mail: [email protected] (A.J.). ORCID

Nam-Kyung Lee: 0000-0002-5359-0687 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Our work is supported by National Research Foundation grants provided by Korean government NRF2016R1D1A1B03931049, NRF-2017R1A2B4010632 (N.-K.L.) and NRF-2015R1D1A1A09057469 (Y.J.).

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ADDITIONAL NOTES Our intention is to provide an operational criterion of irreversibility. The binding energy needed for practically suppressing local arrangements, such as converting a train of adsorbed monomers into a large loop, is somewhat larger than that for suppressing exchange of long chains with the bulk solution. There is no strict criterion at all other than total irreversibility considered in the paper. An operational criterion will depend on the properties considered/measured, on chain length, and, to a lesser extent, on the (realistic) waiting time. b After reaction, the monomers stay at the same distance of the surface. This is the origin of the softer power law resembling the behavior of flexible chains, which is not observed at or close to zc*. More generally, microscopic details of the model, like the mechanism of flexibility, may marginally influence the numerical results. c We also obtained sc = 48.9, 54.6, and 62.2 for l = 60σ, 120σ, and 240σ, with larger zc = 1.3σ and N = 100. The proportionallity constant sc/(lb2)1/3 is about 11 ± 1; scaling is worse for large zc (and shorter chains). d We also do not discriminate according to the number of crossings here. a

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DOI: 10.1021/acs.macromol.7b00748 Macromolecules XXXX, XXX, XXX−XXX

Article

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DOI: 10.1021/acs.macromol.7b00748 Macromolecules XXXX, XXX, XXX−XXX