Irreversible Adsorption of Worm-Like Chains - Macromolecules (ACS

Oct 15, 2015 - Irreversible Adsorption of Worm-Like Chains. Nam-Kyung Lee†‡, Youngkyun Jung¶, and Albert Johner†‡. † Department of Physics,...
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Irreversible Adsorption of Worm-Like Chains Nam-Kyung Lee,†,‡ Youngkyun Jung,¶ and Albert Johner*,†,‡ †

Department of Physics, Sejong University, Seoul 05006, Korea Institut Charles Sadron, Université de Strasbourg, CNRS UPR22, 23 Rue du Loess, 67034 Strasbourg cedex 2, France ¶ National Institute of Supercomputing and Networking, Korea Institute of Science and Technology Information, Daejeon 34141, Korea ‡

S Supporting Information *

ABSTRACT: We present a theory for the irreversible adsorption of semiflexible polymers, described as worm-like chains (WLC) of persistence length S and contour length S, from dilute solution, with a focus on chemisorption (reaction limited adsorption). Early stages are dominated by single chain adsorption. For stiff to moderately flexible chains, shorter than S*∼ S 5/3/b2/3 with b the monomer size, adsorption proceeds out of the first binding point with minimal, flat, adsorption loops of size so ∼ S 1/3 b2/3 by simple zipping in S/s0 steps. For more flexible chains obeying S > S*, adsorption proceeds by multiple zipping from several nucleation points distant along the chain. At a certain typical surface concentration, larger than the 2-d overlap concentration, steric hindrance between incoming polymers and those lying already flat on the surface becomes relevant. The interfacial profile built by stiff loops dangling in the solution comprises an inner layer and an outer layer, where the concentration decreases with the distance z to the surface as ∼1/z and ∼1/z2, respectively. This is in contrast to the ∼1/z4/3 profile predicted for reversible WLC adsorption. The last distant adsorption spots are filled by end grafted chains. We also provide typical adsorption times and a short discussion of physisorption.

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Experiments using strongly binding polymers13 showed that the number of surface-contacts per polymer follows a bimodal distribution, in contrast to the expectation for equilibrium layers. A huge numerical effort was done over the subsequent years.14−16 The latest studies consider a wide range of stiffness and interaction strength.16 A conceptual breakthrough was achieved in a series of papers by O’Shaughnessy and Vavylonis (OV)17−20 who distinguish reaction limited and diffusion limited adsorption. The former applies to adsorption through slow reaction with energy barriers against entering the reaction, where the binding rate upon contact is low, while the latter applies to cases where binding is assured upon first contact, which is expected for hydrogen bonds or ligand/receptor pairs like biotin/streptavidin. Reaction limited adsorption supposes that chain conformations are fully equilibrated under the constraint of existing bonds at any stage and adsorption is ruled by equilibrium statistics, while diffusion limited adsorption involves chain relaxation dynamics. OV put the main focus on chemisorption (reaction limited adsorption). The adsorption kinetics is then ruled by the equilibrium distribution of the loop size n, for a chain near a repulsive wall: p(n) ∝ 1/nα. OV distinguish three cases: (i) α > 2, where the adsorption process

INTRODUCTION Coating solids with polymers is a common way to control their surface properties.1 This can be achieved with little material per surface area as a quantity equivalent to one monolayer is often enough to trigger surface properties. A widespread example is the stabilization of colloidal solutions, against the ever-present van der Waals attractions between alike colloids, by fluffy layers. This prompted early theoretical2 and experimental studies3 of polymer adsorption. Most of the theoretical studies were concerned with fully flexible polymers and layers either in full equilibrium with the solution4 or at least in restricted equilibrium under the constraint of imposed coverage.5 Restricted equilibrium is relevant to adsorption layers compressed against each other. It is then assumed that polymers do not escape from the gap during compression but are otherwise free to reorganize. Simulation studies6 and numerical selfconsistent-mean-field calculations7 provided further insight. In early experimental studies, slow exchange rates with the bulk8,9 are reported for some layers and these findings attributed to metastability. As a matter of fact, slow exchange kinetics is, to some extent, also expected for equilibrium layers, as was shown later.10 The very idea that layers may be out of equilibrium triggered dedicated work. Numerical studies for the irreversible adsorption of a single chain11 were undertaken in Tirrell’s group and their results were in part corroborated by experimental findings.12 © XXXX American Chemical Society

Received: June 17, 2015 Revised: September 30, 2015

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Macromolecules is dominated by adsorption of monomers nearby already adsorbed ones and the polymer zips on the surface, (ii) α < 1, where large adsorption loops dominate, and (iii) 1 < α < 2, where small adsorption loops dominate (zipping) but large adsorption loops form nonetheless before zipping is completed. In regime (iii), called accelerated zipping by OV, zipping proceeds from multiple nucleation sites and this mechanism is found to adequately describe the adsorption of chains swollen by the solvent (self-avoiding walks, SAW) as well as ideal chains (random walks, RW). OV. also consider irreversible adsorption from dilute solution.17,18Reversible vs irreversible graf ted coatings were studied by some of the authors.21 Recently some numerical studies focusing on single chain physisorption (diffusion limited adsorption) were conducted with swollen chains (SAW) and without hydrodynamic interactions,22 in the so-called Rouse dynamics. We will discuss irreversible physisorption in a later part of the paper. Many polymers are not flexible at all scales and are better described by the worm-like chain (WLC) model.23 While adsorption of flexible polymers is fairly understood, adsorption of semi flexible polymers (WLC) is less understood. The main ingredient of the WLC model is the persistence length S , which characterizes the exponential decay of the tangent−tangent correlation function along the polymer. Bending of the polymer entails an energy cost S κ2/2 expressed in thermal units (kBT) per unit polymer length where κ is the (local) curvature. The WLC model proved useful for biofilaments, especially for duplex-DNA (d-DNA). It also describes many synthetic polymers with some intrinsic stiffness. The qualitative behavior of a polymer depends on its length S: short polymer fragments (S≪ S ) are essentially straight if unconstrained, while large ones (S≫S ) are essentially flexible and tend to coil. Quite some theoretical efforts, both numerical24 and analytical,25 went into the elucidation of the equilibrium structure of adsorption layers built from ideal WLC. A complete description of the layer structure at short distance z from the surface was achieved by Semenov.26 The present paper is devoted to the irreversible adsorption of WLC with emphasis on the case of chemisorption.

Figure 1. (a) Correspondence of bulk configurations and configurations touching the surface by any “middle” monomer. (b) Configuration contributing to the partition function Z1, to the tail partition function Zt, to the loop partition function Zl (from left to right). (c) Illustration of the two adsorption mechanisms of WLC: simple zipping with step size s0 (top) and accelerated zipping with multiple nucleation sites (bottom).

chain close to the surface is hence Z1∝ s1/2. The chain close to the surface can now be seen as the union of two tails of length ∼ s matching at any of the ∼ s middle monomers of size b (we do not distinguish between the longitudinal and transverse monomer lengths), Z1∝ Zt2 s, solved by Zt ∝ s−1/4. Therefrom the partition function of a loop of size s can be deduced as for the flexible chains: a loop is the union of two tails matching both in height and orientation Zl ∼ Zt2/(θ·θs). This leads to Zl ∝ s−5/2. For ideal flexible chains following RW statistics the partition functions of tails and loops are simply linked to first passage probabilities and Zt ∝ s−1/2, Zl ∝ s−3/2. These s-dependences are preserved for the WLC in the flexible limit. To summarize the size dependence of the partition functions: s ≪ S : Z1 ∝

■

s , Zt ∝ s−1/4 , Zl ∝ s−5/2

s ≫ S : Z1 ∝ s 0 , Zt ∝ s−1/2 , Zl ∝ s−3/2

CHEMISORPTION OF A WLC Chemisorption corresponds to the case where the surface is repelling the polymer which only attaches through slow chemical reaction. Many reaction attempts are needed before a success and polymer conformations are amply sampled before an attachment. The adsorption rate q for chemisorption hence decomposes into the product of the number of monomers in the very vicinity of the wall, assuming equilibrated conformations, by the reaction rate at contact Q. Equilibrium polymer statistics is reflected by the partition function of loops and tails, which can be obtained at the scaling level following the arguments by Semenov for stiff chains.26 Single Chain Adsorption. Chemisorption is ruled by the equilibrium distribution of loops and tails near a repulsive wall. Specifically we need to know the probability P(s) that given a surface-contact, a monomer a curvilinear distance s apart is in contact with the repulsive surface, see Figure 1b. We first briefly recall the derivation of the loop and tail partition function following ref 26. For a chain of size s approaching the surface by any of its middle monomers, the typical angular fluctuation θ is given by S θ2/s ∼ 1 leading to θ ∼ s/S . The partition function of the

(1)

The polymers can be further swollen by excluded volume interactions. Semiflexible polymers with a persistence length S markedly larger than the monomer size b (chain thickness), e.g., S > 10b, obey excluded volume statistics only if S/b ≳ (S /b)3, deep in the flexible regime. We will hence consider, that excluded volume is not relevant for a single polymer strand of length s in solution (s/b< (S /b)3) and will not comment on the swollen loop regime, which was described by OV.18 Note however that, if the flat polymer would freely arrange on the surface it would be likely swollen by 2-d excluded volume provided S ≫ S which holds in the flexible regime. In the present study, it is assumed that after adsorption the polymer can neither leave the surface nor freely sample 2-d configurations: adsorbed monomers cannot move anymore and pin the polymer. This strictly holds for chemical reaction and holds over experimental times for strong enough monomer binding. Because the loop distribution plays a central role in our study, we also evaluated it by numerical simulation using the simulation package LAMMPS.27 Results are displayed in Figure 2. The asymptotic power laws, represented by dashed lines, B

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Macromolecules

size s. In the following we use the designation P(s) for both cases. The adsorption rate q = Q∫ d(s/b) P(s) is dominated by the minimal loop size so, yielding the rate of minimal loop adsorption q ∼ Qso/b. Zipping on the surface starting out from the first adsorbed monomer proceeds by loops of size so, rather than monomer by monomer. Zipping completes after S/s0 steps only. Zipping by minimal loops hence takes the time: tz∼ Q−1S b/so2. According to OV17,18 for a long chain such strong powerlaw decrease of P(s) corresponds to zipping starting out from the first nucleation (adsorption) point. Indeed, the contribution of new nucleation from monomers distant by ∼ S is ∼ Q (S/ so)−3/2(so/b). Hence the time for a new distant nucleation is tn∼ Q−1S3/2b/so5/2 and is larger than the total zipping time tz provided S > so. From the above we may infer the internal contact probability distribution in the flexible regime (S > S ), P(s) = P(S ) (s/S )α, where the overall power-law dependence ∝ s−3/2 for flexible chains enforces α = −3/2. Then, the distant nucleation time is tn∼ Q−1 S1/2 S b/so5/2 and it is shorter than the zipping time tz∼ Q−1Sb/so2, provided S > S* where S* is the crossover length to the accelerated zipping:

Figure 2. Loop size distributions obtained from the molecular dynamics simulations for chains N = 30 and N = 250. The persistence length of the chains is S ≃ 10.1b for all. (a) The size distribution of the loop formed by first adsorption of a tail of length S/2 = Nb. (b) The size distribution of the smallest internal loop (s < S/2) formed by first internal adsorption of a loop of total length S = Nb. The full distribution (1 < s < S) is, by construction, symmetric with respect to S/2, P(s) = P(S − s) ∝ Zl(s)Zl(S − s). In either case, in scales less than ∼ S the loop statistics follows the predicted scaling law s−5/2 and at longer scales (s ≫ S ), statistics follows s−3/2. The crossover from stiff to flexible behavior is found at s ≈ 2S for (a) and s ≈ S for (b). The loop statistics shows deviations from the simple power law for the largest s values when the connected tail (a) or loop (b) becomes important. Typical configurations taken from simulation snapshots are shown in the panel on the right, where the green color indicates the monomers in contact with the surface.

S* ∼ S5/3/b2/3

(2)

Hence the flexible regime is split into two kinetic regimes: zipping (for S < S < S*) and accelerated zipping with multiple nucleation (for S > S*). The results for zipping time tz and nucleation time tn are summarized in Table 1. For d-DNA we Table 1. Zipping Time tz vs Time for Distant Nucleation tn for Single WLC Adsorption tn tz

nicely fit the data in both regimes for case 2(a) (loop−tail distribution) and 2(b) (loop−loop distribution). The crossover at s ∼ S is narrow in both cases. Figure 2a shows an upturn at large s originating from the short residual tail, P(s) ∝ Zl(s) Zt(S/2 − s) (see also below). Similarly the flattening of the distribution in Figure 2b near s = S/2 is imposed by symmetry as P(s) ∝ Zl(s) Zl(S−s). The latter expressions fully account for upturn/flattening. See Supporting Information for more details on molecular simulations. A stiff loop of size s typically fluctuates away from the surface up to the height zs given by S zs2/s3 ∼ 1. As this distance zs should at least reach the chain thickness b, there is a minimal loop size so ∼ S 1/3b2/3. Smaller fragments (s < s0) starting out parallel from the surface can react over their whole length. For the case of d-DNA (S ≈ 150b), so corresponds to short oligomers so∼ 5b. In the following, we will only consider longer chains obeying S > so. Let us consider the adsorption of the tail formed upon the first surface binding, say of the middle monomer, (see Figure 2). The equilibrium probability for the monomer a distance s away from the initial bond to be within the reaction radius (assumed smaller than or of order b) of the surface is P(s) ∼ 1 for s < so. For larger s, yet smaller than S/2, the probability P is proportional to Zl and reads P(s) ∼ (s/ so)−5/2. If s approaches S/2, this result must be amended to account for the short remaining tail P(s) ∼ (s /so)−5/2 ((S /2 − s)/(S /2))−1/4 , where (S/2−s) > so is assumed. Consider a loop and the probability P′(s) that a monomer at distance s from one loop-end comes within the reaction radius from the surface. Similar arguments lead to the estimate P′(s) ∼ (s/so)−5/2 ∼ P(s), for a small internal loop of

s < S (stiff)

s > S (flexible)

Q−1S3/2S −5/6b−2/3 Q−1SS −2/3b−1/3

Q−1S1/2S 1/6 b−2/3 Q−1S S −2/3b−1/3

obtain S* ∼ 4000b, which corresponds to a contour length in the micron range (b=0.3 nm). On the other hand, for synthetic polymers, S* may approach b. Values for the persistence length S , monomer size b, minimal loop size so and crossover size S* for typical cases are given in Table 2. Stiff synthetic polymers Table 2. Characteristic Lengths Relevant for Single Chain Adsorption with Various Rigidities Assuming WLC Statistics: Synthetic Polymers (SynP), Duplex-DNA (dDNA), Intermediate Filament Vimentin (IF), and F-Actin SynP d-DNA IF F-actin

S (nm)

b (nm)

so (nm)

S* (nm)

5 50 103 17 × 103

0.5 0.3 ∼10 5

∼1 ∼1.5 ∼46 ∼75

∼23 ∼1.5 × 103 ∼2 × 104 ∼3 × 106

(S ≳ 10b) and d-DNA may allow one to study the crossover at S*. Biofilaments like intermediate filaments or F-actin, if considered as WLC without intrinsic curvature and twist, may never be long enough to meet the multiple nucleation regime. The time it takes to adsorb the chain is either the zipping time or the time to bring the largest loop down, whichever is faster. According to the discussion above (see Table 1), we obtain the single chain adsorption times for chemisorption as C

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χ τasd

⎧ −1 S S for S ≪ S* ⎪ Q b S* ⎪ ∼⎨ ⎪ −1 S ⎛ S ⎞1/2 ⎜ ⎟ for S ≫ S* ⎪Q ⎩ b ⎝ S* ⎠

should be understood as the filament length in the surface per unit area. How important is the steric obstacle against adsorption of new chains in the semidilute surface regime? The typical situation is a stiff chain spanning a hole of size ξ. To reach the surface (reaction radius) the section of length ∼ ξ has to bend down by the filament diameter b. This requires an energy of ∼ S (b/ξ2)2ξ. The reduction of the partition function due to steric hindrance becomes hence relevant for S b2/ξ3 > 1. The condition corresponds to concentrations c2 > cster 2 , where we define cster 2 as

(3)

The distinct size dependences of the reaction times eq 3 could be exploited experimentally to assess S*. Adsorption from solution. The adsorbance Γ is defined as the number of monomers connected to the surface per unit area and it is larger than the strict surface concentration c2 (see Figure 1c). In the early stages, the time derivative of Γ dΓ corresponds to the global reaction rate dt ∼ Q (S /b)(so/b)csurf with csurf the concentration of free monomers at the surface. The extra factor S/b stands for the increase of adsorbance per bound polymer, and the factor so/b accounts for the extra monomers brought in the very vicinity of the surface per contact (accelerating the rate). Bringing a middle monomer of a stiff chain (s≪ S ) in the very vicinity of the repulsive surface only marginally restricts its partition function. The reason is quite generic (see Figure 1a,b), independent of the polymer statistics: for a given bulk chain configuration, a configuration in contact is obtained by translation perpendicular to the surface so that the closest monomer comes into contact. There is hence a strict equivalence of bulk configurations and configurations touching the surface by any monomer. The partition functions Z1 and Zbulk only differ by global translational degrees of freedom. According to equilibrium statistics, csurf ∼ bcbulkZ̃ 1/Zbulk, where Z̃ 1 is the partition function for a chain touching the wall by a given monomer. As the ratio of the partition functions Z̃ 1 over Z 1 is ∼b/S, the adsorbance increases with the rate dΓ ∼ Qs ocbulk . dt “Stiff ” Chain Adsorption Regime of WLC. Following OV,18 we now check whether a chain has time to flatten before other chains compete for the adsorption sites. This is obviously the case at low enough bulk concentration, which is hypothesized below. Let us estimate the adsorbance Γ after time τχads, Γ(τχads) ∼ socbulkQτχads ∼ cbulkSb/so. Most chains had time to flatten after time τχads; hence, c2(τχads) ∼ Γ(τχads). The incoming chain has, by hypothesis, time to flatten before competing chains reach the surface overlap concentration c*2 ∼ (S/b)/S2. This holds provided cbulk ≲ so/(b2S2) and in particular for cbulk < 1/(bS2). At the latter bulk concentration of 1/(bS2), foreign polymers start to intrude the sphere of diameter S spanned by a given polymer, which could be termed the overlap bulk concentration c*3 ∼ 1/(bS2) . We hence conclude that polymers adsorbed from dilute solution can freely spread without competing in the early stages. The single chain deposition mechanism is valid until the coverage reaches c*2 , which is far from saturation. We start with a scaling approach, later substantiated by a stochastic geometry description. A chain is locally zipping with a preferred loop size ∼ s0 which fluctuates over the surface up to a height ∼b. Such a loop typically allows to jump over a filament adsorbed flat on the surface.28 As adsorption proceeds above the surface overlap concentration c2*∼ 1/(bS) with random filament orientations, holes of typical size ξ develop. Well above c*2 , ξ becomes independent of the polymer length S while at c*2 , ξ ∼ S. The correlation length ξ obeys the general scaling law, ξ ∼ S f(c/c2*), where f(x) is an unknown scaling function. The requirement that ξ does not depend on S for c2 ≫ c*2 imposes the asymptotic behavior ξ ∼ 1/(c2 b). Here c2b

c 2ster ∼ b−2(S /b)−1/3

(4)

The corresponding correlation length is ξster ∼ so. The concentration cster 2 is higher than the overlap concentration c2* (remember S ≫ so) but lower than the surface concentration at saturation. In the following we first assume that above cster 2 adsorption predominantly proceeds on holes of optimal diameter dopt, measured in the direction of the actual cell cut (see Figure 3). The optimal diameter dopt will be analyzed

Figure 3. (a) Poisson lines with the line density τ = 0.5 (left) and 1 (right). 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). The spanning length d over a polygon is the distance between two consecutive intersection points a1 and a2 along a given line, as illustrated for the highlighted polygon (in dark gray) and the specific line in white. (b) Illustration of the stiff chain adsorption at late stage of adsorption.

below and turns out to only weakly depend on c2. Nonetheless adsorption is slowed down in the late stages. We will show that the optimal size cannot decrease below a certain threshold ∼ (S b3)1/4. Suboptimal adsorption is in principle possible. In practice, however, there is a size dmin below which adsorption is prohibited either by kinetics (limited by the experimental time) or by thermodynamics (the adsorption reaction becoming unfavorable). Adsorption with chain spreading becomes impossible if holes of size dmin become too rare to be connected by the same chain. Scarce remaining adsorption spots can be occupied by chains grafted either by a middle or by an end monomer. We postpone a detailed discussion of steric effects and dopt in their relation with the hole distribution to the end of this section. Let us turn to the late stage of WLC adsorption where holes of size ∼ dmin are filled by spreading chains, which almost results in the ultimate surface concentration c2,∞. In the ultimate grafting regime mostly the absorbance increases rather than the surface concentration. Following OV,18 we consider new coming filaments adsorbing on empty spots (holes), (Figure 3). The surface concentration of these free spots choles is the fast varying quantity in this regime. Other lengths like ξ, dopt are taken roughly as constants ξmin∼ 1/(c2,∞b) and dmin, respectively. Larger and larger loops are needed to span the distance between holes as c2 approaches the saturation concentration c2,∞. Let Ω(s) be the number of loops of size s D

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Macromolecules per unit area. We may interpret the missing surface concentration Δc2 = c2,∞ − c2 as the contribution from all loops of the final distribution Ω(s′) larger than the current s which are yet to form: Δc2 ∼ ∫ Ss (dmin/b)Ω(s′) d(s′/b). We obtain the loop distribution by taking the derivative with respect to s as Ω(s) = −

b2 dΔc 2 dmin ds

P(f ) = Ω(s)

(l 2)1/3

fluctuation zs < z. Hence, we get ∫ zb c(z) dz ∼ ∫ z (s′/ b b)Ω(s′) d(s′/b). Inserting the loop distribution, derivation with respect to z leads to the sought profile: c 2, ∞ (6)

The concentration profile built from irreversible adsorption of ideal WLCs is hence smoother than the one built from reversible adsorption of ideal WLCs26 decaying as ∼z−4/3. The latter decay resembles the decay predicted for both reversible2 and irreversible18 adsorption of flexible polymers swollen in a good solvent z− (3ν−1)/ν with ν≈ 0.588 the swelling exponent. For swollen chains the exponent of the concentration profile is ≈1.30≠ 4/3.29 If the stiff chains are long ((S /(c2,∞ b)2)1/3 < S < S ), the large loops developed in the late stage explore a lateral distance y ∼ s3/2/S 1/2 larger than ξmin, which makes it easier to find a hole and allows for somewhat shorter loops: s ∼ S 1/5/choles2/5. Repeating the above arguments, we obtain the loop distribution Ω(s) ∼ bS 1/2/s7/2 and the outer concentration profile c(z): 1 1 for λ > z > 2 c bz 2, ∞b

(9)

This distribution has a local maximum at the crossover value f* = dmin/(S ξmin2)1/3 and is complemented by a peak at either edge, one at f ∼ 1 contributed by the flat, early adsorbed, chains and another one at f = dmin/S contributed by chains grafted in the ultimate stage. The weight of these two peaks is nonuniversal. In contrast to adsorption at thermodynamic equilibrium there is no self-averaging of the loop distribution along a given chain. Here loops along a given chain are narrowly distributed and do not reconstruct the (wide) global loop distribution. We assumed that the loop distribution is peaked at a single value for each chain. Some recent experimental studies report dynamical heterogeneity of strongly bound polymers due to the diverse surface structure of polymer layers.30 Above we assumed that dopt is almost constant ≈dmin in the late regime. In earlier stages, we must take into account the slight decrease of dopt when adsorption proceeds. Loops formed at some surface concentration c2 and dopt may further decay at a later time when smaller holes become optimal. Already adsorbed chains are competing for the current optimal holes with new coming chains. A decay of older loops into yet smaller ones is expected if the bulk concentration c3 is small enough. In the dilute limit, considered here, early formed loops have hence decayed into very small loops corresponding to the spacing of current optimal holes underneath. It is hence fair to assume that early coming chains lay almost flat in the late stages. This is exploited in the mosaic approach below. Let us turn to the determination of the optimal hole size dopt for adsorption, introduced above. We may consider that a polymer is rigid at the scale of one hole and that its projection on the plane is straight. An array of infinite straight lines drawn randomly on a plane forms a “Poisson mosaic” (see Figure 3). The full characterization of this mosaic is a long-standing problem in mathematics.31 The parameter of the problem is the line length density per unit area c2b introduced above or the associated correlation length ξ = 1/(c2b). Note that ξ, previously defined as a scale (i.e up to a prefactor), is now quantitatively defined. The Poisson lines partition the plane into polygons with a nontrivial distribution. It was shown that thickening the lines into ribbons does not change the distribution, erased small polygons being compensated by the size reduction of larger ones. We can thus describe the adsorption pattern by the Poisson line mosaic below. For an array of Poisson lines (see Figure 3), it is easy to show that intersection points of the array with a random line, or of one line of the array with the other lines of the array, are Poisson distributed. In particular, the spanning lengths d of the new line over the existing polygons is exponential distributed according to

from the surface up to zs ∼ s 3/S . The polymer amount stored per unit area in the sublayer [0,z] is built from loops with the a

c(z) ∼

(8)

s = dmin / f

⎧ S1/2f 1/2 ⎪ for f < f* ⎪ Γbdmin 3/2 P(f ) = ⎨ ⎪ c 2, ∞ dmin for f * < f < ⎪ (Sb2)1/3 ⎩ Γf

(5)

z

Γdminb2

Specializing for the loop distribution in the outer layer and inner layer, we obtain the distribution P(f):

Holes are typically separated by a distance 1/(choles ξmin) along a given straight chain. The typical loop length spanning two optimal holes is at this stage s ∼ 1/(ξmin choles). The distance to surface saturation being defined as δΔc2, its variation ΔΔc2 is related to the actual loop size through δΔc2 ∼ δ[dmin/(bξmin s)]. Applying eq 5 leads to the loop distribution: Ω(s) ∼ b/(ξmin s2). From the loop distribution one can construct the concentration profile. A stiff loop of length s fluctuates away

c(z) ∼

s3

(7)

which crosses over to the inner profile given by eq 6 at z ∼ ξmin. The outer profile is generally expected to develop provided that S > so. The inner profile, in contrast, may have a limited S 3/2

extension b < z < ξmin. Beyond the distance λ ∼ 1/2 the S concentration profile built by stiff loops is cut exponentially 2 ∼e−(z/λ) . In the late stage, that we have just described, the fraction of bound monomers f of an adsorbing chain is directly related to the current loop size s through f = dmin/s. Let P( f) be the probability density for the fraction of bound monomers in an adsorbed chain. Each chain adsorbed with loop size s comprises S/s such loops. The number of chains Ω̃( f) (of length S), with bound fraction f, per unit area is hence linked to the loop distribution Ω(s) through Ω̃(f) = Ω(s)(S/s)−1|d(s/b) /(df)|. After normalization by the total number of adsorbed chains per unit area, Γ/S, we obtain E

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Macromolecules P(d) =

2 −2d / πξ e πξ

polymers adsorb on a Poisson mosaic with a relevant polygon distribution. As in the stiff regime, there is an optimal cell “diameter” given by eq 11 which governs the adsorption process at any stage. In the late stage, adsorption occurs in the holes of minimal “diameter” dmin. The results of OV written for Gaussian chains apply here as well with the minimum hole content ncont ensuring a surface attachment in refs 17and 18 replaced by dmin/b. Besides, there are slight changes due to the introduction of a finite persistence length. The loop distribution b becomes Ω = 2 for (s > S ), which leads to the concentration s S profile build by flexible loops:

(10)

The discussion below will mainly rely on this property. Many other aspects of the polygon distribution are still debated.32−34 The reaction rate summed over the hole distribution is dominated by holes with the optimal diameter dopt. This length dopt is obtained by minimizing α

2Sb2 d3

+

2d πξ

with α≈ 32 being a

constant and is given by dopt 4 = 3παξ Sb2

(11)

The optimal spanning length dopt only weakly depends on the surface concentration c2 as, dopt∝ c2−1/4, with ξ∼ 1/(c2 b). The steric energetic penalty for adsorption in an optimal hole Eopt ∼ (S b2/ξ3)1/4 is proportional to the quartic root of the energy penalty Eξ ∼ S b2/ξ3 for adsorbing on an average hole of spanning distance πξ/2, and hence markedly reduced. Note that dopt ≳ (S b3)1/4 and Eopt ≲ (S /b)1/4. The energetic penalty Eopt is nonetheless increasing with surface concentration as ∝ c23/4 for c2 ≪ 1/b2. This will favor the adsorption by end monomers (as compared to middle monomers) at large enough surface concentration c2/cster ≳ (log S/b)4/3, where steric 2 hindrance dominates over large scale conformational entropy. At the same high surface concentrations c2 (up to details), pairs of optimal holes within reach of one single chain become rare. If filament ends can only adsorb (locally) flat on the surface they experience a fraction of the steric penalty for middle monomer adsorption. When the optimal hole becomes small enough the adsorption rate may become exceedingly slow, which would define a practical maximum surface concentration smaller than the theoretical c2,∞ ∼ 1/b2 for reasonable experimental times. If on the contrary the filament end can adsorb in any orientation grafting does not imply steric hindrance (besides marginal graft/graft interactions) before the surface reaches to the full concentration c2,∞. In both cases, an outer grafted layer will be built. “Flexible” Chain Adsorption Regime of WLC. Irreversible adsorption of a flexible chain has been extensively addressed by OV18 focusing on the excluded volume statistics. As noted by the authors, their arguments also apply to Gaussian chains. Here, we only repeat their single chain results including the dependence on the local persistence length. As is standard for flexible chains we use the number of monomers N rather than the contour length S = Nb. When it comes to high surface concentrations close to a complete monolayer, the role of the local rigidity is analyzed in addition. For large flexible loops (s ≫ S*) the loop statistics eq 1 and the characteristic times in Table 1 show that accelerated zipping prevails. Following the reasoning of OV, we reach the conclusion that an adsorbing chain has time to flatten on the surface without competing with other chains entering from the dilute bulk solution. We hence end up with a monolayer of flattened chains which remains dilute at the scale of monomers. Indeed each polymer is a Gaussian chain composed of Nb/S persistent segments of length S and occupies a disk of radius R ∼ N Sb , which corresponds to the internal density N/R2 ∼ 1/(S b). This is precisely the overlap concentration of the persistent segments c*2 . The typical hole seen by a new incoming chain has hence a diameter S allowing for adsorption without steric penalty. According to the arguments developed above for the stiff regime, steric hindrance becomes relevant above surface concentration cster eq 4. At higher surface concentrations the 2

c(z) =

1 for b Sz

SS > z > S

(12)

The flexible loop profile eq 12 matches with the stiff loop layer eq 7 at z ∼ S .

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SUMMARY AND DISCUSSION We studied irreversible chemisorption of WLC onto a flat surface. Stiff chains zip on the surface while more flexible ones proceed by multiple distant nucleation points. Nonetheless we find that the stiff zipping is accelerated due to stiffness which entails a finite zipping step ∼s0. Furthermore, the crossover from simple zipping to multiple nucleation belongs to the flexible regime. For fairly stiff polymers like d-DNA (S = 50 nm, b = 0.3 nm) the crossover contour length is in the micron range while the minimal loop size s0 is in the nanometer range and zipping is accelerated by about a factor 30, due to stiffness. Fully developed stiff adsorption layers from dilute solution comprise four sublayers, flat polymers closest to the wall followed by a loop layer comprising two power-law regimes c(z) ∝ 1/z and c(z) ∝ 1/z2 and an outer grafted layer where late coming chains end-adsorb on the last empty adsorption spots, distant by more than a polymer radius. For overall flexible chains (S > S ), the stiff loop layer is complemented by a 1 flexible loop layer building the concentration profile c(z) = bSz for S S > z > S . Within the model, the optimal cell “diameter” dopt turns out to play an important role. A typical chemical reaction corresponds to a free energy gain of ∼20kBT per bond and can take place down to small d from the thermodynamic point of view. However, adsorption may be limited by slow kinetics, unless chain ends can react in any orientation. End-grafting in arbitrary orientation allows to saturate the surface. Even if monomer/surface reaction only occurs in locally flat configurations, the barrier against end-grafting is roughly twice smaller than for central monomer reaction which ultimately favors end-grafting over central monomer reaction despite the small fraction (2/N) of ends. As the steric energy barrier increases as ∼ c23/4, adsorption slows down exponentially as t/ ster 3/4 tster∼ e(c2/c2 ) , where tster denotes the crossover time to the regime with steric hindrance. Our study applies to WLCs with no interactions other than steric. As discussed below eq 1, excluded volume is usually irrelevant for the conformation of chains in solution and dangling loops (this is in contrast to annealed 2d polymers). The typical loop size s0 allows an incoming chain to hop over a flat, already reacted, chain. Additional type of interactions either attractive or repulsive between monomers can obviously influence the organization of polymers at the surface. A gentle F

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simulation an end-grafted chain is first equilibrated against the repulsive grafting plane before the monomer/wall attraction is turned on. The main finding is that the chain zips on the surface from the initial grafting point. Because of fast zipping a tension develops and the chain separates in two ”phases”: the corona, which is unaffected by the adsorption process in a first approximation, and the stem, a stretched section linking the corona to the already adsorbed part of the chain. When this structure is developed the corona hardly feels the surface anymore and monomers hence flow to the surface through the stem. This regime, termed central regime, is found to dominate the adsorption kinetics. The adsorption time τϕads∼ ζ N1+ν, with ν the Flory exponent, is found shorter than the Rouse relaxation time tR∼ ζ N1+2ν. The scaling argument for the central regime22 goes as follows: the stretched stem has a length similar to the initial grafted coil extension ∼ b(S/b)ν, the friction of the stretched stem is proportional to its length ∼ ζ b(S/b)ν, and the stem is pulled onto the surface by a tension ∼1/b which determines the adsorption velocity

side by side attraction between monomers may favor local orientational order. A special case arises when polymers are very stiff. If the reaction is only possible in a (locally) flat configuration and the polymer is perfectly rigid and straight, no adsorption with overlap is possible. Adsorption is then described by the Random Sequential Adsorption (RSA) process for needles35,36 which leads to a fractal structure of the flat adsorbate with fractal dimension DRSA = (1 + 4 2 )/(1 + 2 2 ) ≈ 1.74 . It would be interesting to further study this regime in relation with the very stiff chain regime S < so. We only proposed limited numerical simulations to support the analytical equilibrium distribution of loops. Full dynamical (MD) simulations of adsorption layers remain technically challenging, even for standard adsorption models. Chemisorption makes brut force simulation virtually impossible due to the large energy barrier opposing monomer adsorption. Full MD simulation seems not desirable, as polymer relaxation is not directly relevant. In future work we could further exploit simulated loop distributions as inputs to study single chain chemisorption. We restricted ourselves to adsorption from dilute solution, which is common situation. Irreversible adsorption of more concentrated solutions is equally relevant and has been partially addressed in the past.19,37 Dense solutions of long polymers are entangled. One may wonder whether entanglements are important in our system. During chemisorption it is assumed that chain fragments can fully relax their configurations. Most entanglements are transient and do relax over a topological relaxation time, which is somewhat larger than the Zimm time. Yet they do not affect the partition functions and our conclusions. It may happen that an entanglement is trapped inside a loop which further adsorbs. Such a trapped knot would affect the partition functions. From the analysis of the configurations of cycles,38 it is known that cycles much shorter than the knotting length are unknotted. For ideal cycles, the knotting length is about 300 statistical segments38 of length Sb each. Especially for stiff chains, trapped knots are unlikely. Entanglements are more relevant to physisorption of flexible chains, briefly discussed below. During physisorption (diffusion limited adsorption) each monomer/surface contact results in an irreversible binding. The single chain adsorption kinetics is obviously different from that of chemisorption. Nonetheless the large loop structure which develops close to saturation discussed above is independent of the details of the early stages. The large loop distribution and the associated concentration profiles are expected to be the same as for chemisorption. OV have reached a similar conclusion for the adsorption of flexible polymers. The early stages of physisorption involve complex hydrodynamics close to the surface, which depends on details like the surface roughness. Nonetheless the adsorption time of a single chain should be similar to its actual longest relaxation time, which is at worst marginally different from the polymer relaxation time far from the surface. The single chain adsorption times are hence expected to be τϕads ∼ η (S S )3/2 with η the viscosity of the solvent for a flexible chain and τϕads ∼ ζ⊥ S4/S for stiff chains (S < S ) with transverse friction per unit length ζ⊥ of order η. Chemisorption prevails for small chemical rates Q ensuring τχads ≫ τϕads. This simple view is challenged by numerical simulations performed for excluded volume chains without hydrodynamics, which were interpreted by the two-phase model.22 In the

dsads dt

∼

1 . b2ζ(S / b)ν 2

Accordingly the adsorption time is estimated

as τads ∼ ζb (S/b)1+ν which is consistent with the simulation. The authors also describe a last step when the friction of the remaining corona is smaller than that of the stem and it is easier to pull the corona down altogether. In their Rouse description, this happens for small coronas comprising nc monomers, (S/b)ν ∼ nc1+2ν, when chain adsorption is almost completed. Quite remarkably, the scaling description of the central regime holds true even if hydrodynamics is taken into account. The expression of the stem friction is then even more robust. Specializing to the theta solvent (RW statistics) we get τads ∼ ηN3/2 which coincides with the Zimm relaxation time. The central regime is hence not faster than the global chain relaxation. For the excluded volume chain the central regime still looks faster than the Zimm time with ν ≈ 0.588. However, in practice, the central regime never develops for both RW and SAW statistics. Indeed, the friction of the corona is proportional to its radius and reads ∼ηbnνc . The stem friction is as large as the corona friction already in early stages. If hydrodynamics is taken into account we expect the coil to adsorb as a whole rather than to follow the two-phase scenario. In a sense membranes are 2-d analogues of WLCs. It is tempting to try and generalize the chemisorption model to membranes, we intend to do so in the future. Reaction limited kinetics is by definition slower than physisorption. However, hydrodynamics is essential to membrane physisorption. The solvent flow out of an adhesive patch slows down the kinetics of physisorption. The flow also entails a disjoining pressure which might impede the nucleation of distant adhesive patches. The fastest reaction speed ensuring chemisorption is hence expected to be significantly smaller for membranes than for worm like chains.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01303. Technical details of the molecular dynamics simulation (PDF) G

DOI: 10.1021/acs.macromol.5b01303 Macromolecules XXXX, XXX, XXX−XXX

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(29) It was disputed whether ref 3 could measure the exponent 1.30 attributed to equilibrium layers. The fact that the exponent is the same for both reversible and irreversible adsorption, provided the polymer is flexible, clarifies this point. (30) Yu, C.; Granick, S. Langmuir 2014, 30, 14538−14544. (31) Miles, R. Proc. Natl. Acad. Sci. U. S. A. 1964, 52, 901. (32) Calka, P. Ph.D. Thesis, https://tel.archives-ouvertes.fr/tel00448216 2002. (33) Chiu, S. N.; Stoyan, D.; Kendall, W. S.; Mecke, J. Stochastic Geometry and Its Applications, 3rd ed.; Wiley: New York, 2013. (34) Schneider, R.; Weil, W. Stochastic and Integral Geometry; Springer: Heidelberg, Germany, 2008. (35) Tarjus, G.; Viot, P. Phys. Rev. Lett. 1991, 67, 1875. (36) Viot, P.; Tarjus, G.; Ricci, S.; Talbot, J. J. Chem. Phys. 1992, 97, 5212. (37) Guiselin, O. EPL 1992, 17, 225. (38) Frank-Kamenetskii, M. D. Phys. Rep. 1997, 288, 13.

AUTHOR INFORMATION

Corresponding Author

*(A.J.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Our work is supported by National Research Foundation grants provided by the Korean government, (NRF-2012 R1A1A3013044) and (NRF-2014R1A1A2055681). Y.J acknowledges the financial support of a National Research Foundation Grant (NRF-2012R1A1A2007488). The authors thank A. N. Semenov for critically reading the manuscript, especially the first part related to his former work.

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DOI: 10.1021/acs.macromol.5b01303 Macromolecules XXXX, XXX, XXX−XXX