Adsorption of semiflexible wormlike polymers to a bar and their double-chain complex formation

Abstract

We theoretically study pairing (double-strand complexation) of semiflexible wormlike chains (WLC) due to their side-to-side attraction. Considering binding of two WLCs of high stiffness we start with the case of infinite stiffness of one chain which is replaced with a straight bar. A combination of the quantitative transfer matrix approach with scaling arguments in terms of trains, loops of different sizes, tails and supertrains allowed us to characterize all the regimes of semiflexible chain adsorption on a bar. In particular, we predict a self-similar monomer concentration profile c(r) ∝ r^−10/3 near the bar (at distances r below the chain Kuhn length l) at the critical point for adsorption. Such localized critical profile leads to a sharp adsorption transition. Furthermore, we found that supertrains serve as the basic structural elements in WLC complexes leading to bridging, network formation and condensation of semiflexible polymers in dilute solutions. Polymer collapse (precipitation) and redissolution on increasing attraction strength are predicted in qualitative agreement with experiments on aqueous solutions of DNA and F-actin.

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1 Introduction

A semiflexible macromolecule is characterized by high stiffness of its chemical backbone: its persistence length l_p greatly exceeds its thickness d, l_p ≫ d. On the other hand, the total contour length L of the chain can be much larger than l_p. Examples of such polymers range from Kevlar and other aromatic polyamides or synthetic polymers like derivatives of cellulose to helical polymers like poly-γ-benzyl-L-glutamate, surfactant micelles, supramolecular polymers,^1,2 self-assembling peptide tapes and fibrils,³ and biological macromolecules like double-strand DNA, microtubules, F-actin and other protein filaments. High backbone rigidity facilitates adsorption of such polymers on solid surfaces and membranes.^4–8 On the other hand, side-to-side attraction of semiflexible macromolecules can drive their aggregation (mutual ‘adsorption’) leading to formation of linear double-chain or triple-chain complexes,^9,10 supramolecular fibrils and bundles.^11–16

A similar mechanism underlies formation of polyelectrolyte complexes (PECs) in binary solutions of oppositely charged macromolecules (say, A and B).^17–23 Attraction between A and B segments has electrostatic nature in this case; it can be relatively short-range if sufficient amount of salt is added. At high polymer concentrations the two polyelectrolytes (PEs) can form complex coacervate phases,^24,25 while at low concentrations the PECs emerge as finite-size (colloidal) gel particles.^26,27 For example, chitosan, which is a semiflexible polycationic biopolymer of high stiffness, forms PECs with polyanions like polysaccharides and proteins. Such complexes play important role in biological processes and pharmaceutical applications.²⁶ It was also argued that highly stiff PEs should tend to form double-strand complexes in the case of strong enough attraction (Fig. 1).^{17,18,28–31}

Fig. 1 Double-strand complexes of mutually attracting semiflexible chains: (a) two similar worm-like chains, A and B; Δ is the range of attractive interactions between A and B monomers (the potential well around the black chain B is shown with yellow color); d is the range of their hard-core interactions. (b) Two oppositely charged polyelectrolytes (r_D is the Debye screening length). (c) Two chains with complementary associative groups forming reversible bonds (see ladder segment in the middle). The effective rigidity of the chains is characterized by their Kuhn segment l (persistent length l_p = l/2); d < Δ ≪ l. The chain pairing (binding) is stabilized by contact regions (‘trains’) which alternate with ‘loops’; s is the length of a loop and r is its transverse size.

Another example of polymer complexes is provided by heteroassociative polymers bearing many functional groups (stickers) that are able to form reversible bonds (like H-bonds or metal–ligand bonds) between parallel A and B segments (Fig. 1c).³² The transient crosslinks in associative polymer systems enable stimuli-responsiveness of their viscoelastic and rheological behavior, and numerous other remarkable dynamical and processing properties serving as a basis for many industrial and biological applications.^33–35

In the present paper we consider complexation of semiflexible polymer chains using a rather generic and simple Kratky–Porod model (known also as worm-like chain, WLC, model) assuming that polymer conformation can be defined by its backbone trajectory and that the total interaction energy of two chains comes as a sum of short-range interactions of their length elements. The elements do not interact if their distance exceeds the interaction range Δ, Δ ≪ l (here l = 2l_p is Kuhn segment of the chain). Such a formally isotropic local interaction nevertheless leads to strong orientational effects due to polymer chain stiffness. The model can roughly mimic the screened Coulomb interactions of oppositely charged polyelectrolytes. It can also describe the effect of physical bonds between associating groups if the latter are sufficiently flexible, weak and numerous (many stickers per Kuhn segment).

In the next Section 2.1 we define the model in more mathematical terms and show that to a certain extent (at length-scales shorter than l) a pairwise complexation of two WLCs is equivalent to adsorption of an effective WLC on a straight bar (standing for the second WLC in the fully stretched conformation). In Sections 2.2 and 2.3 we derive the master equation for position (R̲) and orientation (t̲) dependent partition function ψ of a semiflexible chain near the bar (at r ≪ l) and obtain this function at the adsorption threshold (corresponding to the critical strength of attractive interaction, u = u*). The distribution of polymer segments in the (R̲, t̲)-space at u = u* is found as well (Section 2.3). The fractal nature of this distribution is revealed using an alternative approach applied both near the bar (Sections 3.1 and 3.2) and far from it (Section 3.3). The segment distributions both below and above u* are considered in Sections 3.4 and 3.5, respectively; the free energy of complexation is analysed in Section 3.5 as well. The main results are discussed and generalized in Section 4. We first highlight the crucial aspects of the adopted approach and the obtained results (Discussion points D1–D5), and then apply them to study the salient features of complexation between two or more semiflexible macromolecules including gelation (network formation), coil-to-globule transition and phase separation in the dilute solution regime (points D6–D9). In particular, we show that the presence of double-strand complexes can lead to a reentrant transition as solvent quality decreases (u grows): from homogeneous polymer solution to a condensed gel phase which then dissolves as the side-to-side attraction (u) gets even stronger (see points D8, D9 of the Discussion). This effect highlights the non-trivial boundary nature of the bundling phenomenon. The results are summarized in the last Section 5.

2 Theoretical methods and basic results

2.1 The model and the theoretical problem

Below we recall some basic properties of stiff WLCs and introduce a theoretical framework to consider their pairwise complexation.

The lowest energy state of a wormlike chain (WLC) is a straight line. The free energy penalty for a deviation from this conformation is due to the chain bending:

where is unit vector tangent to the chain at point s (here s is the curvilinear distance along the chain), R̲(s) is position of point s, l_p = l/2 is the chain persistence length, and T is the thermal energy (temperature in energy units). Eqn (1) corresponds to the Kratky–Porod model,³⁶ which is widely used to study statistical properties of stiff and semiflexible polymer chains.^5–7,37,38 The ideal WLC model is adopted in the present study. Note that intra-chain interactions are neglected here in most cases since for d ≪ l the intra-chain contacts are rare.

Eqn (1) implies that at equilibrium (and in the absence of any constraints)

It means that the typical bending angle of a short segment (of length s ≪ l) is θ ∼ (s/l)^1/2, and its lateral deviation from a straight line is

r ∼ θs ∼ s^3/2/l^1/2 (3)

Let us turn to the system of two WLCs, A and B, which tend to form a complex due to side-to-side attraction between A and B segments (Fig. 1a). If the attraction is sufficiently strong, the typical lateral distance r between the chains should be relatively small, r ≪ l, so that locally the two chains have to stay nearly parallel to each other (Fig. 2a). The relevant correlation length λ along the chain can be defined by the condition that the typical lateral deviation of a λ-segment (with fixed position and orientation of its starting monomer unit) is ∼r: λ ∼ r^2/3l^1/3 (see eqn (3)), hence λ ≪ l. The paired chains can be considered as a sequence of nearly straight segments of intermediate length ΔL, λ ≪ ΔL ≪ l (see Fig. 2a). In each such segment one can choose the axis z along the chains (cf.Fig. 2b), so that locally the curvilinear distance s can be replaced by z-coordinate and conformations of both chains can be described by the vector functions r̲_A(z) and r̲_B(z), where r̲_i (i = A, B) are 2-dimensional projections of position vectors onto xy plane perpendicular to z-axis. In terms of the lateral displacement vector, r̲(z) = r̲_A(z) − r̲_B(z), and the mean trajectory, r̲_m(z) = [r̲_A(z) + r̲_B(z)]/2, the bending energy of both chains reads (cf.eqn (1) and Fig. 2):

Obviously the energy of local interactions between the chains depends on the lateral distance |r̲(z)|, but is nearly independent of r̲_m(z). Hence fluctuations of r̲(z) and r̲_m(z) must be decoupled and therefore, as long as we are interested in the statistics of r̲(z), the first term in eqn (4) can be considered as irrelevant and omitted. The remaining term shows that the statistics of r̲(z) for two closely paired wormlike chains is equivalent to that for one WLC with effective Kuhn length l* = l/2 near a straight bar attracting the WLC.† This analogy is exact in the limit λ/l → 0 (that is, for r ≪ l). For a finite λ/l ≪ 1 the relative error of this approximation is about λ/l since the main relative inaccuracy of eqn (4) is proportional to θ² where θ ∼ (λ/l)^1/2 is the typical angle between the two chains (see the text above eqn (3) and Fig. 2b).

Fig. 2 The paired semiflexible chains, A (green) and B (violet) (global view (a), and local view (b)). Their mean trajectory is shown in (a) as a chain of black segments of length ΔL (only one segment of the mean chain is shown in (b) where it coincides with z-axis). l is Kuhn segment of both chains; r̲ is the local lateral displacement vector between the chains, r ≪ l; λ ∼ r^2/3l^1/3 is the relevant longitudinal correlation length; the angle θ between tangent vectors t̲_A and t̲_B is small. In (c) an effective WLC (A*, cyan) with Kuhn segment l* = l/2 is adsorbed onto a straight bar of thickness d (red) with attractive zone (yellow) of radius Δ around it. The origin of the reference frame is marked in (b) and (c) with white point and sign ‘0’.

Thus, instead of an original complex of two tightly coupled WLCs it is appropriate to consider a simpler problem of adsorption of one effective WLC (A* with effective l*) onto an infinite straight bar (Fig. 2c). Below we present a theoretical approach to obtain the polymer concentration distribution within the latter complex as well as its energy, free energy and other properties. Note that the problem of adsorption of a flexible (ideal Gaussian) chain onto an infinitely thin attractive bar is exactly solvable using the ground-state dominance approximation^39–41 and can be interpreted in terms of length distributions of loops and tails.⁴²

2.2 The transfer-matrix approach

Let us consider an ideal semirigid WLC of length L with one end (at s = 0) free and the second end (at s = L) having orientation t̲ and position R̲ = (x, y, z) near a straight bar coinciding with z-axis (here s is curvilinear distance along the chain contour). In the presence of an arbitrary external potential field U(R̲) (such that the total potential energy of the chain is ) the chain partition function is Ψ = Ψ(R̲, t̲, L). This function in normalized in such a way that ψ = 1 in the case of no field. The obvious initial condition is then

Ψ(R̲, t̲, 0) = 1

Variation of Ψ as L increases can be deduced using the transfer-matrix approach^42,43 leading to the generalized Edwards equation:^5–7,43,44where ∇_t̲² is the Laplacian in the orientational space (t̲), and ∇_R̲ is the gradient in R̲-space (here and below we drop ‘*’ after l). The potential U(R̲) is actually due to interactions between the chain and the bar. For uniform and axially isotropic bar it therefore depends only on the distance r between a short chain segment (of length ds) and z-axis: the potential energy contribution of such segment is U(r)ds. We assume that this interaction is short range: U(r) = 0 for r > Δ, where Δ is much shorter than the Kuhn segment, Δ ≪ l. An example of U(r) is shown in Fig. 3a reflecting the usual trends: attraction at r below Δ and hard-core repulsion at shorter r, r < r₀ = d/2; here the repulsion range r₀ is assumed to be comparable with Δ.

Fig. 3 (a) The polymer/bar interaction model: U(r) is interaction energy per contour length of WLC, r is the distance between a WLC element and the central axis of the bar, Δ is attraction range, d ≡ 2r₀ is hard-core diameter; the effective strength of the attraction can be characterized by u = U_m/T, U_m ≡ |min U(r)|. (b) A view along the bar axis: r̲(z) = (x, y) is projection of the monomer position vector R̲ = (x, y, z) onto the xy-plane (perpendicular to the bar axis z); φ is polar angle of r̲, x = r cos φ, y = r sin φ; t̲_⊥ is projection of the WLC tangent vector t̲ = (t_x, t_y, t_z) onto the xy-plane, t̲_⊥ = (t_x, t_y, 0); t_r and t_φ are radial and azimuthal components of t̲_⊥ in the cylindrical coordinates (r, φ), cf.eqn (7).

Eqn (5) allows to describe adsorption of the WLC on the bar. It can be rewritten using components of t̲ = (t_x, t_y, t_z) and taking into account that the system is uniform in z-direction (since U(R̲) is independent of z), so Ψ is also independent of z: Ψ(R̲, t̲, L) = ψ(r̲, t̲, L), where r̲ = (x, y) is the two-dimensional projection of R̲ on the xy plane. This leads to

where r = |r̲|. Furthermore, making use of cylindrical coordinates, r̲ = (r cos φ, r sin φ) with polar angle φ, and defining the radial (t_r) and azimuthal (t_φ) components of the tangent vector t̲ (cf.Fig. 3b),

t_r = t_x cos φ + t_y sin φ, t_φ = t_y cos φ−t_x sin φ (7)

we transform eqn (6) aswhere ψ = ψ(r, t_r, t_φ, L). Here we took into account that the system is axially symmetric hence ψ is independent of the angle φ: .

For r ≪ leqn (8) can be further simplified by taking into account that the chain must be nearly parallel to the z-axis due to its strong attraction to the bar (cf. the previous section), so that the tangent vector t̲ = (t_r, t_φ, t_z) makes a small angle with z-axis: |t_r| ≪ 1, |t_φ| ≪ 1, t_z ≃ 1. As both t_r and t_φ are small, the orientational Laplacian can be written as

As a result, the master eqn (8) becomes

Returning to eqn (8), note that as its r.h.s. does not explicitly depend on L, its general solution can be written as a sum of terms like

where ψ⁽ⁱ⁾, are eigenfunctions and eigenvalues corresponding to eqn (8). At large L the function ψ is dominated by the term with the largest corresponding to the ground-state function ψ⁽⁰⁾(r, t_r, t_φ) satisfying the eigenvalue equation (the superscript ‘0’ is omitted below):Note that eqn (12) is valid generally, and not only for r ≪ l. An advantage of our approach to consider a WLC near a straight bar is that the adopted approximations are valid for however long chains (in particular, for L ≫ l). As mentioned above the partition function of a long enough chain located near the bar is dominated by the ground state contribution (the ground-state dominance approach³⁹):At low attraction strength, u = |min U(r)|/T (see Fig. 3a), ψ does not grow with L, so . In this case ψ is close to the partition function of a free chain, ψ_free, which is constant independent of L, as follows from eqn (5) with U(R̲) ≡ 0. On the other hand, if u is sufficiently large, u > u*, the ground-state eigenvalue is positive, meaning that a long chain is trapped near the bar (i.e. that the relevant solution of eqn (12) is localized, cf. Section 3.5) and its free energy per length,is negative. Thus, the chain is always adsorbed on the bar for and the critical adsorption point (separating delocalized and adsorbed chain regimes) must correspond to u = u* where vanishes first as u is decreased from the regime of strong attraction.

2.3 Adsorption of WLC on a bar at : quantitative approach for r ≪ l

As mentioned above the chain adsorption (localization near the bar) is driven by its short-range attraction to the bar of strength u (u = |minU(r)|/T, cf.Fig. 3a). Let us first focus on the critical regime (with , cf.eqn (12) and (14)) corresponding to the attraction threshold for the adsorption, u = u*. For r > Δ the interaction potential U(r) just vanishes, so eqn (12) defining the ground-state eigenfunction ψ(r, t_r, t_φ) can be simplified asOn using eqn (9) for r ≪ l we arrive at‡Note that the only parameter entering eqn (16) is l, so all r-dependencies should scale with r/l. Benefiting from an analogy between adsorption onto a bar and on a flat surface,^6,7 it is natural to assume a self-similar dependence of ψ on the distance r for l ≫ r ≫ Δ:

ψ(r, t_r, t_φ) = (r/l)^αg(η, ξ), (17)

where

η ≡  t_r/Θ(r/l), ξ = t_φ/Θ(r/l) (18)

Here the exponent α is a constant to be determined, g(η, ξ) is a function to be found, and an appropriate function Θ(r/l) can be deduced from the structure of eqn (16) leading to:

Θ(r/l) = (r/l)^1/3 (19)

Eqn (19) is in agreement with the results of ref. 6 and 7. Using eqn (16), (17) and (19) we derive the following equation for g(η, ξ):To solve this equation we need boundary conditions. As discussed below (cf. Section 3.5) in the adsorbed state (u > u*) most segments are located close to the bar, being nearly parallel to it. If a segment of length λ ≪ l bends away from the bar, its typical deviation r from the bar and the angle θ its end makes with the bar can be estimated as r ∼ λ^3/2/l^1/2, θ ∼ (λ/l)^1/2 ≪ 1 (see Section 2.1, eqn (3)). Excluding λ from these 2 equations we arrive again at eqn (19) suggesting that chain segments near the bar are aligned almost parallel to it (since Θ = 0 and hence t_r = t_φ = 0 at r/l = 0; note that here we consider the limit Δ/l → 0). A much larger angle at a distance r, θ ≫ Θ(r/l), would require a strong bending of the λ-segment leading to a high elastic energy penalty and, therefore, low probability of such a large angle. Hence, the function g(η, ξ) is expected to strongly decrease at large :§

g(η, ξ) → 0 at ϑ → ∞ (21)

Eqn (20) (with boundary condition, eqn (21)) could possibly be solved analytically in terms of generalized multivariable hypergeometric functions. However, we have got an impression that mathematical theory of such functions is not yet sufficiently developed.⁴⁵ So, we opted to solve eqn (20) numerically using a relaxation method. The relaxation equation associated with eqn (20) iswhere t is an additional variable (fictitious time). At sufficiently long t the ground mode wins and the t-dependence readily becomes exponential, (compare this behavior with eqn (13)). We found that for α₀ > α > α₁, while and therefore eqn (20) is satisfied at α = α₀ and α = α₁. The numerically obtained critical exponents are:

α₀ = 0 ± 5 × 10⁻⁵, α₁ = −2 ± 5 × 10⁻⁴ (23)

What about the dependence of g on angular variables η and ξ in the critical states (α₀, α₁)? For α = 0 the distribution is simply constant, g(η, ξ) = const. However, it is non-trivial (and localized around the origin) for α ≈ −2. The typical profiles of g are shown in Fig. 4 (η-profiles (a) and ξ-profiles (b) for α = −2.0001). Note that ξ-profiles are always symmetric, g(η, ξ) = g(η, −ξ), since the system is achiral. By contrast, the η-profiles are somewhat shifted towards positive η reflecting the fact that conformations (and statistical weights ψ, cf.eqn (17)) with positive and negative t_r are not equivalent: for a given r > Δ it is more favorable to have t_r > 0 since in this case the end fragment of the chain (with s < L near s = L) is closer to the attractive bar thus increasing the probability of contacts with it.

Fig. 4 The distribution of reduced angles η = t_r/Θ, ξ = t_φ/Θ corresponding to the components (t_r, t_φ) of the chain-end tangent vector t̲ in the critical regime (cf.eqn (16)–(20)): (a) g(η, ξ) vs. η for ξ = − 1.6 (red), 0 (black), 1.2 (blue); (b) g(η, ξ) vs. ξ for η = −3.3 (green), −1.6 (red), 0 (black), 1.2 (blue), 2.9 (magenta).

For large ϑ the function g(η, ξ) can be obtained analytically by a simple asymptotic analysis of eqn (20). Indeed, let us assume that g(η, ξ) ≃ g̃(ϑ) ≡ ϑ^−x for ϑ ≫ 1, cf.eqn (21). Then the first term in , eqn (20), is negligible (≃ϑ^−x−2) with respect to the second term (≃ϑ^−x+1), while the third term is exactly zero for g = g̃(ϑ). The remaining second and the forth terms in dictate that x = − 3α. Hence the result is

g(η, ξ) ∝ ϑ^3α, ϑ ≫ 1 (24)

Interestingly, a similar behavior (but with different exponents α) was obtained for WLCs adsorbed onto a permeable membrane.⁷

The monomer density distribution, c(r̲, t̲), in the full space of monomer position R̲ and orientation t̲ is^6,7,42,43

c(R̲, t̲) = c(r, t_r, t_φ) = const ψ(r, t_r, t_φ)ψ^†(r, t_r, t_φ), (25)

where ψ^†(r, t_r, t_φ) = ψ(r, −t_r, t_φ) is the conjugated function. The above eqn (25) is valid in the ground-state dominance (GSD) regime. Note that both ψ and c depend on just one spacial variable r (i.e. these functions are independent of both z and φ) since the system is axially symmetric and uniform along the z-axis, cf.Fig. 2c. Using eqn (17) we get

c(r, t_r, t_φ) = const r^2αg(η, ξ)g(−η, ξ) (26)

For large reduced angles, ϑ ≫ 1 (cf.eqn (21), i.e. for t_⊥ ≫ Θ(r/l)), cf.Fig. 3b, we find from eqn (26) and (24) that (somewhat surprisingly) the c-function does not depend on r:

c(r, t_r, t_φ) ∝ (t_r² + t_φ²)^3α, 1 ≫ t_r² + t_φ² ≫ (r/l)^2/3 (27)

The significance and the physical meaning of these results are discussed in the next section.

3 Critical regimes of WLC adsorption on a bar

3.1 General concepts

As we mentioned before, the present problem of polymer adsorption on a bar is rather similar to single-chain adsorption on a flat surface. The latter phenomenon is well-studied.^{6–8,46–48} For polymer adsorption (of both flexible and semirigid chains) one can distinguish two basic regimes with algebraic dependence of polymer concentration c on the distance x to the surface, c(x) ∝ x^−β: (i) a non-adsorbed state with β = β₀ ≤ 0 (corresponding to the case of a purely repulsive surface) and (ii) critically adsorbed state, β = β₁ ≥ 0. The number of contacts with the surface for a chain of N units located within its own size R ∝ N^ν from the surface is n_c ∝ N^ϕ, where ϕ is the so-called crossover exponent^46–48 related to β: ϕ = 1 − ν(1 − β) if β < 1. For example, adsorption of ideal flexible chains with ν = 0.5 (say, from a theta-solvent) is characterized by the critically adsorbed state with β₁ = 0, ϕ₁ = 0.5, and the non-adsorbed state (for a repulsive wall) with β₀ = −2, ϕ₀ = −0.5 (no polymer contacts with the surface).

Returning to adsorption of a long WLC onto a bar, the general scaling picture described above applies also in this case (with a trivial replacement of distance x with r). It is then natural and tempting to associate the exponents α₀ and α₁ (cf.eqn (23)) with the non-adsorbed (u = 0) and the critically adsorbed (u = u*) states, respectively, and to check if the scaling c∝r^−β holds. To obtain c(r), eqn (26) must be integrated over the ‘angles’ t_r and t_φ. For α = α₀ = 0 the result is c(r) ∝r^α = const, hence β₀ = 0. (Note that the angular factor g(η, ξ) for α = 0 is not only constant asymptotically, but it is also identically constant: g = const is the exact solution of eqn (20) and (15) in this case.)¶ By contrast, the angular integration in eqn (26) for α = α₁ = −2 provides an important factor

(cf.eqn (18) and (19)) leading to the critical exponent β₁ = 10/3:

c(r) ∝ r^2α+2/3 = r^−10/3 (29)

Below we present an alternative derivation of the above law based on a scaling argument.

3.2 Self-similar structure of loops in the proximal layer at u = u*

To begin with let us estimate the attraction strength threshold u* for the critical adsorption of a long WLC (L ≫ l). To fully benefit from attraction to the bar (cf.Fig. 3a) a chain segment of length λ should be located in the attraction region r < Δ. The entropic cost for such confinement is ∼1 per segment of length λ* whose lateral displacement r is small, r ≲ Δ. Using eqn (3) we get

λ* ∼ (Δ²l)^1/3 (30)

The energy gain (in units of thermal energy T) due to attraction of a λ*-segment to the bar is uλ*, so the attraction gain dominates the entropy loss if uλ* ≳ 1. Hence, the WLC adsorption is favorable for

u > u* ∼ (Δ²l)^−1/3 (31)

An analogous scaling argument was used before to obtain a similar estimate of u* for adsorption on a flat solid surface or a membrane.^5–7,49

The typical angle θ* between the tangent vector of an adsorbed λ*-segment of the WLC and the bar is small:

θ* ∼ (λ*/l)^1/2 ∼ (Δ/l)^1/3 (32)

Such a segment is called a ‘train’ below (cf.Fig. 5).

Fig. 5 Polymer chain adsorbed onto a bar: trains, loops and tails. Train is a chain section located near the bar (at r ≲ Δ) which is locally directed along the bar (θ ≲ θ* ∼ (Δ/l)^1/3); loop is a chain fragment between two trains; 0-tail is the chain section between the left end and the first train, L-tail is the section between the last train and the right end.

Considering the critical point u = u*, we note that the free energies of adsorbed and non-adsorbed chains must be balanced since at this point. Therefore, the typical conformation of a long critically adsorbed chain should include ‘loops’ of different sizes in addition to ‘trains’ (see Fig. 5). Note that the attraction energy of one train is E_train ≃Tu*λ* ∼ T. The partition function of a loop of length s ≫ λ* is Z_l(s). To find it we recall that the lateral size of such a loop is (cf.eqn (3))

r(s) ∼ s^3/2/l^1/2, (33)

so the fraction f_Δ of its segments in the Δ-region is negligible, f_Δ ∼ Δ²/r(s)² ∼ (λ*/s)³ ≪ 1 and f_Δs ≪ λ*, so its interaction energy E with the bar is small, E ≪ E_train ∼ T. Therefore, the loop can be considered as a virtually free fragment with the only condition that its second end must be near the bar and oriented nearly parallel to it (within the angle ∼θ*). The corresponding probability, p_loop ∼ (Δ/r(s))²(θ*/θ(s))², where θ(s) ∼ (s/l)^1/2, provides the partition function of a loop of length s:

Z_l(s) = p_loop ∼ (λ*/s)⁴ (34)

The number N(s) of such loops with length equal exactly to s is proportional to Z_l(s) since any chain fragment can form a loop with probability p_loop. Hence the number of monomer units in all loops of size ∼s (say, between s/2 and s) is . Most of these monomers are located at the distance ∼r(s) from the bar, the corresponding area in the transverse (xy) plane is A(s) ∼ r(s)². Therefore, polymer concentration at r ∼ r(s) is . Recalling eqn (33) we thus get

c(r) ∝ r^−10/3 for  l ≫ r ≫ Δ (35)

in agreement with eqn (29). This way we not only justify the transfer matrix approach (Sections 2.2 and 2.3), but also show that the critical exponent α = α₁ (cf.eqn (23)) must be exactly equal to −2. A similar argument to get the critical exponent was proposed for a WLC near an attractive membrane.^6,7 ||

It is natural to expect that c(r) is roughly constant at r₀ < r ∼ Δ. Therefore, a significant fraction p(Δ,l) of polymer (in the region r ≲ l where the chain stiffness does matter)

is concentrated in the Δ-region near the bar: p(Δ,l) ∼ 1/2 since is dominated by the contribution from r ∼ Δ. The above argument shows that at the critical adsorption point, u = u*, the adsorbed structure in the proximal region Δ < r < l can be viewed as a system of loops with fractal size distribution (cf.eqn (34) and (35)). A similar conclusion was drawn for adsorption of WLCs onto membranes.^6,7

3.3 Distal layer

Let us turn to the adsorption regime above the critical point, u ≥ u*, where . At r ≫ l a long WLC (with L ≫ l) can form a distal layer in addition to the proximal (cylindrical) layer at l ≫ r > Δ studied in the previous section. In the distal zone (r ≫ l) the chain rigidity does not matter and, hence, the angular dependence of the partition function ψ can be neglected: ψ(r̲, t̲) ≃ψ(r̲), where ψ(r̲) satisfies (in the GSD approximation valid for L → ∞)** the classical equation for ideal flexible polymers:^6,7Here ∇_r̲² is Laplacian in the xy plane, r̲ = (x, y). For axially symmetric ground state ψ = ψ(r) and . Hence for the only solution of eqn (37) that does not increase exponentially at r → ∞ iswhere h is the terminal size of the layer, and K₀ is the MacDonald function. Using K₀(z) ≃ ln(1/z) at z ≪ 1, we find for r/h ≪ 1:

ψ(r) ≃ const ln(h/r), h ≫ r ≫ l (39)

At the adsorption threshold u − u* = 0⁺ the adsorption free energy vanishes formally implying that h → ∞. To avoid this singularity (which simply means that at u ≤ u* the chain can go away from the bar) we restrict the chain conformational space by demanding that the chain end is located not farther than the coil size, R_coil, from the bar: r < R_coil (here for L ≫ l). This condition implies that if h is larger than R_coil, it must be replaced with R_coil in eqn (39).

Let us now consider the subcritical regime, u < u*. As discussed in point D2 of Section 4, at u = 0 the function ψ = ψ(r̲, t̲) is nearly independent of both position r̲ and orientation t̲ in the distal region, r ≫ l (i.e. the presence of the bar does not affect ψ in this region). On using eqn (25) we find that the same is true for the concentration distribution:

c(r̲, t̲) = c(r, t_r, t_φ) ≃ const, r ≫ l, u = 0 (40)

We are now ready to describe the functions ψ(r̲, t̲) and c(r̲, t̲) for both critically adsorbed (u = u*, ψ = ψ₁ with α = α₁, see eqn (23)) and non-adsorbed (u = 0, ψ = ψ₀ with α = α₀) states. Using eqn (24), (17) and (39) with α = α₁ = −2 and assuming a smooth crossover between proximal and distal regimes (see point D4 of Section 4 for a more detailed discussion) we get ψ = ψ₁ at u = u*:

where h = R_coil and is the angle between the tangent vector t̲ (of the chain end) and z-axis. ††

The position-dependent partition function ψ(r̲) results from orientational averaging of ψ(r̲, t̲):

where d²t is the steric angle element. Using eqn (41) we find

ψ₁(r) ∼ (l/r)^4/3, Δ ≲ r ≪ l (43)

in the proximal region, and

ψ₁(r) ∼ ln(h/r) for h ≫ r ≳ l (44)

in the distal region (see also Discussion point D4).

Turning to the distribution c(r̲, t̲) at u = u*, it is approximately given by c(r̲, t̲) ∼ const ψ(r̲, t̲)² (cf.eqn (25) and (41) and note an agreement of the resultant c(r̲, t̲) with eqn (27) for the proximal regime). Integrating c(r̲, t̲) over orientations we get (using eqn (41))

where c₀ is a constant. Eqn (45) is in agreement with eqn (29) for α = −2 (see also Discussion point D3).

The functions ψ = ψ₀(r̲, t̲) and c(r) for the non-trapped case, u = 0, α = α₀ = 0, can be obtained in a similar way. Using eqn (24), (17) and (40) we get:

ψ₀(r̲, t̲) ≃ 1, c(r) ∼ c₀, Δ ≪ r ≲ h (46)

in agreement with conclusions drawn above eqn (28) (see also point D2 of Section 4).

3.4 Concentration profile below the critical point, at 0 < u < u*

The natural next question is how the functions ψ and c evolve as u is decreased from u* to the athermal point u = 0, that is, for any τ, 0 < τ < 1, where

τ ≡ (u* − u)/u* (47)

measures the deviation from the critical point. Here it is important to note that both limiting functions, ψ = ψ₀ (for u = 0) and ψ = ψ₁ (for u = u*), satisfy the same master eqn (15) outside the bar, in the region r > Δ where U(r) ≡ 0. Although, strictly speaking, the general solution of eqn (15) in this region is a linear combination of an infinite spectrum of independent ψ-functions, the two functions, ψ₀ and ψ₁, play a special role below the critical point (u < u*) as stated in ref. 6: the actual solution, ψ, of eqn (15) in this regime can be approximated as a linear combination of the two basic functions, where C = C(τ) is defined by the effective boundary condition at r = Δ (i.e. at the boundary of the region where eqn (15) applies) which, in turn, is defined by the potential U(r) in the interaction zone r < Δ. ‡‡ It is natural to assume that C is linear in the potential strength u, and therefore is proportional to the deviation τ from the critical value:

C ≃ kτ, τ ≪  1

Thuswhereψ₁(r̲, t̲) is partition function for critical adsorption, and ψ₀(r̲, t̲) ≃ 1 is the contribution of non-adsorbed chain conformations which do not involve any trains, cf.eqn (46). (As argued in point D2 of Section 4, ψ₀ is not affected by the presence of the bar under weak conditions specified there.) The parameter k is important; it is calculated using a different approach (which also validates eqn (48) at 0 < τ ≤ 1) in point D5 of Section 4: k ∼ (l/Δ)². From eqn (48) it is then clear that ψ ≃ψ₁ at low τ ≪ τ_c where

τ_c ≡ 1/k ∼ (Δ/l)² (50)

At τ ∼ τ_c the function ψ is roughly an arithmetic average of ψ₀ and ψ₁ and therefore ψ is dominated by the critical contribution ψ₁ since ψ₁ ≫ ψ₀ at r ≪ h (cp. eqn (41) and (46)). A similar domination applies to the concentration profile c(r) at τ ≲ τ_c. Thus, in this regime both ψ(r,t) and c(r) are approximately defined by eqn (41) and (45) and the critical self-similar structure of loops for r ≪ l (described in Section 3.2) stays intact.

For τ ≫ τ_c the end-partition function is isotropic for r ≳ Λ ≡ τ^−1/2Δ (note that Λ ∼ l(τ_c/τ)^1/2 ≪ l in this regime):

as follows from eqn (41), (46), (48) and (49). (Note that h = R_coil for u ≤ u*). The behavior of ψ at shorter distances, Δ ≪ r ≪ Λ, is described below based on eqn (48), (41) and (46):

The monomer density distribution in the (r̲, t̲) space is c(r̲, t̲) ∼ const ψ(r̲, t̲)² (see text above eqn (45)). Integrating over orientations t̲ we get the concentration profile (a constant common prefactor is omitted):

where h = R_coil. Obviously, the total concentration profile reflects the adsorbed self-similar structure at short r (cf.eqn (29)) and isotropic non-adsorbed virtually free state (corresponding to the last term in eqn (53)) at larger distances (cf.eqn (46)). The polymer concentration strongly increases at short r. The proximal fractal region of aligned loops is preserved, but gets thinner at τ ≫ τ_c: c(r) ∝ r^−10/3 is valid only for Δ ≪ r ≪ l(τ_c/τ)^3/5 ≪ Λ, where Λ = τ^−1/2Δ ≪ l.

Eqn (53) also shows that for τ ≲ τ_c = (Δ/l)² the polymer mass in a cylindrical region of radius r* ∼ l(l/Δ)^2/3 well-exceeding l, r* ≫ l, is dominated by the adsorbed segments located very close to the bar, at r ≲ Δ.

3.5 Concentration profile above the adsorption threshold, u > u* (τ < 0)

In the adsorption regime, u > u* (where τ < 0) the chain gets trapped in a region r ≲ h near the bar. The localization length h depends on u but is independent of the chain length L if it is long enough (R_coil ≫ h). The cut-off length h is related to the adsorption free energy per length, , which is positive for u > u*. In the regime of long chains, , considered below the WLC is really trapped near the bar, so the chain can be considered basically as a sequence of loops and trains (cf.Fig. 5). While at u = u* loops of any size are allowed, at u > u* a loop of length s costs the free energy which is the penalty for desorption of an s-segment from the bar (cf. ref. 6, 7 and eqn (14)), so the statistical weight of a loop, eqn (34), is reduced by the Boltzmann factor suppressing the loops of length . The cut-off length can be estimated as the lateral size r_l of the largest loops of length (the size of its projection onto the xy plane). This size depends on s according to eqn (33) at s ≲ l, and shows the standard Gaussian chain behavior in the semiflexible regime, s ≫ l:Thereforewhere the numerical factors are chosen for consistency with more precise results considered below (cf.eqn (56) and (57)).

For (moderate adsorption with s_m ≫ l) both h and ψ(r) can be obtained in a different way. First we note that ψ(r) at r ≲ l is not affected by such small since self-similar structure in this region consists of short loops with s ≲ l for which the additional factor is close to 1. Therefore ψ(r) is given by eqn (43) for r ≲ l. Furthermore, ψ(r) for r ≫ l can be found using eqn (37) whose solution is given in eqn (38), hence h for is indeed defined by the second line in eqn (55). Using asymptotic behavior of K₀ in eqn (38) we get

For stronger adsorption, , the distal layer (r ≳ l) essentially disappears (it gets exponentially weak), while the proximal cylindrical layer becomes reduced due to suppression of aligned loops with length . The resultant ψ(r) in this regime is given by eqn (43) with the exponential cut-off factor established in ref. 6 and 7

where h is defined by the first line in eqn (55), h ≪ l. The argument of the exponent here comes from a balance between the elastic deformation energy of a half-loop and its desorption free energy (cf.eqn (14)).^6,7

The concentration profiles of polymer segments can be obtained based on eqn (38) and (57) using also eqn (25) and noting that at r ≪ l the orientational (t̲) dependence of the function ψ(r̲, t̲) is concentrated within a small steric angle ∼Θ(r/l)² = (r/l)^2/3 (and that the tail of orientational distribution of ψ at θ ≫ Θ(r/l) is unimportant for both ψ(r) and c(r)). The result for is (cf.eqn (45)):

An alternative way to obtain the concentration profile c(r) is presented in the Discussion point D3 (cf.eqn (80)). For it reads

c(r) ∼ c₀(l/r)^10/3 exp(−2(r/h)^1/2), Δ < r ≲ h (59)

In all the cases the distributions are suppressed beyond the cut-off length h, eqn (55).

It is of interest to obtain the fraction p of adsorbed segments (trains), which is equal to the probability that a segment of the adsorbed chain is located at r < Δ:

(cf.eqn (36)). Using eqn (58) and (55) for we get
where

It remains to establish how the free energy (and h) depend on the adsorption strength u, u > u*. The following relation coming from the theorem on small variations⁵⁰ is useful here (recall that τ ≡ 1 − u/u*, τ < 0 here):

where is the total energy of polymer/bar interactions for a given conformation (here c(r̲) is normalized in such a way that c(r̲)d²r is the total length of chain segments inside the area element d²r), 〈·〉 means conformational average (for equilibrium distribution), and is the adsorption free energy (cf.eqn (14)). Taking into account that ,
and using eqn (63) we obtain (cf. ref. 6 and 7)where p is defined in eqn (60). Let us consider first the regime . In this case eqn (64) and (61) lead towhere C is a numerical constant depending on the shape of the attraction potential U(r), cf.Fig. 3a (C is related to the numerical factors omitted in eqn (64) and (61)). It is important to remind here that all the above equations are valid for . If , the system is nearly critical, hence, in particular, c(r) is approximately given by eqn (45) with h = R_coil (valid for u = u*), leading (after integration in eqn (60)) to p ∼ (l/L)(l/Δ)^4/3 ≡ p₀ in formal agreement with the second line in eqn (61) for . Thusand p is defined by eqn (61) for larger . Then the second term in square brackets of eqn (65) is never important unless L is exponentially large which must not be the case (as discussed in point D2 of Section 4).

Using eqn (64) and (66) with ‘initial’ condition at τ = 0 we get for |τ| ∼ τ_c. Applying then the differential eqn (65) in the region τ < 0, |τ| ≳ τ_c we get

where (cf.eqn (50)) and τ₁ is defined by the condition (cf.eqn (62)):

τ₁ ≃τ_c ln(τ_cL/λ*) (68)

For (i.e. for |τ| ≳ τ₁) the polymer is mostly concentrated near the bar, so eqn (64) and (61) give p ∼ 1 and(recall that 1/λ* ∼ u*). Thus, the adsorbed chain partition function (cf.eqn (13))§§ changes in the region u > u* (τ < 0) asNote double exponential increase of ψ at low |τ| and its further strong exponential increase beyond τ₁.

Now, using eqn (67), (69) and (55) we get the dependence of the terminal length, h = h(τ), for τ < 0:

where

τ* = (Δ/l)^2/3 ∼ λ*/l (72)

corresponds to the crossover, (cf.eqn (69)). Eqn (71) says that the cut-off length h is decreasing with |τ| (i.e. with u) above the critical point u*. Exponential decrease of h from R_coil to l(l/Δ)^2/3 is predicted for τ_c ≲ |τ| < τ₁, followed by power-law decreases from l(l/Δ)^2/3 to l (for τ₁ < |τ| < τ*) and from l to Δ for τ* < |τ| < 1. Therefore, the distal layer gets totally suppressed (h ≲ l) at |τ| > τ*. The concentration profile, c(r), is defined in eqn (58) for |τ| < τ* and eqn (59) for 1 > |τ| > τ*.

The τ-dependence of the fraction p of trains can be found based on eqn (61), (66), (67) and (69):

where

p₀ = (l/L)(l/Δ)^4/3 (74)

To obtain the last line in the above equation we assumed (as before) that the chain is confined within the coil size from the bar and used the relation p ∼ 〈m〉λ*/L, where 〈m〉 is the mean number of train segments obtained in the next section (see eqn (101)) and λ* is the mean train length (thus, 〈m〉 λ* is the mean total length of all segments in contact with the bar).

4 Discussion

D1. Critical adsorption threshold u*

In the previous sections we considered adsorption of an ideal stiff semiflexible macromolecule (WLC of diameter d₁) onto a straight bar (of diameter d₂) assuming that the Kuhn segment l = l₁ of the WLC is much larger than the range Δ of their attractive interactions, l ≫ Δ. Just for simplicity we also assumed that the repulsion range d = d₁ + d₂ is comparable with Δ.

Generally, to compensate for the hard-core repulsion from the bar (at r < d ∼ Δ) an attractive potential of magnitude u_i ∼ 1/Δ may seems to be needed if the typical angle θ between the bar and the interacting polymer segment is θ ∼ 1 (cf.Fig. 3a).¶¶ However, the actual critical adsorption threshold u* is much lower, u* ≪ u_i (cf.eqn (31)) meaning that isotropic attraction (at θ ∼ 1) can be neglected. The reason is that at the critical point the hard-core repulsion is balanced by a strongly anisotropic attraction with typical θ ∼ (Δ/l)^1/3 ≪ 1 (cf.eqn (19) with r ∼ Δ) leading to u* ∼ θ/Δ ∼ 1/λ*. It is due to such a balance between repulsion and attraction that both ψ(r̲, t̲) and c(r̲, t̲) become self-similar at the critical point in the proximal region Δ ≪ r ≪ l.

D2. Athermal case of purely repulsive bar, u = 0

In Section 3.3 we found that at the adsorption threshold, u = u*, the ψ-function (providing the chain-end distribution) follows the logarithmic law far from the bar, r ≫ l. Let us consider the case of no attraction, u = 0. As long as we are interested in large distances, r ≫ l, the WLC can be considered as ‘flexible’, that is, as a chain of N = L/l ≫ 1 Kuhn segments with one end located at a distance r ≳ l from the bar. Let us estimate the upper bound of the chain free energy increment, ΔF, due to repulsion interactions with the bar. Then ΔF is proportional to the mean number of contacts, n_c, between chain segments and the bar: ΔF ∼ Tn_c (by a contact we mean that an l-segment comes close to the bar within a distance ∼Δ). Obviously,where p(n) is the probability that the n-th segment (counted from the fixed end) encounters a contact. Here p(n) can be factorized as p(n) ∼ p_l(n)Δ/l, where p_l(n) is the probability that the n-th l-segment gets within a distance ∼l to the bar, and Δ/l is the conditional probability that in this case the minimal distance between the segment and the bar is ∼Δ. By virtue of the Gaussian statistics of an ideal chain in 2 dimensions p_l(n) is negligible if nl² ≪ r², and for nl² ≳ r² (i.e. n ≳ n₀ ∼ r²/l²) the probability p_l(n) ∼ 1/n. Thus we get using eqn (75):Recalling that the partition function ψ = e^−ΔF/T, we get
where C is numerical constant. Since Δ/l ≪ 1 and r is not exponentially large, the exponent above can be expanded leading to ψ = ψ₀,

ψ₀(r) ∼ 1 + 2C(Δ/l)ln(r/R_coil), l ≪ r ≲ R_coil (77)

Eqn (76) implies that n_c ≪ 1, so that the partition function of a semiflexible loop is not really affected by interactions with the bar. In other words, long polymer loops around the bar show a nearly ideal statistics.

Note that the first term in the above equation corresponds to the partition function of a free chain with no interactions with the bar, ψ_free(r) = 1, while the second term reflects rare contacts with the bar. As Δ/l ≪ 1 is our main small parameter, the second term is small and can be neglected unless R_coil is exponentially large. The condition is: ln(R_coil/l) ≪ l/Δ. To simplify the theory we choose to assume that the chain length is not exponentially large, i.e. that the above condition is satisfied. Hence ψ = ψ₀(r) ≃ 1 for u = 0 at r ≫ Δ.

D3. Fractal structure of weakly to moderately adsorbed chains

In Section 3.2 we considered the fractal distribution of loops in the proximal layer at the critical point, u = u*. Below we show that a similar concept can be used to obtain the concentration profile in the distal region, and to study the statistics of trains.

Let us consider the system just slightly above the adsorption point (u = u*) so that the reduced adsorption free energy is positive and small . In this case the proximal adsorbed structure (at r ≲ l) is virtually the same as at the critical point: it involves trains close to the bar (at r ≲ Δ) and aligned loops (of length s, Δ ≲ s ≲ l) with self-similar length distribution:

N_p(s) ∝ s⁻⁴ (78)

where N_p(s) is the number of aligned loops of length s (cf.eqn (34)). The distal region r ≳ l involves longer loops with s ≳ l. Their length distribution N_d(s) is also self-similar (for l ≪ s ≪ 1/ε), but with a different exponent reflecting their flexible nature:for s ≫ l. Here is free energy penalty for desorption of a loop of length s (see text above eqn (54)). The factor 1/s in eqn (79) reflects the probability that a 2D Gaussian chain of s units forms a loop.

Matching the two distributions at s ∼ l, N_p(l) ∼ N_d(l), and applying the argument of Section 3.2 to obtain the polymer concentration profile c(r) also in the distal region (r ≳ l) based on N_p(s), N_d(s) we get

where is the cut-off length (the terminal lateral size of the adsorbed chain), h ≫ l. Note that eqn (80) are in agreement with eqn (58).

Both short and long loops must have both ends adjacent to train sections which serve as starting and end points for loops. Eqn (78) shows that most loops are short. The typical size of the shortest loops and trains is s ∼ λ* (cf.eqn (30) and the text below eqn (32)). The trains connected by short aligned loops (of length s < l) can be combined together to get a supertrain whose ends are always adjacent with long loops (with s > l). On the other hand, each long loop is located between two supertrains along the chain contour. Thus, a supertrain is a continuous chain section oriented as a whole along the bar and located close to it (in the proximal layer).

Let us estimate the mean contour length, L_st, of one supertrain and their number, N_st. Obviously, the total number of long loops, N_ll, must be close to N_st (strictly, N_st = N_ll + 1). The terminal length of a long loop is s_m ∼ h²/l ∼ 1/ε. Furthermore, eqn (79) shows that

Let L(r₁, r₂) be the total polymer length in the region r₁ < r < r₂. Each long loop of length s > 2l contributes a length ∼ l ln(s/l) in the near-proximal region l < r < 2l (the log-factor here comes from returns of internal l-segments of the loop into the near-proximal zone), henceUsing eqn (79), (81) and (82) we get

L(l,2l) ∼ l ln(h/l)N_ll (83)

As short loops (with s ∼ Δ) and trains dominate by mass in the proximal region, L(0, Δ) must be close to N_stL_st ≃ N_llL_st: L(0, Δ) ∼ N_llL_st. The above relations allow us to estimate L_st:Using eqn (84) and (80) we then get the supertrain length:

L_st ∼ l(l/Δ)^4/3/ln(h/l) (85)

As follows from the scaling law, eqn (29), the masses of trains and supertrains are nearly the same (in particular, close to the critical point), hence the supertrain mass fraction is ∼p (see eqn (60) for the definition of p). At a very low ( that is for 0 < u/u* − 1 ≲ τ₁) it is given by the second eqn (61):The number of supertrains therefore isThis number is nearly proportional to the total adsorption free energy , so that the free energy gain per supertrain is ∼T. By contrast, the potential (attraction) energy of a supertrain, E, is much larger by magnitude than its adsorption free energy F_a (near the critical point u = u*): E ∼ −Tu*L_st ∼ −T(l/Δ)² (cf.eqn (31) and (85)), hence |E| ≫ T. Therefore, the adsorption free energy F_a comes from a subtle balance of the adsorption energy E and confinement entropy of train sections and short loops.

Exactly at the critical point the cut-off length h is replaced by the chain size, R_coil (see text below eqn (39)): for L ≫ l, hence (by virtue of eqn (86) and (87))

i.e. the critical adsorption involves just a few supertrains per chain. Note that the supertrain length L_st ∼ l(l/Δ)^4/3 (here we omitted the log-factor in eqn (85)) is much longer than the Kuhn segment l. Such a chain fragment would adopt an isotropic coil conformation in the free state. By contrast, a supertrain at the critical point is highly stretched along the bar: the ratio of its longitudinal size to lateral size is L_st/l ∼ (l/Δ)^4/3 ≫ 1.

The argument used above to obtain L_st can be applied also beyond the critical point, at u > u*. It shows that eqn (85) stays valid also for τ < 0, |τ| ≲ τ* ∼ (Δ/l)^2/3 corresponding to ε ≲ 1/l: the mean length of a supertrain in this case is

L_st ∼ l(l/Δ)^4/3/ln(h/l), 0 < −τ ≲ τ* (89)

where h = h(τ) is defined in eqn (71). Thus L_st just slightly (logarithmically) grows as |τ| increases from 0 to τ* reaching at |τ| ∼ τ* where (cf.eqn (69)). The number of supertrains N_st = pL/L_st (cf.eqn (87)) changes with τ in a more complicated way: the fraction p of segments in the proximal (r ≲ l) or contact (r ≲ Δ) layer is (cf.eqn (61) where the second term in square brackets can be neglected):where is defined in eqn (62). Using the above equations (including eqn (89)) and we get
Recalling also the τ-dependence of h defined in eqn (71) we finally obtainwhere τ_c ∼ (Δ/l)² and τ₁ is defined in eqn (68). Thus N_st exponentially increases from N_st ∼ ln(L/l) at |τ| ≲ τ_c to the absolute maximum, N_st ∼ (L/l)(Δ/l)^4/3 ln(l/Δ) at |τ| ∼ τ₁, and then it slightly decreases down to N_st ∼ (L/l)(Δ/l)^4/3 at |τ| ∼ τ*.

The evolution of the trapped chain conformation as the attraction strength is increased (at u ≥ u*) is illustrated in Fig. 6 for three characteristic regimes. At the critical point u = u* (Fig. 6a) the large loops forming the distal region dominate by mass; the mean contour length of a large loop is L_ll ∼ L/N_ll, where the number of large loops N_ll is close to the number of supertrains per chain, N_ll ≃N_st ∼ ln(L/l), which is just logarithmic in L. The length of one supertrain is L_st ∼ l(l/Δ)^4/3/ln(L/l), see eqn (85), and the total length of all adsorbed segments L_Δ = pL ≃N_stL_st ∼ l(l/Δ)^4/3 ≪ L, cf.eqn (85) and (88). Furthermore, the average number of trains in the chain is 〈m〉 ≃ L_Δ/λ* ∼ (l/Δ)² ≫ 1 (cf.eqn (96)), hence each supertrain typically contains many (m_st ≃ 〈m〉/N_st ≫ 1) trains. While the fraction of adsorbed segments p ≡ p(Δ,∞) ∼ (l/L)(l/Δ)^4/3 ≪ 1 (cf.eqn (88)) is extremely low, the fraction of adsorbed segments within each supertrain is p(Δ,l) ∼ 1/2, cf.eqn (35) and (36). In the case of moderate adsorption strength (see Fig. 6b), τ₁ ≲ u/u*− 1 ≲ τ*, where τ₁ is defined in eqn (68) and τ* = (Δ/l)^2/3, the cut-off distance h (for long loops) corresponds to the 2nd line of eqn (71), hence h decreases from h ∼ l(l/Δ)^2/3 at u/u* − 1 ∼ τ₁ to h ≃ l at u/u* − 1 ∼ τ*. The mean contour length of a long loop, L_ll, is L_ll ∼ (h²/l)/ln(h/l); it decreases from L_ll ∼ l(l/Δ)^4/3/ln(L/l) to L_ll ∼ l at u/u* − 1 ∼ τ*. Noteworthily, the statistics of the chain for s ≲ l is practically not affected by the change u* → u = u*(1 + τ*), see the text above eqn (78). In the regime of strong adsorption (see Fig. 6c), u/u* − 1 ≫ τ* (where , h ≪ l) the number of supertrains shows an exponential decrease (while the mean supertrain length exponentially increases to keep L_stN_st ≃ L) rapidly reaching the minimal number N_st ≃ 1 and the maximum length L_st ≃ L. Note that for 0 < u/u* − 1 ≲ 1 both the typical train length and the typical length of a short loop (with s ≲ l) are always comparable to λ* (since it is the shortest loops of length ∼λ* that dominate by number and by weight in accordance with eqn (34) and (35)).

Fig. 6 Fractal structure of semiflexible chain adsorbed onto a bar: (a) at the critical attraction point u = u* (τ = 0, ); (b) above it, for moderate excess attraction strength u = u*(1 − τ), τ < 0, τ₁ ≲ |τ| ≲ τ* = (Δ/l)^2/3, ; and (c) for strong attraction, |τ| ≫ τ*, . The main zones around the bar axis: steric repulsion zone of width d = 2r₀ (shown in red); attraction zone (yellow); proximal region involving short aligned loops, r ≲ l (grey); and distal zone extended to the terminal size h of the adsorbed chain, l ≲ r ≲ h (within the light-green bars). Here for part (a), l ≲ h ≲ R_coil for (b) and h ≪ l for (c), cf.eqn (71). The chain can be viewed as an interlacing sequence of supertrains (shown as violet curve parts) and long loops of length s ≳ l (black parts) between the two tails (shown in green only in (a)). Each supertrain is a sequence of trains (of length s ∼ λ* with θ ≲ θ* ∼ (Δ/l)^1/3) connected with short aligned loops of length s, λ* ≲ s ≲ l. The trains and short loops are marked respectively with red and blue arrows.

D4. Chain-end partition function ψ for critical adsorption, u = u*

In the previous point D3 we harnessed the size distribution of loops to obtain the concentration profile c(r) (eqn (80)). The partition function ψ(r) at the critical point, u = u*, can be obtained is a similar way. It can be written as (cf.eqn (48) at τ = 0)where ψ₀(r) ≃ 1 is the statistical weight of non-adsorbed states with no trains (ψ₀ for r ≫ Δ is also approximately equal to the statistical weight of conformations with no intersections at all between the chain and the interaction zone, r < Δ, cf.eqn (77)), and ψ′(r) is the statistical weight for conformations with at least one train.

Let us find ψ′(r) at u = u*. Generally, the WLC conformations relevant for ψ′ involve trains, loops and two tails (cf.Fig. 5). By definition, ψ′(r̲) must be proportional to the probability density ρ(r̲) that the end of the second tail (L-tail) is located at point r̲. The function ρ(r̲) can be deduced from the tail-length distribution P(s) (P(s)ds is the probability that the tail length is between s and s + ds). A tail does not significantly interact with the bar: the interaction energy per tail, E_t, is small, |E_t| ≲ T, since the return probability p_r that a given tail segment (of length λ* located at the curvilinear distance s′ = λ*n′ from the tail root, cf.Fig. 7) intersects the interaction zone, r ≲ Δ, is rapidly decaying with s′: (cf.eqn (3) and (33); note that the factor accounts for an increase of the typical intersection angle with n′). Summing p_r for all λ*-segments with n′ > n we get . Therefore, the total return probability p_rtot is small for large n, p_rtot ≪ 1 for n ≫ 1, hence a tail can be viewed as being virtually free (obeying the ideal statistics) leading to P(s) = const at λ* ≪ s ≪ L. More generally the tail statistical weight is proportional to which, however, equals 1 at the critical point where .

Fig. 7 A tail of length s and its inner point at the curvilinear distance s′ < s from its root (the last train and a part of the last loop are also shown). The curvilinear coordinates along the chain are marked in cyan: the tail root corresponds to L − s, its inner point to L − s + s′ and its free end to L.

Taking into account that the typical lateral size of a tail, r(s), is (cf.eqn (33))

we findwhere and the condition r(s) ≳ r accounts for the fact that short tails with r(s) ≪ r do not contribute to the statistical weight ψ′(r) (more precisely, their contribution is negligible being exponentially small). The first term in the upper line in the r.h.s. of eqn (94) is due to short tails (s ≲ l) which contribute to the proximal layer (r ≲ l), and the second term comes from long tails (s ≫ l) contributing everywhere (both proximal and distal layers). Eqn (94) is obviously in agreement with eqn (43) and (44). It shows that matching the asymptotics of ψ′(r) at r ≪ l and r ≫ l assuming a crossover exactly at r ∼ l is not really accurate for the system we study: it is the long-tail contribution that always dominates ψ′(r) at r ∼ l, the contribution of short tails is weaker by a log-factor formally shifting the crossover to shorter r.

This effect is due to nearly ideal-chain statistics of loops and tails for WLC with l ≫ Δ (see text above eqn (93)). The situation is different in the case of a flexible (Gaussian) chain adsorption onto a bar. The statistics of loops and tails is altered there due to excluded-volume interactions with the bar: a loop or a tail by definition must not enter the volume of the bar. As a result the statistical weights of both loops and tails decrease by log-factors: , (these dependencies come from obvious convolution relations between statistical weights of ideal loops and tails with no interactions with the bar, and Z_loop, Z_tail with excluded volume of the bar). These log-factors for Gaussian chains compensate (in the eventual profiles ψ(r), c(r)) for the analogous factors coming from the self-similar distributions of unperturbed loops and tails.

Note also that using eqn (25) and (94) and taking into account that the short-tail part of ψ(r̲, t̲) is concentrated within a steric angle ∼Θ(r/l)² at r ≲ l (cf.eqn (19) and (41)) we come to the concentration profile c(r) in agreement with eqn (45), (80) and (58). Here again we encounter the same effect: the contribution of long loops dominates c(r) at r ∼ l. Thus, generally, the correct way to establish the proper crossover between proximal (r ≪ l) and distal (r ≫ l) regimes for both c(r) and ψ(r) is to match the short-loop and long-loop branches of loop-length distribution at s ∼ l.

D5. Trains and supertrains below the critical adsorption strength, u ≤ u*

Let us turn to the chain statistics at u ≤ u*. Generally we can classify the WLC states near the bar according to the number m of train segments, m = L_tt/λ*, where L_tt is the total length of all trains and λ* is the average length of a train segment (whose typical energy of interaction with the bar is ∼T at u ∼ u*). The mean L_tt (at a given u) is equal to 〈L_tt〉 = pL (cf.eqn (60)). At u = u* it is equal to(cf.eqn (88)) corresponding to the mean number of train segments (of length ∼λ*, cf.eqn (30))

m₀ ≡ 〈m〉 = pL/λ* ∼ (l/Δ)² (96)

Therefore m₀ is large, but, on the other hand, it is much smaller than the maximum number of such segments in the chain, m₀ ≪ L/λ* (here we consider adsorption of a sufficiently long chain, L ≫ L_tt).

Let us find the probability distribution of m first at u = u*, namely, the probability p₀(m) for the chain to have exactly m ≥ 1 train segments. Below we argue that this distribution is exponential. Indeed, it is clear that if a λ*-segment ending at point L − s is adsorbed, one of the next segments (along the chain) would be also adsorbed with high probability, so that the probability q that no further segment is adsorbed is small, q ≪ 1. Moreover, to demand that there are no trains beyond the point L − s is equivalent to saying that the segment (L − s,L) is a tail (cf.Fig. 7). The tail probability is independent of its length s (provided that s ≫ λ*; cf. point D4), which means that q is constant: it does not depend on s. Moreover, q is obviously independent of whatever is the chain conformation prior to the point L − s. Therefore, the process is Markovian and p₀(m) ∝ (1 − q)^m leading to

p₀(m) ≃ (1/m₀)exp(−m/m₀) (97)

The partition function ψ(r) can be therefore written aswhere the term ‘1’ stands for ψ₀(r) ≃ 1, the partition function with no trains (cf. text below eqn (49)), and , cf.eqn (49), is the partition function reflecting all conformations involving adsorbed segments (trains).||||

The chain potential energy in the state m is E_m = −Tumλ* (omitting a numerical prefactor). At τ > 0 the Boltzmann weights, W_m, of m-states (W_m = e^−E_m/T = e^umλ* = e^(1−τ)m since u*λ* = 1, see eqn (30) and (31)) change due to the additional factor e^−τm destabilizing the states with longer trains and leading to a lower total statistical weight

The last equation is equivalent to eqn (48) where k should be identified with m₀ defined in eqn (96). Eqn (48) is justified this way.

The probability to have exactly m ≥ 1 train segments is

where the numerator in the r.h.s. is the statistical weight of the m-state (cf.eqn (99)). As the chain is delocalized at τ > 0 (i.e. its segments are mostly located far from the bar, at r ∼ R_coil), averaging of p(m|r) over r (to get the averaged distribution p(m)) is equivalent to setting r ∼ R_coil. Eqn (49) and (41) then give , so

p(m) ∼ p₀(m)e^−τm, m ≥ 1 (100)

Hence (cf.eqn (97))Thus 〈m〉 decreases from 〈m〉 ∼ m₀ at τ ≲ (Δ/l)² to 〈m〉 ∼ 1 at τ ∼ Δ/l and to 〈m〉 ∼ 1/m₀ at τ ∼ 1. The latter result means that at τ ∼ 1 just a single train segment (of length ∼λ*) is formed with low probability p₁ ∼ (Δ/l)². This probability can be obtained independently and more directly as

p₁ ∼ N_tp_t (102)

where N_t ∼ L/λ* is the number of λ*-segments in the chain, and p_t is probability that a chosen segment is in the train state, so that it is located within distance Δ from the bar and oriented parallel to it (within the deviation angle θ*): p_t ∼ (Δ²/R_coil²)(θ*)² ∼ Δ²(θ*)²/(Ll). Using eqn (102), (30) and (32) we then get again p₁ ∼ (Δ/l)².

Note that in eqn (101) the averaging is done over all conformations including those with no train at all. By contrast, the mean number of train segments, m̄, averaged over only those conformations that involve a train is

Thus 〈m〉 ∼ m̄ ∼ m₀ at τ = 0.

More generally, eqn (101) and (103) mean that a few supertrains of ∼m̄ λ*-segments are formed with probability . (Note that the number N_st of supertrains is just logarithmic at u = u*, cf.eqn (88); this number is even lower at u < u* since formation of yet another supertrain must be obviously less probable with a weaker attraction. This argument backs the idea of zero or a few supertrains per chain.) Thus, if a supertrain is formed (which is equivalent to demanding that m ≥ 1), its mean length is:

L_st(τ) ∼ λ*m̄ ∼ l(l/Δ)^4/3/(1 + τ/τ_c), 1 ≥ τ ≥ 0 (104)

and the probability to have at least one supertrain is

p_st(τ) ∼ 1/(1 + τ/τ_c) (105)

For τ = 0 the above equations give p_st(0) ∼ 1 and L_st(0) ∼ l(l/Δ)^4/3 in agreement with eqn (85) (recall that the log-factor is neglected in eqn (104)), while for τ = 1 we get L_st(1) ∼ λ*, p_st(1) ∼ (Δ/l)².

Note that the mean potential energy is E = 〈E_m〉 ∼ −T 〈m〉 (1 − τ) ∼ −T(1 − τ)m₀/(1 + τm₀)². This energy is large, |E| ≫ T, for τ ≪ Δ/l, i.e. close to the critical point.

D6. Two WLCs instead of WLC + straight ‘bar’

Having considered WLC adsorption on a bar, let us return to the original problem of two WLCs (A and B chains of length L ≫ l) with local attractive interactions (cf.Fig. 1 and 2). As before here and below (to the end of Discussion section) we assume that attraction is effective only between segments of different chains. As stated in Section 2.1 this problem is equivalent to one WLC with effective Kuhn segment l* near a bar as long as we are interested in distances r ≪ l*. In the general case of two chains with different Kuhn segments, l₁ and l₂, the length l* is defined by the relation

1/l* = 1/l₁ + 1/l₂ (106)

The condition r ≪ l ≡ l* points to the adsorption regime, u > u*, which corresponds to τ < 0, |τ| ≫ τ* (cf.eqn (72)). In this case the chains should form a double-chain complex wherein the chain trajectories are nearly parallel to each other and their lateral separation r is limited by h, r ≲ h, where h ∼ Δ/|τ|^3/2 ≪ l (cf. the 3rd line of eqn (71)). Moreover, for |τ| ≫ τ* the typical distance r is even shorter, r ≲ Δ, while the mass fraction of ‘loops’ of larger lateral size (r ≫ Δ) is small. Short loops (of length s ≪ λ*/|τ|) must show the self-similar size distribution (in the range Δ ≪ r ≪ h): the mass fraction of loops with lateral size ∼ r is

p(r) ∝ r^−4/3 (107)

(cf.eqn (29), (45) and (59) and note that p(r) ∝ r²c(r)). Hence it is the shortest loops of length ∼λ* with r ∼ Δ that dominate by mass.

Generally, the double-chain complex is stable if , where is defined in eqn (67) and (69). In particular, the stability criterion for −τ ∼ τ* = (Δ/l)^2/3 is L ≫ l.

D7. Branching of double chains and gelation at −τ ≫ τ*

Let us consider formation of double-chain complex in the regime of strong attraction, , when the distal layer is suppressed (cf. Section 3.5) and almost all polymer fragments are involved in the supertrain structure whose total length tends to L. Only short ‘proximal’ loops (with s ≪ l) are allowed within this mostly double-chain structure (cf.Fig. 1a). Their length distribution is defined in eqn (78). The number of loops with length ∼s is N_loop(s) ∝ sN_p(s), so

N_loop(s) ∼ N_loop(λ*)(λ*/s)³, s ≲ l (108)

where N_loop(λ*) ∼ L/λ* is the reference number of the shortest loops of size s ∼ λ*.

If two double-chain fragments pass close by, they can form a branching point by chain exchange²⁹ as shown in Fig. 8. An exchange occurs when 2 short loops (11) and (33) transform into 2 transmuted loops (13) and (31) (cf.Fig. 8b). For loops of length ∼ s the exchange probability is significant if both loops are aligned similarly (within the angle θ ∼ θ(s) ∼ (s/l)^1/2) and overlap in space (occupy the same volume v_ov ∼ sr(s)², where r(s) is defined in eqn (33)). The above conditions imply that the two double chains involved in the exchange must intersect at a small angle ∼θ(s). The probability of exchange for two arbitrarily chosen s-loops is therefore

p_ex ∼ v_ovθ²/V (109)

where V is the total volume occupied by the double-chain. Eqn (108) and (109) lead to the following number, N_ex(s), of exchange bridges of length ∼s:Clearly, it is the shortest ‘bridging’ loops of s ∼ λ* that dominate by number, N_ex ∼ N_ex(λ*):

N_ex ∼ n_l(Δ/l)² (111)

where

n_l = L²l/V (112)

is the typical number of overlapping l-segments in the double chain. Each exchange (branching) point brings about the factor ∼2 in the partition function of the system (thus decreasing the total free energy by ∼T), so the total exchange free energy is

F_ex = −T ln Z_ex ∼ −TN_ex ∼ −Tn_l(Δ/l)² (113)

The free energy F_ex due to exchange branching is equivalent to a weak effective attraction of polymer segments (within the second virial approximation) which, just like formation of reversible bonds in solutions of associative polymers,⁵¹ can cause phase separation. The excluded-volume repulsion of double-chain segments results in analogous positive free energy contribution

F_rep ∼ Tn_l(Δ/l) (114)

where Δ/l stands for B/l³, B ∼ l²Δ being the second virial coefficient for steric interaction of 2 segments of length l.*** Obviously F_rep dominates over F_ex, so the double chain remains in the expanded (coil) state for (that is, for −τ ≳ τ*). For l ≪ L ≲ (l/Δ)²l = L_m (i.e. in the regime of a Gaussian coil which is not really swollen by repulsive steric interactions) the coil volume V ∼ R_coil³ = (Ll)^3/2, so we find

N_ex ∼ (L/l)^1/2(Δ/l)² ≪ 1 (115)

The exchange branching therefore can be neglected in this regime (see Fig. 9a).

Fig. 8 Branching by strand exchange between two paired complementary chains (cf.Fig. 1): 1 + 2 and 3 + 4 pairs virtually intersect each other. The red strands (2 and 4) stay intact during exchange; their attractive zones (for the other two chains) are shown in pink and yellow. The exchange involves fragments (11*) and (33*) of length s making a small angle θ ∼ (s/l)^1/2 with each other. The strands 1 and 3 break and recombine in a different way during the exchange: the initial fragments (11*) and (33*) in (a) and (b) are converted into (13*) and (31*) in (c) and (d). The new branching point is marked with grey circle.

Fig. 9 Internal structure of a complex of two complementary semiflexible chains, A (red) and B (blue), for various mutual attractions u, −τ ≡ u/u* − 1: (a) strong attraction (−τ ≳ τ* ≡ (Δ/l)^2/3, cf. sect. D7), (b) moderate attraction (, cf. sect. D8), and (c) weak attraction (τ > 0, cf. sect. D9). In (d) and (e) two distinct modes of branching are shown: by exchange between short loops dominating for −τ ≫ τ_loops ≡ (Δ/l)^14/15 (d), and by chain exchange involving long loops valid for −τ ≲ τ_loops (e), cf.eqn (131). For |τ| ≲ (Δ/l)^4/3 the exchange of long loops leads to collapse of the complex (cf.eqn (132) and (141)). Part (f) shows a gel formed at polymer volume fraction ϕ > ϕ_g ∼ l/L for strong attraction (−τ ≳ τ*), cf.eqn (116). The branching points are indicated with circles. All A-chains are equivalent, as well as all B-chains, different shades of red and blue are used for clarity.

However, the situation may be different in the case of a semidilute solution of WLCs, c > c*, where c is the number of Kuhn segments per volume, and c* ∼ N/R_coil³ ∼ L^−1/2l^−5/2 is the coil overlap concentration (N = L/l, ). In this case the total number of Kuhn segments is cV, so the mean number of overlapping l-segments is n_l ∼ (cV)²l³/V = c²Vl³, hence, by virtue of eqn (111), N_ex ∼ c²VlΔ², and the number of junctions per double-chain is ν = N_exL/(cVl) ∼ cLΔ². A gel (cf.Fig. 9f) is formed in the solution if ν > 1 (as follows from the Flory theory⁵² in the case of high functionality corresponding to a large number of short loops per chain in the present case) leading to the gelation criterion

c > c_g ∼ 1/(LΔ²) (116)

Note that the ratio c_g/c* ∼ l^5/2Δ⁻²L^−1/2 ≫ 1 is large (if L ≲ L_m) and that the polymer volume fraction at the gelation point, ϕ_g ∼ c_gΔ²l ∼ l/L, is small since L ≫ l (semiflexible polymers). Moreover, taking into account that nematic transition in WLC solutions takes place at⁵³

ϕ_in ∼ 10Δ/l (117)

we conclude that the gelation is predicted in the isotropic state if L ≳ l²/(10Δ), while at shorter L it should occur in the nematic regime, ϕ > ϕ_in.

D8. Globule formation at −τ ≪ τ*

Let us turn to the regime of weaker attraction, where and the terminal separation h between interacting chains exceeds the Kuhn segment, h ≫ l. Here the double-chain structure is less perfect: it involves defects like large loops of size ≫ l (Fig. 9b) with two strands (subloops) of length s₁ ≫ l and s₂ ≫ l (cf.Fig. 10a). The partition function of such a loop, Z_loop ∝ (s₁ + s₂)^−3/2, reflects its 3-dimensional and flexible nature. Integrating Z_loop over s₁ and s₂ with s₁ + s₂ ∼ s we get the number of loops of length ∼s,

N_ll(s) ∼ Z_loops² ∝ s^1/2, s_m ≳ s ≳ l (118)

Here

s_m ∼ 1/ε (119)

is the terminal loop length (see Section 3.5). Matching eqn (108) and (118) at s ∼ l we getfor a double-chain of length L. The length fraction of all long loops isLet us assume for simplicity that the tightly paired double-chain fragments (with separation r ≪ l) dominate by mass, so that φ ≲ 0.5. On using eqn (121) and (119) this condition leads to

s_m ≲ l(l/Δ)^8/9, εl ≳ (Δ/l)^8/9 (122)

On the other hand, using also eqn (64) with p ≃ 1 − φ ∼ 1 (where p is the fraction of paired segments) and noting that at τ = 0 we get by integration(cf.eqn (72)), so that the regime we study, (Δ/l)^8/9 ≲ εl ≪ 1, corresponds towhereEqn (123) ensures that s_m ∼ 1/ε is large: s_m ∼ lτ*/|τ| ≫ l. From eqn (120) it is clear that the total number of large loops N_ll is dominated by the largest loops of length s ∼ s_m whose number is

N_ll ∼ L(λ*)²l⁻³(τ*/|τ|)^1/2 (126)

The above equation ensures that N_ll ≫ 1 if L ≫ l(l/Δ)^4/3. The latter condition essentially says that the chain length L must be much larger than the typical supertrain length (cf.eqn (85)); it was already adopted in point D5 (see text below eqn (96)).

Fig. 10 (a) A double chain with a ‘molten’ fragment involving two subloops, s₁ and s₂. (b) Two complementary chains passing by each other at a minimal distance r ∼ l. (c) An alternative paired state of the chains involving a supertrain of size R_st.

Long flexible loops (of length s ∼ s_m) provide an additional channel for crosslinking (branching) of double chain fragments (Fig. 9e). The dominant contribution to the number of junctions (exchange points) due to long loops, N_ex(ll), comes from exchanges between overlapping loops (cf.eqn (110)):

N_ex(ll) ∼ N_ll²v_ll/V (127)

where

v_ll ∼ (s_ml)^3/2 (128)

is the effective (accessible) volume occupied by a long loop, and v_ll/V is the overlap probability for any pair of such loops. Using the above two equations and eqn (126) we get

N_ex(ll) ∼ n_l(τ*/|τ|)^5/2(Δ/l)^8/3 (129)

where n_l is defined in eqn (112). Summing the contributions of short and long loops (cf.eqn (111) and (129)) we get the total number of junctionsObviously, long loops become more important if (recall that τ* = (Δ/l)^2/3)

−τ ≲ (Δ/l)^14/15 ≡ τ_loops ≪ τ* (131)

(cf.Fig. 9e). Comparing the exchange free energy F_ex (cf.eqn (130)) with the repulsion energy, eqn (114), we find that exchange attraction dominates if

−τ ≲ (Δ/l)^4/3 ≡ τ_g (132)

In the regime of eqn (132) the total second virial coefficient becomes negative leading to collapse and gelation of the double-chain polymer coil, that is, to formation of a crosslinked globular complex. The junctions are mostly due to long loops in this regime (cf.eqn (132) and (131)). The collapse stops once all long loops become paired, the corresponding condition is N_ex(ll) ∼ N_ll leading to V ∼ v_llN_ll (note that N_ll ≳ LΔ/l² ≫ 1 at −τ ≲ τ_g, cf.eqn (126), and v_ll ∼ l³(τ*/|τ|)^3/2, cf.eqn (128)). The concentration of the globular complex (and the corresponding volume fraction) therefore are

c ∼ L/(lV) ∼ l⁻³(|τ|/τ_g)², ϕ ∼ cΔ²l ∼ (Δ/l)²(|τ|/τ_g)² (133)

Obviously c is independent of the total chain length L and c ≲ l⁻³ in the regime of eqn (132). Hence polymer volume fraction ϕ ≲ (Δ/l)² is low, ϕ ≪ ϕ_in (cf.eqn (117)), so the globule is predicted to remain isotropic.

Interestingly and somewhat paradoxically, the collapse is predicted here as attraction of polymer chains gets weaker. The reason is that for strong attraction a double-strand (ladder) structure is most favorable (note that we assume a strictly monodisperse system). The double chain complexes do not aggregate further as all attractive interactions (bonds) are already saturated in the paired state (cf.Fig. 1c).

D9. Bridging by double-chain fragments and collapse near and below u*

Let us first consider a pair of long WLCs at the critical point, u = u*, in a volume V. We assume that V ≲ V_coil = R_coil³ = (Ll)^3/2. In the case of WLC adsorption on a straight bar we have shown (see point D3) that the typical length of paired (supertrain) fragments is

L_st ∼ l(l/Δ)^4/3 ≫ l (134)

(see eqn (85) where we omit the log-factor). It is reasonable to assume (as verified below) that for L ≫ L_st the mass fraction of paired fragments is small (see Fig. 9c). The pairing occurs readily once two l-segments (belonging to different chains) come to each other closer than their size, r ∼ l (cf.Fig. 10b). Indeed, as follows from eqn (41) (last line), in this case the statistical weight of a paired state (supertrain in Fig. 10c), , gets larger than the statistical weight ψ₀ of the state with no contacts between the chains, ψ₀ ≃ 1. Such ψ′ leads to double-strand (supertrain) formation with probability .

Let us find the total number of double-strand bridges. To this end it is useful to consider a WLC as a sequence of N = L/L_st supertrain fragments and to find the number n of such overlapping fragments:

n ∼ N²R_st³/V (135)

where R_st ∼ (lL_st)^1/2 is the supertrain size (by virtue of Gaussian statistics of chain fragments at length-scales ≳l). Obviously each pair of overlapping L_st-fragments must give rise to formation (with high probability p) of a double-strand fragment (of length ∼ L_st) bridging the WLCs. The number of bridges is therefore n_br ∼ n, and the corresponding free energy is (in analogy with eqn (113))

F_br ∼ −Tn_br (136)

The mass fraction of paired fragments is ϕ_pf ∼ L_stn_br/L. For V ∼ R_coil³ and n_br ∼ n we thus get (using eqn (135))

ϕ_pf ∼ (L_st/L)^1/2

meaning that ϕ_pf ≪ 1 for L ≫ L_st as stated above.

Using eqn (134) and (135) we get

n_br ∼ n ∼ n_l(Δ/l)^2/3 (137)

where n_l is defined in eqn (112). Comparing eqn (136) and (137) with eqn (114) for the free energy F_rep due to steric repulsion we find that bridging attraction always wins (|F_br| ≫ F_rep) leading to a collapse of the bridged WLCs, that is, formation of a globule with finite density (volume fraction ϕ = LΔ²/V). The equilibrium ϕ can be estimated using the condition that the maximum number of bridges is attained, i.e. that the bridged double-chain fragments occupy most of the chain, n_br ∼ N, leading to

ϕ ∼ (Δ/l)^8/3 (138)

Note that the globule remains isotropic since the nematic transition occurs at a much larger concentration, ϕ_in ∼ 10Δ/l.⁵³ For a binary solution of semiflexible polymers with stoichiometric 1 : 1 composition and with attraction only between segments of different kind (A and B), eqn (138) also defines the density of polymer precipitate at u = u* (τ = 0).

The obtained results remain valid also slightly below u*, at 0 < τ ≲ τ_c ∼ (Δ/l)² (cf.eqn (50)) where the supertrain length remains nearly the same (cf.eqn (104)). Moreover, the approach outlined above can be applied also at τ ≳ τ_c. As τ increases above τ_c the bridging gets weaker. At τ ≫ τ_c the statistical weight for supertrain formation (in the case of a mild overlap of two mutually attracting l-segments, r ∼ l) decreases from to ψ− 1 = ψ_br ∼ τ_c/τ (we neglect log-factors here; cf.eqn (51) and (99)). The typical supertrain length decreases in parallel (cf. point D5 of Section 4, eqn (104)),

L_st(τ) = L_stτ_c/τ (139)

where L_st is defined in eqn (134).

The argument used above to estimate the number of bridges remains valid provided that L_st is replaced with L_st(τ). We must also take into account that now the number of bridges is smaller than n: n_br ∼ ψ_brn since bridges are now formed with low probability ψ_br/(1 +ψ_br) ≃ ψ_br, where ψ_br ∼ τ_c/τ ≪ 1. As a result we get: R_st(τ) ∼ (lL_st(τ))^1/2 ∼ l(l/Δ)^2/3(τ_c/τ)^1/2 and

Eqn (140) are valid as long as L_st(τ) > l, that is, τ ≲ τ* ∼ (Δ/l)^2/3 and, in addition, τ ≳ τ_c. The free energy of bridging, F_br ∼ −Tn_br (see eqn (136) and (140)) dominates the repulsion energy F_rep (eqn (114)) if

τ < τ_gl ∼ (Δ/l)^4/3 (141)

Obviously τ_c ≪ τ_gl ≪ τ*. Thus, we predict that the A/B polymer complex forms a globule at u < u* for 0 < τ ≲ τ_gl. Polymer concentration ϕ in the globule can be obtained using the same criterion as before, i.e. by demanding that the length fraction of bridges, n_brL_st(τ)/L, is about 1/2 (that is, it is close to the maximum value). It leads to

ϕ ∼ (Δ/l)^8/3(τ/τ_c)^3/2, τ_c ≲ τ ≲ τ_gl (142)

In the above τ-range the volume fraction ϕ is still below the nematic transition threshold, ϕ_in ∼ 10Δ/l; moreover, the third-virial term (due to steric repulsion of polymer segments) remains always negligible here. Note that for 0 < τ ≲ τ_c the volume concentration ϕ in the globule is given in eqn (138).

At τ > τ_gl the bridging attraction is weak, so polymer chains always stay in the coil conformation (in particular, a globular complex is not expected in the case of just 2 chains, A and B, considered above). Turning to the case of a dilute or semidilute binary polymer solution (with stoichiometric A/B composition) we predict that the system should stay homogeneous at τ > τ_gl, but phase separate at lower τ (higher u) if its initial concentration is below the threshold defined in eqn (138) and (142).

Thus, we predict that the chains are not paired in dilute solution at τ > τ_gl (that is, for low attraction strength, u < u₁ = u*(1 − τ_gl)). The chain gyration radius R_g in this regime is therefore close to that of an ideal WLC, (for L ≫ l; the condition L ≲ L_m is also assumed). At u just above u₁ the bridging attraction leads to globule formation in the regime of extreme dilution. In this case the gyration radius strongly decreases to the globule size which comes from the condition ϕR_g³ ∼ LΔ², where ϕ is defined in eqn (142) with τ ∼ τ_gl. Hence R_g ∼ L^1/3l^5/9Δ^1/9 at u = u₁+ 0 (note that here R_g ≪ R_g0 since L ≫ λ*). Finally, at u = u₂ = u*(1 + τ_g), where τ_g is defined in eqn (132), the globules get destabilized leading to chain expansion to , where the factor accounts for enhanced rigidity of double-strand (ladder) complexes dominating at u > u₂.

D10. Bundle vs. elastic beam

Let us return to the regime of strong attraction, −τ ≫ τ*. In this case the terminal separation h between WLCs in the double-chain complex is small, h ≪ l (more precisely, h ∼ s^3/2l^−1/2, where s is the typical loop length, , so that h ∼ l(τ*/|τ|)^3/2), hence the double-chain can be considered as a fibril with effective Kuhn length l₂ ≃ 2l₁ (cf. the first term in eqn (4)), where l₁ is Kuhn length of one WLC. Thus, the rigidity is additive here. For a bundle of n chains the result would be l_n ≃ nl₁ in the regime h ≪ l. By contrast, the elastic beam theory applied to bundles predicts a stronger dependence l_n ∼ n²l₁ in the limit of strong interactions of constituent chains.⁵⁴ The difference comes from the fact that the beam theory assumes a solid-like bundle with permanent bonds between the chains, while our model allows for a quasi-free gliding of the chains along their contours (due to a quasi-continuous distribution of interacting sites along the chains). The gliding mechanism should involve exchanges between trains and loops (in particular, consumption of some trains by loops and reverse moves) leading to reptation-like dynamics of chain fragments along the bundle.

D11. The role of tails

Returning to adsorption onto a bar: a typical conformation of an adsorbed chain involves many loops and only 2 tails. Just this notion suggests that the contribution of tails in the segmental concentration profile is not large. To neglect the tails we can demand (this condition is equivalent to u/u* − 1 ≫ τ_c ∼ (Δ/l)², cf. Section 3.5): in this case the terminal tail/loop length is . The adsorbed chain can be roughly considered as a system of blobs (of size s_m for s_m ≳ λ*/τ_c), the number of internal blobs corresponding to loops, n_bi ∼ L/s_m ∼ , being much larger than just two end-blobs corresponding to tails. The tail contribution can be therefore safely neglected at all scales (up to the terminal distance h from the bar, cf.eqn (55)).

However, even at the critical point u = u* (τ = 0) the tail contribution is not so important: in fact, the tail-length (s) distribution is nearly uniform at λ* ≪ s ≪ L (cf. point D4), from which we immediately deduce that the mean masses of all loops and all tails are comparable. More precisely, the tail-length distribution at τ = 0 can be generated just by randomly choosing two points (A and B) on the WLC of length L. (This procedure comes from the notion that at τ = 0 the statistical weight of a tail is independent of its length, and the same is true for the middle segment AB involving loops and trains.) As a result we find that the mean length of each tail, L_t, is L_t ≈ L/3, which is also nearly equal to the length of all loops (corresponding to the segment AB; note that the contribution of trains to its length is negligible if L ≫ L_tt, cf. point D5).†††

Turning to the distribution of the loop and tail segments around the bar, note that the terminal distance h is defined by the coil size at τ = 0, , so h is large for long chains and most of the loop and tail segments are located at r ∼ h. The distribution of loop segments c_l(r) is defined in eqn (45) (the tail contribution is neglected in this equation). As for segments of the tails, their distribution c_t(r) (with scaling accuracy) is similar to the free-end distribution given in eqn (94): , as follows from the ideal WLC statistics of the tails (also note that it is long tails of length s ∼ L that provide the dominant contribution to c_t(r) for all r ≲ h). Recalling that loop and tail masses are comparable we find that:

c_l(r) ∼ c_t(r) for r ∼ h

Noting also that c_l(r) increases faster than c_t(r) as r is decreased in the region r ≪ h (cf.eqn (45) and (94)), we can conclude that c_l(r) ≫ c_t(r) in this region, so tails are subdominant at τ = 0 in the whole range of major interest, r ≪ h.

5 Summary and conclusions

We theoretically studied association of two semiflexible wormlike chains (A and B) driven by mutual short-range attraction of their segments (the interaction energy vanishes at r > Δ) in the case of high chain stiffness, when the chain persistence length l_p = l/2 is much longer than the interaction range Δ. The chains tend to align parallel to each other forming a double-strand complex due to their side-to-side attraction (defined by the parameter u equal to the depth of the interaction potential well divided by the thermal energy T, cf.Fig. 3a). It is shown that the local structure of such complex (at length scales r ≪ l_p) is identical to that of a semiflexible WLC (chain A* with renormalized stiffness defined in eqn (106)) adsorbed onto a straight bar (replacing the second chain B).

We found that the critical strength of attraction, u = u*, corresponding to the adsorption threshold (such that the A*-chain becomes trapped in the vicinity of the bar beyond the threshold, at u > u*) scales as u* ∼ (Δ²l)^−1/3 (see eqn (31)). A similar u* was predicted for adsorption of a WLC onto a solid surface or membrane.^5–7,49

At u = u* the adsorbed structure involves 3 regions: the contact region (at the distance r < Δ from the bar), the proximal cylindrical layer (at Δ < r ≪ l) and the distal corona layer (r ≫ l). At the critical point the contact layer consists of train segments (of typical length s ∼ λ*, λ* ∼ (lΔ²)^1/3) interacting with the bar and aligned parallel to it: the typical angle between a train and the bar is θ ∼ Δ/λ* ∼ (Δ/l)^1/3. Using a scaling argument we established that the proximal layer shows a fractal structure of aligned loops (connecting the trains) implying an algebraic decay of polymer concentration, c(r) ∝ r^−10/3 there. Moreover, using the transfer matrix approach (see eqn (20)) we obtained the orientational distribution of A-chain end segments in the proximal layer: it follows a universal law in terms of the reduced variables η, ξ defined with the r-dependent reference (typical) tilt angle with respect to the bar, Θ(r/l) = (r/l)^1/3 (cf.eqn (17) and (19)). The asymptotically exact orientational distribution profiles are shown in Fig. 4. As for the distal layer, it consists of long ‘isotropic’ loops characterized by another self-similar length distribution (cp. eqn (78) and (79) for ).

Above the critical threshold, at u > u*, the polymer gets trapped by the bar. Depending on the attraction strength u the polymer mass is then dominated by either long loops (for u/u* − 1 ≲ (Δ/l)²) or by the shortest loops and trains (at u/u* − 1 ≫ (Δ/l)² ln(L/L_st)), where L_st ∼ l(l/Δ)^4/3 is the typical supertrain length (see Fig. 6b and c). The loops of intermediate length s (λ* ≪ s ≪ l) are never dominant, i.e. the loops are segregated spatially in two classes according to their sizes. This effect is reminiscent of segregation of chromatin loops between compartments in folded chromosomes.⁵⁵

The net free energy gain on adsorption of a single WLC of length L, εLT, becomes positive at u > u*. At small u/u* − 1 it increases in an exponential fashion ε ∝ exp(S(u/u* − 1)) with high steepness S ∼ (l/Δ)², cf.eqn (67). The fraction p of train sections increases in parallel, it changes from almost zero at u = u* to p ∼ 1 in a narrow region 0 < u/u*− 1 ≲ (Δ/l)² (see Fig. 11 and eqn (73)). Therefore, although the condensation of the A*-chain is continuous, it becomes similar to a first-order phase transition for high stiffness, l ≫ Δ. Similar p-jumps (but with lower steepness) were predicted for adsorption of semiflexible chains onto solid surfaces (S ∼ (l/Δ)^4/3) and permeable membranes (S ∼ (l/Δ)^2/3).^6,7 A qualitatively similar ‘rounded phase transition’ was also predicted for linker-induced aggregation of two semiflexible polyelectrolytes.³¹ Furthermore, we established that the distal layer is suppressed at u/u* − 1 ≳ (Δ/l)^2/3 = τ*. A further increase of the interaction parameter u leads to a shrinkage of the proximal region whose cut-off radius decreases as Δ/(u/u* − 1)^3/2 (see eqn (71)).

Fig. 11 The fraction p = p(Δ,∞), eqn (73), of monomers interacting with the bar vs. the reduced attraction strength, u/u*. The transition width is W ∼ τ_c ∼ (Δ/l)². The steepness is S = 1/W.

Replacing the bar with the second WLC we analysed the equilibrium structure of double-strand complexes and the properties of (dilute or semidilute) binary solutions of the WLCs. In a dilute binary solution at stoichiometric conditions the chains tend to form complexes if monomer attraction is strong enough, u > u*. Moreover, if u/u* − 1 ≫ τ*, the paired chains are always nearly parallel to each other in a double-strand complex (cf.Fig. 9a) and their separation r does not exceed l being typically very short, r ∼ Δ. The double-strand chain remains in the coil state if all attractive interactions (bonds) are already saturated in the paired structure (cf. ladder segment in Fig. 1c). However, at higher concentrations (in the semidilute regime, c > c*) branching of double-chains by strand exchange (cf.Fig. 8 and Discussion point D7) becomes important leading to gelation of the system (cf.Fig. 9f) above some concentration c_g defined in eqn (116). We also found that strand exchange becomes even more important at weaker attraction, u/u* − 1 < τ* (cf.Fig. 9d and e), leading to a collapse of double chains already in the dilute solution regime if u/u* − 1 ≲ (Δ/l)^4/3 (cf.eqn (132)). More generally we established that double-strand complexes form globules or phase separate (in dilute or semidilute regimes) within a finite range of attraction strength u around the critical point:

− (Δ/l)^4/3 ≲ u/u* − 1 ≲ (Δ/l)^4/3 (143)

(see Discussion points D8, D9). Thus, as u increases the polymer component is predicted to first precipitate at u > u₁ and then redissolve at u > u₂, where u₁, u₂ can be deduced from eqn (143). Such a reentrant transition to the solution phase (induced by an increase of the attraction strength u) is reminiscent of redissolution of aggregated semirigid macromolecules (like DNA and F-actin) in aqueous solution upon adding high amounts of multivalent ions which normally promote interchain attraction.^56,57

The side-to-side attraction of stiff polymer chains in solution can generally lead to their precipitation due to formation of a dense phase with chains aligned nearly parallel to each other. However, in some cases such transition does not occur directly, but rather is preempted by formation of finite bundles of aligned chains. The simplest bundle is a double-chain considered in the present paper. Such pairing is expected, for example, if the attraction is provided by transient saturating bonds,⁵⁸ like H-bonds or other directional interactions. The theory of double-chain complexes emerging as solvent quality is decreased and their structural properties elucidated here can provide a useful basis for correct interpretation of scattering experiments on WLC polymer systems.

We thus provide a theoretical framework to study lateral association of dissolved semiflexible polymers with short-range attractive interactions. The obtained results can be applied to study aggregation, bundling and network formation in solutions of synthetic polymers and biopolymers with rigid backbone (like DNA, F-actin⁵⁹ and other protein bio-filaments).

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This research was partially funded by ANR grant ‘MONA_LISA’.

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Footnotes

† In the general case of different chain stiffness, l₁ and l₂, l* is defined in eqn (106).

‡ The physical meaning of the arguments, r, t_r, t_φ, is clarified in Fig. 3b. Both t_r and t_φ are small for r ≪ l.

§ A weaker asymptotic boundary condition like |g(η,ξ)| < C (with some positive C > 0) seems to be enough to get a unique solution as suggested by our numerical results.

¶ Therefore α₀ = 0 is an exact result. Moreover, α₁ = −2 also rigorously comes from eqn (20), , where the operator depends on the parameter α. The reason is that the operators and have exactly the same eigenvalue spectrum since, as follows from the definition of , the second eqn (20), equals to with variable η replaced by −η, where is the transpose of (being real, is also conjugated to ).

|| Note that the scaling law of eqn (34) is asymptotically valid for short loops of length s, λ* ≪ s ≪ l. Furthermore, eqn (35), which follows from eqn (34), defines the contribution of such loops to the concentration profile. No logarithmic prefactors are omitted in these equations; small (vanishing) corrections to these scaling laws decay algebraically rather than logarithmically.

** Note that in the general case ψ(r̲) is defined in eqn (42).

†† Note that ψ₁ ∼ 1 for r ∼ h. This feature can be also deduced from the fact that the chain behaves ‘flexibly’ at r ≫ l, hence the results for flexible chains in the presence of a localized attractive potential are applicable.⁴² More precisely, we can solve eqn (5) in the form of eqn (11): , and note that ψ⁽⁰⁾(R̲, −t̲) is orthogonal to all ψ⁽ⁱ⁾(R̲, t̲) with i > 0 (the latter statement is a generalization of a well-known fact that eigenvectors of a Hermitian operator are mutually orthogonal). Noting also that ψ(R̲, t̲, 0) = 1 and that ψ⁽⁰⁾ and ε⁽⁰⁾ must correspond to ψ⁽⁰⁾ ≃ ψ₁(r) (at r ≫ l), and ε⁽⁰⁾ = 0⁺ at the critical point, we get
Eqn (39) suggests that both integrals above are dominated by the region r ∼ h, so the above equation demands that const in eqn (39) is nearly equal to 1.

‡‡ The prefactor here comes from the notion that ψ ∼ 1 at r ∼ h = R_coil for any u ≤ u*: this is obviously true for u = u* (see the last line in eqn (41)), and for u = 0. Note that while in the latter case (u = 0) comes from eqn (46), it can be also deduced from the fact that the probability that any segment of the chain (whose end is located at a distance r ∼ h = R_coil from the bar) comes close to the bar (within a distance ∼l) is logarithmically small as follows from the Gaussian (flexible chain) statistics which is applicable in this case since all relevant length-scales here are larger than the persistence length l_p = l/2.

§§ Note that this exponential factor, which is most important for u > u*, equals 1 for u ≤ u*, so it was omitted in the previous sections.

¶¶ Such estimate of u_i is hinged on the second virial coefficient, B₂, for interaction of two (nearly) straight λ-segments of the WLC and the bar, Δ ≪ λ ≪ l: B₂ should be negative for attraction to dominate. For a given angle θ a simple calculation gives (omitting numerical factors): B₂ ∼ λ² sin θ[d + Δ(1 − e^U/T)], where the first term in square brackets corresponds to the excluded volume, and the second term - to attraction between segments. Here U ∼ TuΔ/sin θ is the typical attraction energy. The condition B₂ = 0 for θ ∼ 1 and Δ ∼ d then leads to U ∼ T and u_i ∼ 1/Δ.

|||| Note that ψ′(r) is related to the tail-length distribution (see point D4) which is independent of m since mλ* ≪ L (cf. text below eqn (96)).

*** Recall that we consider a long double-chain, L ≫ l, where double-chain fragments pass by each other at arbitrary angles.

††† A more general argument to obtain the tail-length distribution at u = u* is given in Section 7.3 of ref. 7.

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