Adsorption of flexible polyelectrolytes on charged surfaces

Adsorption of weakly charged polyelectrolyte (PE) chains from dilute solution on an oppositely charged surface is studied using the self-consistent mean-field approach. The structure of the adsorbed polymer layer and its excess charge are analyzed in the most important asymptotic and intermediate regimes both analytically and numerically. Different regimes of surface charge compensation by PE chains including partial and full charge inversion are identified and discussed in terms of physical parameters like the magnitude of specific short-range interactions of PE segments with the surface, solvent quality and ionic strength. The effect of excluded-volume monomer interactions is considered quantitatively both in the marginally good and poor solvent regimes.


Introduction
Let us consider a dilute solution of polyelectrolyte (PE) chains. In the presence of an oppositely charged surface the PE chains tend to form an adsorbed layer on it. Polyelectrolyte adsorption is an important phenomenon widely used for practical purposes including the stabilization of colloidal dispersions in numerous biological and industrial applications. 1 The adsorbed layers are generically involved in many technologically important PE systems, including polyelectrolyte multilayers, vesicles, templates, etc. Understanding the structure of PE layers and the physics behind it is a challenging general problem.
It is known that upon adsorption the total charge of the surface plus the adsorbed polymer can be opposite to the initial bare surface charge. 2,3,5,[7][8][9] This phenomenon, known as charge inversion, is important fundamentally and it seems to provide the driving force for the alternate adsorption of positively and negatively charged polymers, leading to the so-called PE multilayer formation. 3,11,12 While the significance of this purely electrostatic driving force has been questioned since other more specific forces can (and often do) contribute to multilayer stability, 10 the very fact that multilayers are typically and easily assembled by charged polymers points to the crucial role of electrostatic effects including the charge inversion phenomenon.
Adsorption of PEs on a charged surface was studied theoretically in ref. 6 for the case of constant surface potential and strongly repulsive short-range interactions of uncharged polymer segments with the surface (nonadsorbing surface). The problem was studied both numerically and by scaling analysis. The predictions 6 concerning the polymer layer thickness and surface excess are generally in agreement with our results (see Discussion, Section 4.3). However, specifically the excess surface charge due to the adsorbed PE was not considered in ref. 6. The latter quantity was analysed by Joanny 7 who developed a theory of polyelectrolyte adsorption from dilute solution on an oppositely charged surface based on a mean-field model rather similar to that adopted in ref. 6. The study 7 was mainly focused on the ideal polymer regimes with insignificant non-Coulombic interactions between polymer segments, discussing briefly the effects of their 3-body excludedvolume interactions and non-electrostatic monomer/surface interactions; the effects of 2-body monomer interactions have not been discussed there. One aim of the present paper is to generalize the theory 7 and to analyse the solvent quality and surface interaction effects in more detail. Another aim is to provide a more complete and quantitative analytical description of the charge inversion effect in the asymptotic regimes of low and high ionic strength and in other regimes. It was found 7 that the excess charge of the adsorbed layer, s ex , is inversely proportional to the Debye length r D at low salt concentration. However, this general relationship has not been fully justified and has been missing a proper prefactor. These gaps are also filled in our study.
PE adsorption and charge inversion phenomena have been studied theoretically more recently in ref. 4, 8 and 9. They used a mean-field approach similar to that of ref. 7, and altogether they generalized the theory to include the monomer and surface interactions. So why do we address the problem again? The point is that the results of ref. 8 and 9 seem to contradict each other: in ref. 8 a full charge inversion was not found (and it was concluded that multilayer formation must rely on non-electrostatic effects), while the regimes of full or even stronger overcharging are predicted in ref. 9. In both cases the results have been obtained mostly numerically: the analytical parts of these studies are more auxiliary and are not entirely satisfactory in our opinion (it was stated 9 that their analytical approach involves 'problematic' approximations). To resolve the issue we considered the same simple mean-field model in a more analytical fashion which allows us to built a more complete picture of possible adsorption regimes and to quantitatively find the degree of overcharging (or undercharging) depending on the conditions. To facilitate this task we identified the basic reduced parameters characterizing the system. This allows us to systematically describe all the regimes in terms of a minimal set of essential parameters. For example, in the case of ideal polymer backbones and purely repulsive surfaces the adsorbed layer structure essentially depends on only one parameter, the reduced ionic strength n.
The model, the basic mean-field equations, and the results of their numerical study highlighting the effects of ionic strength, polymer/surface and excluded-volume interactions on PE adsorption are presented in the next 2 sections. The main results are discussed in detail in Section 4 and the main conclusions are outlined in the last section.

Model and basic equations
In this paper we consider adsorbed layers of long weakly charged flexible PE chains (with a low fraction f { 1 of positively charged units † and a large polymerization index N c 1/f ) in the regime of moderate polymer density in the layer (corresponding to semidilute solution). In this regime the excluded-volume interactions between polymer segments can be described in terms of virial coefficients B and C defining the relevant free energy density where n is the concentration of monomer residues, k B is the Boltzmann constant, and T is the temperature. The corresponding chemical potential at distance z to the surface, depends on the local concentration n = n(z). In what follows we assume the limiting case of dilute bulk PE solution (zero bulk concentration): The surface is negatively charged with density Àes, where s is the number of Àe charges per unit area. The volume density of charge is r = e( fn + c + À c À ), where c + and c À are concentrations of small ions (in particular due to the added salt). In the meanfield approximation (adopted throughout the paper) where F = ej/(k B T), j = j(z) is the electrostatic potential and c b is the bulk concentration of anions (ionic strength). It is assumed that the surface charge is not too high, |F| { 1, so the potential F(z) obeys the Poisson equation in the Debye-Hückel (DH) approximation where F = F(z), n = n(z), and r D is the Debye length, is the Bjerrum length, and e is the dielectric constant of the medium). It is assumed that e does not significantly depend on polymer concentration since the latter is low ( Turning to the polymer concentration profile n(z) we note that it can be described within the mean-field ground-state dominance approximation [13][14][15][16] since PE chains considered here are very long (polymerization index N -N) and flexible (with statistical segment a s ): where a 2 = a s 2 /6, c ¼ ffiffi ffi n p , and is the total effective molecular field (in k B T units). Here E is a constant, the ground-state 'energy' eigenvalue. The validity of the above Edwards equation is hinged on the condition U(z) { 1 which is true since polymer concentration is low, . Note that the ground state eigenvalue E is an important free parameter, E r 0 (see Section 4.2). Eqn (3) and (4) should be supplemented by the boundary conditions at the wall and at infinity. For the potential j(z) they read The boundary condition for the order parameter c(z) at the wall is defined by the polymer/surface interactions. We will separate two cases, namely the case of a steric hard wall when the concentration of the polymer is zero on the surface and the boundary conditions are given by c| z=0 = 0, c| z=N = 0 (6) and the case of an adsorbing wall with specific short-range polymer/wall interactions. The surface concentration of monomers is non-zero in the latter case, c| z=0 = c 0 a 0, and the boundary conditions are: Note that the boundary conditions in eqn 6 are the limiting case of eqn 7 for k -N.
In the general case the solution of eqn (2)-(7) depends on the eigenvalue E correlating with the amount of the adsorbed polymer. The most stable polymer layer corresponds to the minimum of the free energy F including the conformational, electrostatic, ideal-gas and excluded-volume interaction terms given by (per unit surface area): where f id (c + ,c À ) = c + ln c + + c À ln c À À (ln c b + 1)(c + + c À ) + 2c b is the reduced ideal-gas free energy density of small ions. Eqn (2)-(8) can be rewritten in terms of the following nondimensional variables: Here h 0 is the characteristic thickness of the adsorbed layer identified in ref. 7. The system involves the second length-scale, . The ratio of the two lengths defines the dimensionless ionic strength parameter n = h 0 2 /r D 2 .
Eqn (3) and (4) with boundary conditions (5) and (7) in the dimensionless form read: Eqn (11) can be solved in the general way yielding the following integral form The free energy F can be rewritten as Eqn (10) and (11) are formally equivalent to mechanical equations for a 2-dimensional classical particle (coordinatesc,F) with anisotropic mass, m c = 2, m F = À1 in the potential The 'mechanical energy' conservation law yields the following integral: Using boundary conditions at x -N and extinction of the Using eqn (14) we obtain an additional condition at x = 0: for the steric hard wall (c| z=0 = 0), and in the general case, wherec 0 =c(0),F 0 =F(0). In the next two sections we consider the case of dominating electrostatic interactions (i.e., PE chains with negligible B and C interaction parameters).

Repulsive hard wall
We start the analysis of polyelectrolyte layers with the case of a purely repulsive hard wall (k = N) and no excludedvolume interactions. The system of eqn (10) and (11) is then written as This system was solved numerically. We start with the salt free case n = 0. Since the electrical field at the infinity is zero, the adsorbed polyelectrolyte layer must completely screen the surface charge in this case. The reduced density profiles of charged monomers, ñ + (x) =c 2 (x), and the reduced potentialF(x) are shown in Fig. 1. The potential monotonically increases fromF 0 E 2.24 at the wall, and its main change occurs at x B 3 corresponding to the length scale z B 3h 0 . The adsorbed layer is localized near the wall at the same distance. At larger distances the polymer concentration and electric potential decay exponentially:ñðxÞ $FðxÞ $ e À2x ffi ffi The total mass of the adsorbed polymer layer and its free energy depend on the parameter Ẽ. As Ẽ is increased the free energy decreases and its minimum is attained at Ẽ = 0 corresponding to the saturated layer, see Fig. 2. In what follows we consider the saturated layers only, hence we set Ẽ = 0 for n 4 0. The density profiles of charged monomers and the potential profiles for different values of n are shown in Fig. 3. In all cases the potential F passes through a maximum value. It means that the electric field changes its direction inside the layer. With increasing salt concentration the electrostatic attraction between the wall and the polyelectrolyte chains decreases, so the concentration of the polymer in the layer decreases as well. An adsorbed layer is not formed at all if the amount of salt is high enough: n Z n c E 0.78.
In Fig. 4 we show the plots of the surface potentialF 0 and the reduced excess surface charges ex = s ex /s as functions of n. The surface potential increases monotonically with ionic strength, while the excess surface charge attains the maximum value (s ex E 0.008) at n E 0.004. The adsorbed polymer layer overcharges the surface (i.e.,s ex 4 0) for n o 0.015. However, the overcharging effect is very weak and occurs only at low ionic strength (for r D \ 8h 0 ).
Based on eqn (18) we established the asymptotic behavior of the potentialF(x) at x c1 and n { 1 (cf. Section 4.2, eqn (28)). It implies that This asymptotic dependence is shown as a dashed line in Fig. 4b.

Attractive wall
The system of eqn (10) and (11) for the adsorbing wall without volume interactions can be written as A series of profiles for the concentration of charged monomers and the electric potential are shown in Fig. 5-7 fork ¼ 1; 0; À1 and different values of n. In all cases the potential passes through a maximum whose height decreases with the ionic strength in the bulk solution, and so does the polymer concentration at k 4 0. By contrast, the surface potential increases as n is increased (see Fig. 8a).
For anyk 4 0 the polymer layer is not formed for large enough ionic strength, n Z n c ; n c E 1.35 fork ¼ 1. This effect occurs because the electrostatic interaction is screened and the polymer/wall repulsion becomes dominant at high ionic strength.
The degree of overcharging increases both with increasing polymer/wall attraction energy and ionic strength, see Fig. 8b. At low ionic strength (n { 1) the excess charges ex shows the following asymptotic behavior:s ex ' e ffiffiffi n p (cf. eqn (29)), where e is a function ofk which is shown in Fig. 9.
A detailed analysis of the excess charge and the polymer and electric potential profiles at high ionic strength (n c1) is presented in Appendix B.

Effect of excluded-volume interactions
Let us turn to the effect of excluded-volume interactions of polymer segments on their adsorption. Below we consider only the most important case of an indifferent surface (k ¼ 0) focusing on the regimes with negligible contribution of one of the virial coefficients: C = 0 or B = 0.
We start with the case C = 0, B a 0. In Fig. 10 the density of polymer charge and potential profiles are shown. The adsorbed layer thickens with increasing attraction between the monomers or ionic strength. The excess charge also increases with the ionic strength. The plots ofs ex for different values of the second virial parameter are shown in Fig. 11. In the case of attractive interactions between monomers (B o 0) the system of eqn (10) and (11) has a well-defined stable solution if B 4 B c (n). For salt free conditions, the critical value of the second virial parameter is B c E À2.15 (for n = 0). The excess charge monotonically decreases while B is passing from poor to good solvent conditions, and so does the reduced eigenvalue e (cf. Fig. 12). At high ionic strength, n c1, the critical excluded volume parameter is B c C À1/n (cf. Appendix B and eqn (A3)).
Finally we address the effect of three-body interactions, C 4 0. The profiles of ñ + (x) andF(x) obtained in this case using eqn (10) and (11)    The degree of overcharging decreases as the third virial coefficient is increased;s ex also decreases at high ionic strength for C \ 0.5. The reduced thickness h = h/h 0 of the adsorbed layer is plotted in Fig. 15a as a function of the third virial parameter C for the salt free case; h increases with C in accordance with analytical results (cf. Appendix C). The reduced eigenvalue e (cf. Fig. 15b) monotonically decreases with C reflecting the similar behavior of s ex .

The region of validity and essential variables
In this paper we considered the equilibrium structure of PE layers adsorbed on oppositely charged surfaces. The study concerns weakly charged PE chains, f { 1, with very low bulk concentration (dilute solution). The weak charge condition is very important: it allows us to focus on the mean-field solution regime, where the existing well-established theoretical frameworks can be harnessed to cast quantitative results. In Appendix A we specify two main conditions of applicability of the mean-field approach demanding that concentration blobs are Gaussian (eqn (A1)) and electrostatic potential is low (eqn (A2)). It is only for f { 1 that the two conditions are compatible defining a wide range of eligible surface charge densities (s). Note that the second condition also ensures that electrostatic screening effects can be described by the mean field model (r D cl B ): in fact, the relevant length-scale defining the characteristic Debye length r D is the adsorbed layer thickness h 0 (cf. eqn (9)), which is much longer than a ffiffi ffi f p by virtue of eqn (9) and (A2), hence One focus of our study concerns charge inversion and the dependence of the excess charge es ex (that is, the total charge of the surface and adsorbed polymer) and of the layer thickness h on various parameters: bare surface charge es, ionic strength c b , fraction of charged units f, short-range specific attraction of the polymer backbone to the surface (its magnitude is characterized by the parameter k), and virial parameters B and C. We quantitatively described (both numerically and analytically) the effect of monomer interactions considering both the regime where binary interactions (B) are dominating and the Y-regime with significant 3-body interactions (C).
Both s ex and h are generally related to the polymer adsorbance where n(z) is the concentration of PE monomers at distance z from the surface and n max = max n(z): The characteristic thickness of the adsorbed layer h 0 ¼ a 2 sf' B 1=3 (cf. eqn (9)) serves as a natural length-scale for the adsorption structure. Using this length we identified the following essential reduced parameters corresponding to the ionic strength, the monomer/surface and excluded-volume interactions: are the characteristic values of the second and the third virial parameters of non-electrostatic monomer interactions. This way we minimize the number of variables to deal with and obtain universal predictions for the adsorbed layer structure in terms of this minimal set of reduced parameters.
In addition to the Debye-Hückel (DH) approximation for monovalent ions (weak charge, F { 1), we use the ground-state dominance (GSD) approximation to describe the chain statistics. The latter approximation is fully justified for very long (infinite) polymer chains considered in the present paper. In practice the chains are finite, and the GSD approximation is valid for N cN*, where N* B h 2 /a 2 is defined by the condition R coil = aN 1/2 ch, 14,16 meaning that the coil size (Gaussian gyration radius) R coil must well exceed the characteristic adsorbed layer thickness h. For example, h = h 0 in the simplest case of no added salt, no specific monomer-wall attraction and no excluded-volume interactions, so the GSD condition then becomes 7 The same GSD condition (N cN*) also ensures that the concentration of PE segments in the adsorbed layer, n B n max , can be significant (say, semidilute) even if the bulk concentration n b is very small (very dilute bulk polymer solution as considered in the present paper). For example, in the case of no added salt the condition n b /n max { 1 can be formally provided if N|E 0 | c1 which is equivalent to N cN*. It is also worth noting that the regime R coil ch is the most typical for high polymers: in the opposite case R coil { h the polymer coils can be considered similarly to small multi-valent ions. In the latter regime (N { N*) the PE chains are attracted less strongly to the substrate, so the adsorbance G typically decreases for shorter chains. The dependence of the PE adsorption profiles on the chain length (N) was analyzed in ref. 9 using the self-consistent field theory. The N-effect saturates for long chains in reasonable agreement with the theoretical criterion N cN* considered above.

The ground-state eigenvalue and the low-salt limit
The adsorbed layer structure (including the polymer profile n(z)) is defined by eqn (3)    The total free energy of the system (layer + bulk solution) is F(G) À m b G + const, where m b is the monomer chemical potential in the bulk solution (cf. eqn (1)):   The electrostatic potential is not involved in m b since it vanishes in the bulk by definition: j -0 at z -N. The first ideal-gas term in the above equation also vanishes since N -N, and so does the second term (cf. eqn (1)) in the dilute limit n b -0, hence m b = 0.
The adsorbed layer is formed since the total free energy F(G) À m b G decreases with G at low G. The equilibrium state (saturated adsorbed layer) corresponds to the minimum of the total free energy, i.e.

@F @G
¼ m b ¼ 0: Considering F as a functional of concentration profile, F = F [n(z)], we arrive at the general minimization condition dF½n dnðzÞ ¼ 0: Recalling that F = F conf + F int , where F conf is the square-gradient conformational free energy and F int is the free energy of all interactions (including F ex and electrostatic contributions), one can see that eqn (26) is equivalent to the Edwards equation (eqn (4)) with E = 0. ‡ The situation for no salt (n = 0) is different: here, F [n] is defined only for G = s/f : F is infinite for any other G due to violation of the electroneutrality. Therefore F [n] has to be minimized under the side condition ð 1 0 nðzÞdz ¼ s=f   (4)) with E = l/(k B T) a 0. The above equation formally implies that The following corollary can be deduced from the previous point: obviously G must tend to s/f at n -0, that is, G is continuous at n = 0, and so is the whole polymer profile n(z). By virtue of the Edwards equation (eqn (4)), the same statement applies to U(z) À E and fF(z) À E, where F = ej/(k B T). Hence lim ðxÞ ' e for x c1, where e = Àh 0 2 E 0 /a 2 ,F = j/j 0 , Using eqn (11) and taking into account thatc (x) -0 for x c1 we find the asymptotic behavior of the electrostatic potential at low salt concentrations: On using eqn (19) this leads to the asymptotic equation defining the excess surface charge: ; n ( 1: The above result can also be rationalized in the following way: eqn (27) shows that qF/qG = Àk B T(a/h 0 ) 2 e is negative at G = G 0 s/f corresponding to exact charge compensation. Therefore, an increase of PE adsorbance (G 4 G 0 leading to s ex = f (G À G 0 ) 4 0) is favorable: F C const À k B T(a/h 0 ) 2 es ex /f for small s ex . The energy penalty DF es to pay for this excess charge comes from the long-range electric field generated by s ex at z B r D ch 0 . Within the DH approximation its energy is Minimization of F + DF es then gives s ex C seh 0 /r D , which is equivalent to eqn (29). The reduced eigenvalue e also defines the polymer density profile ñ(x) =c(x) 2 in the low salt regime. In the case of no salt (n = 0) we find that ñ(x) decays exponentially at large distances: At finite but low ionic strength (n { 1) the asymptotic behavior of ñ(x) =c(x) 2 can be obtained using eqn (17) with Ẽ = 0 and F(x) defined in eqn (28): , and it is followed by a power-law decay at even larger distances. In this case the low-salt asymptotics, eqn (20), is applicable only for very low n o 10 À3 . The excess charge es ex is positive for n o 0.015, but its value does not exceed 1% of the surface charge: s ex /s o 1%. For n 4 0.015, the adsorbed layer enters the charge undercompensation regime (s ex o 0), and the PE charge disappears completely (s ex = Às) at n = n c E 0.78. The PE layer is not formed at all for n 4 n c . The adsorbed layer thickness stays nearly the same, h B h 0 at 0 o n o 0.6: it is the layer concentration that decreases significantly with n in this regime. Closer to n c the adsorbed layer is characterized by 2 length scales: h 0 (for the proximal region where n(z) is increasing) and h (the layer thickness), the latter is diverging at n c . Note that the ionic strength for the adsorption cutoff, c b ð Þ c ¼ 0:5n c'B À1=3 a À4=3 sf ð Þ 2=3 , is proportional to (sf) 2/3 , so the cutoff ionic strength increases with the polymer charge density ( f ) in agreement with experimental data. 23 It is remarkable that the adsorption criterion n o n c can be rewritten as

The H-solvent regime: ionic strength and specific polymer/surface interaction effects
with const E 0.019. The above condition agrees with the criterion for single chain adsorption transition predicted by Muthukumar. 24 The revealed significant decrease of PE adsorbance G and of the excess PE charge s ex with addition of salt (beyond a very low threshold,  (14) which seems to be reasonable for an indifferent rather than for a repulsive surface). Following Joanny 7 we distinguish the important regime of an indifferent surface where short-range attractive and repulsive interactions of PE segments with the surface are balanced (k = 0). In this case the relative excess charges ex = s ex /s always increases with ionic strength (n), see Fig. 8b. We identified analytically the following asymptotic behaviors of the excess charge (cf. Appendix B): based on eqn (29) with e E 0.344 and eqn (B4). Interpolating the two asymptotics using Padé approximation (with variable ffiffiffi n p ) we find the full dependence on n: The above equation is in good agreement with numerical results in the whole range 0 r n o N (cf. Fig. 16). It shows that the full charge compensation is attained  (15) and (17) of ref. 6.
The effect of polymer/surface interactions (k) can be gathered from Fig. 8b: surface repulsion suppresses PE adsorption, leading to a smaller or negative s ex , and a somewhat thicker but more dilute adsorbed layer. Quite naturally, attractive surface (k o 0) enhances PE adsorption, so the adsorbed layer becomes thinner and denser in this regime. These conclusions are backed by our analytical results described below (cf. Appendix B): in the high salt regime, n c1 (i.e. r D { h 0 ), the layer is characterized by 2 length-scales: short, the Debye screening length r D , and long, D B D 0 h 0 3 /r D 2 defining the layer thickness h C D. The effective extrapolation length D is defined more precisely in eqn (B5) (note that D = D/h 0 ): It shows that D diverges at kD 0 C 1 + 0.5n À1.5 ; for larger kD 0 the PE layer is not formed. For k o 0 and |k|D 0 c1, the layer thickness h decreases as h C D B 1/|k|. While eqn (31) is more precise than the relevant result (eqn (8)) of ref. 7, the two equations fully agree in the limit n -N (note that by definition Àk = 1/d, where d is the surface extrapolation length 7 ). Turning to the excess charge in the high salt regime (n c1), we find that s ex /s C À1 + 2D 0 /D C 1 À 3n À1.5 À 2kh 0 n(1 À 2n À1.5 ).
(32) For 1 ckh 0 4 0, the function s ex = s ex (n) shows a maximum at a large n = n max C (4kh 0 /9) À0.4 with max(s ex /s) C 1-4.61(kh 0 ) 0.6 corresponding to an incomplete charge inversion. At larger ionic strength, n 4 n inv C 0.5(kh 0 ) À1 the excess charge becomes negative entering the charge undercompensation regime. This behavior is in agreement with numerical results of ref. 8 (in the case of 'chemically' repulsive surface). At even higher n 4 n c C (kh 0 ) À1 the adsorbed layer disappears completely (s ex = Às). By contrast, in the case of attractive surface (k = À1/d o 0) the excess charge always increases with the salt amount (cf. Fig. 8b). The full charge inversion is predicted at sufficiently high ionic strength, n 4 n f ' 2 3 k j jh 0 À0:4 At low ionic strength, n { 1, the excess charge is small unless the surface is strongly attractive, k o 0, |kh 0 | c1. In this regime the polymer density profile is nearly exponential, n(z) p exp(À2z/h). The layer thickness and the excess charge are The above equation implies that s ex 4 s for jkj . The polymer density shows significant changes associated with the first two lengths, z* and D. The electrostatic potential profile j(z) involves the two longer lengths, D and r D : As in the other weak adsorption regimes, the potential here changes sign being negative at z B D and positive at z B r D cD. The excess surface charge (cf. eqn (C7)) is positive and low, s ex { s: and it decreases both with B and r D . The above equation is based on the asymptotic behavior of the reduced eigenvalue e established in Appendix C (cf. eqn (C6)). Fig. 12 shows that this analytical prediction is in excellent agreement with our numerical results for B/B 0 \ 1.5. Interestingly, both s ex and h C D are nearly independent of the surface charge s in this regime: As discussed in Appendix C, s ex and h are also nearly independent of the monomer/surface interactions as long as k 4 0 or |kh 0 | o 0.5B/B 0 . For strongly attractive surface, k o 0, The above equations agree with eqn (33) for B = 0. Note that the adsorbed layer thickness h decreases as the monomer/surface attraction gets stronger, but h becomes larger for better solvent quality (larger B) or denser surface charge (higher s), while the excess charge s ex shows the opposite tendencies. The polymer density profile in this regime is given in eqn (C5).
In the high salt regime, n c1, the effect of excluded-volume parameter B is quantitatively equivalent to the effect of monomer/ surface interactionsk ð Þ provided that both parameters are sufficiently small, B { 1/n andk ( 1=n (cf. eqn (B8)). For B þk 4 0, the excess charges ex shows a maximum as a function of ionic strength (n). Such behavior was observed for cationic PEs adsorbed on silica surfaces. 23,25 Conversely, forB þk o 0, a monotonic increase ofs ex and full charge inversion are predicted at high n.
To better illustrate the effect of solvent conditions for n c1 let us consider the case of an indifferent surface (k = 0) and |B| { 1 in more detail. Here the excess charge is (cf. eqn (B6)): s ex ' À1 þ 2 1 þBn 1 À 1:5n À1:5 À Á : In marginal solvent conditions (B 4 0), an overcompensation of surface charge is predicted for n o B À1 (1 À 3B 3/2 ) n inv . In this range of ionic strength the excess charge first increases and then decreases with n showing a maximums ex C 1 À 4.61B 3/5 at n C (4B/9) À2/5 corresponding to a partial charge inversion. The surface charge is undercompensated at a larger ionic strength:s ex o 0 at n 4 n inv . This regime is entered as soon as the Debye length becomes short enough, r D o h*, where the length h* is defined in eqn (35). In poor solvent conditions, B o 0, the excess charge always increases with n. A full charge reversal (s ex 4 s) occurs for n 4 n f with Thus, in a moderately poor solvent the full charge inversion can be easily achieved by increasing ionic strength in the regime n c1. This conclusion is valid both for indifferent or slightly repulsive surfaces. It is backed by our numerical results (see Fig. 11) and agrees with the strong charge inversion effect found numerically in poor solvent conditions. 9 Based on the analytical results fors ex in low and high salt limits (cf. eqn (37)) we propose an analytical interpolation (of Padé type with variable ffiffiffi n p ) valid for any ionic strength n (for k = 0 and |B| { 1): The above equation generalizes eqn (30).

Further remarks
We established that the excess charge in the low-salt regime (r D ch 0 ) is always positive and obeys the general equation (eqn (29)). This result can be compared with the scaling relation The excess charge s ex predicted in Appendix C for C cC 0 at low ionic strength (n { 1) scales as s ex p C À1/3 (cf. eqn (C10)). It is remarkable that the asymptotic analytical equations (eqn (C10) and (C9)) are quantitatively valid for C/C 0 \ 4 as evidenced by the excellent agreement between the numerical and analytical results in this range (cf. Fig. 15b).
In this study we considered weakly charged polyelectrolytes with a small fraction f of charged units. One possible realization is a copolymer with neutral units and a small fraction f of charged units. A more common case is a homopolymer with weakly dissociated groups: each unit of the chain can be charged with probability f. Our theory is applicable in both cases. Note that in the latter case the PE charge is not necessarily constant: it depends on the electric potential. However, for weak potentials required by the DH approximation this dependence can be neglected.
The results on charge inversion considered in this paper may provide some basis for understanding the structure of PE multilayers (PEMs), 3,11,12 but we do not claim a direct relevance of the adsorption results to PEMs. For example, the full charge inversion regime does not necessarily imply that PEM formation is mainly driven electrostatically. We believe however that the quantitative model of PE adsorption considered in this paper and the approach taken to analyse it can be usefully applied to study PEM structure. It is a challenging task to quantitatively assess the role of electrostatic interactions in different regimes of PEM formation. We presume that these interactions always contribute to PEM stability. In fact, as was correctly pointed out in ref. 7, even in the regime of charge undercompensation the electrostatic potential outside the last sublayer of PEM is attractive for the new-coming PE chains forming yet another sublayer. We verified that this feature is generally valid in all the regimes.

Conclusions
1. The adsorbed PE layer structure and charge compensation effect can be described in terms of the minimal set of the following reduced variables reflecting a self-similar structure of flexible PE chains: h = h/h 0 ,s ex = s ex /s, 1= ffiffiffi n p ¼ r D =h 0 ,k ¼ kh 0 , B = B/B 0 , and C = C/C 0 , where h is the layer thickness, s ex = fG À s is the excess charge (in units of proton charge e and per unit area), n is the reduced ion strength, k is the magnitude of specific short-range monomer/surface repulsion (attraction in is the characteristic electro- The reduced parameters depend on the bare surface charge s: n p s À2/3 , k / s À1=3 , B p s 2/3 , and C p s 2 . The following main regimes are distinguished based on the reduced excess charge,s ex , of the PE layer:s ex = À1, no PE layer is formed (the adsorbed charge fG = 0), À1 os ex o 0 charge undercompensation, 0 os ex o 1 partial (incomplete) charge inversion,s ex = 1 full charge inversion, ands ex 4 1 charge inversion with amplification.
2. It is found that in the low salt regime (r D ch 0 ) the PE charge always overcompensates the initial surface charge, and that the excess charges ex is inversely proportional to r D . We established the general asymptotic relations ex C eh 0 /r D , where e is the reduced ground-state eigenvalue involved in the Edwards eqn (4), for n = 0. We calculated e analytically in a number of important asymptotic regimes (fork ) 1, for B c1 and any k, and for C c1) and numerically in intermediate regimes.
For strongly repulsive surfaces we predict the overcharging regime (s ex 4 0) in a very narrow range of low ionic strength, the degree of overcharging being always very small:s ex o 0.01. We also obtained an analytical expression, eqn (B6), for the excess charge in the high salt regime, n c1, and we quantitatively considered the effect of excluded-volume monomer interactions both in the marginally good and poor solvent regimes. All our analytical and numerical results point to the following general trends: excess charge always increases as the solvent quality decreases (lower or negative B), or as the uncharged surface becomes less repulsive or more attractive for PE segments (lower or negative k).
3. The dependence of the excess charge on the ionic strength (parameter n) is more complicated. As is follows from eqn (29) s ex always increases at low n. Its further behavior depends on the parameters k and B: (i) At k 4 0 (repulsive surface) and B 4 0 (marginal solvent) s ex varies with n in a nonmonotonic way reaching maximum (s ex ) max , 1 4 (s ex ) max 4 0, at some n = n max , then decreasing down tos ex = À1 at n = n c . The adsorbed layer disappears at n 4 n c . Such behavior of the PE adsorbance G (nonmonotonic dependence of G = s(1 +s ex )/f on the amount of added NaCl followed by adsorption cutoff) has been observed experimentally for cationic PEs on silica surfaces. 23,25 For strongly repulsive surfaces n max is small, so both the surface coverage G and excess charge s ex mostly decrease with ionic strength (see the dashed curve and the curve fork ¼ 1 in Fig. 8b) in agreement with experimental data on adsorption of weakly charged polycations on mica. 26 (ii) For k 4 0 and B o 0 (poor solvent), s ex monotonically increases with n diverging at the bulk stability threshold n - 4k. In the opposite case,B ok ( 1, the behavior ofs ex is nonmonotonic again like in regime (i).
(iii) For k o 0 (attractive surface) and B o 0, both the excess charge and surface coverage increase monotonically with ionic strength. Such behavior is typically observed for PEs adsorbed onto oppositely charged surfaces with some nonelectrostatic affinity for the polymer. [27][28][29][30] (iv) Finally, for k o 0 and B 4 0, the excess charge increases, reaches a maximum and then decreases down to a plateau value ifB 4k j j, whiles ex monotonically increases up to the plateau at high ionic strength in the opposite case,B ok j j.
The predicted tendencies are in agreement with experimental results 22 showing a decrease of polymer adsorbance G p 1 +s ex with ionic strength c b at a low fraction of charged polymer units ( f = 1%) and an increase of G with c b at higher f = 13% and 30%. (Note thatB k j j ¼ Bs , hence a transition from the regimeB 4k j j toB ok j j is expected as f is increased.) 4. Concerning the effect of surface charge s on charge inversion, we find that normally the reduced excess chargẽ s ex decreases as s is increased. This is always true for indifferent or attractive surfaces. However, in the case of repulsive surfaces (k 4 0) the excess charge can increase with s at low s, sos ex can show a maximum as a function of s. Such behavior in the regimek ( 1, n c1 is implied in eqn (B8).
5. In Y-solvent conditions the full (or stronger) charge inversion (s ex Z s) is expected only with some specific attraction of polymer to the surface (k o 0, |k| Z |k| f , where 1/|k| = d is the surface extrapolation length). The critical surface attraction parameter |k| f always decreases with salt addition following the scaling law |k| f p r 6. The surface layer structure is generally characterized by multiple essentially different length-scales including the PE layer mean thickness h (cf. eqn (24)) and the Debye length r D .
In the case of low ionic strength (long r D ) and significant excluded-volume interactions (B c1 corresponding to high enough surface charge s) the third length z* B h 0 B À1/4 emerged (z* { h { r D ). In Y-solvent conditions with C c1 (this again corresponds to sufficiently high s) two extra lengths are involved: the proximal decay length z* B h 0 C À1/6 showing how fast the effect of surface interactions fades off away from the surface, and the distal decay length x B h 0 C 1/6 characterizing the decrease of polymer density in the region h o z t r D . The characteristic lengths then form the following sequence: We also established (both numerically and analytically) that the adsorbed layer thickness h always increases with ionic strength. This conclusion is in agreement with experimental data. [25][26][27] This condition is compatible with eqn (A1) since f { 1. The same condition (A2) also ensures that the virial approximation is applicable (Cn 2 { 1).
( Appendix B: theory of PE adsorption at high ionic strength Here we consider the adsorption profiles and excess charge for n c 1 at different excluded volumes and surface repulsion parameters, B andk. We start with the simplest case B = 0, k ¼ 0 (ideal backbones, indifferent surface). In this case reduced eqn (4) and boundary conditions arẽ c 00 =Fc,F 00 = Àc 2 + nF, x 4 0, (B1) where prime ( 0 ) means d/dx. The second eqn (B1) can be generally solved (with a givenc) in a standard way yielding (cf. eqn (12) includes the leading correction (for large n). The reduced excess charge is: (note thats ex s ex /s = À1 +s p ands p ¼ Ðñ ðxÞdx G is the reduced charge of the adsorbed PE layer). The first term (=1) is the limiting value obtained by Joanny, 7 and the second term is the main correction. Our analysis shows that other corrections are subdominant (B1/n 3 ). This result indicates that s ex /s o 1, so full charge inversion does not occur in this regime (charge inversion is incomplete).
In the case of poor solvent, B o 0, another restriction, eqn (A3), is applicable. The full charge inversion (s ex = s) is thus predicted for ðBÞ f n ' Àkn 1 À 2n À1:5 À Á À 1:5n À1:5 : where the first equation is approximate: we neglected the c 00 and e terms related to chain entropy in the first eqn (C1). These