Polyelectrolyte Adsorption and Multilayer Formation Jean-Francois Joanny and Martin Castelnovo

3 Polyelectrolyte Adsorption and Multilayer Formation Jean-Francois Joanny and Martin Castelnovo

Abstract

In this chapter, we review our recent theoretical work on the formation of polyelectrolyte multilayers. The first layer is obtained by polyelectrolyte adsorption on a surface with an opposite charge. The adsorbed polyelectrolyte charge is sufficient to invert the charge of the surface. Each subsequent layer is bound to the previous one by the formation of a polyelectrolyte complex. Using a Debye-Hü- ckel-like theory to describe the complexation, we obtain the adsorbance (the adsorbed amount) of each layer and the thickness of the layers.

3.1 Introduction

Recent experiments on polyelectrolyte multilayers, formed by consecutive adsorption of negatively charged and positively charged polyelectrolytes, have shown that, upon addition of a new layer, the number of charges carried by the incorporated polymer is large enough to neutralize the charge of the previous layer and even to invert the sign of the zeta potential (or the global charge of the layer). Neutron experiments also clearly indicate that the newly incorporated layer strongly interpenetrates the previous layer and even the one before but that there is not complete mixing between the layers, each layer keeping its identity [1]. All these results raise a number of theoretical questions that we try to address briefly in this chapter. The first question concerns the first layer of the assembly. It is obtained by adsorption of polyelectrolyte chains on an oppositely charged surface; the adsorption must overcompensate the charge of the surface and the layer must be “strongly anchored to the surface” so that in many cases a polymer such as polyethyleneimine that is different from the polymer used for the following layers is used. Polyelectrolyte adsorption has been studied extensively over recent years and we will summarize the conditions under which charge inversion can occur and the influence of ionic strength on polyelectrolyte adsorption [2]. Starting from the second layer, the new adsorbing layer is formed by adsorption of polyelectrolyte chains of a given charge, interacting with the polyelectrolyte 88 3 Polyelectrolyte Adsorption and Multilayer Formation

chains of the previous layer that have an opposite charge. When they are mixed in solution in water, oppositely charged polyelectrolytes form polyelectrolyte complexes and in most cases precipitate. There is a vast literature on polyelectrolyte complexes, in particular due to the Russian school of Kabanov [3]. Theoretical models have appeared recently that give some predictions for the phase diagram of the complex [4, 5]. We will assume here that polyelectrolyte complexation is the driving force for multilayer formation and that the polymer density inside the multilayer is the density in the dense phase of the polymer mixture (the complex). An important step in the formation of multilayers is the rinsing step between two consecutive adsorptions. This step is aimed at eliminating the non-adsorbed (or the weakly adsorbed) polyelectrolyte. In many experiments, the properties of the multilayer depend for example on the duration of the rinsing step. This indi- cates that the formation of the multilayers is not an equilibrium process and that one should build a kinetic theory which does not seem to be available at the pres- ent time. We will thus limit ourselves here to a thermodynamic theory and im- pose a constrained equilibrium; for example, polyelectrolyte adsorption is as- sumed to be irreversible in the sense that the amount of adsorbed polymer remains constant, even in contact with pure solvent during the rinsing step, and that there is no desorption; however we allow for free equilibration of the chain conformations at each step under this constraint that the adsorbed amount is constant. Another important problem that is not discussed here is the interdiffusion between the layers. It certainly plays a major role in defining some of the properties of the layers when they are used for applications. The properties of polyelectrolyte multilayers are controlled by many parameters such as ionic strength, temperature, pH, the electrostatic charges and the stiff- nesses of the two polymers etc. At each step of the formation, parameters such as the ionic strength can be changed allowing a fine tuning of the multilayer properties. The layer properties also strongly depend on the experimental procedure. In many experiments, the layer is dried at each step, the polymers in the multilayer become glassy and their conformation is frozen. For simplicity, we will assume here that all parameters are fixed during the multilayer formation: there is no change in the salt concentration or in the temperature during the various steps of the formation and that there is no drying of the layers at each step, so that the polymers remain in a liquid state and that a thermodynamic or a constrained thermodynamic equilibrium can be reached. This condition is not fulfilled in most experiments but has been carefully imposed in some recent experiments probing the structure of the monolayers [6]. This short review is organized as follows. In the next section, we give a short summary of the properties of flexible polyelectrolyte in solution. Polyelectrolyte adsorption is considered in the following section and the formation of polyelectrolyte complexes in Section 3.4. All these results are used in Section 3.5 to give a rather rough discussion of the structure of polyelectrolyte multilayers. The last section gives some concluding remarks and discusses some open issues. 3.2 Polyelectrolytes in Solution 89

3.2 Polyelectrolytes in Solution

In this section, we very briefly summarize the solution properties of polyelectrolytes [7]. We first consider a very dilute polyelectrolyte solution in water that we assimilate to a H solvent. Each chain comprises N monomers of size a and a fraction f of the monomers are charged. If the charge is very low, the electrostatic interactions do not play any role and the chain is Gaussian, its end to end distance 1=2 is R0 N a. If the charge is higher, it is stretched by the repulsive electrostatic interactions between monomers. In the presence of a concentration n of added salt, the coulombic interaction between two charges at a distance r is given by the kTl q2 Debye-Hückel potential v r B exp À jr where l is the so-called r B 4pekT Bjerrum length (q is the elementary charge and e the dielectric constant of water) À1 À1=2 and j 8pnlB is the Debye screening length. In the absence of added salt, the Debye-Hückel interaction reduces to the standard Coulomb interaction. In the absence of added salt, the interaction is long ranged and the chains are highly stretched; their size R is given by [8]

2= 1=3 : R Na fB a 1

A weakly charged polyelectrolyte chain is not fully stretched but it canÀ beÁ viewed 2 À1=3 n f lB as an elongated chainÀ ofÁ Gaussian electrostatic blobs of size el a a con- 2 À2=3 f lB taining each gel a monomers as sketched in Fig. 3.1). The size of the blob is such that the electrostatic interaction inside a blob is of the order of kT.As salt is added, the local blob structure of the chain is not affected but if the screening length is smaller than the size of the chain, the chain is no longer fully elongated and bends. It is in general characterized by a persistence length lp which is the length over which it remains stretched. Depending on the theoretical approach, the persistence length is predicted to decrease with the salt concentration as jÀ1 or jÀ2. Experimental results have not allowed us to discriminate between these two results. At very high ionic strength, the electrostatic interaction is short range and is equivalent to an excluded volume f 2 interaction, the corresponding excluded volume parameter is v 4pl . el B j2

Fig. 3.1 Electrostatic blob model. 90 3 Polyelectrolyte Adsorption and Multilayer Formation

When their concentration is increased, polyelectrolyte chains interact strongly and eventually overlap. The electrostatic interaction between chains is much larger than the thermal excitation when the concentration is close to overlap (i. e. when the distance between the chains is of the order of their size). This could in- duce an ordering of the chains but there is no clear evidence for that experimentally. We will thus assume here that semi-dilute polyelectrolyte solutions are a dis- ordered and isotropic liquid. We will view them as interpenetrated chains of blobs that have a local structure identical to that of isolated polyelectrolyte chains shown in Fig. 3.1. The most prominent feature of a polyelectrolyte solution in a scattering experi- ment is a strong peak in the structure factor S q that measures the concentration correlations. At large values of the scattering vector q, the structure factor probes the internal structure of the chains that is roughly the same as in a dilute solu- = 2 n > tion, it thus decays as S q 1 q if q el 1 (characterizing the Gaussian chain = n < structure inside the electrostatic blobs) and S q 1 q if q el 1 (characterizing the elongated structure). At zero wave-vector the structure factor is obtained from the global electroneutrality constraint from the Stillinger-Lovett sum rule S q 01=f . At intermediate wave-vectors the structure factor shows a peak at a wave-vector qÃ; as for neutral polymers in a semidilute solution, the corresponding wave-length 2p=qÃ is the mesh size of the temporary network formed by the overlapping chains. The wave-vector qÃ can be obtained by a scaling argument Ã 1=2 2 1=6 from the radius of a chain in a dilute solution (1) q ca f lB=a . Ãn > 3 > 2 = 1=3 In a concentrated polyelectrolyte solution q el 1orca f lB a the mesh size of the solution is smaller than the electrostatic blob size; the chains are no longer stretched and have Gaussian statistics at any length scale. The effect of the electrostatic interaction is small and the chain structure factor can be calculated from the so-called Random Phase Approximation used for neutral polymers; the position of the peak in the structure factor is given by [9, 10]

Ã 2 2 1=3 q 48 pf lBc=a : 2

When salt is added to a semidilute or a concentrated polyelectrolyte solution, the peak in the structure factor shifts to lower wave-vector and disappears when j ' qÃ. At this point, the electrostatic interactions are sufficiently screened and can be considered as short range. The polyelectrolyte solution behaves as a neutral polymer solution (the polymers having eventually an electrostatic persistence length [7]). The structure factor decreases monotonically with the wave-vector.

3.3 Polyelectrolytes at Interfaces

We now discuss the adsorption of a polyelectrolyte solution on a solid surface that carries an electrical charge per unit area qr with a sign opposite to that of the polymer [2]. The electric field in the very vicinity of the surface is 3.3 Polyelectrolytes at Interfaces 91

E kT=q 4prlB. The field created by the surface decays with the distance from the surface because of the screening by the surface counterions and by the salt molecules. In a first approximation, the electric field can be calculated from the Poisson-Boltzmann equation. In the absence of salt, the screening is due to the counterions. The counterions are confined in the vicinity of the surface within a length k 1= 2prlB which is called the Gouy-Chapman length and the electric 2 field decays with the distance z as E ' . If a sufficient amount of salt is z k added (so that jÀ1 < k), screening becomes dominated by the salt and 2 E ' k exp À jz. In a very low ionic strength solution, a single polyelectrolyte chain is attracted by the electric field of the surface and gets confined within a distance d from the surface. If the surface charge is large enough d < k and the chain feels the surface field. The balance between the electrostatic force and the confinement force for a Gaussian chain gives [11]

2 1=3 d a =f rlB :

This thickness is independent of the chain molecular weight and decreases weakly with the surface charge; it is, in most cases, in the range of a few nano- meters. Note that we have considered here flexible chains and that the thickness of a rigid polyelectrolyte chain adsorbed on an oppositely charged surface would be even smaller [12]. When salt is added to the solution, the structure of the adsorbed chain is not changed as long as the screening length jÀ1 is smaller than the chain thickness d. At higher ionic strength, if there is no short range non-electrostatic attraction between the chain and the surface, the chain desorbs. Adsorption can be induced by a sufficient short range attraction of the chain to the surface. When a dilute polyelectrolyte solution is put in contact with a surface of opposite charge, even if the bulk concentration is rather low, adsorbed chains on the surface overlap. Polyelectrolytes in an adsorbed layer at thermal equilibrium do not develop large loops at very low ionic strength and the thickness of the adsorbed layer is of the order of the thickness d of a single adsorbed chain, as shown by a mean field calculation [13]. The adsorbed chains however have a strong electrostatic interaction between them and it is not clear whether they form a two-dimensional isotropic solution or whether the strong interaction in- duces liquid crystalline or even crystalline order in the direction of the adsorbing plane. Most existing theories predict an overcompensation of the charge of the surface by the adsorbed polymer i.e. the electric charge per unit area due to the adsorbing polymer is larger than the bare charge of the solid surface by dr > 0. The global charge is small in the limit of low ionic strength, a mean field theory based on the Edwards equation for the chain conformation [13] predicts dr=r jd n2 =d2 1 el . This theory is expected to be valid at the scaling level. In the limit of large ionic strength, the polymer behaves as a neutral polymer with excluded volume interactions in the bulk. If the adsorbing surface is a pure 92 3 Polyelectrolyte Adsorption and Multilayer Formation

hard wall with no attractive interactions, there is no adsorption; the adsorption of the monomers to the wall is strongly screened and is not sufficient to compensate the hard wall constraint. If there is a strong enough short range attraction between the adsorbing surface and the monomers, the polymer adsorbs like a neutral polymer. All the results obtained for the adsorption of neutral polymers can be used. The thickness of the layer is the size of the largest loops and is of the order of the size of a polyelectrolyte in dilute solution at the same ionic strength. The monomer concentration profile decays as a power law of the distance z from the solid surface. In a mean field approach

j2a2 c z 4 p 2 2 12 lBf z deff

where deff measures the interaction of the polymer with the surface (both electrostatic and non-electrostatic). A large charge overcompensation is then expected. If the short range attraction exactly compensates the hard wall constraint, the mean field theory predicts on overcompensation dr=r 1; for stronger attraction, the overcompensated charge increases as dr=r j2. One should however keep in mind that the mean field theory based on the Ed- wards equation for the chain conformations ignores the excluded volume correlations and that this result should be only qualitatively correct (one expects a strong overcompensation of the charge that increases with an unknown power of ionic strength). Another limitation of the mean field theory is that it neglects the in- plane concentration fluctuations; other theories [12, 14] that overestimate these fluctuations by assuming that the chains form an ordered array on the surface also predict a strong charge inversion.

3.4 Polyelectrolyte Complexes

When two polyelectrolyte solutions of opposite charges (a polyanion and a polyca- tion) are mixed, the polymers have a tendency to form a dense phase and to separate from the solvent. The dense phase is called a polyelectrolyte complex. Poly- electrolyte complexes have been studied extensively experimentally and show a rich variety of behavior depending on the stoichiometry (the relative molecular weights and charge contents of the two polymers), ionic strength, temperature [3] etc. For simplicity we consider here only symmetric complexes where the two polymers have the same molecular weight N, the same charge fraction f and the same concentration c. We also only consider very large molecular weights; in this limit the two polyelectrolytes precipitate and form a dense complex in equilibrium with pure water; the critical concentration for complex formation is extremely low. Even in this simplified limit, the phase diagram of the polymer mixture in water is not very simple [15, 16]. In general, the backbones of the two polymers are not compatible and repel each other. The repulsion is characterized by a Flory interac- 3.4 Polyelectrolyte Complexes 93 tion parameter v. When the charge fraction is low, the backbone repulsion is dominant and the solution separates into two phases each containing mostly one of the polymers. At high charge fraction, the attractive electrostatic interactions between the polymers dominate and they precipitate to form a complex. In an intermediate range of charge fraction, the equilibrium state can be a mesophase where the two polymers only separate microscopically. A tentative theoretical description of the phase diagram is given in reference [5]. If the two polymers are asym- metric in charge or in mass, more complex structures such as aggregates can form. We only consider here the properties of the complex phase for symmetric polyelectrolyte mixtures in the limit of high ionic strength, which corresponds to most of the experimental results on polyelectrolyte multilayers. The complex is in equilibrium with a very dilute phase (containing salt however and therefore the osmotic pressure of the complex is equal to the osmotic pressure of the simple electrolyte forming the dilute phase). The osmotic pressure difference between the two phases DP has two contributions, an excluded volume contribution and an electrostatic contribution. We will consider here that the two polymers are in a H sol- kT vent and the excluded volume osmotic pressure is written as DP w2c3 ev 3 where c is the total polymer concentration. It is dominated by three body interactions and w2 is the third virial coefficient; w is a volume of the order of the mono- meric volume a3. At the mean field level, the average charge in the polyelectrolyte complex vanishes and the electrostatic energy also vanishes. In order to calculate the electrostatic free energy of the polyelectrolyte complex one must take into ac- count the concentration fluctuations, as one does for example in the classical De- bye-Hückel theory of simple electrolytes [4]. The fluctuation free energy is however different from that of a simple electrolyte because the charges are connected along the polymer chains. An important point is that, in a polyelectrolyte complex, there are free ions due to the added salt but there are also counterions asso- ciated with the two polyelectrolytes. We will suppose that these ions are identical and consider them as added salt. Note however that, in certain experiments, the counterions can be dialyzed out. In the limit where the salt concentration is large, j DP ÀkT n a the electrostatic pressure can be written as el where c = n3 p 2 1 2 c 4 lBf c j=qÃ2 is the relevant correlation length inside the complex (qÃ is the wave-vector characteristic of the charge fluctuations in a dense polyelectrolyte solution given by Eq. (2). The polymer concentration in the complex is then [17]

f 2 c : 5 c nw4=3a2 94 3 Polyelectrolyte Adsorption and Multilayer Formation

3.5 Multilayer Formation

We now use the results presented in the previous sections on polyelectrolyte adsorption and polyelectrolyte complexation to build up a theory for multilayer formation [17]. We start from a positively charged surface. The first layer is obtained by adsorbing a solution of negatively charged polyelectrolyte at high ionic strength. As explained above, there is a strong inversion of the charge and this inversion is higher if there exists a short range non electrostatic attraction between the polymer and the surface. A strong adsorption of the first layer is necessary to anchor the polymer on the surface and in most experimental cases, the first layer is made with a different polymer known for its good adsorption properties. We call C 1 the adsorbance of this first layer. If the short range attraction of the monomers to the surface is large C1 >> 2r=f . The layer is then rinsed. We will suppose that the layer is not dried in any of the rinsing steps and that it is rinsed in water at the same ionic strength as the adsorbing solution. We moreover will make these assumptions for all the following rinsing steps. This ensures that the multilayer is always liquid and never becomes glassy and that it is always in contact with water at the same ionic strength. These restrictions are usually not necessary when multilayers are pre- pared for various applications but they largely simplify the theoretical analysis and they correspond to the recent experiments of Ladam and coworkers [6]. We will also consider that polyelectrolyte adsorption is irreversible, in the sense that when the adsorbed layer is put in contact with the simple electrolyte solution, no poly- C mer desorbs. The amount of polymer in the first layer remains equal to 1 even after the rinsing step. We therefore ignore for the first layer (and for all the consecutive layers) the small desorption that may occur during the rinsing step. We do not however consider the adsorbed layer as frozen and we allow for reequilibra- tion of the local chain conformation during the following steps of the multilayer build-up. Multilayer formation is a non-equilibrium process; we treat here the multilayer as a thermodynamic system under constrained equilibrium: the quantity of adsorbed polymer in each layer remains fixed. After the rinsing step, the surface with the first layer is put in contact with a solution of positively charged polyelectrolytes. The positively charged polyelectrolyte is excluded from the vicinity of the solid surface that has the same electrostatic n charge over a layer of thickness c (the correlation length of the polyelectrolyte complex). It complexes all of the available negatively charged polyelectrolytes of

the first layer by forming a complex of concentration cc given by Eq. (5). The thickness of this complexed layer is L1 2C 1=cc. The surface carries only a quantity C1 of negatively charged polymer and the concentration of negatively charged polymer decays rapidly to zero from the value cc/2 over the complex correlation n length c at the edge of the layer. The positively charged polymer is in equilibrium with a bulk solution and can form loops anchored in the complex and dan- gling into the electrolyte solution. The loop structure is the same as in an adsorbed polyelectrolyte layer and is given by Eq. (4) where the cut-off length is cho- 3.5 Multilayer Formation 95 sen by imposing that at z 0 (at the edge of the complex layer), the concentration matches to that in the complex cc=2. The quantity of polymer in these loops is obtained by integration ! 2 DC 1 À f : = 1 = 6 w2 3 3=2 1 2 2 2=3 n lB a w

The first term is readily obtained from the results on polyelectrolyte complexation of the previous section. It is however independent of ionic strength and the de- pendence on ionic strength is due to the first correction to this result that re- quires a more detailed analysis of the polyelectrolyte complexes. The loops in the electrolyte solution create an overcompensation of the charge of the multilayer (with an effective surface charge f DC). They serve to anchor the next layer by complexation with negatively charged polyelectrolyte. A qualitative plot of the con- centrations in the first two layers is sketched in Fig. 3.2. The build-up process can then be iterated. Each consecutive layer has an adsorbance of order DC and the thickness of all the layers starting from the third one is

w2=3na2 L DC=c : 7 3 c f 2

In this picture the effective surface charge of the multilayer oscillates between the values f DC and Àf DC after deposition of the successive layers (starting from the second layer) and the amount of polymer per unit area in each layer (starting from the third one) increases very weakly with the ionic strength as given by Eq. (6). This is in agreement with the experiments of reference [6].

Fig. 3.2 Concentration profiles of the negatively charged (full line) and positively charged (dash-dotted line) polymers after the formation of the second layer on a positively charged surface. 96 3 Polyelectrolyte Adsorption and Multilayer Formation

3.6 Concluding Remarks

We have summarized in this short review a model for polyelectrolyte multilayer formation which is based on the complexation between consecutive layers. This model is clearly valid for flexible polyelectrolytes only. Even in that case it is rather rough and makes important approximations. The model has been presented for high ionic strength only. Experimentally, the structure of a polyelectrolyte multilayer does not seem to be strongly affected by ionic strength and we believe that the model could be used for weaker ionic strength as well. For weak ionic strength, the electrostatic pressure in a polyelectrolyte complex scales as DP ÀkTqÃ3, the polymer density in a polyelectrolyte 1=3 2=3 lB f complex varies as cc = = and the adsorbance of the layers is 2=3 1=2 1=6 a2 3w8 9 a n lB DC* = = . f 2 3w4 9 We have also ignored in the model the effect of the short range repulsive interactions between the polymer backbones. It has been argued [18] that the short range interaction could lead to the formation of mesophases. The formation of mesophases is studied in detail in reference [5] and they do not seem to be stable at high ionic strength. The experimental results also prove clearly that the multilayer build-up is a non-equilibrium process and not only due to the orientation of lamellar mesophases along the solid substrate. The whole theory is based on linear electrostatics at the level of the Debye- Hückel theory and can certainly not be applied if the local density of charges is too large. It is thus valid only for weakly charged polymers. In the limit of strongly charged polymers, a model based on the formation of ion pairs would certainly be more adequate [19]. A final important limitation is that we have used a quasi-equilibrium approach treating the irreversibility as constraints on the thermodynamics. This should be a reasonable approximation to describe the internal structure of the monolayers but does not explain the fact that the dynamics inside the monolayers seems frozen and that the layers keep their identity. One of the major interest of polyelectrolyte multilayers is their versatility. In any layer, polyelectrolytes can be replaced by other charged objects (surfactants, colloids, proteins etc.) and they seem to be obtained for all kinds of polyelectrolytes. Our model is rather specific for flexible polyelectrolytes. If the polymers are too rigid, interpenetration between different layers is difficult and the complexation mechanism is different. In reference [12], a model has been proposed in the other extreme limit where the polyelectrolytes are rodlike. In this case the charge inversion in the first layer is due to the strong correlations between adsorbed chains that form an ordered array and each layer is treated as adsorbing on an effective surface formed by the previous layers. 3.7 References 97

3.7 References

1 G. Decher, Science 1997, 277, 1232. See 9 V.I. Borue, I. Erukhimovich, Macro- also the other chapters of this book. molecules 1988, 21, 3240. 2 D. Andelman, J.F. Joanny, C.R. Acad. 10 L. Leibler, J.F. Joanny, J. Phys. Paris Sci. Paris IV 2000, 1, 1153. 1990, 51, 545. 3 E. Bekturov, A. Bakauova, Synthetic 11 O. Borisov, E. Zhulina, T. Birhstein, Water Soluble Polymers in Solution, Hütig J. Phys. II Paris 1994, 4, 913. and Wopf Verlag, Berlin 1986; B. Phi- 12 R. Netz, J.F. Joanny, Macromolecules lipp, H. Dautzenberg, K. Linow, J. 1999, 26, 2026. Kötz,W. Dawydoff, Prog. Polym. Sci. 13 J.F. Joanny, Eur. Phys. J.B 1999, 9, 117. 1998, 14, 823. 14 T. Nguyen, A. Grosberg, B. Shklovs- 4 V.I. Borue, I. Erukhimovich, Macro- kii, J. Chem. Phys. 2000, 113, 1110. molecules 1990, 23, 3625. 15 S. Djadoun, R. Goldberg, H. Mora- 5 M. Castelnovo, J.F. Joanny, Eur. Phys. wetz, Macromolecules 1977, 10, 1015. J. E 2001, 6, 377. 16 A. Khokhlov, I. Nyrkova, Macromole- 6 G. Ladam, P. Schaad, J.C. Voegel, P. cules 1992, 25, 1493. Schaaf, G. Decher, F. Cuisinier, Lang- 17 M. Castelnovo, J.F. Joanny, Langmuir muir 2000, 16, 1249. 2000, 16, 7524. 7 J.L. Barrat, J.F. Joanny, Adv. Chem. 18 F. Solis, M. Olvera de la Cruz, J. Phys. 1996, XCIV, 1. Chem. Phys. 1999, 110, 11517. 8 P.G. de Gennes, P. Pincus, R. Velasco, 19 K. Zeldovitch, A. Khokhlov, preprint F. Brochard, J. Phys. Paris 1976, 37, 2000. 1461.