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Formation of Polyelectrolyte Multilayers Martin Castelnovo* and Jean-Franc¸ ois Joanny Institut Charles Sadron (CNRS UPR 022), 6 rue Boussingault, 67083 Strasbourg Cedex, France Received February 14, 2000. In Final Form: May 25, 2000 A mechanism for the formation of polyelectrolyte multilayers is proposed. All of the experiments on such systems show that there is strong interpenetration of consecutive layers. We explain the very strong stability of the multilayers by the complexation between polyelectrolytes of opposite signs. Scaling laws on bulk polyelectrolyte complexation and their applications to the case of multilayers are derived in analogy with the description of neutral polymers under poor solvent conditions. Qualitative agreement is reached with some recent experiments.
1. Introduction Polymer adsorption provides powerful tools for monitoring various interfacial properties such as the stabilization of colloidal suspensions or adhesion and lubrication processes.1 The thickness of an adsorbed polymer layer depends generally on the choice of the substrate and on the chemical nature of the polymer, but it is always limited by the saturation of the adsorbed polymer amount in the layer: the size of the adsorbed layer does not exceed the polymer size in a bulk solution if adsorption proceeds from a dilute solution. Therefore, one cannot use simple adsorption to build up relatively thick films at the molecular level but still of ultrathin thickness at the macroscopic level. Thick films with a rather well-controlled thickness have been obtained with the introduction of the multilayer concept2 based on the sequential adsorption of at least two polymer species having attractive affinities with each other. The first experiments on multilayers made of oppositely charged polyelectrolytes are due to Decher et al.3 This process now seems to be one of the most versatile and efficient ways to build up films with required properties and a well-controlled thickness. Various examples of new applications are, for example, new biologically active filters with included enzymes at different depths or light-emitting thin films and sensors.4 Other geometries can be used to grow the multilayers. Sukhorukov et al. have built up multilayers onto spherical charged colloids and then dissolved the initial substrate by changing pH conditions.5 They obtained this way ultrathin polyelectrolyte shells that seem to have very promising properties as microcapsules. Since the introduction of polyelectrolyte multilayers, there have been a number of experiments devoted to the understanding of the formation mechanisms.6 The internal structure of the multilayers was probed by neutron reflectometry7,8 and shows a strong interdigitation of two (1) Vincent, B. Adv. Colloid Interface Sci. 1974, 4, 193. (2) See for example: Hong, J. D.; Decher, G. Makromol. Chem., Macromol. Symp. 1991, 46, 321. (3) Schmitt, J.; Decher, G.; Hong, J. D. Thin Solid Films 1992, 210/ 211, 831. (4) See, for example, refs inside: Li, M.; Schlenoff, J. B.; Ly H. J. Am. Chem. Soc. 1998, 120, 7626. (5) Sukhorukov, G. B.; Donath, E.; Davis, S.; Lichtenfeld, H.; Caruso, F.; Popov, V. I.; Mo¨hwald, H. Polym. Adv. Technol. 1998, 9, 759. (6) See for example: Fleer, G.; Hoogeveen, N. G.; Cohen Stuart, M. A. Langmuir 1996, 12, 3675. Helm, C. A.; Lowack, K. Macromolecules 1998, 31, 823. (7) Schmitt, J.; Gru¨newald, T.; Decher, G.; Pershan, P. S.; KJaer, K.; Lo¨sche, M. Macromolecules 1993, 26, 7058.
consecutive layers. These neutron experiments have been performed with dry samples and not under the same conditions where the multilayers were grown; they are, nevertheless, relevant from a practical point of view because most of the applications use dried films. The internal structure has been confirmed by the in situ recent experiments of Ladam et al. performed at each step with well-controlled solvent and salt conditions with no drying steps.9 These experiments allow, in principle, quantitative comparisons with models of polyelectrolyte multilayer formation. Although the basic idea of the polyelectrolyte multilayers is fairly simple, the theoretical description is quite complex because of the long range of the Coulombic interaction attaching layers to each other. So far, there have been only a few attempts to model polyelectrolyte multilayers. Solis and Olvera de la Cruz have studied the spontaneous equilibrium layering of a polyelectrolyte mixture close to a charged wall10 due to microphase formation. Although this is quite different from the multilayer formation process, the characteristic equilibrium length scales are similar; we do not believe, however, that this is a reasonable model for polyelectrolyte multilayers. Netz and Joanny propose scaling laws for the multilayer formation in the case of semiflexible polyelectrolytes.11 No strong interdigitation of consecutive layers is then expected because of the rigidity of the chains; the conditions required to build up multilayers are discussed. Our present work can be viewed as complementary to this theoretical description because we focus on flexible polyelectrolytes. The charge overcompensation found in the adsorption of a single polyelectrolyte layer onto a charged wall is the starting point of any model of multilayer formation, because the overcompensated charge allows the adsorption of a new layer of opposite charge. It has been described analytically by one of us in the asymptotic limits of high and low ionic strength.12 We adapt here this calculation to derive scaling laws describing polyelectrolyte multilayers. (8) Lo¨sche, M.; Schmitt, J.; Decher, G.; Bouwman, W. G.; Kjaer, K. Macromolecules 1998, 31, 8893. (9) Ladam, G.; Schaad, P.; Voegel, J. C.; Schaaf, P.; Decher, G.; Cuisinier, F. Langmuir 2000, 16, 1249. (10) Solis, F. J.; Olvera de la Cruz, M. J. Chem. Phys. 1999, 110, 11517. (11) Netz, R. R.; Joanny, J. F. Adsorption of semi-flexible polyelectrolyte on charged planar surfaces: charge compensation, charge reversal, and multilayer formation. Preprint, 2000. (12) Joanny, J. F. Eur. Phys. J. B 1999, 9, 117.
10.1021/la000211h CCC: $19.00 © 2000 American Chemical Society Published on Web 08/22/2000
Formation of Polyelectrolye Multilayers
Figure 1. Sketch of the multilayer deposition technique. The initial substrate is negatively charged, the thin chains represent polycations, and the thicker chains represent polyanions.
The paper is organized as follows. In section 2, we present the results of structural investigations on polyelectrolyte multilayers. We then give in section 3 a simple scaling description of polyelectrolyte complex formation for symmetric chains that we believe is the relevant mechanism for binding each layer to the previous one. This model is applied to polyelectrolyte multilayers in section 4: the adsorption profile of the first polyelectrolyte layer in the limit of high ionic strength is first rederived; the anchoring of the second layer to the first one is then explained by a complexation mechanism. This model provides scaling laws for the single-layer thickness and the total charge of the multilayer. The extension to different conditions such as the low ionic strength limit is discussed in section 4.3. The last section presents some concluding remarks. 2. Internal Structure of the Multilayer Polyelectrolyte multilayers are built up with a rather simple experimental setup sketched in Figure 1. If one starts for example from a negatively charged substrate to grow the film, the first layer is deposited through polymer adsorption by dipping the surface into a polycationic solution (positively charged). The main driving force for this adsorption is, of course, electrostatics. The next step is to rinse the surface with water in order to remove the polymers that are not tightly bound to the substrate. This ensures that no free polycations interact in solution with the other components that will be put in contact with the film. Measurements of the ζ potential of this surface show that there is a charge overcompensation: one can therefore adsorb on the surface a new polymer layer of opposite sign electrostatically. This is done by dipping the substrate into a polyanionic solution (negatively charged). A new rinsing step is needed to obtain an irreversibly adsorbed layer. Again one observes that there is a charge overcompensation. The process for the build up of the next layers is similar, and it seems that there is no limitation on the number of layers.13 Notice that all of the steps of the experiments, including the rinsing steps, have to be done with the same ionic strength and without any drying steps to avoid phenomena not related to the bare build-up mechanism. The in situ experiments of Ladam et al. are done within this spirit.9 They allow a quantitative comparison with our model. The properties of the first layers depend on the substrate used to build up the multilayers. The first layer is usually chosen for its high adsorbance in order to “anchor” strongly (13) Decher, G. Private communication.
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the multilayer to the substrate. After the first four or five layers, the experiments show a linear increase of the multilayer thickness with the number of layers and the ζ potential oscillates symmetrically around the zero value, indicating that the system reaches a kind of stationary regime with universal properties. The thickness of one layer depends strongly on the salt amount but not on the surface charge of the substrate. The only effect of this surface charge in this stationary regime is to shift the overall thickness of the multilayer: it increases for an increasing surface charge. The internal structure of multilayers has been probed mainly by neutron reflectometry.8 When every polyanion layer is deuterated, one expects to see Bragg peaks if the different polymer layers do not interdiffuse into each other. The experiments show, however, that two consecutive layers are strongly interdigitated and one has to label every other polyanion layer to see a long-range order within the multilayer: although the layers are locally interdigitated, the chains do not diffuse from one polyanion layer to the neighboring polyanion layers. The net charge of the multilayer is carried by the last layer, with the preceding layers being globally neutral.4,14 According to the experiments of ref 4, the polyelectrolytes within the multilayer behave locally like bulk polyelectrolyte complexes formed by the same polymers. Our model, based on polyelectrolyte complexation between two polyelectrolyte layers, is therefore consistent with these experimental observations. There are a number of other experimental investigations on various properties of multilayers, but only the experiments just described will be used as a starting point to make a model for the general trends of multilayer formation. Therefore, we will not proceed further in describing those additional studies. 3. Thermodynamics of Polyelectrolyte Complexation Polyelectrolyte complexes are formed by mixing in solution two polyelectrolytes of opposite charges. They have been extensively studied experimentally, in particular by the Russian school of Kabanov.15 When parameters such as the ionic strength or the stoichiometry are varied, a very broad variety of phases have been observed. For the sake of simplicity, we only consider here symmetrical complexes where the polycations and polyanions have the same degree of polymerization and carry the same charge in absolute value. Experimentally, if the concentration is not extremely low, there is a macroscopic phase separation in the solution between a dense neutral phase (with a 1:1 stoichiometry) and a very dilute polyelectrolyte phase. From a theoretical point of view, one has to deal with a multicomponent phase equilibrium: in the solution, there is equilibrium between isolated charged chains, neutral dimeric complexes made of one polyelectrolyte of each sign, trimeric complexes made of one polyanion and two polycations for example, and so on. The dense phase corresponds to the aggregation of the neutral dimeric complexes. A very similar problem has been studied theoretically by Everaers et al. for the structure of polyampholyte solutions.16 We do not need such a level of sophistication to understand the complexation in polyelectrolyte multilayers. We will restrict (14) Farhat, T.; Yassin, G.; Dubas, S. T.; Schlenoff, J. B. Langmuir 1999, 15, 6621. (15) See for example: Philipp, B.; Dautzenberg, H.; Linow, K. J.; Ko¨tz, J.; Dawydoff, W. Prog. Polym. Sci. 1998, 14, 823. (16) Everaers, R.; Johner, R.; Joanny, J. F. Macromolecules 1997, 30, 8478.
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ourselves to the modeling of the dense phase in equilibrium with pure solvent in analogy with the description of globular structures of neutral polymers under poor solvent conditions. This simplified model captures all of the physics required to describe the polyelectrolyte multilayers. Our theoretical description of polyelectrolyte complexes follows the lines of Borue and Erukhimovich.17 Following their approach, in the dense phase, we treat the electrostatic interactions via the random-phase approximation (one loop). We, however, consider here only polymers in a Θ solvent, where the chain statistics is Gaussian, and not in a good solvent. We consider an equilibrium between two phases: a dense phase containing polycations, polyanions, small positive ions, and small negative ions at respective concentrations c+, c-, n+, and n- in a Θ solvent, in equilibrium with a pure solvent with a concentration n of monovalent salt. For a symmetrical complex of stoichiometry 1:1, c+ ) c- ≡ c and n+ ) n- ≡ n˜ . To derive the equilibrium concentrations, one has to calculate the free energies of the system in the two phases. The dense phase is treated within the random-phase approximation (RPA). This approximation is a generalization of the DebyeHu¨ckel theory of strong electrolytes. It is a mean-field theory corrected with fluctuations treated at the Gaussian level. It is accurate for relatively dense systems where concentration fluctuations are small. For example, it is known that neutral polymer melts are well described by RPA, unlike semidilute solutions where correlation effects are missed with this approximation. The free energy per unit volume of the dense phase is derived in appendix A. It reads as
Fdense w2(2c)3 ∆Fdense ) + 2n˜ ln(n˜ a-3) + kT 6 kT
(1)
where, in the limit of very large molecular weights, we have neglected the translational entropies of the polymers. The first term of eq 1 is the three-body interaction for polymer solutions in a Θ solvent. It is added to stabilize the complex; otherwise, the polarization term ∆Fdense/kT would collapse the system into an infinitely dense structure. The second term is the translational entropy of the small ions. The last term is the RPA correction to the mean-field free energy:
∆Fdense 1 ){ξ-3 + κ3(1 - s-1)(1 + 2s-1)1/2} kT 12π
(2)
where we have defined the relevant length scales
κ2 ) 8πlBn˜ q/4 ) 96πlBf 2c/a2 ξ-2 ) 48w2c2/a2
(3)
κ-1 is the Debye screening length associated with the small ions of the dense phase, q/-1 is a screening length associated with the screening of the electrostatic interactions by the charges on the polymers that explicitly takes into account the connectivity of the chains,18 and ξ is the correlation length of a polymer solution in a Θ solvent. The competition between the screening by the free ions and the charged monomers is measured by the parameter s ) κ2/q/2. Notice that, for a simple electrolyte, the RPA is equivalent to the Debye-Hu¨ckel model. The first term (17) Borue, V. Y.; Erukhimovich, I. Y. Macromolecules 1990, 23, 3625.
of eq 2 exists also in the absence of charges along the chains and scales such as w/a3 times the mean-field term w2c3. It can be adsorbed in the three-body interaction by a renormalization of the third virial coefficient in the limit w/a3 , 1. We ignore it in the following. We treat the dilute phase as a simple electrolyte solution for which the Debye-Hu¨ckel approximation is known to be valid. The free energy per unit volume, therefore, reads in the dilute phase as
Fdilute κo3 -3 ) 2n ln(na ) kT 12π
(4)
with the inverse screening length in the dilute phase being given by κo2 ) 8πlBn. The equilibrium concentrations n˜ and c are obtained by balancing the small ion chemical potentials in the two phases and the osmotic pressure inside and outside the complex. As we are dealing with a symmetrical complex, we need not introduce a Donnan membrane potential.19 This would not be the case if the overall net charge of the polymers (without their counterions) were nonzero, leading to an asymmetry between the two phases for the small ions. The equality of the chemical potentials leads to
ln
3κo3 1 ∂∆Fdense n˜ )n 48πn 2 ∂n˜
(5)
The osmotic pressure in the dense phase is obtained by derivation of the free energy Πdense ) n˜ (∂Fdense/∂n˜ ) + c (∂Fdense/∂c) - Fdense. A similar formula is used in the dilute phase. We want to study here complex formation in the high ionic strength limit. In this limit, the dimensionless ratio s is such that s-1 , 1. The Debye-Hu¨ckel approximation is only valid if ) κlB is small. We make a systematic expansion of all thermodynamic quantities as a function of the two small parameters up to first order in and third order in s-1. We first calculate the small ion density n˜ using the chemical potential balance and then write the osmotic pressure difference ∆Π between the two phases. At lowest nonvanishing order, we find 2 3 κo3 -3 ∆Π w (2c) ) s kT 3 24π
(6)
The polymer density inside the complex results thus from a balance between the three-body interaction and an attractive electrostatic pressure. This attractive contribution scales as -kTq/6/κ3 ∼ -f 3c3/2n-3/2a-3. This scaling behavior is qualitatively understood as follows: we introduce the electrostatic excluded volume between charges of the same sign vel ) 4πlBf 2/κo2; in a mean-field approach, the correlation length associated with this interaction is ξel ≡ (a2/12velc)1/2; the free energy associated with the concentration fluctuations calculated within the RPA formalism is of order kT per volume of size ξel; it scales, therefore, as -kTξel-3 in agreement with the first contribution to the RPA calculation. (18) This length can be derived with the following argument. We evaluate the electrostatic energy of a piece of chain of length q/-1 for a dense solution of polyelectrolytes without any counterions. Inside a shell of radius q/-1, there are cq/-3 monomers. The electrostatic energy of a charged monomer surrounded by its neighbors in the shell is simply kTlBf 2cq/-3/q/-1. The number of monomers along the piece of Gaussian chain is q/-2a-2. When this energy is equal to kT, it defines the correlation length of the system. (19) Hill, T. L. An introduction to statistical thermodynamics; Dover Publications: New York, 1986.
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The monomer concentration in the complex is obtained for ∆Π ) 0:
cc ∼ f 2/a2nw4/3
(7)
The structure of the complex is very similar to that of a dense polymer phase in a poor solvent: it is a close packing of blobs of size ξel(c)cc), each blob having an internal Gaussian structure with gblob ∼ (ξel/a)2 monomers. The polymers will form complexes if the attractive energy overcomes the entropy of the free isolated chains, namely, if c/N , ξel-3. This amounts to the requirement that the chains are not composed of a single blob (N . gblob). The last inequality imposes a value for the so-called “saltsuppression” of the complex. This phenomenon is supported by experimental facts in bulk polyelectrolyte complexes.15 All of these results have been obtained by assuming that the ionic strength is large, i.e., that the dimensionless ratio s-1 ) q/2/κ2 is small. This approximation is valid if the salt concentration is large enough, n > f 4/3/a4/3lB1/3w4/9. Below, we need the first correction to the complex concentration when the ionic strength is not too high. It can be calculated by expanding the osmotic pressure up to the next order we find, dropping all numerical prefactors:
cc ∼
(
)
f2 f2 1 a2nw4/3 n3/2lB1/2a2w2/3
In this section, we discuss the adsorption of the first two layers in a polyelectrolyte multilayer focusing on the adsorbed amount, the possibility of charge inversion, and the equilibrium thickness of the layer. We only consider the limit of high ionic strength, which seems to be the most relevant experimentally. For the first layer, we briefly recall the results obtained in ref 12 from a mean-field theory. For the second layer, we use our simplified theory of polyelectrolyte complexes. Within our approach, the global properties of the multilayer can be inferred from the properties of the first two layers. 4.1. Adsorption of the First Layer. In this section we describe the adsorption of a polyelectrolyte solution onto a surface of opposite sign. This problem has been studied by several authors within the mean-field approximation.20 We follow here the approach of ref 12. We consider monodisperse flexible polyelectrolytes in a Θ solvent with some added salt, each chain of length N having Nf positive charges. We assume for the sake of simplicity that the charge distribution is smeared out along the backbone of the polymer. The monomer size is denoted by a. The substrate bears -σ charges/unit area. The simplest way of taking into account the presence of a monovalent salt in the solution is to assume that the electrostatic interactions are screened for distances larger than κ-1. Within the Debye-Hu¨ckel model, the electrostatic potential created by an elementary charge reads
lBkT exp(-κr) e r
a2 ∂2ψ ) (fV(z) - µ)ψ(z) 6 ∂z2
(9)
with lB being the Bjerrum length, lB ) e2/4πkT. The inverse Debye length is related to the salt concentration n by κ2 ) 8πlBn. The polymer concentration profile close (20) See for example: Borukhov, I.; Andelman, D.; Orland, H. Europhys. Lett. 1995, 32, 499.
(10)
where the so-called polymer order parameter is related to the local concentration by c(z) ) ψ2(z). The surface is in contact with a dilute solution of chemical potential µ, which plays the role of the energy of the ground state within the quantum mechanics analogy. The dimensionless electrostatic potential (V(z) ) eφ(z)/kT) is determined selfconsistently from the Debye-Hu¨ckel equation for low surface charges:
∂2V ) κ2V(z) - 4πlBfψ2(z) ∂z2
(11)
Two boundary conditions on the substrate must be added:
1 ∂ψ ψ ∂z ∂V ∂z
(8)
4. Multilayer Formation at High Ionic Strength
φ(r) )
to the surface is described by the Edwards propagator approach in the ground-state dominance approximation.21 This mean-field theory is based on the Schro¨dinger-like equation verified by the polymer propagator. In the limit of infinite chains, this equation reads
|
|
1 d
(12)
) 4πlBσ
(13)
)z)0
z)0
The first boundary condition is the usual one for neutral monomers adsorbing on a surface under the influence of a short-range potential. The short-range potential includes here all of the interactions of nonelectrostatic origin (van der Waals, hydrogen bonding, hydrophobic effect, ...). The extrapolation length d decreases as the strength of the attraction increases. The second boundary condition is standard electrostatics. In the limit of high ionic strength, the typical length scale of variation of the order parameter is much larger than the screening length. Therefore, we can assume that the order parameter is constant in eq 11, and this leads to the effective potential
V(z) ) -
4πlBσ -κz 4πlBf 2 e + 2 ψ (z) κ κ
(14)
The electrostatic potential is written as the sum of an attractive short-range term and a term proportional to the local concentration of the polymer c(z). Following Joanny,12 the short-range term can be incorporated into a new effective boundary condition for the order parameter by integrating eq 10 between 0 and a distance larger than the screening length but smaller than any typical length scale of variation of the polymer concentration. This boundary condition is rewritten as
-
1 1 1 1 ∂ψ ) ) + ψ ∂z deff d del
(15)
where we have introduced the electrostatic extrapolation length del ) κ2δ3. δ is the thickness of one adsorbed polyelectrolyte chain adsorbed onto a charged wall:22 δ-3 ) 24πlBσf/a2. (21) de Gennes, P.-G. Scaling concepts in polymer physics; Cornell University: Ithaca, NY, 1979. (22) Borisov, O.; Zhulina, E.; Birshtein, T. J. Phys. II Fr. 1994, 4, 913.
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The second contribution to the electrostatic potential shows that, at high ionic strength, the screened electrostatic interactions are equivalent to short-range excludedvolume interactions. As mentioned above, the electrostatic excluded volume is
vel )
4πlBf 2 κ2
(16)
The equation for the order parameter now describes the adsorption of a neutral polymer solution with an excluded volume vel adsorbing on a neutral surface under the action of a short-range potential with an extrapolation length deff. This problem has been extensively studied,23 and it is known that the polymer concentration decays as a power law. When the bulk polymer concentration is very small, the concentration profile reads as
c(z) )
a2 3vel(z + deff)2
(17)
Within the mean-field approach, the decay exponent is 2; excluded-volume correlations are important for this problem, and for a polymer in a good solvent, it is known that the concentration profile decays as a power law with an exponent close to 4/3. In the very rough theory that we make here, we keep the mean-field approach, which leads qualitatively to the correct physical picture and allows complete calculations. A scaling theory using good solvent exponents is possible; one, however, has to be cautious for the treatment of the short-range attraction (proximal effect). We can evaluate the total adsorbed amount of the first layer as
Γ1 )
a2 2σ = 3veldeff f
Figure 2. Concentration profile of the first two phases. The thin line represents polycations, while the thicker line represents polyanions. Zone 1: neutral complex made of polyanions and polycations. Zone 2: dangling loops of polyanions carrying the net charge of the multilayer.
(18)
if we neglect the nonelectrostatic part of the adsorbing potential. There is thus a charge overcompensation that will allow the adsorption of a new polyelectrolyte layer of opposite sign in the multilayer. In practice, it seems that the first layer must be chosen in such a way that it ensures a good adhesion of the multilayer on the solid substrate. In the experiments of ref 9, a different polycation (polyethylenimine) is used for the first layer. We believe that the nonelectrostatic interaction is strongly attractive in this case. The adsorbed amount is then larger than that given by eq 18, and the charge overcompensation is even larger. In the following, we make an essential assumption for our build-up mechanism. We suppose that, in the rinsing step and in all of the following steps, the amount of adsorbed polycations of this first layer remains fixed. We thus ignore any desorption. It is an experimental fact that polymer desorption is a very slow process and that, in a reasonable experimental time scale, only a very small fraction of an adsorbed polymer layer exposed to free solvent does desorb. We will also make the same assumption for all of the following layers. We assume that the adsorbed amount is fixed in each layer, but we allow for a reequilibration of the structure of the layers at each step at a fixed adsorbed amount. Note that we impose here an irreversibility in the build-up process. Our description is, therefore, not quite an equilibrium description but rather a succession of dynamically trapped states. (23) See for example: Semenov, A. N.; Bonet-Avalos, J.; Johner, A.; Joanny, J. F. Macromolecules 1996, 29, 2179.
4.2. Adsorption of the Second Layer. We now consider the adsorption of the second layer. The surface with its first adsorbed layer is put in contact with a polyanionic solution. Because the experiments show that there is a strong interpenetration between neighboring layers, our model is based on polyelectrolyte complex formation between the polycations of the first layer and the polyanions of the second layer. The complex cannot form in the very vicinity of the solid surface because of the repulsive interactions between the polyanions and the surface charges. We, therefore, expect a concentration profile as sketched in Figure 2: there is a depletion zone for the polyanions close to the substrate at least of thickness κo-1. In fact, we show in appendix B with a simple model that the thickness of the depletion layer scales as ξel . κo-1. The transition toward a neutral complex structure takes place over the first layer of electrostatic blobs. This is consistent with the interfacial properties of a collapsed structure where the correlation length is ξel.24 The polycation profile decays from the wall to the density of the neutral complex. We will assume this density to be cc. We thus assume that the complexation mechanism is not changed close to a charged substrate and if one of the polyelectrolytes adsorbs. This may only be a rough approximation, but we believe that, even if this is not exact for the formation of the second layer, it will become exact when we consider the formation of layers much further away from the wall, with our aim being essentially to describe the universal behavior of the multilayer. Because the number of available polycations to build up a complex is fixed by the adsorbed amount in the first layer, at a large enough distance from the wall all of the polycations are already involved in a complex and no further complex formation is possible; however, the number of polyanions attached to the layer is not limited because they are in a thermal equilibrium with the bulk solution and the complexed polyanions can make large loops dangling into the solution. This suggests a doublelayer structure, a complex inner layer containing the two polymers with a total monomer density 2cc, and a loop outer layer containing only polyanions. When the polyanion solution is replaced by pure solvent with added salt, the concentration profile of the polyanions in the loop region decays in a way similar to that in an adsorbed polymer layer of neutral polymers with an electrostatic excluded volume vel: (24) Johner, A.; Joanny, J. F. J. Phys II 1991, 1, 181.
Formation of Polyelectrolye Multilayers
c-(z) ∼
a2 vel(z + d)2
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(19)
The length d is imposed by matching this profile with the equilibrium density inside the complex c-(z)0) ) cc. This allows us to estimate the adsorbance in the dangling loops, which is proportional to the net charge of the multilayer (the complexed part being neutral):
∆Γ )
2
∫0∞dzc-(z) ) vaeld ∼ a(cc/vel)1/2 ∼ w12/3
(20)
In our model, the contribution of the loop region to the adsorbed polymer amount is independent of the ionic strength and of the charge of the polymer. The overcompensated charge of the multilayer is thus independent of ionic strength. This result has been obtained using the leading term in the asymptotic expansion for the complex concentration in the high ionic strength limit. If we include the first correction as in eq 8, we obtain ∆Γ ∼ (1/w2/3)(1 - f 2/n3/2lB1/2a2w2/3). We thus predict a very slow increase of ∆Γ with ionic strength. If we now iterate the complexation process and consider that the loops of the second layer are used to complex the polymer of the third layer and so forth, it is clear that the adsorbed amount per new layer is proportional to ∆Γ. In the experiments of ref 9, a weak increase of the amount of polymer in each layer with n is found. The results are analyzed in terms of a power law, and the exponent R verifies 0.05 < Rexp < 0.15. This seems in good agreement with our predictions. The thickness of each layer can also be estimated. If we neglect the transition zone close to the wall, the thickness of the first complexed zone scales as
L1 ∼
Γ1 a2w4/3nσ ∼ cc f3
(21)
It increases with the salt concentration, reflecting the weak swelling of the complex when the attractive interactions are less important. Note that this thickness increases with the surface charge too, because a more important initial polycationic amount leads to thicker complexes. Those two effects are in qualitative agreement with various experiments. The thicknesses of the following complexed layers are estimated in the same way:
L∼
∆Γ w2/3na2 ∼ cc f2
(22)
All of the layers except for the first one are characterized by the same parameters L and ∆Γ. Within our model, the stationary zone where the thickness of one layer is independent of the surface charge is therefore reached at the third layer, while in the experiment it is reached after the first four or five layers. This may come from the fact that our model of complexation close to the charged wall is very simple but also from the fact that usually the polymer of the first layer is different from the polymers forming the bulk of the multilayer. As mentioned in the description of the multilayer structure, the aim of this experimental procedure is to really anchor the multilayer to the substrate. The calculations presented in this paper are done within mean-field assumptions and therefore miss some correlation effects. Because the experimental structures are shown to be rather dense, we expect anyway
the mean-field equations to give a rather good insight of the underlying physics of polyelectrolyte multilayers. 4.3. Discussion. We now discuss the assumptions of our model and some of its limitations: high ionic strength, incompatibility between the polymers, and irreversibility. Ionic Strength. Our model is limited to high ionic strength. At low ionic strength, the range of the electrostatic interactions increases and polyelectrolyte layers interact more strongly with each other, introducing nontrivial correlations. The assumption of a complexation mechanism for the multilayer buildup similar to bulk complexation may no longer be valid. In particular, we expect the effect of the substrate surface charge to propagate over larger length scales. We have not been able to write scaling laws in this regime, although experiments show that the internal structure of the multilayers is not drastically changed: two consecutive layers are still interdigitated, suggesting polyelectrolyte complex formation between consecutive layers. Incompatibility between the Polyelectrolytes. In our model we assumed that the neutral parts of the oppositely charged polyelectrolytes were identical. If this is not the case, there is a new repulsive interaction between the polymers that leads, in general, to phase separation between the backbones in the absence of charges. Experimentally the chemical incompatibility does not seem to be a relevant parameter because multilayers are formed for a large number of polyelectrolyte pairs of opposite signs and universal properties are found. On the theoretical side, the incompatibility can be taken into account through mean-field equations by a Flory-Huggins term Finc/kT ) χc+c- ) χc2 in the free energy. The RPA polarization term is not changed at lowest order in χ. One can build for the multilayer formation a model similar to the one presented here, where the leading repulsive term is now Finc. The net charge of the multilayer is then a decreasing function of the salt concentration, which is in disagreement with experimental observations. If we accept that chemical mismatch can only lead to repulsive contributions in the free energy, any scaling assumption for this repulsive term of the form Frep/kT ) Acδ would give a decreasing net charge with respect to the salt unless δ g 3 so that the third virial coefficient is the first relevant term. This discrepancy is consistent with the fact that only a few charges of opposite signs are sufficient to enhance the compatibility between two chemically different polymers:25 the incompatibility becomes a less important parameter for the description of the charged system. It has been argued by Solis et al. that chemical mismatch may lead to spontaneous layering of a mixture of oppositely charged polymers close to a charged wall by formation of mesophases.10 Using the same formalism (RPA), we found that mesophase formation does not occur in the limit of high ionic strength if s-1 < 1: if the incompatibility is too strong, there is a transition from a homogeneous mixture toward a macroscopic phase separation without any mesoscopic layering. Frozen Dynamics. As mentioned in the description of the internal structure obtained from experiments, there is no long-range diffusion of the polymers in a multilayered film. Although two consecutive layers are strongly interdigitated, a polymer chain does not diffuse much farther than its neighboring layers. This has been shown by labeling every other polyanionic layer and by repeating the same experiment on the sample 1 year later:26 the neutron reflectometry results exhibit no significant (25) Khokhlov, A. R.; Nyrkova, I. A. Macromolecules 1992, 25, 1493. (26) Decher, G. Science 1997, 277, 1232.
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changes. This shows that the dynamics is somehow frozen inside the multilayers: the strong attractive interactions that provide the dense structure of the multilayer may also play the role of potential wells where the polymers are trapped. A similar situation is found with the slowing down of random copolymer dynamics where a glassy, quasi-frozen phase is reached.27 A description of the dynamics of the multilayer assembly is beyond the scope of this work and will be presented elsewhere.
Castelnovo and Joanny
F ) kT
∫drb
Acknowledgment. We are grateful for discussions on the thermodynamics of polyelectrolyte complexation to P. Sens and F. Clement (Strasbourg). This research was supported by the Deutsche Forschung Gemeinschaft through the Schwerpunkt “Polyelektrolyte”. Appendix A: Free Energy of a Mixture of Oppositely Charged Polyelectrolytes within the RPA Mixtures of polymers can be treated within the RPA because concentration fluctuations are small if the amount of solvent is low. This procedure has already been applied to polyelectrolytes by various authors, but they only discuss the structure factors.25 In this appendix we calculate within the RPA the free energy of a mixture of oppositely charged polyelectrolytes both in a Θ solvent. The free energy of the solution reads as (27) Bouchaud, J. P.; Cates, M. E. J. Phys. II Fr. 1993, 3, 1171.
c+ log c+ c- log c+ + n+ log n+ + N N
w2 (c + c-)3 + 6 + lB b)|2 |∇c-(r b)|2 a2 |∇c+(r + dr b′ × + 24 c+(r 2 b) c-(r b) [(f+c+(r b) - f-c-(r b) + n+(r b) - n-(r b))(f+c+(r b′) -
(
n- log n- +
)
∫
}
f-c-(r b′) + n+(r b′) - n-(r b′))]/[|r b-b r ′|]
5. Concluding Remarks We have presented in this paper a mechanism for the formation of polyelectrolyte multilayers at high ionic strength. It is based on the experimental observation that the structure within this system is rather fuzzy, so that two consecutive layers strongly interpenetrate. The strong stability of the film is explained by the complexation of two adjacent layers. We have developed a model for polyelectrolyte complexation in analogy with the description of neutral polymer solutions under poor solvent conditions. This allows us to interpret the structure of a neutral complex as a compact packing of ξel blobs, with ξel being the correlation length associated with electrostatic excluded volume. The complex exhibits Gaussian statistics at length scales smaller than ξel, while at larger length scales the attractive polarization energy dominates. In our model, the experimentally observed stationary regime, where properties independent of the chemical nature of the polymer are found, is reached after the second layer. The predicted layer thicknesses and total charge of the multilayer are in qualitative agreement with the in situ experiments of Ladam et al.9 This model can be viewed as a first step toward a more precise understanding of polyelectrolyte multilayer formation. Its main limitation is the use of mean-field equations that do not take into account some of the correlation effects. We expect these effects not to be crucial because of the dense structure we are trying to describe, where fluctuations can be treated at a Gaussian level through the RPA. Our model for polyelectrolyte adsorption is also rather primitive and certainly requires some improvements. One may evaluate, for example, the influence of the solvent quality on the previous results.
{
(A.1)
with n+ and n- being the concentrations of small ions, including the polyelectrolyte counterions and the salt. The terms of the first line represent the translational entropies of all of the components. The first term of the second line is the third virial coefficient that stabilizes the structure for a Θ solvent (in a Θ solvent the second virial coefficient vanishes), while the following term is associated with the connectivity of the chains with Gaussian statistics. Finally the last term is electrostatic energy. It is convenient to introduce the Fourier components of the concentration fluctuations; the free energy associated with small fluctuations is then written within the RPA up to second order in the fluctuations:
δF
)
kT
1
∑bq 2V
δnq+2 n+
{ ( δcq+2
(
+ δcq-2
q2a2
12c+
q2a2
+
12c-
1
+
Nc+ 1
)
+ w2(c+ + c-) +
)
+ w2(c+ + c-) +
Nc-
4πlB(f+δcq+ - f-δcq- + δnq+ - δnq-)2
δnq-2
+
n+
q2 δcq+δcq-(2w2(c+ + c-))
}
(A.2)
The contribution of the fluctuations to the total free energy is written with the help of the partition function:
∆F kT
∫Dcbbqe-δF[δcbbq]/kT] ∏ b q
≡ -log Z ) -log[
(A.3)
If one lets δF[δc bbq]/kT ≡ (1/2V)∑i,jδcbq,iGbqi,j-1δcbq,j where the index i denotes one component, the free energy of eq A.3 is given by the following integral:
[
]
G ˆ ∫0∞dq q2 log ||2πV ||
V ∆F ) kT 4π2
-1
(A.4)
If we assume a neutral complex where f+c+ ) f-c- and n+ ) n- ≡ n˜ , the determinant of the inverse structure matrix factor G ˆ -1 is written in the limit of infinite chains:
|| || ( )
a2 2 2 (q + ξ-2)((q2 + κ2)q2 + q/4) G ˆ 12c ) 2πV q2n˜ 2 -1
(A.5)
We introduced in this equation the characteristic length scales: κ-1 is the Debye screening length associated with the small ions, and q/-1 is the screening length associated with the specific polymer screening. Finally ξ is the correlation length of a polymer solution in a Θ solvent. They are defined by
Formation of Polyelectrolye Multilayers
Langmuir, Vol. 16, No. 19, 2000 7531
κ2 ) 8πlBn˜ q/4 ) -2
ξ
48πlB(f+2c+ + f-2c-) a2 )
12w2(c+ + c-)2 a2
(A.6)
As for a simple electrolyte, this integral diverges for large wave vectors because of the charges self-energy. This divergence can be removed by subtracting the large wave vector behavior. The final result is
1 ∆F ){ξ-3 + (1 + 2s-1)1/2(1 - s-1)} (A.7) kT 12π
Figure 3. Concentration profile of polyanions when the profile of polycations is imposed.
where we have defined the dimensionless ratio s-1 ) q/2/ κ2. This form for the RPA correction to the mean-field free energy has been already derived by Borue and Erukhimovich17 for a polyelectrolyte solution; the fact that we have a polyelectrolyte mixture essentially changes the definition of the various characteristic lengths. In the main text we only consider symmetrical complexes with f+ ) f≡ f and c+ ) c- ≡ c.
For κL . 1, one can neglect the second term of the righthand side of these equations. The last terms of the righthand side can be incorporated into a new effective boundary condition for the order parameter similar to eq 15. The profile is now given by
Appendix B: Second Layer Profile In this appendix, we use a simplified model to derive the concentration profile of the second layer of the multilayer. We look for the concentration profile of polyanions close to a surface when there is a polycationic preadsorbed layer of uniform density c+ and of thickness L. We impose the constraint that this preadsorbed layer is not affected by the presence of the polyanions. In the real problem, only the total amount of adsorbed polycation is fixed and the density is an equilibrium variable which results from the complexation equilibrium with the polyanions. We only study here the interaction between the polyanion solution and this adsorbed layer. This model allows one to find the characteristic length scales of the second layer buildup. We distinguish two regions in our system (Figure 3): the inner region of the preadsorbed layer, denoted by 1, and the outer region 2. We use the same notations as those in the main text of the paper. The concentration of the polyanionic solution cbulk- is very small. The equations for the electrostatic potential read in the two regions
∂2V ) κ2V + 4πlBf(ψ-2 - c+) ∂z2 ∂2V ) κ2V + 4πlBfψ-2 ∂z2
if 0 e z e L
(B.1)
if L e z e +∞
(B.2)
In the limit of high ionic strength, the term containing the order parameter can be considered as slowly varying compared to the linearized term in the Debye-Hu¨ckel model. The solution of the electrostatic equations is then
V(z) κ2 ) f(c+ - ψ-2) - fc+e-κL cosh κz - σκe-κz 4πlB if 0 e z e L (B.3) V(z) κ2 ) -fψ-2 - fc+e-κz sinh κL - σκe-κz 4πlB if L e z e ∞ (B.4)
2 a2 ∂ ψ) (velψ-2 - µ - velc+)ψ6 ∂z2
if 0 e z e L
2 a2 ∂ ψ) (velψ-2 - µ)ψ6 ∂z2
(B.5)
if L e z e ∞ (B.6)
The variable µ is the chemical potential of the polyanions in the solution. It is related to the bulk concentration by µ ) velcbulk-. It can be seen from these equations that the effect of the preadsorbed layer is to change locally the chemical potential of the polyanions. One can solve these equations by matching asymptotic solutions in regions 1 and 2. In the high ionic strength limit, these equations are identical with the mean-field equations describing polymer depletion or adsorption profiles. Inside the preadsorbed layer, the solution of eq B.5 is a depletion profile close to the wall of a solution of neutral polymers of concentration c- ) c+ + cbulk-; the excluded-volume parameter and the extrapolation length are respectively vel and deff:
(
ψ-(z) ) xc+ + cbulk- tanh
)
z + K′ ξvel
(B.7)
K′ is a constant fixed by the boundary conditions on the wall. As defined in the main text of the paper, ξel is the mean-field correlation length associated with the excludedvolume parameter vel. The outer solution is an adsorption profile of a solution of neutral polymers of concentration c- ) cbulk-:
(
ψ- ) xcbulk- coth
z +K ξvel
)
(B.8)
where K is imposed by matching the outer solution with the inner solution. When cbulk- f 0, one recovers the adsorption profile of eq 17. The profiles are sketched in Figure 3 for cbulk- f 0. We conclude from this crude model that the thickness of the depletion layer scales as ξel. This is a consistent result because the minimal thickness of this layer has to be on the order of the screening length and ξel . κ0-1 in the high ionic strength limit. The depletion zone is also consistent with our complexation model because it corresponds to the first layer of correlation blobs
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Langmuir, Vol. 16, No. 19, 2000
of the collapsed structure. However, this simple model cannot be used to describe the real complexation mechanism: in this case, the preadsorbed layer has to be able to adapt itself to the polyanion profile. One finds therefore two nonlinear and coupled equations. Of all of the
Castelnovo and Joanny
approximations that we tried to solve analytically those equations exclude the specific polymer aspects and so mistreat the complexation mechanism. LA000211H
