Langmuir 2002, 18, 1433-1436
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Model of Ionization Response of Weak Polyacids in a Layered Polyelectrolyte Self-Assembly Dan Finkenstadt*,† and D. D. Johnson†,‡ Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801, and Department of Materials Science and Engineering, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801 Received August 22, 2001. In Final Form: November 19, 2001 We present a simple, quantitative model that predicts the ionization behavior of a weak polyacid as strong polyelectrolyte layers are deposited. The theory treats such layered self-assemblies akin to onedimensional, ionic solids via electrostatics but, importantly, includes local pH response of solution via buffer capacity. The model reduces to a nonlinear, iterative map having a single free parameter and agrees well with experiment. The map shows that steady-state is determined by the polyacid’s initial ionization and that polyelectrolyte charges self-adjust.
Introduction Functionality of polymeric-based multicomposites requires organization and control of the environment at the nanoscale. For a decade, significant design flexibility has been achieved via electrostatic self-assembly to control structural hierarchy, the simplest being layered. Multilayer films may be deposited from solution onto a solid substrate in a layer-by-layer fashion by exposing the substrate alternately to solutions of oppositely charged polyvalent objects, usually polyelectrolytes (PE) or particles having fixed charges.1-8 Recently, multilayers containing a weak polyacid or base, whose charge depends on the local electrostatic environment, have been considered4,9 and hold promise for additional control over molecular organization;4,9-11 the polymer can be weakly or strongly charged, leading to control over film thickness and the possibility to “erase” selectively the films.11 In situ measurements show that fractional ionization of a weak polyacid (poly(methacrylic acid), PMA) embedded within a number, n, of layers (Figure 1) of cationic polymer (positive polyvinylpyridine, QPVP) and anionic polymer (negative polystyrenesulfonate, PSS) oscillates in response to each added electrostatic layer, with an apparent decay with increasing n.12 Similar effects were observed via fluorescence intensity measurements in another polyelectrolyte layered system (see Figure 9 of * To whom correspondence should be addressed. † Department of Physics. ‡ Department of Materials Science and Engineering. (1) Decher, G. Science 1997, 277, 1232-1237. (2) Esker, A. R.; Mengel, C.; Wegner, G. Science 1998, 280, 892-895. (3) Caruso, F.; Caruso, R. A.; Mo¨hwald, H. Science 1998, 282, 11111114. (4) Shiratori, S. S.; Rubner, M. F. Macromolecules 2000, 33, 42134219. (5) Clark, S. L.; Hammond, P. T. Adv. Mater. 1998, 10, 1515-1519. (6) Husemann, M.; Morrison, M.; Benoit, D.; Frommer, J.; Mate, C. M.; Hinsberg, W. D.; Hedrick, J. L.; Hawker, C. J. J. Am. Chem. Soc. 2000, 122, 1844-1845. (7) Xia, Y.; Whitesides, G. M. Annu. Rev. Mater. Sci. 1998, 28, 153184. (8) Huck, W. T. S.; Stroock, A. D.; Whitesides, G. M. Angew. Chem., Int. Ed. 2000, 39, 1058-1061. (9) Yoo, D.; Shiratori, S. S.; Rubner, M. F. Macromolecules 1998, 31, 4309-4318. (10) Mendelsohn, J. D.; Barrett, C. J.; Chan, V. V.; Pal, A. J.; Mayes, A. M.; Rubner, M. F. Langmuir 2000, 16, 5017-5023. (11) Sukhishvili, S. A.; Granick, S. J. Am. Chem. Soc. 2000, 122, 9550-9551. (12) Xie, A. F.; Granick, S. J. Am. Chem. Soc. 2001, 123, 3175-3176.
Figure 1. PE layers are deposited sequentially atop PMA with geometry and dimension (ref 12) shown. Directions of electric fields at PMA associated with each layer are indicated by dark arrows.
ref 13). These environmentally stimulated ionization shifts clearly comprise responses to field changes from the multilayer and hold technological promise as sensors of environmental conditions at the topmost layer. With such intriguing findings, we construct a simple model that includes the main effects at PMA controlling the response: field changes due to deposition and chemistry-driven changes due to buffer-capacity-limited pH response. The model yields quantitatively the (i) oscillatory and asymptotic behavior of the fractional ionization, (ii) dependence of this behavior on PE charges and buffer capacity of solution, and (iii) expected PE charges and their self-adjustment during deposition. We compare to measured fractional ionization versus layer number.12 The Physics and Chemistry Multilayers of chain molecules built using layer-by-layer self-assembly are not stratified into perfect layers but “fuzzy” due to interpenetration between successively deposited layers,1,14-17 as shown using grazing angle specular scattering and X-ray and neutron reflectometry.17 However, the absence of Bragg peaks in scattering data is consistent with a statistical tendency for layers to be stratified. Also, layerlike charge distributions, even if fuzzy,1 lead to roughly constant fields along the layer(13) Klitzing, R. v.; Mo¨hwald, H. Langmuir 1995, 11, 3554-3559. (14) Korneev, D.; Lvov, Y.; Decher, G.; Schmitt, J.; Yaradaikin, S. Physica B 1995, 213/214, 954-956. (15) Kellogg, G. J.; Mayes, A. M.; Stockton, W. B.; Ferreira, M.; Rubner, M. F.; Satija, S. K. Langmuir 1996, 12, 5109-5113. (16) Lo¨sche, M.; Schmitt, J.; Decher, G.; Bouwman, W. G.; Kjaer, K. Macromolecules 1998, 31, 8893-8906. (17) Arys, X.; Laschewsky, A.; Jonas, A. M. Macromolecules 2001, 34, 3318-3330.
10.1021/la015548d CCC: $22.00 © 2002 American Chemical Society Published on Web 01/23/2002
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Figure 2. PMA fractional ionization versus number of polyelectrolyte layers. Circles (squares) are in situ ionization (mass) data obtained experimentally (ref 12). Diamonds and connecting lines are obtained from the two-charge model, eqs 4 and 5.
normal direction. We may then reduce the complexity of the problem by considering an idealization in which PE layers are modeled as infinite planes of uniform charge (Figure 1). Fractional Ionization and Charge Scaling. The fractional ionization, fn, of a polyacid under nPE layers is the ratio of its charge, σ0, to the charge of a 100% ionized polyacid layer, that is, fn ≡ σ0/σ100%. As |σ100%| sets an absolute scale, we define a layer’s relative charge as σ˜ n ≡ σ n/|σ100%|, where σn is the charge on the nth PE layer (Figure 1). Prior to any PE deposition, f0 ) |σ˜ init 0 |, the polyacid’s initial relative charge. We assume that the fractional ionization is in proportion to the electric field at the weak polyacid arising from the entire assembly, relative to the field of a fully ionized polyacid. From Gauss’ law, the electric field for an infinite plane with uniform charge σ is E ) n ˆ (σ/2), where is the dielectric of the medium in which the field is measured, and n ˆ is a plane-normal unit vector pointing away from the surface. For n-layer geometry (Figure 1), the total field at the polyacid is a linear superposition of constant fields given by n
En ) (f0 +
σ˜ l)E100% ∑ l)1
(1)
where E100% ) zˆ (σ100%/2) is the field of a fully ionized polyacid layer, and zˆ is a fixed unit vector along the growth direction. An immediate consequence is that no electrostatic screening occurs for PE layers and for onedimensional, layered systems in general. The field at the polyacid changes magnitude with deposited charge. For the special case of a pair of PE layers with +σ and -σ charges, there is zero electric field change by Gauss’ law. Given our assumption above, that is, fn ) En/E100%, the fractional ionization upon deposition of layer n + 1 is trivially written via eq 1 as
fn+1 ) fn + σ˜ n+1
(2)
So the change of ionization depends only on the most recently deposited layer, and the problem is reduced to counting relative charges. Note that the deposition of positive QPVP increases ionization because its field is in the same direction as that of PMA; conversely, negative PSS decreases ionization. The fractional ionization must then oscillate between high and low values as PE layers are deposited, as is roughly observed12,13 (Figure 2). However, the response of the buffer solution must also be considered. Buffer Capacity. Between each layer deposition, fresh buffer solution is rinsed over the PE layers and pH is
brought back to a constant.12 Buffer solution surrounding the multilayers responds by accepting (donating) hydrogen ions directly from (to) PMA via [-COOH] h [-COO-] + [H+]. The direction in which the reaction proceeds depends on total fractional ionization of PMA compared to some threshold fractional ionization µ. If fn is greater (less) than µ, hydrogen ions are released from (absorbed into) solution and the reaction moves toward the right (left). Thus, local pH near PMA may increase (or decrease) to maintain charge neutrality relative to µ. (Films are clearly permeable to protons, as verified experimentally.13) While local pH above PMA is allowed to change by an amount ∆(pH), the total amount of charge that can be gained or lost is limited by the solution’s buffer capacity C, the amount of charge needed to change the pH by 1 unit.18 This limiting response alters the ionization from the electrostatic-only result, eq 2, by ∆fn+1 ) C ∆(pH). The pH effect is written in terms of quantities already discussed as
(
∆fn+1 ) -C log10
)
fn + σ˜ n+1 µ
(3)
The sign follows because deposition of a large, positively charged layer increases local proton concentration, decreasing pH; similarly, a negative layer increases local pH. So buffer solution acts to limit deviation of fn from µ. The Model When the electrostatic response, eq 2, is combined with the pH response, eq 3, the fractional ionization fn is a first-order, inhomogeneous difference equation, or iterative map, that is,
(
fn+1 ) fn + σ˜ n+1 - C log10
)
fn + σ˜ n+1 µ
(4)
This equation is the central result of this paper. To solve for an fn trajectory, it is necessary to specify (i) f0, the initial fractional ionization of PMA, and (ii) the set of layer charges, σ˜ n. Experimentally, Coulomb repulsion and solution response would prevent the deposition of exceedingly large charges or divergent changes in pH, so the range of validity for this model is limited to parameters that produce fractional ionizations between 0 and 1. The model given by eq 4 may be fit to experiment for a given set of relative charges σ˜ n and for experimentally measured (or numerically fit) values of C and µ. However, for an understanding of the overall effects, we use a simple model for σ˜ n. Two-Charge Model. If each strong PE species (e.g., QPVP and PSS) is deposited in approximately constant amounts, it is justifiable to assume a charge sequence with only two charges in eq 4: layers with positive σ˜ odd ) σ˜ and negative σ˜ even ) -(σ˜ + δσ˜ ). For n > 1, this “twocharge” model is a trivial iterative map given by
σ˜ n+1 ) -σ˜ n - δσ˜
(5)
The first layer of strong PE is deposited onto a weak polyacid that carries an initial fractional ionization f0. This sets the scale of subsequent layer depositions, forcing σ˜ 1 ) f0 to maintain charge neutrality. To compensate for excess positive charge in solution, a slight bias toward excess negative charge on PSS is incorporated via δσ˜ . We reasonably assume that µ ) f0, the sole choice based only (18) See, for example: Reger, D. L.; Goode, S. R.; Mercer, E. E. Chemistry: Principles & Practice; Saunders College Publishing: Orlando, 1993; p 569.
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on charge neutrality. This choice of µ, governing asymptotic behavior according to eq 4, is consistent with the observation that fn becomes centered about f0.12 As a result, the two-charge model, composed of two coupled maps, eqs 4 and 5, depends only on C and f0 (both fixed by experiment) and one free parameter, δσ˜ . This one-parameter model allows a simple explanation of, and fit to, experiment. Comparison to Experiment Xie and Granick12 observed that the fractional ionization of PMA varied from 0.15 to 0.8 as the pH varied from 5 to 7, giving C ) 0.325. In addition, f0 ) 0.479.12 Thus, in the two-charge model, we have µ ) σ˜ 1 ) f0 ) 0.479, all predetermined. For the one free parameter, the excess charge on PSS, we find δσ˜ ) 0.064 leads to very good agreement to experiment (Figure 2), 3.6% average relative error. Notably, the model fits most poorly on the first PSS layer (n ) 2), indicating a smaller charge than predicted by eq 5. The error is propagated into the next few layers by the iterative nature of eq 4 but is “damped out” by buffer solution for larger n. The smaller charge at n ) 2 is reflected in experiment as a smaller initial PSS mass.12 It follows that our model also explains the intensity versus layer data of Klitzing and Mo¨hwald (Figure 9 of ref 13), including their finding that the pH-sensitive dye responds strongly to the charge at the interface of the top layer and buffer solution. From the two-charge model, the QPVP (PSS) charge is 0.479 (-0.543). To compare with experiment, we must estimate charges deposited. Experiment provides the mass densities for each layer, a product of weight-averaged molecular mass, Mw, and number of moles m-2 deposited (which was measured via in situ infrared spectroscopy12). Number-averaged molecular mass Mn is more appropriate for determining charges (proportional to number of mers), so we divide experimental mass densities by Mw/Mn (1.23 and 1.1 for QPVP and PSS, respectively). Assuming strong polyelectrolytes are 100% ionized, the charge is the corrected mass density divided by molecular weight (205 g mol-1 for QPVP, 206 g mol-1 for PSS). Dividing by σ100% ≈ -10 mol charge mm-2 (from the 1 mg m-2 initially deposited PMA and considering f0 * 112) reduces these charges to relative units. The resulting relative charges are (0.51, 0.51, 0.52, 0.48, 0.47) for QPVP and (-0.54, -0.65, -0.62, -0.60, -0.58) for PSS; see ref 19. Averaging these charges to compare to the two-charge model, we find overall agreement of 4% (9%) for QPVP (PSS). As a further contrast and comparison, we used the charges derived from experimental masses in the iteration of eq 4. These “mass data” results (Figure 2) agree well with experiment (2.9% average relative error) but are little better than those obtained from the simple model. Steady-State. Techniques to study iterative maps are known from nonlinear dynamics.20 The complete map for eqs 4 and 5 is plotted (Figure 3) using C ) 0.325, µ ) f0 ) σ˜ 1 ) 0.479 and δσ˜ ) 0.064. The experimental results are also plotted and agree well with theory. The map exhibits a basin of attraction (labeled “period-two attractor”) to fhigh ) 0.726 and flow ) 0.319, in reasonable agreement with experimental values, fhigh ∼ 0.7 and flow ∼ 0.3.12 The fhigh - flow steady-state may be confirmed via λ, the Lyapunov exponent.21 We find λ ) -0.8; negative values signal a stable, attractive trajectory.20 (19) For example, even in the two-charge model, δσ˜ ≈ ∆fn+1)even is layer dependent (0.06, 0.10, 0.13, 0.13, 0.14); so PSS charges are (-0.54, -0.58, -0.61, -0.61, -0.62), reflecting better the observed charges. (20) Jackson, E. A. Perspectives of Nonlinear Dynamics; Cambridge University Press: New York, 1989; Vol. 1, Chapter 4.
Figure 3. Phase plot of ionization from eq 4 for deposition of positive (+, solid line) and negative (-, dashed line) charges from eq 5. The map reads as follows: (1) From f0 on the fn axis, find the intercept on the curve for the oppositely charged layer. (2) Read a new value from the fn+1 axis. (3) Repeat 1 and 2 using this fn+1 as the new fn. (For example, given f0 ) 0.479 for -PMA, f1 ) 0.86 for +QPVP and then f2 ) 0.38 for -PSS.) Experimental ionization is also shown.
The origin of the period-two attractor is most easily seen with a change of variables gn ≡ fn + σ˜ n+1. This transformation reduces the coupled, first-order difference equations, eqs 4 and 5, to a single, second-order difference equation:
( )
gn+1 ) gn-1 - δσ˜ - C log10
gngn-1
(6)
µ2
The period-two asymptotic trajectory occurs when gn+1 ) gn-1, or when µ2 ) gngn-110δσ˜ /C, where asymptotically gn ) fhigh - |σ˜ 1 + δσ˜ | and gn-1 ) flow + |σ˜ 1|, and µ ) σ˜ 1 ) f0 ) 0.479 from initial conditions. Solving this relation for δσ˜ , we find a value of 0.064, in agreement with our best fit value. This shows that µ sets the attraction center for large n (steady-state) and also that δσ˜ (excess PSS charge) is related to µ. Self-Adjusting Mass and Other Predictions. Each overlayer charge is predetermined via the iterative map. If each polyelectrolyte is 100% ionized as assumed in experiment, there is a one-to-one correspondence between charge and mass. Thus, charge neutrality and buffer effects result in self-adjusting mass, in the sense that there is no external control over the mass deposited.12 More complicated charge models than eq 5 are possible,19 perhaps coupling to fn, but a more realistic map is entirely a problem of chemistry and left as an open question. On the basis of eqs 4 and 5, in situ deposition within an electric field should affect the PMA ionization and PE charges, allowing further control over the multilayer. A more interesting case, perhaps for a sensor or switch, is a dual embedding of PMA (initial plus second layer). From Gauss’ law, it is easily shown that fractional ionization of the outer PMA would oscillate, as described earlier. However, due to the response of the outer PMA, the change in field at the n ) 0 PMA layer would be almost zero, because the field of the outer PMA opposes that of subsequent PE layers. The n ) 0 fractional ionization should asymptotically approach µ, with only minor oscillations of order δσ˜ . Conclusions We constructed a simple theory that predicts the observed fractional ionization oscillations of a weak (21) For the model in eq 6, λ for period-two trajectory is λ)
|
(
1 C 10δσ˜ /C C ln 1 + - fhigh - flow + δσ˜ 2 ln(10) µ2 ln(10)
)|
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polyacid embedded in a strong polyelectrolyte self-assembly. The theory includes the effects of electrostatics and buffer capacity and shows that charges are not “screened’ (as often supposed in colloid-science approaches) because layered polycation-polyanion complexes produce constant electric fields. We reduced the theory to a nonlinear, iterative map and introduced a two-charge model to allow one-parameter fits to experiment. By use of known buffer capacity and initial ionization, the map agrees well with measured ionization oscillations and predicts the expected steady-state behavior and polyelectrolyte charges.
Finkenstadt and Johnson
Acknowledgment. We thank Paul Melby for suggesting the second-order map, F. Xie and S. Granick for valuable discussions and comments, and a referee for providing reference 13. Funding is by the Department of Energy at the Frederick Seitz Material Research Laboratory (DEFG02-91ER4539) and the National Science Foundation (DMR-9976550) at the UIUC Materials Computation Center.
LA015548D
