Modeling the Growth Processes of Polyelectrolyte Multilayers Using a

J. Phys. Chem. B 2007, 111, 8509-8519

8509

Modeling the Growth Processes of Polyelectrolyte Multilayers Using a Quartz Crystal Resonator† Mikko Saloma1 ki and Jouko Kankare* Department of Chemistry, UniVersity of Turku, FIN-20014 Turku, Finland ReceiVed: NoVember 7, 2006; In Final Form: January 4, 2007

The layer-by-layer buildup of chitosan/hyaluronan (CH/HA) and poly(L-lysine)/hyaluronan (PLL/HA) multilayers was followed on a quartz crystal resonator (QCR) in different ionic strengths and at different temperatures. These polyelectrolytes were chosen to demonstrate the method whereby useful information is retrieved from acoustically thick polymer layers during their buildup. Surface acoustic impedance recorded in these measurements gives a single or double spiral when plotted in the complex plane. The shape of this spiral depends on the viscoelasticity of the layer material and regularity of the growth process. The polymer layer is assumed to consist of one or two zones. A mathematical model was devised to represent the separation of the layer to two zones with different viscoelastic properties. Viscoelastic quantities of the layer material and the mode and parameters of the growth process were acquired by fitting a spiral to the experimental data. In all the cases the growth process was mainly exponential as a function of deposition cycles, the growth exponent being between 0.250 and 0.275.

Introduction Utilization of thin polymer films in a huge number of applications, such as sensors, physiological coatings, actuators, energy storage devices, displays, etc., necessitates the development of a large armory of characterization methods. Acoustic wave devices offer unique means of studying both the thickness and elastic properties of thin films in situ during the deposition process. One of these devices is the thicknessshear mode (TSM) quartz crystal resonator (QCR) which has become a routine tool when studying the film formation on various surfaces. In most cases QCR is used simply for estimating the amount of material collected on the surface of the resonator. In the simplest case the quartz resonator is connected as a frequencydetermining component in an oscillator. The mass increase of a layer is calculated from the relative frequency decrease by using the Sauerbrey equation derived originally for vacuum1 but extended later for liquid use.2 Often the deviation from the linear Sauerbrey relation has been interpreted as a “viscoelastic effect”, and a further increase in the layer thickness was considered to lead to unreliable results. As a matter of fact, the viscoelasticity has a smaller effect on the results than generally believed. This is especially true when the resonator with its coating is in contact with gas or vacuum. In this case the nonlinearity appears only in the third order of mass. If the layer is in contact with liquid, the effect of viscoelasticity is stronger and depends on the density-viscosity product of the liquid and the complex shear modulus of the layer material.3 In principle, the elastic properties of the layer material have even the first order influence on the estimated mass, but usually the ratio of the shear moduli of liquid and layer is so small that the effect is negligible within the conventional range of the areal mass density. However, if the thickness of the layer is continuously † Part of the special issue “International Symposium on Polyelectrolytes (2006)”. * Corresponding author. E-mail: [email protected].

increased, we are finally coming to the point where the acoustic thickness of the layer approaches the quarter wavelength of the shear wave. If one is measuring the change of the resonant frequency as a function of the increasing layer thickness, this point appears as a minimum and further increase of thickness causes now increase of frequency and in certain cases the resonant frequency may even exceed that of an unloaded resonator. The stationary point is called the acoustic film resonance (AFR).4 With further growth of the layer, AFR is repeated at every odd multiple of the quarter-wave. There are not very many methods which allow a continuous growth of films and simultaneous monitoring of their buildup on a QCR. An early example of these methods is the vacuum evaporation of metals, the original application of a quartz crystal microbalance. Much more recent are the methods where the layer is grown under liquid. Most common of these methods are the electropolymerization of conductive polymers and the layer-by-layer (LbL) deposition of polyelectrolyte multilayers. Presently the buildup of polyelectrolyte multilayers forms an especially interesting problem because of the dual growth mechanism. Some polyelectrolyte systems seem to grow linearly with the number of deposited layers in the LbL process, whereas others grow exponentially, i.e., each bilayer formed by the anionic and cationic polyelectrolyte is always thicker than the previous one by a constant factor. A transition from one growth mechanism to another appears to be induced by the change in the ionic strength5,6 or temperature.7 The change in the mechanism may occur also during the buildup, and this necessitates using methods which allow the characterization of rather thick films. The growth process can be followed by QCR, but so far most studies have been restricted to the thickness range below the acoustic film resonance. A paucity of QCR data on thick layers in the literature is also due to the inability of many commercial instruments to resolve the weak signal of piezoelectric origin from the strong capacitive background of a heavily damped resonator. In this work we show that by recording the QCR response even beyond the points of acoustic

10.1021/jp067344h CCC: $37.00 © 2007 American Chemical Society Published on Web 03/07/2007

8510 J. Phys. Chem. B, Vol. 111, No. 29, 2007

Saloma¨ki and Kankare surface acoustic impedance ζ0, is directly measurable. Its unit is kg m-2 s-1 ) rayl. The layer with a bulk acoustic shear impedance Zf is assumed to be coated on a quartz crystal resonator. The dependence of LAI on the distance y from the resonator surface, corresponding to areal mass density m, is governed by the Riccati equation

()

1 dζ ζ 2 ) -1 jω dm Zf m)

Figure 1. Real and imaginary parts of reduced surface acoustic impedance ϑ0 as functions of phase φ for a hypothetical layer with δ ) 30° and ϑmed ) 0.02(1 + j).

film resonance, valuable information on the growth process and viscoelasticity of films can be obtained. The polyelectrolyte pairs of hyaluronan/chitosan and hyaluronan/poly(L-lysine) have been subjects of several previous studies, not only due to the interesting medical and pharmaceutical properties of hyaluronan but also because of the rapid exponential growth process of the multilayers. The exponential growth is shown to be due to the rapid diffusion of the cationic polyelectrolytes within the layer system,8 hyaluronan only providing an appropriate medium. Due to the general interest in these polyelectrolyte layers, they were chosen to illustrate the use of QCR data for modeling the multilayer growth process. Theory Although the theory of quartz crystal resonators has been presented in various sources (cf. a recent book9), the presentation in this contribution is based on different variables given mostly in a dimensionless form. For improving the readability of this paper, some parts of the theory are briefly reviewed. The theory is based on the use of surface acoustic impedance and not on the relative frequency change and dissipation factor, familiar to those using a quartz crystal microbalance with dissipation monitoring (QCM-D). The main reason is simply mathematical consistency: surface acoustic impedance contains the same information but in the real and imaginary parts of a complex number, allowing coherent analysis by the standard methods of complex calculus. Another reason why the surface acoustic impedance can be considered a more “intrinsic” quantity is its convergence to the bulk acoustic impedance of layer material with the increasing thickness of the layer. The real and imaginary parts of surface acoustic impedance are proportional to the dissipation factor D and relative frequency shift, respectively.10 Acoustic Impedance. It was previously shown11 that a laterally homogeneous layer within a transverse acoustic field can be conveniently described by using the local acoustic impedance (LAI) and its differential equation. This impedance, often denoted as mechanical impedance or load impedance, is the ratio of stress T and particle velocity V

ζ)-

T V

(1)

Because of the phase shift between T and V, LAI is a complexvalued quantity. It is defined everywhere within the transverse acoustic field but only its value at the surface of the resonator,

(2)

∫0y Ff(y) dy

Here Ff is the density of the layer material which generally depends on the distance y from the resonator surface. However, if impedance Zf is constant within the layer, i.e., it does not depend on the distance from the resonator surface, this equation can be solved analytically. Assuming that one surface of the layer corresponding to the total areal mass density mf is in contact with some semiinfinite medium with the bulk acoustic impedance Zmed and the other surface with the resonator, the solution is

ζ(m) )

Zmed + jZf tan[ω(mf - m)Zf -1] jZmedZf -1 tan[ω(mf - m)Zf -1] + 1

(3)

The measurable quantity, surface acoustic impedance ζ(0) or briefly ζ0, is obtained by substituting m ) 0. The behavior of ζ0 as the function of layer properties can be conveniently studied in a dimensionless “reduced” form. We write

Zf ) |Zf| exp(1/2 jδ)

(4)

Here δ is the loss angle, defined on the basis of the complex shear modulus Gf

Zf2 ) FfGf; Gf ) Gf′ + jGf′′; tan δ ) Gf′′/Gf′

(5)

Dividing both sides of eq 3 by |Zf| and substituting m ) 0, all the variables in the equation attain a dimensionless, “reduced” form

ϑmed + j exp(1/2 jδ) tan[φ exp(-1/2 jδ)] ζ0 ) ϑ0 ) |Zf| jϑmed exp(-1/2 jδ) tan[φ exp(-1/2 jδ)] + 1 (6) In addition to the reduced surface acoustic impedance ϑ0, we have defined a dimensionless medium parameter ϑmed and “phase” φ proportional to the areal mass density mf

ϑmed ) Zmed/|Zf|; φ ) ωmf/|Zf|

(7)

There are various ways to present function ϑ0(φ,δ,ϑmed) graphically. If the medium is vacuum or gas, then ϑmed ) 0. Another common medium is a Newtonian liquid, and then ϑmed ) |ϑmed| exp(jπ/4). As a numerical example, we assume that ϑmed ) 0.02(1 + j) and δ ) 30°, which are typical values. We may now calculate both the real and imaginary parts of ϑ0 as functions of φ. In Figure 1 we see that the maxima of the real part, the points of acoustic film resonance, are located close to the odd multiples of π/2. In fact, both the medium and loss angle have influence on the location of maxima. If the medium is vacuum or gas, the points of AFR are located at φAFR ) (n/[cos 1/2δ])(π/2); n ) 1, 3, 5, ... Impedance Spiral. Generally the mass or thickness of the layers is not known a priori, meaning that in practice curves

Modeling Multilayer Growth Processes

J. Phys. Chem. B, Vol. 111, No. 29, 2007 8511

Figure 2. Data of Figure 1 as an Argand diagram. Dotted line, polar line, passes the center of the spiral and intercepts the descending parts of the spiral at the points of acoustic film resonance.

such as those in Figure 1 cannot be drawn. The primary material parameters obtained from the data of QCR measurements during the buildup of films are the real and imaginary components of surface acoustic impedance ζ0. By plotting the imaginary part as the function of the real part, i.e., forming the complex plane representation or Argand diagram of ζ0, we are able to produce a graphical representation which gives both quantitative and qualitative information on the material coated on the resonator. The shape of these graphs can be conveniently studied in their dimensionless form of eq 6 with φ as the dummy parameter. In Figure 2 we have the same data as in Figure 1 but now as an Argand diagram. The graph is a spiralsto be called now an impedance spiralsand its shape is mainly dependent on the loss angle δ. In Figure 3 we see the influence of loss angle on the shape of the spiral when the layer on the resonator is in contact with air, i.e., ϑmed ) 0. At low values of loss angle the spiral has many turns, and it may also extend to the lower right quadrant of the complex plane, corresponding to the range of a positiVe shift of the resonant frequency. At a limiting value of loss angles“critical loss angle”, ca. 41.3162°sthe spiral has a tangent point with the real axis, and at higher values of the loss angle only a negative shift of the resonance frequency is possible. At the highest loss angle 90°, corresponding to a Newtonian liquid, the spiral degenerates into a “fishhook”. Nonzero values of the medium parameter ϑmed have influence on the size of the spiral, not that much on its shape. If the areal mass density of the layer is tending to infinity, i.e., φ f ∞, then ϑ0 f exp(1/2jδ) assuming that δ > 0. This limit does not depend on the type of the semiinfinite medium, i.e., the same limit is obtained from eq 6 irrespective of the value of ϑmed. This limit is the center of the spiral, and consequently we see that the center corresponds to the bulk acoustic impedance of the layer material. The values of the bulk acoustic impedance of the polymer layer and medium determine unequivocally the impedance spiral. Hence if we observe two spirals with the same initial and end points, then we know for sure that at least one of the layers is not homogeneous. The straight line drawn in the Argand diagram from the origin, passing the center and extending to the outmost branch of the spiral is called a polar line (Figure 2). This line has some remarkable properties which are useful in studying the growth processes of films and the corresponding impedance spirals. From the construction of the line, we already know that the

Figure 3. Influence of loss angle δ on the shape of the impedance spiral. The layer is assumed to be in contact with air (ϑmed ) 0).

angle it makes with the real axis is 1/2δ. In the Appendix it is shown that the polar line passes all the vertical points of the spiral, i.e., those points where the real part of surface acoustic impedance reaches its local maxima and minima or in other words the points of acoustic film resonance. In practice, this means that if we are able to reach the first maximum of the real part of surface acoustic impedance in the growth process, we are able to draw the polar line and to obtain an estimate for the loss angle. Growth Models. The character of the growth processes of polymer films depends on the polymer and the method of growing. The electroactive polymer films are usually grown by electrodeposition and the amount of material on the electrode surface depends often linearly on the charge consumed in the process. A similar linear growth model can be observed in the LbL process of many polyelectrolytes. In this case the “linearity” refers to the proportionality between the total thickness of the polymer coating and the number of polymer layers laid on the substrate. In the case of polyelectrolytes the LbL process is actually exponential in the beginning and may change to linear only at the later stages of the process.7 If the polymer coating is done on a QCR surface, the natural way to elucidate the process is to draw the complex surface acoustic impedance as

8512 J. Phys. Chem. B, Vol. 111, No. 29, 2007 an Argand diagram. The shape of the resulting figure does not depend directly on the growth mechanism but gives information on the material and the changes in its elastic properties during coating. Observation of the acoustic film resonance requires a rather thick film. For instance, more than 300 layers of alternate poly(styrene sulfonate) (PSS) and poly(diallyldimethylammonium) (PDADMA) in 0.1 M NaBr at 25 °C are needed before the first maximum of Im ζ0 appears.11,12 If the “full” spiral has to be recorded, thousands of layers are required. This is certainly a prohibitive number, even if an automatic LbL instrument is available. However, certain polyelectrolytes induce a selfaccelerating growth in the multilayer buildup process. One can distinguish two types of films: multilayers whose mass increases linearly with the number of deposition steps and others that grow exponentially. Typical linearly growing films are poly(styrenesulfonate)/poly(allylamine) and PSS/PDADMA, at least at low temperatures and ionic strengths. Hyaluronan/poly(L-lysine), poly(L-glutamic acid)/poly(allylamine), and hyaluronan/chitosan are three well-known examples of exponentially growing films.13-16 The exponential growth process is explained to be caused by the diffusion of at least one of the layer constituents within the entire membrane during each deposition step. In the linearly grown films, this mobility of polyelectrolyte is hindered and limited only to a short interpenetration within a few layers. As a consequence the exponentially grown films are much less structured and less dense than the linearly grown ones. Another obvious consequence is that the rate of the exponential growth process is faster, i.e., fewer deposition steps are needed to reach the thickness required for the appearance of the acoustic film resonance. According to the theory,7 the buildup processes of polyelectrolyte multilayers are inherently exponential, turning linear whenever the diffusion rate is not fast enough for distributing the mobile polyelectrolyte within the entire film. The key idea is that each deposition step leads to a quasi-equilibrium between the concentration of the polymer repeating unit in solution and the composition of the layer. The fast diffusion of species ensures that the composition of the layer corresponds to the polyelectrolyte solution last used for deposition. A consequence is that the impedance spiral is split into a double spiral, one branch corresponding to the anionic, the other to the cationic polyelectrolyte. Separation of the branches depends on the differences in the viscoelastic properties of polyelectrolytes. As long as the growth process is exponential, the bulk acoustic impedance at each branch is constant, leading to two separate spiral centers if the exponential regime extends far enough over the penetration depth of the acoustic shear waves. If the deposition follows the linear mechanism, i.e., the added amount of polyelectrolyte at each step is constant, the proportion of the added polymer to the already deposited material is continuously diminishing. Even though the polymers were completely mixed after each step, each addition would have less and less influence on the acoustic impedance sensed by the resonator and the spiral branches converge to a common center. The spiral branch is called eVen if an equal number of additions of cationic and anionic polyelectrolytes have been made in making the “multilayer”, and odd otherwise. Although it is realized that in many cases no distinct layers are formed in the growth process, just for convenience the successive additions of the cationic and anionic polyelectrolytes is said to form a bilayer. In the model described by Ladam et al.,17 the polyelectrolyte multilayer film is assumed to consist of three zones, denoted by I, II, and III. Zone I is closest to the substrate, in our case

Saloma¨ki and Kankare the resonator surface, and zone III is in contact with the solution. Zone I is composed of one or a few polyelectrolyte layers, and its properties are mostly influenced by the substrate, i.e., in our case the metallized surface of the resonator. It is formed in the very beginning of the deposition process and remains nearly constant during the rest of time. Zone III is formed next at the interface between zone I and solution. In this zone the charges of cationic and anionic polyelectrolytes do not compensate each other and consequently it contains small counterions. It is soft, often rough, and swollen by the solvent. Presumably zone III is the region where diffusion of some polyelectrolytes may occur, giving rise to the exponential growth.7 As the deposition continues, zone II is gradually formed between zones I and III. In this zone the mutual charge compensation of polyelectrolytes is complete, material is stiffer, and diffusion processes are slowed down. Zone II is growing with further deposition, and zone III stays more or less constant. Following the technique described in our previous publication,11 each zone can be represented as a layer matrix Li, and the influence of the zones on the surface acoustic impedance can be written

ζ0 ) (L1L2L3) o Zmed

(8)

Here we have used a short-hand notation for a Mo¨bius transformation in the complex plane

ζ)Loz)

L11z + L12 L21z + L22

(9)

We make two approximations. First of all, we assume that the influence of zone I is negligible. It is very thin, forms in the very beginning, and presumably stays constant during the deposition. We may assume that it makes a negligibly small constant contribution at least in the later stages of the deposition process, and we omit matrix L1 in eq 8. The second approximation, already implied by eq 8, is that despite the obviously blurred borders between zones II and III, their acoustic behavior can be approximated to some extent by two matrices

Li )

(

)

cos[φi exp(-1/2 jδi)] jZisin[φi exp(-1/2 jδi)] ; -1 1 jZi sin[φi exp(- /2 jδi)] cos[φi exp(-1/2 jδi)] i ) 2, 3

(10)

where φi, δi, and Zi refer to the parameters φ, δ, and Zf, respectively, in eqs 4-7. Further assumptions are needed in deciding on the functional dependence of these matrices on the total thickness of the layer, i.e., how the areal mass density is divided between these two zones during the growth process. We introduce a dummy variable b to describe the growth process. The dimensionless parameter b is zero in the beginning of the growth process and increases to the positive direction. We assume that phases φ2 and φ3 of zones II and III, respectively, depend on b meeting the following three conditions:

φ2(0) ) φ3(0) ) 0;

|

∂φ2 ∂b

b)0

) 0; bf∞

φ3(b) 98 constant > 0 (11) The first condition is trivial, the second one tells simply that zone II grows only very slowly in the beginning of the deposition process, and the third condition tells that zone III approaches a constant thickness with continuing deposition.


J. Phys. Chem. B, Vol. 111, No. 29, 2007 8513 number. Then we may write an expansion of eq 14

β ≈ 4(∆+ + ∆- ) +

16 + 3 [(∆ ) + (∆-)3] + ... 3

(15)

and define the aVerage oVercompensation as

∆ ( ) 1/ 4β

Figure 4. Separation of phase φ into zones at different values of thickness parameter b by using functions (12) with b0 ) 1.

There are an infinite number of functions meeting the criteria (11), but we choose somewhat arbitrarily the following functions

{

φ2(b) ) b arctan(b/b0) φ3(b) ) b arccot(b/b0)

(12)

There is only a single parameter b0 which can be used for adjusting the shape of these functions. Figure 4 shows the shape of phases as the function of b when b0 ) 1. Obviously at b0 ) 0 or ∞ there is only one zone present. In the latter case we observe that φ3(b) ) bπ/2 and according to eq 7, b is proportional to areal mass density. Also we see that the odd integer values of b correspond approximately to the points of AFR. In addition to the conditions (11), these functions have the following properties

φ2(b) + φ3(b) ) bπ/2; φ2(b0) ) φ3(b0) ) b0π/4 (13) We emphasize that our choice is completely arbitrary without any physical significance but based only on the general shape of functions which meet the conditions (11) and the necessity to minimize the number of adjustable parameters. The justification for their use arises only from their reasonable fit to the experimental results and the consequent semiquantitative explanation of phenomena. Growth Exponent. If the growth process is exponential, the areal mass density m should follow an exponential dependence on the bilayer number k: m ) A exp(βk).7 The growth exponent β may not depend appreciably on the molecular weight of polymers16 but it depends slightly on temperature.7 Assuming that equilibrium is attained in the coating process, the growth exponent can be written7

β ) ln

xeq+xeq(1 - xeq+)(1 - xeq-)

(14)

where xeq+ and xeq- are mole fractions of cationic and anionic polyelectrolytes (in terms of repeating units) in the entire layers formed with either cationic or anionic polyelectrolyte, respectively, as the last added component. We see that if the polymers are mutually fully “compensated”, then xeq+ ) xeq- ) 0.5, β ) 0, and there is no growth. Suppose now that after each deposition step there is a slight “overcompensation” in the number of repeating units, i.e., xeq ) 0.5 + ∆ where ∆ is a small positive

(16)

It is worth noting that nothing has been assumed about the charge of the polyelectrolyte molecules and ∆( should not be confused with the charge overcompensation φ of Schlenoff et al.18 Equation 14 has been derived on rather general conditions. The most important assumption is that the composition of the layer in equilibrium (or better “quasi-equilibrium”) depends only on the composition of the contacting solution, not on the thickness of the layer. In the very beginning of the deposition process this is not valid because of the substrate effects. For thicker films this is valid as long as the film is thinner than the diffusion layer of the mobile components. Naturally it is also required that each step in the buildup procedure is repeated exactly in the similar way, because the state of the system after the deposition and rinsing steps does not necessarily correspond to a real thermodynamic equilibrium but a kind of steady-state or “quasi-equilibrium”. Reproducibility is ensured only by a fully automated procedure with precise timing. Experimental Section Materials. Poly(L-lysine) hydrobromide (PLL, 39 kDa, Sigma) was used as received. Sodium hyaluronate (HA) (1000 kDa, Acros Organics) was partially depolymerized by heating.19 The solution of HA (1 mg/mL) in 0.15 M NaCl (or 0.5 M) was hydrolyzed thermally at 90 °C for 195 min. The molecular weight of the product was estimated to be ca. 530 kDa.20 on the basis of the Mark-Houwink relation by measuring the intrinsic viscosity of the polymer solution. Chitosan (CHI) (Aldrich, medium molecular weight, 93% deacetylated) was purified21 and the molecular weight estimated as ca. 200 kDa on the basis of the Mark-Houwink relation.22 Chitosan (1 mg/ mL) was solubilized in 10 mM HCl. The amount of added Clwas taken into account when the ionic strength of the solution was adjusted to 0.15 or 0.5 M with NaCl. The HA/PLL coating and rinsing was done in neutral solutions. HA/CHI coating and rinsing were done in solutions with acetic acid in a concentration of 1 mM in each solution. The pH of these solutions was adjusted to 4.5 with 1 M NaOH. The viscosities and densities of the solutions were measured on an Anton Paar AMVn automated microviscometer and DMA45 density meter, respectively. Coating and Measurement. The instrumentation, coating procedure, and measurement have been described in detail in our previous publications.10,11 In short, the quartz crystal (10 MHz, 100 nm gold plating with a chromium adhesion layer, Lap-Tech, Inc., South Bowmanville, ON) was plasma-cleaned, primed with a monolayer of 2-mercaptoethanesulfonic acid, mounted in the holder, measured without liquid loading and then as loaded with the supporting electrolyte, and finally coated with polyelectrolytes in a fully automated system. At each step the polyelectrolyte solution was allowed to stay in contact with the crystal exactly for 15 min, and after the crystal was rinsed with the supporting electrolyte, the QCR measurement was done. These steps were repeated alternately with anionic and cationic polyelectrolytes until data for the wanted number of layers were collected. A new unused crystal was each time taken for coating


Saloma¨ki and Kankare

Figure 5. (a) Argand diagram of the hyaluronan/chitosan multilayer buildup in 0.5 M NaCl at 25 °C: red curve and data points, even branch; black curve and data points, odd branch; open circles and curve, calculated values; filled circles, experimental values. Dashed line is the polar line of the spiral. (b) Logarithm of areal mass density m (in kg/ m2) as a function of the number of bilayers. Areal mass density was calculated from the data of (a) by using the algorithm described in Appendix. Dotted horizontal line at ln m ) -6.8 shows the penetration depth of shear waves.

and measurements. Estimation of the values of the real and imaginary parts of surface acoustic impedance and their covariance matrices was done as described previously.10,11 Results Hyaluronan/Chitosan. The buildup of the multilayer formed by hyaluronic acid and the polysaccharide amine chitosan is reported to be an exponential process, at least at higher ionic strengths.15,16 In the present work two measurements at 25 °C were done, in 0.5 and 0.15 M NaCl. The polymer concentrations were the same, 1 mg/mL in both cases. The impedance spirals measured in 0.5 M NaCl are shown in Figure 5 a. The red curve and data points correspond to the even-numbered layers, i.e., bilayers with the equal number of cationic and anionic polyelectrolytes, whereas the black curve and data points correspond to the multilayers with chitosan as the last added compound. The spirals are very close to each other, showing the structural similarity of polyelectrolytes at this ionic strength. The solid lines are fitted curves obtained by assuming the presence of a single zone and constant bulk acoustic impedance. The data points on the curves marked by open circles are fitted by varying phase φ until minimum distance from the corresponding experimental point (filled circle) is found. The main assumption is that the bulk acoustic impedance is constant within the fitting range but no assumption on the growth processswhether linear or exponentialsis made. The algorithm of the fitting procedure is explained in more detail in the Appendix. Data for all 50 bilayers were used for fitting, and the excellent fit extending up to the spiral center shows that the bulk impedance stays constant during the entire growth process. This means that the values of phase φ and the areal mass densities m derived from the phase values are reliable. In Figure 5b the logarithm of areal mass density is plotted against the bilayer number and we see that between the 4th and 18th bilayer the linear fit is excellent in both branches with the slope β ) 0.275 ( 0.003. The change of the slope after the 18th bilayer and subsequent leveling of the growth curve does not mean cessation of growth but simply reaching the thickness that is out of range of the shear wave. Any increase in m has then an immeasurably small influence on the surface acoustic impedance. However, the unnatural sharpness of the change may be partly an instrumental artifact. The areal mass density at the 18th bilayer

is ca. 250 µg/cm2 or thickness 2.1 µm with the assumed density of 1200 kg/m3. Comparison of Figure 5a with Figure 3 shows that the spirals with the loss angles 50° and 60° resemble the spirals in Figure 5. Indeed, the calculated numerical values of the loss angle δ for the even and odd branches are 53.8° and 54.7°, respectively. The bulk acoustic impedances are 28.3 + 14.3j kRayl and 27.7 + 14.3j kRayl, respectively.23 Using eq 5 and assuming the approximate density of 1200 kg/m3, the corresponding complex shear moduli are essentially equal, 0.50 + 0.68j MPa and 0.47 + 0.66j MPa, respectively. The calculated penetration depth mp of acoustic shear wave is obtained from mp ) |Zf|2/(ωZf′′) ) 1.12g/m2, and its logarithm is shown by a dotted line in Figure 5b. The change of electrolyte from 0.5 M NaCl to a lower ionic strength 0.15 M NaCl has a profound effect. Two hundred layers were deposited on a QCR, and the model was fitted separately by taking either even-numbered or odd-numbered layers. The centers of the well-separated spirals seem to coalesce, meaning that the assumption of a constant impedance within the layer is not valid any more and we have to turn to the double-zone model. Assuming the presence of two zones in the even branch, fitting gives for zone III Z3 ) 9.7 + 8.9j kRayl and for zone II Z2 ) 52.2 + 31.4j kRayl. If we use a plausible density 1200 kg/m3 for both zones, we get the corresponding complex shear moduli G3 ) 13 + 140j kPa and G2 ) 1.4 + 2.7j MPa. The former value indicates a viscous, nearly Newtonian fluid, while the latter value implies a soft viscoelastic solid. Alternatively, the low impedance and high loss angle of zone III may indicate considerable roughness of the layer in contact with liquid. In the same way, fitting to the odd-numbered layers gives two zones with G3 ) 90 + 170j kPa and G2 ) 1.3 + 2.5j MPa. The shear moduli of zone II for the even and odd branches are similar within the experimental accuracy. The same algorithm provides the areal mass densities of the zones as well as bulk impedances. In parts c and d of Figure 6 the thicknesses of the zones of the even and odd branches obtained from masses by assuming the density of 1200 kg/m3 are shown as the function of layer numbers. There is a pronounced difference in the thickness of even and odd branches. At the 20th bilayer, marked by a dashed vertical line, the even and odd branches are nearly equally thick but thereafter the odd branch seems to grow more slowly than the even branch.



Figure 6. (a) Argand diagram of the hyaluronan/chitosan multilayer buildup in 0.15 M NaCl at 25 °C: red curve and data points, even branch; black curve and data points, odd branch; open circles and curve, calculated values; filled circles, experimental values. Arrow shows the 20th bilayer. (b) Logarithm of areal mass density m as a function of the number of bilayers. (c) Thickness of zones II and III in the even branch. (d) Thickness of zones II and III in the odd branch. The thickness values are based on the areal mass density and estimated density of 1200 kg/m3.

The 40th layer is also marked by an arrow in the Argand diagram in Figure 6a, and we see that the data points are strongly accumulated between the 40th and 200th layers when approaching the spiral center. This accumulation aggravates accurate model fitting due to the limited relative precision of measurements close to the spiral center and a proper cautiousness is necessary when interpreting the results from the last 160 layers. Despite the instrumental limitations, the virtual mass densities calculated from these data points have been included in Figure 6. In Figure 6b logarithm of areal mass density is plotted vs the number of bilayers. The graph is strictly linear up to the 20th bilayer, showing the exponential buildup with the growth exponent β ) 0.243 ( 0.003, corresponding to the average overcompensation of about 6%. In Figure 6c the growth regime between the 20th and 40th bilayers may be interpreted to be linear. Hyaluronan/Poly(L-lysine). This is one of the first systems where the exponential growth was observed13 and the mobility of a polyelectrolyte proved by fluorescence labeling.8 More recently this system was thoroughly studied by Porcel et al.24 In the present work this system was built and measured at three different temperatures, 0.5, 25, and 55 °C. The values of surface acoustic impedance measured during the buildup of a 100 layer HA/PLL film at 0.5 °C are shown as an Argand diagram in Figure 7 a. The general shape of the spirals predicts a rather high loss angle when comparing with the models in Figure 3. The spirals seem to converge toward a common center. The even branch of the spiral could be fitted to the single zone model with a reasonable accuracy, giving Z3 ) 36 + 19j kRayl or G3 ) 0.78 + 1.1j MPa. Growth is

exponential up to the 20th bilayer as seen in Figure 7b, giving β ) 0.266 ( 0.004. For unknown reasons the shape of the odd branch does not allow for a good fit. The measurement of the same multilayer at 25 °C gives seemingly strange results (Figure 8). The regular spiral shape is badly distorted, and the centers of the spirals are clearly separated. It seems as if new spirals start to grow from the side of the first spirals. As a matter of fact, this is unambiguous evidence on the change of the growth mechanism from exponential to linear. Both the even and odd branches could be fitted rather well to the double-zone model as can be seen in Figure 8a. The 22nd bilayer has been marked by arrows to Figure 8a and dashed lines to parts b-d of Figure 8. Comparison of parts b-d of Figures 8 shows that the growth rates of even and odd branches up to this bilayer are very similar and exponential, the mean of the growth exponents being 0.250 ( 0.004. After the 22nd layer the growth is more or less linear but the even branch grows faster, resulting finally in the apparent thickness of nearly twice the odd branch. The same polyelectrolyte system was measured also at 55 °C, and the results are shown in Figure 9. Again the 22nd bilayer was the breakpoint where the growth switched from exponential to linear. This point is marked by arrows to Figure 9a and by dashed lines to the other figures. The growth exponents were again quite close, 0.257 and 0.249. Discussion The experimental values of complex shear impedance, modulus, and growth exponents have been collected in Table



Figure 7. (a) Impedance spiral of 50 hyaluronan/poly(L-lysine) bilayers made at 0.5 °C in 0.15 M NaCl: red circle, HA terminating layer; black circle, PLL terminating layer; open circles and red curve, calculated values. (b) Calculated values of logarithm of areal mass density. Calculated penetration depth is marked by a dotted line at ln m ) -6.6.

Figure 8. (a) Argand diagram of the hyaluronan/poly(L-lysine) multilayer buildup in 0.15 M NaCl at 25 °C. For other notation see Figure 6.

1. There are some common features which need discussion. First of all, the range of the growth exponents is very narrow, from 0.245 to 0.275. Picart et al.13 have studied the hyaluronan/PLL films among others by QCM at 5, 15, 25, and 35 MHz using 1 to 12 bilayers. They do not give explicitly the value of the growth exponent, but from their results at 5 and 15 MHz (see Figure 7 in ref 13), we can calculate β ) 0.25 ( 0.01, in excellent conformity with our results.25 The measured frequency shifts at 25 and 35 MHz show leveling at the 11th to 12th bilayers, obviously due to the near by AFR. Porcel et al. measured dried HA/PLL bilayers ellipsometrically, and from their results we can calculate β = 0.26 (see Figure 2 in ref 24). It is remarkable that even though these authors used a spraying technique for coating and dried the films before measurements, the growth exponent has still the same value. The growth exponent for the hyaluronan-chitosan system has been determined previously in 0.15 M NaCl7,15 on a QCR, and the value was 0.33 at an unspecified temperature. The same polyelectrolyte system has been studied also by using surface plasmon resonance, and the growth exponent was in this case 0.48 ( 0.09 in 0.15 M NaCl.16 Inconsistency between the growth exponents is significant. So far this discrepancy remains

unexplained although the differences in the coating procedure cannot be left out of consideration. The growth exponent is connected to the overcompensation by eq 16. The range of β obtained in this work and others implies the range of 6-7% average overcompensation ∆(. This narrow range asks for an explanation, and an obvious common feature is hyaluronate, the common anionic polyelectrolyte. As can be seen in the expression (15), β can be split into separate contributions from cationic and anionic polyelectrolytes. It is tempting to assume that in the present systems the main part of the overcompensation is coming from hyaluronate, the mole fraction of the cationic polyelectrolytes being close to 0.5. However, further experimental evidence is needed to corroborate this assumption. The hyaluronan/chitosan multilayer built and measured in 0.5 M NaCl (Figure 5) is an example of an excellent fit to the model comprising of a single zone and exponential growth regime. When the ionic strength is lowered to 0.15 M (Figure 6), there is a remarkable change in the fitted model. Two zones are now necessary and zone III in contact with medium is a thin layer of a viscous, nearly Newtonian liquid. The increase in ionic strength is known to increase the screening of the permanent



Figure 9. (a) Argand diagram of the hyaluronan/poly(L-lysine) multilayer buildup in 0.15 M NaCl at 55 °C. For other notation see Figure 6.

TABLE 1: Complex Shear Impedance, Modulus, and Growth Exponent of Polyelectrolyte Multilayers zone II multilayer

medium

branch

CH/HA, 25 °C

0.5 M NaCl

CH/HA, 25 °C

0.15 M NaCl

PLL/HA, 0.5 °C PLL/HA, 25 °C

0.15 M NaCl 0.15 M NaCl

PLL/HA, 55 °C

0.15 M NaCl

even odd even odd even even odd even odd

Z′, kRayl

Z′′, kRayl

zone III

G′, MPa

G′′, MPa

52.2 49.2

31.4 30.2

1.45 1.26

2.73 2.48

28.2 23.5 25.9 21.1

14.7 10.6 12.7 9.7

0.48 0.36 0.43 0.29

0.69 0.42 0.55 0.34

charges of polyelectrolytes with a concomitant change of the polymer conformation from rod to coil. It is conceivable that the thin liquid-like zone III in parts c and d of Figure 6 consists of polyelectrolyte chains in a brushlike conformation in 0.15 M NaCl. Increase of ionic strength to 0.5 M shrinks the polyelectrolyte brush, resulting in a seemingly homogeneous single-zone film of Figure 5. Shrinking does not mean decrease in mass. Actually the areal mass density at the 18th bilayer grown in 0.5 M NaCl is about twice the mass of the same bilayer but grown in 0.15 M NaCl (Figures 5b and 6b). Still the film grown at the higher ionic strength is considerably softer, as we see when comparing the shear moduli of the films in Table 1. Up to the 18th bilayer, the growth is strictly exponential and the areal mass densities of the even and odd branches are close as we see in Figures 5b and 6b. In order to study the influence of temperature on the growth process, the system of hyaluronan and poly(L-lysine) was measured at three temperatures. There are numerous studies of this system but, to the best of our knowledge, no reports on the influence of temperature on the formation of multilayers. However, as shown previously in connection with other systems,7 the influence of temperature may be remarkable. In this sense the HA/PLL system does not make an exception. At 0.5 °C the impedance spiral is still quite normal and the singlezone model gives a reasonable fit to the even branch of the spiral (Figure 7). Increase of temperature to 25 °C generates

Z′, kRayl

Z′′, kRayl

G′, MPa

G′′, MPa

growth exponent β

28.3 27.7 9.7 12.9 36.0 14.2 12.9 11.9 10.6

14.3 14.3 8.9 7.7 19.0 3.9 5.4 3.1 3.8

0.50 0.47 0.013 0.090 0.78 0.16 0.11 0.11 0.081

0.68 0.66 0.14 0.17 1.14 0.093 0.12 0.061 0.068

0.275 ( 0.003 0.252 ( 0.004 0.245 ( 0.003 0.266 ( 0.004 0.248 ( 0.002 0.253 ( 0.003 0.257 ( 0.005 0.249 ( 0.004

“child spirals” to both branches of the impedance spiral, indicating a change from the exponential to linear growth regime (Figure 8). The same phenomenon can be seen at 55 °C (Figure 9). Again the point of change is at the 20th to 22nd bilayer. As we see in Figures 8a and 9a, although the even and odd branches are far from each other, still up to the 22nd bilayer the areal mass densities of both branches remain nearly equal as shown in Figures 8b and 9b. This supports the growth mechanism where each deposited portion of polymer is rapidly diffusing through the whole layer forming a homogeneous mixture but different composition depending whether the last added polymer electrolyte was anionic or cationic. If the polymer electrolytes are structurally very different, we may expect that their mixtures have different complex shear moduli depending on the type of the dominant polymer. This explains the widely deviating even and odd branches of HA and PLL but also the close branches of HA and CH in 0.5 M NaCl where both polymers are polysaccharides and probably partially in a globular form. One should not forget that the amount of material in each deposition step is not negligible in the exponential growth regime. With further deposition cycles, the growth mechanism seems to change and simultaneously the areal mass densities of the even and odd branches start to separate. The change in the mechanism means that with increasing thickness the diffusion rate starts to be too slow for distributing the mobile polymers within the entire film during the time span of one deposition step. This leads to



another source of inhomogeneity which is not properly taken into account in our simplistic model. It may also explain the large differences in the apparent thickness of odd and even branches shown in Figures 8c,d and 9c,d. However, it is important to note that the spiral centers of the even and odd branches in Figures 8a and 9a do not coincide. As mentioned previously, this may happen if the growth is exponential up to the virtual end of the process and the acoustic impedances of the anionic and cationic polyelectrolytes are sufficiently different. On the other hand, if the growth turns to linear and this is supposed to be due to the slow diffusion rate of mobile species, zone II within the transverse acoustic field starts to remain unchanged by addition of new layers on top. This leads to the coalescence of the even and odd branches. In the present case the separate spiral centers means that rapid mixing of species still continued at the last layers, although maybe at a somewhat lower growth exponent. This is nicely supported by the results of Porcel et al.24 who observed by confocal laser scanning microscopy that even in the linear growth regime labeled PLL was rapidly diffusing through the whole film. However, our results indicating that the turning point of the growth rate from exponential to linear is at the 20th to 22nd bilayer is at variance with the results of Porcel et al. According to their observations, this transition takes place always after about 12 bilayers in the HA/PLL system. The viscoelastic properties of the HA/PLL multilayer films have been recently measured by using AFM26 and a lowfrequency piezo-rheometer.27 In the former case the result was a static Young’s modulus with the values 40-90 kPa, whereas in the latter case the complex shear modulus in the frequency range 1-103 Hz was obtained. The storage modulus G′ was approximately 30 kPa, conforming to the Young’s modulus 90 kPa of the previous work. However, these values are not in conformity with our measurements which give ca. five times higher values. One reason may be in our case 104 times higher frequency; although if the layer is purely Voigtian, its storage modulus should not depend on the frequency. On the other hand, viscoelastic polymers are seldom purely Voigtian in such a wide frequency range. Especially in the present case the polymer layers are very soft, resembling polymer solutions, and a better model might be some combination of Voigt and Maxwell fluids. The penetration depth of a transverse acoustic wave to a Maxwell fluid is inversely proportional to the square root of frequency in the low-frequency range and consequently the sampling depth at 1000 Hz is considerably deeper than at 10 MHz. Also the thickness in our measurements is much lower. Collin et al.27 report 32 µm for a multilayer of 90 bilayers whereas our results refer to the thickness range below 2 µm. Further studies to solve this discrepancy are obviously needed.

diverge from each other and converge finally toward separate stationary points if the anionic and cationic polymers are viscoelastically sufficiently different. If the growth turns to linear, we should expect coalescent spiral centers but in this case we cannot expect the layer to be homogeneous. For inhomogeneous layers, a simple two-zone model was devised with a minimum number of additional adjustable parameters. The model provides also the complex shear modulus of the layers. Hyaluronan/chitosan and hyaluronan/poly(L-lysine) multilayers were chosen to illustrate the method because these polyelectrolyte multilayers are well-known for their exponential buildup and their studies have been conducted by different techniques. In some cases these polyelectrolyte systems gave a perfect impedance spiral, indicating a single-zone homogeneous material, but in most cases the presence of two zones had to be assumed. The outer zone, i.e., the zone in contact with liquid, had always a lower shear modulus, resembling viscous liquid or rubbery solid. An interesting result is that the growth exponent which characterizes the exponential growth rate is essentially constant in the buildup of these multilayers despite the different coating temperatures, methods, and cationic polyelectrolyte. When assessing the credibility of the models developed in this work, one should take into account not only the rather good fit of the models to the experimental data but also the long linear range in the plots of ln m vs bilayer number. In these plots the areal mass density extends far beyond the Sauerbrey range. This certainly supports not only the exponential growth of these multilayers but, on the other hand, also the physical feasibility of the areal mass densities and viscoelastic parameters derived from the models. Naturally this does not mean that the mathematical form of functions (12) chosen to represent splitting the growth to two zones is physically relevant. It shows only that the shape of the functions within the applied range is appropriate and a single adjustable parameter is enough to describe the variation in the zone splitting. What comes to the range beyond the exponential growth regime, the proposed twozone model may well be too simple. Applying the techniques reported in this work, a more systematic research on the influence of various factors on the growth processes of polyelectrolyte multilayers is in progress in our laboratory and will be a subject of separate publications.

Conclusions

eq 2 is transformed into

The main aim of this paper is to show that by recording the response of a QCR at such high values of layer thickness that the classical Sauerbrey relation is no more valid and acoustic film resonance is observed, one is able retrieve valuable information on the polyelectrolyte multilayers, their growth processes, and stratification. Influence of loading on a QCR is best described by surface acoustic impedance ζ0, local acoustic impedance on the resonator surface. The graphical representation of ζ0 in the complex plane is a spiral. The shape of this impedance spiral is shown to depend not only on the loss angle of the polymer but also on the growth mechanism. If the growth is exponential, the spiral is split into a double spiral where the spiral branches initially

Appendix Polar Line. Some properties of the impedance spiral are conveniently studied by using the differential equation (2). By making a substitution

ζ ) |ζ| exp(jθ)

(| |

(17)

)

dζ ζ 2 j(2θ-δ) e -1 ) jω dm Zf

(18)

and the separation of the real and imaginary parts gives

|| [| |

dζ′ ζ 2 ) -ω sin(2θ - δ) dm Zf ζ 2 dζ′′ )ω cos(2θ - δ) - 1 dm Zf

]

(19)

The phase angle θ ) θre at the maximum of the real part of LAI is obtained when the right side of the first equation of (19)



is equated to zero. The result is θre ) δ. This means that the straight line

ϑ0 ) k exp(1/2 jδ); k > 0

(21)

where L2 and L3 are layer matrices of zones II and III, respectively (cf. eq 10). Phases φ2 and φ3 are assumed to be functions of a mass parameter b and a shape parameter b0 according to eq 12. In certain cases it is enough to consider only one zone in which case we have a single L, b0 ) 0 and 1/2πb ) φ. The differences of the observed and calculated values are denoted by k

k ) ζ0,kobs - ζ0,kcalc

(22)

Let the covariance matrix corresponding to the kth complexvalued data point be Vk. We form a sum p

F)

ωmk ) |Z2|φ2,k + |Z3|φ3,k

(24)

(20)

passes all the vertical tangential points of the spiral in addition to its center (dashed line in Figure 2). In practice, this implies also that if the layer is grown by any meansselectropolymerization, layer-by-layer techniques, spin-coating, etc.sand LAI measured after each addition, and the maximum of the real part is observed, the loss angle of the material is obtained without any need for the thickness of the layer. It is also important to note that this result does not depend on the medium contacting the film but requires constancy of bulk impedance during deposition. Algorithm for Fitting Surface Acoustic Impedance. We have a set of experimental data points {ζ0,iobs:i ) 1, 2, ..., nmax} recorded with an increasing number of layers, and we assume that within a subset {ζ0,kobs: k ) 1, ..., p} the values of bulk acoustic impedances Z2 and Z3 of zones II and III are constant. The calculated values of surface acoustic impedance are then

ζ0,kcalc ) (L2,kL3,k) o Zmed

are calculated according to the definition (7) of φ:

∑ [(Vk-1)11Re2 k + (Vk-1)22 Im2 k + k)1 2(Vk-1)12 Re k Im k] (23)

This function is minimized in terms of p + 5 adjustable parameters, |Z2|, |Z3|, δ2, δ3, b0, and bk, k ) 1, 2, ..., p. One should note that because the data points are complex numbers with the real and imaginary parts, the actual number of data is 2p. Hence the number of degrees of freedom is still p - 5 (or p - 2 in the case of a single zone). The areal mass densities mk

Acknowledgment. Grant 102279 from the Academy of Finland is gratefully acknowledged. References and Notes (1) Sauerbrey, G. Z. Phys. 1959, 155, 206-222. (2) (a) Nomura, T.; Minemura, A. Nippon Kagaku Gaishi 1980, 1621. (b) Nomura, T.; Ijima, M. Anal. Chim. Acta 1981, 131, 97-102. (c) Konash, P. L.; Bastiaans, G. J. Anal. Chem. 1980, 52, 1929-1931. (3) Kankare, J. Langmuir 2002, 18, 7092-7094. (4) (a) Johannsmann, D.; Gru¨ner, J.; Wesser, J.; Mathauer, K.; Wegner, G.; Knoll, W. Thin Solid Films 1992, 210-211, 662. (b) Domack A.; Prucker, O.; Ru¨he, J.; Johannsmann, D. Phys. ReV. E 1997, 56, 680-689. (c) Hillman, A. R.; Brown, M. J. J. Am. Chem. Soc. 1998, 120, 1296812969. (d) Martin, S. J.; Bandey, H. L.; Cernosek, R. W.; Hillman, A. R.; Brown, M. J. Anal. Chem. 2000, 72, 141-149. (e) Lagier, C.; Efimov, I.; Hillman, A. R. Anal. Chem. 2005, 77, 335-343. (5) Ruths, J.; Essler, F.; Decher, G.; Riegler, H. Langmuir 2000, 16, 8871-8878. (6) McAloney, R. A.; Sinyor, M.; Dudnik, V.; Goh, M. C. Langmuir 2001, 17, 6655-6663. (7) Saloma¨ki, M.; Vinokurov, I. A.; Kankare, J. Langmuir 2005, 21, 11232-11240. (8) Picart, C.; Mutterer, J.; Richert, L.; Luo, Y.; Prestwich, G. D.; Schaaf, P.; Voegel, J. C.; Lavalle, P. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12531-12535. (9) Arnau, A., Ed. Piezoelectric Transducers and Applications; Springer: New York, 2004. (10) Kankare, J.; Loikas, K.; Saloma¨ki, M. Anal. Chem. 2006, 78, 18751882. (11) Saloma¨ki, M.; Loikas, K.; Kankare, J. Anal. Chem. 2003, 75, 58955904. (12) Saloma¨ki, M.; Laiho, T.; Kankare, J. Macromolecules 2004, 37, 9585-9590. (13) Picart, C.; Lavalle, P.; Hubert, P.; Cuisinier, F. J. G.; Decher, G.; Schaaf, P.; Voegel, J. C. Langmuir 2001, 17, 7414-7424. (14) Boulmedais, F.; Ball, V.; Schwinte, P.; Frisch, B.; Schaaf, P.; Voegel, J. C. Langmuir 2003, 19, 440-445. (15) Richert, L.; Lavalle, P.; Payan, E.; Shu, X. Z.; Voegel, J.-C.; Picart, C. Langmuir 2004, 20, 448-458. (16) Kujawa, P.; Moraille, P.; Sanchez, J.; Badia, A.; Winnik, F. M. J. Am. Chem. Soc. 2005, 127, 9224-9234. (17) Ladam, G.; Schaad, P.; Voegel, J. C.; Schaaf, P.; Decher, G.; Cuisinier, F. Langmuir 2000, 16, 1249-1255. (18) Schlenoff, J. B.; Dubas, S. T. Macromolecules 2001, 34, 592598. (19) Rehakova, M.; Bakos, D.; Soldan, M.; Vizarova, K. Int. J. Biol. Macromol. 1994, 16, 121-124. (20) Cleland, R. L.; Wang, J. L. Biopolymers 1984, 23, 647-666. (21) Signini, R.; Filho, S. P. C. Polym. Bull. 1999, 42, 159-166. (22) Rinaudo, M.; Milas, M.; Dung, P. L. Int. J. Biol. Macromol. 1993, 15, 281-285. (23) 1 kRayl ) 1000 Rayl )1000 kg m-2 s-1. (24) Porcel, C.; Lavalle, P.; Ball, V.; Decher, G.; Senger, B.; Voegel, J. C.; Schaaf, P. Langmuir 2006, 22, 4376-4383. (25) The lower value of the relative frequency shift at 15 MHz compared with 5 MHz is explained by the high loss compliance and Voigtian character of the layer that increase the medium correction of the Sauerbrey constant (cf. ref 3). Areal mass density is still proportional to the relative frequency shift, and the logarithmic plot gives the growth exponent as a slope. (26) Richert, L.; Engler, A. J.; Discher, D. E.; Picart, C. Biomacromolecules 2004, 5, 1908-1916. (27) Collin, D.; Lavalle, P.; Garza, J. M.; Voegel, J. C.; Schaaf, P.; Martinoty, P. Macromolecules 2004, 37, 10195-10198.

Modeling the Growth Processes of Polyelectrolyte Multilayers Using a

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