Effect of Polyelectrolyte Multilayers on the Response of a Quartz

Anal. Chem. 2003, 75, 5895-5904

Effect of Polyelectrolyte Multilayers on the Response of a Quartz Crystal Microbalance Mikko Saloma 1 ki,† Kari Loikas,‡ and Jouko Kankare*,‡

Department of Chemistry, University of Turku, FIN-20014 Turku, Finland, and Graduate School of Chemical Sensors and Microanalytical Systems

The effect of a polyelectrolyte (PE) multilayers made by a layer-by-layer technique on the response of a quartz crystal microbalance (QCM) is studied by using novel mathematical methods based on the Mo1 bius transformations and their matrix representations in the complex plane. In the first method, the basic properties of the Mo1 bius transformation are used for obtaining the PE bilayer matrix from the QCM impedance measurements taken at four different numbers of layers. In the second method, nonlinear fitting with concomitant error estimation is used for obtaining the elements of the bilayer matrix. The methods are applied to a multilayer composed of 150 bilayers of poly(sodium 4-styrenesulfonate) and poly(diallyldimethylammonium) chloride on a quartz crystal resonator. The structure of the system is discussed, and the bulk acoustic impedance and areal mass density of the bilayer are calculated from the layer matrix. A very versatile and facile technique for the preparation of thin organic films is the sequential layer-by-layer (LbL) deposition of oppositely charged polyelectrolytes, as initially reported by Decher et al.1 A huge number of reports have been published on the use of these techniques for various kinds of polyelectrolytes from purely synthetic polymers to polymers of biological origin. The main advantage of polyelectrolyte multilayers (PEMs) is the possibility of tailoring the surface properties, such as functionality, charge, and wettability, by the choice of the outermost layer, keeping control at the same time over the layer thickness, lateral homogeneity, and low surface roughness, all this by using a very simple fabrication process. Naturally, there is a need to monitor these properties during or after the deposition process. Very often, the PEMs will be used in an aqueous environment, and consequently, the control of the properties should be done in situ. The number of techniques available for these purposes is rather limited. If the polyelectrolytes have spectral absorbance at some UV-vis-NIR wavelength, absorbance measurement is a viable method for estimating the amount of material on a transparent substrate.2,3 In situ ellipsometry can be used for the measurement * Corresponding author. E-mail: [email protected] † Graduate School of Chemical Sensors and Microanalytical Systems. ‡ Department of Chemistry, University of Turku. (1) Decher, G.; Hong, J. D. Makromol. Chem. Macromol. Symp. 1991, 46, 321. (2) McAloney, R. A.; Sinyor, M.; Dudnik, V.; Goh, M. C. Langmuir 2001, 17, 6655-6663. (3) Lukkari, J.; Saloma¨ki, M.; A ¨ a¨ritalo, T.; Loikas, K.; Laiho, T.; Kankare, J. Langmuir 2002, 18, 8496-8502. 10.1021/ac034509z CCC: $25.00 Published on Web 10/04/2003

© 2003 American Chemical Society

of the layer thickness in many cases.4 One of the most powerful methods is the quartz crystal microbalance (QCM) which gives the areal mass density, that is, mass per unit area, of PEM. Additionally, QCM may give also information on the elastic properties of PEM if the electrical impedance measurement and subsequent data treatment are properly done. The experimental techniques presently available for the full characterization of the oscillating thickness-shear mode (TSM) resonator are based either on a freely oscillating quartz crystal, for which the oscillation frequency is determined by the resonant frequency of the crystal itself, or on an electrical impedance measurement in which the resonator is a passive element as the measurement frequency is swept across the resonant region of the crystal and its response is recorded. In the present study, we do not pay attention to the measurement technique because, anyway, the data from different methods can be brought to a common form. Instead, we are concentrating on using the collected data as efficiently as possible. The elastic properties of thin films can be described by the bulk acoustic impedance of the material. This is a complex-valued parameter that depends also on the vibrational frequency used for its measurement. An additional parameter is the areal mass density, that is, mass per unit area of the resonator coating. The primary parameter obtained from the QCM measurement used is the local acoustic impedance (LAI) at the interface of the resonator surface and the film coated on it, also called surface mechanical impedance. This parameter is also complex-valued. Hence, there are three material parameters, that is, areal mass density and the real and imaginary components of bulk acoustic impedance, which characterize the thin film on the resonator surface. However, only two parameters, that is, the real and imaginary components of the local acoustic impedance, are obtained from a single measurement. Consequently, no unique solution is obtained for any of these three unknown parameters unless further measurements are done or new assumptions are made.5 For a long time, areal mass density was the only parameter obtained from the QCM by using the celebrated Sauerbrey equation.6 In this case, an implicit assumption was made that the film was made of fully elastic material without any energy losses. Later, it was realized that the viscoelasticity of the film may introduce errors to the mass determinations. Various models, (4) Harris, J. J.; Bruening, M. L. Langmuir 2000, 16, 2006-2013. (5) Hillman, A. R.; Jackson, A.; Martin, S. J. Anal. Chem. 2001, 73, 540-549. (6) Sauerbrey, G. Z. Phys. 1959, 155, 206-222.

Analytical Chemistry, Vol. 75, No. 21, November 1, 2003 5895

experimental procedures, and data analyses for viscoelastic films have been overviewed by Martin7 and Johannsmann.8 Polyelectrolyte multilayers seem to create special problems when a QCM is applied. There are no problems at low loading with only a few layers on the resonator surface and especially when the QCM measurement is done on a dry sample. Then the simple Sauerbrey equation gives reliable results on the areal mass density. But if there are plenty of layers and especially if the measurement is done in situ under liquid, there are no reasons to believe that a simple change in the resonant frequency gives adequate information on the state and amount of the layer material. The impedance measurement of QCM is becoming a routine tool, and in principle, this method gives more information. However, the full mathematical analysis of local acoustic impedance of thick periodic multilayer systems has apparently not yet been reported, and this is the main purpose of this paper. The application of the theory is illustrated by making and characterizing a multilayer system with 150 bilayers of poly(diallyldimethylammonium chloride) (PDADMA) and poly(sodium 4-styrenesulfonate) (PSS). A novel mathematical method based on the matrix analysis in the complex plane was developed for analyzing the functional dependence of local acoustic impedance on the number of layers and for extracting material parameters from the experimental impedance data. THEORY Mo1 bius Transformations and Matrix Formalism. In a previous publication,9 it was shown that the primary data from a QCM measurement can be presented as a complex-valued quantity called local acoustic impedance (LAI) at the resonator surface, denoted by ζ0. This quantity has also been called mechanical impedance by others,10 but we prefer our denotation because this quantity is closely related to the bulk acoustic impedance. The local acoustic impedance ζ in the bulk phase is defined as the ratio of applied stress, T, to the particle velocity, v, taken with an opposite sign, that is, ζ ) -T/v. We have shown that the onedimensional system of a TSM resonator with an arbitrary, laterally homogeneous surface loading can be described by a nonlinear Riccati equation9

ζ 2 dζ ) -1 dm ˜ Z

()

(1)

from the resonator surface. The main advantage of eq 1 is the simple treatment of boundary conditions, because the local acoustic impedance is a continuous function at the no-slip interfaces. If we cut a thin parallel slice from the layer on the resonator surface and mark the cutting planes by 1 and 2, it can be shown by solving eq 1 that

ω∆m ω∆m - jZ sin Z Z ζ2 ) ω∆m ω∆m -1 + cos -jζ1Z sin Z Z ζ1 cos

Here, ζ1 and ζ2 are local acoustic impedances at the cutting planes 1 and 2, ∆m is the mass per unit area (areal mass density) of the slice, and Z is the bulk acoustic impedance of the slice material, assumed to be constant within the slice. The sign of ∆m is taken to be positive or negative, depending on the relative distances of the cutting planes from the resonator surface. The primary goal in our mathematical treatment is to find an expression for ζ0, the local acoustic impedance at the surface of the resonator, because this is the quantity that is experimentally available. What is interesting in eq 3 is that it shows how we get a solution for ζ0 in a multilayer system. Let us assume that we have a multilayer system with N layers. The outermost “cut plane” is actually the interface against the continuous medium with acoustic impedance Zmed. At this interface, with the no-slip condition, the local acoustic impedance is ζN ) Zmed.9 Proceeding toward the resonator surface, we obtain from eq 3

ω∆mN ω∆mN + jZN sin ZN ZN ζN-1 ) ω∆m ω∆m N N jζNZN-1 sin + cos ZN ZN ζN cos

m ˜ ) jωm ) jω

∫

y

0

F dy; j ) x-1

(2)

Here, F is the density of material, and ω the angular frequency of acoustic vibration. The position variable y stands for the distance (7) Martin, S. J. Faraday Discuss. 1997, 107, 463-476. (8) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501-516. (9) Kankare, J. Langmuir 2002, 18, 7092-7094. (10) McHale, G.; Lu ¨ cklum, R.; Newton, M. I.; Cowen, J. A. J. Appl. Phys. 2000, 88, 7304-7312.

5896

Analytical Chemistry, Vol. 75, No. 21, November 1, 2003

(4)

Using the same kind of transformation, we can proceed iteratively down to the surface plane of the resonator. Equations 3 and 4 represent a Mo¨bius transformation11 or sometimes called linear fractional transformation in the complex plane. Generally, the Mo¨bius transformation is written as a quotient, which can be also represented as a matrix operation

z2 ) where Z stands for the bulk acoustic impedance of the material in the transverse acoustic field and the variable m ˜ is defined as a complex mass impedance, that is, as a cumulative areal mass density between 0 and y multiplied by jω.

(3)

(

)

A11z1 + A12 A A ) A11 A12 o z1 ) A o z1 A21z1 + A22 21 22

(5)

The matrix representation is very convenient, because the successively applied Mo¨bius transformations can be represented as a matrix multiplication. It can be easily shown that

B o (A o z) ) (BA) o z

(6)

Hence, the propagation of transverse acoustic wave in the stratified medium can be described by matrices in an analogous way to the propagation of light in the stratified dielectric films.12 (11) Schwerdtfeger, H. Geometry of Complex Numbers; Dover Publications: Mineola, NY, 1979; p 41. (12) Abele`s, F. Ann. Phys. 1950, 5, 596-640. Born, M.; Wolf, E. Principles of Optics, 5th ed.; Pergamon Press: Oxford, 1975; p 66.

(

If we now denote

ω∆mi ω∆mi jZi sin Zi Zi Ai ) ω∆m ω∆m i i jZi-1 sin cos Zi Zi cos

)

(7)

the iteration scheme suggested by eq 4 can be represented as a matrix product and Mo¨bius transformation

ζ0 ) (A1A2...AN) o ζN

(8)

It is important to note that the final expression in eq 8 is still a Mo¨bius transformation of the form of eq 5. Actually, if we have a layer in which the bulk acoustic impedance varies according to some arbitrary function of the distance from the resonator surface, we can cut the layer into infinitely thin slices and form the product described in eq 8. Hence, the influence of the entire layer can be described as a single matrix. This layer matrix or Mo¨bius matrix has four elements, but as we see from eq 5, the multiplication of each element of the matrix by the same scalar does not change the value of the quotient because of the cancellation. Hence, as a result of this redundancy, the behavior of an arbitrary, laterally homogeneous but vertically nonhomogeneous layer in the transverse acoustic field can be described completely by three complex numbers. It should be noted that if the layer is totally homogeneous, that is, the bulk acoustic impedance is the same everywhere within the layer, the matrices in eq 8 commute, and their product has the form of eq 7, with ∆mi substituted by the total areal mass density of the layer. Then only three real parameters are needed to describe the layer, that is, its areal mass density and the real and imaginary components of the bulk acoustic impedance. On the other hand, if the material of the layer is stiff, so that the modulus of its acoustic impedance is large, and the layer is thin, we may approximate the trigonometric functions by the first terms in their MacLaurin series. Omitting the second-order term Z-2, the layer matrices take then the Jordan canonical form

Ai f

(

1 jω∆mi 0 1

)

(9)

and because these matrices commute, their product becomes simply

A1A2...AN )

(

) (

1 jω(∆m1 + ∆m2 + ... + ∆mN) 1 jω∆m ) 0 1 0 1

)

(10)

Remembering that ζN ) Zmed, the corresponding Mo¨bius transformation gives, finally

ζ0 ) Zmed + jω∆m

mass density contributes in the first approximation only to the imaginary part of LAI, which is directly proportional to the relative change in the resonant frequency of the resonator. In this approximation, the bulk acoustic impedance of the film material has no first-order influence on the result. As shown in our previous publication,9 in the next approximation, both the acoustic impedance of the film and liquid medium have further contributions to LAI. The description of the layers in a stratified medium by layer matrices is by no means a new idea. Granstaff and Martin13 and Behling et al.14 have described the propagation of transverse vibrations by the transmission line method, often referred to as the Mason model.15 However, the layer matrices of the transmission line method are operators in the (T, v) space, whereas in the present work, they are representations of Mo¨bius transformations, as a natural outcome from the solution of the Riccati equation (eq 1). It is this subtle difference that suggests using the special properties of Mo¨bius transformations. One of these special properties is the concept of a fixed point. The fixed point of a Mo¨bius transformation is a point in the complex plane that coincides with its image point, that is,

Aoz)

A11z + A12 )z A21z + A22

(12)

This leads to a second-degree equation

A21z2 + (A22 - A11)z - A12 ) 0

(13)

In the case of the homogeneous layer matrix A of eq 7, the elements of the main diagonal are equal, and we obtain from eq 13

x

zfix ) (

A12 ) (Z A21

(14)

Hence, the fixed point of the layer matrix of a homogeneous layer is the bulk acoustic impedance of the layer material. Only one of the two fixed points (14) is physically acceptable, the condition being 0 < arg Z < π/4. Another way to view the fixed points is to examine the iteration

zk+1 ) A o zk; z0 ) Zmed

(15)

where A is a matrix shown in eq 7 (without subscript i). From the physical viewpoint, this iteration describes the situation as if we were adding one by one layers with mass ∆m and impedance Z on the resonator initially in contact with the medium having impedance Zmed. Obviously, we are finally approaching the situation in which the resonator does not “feel” the original medium, and the system starts to correspond to the resonator in contact with semi-infinite medium with bulk acoustic impedance Z. This

(11)

This is mathematically equivalent to the commonly used Sauerbrey equation in liquid. As we see, the change in the areal

(13) Granstaff, V. E.; Martin, S. J. J. Appl. Phys. 1994, 75, 1319-1329. (14) Behling, C.; Lucklum, R.; Hauptmann, P. Sens. Actuators, A 1997, 61, 260266. (15) Rosenbaum, J. F. Bulk Acoustic Wave Theory and Devices; Artech House, Inc.: Boston, 1988.


5897

Another fundamental property of Mo¨bius transformations is the invariance of the cross ratio. The cross ratio can be defined in terms of four points z1, z2, z3, and z4 in the complex plane.

(z1 - z4)(z3 - z2)

(17)

(z1 - z3)(z2 - z4)

Its invariance means that if points w1, w2, w3, and w4 are the corresponding image points as an arbitrary Mo¨bius transformation is applied to zi’s

{z1, z2, z3, z4} |f {w1, w2, w3, w4}

(18)

we have

(w1 - w4)(w3 - w2) (w1 - w3)(w2 - w4)

)

(z1 - z4)(z3 - z2) (z1 - z3)(z2 - z4)

(19)

The significance of this invariance is in its use for the determination of the Mo¨bius transformation from the measured values of local acoustic impedance. Let us assume that we have the iterative system described by eq 15; i.e., we have some means of adding repeatedly similar layers of material on the resonator surface. Let the first layer be described by matrix N and the corresponding measured LAI by ζ1. We add three times the layer with matrix A which commutes with N and measure LAI each time. We obtain

ζ1 ) N o Zmed; ζ2 ) A o (N o Zmed) ) A o ζ1; ζ3 ) A o ζ2; ζ4 ) A o ζ3 (20) This means that we have mapping Figure 1. Argand diagram of local acoustic impedance. (a) Circles are experimental values of LAI for PSS-PDADMA layers marked at every 10 layers. Line is the locus of matrix iteration of type eq 15 with n as a variable and Meven of Table 1 as the matrix. (a) n ) 2000; (b) The same data but the locus of calculated points drawn up to n ) 300. The layers taken to the data treatment are marked by arrows.

A

{ζ1, ζ2, ζ3} 98 {ζ2, ζ3, ζ4}

Taking now z4 |f z as a variable and w4 |f w as its image point and taking into account mapping (21), we may write the invariance (19) into the form

w - ζ2 z - ζ1 )λ w - ζ3 z - ζ2

can be shown also mathematically, and we have the limiting behavior

lim Ak o Zmed ) Z kf∞

(16)

5898


(22)

where

λ) which is valid for arbitrary Zmed. This means that the trajectory of the local acoustic impedance ζ0 in the complex plane as a function of increasing mass on a resonator is a continuous curvesactually a spiralswhich starts from the bulk acoustic impedance of the medium and approaches the bulk acoustic impedance of the layer material (cf. Figure 1a). The condition is that both the real and imaginary components of Z are positive, which is true for all known materials.

(21)

(ζ3 - ζ2)(ζ2 - ζ4) (ζ1 - ζ3)(ζ4 - ζ3)

(23)

Now w is solved from eq 22, giving

w)

(ζ2 - λζ3)z + λζ1ζ3 - ζ22 (1 - λ)z + λζ1 - ζ2

(24)

This already has the form of a Mo¨bius transformation, and the

corresponding matrix is

D)

where

(

ζ2 - λζ3 λζ1ζ3 - ζ22 λζ1 - ζ2 1-λ

)

What we have now accomplished is a method whereby the layer matrix can be determined from four measurements of LAI. What is needed is a reproducible method for adding similar layers on the resonator surface. One good example is the layer-by-layer deposition of polyelectrolytes. Periodic Multilayers. Periodic multilayers are the result of layering two different polymers, usually oppositely charged polyelectrolytes, alternately on a substrate. In the resulting multilayer system, the layers are generally strongly interpenetrated, and clear borderlines between different layers do not exist. However, a certain degree of periodicity may still exist, and our purpose is to study how this periodicity affects the observed local acoustic impedance. We assume now that we have two oppositely charged polyelectrolytes, A and B, and these polyelectrolytes are alternately coated layer by layer on a resonator, forming a stratified structure ABABAB....AB. Let the number of bilayers AB be n. The layers A and B have their matrices

A)

(

) (

jZ1 sin R jZ2 sin β cos R cos β ;B) jZ1-1 sin R cos R jZ2-1 sin β cos β

)

(26) where

R)

ωm2 ωm1 ; β) Z1 Z2

(27)

with mi and Zi having their obvious meaning. Let the first layer at the outer surface of the layer system be B. The matrix of the bilayer is then M ) AB )

(

cos R cos β - Z1/Z2 sin R sin β

j(Z1 sin R cos β + Z2 sin β cos R)

j(Z1-1 sin R cos β + Z2-1 sin β cos R) cos R cos β - Z2/Z1 sin R sin β

)

(28)

To solve the influence of the layer system on the local acoustic impedance at the resonator surface, we have to evaluate the expression

ζ0 ) Mn o ζmed

(29)

where ζmed is the local acoustic impedance at the outer surface that usually is in contact with a medium, either air (ζmed ) 0) or some liquid (ζmed ) Zmed).One important property of these matrices is their unimodularity; i.e., their determinant is equal to 1. For unimodular matrices, we have12

Mn )

sin(n - 1)θ sin nθ MI sin θ sin θ

cos θ ) 0.5trM

(25)

(30)

(31)

This equation will be used in the reverse mode; i.e., Mn is known and M should be solved. In the multilayer systems, bulk acoustic impedance is not constant but naturally average impedance may be defined. The problem is how the average should be calculated. One obvious way is to define it as the measured LAI at a resonator on which a semiinfinite multilayer system has been deposited. In this case, it becomes as the fixed point of the Mo¨bius transformation of the bilayer, giving from eq 13

Zav )

M11 - M22 ( 2M21

x(

)

M11 - M22 2M21

2

+

M12 M21

(32)

where the sign is chosen in such a way that Zav is in the first quadrant of the complex plane. EXPERIMENTAL SECTION Coating Procedure. The measurements were done on goldcoated polished quartz crystals (International Crystal Manufacturing Company, Inc.) of 10-MHz nominal frequency. The diameter of the crystals was 0.538 in., and the diameter of the gold-coated center was 0.201 in. A chromium adhesion layer (10 nm) was used under the 100-nm gold coating. Before use, the surface of the crystals was cleaned in a plasma cleaner (Harrick Scientific Corp.). The gold surface was modified by keeping the crystal in a 0.001 M solution of 2-aminoethanethiol in water for 60 min. The cationic polymer, poly(diallyldimethylammonium chloride) (PDADMA, MW 100-200 kDa) and the anionic polymer, poly(sodium 4-styrenesulfonate) (PSS, MW 70 kDa) were obtained from Aldrich and used without further purification. The solutions of these polymers were 0.01 M (in terms of monomer unit) in 0.1 M solution of sodium nitrate. A multilayer system with 300 alternate PSS and PDADMA layers was made by using an automated LbL machine. The crystal was kept in a flow cell with ∼300 µL total volume, and solutions were brought into contact with it by using a computer-controlled peristaltic dispensing pump (Ismatec model ISM 834) and a sixposition rotating valve (Omnifit) for choosing the appropriate solution. The entire procedure including coating and rinsing was done by using a 0.1 M aqueous solution of NaNO3 as the supporting electrolyte. The coating procedure was based on the computer-controlled sequences consisting of (1) coating by allowing 1.5 mL of polyelectrolyte solution to flow through the cell; (2) allowing the PE solution to stand in the cell for 15 min; (3) rinsing the cell with 10 mL of 0.1 M NaNO3; (4) allowing the rinsing solution to stand in the cell for 15 min; (5) measuring the impedance (usually only every fifth layer). Hence, we followed the procedure advocated by Ladam et al.16 and Castelnovo and Joanny;17 i.e., the ionic strength was kept constant during the entire deposition process. (16) Ladam, G.; Schaad, P.; Voegel, J. C.; Schaaf, P.; Decher, G.; Cuisinier, F. Langmuir 2000, 16, 1249-1255. (17) Castelnovo, M.; Joanny, J.-F. Langmuir 2000, 16, 7524-7532.


5899

The body of the QCM cell was made of gold-plated copper to ensure good heat conductance and fast temperature stabilization. The crystal was mounted between O-rings and kept under constant tension using springs. The cell was kept in a thermostat at 25 ( 0.03 °C in a stainless steel cylinder in which a slow internal airflow was arranged. The cell temperature was continuously monitored by using a calibrated thermistor. The whole system, including flowing and switching the liquids and the measurement of impedance, was controlled by a program written in LabVIEW (National Instruments). Measurement of QCM Impedance. Measurements of electrical impedance of QCM were done by using a homemade instrument18 based on the principle of double sideband modulation and lock-in detection. The direct connection between LAI ζ0 and electrical impedance Zq is given by the expression

Zq )

(

)

tanc(1/2ζq/ζc) - jζ0/ζq 1 1 - K2 tanc(ζq/ζc) (33) jωC0 tanc(ζq/ζc) - jζ0/ζq

which is mathematically equivalent to the expression given by Lu¨cklum et al.19 but using a different notation. Here, function tanc is defined as the ratio20

tanc z )

tan z z

(34)

and the parameters are as follows: ζq ) ωdqFq; ζc ) xFqjc66(1 + jξ); K2 ) (e262)/[22jc66(1 + jξ)]; ω, angular frequency; Fq, density of quartz (2651 kg m-3); dq, thickness of the resonator; jc66, piezoelectrically stiffened elastic constant for lossless quartz (2.947 × 1010 Nm-2); ξ, loss factor of quartz (7.46 × 10-7 at 10 MHz); e26, piezoelectric stress constant of quartz (9.652 × 10-2 A s m-2); 22, permittivity of quartz (3.982 × 10-11 A2 s4 kg-1 m-3); and C0, geometric capacitance of the resonator. The electrical impedance of the crystal without loading was measured first at the frequency range around the resonant frequency (10 MHz). The measurements were done at 2000 closely spaced frequencies. The thickness and capacitance of the resonator, stray capacitance, and losses of the crystal with its mounting were parameters determined by nonlinear fitting to eq 33. These parameters, except stray capacitance, were considered as typical for the crystal and its mounting, and they were kept constant for the loaded resonator. The measurements for the loaded crystal were done at 500 frequencies because of the wider character of resonance lines. The parameters to be varied for the loaded resonator in the nonlinear fitting were the stray capacitance and the modulus and phase of LAI. The covariance matrix of the real and imaginary parts of LAI was calculated by using the bootstrap technique.21 RESULTS Following Ladam et al.,16 the polyelectrolyte multilayer film can be assumed to consist of three zones, denoted by I, II, and (18) Kankare, J.; Loikas, K. Patent pending. To be published. (19) Lu ¨ cklum, R.; Behling, C.; Cernosek, R. W.; Martin, S. J. J. Phys. D: Appl. Phys. 1997, 30, 346-356. (20) Weisstein, E. W. http://mathworld.wolfram.com/TancFunction.html (21) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, 1992; p 686.

5900


III. Zone I is closest to the substrate, in our case, the resonator surface; II is in the middle; and zone III is in contact with the solution. In accordance with Ladam et al.,16 we make a reasonable assumption that during the growth of the layer system, zones I and III stay constant, while zone II grows, at least when the total number of layers is large enough. Layers in zone III are soft, strongly swollen by the solvent. To this zone, we may attribute our previous results3 which show that at least when the total number of layers is 10 or below, the bulk acoustic impedance is very low, resembling the impedance of liquid. Zone I may consist of layers of impedance similar to those layers within zone II, but presumably thinner and partially influenced by the roughness of the resonator surface. The total matrix of the layer system, corresponding to its Mo¨bius transformation, can be written as the matrix product

P0 ) L1L2L3

(35)

where L1, L2 and L3 refer to zones I, II, and III, respectively. As we introduce n new, mutually identical bilayers of equal thickness and composition corresponding to the layers in zone II, they become localized in zone II, and the new matrix is

Pn ) L1MnL2L3

(36)

where M corresponds to a single bilayer. We introduce another matrix

P′n ) MnP0 ) MnL1L2L3

(37)

There are reasons to believe that matrices P′n and Pn are nearly equal. To be exactly equal, matrices L1 and M should commute.

ML1 ) L1M

(38)

As we have seen, if the bulk acoustic impedances of the materials corresponding to these matrices are equal, the matrices commute. On the other hand, if the layers are very thin (as they actually are), the matrices have nearly the Jordan canonical form (eq 9), and they are approximately commutative, even in the case of unequal acoustic impedance. The strategy of measurement follows now two lines. In the first strategy, we do not make any a priori assumption on the Mo¨bius transformation, relying on the experimental data and the invariance of cross ratio (eq 17). In principle, four measurements of crystal impedance are required to obtain the Mo¨bius matrix. In the second strategy, we assume that the zone II is homogeneous, despite its construction from two polyelectrolytes. The corresponding Mo¨bius matrix is then of the type in eq 26. Starting from one data point corresponding to a suitably small number of layers, the other values of LAI are generated iteratively as in eq 15. The “best” Mo¨bius matrix is then obtained by nonlinear fitting of its parameters. Use of Cross Ratio Invariance. The strategy of measurement is now to introduce successively n new bilayers three times to the system and to make the impedance measurement each time. If the bulk acoustic impedance of liquid is Zmed, we have the local

acoustic impedance at the resonator surface

ζ1 ) L1L2L3 o Zmed ζ2 ) L1MnL2L3 o Zmed ≈ MnL1L2L3 o Zmed ) Mn o ζ1 ζ3 ) L1M2nL2L3 o Zmed ≈ MnL1MnL2L3 o Zmed ) Mn o ζ2 ζ4 ) L1M3nL2L3 o Zmed ≈ Mn o ζ3

(39)

Hence, the Mo¨bius transformation corresponding to matrix Mn effects mapping given in eq 21. The use of cross ratio gives a matrix D (cf. eq 25), which now corresponds to matrix Mn, except that its determinant is not unity. Normalization gives

D′ ) (detD)-1/2 D ) Mn

(40)

We have previously given an expression for the nth power of the bilayer matrix M (eq 30), from which we may solve M.

M)

sin(n - 1)θ sin θ D′ + I sin nθ sin nθ

(41)

To obtain θ, we apply matrix trace to both sides of eq 30, giving

trMn ) trD′ )

sin (n - 1)θ sin nθ trM - 2 ) 2 cos nθ (42) sin θ sin θ

Hence we have

1 θ ) arccos(1/2|trD′|) n

(43)

We have shown that in principle, the bilayer matrix can be solved from the experimental measurements of crystal impedance by using the basic properties of Mo¨bius transformations. However, one should be aware of various pitfalls in this procedure. A very important point is the choice of data used for calculations. For instance, if the four points in the complex plane are collinear and equidistant, the value of λ in eq 23 becomes equal to 1. The element D21 of matrix D becomes 0, and important information is lost. Hence, an important criterion is that the number of bilayers, n, is large enough to reach the curvilinear region and to avoid collinearity of data points. It is also important to choose the data in the region where the relative error of acoustic impedance is small. Even small errors cause large relative errors in 1 - λ, and consequently, large errors in the estimated bulk acoustic impedance. It is also known that the outer polyelectrolyte layers, in zone III, behave differently as a result of the penetration of solvent into the membrane. Hence, it is not feasible to take these first layers as the basis of calculations. How many layers should be discarded is a decision to be made on the basis of experimental results. In our case, we started from 10 to 15 bilayers. In principle, the fourth layer taken to the calculations should be as far from the first layer as possible. In reality, the limit is put by the standard deviation of the LAI components, which increases as the number of layers increases. In the present case, the fourth data point was taken at 235 or 240 layers. Two series of data were chosen according to

the even or odd number of layers. In the former case, the data were taken at 30, 100, 170, and 240 layers, and in the latter case, at 25, 95, 165, and 235 layers. In both cases, we have 35 bilayers as the value of n of eq 36, but the layers are in a different order. In Figure 1, we have the experimental points in the complex plane for even number of layers; i.e., the layers are all bilayers. In Figure 1a, the theoretical locus of LAI has been drawn up to 2000 layers in order to show the convergence of LAI as the spiral toward the bulk acoustic impedance of the layer material. Figure 1b is a magnified excerpt of Figure 1a, giving a better look at the fit of the experimental points to the theoretical curve. The four data points chosen for the estimation of the Mo¨bius transformation have been marked by arrows, and as we can see, they are not collinear. The matrix obtained by applying eqs 25, 40, and 41 is denoted by Meven. If we now take the odd-numbered layers, that is, those layers in which the uppermost layer in contact with solution is the anionic polyelectrolyte, and apply the similar data treatment to layers numbered 25, 95, 165, and 235, we obtain a bilayer matrix Modd. These matrices become

Meven )

(

0.99951 - 0.000452j 70.96 + 2696.5j 4.2998 × 10-9 + 1.4454 × 10-8j 1.00046 + 0.000465j

Modd )

(

0.99952 - 0.000430j 57.56 + 2630.6j -9 -8 4.7725 × 10 + 1.6348 × 10 j 1.00043 + 0.000444j

) )

A bilayer consists of two layers with different materials, and the bilayer matrix Meven is a product of the matrices corresponding to these films. Let these matrices be A and B, as in eq 28. Then Modd is the product of these same matrices but taken in a different order. The commutator of matrices A and B of eq 26 becomes

AB - BA )

(

)

( )

Z2 Z1 1 0 sin R sin β 0 -1 Z1 Z2

(44)

but as we see, the difference Meven - Modd does not possess this form. The conclusion is that the precision of the measurement is not good enough to allow observation of any periodicity in the multilayer. To calculate the bulk acoustic impedance from matrix M, we have two possibilities. Either we may still stick to the assumption that the materials of the bilayer are different, meaning also that the mutual interpenetration has not homogenized bilayers, or alternatively, we may assume that the layers are actually homogeneous and the inequality of the main diagonal elements is due to stochastical errors only. In the former case, the apparent acoustic impedance is obtained as the fixed point of the Mo¨bius transformation (eq 32). In the latter case, we use the simpler eq 14. From matrix Meven, we obtain

Zav ) 382072 + 74895j; Z ) 419300 + 55420j Although there is a slight difference between these values, we are content with Z, because there is no reason to believe that the “bilayer” is not actually a homogeneous layer.The values of bulk Analytical Chemistry, Vol. 75, No. 21, November 1, 2003

5901

Table 1. Values of Shear Modulus, Bulk Acoustic Impedance, and Bilayer Mass Density of PSS-PDADMA Multilayer Made in 0.1 M NaNO3 Solution Obtained by Using Two Different Mathematical Methods method parameter

cross ratio invariance

matrix fitting

unit

Re G Im G Re Z Im Z m (bilayer)

134 36 0.405 0.053 4.24-0.10j

132 ( 2 68 ( 4 0.410 ( 0.004 0.100 ( 0.006 (4.12 ( 0.02)-(0.19 ( 0.01)j

MPa MPa MRayla MRayla µg cm-2

a

Use of Matrix Iteration. Although the method based on the invariance of the cross ratio is mathematically elegant in the determination of the Mo¨bius transformation, it does not easily allow the estimation of errors. In an alternate procedure for determining the Mo¨bius transformation of the bilayer, we assume a priori that the bilayer is actually homogeneous and the matrix can be expressed as the matrix A in eq 26. If the measurements of LAI are repeated N times at the intervals of n layers, the successive measurements are connected by an equation

ζi+1 + i+1 ) Ain o (ζ1 + 1); i ) 1, 2, ..., N - 1 (48)

1 MRayl ) 106 kg m-2 s-1

acoustic impedance for even- and odd-numbered layer systems are not far from each other. Taking the mean value, estimating the density F of the layer material as 1200 kg m-3, and using the equation 2

Z ) FG

(45)

As a result of experimental errors, we have to add the complexvalued stochastic correction terms i, assuming their Gaussian distribution. This is now an expression that can be used as a basis for parameter-fitting. The chi-square function, χ2, becomes N

χ2 )

∑ [(V

-1 i

)11Re2i + (Vi-1)22Im2i + 2(Vi-1)12ReiImi]

i)1

(49) allows us to estimate the shear modulus, G, of the layer material. The results are shown in Table 1. In addition to the bulk acoustic impedance, another parameter that can be extracted from the measurements of electric crystal impedance is the areal mass density of the layers. Assuming that the bilayer is homogeneous, we can use eqs 26 and 27 to derive an equation for m.

m)

1 jω

x

M12 arcsinx-M12M21 M21

(46)

Actually, in the present case, ωm ,|Z|, and we can use a simpler equation

m ≈ M12/jω

(47)

Matrixes Meven and Modd then give

meven ) (4.29 - 0.11j) × 10-5 kg m-2 ) (4.29 - 0.11j) µg cm-2 modd ) (4.19 - 0.09j) × 10-5 kg m-2 ) (4.19 - 0.09j) µg cm-2 The values are close to each other, and we may use their mean for the mass of the bilayer (Table 1). Obviously, the areal mass density should be purely real-valued, but the imaginary part is only ∼2% and can be taken as an experimental error. The density of the bilayer is unknown, but it is commonly estimated22,23 as 1200 kg m-3. This gives ∼35 nm for the thickness of the bilayer. (22) Baba, A.; Kaneko, F.; Advincula, R. C. Colloids Surf., A 2000, 173, 39-49. (23) Caruso, F.; Niikura, K.; Furlong, D. N.; Okahara, Y. Langmuir 1997, 13, 3422-3426.

5902


The 2 × 2 covariance matrices of real and imaginary parts of LAI are denoted by Vi and their inverse Vi-1. The parameters to be optimized by minimizing χ2 are the areal mass density corresponding to n layers, and the real and imaginary parts of bulk acoustic impedance of the layer material. In addition, there is a correction term 1 for ζ1, and the real and imaginary parts of this term are also adjustable parameters in the nonlinear fitting. Altogether, data of 56 films starting from a film of 25 layers and ending at 300 layers taking every fifth layer were included in the nonlinear fitting. The standard deviations of parameters were estimated by using the bootstrap method, and the parameters are shown in Table 1. DISCUSSION Several assumptions have been made in deriving the results. The main assumption is the similarity of individual bilayers in zone II. Without other experimental methods available, the sole, albeit indirect, evidence is given by the reasonably good fit between the observed and calculated data. Progressive increase in the thickness as observed for certain polyelectrolytes24 would presumably yield a Mo¨bius transformation with physically unreasonable matrix elements. It is well-known that the first few layers on the surface of the substrate, zone I or sometimes called “precursor” layers,2 are often considerably thinner. Lo¨sche25 attributed this to the gradual roughening of the film-solution interface as the number of layers is increased. According to their results, the roughening process is equilibrated after 4-5 molecular layers and the layer thickness becomes constant. This was observed for poly(allylamine) and PSS as polyelectrolytes, but there is no reason to believe that PDADMA and PSS would not behave essentially (24) Picart, C.; Mutterer, J.; Richert, L.; Luo, Y.; Prestwich, G. D.; Schaaf, P.; Voegel, J.-C.; Lavalle, Ph. Proc. Nat. Acad. Sci., U.S.A. 2002, 99, 1253112535. (b) Picart, C.; Lavalle, Ph.; Hubert, P.; Cuisinier, F. J. G.; Decher, G.; Schaaf, P.; Voegel, J.-C. Langmuir 2001, 17, 7414-7424. (25) Lo ¨sche, M.; Schmitt, J.; Decher, G.; Bouwman, W. G.; Kjaer, K. Macromolecules 1998, 31, 8893-8906.

in the same way. As a matter of fact, the same phenomenon was directly observed for PSS/PDADMA by McAloney et al.2 by using AFM. In this case, the layers were deposited at various concentrations of NaCl, and remarkable roughening of the tenth and subsequent bilayers was observed at 0.3 M and higher concentrations. On the other hand, zone III is in contact with liquid and presumably partially swollen.16 How far this zone extends toward the bulk of the layer is not known. In this work, it was assumed that the growth of zone II starts after 10 bilayers were laid. Before that, the multilayer system is assumed to be composed mainly of zones I and III. Actually, we do not know for sure the thickness of these layers, and our estimation of 10 bilayers may be rather conservative. Goodness of fit is the only decisive factor. It is reported that the PE layers are strongly interpenetrated and there are no clear borderlines between them.26 It is even said that the “multilayer” factually consists of homogeneous polyelectrolyte complex.27 Despite this, the analysis of impedance measurements was performed as if the layers were separate entities. This was done on purpose in order to see whether the present novel technique would reveal any differences in the Mo¨bius matrices of bilayers of opposite order. There are subtle differences in the elements of matrices Meven and Modd, but these differences are not large enough to yield significantly different impedance values, as one would expect in the case of substantially different elastic properties. Either the polymers are rather similar or the layers are nearly completely intermingled. Anionic polymer PSS consists of a long aliphatic chain with aromatic pendant groups, whereas the cationic counterpart PDADMA is a chain of saturated heterocyclic rings. It is unlikely that materials with such different molecular structures and hydrophobicity/hydrophilicity would yield films with similar elastic properties. Consequently, the results support the anticipated “fuzzy”, nearly amorphous layer structure. Traditionally, the areal mass density has been the most important parameter obtained from the quartz crystal measurements. However, the Sauerbrey equation is usually used for this purpose, and because it is an approximation, the results for thick and viscoelastic films are not reliable. In the present case, no mathematical approximations are used. As seen in Table 1, the real part of the areal mass density of the bilayer is ∼ 4.2 µg/cm2. If we assume that the density is 1200 kg/m3, the thickness is then ∼35 nm. Using 1 M NaCl solution, Dubas and Schlenoff27 have observed a 27-nm thickness of a bilayer using ellipsometry, whereas McAloney et al.2 observed 46 nm/bilayer using AFM. The result is then in line with these results, considering the different supporting electrolyte (NaNO3 vs NaCl), the range of total thickness,and the swollen state of the layers. It should be noted that not only concentration but also type of salt has a strong influence on the thickness of bilayers. This somewhat neglected fact has been under study in our laboratory and will be the subject of a forthcoming publication. The bilayer mass density should be real-valued, but actually, a small imaginary component is invariably obtained. If the nonlinear fitting is forced to accept only real values of areal mass density, the sum of squares remains much higher. However, the imaginary part is relatively small, corresponding to an ∼2° phase (26) Decher, G. In Multilayer Thin Films; Decher, G., Schlenoff, J. B., Eds.; WileyVCH: Weinheim, 2002, Chapter 1. (27) Dubas, S. T.; Schlenoff, J. B. Macromolecules 1999, 32, 8153-8160.

angle, indicating presumably a slight systematic deviation from the assumed model. The real part of the bulk acoustic compliance can be estimated already for very thin layers.3,9 Much thicker film is needed before any reliable estimate for the entire complex acoustic impedance can be obtained. In general, one should take a cautious attitude to the values of acoustic impedance or shear modulus of thin polymer films reported in the literature. Very often, the film used for the determination is too thin to obtain reliable results for the loss modulus. Occasionally, additional assumptions are made about the viscoelastic model of the layer material without firm evidence. These problems in determining the viscoelastic parameters by QCM have been treated by Lu¨cklum et al.28 Even in the present case, when the total thickness of the multilayer is ∼5 µm, the imaginary part of Z (or shear modulus) varies by a factor of 2, depending on the method used for its estimation (cf. Table 1). The real part of shear modulus, or storage modulus, is more accurate. Its value is rather high, approaching the modulus of polymers below their glassy temperature. Unfortunately, the literature on the viscoelasticity of analogous polyelectrolyte layers is scarce, making the comparison difficult. The only elasticity parameter found for the PSS-PDADMA multilayer was measured for the micrometer-sized capsules.29 The static elasticity modulus was 140 MPa, which is very close to the dynamic storage shear modulus obtained in this work, 134 MPa (Table 1). This might be a fortuitous coincidence, but if we assume that this is realistic, it may mean that the material obeys the simple Voigt viscoelastic model with a single relaxation time. This model was for example adopted by Ho¨o¨k et al. for their protein layers.30 The high shear modulus of thick multilayers is in contrast with the previous work, which indicated that thin films measured in the very beginning of the deposition process appeared very soft.3 This underlines the fact that the physical properties of the bulk film are quite different from those of the interfacial layers. Together, these results show that the viscoelastic properties of thin polyelectrolyte multilayers are determined by the gellike outer layers, whereas in thick films, the core dominates the behavior. CONCLUSIONS Our goal in this paper has been to develop methods for studying the influence of multilayer coating on the thicknessshear mode resonator. The two mathematical methods for data treatment developed in this work, the method based on the invariance of cross ratio and the matrix iteration, have their own pros and cons. The success of the cross-ratio method is critically dependent on the choice of the four experimental impedance values. On the other hand, the advantage in using this method is that we do not assume any physical model for the system. Any four points in the complex plane can be used to determine the Mo¨bius transformation if mapping (21) is assumed to exist between the points. The physical relevance is deduced a posteriori, by examining the elements of the Mo¨bius matrix obtained from the experimental results. The fundamental assumption is that the (28) Lu ¨ cklum, R.; Behling, C.; Hauptmann, P.; Cernosek, R. W.; Martin, S. J. Sens. Actuators, A 1998, 66, 184-192. (29) Gao, C.; Leporatti, S.; Moya, S.; Donath, E.; Mo ¨hwald, H. Langmuir 2001, 17, 3491-3495. (30) Ho¨o ¨k, F.; Kasemo, B.; Nylander, T.; Fant, C.; Sott, K.; Elwing, H. Anal. Chem. 2001, 73, 5796-5804.


5903

layers grown between the chosen data points are identical. Any deviation from equality shows up as physically inconceivable matrix elements. In the other method, “matrix iteration”, we assume a priori that the bilayer is homogeneous and the bilayer matrix has the form of eq 7 obtained from the solution of the Riccati eq 1. The advantage is that the method utilizes a larger number of data points, allowing also the estimation of the variance-covariance matrix of parameters. If the basic assumptions on the zone structure of the thick multilayer and commutability of matrices 38 are valid, the method is unique in giving information on the inner structure of a multilayer. No other information, such as thickness or viscoelastic model of the layers, is needed, like in most other methods based on the quartz crystal resonator. The polymers chosen for this work have been commonly used by other workers in various studies, especially for illustrating the properties of multilayers made of

5904


strong polyelectrolytes. The previous works were restricted to rather thin multilayers, at most up to 20 to 30 layers, whereas in the present work, the multilayer system comprises hundreds of layers. This has revealed an unexpected result for the thicker layers: the existence of a “hard core” of the multilayer system, showing a rather high shear modulus. More detailed studies dealing with this phenomenon and its prerequisite conditions are in progress in our laboratory. ACKNOWLEDGMENT Grant no. 102279 from the Academy of Finland is gratefully acknowledged. Received for review May 14, 2003. Accepted August 22, 2003. AC034509Z

Effect of Polyelectrolyte Multilayers on the Response of a Quartz

Recommend Documents