J. Phys. Chem. 1995,99, 14058-14063
14058
Quartz Crystal Microbalance Study of Divalent Metal Cation Binding to Langmuir Monolayers and Langmuir-Blodgett Films of Behenic Acid Michael R. Love11 and Stephen J. Roser* School of Chemistry, The University of Bath, Claverton Down, Bath BA2 7AY, U.K.
Received: February 20, 1995; In Final Form: June 23, 1995@
Divalent metal ions binding to Langmuir-Blodgett (LB) films of behenic acid have been investigated over their complete binding pH ranges. The mass of ions incorporated into the LB films has been measured in-situ by a quartz crystal microbalance (QCM). The corresponding degree of ion binding to the precursor Langmuir monolayer (LM) has been calculated using the one-dimensional Cartesian Poisson-BoltzmannStem (PBS) approach, developed by Bloch and Yun,’ allowing the two systems to be compared quantitatively. The results show that the degree of ion incorporation into an LB film is very similar to that of the precursor LM, even though the metal cations are in vastly differing environments in the two systems.
Introduction An increasing amount of research into organic thin films has been published in recent years since their potential technological applications in such areas as molecular electronics, nonlinear optics, sensors, and electro-optical devices have been realized. If thin films are to have commercial value, then the factors that affect their mechanical, thermal, and chemical stability must be understood in order to reproducibly manufacture zero-defect films. One way of producing high-quality films employs the well-known LB technique,*with arguably the best films coming from the deposition of the simple fatty acids, CH3(CH2),COOH. The quality of fatty acid LB films is determined somewhat by the quality of the precursor LM on water. That is, LB films are built up from LMs which have been compressed and held in the “solid phase”. In this state the molecules are close packed in a distorted hexagonal array, with each molecule covering about 20 A2 of the water ~ u r f a c e . Of ~ course, the exact LM film structure depends, amongst other things, on the presence of any divalent cations in solution and the dissociation state of the fatty acid molecules, governed mainly by the pH of the subphase, all of which is well documented in the literature. This report will investigate the idea that the composition of the LB film mirrors that of the precursor LM. Ion binding to the LM can be modeled using a purely theoretical approach developed by Bloch and Yun’ from electrochemical double-layer theory. They showed that this Poisson-Boltzmann-Stem (PBS) model can accurately describe the binding of calcium and cadmium ions to stearic (n = 16) and arachidic (n = 18) LMs. We demonstrate in this paper that solving the PBS equation for the LM film, calculated using their model, can be used to fit the change in ion binding in a dry LB film.
Experimental Section The apparatus used in these experiments was mostly constructed in-house. The quartz crystal microbalance (QCM) driving electronics are based on a circuit described by Bruckenstein and S h a ~ .The ~ frequency of the uncanned working crystal was compared electronically with a canned 10 MHz standard, the difference measured by a Racal-Dana 1990 frequency meter, which was connected via an IEEE port to a PC controlling data acquisition and analysis. Crystals were @
Abstract published in Advance ACS Abstmcrs, August 15, 1995.
0022-365419512099-14058$09.0010
supplied by Euroquartz Ltd. and were gently cleaned with a cotton bud soaked in Sigma-Aldrich chloroform and then propan-2-01. This simple method of cleaning was totally reproducible from crystal to crystal, but a single crystal was nevertheless used for each set of experiments. The crystal was dipped vertically into the solution, so both sides were active during the deposition cycle. The frequency meter was operated on an accurate averaging cycle, which gave an output every second. Much faster rates of data transfer are possible, but the stability of the sampled frequency is less well-defined. Typically, the uncanned working crystal was stable in and out of solution to f l Hz. The connection from the electronics to the working crystal was kept as short as possible to avoid interference and capacitance effects, and the trough was surrounded by a Faraday cage. All frequencies are reported relative to the 10 MHz standard crystal. The Langmuir trough used was a rectangular single piece of Teflon, 13 cm x 18 cm, with a depth of 0.5 cm and a circular dipping well. Surface pressure was measured with a Nima 9000 pressure transducer and the Wilhelmy plate technique and is reported accurate to f l mN/m. Surface pressure was controlled and maintained by Teflon booms connected to servo motors and operated by an automatic adjustable feedback mechanism. A dc motorized dipping head controlled deposition onto the QCM, and dipping speed could be varied throughout the experiment as appropriate. All troughware was rigorously cleaned with Sigma-Aldrich chloroform and isopropyl alcohol, and the reagents used in the experiment were all analytical quality or better. The water used in the experiments had a resistivity of 2 1 MQ cm and was produced by a Purite Standard StillPlus still and filtration system. The pH of the solution was adjusted with dilute HCl or NaOH and was measured prior to addition to the trough with an electrode supplied by Orion Research Inc. Subphase pH was monitored constantly throughout the measurements. Readings are reported with a validity of f 0 . 0 2 pH units. Monolayers were spread using a Hamilton microsyringe (50 pL) from a 2 x M solution of behenic acid in Sigma-Aldrich HPLC grade chloroform. Evaporation of the solvent was allowed between successive additions. After each experiment the surface was skimmed to remove any collapsed monolayer material that may have disrupted plating and the trough was rigorously cleaned. A series of experiments were performed using an MC12 solution (M = Cd2+,Mn2+, Ca2+, lov3M) over a range of pH 0 1995 American Chemical Society
Divalent Metal Ion Binding to LB Films of Behenic Acid values. This is clearly, at least, a complementary measurement in the case of cadmium to those described using X-ray refle~tion.~ After spreading of the monolayer film onto a subphase at a measured pH and trough temperature, the monolayer was slowly compressed several times at a rate of %3 A2 molecule-' min-' until a reproducible n-A isotherm was obtained. Reproducible isotherms were found at all pH values. After cleaning the crystd carefully and affixing to the dipping head, the crystal was lowered into the solution through the compressed monolayer held at 40 "/m. With a clean gold surface it was found that deposition onto the QCM occurred when the first stroke was both downward and upward through the monolayer. Thus, as has been previously noted? the gold surface can behave both hydrophilically and hydrophobically. In all subsequent experiments the first deposition was on the downward stroke. The speed of dipping was varied as required between experiments. Below pH 5.8 the QCM was dipped into and out of the cadmium subphase at 3.8 c d m i n . However, to produce a macroscopically dry film (as well as reproducible deposition), the speed of dipping had to be reduced around pH 5.8 to 0.9 c d m i n . The films deposited from manganese subphases were dipped throughout at 0.9 c d m i n , and the calcium films likewise at 3.2 c d m i n . In addition the QCM was allowed to gain frequency stability in air before the next dip was made into the subphase. All experiments were performed at room temperature (20 f. 3'7, ensuring that the fatty acid monolayer was always in the S phase at 40 mN/m.' At least 10 bilayers were deposited on both sides of the QCM.
Results and Discussion Brief Overview of the PBS Approach. The partially dissociated LM is negatively charged and therefore attracts any cations in the subphase to the interface, increasing their concentration in the monolayer vicinity. By the same token, the concentration of anions near the LM is depleted with respect to the bulk concentration. The concentration enhancement of cations near the LM causes a larger degree of adsorption of protons or metal ions onto the monolayer, in tum reducing the negative charge on the monolayer and, by that, the concentration of the cations next to it. This leads to an equilibrium between the ions electrically attracted to the interface and those chemically bound to it. A full account of this treatment is given in ref 1, where the authors show that the system can be described mathematically by the PBS equation, which only requires knowledge of the bulk ion concentrations in order to be solved. The Poisson-Boltzmann part of the equation describes the distribution of ions under the Stem layer (the electrically attracted ions in the diffuse part of the double layer) resulting from two competing processes, namely, that of electrical attraction between oppositely charged species versus the entropy of the system. The electrical interactions have minimum energy when the cations are very close to the monolayer, neutralizing its negative charge, thus causing a buildup of cations at the interface and a depletion of anions. The entropy of the system, however, is minimum when the ions are evenly distributed in the subphase. The balance between the two results in a Boltzmann distribution of ions between the interface and the bulk subphase. The chemically bound cations make up the Stem layer, which can be described by considering the chemical equilibrium between carboxylate head groups and cations in solution. In ref 1 Bloch and Yun derive the Grahame equationE
J. Phys. Chem., Vol. 99, No. 38, 1995 14059 for the LM system,
1 -x, 1 -kxH -kxN
+xM
where PO is the Boltzmann concentration enhancement factor at the Stem layer, and UJ = Ao2/R0KJ is a constant for each cationic species determined by the area per fatty acid molecule, Ao, a constant Ro = e2/2coerk~T (where e is the charge on an electron, EO is the permittivity of a vacuum, cr is the dielectric constant of the subphase, kB is Boltzmann's constant, and T is absolute temperature), and KJ, which is the equilibrium binding constant for the species J in the chemical equilibrium
KJ =
JZ+1
rBe-{
where z is the magnitude of the charge on cation J, and r denotes the surface excess (surface concentration) of the component in units of particles/A2. XJ = KJ{J~++)PO~ is denoted the binding ratio of the cation J to the dissociated monolayer molecules, where { } implies the bulk concentration of J in units of particles/A3. In this report J covers H = H+, N = Naf, and M = Cd2+, Mn2+, and Ca2+, with Hf and Na+ being added to the subphase for the purpose of pH adjustment. Therefore, for a general system, POis calculated numerically from the solution of the Grahame equation and is used to find all other parameters of the interface. This involves finding the roots of a polynomial in the seventh power of PO(see Appendix A) and was calculated in this work using the root function of MathCad 4.0. However, an approximate solution can be easily found by using an approximate form of the Grahame equation.' The ratio between the condensation fractions of the two types of ions (monovalent and divalent), e,, is given as
In this report {Naf} is assumed to be 0, and so
Bloch and Yun' showed that when PO>> 1, this equation could be written as
where
The authors describe PoHand PoMas free variables defined, as can be seen, by the bulk concentrations of H+ and M2+, respectively. They should not be confused with PO in the Grahame equation, which is the actual concentration enhancement of the system, which is, in fact, a function of PoH and POM.
Since the dissociated LM is virtually completely complexed with cations, and therefore neutral, the authors could derive a simple expression for the fraction of cation binding per surfactant molecule:
14060 J. Phys. Chem., Vol. 99, No. 38, 1995
Love11 and Roser
where the degree of ion binding, a, is given by 2@BeM+;that 1%
a=-
2
e, + 2
(3)
By substiting 8,from (1) or (2) into (3), using the appropriate equilibrium constant^,^ the composition of the LM can be quantitatively identified, as shown in Figure 1. Bloch and Yun state that equilibrium constants for the long chain fatty acids and their salts are scarce, but for chains with n 2 3 the equilibrium constants are dependent mainly on the polar head of the acid. The values for the equilibrium constants given in ref 9, and in Table 1 as log(k), are in units of l/mole and so must be converted to the units used in this paper, given in Table 1 as log(K). Note that the values quoted for log(k) are the average of all the values given in ref 9 and correspond to the normal chemical equilibrium: k, =
[BeJ(Z-')+ 1
units: l/mol
[Be-Irff1 This is because the fatty acid in the LM is considered to only form 1:1 complexes with the divalent metal ions, (BeM)+. (T in Table 1 is the uncertainty in the values of log(k) and log(K). QCM Studies of LB Films. The use of QCMs to study thin layers of material plated from the gas phase is widespread, and they are commonly used as commercial thickness gauges for vacuum deposition systems. The QCM driving circuitry can be easily redesigned to allow measurements to be made in liquid environments, with a reproducibility approaching that of gas phase sensors. This leads to many potentially exciting experiments involving, for example, adsorption from solution into biosensors,I0 dissolution of polymer films," and studies of ion and solvent doping of electroactive polymer films.12 In addition, for thin, rigid film loadings, a simple relationship between the change in the resonant frequency of the quartz oscillator in the shear mode with mass of adsorbed material applies with excellent accuracy. This is parametrized in the well-known Sauerbrey equation,I3 -2f,2ee f=g where f is the experimentally measured frequency of the oscillator,fo is the bare resonator frequency (10 MHz), Q is the density of the deposited film with thickness E , es is the quartz density (2.648 g ~ m - ~ and ) , p q the effective shear modulus (2.947 x 10" g cm-I s-l for AT-cut quartz). More simply, it can be seen that the change in frequency is given by Af = -2.26 x 108ArnA where AmA is the mass change per unit area on the active faces of the QCM, in g cm-'. Various improvements to this equation have been made over the years, but for the simple, thin, nonviscoelastic films studied in this paper the Sauerbrey equation is more than adequate. This has been confirmed by an earlier study of LB deposition using a QCM.6 The AT-cut of the crystal is made to virtually eliminate any temperatureinduced shifts in fo at conditions around room temperature.
0
3
2
4
5
6
7
a
subphase pH
Figure 1. Model curve (-), from the exact solution of the PBS equation, that fits the LM X-ray reflectivity data from ref 5. Also shown for comparison is the curve produced from the approximate solution of the PBS equation (-) and the original Langmuir type adsorption isotherm fitted by the authors (- - -).
TABLE 1
Hf Na+ CdZf MnZ+ Ca2+ a
8.090 2.449" 5.133 3.965 3.725
4.870 -0.771 1.913 0.745 0.505
0.050 0.039 0.004
8.090 2.449" 4.765 4.229 3.749
4.870 -0.771 1.544 1.009 0.529
0.176 0.089 0.136
Value extrapolated from low concentration data.
In this work we have deposited successive layers of the fatty acid (containing varying degrees of metal ions) vertically onto the QCM using the LB technique. Allowing the QCM to reach frequency stability in air between dips has enabled us to measure the frequency change induced due to the deposition of two bilayers on either side of the QCM per dipping cycle. Hence, we have been able to calculate the mass change per unit area (areal density change) on the crystal oscillator. Any change in the areal density could be a combination of two effects. It should be noted that there will be no net effect due to hydration, as the LB layers have been allowed to dry to equilibrium between dips. Incorporation of metal ions will produce an areal density increase, shown by a resonant frequency decrease. If the ion incorporation invokes a change in the in-plane area of the amphiphilic molecules, then an additional resonant frequency change will be detected on top of that due to the metal ions. A closer packing of the molecules will increase the areal density, again reducing the resonant frequency. Likewise, a looser packing will decrease the areal density, thus increasing the resonant frequency. However, this combined areal density change can be easily resolved into the two contributory components by considering the boundary values for the molecular mass of the LB molecules (see Appendix B). Comparison of LM and LB Film Divalent Ion Binding. The first part of this work14 looked at the binding of cadmium cations to behenic acid LM and LB films across the whole pH binding range, but the Langmuir type adsorption isotherm used to model the binding was empirical in nature and so was prone to experimental inaccuracies. It did show the change in the degree of dissociation of the films but assumed that the concentration of any solute ions at the water-LM interface was equal to that in the bulk. This clearly ignored the electrical attraction of the cations to the negatively charged monolayer and did not distinguish between chemically bound ions and those that were merely electrically attracted to the LM. In this report
J. Phys. Chem., Vol. 99, No. 38, 1995 14061
Divalent Metal Ion Binding to LB Films of Behenic Acid 340
T
330
260
t I
250 4 3
340 330
‘I
-10
-20 4
i
6
5
7
250
4 3
8
I- -20 4
5
6
7
8
subphase pH
subphase pH
Figure 2. Comparison between Cd2+ binding to the LM (- - -) and the LB film (-). The broken line is produced using the value for l o g ( e ) from ref 9. The solid line is produced from the best fit of the data; the resulting log(#) value is given in Table 1.
Figure 4. Comparison between CaZfbinding to the LM (- - -) and the LB film (-), The broken line is produced using the value for log from ref 9. The solid line is produced from the best fit of the data; the resulting log(K):) value is given in Table 1.
340
-
,
80
t 70
t
330 g320 i
160
260
t-10
250
’
3
i
+ -20
4
5
6
7
8
subphase pH
Figure 3. Comparison between MnZ+binding to the LM (- - -) and the LB film (-). The broken line is produced using the value for l o g ( e ) from ref 9. The solid line is produced from the best fit of the data; the resulting log(?;) value is given in Table 1.
that data have been modeled using the less empirical PBS approach and are displayed in Figure 2. We also present PBS fits of manganese and calcium binding to behenic LM and LB films, also studied by us using the QCM, in Figures 3 and 4. The -Af data with the PBS fits are shown in Figures 2-4 in terms of the frequency change per dip and the areal density change per monolayer deposited. The manganese results will be repeated in the near future to obtain a wider spread of the -Af data points. Shown on each of these plots are two PBS ion-binding curves. The broken lines are the PBS fits for M2+ binding to the LM of behenic acid, normalized to fit the areal density change data. These are produced using the values of log(KJ) gained from ref 9 (log(fi’), left-hand column of Table 1). As already mentioned, the authors of ref 1 have shown that these log(e’) values produce curves that model extremely well the ion binding to LMs of stearic and arachidic acid. As can be seen in Figure 1 in the case of cadmium, the log value produces an isotherm that fits the excess scattering density data from X-ray reflection experiments5 using the authors’s value for [Cd2+] = 2.5 x M. The excess scattering density is a direct measure of the amount of cadmium bound to the behenic acid LM. Since ion binding to LMs is only dependent on the head group, then we assume that the log(eMM)values for calcium and manganese are also valid when used in the PBS approach. Normalization of the LM binding curves allows visible comparison to be made with the
(G:)
(e)
best fit isotherms, as shown by the solid lines. The best fit line was gained by calculating the standard deviation of the points weighed by the magnitude of the uncertainty in each. When fitting the solid lines, it is assumed that log(GB) is the same as that for the LM. The best fit values of olg(?): are given in the fourth column of Table 1. If the composition of the deposited LB film mirrors the composition of the precursor LM, then the value for log (?$) should produce a curve that fits the areal density change data. This is the reasoning behind using a monolayer-on-water model to describe the ion binding to the “dry” interior of an LB film. The fact that the LM binding curves do appear to fit the LB mass change data is perhaps surprising when considering the environments of the ions in the two regimes. The LM system involves solvated, freely moving ions binding to a fixed charged boundary due to the electric field created by the charged monolayer molecules, whereas the LB film consists of ions being bound rigidly in a Stern layer between two charged walls in the dry interior of the film. ‘The solid lines in Figures 2-4 show that the shape of the dependence of ion binding to the LB film is the same as that to the LM, the only difference being in the position of the isotherm with respect to the pH axis. The values ascertained for olg&): show that the trend in KM is preserved down the table. That is, for binding to the LM and the LB film, KC^ > KM,, =- Kea, and since KM is given by the equilibrium equation, then, qualitatively, it can be said that in terms of covalency Cd2+ > Mn2+ > Ca2+, as expected. Varying log(KM) changes the absolute position of the PBS curve with respect to the pH axis. Therefore, ion binding to the LB film can be compared to the adsorption of ions onto the precursor LM by comparing the corresponding values of log and log(GB). However, a better measure of the difference between the cation-LM binding and the cation-LB film binding is the amount by which the LM curve must be shifted along the pH axis such that it superimposed the LB isotherm. This pH shift is equal to the difference in pK of the two types of films and can be calculated as follows. Substituting (1) into (3) gives
(kM)
a=
2KM{M2+}P0
KH{H+}
rearranging (4) gives
+ 2KM{M2+}P0
(4)
Love11 and Roser
14062 J. Phys. Chem., Vol. 99, No. 38, 1995 TABLE 3
TABLE 2 cation Cd2+ Mn2+ Ca2+
pPLB 5.293 5.556 5.788
pgL,
u
5.110 5.684 5.800
0.087 0.044 0.065
u 0.025 0.019 0.002
ApK‘ 0.182 -0.128 -0.012
u
-Afmin
0.112 0.063 0.067
where K=
2KM{M2+} KH
Now
( N A = Avogadro’s number). The pK of an acid is the pH at which half the molecules are dissociated, Le. a = 0.5; therefore,
pK = -log[
KP, x NA
]
= -log(KPo) - 3.2203
(7)
The pK of behenic acid in either the LM or LB regime can be calculated by using the appropriate values of log(&) in eq 7. However, the pK equation is difficult to solve numerically since the exact value of PO is a complicated function of [H+] (see Appendix A). It can be solved easily, however, by using the approximate solution of PO,that is, in terms of PoHand PoM. In the approximate .approach PoMis constant for each value of KM, and 8, in ( 2 ) varies only due to the simple reciprocal dependence of PoHon {H+}. Therefore, substituting (2) into (3) leads to
Using the expressions given previously for PoH and PoM, rearranging, and converting {H+} to [H+] gives
cation (HZ)
AoLB(BeH)
-&an
(HZ)
u (AVmolecule) u
CdZ+267.2321.81.7 314.0 1.1 299.5 0.8
Mn2+ 267.0 Ca2+ 266.9
19.14 19.15 19.16
0.12 0.08 0.06
AoLB(Be2M) (AVmolecule) u 18.46 17.55 18.03
0.10 0.06 0.05
is very similar to that of the precursor Langmuir monolayer, even though the metal cations are in vastly differing environments in the two systems. In-Plane Film Structure Inferred from the QCM Measurements. To determine the values of l o g ( e ) , the binding curves had to be fitted to the maxima and minima dictated by the -Af data points, -Afmax and -Affin. Since the composition of the monolayer is known at these extremes, then, using the Sauerbrey equation, the corresponding area per molecule in the LB film, AoLB,can be determined (see Appendix B). These values are given in Table 3. The areas per fatty acid molecule, AoLB(BeH),agree well with those determined by a recent studyI5 using the technique of small angle X-ray reflection from builtup behenic acid layers. The author quotes an area of 18.97 f 3.66 A2 per BeH moleculeI6 compared to our value of about 19.15 A2. The aredmolecule of 18.46 f 0.10 A2 for the anhydrous CdBe2 soap molecules in the LB film (AoLB(Be2M) in Table 3) also agrees well with ref 15, where the author determined an area of 18.44 f 0.10 Az/molecule. The value of 17.55 f 0.06 Az for the MnBez soap molecular area is significantly smaller than expected, but is almost certainly a result of the lack of sufficient data points to satisfactorily determine -Afmax.17 The value for CaBez is also smaller than that for CdBe2 molecules, but 18.03 f 0.05 A2 is not an unreasonable value in that the area per soap molecule would be expected to change for different divalent cations. A consistency in all of the systems studied is that the area per molecule qualitatively decreases when going from the acid to the metal soap in the LB films, as expected. The fact that the experimentally observed molecular area values for BeH and CdBez agree so well with ref 15 is further proof of the usefulness of simple quartz crystal oscillators in the study of thin films.
which leads to pK’ = -log K’ - 3.2203
(10)
where
The difference in pK‘ of the two regimes is given as which can be shown to be
The pK‘ and ApK‘ values calculated from (10) and (1 1) are given in Table 2, along with the uncertainty in each, u. As mentioned earlier, the experimentally measured values of the subphase pH used in Figures 2-4 are accurate to f0.02 of a pH unit. All of the ApK‘ values are close to this range (within their own uncertainties), and therefore it can be stated with some confidence that the degree of ion incorporation into an LBfilm
Summary It has been shown that solving the one-dimensional Grahame equation exactly, and approximately, for the systems under investigation produces ion-binding curves which fit the experimental data. The pK of behenic acid in both the LM and LB films has been calculated, allowing direct comparison to be made between their ion-binding versus pH dependencies. The central finding of this work is that the amount of M2+ ions incorporated in the LB film is very similar to the amount bound to the precursor Langmuir monolayer. We have shown that not only can the QCM allow submonolayer coverages of M2+ ions to be detected but it can also give information on the subsequent in-plane structure of the LB films. In general, the area of the QCM covered by a MBe2 molecule is smaller than the corresponding area per molecule for the BeH species in LB films.
Acknowledgment. S.J.R. would like to thank the Nuffield Foundation for funds to build the equipment, and M.R.L. thanks the EPSRC and IC1 for CASE Award funding.
J. Phys. Chem., Vol. 99, No. 38, 1995 14063
Divalent Metal Ion Binding to LB Films of Behenic Acid
Appendix B
Appendix A Expanding
The Sauerbrey equation13can be written as
Af = SAmA
1
-xM
1 +XH+XN+XM gives
- ~){uHKH{H+>+ uNKN{N~+>
{(PO
+
UMKM{M2+}(Po 2)}[1
+
+ KH{Hf}Po + KN{Na+}Po +
KM{M2f}P:]2} - [ l - KM{M2'}P:I2
=0
which, when further expanded and put in terms of PO,becomes
+ UP; + VP; + WP; + xP,3 + yP; + ZP, +
tP,7
constant = 0 where t = cc2
+ Bc2 + Ac2 + 2Cac v = 2Cab + 2Cc + Cb2 - 2Bc2 + 2Abc + 2Bbc - 2Ac2 + 2Aac + 2Bac + Ca2 - c2 - 3Cc2 w = A a 2 + 2Cb - 6Cbc + Bb2 - 6Cac + 2Ca + 2Ac 4Aac - 4Bbc + Ba2 - 4Abc + Ac2 + Ab2 + 2Bc + 2Bab + 2Cc2 - 4Bac + Bc2 + 2Aab x = 2Bb + 2Aa - 4Bc - 2Bb2 - 4Ac + 2Aac + 2Ab 2Ba2 - 4Bab - 2Aa2 + C + 2Abc - 6Cab - 3Ca2 3Cb2 - 2Ab2 + 2Bbc + 2Ba + 2c + 2Bac + 4Cac 4Aab + 4Cbc - 6Cc y = 2Aab + 2Cb2 - 4Bb + A + B + A b 2 + 2Bc t' 2Ca2 4Ab + Aa2 + 2Ac + 4Cab + 2Bab - 4Aa - 6Cb + Bb2 6Ca - 4Ba + 4Cc + Ba2 z = 1 + 2Bb - 2B + 2Aa + 2Ab + 2Ba - 3 C - 2A + 4Ca + 4Cb constant = 2C + A + B u = 2Cbc
with
A0 A = -{H+}
RO
where S = -2.26 x lo8 cm2 g-' s-' and hmA is the areal density change on the active faces of the QCM in g cm-2. The molecular area, AoLB,in the LB film is therefore determined by the following equation:
B = A0 -{Na+} RO
C = A0 -{M2+}
RO
where F, is the average formula weight of the LB film molecules (grams per mole per molecule), and N A is Avogadro's number. The factor of 4 in the numerator simply corrects for four layers of film deposited per dip. Although the QCM electrodes are not automatically smooth, we do not find the need to introduce a roughness factor, defined as the real electrode aredgeometric area, to correct the mass/ frequency relationship, as the values calculated for the molecular areas using the formula given above are in excellent agreement with those determined independently.I5 Thus, it can be said that we observe an apparent roughness factor of 1.
References and Notes (1) Mati Bloch, J.; Yun, W. Phys. Rev. 1990, 41, 844. (2) Blodgett, K. B.; Langmuir, I. Phys. Rev. 1937, 51, 964. (3) Kjaer, K.; Als-Nielsen, J.; Helm, C. A.; Tippman-Krayer, P.; Mohwald, H. J . Phys. Chem. 1989, 93, 3200. (4) Bruckenstein, S.; Shay, M. Electrochim. Acta 1985, 30, 1295. ( 5 ) Richardson, R. M.; Roser, S. J. Liq. Cryst. 1987, 2, 797. (6) McCaffrey, R.; Bruckenstein, S.; Prasad, P. Langmuir 1986,2,228. (7) Kenn, R. M.; Bohm, C.; Bibo, A. M.; Peterson, I. R.; Mohwald, H.; Als-Nielsen, J.; Kjaer, K. J . Phys. Chem. 1991, 95, 2092. (8) Grahame, D. C. Chem. Rev. 1947, 41, 441. (9) Martell, A. E.; Smith, R. M. Critical Stabiliry Constants; Plenum: to New York, 1977. The values must be multiplied by (U(6.022 x convert to the units used in this report. (10) Ngeh-Ngwainbi, J.; Suleiman, A. A,; Guilbaut, G. G. Biosens. Bioelectron 1990, 5, 13. (11) Hinsberg, W. D.; Kanazawa, K. K. Rev. Sci. Instrum. 1989, 60, 489. (12) Glidle. A.: Hillman. A. R.: Bruckenstein. S. J . Electroanal. Chem. 1961,318,411. (13) Sauerbrev. G. 2.Phvs. 1959, 155. 206. (14) Roser, S: J.; Lovel1,'M. R. J . Chem. SOC., Faraday Trans. 1995, 91, 1783. (15) Musgrove, R. Ph.D. Thesis, University of Bristol, 1991. (16) The authors accurately determined the in-plane distances between molecules (a = 5.70 rfr 0.05 A, b = 7.47 f 0.06 A), but the large uncertainty in the area per molecule arises from their uncertainty in the angle between the a and b dimensions of 117' f 16". (17) The uncertainty in this value of f0.06 AZ/moleculeis also smaller than expected due to the lack of sufficient data points to determine -Afmax satisfactorily. JP950485R
