Langmuir 1989,5, 1176-1180
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Equilibrium Fluctuation Analysis of the Chemical Modulation of the Reactance of Bilayer Membranes Ken-ichi Hongyo, Arnold A. Peterson, Cindy Bruckner, And& Bezegh, Miklds Gratzl, and JiTi Janata* Center for Sensor Technology, Department of Materials Science & Engineering, University of Utah, Salt Lake City, Utah 84112 Received February 14, 1989. In Final Form: April 27, 1989 In contrast to the ac impedance technique, equilibrium fluctuation analysis does not disturb or excite the bilayer membranes during measurement. This technique was employed to determine changes in bilayer electrical properties as caused by changes in their chemical environment. The effect of valinomycin, concanavalin A, yeast mannan, and phloretin on the reactance of lecithin-cholesterol membranes and the influence of valinomycin on the conductance of asolectin membranes were investigated. The results were interpreted by assuming low-pass filter characteristics for the bilayers. The lower frequency limit of the spectra was 8 mHz, which did not allow the determination of individual resistive and capacitive elements. However, the trends in the shift of the sloping part of the spectra were indicating significant effects of chemical modulants on the real part of impedance. Good correlation between the results of ac impedance technique and of fluctuation analysis was found in most studied cases.
Introduction There are two fundamentally different approaches to investigation of dynamic properties of electrochemical systems: deterministic and stochastic. In the forlher, a small amplitude perturbation is imposed on the system under study and the response function is evaluated. This is, of course, the basis of the conventional ac impedance analysis. The second approach uses the spontaneous (thermal) fluctuations as the source of the signal and treats the system as an electrical filter. From the thermodynamical point of view, the principal difference is that the investigations using spontaneous fluctuations are done in the state of true equilibrium. Most studies of bilayer membranes have utilized the deterministic approach by using either dc or ac applied voltage. There are clear advantages in using ac techniques’ because the membrane becomes less stressed (at least a t higher frequencies), and instead of only one resistance value, the entire impedance characteristics can be obtained2y3as functions of frequency. Assuming that the reference electrodes are nonpolarized and that the contacting electrolytes have sufficient conductivity, the applied voltage appears almost entirely across the membrane in either case. Considering the typical thickness of bilayer membranes ( - 5 x lo-’ cm),” the corresponding electric field may be quite significant. For example, an external voltage of 20 mV imposes a potential gradient of about 4 x 104 V cm-l. This may exceed in some cases the dielectric strength of the membrane. In other cases, the membrane’s integrity may be compromised (particularly when a burst of ions is traveling along the membrane that causes the adiabatic dissipation of a significant amount of energy in a small area).5 Important membrane properties (e.g., thickness) are also influenced and modified in certain cases by the externally applied field: Le., by the measurement itself. A stochastic variation of electrical quantities is driven by spontaneous thermodynamic fluctuations,78 which are (1) del Castillo, J.; Rodriguez, A.; Romero, C. A,; Sanchez, V. Science 1966,153, 185. (2) Hongyo, K.; Joseph, J.; Huber, R. J.; Janata, J. Langmuir 1987,3, 827. (3) Ashcroft, R. G.; Coster, H. G. L.; Smith, J. R. Biochim. Biophys. Acta 1981,643, 191.
(4) de Levie, R. J. Electroanal. Chem. 1976, 69, 265. (5) Schmidt, P. P. Biochim. Biophys Acta 1971, 233, 705. (6) Montal, M. Methods Enzymol. 1974,32, 545.
happening always within the bilayer membrane and at its interfaces even in equilibrium state. This process can provide useful information about electrodynamic properties of the membrane. This information is, in principle, equivalent to the real component of impedance (Le., the reactance) obtained by phase-resolved ac measurements. Fluctuations in voltage can be transformed as follows:
where f is frequency, Af is the bandwidth of the analyzer at frequency f and V ( f )is the power spectrum of the voltage fluctuations. This formula is valid as long as the fluctuations originate exclusively from thermal noise. The major advantage of this experimental approach is that the spontaneously fluctuating field does not represent any external perturbation for the membrane investigated. Thus, no significant polarization can occur. These fluctuations of thermodynamic origin are, trivially, also much weaker. A bilayer membrane with a resistance of 10l1a, e.g., may cause an average amplitude square which corresponds to -80 pV a t room temperature or to about a 160 V cm-’ fluctuation in the electric field, if the analysis bandwidth is 3.63 Hz. This bandwidth is used by our spectrum analyzer in the range 0-250 Hz with a “flat top” window function. In fact, the voltage amplitudes are generally even much smaller, due to the the dumping effect of parallel capacitances of the bilayer membrane, which act as filters. Thus, a t higher frequencies, where the bandwidths are proportionally larger (at 0-25 kHz, e.g., 363 Hz), the fluctuating fields are generally even smaller then the above estimate, because the reactance decreases proportionally to the inverse frequency square above the cutoff frequency of the membrane, which is acting as a low-pass filter. The thermodynamic fluctuations cause, indeed, the smallest possible “excitation” voltages that are always inevitably present. In other words, the measurements can be done in the state of a true thermodynamic equilibrium, which is the most stable state of the system, by definition. Thus, the measurements without any external electrical perturbation can provide, in principle, information more relevant to natural bilayer membranes than information (7) deFelice, L. J. Introduction to Membrane Noise; Plenum: New York, 1981. (8) Bezegh, A.; Janata, J. Anal. Chem. 1987,59, 494-501A.
0743-7463/89/2405-1176$01.50/0 0 1989 American Chemical Society
Langmuir, Vol. 5, No. 5, 1989 1177
Equilibrium Fluctuation Analysis of Bilayer Membranes 12
BLM
log Z&
(Q1
10
'
AgIAgCi electrodes
A
8
6
PAR 113
spectrum
preamplifier
0
-1
1 log f
(Hz)
I
computer
I
B
Figure 2. Dependence of the reactance characteristics of PC bilayer on the concentration of valinomycin as determined with fluctuation analysis. a-e: 0,5 X lo", lo-', 5 X lo-', and lo4 M valinomycin concentrations, respectively, calculated with added amounts (nominal values without corrections for adsorption on cell walls and on the membrane surface).
Figure 1. Apparatus of fluctuation analysis. (A) The studied cell. (B)The measuring system: solid line, copper wall; dashed line, magnetizable steel wall, thin arrows; analog signal; thick arrows; digital signal; R1, &, reference electrodesused as electric contacts to the cell. provided by dc or ac techniques. In this paper, the possibility of obtaining information about the chemical modulation of the electrical reactance (or conductance) of phospholipid membranes caused by different stimulants (e.g., valinomycin, concanavalin A) is investigated, using equilibrium fluctuation measurements. The emphasis in this paper is on the development of new methodology rather than on elucidation of mechanism of membrane solute interaction.
Experimental Section The schematic diagram of our measuring system in shown in Figure 1. The cell (Figure 1A)was made of one piece of PMMA, so that the electrical isolation between the two compartments was good. A hole of 1-mm diameter was drilled in the middle wall of the cell, and ita edge was smoothed with acetone. The bilayer membranes were prepared either from 20 mg of egg lecithin (Avanti Chemical) and 20 mg of cholesterol (Sigma) or from asolectin (Associated Concentrates), dissolved in 1mL of n-decane (Aldrich). The membrane was formed according to the following procedure: the edge of the hole was pretreated with the corresponding membrane material and allowed to dry under nitrogen. Then the cell was placed on a mechanically isolated stage inside a solid, double-wall (copper and magnetizable steel) Faraday cage. It was filled with the electrolyte solution (0.1 M KCl for PC bilayers or 1-3 mM CaC12,1mM HEPES for asolectin membranes), and straight Ag/AgCl reference electrodes were immersed into each half-cell, being already connected to the impedance transformer. F d y , the membrane was painted a c m the hole with a fine brush. The spontaneous formation of the membrane was occasionally observed under a stereomicroscope. All membranes were formed at room temperature, and their average lifetime was approximately 5 h. An improper mechanical isolation of the cell could cause a large parasitic peak in the measured spectra around the resonance frequency of the surrounding building, which was about 15 Hz in our case. This interference was practically fully overcome by
6
0
2
4
6
0
1 C
(MxQ')
Reactance of PC bilayer at 1Hz as a function of added valinomycin concentration: c, nominal concentration of valinomycin. Figure 3.
placing the Faraday cage on a heavy steel stage that stood on four hard rubber cubes on the floor of the basement. The double walls of the shielding (Figure 1B) excluded both the electric and magnetic components of any external electromagnetic noise. Straight wire reference electrodes were used instead of coils, to minimize the voltage fluctuations caused by the eventual residual parasitic electromagnetic noise. The thermodynamic voltage fluctuations of typically several microvolt amplitudes were amplified after impedance transformation by XlOOO with a low-noise preamplifier (PAR 113) and analyzed with a computer-controlled (HP S A ) Fourier transform spectrum analyzer (HP 3582A). The data were collected in the range 8 mHz-25 kHz as averaged power spectra of 16 parallel,I" which provided acceptable reliabilitywithin reasonable acquisition time (=lo min). The several, partly overlapping linear spectra obtained in different frequency spans were condensed and transformed by the controlling computer to an amplitude square-log frequency relationship. The algorithms ensured the relative bandwidth (the bandwidth over frequency ratio) to be (9) Gratzl, M.; Janata, J. "Filter Banks for Power Spectrum Estimation with a Logarithmically Uniform Frequency Resolution"; J . Phys. E, in press.
1178 Langmuir, Vol. 5, No. 5, 1989
log (ZQR)e
-1
10
i
4 1
%\
Hongyo et al.
h
I 0
I
2
1
log f (Hz)
4t
Figure 4. Change of the reactance of PC bilayer upon addition of concanavalin A and yeast mannan: a, background; b, +0.02% of concanavalin A; c, +0.04% of yeast mannan.
constant and symmetrical around logarithmically distributed center frequencies. Then, the corresponding log reactance-log frequency function was computed with eq 1and plotted (Figures 2, 4, and 5). The impedance transformer and the cell were separated from each other by the same multiple shielding (steel/copper/steel) as used around the cell (Figure 1B) to avoid any cross-talk or positive feedback. They were placed as close to each other as possible, to minimize the length of the connecting wires. The impedance transformer was ac coupled to the preamplifier (with a 0.03-100000-Hz input bandwidth). A dc analog input coupling and a flat top window function for the FFT were used with the spectrum analyzer. Theoretical Considerations The basic concept behind the approach using fluctuation analysis is that a hypothetical noise generator, corresponding to the Johnson noise generated by any resistor, is considered to be placed in series with that (noiseless) resistor. The generated voltage fluctuation squares are proportional to the resistance of the resistor in question at any frequency, according to eq 1. The composition of these fluctuations of composite circuits follows the rules of impedance composition,’ which means that in a parallel RC the capacitor dumps the amplitudes at different frequencies in such a way that the net fluctuations will finally define exactly the true filter characteristics of the RC through eq 1. With fluctuation analysis, only the real part of the impedance characteristics can be measured, because the phase difference between the “input” (excited by random motions on the atomic scale) and the ”output” signal (the measured fluctuations in voltage) cannot be determined experimentally. Thus, instead of a complex impedance plot a reactance-frequency relationship can be expected as a result of the experiment (examples are shown in a log-log representation in Figures 2, 4, and 5). A lipid bilayer membrane can generally be represented from an electrical point of view by an equivalent circuit consisting of three parallel RCs in series, which correspond to three kinds of constituent layers of different chemical character: the polar head regions (p) at both boundaries of the membrane, the hydrophobic bulk region (h), and two acetyl regions (a) between the bulk and the polar head region^.^ The time constants and resistances of these
I -1
I
I
0
1
log f
IHz)
Figure 5. Dependence of the reactance characteristics of asolectin bilayer on the concentration of valinomycin. a-c: 0 , 2 X and 2 X M nominal valinomycin concentration,respectively.
hypothetical RCs were found to be fairly different for PC/tetradecane bilayers. Consequently, for such membranes the following function could be determined with fluctuation experiments, at least in principle: zRe(f)
=
RP 14- (~TR,C,)’~Z
+
Ra 14- (2~R,C,)~f2 1
Rh + + (2TRhCh)2f2 (2)
where the subscripts correspond to the mentioned different membrane layers. The resistance and capacitance values of the two symmetrical polar head and acetyl regions, respectively, are lumped together for the sake of simplicity. The cutoff frequency of any of the RCs involved is defined as f” = 1/(2aRC) (the time constant tC= RC). In the case of PC/n-decane bilayers, however, the mentioned three membrane layers could not be resolved even with the ac technique, due to slow but steady changes in the membrane parameters during the entire impedance mea~urement.~ Thus, the impedance of such membranes is approximated merely as the parallel combination of the “overall” membrane capacitance, C,, and the chargetransfer resistance, R,. The equivalent electrical circuit then has properties of a low-pass filter, the log reactance-log frequency relationship of which consists theoretically of a flat curve section at the constant level of R, when f > p“ (f” = 1 / ( 2 ~ R & ~is)the “overall” cutoff frequency of the bilayer). A curved transition connects the two straight line sections around p“. The largest contribution to R, is due to the hydrophobic bulk region while C, is determined mostly by the capacitances of the other two pairs of layers in the membrane. In practice, the values of C, and R,, could be resolved from the horizontal and from the sloping part of the recorded spectrum, but the lower frequency limit of our experiments did not allow determination of the flat portion
Langmuir, Vol. 5, No. 5, 1989 1179
Equilibrium Fluctuation Analysis of Bilayer Membranes Table I. Characteristic Parameters of Bilayer Membranes at Different Chemical Modulations As Determined with Fluctuation Analysis valinomy- valinomy- valinomy- valinomyPC cin cin cin cin 1 x 10-7 5 x 10-7 1 x 10-8 5 x io* C," M P,bR F2 9.4 X 2.2 X 7.5 X 7.5 X 2.5 X lo* 10-11 10-12 10-12 10-13 B,' R 2.7 X 1O'O 1.1 X 1O'O 3.4 X loB 4.0 X lo8 1 X lo7 PC concanavalin A yeast mannand c, M 0.02% 0.04% P,R F2 2.0 x 10-13 2.4 x 10-13 1.7 X lo-" B, Q 1.3 X 10" 1.1 x 10" 1.5 x 109 asolectin valinomycin valinomycin C, M 2 x 10-11 2 x 10-10 P,R F2 4.3 X 8.6 X 1.3 X B, 5.9 x 1010 3.0 x 109 2.0 x 108 ~
Modulant nominal concentration. * P = R,Cm2. B = 1/(4GP) = Z,(f = 1). dFirst concanavalin A was added and then yeast mannan to the same solution.
in the studied cases. Thus, to interpret the results, the relationship
zb(fl = 1/ (4r2R,Cm2f2)
(3) must be taken into account, which is valid for f >> p". So, by experimental determination of the sloping part of a power spectrum, only the term P = R,Cm2 can be obtained directly. In cases when only the sloping part of the filter characteristics can be measured, its shifts caused by chemical modulation might be characterized by points taken from each individual curve a t one fixed frequency. However, in such cases it is more correct to use P as a characteristic quantity, because it is invariant with respect to frequency and, in addition, involves the real physical constants of the membrane. In fact, the value of B = 1/(4r2P) equals the reactance a t f = 1, and so P can be determined by linear regression in a log-log representation and by taking then the additive constant term as B. Thus, by using P one can avoid choosing an arbitrary frequency for comparison of shifts. (As long as the low-pass filter approximaion is feasible, the slope of the mentioned linear regression must always be close to -2).
Results and Discussion The power spectra of the equilibrium fluctuation of the pure PC membranes imply a high resistance (larger than 10" Q, Figure 2), which is in agreement with the values obtained by conventional ac measurements.2 Characteristic changes in the fluctuation spectrum are observed after multiple addition of valinomycin. It was not possible to assign the exact numerical values of the individual resistive and capacitive elements, R, and C,, to any reactance curve because the cutoff frequency of the studied membrane is below the lower frequency limit of our instrumentation. The value P as a function of the added amount of valinomycin, however, could be determined (Table I). The (logarithmic) shift of the sloping region of the spectrum is not far from linear over the concentration range G1O4 M (Figure 3), which means that Z,(f=l) = 20 exp(-7.6c) (4) where c is concentration and the units are GQ and pM, respectively. This range is, however, too narrow to draw any conclusions from it about the overall concentration dependence of the membrane reactance. The reaction with concanavalin A does not influence the spectra to a significant extent, but a following addition of
0
-10
-8
-6 log c
(M)
Figure 6. Dependence of the increase in conductanceof asolectin bilayer on the concentration of valinomycin: c, nominal concentration of valinomycin; AG, relative percent increase in conductance (AG = 2 means, e.g., a 100% increase). Solid circles, 1 mM CaC12,pH 5.7; empty circles, 3 mM CaC12,pH 5.7; crosses, 1 mM
HEPES,pH 6.6.
yeast mannan drastically reduces the level of the noise (Figure 4), which means that it increased the real part of admittance of the membrane (for the corresponding values of P, see also Table I). This result is again in good agreement with our previous data, which were obtained with an ac bridge. In agreement with our previous work,2 no significant changes of the noise power spectra were obtained upon addition of phloretin (note that in Figure 3 in ref 2 only the imaginary part of the impedance changed significantly while the conductance, or the real part of impedance that is measured in this work, remained essentially constant). This means that P is nearly constant during the experiment, and consequently, any change in RCtas caused by the addition of phloretin was compensated by an opposite change in C, so that the product R,Cm2 remained essentially unchanged. The chemical modulation of the reactance of asolectin bilayer membranes determined with fluctuation analysis showed also significant changes upon addition of valinomycin in the frequency range below 100 Hz (Figure 5). Such changes have been observed also a t higher frequencies (Figure 6) with the ac impedance technique. At these frequencies, the output voltages would have been already too small for providing reliable results with the technique of fluctuation analysis. In these experiments, nearly the same concentration range has been studied as in those performed on PC membrane, but in this case the resolution in concentration was logarithmically more uniform (see the horizontal axes in Figures 3 and 6, respectively). Here a different type of relationship could be observed: AG(f = 500) = 7.8 X 109c0.9in the linear range of the plot (2 X 10-lo M Ic I2 X lo4 M). AG is the relative increase in conductance (in %), and consequently Z ~ , ( f = 5 0 0=) Zb0(f=500)/(7.8 X 1) (5)
+
where superscript O refers to the pure membrane. Note that eq 4 can also be written in an analogous way: Zh(f=l) = Zh0(f=l) exp(-7.6 X lo%),where the first term is about 20 GQand the unit of c is now M. This difference in type of correlation between PC and asolectin bilayers (cf. eq 4 and 5) can be partly due to the very different frequencies to which they are related. At 1 Hz, a different RC is "active" in bilayer membranes than the one a t 500 Hz (see previous section). In addition, for characterizing PC bilayers, significantly less experimental points were
Langmuir 1989, 5 , 1180-1186
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available; so the linear fit in Figure 3 may be considered as only a first approximation. (In a narrow region, any sufficiently smooth relationship can be approximated by a linear function.) The response of the asolectin membranes was investigated over a concentration interval involving many orders of magnitude (that can be represented more appropriately with a log concentration scale), which provided better resolution and also a better fit to a different function. It should be noted that the observed change in slope below lo4 M in Figure 6 is probably due to an adsorption effect (most of the very small valinomycin amount present in that range is rather adsorbed to the walls of the cell, instead of influencing the impedance properties of the membrane). Conclusions Up to a point, the fluctuation measurement is a useful altemative to the conventional ac bridge measurement. Its most attractive feature is the fact that the experiment is
done without externally applied voltage, under the conditions of a true thermodynamic equilibrium. However, for the same reason the signal is of low power, requiring high-gain amplification and spectrum averaging. These problems render the information acquisition time unacceptably long for practical applications such as chemical sensing. This situation could be improved, in principle, by employing the cross-correlation technique; however, that would require the measurement to be done on a pair of matched membranes. At present, such an approach is not practical. For studies of impedance and chemical modulation of bilayers, however, the approach of fluctuation analysis can be a useful one, as it is shown by the experimental results presented in this work. Acknowledgment. We gratefully acknowledge the support of this work obtained from the Office of Naval Research. Registry No. Valinomycin, 2001-95-8; concanavalin A, 11028-71-0; D-mannon, 9036-88-8;phloretin, 60-82-2;cholesterol, 57-88-5.
A Chromatographic Method for the Determination of Segmental Adsorption Energies of Polymers. Polystyrene on Silica G. P. van der Beek,* M. A. Cohen Stuart, G. J. Fleer, and J. E. Hofman Laboratory for Physical and Colloid Chemistry, Agricultural University, Dreyenplein 6, 6703 HB Wageningen, The Netherlands Received January 19, 1989. In Final Form: April 28, 1989 Polystyrene adsorption on a silica surface was studied by means of adsorption TLC. This is a very sensitive technique for detecting polymer adsorption/desorption transitions in binary solvent mixtures. Such transitions, which occur at a well-defined solvent composition (the so-called critical point), were earlier shown to be related to the segmental adsorption energy. In this paper we derive a relation between the concept of chromatographic solvent strength, as defined by Snyder, and segmental adsorption energies. It turns out that solvent strength data, which are available in the chromatographic literature, can now also serve as a useful source of information for polymer adsorption and adhesion studies. Finally, we give some experimental results for the displacement of polystyrene from silica in two different solvents (carbon tetrachloride and cyclohexane) by various weakly adsorbing low molecular weight displacers. Using tabulated solvent strengths, we determined segmental adsorption energies for polystyrene on silica of l.OkT and 1.9kT, from carbon tetrachloride and cyclohexane, respectively. Introduction Thin-layer chromatography, TLC, is frequently used to analyze polymer samples.'-8 This technique has many advantages: simple instrumentation, high sensitivity, rapidity, and ease of detection of separated bands. TLC can be based on either adsorption (ATLC), size exclusion as in GPC (TLGPC), or dissolution-precipitation (extraction and precipitation TLC). These techniques permit the investigation of all kinds of polydispersity: polydispersity according to molecular weight,14 stereoregularity:* microstructure,'-* and chemical composition in copolymer~.'~~~' It is even possible to separate isotopically different polymers by these techniques.8 We are interested in TLC as a technique to study the adsorption of polymers. Hence, we will exclusively consider ATLC. Polymer adsorption has many technological
* Author to whom correspondence should be addressed.
applications. The mechanical strength of polymer composites depends on the strength of adhesion between the matrix polymer and the reinforcing filler. Polymer adsorption can stabilize dispersions, which is important in products like paints, foods, pharmaceuticals, etc. It can also destabilize colloidal systems, which is useful in water purification and improvement of soil structure. The aim of this paper is to establish a relation between ATLC with polymers and polymer adsorption studies as done by Co(1) Gloeckner, G. Polymer Characterization by Liquid Chromatograp h y ; Elsevier: Amsterdam, 1987. (2) (a) Belenkii, B. G.; Gankina, E. S. J. Chromatogr. 1977, 141, 13. (b) Belenkii, B. G. Chromatography of Polymers; English Ed.; Elsevier: Amsterdam, 1983. (3) Otocka, E. P. Macromolecules 1970, 3, 691. (4) Inagaki, H.Adu. Polym. Sci. 1977, 24, 189. (5) Buter, R.Ph.D. Thesis, University Groningen, 1973, p 133. (6)Miyamoto, T.; Inagaki, H. Macromolecules 1969,2, 554. (7) Gloeckner, G.; Kahle, D. Plaste Kautsch. 1976,23, 338. Donkai, N.; Inagaki, H. Macromolecules 1980,13,1021. (8) Tanaka, T.;
0743-7463/ 8912405-1180$01.50/0 0 1989 American Chemical Society
