Dielectric relaxation spectroscopy and some applications in the pharmaceutical sciences

J O U R N A L O F A publication of the American Pharmaceutical Association and the American Chemical Society

Pharmaceutical Sciences

September 1995 Volume 84, Number 9

RE VIEW A R TlCL E

Dielectric Relaxation Spectroscopy and Some Applications in the Pharmaceutical Sciences

GEOFF SMITH, ALISTAIR P. DUFFY*, JIE SHENA, AND CEDRlC J. OLLIFFt Received January 3, 1 995, from the Department of Pharmaceutical Sciences, *Depatfment of Electronic and Electrical Engineering, and A Department of Applied Physics, De Monthort University, Leicester, LEI 9BH, UK, and fDepartmenf of Pharmacy, Universify of Brighton, Brighton, BN2 4GJ, UK. Accepted for publication May 17, 1995@.

Abstract 0 With a few exceptions, dielectric relaxation spectroscopy (DRS) has been largely neglected by pharmaceutical scientists, despite the potential for this technique as a noninvasive and rapid method for the structural characterization and quality control of pharmaceutical materials. DRS determines both the magnitude and time dependency of electrical polarization (i.e. the separation of localized charge distributions) by either measuring the ability of the material to pass alternating current (frequency domain DRS) or by investigating the current that flows on application of a step voltage (time domain DRS). DRS is thus (i) sensitive to molecular mobility and structure, (ii) non-invasive, and (iii) employs only mild stresses (a weak electromagnetic field) in order to measure the sample properties. The technique covers a broad-band frequency window (from 1 O P to 10 Hz) and therefore enables the investigation of a diverse range of processes, from slow and hindered macromolecular vibrations and restricted charge transfer processes (such as proton conductivity in nearly dry systems) to the relatively fast reorientations of small molecules or side chain groups. The dielectric response provides information on (i) structural characteristics of polymers, gels, proteins, and emulsions, (ii) the interfacial properties of molecular films, (iii) membrane properties, (iv) water content and states of water (and the effects of water as a plasticizer), and (v) lyophilization of biomolecules. This review article details the basis of dielectric theory and the principles of measuring dielectric properties (including a comprehensive account of measurement artifacts), and gives some applications of DRS to the pharmaceutical sciences.

Introduction Dielectric relaxation spectroscopy (DRS) determines both

the time dependency and magnitude of electric polarization processes (i.e., the separation of localized molecular charge distributions) by measuring the rate and extent of polariz-

e Abstract published in Advance ACS Abstracts, July 1, 1995.

0 1995, American Chemical Society and American Pharmaceutical Association

ability of a material placed in a weak electromagnetic electric field. The polarization of a material is reflected at the macroscopic level by the magnitude of the electrical capacitance (i.e., the ability of the material to store electrical charge) and relates to the structure, environment, and mobility of charged molecular groups.

Polarization kinetics are studied either in the frequency domain, by measuring the ability of the material to pass alternating current (frequency domain DRS), or in the time domain, by measuring the transient flow of charge due to an applied step voltage (time domain DRS). Take the example of a frequency domain DRS study of the orientation polarization of permanent dipoles (e.g., water molecules) in time- varying electric fields. At low frequency, the molecular dipoles orientate in the field (i.e. polarize) so as to reduce the electrical potential energy of the material. As the field oscillates, the dipoles follow the change in polarity of the field. However, at sufficiently high frequencies the reorientation of dipolar molecules can no longer keep pace with the alternating field and therefore the dipoles relax to their random orientation. The frequency range over which molecules relax depends, in part, on the rotational freedom of the molecule which, in turn, depends on the environment and physicochemical interactions with neighboring molecules. DRS thus provides a means of determining molecular dynamics and the extent of molecular interactions. The technique covers a broad band frequency window (from to lo1 Hz) and therefore enables the investigation of a diverse range of processes from slow and hindered macromolecular vibrations and restricted charge transfer processes (e.g., proton conductivity in nearly dry systems) to the relatively fast reorientations of small molecules or side chain molecular groups.

There have been many excellent books and reviews on the measurement of dielectric properties and the applications of DRS to aqueous solution^,^^^ biological solutions and cell suspension^,^-^ polymers,lOJ1 and solutions of polyelectrolytes.12 This literature documents the foundations on which DRS has been developed by physiologists, biologists, physi-

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cists, and chemists alike. More recently, however, an interest in DRS has developed among pharmaceutical scientists, who have begun to recognize a potential in the analysis of pharmaceutical system^.^^-^^ Pharmaceutical investigations using DRS have been mirrored elsewhere by a general expansion in dielectric research, fuelled by the recent availability of commercial, computer-controlled equipment de- signed for fast and accurate materials measurement DRS. More than 20 years ago, however, dielectric measurements were taken by manual point-by-point measurements, which restricted the development of applications for DRS. At the same time advances were being made in physical methods, such as NMR, ESR, and IR, that are in essence complemen- tary to DRS as techniques for studying the physical structure of materials.

Although the frequency window for DRS spans some 16 decades, much of the research is confined to the frequency window from to lo7 Hz. This may be a consequence of the technical difficulties that high-frequency measurements entail and the high cost of the instrumentation required. Whatever the reasons, the result is that the applications of DRS in the higher frequency range (from lo8 to 1011 Hz) are yet to be fully exploited. The information derived from studies in the high-frequency region generally relates more to specific dipole reorientation polarizations and can therefore be associated directly with the microscopic structure of the material. In this context, the dipoles act as inherent molecular probes capable of yielding information about the structural properties of the material. The absence of studies in the high- frequency region may also partly explain the lack of published work on the dielectric analysis of water in pharmaceutical systems, a situation which is not reflected in the food industry, where dielectric analysis of hydration is becoming increasingly important. I 6 , l 7

Certainly, the use of both high- and low-frequency DRS in the pharmaceutical sciences is still in its infancy, and yet there is great potential for this technique in the pharmaceutical arena. This review therefore aims to introduce the subject in a brief but concise way and indicate the scope and extent to which DRS may be applied to pharmaceutical problems.

Unfortunately, the measurement and interpretation of dielectric behavior is notoriously mathematical, as models for the dielectric response are necessary in order to interpret the overall macroscopic measurement in terms of the microscopic properties of the sample. This is both unavoidable, and necessary, if meaningful information is to be derived using this technique. However, it is not the intention of this review to present a concise theoretical treatment of dielectric analysis, as it is beyond the present scope and purpose of the paper. This information can be found in a number of books that deal specifically with dielectric theory.18-20 The authors have rather presented the more fundamental aspects of DRS: the basic theory, the measurement technique, and an overview of certain dielectric measurements with examples of some applications to the pharmaceutical sciences. The last section is intended to provide the reader with a flavor of the potential for DRS in the study of pharmaceutical materials and therefore does not pretend to be comprehensive.

A final, but important, note (or warning) to the uninitiated or potential DRS-investigator: The recent availability of automated instrumentation, as with many other techniques (particle sizing being of particular note), can lead to a black- box mentality of the measurement. Such an attitude is potentially dangerous, as measurement artifacts can easily be wrongly associated with the dielectric behavior of the sample; not all of these measurement artifacts are compen- sated for by commercial instrumentation. Data should be interpreted with care, and therefore this review partly con-

centrates on a number of common measurement artifacts that are inherent in dielectric analysis.

Dielectric Theory Defining the Macroscopic Electrical Properties of

Pharmaceutical Materials-The intrinsic macroscopic electrical properties of pharmaceutical materials can be characterized completely by two functions, i.e. the permittivity and conductivity. The latter function encompasses ohmic conductivity and dielectric loss. The magnetic, i.e. inductive, properties are not considered as they approximate those of free space.

Permittivity of Free Space (tv) and Relative Permittivity (E)-According to well-known theories of electrostatic^,^ the permittivity of free space (eV) (In the literature the symbol for the permittivity of free space is often given by go; however, in this text E,, represents the low frequency limit of permittivity and the symbol e, is taken to represent the permittivity of free space.) is given as the constant of proportionality between the applied electric field (E) and the charge stored per unit area (i.e. the dielectric displacement, d = q/A) on the plates of a parallel plate capacitor.

d = c J (1)

The relative permittivity, or dielectric constant (E, some- times written 6, or ), represents the factor by which the capacitance of system to store electrical potential energy (i.e. the dielectric displacement) is increased on substitution of a dielectric in place of free space.

d = eVcE

The above equation states that the charge storage per unit area on the electrodes of a parallel plate capacitor has been increased by a factor of E on substitution of a polarizable material in place of free space. The material increases the electrical storage capacity of the capcitor through the separation (polarization) of localized charge distributions within the material, thereby neutralizing charges at the electrodes which would otherwise have contributed to the external field. The capacitor requires more charge to flow to the plates to maintain the same potential difference between them, and hence the charge stored per unit area is increased. The electrical capacitance of the sample and free space is determined (C, and C,, respectively) and the factor derived from the ratio

= C,lC, (3)

The term dielectric constant, however, is a misnomer, as the polarization of a material varies with the experimental conditions, e.g., temperature, orientation, pressure, the frequency of the applied oscillating field. The term relative permittivity is therefore preferable to the more familiar term, dielectric constant. The polarization (P) of the sample (defined by the dipole moment per unit volume) is derived from the relative permittivity according to

(4) P = ( - 1)cJ

C0nductivit.y (a)-The dc (or ohmic) conductivity describes the dissipation of the potential energy of an applied electric field by the migration of delocalized charge. The potential energy is converted to kinetic energy and subsequently lost as heat due to frictional forces. dc conduction in fully hydrated materials is primarily due to the migration of hydrated ions, whereas conduction in dry or nearly dry systems is due to protonic,Z1,22 ionic, and electronic conduction

1030 / Journal of Pharmaceutical Sciences Vol. 84, No. 9, September 1995

mechanisms.ll Conduction may also occur through dielectric loss (i.e. the loss of ability to store electrical charge), but unlike the dc conductivity of a sample, the dielectric loss is an ac phenomena.

Polarization Mechanisms-The polarization of a material increases the capacity of the measurement system to store electrical potential energy by neutralizing charges a t the electrodes which would otherwise have contributed to the external field. Polarization of a material can occur through (i) orientational polarization, (ii) interfacial polarization, (iii) ionic polarization, and (iv) atomic and electronic polarization.

Orientation (Dipole Reorientation) Polarization-Uneven charge distributions across bonds between atoms result in the formation of permanent dipole moments which respond to the stresses of an applied electric field by reorientating themselves in line with the field, so as to reduce the electrical potential energy of the molecule. A macroscopic dipole moment is generated which neutralizes the external field, thus allowing more charge to accumulate on the electrodes and thereby increasing the charge stored.

Interfacial or Space Charge Polarization (The Maxwell - Wagner Effect)-Interfacial polarization, between electrically dissimilar materials, occurs due to restricted charge transfer a t the interface. The discontinuity results in charge accumulation at the interface, i.e. interfacial or space charge polarization. This phenomena is otherwise known as the Maxwell-Wagner effect after Maxwellz3 and Wagner,24-27 the two scientists who developed the theoretical treatment of the dc and ac electrical properties of heterogeneous materials, respectively.

Cells and liposomes exhibit interfacial polarization owing to the conductivity of the suspending phase and the nonconducting, capacitive nature of the membrane. Very high capacitances are observed as a direct consequence of the thinness of the membrane across which the space charge accumuIates.28 Schwan classified the energy loss due to relaxation of interfacial polarization as the P-d isper~ion .~~

The dielectric response associated with interfacial polarization is useful, in that it enables the deconvolution of the dielectric properties of certain heterogeneous systems30 such as water-in-oil emulsions,31,32 l ipo~omes,3~-~~ microcaps~les,36~3~ or cell suspension^.^^-^^

Interfacial polarization also occurs at the electrode surface, due to the electrode charge transfer r e ~ i s t a n c e , ~ ~ and results in particularly high input capacitances that swamp the sample capacitance (see Electrode Polarization).

Zonic Polarization-The migration of ions (i.e. bound charges) along limited paths, for example in the counterion layers of cells/liposomes,35~4z~43 charged colloids, and other polyelec-

results in localized charge separations and associated relaxation processes which have been classified according to Schwan as the a - d i s p e r s i ~ n . ~ ~ Other ionic polarization mechanisms have also been described. For example, Dissado et al.13,54 characterized a subhertz relaxation in heterogeneous gels as being due to quasi-dc polarization of weakly bound charges that are partially free to move through the gel by charge jumping between discrete clusters. In general, ionic polarization relaxes a t lower frequencies than interfacial polarization.

Atomic and Electronic Polarization-The vibrations of positive and negative charge centers (atoms) relative to each other result in the induced polarization of dipolar materials, whereas the excitation of electron clouds relative to their nuclei results in the induced polarization of both dipolar and nonpolar materials. Atoms can be modeled as oscillators with a damping effect similar to a mechanical spring and mass system which resonate once energy within a particular band of frequencies is input to the system. Below the resonance frequency, atomic and electronic polarization contribute only

Table 1-Classification of the Frequency Ranges of the Dielectric Spectrum over Which Dielectric Phenomena Occur

Frequency Frequency Classification Range Classification Range

Subaudio frequencies < 20 Hz Radio frequencies 3 kHz-30 MHz Audio frequencies 20 Hz-20 kHz Microwave frequencies 0.3-300 GHz

a small amount to the permittivity of the material, whereas above the resonance frequency these contributions disappear.

Relaxation and Resonance Processes-Orientational, interfacial, and ionic polarization processes are thermally damped processes and give rise to relaxation phenomena in the subaudio to microwave frequencies of the electromagnetic spectrum (Table 1). Electronic and atomic polarization processes, however, depend on the inertia of the polarizable species and give rise to resonance phenomena in the IR to UV regions of the electromagnetic spectrum.

Polarization Kinetics, Dielectric Relaxation, and Loss-There is a finite time, following the application of a step voltage to a material, in which equilibrium polarization is established. The kinetics of the process depends on the nature of the charging mechanism. For charging mechanisms that exhibit first-order kinetics, the polarization of the material (P) reaches its final value (Pf) a t a rate proportional to (Pf - PI.

5!!? = K&P, - P) dt (5)

The polarization kinetics in DRS studies are generally characterized by the macroscopic relaxation time, t, (otherwise known as the exponential decay function and equal to the reciprocal of the mean rate coefficient, Kp).

For dipolar reorientation, the time dependence of the process lies in the viscous damping caused by thermal agitation (Brownian motion) and molecular interactions, viz. the kinetics of dipolar orientation depend primarily on the local environment of the dipole and the temperature of the system. For small and relatively simple molecular structures, there is often only one polarization process, characterized by a single time constant. D e b ~ e ~ ~ developed his theory for polar molecules of this type based on ideal gases and dilute solutions. In his analysis, the orientations of polar molecules are represented as spheres rotating in a continuous medium of macroscopic viscosity.

For interfacial polarization, the time dependence of the process is given by the exponential decay function (t = RC), where R is a function of the relative magnitude of the charging and discharging currents, which in turn depend on the resistances of the two materials, and C is a function of the extent of charge accumulation at the interface.

For ionic polarization, the kinetics of the process depend on the charge carrier mobility and the diffusion path length, such that faster processes occur in the case of mobile ions and short diffusion path lengths.

The above processes are thermally damped and therefore, under steady state conditions, a dynamic equilibrium exists between the "ordering" or polarizing effects of the applied field and the randomizing effects of Brownian motion. On removal of the field, the system returns or relaxes to the unordered state. It is for this reason that the dispersion of potential energy in the system (dielectric dispersion) is described as a relaxation process.

If the applied field is oscillating, then the kinetics of the charging process translate to a frequency dependence in the bulk electrical properties. An applied field of sufficiently low frequency will result in the material polarizing in phase with

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-5 0 5 Log frequenc

(normalized to relaxation kquency, 1,)

7 7 7 - r T T ,-- r-

-5 0 5

Log frequenc (normalized to relaxation Yrequency, fo)

-5 0 5

Log frequenc (normalized to relaxatlon iequency. fJ

,/-----

ha

I , , , J , 1 I I I , 1 I I , 1

Relative permitlivity (e)

Figure 1-Typical relaxation data. Relaxations characterized by a single time constant are depicted by the solid lines, whereas relaxations characterized by a symmetrical distribution of time constants are represented by the dashed lines. (A) Frequency dependent real permittivity (i.e. relative permittivity) plotted against log frequency, normalized to the relaxation frequency. (B) Frequency dependent conductivity plotted against log frequency, normalized to the relaxation frequency. (C) Frequency dependent imaginary permittivity (i.e. dielectric loss) plotted against log frequency. Note: relaxations characterized by a distribution of time constants show a broad dispersion curve and smaller absorption maximum in the dielectric loss profile. (D) Complex plane locus or Argand diagram (commonly referred to as the Cole-Cole plot). Relaxations characterized by a distribution of time constants show a semicircle the center of which lies on a point below the abscissa. The Cole-Cole distribution function for the above data is 0.3.

the oscillating field. The energy required to build up the field in one half of the cycle is exactly recuperated in the next half of the cycle, and therefore there is no net absorption of energy. However, as the frequency of the oscillating field increases, a stage is attained whereby the dielectric can no longer respond to the changing conditions and the material relaxes to a random state. The energy input, a t these higher frequencies, is no longer stored as electrical potential energy but is instead dissipated as heat through frictional forces. (The term dispersion is often used interchangeably with term relaxation.) A net absorption of energy occurs and the sample becomes more conductive. Dielectric relaxation is therefore accompanied by a fall in the relative permittivity (Figure 1A) and an associated increase in the conductivity (Figure 1B). For a relaxation with a single time constant ( ~ ~ 1 , the dielectric relaxation decrement in permittivity (AE) and the associated increment in conductivity (Ao) are related through the Kram- ers-Kronig equation:

EVA -- - Ao m

where

(6)

and E , and ti, are the limiting low- and high-frequency permittivities, respectively. This relationship allows one to investigate the relaxation by either the frequency dependent permittivity or the frequency dependent conductivity, in addition to providing a useful means of checking the validity of the measurement.

Complex Permittivity-In the limit of low frequency, the current flowing from the polarization of the dielectric is 90 out of phase with the applied field. However, as the system relaxes toward higher frequencies, a component of the pennit- tivity becomes in phase with the field (i.e. the material becomes lossy) and the permittivity therefore becomes a complex quantity. Empirically the complex permittivity ( E * ) is represented by

E is the real component of permittivity, i.e. the capacitive part, and E represents the dielectric loss, or imaginary permittivity, resulting from the in-phase admittance (Figure 1C). Repre- senting E as a real number and E as an imaginary number serves as a reminder that the two components are indeed out of phase. The magnitude of E* is the vector sum of E and E.

AE = E , - E-


Dielectric Relaxations Characterized by a Single Time Constant-In an oscillating electric field, the complex permittivity of a material, exhibiting a relaxation process characterized by a macroscopic relaxation time, zm, is given by55

A 1 + ioz, * = E , +

where w is the angular frequency of the applied field. The following expressions for the real and imaginary permittivities (derived from eqs 7 and 9) are commonly known as the Debye dispersion formulae:

A E' = +

1 + (wzml2 (10)

(11)

The macroscopic relaxation time (tm) for a Debye-like relaxation can be derived from the frequency at which maximum loss occurs (coo, otherwise known as the relaxation frequency). Thus, differentiation of eq 11 with respect to w and equating to zero gives

-- dE'f - o at w, = I I ~ , do (12)

The relaxation frequency may also be expressed in terms of cycles per second v0): It follows from w = 2xf that fo is related to the macroscopic relaxation time (z,) according to

The characteristic frequency, on which the relaxation is centered, can vary from very low frequencies of the order of lo-' Hz and below for large hindered molecules to frequencies up to 10l2 Hz for small highly mobile molecules.

According to D e b ~ e , ~ ~ the macroscopic relaxation time for dipole reorientation is the time required for the permittivity to fall to lfe of its initial value, and is given by the expression

z, = 4na33- kT (14)

The relaxation time is therefore dependent on the size of the molecular dipole (assumed to be a sphere of radius a ) and the microviscosity of its environment (77). For example, the reorientation of the zwitterionic head group of dipalmitoyl- or dimyristoylphosphatidylcholine membrane vesicles therefore has a larger time constant (z, = 2 x s at 30 "C)35 than the faster reorientation of a smaller molecule such as water in the bulk liquid state (z, = 7.2 x s at 30 0C).35 It is an important fact that the macroscopic relaxation time also varies with the microviscosity, as the relaxation time can be used as an indicator of changes in the environment of the dipole.

Macroscopic (r,) and Microscopic (ri) Relaxation Times for Debye-like Relaxation-As detailed so far, the macroscopic time constant (r,) represents the reciprocal of the mean rate coefficient of the polarization process (eq 5). However, if the frequency dependent dielectric properties are to be interpreted at a molecular level, then it is the microscopic, or intrinsic, molecular relaxation time (ti) that must be derived (only applicable for dipolar relaxation processes).

The differences between the macroscopic and microscopic relaxation times are a consequence of local fields from neighboring dipoles (the internal field) and the resultant cooperativity between the orientations of groups of dipoles.ls Dipoles are not only influenced by the external field but affect one another mutually, and therefore the response to the external field involves a cooperative motion (referred to as "reorientation correlation") that depends on the extent of electrostatic interactions with neighboring dipoles. The microscopic relaxation time, however, is independent of the internal field created by the macroscopic polarization of surrounding molecules and reflects the time required for a given molecule, if fixed and released, to revert to random orientation in the absence of the internal field.55

Fortunately, it has been demonstrated that a process with a single microscopic relaxation time always leads to a single macroscopic relaxation time,56 and therefore the microscopic relaxation time for processes exhibiting Debye-like behavior can be derived by the equation of P o w l e ~ : ~ ~

Non-Debye-like Behavior (Cooperativity and Multiple Relaxation Times)-It is hardly surprising to find that Debye relaxation formulae are not an adequate representation of most relaxation phenomena, as the theory is based on the special case of a gas or dilute solution of spherical dipoles rotating in a homogeneous medium of low dielectric constant. It has been shown, for example, that the dielectric behavior of most pure liquids diverges markedly from ideal Debye-like behavior.58 Dielectric relaxation of most biological materials also exhibits non-Debye-like behavior. Complex systems, like macromolecules and vesicle suspensions, are likely to display cooperative relaxations of participating molecules and side chain molecular groups that result in a broadening of the dielectric loss peak.6J1 In concentrated suspensions, a distribution of relaxation times may arise due to electrical interactions between suspended particles, whereas the dilute suspension may only exhibit a single relaxation time.38 Muscle tissue commonly exhibits a distribution of relaxation times resulting from a distribution in muscle fiber diameter.59 The dielectric dispersion of cell suspensions in the radio frequency range might also be broadened due to contributions of subcellular organelles60 or the relaxation process itself may not be first-order, as is the case for counterion relaxation tangential to the surface of a colloid.61

Relaxations characterized by either long range cooperative reorientations or closely spaced multiple relaxation times will therefore exhibit a broad-frequency relaxation in both the real and imaginary permittivity. Many different functions have been proposed for modeling this non-Debye-like dielectric behavi~r ;~ however, by far the most common function is the empirical one adopted by Cole and Cole.58 Cole and Cole modified the Debye equation to give eq 16, where a is a distribution function reflecting the divergence from Debye- like behavior:

A * = , +

1 + (ioz,)l-" (16)

Plots of E' or E" from the above function against log frequency are shown by the dotted-line curves in Figure 1A,C. However, from these plots alone it is difficult to assess the extent of divergence from Debye-like behavior. A more convenient representation is provided through the complex plane locus, or Argand diagram, where the imaginary part of the dielectric constant is plotted against the real part. This


R R:

CS

A 0

Figure 2-(A) Diagrammatic representation of a material sandwiched between the parallel plates of the measurement cell. (6) The equivalent lumped-circuit model of the macroscopic electric properties of the material (I?, is the macroscopic sample resistance and C, the macroscopic sample capacitance. Note: the elements R, and C, do not include the boundary electrical properties associated with the electrode-material interface, but are rather the bulk electrical properties of the material).

type of representation is commonly referred to as the Cole- Cole plot and leads to the distinctive semicircular profiles shown in Figure 1D.

Sillars62 showed that the area under the 10s curve is proportional to the total concentration of dipoles irrespective of the time constants associated with them. Therefore, as the distribution of relaxation times increases the loss curve becomes broader and the peak height reduces such that the area under the loss curve remains constant (Figure 1C). A measure of the spread of relaxation times can then be defined by the parameter a.63

7rcmax (17) a = area of loss curve

For a Debye-type process characterized by a single relaxation time, the area under the curve equals and therefore a = 1. However, as the distribution of relaxation times tends to infinity, the value of X C , ~ falls such that a becomes less than 1.

The above analyses simply aim to derive parameters that describe the relaxation data without attempting to elucidate the underlying mechanisms responsible for the dielectric behavior and only apply if there is a symmetrical spread of relaxation times. Modeling of the dielectric response should, however, be more informative if any structural information is to be derived.

Measurement of Dielectric Properties The frequency dependent bulk electrical properties of

pharmaceutical materials are known collectively as the admittance (Y), the ability of the material to pass current (The term impedance, 2, i.e. the reciprocal of the admittance, is often used interchangeably with the term admittance). For a material with negligible magnetic properties, the admittance is a complex quantity comprising an element describing the ohmic conductance and dielectric loss and another element describing the capacitive susceptance (i.e. the ability of the material to pass current through capacitive charging). The total frequency dependent bulk electrical properties of the material can therefore be represented by a parallel arrangement of a capacitance (C,) and resistance (R, = l/Gs) (Figure 2). The capacitor models the dielectric properties of the sample, whereas the resistor models the ohmic conductance and dielectric loss of the sample. The two elements are considered in parallel because the application of an emf to the sample results in an equal potential difference across each element, but the total charge flowing is the vector sum of the charge flowing through both elements. As a consequence, and

A 0 Figure 3-(A) Lumped-circuit model for the electrical properties of the measurement cell containing sample, including elements modeling electrode polarization (C, and R, are the bulk electrical properties of the sample). The electrode polarization of the electrode/sample interface is modeled as an interfacial capacitance (G) in series with an interfacial resistance (Rp). (B) The equivalent input admittance to the measurement device, represented by a lumped-circuit capacitance (the input capacitance, C,) in parallel with a lumped-circuit resistance (the input resistance, R,) .

for ease of analysis, the use of the term admittance is advocated in preference to impedance.

Measurement Artifacts-The direct measurement of the capacitance (C,) and conductance (G,) of a pharmaceutical material is not possible. This fact is immediately obvious on considering the system under analysis as a material sandwiched between two electrodes and connected to the measurement device via two cables. The admittance determined by the measurement device ke. the input admittance) is therefore a composite of the individual impedances and admittances making up the circuit. These include contact impedances between terminals and connectors, interfacial impedances between the electrodes and sample in contact with them, and measurement residuals such as admittances due to stray fields and impedances due to circuit inductance. These measurement residuals affect dielectric measurements from low to high frequency, whereas other measurement artifacts, such as transmission line effects and measurement cell losses, occur ostensibly at higher frequencies {from lo3 to 1O1O Hz). Therefore, before interpretations are made concerning the nature of a dielectric relaxation, one must first account for experimental artifacts that are inherent in DRS measurements.

In the sections that follow, an indication has been given of the approximate frequency ranges over which each measurement artifact becomes problematic. These frequency ranges are intended to provide guidance to the reader embarking upon dielectric measurements. They should not, however, be taken as literal, as the frequencies over which measurement artifacts occur as inextricably dependent on the magnitude of the sample admittance and the electrical properties of measurement system (i.e. the test cell and connecting leads).

Electrode Polarization-A discontinuity in charge transfer a t the interface between the sample and electrode is manifest as an interfacial impedance. An inherent boundary resistance (R,) and capacitance CC,) is therefore included in the overall input admittance to the measurement d e ~ i c e l , ~ ~ , ~ ~ (Figure 3).

The effect of boundary resistance {R,) on the resistive part of the input admittance is usually a minor one. However, the polarization capacitance (C,) is particularly disturbing to the apparent sample capacitance, and more so at low frequencies where a very high capacitance will swamp that of the sample and mask low-frequency dispersions. The extent of polarization capacitance is heavily dependent on the composition of electrolyte and occurs due to charge accumulation at the electrode interface. Ionic contamination from organic materials can therefore adversely affect the determination of the low- frequency dielectric properties of the sample unless changes in the magnitude of the electrode polarization capacitance are accounted for.


A B

Figure 4-(A) Lumped-circuit model for the electrical properties of the measurement cell containing sample, including an element modeling the inductance of the measurement cell (C, and RS are the bulk electrical properties of the sample). The measurement cell inductance is modeled as an inductor (L) in series with the sample. (6) The equivalent input admittance to the measurement device, represented by a lumped-circuit capacitance (the input capacitance, C) in parallel with a lumped-circuit resistance (the input resistance, Rl).

Increases in electrode area and surface roughness facilitate charge dissipation, such that the magnitude of the polarization capacitance decreases on increasing the effective surface area for electrical discharge. Electrode polarization is therefore reduced on using platinum electrodes coated with a fine porous layer of platinum b l a ~ k . ~ , ~ ~ , ~ ~ The polarization impedance of platinum electrodes is up to 4 times lower than that of other materials. However, the fine coat of platinum black further reduces the polarization impedance by increasing the effective surface area for electrical discharge. For a review of platinization techniques, see the work of Feltham and S p i r ~ . ~ ~

On increasing the frequency of the alternating field, a decay in the polarization capacitance occurs due to a reduction in the extent of charge migration to the electrode surface. Other factors such as concentrated celb'vesicle suspensions may also decrease electrode polarization, as they create a shadowing effect over the electrode and therefore reduce charge ac- c ~ m u l a t i o n . ~ ~ Electrode polarization effects have been re- ported up to -lo6 Hz.1142366r67 Alternative methods have also been employed to reduce electrode polarization: (i) changing the electrode spacing,l (ii) changing the electrode surface area:8 (iii) using a four-electrode arrangement.l~~~

Measurement Cell Inductance-Inductance may arise from the sample cell and connecting leads, even though the material under test has negligible magnetic properties. The apparent sample capacitance (i.e. the input capacitance to the measurement device) will therefore be lower than expected, as the inductive and capacitive reactance are 180" out of phase and therefore partly cancel one another. The inductive properties of the cell arise from mutual inductance between the electrodes and transmission inductance within the electrodes themselves. By assuming that the mutual inductance of the cell is minimal, then the inductance of the measurement system can be simply modeled as an inductor in series with the pharmaceutical sample (Figure 4A). The above model is then equated to an equivalent parallel arrangement reflecting the input admittance to the bridge (Figure 4B).

In the limit of low frequency (effectively frequencies below -lo7 Hz), the sample capacitance (C,) and sample conductance (G,) are given by the input capacitance (Ci) and input conductance (Gi) to the measurement device:

ci = C, - LG: (18) Gi = G, (19)

The low-frequency input capacitance (CJ therefore shows a direct dependence on both the cell inductance and the sample conductance, whereas the low frequency input conductance (Gi) remains constant. In addition, the effect of cell inductance

A 0

Figure +(A) Measurement cell containing sample, showing the stray fields between the sample and ground. (6) If the sample is predominantly resistive (i.e., Y = i/Rs), then the stray capacitance introduces RC time constants into the overall input admittance to the measurement device. The admittance of the sample may therefore appear frequency dependent even though the electrical properties of the sample are frequency independent.

R

Figure 6-(A) Lumped-circuit model for the electrical properlies of the measurement cell containing sample, including an element modeling the stray capacitance of the measurement system. The stray capacitance is modeled as a lumped- capacitance (C,) in parallel with the sample (G and R, are the bulk electrical properties of the sample). It is important to note that this model is only appropriate if the field lines of the stray field do not first pass through the sample (The cell depicted in Figure 7 satisfies this criterion) (6) The equivalent input admittance to the measurement device, represented by a lumped-circuit capacitance (the input capacitance, C) in parallel with a lumped-circuit resistance (the input resistance, S).

on the input capacitance (CJ is particularly disturbing for conductive samples, owing to the relationship between Ci and the square of the sample conductance. At frequencies above 107-10s Hz, the effects of the cell inductance become frequency dependent and therefore it is more straightforward to use transmission lines to probe the sample admittance.

Stray Field Effects-Any conductor has a certain amount of inherent capacitance, just as it has a certain amount of inherent inductance. This capacitance exists because of stray fields between the conductor and its surroundings, e.g. the ground, anything metallic, or even the operators hand. Stray capacitances are shown schematically in Figure 5A.

The stray capacitance of the measurement system is determined according to Schwan's methodl by considering the measurement system as a lumped parameter network of a capacitor (I?=), representing the residuals in parallel with the sample admittance (Figure 6A). In the limit of low frequency (o - 0; effectively, frequencies below -lo7 Hz) it can be shown that the stray capacitance is given by

(20)

where


(21)

and Ci(H20) is the input capacitance with the measurement cell filled with water, C,(air) is the input capacitance with the measurement cell filled with air, Cair is the capacitance of the air sample in the cell, and ew is the permittivity of water.

Measurement Cell Losses--If the sample is presumed to be predominantly resistive (Figure 5B), then the presence of shunt capacitances (C,) due to stray fields will introduce time constants of magnitude t = RC,. The apparent admittance of the sample will therefore appear frequency dependent, even though the sample properties are frequency independent. The effect of the frequency dependent stray capacitance on the input capacitance will be minimal; however, the effect on the input conductance may be very high, particularly in the case of resistive samples, as leakage currents to ground effectively short-circuit the sample. Measurement cell losses become apparent as the frequency extends into the kilohertz range.

The extent of dielectric loss resulting from the coexistence of stray capacitance (via z = RC,) depends not only upon the environment of the cell (which affects the magnitude of C,) but also varies with the resistance of the sample (Rd. However, the stray fields will be independent of the sample if the field lines passing through the sample are restricted to a zone well within the boundaries of the electrode plates (Figure 7). This type of cell has been employed successfully by many authors up to 500 MHz.67,71-75 Alternatively, the measurement electrodes should be surrounded by a guard ring in order that the stray fields may be ~ont ro l led . ~~~

Measurement Techniques-The bulk electrical properties of the sample are determined by either measuring the ability of the material to pass alternating current (frequency domain DRS) or by investigating the transient flow of charge due to an applied step voltage (time domain DRS).

Frequency Domain DRS-An external alternating field is placed across the measurement cell and the relative phase shift (4) and magnitude of the resulting current (I,) measured with respect to the voltage reference (VJ. The magnitude of the admittance (IYl) is derived from (YI = IJV, and then resolved into the input conductance (Gi) and input susceptance (Si) according to

Gi = IYI cos 4 (23)

where

si = oc, Traditionally, low-frequency measurements (5 lo7 Hz) were

performed by manual bridge balancing techniques. However, these methods are now largely superseded and dielectric measurements in the frequency domain from the subhertz to lower megahertz region are rather carried out using either an impedance analyzer (frequency window from 1 to lo7 Hz) or a frequency response analyzer (from

Essentially, the measurement cell for low frequency DRS (< lo7 MHz) comprises a parallel plate capacitor, although there are many variations on the actual design of the cell. For low-frequency studies, parallel electrodes of variable spacing77 have been employed to account for electrode polarization artifacts (subhertz to -lo6 Hz; see Electrode Polariza- tion) whereas toward higher frequencies (> lo3 Hz) the sample must be enclosed well within the boundaries of the electrodes (Figure 7), so that the measurement cell losses do not depend on the sample resistance (see Measurement Cell Losses) and

to lo7 Hz).

front view of spacer (polystyrene or teflon)

sample cavlty

plan view of spacer sandwiched between stainless-steel electrodes

side view - inlet and over-llow ports for sample

\ stainless-steel eleclrcdes

Note : the assembly may be boned together a1 each corner. using nylon screws, and attached directly 10 the input terminals

of the measurement device.

Figure 7-Sample cell enclosing the material well within the boundaries of the electrodes. This cell design ensures that measurement cell losses (>lo3 Hz) are independent of sample properties since the field lines passing through the sample are confined between the plates of the measurement cell.

can therefore be corrected for. The sample conductance will otherwise appear to increase as the frequency is increased through the kilohertz range and above, even though the sample conductance is itself frequency independent.

Other cell designs (for radio frequency studies) employ pin- type electrodes attached directly to the binding posts of the measurement device so that the cell inductance is kept to a minimum.78 Pin-type electrodes have also been employed in low-frequency measurements but in four-terminal arrange- ments in order to reduce electrode polar i~at ion.~~ The reviews by S ~ h w a n l , ~ ~ provides comprehensive information on the importance of cell design to the accurate determination of dielectric properties.

High-frequency studies (from lo7 to 10l1 Hz) require a fundamentally different approach to the low-frequency methods and instead use measurements on transmission lines to probe the admittance properties of pharmaceutical samples. At high frequencies (> lo7 Hz) the sample/measurement system geometry becomes of comparable dimensions to that of the incident electromagnetic waves, and factors relating t o transmission line theories therefore becomes significant.

A transmission line is simply the medium between the source and some distant point, through which energy is transmitted. This can be either free space, in the case of radio waves emitted from a radio transmitter, or the cables, electrodes, and connectors linking the sample and measurement device in a network analysis test circuit. Transmission line effects arise from the sample having a different impedance to the wires and connectors in the circuit. A discontinuity therefore occurs in the electrical properties of the circuit, and reflected waves of both current and voltage travel from the discontinuity back to the source. There are then two waves on the line, an incident wave and a reflected wave, and the result is a standing wave on the line which is the algebraic sum of the two waves. These conditions apply equally to the current and the voltage waves on the line.

At high frequencies (i.e. short wavelength) and for cables of nominal length, the impedance along each cable will necessarily vary with distance. Under these conditions, the

1036 /Journal of Pharmaceutical Sciences Vol. 84, No. 9, September 1995

input impedance to the measurement device no longer matches the impedance of the sample. Considerable errors are therefore introduced when dealing with lengths of cable of comparable dimensions to the wavelength. When the wavelength is long, the impedance remains constant along the length of the cable and therefore the consideration of transmission line effects is unnecessary. Therefore, traditional low-frequency parallel plate methods for determining sample admittance soon become inadequate as the frequency extends into the megahertz range. However, by measuring the reflection of electromagnetic energy from and/or the transmission through a material, coupled with information on the geometry of the sample, the dielectric properties of the material can be satisfactorily determined. For a detailed explanation of the transmission line theory, see the work of C ~ n n o r ~ ~ and Baden Fuller,80 and for its applications to biological admittance measurement, refer to Schwans review on the determination of biological impedance.l

Time Domain DRS-In general, frequency domain methods rely on the sample being excited with a sinusoidal signal of constant frequency, the measurement being taken, and the measuring equipment moving on to the next frequency. The operation of time domain techniques is fundamentally different.s1-87 The sample is excited by a step or impulse response (an impulse response can be shown by Fourier analysis to contain all frequency components), and the reflection and/or transmission parameters of the response are measured. The basic premise of the method is that if the impedance changes between the sample holder and sample, then some of the incident energy is reflected back. The impedance discontinuity is brought about by differences between the electrical properties (such as the permittivity) of the sample holder and sample.

The capacitive element of the sample introduces time constants which manifest as an exponential increase or decay of voltage and current. Measurment of the bulk signal change (the step change) and the exponential change due to the capacitive charging allows the determination of the circuit parameters and sample admittance. The frequency dependence of the complex permittivity can be determined from the time domain response by applying a Fourier transform. This is most precise if known samples are also measured and used in the conversion.

With time domain DRS, errors are generally introduced at interfaces between transmission line types, such as cables and connectors. In general, the impedances of the cables and connectors will not be the same and there may be imperfec- tions in the dimensions of the test jighample holder, resulting in spurious reflections. These effects lead to artifacts in the response, which need to be removed by careful calibration techniques.

Determination of Intrinsic Electrical Properties from the Sample Admittance-The magnitude of the sample capacitance and sample resistance are dependent both on sample geometry and the intrinsic electrical properties (E , E, and a) of the material comprising the sample. The following equations detail how the intrinsic electrical properties of the material are derived from the sample capacitance and resistance.

The permittivity of free space (ev) and relative permittivity (E) are derived from the electrical capacitance of a vacuum (C,) and the sample (C,) according to the following (after correcting for measurement artifacts such as stray fields):

A D C v = -

A D c, = ,I -

(24)

(25)

where A is the cross sectional area of the sample and D is the thickness of the sample.

Equations 24 and 25 only apply in the case of linear electric fields and are formally equivalent to eqs 1 and 2, as the capacitance is defined as the charge stored per volt (qN) and the field strength is defined as volts per meter (VID).

The dielectric loss ( E ) is manifest as an in-phase admittance and can therefore be derived by modeling the lossy component as a resistor (R,) in parallel with the sample capacitance. As the two elements are in parallel, the total admittance (Y) is therefore given by

Y = iwC, f lIR, (26)

Y = iw(C, - ilwR,) (27)

Representing the total admittance in terms of a complex capacitance (C*) gives

iwC* = iw(C, - i/R,w)

C* = C, - UR,w

(28)

(29)

Dividing by C, (the capacitance of a vacuum) yields an expression in terms of the complex permittivity:

(30)

(31)

The cell constants, in the expressions for C, and R,, cancel and eq 31 reduces to

(32)

Comparing eqs 7 and 32 demonstrates that the dielectric loss (6) is essentially a function of the ac conductivity. However, if the dielectric loss is to be studied in isolation, then the contributions to the imaginary permittivity due to dc conduction (uo) must be subtracted. The true dielectric loss is therefore given by

(33)

The dc conductivity (uo) is derived from the limiting low frequency conductance of the sample (Go) according to

A G , = u - D (34)

Applications of DRS to Pharmaceutical Systems Liposomes and Membranes-There has been a great deal

of interest in liposomes as drug delivery systems both in the past and more recently with experiments on stealth liposomes. A successful liposomal drug delivery system entrapping the active drug will ideally be presented as a freeze-dried formulation so as to provide an acceptable shelf life. Mem- brane stabilization is a critical parameter in freeze-drying liposomes to avoid drug leakage. The process is therefore carried out in the presence of specific cryoprotectants, and often buffer solutions, in order to obtain a reproducibly stable product. The associated mechanisms of membrane cryosta- bilization need to be investigated, as the interactions that produce the optimum product are not fully understood.


( 5, BILAY ER SURFACE

t HYDROCARBONCORE t Figure 8-Rotational motion of the zwitterionic head group of phosphatidylcholine, describing a cone centered on the bilayer normal.

Cryoprotective stabilization could either be affected at the level of the head group or, though unlikely, the hydrocarbon chains, Certainly the stabilization of the membrane is likely to be reflected in the fluidity of the membrane during freezing. In the case of phosphatidylcholine (PC) liposomes, the fluidity of the membrane can be assessed directly by DRS as the membrane possesses a surface dipole-the phosphorylcholine head group. The mobility of this group provides a probe to assess the fluidity of the surface and may therefore give some insight into possible mechanisms of cryoprotectant action and the effect of drugs and salts on membrane properties.

The librational motion of the choline moiety of zwitterionic PC, around the phosphoryl axis, was first observed by Kaatze et al.,86 who commented on a small dielectric relaxation above 50 MHz for dimyristoylphosphatidylcholine in dilute solution. Likewise, Shepherd and BUldtE9 observed a similar dispersion for multilamellar sheets of phosphatidylcholine bilayers. The head groups are assumed to rotate in a manner describing a cone the axis of which lies normal to the membrane (Figure 8). The kinetics of reorientation depend on the environment of the zwitterion and physicochemical interactions with neighboring molecules. Relaxation times between 0.5 and 8 ns have been observed, corresponding to a frequency range

The frequency dependent nature of the dielectric response is described by the Debye equation and the intrinsic molecular relaxation time derived according to Powles (eq 15). The mobility of the cationic group is then determined from zi according tog4

of 20-300 M H z . ' ~ , ~ ~ , ~ ~ - ~ ~

t2 z. = - ' ukT (35)

where is the effective charge separation of the cationic choline group relative to the phosphoryl group and u is the reorientational mobility of the cationic choline group. The microscopic relaxation time constant therefore provides a measure of the fluidity of the interface. The effect of packing constraints on the rotational mobility of the zwitterionic head group of phosphatidylcholine is demonstrated by a number of studies that monitored changes in the relaxation time constant. These studies showed (i) a decrease in the relaxation time constant from 6.02 to 1.01 ns on increasing the temperature from 18 to 60 "Cg2 (The temperature dependence results from a reduction in the viscous damping of head group r o t a t i ~ n . ~ ~ ~ ~ ~ ~ ~ ~ ) ; (ii) a steplike decrease in the relaxation time constant on passing through the gel to liquid-crystalline phase t r a n s i t i ~ n ; ~ ~ . ~ ~ , ~ ~ (iii) that the relaxation time constant for micellar structures (5 = 1 ns), where the available surface area is greater than that in vesicle suspensions, is lower than the relaxation time constant for small vesicles (z = 2 ns),9l thus

demonstrating that the head group is more restricted in the vesicle compared with the micelle; (iv) that the relaxation time constant is lower for solutions of nonassociating zwitterions, such as o-phosphorylethanolamine (z = 0.24 ns, 25 "C) compared to both micellar (t = 1 ns) and vesicular associated zwitterions (z = 2 n ~ ) ; ~ l (v) a reduction in the relaxation time constant on incorporation of cholesterol into dipalmitoylphos- phatidylcholineg5 and egg lecithin bilayers,14 indicating less restricted motion than in the absence of cholesterol.

DRS is therefore capable of providing structural information on liposomal formulations, with the advantage that parameters such as membrane fluidity may be studied without having to include membrane probes as would be the case in studies by ESR.

The membrane activity of drugs can be readily demonstrated by DRS. A membrane is a nonconducting barrier in an otherwise conducting aqueous media and therefore behave like a capacitor that is being charged through the extracellular and intracellular resistances of the surrounding medium. Charging of the membrane is therefore frequency dependent with an exponential time constant of z = RC, where R and C are, respectively, a function of the membrane permittivity and the solution resistance either side of the membrane. Radio frequency DRS studies of the frequency dependent interfacial charging of biological membranes therefore provide a well- established means of deriving the electrical parameters of the membrane and hence characterize its physical properties.1,29,30,61,75 The membrane activity of drugs is demonstrated by changes in the membrane permittivity and intracellular conductivity. Investigation of the passive electrical properties of the red blood cell membrane (membrane conductivity and membrane permittivity) by radio frequency DRS has high- lighted changes to the transport properties of the erythrocyte membrane by the antitumor drug adriamycing6 and the membrane-active antibiotics gramicidin S , polymixin B, vali- nomycin, gramicidin D, amphoteracin B, nystatin, and non- a ~ t i n . ~ ~

At much lower frequencies (0.2-90 Hz) the interfacial charging of phosphatidylcholine membranes results from the electrical discontinuity between substructural regions of the membrane. Deconvolution of the frequency dependent dielectric response is achieved using a dielectric model for the membrane comprising three distinct regions characterized by a capacitor and resistor in parallel. These three regions loosely correspond to the polar head group region, the acetyl region (comprising the carboxyester part of PC), and the hydrophobic region.gs Defining the membrane structure according to the lumped-component electrical network has provided these authors with a means of (i) locating the position of cholesterol in the membraneg9 and (ii) studying the effect that spectroscopic probes have in perturbing membrane properties.'Oo Later Perez and WolfelO1 used the low- frequency interfacial response of the membrane to probe the location of estradiol in PC membranes. The authors used this approach to propose that estradiol locates partially in the polar head group region. Investigation of the effect of sterols on liposome properties is topical, in light of the relatively recent release of the first liposomal amphoteracin B formulation (Ambisome, Vestar).

Further DRS studies of membrane phospholipids have also yielded information on changes in membrane properties on associating with drugs and other solutes. Again, the zwitterionic head group of PC provides an inherent molecular probe for the investigation of the surface of the membrane and the associated water. An increase in the relaxation time constant for head group polarization, upon partitioning of small molecular weight hydrophobic solutes (i.e. tetraalkylammonium ions) into the membrane, indicates the particular sensitivity of DRS in the study of drug-membrane interactions (Table


Table 2-The Effect of 0.1 mM Tetraalkylammoniurn Ions on the Relaxation of Zwitterion Reorientation Polarization in a 10% wlv Suspension of Lecithin Liposomes in 0.1 mM KCI (20 "C, ref 14)

solute z,,, at 20 "C (ns) solute zm at 20 "C (ns) control 4.49 tetraethylammonium 5.68 tetramethylammonium 5.98 tetrapentylammonium 4.94

2). The extent of the increase in the time constant follows the hydrophilicity of the solute and the inverse of its size. Changes in the rotational mobility of this group were also demonstrated by Buchetg2 on incorporating gramicidin A into the membrane.

Dielectric spectroscopic studies of biological and artificial membranes therefore provide a means of studying drug- mediated and penetration-enhancer-mediated effects on membrane properties.

Microcapsules-Drug and ion release from microcapsule suspensions have been investigated by DRS.37J02-104 Micro- capsules by their nature are heterogeneous structures with variations in the thickness and porosity of the capsule wall and polydispersity in capsule size. Drug release varies from capsule to capsule, resulting in an overall distribution of release rates from the suspension. Sekine et al.lo4 have analyzed the dielectric response and developed several formulae relating the dielectric response to a distributed drug release rate. In essence, the microcapsule suspension is treated in the same way as a suspension of cells, i.e. a nonconducting shell entrapping a conductive internal phase and suspended in a conductive external phase. The application of an electromagnetic field results in the shell of the microcapsule charging through the external and internal resistances. Polarization of the shell is therefore time dependent and characterized, very mcuh like the polarization of cell, by an RC time constant. The frequency dependent dielectric response may be analyzed to yield the information on the electrical properties of each phase comprising the microcapsule suspension, i.e., the resistances and capacitances of the internal phase, microcapsule wall, and the external phase. Drug or ion release is then followed by monitoring changes in the dielectric response with time.

In a less comprehensive study of dielectric properties, it has also been shown that the permittivity of microcapsule com- pacts at a fixed frequency may be used to study the aging of micr~capsules .~~~

Emulsions and Microemulsions-DRS is particularly useful for deriving structural information on disperse systems that contain a conductive disperse phase, as is the case with water-in-oil (wlo) emulsions and swollen reverse micelles that contain conductive internal aqueous phases. The noninvasive nature of DRS and the fact that the sample does not require dilution or other means of preparation gives this technique distinct advantages over other methods, such as particle sizing and electron microscopy.

The frequency dependent (-103-107 Hz) permittivity of wlo emulsions with a conductive internal phase can be readily explained by interfacial polarization, as predicted by the various dielectric mixture equations (e.g. Maxwell-Wagner,26 Bruggeman, lo6 and Hanai-Br~ggernan~',~~,~~~ mixture formulae). These formulae have been used to deduce structural information on both wlo and olw microemulsions108 and microernuls i~ns.~~~ Skodvin et a1.ll0 used these mixture formulae to determine the location of drugs (pilocarpine and chloramphenicol) in a variety of oil-in-water emulsions, in addition to assessing the subsequent structural changes that arise from such interactions. High-frequency studies (lo6- 10l1 Hz) have also been employed to find the limiting high- frequency permittivity of macroemulsions from which the

emulsion type and water content may be determined.los It is suggested that this technique may provide the means for on- line process monitoring. Temperature dependent structural transitions in w/o microemulsions have also been elucidated by a combined X-ray diffraction analysis and high-frequency (106-10s Hz) DRS study of the water component in these systems.ll' The dielectric response as a function of temperature indicated the loss of bound water from the interlamellar spaces of the gel state as the temperature was raised, followed by a transition to a swollen micellar phase (i.e., microemul- sion) entrapping water with an extended hydrogen-bonded network.

Cametti et a1.112 observed a high-frequency relaxation in an AOT- [sodium bis(2-ethylhexyl)sulfosuccinate] stabilized wlo emulsion which they attributed to the migration of the anionic surfactant over a restricted distance. Restricted diffusion imparts a fluctuating dipole moment to the emulsion droplet which provides an inherent probe to determine the fluidity of the interface. DRS therefore has applications in the investigation of oil-water interfaces of pharmaceutical disperse systems. This technology also has applications for lecithin-stabilized microemulsions. Lecithin possesses a polarizable zwitterionic head group, and therefore, interfaces incorporating these molecules will also exhibit a surface polarizability. These interfacial dipoles act as inherent molecular probes that can yield information on surface packing and surface viscosity. Other methods for investigating interfacial packing, such as Langmuir film-balance studies, generally investigate the waterlair interface, and therefore, the information derived does not necessarily reflect the physical geometry and structure of the emulsion interface. The dielectric response of the lecithin head group, however, provides a novel in situ approach for studying lecithin- stabilized oil-water interfaces and a direct method for investigating factors affecting emulsion stability. In addition, the speed by which dielectric measurements are taken places this technique in a strong position for adoption as a quality control procedure for the manufacture of emulsions.

Micelles and Liquid Crystals-DRS has provided information on the interactions between inverse micelles of the surfactant AOT in liquid a l k a n e ~ . l l ~ J ~ ~ The relaxation observed between lo3 and lo7 Hz was attributed a single particle rotational diffusion constant and characterized by the Cole- Cole function. At volume fractions approaching 0.3, the Cole- Cole distribution function was found to be significantly less than 1, suggesting the existence of strong interactions between micelles at low volume fractions. At higher volume fractions the authors observed another, much slower process, which they attribute to further more extensive interactions between clusters of micelles and a resultant glass formation.

As with studies on PC membranes, a DRS investigation of the relaxation time constant of other zwitterionic amphipiles may also be employed as a means of investigating surface packing in association colloids and liquid crystals. For example, the reorientation of the choline moiety of phosphatidylcholine about the phosphoryl axis has been studied for L-a- lysolecithin and was shown to increase from 1.54 ns in the micellar phase to 2.22 ns in the lamellar phase.g2

Lims and Franses115 employed DRS (among a number of other techniques) to investigate aqueous dispersions of sodium 44 1'-heptylnony1)benzenesulfonate (SHBS) as vesicles and the much larger liquid crystallites. A dielectric relaxation observed below lo5 Hz (after satisfactorily correcting for electrode polarization using the variable electrode spacing method) was attributed to migration of counterions at the surface of the disperse phase (Schwan's a-dispersion). The authors characterized the response using the Cole-Cole function in order to extract the low-frequency limiting permittivity (and in some cases the relaxation frequency). The dielectric


response provided information on the polydispersity in size, as the relaxation behavior is strongly dependent on particle size. Changes in the relaxation behavior with time for sonicated dispersions was used to indicate the instability of the smaller vesicles and the reversion to back to liquid crystallites.

Particle Sizing and Volume Fraction Determination of Suspensions-Particle sizing by DRS is possible because the central relaxation frequency of a suspension of charged particles (resulting from the polarization of the counterion layer, the so-called a-relaxation) is proportional to the reciprocal of the square of the particle diameter. It is claimed by Sauer116 that the DRS method is rapid with good agreement up to 10% solids. Particle sizing of microcapsule suspensions is also a possibility by this method.lI7

The volume fraction of cells in suspension has also been determined by a dielectric approach, since the magnitude of the dielectric increment due to the p-relaxation (associated with charging of the membrane capacitance through the extra- and intracellular resistances) is given by

(36)

Packed cell volumes up to 0.6 volume fraction have been determined satisfactorily using this method. Dielectric measurements at a fixed frequency have also been used to determine the amount of biomass in suspensions of cultivated yeast.'l*

Alternatively, the amount of material in suspension may be determined simply from the limiting low-frequency/dc conductivity, according to the equation by BruggemanloG that describes the relationship between conductivity and the volume fraction of nonconducting particles, i.e., cells. This approach has been used to determine biomass concentrations in plant cell cultures.11s The technique has applications for the on-line determination of biomass during cell growth.

Gels and Polymers-Hill and Dissado developed a model for the dielectric relaxation based on the cooperative motion of dipoles or bound charges as a cluster type response, deriving the indices n and (1 - m), which reflect, respectively, the structural order of the individual cluster and the extent to which clusters polarize independent of each other.120 They report on the use of their approach as a means of deducing complex structural information about heterogeneous gels of cetostearyl alcoh~l/cetrimide/water.~~ Craig et a1.121 have also used this approach to investigate the structure of solid dispersions of drug in polyethylene glycols of varying molecular weight.

Hydration and Water Content of Pharmaceutical Materials-DRS has contributed toward an understanding of the role of water in areas such as protein hydration and enzyme activity,21,22J22-126 food technology,16J7 lyophiliza- t i ~ n , l ~ ~ J ~ * and polymer hydration.l0 In the area of pharmaceutical technology, the states of water are extremely important parameters with respect to in-use performance characteristics of pharmaceutical materials. However, the pharmaceutical industry is still very limited as to the number of methods it employs for investigating the properties and location of water in pharmaceutically relevant materials. The lack of pharmaceutical applications for DRS in the study of hydration is therefore surprising, given the vast number of dielectric studies that have been carried out on the properties of water and the ease with which these measurements are taken.

One major advantage of DRS, over many other techniques used in the study of water (e.g. NMR and relative humidity), is the direct ability of this technique to differentiate between

the rotational mobility of water of hydration and the rotational mobility of free water. This attribute is of particular importance as the translational and rotational mobility of water determines its availability,129 which in turn affects factors such as the physical, chemical, and microbiological stability of the material.

In the dielectric analysis of water, the molecular dipoles of "free" (or bulk) water relax at frequencies close to 2 x lolo Hz (z, = 8.2 ps a t 25 "C), whereas the molecular dipoles of hydration water relax at frequencies intermediate between bulk water and that of ice vo - lo3 HZ).~ It is assumed that the activation energy for the principle relaxation of bulk water is equivalent 1,o the energy required to break a single hydrogen bond, and therefore, the molecular relaxation time for water is equivalent to the period water must wait until thermal fluctuations result in a molecule being bonded to its neighbor by a lone hydrogen bond.130

For simple solutions of low molecular weight solutes, the effect of solute-solvent interactions on the solvent relaxation time is not dramatic. For example the relaxation times of hydration water in solutions of electrolytes, amino acids, and polyelectrolytes are greater than that of bulk water by a factor of 2-3.126J31,L32 The hydration of small molecular weight organic compounds also induces a positive shift in the relaxation time of water,132-134 which is proposed to support the ~ 1 a t h r a t e l ~ ~ J ~ ~ concept of hydrophobic hydration. The dielectric behavior of proteins and DNA, however, shows a further and more dramatic effect in relation to the relaxation of hydration water. Biomaterials exhibit an additional relaxation time which is nearly always 1 ns (i.e., fo - lo8 Hz).137-141 These materials therefore show two categories of hydration water, one that is strongly associated with the material (internal water) and another that is weakly associated (surface or peripheral water).lZ6 As a result of the discreteness of the dispersion, the amount of internal water may be calculated directly from the decrement in permittivity associated with the relaxation of internal water in the megahertz range.137J4" However, the similarity between the relaxation times of surface water and bulk water means that each relaxation is virtually superimposed on the other. It is possible to deconvolute closely spaced relaxation^'^^ (i.e., separate the contributes from to each relaxation), but the information derived depends on the model used to characterize the system. For example, whatever the model, assumptions have to be made regarding the solute shape, the permittivity of hydration water, and the number of time constants.

A number of indirect approaches for calculating hydration numbers from dielectric data are also available: (i) Hydration numbers for surface water may be derived from the analysis of the limiting permittivity a t frequencies below which bulk water relaxes but above which internal water relaxes. Al- though surface water is still capable of rapid motions about the direction of the dipole vector, it is the direction of this vector that is significantly affected and thus the magnitude of the solvent permittivity is altered dramatically. The contributions to the low-frequency limiting permittivity from surface water and free water may be determined using various mixture formulae and the assumption that surface water exhibits a macroscopic permittivity close to that of ice. Bone123 used this approach to indicate the existence of the two categories of hydration water. (ii) Alternatively, the relaxation time for the reorientation polarization of the solute is determined and the hydrodynamic radius derived from eq 14 (having assumed that the molecule is approximately spherical); the size of the hydration sphere and the total quantity of hydration water is then approximated from the crystal- lographic radius and the hydrodynamic radius. 143~144

DRS has a number of applications for study of hydration of pharmaceutical powders and for the investigation of the

1040 / Journal of Pharmaceutical Sciences Vo/. 84, No. 9, September 1995

content and states of water during processes such as wet granulation and lyophilization. Water content necessarily affects the bulk characteristics of the material, for example the rheology and compressibility, and therefore dielectric methods could be applied in the quality control of raw materials to ensure reproducible processing characteristics. In the wet granulation of powders intended for tableting, the distribution of water across the wet mass can be critical to the ease with which the material undergoes subsequent processing. For example, granules of varying water content may be formed that will either dry, to give a material of variable compressibility, or may fail to spheronize. Dielectric measurements across the powder bed may therefore be used as an in-process tool for determining the critical end point for mixing. Nhuan et al.145 used dielectric methods to study the relaxation time and the activation energies and entropies for water adsorbed to Fractosil2500 and 5000 and Avicell PH- 101. They investigated the adsorption of water covering the surface area of the material by a factor of between 0.5 and 2.5 and witnessed a characteristic threshold in each parameter at a level of hydration equivalent to the first layer of adsorbed water. Studies of hydration dependent dielectric properties (at frequencies between lo3 and lo6 Hz) have been used to highlight the existence of two populations of water on increasing the water content of nearly dry systems.21x22 The dielectric relaxation studied by these workers is characterized by a pH dependence and an isotope effect on switching between deuterated water and normal water. The relationship sug- gests that the effect is due to proton conductivity at the surface of a single macromolecule. Conduction therefore occurs over limited paths and thus results in localized charge separations (i.e., polarization) in the material. The frequency dependence of the process relates to the diffusion path length and the mobility of the diffusing protons. Sakamoto et al.146 developed methods for determining the water content of nearly dry materials at subzero temperatures, by time domain DRS measurements. Their work on silica gel, as a model substrate, showed a multidispersive dielectric relaxation of water associated with the silica gel, that was centred on frequencies distributed between 100 MHz and several gigahertz. Saka- mot0 et al. analyzed their data by differentiating the water into two groups, water which is associated strongly with the gel and water which is only weakly associated. The temperature dependence of the dielectric response indicated that there was a transition-like change of hydration structure around 13 "C that is associated with an increase in the amount of loosely bound water on increasing the temperature. This work highlights potential applications for DRS in the control and optimization of freeze-drying processes.

DRS has also been applied to the study of hydration in biological tissues139 and has been employed in uiuo to directly analyze skin hydration. The technique therefore has potential use for measurements of hydration following the application of occlusive patches to the skin or following the use of penetration enhancers. In addition, the response of the skin to microwave energy is particularly useful for developing and improving methods for hyperthermia therapy.147 Hydration- mediated effects on dielectric properties are also used to probe the role of water in biomaterials. Bone123 studied a medium- frequency relaxation of chymotrypsin (& = 12 MHz) and proposed that the dielectric relaxation was associated with the structural flexibility of the protein and the enhanced polarization of charged protein groups with increasing water content. The dielectric method is therefore suitable for determining the effects of water as a plasticizer in polymeric materials. Studies on hydration water also provide a sensitive probe for phase transition behavior in biological materials. Dielectric analysis of hydrated phospholipid bilayers on passing from the liquid crystal to gel state indicates that water is

more strongly associated in the fluidlike state compared to the gel-state and that head group hydration therefore has an important role in defining the phase transition temperature of the lipid.14*

Protein and Polypeptide Stability in Aqueous Solu- tion-Investigations on the structural conformation of poly- (amino acids) in solution, is by no means a novel use for DRS. However, the development of applications for the technique in this area of pharmaceutical sciences is yet to occur.

Proteins and polypeptides are made up of zwitterionic amino acids and therefore these molecules generally possess a large dipole moment. For polymers, the overall dipole is the vector sum of the individual dipole moments of the repeating polar units. However, in the case of poly(amino acids), the repeating units are highly charged and thus the overall dielectric properties of the molecule are not only associated with dipolar reorientation of the molecule, as a whole, but are also associated with ionic polarization, as the molecule essentially behaves like a polyelectrolyte in solution (see the section on Ionic P o l a r i ~ a t i o n ) . ~ ~ ~ The way in which polypeptides and proteins interact with water is of fundamental importance to the conformational structure/stability of the molecule and associated biological activity. If a poly- (amino acid) is denatured (either on removal of water or following the addition of perturbing agents), then the conformation of the structure changes and a much higher dipole moment results. The dielectric response of the molecule in solution will thus provide information on the structural stability of the molecule exposed to different environments. For example, polyglutamic acid has two pH-dependent con- figurations, an a-helix and a random coil. Takashimalso showed that the coil has a very large dielectric response, whereas the helical form exhibits a much smaller dielectric response. These results are explained in terms of dipole reorientation and counterion polarization of each form in aqueous solution.

I t is evident that there is a close relationship between dielectric properties and the structure of the molecule, which may be exploited in the study of drug stability. A distinct advantage of DRS, over many other techniques, is that the broad-spectrum nature of DRS enables an integrated study of both molecular structure (i.e. conformational stability) and water activity (see the section on Hydration and Water Content of Pharmaceutical Materials). The recent development of peptide drugs and the increasing importance of biotechnology and lyophilization to the pharmaceutical sciences should therefore pave the way for many new applications for DRS.

Summary

DRS is a powerful and versatile technique with many fundamentally different applications that stem from the broad-band frequency window and the variety of polarization processes that can occur over the dielectric part of the electromagnetic spectrum. This review is selective in describing only a limited number of applications for this technique. A selective approach was necessary, in order to accomodate the detail required to understand the nature of the processes being investigated. The applications part of this paper is therefore by no means comprehensive, and there is much work that could not be incorporated. In particular, the review concentrates mainly on aqueous systems such as colloids, suspensions, aqueous solutions, and hydrated powders. How- ever, the dielectric literature also contains a great deal of published work on nonaqueous systems such as investigations on the structural characteristics of polymers. Unfortunately,


this rather large topic, though an extremely important application for DRS, was nevertheless considered outside the scope of this review. Many of these studies differ in approach from those that have been described in this paper, in that the frequency of the applied field is fixed a t a range of discrete frequencies and the temperature is changed, rather than fixing the temperature and altering the frequency of the applied field, as with most spectroscopic techniques. The dielectric response is often used in conjunction with more traditional methods such as rheology and thermal analysis, and therefore this topic could easily form the subject of a separate review. See books by Hedvig'O and Blythe'' for an introduction to the study of polymers by DRS.

Until recently, i t has been necessary for scientists working in the field of DRS to design and establish their own calibrated, measurement system. However, a number of companies are now producing integrated dielectric analysis systems, based on commercial impedance analysers (or frequency response analysers), with a calibrated measurement cell, software, and temperature control. This recent development may signal the start of a new era for dielectric research, with many applications being developed specifically for the pharmaceutical industry. Such an expansion in dielectric analysis should be welcomed by the industry as it seeks out new technologies to keep pace with the ever more stringent regulations being placed on pharmaceutical products by the regulatory authorities. However, the danger for DRS at present is that the current price of this instrumentation could realistically stifle any potential growth in a technique which is still novel to most pharmaceutical scientists.

GLOSSARY

area (m2) Cole-Cole distribution function reflecting the di-

vergence from Debye-like behavior (dimensionless)

Reddish distribution function (dimensionless) capacitance of sample after correcting for measure-

capacitance of air after correcting for measurement

total capacitance input to the measurement device

complex capacitance (F) dielectric displacement (C m-2) distancenength (m) relative permittivity (real permittivity or dielectric

constant) imaginary permittivity/dielectric loss function (di-

mensionless) complex permittivity (equal to the vector sum of c'

and E", dimensionless) dcAimiting low frequency permittivity at frequen-

cies below which relaxation occurs (dimensionless)

ment residuals (F)

residuals (F)

(F)

permittivity of free space (8.854 x high frequency limiting permittivity at frequencies

above which relaxation occurs (dimensionless) increment in permittivity of a material resulting

from polarization (dimensionless) macroscopic field strength (V m-l) frequency of the applied oscillating field (Hz) relaxation frequency (Hz)

F m-I)

GI input conductance to the measurement device, i.e. the conductance of the sample and measurement system ( S or Q-l)

GO 11

k Boltzmann constant (1.38 x J K-I)

dcAimiting low frequency conductivity (S or st-') microviscosity in the neighborhood of a dipole (N S

m-2)

mean rate coefficient for macroscopic exponential polarization (s-l)

KP

L inductance (H) P

4 electrical dharge (C) R, resistance of sample after correct

Dielectric relaxation spectroscopy and some applications in the pharmaceutical sciences

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