Dielectric Theory of Concentrated

Suspensions of

Shell-Spheres in Particular Reference to the

Analysis of Biological Cell Suspensions

Tetsuya HANAI,Koji ASAMIa, nd Naokazu KorzuMi* ReceiveJdune 6, 1979

A theoreticalformulafor interfacialpolarizationis presentedwith whichto expressthe dielectric behaviorof concentratedsuspensionosf sphericalparticlescoveredwith a shellphase. The formula is of a compoundform of Wagner'sand Bruggeman'sequations. A computingschemeis givento carryout numericalcalculationsofthe formulaas a functionofcomplexvariables. Bythe use ofthe computingscheme,numericalcalculationsof the formulawereperformedwith a set ofphaseparameterswhich are in conformitywith biologicalsuspensionsof conductingspherescoveredwith a very thin and nonconductingmembrane. The resultingcomplexplane plots of the complexpermittivity showedcharacteristicprofilesdifferentfromcirculararcsaswellasfromsemicirclese,speciallyat higher concentrations.The procedureof fittingthe theoreticalcurveto the observedrelativepermittivities is proposedso that the membranecapacitanceand the relativepermittivityand the conductivityof the inner phasemaybe evaluated. An exampleofthe curve-fittingis shownwith the dielectricdata observedfor an erythrocytesuspension. KEY WORDS : Interfacial polarization/ Membrane capacitance/ Membrane permittivity / Inner phase conductivity / Plasma phase conductivity/ Erythrocyte suspension/ I. INTRODUCTION It is known that biological suspensions of erythrocytes, bacterial cells, and yeasts show a dielectric relaxation due to the interfacial polarization.1`9) From a dielectric point of view, such suspensions are in a triphasic structure: the conducting cytoplasms covered with the poorly conducting shell phase of lipidic membranes are dispersed in a conducting continuous medium. An elaborate dielectric theory of interfacial polarization for such triphasic systems in concentric structure was proposed by Pauly and Schwan,10> and made it possible to evaluate membrane capacitance, relative permittivity (dielectric constant) and conductivity of the inner plasma-phase from the observed data on dielectric relaxations of biological cell suspensions. The general equation of this theory was derived from an analysis of quasi-electrostatic field, and has a functional form which is also expressed with repeated application of Wagner's equation proposed for diphasic suspensions: the first is for a sphere composed of the spherical inner phase and the concentric shell phase, and the second for the sphere and the outer medium. For diphasic systems such as oil-in-water and water-in-oil emulsions, it was * Th#1#- , aupl, J.AA-: Laboratoryof DielectricsI,nstitute for ChemicalResearch,Kyoto University,Uji, Kyoto. (297)

T. HANAIK, . ASAMaIn, d N. KoIZUMI reported by many workers11-13)that Wagner's equation is in poor agreement with experiments especially at higher concentrations of the suspending particles, and that Bruggeman's equation is satisfactory for representing observed data. The expressions in Wagner's type, therefore, might be insufficient quantitatively also for the theoretical treatment of the triphasic system proposed by Pauly and Schwan especially at higher concentrations of the suspending particles. In this paper a dielectric theory of interfacial polarization is proposed to express the dielectric behavior of concentrated suspensions of spherical particles covered with a shell phase. Since the theoretical expression derived is of a complicated form including complex variables, a computing scheme is presented to obtain the numerical solutions. Some features of the theory are pointed out regarding the type of dielectric relaxation. Finally the procedure of curve-fitting is described with which to evaluate the membrane capacitance of the shell phase, the relative permittivity and the conductivity of the inner phase from observed data.

II. SYMBOLS

s relative permittivity or dielectric constant electrical conductivity Ev permittivity of free space s* complex relative permittivity or complex dielectric constant Es* equivalent complex relative permittivity for a shell-sphere

f

measuring frequency

fo characteristic frequency at which the loss factor shows the maximum D diameter of inner phase d thickness of shell phase

O volume fraction of spheres with a shell in suspension

CM specific membrane capacitance for shell phase.

Subscripts

a outer phase

s

shell phase

i

inner phase

1 limiting value at low frequencies

h limiting value at high frequencies

III. THEORY OF INTERFACIALPOLARIZATIONFOR CONCENTRATED SUSPENSIONOF SPHERESCOVEREDWITH A SHELLPHASE A problem considered here is to derive a complex relative permittivity s* of a concentrated suspension that spheres (the complex relative permittivity si*, the diameter D) covered with a shell phase (es*, the thickness d) are dispersed in a continuous medium (sa*) with a high volume fraction 0, as shown in Fig. 1-A. According to a consideration carried out first by Maxwel114>for quasi-electrostatic field, the equivalent complex relative permittivity ES*(Fig. 1-C) of a diphasic system in concentric structure shown in Fig. 1-B is given by (298)

DielectricTheory of ConcentratedSuspensionosf Shell-Spheres

(A)(B)(C)(D) Fig. 1. Schematicdiagramfor a theoreticalmodel of a concentratedsuspension of shell-spheres.(A) Concentratedsuspensionof spheres(se) coveredwith a shell phase (s,*) dispersedin a continuousmedium (Ea*). (B) Shell structureof a suspendingparticle. (C) A homogeneousphere(@s*a)s an equivalentof the shell sphereshownin (B). (D) Concentratedsuspension of particles(ss*)dispersedin a continuousmedium(s.,*)as an equivalent of the suspensionshownin (A).

=Es*22eEss**++esi*i*-+2((es,s**----seii**))[[DDII((DD++22dd))]]33(1) irrespective of ea of the outer phase. Next an expression must be derived for the relative permittivity (E*)of a disperse system shown in Fig. 1-D, where the spheres (Es*) are suspended in a continuous medium (ea*) with a high volume fraction 0. For diphasic systems such as emulsions, it was found that Wagner's equation15) does not hold in higher concentrations. Hence Hanai16,17)derived an equation in Bruggeman's type extended to complex numbers by means of successive applications of Wagner's equation to the infinitesimally increasing processes in concentration of the disperse phase. In a similar manner, it can readily be shown that complex relative permittivity e* of the present suspension illustrated in Fig. 1-D is given by 1 E*--Es* Ea*V/3 1-0ea*--Es*\5* /=1.(2) Thus the complex relative permittivity e* of the concentrated suspensions can be expressed as a function of ea*, es*, ei*, D, d, and 0, where 6-6--2i fr!v,(3) Ed*--Ea'27r~fўE(4)

Cs--* E-jS27cf,E(5,,)

Si*_ si

12

Xi 71Ev(6)

and

j=/--i.

IV. COMPUTEREXPERIMENTSON NUMERICAL SOLUTIONSOF THE EQUATION

In the present study numerical calculations were performed with a Hewlett-- Packard Model 9810A Programmable Calculator. Since Eq. (2) has the compli- (299)

v

)

T. HANAIK, . ASAMaIn, d N. KoizuMC cated form including a cubic root of the complex variable, the numerical solutions were obtained by the following computing scheme which is within the reach of the present calculator. (a) By the use of subroutine programmes of addition, subtraction, multiplication, and division for complex numbers, E,*of Eq. (1) can readily be calculated numerically provided that the phase parameters ss*, Ei*, D, and d are given. (b) The values of Es* thus obtained are substituted for Eq. (2) to carry out the subsequent calculation for E*. (c) By cubing the both sides of Eq. (2), we have a cubic equation with respect to s* as s*3-3sg*E*2C+(f03-g1s)*(E2+a*--Eg*)]31.E*--Ea*3=O.(7) 11Ed*JJ The coefficients of the cubic Eq. (7) can be determined by substituting Ea*,gs*, and 0. (d) By the u§e of a computer subroutine programme "Root-finder of cubic equation with complex numbers", we obtain three roots of Eq. (7) designated by 63*,33*,and 63*. (e) Among the three roots si*, s2*, and 33*,we have to choose only one solution satisfying Eq. (2) by means of the following criterion. Since Eq. (7) is derived by cubing Eq. (2), respective substitutions of the three roots si*, s2*, and 63* into a function F(E--,k-)101 EE*a----*gssE'*K\16//E*a*)1(/38) are to give F(Ei*)=1+0-=exP (0J), F(E2*)_-- =1+ex2p3(j3'j)(1l(09)) and F(E=3--*) 2--j=-e--x2p3(--.7)3(~11) respectively. Here the exponent 1/3 in Eq. (8) denotes a principal value of cubic roots. When each of the three roots is thus substituted into F(s*) given by Eq. (8), the roots which lead to either Eq. (10) or Eq. (11) should be ruled out, and only one root giving Eq. (9) is adopted as the solution of Eq. (2). (f) The only root is regarded as E* of Eq. (2), giving the values of s, lc, and s" for the concentrated disperse system. V. THE COMPLEXPERMITTIVITYPROFILE OF THE THEORY A general equation of Pauly and Schwan's theory") was characterized by two relaxation times. Under the conditions d«D, ,s<

'YcINot1---0--_00-

\

0.20.3\0.4.\0.5

\0.1

0

1t

0.6 0.70.80.9

0100020003000

4000

6' Fig. 2. Complexplane plots of complexrelative permittivityfor suspensions

of shell-spherescalculatedfrom Eq. (2) (solid curves). The dashed

curves are semicircles.Phase parametersused: so,=80, ira=ei=2.5

mScm-1,si=50, ss=6.5, fcs=0mScm-1,d=50 A, D=3.8 (im.

using a set of phase parameters relevant to biological cell suspensions. Some examples of the results are depicted in Fig. 2. The profiles show marked deviations from circular arcs as well as from semicircles. At low concentrations (0<0.3) the profiles seem to be close to semicircles, the characteristics being the same as the results of Pauly and Schwan's theory. At medium concentrations (0=0.3- '0.5), the profiles may be approximated by circular arcs proposed by Cole and Cole,19>the examples being found in various biological suspensions.2,20'25>At high concentrations (0=0.6-0.8), the profiles are seen to show remarkable deviations from circular arcs. At extremely high concentrations (0= 0.8-0.9), the profiles show very peculiar patterns of which the lower frequency part might be simulated by another relaxation process. As a matter of fact, such high concentrations (0=0.8-0.9) exceed a state for the close-packed structure with uniform spheres, the examples being unavailable in biological suspensions.

VI. PROCEDURETO DETERMINETHE PHASEPARAMETERS BY CURVE-FITTINGBASEDON THE EQUATION Owing to the complicated functional form of Eq. (2), it is impossible to calculate straightforwards the phase parameters such as 0, ss, Si, and in by using dielectric parameters El, en,,ICI,and fo observed. The determination of the phase parameters, however, is possible by fitting the theoretical curve of Eq. (2) to the observed data. Prior to the curve-fitting, it is advisable to make a list of the effects of changing one of these phase parameters on the change in the dielectric parameters predictable from Eq. (2). The procedure is the same as that discussed1s)previously for Pauly and Schwan's theory. The numerical considerations are hereafter restricted to the case of ,rs<10-4 x ira or preferably Ks=0, which is pertinent to biological suspensions. The calculation was made, on a reference state specified by a set of phase parameters relevant to biological suspensions, by changing one of the phase parameters in Eq. (2). The relative variations in the dielectric parameters from those for the (301)

_

·

T. HANAIK, . ASAMarn, d N. Koizunu

Table I. Responseof CharacteristicDielectricParametersto IndividualChangesin Phase Parameters

Caseparameter PhaseVariation [% change]ICIeteh,fo

fromDviaellueecstrfiocrptahreamreefteerrencestatea),%

I0

[+10%]--6.4+

IIes

[-50%]0.00

III

ei [-50%]0.000.00-11+12

IVsi

[-50%]0.000.000.00

3.9-- --33--

1.3-- 7.1+58

1.2 --41

a) The referencestate is specified:ea=ei=80, ,c5=,i=15mS cm-1,es=3, rs=0---10-4Xee, d=50 A, D=0.5 pm, and 0=0.3.

Variation=o`neofphaseparameVt(eaVrlsua/elufoerbythEeqr.(e2fe)areftnecrecshftoaantregthinegrleVfearleunecestate X100

reference state are listed in Table I. The general feature found in the Table is very similar to the case of Pauly and Schwan's theory in that all six terms in the lowerleft part of the Table are zero. Thus the procedure to determine the phase

parameters can be stated as follows:

Step 1 To put tentatively si=ea,

es=3, and Ks=0, provided that sa and

tcc,are given from direct measurements of the continuous medium.

Step 2 To find a proper value of 0 so that the calculated value of ,r5may fit in with the observed value of ,c1. Step 3 To find s, so that the calculated si may fit in with the observed sr. Step 4 To find 6i so that the calculated eh may fit, in with the observed sh,. Step 5 To find ri so that the calculated fo may fit in with the observed fo. If the fitting to si and eh,turned out insufficient after Step 5 was carried out, Steps3, 4, and 5 should be repeated. The numerical values presented in Table I is effective for this reference state. Strictly speaking, such a survey table must be prepared on a reference state pertinent to the respective cases, though the general features in Table I necessary for the curve-fitting procedure is in fact subjected to no alteration.

VII. EXAMPLE OF THE APPLICATION An example of the application of the procedure proposed is shown with a biological suspension. A bovine erythrocyte suspension was prepared by washing the blood thrice with a saline solution after addition of heparin, and finally being suspended in a 50 mM NaC1 solution containing 250 mM sucrose for adjusting the tonicity. The measured relative permittivities are shown in Fig. 3. The respective parameters subjected to Steps 1 to 5 are summarized stepwise in Table II. The associated frequency profiles of the relative permittivity are illustrated in Fig. 3. From this curve-fitting the following phase parameters were evaluated: 0=0.540, ss=4.39, si=69.0 and ri=2.84 mS cm-1. Hence the membrane capacitance was calculated to be CM=0.777 pF cm-2, from a relation c Es(l-----------+d/D~2'(ll~ The erythrocytes measured are, to speak strictly, in a form of depressed sphere (302)

d

Dielectric Theory of Concentrated Suspensions of Shell-Spheres

3000

I--------I-------I-I-I-I----------------------------------I-I-I-----------------------------------------Curve A

2000-- i u 1000--

C \. E, b\\ C B

00

.01

0.11.010

100

Fig. 3.

Frequency

.f/MHz

Frequency dependence of relative permittivity and conductivity for a

bovine erythrocyte suspension and the procedure of curve-fitting, The

hollow circles (-0--) are the measured values. The curves A,

B, C, D, and E correspond to Steps 1, 2, 3, 4, and 5 in Table II,

respectively. The observed values for the outer medium are: ea=77.1,

ra=4.346 mS cm-1.

Table II. Variations of Dielectric Parameters Associated with the Variation of the Phase Parameters Subjected to Each Fitting Step

Curve-fitting procedure

Curve in Fig. 3.

Phase parameterDielectric

parameter

0Ei

,,fiiza) es Si

mS cm-1 mS cm-1 eleh.

Observed value

1.356 2176

71.5 1.63

Step lb) Step 2B Step 3C Step 4D Step 5E

A

0.6

5.00 400

4.3461.099

0.540 5.00 400

4.3461.356

0.540 4.39 400

4.3461.356

0.540 4.39

69.0 4.3461.356

0.540 4.39

69.0 2.8401.356

2681 2475 2176 2176 2176

197

1.79

180

1.83

178 2.07

71.5 2.23

71.5 1.63

a) Since fo is difficult to be assessed in observed data, the fitting was carried out with fin which is the frequency for a half-value point of the entire dielectric dispersion. At lower concentrations (0<0.6), Eq. (2) shows no discernible difference between fo and fin. b) Values of the phase parameters at the starting state (Step 1) are: ea=77.1, Ca= tci=4.346 mS cm-1, es=5, r,=0 mS cm-1, d=50 A, D=4.75 pm, 0=0.6, and ei=400. Owing to exaggerating the difference between Curves A and D in Fig. 3, an extraordinarily large value ei=400 was adopted.

different from perfect sphere which is assumed in the present theory. Such a kind of non-spherical features must be discussed further. Nevertheless, in the present paper, the experimental data of erythrocyte suspensions were cited as a mere example to show the practice of the fitting procedure proposed. ( 303 )

T. HANAS,K. Asruer, and N. Korzrmzr 4000-------------------1---------1------------------,--------1--1-1--1----------------------1 - 3 .0

3000 w,E 2000cn

El from Eq.(2) El from P. S.

-2 .0 E

1000lfrom

Ki from Eq.(2)

-1 .0 P. S.

000 .10.2 030.4 0.5 0.6 0.7 0.8 0.9 1.00 Volume fraction , Fig. 4. Dependenceof limitingrelative permittivityet and conductivity St at low frequencieson volumefractionfor the suspensionsof shell-spherecsalculatedfromEq. (2) and fromPauly and Schwan's theory (P.S.). Phase parameters used for the calculationsare the same as used in Fig. 2.

As readily known from the curve-fitting procedure, the values of 0 and es are closely related to ICIand et respectively, the correlation being characteristic of respective theories. In Fig. 4 are compared the values of xi and Sycalculated from Eq. (2) of the present theory and from Pauly and Schwan's theory with the same values of phase parameters. Marked differences of it1and Etvalues are found between the two theories. The phase parameters obtained by means of the curve-fitting procedure, therefore, are expected to vary from theory to theory used. The curve-fitting based on Pauly and Schwan's theory18) was applied to the observed data of the erythrocyte suspension shown in Fig. 3, the phase parameters obtained being com- pared in Table III. Remarkable differences are found between the two theories with respect to the values of 0, CH, and es as seen in the Table. A number of dielectric

Table III. Comparisonsof PhaseParametersObtainedby Meansof the Curve-Fitting Procedure

Eq. (2) of the present theory0.540 Pauly and Schwan'stheoryb)0.595

Phaseparametersestimateda)

CmICi pF cm-2esElmS

0.777

4.3969.02.84

1.008

5.6969.42.77

cm-1

a) For the estimationfrom the curve-fittingprocedure,the followingvalueswere usedfor both the theories: x1=1.356mScm-1, et=2176, en,=71.5,fo=1.63 MHz, D=4.75 pm, d=50 A, =4.346mScm-1,so,=77.1. b) The curve-fittingprocedureis describedin Reference18.

( 304)

Dielectric Theory of Concentrated Suspensions of Shell-Spheres

data previously reported for biological suspensions await further reconsideration

in

the light of the present theory.

REFERENCES (1) K. Asami, T. Hanai, and N. Koizumi, J. Membrane Biol., 28, 169 (1976). (2) K. S. Cole, "Membranes, Ions and Impulses", Part 1. University of California Press, Berkeley Los-Angeles, California (1960). (3) C. W. Einolf, Jr. and E. L. Carstensen, Biochim. Biophys. Acta, 148, 506 (1967). (4) H. Fricke, H. P. Schwan, K. Li, and V. Bryson, Nature, (London) 177, 134 (1956). ( 5) J. Krupa, B. Kwiatkowski, and J. Terlecki, Biophysik, 8, 227 (1972). (6) H. Pauly, Biophysik,1, 143 (1963). (7) H. P. Schwan, Electrical Properties of Tissue and Cell Suspensions. In "Advances in Biological and Medical Physics", Vol. V, .edited by J. H. Lawrence and C. A. Tobias, Academic Press, New York (1957) pp. 147-209. (8) H. P. Schwan and H. J. Morowitz, Biophys. J., 2, 395 (1962). (9) Y. Sugiura, S. Koga, and H. Akabori, J. Gen. Appl. Microbiol., 10, 163 (1964). (10) H. Pauly and H. P. Schwan, Z. Naturforschg., 14b, 125 (1959). (11) S. S. Dukhin, Dielectric Properties of Disperse Systems. In "Surface and Colloid Science", Vol. 3, edited by E. Matijevic, Wiley-Interscience, New York and London (1971), pp. 83-165. (12) T. Hanai, Electrical Properties of Emulsions. In "Emulsion Science", Chap. 5, edited by P. Sherman, Academic Press, London and New York (1968). (13) L. K. H. van Beek, Dielectric Behaviour of Heterogeneous Systems. In "Progress in Dielectrics", Vol. 7, edited by J. B. Birks, A Heywood Book, London (1967), pp. 69-114. (14) J. C. Maxwell, "A Treatise on Electricity and Magnetism", Third Edition, Chap. IX. Clarendon Press, Oxford (1904). (15) K. W. Wagner, Arch. Electrotech. (Berlin), 2, 371 (1914). (16) T. Hanai, Kolloid Z., 171, 23 (1960). (17) T. Hanai, KolloidZ., 175, 61 (1961). (18) T. Hanai, N. Koizumi, and A. Irimajiri, Biophys. Struct. Mechanism, 1, 285 (1975). (19) K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341 (1941). (20) A. B. Hope, Australian J. Biol. Sci., 9, 53 (1956). (21) A. Irimajiri, T. Hanai, and A. Inouye, Biophys. Struct. Mechanism, 1, 273 (1975). (22) H. Pauly and L. Packer, J. Biophys. Biochem. Cytol., 7, 603 (1960). (23) H. Pauly, L. Packer, and H. P. Schwan, J. Biophys. Biochem. Cytol., 7, 589 (1960). (24) H. P. Schwan, Biophysik, 1, 198 (1963). (25) H. P. Schwan, Determination of Biological Impedances. In "Physical Techniques in biological research", Vol. VI, Electrophysiological Methods, Part B, Chap. 6, edited by W. L. Nastuk, Academic Press, New York and London (1963), pp. 323-407.

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T Hanai, K Asami, N Koizumi

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