The combined total-reflection method in dielectric time-domain spectroscopy

CHEMICAL PHYSICS LETTERS

Volume117.nllmber1

THE COMBINED IN DIELECI’RIC

31 May 1985

TOTALREFLECl-ION MEX-HOD TlMEhDOMAlN SPECTROSCOPY

V-A.

GONCHAROV

Koran

PhyJrcal and

Technrcal 1~111ure

of xhe USSR

Academy

of Screncer. Kazan.

USSR

Received 5 February 1985

method yielding an explicit expression for the complex permirlivily of the sample and Lime-domain analysis of the dala is described.. An algorithm for solving Lhe convolution-ly-pe integral IS developed. Exprimcntal reulls Tor n-bulanol are presenwd.

A modification

of the colaI-reflecbon

allowing a delakd cquauon

1. Ba5ic relation3

p=

The total-reflection method is the most rigorous procedure for dete rmining the complex permittivity allowing for all successive reflection.5 of the test signal ~thin the bounds of a sample. A section of a transmission line loaded with, in general, an arbitrary complex impedance 2 (fig. 1) serves as the measuring cell in this method. The expression for the Sl, coefficient of cell dispersion matrix,

[I -p(E*)‘Q/[l

is the reflection

+B(E*)1/2] coefficient

,

P=Z,/Z,

of the a_ir-dielectric

, (3) in-

terface, p1

=

[Z -

zl(E*)-q/[2

+ z,(E*)-lq

is the reflection coefficient

of the load 2,

(L denotes Fourier-Laplace transform, L [v(t)] = Jr V(f) exp(--sC) dt, s = (Y+ iw, o is the cyclic frequency, a + 0) can be shown to be

is the sample’s complex permittivity (a is its conductivity, ~~ the permittivity of free space), and sl = exp(--2h*I) (y* = w(E’)~/~/c is the propagation constant, c the speed of light, 2 the sample length One of the following variants is usually employed: an open-circuited cell, 2 + - [ 11,

s=CP+Pp117)/(1+PPlv).

s=s,

s=s

11

=

L lyr(Ol /L [Q&)1

0)

(2)

where

‘(P

+GW(l

+m)

I

(6)

or a cell occupymg a certain intermediate section of the matched line, Z = Z, [2], s=szo

= PCi - q)/U

or a short-circuited

-

p2v)

,

(7)

cell, 2 = 0 [3],

(8)

Fig. 1. Functional circuit of the total-reflechon method (timmg circuits are omitted); the sampler produces the analog signal Vi&) + vr(t - zlo/uo).

52

None of equations (6)-(S) gives an explicit expression for e*(h), so that the latter must be computed by numerical methods. The time-domain analysis of the data is still more complicated, and is only possible for an approximate simplified Vti(r) waveform [2].

Volume 117, number 1

However, combining the measurements

of the re-

flected signals for the sample-filled cell when openand when short-circuited drastically changes the situation [4] *(an analogous approach in frequency-domain measurements has been considered in ref. [S] ). Actualiy, eq. (3) gives (1 - p)/(l + p) = fi(e*)‘i2, and we have from (6) and (S), respectively, (1 -S-)/(1

+S,)

=fl(e*)l12

tmh(iy*Q,

(9)

and (1 - SO)/0

+ Soj = P(e*j1’%nh(rr*r)

,

(10)

whence (1 - S,) (1 + S,)/(l

+ S,)

(1 - SO) = tanh2(iy*r)

, (11)

and (1 -S,)(l

-S,)/(l

+S,)(1

+S,)=p%*

-

e*(iwj = L MT)] IL VVjJ ,

(12)

(13)

where

(14) and = Iv;,w

+ v,_mi

* cm0

+ vroch

(15)

(“*” means convolution).

From the expressions for the observable functions of time (see fig. l), V, en(r) = Q.Jt + 2loluo) + V&j, v”“P(r) = F&(t + up,/c> +Vr(f) and Vshort(t) = Vr,.,(t + 2ZO/cj - V&(t), one has V#)

+ V,(r) = -V*&)

+ vamp(r)

‘f=(t)

(16)

and vi,(r) - v,(r) = Pope&j so eqs. (14) and (1s)

&z(t)= w821f30&)*f3&)

-

P”p(0

=f3(r)

,

(17)

take the form I

fW =f2Eso*f20(f)

,

(19)

where, as before, the additronal subscripts “m” and “0” correspond to the cases of 2 + = and Z = 0 (fig. 1). Havmg used the relation e* (w)=

L [G(r)]

(20)

,

where $ (f) = c(t)

+ a/e0

(21)

(Ip(t) = e,&(r) + 4(t), 4(t) is the permanent-dipole response function, e, tie high-frequency perrnittivity, 6 (r) the delta function [6] j, tiile obtains from (13): &j

=~(0G
= G(O*i(r)

t z

s z(t - c’)i(f)

dr’ .

(22)

0

Each of equations (11) and (12) allows one to calculate the complex permittivity from measured So and S, independently, eq. (12) providing the exphcit formula for a computation of e*(iw)

m

31 May 1985

CHEMICAL PHYSICS LEl-l-EFtS

Numerical solution of the latter equation gives one the function (p(t) apd, hence, the quantities e, = a(o), ~=eubJ+MOcp -(t) and p(t) = g(r) - (u/~,,)t_ Parallel with eq. (13), formula (20) can be used for the evaluation of the complex permittivity; if the “net” quantity e(k) = E*(~u) - a/ioeo is required, then (p(t) should be taken instead of G((t). Thus, the considered modification of the totalreflection method allows one to process the data in the frequency, as well as m the time domain in the general form, without any special assumptions about

the incident signal shape I&(?) or the type of dielectric dispersion in the sample under test_ The ueatment procedure reduces to the numerical estimation of the Fourier-Laplace integrals for some known functions of time, and the numerical solution of the convolution-type integral equation - formally, to the same problems as in the lumped-capacitance method (see ref [6], where, m particular, there is an improved algorithm for the numerical Fourier-laplace transformation). In conclusion of this sectron, an interesting feature of eqs. (6x8) is worth noting: [S,(I)

(18)

* Ref. [4] a&. ~ntains explicit integral expressions for the parameters of the Debye-type diektric dispersion via the quantities directly measured in different TDS methods.

+&ml/2 =Q,wI

3

i.e. the joint measurement

of So and S, doubles the effective sample length, thus lowering by a factor of two the low-frequency limit of the single-reflectron method [7,8] _

53

Volume

CHEMICAL

117. number 1

PHYSICS

31 May 1985

LETJYERS

2. On the sdution of the convolution-type equation Retaining the notation of eq. (22) (except for the symbol g(t) which will be changed to p(f)), let us consider an approach, to the numerical solution of equations of this sort, allowing for the features of realf(r) andg(r) functions. We w-illtieatf(t) as an action on a linear system with the transient function p(f), and g(t) as the system’s response for this action. Let t = 0 be the point satisfying the condition that f(t) practrcally vanishes at t < 0 and does not at t > 0 (fip. 2a). As a rule, the physically remable functions of time have “roundings” near r = 0 (the term is t&en from ref. [2]). i.e.f(t) gradually tends to zero along v&h ah its derivatives, as t decreases, so that the initial conditions for solving eq. (22) appear to be uncertain; in particular, this is the ceasein all TDS

measurements

To eliminate

the uncertainty,

a

“gradual” change of the f (f) function at 0 < t < to is usually approximated either by a jump at r = to from 0 to ,F(roj (tte origin of coordinates being transferred to the point fo) or by the section of the tangent to the f(r)-curve in the point to (the intercept of this tangent with the abscissa is then assumed to be the ongin of coordinates [2]). The point to is chosen so thatf(tu) be defined with acceptable accuracy forf(f) experimentally obtained. However, it is evident that the greater the absohrte value off(fo) is, the smaller its relative uncertainty, but, at the same time, the greater part of the action turns out not to be accounted for, and thus, the difference betweeng(tO)/f(tO) and the “genuine” value do(o)is greater. To overcome this contradiction, let us rewrite eq. (22) in the form

where X is an arbitrary parameter. The first two terms in the left-hand side of eq. (23) can easily be seen to represent the response for a partial action f*(t) =f(r), tfo,andthenexttwo, for the complementary one,&(r) = 0, t < to; f(f) - (1 - A)flto), r 2 to. Fig. 2 shows an exemplary course of the response (shaded in fig. 2b) forfA(f) (shaded in fe_ 2a) and for fB(r) in the case of X = 1. An iteration procedure can be constructed for eq. (23):

~(to)lp(“+l’(r) + j tp(“+‘)(t - t’);(t’)

dt’

r0

*0

‘g(r)

-

j-

cp’“‘(t - t’)j(t’)

dt'

cl

+m&%)

(24)

where n = 0, 1,2, ._ is an order of approximation of the unknown (p(t)_Having chosen an arbitrary cp(‘)(t) (for instance, (p(o)(t) = 0), one obtains an equation with respect to cp(l)(r), containing&~(t) out of the convolution integral; this equation can easily be solved numerically, the initial conditions being defined as p(l)(O) = Xg(rO)/f(tO). Having substituted cp’l)(t) in the right-hand side of eq. (24) one computes the next approximation of At), and so on. The resulting {p(“)(r)) can be expected to converge for

certain values of A; it should then converge to I&). as follows directly from (LB), (24).

0

%

0

+o

Fig. 2. Graphical interpretation of eq. (23) for A = 1.

54

A pmcticaI vanant of algorithm (24) for the case of f(t) andg(t) represented by equidistant samples f(q), g(fj) (5=JS, jisaninteger, S the sampling interval) and the trapezoidal approximation off(f) within a particular interval [r,-, Q+~] has been described in ref. [4] and used when processing experimental data in the present work and in the lumped-capacitance method 191. The algorithm can as well apply in the dielectric

Volume 117. number

31 May 1985

CHEMICAL PHYSICS LEITERS

1

method based on the correlation analysis of the tbermal noise [4,10] and, in principle, whenever one deals with the problem of finding a response function of a linear system from its “input” and “output”.

3. Experimental illustration of the combined totalreflection method 2

1

0

A classical-circuit TDS apparatus (fig. 1) has been used to check the method. The transmission line was a length (Z. = 25 cm) of standard 3 mm/7 mm 50 SL coaxial line, and a 15.4 cm section of the same line served as the measunng cell. Fig. 3 represents the observable functions of time for n-butanol at room temperature, and fig. 4

4. Dielectric response temperature (see text).

Fig.

function

3

of n-butanol

at room

Appendix. ACCORU~~II~ for the mismatch of l&e testsignal generator

its “smoothed” dielectric response function so, (t) = 6-1p”61p(r’) dt’ obtained by numerical solution

The analysis of the measuring circuit shown in fig_ 1 in the general case, when the output impedance

of eq. ( 22 1 by the method described m section 2 (for details of the calculation see ref. [4])_ To allow for unwanted repeated reflections from the test-signal generator, zqs. (5A) and (6A) of the appen$ix have been used. The observables V&,,(r) and V&Jr) have been corrected for the open-line edge capacitance before calculating p5 (r).

of the generator, Zg, may differ from Z,, following relation:

_.__A--.__---

.....,.....5;. . P

._. .-... I 210/Yo

. - ---.._.

.._-._-. _.= __--

(1 - s)/(l

+ s) = Ww)L

gives the

V3(01 /L Lt2@)1 .

UN

Here S is, as before, the S,, coefficient of the measurmg cell dispersion matrix (irrespective of the edge load Z),

_..*_____.--. __-.-.

__A... ..---

---‘+ampCt_ open

..‘_” *n.. ‘=___ ---_. _____I--.--.-_ _._ _ --_Lp_;..-.I.-. -.__ _-. _..__..C---___---. .._.._.. ---. .-_. --- --. __.--- ;_y-.L.- -._-_ _____._._.__. .__..zrl.___._ _.,, rrc-.rl __.---.._ I -.-__ .__.A---V ehortct) V;=&(t)

,,,.,.,_,--..

--.

Fig. 3. Vopen(i) and Vhort(t) as functions oi time, directly m easued in the combined totakeflection method; the curves are, respectively, open- asd sbortkrcruted in the A-A-plane. V,“,p(t) and =3(c), sample-fiUed roll attached to the line in the A-A-plane. respectively, open- (z -, -) and short-circuited &= 0) at the edge (experimental curves for n-butanol at room temperature; sampling interval 0.1 ns)-

Volume

P(k)

CHEMJCAL

117. number 1

PHYSICS

= [l - pg exp(-2i7oZo)]

X [l +pg exp(-2i~010)]-1

,

GA)

y. = w/Z0 and pg = (Z. - Z&Z0 +Z& Eqs. (1% (18), (19) and (1A) yield the corrected formula for the complex permittivity: e*(io)

= p2(io)L

The factor p(io)

[g(t)] /L v(t)]

x

IL

1 yopellc0

+ &or& -

Vop&

-

x0

=&P(r)

%/c)lrl

References 84 (1980)

-

(4A)

1

GA)

,

(6A)

and For=(r) =f(O&r)

It is interesting to note that, owing to (1A); eq. (11) remains invariable for any Zg (the factors ofPjio) cancel); however, the latter equation does not permit such an easy transformation to the time domain as eq. (12) does, and requires numerical (iteration) methods to be used when working in the frequency domam.

111 RH. Cole, S. Mashimo and P. Winsor, J. Phys. Chcm.

/cl]

One can also “calibrate” the system having measured fir) and g(r) for the empty (air-fried) cell, and then use relations (13) or (22) with P”(O

31 May 1985

(3A)

_

can be determined expenrnentally as

p(iw) =L [Khorc(f)

LFXIERS

taken instead of g(t) andI( respectively. The use-of properly chosen combinations of the type &4#lr(t + ric) and &Akgalr(t + $1 in place ofY(f) and g”“(r), as a rule, allows one to diminish the sensitive: andfcorr(r) to the ness of the quantitiesg ‘““(r) instrumentation errors of the “air” obser-vables.

786.

PI R. Chahine and TX_ Bose, J. Chem Phys. 72 (1980) 808. 133 B. Gestblom, Chem. Phys. Letters 74 (1980) 333. 141 V-A. Goncharov, article deposited m VINITI (Mosoxv), No. 3579-83 Dep. (1980). A-R von Hippel_ ecL, Diclectic materials and applications t:: VA. Goncharov and Y.D. Feldman, Chem. Phys. Letters 71 (1980) 513. [71 C. Boned and J. Peyxelasse. J. Phys E 15 (1982) 534. 181 B Gestblom and E. Noreland, J. Phyr Chem. 88 (1984). 664. PI V.A. Goncharov and I V. Ovcbkmikov, in: Abstracts of the 3rd AM-Union Conference on the EIechicaI Properties of Molecules. Kazan, USSR, May 1982 (Kazm, 1982) p_ 73; Abstracts of the 5th international Liquid Crystal Conference of Sociakt Countries. Odess. USSR, October 1983, VoL 1, Part II (Odessa, 1983) p. 160.

1101 V.A. Guncharov

and 1-V. Ovchmnikov, Letters 111 (1984) 521.

Chem. Phys