Fluctuation Theory of Light Scattering from Pure WaterAdiel Litan Citation: The Journal of Chemical Physics 48, 1059 (1968); doi: 10.1063/1.1668761 View online: http://dx.doi.org/10.1063/1.1668761 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/48/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theory of Light Scattering from Fluctuations of Membranes and Monolayers J. Chem. Phys. 55, 2097 (1971); 10.1063/1.1676380 Light Scattered from Thermal Fluctuations in Gases J. Acoust. Soc. Am. 47, 64 (1970); 10.1121/1.1974653 Fluctuation Theory of Light Scattering from Liquids J. Chem. Phys. 48, 1052 (1968); 10.1063/1.1668760 Light Scattering of Water, Deuterium Oxide, and Other Pure Liquids J. Chem. Phys. 43, 3881 (1965); 10.1063/1.1696615 Light Scattering by Pure Water J. Chem. Phys. 43, 914 (1965); 10.1063/1.1696871

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 48, KUMBER 3 1 FEBRUARY 19(,8

Fluctuation Theory of Light Scattering from Pure Water

ADIEL LITA="I*,t

Departlllent of Chemistry, University of Oregon, Eugene, Oregon

(Received 3 October 1967)

In the preceding article it has been shown that the conventional fluctuation treat~e~t of l!gh~ scattering, which for a pure liquid yields the Einstein-Smoluchowski-Cabannes. formula, .wIll, m prmczple, ~sually underestimate, but never overestimate, the correct amount of scattenng. In thIs study an exp~esslO~ for th excess scattering over that calculated by the Einstein-Smoluchowski-Cabannes formula, IS. d~nved fo: a simple "iceberg;' model of pure water. The excess scattering is estimated and fou~d to be. wlthm th.e range of experimental error, which agrees with the fact that it has not been observed m prevIOus expen­mental studies.

INTRODUCTION

In the preceding studyl it was shown that the Einstein-Smoluchowski-Cabannes formula for the intensity of light scattered from a pure .liquid will usually underestimate, but never overestlmate, the scattering, because it involves the assumption that fluctuations of the dielectric constant € are caused only by density fluctuations. In the present study it is shown that, in the case of pure water, when the assump­tion is improved, so as to take into account also ~uc~u­ations of € due to fluctuations of the degree of assocIatIOn of the water molecules at constant density, the cal­culated scattering is indeed increased.

In principle, therefore, the experimental scatteringlb

from pure water can exceed the value calculated from the Einstein-Smoluchowski-Cabannes formula. 2 How­ever, a rough estimate shows that an upper bound f~r such an excess scattering is of the order of the expen­mental error, which is in accord with the fact that no excess scattering was observed experimentally by previous workers.3,4

THEORY

The basic result of the fluctuation theory of Smoluchowski, Keesom, and Gans states l that the ratio of the intensity of the light scattered from a pure liquid to the in-tensity of the incident, monoc~romatic and unpolarized light is given by the followmg pro­portionality relation:

Iscatt/ Iincid 0:: V «(11€) 2)T,V,I'[ (6+6p) / (6-7 p) J, where «( 11€) 2)T.v ,I' is the mean square of the dielectric constant fluctuation [a definition of 11€ is given by Eq. (20) of Ref. 1J in a subvolume ~ inside a l~quid which is at a temperature T and chemICal potential p"

* On leave of absence from the Weizmann Institute of Science,

and (6+6p) / (6-7 p) is the Cabannes factor with p denoting the depolarization of the scattered light. This relation leads to the Einstein-Smoluchowski­Cabannes formula when < (I1E) 2)T,v,1' is evaluated by the statistical-mechanical method of either Smoluchowski and Keesom1 or of Einstein,5 which assumes that €

fluctuates only as a result of density fluctuations. The value of the Cab annes factor is usually determined experimentally.

We shall compare the conventional and an improved theoretical evaluation of V«(I1E)2)T,V,1' for a pure model liquid where each molecule can assume two states; a "liquid" state, in which the molecule is not bonded to any other molecule, and an "icy" state, in which it is bonded (the term "iceberg" is often used to designate an aggregate of hydrogen-bonded water molec,":les, it being assumed that the aggregate resembles a mICro ice crystal). Some comments about the model are made in the discussion section.

Let ..7\\ and N2 denote the numbers of "liquid" and "icy" molecules, respectively, which are inside a sub­volume element V, in a certain microscopic state, and let lV be their sum,

(1)

Since this pure model liquid is a single thermo­dynamic component, the conventional approach is to evaluate V «( 11€) 2)T ,v ,I' according to the method of Smoluchowski and Keesom, which assigns to every microscopic state with given V and N the same di­electric constant, E(T, V, N). The conventional result is here denoted by V«(I1E)conv2 ) and is given by6

(2)

where {3= (kT)-1 and K is the isothermal compressi­bility,

K= - V-I(aV /ap)r,N. (3) Rehovoth, Israel. . . . f h d .. f h d I I' 'd 't t Present address: Department of Chemistry, Umverslty of In VIew 0 t e escnptlOn 0 t e mo e IqUl, I California, Santa Cruz, Cahf. seems reasonable to improve the conventional assump-

1 (a) A. Litan, J. Chern. Phys. 48, 1052 (1968). (b) Appendix tion about € and assume instead, that every microscopic of that article. h

2K. J. Mysels, J. Am. Chern. Soc. 86, 3503 (196.4). state with given values of V, lVI, and N2 has t e same 3 J P Kratohvil M Kerker and L. E. Oppenheimer, J. Chern. ~

Phys·. 43,914 (1965).' , 5 A. Einstein, Ann. Physik 33,1215 (1910). 4 G. Cohen and H. Eisenberg, J. Chern, Phys. 43, 3881 (1965). 6 See, e.g., Eq. (6) and Ref. 14 of Ref. 1.

1059

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l060 ADIEL LITAN

dielectric constant f(T, V, Nl, ]\'2), thus allowing for Huctuations of f at constant V and iY due to fluctu­ations ~'Yl and ~;Y2= -~iYl' The evaluation of V«~f)2)1',V,~ subject to the improved assump­tion is equivalent to the conventional evaluation of V«~f)2)1',V'~1'~2 for a two-component liquid,7,8 followed bv a correct accounting for the chemical equilibrium ';hich exists between the species "liquid" and "icy" (at equilibrium, but not in individual microscopic states). The improved expression for V «~f) 2)T ,v,~ is therefore written as

( 4)

where () denotes the fraction of "icy" molecules,

()=N2/N,

Ke= - V-l(aV /ap)r,N,e,

(5)

(6)

and J.t2 is the chemical potential per "icy" molecule. Any quantity in Eqs. (2) and (4) is evaluated at

equilibrium. Moreover, the derivatives in Eq. (2) are, in fact, taken along equilibrium paths and the sub­script eq will be used whenever it is desired to stress this point.

Equations (2) and (4) are now transformed so as to enable a comparison of V«~f)con}) and V«~f);mpr2>.

In the model liquid considered, the dielectric con­stant may be written as

f=f(T, P, () (7)

and (aEjaph in Eq. (2) can be rewritten as follows,

Similarly,

V = VeT, P, N, e) (9)

and

-KV=(~~)T'N,eq =(~~)T'N,e+(~~)T'P'NC~)T,eq' ( 10)

(Along an equilibrium path, () is a function of P and T.) Let ~v denote the difference of partial molecular

volumes between "icy" and "liquid" molecules,

~V= (aV /aN2)r,p,Nl- (aV /aNI)r,p,N2' (11)

Considering V as a function of T, P, N l , and N2 and

7 J. G. Kirkwood and R. J. Goldberg, J. Chern. Phys. 18, 54 (1950) .

8 W. H. Stockrnayer, J. Chern. Phys. 18, 58 (1950).

using Eqs. (1), (5), and (11) it follows that:

(av/aeh'/',.\'=X~v (12)

and then from Eqs. (10) and (6), that

K = Ke- (N / V) ~v( ae/ap)r,Cq. (13)

The desired transformation of Eq. (2) is obtained by substituting into it according to Eqs. (8) and (13) (since the temperature is always held constant, it is henceforth omitted from subscripts of the derivatives),

V«~f)conv2>=p{ Ci)/(!:t (a~tJ x[ Ke-(~)~v (a~)J-I. (14)

In Eq. (4) we first change the variable of differenti­ation jYdNI to ()=N2/(N,+N2),

a a (1-()2a

a (NdNI) a«()/1-() ae (15)

For the purpose of evaluating (aJ.t2/aeh,p, the free energy per molecule of the model liquid is written as

J.t= (1-()J.tI+eJ.t2=J.t2- (l-e)~J.t, (16)

where ( 17)

Differentiation of J.t with respect to () at constant temperature and pressure gives

(aJ.t/a()p = (aJ.tdae) p- (I-e) (a~J.t/ae)p+ ~J.t. (18)

At equilibrium, ~J.t=O, (a}1./ae)p=o, and Eq. (18) reduces to

If ~}1. is considered as a function of the intensive variables, T, P, and (), then, at constant temperature, by a general law of calculus,

The derivative (ae/ap)t:.~ at ~J.t=O is precisely (a()/ap)eq taken along an equilibrium path. Hence,

(21)

and by inserting this result into Eq. (19), we obtain

(d}1.2) (1-() ~v ao P,eq = - (a()/ap)eq .

(22)

Expression (4), with the subscripts T omitted, is now transformed according to Eqs. (15) and (22) and

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LIGHT SCATTERING FROM PURE WATER 1061

N 1 is replaced by N (1-0). This yields

V«(6E)iTllpn=~-1 [(:n:Ke-I-c~rl (~)~ (a(:~t(6V)-IJ (23)

The following expression is obtained from Eqs. (23) and (14):

(rl( aOlap) ('q[ (N IV) 6v( aElap) e+ Ke( aElaO)p J2 V «(6E) impr2)- V «(6E) conv

2) = - [Ke(N I V) 6VKJ ' (24)

where, in the denominator, K appears as a result of a final substitution according to Eq. (13). Some insight into the physical significance of Eq. (24) is now gained by simplifying the square brackets in the

numerator. Alongside the functional dependence (7) of E, we can use another dependence,

E=E(T, V, lV, 0), (25)

and therefore write,

(26)

Now, P=P(T, V,-N,O) (27)

from which, by using general calculus and then substituting according to Eqs. (12) and (6), we deduce that

( a V laO) p .N N 6v (28)

(av lap)O.N VKO

Insertion of this result into Eq. (26) and rearrangement of terms yields

Ke (~) = (~)6V (aE) + KO (aE) ao V.N V ap 0 ao p'

(29)

Making this substitution in Eq. (24) -and reinserting the subscripts T, we finally obtain

V «6E);mpr2)- V «6E)conv2) = -(rI(aElaokv.N2 ( V IN) (6v)-J (aOlapkeqKecl. (30)

Thermodynamic stability requires that 6v and (aOlaph.eq have opposite signs (Le Chatelier's theorem) and that compressibility be always positivt;. Equa tion (30) therefore states that, on the basis of the improved assumption about the values of E in different microscopic states, the value calculated for V«(6E)2)T.V.1' (and hence also for Iscattllincid) is larger than the value calculated by the conventional formula of Einstein, Smoluchowski, and Cabannes, provided (aElaoh.v.N;;roO, i.e., provided E can fluctuate at con­stant lV and V. This conclusion is reached more directly if, instead of using the final formulas for «6EF) in a one- and in a two-component liquid [Eqs. (2) and (4) J, one starts by deriving these formulas and con­siders the difference V«(6ELmp/)- V«(6E)conv2 ) with­out passing to derivatives at constant pressure.

NUMERICAL EVALUATION

The numerical value of V «(6E) impr2)- V «(6E) conv2 )

is now estimated and compared with that of V < (6E) conv2

). This estimate is calculated on the assump­tion that the model liquid for which Eq. (30) has been derived represents adequately pure water. The nature

of the model is discussed and reference to more detailed models for water is made at the beginning of the next section.

We begin with the quantity (aElaoh.v.N appearing on the right-hand side of Eq. (30). For the wave­lengths used in light-scattering measurements, e.g., the sodium-D line, the dielectric constant is equal to the square of the refractive index,

(31)

Now, treating n2 as a function of T, V, lV, and 0, we have

(an2) (an2) (an2) ( ao) aT V.N = aT V.N.O+ ae T,v.N aT V,N'

(32)

Since for liquids which do not exhibit the densitv anomaly of water (an2/aT)v,N::::::;0,9 and since th"e anomaly in water is ascribed to a change of 0 with temperature, we assume that at constant 0,

(an2/aT) V.N ,0::::::;0. (33)

P E. Reisler and H. Eisenberg, J. Chern. Phys. 43, 3875 (1965).

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1062 ADIEL LITAN

The molecules in the "icy" state are assumed to resemble molecules in ice in having a larger partial volume and smaller partial enthalpy than the "liquid" molecules. It follows that both (ae / aph and (ae / aT) J'

are negative and that above the temperature of maxi­mum density of water (TllIax),

I (ae/aT) V,N I > I (ae/aT)], I, 1'> Lax. (34)

Introducing Relations (31), (33), and (34) into Eq. (32), an upper bound for the desired quantity is obtained,

Next, for (ae/aPh at about 10°C we use the vaillell,lf> -1 X to'·lll cm2dYll' l; the tinal conclusion will be shown to dependlitt Ie on the absolute value assigned to this quantity.

Last, the isothermal compressibility of water, K, is takenlO as 4.9X 1O-11 cm2 dyn-1 and Ke is then cal­culated from Eq. (13).

Introducing the above numerical values into Eq. (30), we finally conclude that [note the inequality (38) ]

V «(i1c) illlpn- V «(i1c) conv2)< 5.4X 10-13 (rl cm3. (39)

T> Tmax. Under similar. conditions, i.e., at about 10°C and for the sodium D line,lO n= 1.334, (an/aph= 1.61X

(35) 10-11 cm2 dyn-l and Eqs. (2) and (31) therefore yield

The value of (an2/aT) V,N is obtained from Eisen­berg's studylll: Using Eq. (6) and Identity (1) of Ref. 10 gives

(an2/aT)v,N = -C(n2-1) (n2+2) /3

= -6.WXW-5 deg--1, (36)

the numerical value resulting from using the values 6.204X 10-5 deg-1 for C and 1.780 for n2 at 10°C (both correspond to the sodium D line).

The value of (ae/aT)p, which is required to complete the evauation of the right-hand side of the inequality (35), is estimated from various studies on the structure of water.1l- 15 A value slightly lower than -0.0025 deg--l is indicated by most authors and in one case15

the value is as low as -0.006 deg--l. For the purpose of evaluating an upper bound of the excess scattering and therefore also of (aejaeh,v,N2

, we take

(ae/aT)p= -0.0025 deg--l. (37)

From Relations (35) -( 37) it finally follows that, for the sodium D line

(38)

Proceeding to other quantities on the right-hand side of Eq. (30), the combination (V/N)(i1ii)-1 is equal to the ratio between the specific volume of water (taken as 1 cm3g-1) and the difference between the partial specific volumes of "icy" and of "liquid" water mole­cules (taken as 0.09 cm3g-1~the value of the difference between the specific volumes of ice and of water at the melting pointl6

) •

10 H. Eisenberg, J. Chern. Phys. 43, 3887 (1965). 11 N. E. Dorsey, Properties oj Ordinary Water Substance

(Reinhold Pub!. Corp., New York, 1940), pp. 169, 170. 12 G. Nernethy and H. A. Scheraga, J. Chern. Phys. 36, 3382

(1962) . 13 J. Buijs and G. R. Choppin, J. Chern. Phys. 39, 2035 (1963).

See also J. Chern. Phys. 40, 3120 (1964). . 14 M. R. Thomas, H. A. Scheraga, and E. E. SchrIer, J. Phys.

Chern. 69, 3722 (1965) and Ref. 5 cited therein. ,. A. Eucken, Z. Elektrochern. 53,102 (1940). 16 See p. 467 in Ref. 11.

V «(i1c) con}) = 3.8X W-lliJ-lcm3. (40)

According to Eqs. (39) and (40), the excess light scattering is expected to be less than 1.5% of the value calculated by the conventional approach and, there­fore, within the range of experimental error of ±2%.4,17

We close the numerical evaluation with two re­marks.

(a) Substituting for Ke in Eq. (30) according to Eq. (13), it is seen that the excess light scattering initially increases with the absolute value of (ae/aph, all other quantities held constant, and reaches a maximum at (ae/aph= -!K(V/S) (i1ii)-l= -2.7XW-1O cm2 dyn-l. Even then, V«i1cLlHpr2)­

V«i1c)conv2)<9.2XW-13iJ-1cm3 is still of the order of the experimental error, or smaller.

(b) In calculating the upper bound for V «i1c) illlpr2)­V«i1c)conv2 ) we took (ae/aT)p=-0.0025 deg--l. If the correct value is about -0.006 deg-l,l·' then V«(i1chnpr2)- V«i1c)conv2) is equal to, or smaller than, ~% of V«i1c)conv2

).

To sum up, the estimated magnitude of the excess light scattering from pure water, due to fluctuations in "iceberg" concentration, is too small to be measured by the commonly used experimental techniques.

DISCUSSION

In the theoretical section it is concluded that light scattering from a certain model liquid can exceed the value calculated from the Einstein-Smoluchowski­Cabannes formula. This conclusion is reached by show­ing that improving an assumption on which the Einstein-Smoluchowski-Cabannes formula is based increases the calculated scattering. Following the theoretical section, the numerical value of the excess scattering is estimated for pure water on the assump­tion that the model liquid considered earlier adequately

17 D. J. Cournou, E. L. Mackor, and J. Hijrnans, Trans. Faraday Soc. 60, 1539 (1964).

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LIGHT SCATTERING FROM PURE WATER 1063

represents pure water. The features of that model are now examined.

At the beginning of the theoretical section, the model was introduced by considering two states for each molecule; a "liquid" state, in which the molecule is not bonded to other molecules, and an "icy" state, in which it is bonded. The derivation of the final Eq. (30) applies, however, to any liquid to which an internal parameter IJ can be assigned, at equilibrium as well as in any microscopic state, which satisfies the following thermodynamic relations.

(a) For equilibrium states, IJ is fully determined by the thermodynamic variables which define the state of the liquid.

(b) In a mathematical sense, an increase of IJ by dIJ at constant temperature, volume, and number of molecules, N, induces the same thermodynamic changes which would have resulted if dIJ could be effected ex­perimentally, by withdrawing NdIJ molecules of the so-called "liquid" type and simultaneously adding N dIJ molecules of the "icy" type.

In the entire theoretical section, the only statement made about the "liquid" and "icy" states is, that their partial molecular volumes (i.e., partial molar volumes divided by Avogadro's number) differ by LlV.

The model is therefore not a detailed molecular model and becomes phenomenological if there are means to measure the intensive quantity IJ experi­mentally, say spectrophotometrically,13.14 and to meas­ure, or estimate, the change of the dielectric constant with IJ.

A more detailed model for water in which a molecule can assume more states, e.g., free, bound to other molecules by either one, two, three, or four hydrogen bonds,18 will involve other parameters in addition to IJ. In general, the more detailed the model for a pure liquid, the more parameters the dielectric constant is assumed to depend on, when the conventional approach to light scattering from a pure liquid is improved, and the larger is the value then calculated for V «( Ll€) 2)T. V ,I' and, therefore, for the scattering.

In the theoretical section we studied a model for a pure liquid and chose the concentrations N1/V and NdV of the molecules in their two accessible states as variables which determine the dielectric constant in any microscopic state of a subvolume V. This choice improved upon the conventional choice of just one variable, (N1+N2) IV. Another improvement on the conventional choice is to choose as variables (N1+ N 2 )/V and the energy density, E/V, which leads to the

IS See, e.g., Ref. 14 and references cited therein.

generalized Einstein-Smoluchowski relation.19 Since the pairs N1/V, NdV and (N1+N2)/V, E/V will not be, in general, functions of each other, there arises the question as to which pair should be preferred. In principle, the pair lVI/V, NdV seems to be preferable, because, in liquids which do not exhibit prominent thermal changes of internal structure, (a€jaT)v,N is negligibly small9 and if for the above model (or water) (a€/aT)V,N is larger than for other liquids, it may be ascribed to thermal changes in the ratio of concentra­tions, NdNl' of the two molecular states. The treat­ment would be rigorous only if the dielectric constant would have been expressed as a function of all the variables on which it depends in any microscopic state of subvolume V. The number of these variables is of the order of the number of molecules in V and since it is necessary to restrict oneself to a reasonably small number of variables, one chooses those on which the dielectric constant depends most directly.

To sum up, the conventional treatment of light scattering from a pure liquid makes the assumption that, in any microscopic fluctuation state of a sub­volume V inside the liquid, the dielectric constant €

in V depends only on the total concentration of mole­cules 1V IV. In principle, it is more correct to assume that € in a fluctuation state depends on the individual concentrations of the various molecular forms that can occur in the liquid (e.g., free or aggregated) ; the ratios of these concentrations are fixed at equilibrium by thermodynamic relations, but in a fluctuation state they may deviate from their equilibrium values. Improving the conventional assumption along these lines allows for fluctuations of € at constant total concentration N /V, which can only increase the calculated value of V«(Ll€)2kv,1' and, therefore, also of the scattered light intensity,lb An estimate of this increase for water shows it to be smaller than the present day experimental error.

ACKNOWLEDGMENTS

I am very thankful to Dr. I. Z. Steinberg for sug­gesting the numerical evaluation and to Dr. Jack B. Carmichael and Professor A. Silberberg for helpful criticism. This work was supported in part by a re­search grant from the Heart Institute of the U. S. Public Health Service.

19 See Eq. (7) in Ref. 1. This formula is equivalent to adding the term pl(a./aTh.N2Tv/Cv (v is the molecular volume and Cv, t~e molecular specific heat at constant volume) to V «£i')eonv2 ),

gIven by Eq. (2) of the present study. For water, at the sodium-D ~ine, Eq. (36) gives -6.1 X 10-5 deg-1 for (ae/aTh.N, v/Cv ",,2.4X1Q-s cm3.deg erg-I, and at lODe we therefore have j3-I(a./aT) v. N 2Tv/Cv ""2.5 X 1Q-14pl cm" which is negligible compared to V«&)eonv2 )=3.8XlO-llj3-1 cm3•

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