JOURNAL OF RESEARCH of the Nationa l Bureau of St anda rds - A. Physics and Chemistry Vol. 72A, N o.1, January- Febr ua ry 1968 Density Fluctuations in Fluids Having an Internal Degree of Freedom* Raymond D. Mountain Institute for Basic Standards , National Bureau of Standards , Washington, D. C. 20234 (August 18, 1967) Th e frecluency spec trum of density fluc tuati ons is calculated for a fluid wh ose molec ul es possess an inte rna l d eg ree of fr eedom which is wea kl y co up led to the tr ansla ti onal d eg ree of fr ee dom of the fluid. Irr eversible the rmod ynami cs is u se d to obt ain an e qu ation of motion for the int e rn a l d eg r ee of fr ee d o m. Thi s e quat ion plus th e line arized hydr odynamic e qu ations are solved for th e fre qu ency spec · trum of dens it y flu c tu ations. Th e res ults are co mpar ed with a s imi la r ca lculation involving a fre qu ency depe nd e nt volume viscos it y. T he res ults ar e identi ca l fo r s tru c tur al relaxa ti on but th e re is a differe nce fu r th e rma l re laxation . Th e origin of the difference is di sc usse d a nd th e ma g nitud e of the difference is exa mine d for CC I.I and for CS 2• Key Word s: Bri ll ouin sca llering, dens it y flu ctua ti ons in liquid s, Ray le ig h sca ll e rin g, s pec tr al cli s tr ibution of sca ll e re d li g ht , s tru c tur al relaxation, thermal relaxa tiun, vo lum e vi scos it y. 1. Introduction Th e spec trulll of light sc attered by dens it y Au c tua - tion s in a fluid is propor tional to fre qu ency spec trum of density fluc tuation s [1].1 Th e de velopm e nt of th e gas la se r ha s mad e it po ss ible to m eas ur e th e s pec tra of l ong waveleng th density flu c tua tions and thereby s tud y fluid s in th e fr e qu ency reg ion of a few giga· c ycl es down to esse nti a ll y ze ro cyc l es/sec. For dense fluids this fre qu ency region c orres ponds to s low l y varyin g pro cesses who se time dependen ce may be re asonably exp ec ted to be d esc ribed by th e e quation s of irreversible t hermodyna;ni cs . Wh en c on sidering dens ity fluctuation s th e appropriat e e qu ations ar e the linearized e quation s of hydr ody· nami cs plus two additional re lation s amon g th e vari- ab l es so that a sol ution to the initial value ( or boundary value) prob le m is poss ible. In th is pap er we s hall con sider density fluc tuation s in a flu id whos e mol ecules possess an internal de gree of fr ee dom which relax es toward local e quilibrium with the density and / or th e te mperature [2 , 3]. Thi s is c arried out in sec ti on 2. In sec tion 3 this c alculation is com- pared with an e arlier calc ulation [1] where th e int e rnal d egr ee offr ee dom was a ss umed to re sult in a fre qu ency depe ndent volum e vi sc osity. Th e introdu ction of a fre qu ency depende nt vo lum e viscosity is ba se d on statis ti cal mec ha nical argum e nt s [4 , 5]. Th e co mp a ri son is mad e in two limiting ca ses . In the first case it is a s um ed th at the int ernal vari a bl e depends o nl y on th e loc al te mp e ra tur e (thermal relaxation). It is found that the two c alcu la tion s yie ld similar, but not ide nti ca l, res ults. Th e o ri g in and m ag ni- tud e of the difference ar e examined. Ca rbon te tr a- chloride and c arbon di s ulfid e ar e u se d to illu s trat e th e di sc u ss ion . In the seco nd case it is ass umed that the int e rnal va ri able depends o nl y on the loca l density (s tru c tur al re laxa tion ). Th e two ca lculati ons yie ld ' id e nti ca l res ults. No c ompari so n is mad e wh e n both the rmal re lax a- tion and s tru c tural rela xation a re rr ese nl. 2. Calcula ti on Th e th e rmodynami c s tat eme nt re latin g the para m- e ter s for our s yst e m is [6] dU = TdS + Pi PfJ (1) where U is th e ene rgy,S is th e e ntrop y, T is th e te mp e ratur e, p is th e de nsity , po is th e e quilibrium value of the density, P is th e pr ess ur e, gre prese nts an int e rnal d egr ee of fre e dom and Ag is th e pa rtial de riva - tive of the He lmholtz free energy with res pec t (2) We shall furth er re quir e that Ag= O wh e n th e internal I Fi}!Ufl':5 in braekets iudicati' 111,-, literatuft> rderences a t Ihl' e nd of thi :s pap er. d egre e of fr ee do lll is in l oca l e q u iii bri U III wj t h th e 95
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Density fluctuations in fluids having an internal degree ... · 'I, .. (z) and l./Jdz) are obtained in the same way. In eq (16) bo=(~'Y/s+'Y/ v) /mpo, Co is the low-frequency sound
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JOUR NAL OF RESEARCH of the Nationa l Bureau of Standa rds - A. Physics and Chemistry
Vol. 72A, No.1, Ja nua ry- Februa ry 1968
Density Fluctuations in Fluids Having an Internal Degree of Freedom*
Raymond D. Mountain
Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234
(August 18, 1967)
The fr eclue nc y s pectrum of de ns ity flu c tuations is c al c ula te d for a fluid whose molecules possess a n inte rna l d eg ree of freedom whic h is wea kl y coup le d to the tra ns la ti ona l d egree of freedom of the fluid. Irre ve rs ib le the rmod ynamics is used to obtain a n equ a tion of motion for the inte rn a l d egree of freedo m. This equat ion plus the line arize d h ydrod yna m ic e qua ti ons a re so lve d fo r the frequ e ncy s pec· trum of de ns it y flu ctua tio ns . The res ults a re co mpared with a s imi la r ca lc ul a ti on involving a fr equ e ncy de pe nd e nt vo lum e viscos it y. T he res ults are ide nti ca l fo r s truc tura l re la xa ti on but the re is a differe nce fu r the rma l re laxation . The orig in of th e diffe re nce is di sc ussed a nd the mag nitude of the diffe re nce is e xa mine d for CC I.I a nd fo r CS2 •
Key Word s: Bri ll ouin sca ll e rin g, de ns it y flu c tu a ti ons in liquid s, Ray le igh sca ll e rin g, s pec tra l cli s tr ibution of sca ll e re d li ght , s tru c tura l re laxation , the rm a l re laxa tiun, vo lum e vi scos it y.
1. Introduction
Th e s pectrulll of light scatt e red by de ns it y Au c tua tion s in a fluid is proport ional to freque ncy s pec trum of de nsity flu c tuation s [1].1 Th e de velopm e nt of th e gas laser has made it poss ible to meas ure th e s pec tra of long wavele ngth de ns ity flu c tua tion s and the re b y s tud y fluid s in th e fre que ncy region of a fe w giga · c ycles down to esse nti a ll y ze ro cyc les/sec.
F or de nse fluid s thi s freque nc y region corres pond s to s low l y varyin g processes whose tim e de pe nd e nce may be re asonably expected to be desc ribed by the equation s of irreversible thermodyna;ni cs . Wh e n con sid e r ing de nsity fluctuation s the appropriate equ a tions are the linearized equation s of hydrody· nami cs plus two additional relation s among the variab les so that a solution to the initial valu e (or boundary va lue) prob le m is possible.
In th is pape r we s hall con sider de ns ity flu c tuation s in a flu id whose molec ules possess an internal degree of freedom which relaxes toward loc al equilibrium with the de nsity and/or th e te mperature [2 , 3]. Thi s is carried out in sec tion 2. In sec tion 3 this calculation is compa red with an earlier calc ulation [1] where th e inte rnal degree offreedom was ass umed to result in a freque ncy de pe ndent volum e vi scosity. The introduc tion of a freque ncy de pe nd e nt vo lum e viscosity is ba sed on s ta ti s ti cal mec ha nica l argum ents [4 , 5].
The co mpa ri so n is mad e in two limiting cases. In the firs t case it is a s um ed tha t the inte rna l vari a bl e de pe nd s onl y on th e local te mpe ra ture (th e rm a l relaxation). It is found tha t th e two calcu la tion s yie ld s imil a r, but not ide nti ca l, results. The o ri gin a nd magnitude of th e diffe re nce are exa mined. Carbon te trachlorid e and carbon di s ulfid e are used to illu s trate th e di scuss ion . In the seco nd case it is ass um ed th a t th e int ernal va ri a bl e de pe nd s onl y on th e local d e nsity (s tructura l re laxa tion). The two calculations yie ld ' id e nti cal res ults.
No compari son is made wh e n both thermal re laxation and struc tural re laxation a re rrese nl.
2 . Calculation
The the rmodynami c s tate ment relating the pa rame ters for our system is [6]
dU = TdS + Pi dp +A gd~ PfJ
(1)
where U is the e nergy,S is the e ntrop y, T is th e te mpe rature, p is th e density, po is the equilibrium va lue of th e de nsity , P is th e press ure, gre prese nts a n internal degree of freedom and Ag is the pa rtial de rivative of the He lmholtz free e nergy with res pec t to~ ,
(2)
uf;;re~~~ I :e~rkw.s' POl\son'd b Y lhc Ad v.n cedRese. rchPmjecIS Agencyuflhe Depa ,l men l W e shall furth e r require that Ag= O wh e n th e inte rn a l I Fi}!Ufl':5 in brae kets iudicati' 111,-, literatuft> rderences a t Ihl' e nd of thi :s paper. degree of freedo lll is in loca l eq u iii bri U III wj t h th e
95
den sity and the temperature (whether or not the density and temperature have reached equilibrium values) _ This may be expressed as
(3)
where
(4)
is the local equilibrium value of g_ A kinetic equation for ~ is obtained by applying the
methods of irreversible thermodynamics to the system with the result
(5)
where the kinetic coefficient L is > O. As we are concerned with s mall deviations from equilibrium, Ag is expanded to obtain ag Tt=·-L[A gp( p-po) +Ag,r(T -To) +Agg(g-go)] (6)
whe re Agp = (aAg/ap h. g, e tc. If we expand eq (3) and subs titute thi s in eq (6) we obtain
(7)
The equations of motion for the system are the linearized hydrodynamic equations:
ap/at+potjJ=O, (8)
(9)
and
(10)
+ (a p) [dT + 7 adT] aT p. g at
(13)
where
A similar equation is obtained for the entropy.
dS adS - (as) [d a,dp J +r-- - p+r-at ap T, g at
+ (as) [dT+r~] aT p. g at
+ (as) [_Agpdp_Ag'f'dT]' (14) ag p. T Au A«
Equations (8), (9), (10), (13), and (14) cons titute a complete set of equation s. The next step is to obtain the Fourier-Laplace (space-time) transform of thi s set of equations. The res ult is
zp,..{z) + PolllJ.(z) = Pk , (15)
[C6k2 6.k2 zr ] (z+bok2 )tjJdz) - -+- --1+ p,,(z)
po po zr
rk2 [Ph' (Cil 6.) =tjJh'+-- --- -+- Pk I + zr mpo YPo po
(16)
Here 1jJ = div v, T)" is the shear viscosity, T)l' is the vol- and um e viscosity and A is the thermal conductivity and m is the mass of a molecule. We relate the pressure and entropy to the density, tem perature and g by the thermodynamic relations
(ap) (ap) (a p) dp = apT./1p + aT p. gdT + agp.Td~ (11)
and
dS = G~) dp + G~) p. g dT + G~\. p d~. (12)
The internal variable ~ is e liminated by combining eqs (6), (11), and (12). For the pressure we find
Here we have made use of thermodynamic relationships which are developed in the appendix. The transformed quantities are
pdz) = fdte - ZI L dre'kr[p(r, t) -pill (18)
'I, .. (z) and l./Jdz) are obtained in the same way. In eq
(16) bo = (~'Y/s+ 'Y/ v) /mpo, Co is the low-frequency
sound speed, y is the ratio of specific heats,
/3T is th e iso thermal coefficient of therm al expansion, ~p=-Aep/Aa and 6 =-AeT/Aa. In eq (17) CFToAu~i· and (J = A/poC.·. The quantity C1 re presents the co ntributi on of' the internal degree of freedom to the spec ifi c heat when the density is held constant.
The r ight-hand sides of eqs (16) a nd (17) may be simplified by noting that in equilibrium
and
_1_ dp = C;} dp + C'ci/3T dT mpo ypo y
T"dS _ y - 1 d LT. ------ p +c, . C,. /3TP"
(19)
(20)
This s im plifi ca tion is perm iss ible because we are interes ted in co mputing th e co rre la ti on [u nction (pkp-dz) in terms of the equilibrium corre lation fun cti on (Ipkn.
To obta in (pl,p -dz» we solve eqs (15- 17) for pdz) in te rm s of the initial valu es PI" T" and l./J" . Only the terms in vo lving PI. are of interes t because
3. Comparison With a Calculation a Frequency Dependent, Single ation Time, Volume Viscosity
3.1 . Thermal Relaxation
Using Relax-
We are interes ted in the case wh e re th e relax ing param eter is inde pende nt of the de ns ity, i.e., ~p = O. The correla tion func tion is found to be
whe re
and
(PI,P_I, (z) ) ( IPI,12)
F(z) C(z)
+ z [ ak2 + boF + abok4T + CWT ( 1 - ~) ]
+ abok4 + cak2 (1 - l /y)
28 1-1970-68- 7
(21)
(22)
97
C(z) = z4T (l- CdCv )
+ Z3 [1 + ak2T + bok2T(l- C /Cv)]
+ z2[ak2 + bok2 + abok4T+ C6k2T(l - C1ICp )]
+ z[abok4 + Cijk2(l + ak2T/Y)]
+ aPCijk2/y. (23)
For purposes of comparison le t us recall the corresponding expressions derived earlier for the case of a frequency de pendent volume viscosi ty [I). There
F(z) = z3i+ z2[1 + i(ak2 + bok2 )]
+ z [aF + bok2 + b1k2 + abok4i + Cijk2i( 1 - l /y)]
(24)
and
C(z) = iZ" + i l [1 + i(ak2 + bok2 )]
+ Z2 [ak2 + bok2 + b1k2 + '; (Cijk2 + abok4) ]
[Quantiti es with a tilde refe r to the calc ula tion in ref. 1.1 Here bo is the nonrelaxing part of the kine mati c viscosity, the relaxing part is b, / (1 + iwi)
whe re (26)
Direct co mparison of C and C shows that the two expressions are not equivale nt. Thi s may be reali zed by noting that G contains the combination
while the corresponding terms in Care
If we consider the limiting case where the internal degree of freedom provides the only loss mechanism for tran slational degrees of freedom it is possible to make G and (; into equivalent expressions. In this limit where a = bo = O
(27)
and
A simple comparison shows that if 7'= T(l - C /C .. )
(28)
(29)
then C = G in thi s limit. It also can be shown that F = F under th ese conditions.
In an experiment what is measured is not F(z) /G(z) but the real part of F/G when z = iw:
R F(iw ) e G(iw)' (30)
A co mparison of the complete expressions requires that we co mpare
R F(iw ) 'h R F(iw) e--wJt e - -' G(iw) C(iw)
To do this we write
N2 = - w:l 7' + w [ak2 + bok2 + abok47' + C'f,k2T( 1 - l /y) ]
Examination of these expressions reveals; (a) they are not identical term for term , (b) there will be no sig· nificant difference in their values if
(35 )
is small compared to 1. Using the parameters listed in ref. 1. we find that a
is 6.4 X 10- 4 for CC14 at 20 °C and is 2.0 X 10- 2 for CS2 at 20°C. The wavevector was taken to be 105 em- I. This could be increased to a = 0.2 for CS 2 if we consider scattering angles on the order of 1800 using the He-Ne laser as the light source.
(31) The spectra predicted by eqs (31) and (32) have
and
Re ~( iw ) = NJ5 1 + iVJJ2 •
C(iw) D~ + D~
From eqs (22) and (23) it follow s that
and
+ w [ abok4 + C{jF (1 + (/~2T) ].
Here use has been made of eq (29). The corresponding expressions for NI , etc. , are
been evaluated for CC14 and CS 2 using the parameters listed in ref. 1. For CC4 there is no difference between the two curves. This is not surprising considering the small value of a.
(32) The two curves are not identical for
(33)
98
CS 2 (k=3X 105 e m-I).
The central components are shown in figure 1. The
2 X 109
2 3
w, 108 rad/sec
FIG U RE I. Th e central component of Light scattered from CS, as prer/tcted by eq (82 ), curve A and as predicted by eq (3 /) curve B.
The par<l lll e te r s are take n from tahl l' of ref. I. Th t' wa v(> vec tor /. = 3 X lIP e rn -I ,
't I
i,;
I
I J,.
c urve labeled A was obtained using a frequency depende nt volume viscosity (eq 32). The curve labeled B was ob tain ed using the results of this calculation (eq 31). The difference in the amplitude at W = 0 is due e ntirely to the term
in NI • Such a quantity is missing in N I . It occurs as a result of multiplying the thermal diffusivity and the frequency dependent volume viscosity. We note for w = 0, the present calculation yields the same amplitude as the theory which does not include inte rnal relaxation processes .
The Brillouin components for CS 2 are shown in fi gure 2. The differences are due to the relatively large value of IX and to the different values of Wil T
where WIJ is the Brillouin frequen cy. For curve A, WnT= 78 while for curve B, WuT = 113. The large value of IX is the dominant feature as W/i depe nds most strongly on k when WuT ~ l.
The poss ibility of finding a situation where IX is appreciably greater than 0.2 appears to be re mote. Longer relaxation times exis t for rotational isomers [3], however, CdCv is very s mall so IX is s till mu ch less than unity.
3 .2. Structural Relaxation
N ext le t us examine briefly the other limitin g case whic h involves a structural relaxation rather than a thermal relaxation. A purely s tructural relaxation occurs when g7' = O. The examination is speedil y concluded by observing that if we formally identify Ll in eq (16) with bdT = C~ - C~ (see eq (22) of ref. 1) th en thi s case an d the frequ e ncy depe nd en t volum e viscosity calculation yield id entical results. Also no scal ing of the relaxa tion tim es is needed.
3.6 5 3.70 3.75
W, 1010 rad./sec
FIG URE 2 . The Brillouin componenlS of light scattered from CS 2 liS
predicted by eq (32) , curve A alld as predicted by eq I.~ /) carve B. The paramet ers take n from tublt' uf ref. I. The wClve vector /,= 3 X I O ~' 4' !ll I , The hur i·
I.O nl al li nes indicat e the full width at half maximulll fO I" the components.
99
4. Summary and Discussion
The frequency spectrum of density fluctuations in a fluid containing an internal degree of freedom has been calculated. The procedure used was to obtain an equation of motion for the internal variable g by the methods of irreversible thermodynamics and then to solve the resulting set of coupled equations for the s pectrum of the density fluctuations.
Thi s calculation was compared with an earli e r calc u lation which treated the internal degree of freedo m as a frequency dependent volume viscosity rather than as a separate variable with an equation of motion of its own. Two limiting cases were examin ed. Th e structural relaxation case, g-r = 0 was found to yield identical res ults when co mpared with the earli er calc ul ation. The therm a l re laxatio n case, gp = 0 does not agree exac tl y with th e earli er cal c ul a· ti on. The differe nce res ults fro m the ways the s tresses du e to the intern al varia ble are treated. In the prese nt calcu la t ion the s tresses are additive only to first ord er 1'71- In the earli er calc ul a tion th ey we re s tri c tl y additive.
The equation of motion approach work ed out in thi s paper has an intu itive appea l in that one can see direc tl y how th e intern a l vari a ble ente rs the proble m a nd how it is re lated to th e te mpe ra ture a nd the de nsity. Th e introdu ction of a frequ e ncy depe nd e nt volum e vi scos ity in the case of th ermal relaxa ti on is required by s tatis ti cal mec hani cal co ns ide ra tion s [4. SI and is therefore the preferred way to ha ndle inte rnal degrees of freedom whic h are wea kl y coupl ed to the transla tion al degrees of fr eedom .
Thi s ca lcul at ion could be ada pted to a c he mi cally reac ting syste m. Thi s aspec t of flu c tu a ti on th eory a nd light scatte rin g is di scussed at le ngth in r ece nt publi catio ns [8 , 91 so we shall not di sc uss it here.
5. Appendix
S tarting with eq (1) the differential expreSSIOn for the Helmholtz free e nergy is
dA =- SdT + P2 dp+ A§dg. Po
(36)
The Maxwell relations implied by eq (36) are
137)
a nd
~ (a p ) = A ., §p . Pii a ~ p.-r
In equilibrium Ag= O so that
It follows that
and
where gp =(agla p)1 and f/ =(aglaT)p. If we comb ine these results with eqs (11) and (12) we obtain
(ali) _ (ali) + ~A t:~ - - Po u o"p ap '1. < ap T
(al,i) = (ali) + ~A t:t:T :.IT :.IT p(V' « 0" po" u p. g U (J
- = - -A«b{p (as) (as) ap T. g ap T
(38)
and
(as) = (as) _ A t: ,~ aT. p . g aT p « 0,,/'
Finally we note that
mC5 = (a p) 'Y ap T
and
A most interesting: discussion with C. 1. A. Stegeman provided the stimulus for this work.
6. References
[11 R. D. lv[ountain . J. Res . N BS 70A (Phys. a nd Chem. ), No, 3,
~t (
I
207 (1966). Section 2 of thi s paper contain s a s ummary of the 'j formali sm re lat in g li ght sca tte ring and density Au c tuation s.
[21 Karl F. He rzfe ld and T. A. Litovitz. Absorption and Di spers ion of Ultrasonic Waves (Academic IJress , New York . 1959).
[31 S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North Holland Publishing Co. , Amsterdam. 1962).
[41 1.. I. Komarov, Soviet Phys ics JETP 21 ,99 (1965). [5 1 Robert Zwanzig, 1. Chem. Phys . 43, 714 (1965), [61 V. T. Shmatov , Sovi et Physics JETP 6, 1044 (1958) . 171 R. M. Mazo, 1. Chem . Phys . 28, 1223 (1958). [81 13. 1. Be rne a nd H. L. Frisch. J. Chem, Phy s, 47, 3675 (1967). [9] L. Blum and Z. W. Salsburg, 1. Che m. Ph ys" to be publ ished.