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Density Fluctuations in Fluids Having an Internal Degree of Freedom*

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Density Fluctuations in Fluids Having an Internal Degree of Freedom* JOUR NA L OF RE SEA RCH of the National Bureau of Standards - A. Physics and Chemistry Vol. 72A, No.1, Ja nua ry- February 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 freclu e nc y s pectrum of de nsity flu c tuations is c alc ula te d for a fluid whose molecules possess a n inte rnal d egree of freedom whic h is weakl y couple d to the tra ns la ti ona l d egree of freedom of the fluid. Irre ve rsib le the rmodynamics is used to obtain a n equa tion of motion for the inte rna l d egree of freedo m. This equation plus the lin e arize d h ydrod yna mic e qua ti ons a re solve d for the freque ncy s pec· trum of de ns it y flu ctua tio ns. The results a re compared with a simi la r calc ul ati on involving a fr equ e ncy de pe nde nt volume viscosit y. T he results are id e ntical fo r struc tura l re la xa ti on but the re is a difference fu r the rma l re laxation. The origin of the diffe re nce is di scussed a nd the mag nitude of the diffe re nce is e xa mine d for CC I.I a nd for CS2 • Key Words: Bri ll ouin scall e ring, de ns it y flu c tu a ti ons in liquids, Rayle igh scall e rin g, s pectra l cli s tribution of scall e re d li ght , struc tura l re laxation, the rma l re laxatiun, vo lume vi scosit y. 1. Introduction The compa ri son is made in two limiting cases. In the firs t case it is a s umed that the interna l varia bl e The s pectrulll of light scatt e red by de ns it y Au c tua­ de pe nds onl y on the local te mpe ra ture (therm a l tion s in a fluid is proportional to freque ncy s pectrum relaxation). It is found tha t th e two calcu la tions yie ld of de nsity flu c tuations [1].1 The de velopme nt of th e simila r, but not ide ntical, results. The ori gin a nd magni­ gas laser has made it possible to measure the s pectra tude of th e differe nce are exa mined. Carbon te tra­ of long wavele ngth de nsity flu ctuations and the re b y chloride and carbon dis ulfid e are used to illu strate the study fluid s in th e fr e que ncy region of a few giga· di scussion. In the second case it is assumed that the c ycles down to essentia ll y ze ro cycles/sec. int ernal va ri abl e de pe nds onl y on the local d ensity F or de nse fluid s this freque nc y region corresponds (s tructural re laxation). The two calculations yie ld to s low l y varying processes whose time de pendence 'id e nti cal results. may be re asonably expected to be described by the No comparison is made wh en both thermal re laxa­ equations of irreversible thermodyna;nics. Whe n tion and struc tural re laxation a re rresenl. conside ring density fluctuations the appropriate equations are the linearized equations of hydrody· 2. Calculation namics plus two additional relations among the vari­ ables so that a solution to the initial value (or boundary The thermodynamic state ment relating the param­ va lue) proble m is possible. eters for our system is [6] In this pape r we s hall consider de nsity flu ctuations in a flu id whose molec ules possess an internal degree dU = TdS + Pi dp + A g d~ (1) of freedom which relaxes toward loc al equilibrium with PfJ the de nsity and/or the te mperature [2 , 3]. This is carried out in secti on 2. In section 3 this calculation is com­ where U is the energy,S is the e ntropy, T is the pa red with an earlier calc ulation [1] where the internal te mpe rature, p is the density, po is the equilibrium degree offreedom was assumed to result in a freque ncy va lue of the de nsity, P is the pressure, gre presents a n de pe ndent volume vi scosity. The introduction of a internal degree of freedom and Ag is the partial deriva­ freque ncy de pe nd e nt vo lume viscosity is based on tive of the He lmholtz free energy with respect to~ , stati s ti cal mechanica l arguments [4 , 5]. (2) uf;;re~~~ I :e~ rkw .s' P Ol\s o n'd b Y lh c Ad v .n ce dR ese. r c hPmj ecIS A gen cyu flh e Depa ,l men l W e shall furthe r require that Ag= O wh e n the inte rn a l I Fi}!Ufl':5 in braekets iudicati' 111,-, literatuft> rderences a t Ihl' e nd of thi:s paper. degree of freedolll is in local eq u iii bri U III wj t h the 95 density and the temperature (whether or not the dens­ p ity and temperature have reached equilibrium values)_ + (a ) [dT + 7 adT] This may be expressed as aT p. g at (3) (13) where where (4) A similar equation is obtained for the entropy. is the local equilibrium value of g_ A kinetic equation for ~ is obtained by applying the dS +r--adS - (as)- [d p+r-a,dp J methods of irreversible thermodynamics to the sys­ at ap T, g at tem with the result (5) + (as) [ dT+r~ ] aT p. g at where the kinetic coefficient L is > O. As we are con­ cerned with s mall deviations from equilibrium, Ag is + (as) [_Agpdp_Ag'f'dT]' (14) ag p. T Au A« expanded to obtain ag Equations (8), (9), (10), (13), and (14) constitute a Tt=·-L[Agp( p-po) +Ag,r(T -To) +Agg(g-go)] (6) complete set of equations. The next step is to obtain the Fourier-Laplace (space-time) transform of thi s whe re Agp= (aAg/ap h. g, etc. If we expand eq (3) and set of equations. The result is substitute this in eq (6) we obtain zp,..{z) + PolllJ.(z) = Pk , (15) (7) C6k2 6.k2 zr ] The equations of motion for the system are the lin­ (z+bok2 )tjJdz) - [-+---1+ p,,(z) earized hydrodynamic equations: po po zr ap/at+potjJ=O, (8) (9) 2 =tjJh'+--rk [Ph'--- (Cil-+- 6.) Pk I + zr mpo YPo po and (10) (16) Here 1jJ = div v, T)" is the shear viscosity, T)l' is the vol- and ume viscosity and A is the thermal conductivity and m is the mass of a molecule. We relate the pressure _ [z(Y-l) + z2rTtAugpg.r ] p,(z) and entropy to the density, temperature and g by the pof3T CvO+zr) , thermodynamic relations + [z - (C{ /~,;:2T +ak2] T,,(z) ap) (ap) (a p) dp = (apT./1p + aT p. gdT+ agp.Td~ (11) zr [TUSh' ToA u~p 6 ' ) =--- (Y-l--+ Ph and 1 + zr --+e"zr POf3T c,. -O-CdCv)T".]. (17) dS = G~) dp + G~) p. gdT + G~\. pd~. (12) Here we have made use of thermodynamic relation­ The internal variable ~ is eliminated by combining ships which are developed in the appendix. The trans­ eqs (6), (11), and (12). For the pressure we find formed quantities are d p+r-adp = (afJ)- [ dp+r-adP] pdz) = fdte- ZI Ldre'kr[p(r, t) -pill (18) at ap T. g at 96 'I, .. (z) and l./Jdz) are obtained in the same way. In eq C(z) = z4T (l- CdC v ) (16) bo = (~'Y/ s+ 'Y/ v) /mpo, Co is the low-frequency + Z3 [1 + ak2T + bok2T(l- C /Cv)] sound speed, y is the ratio of specific heats, + z2[ak2 + bok2 + abok4T+ C6k2T(l - C1ICp )] + z[abok 4 + Cijk2(l + ak2T/Y)] /3T is th e isothermal coefficient of thermal expansion, 2 ~p=-A e p/Aa and 6 =-AeT/Aa. In eq (17) CFToAu~i· + aPCijk /y. (23) and (J = A/poC.·. The quantity C1 re presents the con­ tribution of' the internal degree of freedom to the For purposes of comparison let us recall the corre­ specifi c heat when the density is held constant. sponding expressions derived earlier for the case of a The r ight-hand sides of eqs (16) a nd (17) may be frequency de pendent volume viscosity [I). There simplified by noting that in equilibrium F(z) = z3i+ z2[1 + i(ak2 + bok2 )] _1_ dp = C;} dp + C'ci/3T dT (19) 2 2 4 2 mpo ypo y + z [aF + bok + b1k + abok i + Cijk i( 1 - l /y)] and (24) and ---T"dS _ --y - -1 d p +cLT, . (20) C,. /3TP" C(z) = iZ" + i l [1 + i(ak2 + bok2 )] This sim plifi cati on is permissible because we are interested in computing the co rrelati on [u ncti on + Z2 [ak2 + bok2 + b1k2 + '; (Cijk 2 + abok4) ] (pkp-dz) in terms of the equilibrium correlation fun cti on (Ipkn. To obtain (pl,p -dz» we solve eqs (15- 17) for pdz) in terms of the initial valu es PI" T" and l./J" .
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