CC BY
12
UDC 536.4
Thermophysical properties of fluids obtained from equilibrium and nonequilibrium fluctuations
J.V. Sengers1, M.A. Anisimov1*
1 Institute for Physical Science and Technology and Department of Chemical and Biomolecular Engineering, University of Maryland, College Park, MD 20742, USA * E-mail: anisimov@umd.edu
Keywords:
thermophysical properties, fluctuations, isothermal compressibility, osmotic compressibility, interfacial tension, heat capacity, thermal conductivity, diffusion coefficient, Soret coefficient, thermal diffusivity, viscosity.
Abstract. Accurate thermophysical-property information (on thermodynamic and transport properties of substances) is important for science and technology, particularly, for oil and gas industry. Traditionally, thermodynamic properties of interest are obtained through the relations between the equilibrium energy, pressure, temperature, and volume, while the measurements of transport properties require macroscopic fluxes of energy and matter. Such macroscopic measurements have achieved very high quality through efforts of the international thermophysical community. In Russia, B.A. Grigoriev has been a leader in providing accurate thermophysical-property data for fluids over more than 50 years. The purpose of this paper is to update the type of information on thermophysical properties of fluids that can be obtained through noninvasive studies of thermal fluctuations by static and dynamic light scattering. Static Rayleigh light-scattering experiments (the measurements of the intensity of light scattering under a certain scattering angle) can provide the information of the isothermal compressibility (in pure fluids) or osmotic compressibility (in binary fluids). Light scattering from fluid-fluid interface gives the information on the interfacial tension. Decay rates of temperature or concentration fluctuations obtained with dynamic light scattering yield data for thermal diffusivities or diffusion coefficients. In addition, less traditional thermophysical-property information can be obtained from light scattering through measuring the intensities and decay rates of thermal fluctuations in nonequilibrium states. Specifically, the presence of a temperature gradient or a concentration gradient causes a spectacular nonequilibrium enhancement of the temperature or concentration fluctuations, which is linked to various transport properties, such as viscosity and Soret coefficient.
Accurate information on thermophysical property of substances is crucially important for science and technology. Traditionally, thermodynamic properties of interest are obtained through the relations between the equilibrium energy, pressure, temperature, and volume, while the measurements of transport properties require macroscopic fluxes of energy and matter. In the former Soviet Union and Russia, B.A. Grigoriev has been a leader in providing reliable thermophysical-property data for fluids over more than 50 years [1].
The purpose of this paper is to update the type of information that can be obtained by studying thermal fluctuations in fluids through light scattering experiments. Commonly, light-scattering experiments to obtain thermophysical property information have studied thermal fluctuations in fluids in thermodynamic equilibrium states. For example, it is well known that decay rates of temperature or concentration fluctuations can be studied with the aid of Rayleigh light scattering, yielding data for thermal diffusivities or diffusion coefficients [2, 3]. In addition, less traditional thermophysical-property information can be obtained through measuring the intensities and decay rates of thermal fluctuations in nonequilibrium states. Specifically, the presence of a temperature gradient or a concentration gradient causes a spectacular nonequilibrium enhancement of the temperature or concentration fluctuations that can be probed by a variety of optical techniques [4].
The paper is organized as follows. The first section reviews the information that can be obtained from a study of density equilibrium fluctuations in one-component fluids and equilibrium concentration fluctuations in binary fluid mixtures. The next two sections examine the additional information that can be obtained by studying the temperature fluctuations in nonequilibrium states under a temperature or concentration gradient.
Thermodynamic properties from equilibrium fluctuations
Fluctuations of density and concentration.
Statistical thermodynamics [5] gives the following relation between the intensity of molecular density fluctuations (defined as a mean-square deviation from the average equilibrium value of the density ((5p)2^ and the isothermal compressibility
1 f dpT
— —
T p {dP
((5p)2) = ,
(1)
where kB is Boltzmann's constant; T is temperature. The smaller the selected volume, V *, the larger the density fluctuations. The quantity
((Sp)2) V * = kBTp2
(m-) = f^) =
дП )P J V *
I L^L
c Un
= K,
T (oS)
p,T
In a scattering experiment, the probing length
scale A is determined by the wave number q = ^
of radiation, which is the magnitude of the scattering vector q. This vector is the difference between the wave vectors of incident and scattered radiations, q = qo - qs. The scattering vector is governed by the conservation of momentum of photons in the process of scattering. The intensity of scattering (number of photons scattered per second) depends on the scattering wave number [2, 3].
The scattering wave number and, consequently, the probing length scale can be tuned by the wavelength of radiation X and by the angle 8, at which the scattered radiation is observed:
(2)
4n . q =— sin
X { 2
(4)
is independent of the selected volume, thus being
( dP ^
a function of state. Since the condition I — I = 0
Up )t
defines the mechanical stability limit (spinodal) in the substance, the mean-square value of the density fluctuations depends how close the substance to instability. Consequently, one should expect diverging fluctuations of density at the gas-liquid critical point and along the gas-liquid spinodal [6]. For the fluctuations of molecular concentration (c is the number of solute molecules per unit volume) the corresponding equation reads
(3)
where n is the osmotic pressure. The derivative
is known as the «osmotic
compressibility». The osmotic compressibility diverges at the critical point of liquid-liquid demixing in a binary mixture in the same manner as the isothermal compressibility diverges at the gas-liquid critical point of a single-component fluid.
Probing fluctuations by light scattering.
Light scattering is caused by the interaction of phonons with optical inhomogeneities caused by the presence of colloid particles in solution or by thermal fluctuations of number of molecules.
Specifically, in light-scattering experiments, the wavelength of light in scattering media is X = X0nX, where X0 is the wavelength of light in vacuum and nX is the refractive index. Since the scattering angle factor, sin(8/2), varies from 0 to 1, the probing length scale optical inhomogeneities are limited by the wavelength of radiation. For X0 = 488 nm (blue light emanating by an Ar laser) and nX = 1,33 (water), the characteristic value of the probing length scale (at 8 = 180°) is A = 30 nm in the measurement of the static light-scattering intensity. This means that static light scattering (SLS) cannot detect optical inhomogeneities of materials at a scale much smaller than that.
In a dynamic light scattering (DLS) experiment (fig. 1), unlike SLS, the input is the correlation between the numbers of photons (light scattering intensity) scattered at times tx and at t2 [3]. This correlation depends on the relaxation time of the optical-inhomogeneities fluctuations, caused by density/concentration fluctuations, or the characteristic time of Brownian diffusion. In particular, with DLS, one can detect and study the timescale of Brownian diffusion or the relaxation time of fluctuations, which depend on the size of the scatters and viscosity of the medium.
The resolution of scattering techniques depends on the contrast. In light scattering, this is the optical contrast, which is defined as the difference between the refractive indices of inhomogeneities and the average refractive index of the medium.
T
M
Laser
N к:
-Correlator
Fig. 1. Schematics of DLS and definition
of the scattering wave number q: PCS is a photon counting system; correlator is a device that computes the time decay of fluctuations; the scattering volume is shown as the darker area in the center of the flask
The most popular experimental methods to observe and study fluctuations in liquid media are SLS and DLS. The density and concentration fluctuations produce optical inhomogeneities of the medium causing fluctuations of the optical dielectric constant e. The dielectric constant at an optical-frequency range is directly related to the refractive index as e = n2. Therefore, the mean-square fluctuations of the dielectric constant caused by fluctuations of density at constant temperature are
«*>'> -(f I «w>.
scattering, the scattering caused by the fluctuation of anisotropy is partially depolarized.
In SLS, the intensity of scattering (the number of scattered photons per second) and the depolarization ratio are measured at a selected wave number. Since fluctuations of physical properties are proportional to the corresponding thermodynamic properties, the intensity is a direct measure of these properties. In DLS, the measurable quantity is the time-dependent intensity-intensity correlation function of scattered photons. This function provides the information on the relaxation rate of fluctuations.
According to theory, for the vertically polarized incident light, the intensity of light-scattering (extrapolated to zero scattering angle) caused by fluctuation of density in a single component fluid reads
/=/ „((&)>) v v=/o [! j ((sp)>) f *,
(7)
(5)
In a binary liquid solution, the fluctuations of concentration make additional contribution to the fluctuations of dielectric constant:
where V* is the scattering volume and the coefficient Io depends on the intensity of incident light, wavelength of light (as Xo4) and on the configuration of the optical design [2, 3]. Since the absolute value of Io is usually unknown, in practice, the intensity of light scattering is shown in arbitrary units or relative to that of a standard, such as benzene, for which the «absolute» intensity value is established [2]. Since, in accordance with Eq. (1), the mean-squared fluctuations of density is proportional to the isothermal compressibility, the formula (7) can be written in the form:
'(e - •» - '- if )y> [f
= I „
de
° 1Ф
kBTp2KT
(8)
(6)
Moreover, the fluctuations of orientations of anisotropic molecules cause fluctuations of optical anisotropy of the isotropic medium. If the fluctuation inhomogeneities are much smaller than the wavelength of light, the intensity of light scattering does not depend on the scattering angle. Such scattering is known as Rayleigh scattering. The scattering angle determines the wavenumber of scattering, defined by Eq. (4). Scattering of monochromatic vertically polarized light caused by fluctuation of density and concentration is also vertically polarized. In contrast to isotropic light
where /(0 ^ 0) is the light scattering intensity extrapolated to zero scattering angle. The formula (8) is confirmed by accurate static light-scattering experiments. However, this formula, being purely thermodynamic, cannot address relaxation dynamics of the fluctuations. It turns out that in condensed fluid media the fluctuating number of molecules in unit volume can relax via two independent dynamic processes: firstly, through diffusion motion, and, secondly, through the propagation of density waves. The density waves are collective motions of molecules caused by random fluctuations of pressure. These waves can be viewed as «quasi-particles», called
phonons, propagating through the condensed matter with speed of sound. In solids, where the diffusion of molecules is usually negligible, the local density fluctuation can be attributed almost solely to phonons. In liquids, the relative contribution of phonons into the relaxation of density fluctuations depends on their thermodynamic properties.
The separation of density fluctuations in two parts can be illustrated by the following thermodynamic arguments. Consider the density as a function of entropy, S, and pressure, P. Then the fluctuation of density can be represented as
5P(S,p = 5S + № SP.
dS )p \dP/S
(9)
Because of the statistical independence of 8S and SP, (&S8P) = 0 (see further section "Temperature fluctuations in equilibrium"),
(<w)=(f| (m)(io)
dp
dP
f I ^P I - f I ^P
t p ds jp ~ cpv [ p dp/t
one obtains for V = V *:
I(0 ^ o)=io|-^j kjpX =
■ 1• il I
TaP PC
where aP = p
1 ( Ф.
dT
1 i Ф
KT KS _ CP _ Y
Cv
of density. The adiabatic fluctuations of pressure («density waves») are phonons propagating
with speed of sound u = , where A is the
2%
wavelength of sound and ra is the frequency (in radians per second). Photons of light radiation are scattered due to the collisions with phonons in all directions. A light-scattering spectrometer, analyzing light scattering of a certain wave number q, probes only those phonons that have the corresponding wavelength A = 2nq. Collisions of photons with travelling phonons or, in other words, diffraction of photons on the density waves, change the frequency of scattered light due to the Doppler effect. The change of the frequency produces two satellites in the spectrum of the scattered light with frequencies rao ± Ara, symmetrically located on the two sides of the initial light frequency ra0:
Therefore, the density fluctuations can be represented as the sum of isobaric fluctuations of entropy and adiabatic fluctuations of pressure. Since
2% . 0 „ u . 0 Aœ = uq = — sin— = 2œo —sin—.
X 2 c 2
(14)
(11)
(12)
is the isobaric expansivity
and ks = p j is the adiabatic compressibility.
The ratio of these two contributions into the light-scattering intensity is expressed through the ratio of the isobaric and isochoric heat capacities
(13)
and is known as the Landau-Placzek ratio [2].
Spectrum of molecular scattering. The physically different nature of relaxation dynamics of the adiabatic fluctuations of pressure and isobaric fluctuations of entropy determines the spectrum of light scattered by fluctuations
Equation (14) was obtained by Brillouin in 1922, so that the effect is commonly called the Brillouin scattering, and the spectrum satellites called the Brillouin components. In fact, this equation was obtained even earlier, in 1918 by Mandelstam in Russia, but was published later because of the Russian civil war. Consequently, in the Russian literature this effect is called Mandelstam-Brillouin scattering. Experimentally, Eq. (14) was confirmed in 1930 by Gross, also in Russia [2].
The frequency shift in the Brillouin scattering is very small relatively to the frequency of the incident light since the ratio Ara/ra0 is proportional to the ratio of speed of sound in water (about 1,5-103 m/s) and speed of light in water (about 2-108 m/s). The frequency of red light is about 4-1015 Hz and, consequently, the frequency of elastic density waves, probed at the scattering angle of 60°, is about 3-1010 Hz. Therefore, the observation of such a small shift in the light frequency required a highresolution spectrometer, usually a Fabry-Perot interferometer [2]. Nevertheless, for sound propagation, the frequency ~1010 Hz is very high; such sound is called «hyper sound». The speed of hyper sound may be different from the speed of low-frequency («thermodynamic») sound, given by u = kst0'5, because of the existence of relaxation processes.
The Brillouin components are not infinitely sharp lines; they have a certain line width associated with attenuation of the elastic density waves. The line width
5(Дш) = Г Bq2
a, = ■
2M
2up
(
—Л + с + к
3
J_
с
Y
of entropy fluctuation, r = —, is given by
r = DTq2.
This rate determines the half-width of the
Rayleigh component, = r. Since the thermal
diffusivity, DT =
(15)
pCP
the corresponding line
where rB is a kinetic coefficient measured in the same units as the diffusion coefficient and kinematic viscosity, m2/s, is directly linked to the absorption coefficient of sound in a viscous medium, ra, as
(16)
where q is the shear viscosity, k is the thermal conductivity, and q is the «bulk viscosity». The bulk viscosity is phenomenologically defined through the excess sound absorption that cannot be account for the shear viscosity and thermal conductivity.
The adiabatic pressure fluctuations represent only part of the total fluctuations of density. Another part is provided by isobaric fluctuations of entropy. This kind of fluctuations causes the unshifted («central») component (which is known as the Rayleigh line) in the spectrum of molecular light scattering (fig. 2).
Relaxation of the isobaric entropy to the equilibrium value undergoes by diffusion of heat. Consequently, the rate of diffusive relaxation
(17)
width becomes very narrow near the gas-liquid critical point at which the isobaric heat capacity diverges. The intensity of the Rayleigh line and the intensity of two Brillouin components are given by Eq. (12) and (13). At the liquid-gas critical point, the isothermal compressibility diverges, as strongly as CP, thus the ratio also diverges. On the other hand, in almost incompressible solids CP = CV and the central line practically disappears. Remarkably, in liquid water at 4 °C exactly CP = CV, because at this temperature the isobaric thermal expansivity aP = 0, and remains very small around this temperature. Therefore, the central line is practically invisible in water at ambient conditions.
DLS. Even away from the critical point, the rate of relaxation of entropy fluctuations is too slow and the linewidth of the unshifted Rayleigh component is too narrow to be measured with an optical spectrometer because of restrictions associated with finiteness of the aperture. For a typical value of DT = 105 m/s in liquids at ambient conditions, the linewidth of the Rayleigh component lies in the 100 MHz frequency range at the commonly used scattering angle of 90°, for which q2 = 1014 m2. This is two orders narrower than the linewidth and the shift of the Brillouin components. The problem of detecting the linewidth of the unshifted Rayleigh component is even more challenging
Rayleigh peak
7 1
4 1 DTqa
Brillouin peak 1
A \
/ 4
*- Дю = ua ->
Amti-BriUouin peak Ib
Frequency
ro0 - Aro ro0 ro0 + Aro
Fig. 2. Spectrum of isotropic molecular scattering in a single-component liquid
in binary fluids where fluctuations of the concentration cause additional, often dominant, scattering. The relaxation of the concentration fluctuations in binary mixtures is diffusive, like the relaxation of heat. The relaxation rate depends of the coefficient of mutual diffusion, D, as
Г = -
1
t d (q)
= Dq2.
D =
kj _ 6%\\R
is a digital correlator that analyses the correlation of the number of photons scattered at the time t and t + St. The time-dependent correlation function for a single-exponential relaxation process is given by a single exponential as
g 2(t ) -1 = A ехр(-2П ),
(20)
(18)
A practical difference between Eqs. (18) and (17) is that the mutual diffusion coefficient D in binary liquid solutions is usually two orders of magnitude smaller than the heat diffusion coefficient DT. Correspondingly, the linewidth of the Rayleigh component caused by concentration fluctuations lies in the MHz frequency range at the scattering angle of 90°.
Moreover, there are two important examples of an extremely slow diffusion and, consequently, extremely narrow Rayleigh components. The first example is the sharp decrease of the heat diffusion at the gas-liquid critical point in single-component fluids, because of the diverging isobaric heat capacity, and of the mutual diffusion at the fluid-fluid (gas-liquid or liquid-liquid) critical point in binary fluids. The effect of the sharp decrease of the relaxation rate of fluctuations near the critical point is called «critical slowing down». The second example is the light scattering caused by Brownian particles. In accordance with the Stokes-Einstein formula, the diffusion coefficient of Brownian particles is inversely proportional to their size R:
(19)
If the nanoparticles are size-calibrated, Eq. (19) enables one to calculate the shear viscosity through the diffusion coefficient obtained by DLS.
For the particles with the radius of 50 nm, the diffusion coefficient is two orders of magnitude smaller than that in a molecular solution (far away from the critical point) with the same viscosity. As a result, the linewidth of the Rayleigh component caused by critical fluctuations or by Brownian motion is reduced to a MHz, or even to kHz, frequency range.
The physical characteristics of the scattering medium measured in a DLS experiment is the dynamic intensity-intensity autocorrelation function of scattered photons g2(t) = (I(t)I(t + 5t)), where St is a so-called "delay time". A major part of DLS spectrometers
where A is the amplitude of the correlation function, which depends on the intensity and coherence of scattering. The diffusive character of the relaxation is verified by checking whether the rate r is a linear function of q2 as predicted by Eq. (18).
Divergence of the isothermal compressibility at the critical point obtain by SLS. Fig. 3 demonstrates the growth of density fluctuations, which were studied by computer simulations of the lattice-gas model of a single-component fluid. At the critical point, the size and amplitude of the fluctuations diverge. This effect causes the critical opalescence - the light scattering intensity becomes very large.
Since the scattering intensity extrapolated to zero scattering angle is proportional to the isothermal compressibility, SLS is the most accurate way to obtain the character of the temperature dependence of this property. The pioneer results for near-critical carbon dioxide, demonstrated in Fig. 4, were obtained almost 50 years ago!
Interfacial tension near the critical point obtained from fluctuating fluid interface. The vapor-liquid interfacial tension vanishes at the critical point. Fig. 5 presents an accurate approximation for water, provided by the US National Institute of Standards and Technology (NIST). At low temperatures, this property can accurately be obtained by well-developed classical methods, such as the Wilhelmy-plate method. However, in the critical region the interfacial tension becomes so low that only light scattering from the fluctuations of interface can provide reliable data. Fig. 6 demonstrates such data for near-critical xenon.
Osmotic compressibility and diffusion coefficient in a near-critical binary mixture. Like the density fluctuations are responsible for the critical opalescence in single-component fluids, the concentration fluctuations cause the critical opalescence near the critical point of demixing in liquid solutions. The osmotic compressibility of a binary liquid mixture is proportional to the
Fig. 3. Microscopic inhomogeneities in fluids caused by fluctuations of density far away from (a) and very close to (b) the gas-liquid critical point [7]
103
M M
102
счО
Œ
101
100
10-1
O data by Cannell and Lunacek, 1972 • results of the P-V-T measurements [6, 7]
Nul 1,69-10" 0t-1,219
10-
10-:
10-
Fig. 4. Isothermal compressibility (denoted in this figure as pr = kt) of near-critical carbon dioxide above the critical point along the critical isochore as a function T-T
of temperature, t =--, obtained by static
80
60
40
20
10-1 t
100
200
300
400
T,°C
Fig. 5. Vapor-liquid surface tension (c) of water
(https://webbook.nist.gov/chemistry/fluid)
light scattering
0
0
light-scattering intensity. Fig. 7 demonstrates the divergence of the osmotic compressibility extrapolated to zero scattering angle (dashed line). This property obtained for finite size, defined by a finite wave number through Eq. (5), does not diverge. The limiting finite value depends on the scattering wave number. This phenomenon is known as a finite-size effect [7].
The coefficient of mutual diffusion in a binary mixture vanishes at the critical point
of liquid-liquid coexistence. This coefficient can be accurately obtained with DLS measurements. Fig. 8 demonstrates the behavior of the diffusion coefficient of the solution of polystyrene in methylcyclohexane near the critical temperature of demixing. In fact, the measured property vanishes at the critical point only being extrapolated to zero scattering angle, corresponding to zero wave number and, consequently, to a macroscopic length scale. At finite wave numbers the diffusion
• 102
° 101
100
10-
10-
10-:
10-
O Zollwegetal.,1972, reproduced from refs. [6, 7] — theoretical prediction _ ofaxi1-26
10-
10-
10-1
Fig. 6. Vapor-liquid interfacial tension near
the critical point of xenon by light scattering
from the fluctuating interface as a function
T-T of t =-
* 10° Г
1
ii
1 10-1 «
S3
10-'
10-
Anisimov, et al. 2002, reproduced from ref. [7]: □ 0=30° o 0 = 150°
100
\t\
10-1
10-
10-
10-
10-2 104 e = (T - Tc)/T
Fig. 7. Osmotic compressibility of the solution
of polystyrene in methylcyclohexane near the critical temperature of demixing, obtained for two mesoscopic length scales defined by the scattering angles (see the legend)
101'
Q
10-
10-
10-
: Kostko et al., 2007, " reproduced from ref. [7]: i O 30° ■ 90° - " 150° ■ — theoretical prediction
10-
10-
10-
10-2 10-1 (T - TcyT
Fig. 8. Diffusion coefficient of the solution of polystyrene in methylcyclohexane near the critical temperature of demixing: the smaller
the scattering angle (see the legend) the closer the diffusion coefficient to its macroscopic value. The diffusion coefficient vanishing at the critical point is obtained only at zero angle
coefficient reaches its limiting value depending on the value of the scattering angle. This is again a mesoscopic finite-size effect.
The most important advantage of measuring the diffusion coefficient by DLS is the fact that this method is noninvasive: there is no need
to establish a concentration gradient for generating a macroscopic flux of matter. By DLS, transport properties are obtained in an equilibrium system from the rate of spontaneous relaxation of fluctuations. In the next section, we consider the additional information on thermophysical properties that can be obtained from fluctuations in a nonequilibrium steady state (in the presence of temperature and concentration gradients).
Transport properties obtained
from nonequilibrium fluctuations of temperature
and concentration
Temperature fluctuations in a one-component fluid. Since theory of nonequilibrium fluctuation has been developed relatively recently, in following sections the theoretical results are considered in more details. Let's consider a fluid layer between two horizontal conducting plates with temperatures T1 and T2, separated by a distance L. The relevant parameter governing this arrangement is the Rayleigh number Ra defined by
Ra =
apL'gVT vDT '
(21)
where aP is the isobaric thermal expansion coefficient; g - the gravitational acceleration vector; VT - the temperature gradient; v - the kinematic viscosity; DT - the thermal diffusivity. The arrangement is stable without any convection
as long as Ra is smaller than the critical Rayleigh number Rac. Although the theory in this paper can be extended to positive Rayleigh numbers Ra < Rac [8], authors only consider here thermal nonequilibrium states when the fluid layer is heated from above, i.e., T1 > T2, thus only for negative values of the Rayleigh number.
Authors adopt the fluctuating hydrodynamics originally proposed by Landau and Lifshitz [9] and further discussed by Ford and Uhlenbeck [10] for thermal fluctuations in equilibrium, since it can also be extended to nonequilibrium states [4]. In this paper, the diffusive temperature and viscous fluctuations are only considered, and the propagating sound modes are not included. That is, the fluctuations at constant pressure are considered (see Eq. (10)). Moreover, since any temperature fluctuations are small compared to the actual temperature, one can linearize the hydrodynamic equation for the fluctuations. In this approximation the equation for the dependence of the temperature on the time t is given by
PC
—+uW
dt
= VQ,
(22)
Q = - kVT + 5Q,
(23)
PC
^ + uW dt
= kV2T -V5Q.
(24)
T = T0 + ST(r, t); u = 0 + 5u(r, t).
(25a) (25b)
In equilibrium VT = 0 and substitution of Eq. (25a) into Eq. (24) yields
pCP — = kV25T -V5Q. dt
(26)
The correlation function for the random contributions to the heat flux is given by the fluctuation-dissipation theorem:
(Щ (r, t)SQj (rt')) = = 2kBKT025j. 5(r - r ')S(t -1 ').
(27)
If one applies a spatiotemporal Fourier transformation to Eq. (26) and to the resulting correlation function an inverse Fourier transformation, one obtains for the temperature fluctuations [4, 11]:
(ST (q, t )ST (q,0)) = -C- exp(-DTq 2t ),
Pcp
(28)
where p is the density; CP the isobaric specific heat capacity; u - the fluid velocity, and Q - the heat flux. In ordinary hydrodynamics the heat flux Q is related to the temperature gradient VT by the law of Fourier Q = -kVT, where k is the thermal conductivity coefficient. In fluctuating hydrodynamics, one realizes that Fourier's law is only valid on the average, so that
where q is the wave vector of the fluctuations. Hence, we recover the well-known result that the decay rate of the temperature fluctuations in equilibrium is determined by the thermal diffusivity DT = K/pCP.
Temperature fluctuations in nonequilibrium (Tj > T2). If we substitute Eqs. (25) into Eq. (24) with VT ^ 0 we obtain
PC
dST
~dT
+ 5uVT„
= kV 25T -V5Q.
(29)
In contrast to equilibrium, we see that the temperature fluctuations are now coupled to the velocity fluctuations that satisfy a fluctuating Navier - Stokes equation (at constant pressure):
where 8Q represents random molecular contributions to the heat flux, such that on average (5Q) = 0. Substitution of Eq. (23) into Eq. (22) yields a «fluctuating» heat equation:
35u 1 „„
-= vV25u + — V5S,
dt p
(30)
Temperature fluctuations in equilibrium (Tj = T2). We consider temperature fluctuations ST(r, t) around the average temperature T0 and velocity fluctuations 8u(r, t) around the average velocity u = 0 as a function of the position r and time t:
where S is a fluctuating stress tensor, whose correlation function according to the fluctuating-dissipation theorem is given by
5 Sj (r, t )SSu (r', t') =
= 2kBT0pu j + Stf 8 Jk )S(r - r')8(t -1'). (31)
If one applies a spatiotemporal Fourier transformation to Eqs. (29) and (30) and to the resulting correlation function being an inverse Fourier transformation, one now obtains for the temperature fluctuations [4, 11]: (ST (q, t )5T (q,0)) =
= ^ [(1 + At )exp(-q 2t) - Av exp(-vq 2t)]. (32)
pCp
The nonequilibrium amplitudes AT and av are given by
AT — "
A. =■
C__^ (д,УТ0)2
T (v2 - DT 2) DT q6
Cp_jq^Tf
T(v2 -DT2)
(33a)
(33b)
where q„ is the magnitude of the component q„ of the wave vector q that is parallel to the horizontal conducting plates with temperatures tj and t2 (see above), i.e., perpendicular to the temperature gradient vt0. In this paper let's always consider wave vectors as q„, which is the configuration often adopted in experiments [12, 13], so that
At —
A. =
Cp
v (vt0)2
T(v2 - DT2) DT q4
CD
(vt0)2
T (v2 - DT 2) q4
(34a)
(34b)
Many experimentalists measure only the intensity of the nonequilibrium fluctuations:
(st (q)2):
к T
pCp
1 + -
CP
(vt0)2
(35)
T0 DT (v + DT) q
Equation (32) for the temperature fluctuations in a fluid subjected to a temperature gradient was first discovered by Kirkpatrick et al. [14]. Derivation of this result by using fluctuating hydrodynamics was considered by Ronis and Procaccia [15] and by Law and Sengers [16].
The validity of Eq. (32) for the nonequilibrium enhancement of the temperature fluctuations has been confirmed experimentally with great accuracy from light-scattering experiments as shown in fig. 9.
Equation (35) implies that the intensity of the nonequilibrium fluctuations will increase rapidly with smaller values of the wave number, i.e., at longer length scales. This increase cannot go on indefinitely and for large wave numbers the intensity will become suppressed because of gravity effects. Accurate experimental imaging techniques have been developed for measuring these fluctuations at larger wave numbers that cannot be probed by light scattering [4, 17-20].
Applications. From Rayleigh-scattering measurements of the decay rate of the fluctuations in equilibrium one can determine the thermal
diffusivity, DT = ——, in accordance with Eq. (28)
pCF
and infer the thermal conductivity k, assuming the density p and the isobaric specific heat capacity are known [21, 22]. Knowing the value for the thermal diffusivity one may then try in principle to determine the kinematic viscosity v from the time dependence of the correlation function when a temperature gradient is applied in accordance with Eq. (32). This procedure may yield only a limited accuracy since the viscous fluctuations decay much faster than the temperature fluctuations. However, from measuring the intensity of the nonequilibrium fluctuations one should be able to determine
^ 120
20
40
X
- / Experimental
q value, cm-1:
■ 1528
о 1663
/ 1811
□ 2034
if 2122
jr 2255
♦ 2484
2835
— prediction
from Eq. (34) -1-1
20 16 -12 -8 -4 -0
60
80
100
|vr |2
10-10 K2 cm2
10-10 K2 cm2
Fig. 9. Nonequilibrium fluctuation amplitudes AT and Av as a function of for liquid toluene at 40 °C [12]
(vr0)2
0
0
4
q
q
the nonequilibrium enhancement amplitude p
CD
T0 DT {v+DT )
dw
--h uvw
dt
= -VJ,
(36)
J = -pDVw + SJ,
(37)
SJa(r, t )8J0(r', t ') =
= 2kBT0pDl , „
)tp
5aB5(r - r ')S(t -1 '),
(38)
where I —I = — f-^^l = — I is the
v )T P w ^ 5n )T P c ^ 3n,
osmotic compressibility and p. is the exchange chemical potential (conjugate to concentration). Substitution of Eq. (37) into (36) yields
dw
--h uvw
dt
= pDVw -VSJ,
(39)
in Eq. (35) accurately and thus
deduce the value of the kinematic viscosity v. Hence, Rayleigh scattering of nonequilibrium temperature fluctuations provides a method for measuring the thermal diffusivity and kinematic viscosity of fluids simultaneously.
Transport properties from nonequilibrium fluctuations of concentration
Fluctuating equation for concentration.
In general, both temperature fluctuations and concentration fluctuations will be present in mixtures [13, 23]. An important parameter is the Lewis number Le, which is the ratio of thermal diffusivity a governing the decay rate of the temperature fluctuations and the mass diffusion coefficient D governing the decay rate of the concentration fluctuations: Le = a/D. In many liquid solutions Le »1 and coupling between the concentration mode and the heat mode can be neglected. In first approximation the time dependence of the mass fraction w of the solute is then given by
which is the fluctuating equation for the concentration.
Concentration fluctuations in equilibrium
(Vw = 0). Authors consider concentration fluctuations Sw(r, t) around the average concentration value w0 and velocity fluctuations Sw(u, t) around the average velocity u = 0 as a function of the position r and time t:
T = T0 + 5T(r, t); u = 0 + Su(r, t).
(40a) (40b)
In equilibrium (Vw = 0) and substitution of Eq. (37) into Eq. (36) yields
p — = pDV25w -V&7. dt
(41)
where J is the mass diffusion flux of the solute, which according to Fick's law is related to the diffusion coefficient by J = pDVw. Just like Fourier's law (see above), Fick's law is only valid on the average and in fluctuating hydrodynamics Fick's law is replaced by [23, 24]
Just as the temperature-fluctuation correlation function (see Eq. (28)), could be obtained from Eqs. (26) and (27), one finds from Eqs. (37) and (38) for the concentration-fluctuation correlation function:
(Sw(q,t)Sw(q,0))= — | exp(-Dq2t). (42)
P )T_p
Hence, we recover from fluctuating hydrodynamics the well-known result that the decay rate of the concentration fluctuations is determined by the diffusion coefficient D.
Concentration fluctuations in nonequilibrium (Vw ^ 0). Next, we consider a liquid solution with a concentration gradient but in a hydrodynamically stable state without any convection, so that the average velocity (u) = u0 = 0 in accordance with Eq. (40b). Substitution of Eqs. (40) into Eq. (39) now yields
where according to the dissipation-fluctuation term the correlation function of the random contribution to the mass flux is given by [11]
dSw ~dt~
+ 5wVw„
= pDV 25w -V5J.
(43)
Just as in the case of the temperature fluctuations, we see that in nonequilibrium there is a similar coupling between the concentration fluctuations and the velocity fluctuations. Hence, from Eqs. (40), (41), (38) and (43) one obtains for the nonequilibrium concentration fluctuations
(Sw(q, i )Sw(q,0)) = ^ M x
P JTp
\+_L f^w V (q«Vw°)2 л
vD ^Sp
exp(-Dq 2t)
(44)
T, K 298,15 323,15 348,15 373,15
Shadowgraph (this work) О О О О
ref45 + p (this work) О
ref 44 •
P, MPa 0,1 10 20 30 40 Shadowgraph (this work) O O O O O DLS (this work) O
Fig. 10. Pressure (a, b, c, d) and temperature (e, f, g, h) dependencies of the Fick diffusion coefficient (a, e), thermal diffusivity (b, f), kinematic viscosity (c, g) and Soret coefficient (d, h) measured by the shadowgraph method in comparison to literature data and the results from DLS for the mixture containing equal masses of THN and n-C^H^ [20]
Considering wave vectors q = q„ we obtain for the intensity of the nonequilibrium concentration fluctuations
(5w(q)2) =
KT
i+—i —
vD
dw
(Vw0)2
(45)
to be compared with Eq. (35) for the nonequilib-rium temperature fluctuations.
A convenient experimental procedure for establishing a concentration gradient Vw0 is applying a temperature gradient VT0 which induces a concentration gradient through the Soret effect [12, 25, 26]:
Vw0 =- w0(l - w0) ST VT0,
1 +
{w0(l - W0)}2 ST vD
(VT0)2
Applications. Determining the decay rate of the concentration fluctuations in equilibrium from dynamic Rayleigh scattering in accordance with Eq. (42) is a well-established procedure for measuring the diffusion coefficient of liquid mixtures [21, 27]. Measurements of the intensity of the concentration fluctuations in the presence of a temperature gradient should yield accurate values for the amplitude of the nonequilibrium enhancement in Eq. (47) and, hence, for the Soret coefficient ST if a reliable value of the osmotic
compressibility
dw
is available [20, 27]
(46)
where ST is the Soret coefficient. Then the intensity of the nonequilibrium concentration fluctuations is given by
(5w(q)2) = ^ x
(47)
Again, the validity of Eq. (47) for the nonequilibrium concentration fluctuations has been verified by light-scattering experiments [26].
(fig. 10).
Since the amplitude of the nonequilibrium enhancement is proportional to the square of the Soret coefficient, this method could yield accurate values for the Soret coefficient, a very important transport coefficient. Because concentration fluctuations decay much more slowly than temperature fluctuations, measuring nonequilibrium concentration fluctuations is considerably easier than measuring nonequilibrium temperature fluctuations. Hence, with few exceptions [10, 11, 25] almost all currently available experimental studies have dealt with nonequilibrium concentration fluctuations in solutions.
References
1. GRIGORIEV, B.A. Research on thermophysical properties of fluids at the Grozny Petroleum Institute (USSR). International Journal
of Thermophysics, 1990, no. 11, pp. 239-250, ISSN 1572-9567.
2. FABELINSKII, I.L. Molecular scattering of light. NY: Plenum Press, 1968.
3. CHU, B. Laser light scattering: Basic principles and practice. 2nd ed. San Diego: Academic Press, 1991.
4. ORTIZ DE ZARATE, J.M., J.V. SENGERS. Hydrodynamic fluctuations in fluids and fluid mixtures. Amsterdam: Elsevier, 2006.
5. LANDAU, L.D., E.M. LIFSHITZ. Statistical physics. NY: Pergamon, 1959.
6. ANISIMOV, M.A. Critical phenomena in liquids and liquid crystals. London: Gordon and Breach, 1991.
7. ANISIMOV, M.A. Introduction to mesoscopic thermodynamics: Lecture notes. College Park, Maryland: University of Maryland, 2021.
8. ORTIZ DE ZARATE, J.M., J.V SENGERS. Fluctuations in fluids in thermal nonequilibrium states below the convective Rayleigh - Benard instability. Physica A: Statistical Mechanics and its Applications, 2001, vol. 300, no. 1-2, pp. 25-52, ISSN 0378-4371.
9. LANDAU, L.D., E.M. LIFSHITZ. Fluid mechanics. NY: Pergamon, 1959.
10. FOX, R.F., G.E. UHLENBECK. Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamic fluctuations. Physics of fluids, 1970, vol. 13, pp. 1893-1902, ISSN 1070-6631.
11. ORTIZ DE ZARATE, J.M., J.V SENGERS. Fluctuating hydrodynamics and fluctuation-dissipation theorem in equilibrium systems. In: Non-Equilibrium Thermodynamics with Applications / ed. by D. BEDEAUX, S. KJELSTRUP and J.V SENGERS. Cambridge: RSC Publishing, 2016, pp. 21-38.
12. SEGRE, P.N., et al. Rayleigh scattering in a liquid far from thermal equilibrium. Physical Review A, 1992, vol. 45, pp. 714-724, ISSN 2469-9926.
13. LI, W.B., et al. Small-angle Rayleigh scattering from nonequilibrium fluctuations in liquids and liquid mixtures. Physica A: Statistical Mechanics and its Applications, 1994, vol. 204, no. 1,
pp. 399-436, ISSN 0378-4371.
14. KIRKPATRICK, T.R., E.G.D. COHEN, J.R. DORFMAN. Light scattering by a fluid in a nonequilibrium steady state. II. Large gradients. Physical Review A, 1982, vol. 26, pp. 995-1014, ISSN 2469-9926.
15. RONIS, D., I. PROCACCIA. Nonlinear resonant coupling between shear and heat fluctuations
in fluids far from equilibrium. Physical Review A, 1982, vol. 26, pp. 1812-1815, ISSN 2469-9926.
16. LAW, B.M., J.V. SENGERS. Fluctuations in fluids out of thermal equilibrium. Journal of Statistical Physics, 1989, vol. 57, no. 3-4, pp. 531-547, ISSN 0022-4715.
17. BRUYN, J.R., de, et al. Apparatus for the study of Rayleigh - Benard convection in gases under pressure. Review of Scientific Instruments, 1996, vol. 67, pp. 2043-2967, ISSN 0034-6748.
18. TRAINOFF, S.P, D.S. CANNEL. Physical optics treatment of the shadowgraph. Physics of Fluids, 2002, vol. 14, pp. 1340-1363, ISSN 1070-6631.
19. CROCCOLO, F., et al. Nondiffusive decay
of gradient-driven fluctuations in a free-diffusion process. Physical Review E - Statistical, Nonlinear, And Soft Matter Physics, 2007, vol. 76, no. 4, pt. 1, no. 041112, ISSN 1539-3755.
20. WU, W., et al. Simultaneous determination of multiple transport properties over a wide range of temperatures and pressures from the analysis of non-equilibrium fluctuations by the shadowgraph method. Journal of Chemical Physics, 2020, vol. 153, no. 144201-1-12, ISSN 0021-9606.
Определение теплофизических свойств флюидов, находящихся в равновесном и неравновесном состояниях
Дж.В. Сенгерс1, М.А. Анисимов1*
1 Физико-технический институт и Департамент химических и биомолекулярных технологий Мэрилендского университета, США, MD 20742, Колледж-Парк. * E-mail: anisimov@umd.edu
Тезисы. Во всех областях науки и техники, но особенно в нефтегазовой отрасли, критически важно иметь точные данные о теплофизических - термодинамических и транспортных - свойствах веществ. Традиционно необходимые термодинамические характеристики определяют из соотношений равновесной энергии, давления, температуры и объема, тогда как измерения транспортных свойств требуют макропотоков энергии и вещества. Благодаря усилиям мирового научного сообщества такие макроскопические измерения достигли очень высокого качества. В России, например, на протяжении последних 50 лет ведущим специалистом в области определения теплофизических свойств флюидов считается Б.А. Григорьев.
В настоящей статье собрана актуальная информация о современных способах определения теплофизических свойств жидкостей и газов посредством неинвазивных исследований температурных колебаний методами статического и динамического рассеяния света. Эксперименты в области статического
21. LEIPERTZ, A. Transport properties of transparent liquids by photon-correlation spectroscopy. International Journal of Thermophysics, 1988, vol. 9, is. 6, pp. 897-909, ISSN 0195-928X.
22. KRAFT, K., M. MATOS LOPES, A. LEIPERTZ. Thermal diffusivity and thermal conductivity
of toluene by photon correlation spectroscopy: A test of the accuracy of the method. International Journal of Thermophysics, 1995, vol. 16, is. 2, pp. 423-432, ISSN 0195-928X.
23. LAW, B.M., J. C. NIEUWOUDT. Noncritical liquid mixtures far from equilibrium: The Rayleigh line. Physical Review A, 1989, vol. 40, is. 7,
pp. 3880-3885.
24. FOCH, J. Stochastic equations for fluid mixtures. Physics of Fluids, 1971, vol. 14, pp. 893-897, ISSN 1070-6631.
25. VAILATI, A., M. GIGLIO. q Divergence
of nonequilibrium fluctuations and its gravity induced frustration in a temperature stressed liquid mixture. Physical Review Letters, 1996, vol. 77, no. 8, pp. 1484-1487, ISSN 0031-9007.
26. LI, W.B. et al. Light scattering from nonequilibrium concentration fluctuations in a polymer solution. Journal of Chemical Physics, 2000, vol. 112, no. 20, pp. 9139-9150, ISSN 0021-9606.
27. RAUSCH, M.H. Mutual diffusion in binary mixtures of ionic liquids and molecular liquids by dynamic light scattering (DLS). Physical Chemistry Chemical Physics, 2011, vol. 13, no. 20, pp. 9525-9533, ISSN 1463-9076.
рэлеевского светорассеяния (измерение интенсивности рассеяния света под определенным углом рассеивания) могут дать информацию об изотермической (в чистых веществах) либо осмотической (в бинарных смесях) сжимаемости. Рассеяние света на границе раздела жидкостей дает информацию о межфазном натяжении. По результатам измерения методом рассеяния света скорости затухания колебаний температуры или концентрации раствора определяют температуропроводность или коэффициенты диффузии. Вдобавок менее привычные теплофизические данные можно получить путем измерения интенсивности и скорости затухания температурных колебаний для системы в неравновесных состояниях. Колебания температуры или концентрации особенно усиливаются при наличии градиентов температуры или концентрации. Это условие обеспечивает дополнительную возможность определять различные транспортные свойства веществ, такие, например, как вязкость или коэффициент Соре.
Ключевые слова: теплофизические свойства, колебания, изотермическая сжимаемость, осмотическая сжимаемость, межфазное натяжение, теплоемкость, коэффициент диффузии, коэффициент Соре, температуропроводность, вязкость.
