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Transport Behavior of Common, Pourable Liquids
C H A P T E R
6 Transport Behavior of Common, Pourable Liquids: Evidence for Mechanisms Other Than Collisions O U T L I N E 6.1 Transport Properties Expected for Inelastic Collisions of Finite Size Atoms in Liquids
182
6.2 Measurements of Transport Properties for Pourable Liquids
184
6.3 Trends in Transport Properties of Pourable Liquids 186 6.3.1 Mass and Density Effects at Ambient Conditions 186 6.3.2 Ratios of Transport Properties at Ambient Conditions 188
6.3.3 The Temperature Dependence of Transport Properties 189 6.3.4 Effect of Pressure on Transport Properties 195 6.4 Mechanisms of Transport in Liquids
196
References
197
Websites
199
There is no water, so things are bad. If there were water, it would be better. But there is no water. T.S. Eliot, The Waste Land (1922)
The unusual physical properties of liquid water have received considerable attention in the physical sciences (e.g., Chaplin, 2018), due to the importance of this simple fluid to life itself. From a geochemical perspective, the existence of life as we know it on Earth is predicated on its solid being less dense at the freezing point. If this behavior was reversed, as
Measurements, Mechanisms, and Models of Heat Transport DOI: https://doi.org/10.1016/B978-0-12-809981-0.00006-1
181
© 2019 Elsevier Inc. All rights reserved.
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
observed for virtually all other known compounds or elements, the oceans would freeze from the bottom-up and Earth would be an ice world, not a nice world. In great contrast, and as detailed in the present chapter, the transport properties of water are not unusual compared to diverse other substances that flow under any amount of stress (are pourable) near ambient conditions, except near its density maximum at 4 C in regards to pressure (Singh et al., 2017). Why? This question pertains to mechanisms, which must be understood in order to construct a viable microscopic model of heat transport in matter in states other than gaseous. This chapter considers the state of matter commonly referred to as liquid. We probe materials that can be poured near room T and P for several reasons. First, liquids are dense, like solids, and these two phases are lumped together as “condensed matter.” Yet, liquids have a critical point, such that for conditions exceeding this particular temperature and pressure, the liquid phase cannot be distinguished from a gas. Thus, the liquid state constitutes a transitional case, thereby providing a bridge between the relative simplicity of gases and the complexity of solids. Second, transport data on pourable liquids exist in abundance. There are holes and inadequacies but a wide variety of substances have been explored. Reasons include industrial needs, such as transport of foods through tubes or lubricating moving machinery. Progress of chemical reactions involving mass diffusion is another reason. Conduction of heat by fluids is of practical relevance; for example, cooling of automobile engines. The large number of measurements allow general statements to be made, while simultaneously hampering assessment of uncertainties of individual studies. Therefore, the purpose of this chapter is a general description of transport in pourable liquids as a step towards a thorough understanding the microscopic basis of heat transport in solids. Third, substances that flow under any stress are considered. Simple, Newtonian behavior dominating this category simplifies the measurements, makes available data on viscosity easy to analyze and understand, thereby permitting direct comparisons. Molecular compounds are the mainstay of our comparison, along with liquid elements, and certain oils. This chapter does not consider data on fluids or liquids at very high temperature. Although relevant to the Earth and celestial bodies, existence of radiative transfer at high temperatures is a complicating factor. We cover silicate glasses and liquids in Chapter 10, and diffusive radiative transfer in Chapter 11. Pressure measurements of mass selfdiffusivity are accurate and will be explored briefly, but data on thermal conductivity at P are uncertain by B20% (Ross et al., 1984) or more (Hofmeister, 2010). One problem is contact losses (Chapter 4). Another is small samples (Chapter 7). Pressure determinations of viscosity are only briefly touched upon, in the context of the importance of volume to transport properties.
6.1 TRANSPORT PROPERTIES EXPECTED FOR INELASTIC COLLISIONS OF FINITE SIZE ATOMS IN LIQUIDS Chapter 5 provides a new formula for thermal or mass diffusivity (Eq. (5.30)) that explicitly includes only one state variable, temperature. Kinematic viscosity associated
6.1 TRANSPORT PROPERTIES EXPECTED FOR INELASTIC COLLISIONS OF FINITE SIZE ATOMS IN LIQUIDS
183
with colliding spheres differs by a factor of 2/3 (Eq. (5.23)). The end result is a simple inverse dependence of the diffusivities and kinetic viscosity on atomic or molecular mass (m). For liquids, this parameterization is advantageous because the shapes of molecules vary from nearly spherical as in carbon tetrachloride to very long chains of polymers. The other key parameter tied to the substance is the time of interactions during collisions (τ delay) which is certainly temperature-dependent, and likely pressure dependent, but neither was specified in the simple model presented in Chapter 5. Because the equations describing transport properties of liquids must reduce to those of the gas above the critical point, and should also be identical in the limit of point mass constituents, this section adds a few qualifiers to Eq. (5.30) based on well-known characteristics of pourable liquids, and what was learned from examining gas properties in Chapter 5. The adaptations are intended to address crucial differences existing between these states of matter. Available data on the simplest material known, monatomic gas, showed that Dheat was not identical to Dmass due to the former being monitored via flow of the photon gas, whereas the latter involves motion of the atoms themselves, which, unlike photons, are impeded by each other. The difference between the diffusivities systematically increasing with atomic size (Figs. 5.9 and 5.10) underlies this contention. For diatomic and polyatomic gas (Fig. 5.11), the difference between Dheat and Dmass is greater. Hence, lifetimes for mass and heat transport in liquids cannot be identical. Because interactions in liquids are stronger than in gas due to proximity of the constituents, and because molecule shape and orientation during any interactions should have some effect, the lifetime associated with mass diffusion in liquids actually is a combination of several different lifetimes. Let us denote the primary collision of two atoms as being associated with the lifetime τ heat. For mass transport, a secondary collision will have an effect as well. We do not need to account for the far less-frequent tertiary and higher order collisions, because these can be equivalently “lumped” with the secondary collisions. Because inverse lifetimes add, the lifetime associated with mass transport via collisions will be smaller than that with heat. For liquids, the diffusivities associated with inelastic collisions are proposed to follow: 3NRgc T Dheat D (6.1) τ heat 5 uheat τ heat m 3NRgc T Dmass D (6.2) τ mass 5 umass τ mass ; τ mass # τ heat m where N is the number of moles and Rgc is the gas constant. The temperature (T) or pressure dependence of the lifetimes are not specified. The RHS of these equations is general, arising from dimensional analysis, where we have indicated that each property is not only associated with a different lifetime, but also is linked to a different carrier velocity, should the mechanisms differ. Regarding viscosity of gas, two mechanisms exist, idealized as direct and glancing collisions, which represent the forward transfer of momentum and the inhibiting effect of viscous drag. Considering the collisional cross-sections for spheres predicted that kinematic viscosity (ν) should be 2/3 of either diffusivity (Eq. (5.23)), which is observed to nearly
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
hold. For shapes other than spheres, the cross-sections, and therefore the lifetimes of forward momentum transfer and retarding viscous drag will vary from this simple ratio. Because the data reflect both mechanisms, combined, and to preserve the point mass limit, we include the factor of 2/3 while absorbing the complexity into the lifetime: 2NRgc T (6.3) τ drag 5 udrag τ drag ; τ drag Bτ mass νD m Due to the competition, the size of the lifetime associated with viscous flow relative to that of the diffusivities is not obvious, but for small and round molecules the factor of 2/3 should account for the first-order differences. Use of the subscript “drag” anticipates that forward motions are greatly hindered by frictional forces. At high viscosity, diffusion of momentum may be difficult to nearly impossible. Eqs. (6.1) to (6.3) reduce to 5.30 for equal lifetimes and velocities. This meets our stipulation that transport behavior converges above the critical point, where these phases are indistinguishable. The above formulation remains based on molecular collisions as the sole mechanism, while allowing for glancing collisions affecting kinematic viscosity. Other mechanisms are represented by the dimensional formulation, but not by the simple proportionality with temperature. Below, we test the predicted m and T dependencies against available data, recognizing that lifetimes can depend on m or T.
6.2 MEASUREMENTS OF TRANSPORT PROPERTIES FOR POURABLE LIQUIDS Because the purpose of this chapter is to seek trends amongst the data on liquids, we rely on compilations. The transport property best constrained for liquids is Dmass, due to a recent, extensive, and downloadable compilation on diverse molecular compounds by Sua´rez-Iglesias et al. (2015). Their comprehensive review summarizes data at ambient conditions, and various temperatures and pressures, and includes molecular solids and noble gases as well. The figures in this chapter show the accurate Nuclear Magnetic Resonance (NMR) data from this review. The figures average all measurements for any given compound that were made at ambient conditions. Sua´rez-Iglesias et al.’s (2015) compilation for Dmass of diverse pure molecules in the liquid phase by can be considered as inclusive. All transport properties of all three states of H2O are well-studied: see the websites listed at the end of the chapter, Assael et al. (2000), Chaplin (2018), Fokin and Kalashnikov (2008), Ramires and Nieto de Castro (1995), and the compilation of Sua´rez-Iglesias et al. (2015). Online steam tables provide data also on water and water vapor under various conditions. For ice, data sources are James (1968), plus reviews by Weertman (1983) and Fukusako (1990). In contrast, viscosity data shown in the figures are not from thorough and comprehensive reviews, but rather were obtained from compilations that stem from practical needs. The websites listed at the end of the chapter summarize data used by engineers. Much data exist on dynamic viscosity of diverse materials. Because density is commonly measured, kinematic viscosity is easily computed from dynamic viscosity.
6.2 MEASUREMENTS OF TRANSPORT PROPERTIES FOR POURABLE LIQUIDS
185
For liquids, fewer data exist on heat transport. Mostly, thermal conductivity (K) is reported in the websites geared for engineering. Fortunately, heat capacity and density are commonly measured, so that most of these data on K can be converted to thermal diffusivity. However, for liquids, the transient hot-wire method is mostly used, which is considered to be accurate to 5% at ambient conditions (Watanabe, 1996; Watanabe and Kato, 2004). A small amount of radiative transfer exists at 298K, and is a contributing factor to the uncertainty. Ballistic radiation affects the temperature derivative as well. However, many liquids that pour at ambient conditions are not stable at high temperatures, and so the data are limited to low temperatures. The expectedly small amount of radiative transfer is not important to our general arguments. Due to needs of the food industry, data on all three transport properties exist on edible oils, although not all properties have been measured for each of the various types of oil. More importantly, the chemical compositions of the oils are not precisely fixed since these are mixtures, and so the molecular mass is approximate, and the comparison is approximate. Fatty acids have well-defined compositions and tranport properties (Valeri and Meirelles, 1997), but not food oils. We include these complex compounds due to their intermediate viscosities and the number of data, which should average the uncertainties associated with compositional variations. References are: Coupland and McClements (1997), de Freitas-Cabral et al. (2011), Diamante and Lam (2014), Huang and Liu (2009), Xu et al. (2014), and Ya´n˜ez-Limo´n et al. (2005). Data on sugars, syrups, and other foods (Bleazard et al., 1996; Ertl and Dullian, 1973; He et al., 2006; Nguyen et al., 2012; White, 1988) were included for similar reasons. Lubrication is essential to machinery and vehicles. Viscosity data are available for many different oils and greases. To represent this type of material, we include silicone oils, because all three transport properties are well-characterized, plus, these oils have the same building block, chained together. Many physical properties are provided by manufacturer’s websites (listed at the end of the chapter). Data on low molecular mass silicone oils are summarized by Roberts et al. (2017). Thermal conductivity and heat capacity were measured by Sandberg and Sundqvist (1982). Data on large molecular mass oils are given by Koschmieder and Pallas (1974) and Thern and Lu¨demann (1996), where the latter is the source of Dmass values. Last but not least, we examine data on the simplest possible liquids, the elements. All three transport properties of the two metals that are liquid at or very close to ambient conditions (Ga and Hg), are well-studied (Aurnou and Olsen, 2001; Blagoveshchenskii et al., 2015; Meyer, 1961; Schriempf, 1972, 1973). The alkali metals (Cs and Rb) which melt near ambient conditions (at 29 C and 39 C, respectively) are less well-characterized. Sources are Andrade and Dobbs (1952) and the compilations of Vargaftik et al. (1994) and Iida et al. (2006). Specific heats of solid alkali metals were used which are identical at the melting point (Filby and Martin, 1966), but are affected by phase transitions (λ-shaped curves). Mercury, being further from its melting point at 238 C, should not be affected. In contrast, data on the two nonmetals, Br2 and I2, are rather uncertain, as discussed in the review by Touloukian and Ho (1981). Because Cl2 is liquid at T fairly close to ambient, we include these data as well, slightly extrapolated, from the same compilation, recognizing that this is more uncertain. Mass diffusivity data have not been published, to the best of our knowledge. A more detailed investigation than that presented here is warranted. However, the graphical analysis below and comparison with trends for gas transport properties from
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
Chapter 5 establish that modeling collisions during translations insufficiently describes transport properties of liquids.
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS 6.3.1 Mass and Density Effects at Ambient Conditions At ambient temperature and pressure, distinct trends exist for each of the three transport properties with molecular mass, although considerable scatter exists in the data (Fig. 6.1). Molecular compounds, including the complex oils and simple diatomics, define the observed trends: mass diffusivity inversely depends on molecular mass, as expected, whereas thermal diffusivity is approximately constant, and, contrastingly, kinematic viscosity is directly proportional to m. The diatomic molecules describe a trend that is slightly steeper but lower than that of the more complex molecular compounds. The slight increase in Dheat with m which exists for the homogenous diatomics may be due to the large uncertainties for bromine and iodine, and with chlorine data being extrapolated, but a similar increase describes the
FIGURE 6.1 Dependence of measured transport properties of liquids near 298K and ambient pressure on mass of their molecules. Data on Dmass (squares and dotted line) mostly from the compilation of Sua´rez-Iglesias et al. (2015). Data on Dheat (dots and solid line) and on ν (diamonds and dashed line) from websites or publications as described in Section 6.2. Silicone oils (dots with gray interiors) and diatomic homogenous molecules (superimposed 3 or 1 signs) were included in the least squares fits shown. Chlorine viscosity is hidden by the Dheat trend. Large gray symbols: liquid metals, which were not including in the fitting. Open circles: heat transport at 300K, which was not included in the fits, but also appears to be independent of m.
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
187
accurate measurements of the silicone oils. The edible oils cluster together, which probably stems from approximated values for m. Sugary substances have larger ν than the molecular compounds, which seems connected with these compounds flowing, yet being sticky to the touch. Both food types also have higher Dheat but lower Dmass than the trends for the molecular compounds. The metallic elements have Dmass close to values for the molecular compounds, but possess much larger Dheat and lower viscosity. Unlike the molecular compounds and diatomics, transport data for liquid metals do not depend on mass (Fig. 6.1), but instead follow an inverse power law with density (Fig. 6.2). The power-law fits are steep for thermal diffusivity, moderate for kinematic viscosity, and nearly flat for mass diffusivity. Roughly, the trend for kinematic viscosity of the metals approximates the lowest n for molecular solids while the trend for Dmass of the metals approximates the highest values for molecular solids. This behavior suggests an inverse correlation of Dmass with viscosity, which is explored further below. Fig. 6.2 further emphasizes that the thermal diffusivity of the liquid metals differs greatly from Dheat of the molecular compounds, and that density is not a controlling parameter for the compounds, but does have some effect. For molecular liquids, Dheat is constant (Figs. 6.1 and 6.2). In round numbers, the thermal diffusivity equals 0.1 mm2 s21 within 10%, and furthermore occupies a very restricted range of 0.04 0.15 mm2 s21. Such behavior is inconsistent with molecular collisions, for which a strong mass dependence is expected. For Eq. (6.1) to be valid, requires that the lifetime be proportional to the molecular mass. This is reasonable since large molecules may become entangled with each other. Moreover, Dheat for very simple molecular configurations (the diatomics and the silicone oils) increases with molecular mass, suggesting that a simple dependence of lifetime on mass for all three properties does not exist. To explore this further, thermal diffusivity for restricted ranges of m was fit to power laws in density. The results of Table 6.1 show that Dheat increases slightly with ρ, under nearly FIGURE 6.2 Dependence of measured transport properties of liquids at 298K and ambient pressure on density. Filled black symbols and lines depict liquid metals. Least squares fits are listed at the top. Other symbols as in Fig. 6.1. Data sources listed in Section 6.2.
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
TABLE 6.1 Fits of Dheat at 298 300K as a Function of Density for Molecular Liquids Mass Range amu
No. Pts.
Density Range kg m23
Power-Law Fit
Correlation Coefficient
58 64
6
760 1160
0.016883 ρ0.251
0.21
610 1270
0.0007126 ρ
0.77
620 1280
0.029135 ρ
0.34
70 79 86 93 142 170 253 352
9 6 5 3
0.7116
0.1621
20.0673
730 3100
0.13734 ρ
860 4960
0.039303 ρ
0.1312
0.41 0.92
constant molecular mass. Table 6.1 and Figs. 6.1 and 6.2 suggest that a mechanism other than inelastic collisions governs diffusion of heat in liquids composed of molecules. The trend for Dmass extrapolates to the average for Dheat at m 5 1 amu, which reasonably approximates a point mass atom (i.e., monatomic H). The trend for ν extrapolates to a slightly lower value at m 5 1 amu. This behavior is consistent with our model for gas, which indicated ν 5 2/3 Dheat 5 2/3 Dmass, if the cross-sections are described by inelastic collisions of ideal spherical atoms. However, the magnitudes of the material properties for liquids and gas are much different. For example, the lowest value for Dheat of gas is 3.8 mm2 s21 (for radon), which is B20 3 larger than Dheat of liquids. The discrepancy between mass self-diffusivities is even larger. Although kinematic viscosity magnitudes are similar for liquids and gas, this is fortuitous, as follows: The trends of Fig. 6.1 for heat transport and viscous drag are not consistent with collision based transport, although mass transport in molecular liquids appears to be associated with collisions. Liquids, like gas, have Dheat . Dmass, but the discrepancy in magnitude is larger and kinematic viscosity behaves much differently than the two diffusivities. Chemical bonding appears to affect drag because nonmetals have ν . Dheat . Dmass whereas metallic elements have Dheat . ν . Dmass.
6.3.2 Ratios of Transport Properties at Ambient Conditions Understanding how transport properties of liquids are related are needed to build a model. To eliminate the effects of molecular mass and density on the transport properties at STP, mass diffusivity is plotted against kinematic viscosity in Fig. 6.3A. For the molecular compounds, Dmass inversely correlates with ν. This behavior is equivalent to the trends in Fig. 6.1: because Dmass is inversely proportional to m and ν is proportional to m, then Dmass 5 constant/ν. Setting the power during fitting as unity gives: Dmass 5
3 1 2 1000ν
(6.4)
where the constant of 3/2 is uncertain by 5% from least squares fitting. This inverse provides a simple and testable prediction. For bromine, Dmass should be 0.00495 mm2 s21 at STP and for iodine, 0.00025 mm2 s21 is expected.
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
189
FIGURE 6.3 Cross correlations of transport properties for pourable liquids. (A) Dependence of Dmass on ν. (B) Dependence of Dmass on Dheat. Squares and solid line: molecular compounds. Gray interiors show the subset of silicone oils. Triangles and dotted line: liquid metals. Section 6.2 lists data sources.
For molecular compounds, the two diffusivities are uncorrelated (Fig. 6.3B). This result corroborates our inference of different mechanisms controlling their heat and mass transport. The direct correlations of Dmass with ν and of Dmass with Dheat (Fig. 6.3) for the metallic elements are directly related to the trends in Fig. 6.2. Their viscosity represents a substantial component of forward diffusion of momentum, but with more inhibiting drag than occurs for the gases. Likewise, heat transfer in liquid metals involves collisions, but the mechanism differs substantially from that in gases. To probe the lifetimes, ratios of the transport properties are useful (Fig. 6.4). With this approach, a much smaller database is explored, but one that involves well-studied substances. The power laws for the ratios are consistent with the fits in Fig. 6.1 for the molecular compounds and with the metals not depending on atomic mass. For tiny molecules in this subset of the data described in Section 6.2, the ratios converge to 1/2. For collisions of spheres, ratios of unity and 2/3 are expected. Agreement is reasonable, given that neither Dheat nor ν behave as expected for simple, collisional transport. Ratios of liquid metal transport properties follow power laws in density (not shown) as expected from Figs. 6.2 and 6.3. These do not project to an intercept. For liquid metals, Dmass/Dheat is proportional to density, where ν/Dheat goes as ρ0.63.
6.3.3 The Temperature Dependence of Transport Properties Well-studied substances are considered here: water, simple molecular solids, liquid Hg metal, and silicone oils. Because the data on the homogenous diatomics are not very accurate (Touloukian and Ho, 1981) and because these behave similar to more complex molecular compounds, we do not examine the T dependence for this category in detail. For
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
FIGURE 6.4 Ratios of the transport properties as a function of mass. Triangles 5 kinematic viscosity divided by thermal diffusivity: open for the diatomic nonmetals and gray for the elemental metals. Plus signs: mass diffusivity over thermal diffusivity (gray for liquid metals). Circles: silicone oils. Several compounds are labeled. Section 6.2 lists data sources.
bromine, Dheat is nearly constant (50.082 mm2 s21) from freezing to boiling whereas viscosity decreases strongly: ν(T) 5 737200T22.58 mm2 s21. 6.3.3.1 Water, With a Comparison to Ice and Steam For liquid water, the three transport properties depend on T in much different ways (Fig. 6.5). Thermal diffusivity weakly increases with temperature, whereas kinematic viscosity strongly decreases, while mass diffusivity strongly increases. Projected trends for Dheat and ν converge, but both always greatly exceed Dmass. For Dmass, essentially the same second-order polynomial in T describes both the liquid and solid phases. For each of water and ice, Dmass was not well-described by a power law. For each of water and ice, Dheat weakly depends on T and either a linear or a power-law fit suffices for the small ranges of T examined. These measurements involve non-steady-state techniques (hot-wire ˚ ngstrom’s) and are likely influenced by ballistic transport, due to partial transparand A ency of H2O in the infrared (Fig. 2.7). However, near room temperature ballistic augmentation is small, so the order-or-magnitude difference in Dheat represents the effect of freezing on H2O. Similarly, rheologic behaviors of water and ice differ. Water is a Newtonian fluid: nonetheless, υ depends on T in a complex manner. For an accurate fit, four parameters are required for an exponential form (Fig. 6.5), while five parameters are needed for a polynomial fit (not shown). Viscosity of ice is not shown because its behavior non-Newtonian
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
191
FIGURE 6.5
Transport properties of water and other phases of H2O as a function of temperature. Least squares fits have correlation coefficients of 0.99. Arrows: extrapolations for water trends to high T. Thermal diffusivity of water (small circles) and ice (large circles) depend linearly on T, but differ in slope and are discontinuous at freezing. For Dmass, one trend describes both solid and liquid phases (open squares), but the gas (gray squares) differs greatly, although the fit is also a secondorder polynomial. For gas, thermal diffusivity (gray dots and line, with fit listed) and kinematic viscosity (gray diamonds and dashes) are described by similar fits that parallel steam mass diffusivity. Section 6.2 lists data sources; see also Chaplin (2018).
(Weertman, 1983), and thus the creeping behavior of ice is mechanistically distinct. Although creep has been considered in Earth science to be mathematically equivalent to Newtonian viscous flow, this parallelism ignores the grain-size dependence of creep and the often used formulations in the engineering literature (Meyers and Chawla, 2009), see discussion of Hofmeister and Criss (2018). Transport properties for steam are much higher than those of water or ice, and follow simple power laws in T. The observed behavior (Fig. 6.5, gray symbols) with T follows that of the small molecules CO2 and NH3 (Fig. 5.11), whereby Dmass is larger at high T and the powers in T are similar. The behavior and mechanisms of transport in steam are much simpler than in the two condensed phases of H2O. 6.3.3.2 Small Molecule Liquids In comparing the transport properties of molecular compounds, we opted for convenience: this section considers all small molecules on the viscopedia website for which kinematic viscosity as a function of temperature was listed. Mass diffusivity for all of these (Table 6.2) were compiled by Sua´rez-Iglesias et al. (2015). Thermal diffusivity was obtained from various publications and websites described in Section 6.2. Fig. 6.6 shows that small molecules have similar values exist for each of the three transport properties. Moreover, the overall behavior is like that of water and bromine, where ν decreases with temperature, while Dheat is nearly constant, but Dmass increases as T increases. As observed for H2O, Dmass is continuous across the freezing point and is described by a polynomial when the data cover more than a narrow range of 100K. However, the strong curvature seems to be associated with the freezing point. Unlike
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
TABLE 6.2 Properties of Small Molecule Liquids Name
Formula
m amu
ρa kg m23
Dheat mm2 s21
Tfreeze K
Tboil K
Volume m3 mol21
Benzene
C6H6
78
871
0.094a
279
353
0.0896
Toluene
C7H8
92
862
0.102a
178
384
0.106
Methanol
CH4O
32
787
0.105a
175
338
0.0407
Hexane
C6H14
86
655
0.089a
178
342
0.131
Pentane
C5H12
72
621
0.078b
143
309
0.116
a
At 298K. At 300K.
b
FIGURE 6.6 Transport properties of small molecule liquids as a function of temperature. Circles: benzene; dots: toluene; diamonds: methanol; open squares: hexane. Filled squares 5 pentane. Power laws describe kinematic viscosity, which was determined over a small temperature range. For hexane, ν 5 463,300T22.4275. Thermal diffusivity values are nearly independent of T. Mass diffusivity over a wide temperature range is described by a polynomial fit, whereas narrow ranges can be fit to a power law. For benzene, Dmass 5 2.6496 3 10215T4.8211. For toluene and pentane, Dmass was collected at very low T and is continuous across the freezing point.
water, the fits for ν were simpler: kinematic viscosity follows a simple power law over the temperature range explored. In addition, the transport properties follow a prescribed order. Both Dheat and ν decrease according to the order: benzene $ methanolBtoluene . hexane . pentane
(6.5)
The same order holds for 1/Dmass, as expected from its inverse correlation with ν at 298K (Fig. 6.3). This order is controlled by density, except for methanol, which has a much
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
193
lower molecular mass (Table 6.2). The inverse correlation at 298K does not hold at all temperatures. For example, the product ν 3 Dmass equals 1.5 3 1026T1.24 for hexane or is 2.276 3 1027T1.55 for benzene, which have simpler fits for Dmass than do methanol, pentane, or toluene. 6.3.3.3 Silicone Oils Fig. 6.7 shows mass diffusivity measured for four different silicone oils, which are described by a peak in molecular mass, as listed (Thern and Lu¨demann, 1996). Viscosity was inferred from the reported molecular mass, rather than the reported density, which was not entirely consistent with manufacturer’s reports. Polydimethylsiloxane fluids are linear-chain polymers with chemical formulas of (SiO(CH3)2)x. Because the lowestviscosity silicone oil (0.65 cSt) has the shortest chain possible, it alone has a fixed composition of O(Si(CH3)3)2. More viscous oils contain polymers of various lengths and the range of lengths increases with the peak mass, as is evident in the mass distribution plots of Thern and Lu¨demann (1996). Oils from different manufactures may be produced by blending, and so the distributions of mass may differ (Roberts et al., 2017). Therefore, curves for ν in Fig. 6.7 do not correspond exactly to Dmass measurements. In addition, evaporation is significant by 400K for the low viscosity oils (Roberts et al., 2017). Because measurements of ν have been made many times for many different oils, ν vs T is well-constrained, leading to a regular trend in the power-law fits. This may not be the case for Dmass, as suggested by the less regular trend in the power for T. Because of these differences, we do not plot ratios of the transport properties, and focus on the trends, but not the details. The temperature response of the three transport properties is regular and similar to that of water and the small molecule hydrocarbons (Figs. 6.5 and 6.6) which have much lower FIGURE 6.7 Transport properties of silicone oils as a function of temperature. Data from Thern and Lu¨demann (1996) and the Clearco website, see text. Thermal conductivity data of Roberts et al. (2017) indicate Dheat is independent of temperature. The three transport properties were not determined from the exact same substance. Power-law fits are shown, but without the prefactor.
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
m (18 92 amu). Kinematic viscosity and Dmass appear to be inversely correlated whereas ν and Dheat appear to converge at high T. The temperature dependencies are simpler for the more complex silicone oils. However, departures from power laws are observed at the coldest temperatures accessed for 100 cSt oil (Fig. 6.7). 6.3.3.4 Liquid Metallic Mercury Large disparities in values and trends with temperature exist among the three transport properties for Hg (Fig. 6.8), as observed for nonmetals (Figs. 6.5 6.7), although transport properties for these two types of matter differ in many respects. Mass diffusivity with weak curvature over a narrow T range is like that of nonmetals. The weak curvature for Hg was not fit, given the stated uncertainty and that interlaboratory comparisons of suggest uncertainties up to 20% for the tracer methods that are applied to metals (Galus, 1984). The trend for ν of Hg with T is complicated, but similar to a decaying power law, as in nonmetals. Liquid metals offer less resistance to matter flow, which can be attributed to the small size and round shapes of the cations, compared to the larger and less regular shapes of neutral molecules. Thermal diffusivity of Hg depends moderately and linearly on T (Fig. 6.8), rather than being flat and nearly constant, as in the nonmetallic liquids. It is unlikely that atomic motions are involved, due to Dmass being 0.05% of Dheat. One concern is use of containers in laser flash analysis (LFA) measurements. Schreimpf’s (1972) measurements predate three-layer models (Chapter 4). The effect of the supporting walls should be small, and his data were in reasonably agreement with earlier measurements. Thermal conductivity
FIGURE 6.8 Transport properties of liquid mercury as a function of temperature over its entire liquid field. Left axis: thermal diffusivity (dots, with the range and fit from Schriempf, 1972) and mass diffusivity (black squares, shown as 3 1000 measured values with data from Nachtrieb, 1967). Fits have correlation coefficients of 0.99. Uncertainties in Dmass are generally stated to be 5% as indicated. For this reason, only a linear fit was made and diffusivities of Ga, and of Zn and Ag in Hg are also shown (Schriempf, 1973; Galus 1984, respectively). Right axis: kinematic viscosity, compiled by Kozin and Hansen (2013), was fit to a simple power law. For an accurate fit, a fourth-order polynomial is needed.
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measurements involve contact losses as well. It is unlikely that the corrections address all the issues in the older data. Hence, additional measurements of Dheat for liquid metals are needed.
6.3.4 Effect of Pressure on Transport Properties As discussed in Chapter 4, measurements of heat transport at pressure are problematic mostly due to contact losses. We therefore only discuss how viscosity and Dmass respond to pressure. Nuclear magnetic resonance methods used to obtain Dmass for molecular liquids are accurate (Sua´rez-Iglesias et al., 2015), and are amenable to studies at pressure. All the compounds listed in Table 6.2 behave similarly. Fig. 6.9A shows isothermal curves for toluene as a function of P. Isobaric data on Dmass were also plotted as a function of T: these curves are roughly parallel (not shown). However, as discussed in Chapter 1 and Chapter 5, the fundamental variable is volume, not pressure. Isothermal curves for Dmass as a function of volume are parallel at large volumes and essentially linear (Fig. 6.9B). For small volumes, the changes are less rapid, requiring polynomial fits over large ranges of V, whereas small ranges of V and compressed material are exponential. Mass diffusivity for any given liquids is a function of T and V which is affected by the mass of the molecule. At a constant and large volume, the change in Dmass with T is roughly linear, from Fig. 6.9B.
FIGURE 6.9 Mass self-diffusivity and density of liquid toluene collected as a function of pressure and temperature. NMR measurements by Harris et al. (1993), as tabulated by Sua´rez-Iglesias et al. (2015). Each isothermal data set is shown with a particular symbol and line pattern, as labeled. (A) As P increases, isothermal Dmass decays exponentially whereas density increases as a modified power law. (B) As volume increases, isothermal Dmass increases. The data above ambient temperature require quadratic fits. Low T data can be fit to exponentials or power laws, due to lower ranges in V (or P).
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Although a volume dependence was not specified in Eq. (6.2), rarefication certainly affects speeds, causing u to decrease as V increases. Because Dmass increases with V, a strong increase in τ mass is indicated. The trends in Dmass are consistent with a collisional mechanism for mass transport, but one that is severely impeded by strong molecular interactions in the liquid state. It is long known that viscosity increases under compression for pourable liquids except for water below 306K (e.g., Bridgman, 1925). The caused is enhanced frictional drag due to tighter packing of the molecules. The unusual behavior of water stems from its density maximum near 277K, that is, the molar volume of water responds anomalously to pressure around this temperature, and so must viscosity. For measurements of supercooled water and a detailed description of the fascinating changes in viscosity with T and P, see Singh et al. (2017). Hence, viscosity and mass diffusivity for pourable liquids respond in opposite directions to mass, temperature, pressure, and volume, and cannot involve the same mechanism of direct collisions which underlies the kinetic theory of gas.
6.4 MECHANISMS OF TRANSPORT IN LIQUIDS The contrasting behavior of the three transport properties for liquids with basic physical properties and state variables (molecular mass or size, density, temperature, volume, and pressure) indicates that each property is associated with a distinct mechanism. Yet, some ties exist between the transport properties through the motions and interactions of the molecules and through the equation of state (EOS). Forward, direct collisions are essential to move molecules down a compositional gradient. Hence, diffusion of mass in liquids that pour near ambient conditions behaves generally in accord with a collisional model. For molecular compounds, the dependence of Dmass on mass, temperature, and volume (or pressure) is roughly consistent with Eq. (6.2), such that the numerical values are much smaller than in gas. Complexities exist, suggesting that the collisional model for mass transport needs further evaluation. Factors to consider are that the speed of the molecules is largely a thermal effect that is controlled by the EOS, but forward speed is hampered by viscous damping forces. The lifetimes between collisions are shorter due to the compact nature of liquids relative to gases but this may be overshadowed by the lifetimes of the interactions between molecules. Inverse lifetimes add, which makes the important mechanism not obvious. One indication of multiple influences on forward collisions is that mass is an insufficient descriptor (Fig. 6.1) of Dmass for molecular liquids. The scatter in this plot indicates that molecule shape and vibrational or rotational interchanges are important. In liquids, kinematic viscosity does not represent forward motion of momentum, but rather results almost entirely from resistive drag forces. The inverse correlation of Dmass with ν at ambient conditions is a consequence of their contrasting dependence on molecular mass, which was used to represent the size of the molecules. That Dmass and ν are affected in opposite directions by mass, T, V, density, and P results from their contrasting mechanisms of propulsion forward by direct collisions and inhibition of motions by frictional glancing collisions. We reiterate that the equations describing mass diffusion and viscosity, derived by Fick and Newton, respectively, are unrelated.
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Thermal diffusivity is independent of mass and density for molecular liquids (Figs. 6.1 and 6.2), which together show that volume is unimportant. Because Dheat measurements on molecular liquids are subject to ballistic radiative transfer, the temperature dependence, albeit weak, probably is stronger than the intrinsic behavior (Chapter 4). Thus, Dheat is largely independent of T, as well as m, density, and volume, which indicates that P has negligible influence. Yet another mechanism is required for this transport property. For liquids, as in gases, forward and glancing collisions as well as interactions of the molecules (generally vibrational) generate photons. Thermal diffusivity is ascertain by sampling photon flux outward. Metallic liquids similarly show evidence for three different transport mechanisms because their properties also differ greatly. However, transport properties in metallic liquids are governed by density, rather than molecular mass. The mechanisms for flow and mass transport in liquid metals are similar to those in molecular solids, given the results of Section 6.3. However, heat transport in liquid metals differs by being larger and temperature-dependent. The increase should not be due to ballistic transport, since metals are opaque over a wide spectral range. The limited data on liquid metals for Dheat suggest that the mechanism operates like that of the solid state (Criss and Hofmeister, 2017; Chapter 9). The effect of compression on Dmass of toluene suggests that the scatter in Fig. 6.1 for the molecular compounds is related to density and volume effects, as well as molecule shape. Tying all the data together in a model requires a deeper analysis than that presented here, and more data than currently available. We have shown that the mechanisms for heat transport in liquids differ from simple, molecular collisions in gas. Yet, liquids, like gases, will have a photon gas intermingled. Plus, the photon gas is sampled during heat transport measurements in all cases (Chapters 1 and 5). To understand this, we next delve into solids with reliable data on Dheat as a function of temperature. Values of transport properties projected to the point mass are consistent with predictions from the kinetic theory of gas, amended to account for glancing collisions affecting ν. The success in applying this model, which was derived for the unrealistic ideal gas, to pourable liquids stems from this unachievable limit being compatible with the model and to combining many parameters in fitting, which can mask imperfections. If point masses could collide, then these collisions would be elastic, but no time evolution would be possible.
References Andrade, E.Nda C., Dobbs, E.R., 1952. The viscosities of liquid lithium, rubidium and caesium. Proc. R. Soc. A 211, 12 30. Assael, M.J., Bekou, E., Giakoumakis, D., Frienda, D.G., Killeen, M.A., Millat, J., et al., 2000. Experimental data for the viscosity and thermal conductivity of water and steam. J. Phys. Chem. Red. Data 29, 141 166. Aurnou, J.M., Olsen, P.L., 2001. Experiments on Rayleigh-Be´nard convection, magnetoconvection, and rotating magnetoconvection in liquid gallium. J. Fluid. Mech. 430, 283 307. Blagoveshchenskii, N., Novikov, A., Puchkov, A., Savostin, V., Sobolev, O., 2015. Self-diffusion in liquid gallium and a hard sphere model. EPJ Web of Conferences 83, No. 02018. Bleazard, J.G., Sun, T.F., Teja, A.S., 1996. The thermal conductivity and viscosity of acetic acid-water mixtures. Int. J. Thermophys. 17, 111 125. Bridgman, P.W., 1925. The viscosity of liquids under pressure. Proc. Natl. Acad. Sci. 11, 603 606.
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Chaplin, M., 2018. Water structure and science. http://www1.lsbu.ac.uk/water/water_structure_science.html (accessed September 1, 2018). Coupland, J.N., McClements, D.J., 1997. Physical properties of liquid edible oils. J. Am. Oil. Chem. Soc. 74, 1559 1564. Criss, E.M., Hofmeister, A.M., 2017. Isolating lattice from electronic contributions in thermal transport measurements of metals and alloys and a new model. Int. J. Mod. Phys. B 31, No. 175020. de Freitas-Cabral, A.J., de Oliveira, P.C., Moreira, S.G.C., Alcantara Jr., P., 2011. Thermal diffusivity of palm olein and compounds containing β-carotene. Int. J. Thermophys. 32, 1966 1972. Diamante, L.M., Lam, T., 2014. Absolute viscosities of vegetable oils at different temperatures and shear rate range of 64.5 to 4835 s21. J. Food Process. No. 234583, 1 6. Ertl, H., Dullian, A.L., 1973. Self-diffusion and viscosity of some liquids as a function of temperature. AlChE J. 19, 1215 1223. Filby, J.D., Martin, D.L., 1965. The specific heats below 320 C of potassium, rubidium and caesium. Proc. R. Soc. A 284, 83 107. Fokin, L.R., Kalashnikov, A.N., 2008. The viscosity and self-diffusion of rarefied steam: Refinement of reference data. High Temp. 46, 614 619. Fukusako, S., 1990. Thermophysical properties of ice, snow, and sea ice. Int. J. Thermophys. 11, 353 372. Galus, Z., 1984. Diffusion coefficients of metals in mercury. Pure Appl. Chem. 56, 635 644. Harris, K.R., Alexander, J.J., Goscinsa, T., Malhotra, R., Woolf, L.A., Dyamond, J.H., 1993. Temperature and density dependence of the selfdiffusion coefficients of liquid n-octane and toluene. Molec. Phys. 78, 235 248. He, X., Fowler, A., Toner, M., 2006. Water activity and mobility in solution of glycerol and small molecular weight sugars: implication for cryo- and lyoperservation. J. Appl. Phys. 100, No. 074702. Hofmeister, A.M., 2010. Scale aspects of heat transport in the diamond anvil cell, in spectroscopic modeling, and in Earth’s mantle: implications for secular cooling. Phys. Earth Planet. Inter. 180, 138 147. Hofmeister, A.M., Criss, E.M., 2018. How properties that distinguish solids from fluids and constraints of spherical geometry suppress lower mantle convection. J. Earth Sci. 29, 1 20. Available from: https://doi.org/ 10.1007/s12583-017-0819-4. Huang, L., Liu, L.S., 2009. Simultaneous determination of thermal conductivity and thermal diffusivity of food and agricultural materials using a transient plane-source method. J. Food Eng. 95, 179 185. Iida, T., Guthrie, R., Tripathi, N., 2006. A model for accurate predictions of self-diffusivities in liquid metals, semimetals, and semiconductors. Metall. Mater. Trans. B 37, 559 564. James, D.W., 1968. The thermal diffusivity of ice and water between 2 40 and 1 60 C. J. Mater. Sci. 3, 540 543. Koschmieder, E.L., Pallas, S.G., 1974. Heat transfer through a shallow, horizontal convecting fluid layer. Int. J. Heat Mass Trans. 17, 991 1002. Kozin, L.F., Hansen, S.C., 2013. Mercury Handbook: Chemistry, Applications and Environmental. Royal Society of Chemistry, Cambridge, UK. Meyer, R.E., 1961. Self-diffusion of liquid mercury. J. Phys. Chem. 65, 567 568. Meyers, M.A., Chawla, K.K., 2009. Mechanical Behavior of Materials. Cambridge University Press, Cambridge, UK. Nachtrieb, N.H., 1967. Self-diffusion in liquid metals. Adv. Phys. 16, 309 323. Nguyen, L.T., Balasubramaniam, V.M., Sastry, S.K., 2012. Determination of in-situ thermal conductivity, thermal diffusivity, volumetric specific heat and isobaric specific heat of selected foods under pressure. Int. J. Food Prop. 15, 169 187. Ramires, M.L.V., Nieto de Castro, C.A., 1995. Standard reference data for the thermal conductivity of water. J. Phys. Chem. Ref. Data 24, 1377 1381. Roberts, C., Graham, A., Nemer, M., Phinney, L., Garcia, R., Stirrup, E., 2017. Physical properties of lowmolecular weight polydimethylsiloxane fluids. Sandia Report, SAND2017-1242. https://doi.org/10.2172/ 1343365 (accessed 20.12.17). Ross, R.G., Andersson, P., Sundqvist, B., Ba¨ckstro¨m, G., 1984. Thermal conductivity of solids and liquids under pressure. Rep. Prog. Phys. 47, 1347 1402. Sandberg, O., Sundqvist, B., 1982. Thermal properties of two low viscosity silicon oils as functions of temperature and pressure. J. Appl. Phys. 53, 8751 8755.
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Schriempf, J.T., 1972. A laser-flash technique for determining thermal diffusivity of liquid metals at elevated temperatures. Rev. Sci. Instrum. 43, 781 786. Schriempf, J.T., 1973. Thermal diffusivity of liquid gallium. Solid State Commun. 13, 651 653. Singh, L.P., Issenmanna, B., Caupin, F., 2017. Pressure dependence of viscosity in supercooled water and a unified approach for thermodynamic and dynamic anomalies of water. Proc. Natl. Acad. Sci. USA 114, 4312 4317. Sua´rez-Iglesias, O., Medina, I., Sanz, M., Pizarro, C., Bueno, J.L., 2015. Self-diffusion in molecular fluids and noble gases: available data. J. Chem. Eng. Data 60, 2757 2817. Thern, A., Lu¨demann, H.D., 1996. P, T Dependence of the self-diffusion coefficients and densities in liquid silicone oils. Z. Naturforsch. A 51, 192 196. Touloukian, Y.S., Ho, C.Y., 1981. Properties of Nonmetallic Fluids. McGraw-Hill, New York, NY. Valeri, D., Meirelles, A.J.A., 1997. Viscosities of fatty acids, triglycerides, and their binary mixtures. J. Am. Oil. Chem. Soc. 74, 1221 1226. Vargaftik, N.B., Phylippov, L.P., Tarzimanov, A.A., Totskii, E., 1994. Handbook of Thermal Conductivity of Liquids and Gas. CRC Press, Boca Raton, FL (translator: Yu.A. Gorshkov). Watanabe, H., 1996. Accurate and simultaneous measurement of the thermal conductivity and thermal diffusivity of liquids using the transient hot-wire method. Metrologia 33, 101 115. Watanabe, H., Kato, H., 2004. Thermal conductivity and thermal diffusivity of twenty-nine liquids: alkenes, cyclic (alkanes, alkenes, alkadienes, aromatics), and deuterated hydrocarbons. J. Chem. Eng. Data 49, 809 825. Weertman, J., 1983. Creep deformation of ice. Annu. Rev. Earth. Planet. Sci. 11, 215 240. White, D.B., 1988. The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid. Mech. 191, 247 286. Xu, Z., Morris, R.H., Bencsik, M., Newton, M.I., 2014. Detection of virgin olive oil adulteration using low field unilateral NMR. Sensors 14, 2028 2035. Ya´n˜ez-Limo´n, J.M., Mayen-Mondrago´n, R., Martı´nez-Flores, O., Flores-Farias, R., 2005. Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy. Superficies y Vacio 18, 31 37.
Websites Viscosity of liquids http://www.engineersedge.com/fluid_flow/kinematic-viscosity-table.htm http://www.engineeringtoolbox.com/kinematic-viscosity-d_397.html http://www.viscopedia.com/ Thermal conductivity at 25 C http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html Thermal conductivity of various liquids at 300K http://www.engineeringtoolbox.com/thermal-conductivity-liquids-d_1260.html http://www.engineersedge.com/heat_transfer/thermal_conductivity_of_liquids_9921.htm Thermal diffusivity http://www.engineersedge.com/heat_transfer/thermal_diffusivity_table_13953.htm Physical properties of the elements http://periodictable.com/Elements/049/data.html Steam/water tables http://thermopedia.com/content/1150Begal (Reprinted, with permission, from NBS/NRC Steam Tables) https://www.nist.gov/document-12896 (has density as a function of T and P) Physical properties of silicone fluids from manufacturers www.gelest.com www.clearcoproducts.com Thermophysical properties of methanol https://www.thermalfluidscentral.org/encyclopedia/index.php/ Thermophysical_Properties:_Methanol
6 Transport Behavior of Common, Pourable Liquids: Evidence for Mechanisms Other Than Collisions O U T L I N E 6.1 Transport Properties Expected for Inelastic Collisions of Finite Size Atoms in Liquids
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6.3 Trends in Transport Properties of Pourable Liquids 186 6.3.1 Mass and Density Effects at Ambient Conditions 186 6.3.2 Ratios of Transport Properties at Ambient Conditions 188
6.3.3 The Temperature Dependence of Transport Properties 189 6.3.4 Effect of Pressure on Transport Properties 195 6.4 Mechanisms of Transport in Liquids
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References
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There is no water, so things are bad. If there were water, it would be better. But there is no water. T.S. Eliot, The Waste Land (1922)
The unusual physical properties of liquid water have received considerable attention in the physical sciences (e.g., Chaplin, 2018), due to the importance of this simple fluid to life itself. From a geochemical perspective, the existence of life as we know it on Earth is predicated on its solid being less dense at the freezing point. If this behavior was reversed, as
Measurements, Mechanisms, and Models of Heat Transport DOI: https://doi.org/10.1016/B978-0-12-809981-0.00006-1
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© 2019 Elsevier Inc. All rights reserved.
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observed for virtually all other known compounds or elements, the oceans would freeze from the bottom-up and Earth would be an ice world, not a nice world. In great contrast, and as detailed in the present chapter, the transport properties of water are not unusual compared to diverse other substances that flow under any amount of stress (are pourable) near ambient conditions, except near its density maximum at 4 C in regards to pressure (Singh et al., 2017). Why? This question pertains to mechanisms, which must be understood in order to construct a viable microscopic model of heat transport in matter in states other than gaseous. This chapter considers the state of matter commonly referred to as liquid. We probe materials that can be poured near room T and P for several reasons. First, liquids are dense, like solids, and these two phases are lumped together as “condensed matter.” Yet, liquids have a critical point, such that for conditions exceeding this particular temperature and pressure, the liquid phase cannot be distinguished from a gas. Thus, the liquid state constitutes a transitional case, thereby providing a bridge between the relative simplicity of gases and the complexity of solids. Second, transport data on pourable liquids exist in abundance. There are holes and inadequacies but a wide variety of substances have been explored. Reasons include industrial needs, such as transport of foods through tubes or lubricating moving machinery. Progress of chemical reactions involving mass diffusion is another reason. Conduction of heat by fluids is of practical relevance; for example, cooling of automobile engines. The large number of measurements allow general statements to be made, while simultaneously hampering assessment of uncertainties of individual studies. Therefore, the purpose of this chapter is a general description of transport in pourable liquids as a step towards a thorough understanding the microscopic basis of heat transport in solids. Third, substances that flow under any stress are considered. Simple, Newtonian behavior dominating this category simplifies the measurements, makes available data on viscosity easy to analyze and understand, thereby permitting direct comparisons. Molecular compounds are the mainstay of our comparison, along with liquid elements, and certain oils. This chapter does not consider data on fluids or liquids at very high temperature. Although relevant to the Earth and celestial bodies, existence of radiative transfer at high temperatures is a complicating factor. We cover silicate glasses and liquids in Chapter 10, and diffusive radiative transfer in Chapter 11. Pressure measurements of mass selfdiffusivity are accurate and will be explored briefly, but data on thermal conductivity at P are uncertain by B20% (Ross et al., 1984) or more (Hofmeister, 2010). One problem is contact losses (Chapter 4). Another is small samples (Chapter 7). Pressure determinations of viscosity are only briefly touched upon, in the context of the importance of volume to transport properties.
6.1 TRANSPORT PROPERTIES EXPECTED FOR INELASTIC COLLISIONS OF FINITE SIZE ATOMS IN LIQUIDS Chapter 5 provides a new formula for thermal or mass diffusivity (Eq. (5.30)) that explicitly includes only one state variable, temperature. Kinematic viscosity associated
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with colliding spheres differs by a factor of 2/3 (Eq. (5.23)). The end result is a simple inverse dependence of the diffusivities and kinetic viscosity on atomic or molecular mass (m). For liquids, this parameterization is advantageous because the shapes of molecules vary from nearly spherical as in carbon tetrachloride to very long chains of polymers. The other key parameter tied to the substance is the time of interactions during collisions (τ delay) which is certainly temperature-dependent, and likely pressure dependent, but neither was specified in the simple model presented in Chapter 5. Because the equations describing transport properties of liquids must reduce to those of the gas above the critical point, and should also be identical in the limit of point mass constituents, this section adds a few qualifiers to Eq. (5.30) based on well-known characteristics of pourable liquids, and what was learned from examining gas properties in Chapter 5. The adaptations are intended to address crucial differences existing between these states of matter. Available data on the simplest material known, monatomic gas, showed that Dheat was not identical to Dmass due to the former being monitored via flow of the photon gas, whereas the latter involves motion of the atoms themselves, which, unlike photons, are impeded by each other. The difference between the diffusivities systematically increasing with atomic size (Figs. 5.9 and 5.10) underlies this contention. For diatomic and polyatomic gas (Fig. 5.11), the difference between Dheat and Dmass is greater. Hence, lifetimes for mass and heat transport in liquids cannot be identical. Because interactions in liquids are stronger than in gas due to proximity of the constituents, and because molecule shape and orientation during any interactions should have some effect, the lifetime associated with mass diffusion in liquids actually is a combination of several different lifetimes. Let us denote the primary collision of two atoms as being associated with the lifetime τ heat. For mass transport, a secondary collision will have an effect as well. We do not need to account for the far less-frequent tertiary and higher order collisions, because these can be equivalently “lumped” with the secondary collisions. Because inverse lifetimes add, the lifetime associated with mass transport via collisions will be smaller than that with heat. For liquids, the diffusivities associated with inelastic collisions are proposed to follow: 3NRgc T Dheat D (6.1) τ heat 5 uheat τ heat m 3NRgc T Dmass D (6.2) τ mass 5 umass τ mass ; τ mass # τ heat m where N is the number of moles and Rgc is the gas constant. The temperature (T) or pressure dependence of the lifetimes are not specified. The RHS of these equations is general, arising from dimensional analysis, where we have indicated that each property is not only associated with a different lifetime, but also is linked to a different carrier velocity, should the mechanisms differ. Regarding viscosity of gas, two mechanisms exist, idealized as direct and glancing collisions, which represent the forward transfer of momentum and the inhibiting effect of viscous drag. Considering the collisional cross-sections for spheres predicted that kinematic viscosity (ν) should be 2/3 of either diffusivity (Eq. (5.23)), which is observed to nearly
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hold. For shapes other than spheres, the cross-sections, and therefore the lifetimes of forward momentum transfer and retarding viscous drag will vary from this simple ratio. Because the data reflect both mechanisms, combined, and to preserve the point mass limit, we include the factor of 2/3 while absorbing the complexity into the lifetime: 2NRgc T (6.3) τ drag 5 udrag τ drag ; τ drag Bτ mass νD m Due to the competition, the size of the lifetime associated with viscous flow relative to that of the diffusivities is not obvious, but for small and round molecules the factor of 2/3 should account for the first-order differences. Use of the subscript “drag” anticipates that forward motions are greatly hindered by frictional forces. At high viscosity, diffusion of momentum may be difficult to nearly impossible. Eqs. (6.1) to (6.3) reduce to 5.30 for equal lifetimes and velocities. This meets our stipulation that transport behavior converges above the critical point, where these phases are indistinguishable. The above formulation remains based on molecular collisions as the sole mechanism, while allowing for glancing collisions affecting kinematic viscosity. Other mechanisms are represented by the dimensional formulation, but not by the simple proportionality with temperature. Below, we test the predicted m and T dependencies against available data, recognizing that lifetimes can depend on m or T.
6.2 MEASUREMENTS OF TRANSPORT PROPERTIES FOR POURABLE LIQUIDS Because the purpose of this chapter is to seek trends amongst the data on liquids, we rely on compilations. The transport property best constrained for liquids is Dmass, due to a recent, extensive, and downloadable compilation on diverse molecular compounds by Sua´rez-Iglesias et al. (2015). Their comprehensive review summarizes data at ambient conditions, and various temperatures and pressures, and includes molecular solids and noble gases as well. The figures in this chapter show the accurate Nuclear Magnetic Resonance (NMR) data from this review. The figures average all measurements for any given compound that were made at ambient conditions. Sua´rez-Iglesias et al.’s (2015) compilation for Dmass of diverse pure molecules in the liquid phase by can be considered as inclusive. All transport properties of all three states of H2O are well-studied: see the websites listed at the end of the chapter, Assael et al. (2000), Chaplin (2018), Fokin and Kalashnikov (2008), Ramires and Nieto de Castro (1995), and the compilation of Sua´rez-Iglesias et al. (2015). Online steam tables provide data also on water and water vapor under various conditions. For ice, data sources are James (1968), plus reviews by Weertman (1983) and Fukusako (1990). In contrast, viscosity data shown in the figures are not from thorough and comprehensive reviews, but rather were obtained from compilations that stem from practical needs. The websites listed at the end of the chapter summarize data used by engineers. Much data exist on dynamic viscosity of diverse materials. Because density is commonly measured, kinematic viscosity is easily computed from dynamic viscosity.
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For liquids, fewer data exist on heat transport. Mostly, thermal conductivity (K) is reported in the websites geared for engineering. Fortunately, heat capacity and density are commonly measured, so that most of these data on K can be converted to thermal diffusivity. However, for liquids, the transient hot-wire method is mostly used, which is considered to be accurate to 5% at ambient conditions (Watanabe, 1996; Watanabe and Kato, 2004). A small amount of radiative transfer exists at 298K, and is a contributing factor to the uncertainty. Ballistic radiation affects the temperature derivative as well. However, many liquids that pour at ambient conditions are not stable at high temperatures, and so the data are limited to low temperatures. The expectedly small amount of radiative transfer is not important to our general arguments. Due to needs of the food industry, data on all three transport properties exist on edible oils, although not all properties have been measured for each of the various types of oil. More importantly, the chemical compositions of the oils are not precisely fixed since these are mixtures, and so the molecular mass is approximate, and the comparison is approximate. Fatty acids have well-defined compositions and tranport properties (Valeri and Meirelles, 1997), but not food oils. We include these complex compounds due to their intermediate viscosities and the number of data, which should average the uncertainties associated with compositional variations. References are: Coupland and McClements (1997), de Freitas-Cabral et al. (2011), Diamante and Lam (2014), Huang and Liu (2009), Xu et al. (2014), and Ya´n˜ez-Limo´n et al. (2005). Data on sugars, syrups, and other foods (Bleazard et al., 1996; Ertl and Dullian, 1973; He et al., 2006; Nguyen et al., 2012; White, 1988) were included for similar reasons. Lubrication is essential to machinery and vehicles. Viscosity data are available for many different oils and greases. To represent this type of material, we include silicone oils, because all three transport properties are well-characterized, plus, these oils have the same building block, chained together. Many physical properties are provided by manufacturer’s websites (listed at the end of the chapter). Data on low molecular mass silicone oils are summarized by Roberts et al. (2017). Thermal conductivity and heat capacity were measured by Sandberg and Sundqvist (1982). Data on large molecular mass oils are given by Koschmieder and Pallas (1974) and Thern and Lu¨demann (1996), where the latter is the source of Dmass values. Last but not least, we examine data on the simplest possible liquids, the elements. All three transport properties of the two metals that are liquid at or very close to ambient conditions (Ga and Hg), are well-studied (Aurnou and Olsen, 2001; Blagoveshchenskii et al., 2015; Meyer, 1961; Schriempf, 1972, 1973). The alkali metals (Cs and Rb) which melt near ambient conditions (at 29 C and 39 C, respectively) are less well-characterized. Sources are Andrade and Dobbs (1952) and the compilations of Vargaftik et al. (1994) and Iida et al. (2006). Specific heats of solid alkali metals were used which are identical at the melting point (Filby and Martin, 1966), but are affected by phase transitions (λ-shaped curves). Mercury, being further from its melting point at 238 C, should not be affected. In contrast, data on the two nonmetals, Br2 and I2, are rather uncertain, as discussed in the review by Touloukian and Ho (1981). Because Cl2 is liquid at T fairly close to ambient, we include these data as well, slightly extrapolated, from the same compilation, recognizing that this is more uncertain. Mass diffusivity data have not been published, to the best of our knowledge. A more detailed investigation than that presented here is warranted. However, the graphical analysis below and comparison with trends for gas transport properties from
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
Chapter 5 establish that modeling collisions during translations insufficiently describes transport properties of liquids.
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS 6.3.1 Mass and Density Effects at Ambient Conditions At ambient temperature and pressure, distinct trends exist for each of the three transport properties with molecular mass, although considerable scatter exists in the data (Fig. 6.1). Molecular compounds, including the complex oils and simple diatomics, define the observed trends: mass diffusivity inversely depends on molecular mass, as expected, whereas thermal diffusivity is approximately constant, and, contrastingly, kinematic viscosity is directly proportional to m. The diatomic molecules describe a trend that is slightly steeper but lower than that of the more complex molecular compounds. The slight increase in Dheat with m which exists for the homogenous diatomics may be due to the large uncertainties for bromine and iodine, and with chlorine data being extrapolated, but a similar increase describes the
FIGURE 6.1 Dependence of measured transport properties of liquids near 298K and ambient pressure on mass of their molecules. Data on Dmass (squares and dotted line) mostly from the compilation of Sua´rez-Iglesias et al. (2015). Data on Dheat (dots and solid line) and on ν (diamonds and dashed line) from websites or publications as described in Section 6.2. Silicone oils (dots with gray interiors) and diatomic homogenous molecules (superimposed 3 or 1 signs) were included in the least squares fits shown. Chlorine viscosity is hidden by the Dheat trend. Large gray symbols: liquid metals, which were not including in the fitting. Open circles: heat transport at 300K, which was not included in the fits, but also appears to be independent of m.
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187
accurate measurements of the silicone oils. The edible oils cluster together, which probably stems from approximated values for m. Sugary substances have larger ν than the molecular compounds, which seems connected with these compounds flowing, yet being sticky to the touch. Both food types also have higher Dheat but lower Dmass than the trends for the molecular compounds. The metallic elements have Dmass close to values for the molecular compounds, but possess much larger Dheat and lower viscosity. Unlike the molecular compounds and diatomics, transport data for liquid metals do not depend on mass (Fig. 6.1), but instead follow an inverse power law with density (Fig. 6.2). The power-law fits are steep for thermal diffusivity, moderate for kinematic viscosity, and nearly flat for mass diffusivity. Roughly, the trend for kinematic viscosity of the metals approximates the lowest n for molecular solids while the trend for Dmass of the metals approximates the highest values for molecular solids. This behavior suggests an inverse correlation of Dmass with viscosity, which is explored further below. Fig. 6.2 further emphasizes that the thermal diffusivity of the liquid metals differs greatly from Dheat of the molecular compounds, and that density is not a controlling parameter for the compounds, but does have some effect. For molecular liquids, Dheat is constant (Figs. 6.1 and 6.2). In round numbers, the thermal diffusivity equals 0.1 mm2 s21 within 10%, and furthermore occupies a very restricted range of 0.04 0.15 mm2 s21. Such behavior is inconsistent with molecular collisions, for which a strong mass dependence is expected. For Eq. (6.1) to be valid, requires that the lifetime be proportional to the molecular mass. This is reasonable since large molecules may become entangled with each other. Moreover, Dheat for very simple molecular configurations (the diatomics and the silicone oils) increases with molecular mass, suggesting that a simple dependence of lifetime on mass for all three properties does not exist. To explore this further, thermal diffusivity for restricted ranges of m was fit to power laws in density. The results of Table 6.1 show that Dheat increases slightly with ρ, under nearly FIGURE 6.2 Dependence of measured transport properties of liquids at 298K and ambient pressure on density. Filled black symbols and lines depict liquid metals. Least squares fits are listed at the top. Other symbols as in Fig. 6.1. Data sources listed in Section 6.2.
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
TABLE 6.1 Fits of Dheat at 298 300K as a Function of Density for Molecular Liquids Mass Range amu
No. Pts.
Density Range kg m23
Power-Law Fit
Correlation Coefficient
58 64
6
760 1160
0.016883 ρ0.251
0.21
610 1270
0.0007126 ρ
0.77
620 1280
0.029135 ρ
0.34
70 79 86 93 142 170 253 352
9 6 5 3
0.7116
0.1621
20.0673
730 3100
0.13734 ρ
860 4960
0.039303 ρ
0.1312
0.41 0.92
constant molecular mass. Table 6.1 and Figs. 6.1 and 6.2 suggest that a mechanism other than inelastic collisions governs diffusion of heat in liquids composed of molecules. The trend for Dmass extrapolates to the average for Dheat at m 5 1 amu, which reasonably approximates a point mass atom (i.e., monatomic H). The trend for ν extrapolates to a slightly lower value at m 5 1 amu. This behavior is consistent with our model for gas, which indicated ν 5 2/3 Dheat 5 2/3 Dmass, if the cross-sections are described by inelastic collisions of ideal spherical atoms. However, the magnitudes of the material properties for liquids and gas are much different. For example, the lowest value for Dheat of gas is 3.8 mm2 s21 (for radon), which is B20 3 larger than Dheat of liquids. The discrepancy between mass self-diffusivities is even larger. Although kinematic viscosity magnitudes are similar for liquids and gas, this is fortuitous, as follows: The trends of Fig. 6.1 for heat transport and viscous drag are not consistent with collision based transport, although mass transport in molecular liquids appears to be associated with collisions. Liquids, like gas, have Dheat . Dmass, but the discrepancy in magnitude is larger and kinematic viscosity behaves much differently than the two diffusivities. Chemical bonding appears to affect drag because nonmetals have ν . Dheat . Dmass whereas metallic elements have Dheat . ν . Dmass.
6.3.2 Ratios of Transport Properties at Ambient Conditions Understanding how transport properties of liquids are related are needed to build a model. To eliminate the effects of molecular mass and density on the transport properties at STP, mass diffusivity is plotted against kinematic viscosity in Fig. 6.3A. For the molecular compounds, Dmass inversely correlates with ν. This behavior is equivalent to the trends in Fig. 6.1: because Dmass is inversely proportional to m and ν is proportional to m, then Dmass 5 constant/ν. Setting the power during fitting as unity gives: Dmass 5
3 1 2 1000ν
(6.4)
where the constant of 3/2 is uncertain by 5% from least squares fitting. This inverse provides a simple and testable prediction. For bromine, Dmass should be 0.00495 mm2 s21 at STP and for iodine, 0.00025 mm2 s21 is expected.
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
189
FIGURE 6.3 Cross correlations of transport properties for pourable liquids. (A) Dependence of Dmass on ν. (B) Dependence of Dmass on Dheat. Squares and solid line: molecular compounds. Gray interiors show the subset of silicone oils. Triangles and dotted line: liquid metals. Section 6.2 lists data sources.
For molecular compounds, the two diffusivities are uncorrelated (Fig. 6.3B). This result corroborates our inference of different mechanisms controlling their heat and mass transport. The direct correlations of Dmass with ν and of Dmass with Dheat (Fig. 6.3) for the metallic elements are directly related to the trends in Fig. 6.2. Their viscosity represents a substantial component of forward diffusion of momentum, but with more inhibiting drag than occurs for the gases. Likewise, heat transfer in liquid metals involves collisions, but the mechanism differs substantially from that in gases. To probe the lifetimes, ratios of the transport properties are useful (Fig. 6.4). With this approach, a much smaller database is explored, but one that involves well-studied substances. The power laws for the ratios are consistent with the fits in Fig. 6.1 for the molecular compounds and with the metals not depending on atomic mass. For tiny molecules in this subset of the data described in Section 6.2, the ratios converge to 1/2. For collisions of spheres, ratios of unity and 2/3 are expected. Agreement is reasonable, given that neither Dheat nor ν behave as expected for simple, collisional transport. Ratios of liquid metal transport properties follow power laws in density (not shown) as expected from Figs. 6.2 and 6.3. These do not project to an intercept. For liquid metals, Dmass/Dheat is proportional to density, where ν/Dheat goes as ρ0.63.
6.3.3 The Temperature Dependence of Transport Properties Well-studied substances are considered here: water, simple molecular solids, liquid Hg metal, and silicone oils. Because the data on the homogenous diatomics are not very accurate (Touloukian and Ho, 1981) and because these behave similar to more complex molecular compounds, we do not examine the T dependence for this category in detail. For
190
6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
FIGURE 6.4 Ratios of the transport properties as a function of mass. Triangles 5 kinematic viscosity divided by thermal diffusivity: open for the diatomic nonmetals and gray for the elemental metals. Plus signs: mass diffusivity over thermal diffusivity (gray for liquid metals). Circles: silicone oils. Several compounds are labeled. Section 6.2 lists data sources.
bromine, Dheat is nearly constant (50.082 mm2 s21) from freezing to boiling whereas viscosity decreases strongly: ν(T) 5 737200T22.58 mm2 s21. 6.3.3.1 Water, With a Comparison to Ice and Steam For liquid water, the three transport properties depend on T in much different ways (Fig. 6.5). Thermal diffusivity weakly increases with temperature, whereas kinematic viscosity strongly decreases, while mass diffusivity strongly increases. Projected trends for Dheat and ν converge, but both always greatly exceed Dmass. For Dmass, essentially the same second-order polynomial in T describes both the liquid and solid phases. For each of water and ice, Dmass was not well-described by a power law. For each of water and ice, Dheat weakly depends on T and either a linear or a power-law fit suffices for the small ranges of T examined. These measurements involve non-steady-state techniques (hot-wire ˚ ngstrom’s) and are likely influenced by ballistic transport, due to partial transparand A ency of H2O in the infrared (Fig. 2.7). However, near room temperature ballistic augmentation is small, so the order-or-magnitude difference in Dheat represents the effect of freezing on H2O. Similarly, rheologic behaviors of water and ice differ. Water is a Newtonian fluid: nonetheless, υ depends on T in a complex manner. For an accurate fit, four parameters are required for an exponential form (Fig. 6.5), while five parameters are needed for a polynomial fit (not shown). Viscosity of ice is not shown because its behavior non-Newtonian
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
191
FIGURE 6.5
Transport properties of water and other phases of H2O as a function of temperature. Least squares fits have correlation coefficients of 0.99. Arrows: extrapolations for water trends to high T. Thermal diffusivity of water (small circles) and ice (large circles) depend linearly on T, but differ in slope and are discontinuous at freezing. For Dmass, one trend describes both solid and liquid phases (open squares), but the gas (gray squares) differs greatly, although the fit is also a secondorder polynomial. For gas, thermal diffusivity (gray dots and line, with fit listed) and kinematic viscosity (gray diamonds and dashes) are described by similar fits that parallel steam mass diffusivity. Section 6.2 lists data sources; see also Chaplin (2018).
(Weertman, 1983), and thus the creeping behavior of ice is mechanistically distinct. Although creep has been considered in Earth science to be mathematically equivalent to Newtonian viscous flow, this parallelism ignores the grain-size dependence of creep and the often used formulations in the engineering literature (Meyers and Chawla, 2009), see discussion of Hofmeister and Criss (2018). Transport properties for steam are much higher than those of water or ice, and follow simple power laws in T. The observed behavior (Fig. 6.5, gray symbols) with T follows that of the small molecules CO2 and NH3 (Fig. 5.11), whereby Dmass is larger at high T and the powers in T are similar. The behavior and mechanisms of transport in steam are much simpler than in the two condensed phases of H2O. 6.3.3.2 Small Molecule Liquids In comparing the transport properties of molecular compounds, we opted for convenience: this section considers all small molecules on the viscopedia website for which kinematic viscosity as a function of temperature was listed. Mass diffusivity for all of these (Table 6.2) were compiled by Sua´rez-Iglesias et al. (2015). Thermal diffusivity was obtained from various publications and websites described in Section 6.2. Fig. 6.6 shows that small molecules have similar values exist for each of the three transport properties. Moreover, the overall behavior is like that of water and bromine, where ν decreases with temperature, while Dheat is nearly constant, but Dmass increases as T increases. As observed for H2O, Dmass is continuous across the freezing point and is described by a polynomial when the data cover more than a narrow range of 100K. However, the strong curvature seems to be associated with the freezing point. Unlike
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
TABLE 6.2 Properties of Small Molecule Liquids Name
Formula
m amu
ρa kg m23
Dheat mm2 s21
Tfreeze K
Tboil K
Volume m3 mol21
Benzene
C6H6
78
871
0.094a
279
353
0.0896
Toluene
C7H8
92
862
0.102a
178
384
0.106
Methanol
CH4O
32
787
0.105a
175
338
0.0407
Hexane
C6H14
86
655
0.089a
178
342
0.131
Pentane
C5H12
72
621
0.078b
143
309
0.116
a
At 298K. At 300K.
b
FIGURE 6.6 Transport properties of small molecule liquids as a function of temperature. Circles: benzene; dots: toluene; diamonds: methanol; open squares: hexane. Filled squares 5 pentane. Power laws describe kinematic viscosity, which was determined over a small temperature range. For hexane, ν 5 463,300T22.4275. Thermal diffusivity values are nearly independent of T. Mass diffusivity over a wide temperature range is described by a polynomial fit, whereas narrow ranges can be fit to a power law. For benzene, Dmass 5 2.6496 3 10215T4.8211. For toluene and pentane, Dmass was collected at very low T and is continuous across the freezing point.
water, the fits for ν were simpler: kinematic viscosity follows a simple power law over the temperature range explored. In addition, the transport properties follow a prescribed order. Both Dheat and ν decrease according to the order: benzene $ methanolBtoluene . hexane . pentane
(6.5)
The same order holds for 1/Dmass, as expected from its inverse correlation with ν at 298K (Fig. 6.3). This order is controlled by density, except for methanol, which has a much
6.3 TRENDS IN TRANSPORT PROPERTIES OF POURABLE LIQUIDS
193
lower molecular mass (Table 6.2). The inverse correlation at 298K does not hold at all temperatures. For example, the product ν 3 Dmass equals 1.5 3 1026T1.24 for hexane or is 2.276 3 1027T1.55 for benzene, which have simpler fits for Dmass than do methanol, pentane, or toluene. 6.3.3.3 Silicone Oils Fig. 6.7 shows mass diffusivity measured for four different silicone oils, which are described by a peak in molecular mass, as listed (Thern and Lu¨demann, 1996). Viscosity was inferred from the reported molecular mass, rather than the reported density, which was not entirely consistent with manufacturer’s reports. Polydimethylsiloxane fluids are linear-chain polymers with chemical formulas of (SiO(CH3)2)x. Because the lowestviscosity silicone oil (0.65 cSt) has the shortest chain possible, it alone has a fixed composition of O(Si(CH3)3)2. More viscous oils contain polymers of various lengths and the range of lengths increases with the peak mass, as is evident in the mass distribution plots of Thern and Lu¨demann (1996). Oils from different manufactures may be produced by blending, and so the distributions of mass may differ (Roberts et al., 2017). Therefore, curves for ν in Fig. 6.7 do not correspond exactly to Dmass measurements. In addition, evaporation is significant by 400K for the low viscosity oils (Roberts et al., 2017). Because measurements of ν have been made many times for many different oils, ν vs T is well-constrained, leading to a regular trend in the power-law fits. This may not be the case for Dmass, as suggested by the less regular trend in the power for T. Because of these differences, we do not plot ratios of the transport properties, and focus on the trends, but not the details. The temperature response of the three transport properties is regular and similar to that of water and the small molecule hydrocarbons (Figs. 6.5 and 6.6) which have much lower FIGURE 6.7 Transport properties of silicone oils as a function of temperature. Data from Thern and Lu¨demann (1996) and the Clearco website, see text. Thermal conductivity data of Roberts et al. (2017) indicate Dheat is independent of temperature. The three transport properties were not determined from the exact same substance. Power-law fits are shown, but without the prefactor.
194
6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
m (18 92 amu). Kinematic viscosity and Dmass appear to be inversely correlated whereas ν and Dheat appear to converge at high T. The temperature dependencies are simpler for the more complex silicone oils. However, departures from power laws are observed at the coldest temperatures accessed for 100 cSt oil (Fig. 6.7). 6.3.3.4 Liquid Metallic Mercury Large disparities in values and trends with temperature exist among the three transport properties for Hg (Fig. 6.8), as observed for nonmetals (Figs. 6.5 6.7), although transport properties for these two types of matter differ in many respects. Mass diffusivity with weak curvature over a narrow T range is like that of nonmetals. The weak curvature for Hg was not fit, given the stated uncertainty and that interlaboratory comparisons of suggest uncertainties up to 20% for the tracer methods that are applied to metals (Galus, 1984). The trend for ν of Hg with T is complicated, but similar to a decaying power law, as in nonmetals. Liquid metals offer less resistance to matter flow, which can be attributed to the small size and round shapes of the cations, compared to the larger and less regular shapes of neutral molecules. Thermal diffusivity of Hg depends moderately and linearly on T (Fig. 6.8), rather than being flat and nearly constant, as in the nonmetallic liquids. It is unlikely that atomic motions are involved, due to Dmass being 0.05% of Dheat. One concern is use of containers in laser flash analysis (LFA) measurements. Schreimpf’s (1972) measurements predate three-layer models (Chapter 4). The effect of the supporting walls should be small, and his data were in reasonably agreement with earlier measurements. Thermal conductivity
FIGURE 6.8 Transport properties of liquid mercury as a function of temperature over its entire liquid field. Left axis: thermal diffusivity (dots, with the range and fit from Schriempf, 1972) and mass diffusivity (black squares, shown as 3 1000 measured values with data from Nachtrieb, 1967). Fits have correlation coefficients of 0.99. Uncertainties in Dmass are generally stated to be 5% as indicated. For this reason, only a linear fit was made and diffusivities of Ga, and of Zn and Ag in Hg are also shown (Schriempf, 1973; Galus 1984, respectively). Right axis: kinematic viscosity, compiled by Kozin and Hansen (2013), was fit to a simple power law. For an accurate fit, a fourth-order polynomial is needed.
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195
measurements involve contact losses as well. It is unlikely that the corrections address all the issues in the older data. Hence, additional measurements of Dheat for liquid metals are needed.
6.3.4 Effect of Pressure on Transport Properties As discussed in Chapter 4, measurements of heat transport at pressure are problematic mostly due to contact losses. We therefore only discuss how viscosity and Dmass respond to pressure. Nuclear magnetic resonance methods used to obtain Dmass for molecular liquids are accurate (Sua´rez-Iglesias et al., 2015), and are amenable to studies at pressure. All the compounds listed in Table 6.2 behave similarly. Fig. 6.9A shows isothermal curves for toluene as a function of P. Isobaric data on Dmass were also plotted as a function of T: these curves are roughly parallel (not shown). However, as discussed in Chapter 1 and Chapter 5, the fundamental variable is volume, not pressure. Isothermal curves for Dmass as a function of volume are parallel at large volumes and essentially linear (Fig. 6.9B). For small volumes, the changes are less rapid, requiring polynomial fits over large ranges of V, whereas small ranges of V and compressed material are exponential. Mass diffusivity for any given liquids is a function of T and V which is affected by the mass of the molecule. At a constant and large volume, the change in Dmass with T is roughly linear, from Fig. 6.9B.
FIGURE 6.9 Mass self-diffusivity and density of liquid toluene collected as a function of pressure and temperature. NMR measurements by Harris et al. (1993), as tabulated by Sua´rez-Iglesias et al. (2015). Each isothermal data set is shown with a particular symbol and line pattern, as labeled. (A) As P increases, isothermal Dmass decays exponentially whereas density increases as a modified power law. (B) As volume increases, isothermal Dmass increases. The data above ambient temperature require quadratic fits. Low T data can be fit to exponentials or power laws, due to lower ranges in V (or P).
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6. TRANSPORT BEHAVIOR OF COMMON, POURABLE LIQUIDS
Although a volume dependence was not specified in Eq. (6.2), rarefication certainly affects speeds, causing u to decrease as V increases. Because Dmass increases with V, a strong increase in τ mass is indicated. The trends in Dmass are consistent with a collisional mechanism for mass transport, but one that is severely impeded by strong molecular interactions in the liquid state. It is long known that viscosity increases under compression for pourable liquids except for water below 306K (e.g., Bridgman, 1925). The caused is enhanced frictional drag due to tighter packing of the molecules. The unusual behavior of water stems from its density maximum near 277K, that is, the molar volume of water responds anomalously to pressure around this temperature, and so must viscosity. For measurements of supercooled water and a detailed description of the fascinating changes in viscosity with T and P, see Singh et al. (2017). Hence, viscosity and mass diffusivity for pourable liquids respond in opposite directions to mass, temperature, pressure, and volume, and cannot involve the same mechanism of direct collisions which underlies the kinetic theory of gas.
6.4 MECHANISMS OF TRANSPORT IN LIQUIDS The contrasting behavior of the three transport properties for liquids with basic physical properties and state variables (molecular mass or size, density, temperature, volume, and pressure) indicates that each property is associated with a distinct mechanism. Yet, some ties exist between the transport properties through the motions and interactions of the molecules and through the equation of state (EOS). Forward, direct collisions are essential to move molecules down a compositional gradient. Hence, diffusion of mass in liquids that pour near ambient conditions behaves generally in accord with a collisional model. For molecular compounds, the dependence of Dmass on mass, temperature, and volume (or pressure) is roughly consistent with Eq. (6.2), such that the numerical values are much smaller than in gas. Complexities exist, suggesting that the collisional model for mass transport needs further evaluation. Factors to consider are that the speed of the molecules is largely a thermal effect that is controlled by the EOS, but forward speed is hampered by viscous damping forces. The lifetimes between collisions are shorter due to the compact nature of liquids relative to gases but this may be overshadowed by the lifetimes of the interactions between molecules. Inverse lifetimes add, which makes the important mechanism not obvious. One indication of multiple influences on forward collisions is that mass is an insufficient descriptor (Fig. 6.1) of Dmass for molecular liquids. The scatter in this plot indicates that molecule shape and vibrational or rotational interchanges are important. In liquids, kinematic viscosity does not represent forward motion of momentum, but rather results almost entirely from resistive drag forces. The inverse correlation of Dmass with ν at ambient conditions is a consequence of their contrasting dependence on molecular mass, which was used to represent the size of the molecules. That Dmass and ν are affected in opposite directions by mass, T, V, density, and P results from their contrasting mechanisms of propulsion forward by direct collisions and inhibition of motions by frictional glancing collisions. We reiterate that the equations describing mass diffusion and viscosity, derived by Fick and Newton, respectively, are unrelated.
REFERENCES
197
Thermal diffusivity is independent of mass and density for molecular liquids (Figs. 6.1 and 6.2), which together show that volume is unimportant. Because Dheat measurements on molecular liquids are subject to ballistic radiative transfer, the temperature dependence, albeit weak, probably is stronger than the intrinsic behavior (Chapter 4). Thus, Dheat is largely independent of T, as well as m, density, and volume, which indicates that P has negligible influence. Yet another mechanism is required for this transport property. For liquids, as in gases, forward and glancing collisions as well as interactions of the molecules (generally vibrational) generate photons. Thermal diffusivity is ascertain by sampling photon flux outward. Metallic liquids similarly show evidence for three different transport mechanisms because their properties also differ greatly. However, transport properties in metallic liquids are governed by density, rather than molecular mass. The mechanisms for flow and mass transport in liquid metals are similar to those in molecular solids, given the results of Section 6.3. However, heat transport in liquid metals differs by being larger and temperature-dependent. The increase should not be due to ballistic transport, since metals are opaque over a wide spectral range. The limited data on liquid metals for Dheat suggest that the mechanism operates like that of the solid state (Criss and Hofmeister, 2017; Chapter 9). The effect of compression on Dmass of toluene suggests that the scatter in Fig. 6.1 for the molecular compounds is related to density and volume effects, as well as molecule shape. Tying all the data together in a model requires a deeper analysis than that presented here, and more data than currently available. We have shown that the mechanisms for heat transport in liquids differ from simple, molecular collisions in gas. Yet, liquids, like gases, will have a photon gas intermingled. Plus, the photon gas is sampled during heat transport measurements in all cases (Chapters 1 and 5). To understand this, we next delve into solids with reliable data on Dheat as a function of temperature. Values of transport properties projected to the point mass are consistent with predictions from the kinetic theory of gas, amended to account for glancing collisions affecting ν. The success in applying this model, which was derived for the unrealistic ideal gas, to pourable liquids stems from this unachievable limit being compatible with the model and to combining many parameters in fitting, which can mask imperfections. If point masses could collide, then these collisions would be elastic, but no time evolution would be possible.
References Andrade, E.Nda C., Dobbs, E.R., 1952. The viscosities of liquid lithium, rubidium and caesium. Proc. R. Soc. A 211, 12 30. Assael, M.J., Bekou, E., Giakoumakis, D., Frienda, D.G., Killeen, M.A., Millat, J., et al., 2000. Experimental data for the viscosity and thermal conductivity of water and steam. J. Phys. Chem. Red. Data 29, 141 166. Aurnou, J.M., Olsen, P.L., 2001. Experiments on Rayleigh-Be´nard convection, magnetoconvection, and rotating magnetoconvection in liquid gallium. J. Fluid. Mech. 430, 283 307. Blagoveshchenskii, N., Novikov, A., Puchkov, A., Savostin, V., Sobolev, O., 2015. Self-diffusion in liquid gallium and a hard sphere model. EPJ Web of Conferences 83, No. 02018. Bleazard, J.G., Sun, T.F., Teja, A.S., 1996. The thermal conductivity and viscosity of acetic acid-water mixtures. Int. J. Thermophys. 17, 111 125. Bridgman, P.W., 1925. The viscosity of liquids under pressure. Proc. Natl. Acad. Sci. 11, 603 606.
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