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Issue
A&A
Volume 608, December 2017
Article Number L7
Number of page(s) 4
Section Letters
DOI https://doi.org/10.1051/0004-6361/201732182
Published online 05 December 2017

© ESO, 2017

1. Introduction

Understanding the physical parameters of dust aggregates is important in planetary science. Specifically, the thermal conductivity of dust aggregates is key for determining the thermal evolution of planetary bodies, which influences the thermal evolution pathways of both rocky and icy planetesimals (e.g., Henke et al. 2012; Sirono 2017). The thermal evolution and activity of cometary nuclei also depend on the thermal conductivity of icy aggregates (e.g., Haruyama et al. 1993; Guilbert-Lepoutre & Jewitt 2011).

The thermal conductivity of dust aggregates depends on many parameters, and many previous experimental studies have researched the thermal conductivity of dust aggregates with filling factors above 10-1. The thermal conductivity of porous aggregates in vacuum is given by two terms: the thermal conductivity through the solid network ksol, and the thermal conductivity owing to radiative transfer krad. Krause et al. (2011) showed that the thermal conductivity through the solid network ksol is exponentially dependent on the filling factor of dust aggregates φ for 0.15 <φ < 0.54, and concluded that the coordination number of monomer grains C influences the efficiency of heat flux within the aggregates. Sakatani et al. (2016) revealed that ksol is also dependent on the contact radius between monomers rc. The thermal conductivity owing to radiative transfer krad is affected by the temperature of dust aggregates T and the mean free path of photons lp (e.g., Schotte 1960; Merrill 1969). Moreover, lp depends on R and φ when we apply the geometrical optics approximation for the evaluation of lp (e.g., Skorov et al. 2011; Gundlach & Blum 2012).

There are also several theoretical studies on the thermal conductivity of dust aggregates (e.g., Chan & Tien 1973; Sirono 2014; Sakatani et al. 2017). However, no previous research has been conducted on the thermal conductivity of porous aggregates with filling factors lower than 10-1, although Kataoka et al. (2013b) and Arakawa & Nakamoto (2016) revealed that the collisional growth of dust aggregates leads to planetesimal formation via highly porous aggregates with filling factors much lower than 10-1. Therefore, the purpose of this study is to investigate the thermal conductivity and thermal evolution of fluffy dust aggregates in protoplanetary disks.

In this Letter, we calculate thermal conductivity through the solid network ksol for highly porous aggregates with filling factors in the range of 10-2 to 10-1. We use the snapshot data of Kataoka et al. (2013a) to calculate ksol. We then validate our results through a comparison with the experimental data of Krause et al. (2011). We also derive the thermal conductivity owing to radiative transfer krad for porous aggregates of submicron-sized monomers. Our results show that the thermal conductivity of highly porous aggregates is significantly lower than previously assumed.

2. Method

2.1. Arrangement of monomer grains

The arrangement of monomer grains depends on the coagulation history of the aggregates. During initial dust aggregate coagulation in protoplanetary disks, both experimental (e.g., Wurm & Blum 1998) and theoretical (e.g., Kempf et al. 1999) studies have shown that hit-and-stick collisions lead to the formation of fractal aggregates with a fractal dimension D ~ 2, which is called ballistic cluster-cluster aggregation (BCCA; Meakin 1991). Furthermore, Kataoka et al. (2013a) performed three-dimensional numerical simulations of static compression of BCCA aggregates consisting of 16 384 spherical grains using a periodic boundary condition. In this study, we use snapshots of the compressed BCCA aggregates calculated by Kataoka et al. (2013a).

2.2. Temperature structure of the dust aggregate

To calculate the thermal conductivity through the solid network of an aggregate ksol, we have to determine the temperature of each grain in a cubic periodic boundary. We calculated the temperature of each grain using the method of Sirono (2014). Here, we considered one-dimensional heat flow from the lower boundary plane to the upper boundary plane. There are three choices regarding the pair of lower and upper planes, and we calculated ksol from three directions. Then, we averaged these values for each snapshot.

We defined R as the monomer radius and L3 as the volume of each cubic space. The location of the ith grain (xi,yi,zi) satisfies | xi | <L/ 2, | yi | <L/ 2, and | zi | <L/ 2 for i = 1,2,...,N, where N = 16 384 is the number of grains in the periodic boundary. A sketch of a dust aggregate in a cubic periodic boundary is shown in Fig. 1. Here, we assumed that heat flow occurs along the z-direction. The grains located in −L/ 2 <zi < −(L/ 2−R) are on the lower boundary (number 1 in Fig. 1), and the grains located in + (L/ 2−R) <zi < + L/ 2 are on the upper boundary (number 40 in Fig. 1). When the ith grain was located on the lower (upper) boundary, we added a new grain on the upper (lower) boundary. The location of the new grain is (xi,yi,zi + L) if the ith grain is located on the lower boundary (location X in Fig. 1) and (xi,yi,zi−L) if the ith grain is located on the upper boundary (location Y in Fig. 1). We set the temperature of grains located on the lower (number 1 and location Y in Fig. 1) and upper (number 40 and location X in Fig. 1) boundary as T0 + ΔT/ 2 and T0−ΔT/ 2, respectively.

thumbnail Fig. 1

Sketch of a dust aggregate in a cubic periodic boundary. The temperature of grains located on the lower (number 1 and location Y) and upper (number 40 and location X) boundary is set to T0 + ΔT/ 2 and T0−ΔT/ 2, respectively. The temperature of each grain is calculated by solving Eq. (1) simultaneously for each grain.

Heat flows through the monomer-monomer contacts, and for steady-state conditions, the equation of heat balance at the ith grain is given by ∑jFi,j=0,\begin{equation} \sum_{j} F_{i, j} = 0, \label{eq1} \end{equation}(1)where Fi,j is the heat flow from the jth grain to the ith grain, given by Fi,j=Hc(Tj−Ti),\begin{equation} F_{i, j} = H_{\rm c} {(T_{j} - T_{i})}, \end{equation}(2)where Hc is the heat conductance at the contact of two grains, and Ti and Tj are the temperatures of the ith and jth grains, respectively. We considered the contacts not only inside the periodic boundary but also on the side boundaries (e.g., the contacts between numbers 9 and 10 and between numbers 21 and 22 in Fig. 1). The heat conductance at the contact of two grains Hc is (Cooper et al. 1969) Hc=2kmatrc,\begin{equation} H_{\rm c} = 2 k_{\rm mat} r_{\rm c}, \end{equation}(3)where kmat is the material thermal conductivity and rc is the contact radius of monomer grains. The contact radius rc depends on the monomer radius R and the material parameters (Johnson et al. 1971). The heat conductance within a grain Hg is also given by (Sakatani et al. 2017) Hg=(4π3)1/3kmatR.\begin{equation} H_{\rm g} = {\left( \frac{4 \pi}{3} \right)}^{1/3} k_{\rm mat} R. \end{equation}(4)However, we neglected the effect of Hg because Hg is sufficiently higher than Hc for (sub)micron-sized grains. Therefore, the temperature structure of the aggregate in the cubic periodic boundary can be calculated by solving Eq. (1) simultaneously for all N grains, except for lower and upper boundary grains.

2.3. Thermal conductivity through the solid network

After we obtained the temperature structure, we calculated the total heat flow at the upper boundary ∑ upperFi,j, where we took the sum of contacts between the upper boundary ith grain and internal jth grain (for the case of Fig. 1, ∑ upperFi,j = FX,27 + F40,39). The total heat flow at the upper boundary ∑ upperFi,j can be rewritten using the thermal conductivity through the solid network ksol as ∑upperFi,j=ksolΔTLL2.\begin{equation} \sum_{\rm upper} F_{i, j} = k_{\rm sol} \frac{\Delta T}{L} L^{2}. \end{equation}(5)In this study, we discuss ksol as a function of the filling factor φ, and rewrite L using φ as L=(4πN3φ)1/3R.\begin{equation} L = {\left( \frac{4 \pi N}{3 \phi} \right)}^{1/3} R. \end{equation}(6)Therefore, we obtain ksol as a function of φ as follows: ksol=1LΔT∑upperFi,j,=2kmatrcR·(3φ4πN)1/3∑upperTj−TiΔT,≡\begin{eqnarray} k_{\rm sol} &=& \frac{1}{L {\Delta T}} {\sum_{\rm upper} F_{i, j}}, \nonumber \\ &=& 2 k_{\rm mat} \frac{r_{\rm c}}{R} \cdot {\left( \frac{3 \phi}{4 \pi N} \right)}^{1/3} {\sum_{\rm upper} \frac{T_{j} - T_{i}}{{\Delta T}}}, \nonumber \\ &\equiv& 2 k_{\rm mat} \frac{r_{\rm c}}{R} \cdot f {(\phi)}, \label{eq7} \end{eqnarray}(7)where f(φ) is a dimensionless function of φ. Note that the total heat flow at the lower boundary −∑ lowerFi,j is clearly equal to ∑ upperFi,j considering the heat balance.

3. Results

Here, we present the dimensionless function f(φ) for nine snapshots from three runs and three densities (Table 1). We calculated f(φ) in three directions for each snapshot and took the arithmetic mean values. We note that the compressed BCCA aggregates might be isotropic if the number of monomer grains is sufficiently large; thus, we only discuss the mean values of f(φ).

thumbnail Fig. 2

Fitting of the dimensionless function of thermal conductivity f(φ) as a function of the filling factor φ. The green dashed line is the best-fit line, and the magenta solid line represents the simple function f(φ) = φ2.

Table 1

Results of the numerical calculation.

Figure 2 shows the dimensionless function f(φ) as a function of the filling factor φ. The best-fit line given by the least-squares method (green dashed line) is f(φ)=1.18φ2.14.\begin{equation} f {(\phi)} = 1.18 \phi^{2.14}. \end{equation}(8)Hereafter, we use the following more simple relationship between f(φ) and φ (magenta solid line) f(φ)=φ2.\begin{equation} f {(\phi)} = \phi^{2}. \label{eq9} \end{equation}(9)Sakatani et al. (2017) predicted that f(φ) = (2 /π2)Cφ, where C is the coordination number. For highly porous aggregates, the coordination number C is approximately two, and the filling factor dependence on C is weak. Hence, f(φ) would be proportional to φ in the model of Sakatani et al. (2017); however, in reality, f(φ) is approximately proportional to φ2.

In the context of the thermal conductivity of colloidal nanofluid and nanocomposites, Evans et al. (2008) revealed that thermal conductivity is strongly affected by the fraction of linear chains that contribute to heat flow in the aggregates. The contributing grains are called backbone grains, and non-contributing grains are called dead-end grains (numbers 8, 32, and 33 in Fig. 1; Shih et al. 1990). We will discuss the effects of different fractions of backbone and dead-end grains on the thermal conductivity in future research.

By comparing our model to the experimental data of Krause et al. (2011), we can confirm the validity of our model (Fig. 3). The magenta line in Fig. 3 represents the calculated thermal conductivity from Eqs. (7) and (9), the blue curve is the exponential fitting of experimental data, ksol = 1.4e7.91(φ−1) W m-1 K-1 (Krause et al. 2011), and the black dashed line is a model commonly used to study the thermal evolution of planetary bodies, that is, ksol = φkmat (e.g., Sirono 2017). Both experimental (crosses) and numerical data (squares, triangles, and circles) are plotted.

thumbnail Fig. 3

Experimental (crosses) and numerical data (squares, triangles, and circles) of thermal conductivity through the solid network ksol. Our model (magenta line) was compared to the experimental fitting data of Krause et al. (2011, blue curve).

When we consider the dust aggregates of (sub)micron-sized monomers, the contact radius between monomers rc is given by (Johnson et al. 1971) rc=(9πγ(1−ν2)2YR)1/3R,\begin{equation} r_{\rm c} = {\left( \frac{9 \pi \gamma {(1 - \nu^{2})}}{2 Y R} \right)}^{1/3} R, \label{eq10} \end{equation}(10)where γ = 25 mJ m-2, ν = 0.17, and Y = 54 GPa are the surface energy, Poisson’s ratio, and Young’s modulus of SiO2 grains (Wada et al. 2007). We set R = 0.75 μm and kmat = 1.4 W m-1 K-1 to the same values as Krause et al. (2011). Figure 3 clearly shows that our empirical model is applicable to the ksol of porous aggregates with filling factors of φ ~ 0.1. Moreover, our model is applicable not only for φ ≲ 0.1, but also for the range 0.1 ≲ φ ≲ 0.5.

We note that most of the experimental data of ksol were fitted using exponential functions of φ, which pass the material thermal conductivities (e.g., Krause et al. 2011; Henke et al. 2013). However, when we consider the thermal conductivity through the solid network, that is, thermal conductivity limited by the necks between two monomers, ksol must be given by ksol ~ (rc/R)kmat, even for dense dust aggregates whose filling factors are close to unity (e.g., Chan & Tien 1973).

4. Discussion

Finally, we evaluated the total thermal conductivity of porous icy aggregates under vacuum conditions. The thermal conductivity through the solid network ksol is given by ksol=2kmatrcRφ2,\begin{equation} k_{\rm sol} = 2 k_{\rm mat} \frac{r_{\rm c}}{R} \phi^{2}, \end{equation}(11)and thermal conductivity owing to radiative transfer krad is given by (Merrill 1969) krad=163σSBT3lp,\begin{equation} k_{\rm rad} = \frac{16}{3} \sigma_{\rm SB} T^{3} l_{\rm p}, \end{equation}(12)where σSB = 5.67 × 10-8 W m-2 K-4 is the Stefan-Boltzmann constant. We calculated the mean free path of photons in fluffy aggregates of submicron-sized grains lp as follows: lp=1(κabs+κsca)ρmatφ,\begin{equation} l_{\rm p} = \frac{1}{{(\kappa_{\rm abs} + \kappa_{\rm sca})} \rho_{\rm mat} \phi}, \end{equation}(13)where κabs and κsca are the absorption and scattering mass opacities of monomers, respectively, and ρmat = 1.68 × 103 kg m-3 is the material density. Here, the composition of icy dust aggregates is consistent with Pollack et al. (1994). The total mass opacity of submicron-sized monomers κabs + κsca is hardly dependent on the wavelength of the thermal radiation λ = 2.9 × 10-3 (T/ K)-1 m for 10-6 m <λ < 10-4 m, and κabs + κsca is on the order of 102 m2 kg-1 (e.g., Kataoka et al. 2014). Then we set lp = 10-5φ-1 m in this study. We note that for the case of fluffy aggregates of submicron-sized monomers, the wavelength λ is larger than the monomer radius R, even if the temperature is on the order of 103 K. Hence, we cannot apply the geometrical optics approximation to evaluate lp.

Figure 4 shows the ksol of crystalline and amorphous icy aggregates, ksol,cr and ksol,am, and the thermal conductivity owing to radiative transfer krad for aggregates composed of icy monomers with a radius of R = 0.1 μm and temperature of T = 40 K. We set γ = 100 mJ m-2, ν = 0.25, and Y = 7 GPa for icy grains (Wada et al. 2007). The material thermal conductivities of crystalline and amorphous grains, kmat,cr and kmat,am, are given by kmat,cr = 5.67 × 102 (T/ K)-1 W m-1 K-1 and kmat,am = 7.1 × 10-8 (T/ K) W m-1 K-1, respectively (Haruyama et al. 1993).

thumbnail Fig. 4

Comparison between ksol (magenta solid line for crystalline icy aggregates and magenta dashed line for amorphous) and krad (blue dashed line). The monomer radius is R = 0.1 μm, and the temperature is T = 40 K.

For the case of φ < 4 × 10-3, the thermal conductivity owing to radiative transfer krad is higher than the thermal conductivity through the solid network of crystalline icy aggregates ksol,cr, even when the temperature is sufficiently low (T = 40 K). If the total thermal conductivity ksol + krad does not change substantially when crystallization occurs, then the internal temperature of icy planetesimals could still increase after crystallization, which might cause runaway crystallization due to latent heat. In addition, when the temperature of an icy planetesimal increases, sintering can proceed inside the icy aggregate before monomer grains evaporate or melt (e.g., Sirono 2017). We cannot evaluate the contact radius rc from Eq. (10) when aggregates are sintered, and ksol increases linearly as a consequence of the increase of rc. Sintering might also affect the mechanical strength of aggregates (e.g., Omura & Nakamura 2017) and the critical velocity for collisional growth (e.g., Sirono & Ueno 2017). Therefore, the growth pathways of icy planetesimals might be altered by sintering of icy aggregates. Not only icy planetesimals, but also rocky aggregates could experience sintering before growing into dm-sized bodies, which might explain the retainment of chondrules inside fluffy aggregates (Arakawa 2017). We also note that the total thermal conductivity might be controlled by the thermal conductivity due to gas diffusion for the case of fluffy aggregates in high gas density environments (e.g., the innermost region of protoplanetary disks and/or planetary surfaces; Piqueux & Christensen 2009).

In conclusion, we have revealed the filling factor dependence of the thermal conductivity of porous aggregates. We showed that the thermal conductivity of highly porous aggregates is significantly lower than previously assumed. In future work, we will reexamine the growth pathways of planetesimals in protoplanetary disks, and combine this with a density and thermal evolution analysis.

Acknowledgments

We thank the referee Gerhard Wurm for constructive comments. This work is supported by JSPS KAKENHI Grant (15K05266). S.A. is supported by the Grant-in-Aid for JSPS Research Fellow (17J06861).

References

  1. Arakawa, S. 2017, ApJ, 846, 118 [NASA ADS] [CrossRef] [Google Scholar]
  2. Arakawa, S., & Nakamoto, T. 2016, ApJ, 832, L19 [NASA ADS] [CrossRef] [Google Scholar]
  3. Chan, C. K., & Tien, C. L. 1973, Journal of Heat Transfer, 95, 302 [CrossRef] [Google Scholar]
  4. Cooper, M. G., Mikic, B. B., & Yovanovich, M. M. 1969, International Journal of Heat and Mass Transfer, 12, 279 [Google Scholar]
  5. Evans, W., Prasher, R., Fish, J., et al. 2008, International Journal of Heat and Mass Transfer, 51, 1431 [CrossRef] [Google Scholar]
  6. Guilbert-Lepoutre, A., & Jewitt, D. 2011, ApJ, 743, 31 [NASA ADS] [CrossRef] [Google Scholar]
  7. Gundlach, B., & Blum, J. 2012, Icarus, 219, 618 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  8. Haruyama, J., Yamamoto, T., Mizutani, H., & Greenberg, J. M. 1993, J. Geophys. Res., 98, 15 [Google Scholar]
  9. Henke, S., Gail, H.-P., Trieloff, M., Schwarz, W. H., & Kleine, T. 2012, A&A, 537, A45 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  10. Henke, S., Gail, H.-P., Trieloff, M., & Schwarz, W. H. 2013, Icarus, 226, 212 [NASA ADS] [CrossRef] [Google Scholar]
  11. Johnson, K. L., Kendall, K., & Roberts, A. D. 1971, Proc. Roy. Soc. London Ser. A, 324, 301 [Google Scholar]
  12. Kataoka, A., Tanaka, H., Okuzumi, S., & Wada, K. 2013a, A&A, 554, A4 [Google Scholar]
  13. Kataoka, A., Tanaka, H., Okuzumi, S., & Wada, K. 2013b, A&A, 557, L4 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  14. Kataoka, A., Okuzumi, S., Tanaka, H., & Nomura, H. 2014, A&A, 568, A42 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  15. Kempf, S., Pfalzner, S., & Henning, T. K. 1999, Icarus, 141, 388 [NASA ADS] [CrossRef] [Google Scholar]
  16. Krause, M., Blum, J., Skorov, Y. V., & Trieloff, M. 2011, Icarus, 214, 286 [NASA ADS] [CrossRef] [Google Scholar]
  17. Meakin, P. 1991, Rev. Geophys., 29, 317 [NASA ADS] [CrossRef] [Google Scholar]
  18. Merrill, R. B. 1969, Nasa Technical Note, D-5063 [Google Scholar]
  19. Omura, T., & Nakamura, A. M. 2017, Planet. Space Sci., in press [arXiv:1708.08038] [Google Scholar]
  20. Piqueux, S., & Christensen, P. R. 2009, J. Geophys. Res., 114, E09005 [Google Scholar]
  21. Pollack, J. B., Hollenbach, D., Beckwith, S., et al. 1994, ApJ, 421, 615 [NASA ADS] [CrossRef] [Google Scholar]
  22. Sakatani, N., Ogawa, K., Iijima, Y.-i.,Arakawa, M., & Tanaka, S. 2016, Icarus, 267, 1 [NASA ADS] [CrossRef] [Google Scholar]
  23. Sakatani, N., Ogawa, K., Iijima, Y.-i., et al. 2017, AIP Adv., 7, 015310 [NASA ADS] [CrossRef] [Google Scholar]
  24. Schotte, W. 1960, AIChE Journal, 6, 63 [CrossRef] [Google Scholar]
  25. Shih, W.-H., Shih, W. Y., Kim, S.-I., Liu, J., & Aksay, I. A. 1990, Phys. Rev. A, 42, 4772 [NASA ADS] [CrossRef] [Google Scholar]
  26. Sirono, S.-i. 2014, Meteor. Planet. Sci., 49, 109 [NASA ADS] [CrossRef] [Google Scholar]
  27. Sirono, S.-i. 2017, ApJ, 842, 11 [NASA ADS] [CrossRef] [Google Scholar]
  28. Sirono, S.-i., & Ueno, H. 2017, ApJ, 841, 36 [NASA ADS] [CrossRef] [Google Scholar]
  29. Skorov, Y. V., van Lieshout, R., Blum, J., & Keller, H. U. 2011, Icarus, 212, 867 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  30. Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2007, ApJ, 661, 320 [NASA ADS] [CrossRef] [Google Scholar]
  31. Wurm, G., & Blum, J. 1998, Icarus, 132, 125 [NASA ADS] [CrossRef] [Google Scholar]

All Tables

Table 1

Results of the numerical calculation.

In the text

All Figures

thumbnail Fig. 1

Sketch of a dust aggregate in a cubic periodic boundary. The temperature of grains located on the lower (number 1 and location Y) and upper (number 40 and location X) boundary is set to T0 + ΔT/ 2 and T0−ΔT/ 2, respectively. The temperature of each grain is calculated by solving Eq. (1) simultaneously for each grain.

In the text
thumbnail Fig. 2

Fitting of the dimensionless function of thermal conductivity f(φ) as a function of the filling factor φ. The green dashed line is the best-fit line, and the magenta solid line represents the simple function f(φ) = φ2.

In the text
thumbnail Fig. 3

Experimental (crosses) and numerical data (squares, triangles, and circles) of thermal conductivity through the solid network ksol. Our model (magenta line) was compared to the experimental fitting data of Krause et al. (2011, blue curve).

In the text
thumbnail Fig. 4

Comparison between ksol (magenta solid line for crystalline icy aggregates and magenta dashed line for amorphous) and krad (blue dashed line). The monomer radius is R = 0.1 μm, and the temperature is T = 40 K.

In the text

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