Solid–Liquid Thermal Transport and Its Relationship with Wettability

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Solid−Liquid Thermal Transport and Its Relationship with Wettability and the Interfacial Liquid Structure Bladimir Ramos-Alvarado,* Satish Kumar, and G. P. Peterson The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States S Supporting Information *

ABSTRACT: Experiments and atomistic simulations have suggested the existence of a direct correlation between the wetting properties of a surface and heat transfer across it. In this investigation, molecular dynamics simulations of surface wettability and solid−liquid thermal transport were conducted in order to better understand the relationship between the surface chemistry and thermal transport. The wettability transparency of graphene-coated surfaces was considered in order to investigate heat transfer across a complex interface with similar wettability as a bare surface. The results indicate that the relationship between the interfacial heat transfer and wettability is not universal. The density depletion length was found to reconcile the thermal boundary conductance calculations for different bare and graphenecoated silicon surfaces.

T

Ge et al.11 experimentally determined G between water and chemically functionalized aluminum and gold surfaces. The wettability of the surfaces was altered by chemical functionalization using SAMs. G values of 100−180 MW/m2K and 50− 60 MW/m2K were obtained for hydrophilic and hydrophobic surfaces, respectively. Harikrishna et al.12 conducted similar experiments11 but controlled wettability by only using different terminal groups in the alkyl chains, whereas Ge et al.11 used different alkyl groups. Harikrishna et al.12 found a linear relationship between G and the work of adhesion, which is eventually proportional to (1 + cos θ), and consistent with the simulation results by Shenogina et al.8 In this investigation, MD simulations were performed to investigate the relationship between wettability and the interfacial thermal transport in silicon−water systems. Graphene-coated silicon surfaces were considered in order to broaden the perspective of the wettability effect by including a complex system with similar wettability as the bare surfaces. The interfacial liquid structure was characterized by the density depletion length and a remarkable reconciliation of the calculations of G was found for the different systems. Unlike the description of G as a function of the surface wettability, the density depletion presented a universal parameter to explain heat transfer across different silicon−water interfaces with varying wettability. In the limit of null interfacial density depletion, an upper limit of G was found and this value is consistent with experiments for superhydrophilic surfaces. MD simulations of water droplets wettability were conducted for Si(100), Si(111), and the graphene-coated versions of these

hermal transport across solid−liquid interfaces has gained significant importance due to the vast applications of micronanofluidics in areas such as in energy conversion, biomedical devices, sensing in aqueous environments, ultrafast flow delivery, and so forth. The interest in interfacial heat transfer between solids and liquids begun with the early findings by Kapitza1 while studying the superfluidity of helium in cryogenic conditions, but as of today, there is no theory fully capable of explaining the thermal resistance at these interfaces.2 Solid−liquid thermal transport is affected by the affinity between the phases (wettability), but that affinity can also affect the properties of the interfacial liquid, responsible for the heat exchange.3 Early investigations dealt with the wettability effect using molecular dynamics (MD) simulations of simple monatomic systems (Lennard-Jones particles).2,4−6 Wettability was usually introduced by modifying the solid−liquid interaction strength (solid−liquid affinity) in order to later compute the Kapitza length or effective interface thermal resistance thickness (LK). It was found that LK sharply decreased as the solid−liquid affinity increased. More recent numerical investigations use systems with realistic solid structures interacting with water or other complex liquids, where it has become a common practice to calibrate the solid−liquid interaction potentials by matching wettability simulations with experiments.7−10 Merabia et al.7 observed that a hydrophilic gold-water system was more thermally conductive than a hydrophobic gold−octane interface. Shenogina et al.8 found a linear relationship between the thermal boundary conductance, G, and 1 + cos θ, where θ is the contact angle of surfaces chemically functionalized using selfassembled monolayers (SAMs). Acharya et al.9 extended the work by Shenogina et al.8 and demonstrated that the nanoscale surface roughness enhances the interfacial thermal transport. © XXXX American Chemical Society

Received: July 20, 2016 Accepted: August 19, 2016

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Figure 1. Sketch of the heat transfer modeling approach for the (a) silicon−water system and (b) graphene-coated silicon system.

surfaces. The parameters of the simulations, procedures, and results have been previously reported in ref 13. Solid−liquid thermal transport was investigated by nonequilibrium MD simulations in the systems depicted in Figure 1. The silicon− water interaction potentials used to obtain the wettability conditions reported in ref 13. were implemented in the heat transfer simulations. As for the graphene-coated surfaces, the conditions required to observe wettability transparency (same contact angle as the bare surface) were implemented and the solid−liquid interaction potentials were calibrated accordingly. The number of water molecules confined between two silicon slabs was varied in order to enforce similar bulk density and pressure in the region away from the walls, since the minimum pressure required to push water through a slab varies depending on the wettability.14 Thermal energy was added and removed at the same rate in the regions indicated in Figure 1, the heat flow was normalized by the area of the system (heat flux J), and the temperature jump ΔTint was calculated at the interface; see the Supporting Information for a detailed description of the modeling approach. Different heat fluxes were applied to the systems and the thermal boundary conductance was obtained as the slope of the expression J = GΔTint. A linear dependence was found between J and ΔT int , indicating that this investigation was conducted in the linear response regime. Area and length effects were investigated and the dimensions used were those unaffected by the size of the systems. Figure 2 illustrates the characterization of the thermal transport properties of the different surfaces in contact with water as a function of wettability. G was calculated for the hot and cold walls as indicated in Figure S1 (Supporting Information). It was consistently observed that G was greater for surfaces with higher temperature and GHOT/GCOLD ≤ 1.1 was obtained for most of the cases without a noticeable trend, as indicated in Figure S2 (Supporting Information). Murad and Puri15 reported similar wall temperature effects, namely, hot walls are more conductive than cold walls. Thus, the results reported in Figure 2 are the averages and standard deviations of G between hot and cold wall calculations for different realizations. Consistent with the numerical investigation by Shenogina et al.8 and the experimental findings by Harikrishna et al.,12 it was obtained that G ∼ 1 + cos θ for both silicon surfaces, see Figure 2a. It is clear that there is a lack of a universal or anisotropy between the contact angle and G, since

Figure 2. Thermal boundary conductance relationships for different silicon and graphene-coated silicon surfaces. (a) Thermal boundary conductance vs 1 + cos θ (proportional to the work of adhesion) and (b) thermal boundary conductance vs the contact angle.

different curves are obtained for Si(111) and Si(100) surfaces. The main difference between them is the atomic planar density ρs, being the Si(111) surface denser (7.83 nm−2) and more closely packed in a bilayer structure than the Si(100) (6.87 nm−2) with large interstitial spaces along the [111] direction. More noticeable, the two graphene-coated silicon surfaces exhibit a significantly lower G in comparison with the bare surfaces, even though the contact angle is the same. Figure 2b illustrates the direct dependence of G on the contact angle, a functional dependence presented for the first time for silicon and graphene-coated silicon surfaces. As expected, G increases as the surfaces become more hydrophilic, and just as in the previous case, the contact angle does not represent a universal parameter to correlate with the G. The range of the results 3498

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The reconciliation of the contact angle calculations provides an opportunity to test the dependence of G on the surface energy. Figure 3b illustrates that the behavior of G is the same as that for the work of adhesion or 1 + cos θ. This is not surprising but was necessary to verify, because the wettability of the silicon surfaces was found to be crystallographic planedependent (anisotropy). Thus, by finding a unifying parameter able to reconcile the different contact angle calculations, but not the thermal transport, allows to conclude that the work of adhesion is not a universal parameter to explain the thermal transport across solid−liquid interfaces. The contact angle and surface energy correlations support this conclusion. It is evident that having a large work of adhesion, or a hydrophilic surface, promotes an efficient solid−liquid thermal transport. However, the differences observed between the silicon and graphene-coated silicon surfaces calls for further analysis. As a first step, the phonons density of states (DOSs) was obtained for an atomic monolayer in a region unaffected by the liquid (bulk) and for the atomic interfacial monolayers in contact with water. The fast Fourier transform of the velocity autocorrelation function of the solid atoms was used to obtain the DOSs as indicated in ref 19. Figure 4 illustrates the DOSs in

reported in Figure 2 match the experimental results reported by Ge et al.11 and Harikrishna et al.12 in terms of the contact angle and G. The linear relationships previously reported in Figure 2, namely, G ∼ 1 + cos θ and G ∼ 180° − θ, can be partially deducted from the results reported in Figure S3 (Supporting Information), where the value of G is observed to increase as the silicon−water interaction strength (ε) increases and the fact that ε ∼ 1 + cos θ and ε ∼ 180° − θ.16 These two scaling laws originate from a simplification performed on a mean-field model of wettability and have been previously verified for silicon in ref 17. As in the case of the scaling laws depicted in Figure 2, two different linear relationships were found for the wettability of Si(111) and Si(100), being the Si(111) surface more hydrophilic than the Si(100) for the same value of ε, see Table S1 (Supporting Information). The different wettability of the Si(111) and Si(100) planes as a function of ε is understandable, because the lower atomic planar density of Si(100) demands for a larger value of ε to match the wettability of the Si(111) plane. From the comprehensive wettability model reported in ref 13, it is possible to realize that ερs ∼ 1 + cos θ, thus, the work of adhesion is scaled by the energy per unit area determined by the concentration of solid atoms per plane. The application of this new scaling law to the wettability of different surfaces is depicted in Figure 3a, where the

Figure 4. Phonon density of states (DOS) in arbitratry units of atomic layers in the bulk and interface of Si(100) and Si(111) in contact with water.

arbitrary units where the transverse (T), longitudinal (L), acoustic (A), and optical (O) modes are indicated. The DOSs for the bulk monolayer features similar characteristics as those of the bulk material.20 The TA modes for the interfacial monolayers for Si(100) and Si(111) feature similar properties as the bulk with a slight shift in frequency, something also reported in ref 21. As for the LA and LO modes, the bulk properties seem to be fairly preserved. The main difference exists in the high frequency transverse modes (TO). A significant reduction in the DOSs with respect to the bulk occurs on both silicon surfaces, indicating that high-frequency phonons are being transmitted in the in-plane direction. In addition, a more noticeable reduction of the TO modes is observed for the Si(100) than for the Si(111) surface, indicating a strong coupling of these phonon modes at the interface. Goicochea et al.22 found that the low frequency phonons were the most significant modes for heat transfer in a quartz−water interface, whereas a rather weak coupling of high frequency modes was reported, which is in contrast to what is reported in Figure 4. The results depicted in Figure 4 alleviate any concern about an improper characterization of the solid−liquid interactions as an explanation for the different thermal transport between

Figure 3. (a) Reconciliation of the contact angle calculations using the solid−liquid interaction energy per unit area and (b) thermal boundary conductance vs the solid−liquid interaction energy per unit area.

difference in the wettability of different silicon planes reported in ref 17 is no longer observed when the contact angle is characterized by the interaction energy per unit area (ερs). Results for the wettability of graphite reported in ref 18 are also shown and the results for the graphene-coated surfaces were obtained by averaging the contribution of the silicon and graphene surfaces to the work of adhesion. 3499

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The Journal of Physical Chemistry Letters silicon surfaces and reaffirm the conclusion that the work of adhesion is not a universal parameter for describing the solid− liquid thermal transport. The results indicate that high frequency phonons present a strong coupling for silicon and water. This observation may suggest an important effect of the differences found between the interfacial liquid structure of water, which is a parameter strongly affected by the surface roughness of the Si(100) and Si(111) surfaces. In refs 13 and 18, it was indicated that liquid water entrainment was observed in the large intersticia of the Si(100) plane, and in ref 23, the water entrainment was found to be of major importance in the description of the hydrodynamic boundary condition. As in the case of momentum transfer, thermal transport is affected by the effective contact between solid and liquid atoms. A quantification of the equilibrium distance between phases can be obtained by the density depletion length δ=

∫0

ρ (z ) ρ l (z ) ⎤ ⎢1 − s ⎥d z − ⎢⎣ ρsb ρl b ⎥⎦

determined that the liquid layering at the solid−liquid interface had a negligible effect on the liquid thermal properties, but it was suggested that the liquid depletion could have an effect on the interfacial thermal resistance. Tori et al.25 investigated the directional heat transfer contributions to the interfacial thermal transport in simple systems. The roughness of the solid structure was modified in order to alter the corrugation of the energy potential between the solid and liquid phases. Tori et al.25 found a strong contribution of the in-plane energy fluxes, consistent with the DOSs depicted in Figure 4 and from their results, a correlation between the density depletion (concentration of liquid particles and equilibrium distance) and G could be formulated but was not quantified in ref 25. Pham et al.10 used a different definition of the depletion length (position of the first liquid density peak) that suggested a correlation between thermal transport and the equilibrium distance between the phases. While investigating thermal transport in silica−water interfaces, Murad and Puri,26 highlighted the effect of the interfacial water structure on the effectiveness of heat transfer. Murad and Puri26 reported a reduction of the Kapitza resistance (RK) at interfaces where more fluid absorption and adsorption layers were observed. It was suggested that the frequency of intermolecular interactions increases as the layering of interfacial water increases in hydrophilic surfaces; however, it was reported here and in refs 18 and 23 that the solid−liquid interaction strength gives a partial explanation of layering and its effects on interfacial heat and momentum transfer. In a similar investigation Murad and Puri27 added to the previous discussion by observing that pressure also affects the “solid-like” structure of water at the interface, that is, increasing the liquid layering at the interface reduces RK. Therefore, not only the surface chemistry but also the compressibility caused by the confinement or temperature of these nanoscale systems affects the interfacial liquid layering and the solid−liquid thermal transport. It is of utmost importance to highlight that none of the previously discussed investigations proposed or observed a correlation such as that depicted in Figure 5, and to the best of our knowledge this is the first time the density depletion length has been used to explain the solid−liquid thermal transport. The best fit to the data reported in Figure 5 was an exponential function (R2 = 0.99), where the upper limit of the conductance was extrapolated at 195 MW/m2K and the parameter n = −0.5. The upper limit of G when δ → 0 encloses the maximum values of G experimentally reported by Ge et al.11 and Harriskrishna et al.12 for highly hydrophilic surfaces. The interfacial thermal transport between different silicon planes and water was characterized in terms of the wetting properties of the surfaces. The wetting transparency phenomenon was used in order to investigate the relationship between wettability and thermal boundary conductance. The results indicated that the wettability, characterized by the contact angle and the work of adhesion, is not a fundamental parameter to correlate with the interfacial heat transfer. The changes observed in the phonon density of states of the different interfacial planes directed to investigate the interfacial properties of water. It was found that similar to the hydrodynamics of nanoconfined liquids, the density depletion length is a fundamental parameter to explain the interfacial thermal transport.

∞⎡

(1)

where ρs(z) and ρl(z) are the solid and liquid densities along the z direction (thickness) of the water slab, and ρbs and ρb1 are the bulk solid and liquid densities. Figure S4 (Supporting Information) depicts the characteristics and snapshots of the interface of the different surfaces investigated. Figure 5

Figure 5. Reconciliation of the thermal boundary conductance calculated for different surfaces in contact with water using the density depletion length.

illustrates the correlation between the density depletion and the interfacial thermal transport. A unified description of the thermal transport phenomenon is achieved for the different surfaces when the interfacial density depletion is used as a parameter. The water entrainment previously reported for the Si(100) surface18 is responsible for the small δ observed for this surface, hence the larger values of G can be explained due to a better contact between phases (better thermal energy transmission), as reflected also in the phonons DOSs. In other words, if the density depletion is small and water entrainment is observed, the interfacial heat transfer due to high frequency transverse phonon modes increases, as also supported by DOSs of the TO modes in Figure 4. The smaller G of the denser Si(111) plane can be explained due to the larger δ observed for this surface due to the closely packed structure of this plane. Lastly and more importantly, the G for the graphene-coated silicon surfaces is properly matched with the bare surfaces having the same density depletion. A characterization of the solid−liquid potential energy corrugation of these surfaces was reported in ref 23. Xue et al.24 3500

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(16) Sendner, C.; Horinek, D.; Bocquet, L.; Netz, R. R. Interfacial Water at Hydrophobic and Hydrophilic Surfaces: Slip, Viscosity, and Diffusion. Langmuir 2009, 25 (18), 10768−10781. (17) Ramos-Alvarado, B.; Kumar, S.; Peterson, G. P. Hydrodynamic Slip in Silicon Nanochannels. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2016, 93 (3), 033117. (18) Ramos-Alvarado, B.; Kumar, S.; Peterson, G. P. Wettability of Graphitic-Carbon and Silicon Surfaces: MD Modeling and Theoretical Analysis. J. Chem. Phys. 2015, 143 (4), 044703. (19) Duda, J. C.; English, T. S.; Piekos, E. S.; Soffa, W. A.; Zhigilei, L. V.; Hopkins, P. E. Implications of Cross-Species Interactions on the Temperature Dependence of Kapitza Conductance. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84 (19), 193301. (20) Martin, P.; Aksamija, Z.; Pop, E.; Ravaioli, U. Impact of PhononSurface Roughness Scattering on Thermal Conductivity of Thin Si Nanowires. Phys. Rev. Lett. 2009, 102 (12), 125503. (21) Caplan, M. E.; Giri, A.; Hopkins, P. E. Analytical Model for the Effects of Wetting on Thermal Boundary Conductance Across Solid/ Classical Liquid Interfaces. J. Chem. Phys. 2014, 140 (15), 154701. (22) Goicochea, J. V.; Hu, M.; Michel, B.; Poulikakos, D. Surface Functionalization Mechanisms of Enhancing Heat Transfer at SolidLiquid Interfaces. J. Heat Transfer 2011, 133 (8), 082401. (23) Ramos-Alvarado, B.; Kumar, S.; Peterson, G. P. Wettability Transparency and the Quasiuniversal Relationship Between Hydrodynamic Slip and Contact Angle. Appl. Phys. Lett. 2016, 108 (7), 074105. (24) Xue, L.; Keblinski, P.; Phillpot, S. R.; Choi, S. U. S.; Eastman, J. A. Effect of Liquid Layering at the Liquid-Solid Interface on Thermal Transport. Int. J. Heat Mass Transfer 2004, 47 (19−20), 4277−4284. (25) Torii, D.; Ohara, T.; Ishida, K. Molecular-Scale Mechanism of Thermal Resistance at the Solid-Liquid Interfaces: Influence of Interaction Parameters Between Solid and Liquid Molecules. J. Heat Transfer 2010, 132 (1), 012402. (26) Murad, S.; Puri, I. K. Molecular Simulation of Thermal Transport Across Hydrophilic Interfaces. Chem. Phys. Lett. 2008, 467 (1−3), 110−113. (27) Murad, S.; Puri, I. K. Thermal Transport Across Nanoscale Solid-Fluid Interfaces. Appl. Phys. Lett. 2008, 92 (13), 133105.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b01605. Specific details about the modeling approach and methods are thoroughly described, in addition to supporting results to the ones presented here. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the Mexican National Council on Science and Technology (CONACyT) under the scholarship No. 312756.

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REFERENCES

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