Lattice Boltzmann modeling of heat conduction enhancement by colloidal nanoparticle deposition in microporous structures

Drying of colloidal suspension towards the exploitation of the resultant nanoparticle deposition has been applied in different research and engineering fields. Recent experimental studies have shown that neck-based thermal structure (NTS) by colloidal nanoparticle deposition between microsize filler particle configuration (FPC) can significantly enhance vertical heat conduction in innovative three-dimensional chip stacks [Brunschwiler et al., J. Electron. Packag. 138, 041009 (2016)]. However, an in-depth understanding of the mechanisms of colloidal liquid drying, neck formation, and their influence on heat conduction is still lacking. In this paper, using the lattice Boltzmann method, we model neck formation in FPCs and evaluate the thermal performances of resultant NTSs. The colloidal liquid is found drying continuously from the periphery of the microstructure to its center with a decreasing drying rate. With drying, more necks of smaller size are formed between adjacent filler particles, while fewer necks of larger size are formed between filler particle and the top/bottom plate of the FPCs. The necks, forming critical throats between the filler particles, are found to improve the heat flux significantly, leading to an overall heat conduction enhancement of 2.4 times. In addition, the neck count, size, and distribution as well as the thermal performance of NTSs are found to be similar for three different FPCs at a constant filler particle volume fraction. Our simulation results on neck formation and thermal performances of NTSs are in good agreement with experimental results. This demonstrates that the current lattice Boltzmann models are accurate in modeling drying of colloidal suspension and heat conduction in microporous structures, and have high potentials to study other problems such as surface coating, salt transport, salt crystallization, and food preserving.

Figure 1

Illustration of (a) percolating thermal structure and (b) neck-based thermal structure with the necks highlighted in orange color.

Figure 2

Validation of the thermal LBM by the evaluation of the effective thermal conductivity (ETC, ${\lambda }_{\mathrm{eff}}$) of two basic dual-component systems at various thermal conductivity ratio in (a) serial and (b) parallel modes.

Figure 3

Validation of the tricoupled hybrid LBM with drying of colloidal suspension and its nanoparticle deposition in a two-filler-particle system. (a) Simulation setup with indications of the two mesh resolutions and simulated process of colloidal liquid drying and nanoparticle deposition in the finer resolution system. (b) Illustration of experimental process of colloidal liquid drying and neck formation by nanoparticle deposition between filler particles from Ref. [9]; (c),(d) Simulated necks (golden) in the coarser and finer resolution systems. (e) Cross section of simulated necks (golden) in finer resolution system. Simulation results are in lattice units while experimental results are in physical units.

Figure 4

Simulation of the assembly process of filler particles in the cavity between two chip layers in 3D chip stacks using the discrete element method. (a) Initial random distribution of filler particles. (b) Intermediate state during the assembly process. (c) Final assembled compact filler particle configuration (FPC). (d) Extracted cavity with the volume of compact FPC with dimensions (physical units). (e) Extracted FPC for subsequent LBM simulations.

Figure 5

Evaluation of thermal performance of the percolating thermal structures (PTSs). (a) Simulation setups of three different PTSs with similar filler particle volume fraction ($V_{fp}^{n,t}=48.8\%,48.0\%,47.1\%$), with the top and bottom temperatures $T_{\mathrm{top}}=1$ and $T_{\mathrm{bot}}=2$ and four lateral sides adiabatic. (b) Temperature distribution with isolines in black at the slice of $y=30$ in the three different PTSs, and white lines indicating boundaries of filler particles. (c) Heat flux contours at the slice of $y=30$ in different PTSs, with black lines indicating temperature isolines and white lines indicating boundaries of filler particles. All variables are in lattice units.

Figure 6

Drying of colloidal suspension in cavity with filler particle configuration 1 (FPC1), and the resultant neck formation in other two neck-based thermal structures (NTSs). (a) Simulation setup of drying of colloidal suspension in FPC1. (b)–(g) Different stages during the drying process of colloidal liquid in FPC1 and formation of necks by nanoparticle deposition, giving rise to the NTS1. (h) and (i) The resultant neck formation in NTS2 and NTS3.

Figure 7

(a) Normalized colloidal liquid mass and (b) evaporation rate during colloidal liquid drying in three filler particle configurations (FPCs). Inset shows liquid configuration at point P, where the drying rate suddenly decreases since the liquid-vapor interface starts receding into the pores inside the FPC. The dashed blue line shows initial liquid configuration when drying occurs.

Figure 8

Analysis of deposited necks in NTS1. (a) NTS1 counts 261 necks, shown here by removing the particles. Histograms of (b) normalized average neck volume ($V_{\mathrm{neck}}^{n,a}$) and (c) normalized average neck diameter ($d_{\mathrm{neck}}^{n,a}$) of necks between particle and flat substrate and cover (${\mathrm{Neck}}_{p-t,p-b}$), and between particles (${\mathrm{Neck}}_{p-p}$), where the “Distri.” values indicate mean value and standard deviation.

Figure 9

Comparison of the metrics of three filler particle configurations (FPCs) and the deposited necks in neck-based thermal structures (NTSs). (a) Void fraction over the height of system. (b) Neck counts. (c) Neck diameter. (d) Neck volume.

Figure 10

Calculation of effective thermal conductivity (ETC) of the nanocomposite material making the necks. (a) Distribution of nanoparticles of two different sizes at initial state. (b) Final assembled nanoparticle deposition configuration by discrete element method. (c) Extracted compact nanoparticle deposition configuration. (d) Simulation setup of the system with top cover and bottom substrate temperatures $T_{\mathrm{top}}=1$, $T_{\mathrm{bot}}=2$ and lateral sides adiabatic. (e) Temperature distribution with temperature isolines in black and nanoparticle outlines in white at the slice of $y=33$. (f) Heat flux distribution with temperature isolines in black and particle outlines in white at the slice of $y=33$. All variables are in lattice units.

Figure 11

Thermal performances of the neck-based thermal structure (NTS). (a) Simulation setup of NTS1 with the top and bottom temperatures $T_{\mathrm{top}}=1$ and $T_{\mathrm{bot}}=2$ and the four lateral sides adiabatic. (b) Temperature distribution with isolines (black) at the slice of $y=30$ and the white lines denoting the boundaries of the filler particles. (c) Comparison of temperature isolines between NTS1 (green lines) and PTS1 [red lines from Fig. 5] at the slice of $y=30$, with black lines being boundaries of filler particles. (d) Heat flux distribution at the slice of $y=30$, with black lines being temperature isolines and white lines denoting the boundaries of filler particles. (e) Subtraction of heat flux distribution between NTS1 and PTS1 [from Fig. 5] at the slice of $y=30$. All variables are in lattice units.

Figure 12

Comparison of effective thermal conductivity (ETC) between PTS and NTS of both simulations and experiments.