This study explores the complex interplay of factors influencing thermal conductivity enhancement in nanofluids, which are sus- pensions of nanoparticles in base fluids. Nanofluids have emerged as promising materials for improving thermal properties due to the high thermal conductivity of certain nanoparticles. The re- search delves into the phenomenon of thermophoresis, considering temperature-dependent properties of base fluids and nanofluids. It analyzes five particle materials (gold, alumina, titania, copper, and silver) and six types of base-fluids to understand particle migra- tion potential. The study suggests that variations in scaled thermal diffusion factor with temperature and particle size indicate poten- tial for more uniform particle distribution at higher temperatures and smaller sizes. Nanoparticle material and volume fraction also impact migration potential, with certain materials showing ther- mophoretic potential over wider temperature ranges. Additionally, the study investigates Kapitza resistance at the nanoparticle-fluid interface, which significantly a ffects e ffective thermal conductivity. Molecular dynamics simulations and experimental studies have calculated Kapitza resistance for various interfaces, highlighting its temperature-dependent nature. The study concludes by deriving a universal relationship between the Suratman number ( Su) and the Scaled Thermal Di ffusion Factor ( ϕTST), providing insights into whether a system will develop a concentration gradient or remain uniformly distributed. This analysis serves as a valuable tool for predicting and designing nanofluid systems for enhanced thermal conductivity in various engineering applications. keywords: Thermodynamics, Heat Transfer, Nanoparticles, Nanoflu- ids, Thermophoresis, Thermal Conductivity Enhancement, migra- tion, Kapitza Resistance, Interfacial Thermal Resistance ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.2Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS NOMENCLATURE β non-dimensional nanofluid thermal conductivity(-) c concentration (kg /m3) χ ratio of nanolayer thickness to particle radius(-) DBBrownian di ffusion coefficient(m2/s) DDDufour coe fficient (m2/s-K) dpparticle diameter ( m ) DTthermodi ffusion co- efficient(m2/s-K) ε volume ratio for el- lipsoid(-) f subscript for base fluid(-) G Kapitza resistance at solid-liquid inter- face(W/m2- K) γ nanolayer thermal conductivity to par- ticle thermal con- ductivity(-) Jpmass di ffusive flux (kg /m2/s) Jp,cmass di ffusive flux due to concentra- tion gradient(kg/m2/s) Jp,Tmass di ffusive flux due to temperature gradient(kg/m2/s) kBBoltzmann Con- stant( m2kg/s2- K ) kfbase fluid thermal conductivity(W/m-K) kn fnanofluid thermal conductivity(W/m-K)kpparticle thermal conductivity(W/m-K) kpeequivalent particle thermal conductiv- ity(W/m-K) lp particle length ( m ) µTchemical potential ( J /kg ) µn fviscosity of nanofluid(N-s/m2) µfviscosity of base fluid(N-s/m2) nf subscript for nanofluid(-) p subscript for parti- cle(-) ϕ volume fraction of particles(-) ϕeequivalent volume fraction of particles(-) Pr Prandtl number (-) rpradius of particle (nm) ρfdensity of base fluid (kg /m3) ρn fdensity of nano fluid(kg/m3) ρpdensity of nanopar- ticle(kg/m3) STSoret Coe fficient ( K−1) Su Suratman Number ( K−1) Su∗Critical Suratman Number(K−1) σ∗interaction energy (N /m) T temperature ( K ) Tmmelting tempera- ture( K ) VTdrift velocity (m /s) ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.3Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS 1. Introduction Thermal management is a critical aspect in numerous engineering ap- plications, from electronics cooling to solar thermal energy conversion. Traditional heat transfer fluids, while e ffective, often possess limitations in their ability to conduct heat e fficiently. In this regard, nanofluids have emerged as a promising class of engineered materials with the potential to revolutionize thermal management strategies. Nanofluids are sus- pensions of nanoparticles, with diameters typically ranging from 1-100 nanometers, dispersed in a base fluid such as water, ethylene glycol, or oil[1]. The burgeoning field of nanotechnology has ushered in a new era of material design, with nanofluids standing as a prime example. These suspensions of nanoparticles in base fluids o ffer the tantalizing prospect of significantly enhanced thermal properties, particularly thermal con- ductivity. The key innovation lies in the exploitation of the exceptionally high thermal conductivity of certain nanoparticles, such as metals, car- bon nanotubes, or metal oxides. By incorporating these nanoparticles into the base fluid, researchers have observed significant enhancements in the overall thermal conductivity of the resulting nanofluid[2–4]. Researchers have explored the mechanisms by which nanoparticles en- hance thermal conductivity in nanofluids. Factors such as nanoparticle material, size, concentration, temperature and base fluid properties influ- ence thermal conductivity [5]. However, accurately measuring thermal conductivity in nanofluids presents unique challenges due to their size- dependent behavior and potential instabilities, requiring diverse mea- surement techniques. Contact methods o ffer direct interaction but can introduce disturbances, while non-contact techniques provide a more pristine approach[6]. The improved thermal conductivity of nanofluids is attributed to ballis- tic transport, enhanced Brownian motion, and direct interactions among nanoparticles and with the base fluid. The volume fraction of nanopar- ticles is a crucial parameter influencing thermal conductivity, generally increasing with increasing volume fraction but not always linearly due to agglomeration e ffects [2, 7–11]. Temperature also impacts thermal con- ductivity, a ffecting Brownian motion, base fluid properties, and nanopar- ticle properties, necessitating careful consideration in nanofluid design. Discrepancies in experimental measurements are attributed to measure- ment techniques, nanoparticle characteristics, base fluid properties, and temperature variations, highlighting the need for standardized protocols and detailed characterization. Moreover, these discrepancies in published results contribute to the ab- sence of a robust model that can elucidate the physics underlying this en- ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.4Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS hancement [12–15]. Evans et al. [16] demonstrated that e ffective medium theory can predict this enhancement without considering the e ffect of Brownian di ffusion, contrasting with Choi [13] who argued that Brow- nian motion governs the thermal behavior of nanofluids. Eastman [17] proposed ballistic transport as the reason for thermal conductivity en- hancement, while J. A. Eastman and Keblinski [18] identified particle size, agglomeration, particle-fluid interface, and temperature as major contributors. Viscosity and other thermophysical properties can also significantly influence heat transfer enhancement [19, 20]. Experimental observations, such as those by Gao [10], show that thermal conductivity is higher at higher temperatures for certain nanofluids, but Ho and Gao [21] reported reduced enhancement with increased nanoparticle mass fraction due to reduced convection . The enhancement in thermal con- ductivity is experimentally observed to be temperature dependent, Gao [10] showed that thermal conductivity was higher at higher temperature for the nanofluid made from n-Octadecane and Alumina. Ho and Gao [21] reported a lower heater enhancement with the increased nanoparticle mass fraction due to reduced convection. [22–25] suggest that Brown- ian motion plays a dominant role in thermal conductivity enhancement, while others argue that it only a ffects nanoparticles and not heat transfer [26, 27]. When particles are introduced into a base fluid, they exhibit ran- dom movement known as Brownian motion. This motion helps maintain a uniform concentration within the system, counteracting the formation of concentration gradients. However, when a temperature gradient is applied to the nanofluid, a phenomenon called thermophoresis occurs. Thermophoresis causes particles to move towards the colder regions of the system, contrary to the action of Brownian di ffusion. The interplay between these two processes can lead to either a uniform distribution of particles or a non-uniform distribution, with particles potentially mi- grating towards the colder regions. This dominant mode of di ffusion is determined by a thermal di ffusion factor, a dimensionless number that characterizes the prevailing mode between Brownian and thermal di ffu- sion. The discrepancies observed in experimental results regarding the enhancement of thermal conductivity in the base fluid could be attributed to this particle migration phenomenon. 2. Thermophoresis The motion of particle due to the presence of temperature gradient, on top of Brownian motion is known as thermophoresis. Predominantly, the motion of particle occurs from the hot region towards the cold end [28]. The di ffusion defined by Fick’s equation relates the flux of particles /mass due to the concentration gradient. Heat transfer and nanoparticle trans- ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.5Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS port are thermodynamically coupled. heat flux Jq=−µTTD Ddc dx−kdT dx(1) particle flux Jp=−DBdc dx−cDTdT dx=Jp,c+Jp,T (2) The transport forces are thermal (T) and chemical ( µT) [29–33]. DBis Brownian di ffusion coe fficient, DTis Thermal di ffusion coe fficient, DD is the Dufour coe fficient, c is concentration of particles, µTis chemical potential, k is the thermal conductivity, Jp,cand Jp,Tare the di ffusion flux due to concentration and temperature gradient respectively. Thus, the transport of particles can result from the combination of two factors: temperature gradient and concentration gradient. Mechanisms such as sedimentation or buoyancy are not considered here. The thermophoretic motion of particles occurs on top of the Brownian di ffusion, though there is no external field is present [34]. The velocity attained by particles due to themrophoresis is called drift /thermophoretic velocity[35]. VT=−DTdT dx(3) By experimental verification, it was found that this thermophoretic ve- locity in liquid is the modification of the Epstein’s relation and is given by [36] [2] [37] VT=−βµn f ρn f1 TdT dx,β=0.26kf 2kf+kp(4) µn fis the viscosity of the nanofluid, ρn fis the density of nanofluid , kn fand kpare the thermal conductivity of nanofluid and particle respectively. We are using the term nanofluid for the combination of base fluid dispersed with some fraction of nano particles. The relative strength of Brownian di ffusion and thermal di ffusion can be evaluated at zero net mass flux [38] atJp=0 ,1 cdc dx=−DT DBdT dxwhere ST=DT DB(5) Soret coe fficient STis the ratio between the thermal di ffusion coe fficient and the Brownian di ffusion coe fficient, with the units of K−1. Comparing 2 and 3, we can find that DTis Thermal di ffusion coe fficient is given by DT=β ρn fµn f T=0.26kf 2kf+kpµn f ρn fT(6) ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.6Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS Experiments suggests thermal di ffusion coe fficient varies between 1 ∗ 10−8<DT<1∗10−7cm2s−1K−1[39]. Brownian di ffusion coe fficient DBis given by Stokes-Einstein equation [40] DB=kBT 3πµ fdp(7) From 6 and 7 we find Soret Coe fficient ST=3πµ n fµfdpβ ρn fkBT2(8) dpis particle diameter, µfis the viscosity of the base fluid, ρpis the density of particle, kBis Boltzmann constant. The relative e ffect of the thermal di ffusion to the Brownian di ffusion can be written in the non-dimensional form by multiplying Soret coe fficient by temperature. This is known as thermal di ffusion factor. TST=− 1 c∂c ∂x 1 T∂T ∂x  Jp=0=3πµ n fµfdpβ ρn fkBT(9) The expected value from the above relation aligns with the values pro- vided by W ¨urger [41]. The equation 9 do not account for the volume frac- tionϕof the particles in the basefluid. In order to capture the e ffect, ther- mal di ffusion factor is multiplied by ϕto get a scaled non-dimensional parameterϕTST. ϕTST=ϕ3πµ n fµfdpβ ρn fkBT(10) This scaled thermal di ffusion factor will determine determine the dom- inant di ffusion e ffect. WhenϕTST<1, Brownian di ffusion prevents the development of a nanoparticle concentration gradient . When ϕTST>1, then a nanoparticle concentration gradient might develop due to ther- mophoresis. It can be seen from the above equations that scaled thermal diffusion factor and the properties responsible for particle di ffusion are temperature dependent. Thus, we can not assume the properties to be constant over the applicable range of temperature. So it is necessary to provide these properties as a function of temperature and it is shown in table 1. 2.1. Scaled Thermal Di ffusion Factor It can be seen from the equation above 9 that ϕTSTis a function of •Density of nanofluid ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.7Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS Table 1: Temperature dependent properties [42–45] Ethylene Glycol Water Acrylic Acid Formic Acid Caprylic Acid Acetic Acid Para ffin Wax Density, [ g/mL],AB−(1−T/Tc)n A 0.32503 0 .34710 0.34645 0.36821 0.29231 0.35182 0.23837 B 0.25499 0.27400 0.25822 0.24296 0.26676 0.26954 0.25763 n 0.172 0.28571 0.30701 0.23663 0.28020 0.26843 0.274 Tc, K 645 647.13 615 580 692 592.71 745.26 Viscosity, [ cp],log10µ=A+B/T+CT+DT2 A -16.9728 -10.2158 -15.9215 -4.2125 -10.4823 3.8937 -8.5505 B∗1033.1886 1.7925 2.4408 0.97953 2.067 .784 1.6708 C∗10−23.2537 1.7730 3.4383 0.552 1.8423 0.6665 1.5675 D∗10−5-2.448 -1.2631 -2.7677 -0.57723 -1.3722 -0.75606 -1.2341 Thermal Conductivity, [W /m-K], for organic compound log10K=A+B 1−T C2 7, for inorganic K=A+BT+CT2 A -0.5918 0.2758 -1.6101 -0.8626 -1.6624 -1.2836 -1.5198 B - 4.61E-03 0.9742 0.3692 0.9819 0.5893 0.8067 C 645 -5.5391E-06 615 580 692 592.71 745.26 Specific Heat, [ J/mol−K],Cp=A+BT+CT2+DT3 A 75.878 92.053 -18.242 -16.110 70.790 -18.944 151.154 B 6.4182E-01 -3.9953E-02 1.2106 8.7229E-01 1.7647 1.0971 2.7878 C∗10−3-1.6493 -0.21103 -3.1160 -2.3665 -4.1521 -2.8921 -6.1542 D∗10−61.6937 0.53469 3.1409 -2.4454 3.9451 2.9275 5.5249 •Viscosity of basefluid and the nanofluid •Thermal conductivity of the nanofluid and the particles •Diameter of dispersed nanoparticles •Temperature •Volume fraction of particles 3. Concept Discussion and Results It has been discussed that there are several experimentally validated relations for predicting the thermal conductivity, Brownian di ffusion co- efficient and thermal di ffusion coe fficient. But none has combined the results of all the fields in one analysis and tried to capture the impact of temperature and its gradient on the migration of particles in the nanoflu- ids. In order to provide the in-depth analysis on thermophoresis we have captured the temperature dependent properties of the basefluid as well as nanofluid. Five particle materials were included in this study: gold, alumina, tita- nia, copper and silver. The density and thermal conductivity for these particles are listed in Table 2. These properties are considered constant as the variation in density and thermal conductivity for the particle is insignificant within the applicable temperature range. ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.8Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS Table 2: Properties of Nanoparticles [42, 45] Density [ kg/m3],ρpThermal Conductivity [W /m-K], kp Alumina 3950 50 Copper 8960 402 Gold 19300 317.422 Silver 10500 428.227 Titania 4000 11.7 Six types of PCMs were studied. These were modeled only in the liquid phase without solidification. The base fluids are acetic acid, acrylic acid, caprylic acid, ethylene glycol, formic acid and water. Nanoparticles of the five materials listed earlier were paired with these fluids with varying particle size and volume fraction. The potential for particles to migrate or to remain uniformly distributed depends on the size of nanoparticles, volume fraction, particle type, fluid, and operating temperature range. 3.1. Impact of Temperature and Particle Size Variations in thermal di ffusion factor with temperature for copper parti- cles of diameter 5 and 20 nm at 3% volume fraction in all six fluids are shown in Fig 1b and Fig 1c. Another data for several experiments con- ducted using various fluid and particle combination is shown in Fig 1a. It can be seen from these figures, that particles are expected to be more uniformly distributed at higher temperatures and thus, the migration to- wards cold end. This result is supported by experimental and molecular dynamic simulation results of Galliero and Volz [46], Duhr and Braun [47]. This can be attributed to the lower viscosity of fluid at higher temperature, resulting in better mobility of particles and more impact of Brownian di ffusion. When we observe Fig 1b and Fig 1c we see that there is dependency of particle distribution on its size. Tcurve shifts towards the non-uniform zone when particle size changes from 5 nm to 20 nm. The depiction from the figures suggests that bigger particles are expected to be more non-uniformly distributed and towards the cold end. This observation aligns with the results reported by L ¨usebrink and Ripoll [48]. ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.9Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS (a) Experimental Scaled TST. (b) 5 nm diameter (c) 20 nm diameter (d) Silver Particles (e) Alumina Particles (f)ϕTSTvs volume fraction Figure 1: Scaled thermal di ffusion factor dependency on various param- eters 3.2. Impact of Particle Material Nanoparticles of alumina and silver with a diameter of 5 nm and a volume fraction of 2% were investigated. Alumina particles (Fig 1e) ex- hibited potential for thermophoresis across all temperature ranges. In contrast, for temperatures exceeding 300 K, Brownian di ffusion domi- nated for silver particles (Fig 1d) dispersed in water, acrylic acid, formic ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.10Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS acid, and acetic acid. The di fferences in thermal conductivity among various particle types influenced their migration behavior. Alumina, having a relatively low thermal conductivity compared to silver, showed this e ffect. This trend was also observed for other particles with lower thermal conductivity, such as Titania, while particles with higher thermal conductivity, like gold and copper, exhibited potential for uniform distribution. However, the impact of particle density on migration was minor, as the addition of nanoparticles minimally altered nanofluid density, which changes pro- portionally with particle fraction. Hence, fluid density primarily deter- mined nanofluid density. The findings suggest that to compensate for the lower thermal conductivity of alumina and achieve equivalent thermal diffusion to silver, its density would need to increase by 120 times. 3.3. Impact of Volume Fraction The e ffect of volume fraction is shown in Fig 1f. The graph illustrates the change in the scaled thermal di ffusion factor for volume fractions up to 5% using only alumina particles at 310 K as an example. Similar results are observed for other particle types and temperatures. However, an intriguing observation is evident. At lower volume fractions, parti- cles are expected to be uniformly distributed. As the volume fraction increases, there is a tendency for particles to create a concentration gra- dient. Another observation is that at lower volume fraction of particles, the change in ϕTSTis steeper but with the increased volume fraction, it becomes less sensitive. This suggests that there exists an optimal volume fraction of particles that should be added to achieve uniform distribution and consequently, the maximum enhancement in thermal conductivity. Further discussion on the relationship between thermal conductivity en- hancement and particle distribution will be presented in the following section. 3.4. Comparison with Reported Published Data Figure 1a illustrates the trends in ϕTSTfor selected particle-fluid pairs, alongside test conditions from studies on enhancing thermal conductiv- ity in phase change materials [1, 5, 14, 49, 50]. The data from Iyahraja1 and Rajadurai [49] and Kang et al. [1] (black and blue symbols, respec- tively) align with predicted trend lines, suggesting a strong impact of increasing volume fraction from very low concentrations on enhancing thermophoretic migration. Despite silver particles with a 10 nm diame- ter (blue square) and a 0.1% volume fraction, another data point (black square) shifts towards the non-uniformity zone due to a slightly higher 0.4% volume fraction of particles. Additionally, the use of lower thermal ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.11Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS conductivity particles like alumina, as indicated by the red square [5], re- sults in a shift towards higher thermophoretic potential. Fluid viscosity, represented by the dashed green trend line, significantly influences par- ticle migration, as seen in test conditions from Eastman et al. [14] (green square). The commonly used phase change material (PCM), para ffin, shows that dispersing copper nanoparticles with a 10 nm radius may lead to thermophoretic migration [50]. The experimental data suggests that concentration gradients can either build up or be uniformly distributed depending on the fluid and nanopar- ticle pairing. However, the most significant observation is that higher enhancements in thermal conductivity of nanofluids occur when ϕTST is less than 1, indicating uniform distribution. A thermal conductivity enhancement of 54%, 10%, and 24% respectively was measured by Kang et al. [1], Iyahraja1 and Rajadurai [49], Lin and Al-Kayiem [50], all lying in the uniform distribution zone. In contrast, the measured enhancement by Das et al. [5], Eastman et al. [14] was 4%, and both their data points lie in the non-uniformity zone. For instance, experiments by Lin and East- man both using copper nanoparticles show one in the uniform region while the other in the non-uniform region. This disparity is attributed to variations in fluid type, particle size, and operating temperature. If the experiment by Eastman (represented by the green square) were con- ducted at 310 K instead of 300 K, the point would shift towards uniform distribution, potentially increasing the observed enhancement in thermal conductivity. This observation of non-uniform distribution of particles impacting the enhancement in the nanofluid aligns with the findings of Ding and Wen [51], Bahiraei and Hosseinalipour [52]. Additionally, re- turning to Figure 1a, it is evident that the addition of nanoparticles at lower concentrations could significantly increase thermal conductivity, but as the volume fraction of particles increases, the impact lessens, and the concentration gradient begins to build up. This is also supported by Buongiorno et al. [53], who found that adding 22 nm silica particles to water with a 31% volume fraction had almost no impact on thermal conductivity. 4. Interfacial Interaction Energy and Kapitza Resistance Kapitza resistance, also known as interfacial thermal resistance or ther- mal boundary resistance, is a phenomenon that occurs at the interface between a solid and a liquid. It describes the resistance to heat flow across this interface, resulting in a temperature discontinuity or drop. The origin of Kapitza resistance lies in the mismatch between the vibra- tional properties of the solid and the liquid. When heat flows from the ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.12Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS solid to the liquid, the energy carriers must undergo scattering at the interface due to the di fferences in their vibrational spectra and velocities. The presence of Kapitza resistance at the nanoparticle-fluid interface can significantly influence the e ffective thermal conductivity of nanofluids. While the addition of nanoparticles is intended to enhance the thermal conductivity, the interfacial resistance can counteract this e ffect by im- peding heat transfer between the nanoparticles and the base fluid. We define the interaction ( σ∗) energy between the solid-liquid interface using the Kapitza resistance. σ∗=G.A DT(11) G is the Kapitza conductance (inverse of Kapitza resistance) of the solid- liquid interface and A is the surface area of the particle. G is O(108).Kapitza resistance is influenced by nanoparticle size, nanoparticle-fluid interac- tion and nanoparticle concentration. The book on nanofins [54], discusses the impact of fluid and the fin material to enhance the boiling. It shows that though the thermal resistance of silicon nanowires is higher than CNTs but the total thermal resistance in a circuit is lower due to sub- stantially low interfacial thermal resistance. The book also discusses that precipitation of the nanoparticles act as the nanofins and can enhance the heat transfer, but excessive precipitation can have the negative impact on it. The article by Khodayari et al. [55] employs molecular dynamics simu- lations to investigate the Kapitza resistance between nanoparticles and solid surfaces. The key findings reveal that the presence of nanoparticles near the solid-liquid interface increases the Kapitza resistance, hinder- ing heat transfer from the solid surface to the liquid. The nanoparticles contribute significantly to the reduction in heat current, and their dy- namics coupled with the solid surface lead to a shift in the major heat carriers from low to higher frequencies. The study explores realistic and simplified molecular models, varying nanoparticle-nanoparticle and nanoparticle-surface interactions, and employs the thermal relaxation method to determine the interfacial thermal conductance between the nanoparticle-containing suspension and the gold surface. Another work by Dolatabadi et al. [56] presents a comprehensive model for predicting the thermal conductivity of nanofluids, accounting for various factors that influence heat transfer at the nanoscale. The model incorporates the Kapitza resistance (interfacial thermal resistance) between nanopar- ticles and the base fluid, which can hinder heat transfer. It also considers the formation of a nanolayer around the nanoparticles, which can either enhance or reduce thermal conductivity depending on its properties. Additionally, the model includes the e ffects of convective di ffusion due ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.13Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS to Brownian motion of nanoparticles and surface energy with capillary condensation, which can alter the surface properties and thermal con- ductivity. The thickness of the nanolayer is predicted using the BET isotherms and micro /nano-menisci generated pressures. The model is evaluated for four di fferent nanofluids, and the results show good agree- ment with experimental data, providing a comprehensive framework for understanding and accurately predicting the thermal conductivity enhancement in nanofluids for heat transfer and thermal management applications. Pollack [57], Ghasemi and Ward [58], Puech et al. [59] studied the kapitza resistance between the interface of liquid helium and various materials like copper, lead, silver, gold etc was calculated experimentally and it was found that Rk∝T−3. The generalized equation for kapitza resistance for a dispersed nanoparticle in a fluid is Rk=15ℏ3ρpct3 16π5K4 Bρb fcb fFcl ctT3(12) where, ℏ,KB,ρp,cl,ctand cb fare the reduced Planck constant, Boltz- mann constant, density of nanoparticle, longitudinal speed of sound in particle, transverse speed of sound in particle, and speed of sound in the base fluid respectively. Function, F, usually varies between 1.5 and 2.0[56, 57]. Interfacial resistance for various nanofluids like alumina and water, alumina and ethylene glycol and copper-oxide and water was calculated and it was found to vary between 0 .5∗10−8to 1.86∗10−8 m2K/W[56]. Molecular dynamics simulation study also shows that there is a temperature jump across the solid-liquid interface and is character- ized by interfacial resistance and kapitza length [60]. Equilibrium and Non-equilibrium molecular dynamic simulation for the water-graphene interface showed that Rkvaries between 1 .2∗10−8to 1.45∗10−8m2K/W [61]. Another molecular dynamic simulation study showed that Rkis O(10−8) for the silver nanoparticle in water. It also discusses that the heat is dissipated across a 2nm interfacial layer into water [62]. Experimental results have been compared with the E ffective Medium Theory to pre- dict the kapitza resistance, they were found to be in good agreement. The Rkfor the alumina-water interface was 0 .2∗10−8to 5∗10−8m2K/W with a nanolayer thickness of 0.2 nm[55]. Review on nanoscale thermal transport by Cahill et al. [63] shows that kapitza conductance increases at higher temperature and is O(108). This agrees with our analysis above in §3.1 that higher temperature operation tends to provide more uniform distribution of particles. The paper also discusses that the thermal con- ductance for gold and platinum nanoparticles is about 0 .2∗108to 1.3∗108 Wm−2K−1. A benchmark study performed by 30 organizations across the ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.14Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS globe to calculate the thermal conductivity of the nanofluids, using gold ,alumina and silica as the dispersion particles in water used the lower bond of Rkas 10−8m2K/Wto justify their findings [53]. This shows that the validation of experiments needed a discrete value of kapitza resistance. Singh et al. [64] used molecular dynamics simulations to study the ther- mal interfacial resistance between carbon nanotubes and di fferent coolant fluids, including water, ethylene glycol, and a water-ethylene glycol mix- ture. The results showed that the thermal interfacial resistance was low- est for the water-based coolant, followed by the water-ethylene glycol mixture, and highest for pure ethylene glycol. This was attributed to the stronger water-carbon interactions in the water-based coolant, which led to a more ordered molecular layer at the interface and reduced the thermal resistance compared to the other coolants. The findings suggest that water-based nanofluids may be preferable as coolants compared to ethylene glycol-based fluids, due to the lower thermal interfacial resis- tance at the nanoparticle-fluid interface. The interfacial resistance was found to be 2 .13∗10−8, 4.74∗10−8and 7.29∗10−8m2K/Wfor water, ethyl alcohol and 1-hexene respectively. Sarode et al. [65] investigates how the diameter of single-walled armchair carbon nanotubes (CNTs) a ffects the thermal interfacial resistance between the CNT and its surroundings. Us- ing molecular dynamics simulations, the researchers found that thermal interfacial resistance decreases as CNT diameter increases, mainly due to the larger contact area provided by wider CNTs. For CNTs with diam- eters of a few nanometers, the total interfacial thermal conductance first decreases and then stabilizes as the number of walls increases, which is attributed to changes in mechanical strength and adhesive energy. They found that Rkvaried between 0 .7∗10−8and 2.6∗10−8m2K/W. The book by Singh and Banerjee [54] compiles the data for over 25 fluid molecules and their isomers to show that RkisO(10−8) and it is strongly dependent on the solvent (basefluid). 5. Suratman Number Analysis The temperature-dependent nature of properties such as viscosity, den- sity, and thermal conductivity is evident in Equation 9. Monitoring the variation of each property with temperature individually would involve significant bookkeeping e fforts to keep track of all the empirical relations. The distribution of particles has been shown to depend strongly on tem- perature, with the e ffect of temperature on ϕTSTcovering a wide range. Analyzing the data from this figure is challenging because the scaled thermal di ffusion factor can vary widely with temperature. The curve can shift significantly to the left or right depending on the temperature- ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.15Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS dependent properties of the fluid and particles. Therefore, it would be advisable to use a more universal approach that can be applied to each particle-fluid combination. Looking at the equation for the scaled thermal di ffusion factor, Equation 9, we can see terms related to viscosity and inertia. Considering viscous and inertial forces along with the interaction between the solid-liquid in- terface, we can use the Suratman number (Su) for analysis. The Suratman number represents the ratio of surface tension forces to viscous forces in a fluid system. In nanofluids, it plays a crucial role in characterizing inter- facial phenomena and flow behavior due to the presence of nanoparticles and their interactions with the base fluid. Su governs flow pattern tran- sitions, pressure drop, and interfacial phenomena, making it essential for understanding and designing e fficient two-phase flow systems, par- ticularly in microfluidic and nanofluidic applications. Incorporating Su into theoretical models, correlations, and flow regime maps allows for a better understanding and prediction of the two-phase flow behavior of nanofluids, accounting for the e ffects of surface tension, viscosity, and nanoparticle-fluid interactions. Suratman Number ,Su=ρn fσ∗dp µ2 n f(13) Here, Thus, the equation for Suratman number can be written as Su=πGd3 pρn f µ2 n fDT(14) From 14 we see that Su already includes DTin its mathematical repre- sentation. So analyzing thermal di ffusion factor against Su will be easier and can yield some universal relationship. 5.1. Suratman Number vs Scaled Thermal Di ffusion Factor Based on the preceding discussion, we employed Equation 14 to derive a universal relationship between Su and ϕTST. The key parameters in- volved, namely the particle diameter, nanofluid density and viscosity, as well as the thermal di ffusion coe fficient, have been meticulously defined and elaborated upon in §2 and §3. Furthermore, the analysis of Kapitza resistance provides compelling evidence that the resistance ( Rk) is of the orderO(10−8), or equivalently, the thermal conductance ( G) is of the order O(108). The values of Gemployed in our calculations are derived from Table 3. We conducted a comparison using gold and alumina particles, each with a diameter of 10 nm and a volume fraction of 2%, to plot Suver- susϕTST. The results are presented in Figure 2, indicating that data from ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.16Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS Table 3: Thermal Conductance of various Fluids Interfacial /Kapitza Conductance [ Wm−2K−1], G Acetic Acid 1 ∗108 Acrylic Acid 1*108 Caprylic Acid 1.3*108 Ethylene Glycol 1*108 Formic Acid 0.8*108 n-Octadecane 1.3*108 Water 0.5*108 (a) Su vs Scaled TSTfor gold particles (b) Su vs Scaled TSTfor alumina particles Figure 2: Comparison of Su vs Scaled Thermal Di ffusion Factor for gold and alumina particles various fluids collapses onto a single universal curve. This observation suggests a power law relationship, where the slope remains constant while the intercept varies depending on the fluid-particle pairing and their respective properties. ϕTST=a∗Sub(15) In Equation 15, arepresents the intercept, and bdenotes the slope, where bis set to -0.68. We define the critical Suratman number ( Su∗), as the point of intersection where ϕTST=1. Data points to the left of Su∗are expected to exhibit non-uniform distribution, while those to the right should show uniform distribution. This analysis is crucial as it provides insight into whether the initial parameters will lead to the eventual development of a concentration gradient or if the particles will disperse uniformly. To determine the alignment of the experimental points shown in Figure 1a with the predicted region, we have plotted all the experimental data points on the SuvsϕTSTcurve. Two of them are illustrated in Figure 3: ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.17Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS (a) Su vs Scaled TSTfor water silver 1 (b) Su vs Scaled TSTfor EG copper Figure 3: Comparison of Su vs Scaled Thermal Di ffusion Factor for ex- perimental data points in Figure 3b, the experimental data point lies to the left of Su∗, indicating non-uniform distribution, while the point in Figure 3a lies to the right ofSu∗, suggesting the potential for a uniform distribution of particles. Please note that the symbol used to represent the experimental data points is similar to those used in Figure 1a, facilitating easy comparison between the two plots. Based on this plot, we can directly determine the critical Suratman number and ascertain whether the particle-fluid pairing being used positions the point to the left or right of this value. This can be achieved by compiling all the data for all properties as a function of the particle radius, volume fraction and particle thermal conductivity. ϕTST=a∗Sub,for critical Su ϕTST=1 Su∗=1 a1/b ,b is constant ,a=f(r,ϕ,kp) Su∗=172s ko p kp ϕβr roη kpand ko prepresent the particle thermal conductivity and reference ther- mal conductivity in W/m−K, respectively. For carbon nanotubes (CNTs), known for their exceptionally high thermal conductivity, ko pis set at 2000 W/m−K. The variables rand rodenote the particle radius and the ref- erence radius in nm, with robeing 1 nm in this context.After conducting the analysis, the obtained values are η=4.4 andβ=1.4. Consider the scenario depicted in Figure 3a, involving silver particles with a radius of 6 nm and a volume fraction of 0.4%. Using the relationship from Equa- tion 16 and a thermal conductivity value for silver, kp=428 W /m-K, we calculate Su∗=434, aligning with the line of intersection at ϕTST=1. The experimental data point marked by a black square corresponds to a Suratman number of 1 .07×105, exceeding Su∗, indicating it lies within the zone of uniform distribution. A similar analysis can be applied to the case in Figure 3b, involving copper nanoparticles with a radius of 4 nm and a volume fraction of ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.18Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS 0.3%. Using a thermal conductivity value for copper, kp=402 W /m-K, we determine Su∗=50. However, the data point represented by a green square has a Suratman number of 20, below the critical Suratman number, placing it within the non-uniformity zone. By adjusting the particle radius to 3 nm, the new Su∗=14, indicating a move to the zone of uniform distribution. This analysis can aid in predicting system behavior before experiments, o ffering insights into concentration gradient development. One significant benefit of this analysis is its temperature independence, as all temperature-dependent properties are encapsulated in the coe fficients forSu∗. Additionally, the relation remains independent of the fluid used, as fluid impact is accounted for through the Kapitza resistance, which exhibits minimal variation and incorporates the varying fluid-particle interaction e ffects in the coe fficients. 6. Summary and Conclusion The article delves into the importance of thermal management in en- gineering applications and the potential of nanofluids, suspensions of nanoparticles in a base fluid, to enhance thermal conductivity. It em- phasizes how factors like nanoparticle material, size, concentration, tem- perature, and base fluid properties influence thermal conductivity en- hancement. Standardized protocols and detailed characterization are highlighted as necessary to address discrepancies in experimental re- sults. The interplay between Brownian motion and thermophoresis is discussed, impacting particle distribution and, consequently, thermal conductivity enhancement in nanofluids. Thermophoresis in nanofluids is analyzed considering temperature-dependent properties of both the base fluid and the nanofluid. Five particle materials (gold, alumina, tita- nia, copper, and silver) and six types of phase change materials (PCMs) in various base fluids and nanoparticle pairings are examined to under- stand particle migration potential. The analysis indicates that particle material, size, volume fraction, and temperature are crucial in designing nanofluids for enhanced thermal conductivity. Higher temperatures and smaller particle sizes may lead to more uniform particle distribution, underscoring their importance in nanofluid design. The article also discusses the significance of Kapitza resistance in un- derstanding the interfacial interplay between the fluid and the solid nanoparticles. Kapitza resistance at the nanoparticle-fluid interface can significantly impact e ffective thermal conductivity. Molecular dynamics simulations and experimental studies have calculated Kapitza resistance for various interfaces, o ffering insights into heat transfer in nanofluids. The Suratman number analysis is presented as a tool to characterize ©2024, Elsevier Ltd. Bugs, feature requests, suggestions and comments shall be mailed to <elsarticle@stmdocs.in >.19Version: 2 Date: Sunday 12thMay, 2024 Contact: udits@mtu.eduThermophoresis in Dispersed Nanoparticle UditSharma , Jeffrey S. AllenQUICK LINKS interfacial phenomena and flow behavior in nanofluids, aiding in pre- dicting particle distribution patterns based on temperature gradients. The concept of the critical Suratman number provides a valuable tool for preparing experiments in advance, predicting whether the system will develop a concentration gradient or remain uniformly distributed. Determining the critical Suratman number for di fferent particle sizes, vol- ume fractions, and materials allows researchers to anticipate nanofluid behavior and design experiments accordingly. This understanding is crucial for optimizing nanofluid thermal properties for applications like thermal management and energy conversion.