# Mathematics in Nanotechnology

For the past 4 years, the Industrial Mathematics Group at CRM has focussed primarily on applying continuum mathematical models to nanoscale phenomena. Continuum theory may be applied when there is a sufficiently large sample size to ensure that statistical variation of material quantities, such as density, is small. For liquids this suggests a minimum size of around 10nm, although comparison with molecular dynamics simulations indicates results are still accurate down to 3nm for water. In the field of heat flow minimum ranges of 2-5nm are often quoted. Nanoscale is typically described as involving materials with at least one dimension below 100 nm, so there is clearly a range of sizes where continuum theory may be applied to nano phenomena, see [1].

Past and current projects include:

1. Enhanced water flow through carbon nanotubes

Fig. 1: Standard ‘armchair’ configuration CNT, image taken from [2].

Carbon nanotubes have proposed applications in medicine, materials, electronics, solar cells, energy storage, chemistry, optics etc. One application, in textiles, is based on their ability to transport water much faster than predicted by classical theories. A related application is in desalination where CNT membranes have been shown to operate at lower temperatures and with a 6 times greater flow rate than conventional membranes [3].

Our work in this field was based on the observation that water is repelled by the CNT wall, so leaving a ‘depletion layer’. We exploited the well-known bi-viscosity model of non-Newtonian fluid mechanics to show that the depletion layer can explain the observed flow enhancement. By comparison with experiment we found that the viscosity in the depletion layer is approximately that of air or oxygen. The work also provided a physical explanation for the Navier-slip condition between a liquid and a solid [4].

1. Nanofluid flow

A nanofluid consists of a base fluid and a suspension of nanoparticles. There exists a vast number of papers, a journal and many conferences devoted to the study of nanofluids. A great number of research articles demonstrate often remarkable improvements (when compared to the base fluid) in thermal properties, such as greatly increased thermal conductivity and heat transfer. Consequently, they have been proposed as a front runner in the race to cool modern high-performance electronic equipment. However, there are some experiments opposing these impressive properties, most notably the benchmark study [5].

Analyses of nanofluid conductivity have been primarily based on Maxwell’s theory of 1881, which assumes negligible interaction (semi-infinite domain around a single particle) between particles and steady-state conditions. Disagreement between this and experimental results has led to the inclusion of a number of novel physical effects, such as nanolayers (a thin layer of ordered liquid molecules surrounding the particle), particle clustering and nanoconvection. In [6] we show that the enhancement in conductivity may be simply explained by studying a finite domain and unsteady conditions. In a second paper we go on to show that perhaps the most popular mathematical model for nanofluid flow in fact predicts a decrease in heat transfer and that many papers that draw the opposite conclusion involve ‘ad hoc’ assumptions and incorrect parameter values.

1. Melting at the nanoscale

Nanoparticles have many current and potential uses in medicine, materials, energy, electronics etc. They are often exposed to high temperatures and so it is important to understand their thermal response and likely phase change behaviour. This leads to an interesting variation to the standard Stefan problem, since nanoparticles exhibit size-dependent melt temperatures (for example, 2nm diameter gold particles have been observed to melt at 500K below the bulk value).

In [7] we show that the experimentally observed ‘abrupt melting’ can be attributed to a rapid decrease in melt temperature as the particle’s size decreases. Mathematical models often neglect the density variation through the phase change. We demonstrate that this can have a large effect on the melt rate, and that this effect carries through to the macro-scale.

Fig. 2: Transmission Electron Microscope images and explanatory drawings of A) spherical, B) cubical, C) cylindrical nanoforms. Taken from [8]

1. Current and future work

There are many open questions in our work and new areas to explore, currently we are looking into:

i/ Extending the phase change models to include non-Fourier type heat flow (for example with a hyperbolic heat equation); include more size-dependent effects into the melting models, such as size dependent latent heat and surface tension;

ii/ The application of nanofluids in solar energy collectors [9].

iii/ Using our knowledge of diffusion at the nanoscale to investigate the nano-Kirkendall effect, which has recently been used to manufacture various forms of nano-box, see Fig. 2. This work is in collaboration with the Institut Catalá de Nanotecnologia.

References
[1] Myers TG, MacDevette MM, Font F, Cregan V Continuum mathematics at the nanoscale. Journal ofMathematics in Industry 2014

[3] New desalination process developed using carbon nanotubes. Science News 2011.

[4] Myers TG: Why are slip lengths so large in carbon nanotubes? Microfluid Nanofluid 2011, 10(5):1141-1145, doi:10.1007/s10404-010-0752-7.

[5] Buongiorno J, Venerus D, Prabhat N, McKrell T, Townsend J et al (2009) A benchmark study on the thermal conductivity of nanofluids. J Appl Phys 106:094312.

[6] MacDevette MM, Myers TG, Wetton B: Boundary layer analysis and heat transfer of a nanofluid. Microfluid Nanofluid 2014, 17(2):401-412 [doi:10.1007/s10404-013-1319-1].

[7] Font F, Myers TG, Mitchell SL: A mathematical model for nanoparticle melting with density change. Microfluid Nanofluid 2014 [doi:10.1007/s10404-014-1423-x].

[8] Gonzalez et al Carving at the Nanoscale: Sequential Galvanic Exchange and Kirkendall Growth at Room Temperature. Science 9 December 2011.

[9] Cregan V., Myers T.G. Modelling the efficiency of a nanofluid direct absorption solar collector. To appear Int. J. Heat Mass Trans.

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