Supplementary Information. Infrared Transparent Visible Opaque Fabrics (ITVOF) for Personal Cooling

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1 Supplementary Information Infrared Transparent Visible Opaque Fabris (ITVOF) for Personal Cooling Jonathan K. Tong 1,Ɨ, Xiaopeng Huang 1,Ɨ, Svetlana V. Boriskina 1, James Loomis 1, Yanfei Xu 1, and Gang Chen 1,* 1 Department of Mehanial Engineering, Massahusetts Institute of Tehnology, Cambridge, MA 0139 Ɨ These authors ontributed equally to this work *Correspondene and requests for materials should be addressed to either S.V.B. or G.C. ( s: sborisk@mit.edu, ghen@mit.edu)

2 1. Heat transfer model. To evaluate the impat of a loth s IR optial properties on personal ooling, a 1D steady state heat transfer model was adopted, as illustrated in Fig. 1a of the main text. This model ombines a ontrol volume analysis and an analytial formulation of the temperature profile within the loth to analyze heat dissipation from a lothed human body to the ambient environment. Radiative, ondutive, and onvetive heat transfer are all inluded in this analysis. For onveniene, the following denotations are used in this model: 0 surfae of human skin, 1 inner surfae of the loth, outer surfae of the loth, and 3 the ambient environment. The following setions provide a summary of the assumptions, input parameters, and a derivation of the analytial formulas used in this analysis. Assumptions. For onveniene, all assumptions in the model are summarized as follows, (1) In general, the human body an be modelled as a ylinder with a 1 m diameter. In this analysis, the air gap thikness, t a, and loth thikness, t, are assumed to be muh smaller ompared to the diameter. Therefore, urvature effets are assumed to be negligible, thus heat transfer is modelled as 1D transport through parallel slabs. In this analysis, heat transfer is expressed as area normalized heat fluxes. () The human body is assumed to be in a sedentary state with a uniform skin temperature and heat generation. (3) The loth is assumed to over 100% of the human body. () The air between the skin and loth is assumed to be stationary thus onvetive heat transfer is negligible in this region. (5) Air irulation through the loth is negleted. (6) All optial properties are assumed to be gray and diffuse. (7) The skin and environment are assumed to be an ideal blakbody emitter and absorber. (8) An average loth temperature (e.g. mean of T 1 and T ) is assumed for thermal emission by the loth. (9) All radiative view fators are equal to 1. (10) Internal sattering and self absorption effets are negleted within the loth. (11) It is assumed the absorption and emission profile is linear within the loth. This is an approximation of the more rigorous exponential profile that governs absorption and emission when internal sattering is negligible. In the limit of either high loth transmittane or refletane, this is a reasonable assumption due to the linearity of the exponential deay. In the limit of high absorptane, this assumption will no longer be aurate. Despite this, the ooling power and the

3 maximum ambient temperature an still be reasonable predited sine the differene between the inner and outer loth temperatures is expeted to be small. Input parameters. Table 1 shows a list of the input parameters used in this study. In order to determine the total ooling power through the loth, the net heat flux in this analysis an be multiplied by the surfae area of the human body, A. Additionally, the loth is assumed to be partially refletive, transmissive, and absorptive with gray and diffuse optial properties. In onjuntion with Kirhoff s law, the loth s optial properties will adhere to the following relation, ε = α = 1 ρ τ (S1) where ε, α, ρ, and τ are the loth s total hemispherial emittane, absorptane, refletane, and transmittane, respetively. Table 1: Input Parameters Parameter Name Value Parameter Name Value Human body surfae area, A 1.8 m Cloth thikness, t 0.5 mm Heat generation rate, q gen 58. W/m Total emittane of skin 1 Skin temperature, T o C (93 o F) Total emittane of environment 1 Thermal ondutivity of air, k a 0.07 Wm 1 K 1 Conv. heat transfer oeff., h 3 5 Wm K 1 Thermal ondutivity of yarn, k y 0.05 Wm 1 K 1 Air gap thikness, t a mm Cloth porosity 0.15 Cloth refletane, ρ 0 1 Thermal ondutivity of loth, k 0.07 Wm 1 K 1 Cloth transmittane, τ 0 1 Model formalism. In this model, the overall goal is to determine the maximum ambient temperature that an be sustained without ompromising a person s thermal omfort as a funtion of the loth s optial properties. Although a minimum ambient temperature also exists, this is related to personal heating and is thus beyond the sope of this work. The riterion used to evaluate personal thermal omfort is based on the equivalene of the total ooling power with the total heat generation rate of 105 W from the human body. For a given set of material and environmental onditions, the ambient temperature is inreased iteratively until the net ooling power an no longer dissipate the amount of heat generated by the human body. By fixing the skin temperature to be 33.9 o C (93 o F), the primary unknown variables in this model are the inner surfae loth temperature, T 1, the outer surfae loth temperature, T, and the ambient temperature, T 3. Additionally, the air gap thikness, t a, and the onvetive heat transfer oeffiient, h, an also be varied to simulate different environmental onditions (i.e. tight fitting vs. loose fitting loth on different areas of the human body, varying levels of air irulation within the ambient environment, et.)

4 independent of the environment temperature. In order to ompare the impat of the loth s optial properties on personal ooling for various environmental onditions, we onstrain the air gap thikness and onvetive heat transfer oeffiient to ensure a onsistent baseline neutral temperature band is used regardless of the environmental onditions. To aomplish this, we adopt a referene ase that assumes an ambient temperature of 3.9 o C (75 o F), orresponding to the upper limit of a typial neutral temperature band. The refletane and transmittane of the loth are also assumed to be ρ = 0.3 and τ = 0.03, respetively, orresponding to measurements of onventional polyester and otton fabris as shown in Fig. of the main text. Under these onditions, we first hoose the onvetive heat transfer oeffiient and iterate the air gap thikness until the total ooling power exatly balanes the total heat generation rate using the model equations as shown below. In this manner, the maximum ambient temperature for various environmental onditions and onventional lothing is always 3.9 o C (75 o F). Thus, any subsequent improvements an only be attributed to radiative ooling through the loth. In this work, we assume an individual is ooled via natural onvetion, thus the onvetive heat transfer oeffiient has a typial range of 3 5 W/m K with a orresponding air gap thikness of mm. Figure S1: Illustrations depiting the ontrol volume analysis and temperature profile formulation for the heat transfer model. (a) The ontrol volumes hosen in this analysis onsist of CV1 around the human body and CV around only the loth. (b) A shemati illustrating the differential element and energy balane used to derive the temperature profile within the loth. In addition to heat ondution, this analysis inludes radiative absorption and emission.

5 Control volume analysis. The first omponent of the heat transfer model is to identify relevant ontrol volumes (CV) and to apply an energy balane in order to obtain equations that onnet the various heat transfer mehanisms inluded in this model. As shown in Fig. S1a, there are two ontrol volumes that will be used in this study: CV1 is defined around only the human body and CV is defined around the entirety of the surrounding loth. The expressions obtained when applying an energy balane around CV1 and CV are as follows, q + q + τ q 1 ρ q q = 0 (S) CV 1: gen rad, rad,e rad,s ond,a 1 ρ τ q + 1 ρ τ q +q q q = 0 (S3) CV : rad,s rad,e ond,a rad, onv where q gen is the heat generation rate per unit area, q ond,a is the ondutive heat flux between the skin and the loth, q onv is the onvetive heat flux from the loth to the ambient environment, q rad,s is the radiative heat flux from the skin, q rad,e is the radiative heat flux from the ambient environment, and q rad, is the radiative heat flux from the loth. The ondutive, onvetive, and radiative heat flux terms are expressed using Fourier s law, Newton s law of ooling, and the Stefan Boltzmann law as follows, q ond,a = k T T 0 1 a (S) ta onv 3 q = h T T (S5) q q = σt rad,s 0 = σt rad,e 3 (S6) (S7) q rad, T 1 + T = εσ where T 0 is the skin temperature, T 1 is the inner surfae loth temperature, T is the outer surfae loth temperature, T 3 is the ambient temperature, k a is the thermal ondutivity of air, t a is the air gap thikness, h is the onvetive heat transfer oeffiient, σ is the Stefan Boltzmann onstant equal to Wm K. In equation (S8), we assume a mean temperature of T 1 and T to approximate radiative emission by the loth. Additionally, in equations (S), (S3), (S6), and (S7), it was assumed the skin and environment behave like an ideal blakbody with an absorptane and emittane equal to 1. Based on the ontrol volume analysis, we obtain two fundamental equations (S) and (S3) that desribe the various ontributions to heat transfer in this system. Sine there are three unknowns that must be solved for, an additional equation is required in order to omplete this model. Equations (S) and (S3) desribe heat transfer around the human body and the loth, respetively. By dedution, the (S8)

6 remaining equation must desribe the nature of heat transfer within the loth itself. Speifially, by onsidering heat ondution, radiative absorption, and radiative emission, a temperature profile an be derived in order to link the unknown temperatures T 1 and T. Temperature profile of loth. To determine the temperature profile within the loth, heat ondution and radiative heat transfer must be inluded in the heat transfer analysis. If a differential volume element is taken within the loth, as shown in Fig. S1b, the heat equation will take the following form, 1 T k q rad = 0 (S9) x x where k is the loth thermal ondutivity and q rad is the net radiative transfer within the loth. In general, q rad must be determined rigorously using the radiative heat transfer equation in order to aount for all absorption, emission, and internal sattering proesses. For simpliity, we assume internal sattering effets are negligible and only onsider IR refletion at the boundaries of the loth, as will be later shown when determining the expressions for eah radiative heat flux. Additionally, selfabsorption effets are also negleted. Therefore, the net radiative heat transfer will onsist only of inident radiative absorption and outgoing radiative emission as follows, T rad,l' rad,r' rad,s' rad,e' k = q + q + q + q (S10) x x x x x where q rad,l is the radiative emission from the loth to the skin, q rad,r is the radiative emission from the loth to the ambient environment, q rad,s is the absorption of radiation emitted from the skin, and q rad,e is the absorption of radiation emitted from the ambient environment. In general, the analytial form for radiative absorption and emission in the limit of negligible internal sattering will onsist of an exponential deay in aordane to the Beer Lambert law. 1 However, we again simplify the analysis by instead assuming the absorption and emission profile to be linear as follows, qrad,i x = A x+b (S11) where A and B are unknown oeffiients that will depend on the boundary onditions assumed for eah radiative heat flux. In the limit of high absorption, the approximation of a linear absorption and emission profile will be inaurate. Despite this limitation, it is nonetheless expeted that this approximation will provide a reasonable estimation of heat transfer through the loth sine the differene in the inner and outer loth temperature is not expeted to be large, thus inherently making this analysis less sensitive to

7 the absorption and emission profile used. Using equation (S11) and appropriate boundary onditions for eah radiative flux, we obtain the following, 1. Emission from loth to skin: rad,l' rad,l' q x = 0 = q q x = t = 0 rad, q (S1). Emission from loth to ambient environment: rad,r' q x = 0 = 0 q x = t = q rad,r' rad, 3. Absorption by loth from skin: q x = 0 = 1 ρ q rad,s' rad,s q x = t =τ q rad,s' rad,s rad, qrad,l' x = x qrad, t q (S13) rad, qrad,r' x = x t. Absorption by loth from ambient environment: q x = 0 = τ q rad,e' rad,e q x = t = 1 ρ q rad,e' rad,e α q q x = x 1 ρ q (S1) rad,s rad,s' rad,s t α q q x = x t 1 ρ q (S15) rad,e rad,e' rad,e t Upon substituting equations (S1) (S15) into (S10) and using the definition of heat fluxes defined by (S6) (S8), the heat equation will beome, T 1 T+T 1 = ε σ ασt0 ασt3 x kt where a mean temperature of T 1 and T is again used to approximate radiative emission from the loth. Although radiative emission from the loth tehnially depends on the loal temperature T as a funtion of position x, the use of a mean temperature is a reasonable approximation sine T 1 and T are not expeted to be signifiantly different. To determine the temperature profile, all that remains is to integrate equation (S16) and apply appropriate boundary onditions, (S16) T 1 = ε σ ασt α σt x+ C x kt T+T T = ε σ α σt α σt x + Cx + C kt T+T (S17) (S18)

8 The boundary onditions applied in this analysis inludes temperature and heat flux ontinuity at surfae 1 (x = 0) as follows, Tx=0= T 1 (S19) T k x=0 = q x ond,a Upon applying (S19) and (S0) in equations (S17) and (S18), we obtain the final expression for the temperature profile within the loth, 1 T = ε σ α σt α σt x x + T k t k t T+T 1 k T a 0 T a By taking x = t in equation (S1), the following temperature relation is obtained, T = ε σ α σt α σt T T + T t T+T 1 kat k kta (S0) (S1) (S) Therefore, with equations (S), (S3), and (S), we now have a omplete set of equations to desribe heat transfer from a human body overed by loth to the ambient environment. These equations are used to first obtain the air gap thikness, t a, for the previously desribed referene ase with an assumed onvetive heat transfer oeffiient. Following this alulation, we then use the same equations to solve for T 1, T, and T 3 as a funtion of the loth s optial properties and the assumed environmental onditions. From this analysis, we an find the maximum ambient temperature, T 3, whih an be sustained without ompromising personal thermal omfort.

9 . Supplementary Figures: Figure S1: The optial onstants of: (a) polyethylene (PE) and (b) polyethylene terephthalate (PET), more ommonly known as polyester, taken from the literature. For polyethylene, the refrative index, n, is extrapolated from shorter wavelength data. Based on the dispersion of the extintion oeffiient, k, it is expeted the refrative index will also exhibit some dispersion. However, this is assumed to be small and is thus negleted in this study. For polyester, a Lorentzian model was used to fit experimental data from previous studies.

10 Figure S: The visible wavelength extintion, sattering, and absorption effiieny of a single polyethylene fiber. The effiieny fator, Q, is defined as the ratio of the effetive ross setion normalized to the geometri ross setion. The diameter of the fiber is D = 1 μm and the inident light is assumed to be unpolarized. For omputation, the standard Mie theory solutions for an infinitely long ylinder were used. 3 As shown, the absorption effiieny exhibits a similar trend to the total hemispherial absorptane shown in Fig. 6 in the main text. The osillatory behavior is indiative of whispering gallery modes supported by the fiber whih are broadened due to material loss (n = 1.5, k = 5 10 ). In addition, a broad Fabry Perot resonane is also supported by the fiber as indiated by the sattering effiieny, whih inreases from 60 nm to 700 nm.

Figure S3: Numerial simulation results for the IR optial properties of a polyethylene based ITVOF for the ase of a varying fiber diameter (D f = 1 μm, 5 μm, and 10 μm) assuming a fixed yarn diameter

11 Figure S3: Numerial simulation results for the IR optial properties of a polyethylene based ITVOF for the ase of a varying fiber diameter (D f = 1 μm, 5 μm, and 10 μm) assuming a fixed yarn diameter of D y = 50 μm. As before, all simulations assume the fiber separation distane is D s = 1 μm and the yarn separation distane is D p = 5 μm. The spetrally integrated transmittane (τ ) and refletane (ρ ) is shown in eah plot weighted by the Plank s distribution assuming a body temperature of 33.9 o C. Compared to the ase where D y = 30 μm, the overall transmittane is lower, as expeted, due to the ombination of a larger material volume that absorbs more inident IR radiation and a larger number of fibers available to satter inident IR radiation thus inreasing the refletane. However, by reduing the size of the fiber to be D f = 1 μm, whih is far smaller than IR wavelengths, the total transmittane an again be signifiantly enhaned from 0.63 to whih is nearly equal to the ase where D y = 30 μm. Simultaneously, the refletane of the ITVOF is redued from 0.7 to further improving radiative ooling. These results show that reduing the fiber size is far more important than reduing the yarn size. Therefore, this struturing methodology ould potentially be applied to ITVOF that are omparable in size to onventional fabris. The material volume per unit depth for a single yarn is 19 μm for D f = 10 μm, 117 μm for D f = 5 μm, and 5 μm for D f = 1 μm. The optial properties of the ITVOF are again alulated for the wavelength range from 5.5 to μm, whih will provide a onservative estimate of the total transmittane and the refletane.

Figure S: Numerial simulation results for the IR optial properties of an ITVOF blend of polyethylene and polyester with varying volumetri onentrations.

12 Figure S: Numerial simulation results for the IR optial properties of an ITVOF blend of polyethylene and polyester with varying volumetri onentrations. The PE and PET fibers were randomly distributed in the simulation. For all simulations it is assumed D f = 1 μm, D y = 30 μm, D s = 1 μm, and D p = 5 μm. Again, the spetrally integrated transmittane (τ ) and refletane (ρ ) is shown in eah plot weighted by the Plank s distribution assuming a body temperature of 33.9 o C. As shown, a progressive inrease in the volumetri onentration of PET results in an inrease in the spetral absorptane thus dereasing the total transmittane. However, it an also be observed that the spetral refletane is ~0.0 for all ases and exhibits no signifiant variation spetrally further reinforing the point that so long as the fiber is suffiiently small ompared to IR wavelengths, sattering will be minimal. Based on these results, even the highest volumetri onentration of PET fibers (5%PE/75% PET) an provide suffiient ooling to raise the ambient temperature to 6.1 o C due to a ombination of a high total transmittane of 0.78 and a low total refletane of The material volume per unit depth for a single yarn in all ases is equal to μm. The optial properties of the ITVOF are again alulated for the wavelength range from 5.5 to μm, whih will provide a onservative estimate of the total transmittane and the refletane.

13 Referenes: 1. Modest, M. F. Radiative Heat Transfer. (Aademi Press, 003).. Laskarakis, A. & Logothetidis, S. Study of the eletroni and vibrational properties of poly(ethylene terephthalate) and poly(ethylene naphthalate) films. J. Appl. Phys. 101, (007). 3. Bohren, C. F. & Huffman, D. R. Absorption and Sattering of Light by Small Partiles. (WILEY VCH Verlag GmbH & Co. KGaA, 007).

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