Thermal management of flexible wearable

Applied Thermal Engineering 144 (2018) 504–511

Contents lists available at ScienceDirect

Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Thermal management of flexible wearable electronic devices integrated with human skin considering clothing effect ⁎

Yafei Yina,1, Yun Cuia,1, Yuhang Lia,b, , Yufeng Xinga, Min Lia, a b

T

⁎

Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100191, China State Key Laboratory of Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, China

H I GH L IG H T S

analytical heat transfer model is developed to study flexible electronics. • An are considered as porous materials in the model. • Clothes • Design guidelines for flexible electronics considering clothing effects.

A R T I C LE I N FO

A B S T R A C T

Keywords: Flexible wearable electronics Thermal management Clothing effect

The flexible wearable electronic devices can be directly integrated with human skin to monitor the vital signs of human body, which will be covered with clothes. An analytical model is developed in this paper to study the flexible wearable electronic devices integrated with human skin considering clothing effect. The coupling model of Pennes bio-heat transfer equation and Fourier heat conduction equation is adopted to consider the thermal behavior of the whole system. The clothes are modeled as a kind of composite material consisting of clothing fibers and air under the complicated boundary condition including convection, conduction and radiation between the environment. The predicted analytical temperature distribution of the system is validated by finite element analysis (FEA). And the influences of systematic parameters on characteristic temperatures, i.e., the temperature of electronic component and the maximum temperature at the device/skin interface are investigated comprehensively to discuss the design of heat dissipation and the clothing effect. The results presented are able to provide guidelines for the thermal management of flexible wearable electronic devices under real working conditions.

1. Introduction In recent years, flexible electronics have shown wide application prospects in biomedical field [1–8], which attracted great attention among researchers. With the properties of flexibility and stretchability, flexible electronics can be attached to the skin perfectly as wearable electronics and able to maintain their functions under complex deformation conditions like bending, torsion and tension [9–23]. As one of the key issues, thermal management needs to be investigated systematically before flexible wearable medical devices can be taken into practical application to avoid abnormal operating conditions and skin ambustion due to excess temperature [24,25]. Several literatures have studied the thermal management problem of flexible electronics based on the Fourier’s Law [26–32]. Under the

constant input power, the steady-state thermal analyses were developed for micro-scale inorganic light-emitting diodes (μ-ILEDs) by axisymmetric and 3D models [26–28]. While pulsed protocol was adopted by Kim et al. [29] to replace constant power as an effective heat-dissipation method, which was further investigated by Li et al. [30] and Cui et al. [23] to provide detailed parametric discussions and extend the scope of application. Li et al. [32] presented an effective novel way to control the heat flow direction by orthotropic substrate materials to achieve the goal in heat dissipation, which minimizes the adverse thermal effects by preventing the heat into the substrate along the thickness direction. In consideration of the application prospects of flexible electronics in biosensing and long-term monitoring, many researchers further developed the thermal analysis of flexible electronics integrated with skin

⁎

Corresponding authors at: Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100191, China (Y. Li). E-mail addresses: [email protected] (Y. Li), [email protected] (M. Li). 1 These authors contributed equally. https://doi.org/10.1016/j.applthermaleng.2018.08.088 Received 1 July 2018; Received in revised form 14 August 2018; Accepted 23 August 2018 Available online 24 August 2018 1359-4311/ © 2018 Published by Elsevier Ltd.


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research that the multilayer structure is set in quiescent air and the shell fabric of clothes is assumed to have a good performance of windproof. According to the analysis and hypothesis mentioned above, the air gap is modeled as a homogeneous solid plate with an invariable thermal conductivity coefficient and the mass transfer through clothes is also neglected in the analytical model. And the fill material of clothes is modeled as a two-component composite material consisting of clothing fibers and air [47,48], whose effective thermal conductivity is relevant to conductivities of these two ingredients and their volume fractions [49]. Then the analytical model can be simplified as a quartered fivelayer structure, which consists of skin, substrate, encapsulation, air gap and clothes from bottom to top due to the symmetry as shown in Fig. 1(b). The original point is located on the bottom surface of skin, right under the center of electronic component. The positive direction of coordinate z points from skin to clothes while the coordinates x and y lie along the length and width direction of electronic component, respectively. The dimension of electronic component is 2a × 2b × Hele, where a, b and Hele denote the half-length, half width and thickness of the component, respectively. zi denotes the z coordinate value of each layer’s top surface while the subscript i from 1 to 5 represents skin, substrate, encapsulation, air gap and clothes, respectively. A definition of the temperature increase from the core body temperature ΔT is given as ΔT = T (x , y, z )−Tc , where T (x , y, z ) is the temperature at the point (x,y,z) and Tc is the core body temperature. The temperature increases in substrate, encapsulation, air gap and clothes satisfy the Fourier heat conduction equation,

tissue, which is much more complicated than that of the isolated electronics system on account of the influences of metabolic heat generation and blood perfusion inside the skin tissue [33–38]. As a classical model, the Pennes bio-heat equation [39] has been widely used to model the bio-heat transfer problems of skin tissue and investigate thermal damage process of skin [24,25,40]. Through the coupling of the Fourier’s heat conduction and the Pennes bio-heat transfer, Cui et al. [35] developed a one-dimensional thermal analysis for the system of flexible electronic devices, whose in-plane dimensions are much larger than thickness, integrated with skin. The thermal properties of the system under both constant and pulsed input power were fully studied and discussed. Considering devices with small in-plane dimensions which cannot be simplified as one-dimensional models [26,29], Cui et al. [36] and Li et al. [37] further established three-dimensional heat transfer models for different substrate types to investigate the influences of various geometric, material and loading parameters on the characteristic temperatures, i.e., maximum temperatures of device and device/skin interface and discuss the parameter design of thermal comforts. Due to the possible existence of delamination between devices and skin in practical applications, an axisymmetric model was developed by Li et al. [38] to analyze the effects of interfacial thermal resistance on the integrated system of electronic devices and skin, which agreed well with both finite element analysis and experimental measurements. However, all the studies mentioned above only considered devices themselves or the device/skin coupled system, while in the actual application scenario the devices are commonly covered with clothes which probably have some influences on the temperature distribution of the whole system. The clothing effects on thermal response of human body have been taken into consideration in various service conditions [41–45]. The heat generated by device/skin system transfers through the air gap inside of the clothes and clothes to the outer surface of clothes while the outer surface of clothes conducts heat to the environment mainly through convection and radiation [46]. Clothing effect makes it harder to dissipate heat of the whole system, which may result in abnormal operating conditions of devices or skin ambustion. In order to investigate the thermal properties of flexible wearable electronic devices in practical application scene, a three-dimensional heat transfer model containing the air gap and clothes is established in this paper and then validated by the finite element analysis (FEA), where the clothes is modeled as a composite material consisting of clothing fibers and air. Furthermore, the valid heat transfer model is adopted to explore the influences of devices’ systematic parameters, the thickness, porosity of clothes and ambient temperature on characteristic temperatures. This paper is outlined as follows. Section 2 describes the analytical modeling of system while the results and discussion are presented in Section 3. The conclusions are summarized in Section 4.

∇2 (ΔTi ) = 0 (i = 2, 3, 4, 5),

(1)

while the temperature increase in skin layer satisfies the Pennes bioheat equation [39,40]

k1 ∇2 (ΔT1)−ω b ρ b c b (ΔT1 + Tc−Tb) + qmet = 0,

(2)

where k is the thermal conductivity of corresponding layers and ∇2 represents the Laplace operator. ω, ρ, c are the blood perfusion rate, mass density and the specific heat capacity with subscript b denoting blood, qmet represents the metabolic heat generation, and Tb denotes the blood temperature. Since the blood temperature is close to the core temperature generally, the blood temperature Tb is set to be the same as the core temperature Tc [35,36] here and the blood perfusion exists in the whole skin layer. For a healthy person, the metabolic heat generation can be considered as a constant [35,36,40]. Then Eq. (2) can be simplified as,

k1 ∇2 (ΔT1)−ω b ρ b c b ΔT1 + qmet = 0.

(3)

At the bottom surface of skin (z = 0), the temperature is equal to the core body temperature, which gives,

ΔT1 |z = 0 = 0.

(4)

At the top surface of clothes (z = z5), the natural convection heat exchange and radiation heat exchange are taken into consideration, so the boundary condition is given as

2. Analytical modeling A typical flexible wearable device system is composed of skin, device, air gap and clothes as shown in Fig. 1(a). Skin can be regarded as a single layer structure here to simplify the modeling [33] while it is a multi-layer structure in reality [40]. The device consists of encapsulation, substrate and electronic component which can be modeled as planar heat source with heat generation power density Q(W/m2) since its thermal conductivity(∼100 W/m/K) is much larger than that of substrate and encapsulation [36]. On account of the small thickness of air gap, difference between skin surface temperature and clothes inner surface temperature tends to be subtle. So that it’s rational to neglect heat convection of air gap and radiation heat transfer between skin and clothes [46]. There are indeed many factors that affect the heat transfer through clothes like shell fabric, fillers, porosity, thickness, environment factors and so on. In order to reduce complexity of the heat transfer model, a particular condition is considered in the following

−k5

∂ΔT5 ∂z

= Qc + Qr ,

(5)

z = z5

where k5 is the effective thermal conductivity of the bi-component clothing material, Qc and Qr are the natural convection heat and radiation heat, respectively. According to the Effective Medium Theory [49], the effective thermal conductivity satisfies the following equation

va

ka−k5 k −k + vf f 5 = 0, ka + 2k5 k f + 2k5

(6)

which may be rewritten to be explicit for k5,

k5 =

1 [(3va−1) ka + (3vf −1) k f + 4

((3va−1) ka + (3vf −1) k f )2 + 8ka k f ], (7)

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Fig. 1. Schematic diagram of (a) system structure, and (b) the three-dimensional analytical model showing a quarter geometry of the skin/device/clothes system.

which yields

where v and k are volume fraction and thermal conductivity, and the subscripts a and f represent air and fiber, respectively. It is obvious that va + vf = 1. On the basis of the principle of thermal convection and thermal radiation, Qc and Qr can be given as [50,51]

Qc = hc (Tclo−Ta ),

Qr =

4 4 εσ (Tclo,K −Tr,K ),

ΔT2 |z = z2− = ΔT3 |z = z2+ , −k3

∂ΔT5 ∂z

= [h 0 (ΔT5 + Tc−To )]|z = z5 , z = z5

To =

Tr h r + Ta h c , hr + hc

P

= z = z2−

⎧ 4ab , (x , y ) ∈ Ω , ⎨ 0, (x , y ) ∉ Ω ⎩

(9)

where Ω is the region of electronic component at the encapsulation/ substrate interface, i.e., Ω = {(x , y )| 0 ⩽ x ⩽ a, 0 ⩽ y ⩽ b} . P is the input power of the heat generation of electronic component. At other interfaces including skin/substrate interface (z = z1), encapsulation/air gap interface (z = z3) and air gap/clothes interface (z = z4), both the temperature and heat flux are continuous, which give

ΔTi |z = zi− = ΔTi + 1 |z = zi+ , −ki

(10)

= 1, 3, 4).

(11)

(12)

(13)

An assumption is made here that the surrounding environment is the ideal emitter, then the radiation temperature is equal to the ambient temperature [46]. Meanwhile the ambient temperature is regarded as the clothes temperature when the linear radiation coefficient is calculated since the temperature difference between them is generally slight [37,38,52]. According to the assumptions above, the following equations can be obtained,

Tr = Ta, To = Ta,

(14)

3 hr = 4εσTa,K ,

(15)

∂ΔTi ∂z

= −ki + 1 z = zi−

∂ΔTi + 1 ∂z

(i z = zi+

(17)

Due to the metabolic heat generation in Eq. (3), the natural convention and the heat generation of electronic component in Eqs. (11) and (16), both the above governing equations and boundary conditions are nonhomogeneous, which brings difficulty to solve the partial differential equations. Based on the method of linear superposition, ΔT can be solved as a summation of the following two subproblems [36,37]: (1) Problem I satisfies nonhomogeneous governing equations, natural convection and radiation boundary conditions without planar heat source at the encapsulation/substrate interface. The solution for problem I corresponds to the temperature increase caused by the metabolic heat generation and the natural convection and radiation boundary conditions. (2) Problem Ⅱ satisfies homogeneous governing equations and planar heat source at the encapsulation/substrate interface with adiabatic boundary condition at the outer surface of clothes. The solution for problem Ⅱ corresponds to the temperature increase due to the heat input of electronic component.

where h0 is the total heat transfer coefficient, and To is the operating temperature, which can be given as

h0 = hr + hc,

z = z2+

∂ΔT2 ∂z

(16)

where hr is the linear radiation coefficient, which is related to the clothes temperature and the radiation temperature [46]. In this case, the boundary condition can be expressed as

−k5

+ k2

(8)

where hc is convective heat transfer coefficient. Tclo, Ta and Tr are clothes surface temperature, ambient temperature and radiation temperature, respectively. The subscript K means the temperature value calculated in Eq. (9) is under Kelvin temperature scale, and Tclo = T5 |z = z5 . ε is the emissivity and σ is the Stefan-Boltzmann constant. In order to express Qc and Qr in a unified compact form, Qr can be linearized as

Qr = hr (Tclo−Ta ),

∂ΔT3 ∂z

2.1. Solution for problem I In brief, problem I describes a condition that the electronic component is out of work. Since metabolic heat generation and the natural convection and radiation are permanent and homogeneous, problem I can be regarded as a one-dimensional heat transfer problem along the

At the encapsulation/substrate interface (z = z2), the temperature is continuous and the heat flux satisfies the surface heat source condition, 506


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equations for ΔT are given by

thickness direction. Then the governing equations for ΔT are given as follows:

k1 ∇2 (ΔT1)−ω b ρ b c b ΔT1 = 0,

d2ΔT1 k1 −ω b ρ b c b ΔT1 + qmet = 0, dz 2

(18)

d2ΔTi dz 2

(19)

= 0 (i = 2, 3, 4, 5).

for skin layer and Eq. (1) for the other 4 layers. The boundary and continuity conditions in Eqs. (4), (16), (17) keep the same while the boundary condition on the outer surface of clothes becomes homogeneous, i.e.,

The boundary and continuity conditions remain unchanged except Eq. (16), which is rewritten as

ΔT2 |z = z2−

dΔT3 = ΔT3 |z = z2+ , −k3 dz

z = z2+

dΔT2 + k2 dz

−k5

z = z2−

⎧ ΔT1,1 = A1,1 exp(z η k1 ) + B1,1 exp(−z η k1 ) + qmet η , ⎨ ΔTi,1 = Ai,1 z + Bi,1 (i = 2, 3, 4, 5) ⎩

ΔT ̂(α, β , z ) =

qmet

D=

+

(z 4 − z3 ) k4

η k1 , k1 k2

F=

+

E=

(z3 − z2 ) k3

+

(z2 − z1 ) ⎤ k2 ⎦

η

(26)

d2ΔT1 ̂ −((α 2 + β 2) + η k1 )ΔT1 ̂ = 0, dz 2

(27)

d2ΔTi ̂ −(α 2 + β 2)ΔTi ̂ = 0, dz 2

(28)

−

dΔT ̂ k5 ∂z 5

dΔT ̂ ΔTi |̂ z = zi− = ΔTi +̂ 1 |z = z1 , −ki dz i

z = z2+

z = z5

+ k2

dΔT2̂ dz

= z = z2−

P sin(αa) sin(βb) , 4abαβ

= h 0 ΔT5 |̂ z = z5 ,

z = zi−

= −ki + 1

̂ 1 dΔTi + dz

z = zi+

(i = 1, 3, 4). (29)

The solutions of Eqs. (27) and (28) take the form of

ΔT1,2̂ (α, β , z ) = A1,2 (α, β ) exp(z (α 2 + β 2) + η k1 ) + B1,2 (α, β ) exp(−z (α 2 + β 2) + η k1 ) ΔTi,2̂ (α, β , z ) = Ai,2 (α, β ) exp(z α 2 + β 2 ) + Bi,2 (α, β ) exp(−z α 2 + β 2 ) (30) where A and B are undetermined coefficients, the numbers in the subscripts after comma means that these are the parameters in problem II and the numbers before comma represent skin, substrate, encapsulation, air interlayer and clothes, respectively from 1 to 5. The undetermined coefficients above can be determined by the boundary and continuity conditions in Eqs. (29) as

+ k2.

z (z − z ) h 0 k2 ⎡ k2 − 2 k 1 ⎤ 2 ⎣ 3 ⎦

+ cosh(z1 η k1 )(Tc−To) ⎤ ⎦

ΔT (x , y, z ) cos(αx ) cos(βy )dx dy,

dΔT ̂ ΔT2 ̂|z = z2− = ΔT3 |̂ z = z2+ , −k3 dz 3

H = CD cosh(z1 η k1 ) + h 0 sinh (z1 η k1 ) qmet

+∞

ΔT1 |̂ z = 0 = 0,

(z − z ) (z − z ) z h 0 k2 ⎡ k 3 − 3 k 2 − 2 k 1 ⎤ 3 2 ⎣ 4 ⎦ z G = h 0 k2 k5 + k2 5

I = Dh 0 ⎡ (cosh(z1 η k1 )−1) ⎣

+∞

∫0 ∫0

and the boundary and continuity conditions in Eqs. (4), (16), (17) and (25) become

where (z5 − z 4 ) k5

(25)

the governing equations then become

(21)

1 (−CD + h 0 ) e 1 − h0 − h 0 (Tc − To ) ⎧ η η 1 A1,1 = 2 ⎪ H ⎪ q q (CD + h 0 ) e z1 η k1 met − h 0 met − h 0 (Tc − To ) ⎪ η η 1 = − B 1,1 ⎪ 2 H ⎪ I A2,1 = − H ⎪ ⎪ ⎪ B2,1 q ⎪ (C + h 0 z1 ) D (cosh(z1 η k1 ) − 1) met + (h 0 z1 D cosh(z1 η k1 ) − h 0 sinh (z1 η k1 ))(Tc − To ) η ⎪ = H ⎪ ⎪ k I A3,1 = − k2 H 3 ⎨ q (C + E ) D (cosh(z1 η k1 ) − 1) met + (ED cosh(z1 η k1 ) − h 0 sinh (z1 η k1 ))(Tc − To ) ⎪ η ⎪ B3,1 = H ⎪ k I ⎪ A 4,1 = − k2 H 4 ⎪ q ⎪ (C + F ) D (cosh(z1 η k1 ) − 1) met + (FD cosh(z1 η k1 ) − h 0 sinh (z1 η k1 ))(Tc − To ) η ⎪ B4,1 = H ⎪ k I ⎪ A5,1 = − k2 H 5 ⎪ q ⎪ GD (cosh(z1 η k1 ) − 1) met + ((G − C ) D cosh(z1 η k1 ) − h 0 sinh (z1 η k1 ))(Tc − To ) η ⎪ B5,1 = H ⎩ (22) ,

C = h 0 k2 ⎡ ⎣

= h 0 ΔT5 |z = z5 , z = z5

Through the Fourier Cosine transform along x and y directions,

(20)

where η = ω b ρ b c b , A and B are undetermined coefficients, the numbers in the subscripts after comma means that these are the parameters in problem I and the numbers before comma represent skin, substrate, encapsulation, air gap and clothes, respectively from 1 to 5. The undetermined coefficients above can be determined by the boundary and continuity conditions in Eqs. (4), (11), (17) and (20) as η k qmet

∂ΔT5 ∂z

= 0.

The solutions for Eqs. (18)–(19) can be expressed as follows:

−z

(24)

(23)

2.2. Solution for problem II When the electronic component is maintained in normal working condition, the planar heat source only exists in certain area at the encapsulation/substrate interface. Therefore, problem II is a three-dimensional heat conduction problem. The homogeneous governing 507


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the Eq.(34), which gives

⎛ ⎞ ⎛ ⎞ h0 h0 ⎧ ⎜W1 + W2 2 2 ⎟ cosh(ξ5) + ⎜W2 + W1 2 2 ⎟ sinh (ξ5 ) α + β k5 ⎠ α + β k5 ⎠ ⎪ P sin(αa) sin(βb) ⎝ ⎝ ⎪ A1,2 = 8abαβ α2 + β2 ⎛ ⎞ ⎛ ⎞ h0 h0 ⎪ ⎜Z1 + Z2 2 2 ⎟ cosh(ξ5) + ⎜Z2 + Z1 2 2 ⎟ sinh (ξ5 ) α + β k5 ⎠ α + β k5 ⎠ ⎝ ⎝ ⎪ ⎪ B1,2 = −A1,2 ⎪ A1,2 (m1 cosh(ξ1) + sinh(ξ1 )) ⎪ A2,2 = 2 2 ⎪ e z1 α + β ⎪ A1,2 (m1 cosh(ξ1) − sinh(ξ1 )) B2,2 = − ⎪ 2 2 e−z1 α + β ⎪ P sin(αa) sin(βb) ⎪ A1,2 (X1 + X2) − 8abαβ α2 + β2 ⎪ A3,2 = 2 + β2 ⎪ z α 2 e k3 ⎪ P sin(αa) sin(βb) ⎪ A1,2 (X1 − X2) − 8abαβ α2 + β2 B3,2 = − ⎨ 2 2 e−z 2 α + β k3 ⎪ ⎪ P sin(αa) sin(βb) A1,2 (Y1 + Y2) − (k3 cosh(ξ3) + k 4 sinh (ξ3 )) ⎪ 8abαβ α2 + β2 A = ⎪ 4,2 2 2 e z3 α + β k3 k 4 ⎪ P sin(αa) sin(βb) ⎪ A1,2 (Y1 − Y2) − (k3 cosh(ξ3) − k 4 sinh (ξ3 )) ⎪ 8abαβ α2 + β2 B4,2 = − ⎪ 2 2 e−z3 α + β k3 k 4 ⎪ P sin(αa) sin(βb) ⎪ A1,2 (Z1 + Z2) − (W1 + W2 ) 8abαβ α2 + β2 ⎪ A = 5,2 2 2 ⎪ e z 4 α + β k3 k 4 k5 ⎪ P sin(αa) sin(βb) ⎪ A1,2 (Z1 − Z2) − (W1 − W2 ) 8abαβ α2 + β2 ⎪ B5,2 = − 2 + β2 ⎪ − z α e 4 k3 k 4 k5 ⎩

ΔT2,2 (x , y, z2) =

ΔTele =

W1 = k 4 (k3 cosh(ξ3) cosh(ξ 4 ) + k 4 sinh(ξ3) sinh(ξ 4 )) W2 = k5 (k 4 sinh(ξ3) cosh(ξ 4 ) + k3 cosh(ξ3) sinh(ξ 4 )) X1 = k2 (m1 cosh(ξ1) cosh(ξ2) + sinh(ξ1) sinh(ξ2 )) X2 = k3 (m1 cosh(ξ1) sinh(ξ2) + sinh(ξ1) cosh(ξ2 )) Y1 = k3 (X1 cosh(ξ3) + X2 sinh(ξ3 )) Y2 = k 4 (X2 cosh(ξ3) + X1 sinh(ξ3 )) Z1 = k 4 (Y1 cosh(ξ 4 ) + Y2 sinh(ξ 4 )) (32)

Then the temperature increase in each layer can be obtained by the inverse Fourier Cosine Transform of Eq. (30), i.e.,

∫0 ∫0

∞

ΔT (̂ α , β , z ) cos(αx ) cos(βy )dα dβ .

∫0 ∫0

∞

8 π2

∞

∞

A1,2 sinh(ξ1 )dα dβ .

A1,2

X2 sin(αa) sin(βb) dα dβ . k3 αβ

(38)

Tele = Tc + A2,1 z2 + B2,1 + ΔTele,

(39)

max Tskin

(40)

= Tc + A2,1 z1 + B2,1 +

max ΔTskin .

(41)

(42)

Thus, the thickness of clothes can be obtained according to its effective thermal conductivity and the ambient temperature. The volume fraction of air in clothes layer, or porosity in other words, is 75% [47]. The outer surface of the clothes has a natural convection boundary with the coefficient of heat convection hc = 15 Wm−2 K−1 [56] and a ra3 diation boundary with the linear radiation coefficient hr = 4εσTa,K , where ε = 0.95 and σ = 5.67 × 10−8 W/(m2K4) [50–52]. The core body temperature Tc = 37 °C is applied at the bottom surface of the skin [36,40]. The ambient temperature is set as 25 °C, i.e., Ta = 25 °C, Ta,K = 298 K [36]. The metabolic heat generation in skin tissue is 368 Wm−3 [40]. The material properties of blood are ρ b = 1069 kg/m3 , c b = 3659 J/(kg·K) andω b=0.0005 mL/(mL·s ) [28,33]. In ABAQUS, the blood perfusion is converted into several discrete surface film conditions uniformly in skin layer. The number of the discrete surface film conditions n is large enough to ensure the convergence. In each surface

⎡ A2,2 (α, β ) exp(z α 2 + β 2 ) ⎤ ⎢ ⎥ cos(αx ) cos(βy )dα dβ . ( , ) exp(−z α 2 + β 2 ) ⎥ ⎢ ⎣+ B2,2 α β ⎦ (34)

∫0 ∫0

∞

T (x , y, z2) = Tc + A2,1 z2 + B2,1 + ΔT2,2 (x , y, z2),

H5 = R cl k5.

In the overall system, the temperature of electronic component and the maximum temperature at the skin/substrate interface are the characteristic temperatures of our main concern, which has been emphasized in Section 1. The maximum temperature increase at the skin surface can be obtained by setting (x,y,z) = (0,0,z1) in Eq. (34) as max ΔTskin =

∞

∫0 ∫0

where Rcl is the thermal resistance of clothes. In addition, the thickness, thermal resistance and thermal conductivity of clothes satisfy the following relation

(33)

ΔT2,2 (x , y, z ) ∞

16 π 2 [2ab + (a + b) Hele ]

R cl = (1.87−0.04To) × 0.155 m2K/W,

For example, the temperature increase distribution in the substrate layer is obtained by

4 = 2 π

(36)

In order to validate the analytical model, a three-dimensional finite element analysis (FEA) is performed by using ABAQUS software. All the calculating parameters are listed as follows if not specified. The inplane dimension of the whole system is 20 mm × 20 mm, which is large enough for a converged result. The electronic component is modeled as a tridimensional body heat source with the dimension 2a × 2b × Hele, where the in-plane dimension is 100 μm × 200 μm and thickness is 6.5 μm [29,30]. The thermal conductivities of skin, substrate, encapsulation, air and clothing fiber (cotton as a typical material) are 0.47 Wm−1 K−1, 0.6 Wm−1 K−1, 0.2 Wm−1 K−1, 160 Wm−1 K−1, 0.026 Wm−1 K−1 and 0.05 Wm−1 K−1, respectively [29,30,33,52,53]. And the thicknesses of skin, substrate, encapsulation, air gap are taken as 4 mm, 0.2 mm, 7 μm, 1 mm, respectively [29,30,33,54]. For clothes, its thickness is related to the clothing thermal resistance [46]. According to R De Dear [55], a statistically significant relationship between befitting clothing thermal resistance and operating temperature is

ξi = (z i−z i − 1 ) α 2 + β 2 (i = 2, 3, 4, 5)

∞

X2 cos(αx ) cos(βy )dα dβ . k3

Based on the method of linear superposition, the temperature distribution at the encapsulation/substrate interface, the temperature of electronic component and the maximum temperature at the skin/substrate interface can be obtained by

ξ1 = z1 α 2 + β 2 + η k1

4 ΔT (x , y, z ) = 2 π

A1,2

3. Results and discussion

α2 + β2 + η k1 α 2 + β2

Z2 = k5 (Y2 cosh(ξ 4 ) + Y1 sinh(ξ 4 ))

∞

(37)

where k1 k2

∞

∫0 ∫0

Due to the large thermal conductivity of the electronic component (∼150 Wm−1 K−1) and the thin thickness of encapsulation layer (∼7 μm) [29,30], the temperature in the region of electronic component {(x , y, z )| 0 ⩽ x ⩽ a, 0 ⩽ y ⩽ b, z = z2} tends to be unified. As a result, the temperature inside this region could be approximated by averaging the surface temperature increase at the encapsulation/substrate interface over the total surface area of electronic component as

(31)

m1 =

8 π2

(35)

The temperature distribution at the encapsulation/substrate interface outside the component region can be obtained by setting z = z2 in 508


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With Clothes

60

With Clothes

60

FEA

55

Without Clothes

Temperature (oC)

Temperature (oC)

FEA

50 45 40 35 -1000

-500

0

500

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film condition, the sink temperature is the blood temperature which is set to be core body temperature and the coefficient is ωbρbcb/n. The method above has been proved effective in several previous articles [35,36,38]. The temperature distributions at the interface of encapsulation and substrate with the heat generation power P = 5 mW from the threedimensional FEA and the analytical model are shown in Fig. 2 (a) and (b), which are the temperatures along x direction and y direction, respectively. The analytical results agree well with FEA, which validates the method to simplify the electronic component as a planar heat source. As shown in Fig. 2, the maximum temperature occurs inside the region of electronic component and keeps as a uniform value at around 59.6 °C just as the analysis above. Outside the component region, the temperature decays rapidly as away from the boundary of electronic component and then begins to flatten gradually. Besides, temperature

distribution without clothes effects is also shown by a dashed line in Fig. 2, which presents a similar trend and a lower temperature. Fig. 3(a) shows the characteristic temperatures, i.e., the temperature of electronic component and the maximum temperature at the device/skin interface versus the in-plane size of component. The length 2a of one side of the component is fixed to be 100 μm while the length 2b of the other side varies from 100 μm to 500 μm. With the increase of the component’s in-plane size, both characteristic temperatures drop. The component’s temperature decreases from 68.2 °C to 50.0 °C while the maximum temperature at the device/skin interface decreases from 43.9 °C to 42.8 °C. For a heat source, i.e., the electronic component, with fixed size of 100 μm×200 μm×6.5 μm, the influences of the substrate and encapsulation thicknesses on these two characteristic temperatures are presented in Fig. 3(b) and (c). Both characteristic temperatures show the same trend with the substrate and encapsulation

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Acknowledgement

thickness, i.e. a thicker substrate or encapsulation is beneficial for the heat dissipation of the system. The clothing effects are fully investigated in Figs. 4 and 5. Fig. 4(a) and (b) show the two characteristic temperatures versus the clothes thickness with various ambient temperatures, respectively, while Fig. 5(a) and (b) show the two characteristic temperatures versus the clothes porosity. The effects of thickness and porosity on both characteristic temperatures present a consistent trend. Specifically speaking, both characteristic temperatures rise as the clothes thickness or the porosity increases, and the lower the ambient temperature is, the more obvious the increase of the characteristic temperatures are. By their nature, what affects these two characteristic temperatures is the clothes thermal resistance, which is related to the thickness and efficient thermal conductivity of the clothes. On one hand, thermal resistance is proportional to the thickness. On the other hand, the porosity increases, resulting in a decrease in the efficient thermal conductivity, and hence thermal resistance increases as well.

The authors acknowledge the supports from the National Basic Research Program of China (Grant No. 2015CB351900), the National Natural Science Foundation of China (Grant Nos. 11502009, 11772030), the Science and Technology Foundation of China Aerospace Science and Industrial Corporation, and Opening fund of State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University (SV2018-KF-13). References [1] D.H. Kim, J. Viventi, J.J. Amsden, J. Xiao, L. Vigeland, Y.S. Kim, J.A. Blanco, B. Panilaitis, E.S. Frechette, D. Contreras, D.L. Kaplan, F.G. Omenetto, Y. Huang, K.C. Hwang, M.R. Zakin, B. Litt, J.A. Rogers, Dissolvable films of silk fibroin for ultrathin conformal bio-integrated electronics, Nat. Mater. 9 (2010) 511–517. [2] D.H. Kim, N. Lu, R. Ma, Y.S. Kim, R.H. Kim, S. Wang, J. Wu, S.M. Won, H. Tao, A. Islam, K.J. Yu, T. Kim, R. Chowdhury, M. Ying, L. Xu, M. Li, H.J. Chung, H. Keum, M. McCormick, P. Liu, Y.W. Zhang, F.G. Omenetto, Y. Huang, T. Coleman, J.A. Rogers, Epidermal electronics, Science 333 (2011) 838–843. [3] W.H. Yeo, Y.S. Kim, J. Lee, A. Ameen, L. Shi, M. Li, S. Wang, R. Ma, S.H. Jin, Z. Kang, Y. Huang, J.A. Rogers, Multifunctional epidermal electronics printed directly onto the skin, Adv. Mater. 25 (2013) 2773–2778. [4] Y. Chen, B. Lu, Y. Chen, X. Feng, Breathable and stretchable temperature sensors inspired by skin, Sci. Rep. 5 (2015) 11505. [5] Y. Chen, S. Lu, S. Zhang, Y. Li, Z. Qu, Y. Chen, B. Lu, X. Wang, X. Feng, Skin-like biosensor system via electrochemical channels for noninvasive blood glucose monitoring, Sci. Adv. 3 (2017) e1701629. [6] L. Tian, Y. Li, R.C. Webb, S. Krishnan, Z. Bian, J. Song, X. Ning, K. Crawford, J. Kurniawan, A. Bonifas, J. Ma, Y. Liu, X. Xie, J. Chen, Y. Liu, Z. Shi, T. Wu, R. Ning, D. Li, S. Sinha, D. Cahill, Y. Huang, J.A. Rogers, Flexible and stretchable 3 omega sensors for thermal characterization of human skin, Adv. Funct. Mater. 27 (2017) 1701282. [7] Y. Zhang, R.C. Webb, H. Luo, Y. Xue, J. Kurniawan, N. Cho, S. Krishnan, Y. Li, Y. Huang, J.A. Rogers, Theoretical and experimental studies of epidermal heat flux sensors for measurements of core body temperature, Adv. Healthc. Mater. 5 (2016) 119–127. [8] J. Song, X. Feng, Y. Huang, Mechanics and thermal management of stretchable inorganic electronics, Natl. Sci. Rev. 3 (2016) 128–143. [9] R.H. Kim, M.H. Bae, D.G. Kim, H. Cheng, B.H. Kim, D.H. Kim, M. Li, J. Wu, F. Du, H.S. Kim, S. Kim, D. Estrada, S.W. Hong, Y. Huang, E. Pop, J.A. Rogers, Stretchable, transparent graphene interconnects for arrays of microscale inorganic light emitting diodes on rubber substrates, Nano Lett. 11 (2011) 3881–3886. [10] R.H. Kim, H. Tao, T.I. Kim, Y. Zhang, S. Kim, B. Panilaitis, M. Yang, D.H. Kim, Y.H. Jung, B.H. Kim, Y. Li, Y. Huang, F.G. Omenetto, J.A. Rogers, Materials and designs for wirelessly powered implantable light-emitting systems, Small 8 (2012) 2812–2818. [11] N. Lu, D.H. Kim, Flexible and stretchable electronics paving the way for soft robotics, Soft Robot. 1 (2013) 53–62. [12] H. Cheng, S. Wang, Mechanics of interfacial delamination in epidermal electronics systems, J. Appl. Mech-T. ASME 81 (2014) 044501. [13] S. Wang, Y. Huang, J.A. Rogers, Mechanical designs for inorganic stretchable circuits in soft electronics, IEEE T. Comp. Pack. Man. 5 (2015) 1201–1218. [14] N. Lu, S. Yang, Mechanics for stretchable sensors, Curr. Opin. Solid St. M. 19 (2015) 149–159. [15] W. Dong, L. Xiao, C. Zhu, D. Ye, S. Wang, Y. Huang, Z. Yin, Theoretical and experimental study of 2D conformability of stretchable electronics laminated onto skin, Sci. China Technol. Sc. 60 (2017) 1415–1422.

4. Conclusions A three-dimensional analytical heat transfer model, validated by finite element analysis, is developed to predict the temperature distribution in the system of flexible wearable electronic device integrated with human skin considering the influence from the clothes thermal resistance and the air gap under clothes. This model accounts for the Pennes bio-heat transfer equation for the skin tissue and the Fourier heat conduction equation for device, air gap and clothes. The characteristic temperatures, i.e., the temperature of electronic component and the maximum temperature at the device/skin interface, are obtained analytically. The influence of the devices’ systematic parameters, the thickness, porosity of clothes and ambient temperature on characteristic temperatures are fully studied. It shows that a bigger electronic component, a thicker substrate or a thicker encapsulation are helpful to reduce the characteristic temperatures. While the thickness and porosity of clothes can also affect the characteristic temperatures, that is both characteristic temperatures decrease as you wear thinner or less fluffy clothes. The results presented in this paper can provide guidelines for the thermal management of flexible wearable electronic devices under some specific working conditions. Due to the simplification of heat convection and radiation heat transfer of air gap and mass transfer through clothes, the analytical model proposed in this article is just valid based on the premise of quiescent air and windtight clothes and a more thorough model remains to be developed for a wider applicability.

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