IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 7, NO. 9, SEPTEMBER 2017 1459
Hotspot Size Effect on Conductive Heat SpreadingHongtao Alex Guo, Kris F. Wiedenheft, and Chuan-Hua Chen
Abstract— Solid heat spreaders, particularly those made ofcopper or graphite, are often benchmark solutions for hotspotthermal management. In this paper, we present exact andapproximate analytical solutions of steady-state hotspot coolingwith a planar heat spreader, which is subjected to adiabaticconditions except for a hotspot centered at the top surface anda constant temperature at the bottom surface. The approximatesolution bridges exact solutions at two limits of hotspot size:infinitesimal hotspot at the center and uniform heat flux acrossthe spreader. The approximate solution accounts for variablehotspot size and anisotropic thermal conductivity in a compactform, which is useful for estimating thermal parameters such asconduction shape factor and effective thermal conductivity.
SOLID heat spreaders are frequently used for hotspot ther-mal management in electronic packaging [1]–[3]. These
solid spreaders are made of a variety of materials includingcopper and graphite [1], with either isotropic or anisotropicthermal conductivity [4]–[6]. Due to their simplicity, solidspreaders are also useful models for more complex systemssuch as vapor chambers, for which effective thermal conduc-tivities are frequently reported [7]–[9]. Analytical solutionsfor conductive heat spreading have been reported for manygeometrical and thermal configurations, usually in the form ofFourier series [3], [10]. To simplify calculations, approximatesolutions have been proposed, typically in the form of apolynomial (see [11]). Although the series solutions and thepolynomial approximations are very accurate, their mathemat-ical complexity often obscures the physics.
In this paper, we present an approximate solution for steady-state hotspot cooling on a solid heat spreader. Unlike priorwork, our approximation is basically a composite solutionjoining two limits in terms of hotspot size: infinitesimalhotspot at the center and uniform heat flux across the spreader.The approximate solution bridges the exact solutions at thesetwo limits to account for the hotspot size effect for both
Manuscript received October 24, 2016; accepted May 4, 2017. Date ofpublication May 31, 2017; date of current version August 31, 2017. Thiswork was supported in part by the National Science Foundation under GrantCBET-12-36373 and in part by the Intel Corporation. The work ofK. F. Wiedenheft was supported in part by the NSF Gradate ResearchFellowship under Grant DGF-11-06401 and in part by the NSF ResearchTriangle MRSEC under Grant DMR-11-21107. Recommended for publicationby Associate Editor G. Refai-Ahmad upon evaluation of reviewers’ comments.(Corresponding author: Chuan-Hua Chen.)
The authors are with the Department of Mechanical Engineering andMaterials Science, Duke University, Durham, NC 27708 USA (e-mail:[email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCPMT.2017.2704419
Fig. 1. Problem setup for hotspot cooling on a solid heat spreader. Circulardisk of height H and radius L is subjected to a heat flux of q0 localized ona circular area of radius a on the top, and constant temperature T0 at thebottom. Rest of the heat spreader is adiabatic.
isotropic and orthotropic media. With straightforward physicalinterpretations, our approximation can be used to deducethermal design parameters such as conduction shape factorand effective thermal conductivity.
II. PROBLEM SETUP
The solid heat spreader in Fig. 1 is a circular disk witha height H and a radius L. A hotspot with a radius aand a constant heat flux q0 is located at the center of thetop surface. The bottom surface is isothermal at T0. Therest of the heat spreader surface is adiabatic. The originof the cylindrical coordinate system is located at the centerof the bottom surface. We will discuss the axisymmetricsolutions for both isotropic and orthotropic spreaders, andderive exact and approximate solutions for the temperaturerise �T = T (r, z) − T0. We will mainly study the maximumtemperature rise at the center of the top surface
�T̂ ≡ T (0, H ) − T0 (1)
which is arguably the most important indicator of the coolingperformance. Toward the end, we will extend the conclusionsfrom the maximum to average temperature and from cylindri-cal to Cartesian system.
III. COMPOSITE SOLUTION
To understand the effect of the hotspot size, it is helpfulto first examine two limiting cases. In the limit of uniformheat flux across the spreader with a = L, the largest possiblehotspot radius, the temperature rise on the top surface isuniform and given by Fourier’s law
�T |a=L = q0 H
k. (2)
In the limit of an infinitesimal hotspot with a → 0, the hotspotradius is the only relevant length scale in an essentially semi-infinite medium, so the temperature rise scales with a instead
�T |a→0 ∼ q0a
k= a
H
q0 H
k(3)
where ∼ denotes an order-of-magnitude scaling relation. For ahigh-aspect-ratio heat spreader with L/H � 1, the following
1460 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 7, NO. 9, SEPTEMBER 2017
composite solution reduces to (2) and (3) in the respectivelimits of hotspot radius:
�T (a) ∼ (1 − e− a
H)q0 H
k= �
( a
H
) q0 H
k(4)
where � is a geometrical parameter for heat spreading
�(ξ) ≡ 1 − e−ξ . (5)
The heat spreader size L does not enter the approximatesolution explicitly, except for the high-aspect-ratio restrictionwhich ensures the recovery of the uniform heat flux limitas a → L.
More generally, the composite solution can be constructedas follows:
�T (a) ≈ �(ξ)�T |a=L, ξ = �T |a→0
�T |a=L(6)
where the argument ξ is given by the ratio of the twolimits. The function �(ξ) represents the hotspot size effect byabsorbing the geometrical correction to q0 H/k, the Fourier’slaw solution in the case of uniform heat flux across thespreader. This composite solution turns out to be reasonablyaccurate for any hotspot size, as long as the heat spreader has ahigh aspect ratio as specified below. The geometrical function�(ξ) is readily linked to the conduction shape factor [12]
S ≡ πa2q0
k�T≈ πa2
H�(ξ). (7)
IV. ISOTROPIC MEDIUM
For an isotropic medium with a constant thermal conductiv-ity k, the steady-state axisymmetric distribution of temperatureT (r, z) is governed by the heat conduction equation
∂2T
∂r2 + 1
r
∂T
∂r+ ∂2T
∂z2 = 0 (8)
subjected to the following boundary conditions:T |z=0 = T0
∂T
∂z
∣∣∣∣z=H
={
q0/k, 0 ≤ r ≤ a
0, a < r ≤ L
∂T
∂r
∣∣∣∣r=L
= 0. (9)
A. Exact Solution
Using separation of variables, (8) is solved as
T (r, z) = T0 + q0 H
k
a2
L2
z
H+ q0a
k
∞∑
n=1
An J0(λnr)sinh(λnz)
cosh(λn H )
(10)
where J0 and J1 are the Bessel functions of the first kind oforder 0 and 1, respectively, and
An = 2J1(λna)
λ2n L2 J 2
0 (λn L), λn = z1n
L
Fig. 2. Hotspot size effect on a cylindrical heat spreader with an aspect ratioL/H = 10. Approximate solution (13) and exact solution (11) of the maxi-mum temperature rise �T̂ overlap with each other in the limits of a/L → 0and a/L → 1.
where z1n represents the nth positive root of J1. Accordingto (10), the maximum temperature rise at the hotspot center is
�T̂ = q0 H
k
a2
L2 + q0a
k
∞∑
n=1
An tanh(λn H ). (11)
The exact solution (11) is plotted in Fig. 2. For all exactsolutions plotted in this paper, the truncation error is keptbelow 0.1%. The large and infinitesimal hotspot limits areapparent, signified by �T ∼ a0 and �T ∼ a1, respectively.In fact, (11) is arranged such that the first and second termscorrespond to the limits represented by (2) and (3), respec-tively. However, the series solution in the second term is toocomplex for the hotspot size effect to be apparent.
B. Approximate Solution
A more convenient form of the infinitesimal limit is obtainedby approximating the heat spreader as a semi-infinite medium,as discussed in [13, Ch. 8.2.III]
�T̂ |a→0 → q0a
k
∫ ∞
0
J1(λa)
λdλ = q0a
k= a
H
q0 H
k(12)
where λ is the continuous eigenvalue in the semi-infinitedomain. The scaling relation in (3) turns out to be exactfor the maximum temperature rise in a cylindrical system.Following (6), the approximate solution for the maximumtemperature rise is constructed as
�T̂ ≈ (1 − e− a
H)q0 H
k= �
( a
H
) q0 H
k. (13)
This approximation is much more compact compared to theirexact counterparts in (11), but it is reasonably accurate as longas the aspect ratio satisfies L/H ≥ 3.
In Fig. 2, the approximate solution is plotted againstthe exact solution for a given aspect ratio of L/H = 10. Theapproximate solution nearly overlaps with the exact one at thetwo limits of hotspot size. A small error is introduced that
GUO et al.: HOTSPOT SIZE EFFECT ON CONDUCTIVE HEAT SPREADING 1461
Fig. 3. (a) Hotspot size effect on heat spreaders with different aspectratios (L/H ). The exact and approximate solutions follow (11) and (13),respectively. (b) Relative error of the approximation ε peaks around a/H = 1.The error is within 7% for high-aspect-ratio heat spreaders with L/H ≥ 3.
peaks at intermediate hotspot sizes. The approximation errorε is assessed by
ε = |�Texact − �Tapprox|�Texact
. (14)
The approximate solution (13) slightly underestimates theexact �T̂ , but the error is less than 7% as long as the aspectratio L/H ≥ 3, which is the case for most heat spreaders.
For different aspect ratios, the exact and approximate solu-tions for �T̂ (a) are plotted in Fig. 3(a), and their differenceis plotted in Fig. 3(b). The maximum error occurs arounda = H , an intermediate hotspot size between the two limits.If the aspect ratio is too small (L/H < 3), the approximatesolution (13) is still accurate at the infinitesimal limit, but theerror introduced at the large-hotspot limit may exceed 7%.
V. ORTHOTROPIC MEDIUM
For an orthotropic medium in the cylindrical coordinatesystem, the axisymmetric governing equation becomes
kr
(∂2T
∂r2 + 1
r
∂T
∂r
)+ kz
∂2T
∂z2 = 0 (15)
where kr and kz are the radial and axial thermal conductivities,respectively. The boundary condition is the same as (9), except
Fig. 4. Hotspot size effect on heat spreaders with different thermalconductivity ratios (kr /kz ) at a given aspect ratio L/H = 10. The exactand approximate solutions follow (18) and (21), respectively.
for the consideration of anisotropy in the imposed heat fluxwith
∂T
∂z
∣∣∣∣z=H
={
q0/kz, 0 ≤ r ≤ a
0, a < r ≤ L .(16)
A. Exact Solution
Using separation of variables again, (15) is solved as
T (r, z) = T0 + q0 H
kz
a2
L2
z
H
+ q0a√kzkr
∞∑
n=1
An J0(λnr)sinh(
√kr/kzλnz)
cosh(√
kr/kzλn H )(17)
giving rise to
�T̂ = q0 H
kz
a2
L2 + q0a√kzkr
∞∑
n=1
An tanh(√
kr/kzλn H ). (18)
The solution for orthotropic medium can also be obtained fromthe isotropic solution (10) with an isotropic conductivity
k̃ = √kr kz (19)
using the following transformations [10]:
r̃ =(
kz
kr
) 14
r z̃ =(
kr
kz
) 14
z q̃0 =(
kr
kz
) 14
q0. (20)
B. Approximate Solution
Applying the above transformation to (13), the approximatesolution to the orthotropic heat spreader becomes
�T̂ ≈ (1 − e
−√
kzkr
aH
)q0 H
kz(21)
which approximates the exact solution (18) with an errorof less than 7%, as long as the modified aspect ratio(kz/kr )
1/2 L/H ≥ 3. The exact and approximate solutions tothe orthotropic problem are plotted in Fig. 4. When Fig. 4
1462 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 7, NO. 9, SEPTEMBER 2017
Fig. 5. All the curves in Figs. 3(a) and 4 with various aspect ratios (L/H )and conductivity ratios (kr /kz ) collapse onto a single curve according to (21).
is compared to Fig. 3(a), it is apparent that the anisotropicconductivity is effectively modifying the aspect ratio of theheat spreader.
In accordance with (21), an effective hotspot size(kz/kr )
1/4a normalized by an effective spreader height(kr/kz)
1/4 H can be used to collapse all the curvesin Figs. 3(a) and 4 onto a single curve in Fig. 5. We emphasizethat the master curve according to (21) is far from apparentby direct examination of the Fourier–Bessel series solutionin (18).
C. Effective Isotropic Conductivity
As an example application of the approximate solution,we use it to assess the effective isotropic conductivity of anorthotropic heat spreader. The goal is to find an isotropicheat spreader with keff that spreads the same amount ofheat (q0πa2) with the same maximum temperature rise (�T̂ ).Equating the approximations (13) and (21), the effectiveisotropic conductivity is approximately
keff ≈ 1 − e− aH
1 − e−√
kzkr
aH
kz. (22)
The exact value of keff can be similarly obtained by equatingthe exact solutions (11) and (18).
Both the exact and approximate solutions of the effectivethermal conductivity are plotted in Fig. 6. The approximatesolution closely follows the exact solution, validating thecompact approximation in (22). The good match holds aslong as the high-aspect-ratio requirement for approximations(13) and (21) are both satisfied. The limit of uniform heat flux(a = L) is represented by a horizontal line independent ofthe radial conductivity, since the heat conduction in this limitis only in the axial (z) direction. The limit of infinitesimalhotspot (a → 0) approaches a diagonal line with keff =k̃ = (kr kz)
1/2, which can be obtained as a limit of (22). Theslight difference from the 1/2 power law results from the finitehotspot size.
Fig. 6. Effective isotropic thermal conductivity (keff ) of an orthotropic heatspreader with an aspect ratio L/H = 10. The approximation solution (dottedlines) follows (22), and the exact solution (solid lines) is obtained byequating (11) and (18).
VI. AVERAGE TEMPERATURE
The same idea for constructing the approximate solution canbe applied to the average temperature rise over the hotspot area
�T̄ ≡∫ a
0 T (r, H )2πrdr
πa2 − T0. (23)
According to (10), the average temperature rise for an isotropicheat spreader is
�T̄ = q0 H
k
a2
L2 + q0a
k
∞∑
n=1
An tanh(λn H )2J1(λna)
λna. (24)
The infinitesimal limit can again be obtained from the semi-infinite solution [13]
�T̄ |a→0 → q0a
k
∫ ∞
0
2J 21 (λa)
aλ2 dλ = 8
3π
q0a
k= 8a
3π H
q0 H
k.
(25)
Following (6), the average temperature rise is approximately
�T̄ ≈ (1 − e− 8
3πaH
)q0 H
k= �
(8a
3π H
)q0 H
k. (26)
The approximate and exact solutions for the average tem-perature are plotted in Fig. 7. When L/H ≥ 3, the maximumapproximation error for the average temperature �T̄ is 12%,which is larger than the 7% error bound for the maximumtemperature �T̂ . Unlike the exact solution for the maximumtemperature that quickly approaches q0 H/k in the large-hotspot limit (a → L), the average temperature is appreciablysmaller than q0 H/k until the hotspot radius is exactly equalto the spreader radius (a = L).
VII. CARTESIAN SYSTEM
The approximate solution can be easily extended to theCartesian coordinate system, where a square heater is sub-jected to the same boundary condition as in Fig. 1, except forthe following change in geometry: The square heat spreaderhas a height H and an area of 2L ×2L, and the square hotspot
GUO et al.: HOTSPOT SIZE EFFECT ON CONDUCTIVE HEAT SPREADING 1463
Fig. 7. Average temperature rise T̄ of a circular hotspot on a cylindricalheat spreader with an aspect ratio L/H = 10. The approximate solution (26)closely follows the exact solution (24).
Fig. 8. Hotspot size effect on a square heat spreader with an aspect ratioL/H = 10. The horizontal axis is the half-width a of a square hotspotnormalized by the half-width L of a square spreader. The approximatesolution (28) for the maximum temperature rise �T̂ closely follows the exactsolution (31). The approximation solution (29) for the average temperature�T̄ closely follows the exact solution (32).
at the top center has an area of 2a × 2a. For an isotropicmedium, the exact solution is detailed in the Appendix. Theapproximate solutions above, including (13) and (26), stillapply with an equivalent circular radius [6]
ae = 2√π
a (27)
which converts the square hotspot with a half-width a to acircular one of the same area. Accordingly, the approximatesolutions for the maximum and average temperature rises takethe form of
�T̂ ≈ q0 H
k
(1 − e− ae
H) = q0 H
k
(1 − e
− 2√π
aH
)(28)
�T̄ ≈ q0 H
k
(1 − e− 8
3πaeH
) = q0 H
k
(1 − e
− 163π
√π
aH). (29)
For an orthotropic heat spreader that is transverselyisotropic (kx = ky = kxy �= kz), the approximate solution (21)
is still applicable with kz/kxy as the conductivity ratio and ae
as the equivalent radius.In Fig. 8, the approximate solutions using the equivalent
radius agree very well with the exact solutions. For high-aspect-ratio heat spreaders with L/H ≥ 3, the error ofapproximation is within 7% for the maximum temperatureand within 14% for the average temperature. At the sameaspect ratio L/H and hotspot size a/L, the temperature rise isslightly higher for the square heater in Fig. 8 compared to thecircular heater in Figs. 2 and 7, mainly because of the largerheat load (4a2q0 instead of πa2q0).
VIII. CONCLUSION
For a solid heat spreader subjected to a localized heat flux onone side and a constant temperature on the other side (Fig. 1),we have developed an approximate solution in a compactform (6) that accounts for the variable hotspot size for bothisotropic and orthotropic media. The approximate solution isa composite solution that bridges the exact solutions at twolimits: infinitesimal hotspot at the center (3) and uniform heatflux across the heat spreader (2). The simple approximationis particularly useful for quick estimation of parameters inthermal design and analysis, e.g., the heat spreading parameter� that accounts for the hotspot size effect and the effectiveisotropic conductivity keff that characterizes an orthotropicheat spreader. These approximate parameters are used in [14]for the interpretation of hotspot cooling experiments.
APPENDIX
EXACT SOLUTION IN CARTESIAN SYSTEM
For the Cartesian heat spreading problem in Section VII,the origin of the coordinate system is located at the bottomcenter of the heat spreader (similar to Fig. 1), and the exactsolution takes the form of
T (x, y, z) = T0 + q0 H
k
a2
L2
z
H+ q0a
k
∞∑
m=0
∞∑
n=0m+n �=0
Cmn
× cos(mπx
L
)cos
(nπy
L
) sinh(γmnz)
cosh(γmn H )(30)
where
Cmn =
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
2
nπγmn Lsin
(nπa
L
), m = 0, n ≥ 1
2
mπγmn Lsin
(mπa
L
), m ≥ 1, n = 0
4
mnπ2γmnasin
(mπa
L
)sin
(nπa
L
), m ≥ 1, n ≥ 1
and
γmn = π√
m2 + n2
L.
1464 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 7, NO. 9, SEPTEMBER 2017
Accordingly, the maximum and average temperature rises are
�T̂ = q0 H
k
a2
L2 + q0a
k
∞∑
m=0
∞∑
n=0m+n �=0
Cmn tanh(γmn H ) (31)
�T̄ = q0 H
k
a2
L2 + q0a
k
×[ ∞∑
m=1
C2m0tanh(γm0 H )
γm0 L2
2a
+∞∑
n=1
C20n tanh(γ0n H )
γ0n L2
2a
+∞∑
m=1
∞∑
n=1
C2mn tanh(γmn H )
γmn L2
4a
]
. (32)
The infinitesimal limit follows the semi-infinite solution in[3, Sec. 4.4.1]
�T̂ |a→0 → 4 sinh−1(1)
π
q0a
k= 0.9945
q0ae
k(33)
�T̄ |a→0 → 4 sinh−1(1) − (45/4 − 4)/3
π
q0a
k= 0.9881
8q0ae
3πk(34)
where a is the half-width of the square hotspot and ae is theeffective radius defined in (27).
REFERENCES
[1] X. C. Tong, Advanced Materials for Thermal Management of ElectronicPackaging. New York, NY, USA: Springer, 2011, ch. 9.
[2] R. Mahajan, C.-P. Chiu, and G. Chrysler, “Cooling a microprocessorchip,” Proc. IEEE, vol. 94, no. 8, pp. 1476–1486, Aug. 2006.
[3] M. M. Yovanovich and E. E. Marotta, “Thermal spreading and contactresistances,” in Heat Transfer Handbook, A. Bejan and A. D. Kraus, Eds.New York, NY, USA: Wiley, 2003, pp. 261–393.
[4] D. P. Kennedy, “Spreading resistance in cylindrical semiconductordevices,” J. Appl. Phys., vol. 31, no. 8, pp. 1490–1497, 1960.
[5] P. Hui and H. S. Tan, “Temperature distributions in a heat dissipationsystem using a cylindrical diamond heat spreader on a copper heat sink,”Jpn. J. Appl. Phys., vol. 75, no. 2, pp. 748–757, 1994.
[6] Y. S. Muzychka, M. M. Yovanovich, and J. R. Culham, “Influ-ence of geometry and edge cooling on thermal spreading resistance,”J. Thermophys. Heat Transf., vol. 20, pp. 247–255, Apr. 2006.
[7] I. Sauciuc, G. Chrysler, R. Mahajan, and R. Prasher, “Spreading in theheat sink base: Phase change systems or solid metals??” IEEE Trans.Compon. Packag. Technol., vol. 25, no. 4, pp. 621–628, Dec. 2002.
[8] Y.-S. Chen, K.-H. Chien, C.-C. Wang, T.-C. Hung, Y.-M. Ferng, andB.-S. Pei, “Investigations of the thermal spreading effects of rectangularconduction plates and vapor chamber,” J. Electron. Packag., vol. 129,pp. 348–355, Sep. 2007.
[9] H. Shabgard, M. J. Allen, N. Sharifi, S. P. Benn, A. Faghri, andT. L. Bergman, “Heat pipe heat exchangers and heat sinks: Opportuni-ties, challenges, applications, analysis, and state of the art,” Int. J. HeatMass Transf., vol. 89, pp. 138–158, Oct. 2015.
[10] D. W. Hahn and M. N. Ozisik, Heat Conduction, 3rd ed. New York,NY, USA: Wiley, 2012, ch. 15.
[11] M. M. Yovanovich, J. R. Culham, and P. Teertstra, “Analytical modelingof spreading resistance in flux tubes, half spaces, and compound disks,”IEEE Trans. Compon., Packag., Manuf. Technol., A, vol. 21, no. 1,pp. 168–176, Mar. 1998.
[12] T. L. Bergman, F. P. Incropera, D. P. DeWitt, and A. S. Lavine,Fundamentals of Heat and Mass Transfer, 7th ed. New York, NY, USA:Wiley, 2011, ch. 4.
[13] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed.Oxford, U.K.: Clarendon, 1959, ch. 8.
[14] K. F. Wiedenheft et al., “Hotspot cooling with jumping-drop vaporchambers,” Appl. Phys. Lett., vol. 110, no. 14, p. 141601, 2017.
Hongtao Alex Guo received the B.E. degreein thermal energy and power engineering fromBeihang University, Beijing, China, in 2015, andthe M.Eng. degree in mechanical engineering fromDuke University, Durham, NC, USA, in 2016, wherehe is currently pursuing the Ph.D. degree with theDepartment of Mechanical Engineering and Materi-als Science.
He is currently with the Microscale Physicochem-ical Hydrodynamics Laboratory, Duke University.His current research interests include phase-change
heat transfer, interfacial electrohydrodynamics, and impulsive biomechanics.
Kris F. Wiedenheft received the B.S. degree inmechanical engineering from North Carolina Agri-cultural and Technical State University (NC A&T),Greensboro, NC, USA, in 2015. He is currentlypursuing the Ph.D. degree with the Departmentof Mechanical Engineering and Materials Science,Duke University, Durham, NC, USA.
He was a Mechanical Technician with the Pre-stone Products Research and Development Labora-tory, Danbury, CT, USA, from 2007 to 2011. Hewas an Engineering Intern with BASF, Wyandotte,
MI, USA, in 2013 and also with the Jet Propulsion Laboratory, Pasadena,CA, USA, in 2014. His current research interests include superhydrophobicstructures and phase-change heat transfer.
Mr. Wiedenheft is a member of the National Society of Professional Engi-neers. He was a recipient of the Namaskar Award for Engineering Excellencefrom NC A&T, the Dean’s Graduate Fellowship from Duke University, theB. M. Goldwater Scholarship, and the NSF Graduate Research Fellowship.
Chuan-Hua Chen received the B.S. degree inapplied mechanics from Peking University, Beijing,China, in 1998, and the Ph.D. degree in mechanicalengineering from Stanford University, Stanford,CA, USA, in 2004.
He was a Post-Doctoral Associate with PrincetonUniversity, Princeton, NJ, USA, and a ResearchScientist with Rockwell Scientific Company,Thousand Oaks, CA, USA. Since 2007, he has beenan Assistant Professor and a Hunt Faculty Scholarof Mechanical Engineering and Materials Science
with Duke University, Durham, NC, USA, where he directs the MicroscalePhysicochemical Hydrodynamics Laboratory. In 2014, he was promoted toAssociate Professor.
Dr. Chen was a recipient of the NSF CAREER Award and the DARPAYoung Faculty Award for his research integrating physicochemicalhydrodynamics and interfacial engineering.
