Optimizing the design of composite phase change materials for high thermal powerdensityMichael T. Barako, Srilakshmi Lingamneni, Joseph S. Katz, Tanya Liu, Kenneth E. Goodson, and Jesse Tice

Citation: Journal of Applied Physics 124, 145103 (2018); doi: 10.1063/1.5031914View online: https://doi.org/10.1063/1.5031914View Table of Contents: http://aip.scitation.org/toc/jap/124/14Published by the American Institute of Physics

Optimizing the design of composite phase change materials for high thermalpower density

Michael T. Barako,1,2,a) Srilakshmi Lingamneni,2 Joseph S. Katz,3 Tanya Liu,2

Kenneth E. Goodson,2 and Jesse Tice11NG Next Basic Research Laboratory, Northrop Grumman Corporation, Redondo Beach, California 90278, USA2Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA3Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

(Received 2 April 2018; accepted 23 September 2018; published online 10 October 2018)

Phase change materials (PCMs) provide a high energy density for thermal storage systems but oftensuffer from limited power densities due to the low PCM thermal conductivity. Much like their elec-trochemical analogs, an ideal thermal energy storage medium combines the energy density of athermal battery with the power density of a thermal capacitor. Here, we define the design rules andidentify the performance limits for rationally-designed composites that combine an energy densePCM with a thermally conductive material. Beginning with the Stefan-Neumann model, we estab-lish the material design space using a Ragone framework and identify regimes where hybrid conduc-tive-capacitive composites have thermal power densities exceeding that of copper and other highconductivity materials. We invoke the mathematical bounds on isotropic conductivity to optimizeand define the theoretical limits for transient cooling using PCM composites. We then demonstratethe impact of power density on thermal transients using copper inverse opals infiltrated with paraffinwax to suppress the temperature rise in kW cm−2 hotspots by ∼10% compared to equivalent copperthin film heat spreaders. These design rules and performance limits illuminate a path toward therational design of composite phase change materials capable of buffering extreme transient thermalloads. Published by AIP Publishing. https://doi.org/10.1063/1.5031914

I. INTRODUCTION

Thermal energy storage systems employ the high energydensity of phase change materials (PCMs) to buffer transientheating events and provide nearly isothermal energystorage.1–11 However, many organic (e.g., hydrocarbons) andinorganic (e.g., salts) PCMs have a limited achievable powerdensity due to their low thermal conductivity. This limitationthrottles the rate of heat absorption and release, minimizesthe thermally-accessible PCM volume, and increases thesuperheat temperature at the PCM interface. The result is thatmany PCMs fail to accommodate the increasingly aggressiverequirements for the rate of heat delivery and extraction insystems with intense heat fluxes and/or extreme thermal tran-sients, such as pulsed electronics on aerospace platforms.12

Much like electrochemical energy storage, an ideal thermalenergy storage system combines a large capacity with a fastcharge/discharge rate to maximize heat delivery and extrac-tion from the PCM.11,13–16 This unique combination offerscritical advantages to technologies in both thermal manage-ment, where the addition of thermal capacitance enablesarchitectures designed for time-averaged loads instead ofpeak loads, and thermal batteries, where the addition ofthermal conductance increases the rate of energy delivery andextraction.

While new PCMs have been developed with enhancedenergy density,17–19 most efforts to improve thermal power

density focus on increasing the composite thermal conductiv-ity using combinations of thermally conductive materials andtraditional PCMs. Many transient heat sink designs are builtaround organic PCMs such as paraffin wax due to their highlatent heats and chemically-tunable melting temperatures.8,20

Hydrocarbons, salts, and other thermally insulating PCMsare converted into composites by either dispersing nanoparti-cles (e.g., carbon nanotubes) into the PCM or infiltratingporous conductors (e.g., carbon foams) with thePCM.4,7,21–28 However, most of these composites do nothave deliberately designed morphologies, and instead theyrely on forming percolation networks or having continuousbut disordered foam-like morphologies to conduct and dis-tribute heat. This causes most composite thermal conductivi-ties to remain below ∼3Wm−1 K−1 for nanoparticledispersions and ∼10Wm−1 K−1 for PCM-filled foams due tolimited control over morphology and a limited range ofaccessible volume fractions. In such cases, the increase inthermal capacitance from the PCM fails to compensate forthe reduction in thermal conductivity compared to conven-tional materials used in steady state thermal management(e.g., copper, graphite, heat pipes). Some of the most promis-ing composites use high density graphitic foams,7,27,29 butdeterministic material architectures are required to maximizethe performance. By identifying and optimizing the compos-ite landscape, it is possible to optimize the tradeoff betweenthermal conductivity and thermal capacity in PCM compos-ites.16 All transient thermal architectures seek to minimizethe effective thermal impedance of the system, whichincludes both capacitive and conductive components. The

a)Author to whom correspondence should be addressed:mbarako@NorthropGrummanNext.net

JOURNAL OF APPLIED PHYSICS 124, 145103 (2018)

0021-8979/2018/124(14)/145103/13/$30.00 124, 145103-1 Published by AIP Publishing.

incorporation of a PCM-based thermal component ratherthan a single-phase component is only effective if it increasesthe cooling capacity (the principal figure-of-merit11,13) rela-tive to the transient performance of the component that it isreplacing, such as a heat pipe or graphitic heat spreader. Thisnecessitates the establishment of a material design space andmaterial design rules that consider both the morphology andcomposition of the composite to quantify the efficacy of bothexisting and novel PCMs.

These rules represent an important shift away from disor-dered composites and toward materials with deliberate andoptimized designs. By combining a thermally conductivematerial with an energy dense PCM in a rational materialarchitecture, composites can be designed to achieve higherpower densities and transfer higher intensity heating loadsthan either of their individual constituent components.16 Ourwork aims to move beyond the mixed component approachto PCM composites and toward the deterministic design of het-erogeneous material architectures30 having tunable, predictable,and desirable properties. We begin by combining effectivemedium theory with the Stefan-Neumann formalism to definea framework for the design and optimization of compositePCMs, and we identify the design domains where thermalpower densities exist beyond those of high conductivitysingle-phase materials. We use our “materials-by-design”30,31

principles to establish the mathematical performance limitsfor uniform thermal energy storage composites, which rep-resent a critical normalization metric for assessing novelPCMs. By re-imagining the material design space, weprovide a quantitative connection between material designand thermal dynamics required to make informed materialdecisions and enable a more rational design of transientthermal systems.

II. MATERIAL DESIGN SPACE

The Stefan problem32,33 is commonly used to modelsolid-liquid phase change processes and it originated in the19th century to describe freezing phenomena in sea ice. Ituses a modified linear heat diffusion equation to model thenonlinearity of a system undergoing phase change. In thisframework, the system is partitioned into liquid (T > Tm) andsolid (T < Tm) single-phase linear domains separated by aboundary (T = Tm) that moves at a rate governed by anenergy balance at the interface, where Tm is the meltingtemperature. The framework, applications, and solutions tothe Stefan problem under different configurations have

been extensively reviewed,34–38 but only a few analyticalsolutions and semi-analytical approximations exist for one-dimensional geometries with simple boundary conditionsas shown in Table I. These solutions provide a basis forgenerating design rules and correlating material propertiesto performance.

Lu,15 Shamberger,13 and Shao et al.11 each provide aset of PCM design rules derived from the solutions to theone-dimensional Stefan-Neumann problem in Cartesiancoordinates. These design rules correlate the PCM proper-ties to the thermal response of the system with the objectiveof either minimizing the temperature rise under a constantheat flux boundary condition or maximizing the heatabsorption rate under a constant temperature boundary con-dition. Both scenarios are functionally equivalent in thatthey minimize the apparent thermal impedance of thesystem, which is challenging to precisely define due to thenonlinearity of the phase change process. However, thematerials selection and design criteria are generally insensi-tive to the type of boundary condition13 and provide auseful framework for a first-order material design optimiza-tion to minimize thermal impedance.

The general strategy to develop these design rules is tosolve the governing heat diffusion equation(s) for heatabsorbed (or, equivalently, the temperature rise) for a givensystem at the location of interest and collect the materialdesign parameters into a single figure-of-merit ηeff as definedin Sec. II A. In Secs. II B and II C, we describe the thermalproperties of composites as a function of the volume fractionand morphological arrangement of the components usingeffective medium theory. We then identify the materialdesign space and design boundaries in Sec. II D based onthese composite properties using a Ragone framework16

(see Fig. 2), which is commonly used to show the landscapeof electrochemical energy storage for the coupled optimiza-tion of energy density and power density to create highcapacity storage systems with fast charge/discharge rates. InSec. II E, we integrate the structure-property relations (effec-tive medium theory) into the property-performance relation(the figure-of-merit) to define an objective function that weanalytically optimize. We then generalize this optimizationmethod in Sec. II F to broaden the applicability of ouranalysis to arbitrary system configurations.

A. Thermal energy storage figure-of-merit

In a semi-infinite medium with a planar heat source, thefigure-of-merit ηeff is approximately proportional to the heatabsorption capability of the medium, or the “cooling capac-ity,”13 and is given by an effusivity-like combination of theeffective thermal conductivity keff and the effective volumet-ric thermal energy density Eeff:

ηeff (ΔT) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikeffEeff (ΔT)

p, (1)

where the subscript eff refers to the effective properties ofthe medium. In this one-dimensional planar geometry, ηeffgives equal weight to the conduction and storage terms and

TABLE I. Representative analytical and approximate solutions to theone-dimensional Stefan-Neumann problem.

Heatsource

Boundarycondition

Representativesolutions

Solutiontype

Plane Temperature 13, 14, and 33 AnalyticalHeat flux 15, 33, and 39 Approximate

Line Temperature 14 and 33 ApproximateHeat flux 33 and 40 Analytical

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provides a first-order approximation for more complexgeometries. However, in other configurations (e.g., cylin-drical coordinates), the Stefan-Neumann solution may giveasymmetric weight to the conductive and capacitive terms(see Sec. II F). Under both constant heat flux and constanttemperature boundary conditions, Eq. (1) becomes ourobjective function for material optimization within thedesign space.

Composite materials provide an avenue to decouplingthe terms in Eq. (1), where a conductive component isdesigned to achieve the highest possible keff for the givenvolume fraction (see Sec. II C), while the PCM is designedto provide the highest possible latent heat (the primary com-ponent of Eeff, see Sec. II B) at an appropriate melting tem-perature Tm. In the context of thermal energy storage, thisdefines the relationship between the ability to store heat (i.e.,heat capacity and latent heat, when applicable) to the abilityto transfer heat (i.e., thermal conductivity). Low conductivityPCMs can be combined with high conductivity single phase

materials to access performance regimes that are inaccessibleusing either material alone (see Sec. II D).

The intrinsic material properties of interest are the thermalconductivity k, the volumetric sensible heat capacity CV, andthe latent heat H (or, equivalently, the volumetric latent heatHV= ρH, where ρ is the mass density). In this work, we demon-strate these composite design principles using two commonlyused materials in electronics thermal management: paraffinwax (Sigma Aldrich, Tm= 53–57 °C, kPC= 0.25Wm−1 K−1,CV,PC= 2.25 × 106 J m−3 K−1, HV,PC= 1.9 × 108 J m−3), whichis a PCM with a tunable melting point of ∼55 °C, and copper(kC = 400Wm−1 K−1, CV,C = 3.4 × 106 J m−3 K−1). The sub-scripts C and PC refer to the conductive material and thephase change material, respectively, and the subscript Vdenotes volume-specific quantities. We assume that all prop-erties are both temperature-independent and the same forthe solid and liquid phases in the PCM. For this particularcombination of materials, the two composite properties ofinterest (keff and Eeff ) are strongly decoupled such that

FIG. 1. Example binary phase diagrams for different composite morphologies and component properties. (a) Phase diagram for thermal conductivity andenergy density scaling with volume fraction and (b) figure-of-merit for PCM composites having different morphologies for a thermal conductivity ratioκ = 1000 (e.g., copper and paraffin wax). (c) Phase diagram and (d) figure-of-merit for a thermal conductivity ratio κ = 4 (e.g., copper and indium). The one-dimensional upper bound is given by Eq. (4) and the three-dimensional upper bound is given by Eq. (5). The figure-of-merit is normalized to the figure-of-meritfor the pure conductor (ηC ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kCCV,CΔTp

, where ΔT = 1 K). Inverse opals become discontinuous for Φ < 0.06, and the original data can be found fromRef. 42. All plots are shown with the ratio HV,PC/(CV,CΔT) =HV,PC/(CV,PCΔT) = 100 (i.e., the energy stored by the sensible heat capacity per degree is ∼1% ofthe energy stored by the latent heat).

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nearly all conduction is facilitated by the copper (kC≫ kPC),whereas nearly all of the energy stored is in the paraffinwax (HV,PC≫ CV,effΔT) for small temperature excursions(ΔT ≈ 0) near the melting temperature Tm.

When the components of a composite are uniformly dis-persed with suitably small length scales (relative to thethermal diffusion time scales) to maintain local thermal equi-librium, the entire volume can be approximated as a homoge-neous effective medium. In Secs. II B and II C, we applyeffective medium theory to connect the composition andmorphology of a given composite to the effective propertieskeff and Eeff.

B. Composite energy density

The effective volumetric thermal energy density Eeff isan intrinsic property that includes both the sensible heatcapacity terms and the latent heat of phase change. In aneffective medium, Eeff is given by a volume-weighted linearcombination (i.e., a rule-of-mixtures) of the thermal storageproperties of the individual phases:

Eeff (Φ, ΔT) ¼ (CV,CΔT)Φ

þ (CV,PCΔT þ HV,PC)(1� Φ), (2)

where Φ is the volume fraction of the conductive material(and the PCM has a complementary volume fraction 1−Φ),CV is the sensible volumetric heat capacity, ΔT is the temper-ature rise, and HV is the volumetric latent heat of phasechange. The analysis performed by Lu15 includes designrules and thermal response characteristics that consider con-tributions from both the sensible and latent heat that areadded together. Equation (2) follows from this assumptionthat for a mean temperature rise ΔT that crosses the meltingtemperature Tm, we can approximate the energy density as alinear combination of the sensible heat capacity and thelatent heat. In most cases, the sensible heat capacity terms inEq. (2) can be neglected since PCMs are often selected withHV≫CVΔT and phase change thermal storage systems arenecessarily designed to achieve small temperature changes(CVΔT→ 0). To establish baseline material’ design rulesthat are independent of changes in temperature, we apply theaforementioned assumption to Eq. (2) to approximate theenergy density as being derived entirely from the latent heatof the PCM:

Eeff (Φ) � HV,PC(1� Φ): (3)

For much of the remaining analysis, we will employ thisapproximation noting its divergence with increasing tempera-ture rise. However, Eq. (2) remains useful for material selec-tion and benchmarking when comparing PCM composites tosingle-phase materials which only have capacitance derivedfrom sensible heating. It is also important to consider theeffects of sensible heating in predicting the real temperatureresponse of a transient system that necessarily experiences anonzero temperature rise. While the energy storage term Eeff

is only a function of the relative properties and amount of theconstituent phases, the effective thermal conductivity keff

additionally depends on the arrangement of the constituentphases within the composite.

C. Composite thermal conductivity

The effective thermal conductivity keff of a composite isan extrinsic property that is closely correlated to the geomet-ric arrangement of phases since the morphology determinesthe path along which heat travels. The anisotropic (i.e., one-dimensional) limit of conductivity in a binary compositekeff,max(1D) corresponds to a linear scaling with volumefraction and corresponds to the two phases being aligned inparallel:

keff,max(1D) ¼ kCΦþ kPC(1� Φ), (4)

where kC and kPC are the thermal conductivities of the con-ductive material and the phase change material, respectively.In isotropic porous media, it is not possible for the entirevolume to simultaneously contribute fully to conduction ineach direction. Therefore, the thermal conductivity scalessublinearly with volume fraction, and the exact functionalform depends on the morphology. The theoretical maximumfor the effective isotropic thermal conductivity keff,max isdefined by Hashin and Shtrikman using a variationalapproach.41 For a binary phase change composite, this isgiven as

keff,max ¼ kC2kCΦþ kPC(3� 2Φ)kC(3� Φ)þ kPCΦ

: (5)

Equation (5) establishes an upper bound for the effective iso-tropic conductivity in terms of the volume fraction and theconductivity of the constituent phases. The thermal conduc-tivity, energy density, and figure-of-merit scaling withvolume fraction are shown in Fig. 1 for different compositemorphologies.

D. Thermal energy Ragone framework

The relationship between energy density and powerdensity for energy storage systems is often placed in thecontext of a Ragone framework. While most commonlyassociated with electrochemical energy storage, we adoptthis framework for thermal energy systems where we con-sider both the capacity for storing thermal energy and therate of heat delivery and extraction. In the thermal Ragoneplot shown in Fig. 2, there is a clear distinction betweenmaterials that are intrinsically energy dense such as PCMsand materials that are intrinsically thermally conductivesuch as metals. Composite engineering enables materials toaccess the property space in between pure materials andultimately to achieve higher figures-of-merit than either ofthe pure materials alone. The morphology of the compositedetermines the shape of the curve that connects the constit-uent materials, while the composition of the compositedetermines the position along the curve. The limiting casesfor thermal conductivity [Eqs. (4) and (5)] define the upperboundary of the material design space, which can be opti-mized to identify the maximum possible performance for a

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composite comprised of a specific combination of PCMand conductor.

Figure 2 illuminates a critical shortcoming of manyPCM-based transient thermal management schemes. Phasechange composites are often benchmarked against only thepure PCM endmember and not the conductive phase (thecomplementary endmember in a binary composite). Usingthe copper/paraffin composite design space as an example,we find that ηeff can be as much as 20 times larger than pureparaffin but only ∼4 times larger than pure copper for smalltemperature excursions about Tm. The composite perfor-mance can be further improved using materials that each con-tribute significantly to both keff and Eeff. For example, lowmelting point metal alloys (e.g., InxGa1-x) can have thermalconductivities between ∼101 and 102Wm−1 K−1.Composites of such PCM alloys and high conductivity solidmetals will have increased effective thermal conductivitieskeff due to significant conduction occurring in both compo-nents. As the allowable temperature range increases, the sen-sible heat capacity term becomes increasingly significant,and the design optimization shifts to favor the conductivephase over the PCM. The figure-of-merit given by Eq. (1) isderived from thermal performance, although practical engi-neering contexts may also include considerations for size,weight, power, reliability, and/or cost.

E. Optimizing the figure of merit

We can combine Eqs. (1), (3), and (5) to define the theo-retical maximum figure-of-merit ηeff,max for any isotropiceffective medium for thermal storage that only depends on

kC, kPC, and Φ:

ηeff,max(kC, kPC, HV,PC, Φ)

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikCHV,PC

2(kPC � kC)Φ2 þ (2kC � 5kPC)Φþ 3kPC(kPC � kC)Φþ 3kC

� �s,

(6a)

or, equivalently, in terms of the thermal conductivity ratioκ = kC/kPC

ηeff,max(κ, Φ)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikCHV,PC

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(1� κ)Φ2 þ (2κ � 5)Φþ 3

(1� κ)Φþ 3κ

s: (6b)

Equation (6) is evaluated over the domain Φ ∈ [0,1] formaterial combinations having different thermal conductivityratios and is shown in Fig. 3 (normalized to

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikCHV,PC

p).

When kC≫ kPC (i.e., κ→∞), there is a well-definedmaximum that occurs due to the complementary functionalityprovided by each phase. As kPC approaches kC (i.e., κ→ 1),the figure-of-merit increases and the net contribution of theconductive phase diminishes, which shifts the peaktoward larger PCM volume fractions until the functionbecomes monotonic over the domain. Setting the derivativeof Eq. (6) to zero, we write an analytical expression forthe optimized volume fraction Φopt as a function of theconstituent conductivities:

Φopt(κ) ¼ 6κ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6(4κ2 þ 4κ þ 1)

p2(κ � 1)

: (7)

FIG. 2. Thermal Ragone plot forcopper/paraffin and copper/galliumcomposites at ΔT = 1 K. The thermalenergy density is primarily derivedfrom the latent heat of the phasechange process, whereas the thermalpower density is primarily derivedfrom the effective thermal conductiv-ity. The thermal energy density Eeff iscalculated using Eq. (2) and is normal-ized per Kelvin. Materials indicated by(PCM) include contributions from thephase change process, whereas theother materials only include sensibleheat contributions. The thermal con-ductivity scaling with volume fractiondepends on the morphology of thecomposite and is bounded either by theserial and parallel arrangement ofphases (the anisotropic limits) or bythe Hashin-Shtrikman criteria41 (theisotropic limits). The figure-of-meritηeff is given by Eq. (1).

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The unconstrained solution produces two roots, but since3κ/(κ-1) > 1 for κ > 1, the positive root of Φopt is alwaysgreater than unity and is therefore discarded. The physicalsolution (i.e., the negative root) is given by Eq. (7), which isa monotonically increasing function of κ that converges toΦopt ¼ 3� ffiffiffi

6p � 0:55 as κ→∞. This is consistent with the

estimate of Φ∼ 0.5 provided by Shamberger and Fisher16 formaterial combinations with high κ and additionally accountsfor the isotropic limit for conduction in a porous medium.Figure 3(b) shows the solution to Eq. (7) as a function of thethermal conductivity ratio κ. For large values of κ, thermalconduction is determined almost entirely by the conductivephase, and the optimized composite becomes insensitive tokPC. If κ is reduced (i.e., increased kPC), the PCM providesan increasing contribution to the overall thermal conduction.The diminishing need for the conductive phase causes Φopt

to shift toward zero, and the combination of increasing keff

(due to kPC) and increasing Eeff (due to decreasing Φopt)causes an overall increase in the figure-of-merit ηeff,max.In the limit where κ is less than 1þ ffiffiffi

3p

=ffiffiffi2

p � 2:2, thenΦopt = 0 (as bounded by the domain Φ ∈ [0,1]) and there isno enhancement in ηeff,max by including a conductive phase.This transition occurs in the regime where the thermal con-ductivity of the PCM is comparable to the conductive phase,and the marginal increase in keff due to the inclusion of aconductive phase does not offset the reduction in energydensity by removing PCM.

F. Generalizing the figure-of-merit

The optimization strategy presented here can be broadlyapplied to different system geometries and material designconstraints. In general, the material design objectives willtend toward the maximization of both keff and Eeff, althoughnot necessarily with equal weights. Equation (1) representsone specific figure-of-merit that does give equal weight tothe conduction and storage terms as derived from the solutionto the one-dimensional Stefan-Neumann problem for a planarheat source. This one-dimensional approximation is usefulfor establishing a general composite design methodology formore complex geometries and/or operating conditions wheresolutions may not be easily obtained. As a first-order approx-imation, the correlation between the figure-of-merit given byEq. (1) and the real thermal response will diverge as thesystem configuration moves away from any of the keyassumptions or boundary conditions (one-dimensional,planar heat source, semi-infinite far-field boundary).Different system configurations may not concisely isolate keffand Eeff and/or have performance metrics that give asymmet-ric weight to the conductance and capacitance terms. Forexample, cylindrical systems with line heat sources yieldsolutions to the Stefan problem where the material propertiesare embedded as arguments in exponential integral func-tions40 and more heavily weight the conductive terms ratherthan the capacitive terms (due to radial spreading). The sim-plicity and symmetry of Eq. (1) provides a compact contextfor developing material selection rules and identifying criticalasymptotes but does not quantitatively predict the thermalresponse in more complex systems. In such cases, the sameoptimization methodology from the preceding sections canbe more accurately employed using a figure-of-merit that cor-responds to the system-specific Stefan-Neumann solution orequivalent heat transfer model. Other solution forms tend tosignificantly increase the complexity of the preceding analy-sis but provide a more accurate prediction of the absolutetransient thermal response of the system.

Approximate and numerical solutions may be required togenerate design rules for more complex systems when analyt-ical solutions are not available. Furthermore, our assumptionof a homogeneous composite may be relaxed to include het-erogeneous composites with spatially-varying properties(e.g., Ref. 14). Our optimization uses a thermal conductivitymodel for the isotropic limit of a homogeneous composite,but equivalent thermal conductivity models having the appro-priate scaling with volume fraction can replace Eq. (5) tomore accurately represent the composite morphology. The

FIG. 3. Optimization of the figure-of-merit using the theoretical upperbound for an isotropic binary composite. (a) The phase diagram for thefigure-of-merit ηeff,max is calculated using Eq. (6) and is a function of thethermal conductivity ratio κ = kC/kPC. (b) The optimized volume fractionΦopt is calculated using Eq. (7) and remains approximately constant whenthe thermal conductivity of the PCM is negligible and shifts toward zero asthe PCM conductivity becomes increasingly significant to the effectivethermal conductivity.

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approximations and assumptions that go into the derivationof the figure-of-merit and the thermal conductivity modelssuggest that our analysis provides general materials’ designguidelines but requires computational models to validate per-formance in any real system. For example, expansion andcontraction of the melting process and mass transport associ-ated with the liquid phase cause real changes to the compos-ite morphology that are not accounted for in our model, andany deviations from either the one-dimensional or semi-infinite approximations will further introduce error. The accu-racy of both the heat transfer solution and the thermal con-ductivity model will directly impact the accuracy of thematerial design rules and optimization and must be carefullyevaluated to make appropriate material decisions.

The manufacturability of PCM composites further con-strains the morphologies and volume fractions that can beproduced experimentally. The isotropic limit given by Eq. (5)corresponds to a morphology of dilute spherical PCM inclu-sions within a solid conducting matrix and was originallyderived by Maxwell.41,43 At Φopt = 0.55, the conductivematerial has closed-cell morphology with isolated PCMembedded inside, which may be difficult to fabricate.Traditional nanocomposite fabrication relies on either dis-persing solid particles in a liquid matrix or infiltrating aporous solid with a liquid interstitial material. Since an open-celled morphology is required for PCM infiltration, a morecomplicated fabrication process would be required toco-deposit the PCM and the conductive phases to constructthe optimized composite described by Eqs. (5)–(7), which isdefined by a dilute and uniform arrangement of PCM spheresembedded in a conductive matrix.43 Composite morphologiesthat use uniform spherical inclusions in a continuous matrixexhibit some of the highest thermal conductivities across awide range of volume fractions (see Fig. 1) and remainwithin a few percent of the theoretical limit for Φ > 0.5.Armed with these composite design rules, we can nowdesign experiments to demonstrate the utility of a simplifiedfigure-of-merit in selecting materials to mitigate extremethermal transients.

III. EXPERIMENTAL

A. Paraffin-infused copper inverse opals

Inspired by the Maxwell-like morphology of dilute sphericalinclusions in a matrix, we create high figure-of-merit compos-ite thin films (∼6 μm thick) using high conductivity copperinverse opals (CuIOs) infiltrated with high energy densityparaffin wax. The complete synthesis protocol for porousCuIOs is reported in the literature.42 In brief, monodispersepolystyrene spheres are crystallized into a close-packed,face-centered cubic (FCC) lattice (an “opal” of volume frac-tion 1 − Φ = 0.74) and the interstitial volume is filled with acontinuous network of metal (an “inverse opal” of volumefraction Φ = 0.26) using electrodeposition. After dissolvingthe sacrificial spheres, the now open-celled CuIO is infil-trated with paraffin wax (Sigma Aldrich, melting tempera-ture Tm,paraffin = 53–57 °C) using lateral capillary wicking.The solid wax is placed at the edge of the CuIO at roomtemperature, and by heating the substrate beyond Tm,paraffin

on a hot plate (set to ∼70 °C), the molten wax is pulled lat-erally into the CuIO by capillary forces to form the CuIO/paraffin composites shown in Fig. 4.

The thermal conductivity of CuIOs converges to Eq. (5) inthe dilute sphere limit given by Maxwell (Φ > 0.5), and there isadditional suppression in keff as Φ decreases. The effectivethermal conductivity for close-packed CuIOs with Φ = 0.26and κ≫ 1 is keff = 64Wm−1 K−1.42 When combined withparaffin wax (volumetric latent heat HV,PC= 1.9 × 108 J m−3),the figure-of-merit is calculated from Eqs. (1) and (2) to beηeff = 9.6 × 104 J m−2 K−1/2 s−1/2 at ΔT = 1 K.

Despite approaching the scaling limits of keff, continuousand open-celled FCC inverse opals only exist for 0.06 <Φ <0.26 (constrained by the minimum required for continuity atΦ = 0.06 and the hard sphere FCC packing limit Φ = 0.26)and are lower density than desired in this design framework.Other morphologies can be used to increase Φ toward Φopt

without a significant reduction in thermal conductivity fromMaxwell’s effective medium. Amorphous but open-celledarrangements of PCM spheres can exist at larger values of Φ(e.g., Ref. 44), but it is more difficult to predict and tune theexact morphology and thermal conductivity. Alternatively,composites could be realized using PCM emulsions that arequenched into a porous, amorphous packing (with Φ≈ 0.5)and the interstitial volume filled with metal.

The material design rules also establish requirements forthe characteristic length scales of the composite derived fromthe thermal time scales.16 One critical assumption in effectivemedium theory, which approximates a heterogeneousmedium as being homogeneous with a single set ofappropriately-averaged properties, is that the characteristic

FIG. 4. Synthesis of copper inverse opals (CuIOs) infiltrated with paraffinwax. (a) Crystalline CuIOs with 1 μm pore diameters are deposited usingelectrodeposition around sacrificial opal templates following the protocol inRef. 42. [(b) and (c)] Paraffin wax is placed in along the edge of the CuIOsand wicks into the interstitial volume. [(d)–(g)] Scanning electron micros-copy images of both unfilled and paraffin-filled CuIOs.

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length scales of the components are suitably small and uni-formly dispersed. If the PCM length scales are large com-pared to the thermal diffusion length within the PCM, heatcannot access the entire local volume of PCM, which reducesthe apparent capacitance of the composite. These diffusionlengths establish an upper bound on the characteristic lengthscales and set a minimum requirement for utilizing the entirevolume of available PCM. By taking the materials’ selection,morphology, and length scales into design considerationtogether, these design rules and approaches can be used toapproach the theoretical limits of thermal energy storage.

B. Experimental test devices

We implement CuIO/paraffin wax composites as thincapacitive heat spreaders above 50 μm × 50 μm hotspotsdriven by transient high power density loads. We fabricatehotspot test devices (see Fig. 5 and Appendix) to demonstratethe cooling capacity of these PCM composites. Each 1 cm ×1 cm chip consists of a fused silica substrate with twophotolithographically-patterned serpentine resistors (occupy-ing a 50 μm × 50 μm region or an area Ahotspot = 2500 μm2)that simultaneously produce Joule heating and serve as four-point electrical resistance thermometers. These resistors arepassivated using 250 nm-thick SiO2 from a combination oflow temperature atomic layer deposition and chemical vapordeposition (details are given in Refs. 24 and 45) to isolatethe device layer from the electrically-conductive PCM

composite. Each chip contains two identical structures thatare symmetric about the center of the chip and separated by4 mm. A metal seed layer (5 nm Ti + 50 nm Au) is evaporatedthrough a shadow mask onto the SiO2 passivation over a5 mm × 5mm area centered above the hotspots. This seedlayer serves as the cathode for the electrochemical depositionof the CuIO and defines the area of the chip occupied by theCuIO/paraffin wax composite heat spreader.

In a typical measurement, a step function current sourcedrives the device to produce an approximately constant inputpower, i.e., a constant heat flux boundary condition. We usea current source (Keithley 6221) to generate a constantcurrent I beginning at t = 0, which for the small changes indevice resistance R (used to monitor temperature) can beapproximated as a constant rate of heat generation q = I2R(where R is the steady state resistance) in the serpentineregion. The transient temperature response is monitored byrecording the time-varying device voltage at 1 kHz using aNational Instruments Data Acquisition System (NI DAQMX9220) and using the linear correlation between resistanceand temperature in the metal. Each device is independentlycalibrated as a temperature sensor by placing the device in anoven and measuring the resistance as a function of tempera-ture, which is then fit using a linear regression. The deviceshave a room temperature resistance of ∼145Ω and a mea-sured temperature sensitivity of ∼0.4ΩK−1 (calibrated foreach unique device). The driving current is chosen between5 and 20 mA to produce heating loads up to ∼3 kW cm−2.

FIG. 5. Electrothermal metrology used to characterize thin film transient heat spreaders with phase change materials (PCMs). (a) The PCM sample is depositedabove a passivated metal thin film resistor that serves as both a heater and a thermometer. (b) A step function current input generates heat at the serpentine resis-tor and the temperature of the device is monitored as a function of time. (c) A profilometer scan shows the uniformity of a CuIO/paraffin sample. Photographsshow the measurement device (d) as-fabricated, (e) with bulk paraffin wax, and (f ) with a thin CuIO/paraffin composite.

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We use the applied current and measure voltage at steady statetemperature V(TSS) to calculate the input device heat fluxq00 = I × V(TSS)/Ahotspot and assume this value to be constantover the duration of the measurement, which is a conserva-tive upper bound for heating rate. In some measurements, weobserve the evolution of the phase change process and thepropagation of the melting front using a camera (with amacro lens) mounted above the device.

IV. RESULTS AND DISCUSSION

Three different material systems are considered for theheat spreader as shown in Table II: thick volumes (>500 μm)of pure paraffin, thin films (1.5 μm and 3.8 μm) of electrode-posited copper, and thin films (∼6 μm) of CuIO/paraffindescribed in Sec. III A. Three devices are used fornominally-identical CuIO/paraffin composites to verify con-sistency and repeatability. Reference data are obtained fortwo pure samples of thick paraffin wax and two pure samplesof thin solid copper. The use of solid copper represents aconventional high performance material against which thecapacitive system is evaluated.

The temperature responses for each material system areshown in Fig. 6. First, we measure the steady state thermalresistance Rth by applying different heat loads Q and measur-ing the steady state temperature rise ΔTSS of the device. Thethermal resistance is equal to the proportionality constantbetween ΔTSS and Q. The native device (without anyheat spreader) has a thermal resistance of Rth≈ 6700 KW−1

that corresponds primarily to conduction into the SiO2

substrate with a small contribution from conduction into theair above the device. This resistance is reduced toRth,paraffin≈ 6000 KW−1 after adding a thick slab of paraffinwax on the surface. The 6 μm-thick CuIO/paraffin compositereduces the steady state thermal resistance of the device toRth,CuIO/paraffin≈ 1200 KW−1, an ∼80% reduction comparedto the pure paraffin since it provides a high conductivitylateral heat transfer path from the heater, which behaves likea point source in this geometry. The 3.8 μm-thick copperfilm has a thermal resistance that is approximately half of theCuIO/paraffin composites. However, the 1.5 μm-thick copperfilm has a comparable thermal resistance to the CuIO/paraffincomposite thin films and is ideal for comparing the transient

thermal response between two material systems havingsimilar thermal resistances but significantly different thermalcapacitances.

The transient thermal response is measured by applyinga step function current and recording the voltage across thedevice, which is linearly proportional to temperature (sinceresistance varies linearly with temperature in metals at thetemperatures used in this work). The resulting traces showthe time-varying temperature in Figs. 6(a)–6(c) for differentapplied heating loads. We define the rise time τ as theelapsed time before the temperature rise ΔT reaches 95% ofthe steady state temperature rise ΔTSS. All devices are keptbelow 110 °C to prevent catastrophic failure ortemperature-induced material changes in the metal lines. Forthe thick paraffin devices, this temperature threshold is metfor modest heat fluxes of ∼600W cm−2 due to the largethermal resistance of the system. The low thermal conductiv-ity confines the heat to a small volume above the device,which creates a large temperature gradient near the deviceand a short rise time (τparaffin = 240 ± 20 ms) due to the ineffi-ciently utilized thermal capacitance. This highlights one ofthe key shortcomings of many PCM-based heat sinks forhigh power density applications since only a small fraction ofthe available PCM volume is thermally accessible when thethermal conductivity is low. The 1.5 μm-thick copper film, incontrast, has a much lower thermal resistance and can sustainheat fluxes that approach 3000W cm−2 with a rise timeof τCu= 530 ± 30 ms. By moving to the CuIO/paraffin com-posite heat spreader, approximately the same thermal resis-tance is achieved but the rise time increases by 60% to τCuIO/paraffin = 850 ± 50 ms. This suppression in the thermal tran-sient by a factor of ∼1.5 for the CuIO/paraffin composite islower than the estimated performance increase of ∼3 from thefigure-of-merit. This discrepancy is attributed to the complexityof the system geometry relative to the one-dimensional, semi-infinite system that we used in our original derivation. Whileuseful for assessing material design and selection rules, thefigure-of-merit should be derived for the appropriate geometryto accurately predict the transient thermal response. For ourgeometry, a one-dimensional cylindrical geometry might leadto a better predictive model for the transient response but is sig-nificantly more complicated to use for material benchmarking.The inherent simplicity and symmetry of Eq. (1) provides anaccessible first-order approximation for material evaluation anda relative thermal response, but better geometric representationsare required to predict the absolute thermal response in morecomplex systems.

This capacitive effect is directly observed in Fig. 6(e),where we overlay the normalized temperature rise forthree material systems (two CuIO/paraffin composites and a1.5 μm-thick copper film). Each color consists of elevenseparate traces taken at different heating inputs (fromQ00 ≈ 500–3000W cm−2). The copper film (small τCu) experi-ences a much more rapid rise to steady state temperaturecompared to the CuIO/paraffin composites (large τCuIO/paraffin)at all applied heating power loads. The rise time appears tohave a small and statistically insignificant monotonic increasewith the applied heating load, which we hypothesize to becaused by the nonlinearity of the phase change process in the

TABLE II. Devices measured in the present work.

Material

Figure of Meritηeff (ΔT = 1 K)( J m−2 (K s)−1/2) Sample #

Thickness(μm)

Steady Statethermalresistance(K W−1)

Air 4.7 × 100 1 Semi-infinite 6706Paraffin 6.9 × 103 1 >500 5966

2 >500 6030CuIO/Paraffin

9.6 × 104 1 6 12472 6 12033 6 1202

Copper 3.7 × 104 1 1.5 1279

2 3.8 654

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system. These characteristic curves illustrate the temporalregime that differentiates capacitive heat spreaders frompurely resistive heat spreaders. The capacitive heat spreadingeffect is directly monitored using a high speed camerathrough a macro lens and observing the propagation of themelting front. Since the lateral diffusion length is muchlarger than the size of the hotspot, Fig. 6(f ) shows that thedevice behaves as a point source of heat and the meltingfront diffuses radially away from the hotspot.

As the temperature rise increases, the sensible heat(CvΔT) provides an increasing contribution to the totalenergy density Eeff as given by Eq. (2). This diminishes thebenefits provided by the latent heat of the PCM and reducesthe overall design space where a composite can outperformthe high-conductivity single-phase material as shown inFig. 7. For a given composite with thermal conductivity keffand conductor volume fraction Φ, a crossover temperaturerise ΔT* where the pure conductor outperforms the

FIG. 6. [(a)–(c)] Time-varying temperature response of the three material systems to different magnitudes of step function heating loads. The solid lines showdata that have been filtered using a low-pass filter, and each plot includes one trace overlaid above unfiltered data (shown in a lighter color) to show a represen-tative raw signal. (d) The steady state temperature response is shown for different magnitudes of step function heating. The thermal resistance is defined as thelinear scaling coefficient between temperature rise and heating load. (e) The dynamic temperature response is shown for two devices of 6 μm-thick CuIO/paraf-fin (shown in different shades of blue) and the 1.5 μm-thick copper thin film. The temperature rise is normalized to the steady state temperature rise, and overthe range of ∼500–3000W cm−2 applied heat flux the data all collapse onto a single curve for each material system. Each color contains 11 separate traces,each taken at a different applied heat flux. The inset plot shows the time to reach 95% of the steady state temperature rise, which is observed to be independentof the applied heat flux. The CuIO/paraffin samples exhibit rise times that are twice that of the solid copper due to the thermal capacitance of the PCM.(f ) Images taken using a high speed camera to observe the melting propagation in a CuIO/paraffin composite at a power load of 70 mW.

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composite can be defined by equating the compositefigure-of-merit ηeff(ΔT) defined by Eqs. (1) and (2) to theconductor figure-of-merit ηC(ΔT):

ΔT� ¼ keffHV,PC(1� Φ)e2C � e2eff

, (8)

where e2 = kCV is the thermal effusivity. The temperaturewindow increases as the intrinsic material properties of thecomposite increase (keff, HV,PC, eeff ) and as the compositemorphology becomes optimized (keff, Φ). For a given com-posite of known volume fraction and effective thermal con-ductivity, Eq. (8) defines the minimum temperature rise,where ηC > ηeff. This analysis can be generalized to identifythe crossover temperature where the solid conductor outper-forms the optimized composite, where Φ is chosen to maxi-mize keff and ηeff. Recall that in Sec. II D, we determined theoptimized composite using Eq. (3) as an approximation forenergy density, which neglects sensible heat contributions byletting CVΔT→ 0. If instead the temperature-dependent energydensity given by Eq. (2) is used, the same analysis can be con-ducted for the theoretical composite limit. Alternative forms ofthe thermal conductivity scaling with volume fraction can alsobe used to establish local optima when the composite morphol-ogy is independently defined. The temperature-dependentfigure-of-merit scaling with volume fraction is shown in Fig. 7.This suggests that the material design rules must also consider

the operating temperature range to assess the efficacy of aPCM heat sink compared to a single-phase material and high-lights the difficulty in enhancing a low-conductivity PCM tocompete with conventional steady state thermal architecturesfor high power applications.

In low power applications of PCM thermal buffering,there is often a characteristic plateau in the transient tempera-ture response that often occurs due to large volumes of PCMand low intensity heating such that only weak temperaturegradients are present. The result is nearly isothermal heatabsorption/rejection during a window of finite time. In highpower density applications with semi-infinite geometries,there are much stronger temperature gradients that lead toappreciable superheat temperatures at the source. The resultis a suppressed characteristic thermal response rather than themore familiar isothermal plateau.

The majority of the material design rules in this workdescribe binary composites where one component exclu-sively provides thermal conductivity (e.g., copper) and theother component provides thermal energy storage (e.g., paraf-fin). In an ideal composite, both components contributeappreciably to the energy storage and the thermal conductiv-ity. We can approach this ideal composite using phasechange liquid metals with high thermal conductivity in placeof organic PCMs. For example, gallium can be used as aPCM (Tm,gallium = 30 °C, HV,gallium = 4.73 × 108 J m−3) andalso provides a thermal conductivity two orders of magnitude

FIG. 7. [(a)–(c)] Thermal Ragone plots for different temperature rises. The design space where copper/paraffin composites outperform pure copper (denoted bygreen region) shrinks as the magnitude of the temperature rise increases owing to the increasing contribution of the sensible heat capacity to total energydensity relative to the latent heat of the PCM. (d) The figure-of-merit scaling with average temperature rise when considering contributions from the sensibleheat capacity to the energy and power densities. (e) In the isotropic limit for copper/paraffin wax composites, the optimized conductor volume fraction shiftstoward unity as the temperature rise increases.

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higher than paraffin wax (kgallium = 40Wm−1 K−1). Whilepure gallium has a high figure-of-merit, this can be furtherincreased by forming a composite between gallium and ahigher conductivity metal as shown in Fig. 2. Furthermore,low melting point metal alloys made from gallium, indium,and/or tin can be formed to tune the melting point of thecomposite while retaining a much higher thermal conductiv-ity than most organic PCMs. These metals represent animportant direction moving forward in creating capacitivethermal management for high power density heat sources.

V. CONCLUDING REMARKS

We use a Ragone landscape to establish the design spaceand performance limits of phase change composites for highpower transient thermal management by combining effectivemedium theory with solutions to the Stefan-Neumannproblem. We used the mathematical bounds from Hashin andShtrikman to identify the isotropic limit of phase changecomposites for transient thermal management. These designrules represent a methodology that can be generalized for dif-ferent geometries and operating conditions by solving theappropriate Stefan-Neumann problem, collecting the materialproperties into a system-specific figure-of-merit, and usingstructure-property relations to define and optimize the mate-rial design space. We then experimentally demonstrated thecapacitive suppression of temperature rise in on-chip hotspotsusing thin film “thermal capacitors” inspired by this optimi-zation. Compared to the equivalent solid copper thin film,the capacitive composites increase the time constant by afactor of two despite having the same thermal resistance.

The design rules suggest that low melting point metalscan provide significant performance improvements over par-affin wax and comparable organic PCMs due to their higherintrinsic thermal conductivity. Composites of copper andgallium have one of the highest available figures-of-merit [upto 2.1 × 105 J m−2 (K s)−1/2, see Fig. 2] and can operate attemperatures relevant for electronics cooling applications. Weanticipate a significant increase in cooling capacity forcopper inverse opals infiltrated with eutectic metal PCMs.Furthermore, the melting temperature of alloys, such asInxGa1-x, can be tuned using the composition, which adds anadditional engineering control to match application require-ments. More advanced PCMs may be engineered at the atomiclevel to maximize the latent heat and push the limits of PCMenergy density. The metrics used in this work provide a criticaltool to make informed material design decisions for transientthermal architectures, but additional work is required to estab-lish the relationship between a more complex heating signaland the dynamic system response required by most engineeringapplications. This work demonstrates that a materials-by-designapproach to composite engineering can be used to unlock thefull potential of phase change materials for transient thermalmanagement of high power density heat sources.

APPENDIX: MICROFABRICATION PROCESS

The heater/thermometer device fabrication consists of aprimary lithography step, metal deposition, passivation layerdeposition, and a final lithography/etch step to define the

passivation layer openings. First, bare SiO2 substrates arecoated with LOL-2000 at 3000 RPM for 60 s and baked for5 min at 170 °C on a hot plate, which creates a ∼200nm-thick layer. Next, 1 μm of Shipley Megaposit SPR3612 isspin-coated onto the wafers. This resist is exposed in anASML PAS5500 stepper and developed using MF-26A. Atitanium adhesion layer and a platinum metal layer are depos-ited by electron beam evaporation. The wafers are thensoaked in acetone and subjected to ultrasonic cleaning to liftoff the unwanted metal, followed by rinsing with methanoland isopropanol before drying. This leaves behind thedesired metal patterns on the wafer surface. The passivationlayer consists of a SiO2 trilayer deposited over the metal fea-tures. First, a conformal coating of 40 nm SiO2 is depositedusing a plasma enhanced atomic layer deposition methodwith a Cambridge Nanotech Fiji system. Next, a 170 nmcoating of SiO2 is applied using a PlasmaTherm VersalineHDPCVD system, followed by another coating of 40 nmALD SiO2. Another 1 μm layer of Shipley MegapositSPR3612 is spin-coated onto the wafers, exposed, and devel-oped (as described above) to define openings in the passiv-ation layer for electrical contacts. The trilayer SiO2 is etchedusing buffered oxide etch, and the resist is stripped. Thisprocess has been described previously in Ref. 45.

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