ieee transactions on antennas and propagation, vol. … · these results attest to the importance...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 11, NOVEMBER 2017 6035

Electromagnetic Scattering From IndividualCrumpled Graphene Flakes: A Characteristic

Modes ApproachKalyan C. Durbhakula, Student Member, IEEE, Ahmed M. Hassan, Member, IEEE, Fernando Vargas-Lara,

Deb Chatterjee, Senior Member, IEEE, Md. Gaffar, Jack F. Douglas, and Edward J. Garboczi

Abstract— Graphene flakes (GFs) in real composites are rarelyperfectly flat, and often exhibit complicated crumpled shapes.Therefore, the goal of this paper was to quantify the elec-tromagnetic scattering characteristics of individual crumpledGFs with shapes resembling those found in real composites.The extinction cross sections of tens of GFs, with differentsizes and various levels of crumpleness, were calculated usingmultiple independent solvers. The results show that resonancesin the extinction cross section spectrum decrease in amplitudeas the GFs become more crumpled. Moreover, some crumpledGFs exhibited a broader resonance than that of perfectly flatGFs. To explain these results, we used a characteristic modeanalysis to decompose the graphene surface currents into a setof fundamental currents or modes. For perfectly flat square GFs,the vertical and horizontal modes were found to overlap andresonate at the same frequencies. However, as the GFs becamemore crumpled, the horizontal/vertical symmetry broke downcausing the corresponding modes to separate and resonate atdifferent frequencies leading to an overall broader bandwidth.These results attest to the importance of modeling the exact shapeof GFs to accurately characterize their electromagnetic response.

Index Terms— Characteristic mode analysis (CMA), crumpled,electromagnetic response, graphene flakes (GFs).

I. INTRODUCTION

GRAPHENE, a 2-D monolayer of sp2 hybridized carbonatoms arranged in a honeycomb lattice, has received

Manuscript received February 3, 2017; revised August 13, 2017; acceptedSeptember 3, 2017. Date of publication September 14, 2017; date of currentversion October 27, 2017. This work was supported in part by the NIST“Multi-Scale Computational Modeling of Carbon Nanostructure Composites”under Grant 70NANB15H285, in part by NIST Project “Carbon Nanocom-posite Manufacturing: Processing, Properties, Performance,” and in part byNSF CRI Award 1629908 “II-NEW: Experimental Characterization andCAD Development Testbed for Nanoscale Integrated Circuits.” The work ofF. Vargas-Lara was supported by NIST under Award 70NANB13H202 andAward 70NANB15H282. (Corresponding author: Ahmed M. Hassan.)

K. C. Durbhakula, A. M. Hassan, D. Chatterjee, and M. Gaffarare with the Department of Computer Science and Electrical Engineer-ing, University of Missouri–Kansas City, Kansas City, MO 64110 USA(e-mail: [email protected]; [email protected]; [email protected];[email protected]).

F. Vargas-Lara and J. F. Douglas are with the Materials Science andEngineering Division, Material Measurement Laboratory, National Insti-tute of Standards and Technology, Gaithersburg, MD 20899 USA (e-mail:[email protected]; [email protected]).

E. J. Garboczi is with the Material Measurement Laboratory, Applied Chem-icals and Materials Division, National Institute of Standards and Technology,Boulder, CO 80305 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/TAP.2017.2752218

significant interest for its combination of unique chemi-cal and electronic structure. Graphene exhibits exceptionalmechanical, thermal, and electrical properties which led tothe incorporation of graphene flakes (GFs) in a plethora ofcomposites [1]–[4]. However, all GFs undergo fluctuations andcannot be perfectly flat. Even those GFs are designated as flatexperience crumpleness on the order of tenths of nanometersdue to thermal fluctuations of the carbon atoms [5].

In composites, GFs are more crumpled, and they tend toexhibit complex crumpled shapes [6]–[8]. In addition to theircrumpled shapes, different GFs also show randomness in theirorientation and are rarely perfectly aligned with each otherinside the composite.

Deng and Berry [6] discussed the factors that can lead toripples, wrinkles, or crumples in GFs. These factors includethermal vibrations, differences in the thermal expansion coef-ficient of the GFs and the substrate on which it is grown,solvent surface tension during the transfer process of the GFsfrom the substrate to its new host, and the evaporation processof any trapped solvent [6]. Moreover, recent advancements inthe fabrication technology allow the fabrication of GFs withcontrolled shapes [9], [10]. Since GFs are typically crumpled,the effect of the crumpleness of the GFs on their mechanical,thermal, and electrical properties has attracted wide attention.For example, GFs with wrinkles or crumples were found toreduce the thermal conductivity by 27% in comparison towrinkle-free GFs [11]. All the previous studies indicate to thesignificance of understanding how the properties of crumpledGFs differ from perfectly flat ones. Therefore, the goal of thispaper is to study the electromagnetic scattering characteristicsof crumpled GFs compared to perfectly flat GFs.

The electromagnetic scattering characteristics of flatGFs have received rising interest in the last decade.Llaster et al. [12] studied the scattering properties of graphene-based nanopatches placed in free space or on a substratecompose of silica or silicon. Their computational studiesshowed that the resonance in the extinction cross sectionspectrum of the GFs shifts to lower frequencies and the peakamplitude decreases when it is placed on a substrate [12].Costa et al. [13] analyzed the absorption cross section of flatgraphene nanoantennas of different sizes, chemical potential,and temperature. They also studied flat canonical shapes liketriangular and circular GFs and they showed that the shape of

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6036 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 11, NOVEMBER 2017

the GFs has a substantial effect on the peak amplitude and aweak effect on the frequency of the resonance in the extinctioncross section spectrum. They also showed that the chemicalpotential of a GF could be used to tune both the frequencyand the peak amplitude of the resonances for all grapheneshapes. Most of the previous studies simulated the graphenestructure using frequency domain solvers such as the methodof moments (MOM) or the finite-element (FE) analysis whichare suited for modeling the frequency dispersive propertiesof graphene. However, Nayyeri et al. [14], [15] representedthe surface conductivity of graphene as a series of partialfractions to allow the simulation of various graphene-basedstructures using the finite difference time domain method. Sev-eral simplified models were developed to reduce the compu-tational complexity of the full-wave electromagnetic analysisof GFs [16], [17]. For example, Cao et al. [16] developedthe partial element equivalent circuit method for a perfectlyflat graphene patch. Raeis-Zadeh and Safavi-Naeini [17] usedthe generalized multipole technique to study the electromag-netic scattering and the field enhancement factor of isolatedgraphene disks and graphene nanodimers. In all the previousstudies, the GFs were perfectly flat, and no curvature or crum-pleness was incorporated in the graphene shapes studied.To the best of our knowledge, the cloaking application is theonly case where curvature in the GFs was included [18], [19].In that case, nonflat GFs were wrapped around a cylindricaltarget to reduce its electromagnetic signature [18], [19]. Thispaper is the first time that the electromagnetic responses ofcrumpled GFs are simulated. Moreover, this is the first timethat the effect of graphene crumpleness on the electromagneticresponse has been quantified.

The basic assumptions of our study were that all the GFswere embedded in free space, and we limit our simulationsto maximum frequency of 4 THz, below which the intrabandconductivity dominates the graphene response [12], [13], [20].Expressions for the intraband conductivity were obtainedusing the Kubo formula, and the scattering characteristicsof the GFs were calculated using classical electromagneticsolvers and assuming the following time dependence (e jωt). Toexplain the electromagnetic response of these crumpled GFs,we used the characteristic mode analysis (CMA) developedby Garbacz [21] and Harrington and Mautz [22]–[24]. In theCMA analysis, the surface current on a GF is decomposedinto a set of fundamental modes that are calculated via aneigenvalue problem. By studying how the behavior of thesemodes varies with the level of crumpleness, we explain theelectromagnetic response of the GFs. In summary, this paperhas two main goals. First, we calculate the electromagneticresponse of crumpled GFs using classical electromagneticsimulations. Second, we use the CMA to quantify and validatethe correctness of the calculated electromagnetic response.

This paper is organized as follows. Section II describes thecoarse-grained molecular dynamics (MD) model that we usedto generate the crumpled GFs, how alignment was achieved,and what surface conductivity model was used for the GFs.Section III presents the numerical results and Section IVexplains these results using the CMA. Finally, Section Vsummarizes the conclusion.

Fig. 1. (a) Graphene coarse-grained model. In the upper panel, we representthe GF as a square lattice of 102 soft spheres (gray spheres) connected bysprings (light blue cylinders) and (b) GF representative snapshot, defined bythe centers of the spheres in (a), after the spheres reached thermal equilibrium.

II. CRUMPLED GRAPHENE FLAKES MODEL

A. Graphene Flakes Morphologies

Realistically shaped GFs are generated using a coarse-grained model based on MD simulations. The coarse-grainedmodel has been previously used to study the transportproperties of 2-D polymers [25], the interactions of DNA-based nanomaterials [26], and the electrostatic properties ofGFs [27]. The details of the model are presented in ourprevious work [27]. However, for completeness, the model isbriefly described in this section.

In the coarse-grained model, each GF is represented as asquare lattice of N × N soft beads, shown as gray spheresin Fig. 1(a), connected by springs, shown as light bluecylinders in Fig. 1(a). The model is termed coarse-grainedsince each bead in Fig. 1(a) corresponds to many carbonatoms to reduce computational complexity. The beads werethen allowed to move, until they reached equilibrium via theMD simulations summarized in the Appendix. The centers ofthe spheres were then used to construct the equivalent GFs,as shown in Fig. 1(b), which were forwarded to the electro-magnetic solvers discussed in the following section. In this

DURBHAKULA et al.: ELECTROMAGNETIC SCATTERING FROM INDIVIDUAL CRUMPLED GFs 6037

paper, the spheres had a diameter of 1 μm, and different sizeswere achieved by varying N . The spheres overlap slightly.Therefore, for a GF with N × N spheres, the equivalent GFreconstructed from the center of these spheres has a surfacearea of 0.96 N× 0.96 N = 0.92 N2μm2. Three GF sizes wereconsidered: N = 10, N = 20, and N = 50.

The waviness or average curvature of 1-D wire-like struc-tures like polymeric chains and carbon nanotubes is typicallyquantified using the persistence length [28], [29]. The persis-tence length, P , is a measure of how hard it is to bend a 1-Dchain. Therefore, the larger the persistence length, the moredifficult it is to bend the 1-D chain and the straighter it appears.The persistence length, P , of an ensemble of 1-D chains canbe calculated using the standard definition [28]

P = 〈Re · L1〉〈L1〉 (1)

where 〈〉 refers to the average over the ensemble of 1-D chains,Re is the end-to-end vector of the 1-D chains, and L1 is thevector tangent to the first segment of each 1-D chain.

To the best of our knowledge, there is no quantitativeparameter similar to the persistence length that can be usedto quantify crumpleness in the semiflexible 2-D polymer-likeGFs considered in this paper. Therefore, we introduce the newparameter, persistence area (PA), as the 2-D extension of theclassical 1-D persistence length. To calculate, PA , we dividedeach GF into N vertical 1-D chains [e.g., chain 1 is formedby bead number (1, 2, 3 . . . , 10) and chain 2 is formed bybead number (11, 12, 13, . . . , 20)] as labeled in Fig. 1 andwe calculated the persistence length of these chains whichwe termed Py . Py is a single value that describes the overallnonstraightness or curvature of these N 1-D vertical chains.Then we divided the GF into N horizontal 1-D chains [e.g.,chain 1 is formed by bead number (1, 11, 21, . . . , 91) andchain 2 is formed by bead number (2, 12, 22, . . . , 92)] aslabeled in Fig. 1 and we calculated the persistence lengthof these 1-D chains which we termed Px . Px is a singlevalue that describes the overall nonstraightness or curvature ofthese N 1-D horizontal chains. Using Px and Py , PA can beexpressed as

PA = Px Py

Lx L y(2)

where Lx and L y are the lengths of the horizontal and vertical1-D chains composing the GF, respectively. If the GF isperfectly flat, Px = Lx and Py = L y . PX and Py decreaseas a GF becomes more crumpled, whereas L X and L y willstay constant leading to an overall decrease in the value ofPA . Therefore, the values of PA range from 1 to 0, being 1for perfectly flat surfaces and less than one for crumpled GFas shown in (2).

Two levels of crumpleness were considered in this paper:highly crumpled (HC) and slightly crumpled (SC). The HCGFs had a PA = 0.28 and the SC GFs has a PA = 0.64.Examples of a perfectly flat, an SC, and an HC GF are shownin Fig. 2.

Fig. 2. Different GFs represented from left to right. (a) Flat. (b) SC.(c) HC GF.

B. Electromagnetic Analysis of Graphene Flakes

All the GFs simulated in this paper were simulated as thinsheets with a surface conductivity described by the followingdrude model:

σ(ω) = 2e2kB T

π�2 ln

[2 cosh

(μc

kB T

)] − j

(ω − jτ−1)(3)

where e is an electron charge, kB is the Boltzmann’s constant,T is the temperature, h̄ is the reduced Planck’s constant,μc is the chemical potential, j is the

√−1, ω is the radialfrequency, and τ is the relaxation time. Equation (3) representsthe intraband portion of the surface conductivity and, therefore,is only valid in the microwave and terahertz range wherethe interband term is negligible. By assuming zero chemicalpotential μc = 0, the surface conductivity in (3) can beexpressed as

σ(ω) =(

2 ln(2)e2kB T

π�2 τ

)1

1 + jωτ(4a)

σ(ω) = σ0

1 + jωτwhere σ0 = 2 ln(2)e2kB T τ

π�2 . (4b)

The distributed surface impedance of the GF can beexpressed as the inverse of the surface conductivity as

ZG(ω) = 1

σ(ω)= 1

σo+ jω

τ

σo= RG + jωLG (5a)

RG = π�2

2 ln(2)e2kB T τLG = π�

2

2 ln(2)e2kB T(5b)

where RG and LG are the resistive and inductive components,respectively, of the distributed surface impedance ZG . In thispaper, we assume that the conductivity σ and the surfaceimpedance ZG do not vary with the level of crumplenessin order to isolate the effect of variations in shape on theelectromagnetic response of GFs.

The GFs considered in this paper exhibit highly complexshapes and, therefore, their electromagnetic response is notuniform in all directions for a given incident direction. In thispaper, we calculate the total extinction cross section (Cext)defined as the summation of: 1) the absorbed power and2) the power carried by the scattered fields in all directions,normalized with the power density of the incident wave [29].

Therefore, Cext represents the total signature of the GF fora given incident direction and it is calculated as follows:

Pext = 0.5Re

(∫I ∗.Ei ds

)(6a)

Cext = Pext

|Ei |2/2η(6b)


Fig. 3. Two SC GF that are (a) randomly oriented and (b) aligned such thattheir major axis/dimension is parallel to the x-axis.

where Pext represents the sum of the absorbed power andthe scattered power in all directions around the GF. Thesymbol (∗) represents the conjugate transpose of the totalsurface current (I ) and Ei represents the incident electric field.The dot product in (6a) is integrated over the surface of the GFto yield the total extinction power. The parameter η representsthe intrinsic impedance of free space which is the mediumwhere the GFs are embedded.

In (3)–(5), it was shown that the relaxation time τ has astrong effect on the distributed surface impedance of grapheneZG , especially on the resistive part RG . From (5b), it isclear that RG is inversely proportional to τ and, therefore,higher τ values will lead to a decrease in RG and a reductionin the losses in the GF. Reported values of τ vary from0.1 ps, [12], [13], [16], [17], to 1 ps, [16], [30]–[34]. Thiswide variability in the value of the relaxation time can beexplained, in part, by the variations in the quality of thegraphene fabricated. Higher τ values typically correspond tohigher quality GFs containing a lower number of defects,fabricated using mechanical exfoliation [35]. For this season,we study two values for τ in the following sections, to coverthe range seen in practice: τ = 0.1 ps and τ = 1 ps.

C. Alignment of Graphene Flakes

The generated GFs are randomly oriented and, therefore,to isolate the effect of shape on the extinction cross sectionthey need to be aligned with the electric field polarization ofthe incident wave. The alignment is performed by rotating theGFs until their radius of gyrations tensor, G, is diagonal andGx x ≥ Gyy ≥ Gzz [29]. This gyration tensor approach isequivalent to fitting the GF within the smallest possible tri-axial ellipsoid and then rotating the GF until the major axisof the ellipsoid is aligned with the x-axis [29]. To clarify thealignment process, Fig. 3(a) shows two randomly oriented SCGFs. Fig. 3(a) clearly shows that the two GFs have randomorientations and they are aligned parallel to two differentdirections. Fig. 3(b) shows the same two GFs after they werealigned such that both GFs have their major axis parallel tothe x-axis. Unless otherwise specified, the incident wave will

Fig. 4. Extinction cross section of an SC GF calculated using the independentsolvers COMSOL (solid blue curve) and FEKO (dashed red curve).

always be in the negative z-direction, and the electric field willbe polarized in the x-direction for maximum coupling withthe aligned GF. The extinction cross section of both randomlyoriented and aligned GF will be presented in the followingsection.

III. NUMERICAL RESULTS

Due to the complex nature of the shapes studied in thispaper, we performed the simulations twice using two inde-pendent commercial solvers: 1) the MOM-based FEKO1 [36]and 2) the FE-based COMSOL [37]. Fig. 4 shows the Cext ofa 92 μm2 SC GF based on a relaxation time τ = 0.1 ps in thegraphene surface conductivity equations (3)–(5). Fig. 4 showsone example of the excellent agreement we obtained betweenthe Cext calculated using FEKO and the Cext calculatedusing COMSOL. Other GFs shapes and sizes showed similaragreement, which tends to validates our numerical results.

To investigate the effect of crumpleness on the electro-magnetic response, Fig. 5 shows the Cext of a flat, SC,and HC represented by blue, orange, and light yellow lines,respectively. All the three GFs are of equal size 92 μm2,equal conductivity, and equal relaxation time τ = 0.1 ps.Moreover, they were all aligned with the x-axis and, therefore,they only differed in their shape due to their different levelof crumpleness. In both FEKO and COMSOL, each GF wassimulated as a surface carrying an impedance as defined by (5).Fig. 5 shows that the SC and HC GFs are both resonating ata frequency similar to that of flat graphene. However, as weprogress from the flat to the HC GF, the peak amplitude ofthe resonance decreases as the level of crumpleness increases.

To confirm this finding, we simulated ten different GFs,in the SC category, to cover a wide range of shape variations.All the shapes had the same area, 92 μm2, and a relaxationtime, τ = 0.1 ps. Fig. 6(a) shows the extinction cross

1Certain commercial equipment, instruments, or materials are identifiedin this paper to foster understanding. Such identification does not implyrecommendation or endorsement by the National Institute of Standards andTechnology, nor does it imply that the materials or equipment identified arenecessarily the best available for the purpose.


Fig. 5. Extinction cross section of (a) flat (dashed blue line), (b) SC (dashedred line), and (c) HC GF (solid yellow line).

Fig. 6. Extinction cross section of ten 92 μm2 SC. (a) Randomly oriented.(b) x-axis aligned GFs. In both subplots, the extinction cross section of a flatGF for comparison. A relaxation time τ = 0.1 ps is assumed.

section of these randomly oriented GFs, whereas Fig. 6(b)shows the same GFs after they are aligned with the x-axis.Fig. 6(a) shows a wide variation in the peak amplitudes and theresonance frequencies of the randomly oriented GFs. However,when all the GFs are aligned with the x-axis, they all exhibitedvery similar values of Cext, as shown in Fig. 6(b). Therefore,Fig. 6(b) indicates that, for the level of crumpleness studiedin Fig. 6, Cext is more sensitive to the orientation of the GFsthan to its exact shape provided that all the GFs had approxi-mately the same level of crumpleness. Moreover, in Fig. 6(b)the aligned GFs resonated at a frequency very similar to thatof a flat GF. The SC GFs generated a lower value Cext than didthe flat GF both when they were randomly oriented, as shownin Fig. 6(a), and when they were aligned, as shown in Fig. 6(b).The importance of the results in Fig. 6 is that they showthat the crumpleness of a GF has a strong effect on theirelectromagnetic response and, therefore, it needs to be takenexplicitly into consideration when interpreting experimentalmeasurements. For example, in many experiments the electro-magnetic response from crumpled GFs is typically collected toinfer their conductivity. Once Cext is experimentally collected,the conductivity of a single GF is typically backtracked usingvarious effective medium approximations. But if the GFs areassumed to be flat and their crumpleness is neglected, thentheir conductivity will be underestimated because the results

Fig. 7. Extinction cross section of ten 92 μm2 SC. (a) Randomly orientedand (b) x-axis aligned GFs. In both subplots, the extinction cross section ofa flat GF for comparison. A relaxation time τ = 1 ps is assumed.

in Fig. 6(b) show that the crumpled GF generate lower valuesof Cext than do flat GF with the same conductivity.

Fig. 7 repeats the simulations in Fig. 6, using the sameGFs and the same orientations, previously described, but fora different relaxation time τ = 1 ps as in [16] and [30]–[34].In comparison to Fig. 6, Fig. 7 shows narrower resonancesand larger peak amplitudes. This change in the resonancescan be explained by the fact that the distributed resistanceRG in (5) is inversely proportional to the relaxation time τ .Therefore, increasing the relaxation time τ reduces the lossesin the GF, which increases the quality factor of the resonanceleading to larger peak amplitudes and narrower resonances.The resonance frequencies in Fig. 7 were, however, similarto those of Fig. 6 since the relaxation time does not affectthe distributed inductance LG which controls the resonancefrequency. All the GFs used in Figs. 6 and 7 are SC GFs.The value of the persistence area, PA, remains the sameregardless of shape and size of GFs and it only depends on thelevel of crumpleness. Therefore, PA = 0.64 for all SC GFsin Figs. 6 and 7.

All the previous figures showed results for GFs with thesame size, 10 × 10 beads that correspond to a surface areaof 92 μm2, as described in Section II. To investigate the effectof crumpleness on GFs with different sizes, Fig. 8(a) shows theaverage peak amplitude of the first resonance for aligned GFsof sizes: 10×10 spheres, 20×20 spheres, and 50×50 spheresthat correspond to a surface area of 92, 369, and 2304 μm2,respectively. For each size, three categories are considered:flat, SC, and HC. For each crumpleness category and size,the average peak amplitude is calculated from ten differentshapes to capture a wide range of shape variations. Finally, tworelaxation times τ = 0.1 ps and τ = 1 ps were tested. Fig. 8(a)shows that the average amplitude of the first resonance isapproximately ten times lower for the case of τ = 0.1 psthan for the τ = 1 ps for all sizes and crumpleness categories.Regarding the three crumpleness levels, it is clear that theaverage peak amplitude of the first resonance decreases as theGF become more crumpled for all sizes and relaxations times.Fig. 8(a) indicates that the peak amplitude of Cext can be usedto infer the level of crumpleness of aligned GF flakes in anexperimental sample. The first resonance frequency of a GFis strongly dependent on the size or the surface area of theGF but weakly dependent on the level of crumpleness. Thatis, as long as all the GFs have the same surface area they


Fig. 8. (a) Average amplitude of first resonance of flat (solid black line),SC (solid blue line), and HC (dashed red line) GFs. Each data point for theSC and HC categories is calculated from the average of ten different shapes.(b) Average frequency of the first resonance of flat (solid black line), SC(solid blue line), and HC (dashed red line) GFs. Each data point for the SCand HC categories is calculated from the average of ten different shapes.

will exhibit a similar first resonance regardless of their levelof crumpleness. Fig. 8(b) shows the average first resonancefrequency versus the size of the GFs for the three categories,flat, SC, and HC. Fig. 8(b) shows that the 92 μm2 flat, SCGFs, and HC GFs resonates on average at 1.083, 1.1068, and1.1838 THz, respectively. Similarly, the average first resonancefrequency for the 369 μm2 GFs occurs at the similar valuesof 0.7183, 0.7615, and 0.7236 THz for the flat, SC, and HCcategories, respectively.

Therefore, Fig. 8 shows the potential of electromagneticwaves for the nondestructive evaluation of the geometricalproperties of graphene devices and composites. In the follow-ing section, the CMA will be used to understand more clearlyhow the fundamental modes of the GF vary with the shape.

IV. CHARACTERISTIC MODE ANALYSIS

A. Basic Theory of CMA

CMA has been used extensively in antennadesign [38]–[40]. Recently, CMA was also employed toquantify the electromagnetic scattering characteristics of car-bon nanotubes with realistic shape [29]. The basis of the CMA

Fig. 9. (a) Total extinction cross section and (b) modal significance of a10 μm × 7 μm flat GF. A relaxation time τ = 1 ps is assumed. Modes 1, 2,and 3 in subplot (b) are represented by solid blue line, dashed red line, anddotted black lines, respectively.

is to solve an eigenvalue problem to extract the fundamentalmodes, also termed eigen-currents, for each GF. Theeigenvalue problem typically requires the calculation ofthe MOM impedance matrix, Z , and solving the followingeigenvalue equation:

X Jn = λn R Jn (7)

where X and R are the imaginary and real part, respectively,of the MOM impedance matrix Z , Jn are the modes or eigencurrents, and λn are the eigenvalues. The eigenvalue prob-lem (7) is solved, for each frequency, using the proce-dure detailed by Harrington and Mautz [22]. In this paper,the MOM impedance matrix was extracted from the FEKOsolver. The accuracy of FEKO’s MOM impedance matrixhas been reported in a wide variety of applications [36].The characteristic modes were then extracted from FEKO’sMOM impedance matrix using an in-house code based onHarrington et al.’s formulation detailed in [26]. The validity ofour in-house code was described in our previous work wherethe CMA of CNTs with realistic shapes was performed [29].

The total surface current, I , flowing on the surface of a GFcan be expressed in terms of the eigenvalues λn and the eigencurrents Jn as follows:

I =∑

n

Vn Jn

(1 + jλn)(8a)

Vn =∫∫

Jn · Eids (8b)

where Vn is termed the modal excitation coefficient, whichdescribes the level of coupling between the incident electricfield and the mode Jn . If Vm is large for a particular mode m,this means that the incident field is capable of exciting themode m and that it will be strongly expressed in the total cur-rent I . Moreover, the modes also depend on the magnitude of1/|1 + jλn| which is termed the modal significance MSn [38].The parameter MSn varies with frequency, but it does not varywith the excitation or the incident wave properties, since itrepresents the relative influence of the mode at a particularfrequency. Therefore, if mode m has a high MSm at a particularfrequency, it will dominate the other modes with lower MSn

values. This dominance means that the total current on theGF, at this frequency, will be composed primarily of a scaledversion of the eigen current Jm .


Fig. 10. Eigen currents of a 10 μm × 7 μm flat GF. (a)–(f) Correspond to modes 1–6, respectively.

B. CMA Analysis of Flat GF

To explain the results in the previous section, we start byexamining the CMA of a flat GF. All the GFs studied inthe previous section, even the crumpled ones, where squareshaped meaning that the number of carbon atoms acrosseach of the four edges was the same. However, as thelevel of crumpleness increases, the geometrical symmetrybreaks down causing the horizontal and vertical modes todeviate. Therefore, we start by studying a flat rectangular GF,10 μm ×7 μm, as a simple geometry where the vertical andhorizontal directions differ. In all the results in this section,we use the relaxation time τ = 1 ps since it exhibits narrowerresonances than the τ = 0.1 ps and, therefore, adjacent modeswill be easier to resolve. Also, unless otherwise specified,the excitation will be a plane wave propagating in the negativez-direction and the electric field will be polarized along thex-axis. Fig. 9(a) shows the extinction cross section Cext ofthe 10 μm × 7 μm flat GF, where it shows a clear resonanceat 0.9779 THz. This resonance frequency is different fromthe cases in Section III, because all the GF in Section IIIhad a surface area of 92 μm2. Fig. 9(b) shows the modalsignificance MSn of the first three modes of the 10μm×7 μmflat GF. These three modes resonate at 0.9779, 1.267, and1.363 THz. The resonance frequency of mode 1, in Fig. 9(b),closely resembles the resonance frequency of Cext, in Fig. 9(a),indicating that the resonance in Cext is primarily due tomode 1. Moreover, Fig. 9(a) does not show resonances thatcorrespond to the resonance frequencies of modes 2 and 3 fora square GF. To understand why mode 1 was excited andmodes 2 and 3 were not excited by the incident wave in Fig. 9,Fig. 10 shows the first six modes, Jn , of the 10μm × 7 μm

flat GF. For a mode to be excited, it has to exhibit a largemodal excitation coefficient Vn as described by (8b). This isthe term in the CMA which describes the coupling betweenthe incident electric field and the mode. A large Vn is obtainedwhen the direction of the mode or eigen current, throughoutthe GF, is aligned with the direction of the incident electricfield. This alignment will lead to a large dot product in (8b)and, therefore, a large modal excitation coefficient Vn for thatmode. This was the case for mode 1 in Fig. 10(a) where it isclear that the eigen currents were pointing in the x-directionthroughout the majority of the GF which is the same directionas the incident electric field. Since this mode is excited, we seea resonance in Cext in Fig. 9(a) at the same frequency wherethe modal significance for mode 1, MS1, exhibits a resonance.

However, a mode will not be excited if it exhibits a lowmodal excitation coefficient Vn . This can happen primarily intwo ways. In the first case, the eigen current is perpendicularto the incident electric field throughout the majority of theGF. This will cause the dot product in (8b) to fall to zero.For example, in mode 2 in Fig. 10(b) the eigen currentsare oriented parallel to the y-axis throughout the majority ofthe GF perpendicular to the x-aligned incident electric field.One way to excite mode 2 is to rotate the direction of theincident electric field to the y-direction. In this case, the dotproduct between the incident electric field and the secondeigen current, J2, will have a large value and, therefore, themodal excitation coefficient V2 will be large. In this case,mode 2 will be excited, mode 1 will be suppressed, and theresonance of Cext in Fig. 9(a) will shift to 1.267 THz, whichis the resonance frequency of the second mode, as shownin Fig. 9(b). The second case that can exhibit a low modal


Fig. 11. Eigen currents of a 10 μm × 7 μm C-shaped GF. (a)–(f) Correspond to modes 1–6, respectively.

excitation coefficient, Vn , is when the eigen-current arrows arealigned with the general direction of the incident electric fieldbut with varying polarity, i.e., pointing in the same direction asthe incident electric field over parts of the GF and pointing inthe opposite direction over other portions. Therefore, the dotproduct in (8b) will be positive over some portions of the GFand negative over other portions and when the contributionsfrom all segments of the GF are summed up, they will cancelcausing the modal excitation coefficient to drop to zero. Thisis the case for mode 4 in Fig. 10(d), where the eigen currentsare oriented to the left (negative x-direction) over half of theGF and to the right (positive x-direction) over the other half,which will cancel out the integration in (8b) leading to anegligible V4. Fig. 10(c), (e), and (f) represents modes 3, 5,and 6, respectively. These modes will not be excited, becausethey exhibit low modal excitation coefficients; V3, V5, andV6; similar to V2 and V4 previously described. However, thesehigher order modes are presented to show the complex patternsthat can be exhibited by the interaction of the horizontal andthe vertical modes on a rectangular-shaped GF. Formulas thatdescribe the eigen currents or modes for rectangular perfectlyconducting patches have been reported by Wu and Su[41].These formulas show qualitative agreement with the modebehavior shown in Fig. 10 calculated from a rectangular GF.

C. CMA Analysis of a C-Shaped GF

As an example of a case where the higher order modes canbe excited, Figs. 11 and 12 study the same 10μm × 7 μmGF bent into a C-shape. In Figs. 10 and 11, the straightedge is 7μm long, and the curved edge is 10μm long. Thefirst six modes of the C-shaped GF are shown in Fig. 11.By comparing Figs. 10 and 11, it is clear that the modes didnot vary significantly from bending the GF into a C-shape.The only difference is that the modes in Fig. 11 are impressedon a curved surface instead of the flat surface in Fig. 10.

Fig. 12. (a) Total extinction cross section using two independent solversCOMSOL and FEKO and (b) modal significance of a 10 μm × 7 μmC-shaped GF. A relaxation time τ = 1 ps is assumed. Modes 1–5 in (b) arerepresented by a solid blue line, a dashed red line, a dotted black line, a solidgreen line, and a (+) brown line, respectively.

Using the same excitation as the flat GF in Fig. 9, the firstresonance of the C-shaped graphene is obtained at 1.64 THz,as shown in Fig. 12(a) which is different from the resonancefrequency of the flat 10μm × 7 μm GF in Figs. 9 and 10.The reason for this difference in the resonance frequencyis that in the C-shaped case the first three modes were notcoupled with x-polarized incident plane wave and, therefore,they yielded low values for V1, V2, and V3 as defined in (8b).The first eigen-current in Fig. 11(a) is not coupled, becauseit is maximum at the center of the C-shaped GF, where it isdirected parallel to the z-axis and, therefore, it is perpendicularto the x-polarized incident field. The second eigen currentin Fig. 11(b) is parallel to the y-axis and, therefore, it isalso perpendicular to the x-polarized incident plane wave.Similarly, the third eigen current in Fig. 11(c) does not couplewith the incident wave. In the case of the flat GF, the fourtheigen current in Fig. 10(d) was oriented toward the negativex-axis for half of the GF and toward the positive x-axis for theother half. Hence, the modal significance V4 in (8b) canceledout. However, bending the GF into a C-shape caused the fourtheigen current to point toward the positive x-axis throughout


Fig. 13. Modal significance of the first three modes of a single (a) flat, (b) SC, and (c) HC. A relaxation time τ = 1 ps is assumed. In three subplots,modes 1–3 are represented by a solid blue line, a dashed red line, and a dotted black line, respectively.

the GF, as shown in Fig. 11(d), leading to a significant V4 anda corresponding resonance in Cext, as shown in Fig. 12(a). It isimportant to emphasize that the C-shaped GF had the samearea, conductivity, and excitation as the flat rectangular GF.Therefore, the results in Figs. 9–12 indicate the importanceof studying the shape and orientation of a GF, and theyalso show that the usefulness of the CMA in explaining theelectromagnetic scattering characteristics of GFs.

Furthermore, the results in Figs. 11 and 12(b) validate ourCMA procedure. That is, the CMA in Figs. 11 and 12(b)shows that modes 1–3, which resonate at a frequency below1.64 THz, should not contribute to the extinction cross sectionfor this specific excitation. However, mode 4, which resonatesat 1.64 THz, should contribute to the extinction cross section.The extinction cross sections, calculated using both FEKOand COMSOL, follow precisely the CMA predictions byshowing no resonances below 1.64 THz and showing the firstresonance at precisely 1.64 THz. The close agreement betweenthe extinction cross section calculated using the independentsolver COMSOL in Fig. 12 and the CMA predictions validatesour CMA procedure.

In summary, the significance of the results in Figs. 9–12 isthat they show the fundamental modes that a scatterer like aGF can exhibit. Each mode has a different current distributionand, therefore, a different electromagnetic response. If theelectromagnetic response from a specific mode is desired,the shape/size of the GF and the incident excitation can betuned such that only this mode is excited. Therefore, the modeanalysis in Figs. 9–12 provides information that can be used todesign grapheme-based sensors with optimized performance.

D. CMA Analysis of Crumpled GFs

Moving on to more complex shapes, we perform the CMAof a 92 μm2 square shaped flat GF, a 92 μm2 SC GF, anda 92 μm2 HC GF. The MSn for the first three modes isshown in Fig. 13(a)–(c) for a flat, an SC, and an HC GF.It is important to emphasize that the modal significance MSn

does not depend on the direction of the incident wave or thedirection of the incident electric field. However, MSn shows,at each frequency, the relative importance of each mode whichis controlled only by the shape and material composition of

Fig. 14. Eigen currents of HC Graphene. (a) Mode 1. (b) Mode 2.

the scatterer. The dependence on the excitation is entirelydetermined by the modal excitation coefficient Vn in (8b).Therefore, one of the main advantages of the CMA is thatit isolates the dependence on the incident excitation in Vn .For the flat GF in Fig. 13(a), modes 1 and 2 overlap due tothe symmetry in the square-shaped GF. That is mode 1 willexhibit horizontal currents similar to Fig. 10(a) and mode 2will exhibit vertical currents similar to Fig. 10(b) but bothmodes resonate at the same frequency because the GF has thesame length and width. In Fig. 13(b), modes 1 and 2 start toseparate as the GF becomes crumpled and the symmetry inthe vertical and horizontal direction breaks down. Similarly,in Fig. 13(c), the GF becomes more crumpled, diminishing thevertical and horizontal symmetries, and causing the separationbetween mode 1 and mode 2 to increase.

To highlight the separation of the horizontal and verticalmodes, Fig. 14(a) and (b) shows modes 1 and 2 for an HC GFin Fig. 13(c). From Fig. 14(a) and (b), it is clear that the firstmode has a general horizontal orientation and the second modehas a general vertical orientation. However, the crumplenessbroke the symmetry leading to small differences in modes1 and 2 shown in Fig. 14(a) and (b), respectively. Fig. 15 showsthe extinction cross section of a HC GF due to three differentelectric field directions. This is the same HC GF that was usedin Figs. 13(c) and 14. For the HC GF, the horizontal mode1 and the vertical mode 2 are well separated. The incidentelectric field is oriented parallel to the x-axis in Fig. 15(a),parallel to the y-axis in Fig. 15(b), and at a 45° angle withthe x-axis in Fig. 15(c). Therefore, in Fig. 15(c) the excitationhas both an x and y electric field component. The resonance


Fig. 15. Total extinction cross section of crumpled GF when incident field is along (a) x-axis, (b) y-axis, and (c) at 45° with x-axis.

in the Cext in Fig. 15(a) is at 1.163 THz showing that only thehorizontal mode 2 was excited. In Fig. 15(b), the resonanceis at 1.076 THz showing that only the vertical mode 1 wasexcited. Fig. 15(c) shows a broader resonance than bothFig. 15(a) and (b) since the electric field in this case hadboth a horizontal and a vertical component and, therefore,it was capable of exciting both the horizontal and verticalmodes 1 and 2. Since these two modes resonated at twodifferent frequencies, a broad resonance in Cext is achievedwhich covers the two resonance frequencies in Fig. 13(c).The separation of mode 1 and mode 2 can explain, in part,the broadband response in the Cext of the randomly orientedHC GF in Figs. 6(a) and 7(a).

The significance of the results in Fig. 15, is that it canalso lead to a more accurate interpretation of the electro-magnetic response from GF. For example, if the GF in anexperimental sample generate a broadband response similarto that in Fig. 15(c) but they are assumed to be perfectly flatthen the relaxation time, τ , in the conductivity expression (4b)might be interpreted to be larger than what it actually is. Thisoverestimation of τ stems from the fact that the larger therelaxation time, the broader the resonance, as can be seenfrom comparing Figs. 6(b) and 7(b).

On the other hand, the results in Fig. 15 show that crum-pleness is an additional degree of freedom that can be used todevelop sensors with a desired electromagnetic response. Forexample, if sensors with varying bandwidth are desired, thenwe can introduce crumpleness in varying degrees to a GF tovary the separation between the vertical and horizontal modessimilar to Fig. 15(c). The results in Fig. 15 demonstratedthe versatility of CMA in studying and using crumpleness toachieve a desired electromagnetic response.

V. CONCLUSION

In this paper, we studied the electromagnetic responseof individual crumpled GFs and how this electromagneticresponse varies with the following parameters: 1) the levelof crumpleness, 2) the size, and 3) the relaxation time inthe intraband drude conductivity model of the GFs. GFs withdifferent levels of crumpleness were generated using a coarse-grained MD simulations calibrated to generate shapes similarto those found in actual samples. The simulations showed thatcrumpled GFs with random orientation and random shapesexhibited wide variations in their total extinction cross section

spectrum. However, when GFs with different shapes but withthe same size and the same level of crumpleness were aligned,they showed similar total extinction cross sections. Theseresults show that, for the same level of crumpleness, the elec-tromagnetic response is more sensitive to the orientation ofthe GFs than to its exact shape. When the total extinctioncross section of crumpled GFs was compared to that of aflat GF, with the same size and conductivity, it was clear thatthe peak amplitude of the resonance decreased as the level ofcrumpleness increased.

The results also showed that for certain orientations ofcrumpled GFs, broadband response is exhibited in comparisonto a perfectly flat GF. These results are explained usingCMA, which calculates the fundamental modes that a GFcan exhibit. CMA quantifies the dependence of each modeon the frequency of the incident wave and also determineswhich modes will be excited due to a specific electric fieldpolarization. Using this information, we presented severalexamples by which the incident electric field or the shape ofthe GFs can be varied to excite a certain mode and, therefore,achieve a resonance in the total extinction cross section at adesired frequency. The CMA also shows that as square GFsbecome more crumpled, the symmetry between the horizontaland vertical directions breaks down, causing the resonancefrequency of the horizontal and vertical modes to separate. If aGFs is oriented in a certain way, with respect to the incidentelectric field, both the horizontal and vertical modes can beexcited explaining the broadband response achieved in someof the results.

In summary, the main contribution of this paper is that itshows that it is important to explicitly account for crumplenessin GFs when interpreting experimental measurements fromgraphene-based devices and composites. The assumption thatthe GFs are perfectly flat, when they are actually crumpled,can lead to erroneous conclusions about the properties, such asthe conductivity, of the GFs. On the other hand, this paper alsoshows that varying the crumpleness in a GF has the potential tobe used, in conjunction with CMA, as a new degree of freedomto develop novel graphene-based devices and sensors.

Extreme levels of crumpleness can affect the electron trans-port properties of the GF. This disturbance in the electrontransport properties can affect the relaxation time in thedrude model expression for the GF’s surface conductivity.However, to the best of our knowledge, there is no reported


data in the literature that correlate variations in crumplenesswith variations in the relaxation time. Moreover, GFs withhigh levels of crumples can exhibit nonhomogeneous conduc-tivities. The variations in the relaxation time and the surfaceconductivity of a GF with respect to the level of crumplenesswill be the focus of our future work. Moreover, GFs with non-homogeneous conductivities will also be studied to investigatehow the characteristic modes of GFs with nonhomogeneousconductivities will differ from those of GFs with homogeneousconductivities.

APPENDIX

COARSE-GRAINED MOLECULAR

DYNAMICS GRAPHENE MODEL

GFs with different crumpleness levels were generated bymodulating the forces between the spheres representing theGFs shown in Fig. 1(a). The steric repulsion among allthe spheres was obtained by their interaction through aWeeks–Chandler–Andersen potential [42]

UW C A(r)

= UL J (r) − UL J (rC ) − (r − rC )dUL J (r − rC )

dr

∣∣∣∣r=rC

(A1)

where r is the separation between two spheres, and rc is acutoff distance for the potential. The parameter ULJ is theLennard-Jones (LJ) potential

UL J (r) = 4ε

[(σ

r

)12 −(σ

r

)6]

if r < rc. (A2)

In (A2), σ is the length LJ parameter, and we set rc = 21/6σ ,σ is the center to center distance between two adjacents, ε isthe LJ energy parameter and, in this paper, we fix σ = 1 μmand ε = 1 kJ/mol. Our GFs model is an extension ofthe popular Kremer and Grest model [43] of semiflexiblepolymeric chains [42] to a 2-D polymer counterpart.

To obtain the connectivity along the GFs, the nearest beadsalong the square lattices were linked by a finitely extensiblenonlinear-elastic potential (UFENE)

UFENE(r) = −kFENE R2o

2Log

[1 −

(r

Ro

)2]

(A3)

where kFENE = 30ε and R0 = 1.5σ .To control the intrinsic stiffness of the GFs, we utilized two

angular potentials . First, we introduce a linear potential alongthe x and y directions

Ulinear−x = klinear−x (1 + cos(θ)) (A4)

Ulinear−y = klinear−y(1 + cos(θ)) (A5)

where θ is the angle formed by three consecutive spheres alongeach direction, i.e., in the x-direction, the triplet of beadsnumbered (1, 11, 21) or (11, 21, 32), or in the case of they direction, (1, 2, 3) or (2, 3, 4) as labeled in Fig. 1.

In this paper, we only considered the isotropic caseklinear−x = klinear−y = klinear. The second angular potentialwas given by the expression

Uperp(α) = kperp

2

(α − π

2

)2(A6)

where α is the angle formed by three neighboring beads;two beads that belong to one chain, and one bead ofthe nearest chain, i.e., the triplet (1, 2, 12) or (1, 11, 12)or (2, 12, 11) or (2, 1, 11) (see Fig. 1). We vary klinear andkperp to generate SC and HC GF, as shown in Fig. 2.

The spheres composing the GF were allowed to moveby integrating the Newtownian equations of motion forthe interaction potentials above by using a two level,three-cycle velocity-Verlet version of the rRESPA multi-ple time-step algorithms considering NVT ensembles [44].The temperature T was controlled using the Nose–Hoovermethod [45], [46]. The simulations were carried out using theprogram LAMMPS [47]. The GF were only extracted aftera time >106 time steps to ensure that the GF spheres hadreached equilibrium.

In this paper, we used klinear = kperp = 0.1 ε, to generateGFs with a PA = 0.28 which was termed as HC GF andwe used klinear = kperp = 10.0 ε, to generate GF having aPA = 0.64 that was termed as SC GF.

ACKNOWLEDGMENT

Partial contribution of NIST–not subject to U.S. copyright.

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Kalyan C. Durbhakula (S’15) received the B.Tech.degree in electronics and communication engi-neering from the MLR Institute of Technology,Hyderabad, India, in 2013, and the M.S degreein electrical engineering from the University ofMissouri–Kansas City (UMKC), Kansas City, MO,USA, in 2015, where he is currently pursuing thePh.D. degree in electrical engineering.

His current research interests include nanoelec-tromagnetics, computational electromagnetics, mul-tilayered Green’s functions, characteristic mode

analysis and EMI/EMC analysis.Mr. Durbhakula was a recipient of the UMKC’s School of Graduate Studies

Research Grant award in 2016.

Ahmed M. Hassan (S’07–M’12) received the B.Sc.(with Highest Hons.) and M.Sc. degrees in elec-tronics and communications engineering from CairoUniversity, Giza, Egypt, in 2004 and 2006, respec-tively, and the Ph.D. degree in electrical engineeringfrom the University of Arkansas, Fayetteville, AR,USA, in 2010.

From 2011 to 2012, he was a Post-DoctoralResearcher at the Department of Electrical Engineer-ing, University of Arkansas. From 2012 to 2015, hewas a Post-Doctoral Researcher at the National Insti-

tute of Standards and Technology, Gaithersburg, MD, USA. He is currentlyan Assistant Professor with the Computer Science Electrical EngineeringDepartment, University of Missouri–Kansas City, Kansas City, MO, USA. Hiscurrent research interests include nanoelectromagnetics, bioelectromagnetics,nondestructive evaluation, experimental microwave, and terahertz imaging.

Dr. Hassan was a recipient of the 2017 University of Missouri–Kansas CityCSEE Excellence in Teaching award, the 2014 Outstanding Poster Award inthe 21st Annual NIST Sigma Xi Postdoctoral Poster Presentation, and the2007 Doctoral Academy Fellowship at the University of Arkansas.


Fernando Vargas-Lara received the B.S. degree inphysics from National Polytechnic School, Quito,Ecuador, in 2003, and the Ph.D. degree in physicsfrom Wesleyan University, Middletown, CT, USA,in 2013.

Since 2013, he has been a Post-DoctoralResearcher with the National Institute of Stan-dards and Technology (NIST), Materials Science andEngineering Division, Gaithersburg, MD, USA. Hiscurrent research interests include the self-assemblyof DNA-based structures, transport and structural

properties of polymers and complex shaped particles, and electromagneticproperties of carbon nanotube and graphene composites.

Dr. Vargas-Lara is a member of the American Physics Society and the Amer-ican Chemistry Society. He received the Material Measurement LaboratoryDistinguished Associate Award, NIST, Gaithersburg, MD, USA, in 2015, andthe Soft Matter Poster Prize at the Boulder Summer School: Polymers in Softand Biological Matter, University of Colorado Boulder, Boulder, CO, USA,in 2012.

Deb Chatterjee (M’99–SM’16) received theB.Tech. degree in electrical engineering fromJadavpur University, Kolkata, India, in 1982,the M.Tech. degree from IIT Kharagpur, Kharagpur,India, in 1984, the M.A.Sc. degree from ConcordiaUniversity, Montréal, QC, Canada, in 1992, andthe Ph.D. degree from the University of Kansas,Lawrence, KS, USA, in 1998.

He was an ONR-ASEE Summer Faculty Fellowwith the Naval Research Laboratory, Washington,DC, USA, from 2009 to 2014. He is currently an

Associate Professor with the Department of Computer Science and ElectricalEngineering, University of Missouri–Kansas City, Kansas City, MO, USA. Hiscurrent research interests include asymptotic techniques, multilayer Green’sfunctions, characteristic mode theory and applications, electronic packaging,and microwave imaging.

Md. Gaffar received the B.Sc. degree in electri-cal engineering from the Bangladesh University ofEngineering and Technology, Dhaka, Bangladesh,in 2009, and the Ph.D. degree from the Schoolof Electrical and Computer Engineering, PurdueUniversity, West Lafayette, IN, USA, in 2015.

His current research interests include computa-tional electromagnetics and semiconductor physics.

Dr. Gaffar received academic awards in recog-nition of his research achievements, including theBest Poster Award (among all groups) and the Best

Project Award in Communication and Electromagnetic in EEE UndergraduateProject Workshop 2009. His research has been recognized by the IEEEInternational Microwave Symposium Best Student Paper Finalist Awardin 2013 and 2015, and the 2014 IEEE International Symposium on Antennasand Propagation Honorable Mention Paper Award.

Jack F. Douglas received the B.Sc. degree in chem-istry, the M.S. degree in mathematics from VirginiaCommonwealth University (VCU), Richmond, VA,USA, in 1979 and 1981, respectively, and the Ph.D.degree in chemistry from the University of Chicago,Chicago, IL, USA, in 1986, under the direction ofProf. Karl Freed.

He was an IBM Graduate Fellow from 1985 to1986. He was first a NATO Fellow at the CavendishLaboratory, Cambridge, U.K., under the directionof Prof. Sam Edwards and afterwards a National

Research Council Post-Doctoral Fellow at the National Institute of Stan-dards and Technology (NIST), Gaithersburg. MD, USA. He was a ResearchScientist at the Polymers Division, NIST, where he was an NIST Fellow.He has authored 400 papers covering a broad range of topics in statisticalphysics (transport properties of suspensions, percolation theory, random andself-avoiding walks, random surfaces, protein dynamics, the physics of worm-like polymers such as DNA, phase separation in polymer blends and solu-tions, self-assembly of block copolymers, polymer nanocomposites, rubberelasticity, self-assembly and molecular binding, crystallization under far fromequilibrium conditions, the dynamics of grain boundaries, the dynamicsof nanoparticles and thin films, photopolymerization, renormalization grouptheory, path-integration, and fractional calculus, surface interacting polymers,dewetting of polymer films, supercooled liquids and glasses, and entanglementinteractions in polymer fluids.

Dr. Douglas was a recipient of Phi Kappa Phi and Sigma Xi, and theAmerican Chemical Society Senior Award at VCU, and a Bronze Medal—Department of Commerce. He was an APS Fellow and did a 5-year rotationas an Editorial Board Member of Physical Review Letters.

Edward J. Garboczi received the B.S. (HighestHons.) degree in physics and the Ph.D. degreein condensed matter physics (theory) from Michi-gan State University, East Lansing, MI, USA,in 1980 and 1985, respectively.

From 1985 to 1988, he was a Research Physicistat Armstrong World Industries, Inc., Lancaster, PA,USA. From 1988 to 2014, he was a Physicist atthe National Institute of Standards and Technology,Materials and Structural Systems Division, Gaithers-burg, MD, USA, where he was a Group Leader, and

a Fellow. In 2014, he joined the Division of Applied Chemicals and MaterialsDivision, NIST Boulder. His current research interests include computationalmaterials science, porous materials, electrical properties of nanocomposites,and 3-D particle shape analysis of a wide range of particles including gravel,sand, cement, chemical explosives, lunar soil, and metal powder for additivemanufacturing.

Dr. Garboczi is a Fellow of the American Concrete Institute, the AmericanCeramic Society (ACerS). He was a recipient of the ACI Robert E. PhilleoAward, the ACerS Della Roy Lecture Award, the ACerS Edward C. Henry BestPaper Award (Electronics Division), and a Silver Medal from the Departmentof Commerce for the creation of the Virtual Cement and Concrete TestingLaboratory.

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