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Computationally efficient modeling and analysis of conductivity and sensitivity of CNT/polymer composites Rahman, Rubaiya 2015

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Computationally Efficient Modeling and Analysis of Conductivityand Sensitivity of CNT/Polymer CompositesbyRubaiya RahmanM. Sc., Bangladesh University of Engineering and Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2015c© Rubaiya Rahman, 2015AbstratIn this PhD thesis, an eient model of ondutivity for arbon nanotube (CNT)/po-lymer omposites is developed, onsidering the eet of intertube tunneling throughpolymers, and the eletromehanial properties of the omposites are estimated. Thestatistial nature of intertube distane is rst investigated through numerial analysisfor both two dimensional and three dimensional networks of CNTs. Considering theintertube distane as the key parameter of tunneling eet, analytial models aredeveloped for tunneling ondutivity at the CNT juntions and the overall ondu-tivity of the omposites. The model of omposite ondutivity provides signiantlylower omputational ost as ompared to the numerial resistive network models, withreasonable auray. By inorporation of eletron tunneling eets, this model alsoprovides loser approximation to experimental results in omparison to the modelsbased on perolation theory, whih are highly relevant for ller/polymer ompositeappliations designed around the perolation threshold.Using the ondutivity model, the sensitivity of the omposite lms is estimatedin presene of an organi gas. The hange in the lm resistane due to gas absorptionis investigated for dierent CNT and gas onentrations. From the phase of reetedradio frequeny (RF) signal, the wireless gas sensitivity is estimated for a losslesstransmission system terminated with a omposite lm as the load. Films with lowerller onentration is found to have higher gas sensitivity and higher wireless sensitiv-ity within a low range of gas pressure. This work is useful for design and developmentiiAbstratof biohazard gas sensors for real-time remote monitoring.Furthermore, the sensitivity of the omposite lms is estimated under the ap-pliation of tensile strain. The inuene of varying lm thikness on the intertubedistane in omposite lms is analyzed numerially. Then our analytial model is em-ployed to estimate the omposite ondutivity and lm sensitivity under mehanialstrain. The partial alignment of CNTs introdued by the lm thikness less than theCNT length is observed to have signiant inuene on the omposite ondutivityand strain sensitivity speially at low CNT onentration. The numerial results areompared with literature reports and experimental results. This work is helpful forstrain sensing and strethable swithing appliations.iiiPrefaceChapters 25 are based on works under the supervision of Professor Peyman Servati.For Chapter 4, I have collaborated with Dr. Saeid Soltanian, who worked as a researchassociate at the Flexible Electronics and Energy Lab (FEEL).Unless otherwise stated, for all chapters and corresponding papers, I conductedthe literature survey on related topics, identified the potentials and challenges, andperformed simulations, experiments, and analyses. I wrote all the paper drafts forwhich I am the first author. My supervisors guided the research, validated the anal-yses, and gave comments on improving the manuscripts. Dr. Soltanian helped mefor the experimental setup and training on relevant instruments and revising onemanuscript.Publications related to Chapter 2:• R. Rahman and P. Servati, Effects of intertube distance and alignment ontunnelling and strain sensitivity of nanotube/polymer composite films, Nan-otechnology, vol. 23, pp. 0311, Jan 2012.• R. Rahman and P. Servati, Analytical modeling of the transparency and sheetresistance of SWNT thin films, in 23rd Annual IEEE Photonics Society Meet-ing, Denver, CO, USA, Nov, 2010.ivPrefacePublications related to Chapter 3:• R. Rahman and P. Servati, Efficient analytical model of conductivity of CNT/polymer composites for wireless gas sensors, IEEE Trans. on Nanotechnology,vol. 14, pp. 112, Jan 2015.Publications related to Chapter 4:• R. Rahman, S. Soltanian, and P. Servati, Coupled effects of film thickness andfilter length on conductivity and strain sensitivity of carbon nanotube/polymercomposite thin films, submitted.• R. Rahman and P. Servati, Orientation of fillers in CNT/polymer compositeinterfacial layer for enhancing charge transportation, in Society for InformationDisplay (SID) International Symposium, Vancouver, BC, Canada, May 2013.vTable of ContentsAbstrat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPrefae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Carbon nanotube, its network and omposites . . . . . . . . . . . . . 11.2 Need for predition of omposite behavior . . . . . . . . . . . . . . . 51.3 Perolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Researh objetive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12viTable of Contents2 Analysis of 2D network of CNTs . . . . . . . . . . . . . . . . . . . . 142.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Constrution of 2D mirostruture ell . . . . . . . . . . . . . . . . . 152.3 Intertube distane analysis . . . . . . . . . . . . . . . . . . . . . . . 172.4 Tunneling resistane analysis . . . . . . . . . . . . . . . . . . . . . . 232.5 Strain sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Analysis of unonned 3D network of CNTs . . . . . . . . . . . . . 373.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Role of tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Constrution of 3D mirostruture unit ell . . . . . . . . . . . . . . 443.4 Analytial model of tunneling ondutivity . . . . . . . . . . . . . . 463.5 Analytial model of omposite ondutivity . . . . . . . . . . . . . . 503.6 Sensitivity of gas sensors . . . . . . . . . . . . . . . . . . . . . . . . 603.7 Wireless sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Analysis of onned 3D network of CNTs . . . . . . . . . . . . . . . 744.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Constrution of numerial samples of nanoomposite . . . . . . . . . 764.3 Analysis of intertube distane . . . . . . . . . . . . . . . . . . . . . . 784.4 Analysis of ondutivity . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Experimental veriation of thikness eet on ondutivity . . . . . 844.5.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5.2 Measurement and results . . . . . . . . . . . . . . . . . . . . 894.6 Analysis of strain sensitivity . . . . . . . . . . . . . . . . . . . . . . . 92viiTable of Contents4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.1 Model for compressive strain application . . . . . . . . . . . . 1045.2.2 Application in non-uniform swelling of films . . . . . . . . . . 1045.2.3 Model for composite fibers . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106viiiList of Figures1.1 Shemati diagram of single walled and multiwalled arbon nanotubes [8℄ 31.2 Illustration of the relation between the hirality and nanotube indies.Chiral vetor Ch = na1 +ma2 is dened on a graphene sheet by unitvetors a1 and a2 and the hiral angle with respet to the zigzag axis [9℄ 31.3 Shemati diagram of (a) non-perolating lusters and (b) perolatingnetwork of stik-like llers onneting the left and right ends of thematrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Illustration of eletron transfer through a CNT network via CNT pathsand tunneling gaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Illustration of the hierarhial organization of work done in the thesis. 122.1 Sample of CNT/polymer omposite thin lm as a 2D square ell witheletri eld applied on it and a perolating luster shown within thedashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Shemati diagrams of a CNT pair with tunneling gap between them.The resistane along the CNTs are denoted as Rc and the tunnelingresistane as Rt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Intertube distane for varying CNT onentration in a random 2Dnetwork. The inset shows the zoomed in gure of one data point witherror bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19ixList of Figures2.4 The log-log plot of the intertube distane for varying CNT onentra-tion and the liner t . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Shemati diagram of CNT 2D-network with θµ = 90◦ and θµ = 15◦are shown in (a) and (b) respetively. Figure () shows the intertubedistane for varying alignment order of CNTs at dierent volume fration. 222.6 Intertube distane at varying CNT aspet ratio for a planar CNT on-entration of 4.59/µm2. . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Intertube tunneling resistane of CNT/polymer omposites at varyingorientation ut-o angles of CNTs for polymers with dierent tunnelingbarrier height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.8 Intertube tunneling resistane at varying orientation ut-o angles ofCNTs for dierent CNT volume fration. . . . . . . . . . . . . . . . . 262.9 Intertube tunneling resistane at varying CNT aspet ratio for a planarCNT onentration of 4.59/µm2. . . . . . . . . . . . . . . . . . . . . 272.10 Tunneling resistane hange ratio at small mehanial strains for dif-ferent CNT onentration. . . . . . . . . . . . . . . . . . . . . . . . . 282.11 Tunneling resistane hange ratio at large mehanial strains for dif-ferent CNT onentration. . . . . . . . . . . . . . . . . . . . . . . . . 292.12 Tunneling gauge fator as a funtion of CNT volume fration at smallmehanial strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.13 Tunneling resistane hange ratio at small mehanial strain for dier-ent orientation ut-o angles. . . . . . . . . . . . . . . . . . . . . . . 312.14 Tunneling resistane hange ratio at large mehanial strain for dier-ent orientation ut-o angles. . . . . . . . . . . . . . . . . . . . . . . 32xList of Figures2.15 Tunneling gauge fator as a funtion of orientation ut-o angle ofCNTs at small mehanial strains. . . . . . . . . . . . . . . . . . . . . 332.16 Tunneling resistane hange ratio at small strain for dierent aspetratio of CNTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.17 Tunneling resistane hange ratio at large strain for dierent aspetratio of CNTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.18 Tunneling gauge fator and resistane hange ratio at small strain fordierent aspet ratio of CNTs. . . . . . . . . . . . . . . . . . . . . . . 363.1 A 3D ubi sample of CNT/polymer omposite. . . . . . . . . . . . . 433.2 A shemati diagram of (a) top view of the 3D CNT network witheletri eld applied on it. The perolating luster is shown within thedashed line, and (b) a CNT pair with tunneling gap dt between them.The resistane along the CNTs are denoted as Rc and the tunnelingresistane as Rt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Tunneling ondutivity at varying intertube gap for dierent polymers.Our model is ompared with the numerial data of literature [26℄ . . 493.4 Shemati diagram of ondutive pathways with tunneling juntionsaross the CNT/polymer omposite sample and its representative re-sistive network model. . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Intertube distane for varying CNT onentration for a random 3Dnetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 MWNT/epoxy omposite ondutivity at varying CNT onentration.Our model is ompared with the literature data based on perolationmodel [21℄, numerial simulation [32℄, and experimental results [26, 67℄. 573.7 Shemati diagram of hange in intertube distane due to lm swelling. 61xiList of Figures3.8 Inrease in lm thikness at varying gas onentration for lms withdierent CNT onentration. . . . . . . . . . . . . . . . . . . . . . . . 633.9 Inrease in intertube distane at varying gas onentration for lmswith dierent CNT onentration. . . . . . . . . . . . . . . . . . . . . 643.10 Composite ondutivity at varying gas onentration for lms withdierent CNT onentration. . . . . . . . . . . . . . . . . . . . . . . . 653.11 Resistane hange ratio of lms with dierent CNT onentration atvarying gas pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.12 Reetion (S11) phase for varying real part of load impedane. . . . . 713.13 Reetion (S11) phase of lms with dierent CNT onentration atvarying gas pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1 Illustration of (a) a 3D ubi sample of CNT/polymer omposite and(b) a randomly oriented ller making orientation angles with X-Y andX-Z plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Shemati diagram of CNT/polymer omposite lms with dierentthiknesses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 Intertube distane at varying CNT onentration for CNT/polymeromposite lms with dierent thikness. . . . . . . . . . . . . . . . . 814.4 Intertube distane at varying sample lm thikness for dierent CNTonentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5 Intertube distane at varying alignment order of CNTs for dierentCNT volume fration. . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6 Condutivity of (a) MWNT/epoxy and (b) MWNT/PMMA ompos-ites at varying lm thikness to ller length ratio for dierent CNTvolume fration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xiiList of Figures4.7 Composite ondutivity at varying alignment order of CNTs for dier-ent CNT volume fration. . . . . . . . . . . . . . . . . . . . . . . . . 864.8 (a) Dropasted samples of SWNT/DMF solution. (b) The SEM imageof a SWNT/DMF sample shows good dispersion. . . . . . . . . . . . 874.9 (a) SWNT/PMMA omposite lm samples with dierent lm thik-ness having same CNT onentration. (b) The SEM image of a SWNT/PMMAlm with its surfae and edge. The CNTs look well distributed in thepolymer matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.10 (a) Composite lm thikness prole at the omputer monitor and (b)the extrated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.11 Change in lm thikness with the spin speed at dierent CNT weightfrations. The error bars are inluded by using std. dev. of thethikness data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.12 Experimental data of SWNT/PMMA omposite ondutivity at vary-ing lm thikness for dierent CNT weight fration. . . . . . . . . . . 914.13 Intertube distane at applied strain, for a CNT onentration φ =1.14% and dierent lm thikness. . . . . . . . . . . . . . . . . . . . . 934.14 Condutivity of MWNT/epoxy omposite lms (φ = 1.14%) at appliedstrain for dierent lm thikness. . . . . . . . . . . . . . . . . . . . . 944.15 Resistane of MWNT/epoxy omposite lms (φ = 1.14%) at appliedstrain for dierent lm thikness. . . . . . . . . . . . . . . . . . . . . 954.16 Resistane hange ratio of MWNT/epoxy omposite lms (φ = 1.14%)at applied strain for dierent lm thikness. . . . . . . . . . . . . . . 96xiiiList of AbbreviationsCNT Carbon NanotubeDMF DimethylformamideGF Gauge FatorICCG Inomplete Cholesky Conjugate GradientITO Indium Tin OxideMC Monte CarloMIM Metal-Insulator-MetalMWNT Multi-Walled Carbon NanotubeNT NanotubeOLED Organi Light Emitting DiodePDMS PolydimethylsiloxanePMMA Poly(methyl metharylate)PVP PolyvinylpyrrolidoneRF Radio FrequenySBS Sik Building SyndromeSEM Sanning Eletron MirosopeSHM Strutural Health MonitoringSWNT Single Walled Carbon NanotubeTC Transparent CondutorVOC Volatile Organi CompoundxivAcknowledgmentsFirst of all, I would like to express my deep and sincere gratitude to my supervisor,Dr. Peyman Servati, for his invaluable support and guidance throughout my PhDprogram. I am indebted to him for his time, patience, and effort on my research,validating the analysis, editing the manuscripts, and providing critical suggestionsand comments. I would like to thank our research associate, Dr. Saeid Soltanian forhis great help and guidance for the experimental setup, instrumental training, andresult analysis.I would also like to thank Professors Lucas Chrostowski, Karen Cheung, andFarrokh Sassani for serving as the Chairs during my PhD qualifying, departmental,and university examinations, respectively. I am grateful to Professors John Madden,Frank Ko, and Shahriar Mirabbasi for their time and effort in evaluating my workand providing valuable feedback and suggestions. Thanks to Professors FariborzTaghipour and Mu Chiao for providing critical feedback as the university examiners,and Professor Ash Parameswaran for his important review as the external examiner.Great thanks go to my supportive and knowledgable colleagues at Flexible Electronicsand Energy Lab (FEEL). Finally, I would like to thank University of British Columbia(UBC) and Natural Sciences and Engineering Research Council of Canada (NSERC)for their financial support throughout my PhD studies.xvDediationTo my familyxviChapter 1Introdution1.1 Carbon nanotube, its network and ompositesDue to the remarkable thermal, eletrial, mehanial and optial properties of arbonnanotubes (CNT), CNTs have attrated inreasing attention of researhers sine theirdisovery by Iijima in 1991 [1℄. In reent years, lms made from network of CNTs andtheir polymer omposites have attrated onsiderable researh interest due to theirpossible use in future nanoeletronis. They have been widely investigated over thelast few deades for multifarious appliations inluding large-area exible eletronis,strain sensors, bio-sensors, transparent eletrodes, transistors and photovoltai (PV)devies [25℄. Dierent appliations of CNTs relate to their dierent spei proper-ties. Hene it is important to have an overview of their properties before we exploretheir potential appliations.CNTs are basially allotropes of arbon making hexagonal networks of arbonatoms. They an be thought of as one or more layers of graphite rolled-up into aylinder. The ylindrial strutures an be of approximately 1 to 80 nm in diameterand 1 to 100 mirons in length [6, 7℄. They an be ategorized into two types de-pending on the number of graphene layers in eah ylindrial struture- single-wallednanotubes (SWNTs) and multi-walled nanotubes (MWNTs). SWNTs onsists of onlyone single layer of graphene ylinder whereas the MWNTs have two or more layers, asshown in Fig. 1.1. The SWNTs an be either metalli or semionduting depending1Chapter 1. Introdutionupon the hirality and tube diameter. The hirality is determined by the sheet dire-tion in whih the graphite sheet is rolled to form the ylinder. It an be understoodfrom Fig. 1.2.If the lattie onstant in the graphite sheet is a, then the diameter DCNT and thehiral angle ΘCNT of the CNT an be obtained by a pair of indies (n,m) using thefollowing relations-DCNT =a√m2 +mn + n2π , (1.1)ΘCNT = arctan[ −√3n2m+ n], (1.2)The relation between n and m determines the hirality and eletrial property of theCNT. The hirality an be of three types. For a given SWNT, if n = m i.e., thehiral angle ΘCNT = 30◦, then the nanotube is of armhair type. If n = 0 or m = 0i.e., ΘCNT = 0◦, then the nanotube is alled zigzag type. And for any other values ofn and m and hiral angles between 0◦ and 30◦ the CNTs are of hiral type. If n = mthen the CNT is metalli. If n−m is a multiple of 3, then the tube is semiondutingwith a very small bandgap, hene nearly metalli. For any other relation of (n,m),the nanotube is a moderate semiondutor with a bandgap inversely proportionalto the tube diameter [10℄. The MWNTs are formed of either nested CNT shellsor innamon roll like struture that leads to their metalli harateristis. Boththe metalli SWNTs and MWNTs an transport eletrons without sattering, i.e.,ballistially due to their nearly one dimensional eletroni struture whih enablesthem to arry high urrents with negligible heating [2℄. In general, individual CNTsan have the eletri-urrent-arrying apaity up to approximately 1000 times higher2Chapter 1. Introdutionc1f1Figure 1.1: Shemati diagram of single walled and multiwalled arbon nanotubes [8℄Armchair tubule (4,4)a1a2Zigzag tubule (7,0)na1ChĬCNTma2Chiral tubule (4,3)Two-dimensional graphene sheetFigure 1.2: Illustration of the relation between the hirality and nanotube indies.Chiral vetor Ch = na1 +ma2 is dened on a graphene sheet by unit vetors a1 anda2 and the hiral angle with respet to the zigzag axis [9℄3Chapter 1. Introdutionthan opper [11℄. This indiates the potential of CNTs as interonnets in large-saleintegrated nanoeletroni devies. When multiple number of CNTs form a network,the high aspet ratio of CNTs leads to one of the smallest perolation thresholdsamong lms made from perolating stiks [12℄. If introdued in an insulating polymermatrix even at a very low onentration, the CNTs form an eletrial perolatingnetwork that turns the insulating matrix to a onduting one [13℄. This opens upthe potential of CNT/polymer omposites as materials for ondutive lm, swithingappliation, resistive sensor, et.Due to the strong arbon-arbon bonds in graphite, CNTs posses the highestmehanial strength as ber. Several studies report that MWNTs are extremelyexible and resistant to frature when subjeted to large strain [14℄. Thus CNTsan be applied as reinforement material in omposites, as well as for strain sensingwhere high sensitivity and repetitive straining are onerned. On the other hand,the low density (1.3 g/m3) of CNTs failitates the lighter weight of materials whenintrodued with CNTs while inreasing the durability.The geometry of the CNT also oers advantage in many appliations. Nanosen-sors based on CNT and its polymer omposites are thousands of times smaller thaneven MEMS sensors and onsume less power. Again, ompared to the silion basedpiezoresistive sensors, they are less sensitive to the temperature variation [2℄. Thesefeatures make them highly suitable as implantable sensors espeially in biomedialsensing appliations.Besides the exellent eletrial and mehanial properties, CNT and its ompositeshave proved superiority over many other materials in improving the optial proper-ties of devies. Through proper design and fabriation, thin lms made of CNTsand its omposites an exhibit high ondutivity and transpareny at the same time.4Chapter 1. IntrodutionCNT/polymer omposite nanobers an potentially replae the omparatively ostlyindium tin oxide (ITO) lms used as transparent ondutive (TC) eletrodes in thinlm solar ells [3, 4℄. Also CNT/polymer omposites have been reported to demon-strate exellent eld enhaning apability when applied in organi light emittingdiodes (OLEDs) as a harge transportation layer [5℄. That an result in a lower op-erating voltage and improved emission in OLED devies, thus reduing the operatingost of OLED appliations, i.e. large area and exible panels, solid state lighting,et.1.2 Need for predition of omposite behaviorDespite all the attrative features of CNTs and CNT/polymer omposites with abroad range of appliations in nanoeletroni devies, there is still a gap in theoreti-al understanding of the properties in CNT/polymer omposite lms. There has notbeen suient progress in the understanding of the harateristis of random networksformed by suh tubes. To obtain the highest benet from the exeptional propertiesof the CNTs in its omposite appliations, we need to gain a thorough knowledgeabout the eletrial and mehanial harateristis of its omposites and the ondu-tion mehanism, load transfer mehanism, et. through them. The theoretial andomputational predition of the eletrial, mehanial or optial behavior of arbonnanotube/polymer omposites is of ruial importane before they are used in realappliations. This an be done by omprehensive analytial and numerial study.Through modeling and simulation of arbon nanotube/polymer omposites we anobtain initial guidelines for the development of nanoomposite appliations that anhelp in minimizing the ost and time and optimizing the sope of real experiments.Sine the eletrial property of the CNT/polymer omposites plays the most in-5Chapter 1. Introdutionuential role in determining the ondutivity and sensitivity of the omposite devies,the understanding of the ondution mehanism in CNT omposites is ruial. Therole of dierent parameters on the ondutivity and other eletrial properties areyet to be revealed thoroughly. Computational analysis and modeling of the ondu-tivity an help in this regard. However, proper seletion of mathematial models andomputational methods is highly important sine it relates to the eieny of ompu-tation in terms of ost, time and performane. The issue of modeling and numerialanalysis will be addressed in this work. Before we disuss the ondutivity modeling,we need to know the dierent mehanisms of ondution in the CNT/polymer om-posites that have worked as the basis of modeling over the last deade. The followingtwo setions disuss these mehanisms and the onventional models based on them.1.3 PerolationIn a CNT-polymer omposite, the CNTs at as ondutive llers in the insulatingpolymer matrix. In a ondutor-insulator omposite, if we gradually inrease thenumber of ondutive llers, it builds up disrete lusters of llers. As the numberof llers inreases, the number of the lusters also inrease and their size gets bigger.After reahing a ertain ller volume fration the lusters get onneted and formsa ontinuous ondutive path aross the insulating polymer matrix. This volumefration of llers is known as the perolation threshold, φc. At this ller onentration,the eletrial ondution eetively begins through the perolating ller network andthe omposite makes transition from the insulating to ondutive state. This isillustrated in Fig. 1.3 for stik-like llers.Various analytial and numerial approahes an be found in literature [1517℄ forestimation of CNT omposite ondutivity and sheet resistane apart from the exper-6Chapter 1. Introdution( )a Electrodes(b)Figure 1.3: Shemati diagram of (a) non-perolating lusters and (b) perolatingnetwork of stik-like llers onneting the left and right ends of the matrix.imental works. Most of the works have been done within the framework of traditionalperolation theory [18, 19℄ assuming the CNT llers to form a perolating networkaross the matrix. Perolation theory is well established for disordered systems. Ithas found appliations in a wide variety of elds, suh as polymers siene, ompos-ites and porous media, information tehnology, wireless ommuniations, medial orbiologial studies, speies evolution, et., and now is being used for the analysis ofeletrial ondution in nanoomposites for more than two deades [20℄. Most of theperolating systems require omputationally omplex simulation by the Monte Carlo(MC) method- a omputational algorithm that rely on repeated random sampling toobtain numerial results. Traditionally for a perolating network the ondutivity isgiven by [21℄,σ = σo(φ− φc)tforφ > φc, (1.3)7Chapter 1. Introdutionwhere t is the ritial exponent, φ the volume fration of the ondutive ller, φc theperolation threshold, and σo the eletrial ondutivity of the ller. Here, t dependson dimensionality of the ller network and is ideally independent of the aspet ratio ororientation of the llers, whereas φc is a funtion of both aspet ratio and orientationorder of the llers [17℄. Usually both these parameters are determined experimentally.Several parameters inluding type, aspet-ratio, alignment order, synthesis method,surfae modiation of CNTs as well as polymer type and dispersion method inuenethe maximum ondutivity of the omposite for a given CNT onentration [22℄. Mostof the experimental results suggest an exponential inrease of lm ondutivity withthe inrease of CNT onentration [12, 23℄. Usually larger aspet ratio and randomisotropi orientation of tubes lowers the perolation threshold, and thus enhane theondutivity of a CNT thin lm [12, 24℄.1.4 TunnelingTunneling is the seond mehanism of eletri ondution through a ller networkwhen the physially onneted perolating network is still absent. The measuredresistane of the CNT/polymer lm arises from both the intrinsi resistane alongthe CNTs and the intertube juntion resistane between CNT pairs along ondutionpath. Again, the juntion resistane onsists of resistane due to diret CNT-CNTontat and resistane that arises from eletron tunneling. Here by tunneling re-sistane, we mean the resistane between those CNT pairs whih are not at diretontat, but are separated within eletron tunneling gap. Ideally in a perolating net-work the tube-tube ontats are made by intimate ontat, that means they touheah other. But in pratie in ase of most of the juntions of a ller network ina CNT/polymer omposite, a thin layer of the insulating polymers (or surfatant)8Chapter 1. Introduction !"#$%& '( )*$&+,-$/+*- +  !"#$% &'() *++#,-$.-!/0 /1 %+%-$/0 -$.0,1%$ -3$/#"3 . 456 0%-7/$8 9!. 456 :.-3,.0; -#00%+!0" ".:,':$%9%0-, ;!$%- /0-.- <%-7%%0 -3% 456, =>?℄' 63#, -3% %+%-$! /0;#-!/0 .- 456A456 B#0-!/0, /#$, -3$/#"3 -#00%+!0" #0;%$ .::+!%; 9/+-."% .0; 8%%:, /0-!0#!-C/1 -3% %+%-$/0 -$.0,1%$ .$/,, -3% :/+CD%$ D.-$!EF ., ,3/70 !0  !"' &'(' *0 1.-F$%%0- ,-#;!%, ,#""%,- -3.- -#00%+!0" :+.C, -3% D.B/$ $/+% !0 ;%-%$D!0!0" -3% %+%-$!A.+ /0;#-!9!-C /1 . 456G:/+CD%$ /D:/,!-% =>?F >H℄' 63% -3!0 !0,#+.-!0" +.C%$ /1:/+CD%$ <%-7%%0 -3% 456, .-, ., -3% -#00%+!0" <.$$!%$F #0+%,, -3% +.C%$ -3!80%,,"/%, 3!"3%$ -3.0 0D =>I℄' 63% %+%-$!.+ /0;#-!9!-C /1 !0;!9!;#.+ 456, 9.$!%,7!-3!0 -3% $.0"% /1 -/ JGD =>KF>L℄F 73%$%., D/,- /1 -3% D%.,#$%; /D:/,!-%/0;#-!9!-!%, $.0"% 1$/D -/ JGD .- /$ .</9% :%$/+.-!/0 -3$%,3/+;, =MNFM&℄'63!, !0 -#$0 !0;!.-%, -3.- -3% -#00%+!0" $%,!,-.0% :+.C, . ;/D!0.0- $/+% !0 -3%/D:/,!-% $%,!,-.0%F !0 :.$-!#+.$ 1/$ O+D, -3.- 3.9% +/7%$ /0%0-$.-!/0 /1 -#<%,F., !, -3% .,% 1/$ D.0C :$.-!.+ .::+!.-!/0,F !'%' -$.0,:.$%0- /0;#-/$,F ,-$.!0,%0,/$,F %-' 6#00%+!0" $%,!,-.0% 3., <%%0 $%:/$-%; -/ "$%.-+C !0P#%0% /D:/,!-%/0;#-!9!-C .0; ,-$.!0 ,%0,!-!9!-C =>IF M>℄'Lare what and dots are what?Figure 1.3: Illustration of electron transfer through a CNT network via CNT pathsand tunneling gaps. The electrons are shown by black dots and the CNTs by bluelines in this schematic diagram.prevents direct contact between the CNTs [25]. Thus the electric conduction at CNT-CNT junctions occurs through tunneling under applied voltage and keeps continuityof the electron transfer across the polymer matrix, as shown in Fig. 1.4. In fact ,recent studies suggest that tunneling plays the major role in determining the electricalconductivity of a CNT/polymer composite [25, 26]. The thin insulating layer ofpolymer between the CNTs acts as the tunneling barrier, unless the layer thicknessgoes higher than 1.8 nm [27]. The electrical conductivity of individual NTs varieswithin the range of 1 4 to 107 S/m [28, 29], whereas most of the measured compositeconductivities range from 10−5 to 102 S/m at or above percolation thresholds [30,31].This in turn indicates that the tunneling resistance plays a dominant role in thecomposite resistance, in particular for films that have lower concentration of tubes,as is the case for many practical applications, i.e. transparent conductors, strainsensors, etc. Tunneling resistance has been reported to greatly influence compositeconductivity and strain sensitivity [27, 32].9Chapter 1. Introdution1.5 MotivationIn spite of being widely used in dierent multiphase systems, the perolation modelhas some limitations when applied in CNT/polymer omposites. Firstly, the pero-lation and other assoiated models allow estimation of eletroni behavior for CNTonentrations above the perolation threshold, and the perolation threshold needsto be known beforehand through experiments. This limits it's appliability to only theller onentration region above the perolation threshold. It would be more usefulif the model an estimate the omposite ondutivity below and above the perola-tion threshold region, and thus an estimate the perolation threshold for a givenCNT aspet ratio, alignment or polymer type before onduting the experiments.This is quite important for many novel uses of CNT omposites that use lower thanperolation or lose to perolation onentrations. Seondly, it assumes the llers toform a perolating network with intimate ontats between CNTs at the juntions.Hene, the ondutivity arising from tunneling, a major omponent of ondutivityas found in reent studies, is not taken into aount [21℄. Tunneling ondution is aprevalent ase in polymer/CNT omposites, and beomes more prominent when theller onentration is low or near the perolation threshold region.Sine the perolation models ignore the tunneling eet, we need to onsider alter-native methods that an inlude this eet in the ondutivity estimation of the CNTomposites. Resistor network models have been used suessfully in some numeri-al analysis [26, 33, 34℄ to simulate the omposite mirostruture of nanoomposite.These models an work better than the perolation models sine we an onsiderthe tunneling ondution between CNTs while using them. In partiular, the resultsby [26,32℄ inludes the eet of eletron tunneling in the omposite ondutivity. Butthis method has some limitations too. Due to the multi-step alulations and huge10Chapter 1. Introdutionnumber of omputations involved in eah step, these numerial models beome notonly laborious to apply, but also require onsiderable omputer resoures inludingmemory and CPU time. Even for a small mirostruture simulation, the 3D resis-tor network model involves high omputational ost [21℄. Hene, its appliation innumerial studies is still very limited.Considering all these fats, it is obvious that we are still in need of a model thatan provide ondutivity estimation in both above and below the perolation thresh-old regions, an predit the perolation threshold without arrying out experiment orextensive numerial simulation, inludes the tunneling eet as the major mehanismof eletrial ondution, and an ahieve all these goals inurring low omputationalost. Also the role of many parameters i.e., CNT onentration, alignment, aspetratio, omposite lm thikness below and above the ller length, et., that inuenethe nanoomposite ondutivity and sensitivity, are yet to be revealed thoroughly. Adetailed and systemati analysis of the eet of these parameters an help in deeperunderstanding of the omposite behavior, solve urrent problems in the experiments,and failitate with eient design and optimization of nanoomposite devies. Thesereasons worked as the motivation of this thesis to develop an analytial model of on-dutivity for CNT/polymer omposites and analyze the eet of dierent parameterson ondutivity and sensitivity.1.6 Researh objetiveIn this work, the researh goal is primarily to develop a omputationally eientmodel of ondutivity for CNT/polymer omposites. The model inludes the ele-tron tunneling eet in the ondution mehanism. Also it is able to estimate thenanoomposite ondutivity in a wide range of CNT onentration, both below and11Chapter 1. Introdution2D network 3D unconfined network 3D confined network for Chapter 2 for Chapter 3 for Chapter 4Figure 1.5: Illustration of the hierarhial organization of work done in the thesis.above the perolation threshold. Seondly, this thesis aim to study the inueneof dierent parameters, i.e., CNT onentration, aspet ratio, alignment, ompositelm thikness, et., on the omposite ondutivity and tunneling resistane for bothtwo-dimensional and three-dimensional CNT networks. Finally the strain sensitivityand gas sensitivity of the omposite lms will also be analyzed in terms of dierentparameters in the ontext of the model developed.1.7 Thesis outlineThe researh is presented in ve hapters in this thesis. This hapter provided thebakground, motivation and objetives of this work. The detailed literature reviewalong with the orresponding topis of work is given in the three following hapters.Chapter 2 desribes the work on two dimensional CNT networks. The eet of dif-ferent parameters on the intertube distane in a 2D CNT network is studied and amodel of intertube distane is presented in this hapter. The tunneling resistane andstrain sensitivity are also analyzed for quasi-2D lms of CNT/polymer omposites.12Chapter 1. IntrodutionIn Chapter 3, the work moves to three dimensional unonned CNT networks. Here,models developed for intertube distane, tunneling ondutivity, and omposite on-dutivity. The work here introdues a low-ost analytial approah of ondutivityestimation of omposite lms. The models are veried with other results in literature.After that, gas sensitivity of CNT/polymer based sensors is estimated and analyzedfor dierent parameters. Chapter 4 ontinues work on CNT 3D network with on-ned thikness. It provides study on eet of lm thikness ompared to the llerlength on ondutivity and strain sensitivity of omposites. The results are veriedwith experiments. Finally, the ontributions of this entire researh is summarized inChapter 6, along with a brief disussion on the future diretion of this work.13Chapter 2Analysis of 2D network of CNTs2.1 IntrodutionThe exellent eletrial, optial and mehanial properties of CNTs have made thema potential andidate for nanoomposite thin lms appliations. When introduedin a thin polymer matrix, the CNTs form a quasi-two-dimensional network over thepolymer lm. If the CNT onentration is above the perolation threshold, this 2DCNT network works as a thin onduting layer that an be applied as thin lm on-dutors in appliations like large-area exible eletronis, strain sensors, transparenteletrodes in solar ells, et. When embedded in polymer lms, the high mehanialstrength of the CNTs makes them suitable to reinfore the polymers and enhanethe durability of the thin omposite lms. As reported by Yu et al. [35℄ the Young'smodulus of individual multiwalled nanotubes (MWNT) varies within 0.27−0.95TPa,whereas for single walled nanotubes (SWNT), this value is found to be higher, around0.32−1.47TPa [36℄. CNTs also have been reported to enhane the piezoresistive prop-erty of the omposite lms and show higher sensitivity than the onventional strainsensors [37℄.In a thin CNT/polymer omposite lm, the piezoresistive property is mainlyinuened by the tunneling ondution through the CNT network. The tunnelingurrent in turn depends on the distane between neighboring tubes. Therefore theintertube distane in a CNT network plays ruial role in determining the tunneling14Chapter 2. Analysis of 2D network of CNTsondutivity, piezoresistivity, and onsequently the sensitivity of the omposite lms.Many parameters, i.e., the CNT onentration, alignment order, aspet ratio, et.inuene the intertube distane. In this hapter, a Monte Carlo based statistialmethod is used to investigate the role of intertube distane and alignment on thetunneling resistane and strain sensitivity of CNT/polymer omposite thin lms,so as to provide a deeper understanding of the ondution mehanism through theomposite lms. Considering the widespread appliation of omposite thin lms,we onduted the numerial analysis on a 2D CNT network here. Here, we haveused randomly generated 2D samples with dierent onentration φ, length LCNT ,diameter DCNT , and alignment angle θ of nanotubes and polymer tunneling barrierheight λ to investigate their role on the tunneling resistane Rt and the sensitivity tomehanial strain, i.e. gauge fator GF . These results are ritial for understandingthe ondution mehanisms in nanoomposite lms.2.2 Constrution of 2D mirostruture ellTo predit the eletrial properties, we built numerial mirostruture unit ells asrepresentative elements of CNT/polymer nanoomposites. A thin lm of CNT/polymeromposite is simulated as a square quasi-two-dimensional network of CNT llers inpolymer matrix. We made the following assumptions to simplify the alulation andto have onsistent model:• We assumed an ideal state of uniform dispersion of CNTs. Therefore we ne-gleted the aggregation of CNTs in the omposite strutures.• We onsidered the CNTs as soft-ore and penetrable ylinders with given lengthand diameters.15Chapter 2. Analysis of 2D network of CNTsXYș0 2 4 6 8 10X (µm)Y (µm)Figure 2.1: Sample of CNT/polymer composite thin film as a 2D square cell withelectric field applied on it and a percolating cluster shown within the dashed line.• CNTs are simulated as stick-like fillers, hence the effect of tube bending andcurliness is neglected here.In the sample cell, we placed CNTs like stick-shaped fillers in a single layer.Their position and orientation were chosen randomly. The concentration of CNTsare determined by the filler volume fraction φ, which is the ratio of total volume ofCNTs to the volume of composite cell. For a quasi-2D cell containing N no. of CNTswith length LCNT and diameter DCNT , and with cell length, width, and thickness ofLx, Ly, DCNT respectively, the volume fraction can be calculated as,φ =V olume of CNTsV olume of Cells=N × LCNT × pi/4×D2CNTLx × Ly ×DCNT. (2.1)16Chapter 2. Analysis of 2D network of CNTsThe alignment order was determined by the orientation angle θ- the angle the llermakes with X axis. The diretion of applied eld and eletrial ondution is assumedalong X-axis, determined by loation of eletrodes. The perolating luster whihbuilds up a onduting bridge from the left eletrode to the right is shown withinthe dashed line in Fig.1.1. Monte-Carlo proedure of large number of simulation hasbeen performed to obtain the average value of intertube distane, tunneling resistaneand other numerial data. As the length and diameter of CNTs we used LCNT =1µm,DCNT = 1nm for 2D network, thus the aspet ratio being 1000. It is importantto note that, the results of simulations an be inuened by the the size of the unitell. Smaller ell size is helpful in reduing the omputational ost but the results itgenerates are very unstable. For this work we used Lx = Ly = 10×LCNT for the 2Dunit ells. This ell size is big enough to ahieve numerial onvergene and stabilityin the results as onrmed by literature [37℄.2.3 Intertube distane analysisIn our study, we indiated the intertube gaps as a ruial parameter for eletrialproperties of omposites. Any hange in the CNT onentration, alignment, aspetratio, or sample lm thikness hanges the average distane between the neighboringtubes in a sample. The ondutivity and strain sensitivity of a omposite lm stronglydepend on the tunneling resistane at the CNT juntions, hene on the intertubedistane too. Therefore an analysis of eet of all these parameters mentioned aboveon the average intertube distane is extremely neessary.A shemati diagram of a CNT pair with a tunneling distane dt is shown in Fig.2.2, where eletron ondution ours between two tubes through the minimum inter-tube gap and also through the tubes themselves. It illustrates the series resistanes17Chapter 2. Analysis of 2D network of CNTsPolymer CNTs Rc Rc Rt Rc= CNT resistance Rt= Tunneling resistance  dt Figure 2.2: Shemati diagrams of a CNT pair with tunneling gap between them.The resistane along the CNTs are denoted as Rc and the tunneling resistane as Rt.18Chapter 2. Analysis of 2D network of CNTs0 0.005 0.01 0.0150123456789Volume fraction, fIntertube distance, d (nm)1.4 1.5 1.6 1.7x 10- (nm)2.4999995 2.5000005 L=1 um, D=1 nm2D network. dist.xls noidata pointd dmin_phiErr.figCNT_netfigures_filec2f3x1 dmin_phiErrFigure 2.3: Intertube distane for varying CNT onentration in a random 2D net-work. The inset shows the zoomed in gure of one data point with error bar.representing tunneling and ondution along CNT segments, denoted by Rt and Rc,respetively. Sine tunneling ours only between nearest neighbor CNTs, thereforethe minimum distane between any arbitrary pair of tubes determines the tunnelingprobability at that juntion.We numerially generated random samples of CNT network using MATLAB withwide variation in the sample parameters. The parameters onsidered here are vol-ume fration φ, aspet ratio AR and alignment angle θ of CNTs. For eah set ofparameters, the average minimum intertube distane d is alulated by Monte Carloproedure with over 1000 randomly generated samples following the method [21℄.The statistial eet of CNT onentration on intertube distane is shown inFig.2.3 for 2D CNT networks with random orientation order. We an see with in-reasing ller onentration, d dereases in a power law manner, thus enhaning the19Chapter 2. Analysis of 2D network of CNTs-2.6 -2.4 -2.2 -2  -1.8 -1.6 -1.4 -1.2-4.5-4-3.5-3-2.5-2-1.5-1-0.50log ( )  Simulated Data   Linear fit10-2.6 10-2.4 10-2.  10-2.0 10-1.  10-1.6 10-1.2 10-1.  10-2.5 10-4.5 10-0.5 10-1.5 10-4.0 10-3.5 10-3.0 10-2.0 10-1.0   Intertube distance, d  (nm)      Volume fraction, ϕ Figure 2.4: The log-log plot of the intertube distane for varying CNT onentrationand the liner tprobability of eletron tunneling from one tube to other. Eah of the simulationdata for d is evaluated from 1000 randomly generated sample. The inset shows thezoomed in gure of one of the data points. From the error bars, we an understandthe estimation of d is reasonably aurate.For a randomly oriented CNT network, we propose here a relation between d andφ that readsd ∝ φ−η = γφ−η, (2.2)whereas γ and η are funtions of CNT aspet ratio, orientation angles, lm thiknessand dimensionality. Both of these parameters are ideally independent of polymertype. From a large number of simulations we numerially extrated the equation20Chapter 2. Analysis of 2D network of CNTsparameters as γ = 1.345 × 10−8nm and η = 3.0109 for 2D CNT network. Fig. 2.4shows the linear t to the logarithmi plot of d−φ used for the parameter extration.CNTs an be randomly distributed or partially aligned to the applied eld dire-tion. For 2D network, we simulated the intertube distane for dierent orientationangle θ of CNTs. The overall tube orientation in the sample spae is measured herein terms of a ut-o angle θµ [24℄ with respet to the X axis. The orientation angleof an arbitrary tube an take a statistially random value between +θµ and −θµ,where 0◦ ≤ θµ ≤ 90◦. So, at θµ = 0◦ the tubes are ompletely aligned towards Xaxis whereas, at θµ = 90◦, they are distributed with random orientation. Figure2.5 illustrates two 2D samples with two dierent alignment order of CNTs, one withomplete randomness (θµ = 90◦), the other with partial alignment (θµ = 15◦). It alsodemonstrates the simulated result for d as θµ is varied from 10◦ to 90◦ for dierentonentrations of CNTs. We an see, as the ut-o angle dereases from 90◦, thellers get more aligned with the X axis, and the average intertube gap inreases.This gure shows that the value of d is the lowest at random orientation of tubes forany level of onentration.Along with the eet of onentration and orientation of the CNTs, we also studiedthe eet of hanging CNT aspet ratio on the intertube distane for a 2D randomCNT network. For varying aspet ratio, we hanged the CNT length, keeping CNTdiameter at 1 nm and onentration at 4.59 NTs per µm2. As the length and aspetratio of CNTs inreases the overall intertube gap is lowered, hene d is found toderease with a power law. This result is shown in Fig. 2.6.21Chapter 2. Analysis of 2D network of CNTs        20 30 40 50 60 70 80 90x 10  φ=2.36e-3φ=3.14e-3φ=3.93e-3φ=5.5e-3φ=7.85e-310 20 30 40 70 80 9050 60d  (nm) (degree)     (a)  = 90º  (b)  = 15º      (c)   Figure 2.5: Shemati diagram of CNT 2D-network with θµ = 90◦ and θµ = 15◦ areshown in (a) and (b) respetively. Figure () shows the intertube distane for varyingalignment order of CNTs at dierent volume fration.22Chapter 2. Analysis of 2D network of CNTs !"#$%& '( )*"+,-.- /0 '1 *%$2/&3 /0  45- (nm)1000 1500 2000 2500 300000.010.020.03Aspect ratio o f CNTd (nm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ormattingissue?Figure 2.6: Intertube distance at varying CNT aspect ratio for a planar CNT con-centration of 4.59/µm2.2.4 Tunneling resistance analysisIn the intertube distance analysis, we have seen that the intertube gap d decreasesas the concentration, aspect ratio or degree of randomness of the CNTs increases.The reduced tube-tube distance narrows the tunneling barrier width and enhancesthe probability of tunneling. When the minimum distance between two arbitrarytubes decreases to the maximum allowed tunneling gap, then electron can tunnelbetween these two tubes. Thus, an electrical-percolation network in the compositesis formed, although a geometrically connected physical-percolation network is stillabsent. Thus we found the intertube distance as a key parameter influencing theelectrical properties of the composites.The intertube tunneling resistance, the major component of the CNT/polymer23Chapter 2. Analysis of 2D network of CNTsomposite resistane, is aeted by dierent fators, inluding the overlapping areaat the CNT-CNT juntion, CNT diameter, thikness of the polymer layer betweenadjaent CNTs, and polymer properties. Assuming a square tunneling barrier heightof λ, the tunneling resistane between two neighboring CNTs an be approximatedby [32, 38℄Rt =h2dAe2√2mλexp(4πdh√2mλ), (2.3)where m is the mass of eletron, e the elementary harge, h the Plank's onstant, dthe distane between CNTs, and A the ross setional area of the juntion. Smallerdiameter CNTs usually result in smaller overlapping area A and higher Rt. The rosssetional area A may vary with bending and alignment of the tubes. For simpliity,we approximated this area at the overlapping position by A = D2CNT .Using equation (2.2) for CNT 2D network along with (2.3) we estimated the eetof CNT loading on tunneling resistane for a thin omposite lm. We proposed amodel of the intertube tunneling resistane for a randomly oriented CNT network by,Rt = aφ−ηexp(bφ−η), (2.4)where, a = h2γAe2√2mλ and b =4piγh√2mλ. a and b depend on the polymer type and allthose CNT parameters that inuene d. Hene, in a CNT/polymer omposite, thetunneling resistane Rt depends not only on the intertube distane, but also on thepolymer type.The tunneling energy barrier height λ varies widely depending on the polymerused. Poly(methyl metharylate) (PMMA), polydimethylsiloxane (PDMS), polyethy-24Chapter 2. Analysis of 2D network of CNTs10 20 30 40 50 60 70 80 9010410610810101012θµ (degree)Rt (Ω)  λ=0.17 eVλ=0.5 eVλ=1.5 eVλ=3.0 eVλ=4.5 eVFigure 2.7: Intertube tunneling resistane of CNT/polymer omposites at varyingorientation ut-o angles of CNTs for polymers with dierent tunneling barrier height.lene, polystyrene and some epoxies are examples of polymers used in CNT omposites.Some of these polymers have low tunneling barrier (e.g. for PMMA λ = 0.17eV [39℄),whereas some have relatively high value of λ (e.g. for epoxy λ = 1.5eV [32℄).We varied the orientation and alignment of CNTs by hanging the ut-o angleθµ, and observed its eet on tunneling resistane for polymers with dierent λ.The CNT volume fration was kept xed at φ = 0.0055. We used the simulatedresult of d as shown in Fig. 2.5 at dierent ut-o angles in (2.3) to alulate theorresponding Rt. The result is demonstrated in Fig. 2.7. It shows that Rt is higherfor a polymer with higher λ and lower θµ, i.e. a higher alignment of CNTs in thediretion of ondution. At lower θµ, i.e., higher CNT alignment, the probability ofmaking tunneling CNT-CNT juntions dereases and hene the perolation thresholdgoes higher [17℄. This atually results from the larger intertube gap at lower θµ as25Chapter 2. Analysis of 2D network of CNTs10 20 30 40 50 60 70 80 901051010101510201025θµ (degree)R t (Ω)  φ=2.36e-3φ=3.14e-3φ=3.93e-3φ=5.5e-3φ=7.85e-3Figure 2.8: Intertube tunneling resistane at varying orientation ut-o angles ofCNTs for dierent CNT volume fration.shown by Fig. 2.5, and hene inreases Rt.We an tune the tunneling resistane and onsequently the thin lm resistaneby varying both the orientation and onentration of CNTs in the omposite lm.The eet of CNT onentration and orientation order on the tunneling resistaneis shown in Fig. 2.8. We kept the tunneling barrier potential of the polymer at 1.5eV. We an see Rt dereases with inreased ut-o angle, i.e., dereased alignment ofthe tubes for all range of CNT volume fration. For any given alignment order, thetunneling resistane is lower for higher volume fration of CNTs, as demonstrated bythis gure.Finally, we present the eet of hanging CNT aspet ratio on the intertubetunneling resistane. Here, we varied the CNT length with the same diameter keepingthe CNT onentration at 4.59 NTs per µm2, and alulated the tunneling resistane.26Chapter 2. Analysis of 2D network of CNTs1000 1500 2000 2500 300001000200030004000500060007000Aspect ratio of CNTRt (Ω)Figure 2.9: Intertube tunneling resistane at varying CNT aspet ratio for a planarCNT onentration of 4.59/µm2.With the inreasing aspet ratio of CNTs the overall intertube gap is lowered, henethe tunneling resistane falls exponentially. This result is shown in gure Strain sensitivity analysisWhen tensile strain is applied on a CNT/polymer omposite lm, the lm dimensionand the orientation angle of most of the tubes in the lm is modied, whih inturn hanges the intertube distane and the tunneling resistane. Consequently, thisaets the overall lm resistane. The hange in lm dimension is determined by theapplied strain ǫ and the Poisson's ratio of the polymer ν. The sensitivity of a thinlm strain sensor is presented as the gauge fator GF , desribed by27Chapter 2. Analysis of 2D network of CNTs0 0.2 0.4 0.6 0.8 (%)∆  Rt / Rt  φ=3.93e-3φ=4.71e-3φ=5.5e-3φ=7.85e-3Figure 2.10: Tunneling resistane hange ratio at small mehanial strains for dierentCNT onentration.GF = △Rf/Rf△Lf/Lf= △Rf/Rfǫ . (2.5)Here Rf is the lm resistane and Lf is the initial length of the lm before the strainis applied. For a given strain ǫ, the higher the resistane hange ratio △Rf/Rf , thehigher the gauge fator and the better the sensitivity.Sine the lm resistane is mostly dominated by tunneling resistane, and thehange in tunneling resistane is in turn governed by intertube distane variation withapplied strain, tunneling resistane hange ratio an be used as a powerful indiatorof strain sensitivity of a CNT/polymer omposite thin lm. We simulated the valuesof tunneling resistane hange ratio △Rt/Rt for dierent mehanial strains appliedon a CNT/polymer quasi-2D thin lm using the ber reorientation model [40℄. Theposition and orientation angle of the CNTs is hanged with the applied strain during28Chapter 2. Analysis of 2D network of CNTs3104105106 / Rt φ=1.57e-3φ=2.36e-3φ=3.14e-3φ=3.93e-3φ=4.71e-3φ=5.5e-3φ=7.85e-320 40 60 80 100 120 140 16010010110210Strain (%)∆  Rt / R Figure 2.11: Tunneling resistane hange ratio at large mehanial strains for dierentCNT onentration.the simulations.We estimated the tunneling resistane hange ratio △Rt/Rt both for small andlarge mehanial strains, sine dierent range of strain is used in dierent sensingappliations. Figs. 2.10 and 2.11 show eet of CNT onentration on △Rt/Rt forsmall strain (ǫ ≤ 0.01) and large strain (0.2 < ǫ < 1.5) respetively, for a polymerproperties with λ = 1.5eV and ν = 0.4. We an see, at lower strain levels, △Rt/Rtinreases almost linearly with the strain, whereas for large strain it inreases in anon-linear fashion. A lower CNT onentration shows to yield a higher △Rt/Rt andhene a higher sensitivity whih agrees with the result reported in [32℄. This indiatesthat the tunneling resistane hange ratio demonstrates a behavior similar to the lmresistane hange ratio under mehanial strain for elasti sensors.However, the result for very high applied strains (0.2 < ǫ < 1.5) in highly streth-29Chapter 2. Analysis of 2D network of CNTs2 3 4 5 6 7 8x 10-3101102103104Volume fraction, φGF tFigure 2.12: Tunneling gauge fator as a funtion of CNT volume fration at smallmehanial strains.able rubber omposites appears to be dierent, as shown in Fig. 2.11. This gureillustrates an exponential inrease in △Rt/Rt with inreasing strain up to a ertainlimit, then △Rt/Rt saturates for higher CNT onentrations, and falls for lower CNTonentrations. This hange in the △Rt/Rt pattern may result from the narrowingwidth of the lm at high strains and the orresponding hange in the tube positions.As the strain goes higher, the CNTs in the longitudinal diretion are displaed tohave a larger distane. But at the same time the width of the polymer lm dereasesaording to its Poisson's ratio, and brings the CNTs lying in the same ross-setionloser to eah other. Therefore, these two eets play simultaneously in the redis-tribution of the tubes and in determining the d, Rt and △Rt/Rt under a partiularmehanial strain. The drop in the △Rt/Rt after a ertain strain may be ausedby the dominant eet of the hange in the lm width and the minimum intertube30Chapter 2. Analysis of 2D network of CNTs0. Rt / Rt θµ= 90 degθµ= 75 degθµ= 60 degθµ= 45 degθµ= 30 degθµ= 15 deg0 0.2 0.4 0.6 0.8 (%)∆  R Figure 2.13: Tunneling resistane hange ratio at small mehanial strain for dierentorientation ut-o angles.distane. Sine at lower strain levels, △Rt/Rt inreases linearly with the strain, thetunneling gauge fator GFt appears to be onstant for a given CNT onentration.The gauge fator for low range of strain is shown in Fig. 2.12. We an see, the gaugefator dereases as the ller onentration inreases in small strain range. However,for the large applied strain, the GFt is a funtion of both the CNT onentrationand strain, hene should have dierent values for dierent CNT onentration eventhough the applied strain is same.Figs. 2.13 and 2.14 illustrate the eet of CNT alignment on the strain sensitivityat low and very high applied mehanial strains, respetively. At low strains, the△Rt/Rt shows linear inrement with inreasing strain. Thus it results in a onstantvalue of gauge fator for a given alignment order in the low strain region, as shownin Fig. 2.15. We an see that as the ut-o angle θµ is dereased from 90◦ to 30◦,31Chapter 2. Analysis of 2D network of CNTs300400500600 Rt / Rt θµ= 90 degθµ= 75 degθµ= 60 degθµ= 45 degθµ= 30 degθµ= 15 deg0 50 100 1500100200Strain (%)∆  R Figure 2.14: Tunneling resistane hange ratio at large mehanial strain for dierentorientation ut-o angles.i.e. as the CNTs are more aligned, we get inreased sensitivity, i.e. higher GFt, butas θµ goes even lower from 30◦ to 15◦, the sensitivity dereases. This result is quitereasonable, beause when the CNTs are not aligned with the strain diretion thenany hange in the strain will hange both the linear and angular distane between thetube pairs, thus aeting the △Rt/Rt . At dierent applied strains, the CNTs will bereoriented with a new θµ and d, leading to inremental hange in d and Rt. From Fig.2.5 we an see that near the random orientation order, i.e. θµ = 90◦, the inrementalhange in d with θµ is low, but it is higher as the θµ dereases, i.e., the tubes are morealigned. The higher the inremental hange in d, the higher the hange in Rt and thehigher the sensitivity. But when the tubes are arranged in omplete alignment, withthe applied strain there will be no hange in the angular distane of the tubes. Sothe lower inremental hange in intertube gap will result in lower resistane range,32Chapter 2. Analysis of 2D network of CNTs20 30 40 50 60 70 80 901234567 θµ (degree)GFtFigure 2.15: Tunneling gauge fator as a funtion of orientation ut-o angle of CNTsat small mehanial strains.hene lower strain sensitivity. Therefore the orientation order of CNTs is ruial indetermining the optimum sensitivity for a given strain range. However, this result oflow strain eet onits with the result in [32℄, whih reports lower resistane hangeratio at higher alignment, possibly due to elimination of hange in angular distanebetween tubes. When large strain is applied, as shown in gure 2.14, we see that,initially △Rt/Rt inreases non-linearly with inreasing strain, and after reahing amaximum value it begins to drop. For lower θµ, this drop in △Rt/Rt begins at lowerstrain values. The probable ause behind this result at high strain regime may bethe eet of the lm dimension on the distribution of the tubes. However, our resultagrees with the experimental reports by [41℄.After CNT onentration and orientation study, we examined the eet of dierentCNT aspet ratio on the strain sensitivity of CNT/polymer thin lm. The results33Chapter 2. Analysis of 2D network of CNTs0. Rt / Rt AR= 1000AR= 1500AR= 20000 0.2 0.4 0.6 0.8 (%)∆  R Figure 2.16: Tunneling resistane hange ratio at small strain for dierent aspetratio of CNTs.of strain appliation are shown in Figs. 2.16 and 2.17 for low (ǫ ≤ 0.01) and high(20 < ǫ < 150) strain values, respetively. The orresponding gauge fator for lowstrain range is shown in Fig. 2.17. Keeping the diameter at 1 nm, we hanged thelength of the tube to 1µm, 1.5µm, and 2µm to obtain aspet ratio of 1000, 1500,and 2000, respetively. The volume fration is kept xed at φ = 0.55%. From thegures we nd that △Rt/Rt inreases linearly with strain for lower strains, but anon-linear monotonous inrease is found at higher strains. For all range of strain,the lm having CNTs with a lower aspet ratio values shows a higher △Rt/Rt andGFt. This in turn indiates that under tensile strain, the hange in intertube distaneand tunneling resistane is more signiant for llers with lower aspet ratio. Theseresults are helpful for optimization of CNT length and diameter while designing asensor for a partiular sensitivity and strain range.34Chapter 2. Analysis of 2D network of CNTs4050607080 R t / Rt A R=1000AR=1500AR=20000 50 100 1500102030Strain (%)∆  R Figure 2.17: Tunneling resistane hange ratio at large strain for dierent aspetratio of CNTs.2.6 SummaryIn this hapter we have numerially studied the eet of dierent parameters (CNTonentration, aspet ratio, alignment order) on the intertube distane of a 2D CNTnetwork. Monte-Carlo based statistial simulation was used over 1000 samples foreah set of parameter variation. Based on the numerial results, we proposed amodel of intertube distane that an be employed for tunneling resistane, ondu-tivity or sensitivity estimation of omposite thin lms. The tunneling resistaneof the CNT/polymer omposite lms was then alulated and analyzed for all theparameters mentioned above. In addition the eet of polymer barrier height for dif-ferent polymers was also studied. Finally, the strain sensitivity of the CNT/polymeromposite thin lms was estimated qualitatively based on the tunneling resistane at35Chapter 2. Analysis of 2D network of CNTsFigure 2.18: Tunneling gauge factor at small strain for different aspect ratio of CNTs.different parameter variations. The numerical analysis presented in this chapter isimportant for optimization of the composite fabrication for specific application suchas flexible electronics and sensors.36Chapter 3Analysis of unonned 3D network ofCNTs3.1 IntrodutionGas sensors play an important role in assessing the environmental and health risks bydeteting gases that an be harmful. The inreasing need for environmental safety andhealth monitoring, indoor air quality monitoring, early detetion of possible aiden-tal leakage of harmful substanes, and analytial work at laboratories and hospitalshave made gas sensing highly relevant for residential and industrial appliations. Forexample, sensors of arbon monoxide, oxygen and ammonia have been widely usedin monitoring air quality in industrial proesses [42, 43℄, spoilage detetion in foodand mediine pakages [44,45℄, monitoring of environment of ombustion engines [46℄,detetion of limate hange due to partile formation in the environment to preventaidiation and other human health issues [47℄.A good gas sensor should posses both high sensitivity so that it detets the gasleakage early and aurately, and good seletivity so that it does not give false alarms[48℄. In addition, the gas sensors should also be able to respond to the wide range ofgas onentrations onsidering the atual eld senario where dierent gases an havedierent onentration threshold for safety. An important role of gas sensors is airquality ontrol. Proper ventilation and indoor air quality is of high interest speially37Chapter 3. Analysis of unonned 3D network of CNTswhen people stay longer time in an enlosed spae, i.e., oes, airraft abins, et.They are usually exposed to dierent gases and vapors whih an ause disomfort andhealth risks if it exeeds a ertain onentration levels. Earlier only the informationfrom arbon dioxide (CO2) sensors was onsidered to be the most important fatorfor assessment of omfort level. But reent studies have shown that gases generatedfrom volatile organi ompound (VOC) soures, suh as, building paints, furniture,photoopy mahines, even people staying or working indoor (bioeuents), et., havestronger inuene on person's omfort and health safety [49℄. Long-term exposureto VOCs in the indoor environment an ontribute to sik building syndrome (SBS)that inludes irritation of the eyes, nose, throat or skin, neurotoxi or general healthproblems, et. Therefore, sensitivity towards organi vapors and gases (i.e., methylenehloride, benzene, and aetone) has gained signiant importane in researh for loalenvironmental ontrol [50, 51℄.There is a wide variety of gas sensors with dierent working priniples and vari-ous sensing devies. The resistive metal-oxide gas sensors omprise a signiant partin the present market [52℄. But the maximum performane of this kind of sensorsusually requires operation at elevated temperature leading to higher power onsump-tion. This limits its appliation at room temperature. In reent years, sensors madeof polymer omposites have drawn muh attention sine they oer sensing apabil-ities at room temperatures [53℄. Again, sensors based on nanosale properties havebeen proven superior ompared to marosale sensors in terms of sensitivity, sele-tivity and smaller size, mainly attributed to higher ative area and surfae to volumeratios of the nanosale materials [54℄. Partiularly for appliation like remote moni-toring of health and environment, and national seurity, there is always a quest forsmaller devies apable of moleular level sensing and monitoring [52℄. In this re-38Chapter 3. Analysis of unonned 3D network of CNTsgard, nanowires and nanobers an work as building bloks of ative materials ingas sensors. Among the nanostrutures, arbon nanotubes (CNTs) have attratedonsiderable attention as a model nanostruture due to their attrative eletroni,thermal and mehanial properties inluding high eletrial and thermal ondutiv-ity, high tensile strength, large aspet ratio and surfae area, et. The large surfaearea aused by the hollow ylindrial strutures and outer sidewalls gives CNTs highgas absorptive apaity [52℄. Appliations of CNTs have been reported in literatureas humidity sensors as well as sensors of many other gases (e.g., ammonia, nitrogendioxide, dimethyl methylphosphonate) [55℄- [61℄.As disussed in the previous hapters, CNTs an form an eletrial perolating net-work when introdued in an insulating polymer matrix even at a very low amount.When this kind of CNT/polymer omposites interat with any gas, even at roomtemperature, an hange their eletrial properties with fast response and good re-versibility. Besides the sensitivity of the CNT/polymer omposites have been provedto be better than the pure polymer or the CNTs alone [62℄. These omposites possessother improved sensing harateristis, suh as enhaned seletivity, lower detetionlimit, extended detetion apaities to a number of gases, and highly salable man-ufaturing proesses [53℄. Films made of these nanoomposites an be implementedin a wireless transmission system to provide real-time sensitivity data by hange inreetion wave phase. Due to the ease of integration with a system, the nanosalesensors have high potential to be implemented in wireless ommuniation networksfor real-time remote monitoring, for example real-time physiologial sensing of bio-hazard material detetion using personal mobile stations and internet servies. Theintegration of sensors with ative wireless devies with data proessing, ommunia-tion omponents and a power soure along with the inherent sensing omponents an39Chapter 3. Analysis of unonned 3D network of CNTsopen up vast opportunities for a variety of sensor systems.Although several works have been reported on CNT/polymer omposite basedgas sensors and wireless sensors [51,54,62℄, a key limitation for arrying out detailedanalysis is the omputational ost for modeling of omplex CNT/polymer lms andtheir response to gases as sensors. All the parameters that inuene the ompositeondutivity, suh as, CNT onentration, length, diameter, CNT orientation in theomposite, as seen in Chapter 2, an have impat on the gas sensitivity of ompositelm sensors too. Besides, some other fators, i.e., surfatants used for funtional-ization, polymer and gas types, omposite lm thikness, et. also inuene theswelling rate of the lm at gas exposure and the resulting sensitivity. To design agas sensor eiently and ahieve the highest sensor performane, it is of immenseimportane to understand the inuene of all these fators, many of whih is yet tobe revealed. Dierent polymers, i.e., polystyrene, polyvinylpyrrolidone (PVP), poly-methyl metharylate (PMMA) and their opolymers have been found to demonstratedierent levels of sensitivity when used with CNTs as sensing materials [52,53℄. Sinethe energy barrier height diers in dierent polymers and inuenes the tunnelingondutivity and sensitivity in the omposites, the inorporation of this parameterin the omposite ondutivity models is neessary but still not addressed in liter-ature. Also the eet of varying CNT onentration on the sensitivity of wirelessgas sensors has not been addressed despite of it's importane in sensor design andoptimization. A thorough theoretial understanding an be obtained by analytialand simulation researh that an help not only to explain the observed experimentalresults with in-depth mehanism, but also to estimate the sensor behavior beforearrying out the atual experiment. Sine the omposite sensitivity is inuened byomposite resistane, for analytial study of sensitivity we also need a well-built an-40Chapter 3. Analysis of unonned 3D network of CNTsalytial model of omposite ondutivity. Sine the theoretial perolation modelsignore the tunneling eet, and the numerial models inur too muh omputationalost, as mentioned in Chapter 1, an analytial model inluding tunneling eet isstill in need for omputationally-eient estimation of omposite ondutivity andsensitivity.In this hapter, we present an analytial model of the ondutivity for CNT/polymeromposites onsidering 3D CNT networks in polymer matrix. Then using this model,we study the sensitivity of passive wireless gas sensors made of CNT/polymer om-posites for bio-hazard vapor detetion. We develop analytial models of tunneling andomposite ondutivity as alternatives for omputationally ostly numerial models.We inlude the eet of eletron tunneling through polymers with dierent energybarrier heights in the ondutivity models in order to estimate the sensitivity. Weonsider multiwalled arbon nanotubes (MWNTs) as the llers inside the polymermatrix. Our sensitivity estimation is based on the tunneling ondutivity of ompos-ites whih we developed from MIM [63℄ theory. For the sensitivity study we onsiderdihloromethane (CH2Cl2) gas sine it is one of the VOC gases whih is highlydangerous to human health and needs preise monitoring and ontrol for hazard pre-vention. As the polymer, PMMA is onsidered sine it has high potential as sensormaterial due to its bio-ompatibility, resistane to long exposure to temperature, andappliation in medial implants, bone ements, et. [64℄. The gas sensitivity is esti-mated numerially from the resistane hange of the omposite thin lms due to theexpeted hange in tunneling urrent during gas absorption. The wireless sensitivityis alulated from the hange in the reeted wave phase in a wireless transmissionsystem.41Chapter 3. Analysis of unonned 3D network of CNTs3.2 Role of tunnelingThe gas sensitivity of a CNT/polymer omposite based sensor is governed by theeletrial properties of the omposite. When a omposite sample is under an ap-plied eletri eld, the ller juntions along the ondutive ller network enountereld emission tunneling. Generally, in a ondutor-insulator omposite, if we grad-ually inrease the number of ondutive llers (e.g. CNTs), after reahing the llervolume fration of perolation threshold, the eletrial ondutivity of the ompos-ite takes a sharp upturn. Sine the CNTs rarely make intimate ontats in realexperimental samples and ondution ours mainly by tunneling urrents at thejuntions, the perolation threshold here means eletrial perolation threshold dueto tunneling, not the physial perolation threshold. Obviously the eletrial per-olation happens before the atual physially onneted perolating network formswith the inrease of llers. The eletron tunneling eet through polymers beomesmore prominent at low ller onentration region near the perolation threshold.Any physial hange in the omposite, suh as expansion or swelling due to preseneof gas, auses hange in the distane between neighboring llers, thus aets thetunneling urrent through it. Consequently, the overall ondutivity and ompositesample resistane is hanged. Thus by measuring the hange in the resistane wean estimate the hange in the gas pressure surrounding the omposite sample. Thustunneling plays a major role in determining the omposite ondutivity and sensitiv-ity to gas pressure of a CNT/polymer omposite [25, 26℄, in partiular at low lleronentration.In the previous hapter, we have seen that the tunneling resistane between CNTsin a juntion is strongly inuened by the tunneling intertube distane dt. Again theintertube distane dt is inuened by the CNT onentration, aspet ratio, alignment,42Chapter 3. Analysis of unonned 3D network of CNTs510152025Z (µm)0 510 1520 2505101520250X ( µm)Y ( µm)Figure 3.1: A 3D ubi sample of CNT/polymer omposite.sample lm thikness, level of gas absorption by the omposite sample, et. Besides,the energy barrier height of the polymer, the type of CNT, synthesis method, surfaemodiation of CNTs, et. are fators that plays role in the omposite ondutivity,although don't have inuene on dt. In this work, we used numerially generated3D omposite samples with varying volume fration φ of CNTs, then applied MonteCarlo based statistial method to alulate the average intertube distane. Then wethe developed analytial models of the tunneling ondutivity between two CNTsand the omposite ondutivity for a network of CNTs in a polymer matrix, andnally estimated the omposite ondutivity and sensitivity for varying levels of gasabsorption using our analytial model.43Chapter 3. Analysis of unonned 3D network of CNTs3.3 Constrution of 3D mirostruture unit ellA three dimensional mirostruture sample of CNT/polymer omposite is simulatedwhere CNTs are assumed as multiwalled arbon nanotubes (MWNTs) and are ran-domly dispersed in the polymer matrix. Here, CNTs are onsidered as penetrablesoft-ore ylinders with length LCNT = 5µm and diameter DCNT = 50nm, heneaspet ratio AR of 100. The sample as shown in Fig.3.1, is simulated as a ubi unitell with a length of 25µm on eah side. The position and orientation of the stik-likellers is randomly hosen following the method by Hu et al [26℄. The onentrationof CNTs are determined by the ller volume fration φ, whih is the ratio of totalvolume of CNTs to the volume of omposite ell. For a 3D ell of size (Lx×Ly ×Lz)and ontainingNCNT no. of CNTs with length LCNT and diameterDCNT , the volumefration an be alulated asφ = Volume of CNTsVolume of cell =NCNT × LCNT × (π/4)D2CNTLx × Ly × Lz. (3.1)As we did in our 2D samples in Chapter 2, we varied the CNT onentration byhanging the stik numbers in the unit samples. Monte-Carlo proedure has beenperformed to obtain the average value of intertube distane and other numerialdata, eah data from 100 randomly generated samples. Sine the smaller ell sizean generate unstable result even though it redues the omputational ost, we hoseLx/LCNT = 5 for 3D unit ells. This ell size is big enough to yield suiently stableand onverged results as onrmed by literature [37℄.44Chapter 3. Analysis of unonned 3D network of CNTs                                                              Polymer CNTs Rc Rc Rt Rc= CNT resistance Rt= Tunneling resistance  dt 0 5 10 15 20 250510152025X (mm)Y (mm)Electrode 2Electrode 1Y (mm) (a)   (b)   Figure 3.2: A shemati diagram of (a) top view of the 3D CNT network with eletrield applied on it. The perolating luster is shown within the dashed line, and (b)a CNT pair with tunneling gap dt between them. The resistane along the CNTs aredenoted as Rc and the tunneling resistane as Rt.45Chapter 3. Analysis of unonned 3D network of CNTs3.4 Analytial model of tunneling ondutivityIn a CNT network, starting from a low onentration, as the onentration of CNTsinreases, the intertube distane dereases, and thus enhanes the probability ofeletron transfer under an applied eletri eld. The 2D top view of the 3D numerialsample is shown in a shemati diagram is shown in Fig.3.2(a), with an applied eletrield aross it. The diretion of applied eld and eletrial ondution here is assumedalong the X-axis, as we did in our 2D network study. The perolating lusters thatbuild up onduting bridges from the left eletrode to the right one transfers theeletrons aross the sample. One of suh lusters is shown within the dashed line.If voltage is applied between two neighboring tubes, the eletron nds the shortestdistane between these two tubes and an tunnel through it if the intertube gapis below the maximum allowed tunneling gap, i.e., within an approximate range of1.8 nm as reported in literature [27℄. The applied voltage hanges the shape of theenergy barrier aused by the polymer and exerts a driving fore on the eletronsto tunnel through the barrier [25℄. This results in a small urrent at the juntion.These small juntion urrents sums up to generate the total urrent in the ompositesample. A pair of neighboring CNTs is shown in Fig. 3.2(b) where eletron tunnelingours between two tubes through the minimum intertube gap dt. The resistane ofthe eletron's pathway is represented by a series resistane omprising the tunnelingresistane Rt, and CNT resistane Rc.The tunneling urrent It through a single juntion an be expressed as [25℄,It ∝∫ eV0ρ1(r, eV, E)ρ2(r, eV, E)T (r, eV, E)dE. (3.2)Here, ρ1(r, eV, E) and ρ2(r, eV, E) are the density of states of the two neighboringMWNTs at loation r, energy E (with respet to their individual Fermi levels) and46Chapter 3. Analysis of unonned 3D network of CNTsapplied voltage V. T (r, eV, E) is the tunneling transmission probability for eletron.For a retangular tunneling barrier the transmission probability an be estimatedby [65℄T = exp(−4πdth√2mλ). (3.3)Here, λ = V −E is the height of the energy barrier, dt is the width of the energybarrier, i.e. thikness of the polymer at tunneling juntion determined by intertubegap, m is the mass of eletron and h is Planks onstant. We an simplify Eq. (3.2)as [66℄,It = cρ1ρ2e−βdt , (3.4)where β is given byβ = 2π√2mλh . (3.5)The tunneling urrent in a juntion is proportional to the ross-setional area of thetunneling path At, whih an be approximated byAt = πDCNT 2/4, (3.6)where DCNT is the diameter of the CNTs. We an rewrite Eq. (3.4) asIt = z1e−βdt , (3.7)where z1 is a parameter dependent on tunneling path ross-setional area, appliedvoltage and energy level of the neighboring CNTs. Hene the tunneling urrentdensity an be derived by47Chapter 3. Analysis of unonned 3D network of CNTsJt = It/At = z2e−βdt , (3.8)where z2 is independent of tunneling path ross-setional area and CNT diameter.Again the tunneling urrent density through a single juntion depends on the tunnel-ing ondutivity σt at the juntion area, applied voltage between neighboring tubesV and the tunneling distane dt by the following,Jt = σtV/dt. (3.9)Therefore we an derive the tunneling ondutivity σt asσt ∝ Jt = ze−βdt , (3.10)where z is a independent of tunneling path ross-setional area, applied voltage orenergy level of the neighboring CNTs, but depends on the CNT ondutivity. Asseen from Eq. (3.10), the tunneling ondutivity between two neighboring CNTsdepends highly on the distane between them and dereases exponentially as thedistane inreases. On the other hand the ondutivity σt inreases as the two CNTsin a juntion ome loser. When the two CNTs touh eah other, i.e., at dt = 0, σtreahes its highest possible value, i.e., the intrinsi ondutivity of the CNTs, σCNT .So for CNTs at intimate ontat, we an put σt = σCNT for dt = 0 in Eq. (3.10) andobtain,z = σCNT , (3.11)Hene we derive,σt = σCNT e−βdt , (3.12)48Chapter 3. Analysis of unonned 3D network of CNTs100t (S/m)10-5uctivity, V10ing conduPresent model, O= 0.5 eVPresent model, O= 0.75 eVPresent model O= 1 eV10-Tunnel  ,   Present model, O= 1.5 eVPresent model, O= 2 eVHu et al numerical O= 0 5 eV0 0 5 1 1 5 2 2 5 3   . ,  .  Hu et al. numerical, O= 0.75 eVHu et al. numerical, O= 1 eV. . .Intertube distance, dt (nm)Figure 3.3: Tunneling ondutivity at varying intertube gap for dierent polymers.Our model is ompared with the numerial data of literature [26℄Now from the Eq. (3.5) we an see the exponential term β varies with the energybarrier height λ of the polymer. For simpliation we an write,β ∝√λ = r√λ, (3.13)where r is independent of the polymer type. Using these derivations, we an rewriteour model for tunneling ondutivity for a random 3D network of a MWNT/polymeromposite asσt = σCNT e−r√λdt . (3.14)We applied our proposed model in Eq. (3.14) to examine tunneling ondutivity inMWNT/polymer omposites with dierent polymers i.e., varying λ and with dierent49Chapter 3. Analysis of unonned 3D network of CNTsintertube gaps, and ompared with literature reports as shown in Fig.3.3. We haveassumed the ondutivity of the MWNT with average length LCNT = 5µm anddiameter DCNT = 50nm as σCNT = 104S/m [26℄. For our 3D omposite model wedetermined r = 9.7692nm−1eV −0.5. For dierent polymers the energy barrier heightλ an vary within a wide range (e.g. for PMMA λ = 0.17eV and for epoxy λ = 1.5eV )[38℄. As we an see for polymers with higher λ we obtain lower ondutivity for thesame intertube distane. Our result shows lose agreement with literature reports[26℄.3.5 Analytial model of omposite ondutivityGenerally there are multiple numbers of perolating lusters aross a omposite sam-ple rather than a single luster as shown in Fig.3.2(a). Together they form a on-dutive ller network that arry the harges aross the sample when the sample isunder potential dierene. The ondutive ller network in a omposite sample anbe onsidered as a ombination of multiple parallel ondutive pathways eah madeof a series of CNTs onneted by juntions along the applied eletri eld. Eahpathway arries a urrent determined by the number of CNTs, their juntions in thatpathway and the intertube gaps in those juntions. The urrent of a pathway islimited by the tunneling urrent of the juntion in that pathway with the maximumtunneling tube-tube gap. We an onsider eah single ondutive pathway as a seriesombination of a number of tunneling resistane Rt and CNT resistane Rc and theparallel ombination of these pathways forms the entire ondutive network. Thusthe urrents of all the pathways add up to generate the total urrent through thesample.A shemati diagram of a resistive network model representing a omposite net-50Chapter 3. Analysis of unonned 3D network of CNTsRc11 Rt11 i=1 i=2 i=N j=1 j=Mi k=Li k=1 Rt1j Rc1k i=1 Rc21 Rt21 Rt2j Rc2k i=2 RcN1 RtN1 RtNMi RcNLi i=N Ii Electrode 1V = Vf  Electrode 2           V = 0  Figure 3.4: Shemati diagram of ondutive pathways with tunneling juntionsaross the CNT/polymer omposite sample and its representative resistive networkmodel.51Chapter 3. Analysis of unonned 3D network of CNTswork is shown in Fig.3.4 that onsists of N number of parallel ondutive pathways,eah pathway having L number of CNTs and M number of tunneling juntions. Theondutive pathways are denoted by index i, the resistane at intertube tunnelinggaps by index j, and the resistane inside the CNT segments by index k. Thus theresistane present at the jth tunneling juntion of the ith pathway is denoted byRtij , and the resistane at the kth CNT segment of the ith pathway is denoted byRcik. The other parameters follow the similar fashion. Sine dierent pathways mayhave dierent numbers of CNTs and juntions, for the ith pathway the number oftunneling and CNT resistanes are denoted as Mi and Li respetively. Thus, thetotal resistane present in ith pathway an be written as,Ri =Mi∑j=1Rtij +Li∑k=1Rcik. (3.15)If a voltage applied aross a omposite lm is Vf , and Ii is the urrent in the ithpathway, then the total urrent through the lm If should be the aumulation ofthe urrents passing through all the N number of parallel ondutive pathways alongthe voltage bias diretion,If =N∑i=1Ii, (3.16)where,Ii =VfRi= Vf∑Mij=1Rtij +∑Lik=1Rcik. (3.17)Now, the tunneling resistane at a juntion with intertube gap dt and ross-setional area of tunneling urrent At an be alulated from the tunneling ondu-tivity at that juntion based on our model at Eq. (3.12),52Chapter 3. Analysis of unonned 3D network of CNTsRt =1σtdtAt= 1σCNT e−βdtdtAt. (3.18)Similarly, the CNT segment resistane with segment length dc and ross-setionalarea of the CNT Ac is an be alulated by Ohm's law,Rc =1σCNTdcAc. (3.19)Using Eq.s (3.16), (3.17), (3.18), and (3.19), we an write,If =N∑i=1Vf∑Mij=1Rtij +∑Lik=1Rcik(3.20)=N∑i=1Vf∑Mij=11σCNT e−βdtijdtijAtij +∑Lik=11σCNTdcikAcik, (3.21)If the sample CNT/polymer omposite lm with length Lf and ross-setional areaAf , has a total urrent If owing through it while the applied voltage aross it is Vf ,we an alulate the ondutivity of the omposite byσcomp =IfVfLfAf= LfAfVf(N∑i=1Ii) (3.22)= LfAf(N∑i=11∑Mij=11σCNT e−βdtijdtijAtij +∑Lik=11σCNTdcikAcik). (3.23)The ross-setional area of both the tunneling urrent and the urrent through theCNT segments an be approximated by,At = Ac = πDCNT 2/4 = A. (3.24)Hene the omposite ondutivity beomes,σcomp = σCNTLfAAf(N∑i=11∑Mij=1 dtijeβdtij +∑Lik=1 dcik). (3.25)53Chapter 3. Analysis of unonned 3D network of CNTsIt an be notied in Eq. (3.25) that any little hange in dt have a pronouned eetin the σcomp as ompared to the dc due to the presene of the exponential term eβdt ,hene should dominate over the other term. As it has been already reported fromexperimental results in literature that the tunneling plays the major role in determin-ing the eletrial properties of the CNT/polymer omposites [25, 26℄, hene we anexpet the omposite ondutivity to follow the pattern of tunneling ondutivity,that is to follow an exponential relationship with the intertube tunneling distane dtand be proportional to the intrinsi CNT ondutivity σCNT . So for simpliation ofEq. (3.25) we an approximate the omposite ondutivity by,σcomp = αe−δ, (3.26)where α should be proportional to the σCNT and δ should be a funtion of intertubetunneling distane dt. Now, the dt on eah pathway varies from juntion to juntionbut on average it dereases as the CNT onentration φ inreases. For a sample withuniformly distributed llers we an use the average tunneling intertube distane dto estimate the ondutivity instead of individual dtij at eah (i, j)th juntion. Thenumber of parallel ondutive pathways (N) in the sample, the number of tunnelingjuntions (M) and CNT segments (L) in eah pathway depend on the CNT onen-tration. As the CNT onentration inreases, there is more parallel pathways allowinghigher eletron transfer aross the sample, and more juntions on eah pathway pro-viding routes with shorter length and lower resistane. Thus as the φ goes higherit redues the average tunneling distane d at the neighboring CNTs eventually in-reasing the omposite ondutivity σcomp. Hene δ should be proportional to d andwe an rewrite the Eq. (3.26) asσcomp = σCNT e−(sβd). (3.27)54Chapter 3. Analysis of unonned 3D network of CNTsHere, the average intertube distane d an vary with the CNT onentration, llerorientation, lm thikness, et. β inludes the eet of the polymer energy barrierheight as in the tunneling ondutivity. The exponent s inludes the eet of lleronentration on the number of juntions and the number of parallel pathways. It isto note that, the ondutivity model in Eq. (3.27) assumes an ideal insulating matrixwith absolutely no ondution through it. In pratie the polymer insulators haveondutivity around 10−12S/m [27℄. We an onsider the matrix resistane as a resis-tane parallel to the CNT network resistane, thus inlude the matrix ondutivityin order to estimate the omposite ondutivity,σcomp = σCNT e−(sβd) + σmatrix. (3.28)For estimation of the average intertube distane d, we numerially modeled it asa funtion of CNT volume fration φ, as we did in our previous work for 2D networkin Chapter 2. Tunneling ours only between nearest neighbor CNTs, therefore theminimum distane between any arbitrary pair of tubes determines the tunnelingprobability. In this work, we have used a Monte-Carlo proedure of 100 simulationsto estimate the average minimum distane d between any arbitrary pair of tubeswithin tunneling range in a 3D CNT network, and repeated it for varying CNTonentration. The result is shown in Fig.3.5. It shows with inreasing CNT volumefration φ the intertube distane d dereases with a power law. We propose here arelation between d and φ for a randomly oriented 3D network of CNTs asd ∝ φ−η = γφ−η. (3.29)The values of γ and η an vary with the aspet ratio or the alignment order ofthe llers. For the randomly distributed CNTs with AR=100, we obtained γ =55Chapter 3. Analysis of unonned 3D network of CNTs0.01 0.02 0.03 0.04 0.05012345678910CNT volume fraction, IIntertube distance, d (nm)Numerical modelSimulation dataFigure 3.5: Intertube distane for varying CNT onentration for a random 3D net-work.1.1771X10−3 nm and η = 1.4972 from the linear t of the exponential urve. Sinethe parameter s is determined by the number of pathways and juntions, it shouldinrease with the CNT onentration φ. Here we assume,s = mφn. (3.30)The values of m and n depend on the distribution of the juntions and pathwaysover the sample and should be independent of the ller aspet ratio or alignmentorder. For an ideal omposite sample with uniform distribution of llers we obtainedm = 91.7487 and n = 1.0915 from our numerial simulation.To verify our proposed model, we applied Eq. (3.28) for estimation of ompositeondutivity at dierent CNT onentration for dierent polymers. We have assumedthe ondutivity of the MWNT for this work as σcnt = 104S/m [26℄ as we did for56Chapter 3. Analysis of unonned 3D network of CNTs0 0.01 0.02 0.03 0.04 0.05 0.0610-410-2100102CNT volume fraction, fComposite conductivity, scomp (S/m)  Hu et al,percolation modelHu et al, num with tunnelingHu et al, experimentalNCT Co., experimentalpresent modelref[20]= HuNanotech19, perc modelref[31]= HuCarb48, num with tunnref[25]= HuActMat56, exptref[66]= NCT Co., exptFigure 3.6: MWNT/epoxy omposite ondutivity at varying CNT onentration.Our model is ompared with the literature data based on perolation model [21℄,numerial simulation [32℄, and experimental results [26, 67℄.57Chapter 3. Analysis of unonned 3D network of CNTstunneling ondutivity estimation. Fig. 3.6 shows the ondutivity of MWNT/epoxyomposite obtained from our model and ompares this result with dierent numerial,theoretial and experimental results in literature [21, 26, 32, 67℄. We have used λ =1.5eV for epoxy [32℄. Below we ompare our results based on our proposed modelwith the other models and results in literature.Among the theoretial models, the traditional perolation model and its modiedversions have been popularly used in literature [21℄ whih inlude the eet of CNTaspet ratio on omposite ondutivity, but do not inlude the eet of tunneling.Our model takes into aount the tunneling eet and generates results with betteragreement with the experimental data in omparison to results by modied perola-tion model [21℄, as an be seen from Fig. 3.6. Again, the perolation models allow theondutivity estimation only for CNT onentrations above the perolation threshold,and the perolation threshold needs to be known beforehand through experiments.But our model does not limit it's appliability to any partiular ller onentrationregion. We an estimate the omposite ondutivity below and above the perolationthreshold region, and thus an estimate the perolation threshold for a given CNTaspet ratio, alignment or polymer type before onduting the experiments.As disussed in the Chapter 1, various numerial models also have been reportedin literature so far to predit the eletrial ondutivity in omposites. The numer-ial resistor network model has been used in many studies [21, 26, 33, 34℄ and hasthe advantage that it an inlude the tunneling eet. But it's drawbak is thehigh omputational ost it inurs [21℄. For example Hu et al applied a numerialmethod for ondutivity estimation and inluded the tunneling eet [32℄. This pro-edure requires multiple step, eah step involving omputational ost. In the rststep, the number of juntions and CNT segment lengths between juntions are alu-58Chapter 3. Analysis of unonned 3D network of CNTslated numerially, then the tunneling resistane at eah juntion is alulated. Theurrent through the entire omposite lm is then alulated by an iterative equa-tion solver (Inomplete Cholesky Conjugate Gradient method- ICCG.). Finally, themarosopi eletrial ondutivity of the omposites is estimated using the Ohm'slaw. Sine this numerial method based on resistor network model needs to ountthe number of juntions and alulate the large number of CNT segment lengths be-tween juntions, it involves a high omputational ost even for a small sized samples,thus limiting its appliation. As seen from Fig. 3.6, our proposed analytial modelprovides a lose t to numerial results. Sine the method we adopted for the modelis mostly analytial, it signiantly redues the ost of this omputation. We use sta-tistial numerial analysis only for alulation of intertube distane, while the rest ofour method is analytial. The tting parameters were extrated from the numerialresults (i.e. γ, η) and they an vary for samples with dierent CNT length, diameter,or alignment order. The result of this numerial model for intertube distane d as afuntion of CNT onentration is then used to estimate the tunneling ondutivity,omposite ondutivity and sensitivity with our analytial model for any CNT on-entration. Unlike the numerial models in literature, there is no need for repeatedalulation of CNT juntion numbers, tunneling resistane at eah juntion, totalurrent through sample, et. for eah CNT onentration while our model is used,thus the omputational ost is saved at eah iteration.If we ompare our results with the experimental results in literature, from Fig. 3.6we nd that, at low ller onentration, we get a wide range ondutivity from thereported experimental data around the values estimated by our model. There an bemany possible reasons why the experimental data vary for the same CNT/polymeromposites. Dierent experiments are done at dierent environments and may use59Chapter 3. Analysis of unonned 3D network of CNTsdierent solvents or surfatants. The CNTs with the same length and diameter syn-thesized by dierent methods may vary a little in properties. Also in the numerialsimulation we onsider all the CNTs with the same AR, whereas in pratie the as-pet ratio of the CNTs even in the same bath in a experiment varies. Finally, unlikethe ideal ase of numerial simulation where the CNTs are uniformly distributedover the sample area, the CNT distribution in the samples in experiments are non-ideal and an vary in a wide range from sample to sample and from experiment toexperiment. However, our model estimates omposite ondutivity at low ller on-entration within the ondutivity range reported in the literature. At higher CNTonentration, our model slightly overestimates the ondutivity possibly due to un-avoidable presene of CNT aggregates in the experimental samples. The assumptionswe made in Eq. (3.24), that the average tunneling ross setional area is A, an alsohave impat on ondutivity estimation, speially at higher CNT onentration. Be-ause higher onentration auses higher agglomeration and more bending in eahCNT whih is not onsidered in our analytial model.3.6 Sensitivity of gas sensorsThe working priniple of a CNT/polymer omposite lm gas sensor is based on thehange in the lm resistane upon gas absorption. Hene our omposite ondutivitymodel an be applied to estimate the sensitivity of the gas sensors by numerial al-ulation of the the average intertube distane for tunneling urrent at the presene ofdierent gas onentrations. As the gas onentration surrounding the CNT/polymerlm inreases, it rises the partial pressure of the gas, ausing more gas absorptionby the sensor lm and inreasing volume of the lm. For very thin lms, the ex-pansion in the lateral diretion of the polymer lm is negligible as ompared to the60Chapter 3. Analysis of unonned 3D network of CNTsBefore swelling After swellingFigure 3.7: Shemati diagram of hange in intertube distane due to lm swelling.expansion in the vertial diretion, i.e. along the lm thikness [54℄. Hene in oursimulation work we have onsidered mainly the lm thikness hange as the eetof the presene of organi gas. The polymer lm swelling due to gas absorption willause the llers inside the polymer to move from their respetive positions inreasingthe gap between the neighboring tubes, as shown in the Fig.3.7. If the inrease inthe omposite lm thikness is uniform over the sample volume, it auses a uniformhange in the ller positions leading to higher average intertube distane. This ineet lowers the tunneling urrents through the juntions, dereases the ompositeondutivity, and inreases the overall lm resistane. Thus using our model wean estimate the hange in the omposite lm resistane numerially and alulatethe gas sensitivity. As mentioned in the introdution of this hapter, we onsid-ered MWNT/PMMA omposite lm as the bioompatible gas sensing material anddihloromethane (CH2Cl2) as the biohazard gas for sensitivity estimation throughour simulation work.In this hapter, we numerially studied how the initial ller onentration haveinuene on the hange in lm thikness, intertube distane, resistane, pressuresensitivity of gas in MWNT/PMMA omposite lms and also wireless sensitivitywhen the omposite lm is integrated with a passive wireless system for remote61Chapter 3. Analysis of unonned 3D network of CNTsmonitoring.When a polymer omposite lm is exposed in organi vapor, the lm swellingdue to gas absorption auses lm volume hange leading to hanges in the lleronentration inside the omposite lm. The ller onentration is diretly relatedto the volume of the omposite sample. Here, we measure the gas onentrationby partial pressure Ps of the gas. As the onentration of the gas inreases, Psgoes higher, the lm thikness t should inrease ausing the ller onentration φ toderease. The hange in the omposite lm thikness in presene of an organi gasdepends on the many fators, i.e., type of the polymer and gas, partial pressure ofthe gas, initial thikness of the lm, et. They an be related by the Flory-Hugginstheory and the modied Raoult's law [68℄,log(t− t0t)+(t0t)+ χ(t0t)2= log( PsP sats), (3.31)where t0 is the original thikness of the lm in absene of the gas, i.e. at Ps = 0,t is the lm thikness after swelling at the partial pressure of the gas Ps, P sats isthe saturated vapor pressure of the gas, and χ is the Flory-Huggins polymer-solventinteration parameter. Both P sats and χ are temperature dependent parameters andvaries with dierent polymers and solvent vapors. For our simulation work withPMMA polymer and CH2Cl2 gas, we numerially solved the thikness inrease ratet/t0 of Eq. (3.31) from the experimental data provided by Yoon [54℄. In order toobserve the eet of ller onentration on lm swelling, we used dierent simulatedsamples with varying initial CNT onentration, φ0 = 10%, 8%, 5%, 3%. The initialresistane for all these samples were kept same at R0 = 50Ω, and the lm thiknesswas adjusted aordingly. The result is shown in Fig. 3.8. As we an see fromthe gure, the lm thikness grows exponentially with the gas onentration, and the62Chapter 3. Analysis of unonned 3D network of CNTs0 50 100 150 2000123456789Partial Pressure of gas, Ps (Torr)Increase in film thickness, ' t (Pm)I0=0.1I0=0.08I0=0.05I0=0.03Figure 3.8: Inrease in lm thikness at varying gas onentration for lms withdierent CNT onentration.thikness inrease is higher for lms with lower φ0. We an explain it as, for lms withhigher ller onentration, there is less free volume for polymer matrix, and hene theswelling of the polymer for the same gas onentration is less than lms with lowerller onentration. As the lm thikness inrease with the inreasing gas pressure,the volume of the lm inreases while the total number of CNTs in the lm remains thesame. Thus the inrease in lm thikness dereases the CNT onentration ausinglarger tube-tube gap. The CNT onentration φ at the gas partial pressure Ps isinversely proportional to the swollen lm thikness and proportional to the initialCNT onentration φ0, thus an be related to the initial lm thikness and lleronentration byφ ∝ 1t =φ0t0t . (3.32)63Chapter 3. Analysis of unonned 3D network of CNTs0 50 100 150 20000. Pressure of gas, Ps (Torr)Change in intertube distance, ' d (nm)I0=0.1I0=0.08I0=0.05I0=0.03Figure 3.9: Inrease in intertube distane at varying gas onentration for lms withdierent CNT onentration.Now, the average intertube distane d is inuened by the ller onentration, hened will hange as the onentration of CH2Cl2 gas varies. We used our numerialmodel (3.29) to determine the hange in the intertube distane d as,∆d = d0(φ0φ − 1) = d0(tt0− 1), (3.33)where d0 is the initial intertube distane before lm swelling. We alulated d atvarious gas onentration and for lms with dierent φ0. Our results are shown inFig. 3.9.The inreased intertube distane due to lm swelling dereases the tunnelingurrent through the tunneling juntions. As a result the urrent through eah path-way dereases, thus sets the lm ondutivity to a lower level. We determined the64Chapter 3. Analysis of unonned 3D network of CNTs0 50 100 150 2000500100015002000250030003500Partial Pressure of gas, Ps (Torr)Electrical conductivity of composite, Vcomp (S/m) I0=0.1I0=0.08I0=0.05I0=0.03Figure 3.10: Composite ondutivity at varying gas onentration for lms withdierent CNT onentration.MWNT/PMMA omposite ondutivity σcomp at the inreased intertube distane inpresene of the CH2Cl2 gas using our ondutivity model (3.27). The results areshown in Fig. 3.10.From the ondutivity and thikness data obtained from our simulation workwe alulated the swollen lm resistane R at dierent gas onentration. Then wedetermined the resistane hange ratio ∆R/R0 = R−R0R0 to estimate the sensitivity ofthe lm in presene of CH2Cl2 gas, as shown in Fig. 3.11. For our simulation work,we kept the initial resistane of the lms at Ps = 0 as R0 = 50Ω whih is used inthe next setion as the harateristi impedane of a wireless transmission systemfor wireless sensitivity estimation. The inreased intertube gaps due to presene oforgani gas auses inreased resistane for lms with all CNT onentrations. Fromthe results it an be seen that the hange in resistane inreases exponentially as the65Chapter 3. Analysis of unonned 3D network of CNTs0 50 100 150 200050100150200250300350400450Partial Pressure of gas, Ps (Torr)Resistance change ratio, DR/R0 (%)  f0=0.1f0=0.08f0=0.05f0=0.03f0=0.1, expt, Ref [13]expt., Yoon ref[54]= Yoon et alFigure 3.11: Resistane hange ratio of lms with dierent CNT onentration atvarying gas pressure.gas onentration inreases, thus from the ∆R/R0 value we an sense the pressureof the gas present. We have ompared our sensitivity result for φ = 10% withthe experimental data of Yoon et al [54℄. Our result shows reasonable agreementwith Yoon's data. Sine our ondutivity model is already veried for a wide rangeof CNT onentrations, and the gas sensitivity (∆R/R0) data are alulated usingthat model, results for all these CNT onentrations should provide quantitativeinformation within an aeptable range. As we an see that the ∆R/R0 for thesame gas pressure is higher for lms with lower φ, hene the lms with lower CNTonentration proves to be more sensitive than the higher onentrated lms. Itis aused by the larger hange in the intertube gaps in lms with lower φ due tothe more free volume oupied by the polymer. However, the ller onentrationshould be suiently above the perolation threshold so that the lm does not turn66Chapter 3. Analysis of unonned 3D network of CNTsto insulating state while swollen and the resistane keeps in measurable range. It anbe understood from the relation below whih we derived from Eqs. (3.27-3.32) forthe lm resistane at a given partial pressure of gas,R(Ps) =1t(Ps)σCNTexp[βmγ( φ0t0t(Ps))n−η]. (3.34)From this relation we an see, only t is dependent on partial pressure of gas. For agiven initial lm thikness, the initial lm resistane in absene of gas is determinedby the initial ller onentration. But at presene of a gas the lm resistane isdetermined by both the initial ller onentration and the lm thikness swellingratio whih in turn depends on the pressure of the gas and polymer solvent interationparameter. Hene, to keep the resistane in a measurable range, the design windowof gas sensors with dierent polymers should have dierent sets of limits on the initialller onentrations and partial pressure of gas.It is also to note that, the larger hange in resistivity of the lms with lowerCNT onentration is aused by the larger hange in lm thikness. At a ertainpressure of gas, the swelling rate of the polymer should remain the same irrespetiveof the ller onentration. Thus the lms with lower φ and higher free volume willpotentially take more time to expand to the nal lm thikness, although givinghigher sensitivity. Hene both the sensitivity level and response time should beonsidered while hoosing the ller onentration for sensor fabriation, dependingon the design requirement in dierent appliations.As we mentioned earlier, the omposite sensitivity is mainly governed by the tun-neling eet, hene tunneling resistane an be onsidered playing dominant role inthe lm resistane and sensitivity. But we need to keep in mind that the degree ofdominane of tunneling resistane does not remain the same over all the ller onen-67Chapter 3. Analysis of unonned 3D network of CNTstration range. At low CNT onentration near the perolation region, the tunnelingworks as the main mehanism of ondution, hene the omposite shows higher rateof hange in resistane, i.e. higher sensitivity. But at high CNT onentration, asthere are more parallel ondutive paths through the CNTs, and the average inter-tube distane beomes very small, the ondution through the CNTs preedes thetunneling ondution. This an be understood from the lower resistane hange, i.e.,lower sensitivity for lms with higher CNT onentration. These results simply re-onrm the fat that the tunneling mehanism is working as the key to the ompositesensitivity.3.7 Wireless sensingAs the gas sensor response in this work is based on the hange in lm resistane, wean apply this resistane hange to alulate the hange in the load impedane on alossless transmission line, and from the phase of the reeted wave we an estimatethe wireless sensitivity of the lm [69℄. Here, the sensitivity of gas sensor as a passivewireless sensor has been estimated in terms of hange in the phase of satteringparameter (S11phase). Sattering parameters omes into play when there is a multi-port network to deal with, whih is a two-port wireless transmission system in thisase. In a two-port wireless transmission system, the voltage reetion oeientis widely used to desribe how muh of an eletromagneti wave is reeted in thetransmission medium by an impedane. Here, rst the reetion oeient Γ isalulated from the transmitted and reeted voltage ratio. Then the phase of thereetion oeient is measured by sattering parameter S11 in order to analyze thewireless sensitivity of that two-port system for varying parameters.If a lossless transmission line is terminated with a load impedane ZL = a + jb,68Chapter 3. Analysis of unonned 3D network of CNTsand there is a transmitted wave on the line moving in +x diretion reating a reetedwave towards the −x diretion, then the voltage aross the line an be given by [70℄,V = V +e−jξx + V −ejξx, (3.35)where V + is the voltage amplitude of the inident wave, V − is the voltage amplitudeof the reeted wave, and ξ is the phase onstant of the lossless line. The voltagereetion oeient Γ an be obtained from the reeted to transmitted voltage ratio,Γ = V−V + =ZL − Z0ZL + Z0, (3.36)where Z0 is the harateristi impedane of the transmission line, typially 50Ω andfor this work with a lossless system is onsidered as Z0 = R0 + 0j [54℄. Sine ZL isa omplex number, the voltage reetion oeient Γ will have both magnitude andphase information,Γ = |Γ|ejΘ. (3.37)Putting the omplex values of ZL and Z0 in Eq. (3.36) we get,Γ = a + jb− Z0a + jb+ Z0(3.38)= (a− Z0 + jb)(a+ Z0 − jb)(a + Z0 + jb)(a + Z0 − jb)(3.39)= a2 + b2 − Z20 + 2jbZ0a2 + b2 + Z20 + 2aZ0(3.40)= a2 + b2 − Z20a2 + b2 + Z20 + 2aZ0+ 2jbZ0a2 + b2 + Z20 + 2aZ0(3.41)= A+ jB. (3.42)69Chapter 3. Analysis of unonned 3D network of CNTsThus the magnitude and phase of Γ an be determined from the load impedane by,|Γ| =√A2 + B2 (3.43)=√(a2 + b2 − Z20)2 + 4b2Z20(a + Z0)2 + b2, (3.44)andΘ = cos−1 A√A2 +B2(3.45)= cos−1 a2 + b2 − Z20√(a2 + b2 − Z20)2 + 4b2Z20. (3.46)We an denote the phase of the reeted wave Θ as the sattering parameter S11 phaseto indiate the hange in phase of the reeted wave with respet to the transmittedwave at the same port terminated with the load impedane. A higher S11 phase fora given ZL indiates better wireless sensitivity of that load. It has been reportedin literature that as long as the imaginary part of the load impedane is low, thereeted wave phase shows large phase shift with small hange in the real part ofthe load impedane near the harateristi impedane [54℄. This is illustrated in Fig.3.12 for a varying range of a and b = 0.1.If the omposite lm is used as the load impedane in a lossless transmissionsystem, the load impedane will inrease due to the inreasing lm resistane athigher gas onentration, leading to hange in Θ aording to the Eq. (3.46). Theimaginary part b of the load impedane is a funtion of frequeny. As long as thefrequeny is kept onstant the hange in b is negligible, although the real part a keepshanging with hange in temperature, lm swelling due to gas absorption or otherparameters [54℄. Hene, the hange in S11 phase for lm swelling ours mainly dueto the hange in the real part of the load impedane, i.e. the resistane, while the70Chapter 3. Analysis of unonned 3D network of CNTs45 46 47 48 49 50 51 52 53 54 55-180-150-120-90-60-300Real part of impedence, a (:)S11 phase (degree)Z0=50 :ZL=a+jbb=0.1Figure 3.12: Reetion (S11) phase for varying real part of load impedane.50 100 150 200-5-4-3-2-10Partial Pressure of gas, Ps (Torr)S11 phase (degree)I0=0.1I0=0.08I0=0.05I0=0.03Figure 3.13: Reetion (S11) phase of lms with dierent CNT onentration atvarying gas pressure.71Chapter 3. Analysis of unonned 3D network of CNTsimaginary part an be onsidered a onstant at a given frequeny. For S11 phaseestimation in our simulation work, we used b = 0.1 for 400 MHz frequeny [54℄. Forvarying gas onentration, we alulated the S11 phase for the hanging lm resistaneand ompared the results for lms with dierent CNT onentration.As demonstrated in Fig. 3.13, we an nd that the reetion phase S11 is higherfor lms with lower φ at a given gas pressure, and as the gas onentration inreasesthe reetion phase beomes loser for lms with dierent ller onentration. Thusthe wireless sensitivity of MWNT/PMMA omposite lms for CH2Cl2 gas pressuredetetion is high for low φ lms as long as the gas pressure is kept within a lower range(≤ 100 Torr). However, sine the dierential phase dierene (△S11phase/△Ps)of the omposite lms with higher φ at all gas onentration is higher than thelower φ lms, higher ller onentration may be more useful for remote metering andregulation of the gas pressure.3.8 SummaryIn this hapter, we have proposed an eient analytial model for the ondutiv-ity of CNT/polymer omposites inluding the eet of CNT onentration and eldemission tunneling through the polymers. This analytial model demonstrates bet-ter auray than the traditional perolation models in ondutivity estimation andsaves tremendously in omputational ost inurred by numerial resistor networkmodels. Thus, this model paves the way for a omputationally-eient estimation ofCNT/polymer omposites ondutivity with a reasonable auray over a wide rangeof CNT onentration. The model have been used to estimate the hange in resistanedue to lm swelling in presene of organi gas, thus measure the gas sensitivity of thelm. The hange in reetion phase was alulated to determine the sensitivity of72Chapter 3. Analysis of unonned 3D network of CNTsthe lm used as passive wireless gas sensor. The lms with lower ller onentrationexhibited higher gas sensitivity at any given gas onentration and higher wirelesssensitivity within a low range of gas pressure. This analytial model an failitategaining deeper understanding of the sensing ability at the mirosopi level whihan help in improving the design and tuning of wireless gas sensors to ahieve betterseletivity and sensitivity.73Chapter 4Analysis of onned 3D network ofCNTs4.1 IntrodutionDue to the ever-inreasing number of engineering strutures, strutural health mon-itoring (SHM) have beome more important over time for early detetion of damageand haraterization strategy of strutural systems. To determine the urrent stateof a system health, the SHM proess requires to observe the system over time usingperiodially sampled dynami response measurements from an array of sensors. Thesensors provide the data that is needed for the analysis of damage-sensitive features.Hene, strain sensors have been widely used for deades for this purpose. As wementioned in Chapter 2, the tunneling ondution in the CNT omposites leads tosuperior piezoresistive property of these omposites and make them highly suitablefor strain sensing than the onventional strain sensors [37℄. Due to the high mehani-al strength of CNTs, the embedding of CNTs in a polymer helps in the reinforementof the omposite lms whih is neessary for repetitive strain/stress exertion. Thinlms of CNT/polymer omposites have been reported to demonstrate great potentialin strain and pressure sensing appliations, i.e., strutural health monitoring, damageor deformation detetion, uid ow sensing, artiial skin, et. [7174℄. To identifyloal strutural damage and to investigate strutures at the omponent level, a large74Chapter 4. Analysis of onned 3D network of CNTsnumber of sensors need to be installed. Thinner omposite lms an help in ostredution of the sensors and allow installation of large number of sensors in a singlemonitoring system. In a CNT/ polymer omposite lm, when the lm thikness isredued below the ller length, it introdues some alignment of the llers whih anpotentially aet the average intertube distane and eventually the lm ondutivityand sensitivity. Although there has been reports on the eet of lm thikness on theomposite ondutivity [75, 76℄, the omparative eet of lm thikness below andabove the ller length has not been reported yet. Hene a systemati study on therelative eet of lm thikness and CNT length on the omposite ondutivity andsensitivity is still in need.In this hapter, we present our experimental and numerial study of the rela-tive eet of lm thikness and ller length on the eletromehanial properties ofCNT/polymer omposites. We numerially generated 3D omposite samples of dif-ferent thikness t and CNT volume fration φ. The sample lm thikness is hangedbelow and above the length of CNTs to observe the eet of ller alignment on theintertube gaps introdued by the sample thikness. We applied Monte Carlo basedstatistial method to alulate the average intertube distane d. Then the ompos-ite lm ondutivity was estimated using our analytial model developed in the lasthapter. Finally, the strain sensitivity of the lms with dierent thikness under ap-plied strain in longitudinal diretion was estimated and analyzed. In order to verifythe numerial results on the behavior of lm ondutivity with the varying lm thik-ness and ller alignment, we arried out experiments with SWNT/PMMA ompositelms and ompared the experimental results with the numerial ones.75Chapter 4. Analysis of onned 3D network of CNTs4.2 Constrution of numerial samples ofnanoompositeA three dimensional mirostruture sample of CNT/polymer omposite is simulatedas we did in the last hapter. The CNTs are assumed as multiwalled arbon nanotubes(MWNTs) and randomly dispersed in the polymer matrix. Here, CNTs are onsideredas penetrable soft-ore ylinders with length LCNT = 5µm and diameter DCNT =50nm, hene aspet ratio AR of 100. The sample, as shown in Fig.4.1(a), is simulatedas a ubi unit ell with a length of 25µm on eah side.The position and orientation of the stik-like llers is randomly hosen followingthe method by Hu et al [26℄. The onentration of CNTs are determined by theller volume fration φ, whih is the ratio of total volume of CNTs to the volumeof omposite ell. The alignment order was determined by two orientation angles- θand ψ, the angles that the llers makes with the X-Z and X-Y planes respetively,as shown in Fig.4.1(b). We varied the CNT onentration by hanging the stiknumbers in the unit samples. The ut-o angles θµ and ψµ are the maximum anglebetween an arbitrary ller and the X-Z and X-Y planes respetively. The averagevalue of intertube distane and other numerial data are obtained by Monte-Carlosimulations over large number samples. We varied the thikness of the samples byhanging the length of Lz while keeping the Lx = Ly of the same length. The ellsize is kept suiently large (Lx = Ly = 5 × LCNT ) in order to ahieve numerialonvergene and stability in the results [37℄.76Chapter 4. Analysis of onned 3D network of CNTs0 510 1520 2505101520250510152025X (Pm)Y (Pm)Z (P m)(a)(b)Figure 4.1: Illustration of (a) a 3D ubi sample of CNT/polymer omposite and (b)a randomly oriented ller making orientation angles with X-Y and X-Z plane.77Chapter 4. Analysis of onned 3D network of CNTs4.3 Analysis of intertube distaneWe onduted a numerial analysis on the intertube distane of the numerial om-posite samples as we did in the previous two hapters. Here we varied both theller onentration and the sample lm thikness over a wide range. We used aMonte-Carlo proedure of 100 simulations to estimate the average minimum distaned between any arbitrary pair of tubes within tunneling range in 3D CNT networksfor eah ombination of CNT onentration and sample thikness. For a given lmthikness, as the ller onentration in a omposite sample inreases, it dereasesthe inter-ller gaps. If an eletri eld is applied aross a CNT/polymer ompositesample, the dereased intertube distane at higher CNT onentration enhanes theprobability of eletron transfer between neighboring tubes. In this hapter too, thediretion of applied eld and eletrial ondution is assumed along the X-axis.The ller orientation is inuened by the lm thikness. For samples with thik-ness greater than the length of the CNTs, the llers an be oriented in any orderinside the 3D unit ells. Hene both the ut-o angles θµ and ψµ are set at theirhighest value, i.e. 90◦ and the llers an ahieve omplete randomness in orientation.If the sample lm thikness is dereased below the ller length, that will limit therandomness of the llers and introdue some alignment, thus redue the ut-o an-gle ψµ to a value below 90◦. The shemati diagram in Fig. 4.2 demonstrates howthe alignment order of the CNTs is hanged when the lm thikness is dereasedbelow the CNT length, keeping the number of CNTs per unit volume unhanged.The hange in the CNT orientation aets the average distane between tubes andonsequently the tunneling urrent through the juntions. Thus the alignment eetintrodued by the lm thikness an inuene the overall ondutivity and sensitivityof the omposite lm.78Chapter 4. Analysis of onned 3D network of CNTs202020Figure 4.2: Shemati diagram of CNT/polymer omposite lms with dierent thik-nesses.The results of our numerial simulation of the intertube distane for varying CNTonentration and lm thikness is shown in Fig. 4.3. Sine the CNTs used in oursimulation have length of LCNT = 5µm and diameter of DCNT = 50nm, we variedthe lm thikness from 15µm to 50nm to study eet of lm thikness both belowand above the CNT length. From the logarithmi plot we an see that, for alllm thikness t, the intertube distane d dereases in a power law manner as theCNT onentration inreases. At any CNT onentration, the d has higher value atlower lm thikness, exept when t is lose to CNT length LCNT = 5µm. Fig. 4.4demonstrates the hange in d with thikness. Our simulation results show that whenthe lm thikness t dereases from 15µm to 5µm the intertube distane inreasesgradually. But as t drops slightly from 5µm, d starts dereasing, keeps lower valuefor a while, then again starts inreasing with the dereasing t. This behavior opensup a question of the eet of the partial alignment on d. Before the t is dereasedto 5µm, the llers are allowed to have full randomness whereas right after the t is79Chapter 4. Analysis of onned 3D network of CNTsredued below 5µm, the llers starts aligning to the X-Y plane gradually making ψµless than 90◦. Thus at t = 50nm the omposite lm beomes almost a quasi-2D planemaking ψµ ≈ 0◦. We alulated the ut-o angle ψµ for t ≤ 5µm and re-plotted theorresponding values of d in Fig. 4.5. It shows that the intertube distane reahesthe lowest value at around ψµ = 45◦. Therefore, the minimum intertube distane anbe found at partial alignment, not in ompletely random or ompletely aligned CNTnetwork. This behavior is onsistently shown by samples of dierent onentrations,whereas the drop in the intertube gap at partial alignment is more pronouned atlower onentration of llers. For example, when ψµ is dereased from 90◦ to 53◦,the intertube distane dereases from 0.47nm to 0.05nm for φ = 7% onentration,whereas for the same hange in alignment, the drop in the intertube distane forφ = 1.42% is from 7.61nm to 2.026nm. This behavior an be explained in ontextof CNT-CNT juntions that build up the ondutive pathway for perolation. In ahighly onentrated random CNT network, the number of nanotube juntions is high.When a ller is aligned slightly, it may lose ontat with one neighbor ller. But sineeah ller is surrounded by many others, it an make ontat with another neighboringller. Thus the average intertube distane does not get signiantly aeted by thepartial alignment of llers, and the perolation ontinues. However, higher alignmentauses loss of juntions and disontinuity in the network, and results in inreasingintertube gap. In ase of low CNT low onentration near perolation threshold, slightalignment of the llers allows less number of llers to build the onduting bridgesby reduing gaps between the neighboring llers along the diretion of ondution.However, at higher alignment, the low onentrated CNT network also experienesthe loss of juntions like the highly onentrated networks and d inreases.80Chapter 4. Analysis of onned 3D network of CNTs102 101 (nm)100stance, dt=50 nmt=100 nm10-1rtube dis t=300 nmt=500 nmt=1 Pm 1010-2Inter  t=3 Pmt=5 Pm10-2 t=10 Pmt=15 Pm10-1.5 10-1.15CNT volume fraction, IFigure 4.3: Intertube distane at varying CNT onentration for CNT/polymer om-posite lms with dierent thikness.4.4 Analysis of ondutivityIn order to estimate the eletrial ondutivity of the omposite samples of dier-ent ller onentrations and lm thikness, we used the simulated data of intertubedistane and applied our ondutivity model that we developed in the last hapter,σcomp = σCNT e−(sβd) + σmatrix. (4.1)As before, the intrinsi ondutivity σCNT is assumed as 104S/m for MWNTs withAR=100. For simulation work, we onsidered epoxy and PMMA as the polymer withenergy barrier height λ = 1.5eV and λ = 0.17eV respetively [32, 39℄.We varied the sample lm thikness for eah ller onentration to study the eetof lm thikness on omposite ondutivity. The results are shown in Fig. 4.6 for81Chapter 4. Analysis of onned 3D network of CNTs0 5 10 1505101520Composite film thickness, t (mm)Intertube distance, d (nm)  f=1.42%f=2.00%f=3.00%f=4.00%f=5.00%f=6.00%f=7.00%Figure 4.4: Intertube distane at varying sample lm thikness for dierent CNTonentration.both MWNT/epoxy and MWNT/PMMA omposites. We plotted σcomp in terms oft/LCNT to understand the relative eet of ller length and lm thikness. As the lmthikness is redued from 15µm to 5µm, making t/LCNT from 3 to 1, the ondutivitydereases. But just below t/LCNT < 1 it inreases sharply whih is due to thelowered d at partial alignment, as shown by our results on intertube distane. Aftert/LCNT is dereased to a ertain extent, σcomp starts dropping again with lower lmthikness. Fig. 4.7 shows the ondutivity plot for the orresponding ut-o anglesψµ at dierent t/LCNT . It shows the highest σcomp is ahieved at around ψµ = 45◦.Thus our simulation shows that for a given ller onentration and aspet ratio thehighest omposite ondutivity is obtained at slightly aligned network rather thanat isotropi network. Although this result ontradits with many reports laimingthat maximum ondutivity and minimum perolation threshold should our at82Chapter 4. Analysis of onned 3D network of CNTs10 20 30 40 50 60 70 80 9010-1100101102Cut-off angle, ym (degree)Intertube distance, d (nm)  f=1.42%f=2.00%f=3.00%f=4.00%f=5.00%f=6.00%f=7.00%Figure 4.5: Intertube distane at varying alignment order of CNTs for dierent CNTvolume fration.83Chapter 4. Analysis of onned 3D network of CNTsomplete isotropy [77, 78℄, it agrees with the experimental result by Du et al [24℄.Our result indiates that with omposite lms slightly thinner than the ller length,ausing partial alignment of the llers, we an attain signiantly higher ondutivityin a omposite sample with lower CNT onentration in omparison to a samplewith higher CNT onentration and random alignment. The rise in ondutivityat partial alignment is more prominent in samples with low CNT onentration.However, at higher alignment the ondutivity drops for all CNT onentration sinethe onduting pathway beomes disontinuous.4.5 Experimental veriation of thikness eet onondutivityTo verify our numerial results, we onduted experiments on CNT/polymer ompos-ite lms with varying CNT onentration and lm thikness. Single walled arbonnanotubes were bought from Cheap Tubes In. The CNTs had purity over 90% withaverage length of 30µm and outer diameter of 1.5 nm. As the polymer we usedPMMA with moleular weight of 35 × 104. Dimethylformamide (DMF) was used asthe ommon solvent for both SWNT and PMMA.4.5.1 PreparationAt rst, CNT/DMF dispersion of 0.5 mg/ml onentration was prepared by ontin-uous soniation of 30 minutes using Sonis VCX 750 ultrasoni proessor equippedwith 6 mm tip. We added polyvinylpyrrolidone (PVP) as the surfatant in order toobtain uniform dispersion. The PVP was added in 1:3 weight ratio to CNTs. TheCNTs were well dispersed in the DMF. We made some dropasted samples of the84Chapter 4. Analysis of onned 3D network of CNTs0 1 2 310-1510-1010-5100105Film thickness to CNT length ratio, t/LCNTComposite conductivity, Vcomp (S/m)I=0.57%I=0.68%I=0.85%I=1.14%I=1.42%I=2.00%I=3.00%I=4.00%I=5.00%I=6.00%I=7.00%0 1 2 310-1510-1010-5100105Film thickness to CNT length ratio, t/LCNTComposite conductivity, Vcomp (S/m)I=0.57%I=0.68%I=0.85%I=1.14%I=1.42%I=2.00%I=3.00%I=4.00%I=5.00%I=6.00%I=7.00%(a) MWNT/epoxy composite (b) MWNT/PMMA composite Figure 4.6: Condutivity of (a) MWNT/epoxy and (b) MWNT/PMMA ompositesat varying lm thikness to ller length ratio for dierent CNT volume fration.85Chapter 4. Analysis of onned 3D network of CNTs105m)I=0.57%I=0.68%100V comp (S/m I=0.85%I=1.14%I=1.42%-5uctivity, V I=2.00%I=3.00%I=4.00%10osite cond I=5.00%I=6.00%I=7.00%10-10Comp o0 10 20 30 40 50 60 70 80 90Cut off angle \ (degree)- , PFigure 4.7: Composite ondutivity at varying alignment order of CNTs for dierentCNT volume fration.CNT/DMF solution and used sanning eletron mirosope (SEM) image of themin order to be onrmed about the dispersion. Fig. 4.8(a) shows some dropastedsamples and Fig. 4.8(b) shows the SEM image of one of the samples. We an seeCNTs are well dispersed in the sample and almost no agglomeration present there.After preparing the CNT/DMF solution, dierent amounts of this solution andPMMA powder were mixed in order to vary the CNT onentration in the nalCNT/PMMA omposite lms. The amount of CNT/DMF solution was varied be-tween 5-40 ml while mixing with 1 gm of PMMA powder. The mixture was rststirred overnight at room temperature in a losed ontainer to obtain a uniformsolution. Then the solution was again stirred in an open ontainer at 80◦C forfew hours to evaporate the DMF and attain the desired visosity. In the nalCNT/PMMA/DMF omposite solutions after evaporation, the PMMA onentra-86Chapter 4. Analysis of confined 3D network of CNTs1 µm1 cm(a) (b)Figure 4.8: (a) Dropcasted samples of SWNT/DMF solution with silver paste con-tacts. (b) The SEM image of a sample of SWNT/DMF dispersion.tion was kept at 1/6 gm/ml to attain the same viscosity for all solutions with varyingCNT concentrations. Uniform viscosity is important to control the film thicknesswhile changing the spin speed for samples with varying CNT concentrations. Inthe next step, the CNT/PMMA/DMF composite solution was spincoated on glasssubstrates. For each batch of composite solution with different CNT concentrations,we varied the spin speed from 700 to 2000 rpms in order to vary the coated filmthickness within a range of 3-60 µm. The spincoated samples were then degassed ina vacuum chamber in order to make sure that no air bubble stays inside the film.This step also ensured the uniformity of the film surface and equal thickness over thelargest possible area of the sample. Finally the composite films were dried in air at50◦ C. In the final CNT/PMMA composite films, the CNT concentration varied from0.24 to 1.5% in weight fraction.In order to examine the CNT distribution inside the composite film we again tookSEM image of some composite film samples. Fig. 4.9(a) shows some SWNT/PMMA87Chapter 4. Analysis of confined 3D network of CNTs1 cm(a)10 µm(b)Figure 4.9: (a) SWNT/PMMA composite film samples with different film thicknesshaving same CNT concentration. (b) The SEM image of a SWNT/PMMA film withits surface and edge.88Chapter 4. Analysis of confined 3D network of CNTscomposite film samples with different film thickness having same CNT concentration.From the SEM micrograph of a typical composite sample, as shown in Fig. 4.9(b),we can see the SWNTs are almost uniformly distributed in the PMMA matrix.4.5.2 Measurements and resultsLarge number of samples with different thickness and CNT weight fraction were mea-sured for conductivity and thickness. The film thickness of the samples was measuredusing profilometer. Although the films had higher thickness around the sample edge,most of the surface area showed uniformity in thickness. The profilometer reading ofa typical sample film with 0.24% SWNT concentration is shown in Fig. 4.10. Thethickness profile shows a large bump at the film edge, but almost equal thicknessover the rest of the film surface, giving an average thickness of 13.4 µm and standarddeviation of 0.1727 µm. Thus for each sample, different locations were chosen formeasurement in order to calculate the average thickness and compare the standarddeviation values. The change in the film thickness with different spincoating speedfor solutions with different CNT concentrations is shown in Fig. 4.11. The error barsindicate reasonable accuracy of the measured data.For measuring the composite film conductivity we used four point probe method.Our conductivity measurements of films with varying thickness and different CNTweight fractions are shown in Fig. 4.12. It was observed that for low CNT concen-trations, i.e. CNT weight fraction 0.24%, 0.26%, 0.28%, the film conductivity risesimmediately after the film thickness is decreased below 30 µm, which is the averagelength of the SWNTs here. But it drops after t is reduced further. Thus our ex-perimental results demonstrate similar pattern of conductivity change with thevarying film thickness. It is interesting to note that the film with 0.26% CNT89Chapter 4. Analysis of confined 3D network of CNTsTip position (mm)Film thickness (um)Figure 4.10: Composite film thickness profile for a typical sample with 0.24% CNTconcentration.600 800 1000 1200 1400 1600 1800 2000 22000510152025spin speed (rpm)Composite film thickness, t ( Pm)  m = 0.24%m = 0.262%m = 0.306%m = 0.365%Figure 4.11: Change in film thickness with the spin speed at different CNT weightfractions. The error bars are included by using std. dev. of the thickness data.90Chapter 4. Analysis of onned 3D network of CNTs0 10 20 30 40 50 60 7010-410-310-210-1100101Composite film thickness, t (mm)Composite conductivity, s (S/m)  m=0.24%m=0.26%m=0.28%m=0.38%m=0.5%m=0.99%m=1.5%Film thickness to CNT length ratio, t/L 1 2 1.3 1.6 2.3 0.3 0.6 Figure 4.12: Experimental data of SWNT/PMMA omposite ondutivity at varyinglm thikness for dierent CNT weight fration.weight and 18.9µm thikness is showing higher ondutivity than the lm with 0.28%CNT weight and 34µm thikness despite having lower ller onentration and lowerlm thikness. This result is signiantly important in ost redution and designoptimization of omposite devies based on thin lms. However, similar inrease inthe ondutivity of the lms with thikness around 30µm were not observed for thehigher CNT weight fration. Thus our numerial predition of the relative inueneof lm thikness and ller length on the omposite ondutivity is onrmed by theexperimental results.In the experimental samples the CNTs varied in length, making volume frationestimation diult. Hene we used weight fration (m%) of CNTs as the CNTonentration parameter here. However, for low CNT onentration, as used in theexperiment, the weight fration and the volume fration give very lose values. In91Chapter 4. Analysis of onned 3D network of CNTsgeneral, we an onvert the volume fration φ to weight fration m by the followingrelation,m = 11 + (ρp/ρc)× (1/φ− 1), (4.2)where, ρp and ρc are the density of the polymer and CNT respetively. Sine duringthe experiments it is not pratially feasible to have samples with all the CNTs of sameaspet ratio, hirality, or uniformity in distribution, as assumed in our simulationwork, the parameters vary between the experimental and the numerial study. This iswhy from our experimental results it is diult to verify the quantitative informationprovided by our simulation work based on our ondutivity model. Nevertheless, ourexperimental results onrm the qualitative trend as shown by our numerial result,whih is important for designing the experiments before arrying them out.4.6 Analysis of strain sensitivityWhen tensile strain is applied on a CNT/polymer omposite lm, the hange inthe lm dimension auses hange in the orientation angle of most of the tubes inthe lm, whih in turn hanges the intertube distane and the tunneling resistane.Consequently, this aets the overall lm resistane. The hange in resistane in aperolating network of a thin lm an be used to sense the applied strain whih anbe very helpful in many pratial appliation, suh as, early detetion of struturaldamage, streth or deformation reading of artiial skin, et. We have numeriallystudied the eet of uniaxial strain on the sensitivity of omposite lms of varyingthikness. The hange in lm dimension is determined by the applied strain ǫ andthe Poisson's ratio of the polymer ν. The sensitivity of a thin lm strain sensor is92Chapter 4. Analysis of onned 3D network of CNTs40 t=100 nmt 300t 53035(nm)=  nmt=500 nmt=1 Pm23nm)= Pmt= 7 Pm25ance, d (t=3 Pmt=4 Pmt 51d (n1520ube dista = Pmt= 7 Pm0 10 20H (%)10Intertu050 10 20 30 40 50Strain, H (%)Figure 4.13: Intertube distane at applied strain, for a CNT onentration φ = 1.14%and dierent lm thikness.measured in terms of the gauge fator GF ,GF = △R/R0△Lf/Lf0= (R −R0)/R0(Lf − Lf0)/Lf0= △R/R0ǫ . (4.3)Here R0 and Lf0 are the initial resistane and length of the lm when no strainis applied, and R and Lf are the resistane and length of the lm when strain isapplied, respetively. For a given strain ǫ, higher resistane hange ratio △R/R0indiates higher gauge fator and better sensitivity. Sine we observed from ourresults in the previous setion that the lm thikness and ller length ontribute tothe intertube distane and omposite ondutivity, the hange in the lm thiknessshould also have impat on the gauge fator of the lm.For dierent lm thikness t and mehanial strain ǫ, we numerially alulatedthe average intertube distane using the ber reorientation model [40℄. We onsidered93Chapter 4. Analysis of onned 3D network of CNTs103)100mp (S/m)10-3vity, Vcomt 1 m10-6conductiv = Pt=3 Pmt=4 Pm10-9mposite c t=5 Pmt=7 Pm12Com0 10 20 30 40 5010-Strain, H (%)Figure 4.14: Condutivity of MWNT/epoxy omposite lms (φ = 1.14%) at appliedstrain for dierent lm thikness.epoxy as the polymer for our numerial study and the Poisson's ratio for it as ν = 0.4[38℄. We then obtained the lm ondutivity applying our analytial model. Fromthe ondutivity and lm thikness we alulated the lm resistane and resistanehange ratio for varying strains. The lm thikness was varied from 100nm to 7µm.With hanging t, the number of CNTs in the unstrained (ǫ = 0) omposite samplewas adjusted to keep the CNT onentration at φ = 1.14%. The results are shown inFigs. 4.13-4.16. The hange in the average intertube distane d with the inreasingstrain is shown in Fig. 4.13. The inset gure provides a loser view of the urvesfor t = 5 and 7µm. As we an see, the intertube distane inreases almost linearlywith the strain for almost all lm thikness exept when t takes value lose to llerlength L, i.e. for t = 5 and 7µm. At these lm thiknesses we nd that the initialintertube distane at ǫ = 0 is higher than the initial d values for t = 4µm lm, unlike94Chapter 4. Analysis of onned 3D network of CNTs1015e, R (:) t=1 Pmt=3 Pm1012sistance t=4 Pmt=5 Pm109te film re t=7 Pm106Composit103C0 10 20 30 40 50Strain, H (%)Figure 4.15: Resistane of MWNT/epoxy omposite lms (φ = 1.14%) at appliedstrain for dierent lm thikness.the trend shown by other urves. Again, with inreasing strain, the values of d fort = 5 and 7µm drop with a power law and does not inrease until the strain goeshigher than 25%. This exeptional trend of d shown by lms with t/L ≈ 1 an beexplained by the eet of partial alignment of llers. For lms with t = 5 and 7µm,the CNTs in unstrained lm an be oriented with full randomness in all X, Y, andZ axes diretions. The appliation of small longitudinal strain (ǫ < 25%) aligns thellers slightly towards X axis diretion hanging the ut-o angle ψµ from 90◦ to alesser value. The partial alignment brings the overall CNTs loser and redues d. Butwith higher strain as the CNTs beome more aligned, they get disonneted at thejuntions and the average intertube distane inreases. For lower lm thiknesses,i.e., t < 5µm or t/L < 1, the CNTs are already slightly aligned towards the X-Y planein the unstrained lms. The appliation of strain aligns them further and auses the95Chapter 4. Analysis of onned 3D network of CNTs350400)t=1 Pmt=3 Pm250300R/R0 (% t=4 Pmt=5 Pmt=7 Pm200e ratio, 'R0t=5 Pm100150e change-50R/R0 (%)t=7 Pm050sistance100'Rc4-100-50ResFig0 1 2-H (%)0 10 20 30 40 50Strain, H (%)Figure 4.16: Resistane hange ratio of MWNT/epoxy omposite lms (φ = 1.14%)at applied strain for dierent lm thikness.intertube distane to inrease.The hange in lm ondutivity due to the hange in intertube distane withapplied strain is shown in Fig. 4.14. We an see the ondutivity towards the lon-gitudinal diretion dereases as the strain inreases for most of the lm thiknessesexept for t = 5 and 7µm, as expeted from the intertube distane pattern. However,for t = 5 and 7µm, the ondutivity starts from a low value at unstrained lms, thenrises drastially with the appliation of strain smaller than 10%. It keeps inreas-ing till (ǫ < 25%), then dereases with further strain due to loss of juntions. Theorresponding lm resistane were alulated whih is shown in Fig. 4.15. Theseresults shows potential of thin omposite lms with t/L ≈ 1 for the appliation ofstrethable swithes. The ller onentration an be optimized along with the thik-ness of the lm to obtain ertain design window for spei swithing requirements96Chapter 4. Analysis of onned 3D network of CNTsor polymer-ller ombinations.Finally we alulated the resistane hange ratio △R/R0 to estimate the strainsensitivity for dierent lm thikness whih is shown in Fig. 4.16. We an see that fort < 5µm, with inreasing ǫ the lm resistane inreases resulting in higher resistanehange ratio △R/R0. The △R/R0 values are higher for thinner lms at any strainindiating higher gauge fator. However, for t = 5 and 7µm, as shown in the insetgure, the R/R0 dereases from zero to negative values with inrease of ǫ and quiklyreahes the minimum value of −100% at (ǫ ≈ 3%). From these results we an saythat lms with t < L shows good sensitivity to a broader range of strain, and thesensitivity gets higher for thinner lms. However, sine the ondutivity goes lower asthe lm gets thinner, a trade-o is needed to keep the resistane in measurable rangewhile designing the sensors. The lms with t = 5 and 7µm, with sharp hange inR with strain, show high potential in sensor appliation within smaller strain range.However, sine thiker lms have more robustness and higher potential for repeatedstrain appliation, sensor designing will need onsideration of all these parametersdepending on the strain requirements.4.7 SummaryIn this hapter, we have studied the relative eet of lm thikness and ller lengthon the ondutivity and strain sensitivity of CNT/polymer omposite lms. We nu-merially determined the hange in intertube distane for dierent lm thikness andCNT onentrations. Then the ondutivity of the omposite lms was estimatedapplying our analytial model. The partial alignment of llers introdued by the lmthikness lower than the ller length has shown to have signiant inuene on the av-erage intertube distane of the llers ausing remarkable impat on lm ondutivity97Chapter 4. Analysis of onned 3D network of CNTsand resistane hange ratio. Our results show that the minimum intertube distaneand highest ondutivity is ahieved at partial alignment of CNTs, speially at lowCNT onentrations. The numerial results were supported by the experimental re-sults. The sensitivity under longitudinal strain showed unusual trend at partial lleralignment when the lm thikness was lose to the ller length, indiating potentialappliation in strethable swithes and low range high sensitivity strain sensors.98Chapter 5Summary and future workIn this final chapter, we summarize our results and highlight the contributions of thisthesis in Section 5.1. In the next section we also propose ideas for continuation ofthis work in future research.5.1 Summary of resultsChapters 25 contain the major work carried out in my PhD research along with theresults. In the following, we briefly review the main results of each chapter.In Chapter 2, we have focussed on the behavior of two dimensional network ofCNTs. We numerically studied the effect of CNT concentration, alignment and as-pect ratio on the intertube distance and proposed a numerical model of it. Then weestimated the tunneling resistance in CNT/polymer thin films using a Monte-Carlobased statistical simulation. For thin films we considered quasi-2D CNT network.The CNT and polymer properties can be modified to obtain the desired resistanceand sensitivity to strain. We calculated the tunneling resistance change ratio andgauge factor for a wide range of external mechanical strain to estimate the effect onstrain sensitivity under different parameter variation (CNT concentration, aspect ra-tio, alignment order). Our results demonstrate that higher sensitivity can be obtainedfor lower concentration and aspect-ratio of CNTs and at a specific partial alignmentwith respect to conduction direction, rather than a random network. The choice of99Chapter 5. Summary and future workpolymer and its energy barrier height play a critical role in determining the role oftunneling and percolation of NTs in the overall conductance and piezoresistance ofthe composite film. Our numerical analysis done in this chapter is a step towardunderstanding the role of intertube tunneling in determining the tunneling resistanceand qualitative estimation of strain sensitivity of a CNT/polymer composite thinfilms. Our study on quasi-2D CNT networks and its polymer composites can beutilized for design optimization and cost-effective fabrication of composite thin filmapplication such as flexible electronics and sensors.In Chapter 3, we moved towards the study of three dimensional network of CNTsfor analysis of conductivity and gas sensitivity of CNT/polymer composites. Firstwe proposed a numerical model of intertube distance for 3D CNT network. Then wedeveloped an efficient analytical model of conductivity of CNT/polymer compositesincluding the effect of CNT concentration and field emission tunneling through thepolymers. Here are the key features and advantages of our conductivity model ascompared to other models existing in the literature:1. The model presented in this work incorporates the electron tunneling effectbetween close fillers that are not in intimate contact. This is a prevalent casein polymer/CNT composites, but it is ignored in theoretical percolation mod-els and its modified versions. Thus, our model provides better accuracy ascompared to well-published percolation and associated models.2. The percolation and other associated models allow estimation of electronic be-havior for CNT concentrations above the percolation threshold, and the perco-lation threshold needs to be known beforehand through experiments. But ourmodel does not limit it's applicability to any filler concentration region. We canestimate the composite conductivity below and above the percolation threshold100Chapter 5. Summary and future workregion, and thus can estimate the percolation threshold for a given CNT aspectratio, alignment or polymer type before conducting the experiments. This isquite important for many novel uses of CNT composites that use lower thanpercolation or close to percolation concentrations.3. Our model provides a low-computational-cost method of conductivity estima-tion of composite films as compared to other published numerical models. Al-though some numerical models reported in literature include the tunneling ef-fect, their application is laborious and can be handled only for small samplestructures due to their high computational cost. Our analytical model dras-tically reduces the cost while providing the same accuracy as the numericalmodels.4. Inclusion of the intertube distance is another achievement in our conductivitymodel. Since the tunneling current through the junctions are playing the mainrole in determining the composite conductivity, and it depends on the distancebetween neighboring tubes, the average intertube distance works as the keyparameter in determining the conductivity and sensitivity of the compositefilms. Any change in the CNT concentration, alignment order, aspect ratio, filmswelling or stretching affects the intertube distance, in turn affects the tunnelingcurrent and the overall composite conductivity. Hence, establishing the relationbetween the intertube distance and composite conductivity becomes important,and this is done in our model. In our work, we already proposed a numericalmodel relating the average intertube distance and CNT concentration, whichallows us to estimate the composite conductivity for a wide range of CNTconcentration values. The change in intertube distance for variation in otherparameters, i.e., CNT alignment order, aspect ratio, film swelling, etc. can be101Chapter 5. Summary and future workeasily evaluated numerically with low computational cost. Then, we can applyour model to estimate the composite conductivity and sensitivity for all thesedifferent conditions of films without incurring much computational cost usingour analytical method. This in effect is a new hierarchical method for estimationof film properties that provide accuracy while having ease of computation.The model have been used to estimate the change in resistance due to film swellingin presence of organic gas, thus measure the gas sensitivity of the film. The change inreflection phase was calculated to determine the sensitivity of the film used as passivewireless gas sensor. Our wireless sensitivity results (S11phase) provides importantqualitative information for wireless sensing. Firstly, it shows that there is a change inthe reflected wave phase as the gas concentration increases, thus proves the potentialof the composite films sensor for wireless applications. Secondly, it reveals that thephase of the reflected wave gets higher for films with lower CNT concentrations for agiven range of gas pressures. Thus, the lower filler concentration shows higher wirelesssensitivity as well as higher gas sensitivity. Thirdly, our results show that for a gasconcentration higher than 150 Torr, the difference between the S11 phases for filmswith different CNT concentrations becomes negligible. Thus, the wireless sensitivity(S11) of the films with different CNT concentrations becomes indistinguishable athigh gas concentration values, whereas their gas sensitivity (4R/R0) at higher gaspressure gives more distinguishable values. This indicates that the films with higherfiller concentration with higher differential phase difference (4S11phase/4Ps) maybe more useful for remote metering and regulation of the gas pressure. Finally,since our results shows that the rate of S11 phase change becomes negligible afterthe gas pressure of 200 Torr, it indicates that there is a range of gas concentrationwithin which the wireless sensitivity of gas can be measured. The knowledge of the102Chapter 5. Summary and future workqualitative trend provided by these results can help in setting a cost-effective designwindow of parameters for wireless gas sensors before carrying out the experiments.In Chapter 4, we continued our study on 3D network of CNTs. Here we focussedon the relative effect of film thickness and filler length on the the electromechani-cal properties of CNT/polymer composites. We analyzed the conductivity and strainsensitivity of CNT/polymer composite films by varying the film thickness over a widerange below and above the filler length. The main goal was to observe the compositebehavior at the filler alignment introduced by the film thickness lower than the fillerlength. The partial alignment of fillers has shown to have significant influence onthe average intertube distance of the fillers causing remarkable impact on film con-ductivity and resistance change ratio. Our results on CNT alignment effect showsthat higher conductivity in composites can be obtained at low CNT concentrationby aligning the fillers slightly to the direction of conduction, rather than by increas-ing filler concentration and keeping the network random. It indicates that partialalignment of CNTs will allow low cost composite fabrication with low CNT loadingto achieve the same conductivity offered by highly on concentrated and randomlydistributed CNT composite samples. This finding has significant importance for par-ticular application, i.e. transparent conductors made of CNT/polymer compositesfor solar cells where low filler concentration is desired to ensure higher transparencyand the conductivity then needs a trade-o. Our study on composite film thicknessshows that conductivity works as a function of film thickness when the thickness goesbelow filler length, and higher strain sensitivity can be obtained with a thinner filmkeeping the same CNT concentration. Our results on composite films with thicknesslose to the filler length reveals an exceptional trend of conductivity with the changingstrain, that shows potential of this kind of films for stretchable switching applications103Chapter 5. Summary and future workand low strain sensing applications. Our results presented here is broadly applicableto conductive, stick-like filler networks in an insulating matrix.5.2 Future workIn this section we shed light on the potential future works that can be extended fromthe findings of this thesis.5.2.1 Model for compressive strain applicationThe analytical model of conductivity proposed in Chapter 3 of my thesis provides alow-computational-cost alternative for not only conductivity estimation of nanocom-posites, but also estimation of many properties of composites that depend on tun-neling conduction, such as, sensitivity to strain, swelling and bending applications.However, since the model is based on the assumption that the CNTs as soft-corepenetrable cylinders, the application of our model becomes limited. Although it canpredict the electrical behavior of composites under tensile strain, it is unable to tacklewith the prediction when the sensor is subjected to compressive strain. In future workthe model can be developed with hard-core bendable filler objects so that the geo-metric change of the fillers and the corresponding change in the junction areas as aneffect of mechanical compression can be included. Then the model will be able todeal with both the tensile and compressive strains.5.2.2 Application in non-uniform swelling of filmsOur work on the gas sensitivity of the composites assumes that the swelling of thecomposite films is uniform throughout the film thickness which is practically truefor thin films. For our case, we considered thin films that keep in micrometer range104Chapter 5. Summary and future workeven after swelling. This assumption leads to the uniform change in the intertubedistance of the fillers in the polymer matrix and the resulting film resistance is usedfor sensitivity estimation. However, for some applications thicker films may needconsideration. The change of non-uniform swelling increases as the film thicknessgets higher. To include the effect of higher swelling near the surface/edge than theenter of the film, we need to consider graded swelling. In that case, both the gasabsorption rate and swelling ratio would decrease as we move from the surface layerto the inner layers of the film. The grading can also vary with the type of the polymer.In future work, our model can be used as the base model required for implementationof gradation models for application in non-uniform swelling of composite films.5.2.3 Model for composite fibersDue to the high mechanical strength of the CNTs, the CNT/polymer compositeshave high potential for the application of composite fibers and yarns. The geometricfeatures of a composite fiber, along with the properties of the CNT and the polymer,can influence the electrical and mechanical behavior of the fiber. The concentration,aspect ratio, alignment and type of the CNTs play role in determining the fiber con-ductivity. In addition, change in the fiber diameter can influence the filler alignmentinside the fiber, thus can affect the intertube distance and the corresponding tun-neling conductivity and overall resistance of the fiber. Numerical simulation can bedone to model the intertube distance for different parameters. Then our conductivitymodel can be applied to estimate the fiber conductivity and other electromechanicalproperties. Based on the fiber properties, the behavior of composite yarns can bestudied as a future work.105Bibliography[1℄ S. Iijima, Helial mirotubules of graphiti arbon," Nature, vol. 354, pp. 5658,1991.[2℄ N. Sinha, J. Ma, and J. T. W. Yeow, Carbon Nanotube-Based Sensors," Journalof Nanosiene and Nanotehnology, vol.6, pp. 573–590, 2006.[3℄ C. Park, Z. Ounaies, K. A. Watson, R. E. Crooks, J. Jr Smith, S .E. Lowther,J. W. Connell, E. J. Siohi, J. S. Harrison, and T. L. St. Clair, Dispersion ofsingle wall arbon nanotubes by in situ polymerization under soniation," Chem.Phys. 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