The nomenclature of unidirectional carbon-fiber plastics for functional purpose [9, 31, 32].
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"59030",title:"Functional Materials for Construction Application Based on Classical and Nano Composites: Production and Properties",doi:"10.5772/intechopen.73393",slug:"functional-materials-for-construction-application-based-on-classical-and-nano-composites-production-",body:'Polymeric composite materials (PCMs), combining low density, high elasticity modulus and strength, are now widely used in various fields of industry: aircraft building, automotive industry, construction, sports industry, medicine, etc. Modification of the surface of the reinforcing fiber and liquid polymer binder (PB) to improve the physico-mechanical and operational properties of the resulting classical PCMs and nano-modified (NM) PCMs is an actual task of polymer materials science [1, 2].
This task is realized both as a complex as well as separately by various methods of modification: physical—in the form of ultrasonic (US), chemical and combined physical and chemical. The aspects of designing the technological process for the production of functional classical PCMs and NMPCMs of structural design on the basis of reactoplastics are no less relevant [3]. Moreover, exactly US treatment is the dominant method of physical modification, which is simultaneously aimed at the intensification of many technological operations for the production of such materials, as well as improving the physico-mechanical and operational characteristics of the products and structures obtained [4].
A number of studies are devoted to the issues of preparation, enhancement of physico-mechanical and operational properties and features of the use of functional reactoplastic NMPCMs. In particular, the directions in which NMPCMs production technology is developing, including economic aspects of implementing their formation nanotechnologies, were described [5]. Among a wide range of means used to produce NMPCMs, it is preferable to use technical means implementing the method of low-frequency US cavitation [6]. The advisability of such choice is primarily due to the need for a uniform (homogeneous) distribution of nanoparticles introduced into the liquid oligomer. At the same time, this process is actively hampered by the physico-chemical nature of the nanoparticles, which are characterized by high surface energy.
This, in turn, causes the mutual attraction of nanoparticles, which leads to their coalescence and aggregation upon incorporation into the liquid oligomer. It was shown that the conditions necessary to create functional nanomaterials are small size and a distribution of carbon nanofiller particles as uniform as possible in the liquid polymeric matrix [7]. The efficiency of introducing the nanoparticles into the liquid polymer medium depends on not only the dose but also the mixing parameters [8].
One of the options for improving the physical and mechanical properties of carbon PCMs is the creation of carbon plastics combined filling [9]. This is accomplished by modifying the surface of a continuous carbon fiber and a liquid PB. The chemical modification of the last one is carried out by incorporating and subsequent uniform distribution in the bulk of the liquid PB of ultradispersed carbon nanoparticles. As the latter, for example, fullerenes, astralenes, CNTs and others are used.
Thus, the above brief analysis shows that the most relevant areas of research in the field of polymer nanotechnology in relation to functional nanocomposites are, in particular, the following:
selection of the most effective types of carbon nanofillers for the development of functional PCMs using the example of carbon plastics;
determination of the optimal concentration of nanoparticles in the polymeric matrix;
choice of effective methods for dispersing nanofillers during the production of functional NMPCMs based on reactoplastics;
creation of carbon plastics combined filling;
production of functional hybrid PCMs (HPCMs) based on reinforcing fabric with NM filler;
production of functional NM carbonic composites with increased electrical conductivity and crack resistance.
The above aspects are briefly presented in this chapter.
As the dominant method of physical modification of liquid polymer media and reinforced classical PCMs and NMPSMs of functional purpose, US low-frequency cavitation is used in most cases. The main parameters of this action are the frequency, amplitude, intensity, pressure, temperature and volume of the liquid medium (PB) being treated. The set of interdependent optimal parameters of cavitation processing, as a rule, is set experimentally in each specific case.
In the optimum, the resulting set of parameters of US cavitation processing leads to an increase in the physico-mechanical and operational properties of solidified oligomers and reinforced PCMs based on them [10]. The amount of hardening depends on the particular type of oligomer to be processed and can vary from 40 to 50% for classical hardened reactoplastics or may increase for several times (depending on the type of nano-modifier used) as compared to the initial composite for NMPCMs [8, 11].
The influence of the developed modes of low-frequency US cavitation treatment on the operational properties of reactoplastic PCMs was investigated in comparison with the known methods. An effective range of interrelated processing parameters has been found [11]. The obtained experimental results confirmed the prospects of realization of such a modification of liquid oligomers by the example of epoxy oligomers (EOs) and epoxy compositions (ECs).
It was found that such a modification is effective not only in the low-frequency, but also in the mid-frequency US ranges, both individually and in combination [12]. Variable overpressure during the US treatment is also a promising method. In particular, this is confirmed by the results of molding epoxy couplings with shape memory effect. An increase in intensity and a reduction in time was also established, both for a separate operation of sonification the liquid polymeric composition and for the manufacture of the entire product as a whole [13].
The purpose of using US modification of reinforced PCMs is to achieve a range of positive results upon completion of such a modification.
The first positive result is the US activation of the surface and the structure of the reinforcing fibrous macrofiller to improve its wettability of the liquid EC. The second positive result is the degassing of the structure of the reinforcing macrofiller immediately before it is impregnated. The third positive result is an increase in the productivity (not less than 2–2.5 times) of operations of sonification, impregnation, dosed application of liquid EC and winding while maintaining the stability of the properties of the final PCM.
Another positive result is the stabilization of the content of the PB in the impregnated woven filler, with variation in the speed of its stretching in the impregnation and dosing operations [14]. Finally, the effective parameters of US treatment contribute to increasing the deformation-strength and adhesion characteristics of PCMs, to lower the level of residual stresses, to increase the durability and to reduce the cumulative hardening time [15].
As a result of practical use of the methods of US cavitation processing, the designs of the impregnation, dosing and winding units on the serial impregnating and drying equipment were improved. The above mentioned results once again confirm the choice of US treatment as the main method of physical modification of liquid (including polymeric) media and reinforced fibrous PCMs of functional purpose at the main stages of their production.
Despite the existence of a huge range of US cavitation processing tools, the technical means used for US modification of liquid polymeric media and reinforcing fillers based on them can be conditionally divided into US concentrators, speed transformers and radiating plates.
US concentrators (dispersants) are effectively used primarily for processing liquid polymeric media, including when nanomodifiers are introduced into them. Radiation plates are used mainly for processing impregnated woven fibrous fillers up to 2000 mm wide.
Despite on the difference in the objects of application, for both of the above mentioned types of US technical means it is necessary to determine the effective design and technological parameters [16]. Otherwise, the appearance of defective areas of the resulting final PCMs (both classical and NM) is possible.
In this case, during the treatment of impregnated tissue, the US cavitators are based on piezoceramic transducers with a radiating plate that undergoes bending vibrations [17]. In addition, it is necessary to eliminate the non-uniformity of bending vibrations of radiating rectangular plates by calculation-experimental methods [16, 17]. Otherwise, there is a high probability of obtaining defective PCMs.
The tasks of designing the technology and equipment for the production of classical PCMs and NM functional PCMs are aimed at identifying and studying the interrelations between the structural, mechanical and geometric parameters of products, on the one hand, and the technological factors of their production, on the other. For example, obtained analytically kinetic equations of longitudinal and transverse impregnation of oriented and woven fibrous fillers with liquid PBs make it possible to predict the impregnation time and the speed of broaching the fibrous filler through the impregnating bath, and also to design its dimensions [18]. Design of the optimal tension force of impregnated fibrous fillers in circumferential winding allows the study of experimental results on the influence of technological impregnation regimes on the strength of impregnated and cured fibrous fillers [19, 20].
Moreover, in order to minimize material and time costs, as a rule, the methodology of structural and parametric modeling of design and technological parameters of technology and equipment (tools) for sonification of liquid polymeric media and for producing reactoplastic PCMs is used [3]. In turn, the geometric parameters of the model of oriented classical PCMs, which are considered as structurally inhomogeneous media, are the basis for designing technological manufacturing parameters and predicting the stress-strain state of solidified classical PCMs [21, 22, 23]. Figure 1 shows an innovative scheme for producing of unidirectional reactoplastic classical PCMs and NMPCMs for functional purposes using low-frequency US treatment.
Scheme of impregnation and dosed deposition of PB on long-length fibrous material: 1—impregnating bath; 2—liquid PB; 3—envelope roll in impregnating bath; 4—long-measuring fibrous material; 5—bobbin with a dry fibrous material; 6, 9—enveloping rollers; 7—squeezing rollers; 8—drying chamber; 10—take-up bobbin; 11, 12—a pair of working US instruments (emitting US plates); 13, 14—US generators; 15—magnetostrictive transducer; 16—waveguide-concentrator US; 17—reservoir with epoxy resin; 18—reservoir with curing agent of epoxy resin; 19—the tap; 20—thermoregulating cell; F1 and F2 are the forces of pressing working tools 11 and 12 against the impregnated fibrous material; α1 and α2 are the slopes of the working tools 11 and 12 to the surface of the impregnated fibrous material; v—broaching speed.
For example, the investigated traditional structural scheme for impregnation and dosing of a polymer (epoxy) binder on a long fibrous filler using US modification (Figure 1) can be conveniently divided into such separate structured blocks: I—the block for sonification the EO and preparing an impregnation EC based on it; II—the block of “free” impregnation of oriented (woven) fibrous filler with liquid EC; III—the block of dosed application of liquid EC to impregnated fibrous filler.
The main stages of innovative production of unidirectional reactoplastic PCMs for functional purposes using US are as follows (Figure 1). Prior to the impregnation step, US treatment of the epoxy resin in reservoir 17 is carried out using an US concentrator 16. The US concentrator 16 is connected to a magnetostrictive transducer 15, which is powered by an US generator 14 (block I in Figure 1). Depending on the selected geometry of the US concentrator 16, a certain value of the amplitude A and intensity Io of the US vibrations introduced into the liquid epoxy resin is obtained at the output. In the case of preparing a NM EC, for example, CNTs are added thereto. Controlled parameters of US processing are: time τ, temperature T, amplitude A and intensity Io.
After US treatment of the epoxy resin, a hardener of epoxy resin enters the reservoir 17 from reservoir 18. Here, the epoxy resin is mixed with its hardener with the same US concentrator for a few seconds, resulting in an impregnating composition. Next, the tap 19 is opened and the impregnating composition is fed to the bath 1. After this, the dry long fibrous material 4 is rolled up from the bobbin 5, which, after passing the enveloping roll 6, enters the bath 1, where it is impregnated with the liquid PB 2, which has already been treated with US (block II in Figure 1). In this case, several structural forms of the placement of long fibers in the cross section of the impregnated filler, which affect the kinetics of the impregnation process, are possible.
After exiting the impregnation bath 1, a preliminary uncontrolled application of the PB 2 to the long fibrous material 4 is provided. This material in the dosing zone (block III in Figure 1) is processed on both sides by US instruments in the form of US rectangular plates 11 and 12 emitting US having individual drives from the US generator 13. These plates 11 and 12 contact the material 4, which was impregnated and processed by the edge of the plate edge, with varying pressing forces F1 and F2, respectively. The US vibrations propagate both along the width and along the length of the plates 11 and 12. Moreover, the tools 11 and 12 are disposed with respect to each other along the length of the material 4, processed on both sides relative to it, and at different angles of inclination α1 and α2 to the plane of the material. In turn, working US instruments 11 and 12 consist of several successively located structural elements of various shapes and sizes made of different materials.
Varying the content of the PB, uniformity of its distribution in the material, and removal of the excess PB are achieved by adjusting the angle of inclination α1 and α2 of the radiating plate to the surface of the material 4, by varying the power (intensity I1 and I2) applied to the transducers, and by dosing the pressing forces F1 and F2. The final spinning of the PB from the impregnated material is carried out by a means of squeezing the material made in the form of two rolls 7. The impregnated and squeezed material then enters the drying chamber 8, and after drying passes through the envelope roller 9 and is wound onto the take-up reel 10. When using the developed innovative technology and equipment, the dependence of the deposit on the speed of drawing the impregnated material is largely eliminated, as well as the degassing of the air from the structure of the impregnated material due to the contact US action.
Then, within the synthesis framework, only the above mentioned enlarged blocks I–III and their constituent structural elements are analyzed, as well as structural and technological interrelations between them. Implementation of the developed approaches to the use of effective US devices allows obtaining a wide range of practically defect-free classical PCMs and NM PCMs for functional purposes. Effective hardware for connection and repair of polyethylene pipelines using US modification and heat shrinking are studied in [24, 25, 26, 27, 28]. The epoxy-glue compositions and banding, fiberglass tape impregnated with epoxy compositions, thermistor couplings with shape memory and surface-treatment methods for polyethylene pipes are investigated. The results obtained in these studies are another example of the creation of functional materials for construction application based on classical reactoplastics and thermoplastics used in gas-pipeline repair.
A set of specific requirements to the constructional materials based on PCMs, which are used in highly loaded structural elements, are set [29, 30]. Among them are the simultaneous provision of high strength and rigidity; resistance to alternating dynamic loads; small mass; high long-term strength. No less important indicators are heat resistance and corrosion resistance, while ensuring a high degree of reliability of the designed construction as a whole. The above complex of requirements is satisfied, for example, by classical and NM PCMs in the form of carbon plastics based on thermoset (epoxy) matrices [31, 32].
For example, the spectrum of application of NMPCMs based on carbon fibers and HD modified with nanoparticles (CNTs) is highly loaded materials, products and structures intended primarily for use in chemical, shipbuilding, machine, engineering, building, aviation, rocket and aerospace and other industries [33, 34, 35, 36]. The advantage of using carbon plastics in comparison with traditionally used metals, fiberglass and organic fibers, is that it provides increased compressive strength, high elasticity modulus and fatigue strength, low creep and dimensional stability of the molded article due to the low temperature coefficient of linear expansion. Not less important advantages are high indicators of chemical and radiation resistance, as well as good workability (manufacturability) [37]. The above mentioned set of advantages also determines the advisability of processing carbon plastics on serial production equipment at low labor and energy costs.
It should be noted that in the CIS countries, a wide range of unidirectional structural carbon plastics for functional purposes was created (Table 1) [9, 31, 32].
Physico-mechanical properties | Structural carbon-fiber brand | |||||
---|---|---|---|---|---|---|
KMU-11 m | KMU-31n | KMU-41 | KMU-4e | KMU-5e | KMU-6-36 | |
Tensile strength σt∙10−1, MPa | 80 | 75 | 70 | 100 | 80 | 70 |
Compression strength σcs∙10−1, MPa | 60 | 60 | 60 | 100 | 80 | 70 |
Shear strength τs∙10−1, MPa | 5.9 | 5.5 | 5.5 | 8.0 | 6.0 | 5.7 |
Elasticity modulus E ∙10−1, MPa | 14,500 | 15,000 | 14,000 | 12,000 | 12,000 | 12,500 |
Fatigue strength σ−1∙10−1 based on I07 cycles, MPa | 50 | 45 | 50 | 70 | 55 | 50 |
Stress coefficient K1c∙10−1/0.31, MN∙m3/2 | — | 68 | 126 | — | — | 94 |
Maximum operating temperature, °С | 180 | 100 | 150 | 150 | 80 | 150 |
In the CIS countries, aircraft component parts and units are usually manufactured by an autoclave solvent polymer impregnation processing technique [38]. However, foreign companies usually use nonsolvent processing technique. Molding without autoclave (e.g. infusion molding or vacuum and pressure impregnation) sufficiently decreases the PCM production costs.
In the reinforcement plane, the properties of the PCMs are determined mainly by the properties of the reinforcing material. And in the perpendicular plane, the properties of the PCM are largely limited by the strength characteristics of the PB and the adhesion value between the PB and the reinforcing filler. This often leads to the destruction crack initiation and propagation in the interlayer space of PCM under the influence of normal and tangential loads arising during the operation of products.
The growth of such cracks under the action of alternating, static and shock loads leads to a catastrophic destruction of PCM components. Therefore, the low impact strength of carbon composites significantly reduces the area of their possible application. The standard method for increasing the crack resistance is to introduce a thermoplastic polymer into the epoxide matrix which is soluble in it, for example, a polysulfone, whose concentration exceeds 20% (by weight) [38]. At the same time, the viscosity of the PB becomes extremely high, which makes this method unacceptable for the manufacture of PCM by infusion methods.
Especially emphasized should be the promising constructional NM carbon composite of brand VKU-18tr [39]. It is produced on the basis of the polymeric binding grade ENFB-2 M, an equiprobable fabric of Porcher Ind. (art. 3692). Nanoparticles of astralene of NTS brand in the amount of 0.5% of the mass fraction of the cured polymer matrix are used as the nanomodifier of the liquid PB. Carbon plastic of the brand VKU-18tr was compared with its closest analog in the form of carbon-plastic KMU-4-2 m-3692, obtained on the basis of the same components, but without the astralean nanoparticles.
The results of comparative tests of these carbon plastics are given in Table 2.
Property | Reinforcement direction | Material | Improvement of carbon nanofiber properties, % | |
---|---|---|---|---|
Carbon-fiber plastic VKU-18tr | Carbon-fiber plastic KMU-4-2M-3692 | |||
Tensile strength, MPa | [0°] | 780 | 750 | 4 |
[90°] | 600 | 550 | 8 | |
[0°, 90°, ±45°] | 640 | 490 | 23 | |
Tensile elasticity modulus, GPa | [0°] | 65.4 | 64 | — |
[0°, 90°, ±45°] | 55.5 | 45 | — | |
Shear strength, MPa | [0°] | 768 | 700 | 10 |
[90°] | 660 | 600 | 10 | |
[0°, 90°, ±45°] | 450 | 405 | 11 | |
Interlayer shear strength, MPa | [0°] | 46 | 42 | 10 |
Shear strength in sheet plane, MPa | [±45°] | 93 | 82 | 13 |
Operating temperature, °С | — | 170 | 120 | 42 |
Tropical stability after exposure in a tropical chamber for 3 mo., percent retention of compression properties | [0°, 90°, ±45°] | 96 | 55 | 41 |
The results of Table 2 indicate that the VKU-18tr NM carbon plastic surpasses the number of operating parameters of its analog material, for example, strength, including at higher temperatures. So, for example, at a temperature of 170°C, the strength of the VKU-18tr NM carbon plastic produced by compression is 54% higher. With an interlayer shifting, this excess is 32%, and with a bending, 20%. The same trend is observed with an increase in temperature: hardening at room temperature is from 10 to 15%, and at a temperature of 170°C it increases to 30–50%. Even more, as the temperature rises, the strength of the VKU-18tr NM carbon plastics increases under compression.
The above confirms the urgency of developing new, cost-effective and technologically advantageous methods for modifying the surface of carbon fiber and liquid PB. The ultimate goal of such a modification is to improve the physico-mechanical and operational properties of carbon composites.
One of the promising directions for improving the physico-mechanical and operational characteristics of elements of load-bearing unit made from reinforced reactoplastic PCMs is the creation of functional reactoplastic hybrid NMPCMs (HNMPCMs). The latter are obtained by incorporating nanoparticles into the structure of a liquid polymeric matrix and a reinforcing macrofiller [40, 41]. In investigations it was established that the nanomodification of the liquid PB, carried out with the help of optimal US action prior to impregnation of the reinforcing macronutrient, improves the processability of the PB and the strength of the solidified polymeric matrix [7, 37].
This, in turn, favorably affects on the hardening of the obtained NMPCMs, especially under the action of tensile loads [9]. At the same time, the nanomodification of only the liquid polymeric matrix does not have a significant effect on the strength characteristics of the PCM structural elements, the functional purpose of which is mainly in the perception of external force loads. This can be explained by the fact that when impregnating the reinforcing strands of the macrofiller with the NM liquid polymer matrix, the nanoparticles do not enter the interfiber space of the reinforcing macrofillers and filtering out on their boundaries.
Figure 2 shows a structural model of a functional HNMPCM, the reinforcing fabric of which contains fibers coated with CNTs [40]. The connection of CNTs grown on the surface of each individual fiber during interaction between parallel reinforcing fibers and transverse layers of reinforcing fabric is shown.
Scheme of a functional hybrid nanocomposite with grown CNTs on monofilaments [40].
The search for new efficient technologies has already led to original methods. One of them is growing of CNTs on monofilaments of reinforcing fabrics (Figure 3).
SEM image of glass fabrics and carbon fabrics used in laminate production [40]: a—without CNT; b—with CNTs grown along the radius on the surface of the fiber, as well as individual fibers coated with CNTs; resolution on the lower images of a and b ~20 μm.
With this method, the classical technology of manufacturing composite products [40], practically is not changed, which is an undoubted advantage of the developed technology [41]. The principal possibility of growing CNTs on reinforcing fibers before their subsequent impregnation with a liquid PB was investigated. For the study, dry fiberglass and dry twill rope twill weave were used (type Twill of the brand: 3k, 2 × 2 Twill Weave Carbon Fiber Fabric, 5.7 Oz/Sq Yd, 50″ wide, .012″ thick and 3k, 2 × 2 Twill Weave). In Figure 3a and b shows the surface of the fiber before and after application of CNTs.
Growth of CNTs on carbon fibers was carried out by pyrolysis of gases (CVD process) [41]. The length of CNTs, as a rule, was much greater than the distance between the layers of tissue (~10 μm) and between the fibers in strands (~1–5 μm). CNTs grown on fiber reinforcing fibers were studied using a scanning electron microscope (SEM). It was found that the CNT grew uniformly and densely in the radial direction on the surface of each individual fiber in the reinforcing fabric (Figure 3b).
Morphology (uniformity of growth, distribution along the fiber and throughout the laminate), the diameter of CNTs and the uniformity of their distribution were evaluated using a transmission electron microscope (TEM). The TEM analysis of multilayer photos showed that CNTs grown on fiber reinforcing fibers have an outer diameter of ~17 ± 2 nm and eight concentric layers (Figure 4).
TEM images of CNTs grown on a carbon fiber with a low (a) image resolution and a single high-resolution CNT (b) (concentric layers in the CNT structure are shown) [41].
As a result of impregnation with a polyester binder reinforcing NM fabric, a new structure is obtained—a HNMPCM, where nanoadditives are present not only in the polymeric matrix, but also in fibers. It is called free reinforced plastic (FFRP). When studying the wettability of CNTs with a reactoplastic polyester adhesive, it was found that CNTs are easily impregnated with such binding capillary method (free impregnation method) [41]. Improved adhesion between CNTs and reactoplastic material creates a strong functional nanocomposite for power purposes. Thus, due to the synthesis of CNTs directly on macrofibers, the solidified polymer fixes uniformly distributed CNTs on the surface of dry macrofibers, creating a functional nanocomposite of increased strength (the shear strength of HNMPCMs based on the polyester matrix increased by 60%).
It is known that one of the main problems during the process of making samples was the impregnation by PB of reinforcing woven layers, which contain single CNTs. Images obtained from optical and scanning electron microscopes demonstrate the presence of CNTs throughout the entire laminate section and the absence of differences between laminates with and without CNTs (Figure 5).
Hybrid composite samples based on a nanocomposite with a CNT content of 0.2% [41]: a—three-layer laminate after trimming; b—SEM image of intersecting tissue layers impregnated with a nanocomposite binder; c—SEM image of the section of composites without CNTs; d—cross section of reinforced CNT hybrid NMPCMs.
The PB under impregnation under the influence of vacuum completely impregnated all reinforcing macrofibers together with CNTs. The sections of the laminate without CNTs (Figure 5c) and with CNTs (Figure 5d) at the same scale indicate a qualitative structure and a uniform impregnation of the hybrid layers with a PB. This is confirmed by the absence of voids in the structure and the quasiregular distribution of the fibers after vacuum impregnation.
At the same time, the distance between macrofibers in the HNMPCM is slightly larger than in the base sample due to the expansion of the fibers by growing CNTs. Thus, the creation and study of functional HNMPCMs has shown that the application of the method of gas-phase growth of CNTs ensures their uniform distribution on reinforcing macrofibers of the filler, and vacuum impregnation with a PB reliably binds CNTs with reinforcing macrofibers.
In the past 10 years in the CIS countries the possibility of using CNTs for additional reinforcement of PCM has been intensively studied. As a result of these studies, HNMPCMs are created, in which CNTs are contained along with a reinforcing macrofiller, carbon or glass fiber [33, 42]. These studies provide the basis for the creation of functional NM carbonocomposites with increased crack resistance. According to the data of the studies of a number of investigators, CNTs exert maximum influence on the value of the ultimate strength at interlayer shear and the fracture toughness of reinforced PCMs [43, 44, 45].
At the same time, the characteristics of PCM, which depend on the properties of the reinforcing filler (elasticity modulus and tensile strength), change slightly. Still, it should be noted that the results of the study of the effect of CNTs on the crack resistance of the HNMPCMs, obtained by various authors, differ significantly. The use of PB modified by CNTs in the composition of the HNMPCM allows increasing the crack resistance coefficients G1c and G2c by an average of 15–40% [46, 47, 48, 49]. At the same time, it is possible to obtain an increase in the crack resistance coefficient up to 120–180% if a thermoplastic polymer is incorporated into the PB together with the CNTs, which acts as a non-covalent CNT modifier [50, 51].
When using films based on a partially cured EC filled with CNT [40], or a mixture of CNT and thermoplastic polymer [52, 53], it is possible to increase the coefficient of crack resistance by 2–3 times. The resulting films were placed between the layers of the reinforcing filler. In addition, the crack resistance is affected by the method of obtaining HNMPCM [52]. In the first embodiment, the nanofiller was functionalized with silane groups. Before the winding on the reinforcing fiberglass, a PB (EC) modified with short CNTs and nanofibers was applied, after which the coefficient of crack resistance of the obtained monolithic HNMPCM sample was determined.
In the second variant of the winding, two separate plates were made on the basis of a modified PB, which were then glued together with a PB containing a functionalized nanofiller. The crack propagated along the gluing layer. Despite the practically identical composition of the modified binder, in the first case the crack resistance coefficients in the reinforcement direction G1с and in the perpendicular direction G2с increased by 5–14% compared to the initial values (monolithic plate based on the unmodified PB), and in the second by 79–109%, respectively. The influence of the degree of aggregation of CNTs and the type of functional groups covalently attached to their surface on the crack resistance of HNMPCM obtained by the vacuum forming method was investigated [38]. A binder based on an epoxy amine resin and an amine curing agent was used.
To regulate the rheological properties of the binder, an unsaturated polyester resin modifier and a radical polymerization initiator was used. The modification of the chemical structure occurred according to the type of interpenetrating polymer networks. The method of preparation of the concentrate (1.5% by weight) of CNTs “Taunit-M”, covalently functionalized with carboxyl (CNT-COOH) and amide groups (CNT-CONH2) in the epoxy amine oligomer, is described by [54]. Four types of concentrates of functionalized CNTs (COOH-1, COOH-2, СОNH2-1 and СОNH2-2) were used in the epoxy amine resin, which differed in the types of functional groups and the degree of aggregate dispersion. The aggregate size in concentrates COOH-2 and СОNН2-2 was 3–20 μm, and in concentrates СООН-1 and СОNН2-1 does not exceed 3–5 μm. As the reinforcing filler, a uniform fabric firm Toray was used: weav-twill, thickness of the bundle 3k (3000 filaments).
Six layers of filler were used to assemble the pack, each of which was applied with a previously prepared binder in equal proportions in such a way that its mass fraction in the finished cured sample was 36–40%. It was found that the modification of the epoxy binder by carboxylated and amidated nanotubes leads to a decrease in the glass transition temperature Тg (maximum of 8°C). And with an increase in the degree of aggregation, the difference between the glass transition temperatures Тg in the modified functionalized CNT (FCNT) and the control sample decreases. The maximum hardening of the modified epoxy binder was 20%. The use of a binder modified with well-dispersed carboxylated nanotubes makes it possible to increase the value of the energy of interlayer destruction of carbon-plastic GIR by 63% as compared to the initial sample. The likely reason for the increase in GIR is the change in the propagation path of the crack in the modified FCNT samples from the HPCM.
For HNMPCMs prepared using a binder with uniformly distributed CNTs functionalized with amide groups, an increase in the GIR value compared to the starting plastic is 41%. Aggregation of carboxylated FCNT leads to a decrease in the effect—an increase in the GIR value is only 17%. Figure 6 shows the results of a fractographic study of the surface of the HNMPCM after separation of the sample with a COOH-1 concentrator and a sample of reinforced plastic made using a SEM at various magnifications.
Results of fractographic investigation of the surface of the HNMPCM after separation, obtained with the help of SEM [38]: a, c, and e—control sample; b, d, and f—sample of HNMPCM with COOH-1 concentrate.
As can be seen from Figure 6, on a scale of up to 10 μm, the pattern of destruction of the initial and modified sample is practically the same. However, on a larger scale, there are significant differences: a large number of “pyramidal” formations are observed on the surface of a modified sample with FCNTs. In addition, the number of fibers destroyed during the stratification of the sample increases. It was found that when modifying the epoxy binders of the FCNT, the values of the elasticity modulus E and the tensile strength σt of the HNMPCM decrease on average by 35 and 25%, respectively. This is explained by a decrease in the volume fraction of the reinforcing fiber in the samples of HNMPCM (an average of 20%) compared to the control sample.
In recent years, the efforts of researchers in the field of polymer nanotechnology are concentrated in the direction of the creation of functional constructional PCMs of a new type. In such PCMs, the CNTs themselves are used as the reinforcing element. In particular, one of the most important functions of such PCMs is a high level of electrical conductivity [55]. So, it is shown that the conductivity of the CNT-filled composites depends not only on CNTs type, concentration and polymer matrix composition, but also on the nanocomposite production method.
For example, for materials that can be manufactured by pressure die casting methods, the task of creating nanocomposites that combine a high level of functional and physico-mechanical properties can be solved by using multiphase polymer matrices and extruders. This is necessary to ensure a high level of shear stresses. In particular, the authors of the paper used the following method [56]. An array of vertically oriented CNTs that were grown on an alumina substrate was sheared off the substrate and then compressed in a direction perpendicular to the CNT orientation axis. Then the resulting deformed nanofiller was impregnated with ECs, which are used for the manufacture of PCM and cured.
It is shown that the electrical conductivity of such a deformed nanocomposite in the CNT orientation direction is 10 Sm/cm at a CNT concentration of about 6%. Other authors reached significantly larger values of the electro-physical and physico-mechanical characteristics [57]. In contrast to the method described in [56], they stretched an array of vertically oriented CNTs in a direction perpendicular to their growth. The resulting stretched sheets were impregnated with an EC by vacuum infusion. For PCM, obtained on the basis of 1750 stretched sheets oriented in mutually perpendicular directions, the following physico-mechanical and operational (electrical conductivity) characteristics were obtained: tensile strength of 117 MPa, elasticity modulus of 7.45 GPa, electrical conductivity of 2205 Sm/cm at concentration of CNTs at the level of 8%.
Other researchers demonstrated a different way of obtaining PCM with a high level of functional and physico-mechanical properties [58, 59]. It is declared that this method can be used in the future for large-scale industrial production. According to this method, an array of vertically grown CNTs (7–9 nm in size) was subjected to orientational stretching and impregnation with a solution of bismaleinimide resin (Figure 7).
Scheme of the technological process of obtaining a complex filament from the CNT′s array [58].
The complex filament was collected by successively stacking several layers of oriented and impregnated nanotubes. From the obtained filaments, a prepreg was prepared, which was then pressed (the proportion of PCM being 50–55%).
Figure 8 shows a comparison of the dependence of the tensile strength of PCM on its elasticity modulus for materials reinforced with standard carbon fibers and CNTs.
Comparative dependence of the tensile strength on the value of the elasticity modulus for materials reinforced with standard carbon fibers and CNTs [58].
□—high-strength plastics; ▪—high-modulus plastics; •—developed plastics.
It was studied that such HNMPCMs due to the use of CNT as a reinforcing filler are characterized by a complex of high functional characteristics. In particular, the tensile strength is 3.8 GPa, elasticity modulus of 293 GPa and the electrical conductivity reaches 1230 Sm/cm. At the same time, during the production of such materials, it is advisable to use extruders in point of view of the need to obtain high shear stresses during the shaping of the such HNMPCMs.
Taking into account the fact that the density of the PCM is only 1.25 g/cm3, this material is extremely promising for use in the design of aviation and space technology. Also, a high level of electrical conductivity ensures the stability of the resulting HNMPCM to the effect of a lightning discharge. In addition, one of the possible ways to impart electrical conductivity to classical PCM based on reinforcing fillers is the use of CNTs decorated with metal nanoparticles [55, 60].
Thus, based on the foregoing, it can be concluded that the electrical conductivity of nanocomposites with carbon nanotubes is influenced by: the type of nanotubes, the composition of the polymeric matrix, and the technologies for obtaining the NMPCMs, including US treatment for modifying liquid polymeric media [61]. The last one forms the distribution of CNTs by the bulk (volume) of the composite and affect the value of the contact resistance between CNTs. At the same time, the technology of obtaining composite materials from a stretched array of vertically oriented nanotubes makes it possible to produce functional PCMs with a record level of electrical conductivity and physico-mechanical properties.
The results of the studies described in this chapter confirm the effectiveness of physical, chemical and combined physico-chemical methods of modification as a basic direction of improving the technological and operational characteristics of classical PCMs and NM liquid polymer media and reinforced PCMs of functional purpose on their basis. The choice of the dominant method of physical modification of liquid polymer media and reinforced classical PCMs and NM PCMs of functional purpose in the form of US low-frequency cavitation has been experimentally confirmed.
The effective spectrum of interrelated regime parameters of US processing is characterized, which is established exclusively by an experimental method. The efficiency of creation of carbon plastics of combined filling is shown, as well as the prospects of creating HNMPCMs on the basis of reinforcing fabric with NM filler, which is used as CNT. Methods for obtaining functional NM carbonocomposites with improved physico-mechanical and operational properties, in particular, with increased strength, electrical conductivity and crack resistance are described. Potential applications of such materials are briefly described.
Further directions of research in the field of creation of reinforced PCMs on functional purpose are improving the properties of nanomodifiers used in the form of CNTs, improving the technology of de-agglomeration and subsequent alignment of nanocomposite components, and developing innovative methods for the synthesis of carbon plastics of combined filling and hybrid coal composites.
In statistical physics only a few problems can be solved exactly. For complex problems, numerical methods can give exact results for problems that could only be solved in an approximate way. Numerical simulation can be a way to test the theory. The numerical results can be compared to the experimental results. The numerical simulation is placed between the fundamental and the experimental treatment; it has a quasi-experimental character (numerical experience). For problems of statistical physics, the most widely used simulation methods are the Monte Carlo method and the molecular dynamics method.
The first Monte Carlo simulation (MCS) was proposed by Metropolis et al. in 1953 [1]. The second Monte Carlo simulation was proposed by Wood and Parker in 1957 [2]. The obtained results were in good agreement with the experimental results of Bridgman [3] and those of Michels et al. [4]. In this method we attribute a series of initial positions chosen randomly to a system of N particles interacting through a defined potential. A sequence of particle configurations is generated by giving successive displacements to particles; we only retain configurations to ensure that the probability density is that of the chosen.
Molecular dynamics simulation (MDS) has been first introduced to simulate the behavior of fluids and solids at the molecular or atomic level. MDS was used for the first time by Alder and Wainwright in the late 1950s [5, 6] to study the interactions of hard spheres. The principle is the resolution of equations of motion for a hard sphere system in a simulation cell. The basic algorithm is Verlet’s algorithm [7].
In this chapter, we will present techniques of numerical simulations using the Monte Carlo method. We will present an application on the gas phase during plasma-enhanced chemical vapor deposition (PECVD) of thin films. The application concerns collisions between particles. Particles are in Brownian motion. Collisions, elastic or inelastic, are considered to be binary. Non-elastic collisions result in effective chemical reactions.
In Section 2, we cite some MCS and MDS works on PECVD processes. Section 3 presents general rules on numerical simulation methods. Section 4 presents how to simulate a physical problem using MCS? We present the Metropolis algorithm as a scheme to trait random configurations and different modules related to elaborate an MCS code. In Section 5, we apply the MCS on SiH4/H2 gas mixture during a PECVD process. Finally the conclusion summarizes the contents of the chapter.
The PECVD is the most widely used technique to produce hydrogenated amorphous silicon thin films (a-Si:H) for solar cells and for film transistors and electronic devices [8, 9]. Reactions during plasma deposition are complex and are not understood completely.
Gorbachev et al. [10, 11, 12] have developed a model that is based on chemical reactions and different processes in a PECVD reactor. The model takes into account the formation of SinHm oligomers (n ≤ 5). It presents a simulation of the growth of the films. Gorbachev et al. found that Si2H5 and Si3H7 strongly influence the growth of the film [11].
Valipa et al. [13] calculated the β reactivity of the SiH3 radical on a surface of a silicon lattice plane during the growth of a-Si:H using MDS. The mechanisms of physical and chemical interactions of low temperature plasmas with surfaces can be explored using MDS [14].
For a CH4/H2 mixture, Farouk et al. used the Monte Carlo method (PIC/MC); they calculated the ionization rate of the plasma and the deposition rate of the thin layer [15]. Rodgers et al. [16] have developed three-dimensional Monte Carlo simulations of diamond (100) surface CVD. Other works on MCS are in [17, 18, 19].
In our previous works [20, 21, 22, 23, 24], we were interested in the study of the gas phase and the interaction of plasmas with the surface, for SiH4/H2 and CH4/H2 gas mixtures during PECVD processes. The used numerical simulation techniques were MCS and MDS. To complete the studies, we used the fluid model [25].
The starting point of numerical simulation is a physical phenomenon; its purpose is to obtain useful physical results. Between these two points, several steps can be identified. These steps are general and they are applicable for MCS. The steps can be summarized as follows:
The physical phenomenon must be defined by the description of the dominant domain of physics. The main assumptions and simplifying approximations are necessary to understand the physical phenomenon and the design of the first model.
Mathematical model requires a mathematical formulation of the problem. It may be a problem of elements or discrete object or a problem of a continuous medium; it may be a spatiotemporal problem or frequency problem and may be a deterministic or probabilistic problem.
It would be interesting to know the mathematical equations that govern the phenomenon:
The forces between particles and elements
The potential interaction
The determination of a time scale
The determination of a length scale
Definition of constant magnitudes of motion and equilibrium magnitudes
Continuity equations, balance equations, transfer equations, etc.
The MCS technique has been chosen for this work; knowing its basic algorithm is necessary for elaborating the simulation. This step requires some actions:
Validation of the model on simple cases
Simulation calculation on complex phenomena
The MCS is based on a probabilistic process with a random choice of configurations and samples of the situation of the physical system. The two pedagogical examples most cited in the literature are the integration of a single variable function and Ising’s model of spin. In the following subsection, we define the integration of a single variable function. We introduce the Ising model at the end of Section 4.2.2.
Calculation of the definite integral for a function f(x) of a single variable x on domain {a, b} has been proposed (Figure 1):
The integral of a function f(x).
Let:
Let xi and yi be real random numbers (i = 1, 2,…, N), and let H be a real number greater than the f(x) for x belonging to the domain {a, b} (or x ∈ {a, b}).
Let r1 and r2 be two random numbers belonging to the domain {0, 1} according to a uniform distribution law. Generators (e.g., Ran, RANDOM, RANDUM, or other IMSL mathematical libraries) of random numbers can be used:
where xi and yi are random numbers (xi ∈ {a, b} and yi ∈ {0, H}).
The Monte Carlo (MC) method is based on a probabilistic process. Let N be the total number of cases chosen (possible cases). It is necessary to count the number of favorable cases (or the number of points below the curve y = f(x)); let yi ≤ f(xi)). The number of favorable cases is Nfav. When N➔∞, the value I of the integral is [26]:
An example [26] is the calculation of the value π by calculating the integral I on a quarter circle of unit radius (R = 1.0). The pairs of random numbers (xi, yi) satisfying the condition: xi2 + yi2 ≤ 1. The function f(x) is equal to
We take a = 0.0, b = 1.0, and H = 1.0.
For different values of N, we show that the numerical solution tends to π = 4I.
Although this integral is simple, it shows the strength and simplicity of the method. The technique can be generalized for the integration of multivariate functions.
We note that integration by the MC method is based on:
The choice of random configurations according to a uniform distribution law
Each configuration chosen is either favorable or unfavorable (the “or” is exclusive).
For statistical physics problems, the probabilistic choice of configurations is not always deterministic; the favorable and unfavorable cases are not exclusive. According to the Metropolis algorithm [26, 27], the steps of the simulation are:
Choice of a simulation cell of adequate shape to the studied phenomena. The size of the simulation cell is related to a scale of length characteristic of the forces and interaction potential of the studied phenomenon. This cell may contain Npc particles (and/or elements).
Choice of an initial configuration that responds to some physical and thermodynamic properties. The total or internal energy of the system is Ei.
Infinitesimal random displacement of a particle (or element of the system) and calculation of the new internal energy of the system Ef. This displacement is related to the physical magnitudes: time scale and length scale. The physical system tends toward a minimization of the internal energy of the system with some fluctuation. Let ΔE = Ef-Ei the fluctuation.
If ΔE ≤ 0; the new configuration is retained (favorable) and the different averages can be obtained; go to step (c).
If ΔE > 0; a random number ε is chosen such that 0 < ε < 1. Let the probability Pr equal to: Pr = exp. (−ΔE/kBT) (where kB is the Boltzmann constant and T is the temperature).
If ε < Pr, accept the move and in any case go back to step (c) for a new choice of an infinitesimal displacement (new configuration). Note that if such a trial move is rejected, the old configuration is again counted in the averaging with probability Pr.
Figure 2 shows how to choose between the selected configurations. Let ε be a random number following a uniform law; If ε1 ≤ Pr the configuration is retained, and if ε2 > Pr the configuration is rejected.
Configuration choice according to Metropolis scheme.
Numerical simulation using the MC method is a very important tool for the study of static properties. The basic algorithm is based on probability notions. Understanding of the distribution function and/or interaction potentials is the heart of the calculation.
In equilibrium statistical physics, the system has a certain probability that can be in any states. The probability of being in a state μ with energy H(μ) is given by the Boltzmann distribution P(μ):
where T is the absolute temperature and kB is called Boltzmann’s constant. It is conventional to denote the quantity (kBT)−1 by the symbol β. The normalizing factor Z, or partition function, is given by:
The average of a quantity Q fora system in equilibrium is:
The internal energy U, is given by:
which can be written in terms of a derivative of the partition function:
From thermodynamics we have expressions for the specific heat C, the entropy S, and the Helmholtz free energy F:
or
and
and
We can calculate other parameters affecting the system.
The Monte Carlo method is an excellent technique for estimating probabilities, and we can take advantage of this property in evaluating the results. The simplest and most popular model of a system of interacting variables in statistical physics is the Ising model. It consists of spins σi which are confined to the sites of a lattice and which may have only the values (+1) and (−1). These spins interact with their nearest neighbors on the lattice with interaction constant J; they can interact with an external magnetic field B coupling to the spins. The Hamiltonian H for this model is [26]:
The Ising model has been studied in one and two dimensions to obtain results of thermal properties, phase transition, and magnetic properties [26, 27, 28]. For chosen values of J and/or B, different steps may be taken for the calculations (simulation cell, initialization, configurations, boundary conditions, calculation algorithms). For any configuration, each spin takes the two possible directions. The detail of the calculation procedure is not the purpose of this chapter.
We give a system of N particles (atoms, molecules, ions or particles) placed in a cell of fixed volume, generally of cubic form. The initial positions may, depending on the case, be distributed randomly according to a certain law (uniform or otherwise) or have a given symmetry. In a fluid, a gas, or a plasma, the particles may have random positions in general; in a solid or surface, with a crystal structure, the particles take ordered positions. The choice of random initial positions allows great freedom on the choice of the number of particles in the cell.
At the first step, the particles are given velocities that are generally selected to have a zero total momentum. If the system is in thermodynamic equilibrium, the initial velocities will be randomly chosen according to a Maxwell-Boltzmann law. In the general case, the velocity distribution is according to the problem dealt with. All other phase properties can be initialized to the particles; the main thing is the conservation of the total quantities of the system.
The particles interact with each other according to chosen interaction potentials. Since the interaction potentials are specific for each “numerical experiment,” the main part of the work consists in calculating the interaction energies for each proposed configuration.
The choice of interaction potentials is directly related to the mathematical formulation of the problem according to the state of the medium: fluid, gas, plasma, or solid. It can be Lennard-Jones potential, Coulomb potential, Debye potential, Morse potential, Stillinger-Weber potential, Born-Mayer potential, Moliere potential, or others.
In general, two main boundary conditions are used: periodic boundary conditions (PBC) and minimum image convention (MIC) [29].
To minimize the surface effect, periodic boundary conditions (PBC) [30] are invariably imposed. The simulation cell is reproduced throughout the space to form an infinite mesh. We can simulate the properties of an infinite system. The particles that we follow are in the central cell; if a particle crosses a wall with a certain velocity, its image returns with the same velocity by the opposite wall. Under these conditions, the number of particles in the central cell, and consequently the density, is constant. These conditions also allow the conservation of the energy and the momentum of the system and do not introduce periodic effects (because of the interaction between particles).
According to the hypotheses and according to the geometry of the problem, other boundary conditions are proposed [26]. For example, in order to model thin films, the simulation cells are longitudinal and parallel to the film; one uses PBC in the directions parallel to the film. In the direction normal to the film, free edge boundary conditions can be used. In such cases, it may be appropriate to also include surface fields and surface interactions. In this way, one can study phenomena such as wetting, interface localization-delocalization transitions, surface-induced ordering and disordering, etc.
The core of the program includes calculating the potential energies of particle configuration and particle collisions. The interactions and collisions between particles can be elastic or inelastic; they can be binary or collective. For computation, the interaction energy of a particle with its neighbors is carried out by refocusing a base cell on the particle. This particle only interacts with particles in this region. This is called the “minimal image convention” (MIC) [1].
Generally, a RANDOM generator of real random numbers ri belonging to the domain {0, 1} (or ri ∈ {0, 1} is available. This distribution law is uniform.
To have a real random number xi belonging to the domain {a, b} (or xi
To have a real random number xi belonging to the domain {a, b} (or xi ∈ {a,b}) according to a formula (or law) of nonuniform distribution f(x), a histogram technique is used. Let Nm be the number of intervals. If the mesh is regular (Figure 3):
Random number selection according to f (x) distribution.
We define:
We define the sequence:
and the sequence:
Hence each real random number ri belongs to the domain {0, 1} (where ri ∈ {0, 1}) (according to the uniform law); this number belongs to the domain {rxj-1, rxj}. It corresponds to a random value xran of the domain {xj-1, xj}; this number satisfies the formula (or the law) of nonuniform distribution f(x).
This technique can be generalized for a nonuniform distribution law f(x) with an irregular mesh Δxi, or with tabular data f(xi) with i = 1,…, m.
The technique can be generalized, too, for a discrete distribution law f(i) with i = 1,…, m.
In the literature, the reader can find simple algorithms for the choice of random numbers of some simple functions (Gaussian, etc.).
It is necessary to find some parameters allowing the control of the smooth course of the evolution of the system. We must look for the constants of movement. For example for an isolated system, we have the conservation of the total energy and the quantity of matter.
By using the numerical simulation, it is possible to calculate many spatiotemporal quantities F(r,t). These quantities can be positions, speeds, kinetic moments, particle energies, concentrations, transport coefficients, etc. It would then be possible to calculate all other quantities related to F(r,t).
For the calculation of the averages, one can note the quantities on the space, on the time or on both. The histogram methods can be used. Static or dynamic distribution functions and spatial or temporal correlation functions can be calculated. It should be noted that the SMC is much more adequate for static properties because of the probabilistic choice of configurations.
Any calculated function or parameter F(r,t) can be used for another application in another calculation program.
In the MCS model discussed extensively in this chapter, it’s more about collisions between particles. It’s particle-particle MCS or PP-MCS. In many problems of physics, the general idea is the same, but the applications and proposed models are numerous.
Other MCS models, named particle-in-cell MCS (PIC-MCS), are based on particle-cell interactions. In these last models, we also use a probabilistic choice of configurations and small variations in the state of the system (following the Metropolis algorithm); the interaction is between the particle with a cell, a mesh, or a drop. The parameters and variables of the cell, although local and instantaneous, are macroscopic. These parameters and variables can be thermodynamic, fluid, or electromagnetic. An example of the model based on PIC-MCS is described by Mattei et al. [31] for simulation of electromagnetic particle-in-cell collision in inductively coupled plasmas. Several works can be found in the literature on this same line of work. Other MCS models using particles may be considered. [32].
For statistical physics problem solving (such as thin film deposition problems), MCS models use experimental, numerical, or theoretical data from other methods and models. Models can be improved to hybrid models. In the hybrid models, connections between two modules can be realized. The first module is MCS; the second module is fluid, electromagnetic, or other. An example of a three-module hybrid model is presented by Mao and Bogaerts [33] to study gas mixtures in PECVD system. The three modules are MCS, fluid, and electromagnetic. The first module EM calculates the electromagnetic fields by solving Maxwell equations. These fields are used as inputs in the module MCS, where the electron density, electron temperature, electron energy distribution function, and electron impact reaction rates can be computed with a Monte Carlo procedure. Subsequently, the module fluid calculates densities and fluxes of the various plasma species (i.e., heavy particles and electrons) with continuity equations and the electrostatic field with Poisson’s equation. This electrostatic field is used as input again in the EM. This cycle is iterated until convergence. The schematic of the hybrid model is given in Figure 4.
Schematic of a hybrid model of three modules used to study gas mixtures in the PECVD [33].
To solve statistical physics problems with evolutions as a function of time, kinetic models of MCS (kMCS) are used. Using kMCS, Battaile and Srolovitz [17] described kinetic phenomena of the diffusive motion of a single interstitial atom in a close-packed metal crystal. The motion of the interstitial atom is usually limited to two types: vibration of the atom around the center of the interstitial hole in which it resides and hops to nearest-neighbor interstitial sites. The atom can hop into any of the nearest-neighbor interstitial sites; it executes a random walk. In an MC simulation of this diffusion process, the new position of the interstitial atom is chosen at random from a list of the adjacent interstitial sites.
Other CVD and PECVD works on MCS are presented in Ref.s [15, 34, 35, 36, 37, 38]. They show how MCS methods can study properties of gas mixtures and properties of the growth of thin films.
In this section, we present an example of PP-MCS of collisions and reactions in gas phase of SiH4/H2 mixture used in PECVD process. Some paragraphs have been treated in previous works [21, 24].
We use a MCS to study collisions and chemical reactions in gas phase of SiH4/H2 mixture used in the PECVD process. In this phase, important reactions have been identified that contribute to the production and the consumption of hydrogen (H), silylene (SiH2), and silyl (SiH3). The hydrogen consumption reactions SiH4 + H → SiH3 + H2 and SiH3 + H → SiH2 + H2 are found to play a central role in deciding the distribution of hydrogen [39].The plasma chemistry indicates that H atoms and SiH3 radicals play an important role in the a-Si:H deposition process [40]. Experimentally, it is generally accepted that SiH3 radicals dominate a-Si:H and μc-Si film growth from SiH4 plasmas in the PECVD; it is the key precursor of a-Si:H deposition [41]. The proposed MCS allowed to get the ratio SiH2/SiH3 and mean value of densities of species. It provides information on SiH4 dissociation and on the production of SiH3, H, SiH2, and Si2H6 and other important parameters.
The plasma in the PECVD reactor is weakly ionized. For our study, the mixture gas contains 22% of SiH4 and 78% of H2; the pressure is 100 mtorr, the temperature of the gas ranges from 373 to 723 K, the electron temperature is about 2.5 eV, and the electron density is 3. 108 cm−3. The process is considered to be stationary. We take into account electrons and eight neutral species (SiH4, SiH3, SiH2, H, H2, Si2H6, Si2H5, SiH). Reactions taken into account include seven electron-neutral and 14 neutral-neutral reactions. Table 1 shows the 21 reactions and rate constants Kreac. At low temperature, the neutrals interact occasionally with each other and move under the effect of thermal agitation; their velocity distribution function is Maxwell-Boltzmann distribution. Electrons have the mean velocity with kinetic energy Te.
Symbol | Reactions | Kreac (cm3/s) |
---|---|---|
R1 | SiH4 + e→SiH3 + H+e | k1 = 3 × 10−11 [42] |
R2 | SiH4 + e→SiH2 + 2H + e | K2 = 1.5 × 10−10 [42] |
R3 | SiH4 + e→SiH + H + H2 + e | K3 = 9.34 × 10−12 [42] |
R4 | SiH4 + e→SiH2 + H2 + e | K4 = 7.19 × 10−12 [42] |
R5 | H2 + e→2H + e | K5 = 4.49 × 10−12 [42] |
R6 | Si2H6 + e→SiH3 + SiH2 + H + e | K6 = 3.72 × 10−10 [42] |
R7 | Si2H6 + e→SiH4 + SiH2 +e | K7 = 1.1 × 1010× (1.(1./(1. + (0.63 × P)))) [43] |
R8 | SiH4 + H→SiH3 + H2 | K8 = 2.8 × 10−11 × exp.(−1250/T) [44] |
R9 | SiH4 + SiH2→Si2H6 | K9 = 1.1 × 1010 × (1.−(1./(1. + (0.63 × P)))) [43] |
R10 | SiH3 + SiH3→SiH4 + SiH2 | K10 = 0.45 × 1.5 × 10−10 [44] |
R11 | SiH4 + Si2H5→SiH3 + Si2H6 | K11 = 5 × 10−13 [42] |
R12 | SiH3 + H→SiH2 + H2 | K12 = 2 × 10−11 [44] |
R13 | SiH3 + Si2H6→SiH4 + Si2H5 | K13 = 4 × 10−10 × exp. (−2500/T) [44] |
R14 | SiH2 + H→SiH + H2 | k14 = 2 × 10−11 [44] |
R15 | Si2H6 + H→Si2H5 + H2 | K15 = 0.66 × 2.4 × 10−10 × exp. (−1250/T) [43] |
R16 | Si2H6 + H→SiH4 + SiH3 | K16 = 0.34 × 2.4 × 10−10 × exp. (−1250/T) [44] |
R17 | SiH + H2→SiH3 | K17 = 2 × 10−12 [43] |
R18 | SiH2 + SiH3→Si2H5 | K18 = 3.77 × 10−13 [43] |
R19 | SiH2 + H2→SiH4 | K19 = 3 × 10−12 × (1. + (1./1. + (0.03 × P))) [43] |
R20 | 2SiH3→Si2H6 | K20 = 0.1 × 1.5 × 10−10 [43] |
R21 | SiH4 + SiH→Si2H5 | K21 = (1.−(1./(1. + (0.33 × P)))) × (6.9 × 10−10) [43] |
List of gas phase reactions and corresponding rate constants [24].
Let
And chemical reaction for the production of A is as:
Rate production and consumption for any species A are taken as:
The MCS is based on binary collisions at the microscopic level. Elastic collisions are between all particles, and inelastic collisions (or effective collisions) are those that result in a chemical reaction. A chemical reaction needs a collision involving at least two particles (atoms, ions, electrons, or molecules). According to kinetic theory, gases consist of particles in random motion. These particles are uniformly distributed in a cell which has a parallelepiped form of sizes Lx, Ly, and Lz (Figure 5). These particles move in a straight line until they collide with other particles or the walls of their container. Dimensions and volume of Monte Carlo cell must take into consideration the mean free path of species.
Form of the simulation cell.
Let ni be the density of neutral spice i (i = 1,…, 8). The first particle i is randomly chosen according to a probability of neutral species Prsp,I (nonuniform discrete distribution) given by:
The chosen particle takes randomly three components of space in cell ri(xi, yi, zi) according to the normal distribution (nonuniform distribution). It takes also randomly three components of velocity vi (vxi, vyi, vzi) according to Maxwell-Boltzmann distribution.
Let ni and nj be the densities of species i and j in the gas and Vij the relative velocity between the two species i and j.
According to the kinetic theory of gases, we have for an incident particle i on a target particle j the average collision frequency νij as:
where <sij> is the cross section of the particle j.
The mean free path <λι> of species i is:
The time between two collisions τij is then:
For chemical effective reactions (inelastic collisions) between two reactive species i and j giving products i’ and j’, the rate constant reaction verifies [45]:
General rules of collision theory are applied:
The new velocities of the colliding particles are calculated using conservation of energy and momentum for elastic collisions.
Conservation of total energy as isolated system.
Movement of the center of mass and relative motion around the center of mass.
The reader can refer to some fundamental physics books that deal with general notions of collisions and corresponding parameters [45, 46, 47, 48].
The plasma in the PECVD reactor is weakly ionized. At low temperature, particles interact occasionally with each other and move under the effect of thermal agitation. In reality, only a small fraction of collisions are effective (result in a chemical reaction) [21].
In our MCS, after traveling a random walk given by a Gaussian distribution, the first chosen particle collides with a second particle (molecule, atom, radical, or electron). The last particle j is randomly chosen according to a (i-j) collision probability Prcol,j (nonuniform discrete distribution) given by:
where
The activation energy is given by:
where the pre-exponential factor is assumed to be the collision frequency factor and Kreac is the rate constant of the gas phase reaction.
The two colliding particles (e.g., the electron and SiH4 molecule) can interact by several reactions (R1, R2, R3, and R4 in Table 1); we choose randomly one of gas phase reactions occurring according to a, nonuniform discrete distribution reaction probability Prreac (i,j):
where
All chemical systems go naturally toward states of minimum Gibbs free energy [21, 24]. A chemical reaction tends to occur in the direction of lower Gibbs free energy. To determine the direction of the reaction that is taking place, we use the old and new values of Kreac and the equilibrium constant with reactants and product concentrations. Each set of binary collisions can be related or converted into time. As cited in section (a), Table 1 gives gas phase reactions and corresponding rate constants used in this MCS.
To continue the simulation, after the elastic collision, particle i takes new values of components velocity and new mean free path; mean free path is taken from a normal (nonuniform) distribution (Gaussian distribution). If the collision is inelastic, we have to take a new particle.
From Metropolis algorithm, the scheme of this MCS is as follows:
Choices of particle of spice i with random position, velocity, and mean free path; periodic boundary conditions are used to keep particles in the elementary cell.
Choices of random collision with a spice j.
Study of collision type (elastic, inelastic). If the collision is elastic the particle i move with a new velocity and mean free path, and we return to step (b). If the collision is inelastic particles i and j give new particles i’ and j’, according to Metropolis scheme, and we return to step (a) or (b). Periodic boundary conditions are used to keep particles in the elementary cell.
At each step, we can note the different statistics.
Once the species are selected for the simulation model, an estimate of species densities should be made. Following the model of interaction and collisions between particles (binary, collective, etc.), a first choice of the minimum number Ni of particles of each species is made. A first estimate of the sizes (Lx, Ly, Lz) of the elementary cell is made.
The study of the types of interaction potentials and the calculation of the approximate values of the force ranges, the kinetic energies, the internal energies, and the energies of activation make it possible to correct the minimal numbers Ni of particles and the sizes (Lx, Ly, Lz) of the elementary cell.
Let kp be the number of a species, kp = 1,…, 9. The minimal numbers Qnp(kp) and the sizes (Lx, Ly, Lz) have to be discussed for statistical calculations.
For numerical programming, according to the programming language used and according to the size (or the computational capacity) of the computer, it is necessary to find a judicious choice of the tables of integer or real values and which values would be useful to save all during simulation. Let Ncol,m be the maximum number of elastic collisions per particle, and let Ncycle be the number of cycles to average the simulation calculations.
For this MCS, the numerical chosen values are in Table 2.
Cell dimensions and steps for collisions | Number of species Kp | Initial number of particles in cell | ||
---|---|---|---|---|
Lx (m) | 4.68 10−6 | 1 | Qnp(SiH4) | Qnp1 |
Ly (m) | 4.68 10−6 | 2 | Qnp(SiH3) | 10 |
Lz (m) | 20.0 10−3 | 3 | Qnp(SiH2) | 10 |
4 | Qnp(H) | 10 | ||
Ncol,m | 500 | 5 | Qnp(H2) | Qnp5 |
Ncycle internal cycle | 2000 | 6 | Qnp(Si2H6) | 10 |
Ncycle external cycle | 200,000 | 7 | Qnp(SiH) | 10 |
8 | Qnp(Si2H5) | 10 | ||
9 | Qnp(e) | Qnp9 |
Used quantities and parameters in calculations for the gas temperature Tg = 520 K.
For radicals (e.g., SiH3), particle numbers Qnp(k) are very small; we take Qnp(k) = 10. These numbers cannot take value 1 or 0, even if a species k is in trace form in the gas. The value 0 for a species k means that any other species k’ does not make a collision with the species k; and the value 1 means that we have no collisions between particles of the same species in the cell.
Qnp1, Qnp5, and Qnp9 are calculated from the volume of cell, the pressure, the temperature, and the total number of particles in the cell (Qnp1 = 0.81187824 * 109; Qnp5 = 0.20296956 * 109; Qnp9 = 131).
As we have chosen a stationary regime, we must reach the values and properties at equilibrium. The results of the simulation show this trend. In MCS, averaged values, distribution functions, autocorrelation functions, and correlation functions can be calculated. To ensure rapid convergence of calculations, it would be useful to look for statistically symmetric (or stationary or unsteady) parameters [26, 50].
As an example for our MCS calculation, we have:
The number of Si2H6, SiH, and Si2H5 particles reaching the surface is negligible.
Let Ns,i and Ns, H2 be the densities of a species i and H2 reaching the surface. The ratios Ns,i/Ns, H2 are too small (Table 3).
Let Ns,i be the density of a species i reaching the surface and Nv,i the density of same species i in volume. The ratios Ns,i/Nv,i are too small (Table 4); the surface effect is negligible.
The reactions begin with the dissociation (consumption) of H2 and SiH4 by R5, R1, and R2 reactions.
The production of SiH3 is done by R8, and then there is production of SiH2 by R12.
The reaction R2: SiH4 + e → SiH2 + 2H + e plays the central role in SiH4 dissociation by electron impact [24]. This result is compatible with [39].
The second important chemical reaction in the SiH4 dissociation is R1: SiH4 + e → SiH3 + H + e [24]. This result is compatible with that of Perkins et al. [51] and that of Doyle et al. [52].
Type | H2 | SiH4 | H | SiH3 | SiH2 |
---|---|---|---|---|---|
Ns,i/Ns, H2 | 1 | 0.23 | 1.67 10−4 | 8.60 10−5 | 9.86 10−6 |
Ratios Ns,i/Ns, H2 of particles reaching the surface compared to H2.
Type | SiH4 | SiH3 | SiH2 |
---|---|---|---|
v, j | 6.695 10−6 | 7.965 10−6 | 775 10−6 |
Ratios Ns,i/Nv,i of particles reaching the surface compared to volume.
MCS is a widely used method in statistical physics to study thermodynamic, structural, or phase properties. It is based on random and probabilistic processes. The purpose of this chapter is to present the technique for general use in physics for the study of thin film deposition problems. The technique can be generalized to other fields of science: biology, economics, transportation, and social sciences.
We started by presenting general rules for numerical simulation methods. Metropolis algorithm has been considered as the basic algorithm. After, we presented the different steps for the realization of a MCS code. We chose the particle-particle model MCS (PP-MCS) to explain the different steps and procedures to be applied in the deposition of thin layers by PECVD processes. We have shown that this technique can be generalized to the particle-in-cell MCS (PIC-MCS) case or kinetic MCS (kMCS), as it can be joined with other modules to give hybrid models. It is important to know how to choose random configurations from the laws or probability distributions in the system.
A numerical application is presented for collisions in a SiH4/H2 gas mixture in the PECVD process. A preliminary work of determination of the chemical reactions between molecules and radicals is made. A choice of the simulation cell is made, and the definition of the probabilities of the collisions between peers is made. The Metropolis algorithm makes it possible to follow the various elastic and inelastic collisions; it also makes it possible to make the statistics of the interactions with the surface. The results are compatible with [39, 51, 52].
Other questions may be asked to account for molecular ions, surface and volume kinetics, or thin film formation. The techniques and different models of the MCS (PP-MCS, MCS-PIC, kMCS) allow taking care of these questions.
The interconnection of the MCS with other models (MDS, hybrid model, fluid model, electromagnetic model, etc.) would allow answering more questions. The methods can be applied to other specialties than the physical sciences.
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\n\nPolicy last updated: 2016-06-08
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