Binary Reaction Research Articles

Mixed Convection and Thermally Radiative Flow of MHD Williamson Nanofluid with Arrhenius Activation Energy and Cattaneo–Christov Heat-Mass Flux

In this paper, we explored the impact of thermally radiative MHD flow of Williamson nanofluid over a stretchy plate. The flow in a stretchy plate is saturated via Darcy–Forchheimer relation. Cattaneo–Christov heat-mass flux theory is adopted to frame the energy and nanoparticle concentration equations. Additionally, the mass transfer analysis is made by activation energy and binary chemical reaction. Activation energy is invoked through the modified Arrhenius function. The intention of the current investigation is to enhance the heat transfer rate in industrial processes. The non-Newtonian nanofluids have more prominent thermal characteristics compared to ordinary working fluids. The governing models are altered into ODE models, and these models are numerically solved by applying the MATLAB bvp4c algorithm. The graphical and tabular interpretations have scrutinized the impact of sundry distinct parameters. The fluid speed escalates for enhancing the Richardson number, and it falls off for higher values of the Weissenberg number. It is noticed that the fluid temperature declines for higher values of the Brownian motion parameter and it grows for larger values of the thermophoresis parameter. The activation energy enriches the heat transfer gradient and suppresses the local Sherwood number. Additionally, the more significant heat transfer gradient occurs in heat-absorbing nonradiative viscous nanofluid and a smaller heat transfer gradient occurs in heat-generating radiative Williamson nanofluid. Also, we noticed that a higher heat transfer gradient appears in the Fourier model than in the Catteneo–Christov model. In addition, the comparative results are confirmed and reached an outstanding accord.

Deficiency zero for random reaction networks under a stochastic block model framework

Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper Prevalence of deficiency zero reaction networks in an Erdős-Rényi framework, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with n species, with an edge between two arbitrary vertices occurring independently with probability \(p_n\), we established the threshold function \(r(n)=\frac{1}{n^3}\) such that the probability of the random network being deficiency zero converges to 1 if \(\frac{p_n}{r(n)}\rightarrow 0\) and converges to 0 if \(\frac{p_n}{r(n)}\rightarrow \infty \), as \(n \rightarrow \infty \). With the base Erdős-Rényi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters \(\alpha _{i,j}\), with \(i,j\in \{0,1,2\}\) enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as “closed” networks with very few inflow and outflow reactions, or “open” networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters \(\{\alpha _{i,j}\}\), we establish a threshold function \(r(n,\{\alpha _{i,j}\})\) such that the probability of the random network being deficiency zero converges to 1 if \(\frac{p_n}{r(n,\{\alpha _{i,j}\})}\rightarrow 0\) and converges to 0 if \(\frac{p_n}{r(n,\{\alpha _{i,j}\})}\rightarrow \infty \).

Flow and heat transfer over a permeable moving wedge in a hybrid nanofluid with activation energy and binary chemical reaction

PurposeThe analysis of boundary layers is needed to reflect the behaviour of fluid flows in current industrial processes and to improve the efficacy of products. Hence, this study aims to analyse the flow and heat transfer performance of hybrid alumina-copper/water (Al2O3-Cu/H2O) nanofluid with the inclusion of activation energy and binary chemical reaction effect towards a moving wedge.Design/methodology/approachThe multivariable differential equations with partial derivatives are converted into a specific type of ordinary differential equations by using valid similarity transformations. The reduced mathematical model is elucidated in the MATLAB system by using the bvp4c procedure. This solution method is competent in delivering multiple solutions once appropriate assumptions are supplied.FindingsThe results of multiple control parameters have been studied, and the findings are verified to provide more than one solution. The coefficient of skin friction was discovered to be increased by adding nanoparticles volume fraction from 0% to 0.5% and 1%, by almost 1.6% and 3.2%. Besides, increasing the nanoparticles volume fraction improves heat transfer efficiency gradually. The inclusion of the activation energy factor displays a downward trend in the mass transfer rates, consequently reducing the concentration profile. In contrast, the increment of the binary reaction rate greatly facilitates the augmentation of mass transfer rates. There is a significant enhancement in the heat transfer rate, approximately 13.2%, when the suction effect dominates about 10% in the boundary layer flow. Additionally, the results revealed that as the activation energy rises, the temperature and concentration profiles rise as well. It is proved that the activation energy parameter boosts the concentration of chemical species in the boundary layer. A similar pattern emerges as the wedge angle parameter increases. The current effort aims to improve the thermal analysis process, particularly in real-world applications such as geothermal reservoirs, chemical engineering and food processing, which often encountered mass transfer phenomenon followed by chemical reactions with activation energy.Originality/valueThe present results are original and new for the study of flow and heat transfer over a permeable moving wedge in a hybrid nanofluid with activation energy and binary chemical reaction.

Transient rotating nanofluid flow over a Riga plate with gyrotactic micro-organisms, binary chemical reaction and non-Fourier heat flux

Bioconvection for rotational flow is conceived to provide stability to improved thermal transportation for nanofluid flow over Riga plate design. Nanoparticles are considered due to their unusual characteristics like extraordinary thermal conductivity, which are significant in heat exchangers, advanced nanotechnology, electronics, and material sciences. Cattaneo–Christov theory and activation energy are incorporated. The unsteady three dimensional partially differentiate formulation is simplified in the form of two independent coordinates (ζ,η). For steady-state solution (ζ=1), Galerkin discretization in used to employ finite element simulation in MATLAB environment. The unsteady parameter rotating parameter, thermophoresis, and Brownian motion parameter escalated the nanofluid temperature field. Modified electromagnetic parameter MH accelerated the primary flow velocity and activation energy augmented the volume fraction of nanoparticles in the boundary layer region. The larger modified Hartmaan number MH reduces the coefficient of skin friction in primary direction but the magnitude of coefficient of skin friction in secondary direction is augmented. The local Nusselt number Rex1/2Nux is directly proportional to MH but it is inversely related to β. The higher values of Brownian motion and thermophoresis boosted the temperature. An excellent accord among the present and previously existing solutions is establishes the validity of the current findings.

Amine-functionalized fumed silica for CO2 capture through particle molecular layer deposition

Solid adsorbent materials for CO2 capture have received increasing attention due to the high regenerative energy requirements and physical limitations of traditional liquid systems. Supported amines have become widely-researched materials due to their ambient adsorption capabilities and low regeneration temperatures. Here, Particle Molecular Layer Deposition (MLD) is introduced as a novel loading method for amine functionalization to improve the regeneration stability and adsorption capacity of supported amine sorbents. This paper confirms two MLD chemistries: (3-aminopropyl)triethoxysilane (APTES) and N1-(3-trimethoxysilylpropyl)diethylene triamine (TMPTA). Both precursors were deposited at 150 °C in a binary reaction with water. Adsorption capacity of the amine functional groups increased to ~0.005 mmol/m2 as the number of MLD cycles increased. Low sorbent regeneration temperatures and stable regeneration of active sites over 25 cycles were demonstrated by both the APTES and TMPTA functionalized sorbent. This is the first demonstration of an MLD process used to functionalize a sorbent for CO2 capture. This is also the first use of TMPTA for MLD.

Nonlinear radiation on Maxwell fluid in a convective heat transfer with viscous dissipation and activation energy

AbstractThe present analysis addresses linear and nonlinear radiation effects in hydrodynamic viscous Maxwell fluid flow on a unidirectional stretching surface through viscous dissipation. The relaxation effect is considered in the mathematical model, which elucidates mass transport mechanisms under binary chemical reaction and activation energy. Mathematical modeling contains nonlinear partial differential equations using boundary conditions. Appropriate transformations convert the partial differential equations into ordinary differential equations. Numerical solutions for regular differential equations are brought by Runge–Kutta–Fehlberg numerical quadrature and a shooting method with a tolerance level of 10−9. The influence of physical variables, such as Deborah relaxation number, rotation parameter, Biot number, activation energy parameter, reaction rate parameter, Eckert number, and Prandtl number are investigated. Increasing the Biot number improves the temperature region in the boundary layer. With high rotation, the increasing Deborah number enhances the fluid temperature substantially throughout the boundary layer.

Influence of Chemical Reaction and Arrhenius Activation Energy on Hydromagnetic Non-Darcian Casson Nanofluid Flow with Second-Order Slip Condition

This article investigates the combined effect of second-order velocity slip, Arrhenius activation energy and binary chemical reaction on reactive Casson nanofluid flow in a non-Darcian porous medium. The governing equations of the problem were first non-dimensionalized and later reduced to ordinary nonlinear differential equations by adopting a similarity transformation. The emerging nonlinear boundary value problem was solved by using Galerkin weighted residual method (GWRM). The obtained results were compared with those found in the literature to validate our method. The impact of pertinent parameters on the velocity component, temperature distribution and concentration profile are presented using graphs and were discussed. The computational results show that an increase in second order slip parameter significantly results to an increase in the temperature as well as nanoparticle concentration profiles, while it reduces the velocity profile.

Entropy generation optimization and activation energy in flow of Walters-B nanomaterial

In present research, we concentrated on the characterization of Walters-B nanofluid flow to investigate the irreversibility mechanism. Energy equation incorporated with radiation effects and heat generation phenomena. Influence of activation energy is discussed using modified Arrhenius energy term along binary chemical reaction. The consequences of thermophoresis, Brownian motion and viscous dissipation on fluids velocity, temperature of fluid particles and concentration of involved chemical species. Set of ordinary differential equations are obtained by implementing appropriate similarity variables. Governing mathematical model is solved using homotopy analysis method. Flow behavior is analyzed through Nusselt number, coefficient of drag force and mass transfer rate.

Significance of Brownian motion and thermophoresis influence on dynamics of Reiner-Rivlin fluid over a disk with non-Fourier heat flux theory and gyrotactic microorganisms: A Numerical approach

Bioconvection for rotational flow is conceived to provide stability to improved thermal transportation for Reiner-Rivlin nano fluid over a disk with multi-slips. The nonFourier heat flux, binary chemical reaction, magnetic force, and activation energy are incorporated. A system of nonlinear differential equations in coupled form is acquired through the fundamental relations of Reiner-Rivlin fluids. The Runge-Kutta method of fourth-order is used to solved differential equations in MATLAB environment. The impact of various parameters are discussed and drawn physically with the help of graphs. The Reiner-Rivlin fluid parameter, magnetic parameter, thermophoresis, and Brownian motion parameter escalated the nanofluid temperature field. The electromagnetic parameter and Reiner-Rivlin fluid parameter decelerated the primary flow velocity and activation energy augmented the volume fraction of nanoparticles in the boundary layer region. An excellent accord among the present and previously existing solutions is establishes the validity of the current findings.

Rotating flow assessment of magnetized mixture fluid suspended with hybrid nanoparticles and chemical reactions of species

The current study characterizes the effects of Hall current, Arrhenius activation energy and binary chemical reaction on the rotating flow of hybrid nanofluid in two double disks. By the use of suitable similarity transformations, the system of partial differential equations and boundary conditions for hybrid nanofluid are transformed to ordinary differential equations which are solved through optimal homotopy analysis method. The intensified magnetic field and hybrid nanofluid performances are represented in three dimensional model with flow, heat and mass transfer. Radial velocity decreases and tangential velocity increases with the Hall parameter. Temperature rises with high values of rotation parameter while it decreases with the Prandtl number. Nanoparticles concentration enhances with the increments in Arrhenius activation energy parameter and stretching parameter due to lower disk. There exists a close and favorable harmony in the results of present and published work.

Irreversibility analysis in natural bio-convective flow of Eyring-Powell nanofluid subject to activation energy and gyrotactic microorganisms

The developing interest in thermal engineering and nano-technology presented the idea of nanoparticles with enhanced thermal mechanisms and attained convinced novel applications in heat transportation systems, solar energy, micromechanics, air conditioning systems, cooling and heating facilities, bio-medical applications like cancer treatment, artificial heart surgery etc. Beside this, the phenomenon of entropy generation is another fascinating research area which used to improve the assessment of heat transportation with optimized procedure. The entropy generation in bio-convective flow of Eyring-Powell nanofluid over permeable stretched surface of cylinder is scrutinized in this research. The effects of magnetic field, thermal radiation and viscous dissipation are considered. Concentration communication is obtained in view of activation energy associated with binary chemical reaction. Furthermore, properties of Brownian diffusion and thermophoresis of nanoparticles are accounted. Suspended nanoparticles are made stable through mutual effects of bioconvection and buoyancy forces. Total entropy generation rate is modeled through second thermodynamics law. The governing flow model is obtained with the help of boundary layer approximations. Dimensional model is transformed into non-dimensional one by transformations and then tackled by Newton built-in shooting procedure. Impact of different influential variables on entropy generation, velocity, temperature, concentration, Bejan number and motile density of microorganisms are analyzed via graphs. Numerical values of skin friction coefficient, density, Sherwood and Nusselt numbers are tabulated and examined. Major findings are listed at the end. Since this analysis is based on theoretical flow assumptions, therefore the flow parameters specify the constant range like 0.0⩽Γ⩽6.0, 1.0⩽γ⩽4.0, 0.1⩽Ha⩽1.2, 0.2⩽λ⩽0.6,0.2⩽Rd⩽1.7, 0.1⩽K⩽3.0, 0.1⩽Pr⩽4.5, 0.3⩽Nb⩽1.8, 0.2⩽Nt⩽1.7, 0.0⩽E1⩽1.0, 1.0⩽Ec⩽4.0, 0.2⩽σ⩽1.4, 0.4⩽Sc⩽1.7, 1.0⩽δ⩽2.5, 0.5⩽Pe⩽2.0, 0.5⩽Ω°⩽2.0, 0.1⩽Br⩽1.0,0.1⩽Lb⩽1.5. The results show that fluid velocity decreases with curvature parameter and porosity constant. A decreasing variation in nanofluid temperature due to curvature parameter is examined. The entropy generation and Bejan number decline Brinkman number, motile density diffusion parameter and mass concentration diffusion variable. Furthermore, it is observed that skin friction coefficient increases with curvature parameter and porosity constant.

Irreversibility analysis in Darcy-Forchheimer flow of viscous fluid with Dufour and Soret effects via finite difference method

Background and objectiveHere irreversibility analysis in unsteady Darcy-Forchheimer flow of viscous fluid by a stretched sheet is examined. Lorentz force effect is considered. Energy attribution is discussed through thermodynamic first law with Joule heating, radiation and dissipation effects. Entropy generation is calculated. Thermo diffusion and diffusion-thermo characteristics are also examined. Furthermore binary chemical reaction is addressed. Significance of entropy generation and heat transfer rate is considered. Here our main aim is to discuss the entropy rate and heat transfer analysis. The recommended model is pertinent for heat exchangers, entropy optimization, biomedicine, two-phase flows, polymers, thermal and solutal transportation, fuel cells and geothermal energy system. MethodsPartial differential equations (PDEs) are altered into dimensionless form through appropriate variables. The resulting governing equations are then solved numerically by using finite difference method (FDM). ResultsInfluence of sundry variables on velocity, temperature, entropy optimization and concentration are deliberated. Skin friction coefficient and Sherwood and Nusselt numbers are scrutinized. An improvement in velocity field is noticed for Reynold number. An increment in magnetic parameter has reverse trend for temperature and velocity. Temperature is augmented against higher radiation and Dufour number. Concentration has opposite characteristics with variation of reaction parameter and Soret number. ConclusionsSignificance enhancement in velocity gradient is seen for higher magnetic and porosity variables. An increment in Forchheimer number improves the surface drag force. Magnitude of Nusselt number (heat transfer rate) reduces for higher Reynold number. An improvement in radiation declines the heat transfer rate. An augmentation in radiation parameter improves the temperature and entropy generation rate. Entropy generation rate is augmented versus Reynold number and radiation parameter. Entropy optimization and surface drag force have similar effects for magnetic parameter.

An assessment of the mathematical model for estimating of entropy optimized viscous fluid flow towards a rotating cone surface

Entropy optimization in convective viscous fluids flow due to a rotating cone is explored. Heat expression with heat source/sink and dissipation is considered. Irreversibility with binary chemical reaction is also deliberated. Nonlinear system is reduced to ODEs by suitable variables. Newton built in shooting procedure is adopted for numerical solution. Salient features velocity filed, Bejan number, entropy rate, concentration and temperature are deliberated. Numerical outcomes for velocity gradient and mass and heat transfer rates are displayed through tables. Assessments between the current and previous published outcomes are in an excellent agreement. It is noted that velocity and temperature show contrasting behavior for larger variable viscosity parameter. Entropy rate and Bejan number have reverse effect against viscosity variable. For rising values of thermal conductivity variable both Bejan number and entropy optimization have similar effect.

Investigation of binary chemical reaction in magnetohydrodynamic nanofluid flow with double stratification

This article addresses MHD nanofluid flow induced by stretched surface. Heat transport features are elaborated by implementing double diffusive stratification. Chemically reactive species is implemented in order to explore the properties of nanofluid through Brownian motion and thermophoresis. Activation energy concept is utilized for nano liquid. Further zero mass flux is assumed at the sheet’s surface for better and high accuracy of the out-turn. Trasnformations are used to reconstruct the partial differential equations into ordinary differential equations. Homotopy analysis method is utilized to obtain the solution. Physical features like flow, heat and mass are elaborated through graphs. Thermal stratified parameter reduces the temperature as well as concentration profile. Also decay in concentration field is noticed for larger reaction rate parameter. Both temperature and concentration grows for Thermophoresis parameter. To check the heat transfer rate, graphical exposition of Nusselt number are also discussed and interpret. It is noticed that amount of heat transfer decreases with the increment in Hartmann number. Numerical results shows that drag force increased for enlarged Hartmann number.

Analyzing the impact of induced magnetic flux and Fourier\u2019s and Fick\u2019s theories on the Carreau-Yasuda nanofluid flow

The current study analyzes the effects of modified Fourier and Fick's theories on the Carreau-Yasuda nanofluid flow over a stretched surface accompanying activation energy with binary chemical reaction. Mechanism of heat transfer is observed in the occurrence of heat source/sink and Newtonian heating. The induced magnetic field is incorporated to boost the electric conductivity of nanofluid. The formulation of the model consists of nonlinear coupled partial differential equations that are transmuted into coupled ordinary differential equations with high nonlinearity by applying boundary layer approximation. The numerical solution of this coupled system is carried out by implementing the MATLAB solver bvp4c package. Also, to verify the accuracy of the numerical scheme grid-free analysis for the Nusselt number is presented. The influence of different parameters, for example, reciprocal magnetic Prandtl number, stretching ratio parameter, Brownian motion, thermophoresis, and Schmidt number on the physical quantities like velocity, temperature distribution, and concentration distribution are addressed with graphs. The Skin friction coefficient and local Nusselt number for different parameters are estimated through Tables. The analysis shows that the concentration of nanoparticles increases on increasing the chemical reaction with activation energy and also Brownian motion efficiency and thermophoresis parameter increases the nanoparticle concentration. Opposite behavior of velocity profile and the Skin friction coefficient is observed for increasing the stretching ratio parameter. In order to validate the present results, a comparison with previously published results is presented. Also, Factors of thermal and solutal relaxation time effectively contribute to optimizing the process of stretchable surface chilling, which is important in many industrial applications.

A numerical model for analysis of binary chemical reaction and activation energy of thermo solutal micropolar nanofluid flow through permeable stretching sheet: nanoparticle study

The mechanism of nanofluid to improve heat transfer features has received great consideration due to their wide applications in chemical engineering and industry. In light of these facts, a numerical simulation for the flow of a micropolar nanofluid with suspended nanoparticles has been analyzed past a permeable stretching sheet with non-uniform heat source/sink, Binary chemical reaction and activation energy. In modeling micropolar nanofluid quantifies and qualifies the thermal phenomena caused by convective heat transfer in the presence of non-uniform heat source/sink and reaction rate. The formulated equations are altered to ordinary differential equations by employing similarity transformations which are then solved by utilizing shooting technique and RKF-45 method. The potentialities of all the representatives are put into graphs and are elucidated. Furthermore, the skin friction coefficient and Nusselt number in the boundary layer regime, are exhibited through graphs and tables and are deliberated with proper physical justification. The significant outcomes of the current investigation are that increment in the suction parameter declines the flow velocity and temperature while the injection is uplift the temperature. The skin friction factor is trigger considerable decrease with the stretching parameter. The heat transfer rate increases with the increased values of the radiation parameter.

Impact of Binary Chemical Reaction and Activation Energy on Heat and Mass Transfer of Marangoni Driven Boundary Layer Flow of a Non-Newtonian Nanofluid

The flow and heat transfer of non-Newtonian nanofluids has an extensive range of applications in oceanography, the cooling of metallic plates, melt-spinning, the movement of biological fluids, heat exchangers technology, coating and suspensions. In view of these applications, we studied the steady Marangoni driven boundary layer flow, heat and mass transfer characteristics of a nanofluid. A non-Newtonian second-grade liquid model is used to deliberate the effect of activation energy on the chemically reactive non-Newtonian nanofluid. By applying suitable similarity transformations, the system of governing equations is transformed into a set of ordinary differential equations. These reduced equations are tackled numerically using the Runge–Kutta–Fehlberg fourth-fifth order (RKF-45) method. The velocity, concentration, thermal fields and rate of heat transfer are explored for the embedded non-dimensional parameters graphically. Our results revealed that the escalating values of the Marangoni number improve the velocity gradient and reduce the heat transfer. As the values of the porosity parameter increase, the velocity gradient is reduced and the heat transfer is improved. Finally, the Nusselt number is found to decline as the porosity parameter increases.

Bio-convective micropolar nanofluid flow over thin moving needle subject to Arrhenius activation energy, viscous dissipation and binary chemical reaction

The study of nanofluid dynamics considering the bioconvection effects has significant applications, because nanofluid has remarkable capabilities for thermal transportation in comparison with pure fluids in various industrial and engineering applications. Keeping in view, such incredible characteristics the main objective of this work is to examine the influence of bioconvective micropolar nanofluid flow over a thin moving needle comprising of gyrotactic microorganism. The flow is also exposed to Arrhenius activation energy, binary chemical reaction and viscous dissipation. The governing equations are transformed to a dimensionless form by making use of a set of suitable variables and solved afterwards by employing homotopy analysis method (HAM). As main outcomes it is established in this work that, flow profiles decline with growing values of buoyancy ratio parameter, material parameter, volume fraction and bioconvection Rayleigh number. Thermal profile grows up with higher values of volumetric fraction, Brownian motion, Eckert number and thermophoretic parameter. Concentration profiles decline with growing values of thermosphoretic parameter, Brownian motion, Lewis number, chemical reaction parameter and grow up for increasing values of activation energy parameter. Moreover, a fine agreement between current results and the results available in literature has also established in current work.

Features of 3D magneto-convective nonlinear radiative Williamson nanofluid flow with activation energy, multiple slips and Hall effect

In this article, the impacts of Hall current and Arrhenius activation energy on three-dimensional hydromagnetic Williamson nanofluid flow past a slendering stretching sheet in the presence of multiple slips, viscous dissipation, Joule heating and binary chemical reaction is analyzed. The presence of nonlinear thermal radiation and nonlinear mixed convection is also taken into consideration. The dimensional governing equations are transformed into non-dimensional ordinary differential equations by using some suitable similarity transformation. The resulting coupled and highly nonlinear boundary value problem is then solved numerically by shooting technique based on Runge-Kutta-Fehlberg method. The behaviors of concentration, temperature and velocity distributions w.r.t. the various controlling parameters are illustrated graphically. However, the numerical values of local skin-friction coefficients, local heat and mass transfer rates are explained and presented in tabular form. Furthermore, a result validation is performed to check the accuracy and correctness of the obtained results by comparing the results with previously published results for some limited case of the present problem and an excellent agreement is found between the results.

Slip-free multiplication and complexity of dislocation networks in FCC metals

During plastic deformation of crystalline solids, intricate networks of dislocation lines form and evolve. To capture dislocation density evolution, prominent theories of crystal plasticity assume that 1) multiplication is driven by slip in active slip systems and 2) pair-wise slip system interactions dominate network evolution. In this work, we analyze a massive database of over 100 discrete dislocation dynamics simulations (with cross-slip suppressed), and our findings bring both of these assumptions into question. We demonstrate that dislocation multiplication is commonly observed on slip systems with no applied stress and no plastic strain rate, a phenomenon we refer to as slip-free multiplication. We show that while the formation of glissile junctions provides one mechanism for slip-free multiplication, additional mechanisms which account for the influence of coplanar interactions are needed to fully explain the observations. Unlike glissile junction formation which results from a binary reaction between a pair of slip systems, these new multiplication mechanisms require higher order reactions that lead to complex network configurations. While these complex configurations have not been given much attention previously, they account for about 50% of the line intersections in our database.