Nature inspired optimal design of heat conveying networks for advanced fiber-reinforced composites

Abstract

Keywords: Heat transfer; fiber reinforced composites; optimal design; nature inspired solutions; MEMS devices

Introduction

Biological tissues are mostly presented by composite materials reinforced by fibers or tubes conveying biological fluids to and from the live cells (Fung, 1981). In animal tissues

Figure 1 Examples Of Arterial Vasculature

when R / L ≤ 1 , the solution of the optimization problem (3) for the porous flow is also given by (1). The relationship (2) also follows from the minimal energy loss principle (Rosen, 1967). As is was shown in numerous measurements, the correspondence between the theoretical and statistical data is very good in mean values, while some dispersion proper to biomedical data is presented. Detailed computations on the optimal branched angles and some configurations with deviation from the optimal ones revealed the increase in the energy loss in the most deviated stated does not exceed 10% (Kizilova, 2004b). Transportation networks also serve for strength of the organ and organisms. The blood vessel vasculatures with different asymmetry of their branching support the shape and volume of soft inner organs (spleen, kidneys, liver, etc.) (Fig.2 a,b) as well as systems of veins keep the plant leaves unfolded and strong enough against wind, rain and other mechanical loads (Fig.2b).

As it was shown in numerous experimental studies and observations, geometry of the pipelines is determined by certain relationships between the diameters of the pipes in the bifurcations (Murray, 1926a; 1926b), and between the diameters and branching angles (Rosen, 1967; Weibel, 1963; La Barbera, 1990; Kizilova and Popova, 1999; McCulloh, Sperry and Adler, 2003)

It is amazing the principles of construction of fluidconveying networks in animal and plant tissues and organs are the same, though the blood vessels and airways are soft and distensible, while the conducting elements in plants are rigid and possess porous walls and partitions (Kizilova, 2008). Theoretical substantiation of (1) is based on Murray’s model of the optimal tube providing steady flow of a viscous fluid at total minimal energy expenses W so that

where Q=const, Z = 8η L / (π R 4 ) . Solution of the optimization problem (3) gives the relation Q ~ R 3 for the optimal tube. It means for the bifurcation of three optimal tubes with diameters d 0,1,2 the relationship (1) follows from the mass conservation law. Since in the Poiseuille flow the wall shear stress (WSS) is

Organs Supplied By Branched System Of

TUBES WITH ξ = 0.8 (A) AND VENATION OF THE LARGER (B) AND THE SMALLER (C) LEAF VEINS

in the optimal tube with Q ~ R 3 the WSS will be constant. It implies when the WSS is maintained at some constant level during the vessel growth and development, the optimal vessel will grow. It is approved the mechanosensory cells in the innermost endothelial layer of the blood vessels can estimate the WSS and transfer information on it into the inner layer composed of active smooth muscle cells (Zaragoza, Márquez and Saura, 2012). In that way, the blood vessel segments which are locally optimal to the steady flow can be developed. It is interesting; the relationship (1) is also a necessary condition of global optimality of the binary system of tubes (Chernousko, 1977; Kizilova, 2005a). Murray’s law (1) has been generalized for the case of the steady viscous flow through the rigid tubes with permeable walls (Kizilova, 2005). It was shown,

In the solid tissues like bones and teeth the families of trabeculae orthogonal to the loaded surface (i.e. elongated according to the directions of maximal compression) and the orthogonal family of trabeculae located according to the directions of maximal extension in the tissue effectively work against extension and compression loads, while the nonworking substance is dissolved and deleted producing the lightweight design (Fung, 1981). Main principle of biological growth is connected with elongation of cells and extracellular reinforcing structures (trabeculae in bones, sclerenchyma and collenchyma in plants and others) according to principals of the stress tensor at given external load. When the external load varies changing the stress field, the reinforcing system is remodeled keeping the optimal density and orientation for new 637

molecular motors (Kizilova, 2011). Probably, the branched fibers allow distribution information and cargo carried by the molecular motors more uniformly, just as along the side roads without jumping between the fibers.

load conditions by active biological feedbacks. As a result, the uniform pattern of trabeculae in infant bones transform into clear body-specific orthogonal families of trabeculae, as well as tree trunks demonstrate straight or spiral grains depending on the permanent wind load (Leelavanichkul and Cherkaev, 2004). The corresponding theoretical model of bone as adaptive material has been developed (Cowin, 1989), but its practical implementation into the strategies of the in vivo growth control for tissue engineering purposes or into the smart materials with stress-dependent properties remains a challenge (Kizilova, 2012). It is important that the principles of biological growth discovered in animals and plants are the same in spite of the phylogenetic development of plants and animals had been separated since early stages of evolution when both types of live matter were presented by single animal and single plant cells and did not possess any macroscopic transportation and reinforcing structures. Most biological tissues are composite materials formed by layers with different properties, viscoelastic solid porous structures and fibre reinforced materials with different structural patterns (Fung, 1981). The layered structures composed according to the external load can be found in cartilage. Coordinated growth of skeletal muscles and bones is provided by extension of the bones and muscles which are in parallel connection. The bones, cartilage, ligaments and other collagen structures possess piezoelectric properties, and electric fields generated in the loaded collagen fibers strongly influence growth direction and intensity (Fukada and Yasuda, 1957; Avdeev and Regirer, 1985). In human skin the collagen fibers provide asymmetry in the skin distensibility depending on its natural loading. The collagen fibers in dermis and epidermis are oriented according to principals of stress tensor and provide maximal distensibility along the so-called Langer’s lines and maximal rigidity along the orthogonal families (Langer, 1861). The same orientation of fibers has beet detected in the vascular cambium of trees (Kramer, 2002). Orientation of wood grains on the debarked surface of trunks and stems correspond to two families of fibers oriented according to the stress field. The outer layer of blood vessels (adventitia) is reinforced by two oppositely directed spiral families of collagen fibers (Holzapfel, Gasser and Ogden, 2006). The chords preventing the heart valves from outwards movement are also branched structures distributing the load uniformly along the leaflets of the valves. It is important the fibers in skin and other collagen tissues possess branching structure and are weaved in the textures of different density, anisotropy and strength (Fig.3). The branching fibers have been found at the micro scale in tendon (Birk, et al., 1989), epithelium (Brownfield, Venugopalan and Lo, 2013), myocardial and many other tissues. In the myocardial tissues the branching are formed by myofibrils, not by collagen fibers. In that way, branching structures are proper to different types of cells and proteins. The branched nanostructures form cytoskeleton and serve for strength of animal cells and active intercellular transport by

Figure 3 The Patterns Of Branching Collagen Fibers In Human

ARTERIAL WALL OF ELASTIC TYPE (A) AND IN THE SKIN (B) In the tissues, organs and organisms of animals and plants the reinforced structured are loaded by pointed forces, pressures and shear forces. In the presence of the gravity field the efficiency of the branched rigid structures is an important constituent of the strength and durability of the general construction. According to the Schwendener’s theory, the shape of plants is based on the concept of maximum strength (Schwendener, 1874; Schwendener, 1878). The mechanical structure of plants is determined by their ability for gravity recognition. Positive gravitropism of shoots and negative gravitropism of roots are determined by sedimentation of statoliths in the gravity field and polar transport of the plant hormone auxin. Distribution of the small branches in the crown is determined by maximizing its effective leaf area (Honda, 1978). Total leaf mass ML is related to the diameter D of trunk as ML~D2 (Niklas and Spatz, 2004). Therefore, the relations between different organs in plants are determined by mechanical (stress-strain) and hydraulic (water supply) factors. The principles of reinforcement are similar in the plant and animal tissues and based on minimal total energy expenses at a given body mass/volume and external load. The natureinspired principles of reinforcement by micro and nanofibers in cells, tissues and organs can be used for elaboration of novel functional composite materials with optimal structural and transport properties, as well as in MEMS, fuel cells, micro heaters/coolers and other advanced technological units. Since bifurcating fibers, nano and microtubes are proper to live nature, their branching patterns deserve detailed consideration from the mechanical point of view. Recently significant attention is paid for manufacture the branched micro and nanofibers for technical applications. The electrospun nanofibers with different density and length of the side branches has been elaborated (Yarin, et al., 2005; Gevorkyan, 2014). Chitosan fibers that have been found important as biodegradable scaffolds for tissue engineered skin 638

production can also be synthesized in the branched forms, which influence their degradation rate, adhesively for proteins and cells (Aggarwal and Matthew, 2009). The synthesized networks of branched nanofibes and morphology of the single branch are presented in Fig. 4a and Fig.4 b accordingly. Novel technologies allow obtaining the super strong multi-walled carbon nanotubes with single and several side branches (Fig.5.a). Since the synthesized networks of branched nanotubes, fibers and ribbons can be used for reinforcement in the advanced composites, the high heat, electric charge and mass conductivity of such structures can be used. The resulted manufactured materials would possess unique high thermal conductivity or charge conductivity properties as well as high strength provided the reinforcing network has certain optimal design based on the laws of reinforcement in live nature.

− The matrix material is viscoelastic/viscoplastic and serves for stress redistribution preventing crack generation or sticking crack propagation at fibers/tubes; − The embedded tubes serve for reinforcement of the material as well as for mass and heat transport purposes; − The branching fibers/tubes are commonly used at macro, micro and nano scales; − Multi-criteria optimization of the network of reinforcing tubes for both mechanical strength and heat/mass conductivity must be done, which is the subject of the present work.

Optimal Design Of The Networks Of HEAT Conducting Fibers

The hear flux in the micro and nanofibers can be described by Guyer-Krumhansl equation in the form

At τ = 0, χ = 0 (4) transforms into the Fourier law for stationary heat flux. a

Let introduce the characteristic values t * ,L,T* ,q* for the time, space, temperature and heat flux values. Then (4) can be rewritten in the non-dimensional form

Figure 4 Images Of The Network Of

BRANCHED NANOFIBERS (FROM GEVORKYAN, ET AL., 2014) (A) AND OF THE SINGLE Y-SHAPED JUNCTION OF NANOFIBERS (FROM BOSKOVIC, ET AL., 2004) (B)

where the nondimensional valued are marked by circle upper script. When τ << t * the relaxational effects may be neglected. At the small space scales the last term in the left hand site (5) is negligible in comparison to the terms in the right hand side part of (5), so (5) reduces to the Poiseuille type equation (Alvarez, Jou and Sellitto, 2009)

A Y-Junction Nanotube With TWO SIDE

BRANCHES (A) AND A NETWORK OF THE BRANCHED TUBES (B) (FROM HEYNING, 2005)

where the nondimensional heat flux qo is analogous to fluid velocity driven by the temperature gradient, which is analogous to the pressure drop for the fluid flow,

Summarizing the above presented data on fiber reinforcement in nature, one can conclude both plants and animals use the same main principles of mechanical construction of their tissues and organs, namely − Reinforcement by relatively rigid fibers/tubes located according the principles of the stress tensor at given external mechanical load;

Figure 6 Branching Networks For

SYMMETRICAL (A) AND MINIMAL TOTAL ENERGY EXPENSES (B) DESIGNS

The Bifurcation In B,C Perpendicularly To The Plane 0XY

In that way solution of the optimization problem (3) for the branching network of the heat conducting micro or nanowires and the relationships (1), (2) must be fulfilled. Like for the fluid flow case, the conditions of local optimality of the tube/wire will coincide with necessary conditions of global optimality of the network in the meaning of the minimal total energy expenses for the fluid/heat flux and structural support of the network (material and other expenses). It gives example of the functionally perfect nature inspired design with optimal branching angles α1 ,α 2 (Fig.6 b) instead of geometrically

The criterion (7) must be important for rigid branches of trees, bushes and shoots (Zamir and Medeiros, 1982), while for the leaf branches the total lateral surface provided the fluid delivery to the distributed customers (live cells) can be more important.

M = σ J / h , where for the uniform circular bar h is the radius of the cross section. Since the maximal bending moments are produced in the cross sections at maximal distance to the applied forces, for the three bars composing the bifurcations the maximal stresses will be reached at the section O of the first bar, and the section a of the second and thirds bars. Then the restriction on the maximal stress will give the inequalities

Optimal Design Of Loaded Y-Shaped Fibers

Let us consider three bars of circular cross sections composed a bifurcation OABC (Fig. 6). The lines OA, AB and AC belong to the same plane, the coordinates of the points are O(0,0), A(x,0), B(a,b1), and C(a,-b2), the diameters and lengths of the bars are d 0 ,d1 ,d 2 and L0 ,L1 ,L2 correspondingly. The bar OA is rigidly clamped at the cross section x=0, y=0, the

perpendicularly to the plane 0xy and applied in the points B,C (Fig.7). Let us find out the branching design when at the given volume of the bifurcation

M 2max , and M 0max will be given by minimal diameters, so we can come from (9) to the equalities

where the maximal stress σ max in the bifurcation is restricted by some critical value σ max ≤ σ * .

χ = X / a at different area geometry and force distributions

in (7) and (8) gives the following criteria in the nondimensional form

correspond to the criteria (8) and (7) accordingly. Optimal location X of the bifurcation for the given geometry (a, b1, b2) and mechanical load (F1, F2) can be found from the conditions /

The branching ratio K, the optimal Murray parameter µ and the optimal bifurcation angles α1,2 =

can be computed then and compared to the measured values presented in the previous chapter. Since the real load on the branching plant structures includes own body mass, the payload (leaf mass), the wind, rain and snow load, the force

distribution ( F1 ,F2 ) must be insignificant and only the force asymmetry f might be important in connection of development the symmetric or quite asymmetric branches. Geometry of the bifurcation can also be described by relative parameters (b1 + b 2 ) / a ∈]0,2[, b1 / b 2 ∈]0,1[ .

Results And Discussions

1,2 1,2 Direct computations by (10), (11) at known give algebraic equation for determination the optimal location of the branching point x=X. Numerical computations have

Figure 8 Location Of The Optimal

1 2 1 2 and . non-symmetric areas Three values of the force asymmetry f have been chosen: f=0;2;0.5. Due to the symmetry the values

AT b1 + b 2 = a (A), b1 + b 2 = a / 2 (B), b1 + b 2 = 2a (C).

Square, Rhomb And Triangle Signs

CORRESPOND TO THE FORCE ASYMMETRIES F=1, F=2 AND F=0.5 ACCORDINGLY

β1 = 0.1;0.2;0.3;0.4;05 have been used. Location of the optimal bifurcation point A in dimensionless coordinate

Figure 9 Location Of The Bifurcation Point

ON THE AREA x ∈ [0,a],b ∈ [ − b 2 ,b1 ] AT DIFFERENT BIFURCATION ASYMMETRY β1 ∈ [0.1,0.5] AND

AREA Geometry

b1 + b 2 = a (A), b1 + b 2 = a / 2 (B), b1 + b 2 = 2a (C) If we compare the branching angle α optimal for the stress minimization in the structure and the branching angle α * computed for the same diameters from (2) and optimal for the fluid delivery along the branch, we shall obtain quite good correlation between them (fig.10b). It means both optimal solutions are quite close to each other. Taking into account the computed influence of small deviations of location of the branching point A in the area which corresponds to ±5% additional energy lost (Kizilova, 2004b), in nature the scatter of the data around the line α * = α (solid line in fig.10b) correspond to rather small energy lost compensated by optimality to some other external conditions or internal properties.

given geometry (fig.7). The computed dependency α (K) where α = α1 + α 2 is presented in fig.10a. Three sets of data corresponded to different geometry of the area are clearly visible. Inside each set the three sets correspondent to different force asymmetry are clear separated only in the case b1 + b 2 = 2a with bigger branching angle α and branching coefficient K. There is quite good approximation of the general data K = k1 exp(k 2α ) (R2=0.688) depicted in fig.10a by the solid line. The computed dependence is very close to those measured on the plant leaves (Kizilova, N., 2004a).

found (fig.11b), while the shorter branch exhibits some noticeable scatter around the exponential averaged values (straight line in fig.11b) depending on the applied forces and initial branch asymmetry. The data measured on the vascular beds demonstrated the same dependence, as if the main daughter branch follows the diameter of the parent branch, while the smaller daughter branch has more freedom for branching and, therefore, the bigger scatter.

b FIGURE 10 DEPENDENCIES α (K) (A) AND α * (α ) (B). THE SQUARE AND RHOMB SIGNS IN (B) CORRESPOND TO THE LONGER (AC) AND SHORTER (AB) RODS ACCORDINGLY The dependence µ (d1 / d 2 ) presented in fig.11a is similar to the measured dependencies µ (d 0 ) and µ (ξ ) (Kizilova, N., 2004a). The thicker the main branch, the closer the optimality coefficient to 1, while the small branches demonstrate bigger scatter around the optimal value. In the experimental data 0 < ξ < 1 , while in fig.11b d1 / d 2 could be bigger than 1, because in the cases when the shorter rod is loaded by the bigger force, in the optimal case it is thicker than the less loaded longer branch. In this cases diameter ratios of the shorter and longer branches may give values d1 / d 2 >1.

Beams

In that way, the computed configurations of the optimal bifurcating fibres experienced minimal internal stress at given asymmetric load can be used for reinforcement of the tissuelike engineered composites in the woven or layered (fig.3) patterns, as well as 3D structures reinforcing convex shells (containers, capsules, roofs, pavilions, panels, etc).

Quite strong dependence α 2 = κ1 ln(M 0 ) + κ 2 (R2=0.864) of the branching angle of the longer branch on the total bending moment M0 appeared in the main rod has been

problem for the stationary fluid flow in rigid cylindrical tube when the total energy expenses for the viscous flow and metabolism are minimal gives the Murray’s law. In that way, transportations networks in live nature are optimal pipelines provided minimal energy costs for transport and metabolism. Solution of similar optimization problem for the fluid percolation through the cylindrical tube with permeable wall at the assumption of the long thing tubes (d/L<<1) has the same form (Kizilova, 2005). As is was shown in the present paper, the optimal rigid Y-shape rods fastened at the beginning of its parent rod and loaded by non-symmetric forces reveal the distributions between diameters, branching angles and lengths that posses certain regularities similar to those obtained on the measured data. Basing on the theoretical results, the obtained regularities are proposed for fabrication of the branching structures of nano/microtubes as reinforcing systems for the composite materials with optimal properties. Those materials will provide multicriteria optimization of their mechanical (strengthening), heat and flow conductivity (transportation) properties. Due to similarity of the solutions of both the mechanical and transportation problems, significant economy of the materials and lightweight design could be reached, which is especially important for the micro heaters/coolers, microfluidic separators/homogenizators, fuel cells, artificial cells and tissues, microengines and other MEMS units.

The branched structures composed from nanotubes are perspective for optimal reinforcement of microscopic objects like artificial cells, tissue substitutes, MEMS units, fuel cells and others. Modern technologies allow synthesis of carbon, metal, polymer and other branched Y-shaped conjugations of nanotubes that can be used for simultaneous strengthening of the unit and delivery and distribution of macro- and nanofluids through them. The aerosol technique based on spray of a catalyst-precursor solution composed of metal salts in water directly into a furnace is a low-cost technology for obtaining Y-shape nanotubes and more complex branched structures of them (Heyning, Bernier and Glerup, 2005). The Y-shaped carbon nanotubes can be obtained by the arc discharge method (Osvatha, Koosa, Horvatha, 2003) and used for the reinforcement and heat conductivity purposes. The Y-shape TiO2 nanotubes have been obtained by multi-step sonoelectrochemical anodization method (Mohapatra, 2008). Being embedded into a viscoelastic matrix with needed thermomecanical or electromechanical properties, the structures form new composites reinforced by a branching network of tubes. Many micro-units like liquid-based microcoolers and heaters, fuel cells, artificial cells, molecular motors, lab-on-a-chip need permanent delivery of the working substances and taking away the products of reactions/decay, assimilates, and useful produced substances that can be fulfilled by the same elements which provide strengthening. It is a reasonable way for economy of the material and lightweight design of the micro- and nanosystems by double exploitation of the same system whose design provides optimality for both mechanical and transportation properties. The diameters of the nanotubes in the manufactured Yshape junctions are usually constant or uniform dependently on the material, and the branching angles are determined by the technological conditions and could be far from optimal ones in the above discussed meaning. Recently novel approaches for the controlled branching of the nanotubes by nucleation their lateral surface with a catalyst and, therefore, initiation of the branched growth have been proposed (Gothard, 2004). This will allow manufacturing of the branched structures of nanotubes as reinforcing structures that provide multicriteria optimization of the mechanical, heat and flow conductivity properties of the corresponding composite materials.

Nomenclature

d 0 - diameter of the parent branch; d1,2 - diameters of the daughter branches; F – force; h - distance to the axis; J - moment of inertia; L – length;

M – moment of force; R – radius; Q - volumetric flow rate; r q - heat flux; T – temperature; Z - Poiseuille resistivity for the steady flow; α1,α 2 - branching angles of the daughter branches;

Conclusions

Natural materials in tissues and organs of plants and animals are mostly presented by fiber reinforced composites. The reinforcing fibers, from nano to macro scales, are branched systems of tubes or rods that exhibited certain geometrical regularities between the diameters and branching angles at the bifurcations, diameters and lengths in the general network. Statistical analysis of the measurement data obtained on the vascular beds of human and animals, as well as tree branches and leaf venation systems revealed the same regularities in their geometry. Solution of the optimization

α = α1 + α 2 ; η - fluid viscosity; λ - thermal conductivity; µ=

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σ - stress; τ - relaxation time; τ w - wall shear stress (WSS); χ - parameter in the heat equation.

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