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AIP ADVANCES 4, 017109 (2014)

Development of a general method for obtaining the geometry of microfluidic networks Mohammad Sayed Razavi,1,a Ebrahim Shirani,2 and M. R. Salimpour1 1

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran 2 Department of Engineering, Foolad Institute of Technology, FooladShahr 84916-63763, Isfahan, Iran (Received 9 October 2013; accepted 16 December 2013; published online 3 January 2014)

In the present study, a general method for geometry of fluidic networks is developed with emphasis on pressure-driven flows in the microfluidic applications. The design method is based on general features of network’s geometry such as cross-sectional area and length of channels. Also, the method is applicable to various cross-sectional shapes such as circular, rectangular, triangular, and trapezoidal cross sections. Using constructal theory, the flow resistance, energy loss and performance of the network are optimized. Also, by this method, practical design strategies for the fabrication of microfluidic networks can be improved. The design method enables rapid prediction of fluid flow in the complex network of channels and is very useful for improving proper miniaturization and integration of microfluidic networks. Minimization of flow resistance of the network of channels leads to universal constants for consecutive cross-sectional areas and lengths. For a Y-shaped network, the optimal ratios of consecutive cross-section areas (Ai+1 /Ai ) and lengths (Li+1 /Li ) are obtained as Ai+1 /Ai = 2−2/3 and Li+1 /Li = 2−1/3 , respectively. It is shown that energy loss in the network is proportional to the volume of network. It is also seen when the number of channels is increased both the hydraulic resistance and the volume occupied by the network are increased in a similar manner. Furthermore, the method offers that fabrication of multi-depth and multi-width microchannels should be considered as an integral part of designing procedures. Finally, numerical simulations for the fluid flow in the network have been performed and results show very good agreement C 2014 Author(s). All article content, except where otherwise with analytic results.  noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4861067]

I. INTRODUCTION

In the recent years, the emergence and development of microfabrication technology have made it possible to integrate microfluidic components for different applications.1–4 Miniaturization of biological and chemical processes is essential to construct complex networks of microfluidic components.5, 6 Microfluidic devices such as micro heat- exchanger, micro reactor and micro array offer many advantages over conventional scales including efficiency, speedup in processes and decreasing cost and sample volume. Transport of liquids in the microchannel is an integral part of microfluidic design. Measurement of flow, hydraulic resistance and energy loss are essential not only for prediction of the system’s performance but also for optimal designing of microfluidic component such as micro-pumps and micro-valves. Microfluidic applications mostly use pressure-driven based networks for complex biochemical and pharmaceutical processes. As the number of processes increases, the configuration of the network becomes more complicated and precise control of fluid flow is more challenging. Computational a Corresponding author: Mohammad Sayed Razavi. E-mail: [email protected]

2158-3226/2014/4(1)/017109/14

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Fluid Dynamics (CFD) has been considered as a useful tool for obtaining the details of phenomena related to fluid flow in the microchannels.7 However, CFD based methods cannot provide simple and sufficient information to optimize the configuration of microfluidic networks and simple analytical methods are more capable to provide insightful information. In order to achieve design methods that maximize performance and to determine flow-rate, energy loss and entropy generations of the network, analytical models describing transport phenomena in the microfluidic networks can be used. In the physics of pressure driven flows, hydraulic resistance is of great importance.8 Based on the hydraulic resistance concept, the significant feature of fluidic networks including flow rate and energy loss of the whole system can be obtained easily without comprehensive numerical simulations. Also, the hydraulic resistance concept is used to design hydrodynamic trapping applications.9 Using hydrodynamic resistance of microfluidic networks, particles, cells and droplets can be sorted hydrodynamically.10–13 As a result of advances in the microfabrication technology, microchannels with different cross sectional shapes are fabricated via different fabrication techniques such as lithography, dry and wet etching. Considering fabrication techniques, different cross-sectional geometries such as circular, rectangular, triangular and trapezoidal geometries have been fabricated effectively. The fluid flow and energy loss depend on the shape of cross-section. Also, the most of analytical studies have been performed for simple geometries such as circular and rectangular geometries. Increasing the ratio of surface area to volume is necessary to achieve efficient microfluidic devices. For instance, in the proteomic reactor where proteins are broken into peptides by enzymes, the precise design of microfluidic network is crucial to have complete reaction.14 Proteins are immobilized on the surface of the wall and enzymes are transported by fluid flow. Using arrays of microchannels with larger surface area helps to have complete reactions. Also, in the micro heat-exchanger, cooling and heating are directly proportional to surface area. Recently, physiological principles in the vascular system have served as a proper procedure for designing branching networks.15–18 The optimal relationship between branches in the vascular system is known as Murray’s law. Based on the principle of minimum work and assuming that natural organisms must have achieved an optimum arrangement to minimize losses, the relationship between diameters of mother and daughter branches was derived by Murray over 80 years ago.19 Murray’s law states the cube of the diameter of mother branch is equal to the sum of the cubes of the diameters of daughter branches.20 Also, it has been shown that most of the branching systems in the nature such as vasculatures and plants obey Murray’s law.21–23 Murray’s law is also in agreement with the constructal theory which states “For a finite-size flow system to persist in time (to survive) its configuration must evolve (morph) in time in such a way that it provides easier flow access to its currents”.24 Moreover, some studies have tried to generalize Murray’s law to obtain a general rule for designing microfluidic networks.25 Therefore, development of a theoretical model for efficient design of microfluidic devices plays an important role in progressing proper integration and miniaturization of microfluidic components. Despite numerous advances in manufacturing techniques, few studies have been performed to realize the optimum design of microfluidic networks. In this paper, we develop a general method for designing the geometry of microfluidic networks using the constructal theory. We introduce the governing equations. In addition, a general formulation for hydraulic resistance for different shape of cross section is proposed. Also, a unified method for optimizing the architecture of microfluidic network is developed. Deviation from optimal design is also considered. Moreover, we investigate the variation of total volume and total hydraulic resistance of the network with increasing of its branches. Finally, numerical simulation is performed to assess how the design method is reliable for practical systems.

II. METHODS A. The governing equations

In the Microfluidic devices, the Navier-Stokes that are partial differential equations non-linear equations are used to obtain fluid flow field. In general, the Navier-Stokes equations cannot be solved

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analytically; instead numerical simulations must be used. However, for simple geometries analytical solutions can be obtained using methods such as direct integration, Fourier transforms, separation of variable and perturbation method. The Navier-Stokes equations in the operator form are expressed as: ∂ u¯ ¯ u¯ = −∇ p + η∇ 2 u¯ + ρ u.∇ (1) ρ ∂t where, u¯ denotes the velocity field; p is pressure; and ρ and η are the density and viscosity of the fluid flow, respectively. B. Hagen-Poiseuille law

The Poiseuille flow describes pressure driven flows through the tubes with circular crosssections. By assuming that the flow is fully developed, laminar and steady, volumetric flow rate is expressed as Q=

π R 4 p 8η L

(2)

where, R and L are the radius and length of the tube, respectively; and p is the pressure drop. Fluid flows in the microchannels are well approximated by Hagen- Poiseuille flow where the conduit is circular, perfectly straight and long. This approximation is accurate if R/L  1 and R/L  1/Re;  where Re is the Reynolds number (Re = 2ρU R η).26 The volumetric flow rate can be rewritten in terms of the cross-sectional area ( A = π R 2 ) as: Q=

p A2 8π ηL

(3)

Therefore, analogous to definition of electric resistance,27, 28 hydraulic resistance can be defined as: RH =

ηL p = 8π 2 Q A

(4)

Using Eq. (4) the energy loss due to friction is calculated as:29 W = Qp = R H Q 2

(5)

C. Generalization of hydraulic resistance

Obtaining unified form of the hydraulic resistance is critical to move toward optimization of microfluidic networks regardless of the shape of the cross-section. Hydraulic resistance depends on the area and perimeter of the channel’s cross-section. The hydraulic resistance dependence on the area and perimeter of cross-section can be described by using the unit hydraulic resistance which is given by: ηL (6) A2 Where, L and A denote the length and cross-sectional area, respectively. Hydraulic resistance is related to unit hydraulic resistance using geometric correction factor (β) as: R ∗H =

ηL (7) A2 It has been shown that β is a function of conduit’s shape.30 The compactness is defined as the ratio of square of conduit’s perimeter to cross-sectional area. Also, β is expressed as a function of the compactness. For a conduit with an arbitrary shape of cross-section the compactness is expressed as: R H = β R ∗H = β

C=

P2 A

(8)

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TABLE I. Values of compactness and geometric correction factor for different cross-section shapes.31 Cross-section

Shape

Circle

β



2C

4(W +H )2 W ×H

Rectangle

Triangle

C



P2 , P/2(P/2−A)(P/2−B)(P/2−C)

22 7 C

P = A+ B +C

4(A+B+(A−B)/cos α)2 (A+B)(A−B) tan α , α

Trapezoidal

C=

Arbitrary

= 54.8

P2 A

25 17 C



65 3



+ 40 173

18 5 C

− 35

C

where, P and A are the perimeter and cross-sectional area of the conduit, respectively. Alternatively, compactness can be assumed as the ratio of the square of the surface area of a conduit with unit length, i.e. (P × 1)2 to the volume of the conduit, i.e. (A × 1). Therefore, the compactness is a dimensionless parameter which can be used to quantify the value of the surface to volume phenomena. For common shape of conduits in the microfluidic applications including circular, rectangular, triangular and trapezoidal, the geometric correction factor (β) is a linear function of compactness.31 Despite emphasis on microfluidic applications, the results are applicable to all laminar flows. Some of the relations have been derived analytically and the other obtained numerically. The compactness, geometric factor and their relations are summarized in Table I. D. Optimization of dichotomous networks

Based on constructal theory, fluid flow configurations must evolve such that they provide easier access to flow In the constructal design of microfluidic networks the challenge is to determine the ratios of two consecutive cross-sectional areas (α = Ai+1 /Ai ) and lengths (γ = L i+1 /L i ) which minimize flow resistance. For a tree-shaped construct (see Fig. 1) by constraining the total volume occupied by the tree-shaped construct in the space (Ai L i + 2Ai+1 L i+1 = const), the hydraulic resistance of two successive generations, i.e. ith and (i + 1)th generation can be minimized (see Fig. 1). By using Eq. (7), continuity equation (Q i = 2Q i+1 ) and analogy between electrical circuit and hydraulic circuit (see Fig. 2), the total hydraulic resistance of a tree-shaped construct is calculated as: R H,T = R H,i +

R H,i+1 2

(9)

Therefore, the total hydraulic resistance of a tree-shaped geometry can be written as: R H,T = βη(

Li L i+1 + ) 2 2 Ai 2Ai+1

(10)

Minimizing R H,T subjected to constant volume constraint (Ai L i + 2Ai+1 L i+1 = const) results in (for details see Appendix A).  (11) α = Ai+1 Ai = 2−2/3

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FIG. 1. Schematic diagram of a tree- shaped network, a) network with n generations b) area occupied by tree-shaped construct in the plane.

FIG. 2. The similarities between fluid flow and flow of electricity: a) Poiseuille flow in a conduit and equivalent hydraulic resistance of conduit b) tree-shaped construct and equivalent hydraulic resistance of tree-shaped construct.

It means that the optimal ratio of consecutive cross-sectional area is a universal constant equals to 2−2/3 ∼ = 0.6. Similarly, optimization of the hydraulic resistance subjected to the constraint of constant area (see Fig. 1(b)) results in (see Appendix A 1)  γ = L i+1 L i = 2−1/3 ∼ (12) = 0.794 Therefore, the ratios of consecutive length and cross-sectional area in the tree-shaped networks that minimize hydraulic resistance under constraints of the constant volume and area are universal constants and are equal to Ai+1 /Ai = 2−2/3 and L i+1 /L i = 2−1/3 , respectively. E. Extension of the optimization to more general shape of the networks

The optimal geometry of Y-shaped networks can be generalized to more complex geometries. For a branching system which m parallel branches in ith generation spilt into n child branches in (i + 1)th generation, the total hydraulic resistance is calculated as (see Appendix B) ⎛⎛ ⎞−1 ⎛ ⎞−1 ⎞ m n 2 2   Ai+1, j Ai, j ⎜ ⎠ +⎝ ⎠ ⎟ R H,T = βη ⎝⎝ (13) ⎠ L i, j L i+1, j j=1 j=1 where Ai, j and L i, j are cross-sectional area and length of jth branch in ith generation. Similar to dichotomous networks, minimization of Eq. (13) subjected to the constant volume constraint

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m j=1

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Ai, j L i, j +

n j=1

AIP Advances 4, 017109 (2014)

Ai+1, j L i+1, j = const) results in (see Appendix B): m 

3/2

Ai, j =

j=1

L i,1 1/2 Ai,1

=

L i,2 1/2 Ai,2

= ... =

n 

3/2

Ai+1, j

(14)

j=1

L i,m 1/2 Ai,m

,

L i+1,1 1/2 Ai+1,1

=

L i+1,2 1/2 Ai+1,2

= ... =

L i+1,n 1/2

Ai+1,n

(15)

F. Murray’s law

In the vasculatures, there are two energy loss terms. The first loss is the energy required to pump fluid through the conduit to overcome viscous losses (W). The second term is related to the cost function at which energy is used up by the blood vessels by metabolism or the energy destroyed by elastic tubes.20, 32 The cost function is assumed to be proportional to the volume of conduit. So the global energy loss in the conduit by using Eqs. (5) and (7) can be expressed as: E global = β

ηL 2 Q + (AL) A2

(16)

where is metabolic rate. The key to achieve maximum global performance is minimizing global energy. Minimization of Eq. (16) with respect to cross-sectional area (A) results in:

1/2 3/2 Q= A (17) 2βη daughter branches in For a branching system at which m branches in ith generation 

spilt into n

m n th (i + 1) generation using Eq. (17) and conservation of mass j=1 Q i, j = j=1 Q i+1, j results in: m  j=1

3/2 Ai, j

=

n 

3/2

Ai+1, j

(18)

j=1

Moreover, energy loss due to friction can be calculated using Eqs. (5), (7) and (17) as

W = V, V = AL 2

(19)

The above equation indicates that energy loss is proportional to volume of conduit. So whenever the volume is constant, the energy dissipation is also constant. Also, Eq. (19) shows that energy loss in the conduits of network is related to the volume of network. Therefore, if a network obeys optimal geometry, the energy loss is proportional to the network volume. By using Eq. (17) and conservation of mass at the bifurcation for a tree shaped construct (Q i = 2Q i+1 ), the ratio of consecutive cross-section area is expressed as:  α = Ai+1 Ai = 2−2/3 (20) For circular tubes Eq. (20) in terms of the tube diameter is written as:  Di+1 Di = 2−1/3

(21)

The above relationship is known as Murray’s law. Comparing Eqs. (11) and Eq. (14) to Eqs. (20) and (18) reveals that the result of two different approaches, i.e. minimization of global loss of a biological fluid,  Eq. (16), and hydraulic resistance of tree-shaped constructs, is the same. Therefore, α = Ai+1 Ai = 2−2/3 provides the optimal pattern for geometry of tree-shaped fluidic networks not only for non-living fluids but also for biological fluids.

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G. Deviation from optimal geometry of Y-shaped networks

Deviation from optimal geometry of a Y-shaped network can be measured by comparing non-optimal geometry to optimal one. The optimality ratio (ϕ) is defined as the ratio of nonoptimal to optimal hydraulic resistance. It is assumed that for optimal geometry and non-optimal one, the ratio of length of the mother branch (L i ) to the length of daughter branches (L i+1 ) are fixed, i.e. γ = L i+1 /L i = const. However, the cross-sectional areas of the mother branch ( Ai ) and daughter branches ( Ai+1 ) can be varied so that the volume occupied by the construct in the space preserved constant, i.e Ai L i + 2Ai+1 L i+1 = const. Consequently, for the optimal geometry and the non-optimal one the ratio of lengths is the same. However, for optimal geometry the ratio of cross-sectional areas follows α = Ai+1 /Ai = 2−2/3 and for the non-optimal one, α is arbitrary. So by using α = Ai+1 /Ai = 2−2/3 as the feature of optimal architecture, ϕ is written as ϕ=

RH R H,opt.

=

(1 +

γ )(1 2α 2

+ 2αγ )2 1 3

(1 + 2 γ )3

(22)

Moreover, the critical ratio of lengths (γc ) is obtained by maximizing ϕ with respect to γ ∂ϕ = 0) and solving for γ . Therefore, the critical ratio of the lengths is written as a function of α ( ∂γ as follows 1

1 1 − 6 · 2 3 α 2 + 8α 3 γc = 2 2 13 − 3α + 2 43 α 3

(23)

Eq. (23) shows the ratio of lengths at which hydraulic resistance is maximized at each value of the ratio of cross-sectional area. Therefore, for each ratio of cross-sectional area, except α = Ai+1 /Ai = 2−2/3 which is the feature of optimal geometry, there is a critical ratio of lengths which can be obtained using Eq. (23). H. Extension to microfluidic networks

Arrangement of channels in parallel and series makes complex network. In addition to each channel evaluating total hydraulic resistance of the channel’s network is crucial to obtaining total flow rate and energy loss. For dichotomous tree-shaped networks (see Fig. 1(a)), i.e. one branch splits into two identical daughter’s branches, the total hydraulic resistance is expressed as:   (χ )n+1 − 1 R2 Rn 1 γ R1 + + · · · + n = R0 1 + χ + (χ )2 + · · · + (χ )n = ,χ= R T = R0 + 2 4 2 χ −1 2 α2 (24) Using Eqs. (11) and (24) and sum of the geometric series, the ratio of the total hydraulic resistance to the hydraulic resistance of mother branch is written as:  1/3 n+1 −1 2 γ RT (25) = 1/3 R0 2 γ −1 By combining Eq. (24), (11) and (12), Eq. (25) can be rewritten as RT =n+1 R0

(26)

Analogous to the hydraulic resistance, the total volume of networks can be written as VT = A0 L 0 + 2A1 L 1 + 4A2 L 2 + · · · + 2n An L n = V0 (1 + 2αγ + (2αγ )2 + · · · + (2αγ )n ) (27) Using Eqs. (27) and (11) and sum of geometric series, the ratio of the total volume to the volume of mother’s branch is given as:  1/3 n+1 −1 2 γ VT (28) = 1/3 V0 2 γ −1

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FIG. 3. a) Three-dimensional surface of optimality ratio with respect to ratios of lengths and cross-section areas, b) The optimality ratio with ratio of cross-section areas, c) The optimality ratio with ratio of lengths.

Again, replacing γ with Eq. (12) yields: VT =n+1 V0

(29)

As can be seen, the ratio of the total volume of network to the mother’s branch one is the same as the ratio of total hydraulic resistance of the network to the mother’s branch one as the ratio of surface areas obeys optimal pattern, i.e. α = Ai+1 /Ai = 2−2/3 . Also, if the ratio of lengths is equal to γ = L i+1 /L i = 2−1/3 the hydraulic resistance of the network or the total volume occupied by the network is (n + 1) times larger than branch one. I. Numerical simulations

A network with three generations (n = 3) of microchannels with square cross-sectional is selected. The side and length of mother’s branch are 250 (μm) and 2500 (μm), respectively. The ratios of cross-sectional and lengths have been chosen α = 2−2/3 and γ = 2−1/3 , respectively. A 3-D Finite Element (F.E) code is used to solve the Navier-Stokes equations. Mesh refinement is used to assure reliability of the results. The pressure at the outlet is assigned while different flow rates are applied at the inlet. Using numerical simulations, the pressure at the inlet of the network is computed. Finally, the hydraulic resistance of the network (R H = p/Q) is evaluated. The simulations are performed for different Reynolds numbers of 1, 10,100, 200, 400, 600, 800 and 1000. III. RESULTS

Fig. 3(a) illustrates the variation of the hydraulic resistance with respect to the ratio of crosssectional area (α) and the ratio of length (γ ). From Fig. 3(b), it is obvious that for α = 2−2/3 the hydraulic resistance is minimum, regardless of the value ofγ . However, the deviation from α = 2−2/3

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FIG. 4. Variation of critical length ratio versus ratio of cross section areas.

FIG. 5. The maximum optimality ratio as a function of ratio of cross-section areas.

leads to increase of resistance especially as α decreases. In addition, according to Fig. 3(c), deviation from α = 2−2/3 ∼ = 0.63 leads to variation of the optimality ratio with the ratio of length. In fact, for each α there is a critical γ which maximizes hydraulic resistance. Moreover, it is clear that the critical ratio of length (γc ) which maximizes hydraulic resistance depends on the ratio of cross-sectional area. The relationship between γc and α is plotted in Fig. 4. From this figure it is seen γc varies from 0.4 to 0.9. To investigate the effect of γc on optimality ratio, Fig. 5 is provided. This figure illustrates the variation of ϕ at γc as a function of α. As shown, for γ > 2−1/3 , the effect of deviation from α ∼ = 0.63 is less than 20%. However, for γ < 2−1/3 , the deviation can increase the hydraulic resistance more than 700%. Fig. 6 depicts variation of the total hydraulic resistance or the total volume with respect to the generation number (n). For the case of optimal values of the ratio of length and cross-sectional

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FIG. 6. The ratio of total volume to mother’s volume or the total hydraulic resistance to mother’s hydraulic resistance with generation number.

area, i.e. α = 2−2/3 andγ = 2−1/3 , it is seen that the total resistance and volume increase linearly with the generation number (n) in a similar way. On the other hand, for α = 2−2/3 and γ > 2−1/3 , the total hydraulic resistance or the total volume grows rapidly. However, for α = 2−2/3 and γ < 2−1/3 the total hydraulic resistance approaches to a constant value of (1 − 21/3 γ )−1 as n → ∞. The present study is validated using numerical simulations. Fig. 7 shows the difference between numerical simulation and analytical solution with Reynolds number. For this purpose, “Difference” is evaluated as Di f f er ence = (R H,num. − R H,theo. )/R H,theo. × 100. The Reynolds number (Re = ρU D H /η) is also computed for the mother branch using the hydraulic diameter (D H = 4A/P). From this figure, it is evident that high Reynolds number leads to large errors. However, for Re