The precise microfluidic design control enables to fabricate exactly-reproducible microfluidic channel from computer-designed photo masks. 57 These designs can be optimized prior to the actual device fabrication by predicting the fluid flow by computational fluid dynamics (CFD) and therefore enabling simulation-based rapid prototyping. 131
The description of the dynamics of Newtonian fluids are based on the conservation of energy, mass and momentum. 213,214 This leads to a set of two partial differential equations (PDE), the so-called Navier-Stokes equations. 215,216 These PDEs can be solved numerically by computational fluid dynamics (CFD) simulations. 217 For this task, the finite element methods (FEM) is the most common tool that enables highly accurate modeling by handling complex meshes to describe the fluid dynamics within the flow geometry. 218-220 These analyses have been performed using the software COMSOL Multiphysics (v4.2a).
Assuming isotropic conditions, i.e. omnidirectional forces, a system of second degree non-linear partial differential equations can be established, the Navier-Stokes equations. 215,216
Which contains the density , the dynamic viscosity , the velocity vector , the volume force vector (i.e. gravity), the absolute temperature and the pressure .214
The first equation describes the velocity field of an incompressible Newtonian fluid in a finite element. 214 The second equation is the continuity equation which is yielded by assuming an incompressible fluid and taking the conservation of mass into account. This approximation is in good agreement with reality when the temperature variation are small and the density is constant or nearly constant. 214 This is the case for liquids in a microfluidic channel.
Consequently, the fluid mass that enters a finite volume element in a given time ∆t is exactly equal to the exiting fluid mass. 214,221
The flow in microfluidic devices is laminar which is why stationary conditions can be assumed. The Navier-Stokes equations for this case is given by
The term reflects the stress forces per volume unit due to the pressure gradient and the viscosity .222 The expression represents the inertial term. The Reynolds number (Re) is a fundamental dimensionless number that describes the ratio between these inertial and viscous forces. 10,214,222 It is very useful for the prediction of fluid flow and defined by:
with a velocity scale , the characteristic length of the geometry , the density , the time t and the viscosity of the fluid .
The flow is laminar at low Reynolds numbers (Re < 1) which is typically the case for example at very small geometries, very low flow rates and/or high viscosities. 10 The flow becomes turbulent in the opposite case, i.e. with larger geometries, increasing flow rates and/or decreasing viscosities. 10,213 The laminar flow at low Reynolds numbers is time-symmetrical which is reflected by the absence of a time variable in the Navier-Stokes equations in contrast to the above Navier-Stokes equations.
The dimensionless Weber (We) number is the ratio between the inertia of the fluid compared to its surface tension. 10,213 It is expressed by the following expression:
with the density of the fluid , the velocity , characteristic length (i.e. droplet diameter) and the surface tension .
The Deborah (De) number is the dimensionless ratio that expresses elastic effects by relating the polymer relaxation time to the relevant flow time scale .10 It is given by
The Deborah number is very useful for the charaterization of a fluid’s response to a stimulus of a given duration. The time scale of such a stimulus could be given by the fluid’s flow through a microchannel tapering, or its passing of a tight nozzle geometry, at a given flow rate. 10
The relation between the Reynolds and Deborah number is defined as the Elasticity (El) number that describes the relative importance of elastic to inertial effects and is given by10
with the shortest dimension that sets the shear rate and expresses the. Consequently, this dimensionless number depends only on the material properties and the geometry, but it is flow rate independent. 10
The equivalent pressure of the surface tension is given by the Laplace equation. 223
with the contact angle between the fluid and the surface and the channel diameter . This shows that the surface tension is inversely proportional to the microchannel dimension.
3.2.2 The No-Slip condition
The surface to volume ratio in microfluidic channels is very high which is why surface properties have a great influence on the fluid dynamics. 224,225 These hydrodynamic interaction between solids and liquids can be described using the no-slip condition. 226,227 The velocity vector of a flowing fluid is assumed to be zero at the wall. This condition remains valid in the sub-micron range which has been shown by moleculardynamic calculations that assumed a hydrodynamic wall that corresponds to a monomolecular layer of fluid molecules resting on a solid wall. 228,229 The no-slip condition is also in good agreement with aqueous samples that flow in PDMS-based microfluidic channels which have been studied in this thesis. 230-232
Microscale particle image velocimetry- (µPIV) and surface force apparatus (SFA) experiments have shown that the velocity close to the wall is not exactly zero, a velocity
component remains, when hydrophilic species are in contact with strongly hydrophobic or nanostructured materials. 231,233
This is why a slip length is defined that considers this velocity deviation. 225,227,232 This slip length is a depth below the wall surface when extrapolating the linear velocity component . A -value of infinity corresponds to perfect slip, while a value of zero corresponds to the no-slip condition. Depending on the experimental conditions, this depth corrects the deviation and it can range between a few molecule diameters up to multiple micrometers. 228 3.2.3 Convection and Diffusion
A widely used principle for the fast and diffusive mixing of species is hydrodynamic focusing. 46,129,132,134,140,234,235 This can be achieved in a cross-like channel geometry similar to Fig.10 which shows a CFD simulation of a hydrodynamically focused stream of a given concentration (red) with another miscible fluid with the concentration zero (blue).
Figure 10 Example of hydrodynamic flow focusing. The CFD-simulation (COMSOL Multiphysics v4.2a) shows convective transport of a species due to the laminar flow of a Newtonian fluid that is coupled with the diffusive species transport. The color bar describes the transition from high (red) to low (blue) concentrations of the species.
The mixing time depends on the diffusion coefficient of the species and the width of the focused stream which can be controlled by adjusting the flow rate ratio between main and side channels. The mixing is diffusion based because of the laminar flow profile inside the small microchannels. The diffusion of a species with the concentration of the species for the general non-conservative case is given by
with diffusion coefficient , the reaction rate expression for the species and the velocity vector .214 The mixing and diffusion times are given by the equations:
with
with the kinematic viscosity which is the ratio of the (dynamic) viscosity and the density . These equations also show the characteristic length linearity to the mixing time
and proportionality to the square of the diffusion time .
While the diffusive species exchange is not effective on the centimeter scale, the small microfluidic channels enable effective diffusion based experiments at very low Reynolds numbers within very short times (10 µs). 129,226,236
A very important ratio for the description of transport phenomena is the dimensionless Péclet (Pe) number. It describes the ratio between advective transport (i.e. fluid flow) and diffusive transport (i.e. diffusion) and is defined as10
with the characteristic length , the characteristic velocity and the diffusion coefficient D. The Péclet number is the product of the dimensionless Reynolds (Re) and Schmidt (Sc) numbers of which the latter is defined by the ratio between the viscous diffusion rate and the molecular (mass) diffusion rate. The Schmidt number is given by: 10
with the kinematic viscosity and the mass diffusivity .
Experiments in microfluidic take advantage of the laminar and non-turbulent flow conditions that are enabled through the small channel dimensions which results in low Reynolds and Péclet numbers. 9,10 This fact leads to advantageous experimental conditions such as laminar diffusion-based hydrodynamic focusing and stationary continuous flow which projects dynamic processes to a fixed position along the flow direction. This fact makes even very fast kinetics or the shearing of samples locally stable and time-symmetric. The resulting steady state condition enables mapping scans of reactions or shear profiles with suitable in situ probing techniques like, i.e. small-angle X-ray scattering. 9,13,46,129,135,230 The microchannel design remains adjustable, due to rapid prototyping and computer aided design, to the experimental needs which can range from single molecules to cell studies (see chapter 1.3).
131,237,238
As an example, fast reactions kinetics can be controlled with great precision due to the microchannel design. 92,239 Another example describes actin fibers that can be fully elongated by suppressing Brownian motion which is of high interest for the modeling of in vivo processes of semiflexible biopolymers (i.e. DNA or proteins) or dilute solutions of block copolymers. 240
Next to defined mixing scenarios and concentration gradients on the micron scale, 9,15 microfluidic devices are also well-suited for the generation of defined shear and elongational force profiles. 13,241-246