'Multilevel Based All-At-Once Methods in PDE Constrained Optimization with Applications to Shape Optimization of Active Microfluidic Biochips
|Projektträger:||DFG (Deutsche Forschungsgemeinschaft)|
|Projektverantwortung vor Ort:||Prof. Dr. R. Hoppe, Prof. Dr. K. Siebert, Prof. Dr. A. Wixforth|
|Beteiligte Wissenschaftler der Universität Augsburg:||
Prof. Dr. K. Siebert
Prof. Dr. A. Wixforth
a) Prof.Dr. Ronald H.W. Hoppe
geb. 10.04.1951, Nationalität: deutsch
Adresse: Universität Augsburg
Lehrstuhl für Angewandte Analysis/Numerik
Institut für Mathematik
Universitätsstr. 14, D-86159 Augsburg
Tel.: 0821-598-2194, Fax: 0821-598-2339
Address: University of Houston
Department of Mathematics
651 Ph. G. Hoffman
Houston, TX 77204-3008, U.S.A.
Tel.: +1-713-743-3452, Fax: +1-713-743-3505
b) Prof.Dr. Kunibert Siebert
geb. 09.05.1963, Nationalität: deutsch
Adresse: Universität Augsburg
Lehrstuhl für Angewandte Analysis/Numerik
Institut für Mathematik
Universitätsstr. 14, D-86159 Augsburg
Tel.: 0821-598-2190, Fax: 0821-598-2339
c) Prof.Dr. Achim Wixforth
geb. 26.05.1956, Nationalität: deutsch
Adresse: Universität Augsburg
Lehrstuhl für Experimentalphysik I
Institut für Physik
Universitätsstr. 1, D-86159 Augsburg
Tel.: 0821-598-3300, Fax: 0821-598-3225
This project within the area of PDE constrained optimization
focuses on the development, analysis and implementation of
optimization algorithms that combine efficient solution techniques
from the numerics of PDEs, namely multilevel iterative solvers,
and state-of-the-art optimization approaches, the so-called
`all-at-once' optimization methods. It is well-known that
multilevel techniques provide efficient PDE solvers of optimal
algorithmic complexity. On the other hand, optimization methods
within the all-at-once approach, such as sequential quadratic
programming (SQP) methods and primal-dual Newton interior-point
methods, have the appealing feature that in contrast to more
traditional approaches, the numerical solution of the state
equations is an integral part of the optimization routine. This is
realized by incorporating the PDEs as constraints into the
optimization routine. These strategies allow to save a
considerable amount of computational work compared to methods that
treat the PDE solution
as an implicit function of the control/design variables.
Moreover, the proper combination of multilevel techniques and optimization algorithms makes it possible to extract essential structural information from the originally infinite dimensional optimization problem. This can not be done with respect to a single grid. We aim to develop and analyze multilevel preconditioners for optimization subproblems arising in SQP and primal-dual Newton interior-point methods including strategies to control the level of inexactness allowable in optimization subproblems, when using iterative subproblem solvers. Moreover, we will investigate strategies to use multilevel methods for detection of negative curvature and in path following methods.
The developed PDE constrained optimization algorithms will be
applied to the optimal design of microfluidic biochips with
emphasis on the optimization of the geometry of the devices (shape
optimization) in order to achieve an optimal operational behavior.
Here, the state equations consist of a system of partial
differential equations describing the transport of microfluids
driven by piezoelectrically agitated surface acoustic waves.
During the last couple of years, such biochips have attracted a considerable amount of interest, since pharmacology, molecular biology, and clinical diagnostics require the precise handling of precious, tiny samples and costly reagents in amounts of nanoliters. Biochips can transport such volumes and perform biochemical analysis of the samples. Microfluidic biochips and microarrays are used in pharmaceutical, medical and forensic applications as well as in academic research and development for high throughput screening, genotyping and sequencing by hybridization in genomics, protein profiling in proteomics, and cytometry in cell analysis. Traditional technologies rely on fluorescent dyes, radioactive markers, or nanoscale gold-beads based on positive hybridization processes. However, these methods only allow a relatively small number of DNA probes per assay, and they only yield endpoint results and do not provide information about the kinetics of the processes. With the need for better sensitivity, flexibility, cost-effectiveness and a significant speed-up of hybridization, the current technological trend is obtained by the integration of the microfluidics on the chips itself.
A new type of nanotechnological devices are surface acoustic wave driven microfluidic biochips. The experimental technique is based on piezoelectrically actuated surface acoustic waves on the surface of a chip which transport the droplet containing probe along a lithographically produced network channels to marker molecules placed at prespecified surface locations. Hence, these microfluidic biochips allow the in-situ investigation of the dynamics of hybridization processes with extremely high time resolution.
The scientific merits of the proposal are the design, analysis and
implementation of efficient algorithmic tools for a class of
challenging problems in nonlinear optimization and the
demonstration of their performance by the application to a
real-life problem that has a significant impact on material
sciences and life sciences.
The broader technological impact of the project is a better design of microfluidic biochips with an improved performance for biochemical applications.
PDE Constrained Optimization
Optimization problem with constraints given by partial differential equations (PDE) can be written as follows1b) corresponds to the PDE, whereas refer to the sets of admissible states and design variables, respectively.
The discretization of PDE constrained optimization problems typically gives rise to large-scale, equality and inequality constrained nonlinear programming problems of the form2b) with comprises the discretized state equation and possibly further equality constraints , whereas (2c) with represents inequality constraints on the state and design variables.
Biochips, of the microarray type, are fast becoming the default
tool for combinatorial chemical and biological analysis in
environmental and medical studies. The goal of this project is to
develop, analyze and implement state-of-the-art optimization
techniques for an improved design of these devices.
Programmable biochips are miniaturized biochemical labs that are physically and/or electronically controllable. The technology combines digital photolithography, microfluidics and chemistry. The precise positioning of the samples (e.g., DNA solutes or proteins) on the surface of the chip in picoliter to nano liter volumes can be done either by means of external forces (active devices) or by specific geometric patterns (passive devices).
During the last couple of years, such biochips have attracted a considerable amount of interest, since pharmacology, molecular biology, and clinical diagnostics require the precise handling of precious, tiny samples and costly reagents in amounts of nanoliters. Biochips can transport such volumes and perform biochemical analysis of the samples. Nanoliter fluidic biochips and microarrays are used in pharmaceutical, medical and forensic applications as well as in academic research and development for high throughput screening, genotyping and sequencing by hybridization in genomics, protein profiling in proteomics, and cytometry in cell analysis . Traditional technologies rely on fluorescent dyes, radioactive markers, or nanoscale gold-beads based on positive hybridization processes. However, these methods only allow a relatively small number of DNA probes per assay, and they only yield endpoint results and do not provide information about the kinetics of the processes. With the need for better sensitivity, flexibility, cost-effectiveness and a significant speed-up of hybridization, the current technological trend is obtained by the integration of the microfluidics on the chips itself.
A substantial amount of research activity in this direction is in the area of microchip capillary electrophoresis (cf., e.g., [3,81,110]). Here, gel-filled capillary channels are placed on micromachined glass or polymer microreplicated chips. More recent and novel nanotechnological devices are surface acoustic wave driven nanoliter fluidic biochips and pressure-driven nano-chambered, multi-channel microfluidic biochips. The former technique is based on piezoelectrically actuated surface acoustic waves on the surface of a chip which transport the droplet containing probe along a lithographically produced network to marker molecules placed at prespecified surface locations (cf., e.g., [13,114,139,140]). The latter one adequately combines digital photolithography, microfluidics and chemistry (cf., e.g., [38,39,40,89,90,110,146]). These nanoliter fluidic biochips allow the in-situ investigation of the dynamics of hybridization processes with extremely high time resolution.
By changing the surface chemistry appropriately, a fluidic network is produced on the chip: Without mechanical tools the chip is equipped with paths on which samples (and reagents) propagate as if on tracks. This is done lithographically by a lateral modulation of the wetting properties of the free surface which leads to pronounced hydrophilic and super-hydrophobic regions with significantly different wetting angles (cf., e.g., [27,51,73,95,99]). Small amounts of reagents are confined to these tracks in contrast to mechanical barriers used in conventional microfluidics.
The active devices which will be considered in this project are nanoliter fluidic biochips where the core of the technology are nanopumps featuring surface acoustic waves generated by electric pulses of high frequency. These waves propagate like a miniaturized earthquake (nanoscale earthquake) and in this way transport liquids along the surface of the chip.
Figure 1 below gives an illustration of a nano titration chip. On the fluidic network a small portion of titrate solution (middle) is separated from a larger volume (right). Surface acoustic waves transport this quantity towards the analyte (left) at the reaction site. Once a critical concentration is attained, it can be either detected by a change of the color of the analyte or a change of the conductivity. In the latter case, this can be easily measured by a sensor that is integrated on the same chip.
Surface Acoustic Waves (SAW) have been used for a long time in
high frequency applications (cf., e.g., [21,101,105], and ).
Using SAW-principles it is now possible to combine
microelectronics and biochemistry. Modern semiconductor technology
enables the cost-effective production of devices that unify
biological functionality, sensors and pumps for the transport of
samples. These devices can be easily integrated in electronic
systems like those that are used in point-of-care
diagnostics (see [4,116,117,126]).
The micropump consists of a piezoelectric substrate equipped with so-called interdigital transducers on the surface. Radio-frequency signals are fed into those transducers and are converted to a deformation of the crystal underground. In this way, a mechanical wave is launched across the surface with wavelengths in the range of a few microns and amplitudes about only a nanometer. Liquids on the surface are subject to the vibrating force and absorb parts of the energy. The absorbtion of energy for various frequencies depends on the density and viscosity.
Surface acoustic waves of larger amplitudes move liquid droplets as a whole whereas low power surface acoustic waves induce some sort of an internal streaming. The latter case enables to construct surface acoustic wave based nanomixers. If the frequency of the surface acoustic wave is changed, different streaming patterns are induced and superimposed within the droplet that leads to a homogeneous blend of the water and the probe much faster than by more conventional diffusion type microfluidic mixing techniques.
Figure 2 illustrates the effect of nanomixing in case of the dissolution of a fluorescent dye deposited on the chip surface with agitation (acoustically induced mixing) and without agitation.
By using surface acoustic wave nanopumps, different reagents can
be efficiently mixed, separated or moved to different reaction
sites on the chip. Compared with conventional micro titer plates,
the respective volumes are reduced by several orders of
Previous work of the project leaders
Primal-dual Newton interior point methods
Discretization of the objective functional, the state equation and
the equality constraints by finite elements with respect to a
of the computational
domain leads to the finite dimensional constrained
minimization problem: Find
Shape optimization of electrorheological shock absorbers
Electrorheological fluids are concentrated suspensions of small
electrically polarizable particles with diameters in the range of
micrometers dissolved in nonconducting silicon oils. The
rheological effect is based on the fact that under the influence
of an outer electric field the particles form chains along the
field lines and then aggregate to form larger and larger columns.
The impact on the macroscopic scale consists in a rapid change of
the rheological properties which happens within a few
milliseconds. The viscosity increases in the direction orthogonal
to the electric field such that the character of the fluid changes
from liquid to almost solid. Under the action of large stresses,
depending on the electric field strength (field dependent yield
stress), the columnar structures break such that the viscosity
decreases and the fluid behaves less anisotropic. The process is
reversible, i.e., the viscosity decreases with decreasing field
strength and the fluid behaves again like a Newtonian fluid for
vanishing outer electric field.
Therefore, electrorheological fluids are used in all technological processes where a controlled power transmission plays a significant role. The field of applications ranges from automotive shock absorbers and actuators in hydraulic systems to tactile devices for virtual reality .
In particular, we have been concerned with the optimization of the shape of the walls of an ERF shock absorber (cf. Figure 3 (left)) in a vicinity of the inlet and outlet boundary of the ERF transfer ducts (cf. ). A schematic diagram of such a shock absorber is shown in Figure 4 (left). In contrast to conventional shock absorbers, where the fluid chambers are filled with hydraulic oils, both in the compression and in the rebound stage, ERF shock absorbers have a much wider characteristics (damper force as a function of the velocity of the piston; cf. Figure 3 (right)). Therefore, ERF shock absorbers offer the best compromise between safety and comfort for a wide spectrum of road conditions.
The performance of the shock absorber does not only depend on the
applied voltage and the velocity of the piston, but also on the
geometry of the device. In particular, the geometry of the inlet
and outlet boundaries of the ducts play a decisive role. In
extreme cases, cavitation due to high pressure variations
may occur which negatively effects the damper characteristics.
Therefore, given a prescribed pressure profile , the
optimization issue is to design the geometry in such a way that
pressure variations are minimized. Due to axisymmetry, the computational
domain reduces to the right part of the fluid chamber.
The inlet and outlet boundaries are represented
by B-splines using the de Boor control points
as design variables (cf. Figure 4 (right)).
Consequently, the computational domain depends on the choice of
the design variables, i.e.,
In the stationary case, the fluid flow is described by the equations
We note that the differences between the initial configuration and
the optimized configurations are small, but give rise to
reductions in the pressure variations between 10 and 20 %,
depending on the operating conditions.
2.2.4 Modeling and simulation of active biochips
22.214.171.124 Piezoelectrically actuated acoustic surface waves
Mathematical models for SAW biochips are based on the linearized
equations of piezoelectricity as given by
For an SAW chip based on lithium niobate (LiNbO) of length mm and height mm and with an operating frequency of Figure 6 displays the amplitudes of the electric potential which is in the range of nanometers.
The SAWs are strictly confined to the surface of the substrate.
Their penetration depth into the piezoelectric material is in the
range of one wavelength. The velocity of the SAW is independent of
the applied frequency. In the case of
LiNbO, the SAW velocity is given by
. Thus, for an excitation at the frequency MHz
the theoretical wavelength of the SAW is given by
. The simulations show
the same wavelength for the SAW.
The excitation of an IDT on the surface of a piezoelectric material leads to the generation of bulk acoustic waves (BAWs) as well as surface acoustic waves. These bulk waves can also be observed in Figure 6. Technologically, they are desirably employed in solid-state circuits . However, for SAW devices their presence is unwanted, since the interference of BAWs with SAWs can lead to a complete loss of functionality of the device. The used approach is sufficiently general to simulate every kind of piezoelectric resonator. In Figure 7 we have used an ° cut of LiNbO to generate a strong bulk acoustic wave at frequency .
Rayleigh surface waves characteristically show an elliptical displacement, i.e., the displacements in the - and -direction are ° out of phase with each other. Additionally, the amplitude of the surface displacement in the -direction is larger than the one along the SAW propagation axis . The simulations reveal these properties (see Figures 8 and 9). In Figure 8, the displacements in the - and -direction for a certain surface area are depicted. The -displacements are flipped vertically for easier comparability.
In Figure 9 a certain surface area is magnified and the vectors indicate the surface displacements.
Modeling and simulation of microfluidic flows on biochips
The modeling of the micro-fluidic flow is based on the compressible Navier-Stokes equations (see [170,42]). In the model, compressible and non-linear effects are the driving force of the resulting flow. We consider the following situation: We start with a given equilibrium position of the flow field together with a given displacement describing the actual position of the time-dependent domain , occupied by the fluid. The displacement will be given by the modeling of active biochips described in the previous section. In the presented approach, we neglect the damping effect of the fluid on the solution of the piezoelectric equations.
We are now looking for the velocity , pressure and density such that
In order to close the system we prescribe initial values
The Navier-Stokes system given above is not solved directly, mainly because the underlying physical processes take place on extremely different time scales. The elastic displacement of the substrate traversed by SAWs is a process with a time scale of nanoseconds, while the resulting flow due to acoustic streaming reaches an equilibrium on a time scale of milliseconds.
We use a technique from physics known as approximation theory to derive the model. First, we consider an expansion of the unknowns , and in a parameter , to be chosen appropriately:
In a second step, we then collect all second order terms while considering the solution of the first problem as given data:
Contrary to the first order acoustic equations, we expect second order velocity and pressure to have nonzero mean value in time, as the data of the problem exhibit nonzero mean values. This is the mathematical background of the effect known in physics as acoustic streaming, which describes the creation of stationary flow patterns in reaction to a high frequency acoustic wave.
We now assume that the excitation is a harmonic process with period . This period, being on the order of nanoseconds, is small compared to the time scale of the relaxation processes in acoustic streaming. Applying the time-averaging operator
Note that the boundary condition and the right hand side are compatible, since
Subproblem 1: Simulation of the damped acoustic field
The mathematical structure of the acoustic problem (after suitable scaling) is
This system is discretized by finite elements in space and the implicit Euler or Crank-Nicholson scheme in time. In the -th time-step this then results in a linear system of the form
As in the modeling of the piezoelectric material above, we also have the possibility of assuming a time harmonic dependency of the solution if given boundary data is time harmonic. In this situation we make the ansatz
Concerning the difference between the time discretisation and the harmonic ansatz just presented, the numerical effort to solve these is also comparable. The harmonic ansatz has the bonus of yielding precise values for the right hand sides and boundary values of the acoustic streaming problem treated below. However, it is naturally not applicable to any problem involving nonharmonic driving forces, e.g. a SAW device operated in pulsed mode.
Subproblem 2: Simulation of Acoustic Streaming
The problem of acoustic streaming has the mathematical form
This is a Stokes system with non-homogeneous divergence constraint and can efficiently be solved using a stable finite element discretization such as the Taylor-Hood element.
In this section we present numerical results for a 2d problem. The solvers for the damped acoustic field and for the acoustic streaming are also implemented within ALBERTA. We have used following data:
The problem was discretised by the Finite Element Method using the ALBERTA package developed as a joint project with Kunibert Siebert, Alfred Schmidt, and others. The data of the first 2D test problem:
The acoustically driven micro- and nanofluidic has over the last
four years been developed in the research group of Prof. Wixforth.
Based on this technology, many different applications have been
devised which even led to the foundation of a start up company
(Advalytix, Brunnthal). Since then , a close cooperation between
the research group and the company has been established, including
many more cooperation partners working in this field. Mostly, we
have been concentrating on so-called planar microfluidic chips, an
example of which is given in section 2.1.2. of this proposal.
Here, surface acoustic waves (SAW) are used to interáct with small
amounts of fluids at the surface of a planar chip. Chemical
instrumentalization of parts of the chip's surface is employed to
laterally modulate the wetting properties of the surface, hence
providing "virtual tracks" for the fluid, if actuated by the SAW.
A more or less detailed understanding of the physics behind the
interaction between the fluid and the SAW has been reached
[98,82]. Based on the planar microfluidic technology,
quite complex programmable biochips have been demonstrated. The
most advanced chip, so far, is probably a system, on which a
sub-microliter polymerase chain reaction (PCR) has been
successfully demonstrated .
Aims of the project
The goal of the project is
Primal-dual Newton interior-point methods
The interior-point aspect of primal-dual Newton interior-point methods is that classical barrier methods can be used to transform constrained problems to unconstrained ones. They typically give rise to parameterized families of approximate subproblems. In particular, coupling the inequality constraints by logarithmic barrier functions gives rise to
Primal-dual active set strategies
Active set strategies are iterative schemes for the solution of inequality constrained optimization problems, where at each iteration step the solution of an equality constrained subproblem is required by identifying a set of active constraints (cf., e.g.,  and the references therein). In case of bilateral box constraints on the design variables
In view of (32), the active set strategy can be
interpreted as the discrete version of a semismooth Newton method
in function space [118,210]. Based on this
interpretation, it is the aim of the proposal to develop, analyze
and implement Newton multigrid methods as well as full nonlinear
Specific issues that will be addressed include the
Adaptive mesh refinement
In this project, we plan to investigate different target quantities and study their impact on the mesh adaptation and solution accuracy. We emphasize that the inequality constraints introduce a nonlinear and even nonsmooth behavior to the problem. Due to this fact there are several impacts on the (a posteriori) error estimator based mesh adaptivity:
Shape optimization of microfluidic biochips
A current trend in the design of active SAW-driven microfluidic
biochips is the production of plastic chips with a network of
channels and reservoirs that are placed on top of the SAW device.
The channels and reservoirs are milled out of the plastic plate and
a plastic foil is finally vapor deposited on top of it. Figure
15 shows such a plastic chip with two interconnected
reservoirs featuring capillary barriers at its inflow/outflow
boundaries for the purpose of a controlled filling process (see
Figure 17). The chips can be used in various
biomedical applications such as the early diagnosis of liver damage
(e.g., hepatitis) by measuring enzyme activities in the blood serum
of patients. The concentration of the enzyme is determined
indirectly by measuring the optical density. For doing the
biochemical analysis, certain reagents must be transported to the
probe in a controlled way along the channels of the network to the
reaction chamber. The transport of the fluid and the precise dosage
are realized by the piezoelectrically actuated SAWs and by
the design of the capillary barriers.
For the mathematical description of the transport process and the filling of the reservoirs, we will provide a two-phase flow model (air/liquid 1) and a multi-phase flow model (air/liquid 1/liquid 2/liquid 3). Here, liquid 1 is supposed to contain the reagents, whereas liquid 2 is carrying the probe and liquid 3 results from the mixing of liquid 1 and liquid 2. We will consider the optimization of the channel geometry and the optimal positioning of the interdigital transducers (IDT), the optimization of its apertures and of the frequency of the SAWs. At a later stage, we will be concerned with the shape optimization of the reservoir as well as with the optimal positioning of the capillary barriers and the optimization of its apertures which requires the use of Signorini type boundary conditions for modeling the stopping properties of the capillary barriers.
For the efficient numerical solution of these optimization problems, we will use the multilevel based all-at-once methods to be developed in this project.
Optimization of the geometry of the channel
We assume the position of the IDTs and the frequency of the SAWs
to be fixed. Hence, in a pre-processing step we are able to
compute the displacement on the walls
traversed by the SAWs using the methods
as described in 126.96.36.199. The walls of the
channel are represented by a spline surface involving the product
of cubic spline blending functions with respect to an array
of control points serving as the design variables. Due to
technical reasons, the design variables are subject to bilateral
. The computational domain , which is the flow
region inside the channel, thus depends on the choice of the
design variables, i.e.,
. The design
objective is to achieve an optimal pumping rate. The state
equation is given by the two-phase air/liquid 1 flow model.
Optimization of the position and aperture of the IDT and of the
frequency of the SAW
In this case, we keep the geometry of the channel
fixed, but choose the position
of the IDT (e.g.,
monodirectional IDT in the channel or IDT at the channel corner),
its aperture and the frequency of the SAW as design
. The design
objective and the state equation are the same as before. As far as
the inequality constraints are concerned, suitable bounds for the
Shape optimization of the reaction chamber
Figure 16 displays the fluid flow into the reservoir with two capillary barriers (see also Figure 17) at various stages of the filling process. In particular, Figure 16 (left) shows a snapshot of the first exit time of the (green) dye, whereas Figure 16 (middle) represents a snapshot of the first time instant where the (green) dye covers the outermost edges of the wall of the reaction chamber.
Experimental evidence suggests that the diamond-like geometry of
the reservoir is not optimal for rapid filling.
At a later stage of the project, we will consider the optimization of the geometry of the reservoir in order to achieve the filling of the chamber in minimal time . The mathematical description of the filling process requires the use of a multi-phase flow model in not necessarily circular channels and reservoirs, where the capillary barriers can be modeled by Signorini type boundary conditions
Structural optimization of the capillary barriers
For a fixed geometry of the reservoir, another design objective is to determine the positions and the apertures of the capillary barriers in order to minimize the filling time. The state equations are the same as before including Signorini type boundary conditions for the capillary barriers, whereas the design variables are given by .
In this project, we plan to investigate a combination of acoustic
nanopumps and conventional microfluidic systems, i.e., three
dimensional complex systems of channels and reactors, in which
biological assays can be performed. Such 3d biochips usually
consist of cheap substrate materials like, e.g., plastic, silicon,
or glass. Deep etching, molding or hot embossing techniques are
employed to fabricate the channels and reactor compartments in
such 3d microfluidic chips. However, most of these chips rely on
external pumping or integrated electro-osmotic actuation schemes,
that cause additional costs and problems. Moreover, mixing in such
devices is usually a difficult task because of the laminar flow at
low Reynolds numbers. Here, we have recently successfully
demonstrated  that the coupling of SAW into a 3d
microfluidic chip leads to pronounced and efficient mixing, like
in capillary gaps  or in open droplets .
Here, the SAW on a piezoelectric substrate is coupled into the
chip substrate (plastic in the latter case), where it is converted
into a wave in the substrate material, and then coupled into the
fluid in the channel, where it
induces acoustic streaming and mixing.
The coupling of SAW into different substrate materials and finally into the fluid in a 3d channel requires a detailed knowledge and understanding of the sound propagation in such "layered" systems. Not only diffraction and mode conversion has to be taken into account, also the complex shape of the channels and reactor compartments need to be considered.
A typical example of such a 3d biochip is given in Fig. 1, where we show a microfluidic reactor for a biological assay. It holds an inlet and an outlet (filled with blue colored water) and a reaction chamber (filled with yellow fluid). The inlet is coupled to an SAW pump on a piezoelelectric chip underneath the plastic chip (not visible in the figure), which - if activated - pumps the reagent (blue) into the reaction chamber. There, it is stirred by another SAW coming from below the yellowish region (Fig. 18).
In this figure, another important functional block of the
investigated microfluidic system is already visible: To prevent
premature mixing of the two reagents and to be able to meter a
certain amount of "blue" reagent into the "yellow" reactor, a
capillary barrier has been employed at the connection between the
inlet and the reactor, and the outlet and the reactor. This
capillary barrier should withhold the reagent 1 without the
activation of the SAW pump, and should open under the influence of
the SAW induced acoustic streaming. In that sense, it represents a
pressure driven valve, which needs to be optimized for optimum
functionality. This optimization is an important task to be solved
in this proposal.
To be able to precisely meter a certain amount of fluid, employing the SAW pump, a detailed understanding of the flow profile, the velocity fields and hence the pressure in the channel is necessary.
As has already been shown, the acoustic coupling between SAW and fluid leads to a streaming velocity field, that results in an angle between the surface of the piezoelectric chip and the acoustic "jet" into the fluid. This "jet can be nicely visualized by using a bit of ink, as shown in Figure 19.
An SAW is coupled from a piezoelectric substrate (LiNbO3) through
a plastic substrate into the fluid. The resulting acoustic
streaming under an angle is visualized by a little bit of ink in
Recent experimental investigations have resulted in a modified chip layout, taking into account this "refraction" angle. It could be shown that an oblique inlet channel (tilted under exactly this angle) leads to a much higher streaming velocity than just coupling the ultrasound into a parallel channel. First experiments revealed an increase of the flow velocity by about a factor of five, even in a non-optimized channel geometry. Again, the optimization of the channel geometry with respect to the maximum flow velocity is an important scope of the present proposal.