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Hello and welcome back to this 30th lecture of Biomicroelectromechanical Systems.
Just quickly preview what we did last time. We talked about these famous Navier-Stokes
Conservation of mass and Conservation of momentum equations, tried to derive them from the elementary
control volume concept and then actually tried to go ahead and scale down both of these equations
and tried to figure out how they apply to microfluidics.
Let us actually go ahead and just reiterate a little bit of what the conclusions were
out of the scaling down. We tried to first take the conservation of momentum equation
and the scale down version of this equation came out to be dui star by dxi star equals
0. and this essentially indicated and Let me just reiterate the way these expressions
came into picture. xi would be related to a scaled distance at that particular scale
times of xi star, the xi star is a dimensionless number. Similarly, ui and uj both were equated
to a scaled velocity at that particular scale, times of ui star and same scale velocity times
of uj star respectively. Also, we saw that the time star of the non-dimensional
quantity in time in this particular scale would be represented as the original time
divided by the distance by u, the velocity and the pressure star which is again a non-dimensional
quantity in pressure in this particular scale was represented by the term P - this is the
absolute value of pressure divided by eta U by D. As we all know, eta is the viscosity
and it is actually scale independent. So, it remains constant even if you go to the
micron scale; you have to assume though that the equations are that of continuity equations
or continuously maintained for the viscosity to be independent of time and constant property
associated with the medium. So, because our micro flows are mostly in
this domain where the continuum is still assumed and all the modeling is done using the continuity
equations, it is almost always obvious to assume the viscosity to be constant at that
particular scale. Therefore, the pressure star that means P star which is essentially
the dimensionless number in the Navier-Stokes equation would be represented as P the absolute
value of the pressure divided by nU by D and so we derived at the second equation the conservation
of momentum equation and we figured out that rho U D by eta times of del ui star by del
t star plus uj star times of del ui star divided by del xj star would be equal to minus del
P star by del xi star plus del 2 ui star divided by or del x with respect to del xj star square.
Which means that This essentially is Reynolds number Re and this we found out last time
while scaling down and Re is more or less, less than 100 in all microfluidic devices,
very often less than 0.1. It almost always gives you an opportunity to write down minus
del P star by del xi star, the right hand side of this particular equation, plus del
2 ui star divided by del xj star square equal to 0 because the left hand side here is essentially
dominated by the lesser magnitude of the Reynolds number. So, irrespective of whatever this
quantity here is, the overall size of the quantity in the LHS because of the low value
of Reynolds number is very small in comparison to this quantity inside the brackets.
So, because it is overall negligible, we can assume this RHS part of the equation to be
equal to 0. What are our conclusions from this particular statement? One conclusion
which comes out is that at this particular scale, the momentum equation becomes time
independent. Therefore, it is obvious to assume that if suppose you have 2 or 3 flows which
are introduced into a same micro channel and we assume low value of Reynolds number, due
to which the left hand side of this continuity equation or momentum equation becomes 0, the
obvious conclusion would be that the pressure driven flow - number 1 and number 2 is that
it is time independent. Therefore, if you could actually run past both these different
types of flows from the outlet end back into the inlet end or you reverse the flows in
time, they should be able to emanate as original inlets to the particular channel.
This is illustrated in this particular example here. If you see again I would just like to
reiterate that this slide was shown before. That in microfluidics very often there are
situations like this here where you are seeing that you have these different dyes which are
flowing into this particular channel; you have about 1, 2, 3, 4, 5, 6, 7 different dies
here. As you see, as this flow happens past this
micro channel here and goes for a certain distance. We are able to extract the dyes
independent in the same cover as they were introduced originally at the other ends. Therefore,
it is really the momentum equation is really time independent assuming that this were to
be true and this actually is a simulation result done by the Whitesides group at Harvard.
If you assume these flows to be flowing in the reverse direction then you should be able
to get these flows out as independent dyes or as independent colours.
This is in fact, the essence of the scaled down form of the momentum equation in this
particular scale. The domain of microfluidics therefore, is a very novel and interesting
domain wherein it is a really very hard for flows laid out together without any diffusional
effects to mix. However, because of this particular deliberation
that has been just shown in the last slide and the time independence of the momentum
equation all the mixing which happens at the micron scale takes place because of a concentration
gradient and is diffusional in nature. Basically, that gives us a feel that it may
be impossible sometimes to flow two fluids together in a micro channel and mix them,
if the 2 flows have equal concentrations and there is no concentration gradient which is
established between the flows; that is number 1. Number 2 is that it becomes a very interesting
paradigm in microfluidics to design intelligent systems which could be actually, suppose there
is also existence of diffusion in the parallel flows so which could be actually promoting
diffusion and reducing the diffusion time in a manner that it becomes lesser than the
residence time of the flows and the flows start mixing. So, these are 2 interesting
paradigms in microfluidics.
So, I would like to go ahead and actually see some of the effects of Reynolds numbers
on the different flows particularly within micro channels and I call this slide, transition
to turbulence within the micro channels. If you see here, there are different Reynolds
number values at which we plotted the U by U maximum, which is essentially the velocity
at a certain position divided by velocity at the center which is the maximum velocity
and this is plotted with respect to x by D, x is the position at which the local velocity
is u and D is essential the diameter of the particular channel.
As we see here, the Reynolds number definitely is a very good predictor of when the turbulence
will take place. The plot is between the velocity ratio at a point within the micro channel
and the ratio of the distance of that point from the center. Conclusions from this are
that if you see this particular plot here, it is basically at a very high Reynolds number
value, 2770 that you see a fairly a large scatter in the velocity. Therefore, you can
see that you know the if you see this scatter can be also represented by this span of the
parabola, an inverted parabola. and so if The span is much more in this case as compared
to the low Reynolds number cases from 238 to almost 1980 when the scatter is not that
high and which also shows that probably at 2770, there is a transition or between 2770
and 1980, there is definitely a transition into the turbulent zone.
Therefore, there is no control really on U after this higher Reynolds number value is
reached whereas, at a lower Reynolds number value there is still some control and you
can really say that the flows have not yet turned into a eddies or vortices or it has
not really turned into turbulent flows and that is what this graph would indicate. Therefore,
the flows shows fairly small scatter between the velocity profiles for Re values between
238 and 1980 and a wide scatter at 2770 which means that the transition is expected in the
region 1000 to 2000. Which is also true otherwise that there is
a transition definitely between Reynolds number 1000 and 2000 whether it is the macroscopic
counterpart or the micro channel.
One more interesting paradigm to be evaluated in microfluidics is related to the effects
which can come at the entrance of a micro channel. As we know that as a channel is approached
by flow of a constant velocity and pressure driven flow at the moment after the flow enters
the channels, there is always some time which the flow takes to get fully developed where
this parabolic profile of the flow gets created within the micro channel.
Now, in cases with in channel In cases of micro channels though this length which is
also known as the entrance length where the flow profile or whatever the flow profile
takes to get developed fully really also is a kind of function of the Reynolds number
and it is very critical to design micro channels by assuming what really would be the entrance
length over and above which the flow is fully developed within the micro channel.
Let us explain this even by saying that there are a lot of fundamental differences in the
physics of flows between micro and macro scales and most of the fabrication techniques as
you know are typically two and half D. Therefore, one might think that the flow 2-D is independent
of the of the Z direction. However, the aspect ratios that are normally realized are sometimes
1 and Z becomes a critical direction for these kinds of flows.
When a fluid flows from a large vessel or reservoir into a small one of a constant cross
section which means the velocity of approach of this particular flow is almost constant
as this approaches this small channel here. The flow takes or the flow profile requires
a certain distance to get fully developed. So, this is also known as the entrance length.
Various people have tried to experimentally and otherwise calculate what this entrance
length would be and the best experiment that was proposed by Shah and London was that the
amount of length needed for centerline velocity to develop 99 percent of the full velocity
is actually given by this expression here. Le, Le is the entrance length divided by Dh,
Dh is the diameter of the particular channel is equal to 0.6 divided by 1 plus 0.035 Re
Reynolds number of that particular diameter or cross section and plus 0.056 Re Reynolds
number at that particular cross-section. So, entrance length is then a function of
the Reynolds number very critically and also the cross section diameter of that particular
micro channel. Le is normally about 60 percent of the hydraulic diameter in case the Reynolds
number is low because then there is the 1 is prominent over this term 0.035 Reynolds
number or 0.056 Reynolds number. However, in cases where this Reynolds number
becomes a little bigger, the entrance length effects are felt more properly. This is the
hydraulic diameter cross-section diameter or the hydraulic diameter. In cases where
the Reynolds number becomes little bigger, the contribution from this term and this term
is significantly enhanced and therefore, the entrance length becomes less than 60 percent
of the hydraulic diameter. What does it mean really? It means that suppose
in case of a micro channel, when the Reynolds number is comparatively low at the entrance
length has significant amount in terms of 60 percent of the hydraulic diameter as opposed
to macro scale where Re is more and this is much lesser than 60 percent.
So, this shows that whenever you want to design micro channel, you almost have to be careful
always about selecting the entrance length before the perturbations or structures that
you would like to create within the micro channel to cause certain physical phenomena
would start to be placed. You need to wait up till the whole entrance length which is
needed for developing the flow fully is traversed before the flow can get into region were significant
differences can be made to it by changing the physical form or shape of the micro channel.
So definitely, entrance effects are very critical as far as microfluidics is concerned.
So, this brings us to an end of the theoretical analysis that was needed to understand microfluidics.
I would like to go ahead now today and try to get into a little more different area as
to what would happen really if the continuum assumptions failed and then would be talking
about things which are more related to inter molecular forces. Scaling from microns to
nanometers it is very important that we consider altogether different approach which is not
driven by continuity and which is more driven by inter molecular forces between the molecules
which are present in a very small nanometer cube size probably control volume.
After looking into that aspect really, I would like to go ahead and start with the topics
of the actual technology that microfluidics can be put to and that can be directly applied
to as in BIOMEMS wherein we would start realizing these active devices like Microvalves, Micromixers,
Micropumps, etcetera. The idea is that after we demonstrate all these on a fundamental
level we will go ahead and assemble these units together on a single BIOMEMS platform
which would do something useful and important. So, the last part of the course would be probably
dedicated to reviewing some of the papers and articles which have been in this area
where these small fundamental concepts are introduced and integrated together to form
integrated systems which would do total chemical analysis.
Let us actually go ahead and see what happens when the scale changes to a little bit lower
and as I told you before in one of my earlier lectures that continuum assumptions fail beyond
a certain scale. Let us say for example, if you have a small control volume which is about
1 nanometer cube in dimensions and you assume or you try to find out what is the density
or velocity of molecules which are within this 1 nanometer cube.
There is a tendency of the molecules to rush past the boundaries of these very randomly
with time resulting in a change in density with time or may be a change in viscosity
with time. So, all these properties which were assumed to be averaged out properties
as in case of continuum mechanics or continuum fluid mechanics would actually boil down to
a time dependent property when you go to this particular scale. Therefore, whatever the
presumption Navier-Stokes equations made in assuming that the density is constant, the
flow is incompressible or even if the velocity is really only dependent on space and even
if it is dependent on time, it does not change at a certain instant of time.
These assumptions were made in order to derive the Navier-Stokes equations. However, that
is not true at that small scale of 1 nanometer cube or so. Therefore, inter molecular forces
would play a major role there in order to figure out what is going on in terms of interactions
between the molecules etcetera. Intermolecular Forces.
. What are Intermolecular Forces? If you look
at the different states of matter you can really categorize matter into solids, liquids
and gases. These are all based on the interaction among the different states depending on the
forces between molecules of these particular states which comprise the matter.
These interaction forces can be very well explained by a model of potential energy of
two such molecular systems which are close by at a distance r and this is also known
as the Lennard Jones potential model. It is a non-dimensional equation and we have scaled
down factors there. You have a skill factor for distance between let us say 2 molecular
systems and so without going into the details of derivation of such an equation I would
just like to illustrate because most of the stimulation work which is at this nanometrics
scale as far as fluidics is concerned is more using this model of Lennard Jones potential.
Let us say, you have two systems i and j and that means 2 molecular systems of this particular
notation and they are separated by a distance r. So, the potential that would happen between
2 of these i and j systems is equal to 4 epsilon cij r by sigma to the power minus 12 minus
dij r by sigma to the power of minus 6. Let me just go ahead and explain what all
these different terms are. Epsilon is essentially the scaled energy; this is the scaled energy,
the characteristic energy scale we call it, characteristic energy scale. Sigma is the
characteristic distance scale or the length scale; this is the length scale.
By now, you are probably familiar because we did a scaling down in the Navier-Stokes
equation of conservation of momentum. You probably are aware of what these different
scales mean. It is characteristic distance or characteristic energy value at the scale
at which all these different forces or potentials are considered.
r by sigma is a non-dimensional term. Similarly, dij by epsilon again is a non-dimensional
term because it is a comparison of the potential with respect to an energy scale that means
a value of energy at that particular scale in which all these experiments or all these
equations is being used for studying the intermolecular behaviour.
What is more interesting here is that there are 2 different terms and dependencies on
r by sigma. 1 is to the power of minus 12 and another is to the power of minus 6. This
minus 12 term would signify the pairwise repulsion that exists between 2 molecules and this repulsion
as you all know are because of inter-electronic repulsion between the 2 molecular systems.
There are orbitals which contain electrons which has a probability of having an electron
or having bunch of electrons. When 2 of such molecular systems come close
by, there is repulsion between the electrons on both these systems which would cause them
to have a positive potential. It takes some work for getting them any closer than the
characteristic distance r in which they are placed. So, that is what the r minus 12 dependence
would signify and then there is r minus 6 dependence which is a mildly attractive potential
and this is due to Van der waals interaction forces.
So, what are really Van der waals forces? They are basically attractions formulated
between 2 such molecular systems between the electrons on one and the nucleus on the other
and similarly, the electrons on the other and the nucleus on the first one. That is
what Van der waals interaction forces are and they are principally attractive in nature.
However, they are very weak forces because the attraction between the nucleus on the
first and the electron on the second has to go through an electronic layer. I mean it
has to shield by electronic layer it is shielded by electronic layer which is in between. Therefore,
r by sigma to the power minus 6 dependency is a pretty much weak force in comparison
to the minus 12 dependency, which is a strong force.
Therefore, we can easily say that the potential is contributed by the pairwise repulsion and
the pairwise attraction given by these 2 terms here - minus 12 and minus 6. Also what is
interesting for me to tell you is that as you know force between these 2 systems i and
j is also represented by a negative gradient of the potential between the 2 systems.
We can calculate Fij the force as minus dVij by dr where r is the intermolecular distance
between these 2 molecular systems. Fij(r) essentially is minus dVij(r) by dr and if
I actually differentiate this Vij term here with respect to r, we get 46 epsilon by sigma
cij r by sigma to the power of minus 13 which is a pairwise repulsion force probably and
minus dij by 2 r by sigma to the power of minus 7 which is the pairwise Van der waals
attraction force. In this particular scale, I can also assume
that the corresponding timescale would be represented by this quantity here sigma root
over m by epsilon, m by epsilon again is mass per unit energy which has the dimensions of
square of velocity and a typical characteristic velocity at this scale could be represented
by the mass at this scale divided by the energy at this particular scale.
Therefore, under root the mass per unit energy gives you the velocity scale and dividing
the distance sigma by the velocity scale would give you an indication of the timescale - what
kind of times or characteristic times would exist in this particular scale.
What is also interesting here is that if you really plot the dimensionless parameters V(r)
by 4 epsilon and F(r) by 48 epsilon times sigma which is, if you just see back in this
equation nothing, but this whole dimensional term in one case, the force divided by 48
epsilon sigma and this whole dimensional term on another case which is Vij, the potential
function divided by 4 epsilon. These are 2 non-dimensional terms and what
is interesting here to find out is that there are 2 curves which come out. One is for the
potential function and of course, these dimensionless numbers are plotted with respect to the separation
distance r by sigma. So, what we find out here is that the behaviour in both the cases
are pretty much similar except the fact that the force value has a more deeper well in
comparison to as of the potential value has much more deeper well in comparison to the
force value here. What is also interesting here is to note that
beyond a certain distance, let us say this particular distance where it really has the
peak point, the forces try to gain prominence of the Van der waals attraction. So, this
region here is the repulsive force. If you have to see the dimensionless quantity in
this particular example F(r) sigma by 48 epsilon is continuously going down which means that
if you look at the dimensionless quantity here which is let us say cij r by sigma to
the power of minus 13 minus dij by 2 r by sigma to the power of minus 7. This going
down would signify that this term here, the second term here is starting to dominate and
which means that this is the pairwise repulsion this is the pairwise attraction - the Van
der waals attraction. So, the Van der waals attraction is continuously
dominating or it is trying to dominate, but it is being superposed by a higher value of
r by sigma when it is below the certain separation distance. After it crosses certain characteristic
separation distance, this component here really predominates which is actually this point
here, where the pairwise attraction becomes more than the pairwise repulsion after which
again there is a slow prominence or a slow gain of the first term where it asymptotically
goes all the way to 0. If you draw the potential function, it also
shows similar behaviour and the point where the forces have really changed from being
perfectly repulsive to slightly attractive here. The potential function is actually dipping
down to its minima at that particular point where there are probably close to 0 forces.
There are no forces of any attraction or repulsion and the attraction, repulsion is kind of balanced
each other and both the terms in the dimensionless quantity are equal to each other.
So, if you really look at these Lennard Jones functions in the case of diatomic gases particularly
or in case of certain gaseous states, the values of various parameters available for
a limited number of molecules have only been studied and summarized and specially they
have been done so more for gases wherein you can find out the characteristic energy scale
per unit, Boltzmann constant and also the characteristics distance scale. Mind you,
there are only a few systems in the world which have been really modelled and studied
for finding out these different energy scales or distance scales etcetera.
Some of them are illustrated here in the table like for example, in case of air, the energy
scale would be typically energy scale per unit Boltzmann constant k would typically
be equal to about 97 kelvin, for N2 nitrogen it would be about 91.5, CO2 it is 190, oxygen
it is 113, Air 124, so and so forth and in terms of distance scale in nanometer range,
the air molecules have probably a characteristic distance of 0.362 which goes all the way to
about 0.342 for Argon. These have been of course, calculated by assuming
cij and dij parameters to be both unity. That is how these values have been really calculated.
The timescale for these kinds of energy and distance scales come out to be about 2.2 picoseconds.
These are some of the characteristic dimensions which are available at the domestic scale
that we are talking about. The plot again here as you are seeing here has been shown
only for these few systems. What is interesting here what I did not discuss before is that
it is basically the potential energy scaled by a factor of 1 by 4 epsilon and the force
scaled by sigma by 4 epsilon. The effects of these constants in the table
above here really is to not change the basic characteristic, but to actually shift it by
adding a dc bias. So, the whole curve can shift up and down if these characteristic
parameters epsilon by k and sigma would change without really changing the behaviour or the
trend that the dimensionless number would follow with respect to r by sigma. So, that
is all about Lennard Jones potential models.
If you look at the states of matters, what are the differences in terms of forces? In
a solid, we can assume all the molecules being densely packed and held by strong repulsive
forces. The pairwise repulsion is very high in some solids and as a result of which putting
a particular atom from its position out of that is highly difficult because it is being
repelled from all these sides by equal amount of high repulsive forces. So, it remains in
its state very firmly bound in where it is without really being able to get liberated
from that particular state. So, moving an atom from one neighbourhood
to another becomes an immense problem. For a molecule to kind of leave its neighbourhood
and join another neighbourhood you would need to provide energy to the system and this energy
would be in terms of thermal energy. If you have bond vibrations which are happening between
these molecular system, after a while the kinetic energy of the molecules would be sufficient
enough for it to leave one neighbourhood to another and that is also known as the melting
temperature or the melting point of the particular solid where the molecules are just about capable
to move between neighbourhoods and it has sufficient amount of kinetic energy for doing
that. Beyond the melting temperature of course,
the average molecular thermal energy becomes high enough so that the molecules are able
to vibrate freely, go over large distances and you know between one neighbourhood to
another. If the distance that they are traversing of distance scale that they are traversing
is till stigma they are also called liquid state material.
But then if the distance becomes almost an order of magnitude more like about 10 sigma
or so, they are known as gaseous states. In that kind of a situation where the inter molecular
distances are more than about 10 sigma, sigma being the distance scale in the solid state,
these are essentially then known as the boiling temperature and the species are known as gaseous
species. In a nutshell, there are 3 different configurations
to look at. Solid state, the atoms are firmly bound, there are huge amount of repulsive
forces to an atom in its pocket from all its neighbours and movement between neighbourhoods
becomes an impossibility. In a liquid state just about when kinetic energy is applied
and it has been able to lift to a state where the atom can from move from one neighbourhood
to another; the distances moved is still about same range sigma, but the movement just becomes
possible is known as liquid state or the melting point of the material. At a point when this
sigma increases to about 10 sigma or so and there is total decontrol on the molecule from
its current location, it reaches the boiling point, it is known as a gas phase.
Therefore with this kind of a concept in mind, let us actually look at what happens when
the continuum fails. We would like to do this - the molecular approaches to estimate flows
particularly when there is failure of the continuum approach and so the only other approach
which probably emerges for solving liquid flows is called molecular dynamic simulation
which talks about all these inter molecular forces as I have been telling before - inter
molecular forces and which can be established from the Lennard Jones potential model.
The MD technique is in principle very straightforward and is an application of this Newton's second
law. What is Newton's second law? The product of mass and acceleration is equal to the total
amount of force that a particular system should have. The same can be translated into molecular
systems as well. However, we have to somehow ascertain how
to find the acceleration of this kind of a molecular system and we already know that
from Lennard Jones potential. We can find out the average force between 2 molecular
systems i and j separated by a distance of r. That is what we are doing here, the product
of mass and acceleration of each particle is equated to the sum total of forces because
of Lennard Jones equations and the technique really begins with collection of molecules
in space. Each molecule has a random velocity assigned to it as per the Boltzmann velocity
distribution and the molecular velocities are integrated forward in time to arrive at
the new molecular positions. If you see the Intermolecular Forces really
which are equated to mi d2 ri by dt2 at time t, let us say time instance t is equal to
minus d by dr, a negative gradient of potential, Lennard Jones potential. sigma i not equal
to j and these are that systems all systems of the likes i and all systems of the likes
j that 2 molecule systems between which there is a force of attraction and repulsion.
So, it is the negative gradient with respect to r of Vij and here the assumptions that
we use to simplify is that if you consider a molecular system, it is a huge amount or
huge number of molecules. 1 mole as you know is about 10 to the power of 23 molecules.
Therefore, if suppose you are talking about a certain radius or if you are talking about
a molecule system where there are 1000s of these molecules which are inter playing and
there are inter molecular forces between these 1000s of molecules, there would be a tendency
of the computations to go extensively at a very high rate and at sometimes so essentially
the whole idea that you know in a molecular system like this.
If you have 1 molecule surrounded by 1000s or actually millions of molecules, there are
intermolecular forces between this 1 molecule of interest and its neighbourhood which comprises
of millions of such forces and it makes the whole system computationally extensive and
the solutions may not converge with time and there may be issues related to more number
of computations. To solve this problem, you really need to
consider an effective radius within which you can consider the interactions. It is called
the cut off rate radius. If radius is exceeded or the radius goes beyond the cut off radius,
we assume that the molecular forces arising out of molecules which are outside this cut
off radius is 0. You terminate the calculations at a certain intermolecular distance beyond
which we assume that the distances are too large for the forces to be really effective
in the overall calculation. Therefore, this Lennard Jones function really
changes into Vij ri minus rj, vector rj is essentially the vector radius where there
would be a forced cut off. This is the cutting radius or cut-off radius. Any point below
this would be effectively considered in the calculation is here. It is kind of a truncated
solution. Truncated solution of the force between the radius with cut off radius of
let us say rj in this particular case is given by minus d by dr of sigma i not equal to j,
that means these are 2 different molecular systems, Vij ri minus rj mod.
This is equated to by the Newton's second law mi del2 ri vector by dt2. This is the
acceleration at time t and the forward integration of this twice successively would give you
the new position vector ri from which you would again recalculate this. Consider it
is a criterion to define new cut off radius. Therefore, from position to position you can
have a very good estimate of the velocity, the position vectors and the acceleration
vectors of these individual particles based on this force, change of momentum approximation.
Therefore, the averages of velocities, let us say, are calculated by if you have a knowledge
of all positions by the forward time integration and you want to find the x velocity. All the
x positions, the d by dxi by dt, you make a sigma of all these and calculate an average
velocity, an average density, an average of other properties, etcetera Mind you because
these positions are changing with time, these quantities would also be changing with time.
Even though continuum assumption is not really validated in this particular case, the averages
keep on varying based on the interactions - the molecular interactions in that small
control volume. Therefore, these are very appropriate, very
accurate technique to simulate the motional properties of fluids at the nanometer scale;
these are called molecular dynamic simulations. In fact, such simulations can be used to study
lot of properties like combustion, fluid mechanics in general, then lot of surface-based interactions
with charged molecules or interactions between different charged molecules in a system, so
on and so forth. They are very useful as a technique to predict the position vectors,
velocity average properties, etcetera time variant essentially.
So, this gives you an estimate of how from continuum to a scale where continuum does
not hold you could to change your approach to get the different solutions. Now, I would
like to go more towards the application side and describe these different kind applications
of these micro scale fluidic techniques into realizing engineering products like Microvalves,
Micromixers, Micropumps etcetera.
Let us study these one by one. I would like to first start by looking at Micromixers.
If you look at really the characterization of micro flows, the first property which comes
out as we have seen earlier is the low value of Reynolds number. Of course, it is the ratio
between inertial and viscous forces again. You can mathematically define it as rho vd
by mu, rho is the density, v is velocity, d is the dimension at that particular scale
and mu is the viscosity of the medium. The way you calculate d in Reynolds number
is essentially four times area by parameter and it is also known as the hydraulic diameter
of particular section. Based on the different ranges that this Reynolds number kind of comes
into ranges of values you could categorize flows into many types.
This table here illustrates the corresponding range of Reynolds numbers with the description
of the flow. You are looking Re between 0 and 1 which is mostly the case in the microfluidics.
The flows are highly viscous, they are highly laminar and they are as if they execute creeping
motion as they are moving in a very tightly packed, close pack channel or creeping in
a closed pack channel without really interfering much into each other's path; the molecules
move in perfect streamlines parallel to each other.
For Reynolds number of 1 to 100, the flow still remains very laminar and there is of
course, a very strong dependency on the Reynolds number in this particular scale. Suddenly,
the flow properties may demonstrate or change in behaviour with very minute change in Reynolds
number. If you go slightly above and consider the
Reynolds number between 100 and 1000, the flows again are very laminar, they have not
yet changed the genes. However, here in this particular range you could really have a very
good boundary layer which is formulated. If you may recall boundary layer again is the
layer which is separating the fully developed flow from the flow which is made up of these;
the flow which is actually having shear stress or predominated by shear. Therefore, this
boundary between the fully developed part and the sheared part of the flow is what laminar
boundary layer rapidly is and this boundary layer formulates whenever very close to a
fixed surface maybe a channel or some kind of flat plate or flow over a surface.
So, essentially this Reynolds number range of 100 to 1000 would inculcate a prominent
boundary layer and here most of the theorizing of boundary layer is very useful. If you change
the Reynolds number a little bit from 1000 plus all the way to about 10000, there is
a slow transition which takes place in the flow from laminar to turbulence, but then
this is really a transition. You have instances where the flow is slightly laminar or instances
where the flow is turbulent and is slowly changing the behaviour. After a certain value
let us say 10 to the power 4 or so on up to about 10 to the power 6, the flow becomes
fully turbulent. Of course, in this particular range there
is some Reynolds dependence which happens particularly from 10 to the power 4 to 10
to the power 5 Reynolds number. Beyond which all the way up to infinity not only flow is
turbulent, but it is having very less dependency on the Reynolds number and that is probably
because the flow really gets defined by the local eddies and vortices more than the overall
bulk flow. That is essentially what these different categories of flows are with respect
to different values of the Reynolds numbers. What I would like to reiterate here is that
as I told you before in the 1 to 100 case or even less than one case as is mostly the
case in all the microfluidics devices, the mixing behaviour or the mass transport between
the mixing inter layers and typically is not really dependent on the momentum of the flows.
There are no eddies or no vortices which would cause mass transport between the 2 mixing
inter layers just by virtue of motion.
You need something else to drive the flow up and down a particular direction and what
could be more appropriate than a concentration gradient. Therefore, the diffusion approximation
comes into picture. Most of the micro scale flow mixing takes place probably by diffusion.
If you look at the diffusion equations really, the diffusion flux because of the existence
of this concentration gradient here dc by dx, they are proportionate to each other.
Diffusion flux is proportional to the negative gradient of concentration with respect to
the perpendicular distance. The proportionality constant really is D here,
the diffusion constant which can be varied for different states of the matter. For example,
diffusion constant is very low if you consider solid. You see this scale below here, diffusion
constant is in the range of about 10 to the power minus 10 in case of solids; in case
of polymers and glasses this constant increases a little bit about 10 to power minus 8, but
still is very low. Mostly liquids have a diffusion constant in the range of 10 to the power of
6 to minus 4 in this particular region and therefore, it is of some significance, the
diffusion is of some significance as for as liquid is concerned.
But in case of gases, this constant is very high. It is about 10 to the power minus 2
to 1 centimeter square per second. Therefore, you have to be very careful about the diffusion
as you go to gas flows. However, most microfluidics being single phase and mostly liquid dominated,
the diffusion constants that we will be considering in all our calculations are in the range of
about 10 to the power of minus 6 to minus 4.
The mass transport Mt across the boundary between 2 fluids is also equal to the flux
times of the area of interface between the 2 mixing fluids. You have let us say 2 fluids
running parallel and there interfacial area is A and flux is 5. The amount of mass transport
Mt really is equal to 5 times A, 5 being the mass transport per unit area or the flux of
mass per unit area. Therefore, interfacial area in such cases becomes highly prominent
especially when mixing is concerned. If you have somehow a mechanism wherein you can accommodate
2 flows over a larger amount of area, automatically their mass transport would be more. Sometimes
it becomes architectural promising to design micro-mixers where this area of interface
between the 2 flows is increased resulting in some mass transport.
So, if area is more, then the mass transport Mt is more. Also of special significance is
an equation which I am not going to prove here though, but then I just borrowed it from
normal diffusion kinetics and here it talks about the time of diffusion. The time of diffusion
is essentially equal to d square by 2D, d being the path length over which the 2 flows
are going for which they are diffusing at the end or throughout as they are going with
respect to each other and big D here is the diffusion coefficient.
If you have a length L of channel over which these 2 flows are mixing and going as they
move along the length L is nothing, but the path Ll the path d or path length d. If the
length is more, then in that case the time of diffusion tau is automatically more. This
length is essentially the cross sectional length when we are talking about 2 flows which
are going in a particular channel of the cross sectional length L.
So, it almost makes sense to assume here that the time of diffusion will reduce, if the
d value that is the cross-sectional length value 2 flows is reduced. If I am able to
somehow create shorter laminas of these flows and thus the cross sectional length of the
flow reduces by division of d into let us say n where n in the number of laminas. Then
automatically the time or diffusion reduces as the square of the reduction in the d and
it is very important understanding that one must have in order to design what you call
passive micro-mixers, where the whole idea is how the diffusion length can be split up
into smaller values so that time of diffusion would reduce.
The whole essence of microfluidic structures or microfluidic architectures is how the time,
where these 2 flows have been introduced from one side and side within the chip is going
to be more than the time that the fluidics take to diffuse into each other. At the end
of the day when it emerges out from the other side of the chip they are fully mixed.
So, range d values for different states are indicated here in this particular illustration
and that is a little bit of fundamentals of how mixing would take place at the microscopic
length scale.
What really would be considered when we talk about design of micro-mixers? The number one
problem that we are left with this is a small value of the Reynolds number and therefore,
the diffusive mixing. Design considerations should be definitely based on somehow enable
to promote a diffusion between 2 layers and essentially which means faster mixing time,
general requirements, a small device size and also you should have integration ability
particularly complex systems with the overall setup. Fast mixing time again can be achieved
by, as I told you before, decreasing the path length. The path length means the cross sectional
path, the path that diffusion happens along and is perpendicular to the direction of the
actual flow of these fluids and you can also have more accurate or better mixing and faster
mixing by increasing the interfacial area between the two or more mixing streams.
So, smaller the mixing channel length faster would be the mixing process. However, the
desired high throughput, particulate infield and high driving pressures do not tolerate
too small channels; that is the unfortunate part. Most of the flows in microfluidics as
you know are pressure driven and are therefore, too small channel would have too high resistance
and therefore, the yield at the end of the channel of the flow would be so small that
it is very negligible. So, you have do design something wherein you
can still be able to get substantial amount of throughput, but at the same time the mixing
can be faster. Therefore, lamination is something that comes into picture automatically that
means, you split up let us say 2 streams of 2 different dyes or 2 different colors which
you want to mix into multiple streams stacking with each other. If you have n such splits,
what would happen is that you have one stream with a diffusion length d or cross sectional
length or 2 streams going into a cross sectional length L or d, which is also the diffusion
length and in another instance after n splitting, the cross sectional is simply reduced by d
by n. We are going to investigate this a little
bit further as we do mixer designing. Therefore, the first solution really is to split n sub
streams and rejoining them again in a single stream so that the mixing time reduces by
a factor of n square. This basically is the principle of lamination mixers.
So, this brings us to an end of this particular lecture. We have a take-home message that
mixing essentially is diffusional, it is dependent on the interfacial area, it is dependent on
the diffusion length, time of diffusion essentially has to be kind of lesser than the time of
residence for the flow to be effectively mixing with each other and overall requirements of
the micro-mixers should be it has a faster mixing time, small device size, integration
ability particularly in a higher order complex system. We will consider these aspects in
little more detail in the next lecture. Thank you.