Microfluid Mechanics/Models for Gases and Liquids

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Flowing medium
A fluid is composed of atoms and molecules. Depending on the phase of the fluid (gas,liquid or supercritical), the distance between the molecules shows orders of magnitude difference, being the largest in the gas phase and shortest in the liquid phase. As the distance between the molecules or the mean free path of the flowing medium approaches to the characteristic size of the flow device, the flow can not be treated as continuum.

In a solid, molecules form a regular lattice and oscillate around an equilibrium point. At this state, there is a strong attraction between the molecules and the kinetic energy of the molecules can not overcome this force in this phase of the matter. When enough energy is given to the molecules, e.g. by heating it, the matter melts and consequently becomes a liquid. The molecules gain kinetic energy as a result of added heat and start to move around in an irregular pattern. However, the density of liquids and solids, in other words the mean molecular distances at these two phases do not differ much from each other. When the liquid vaporizes and turns into the gas phase, the density drastically drops as the molecules starts to move freely between the intermolecular collisions.



Gas as a flowing medium
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Macroscopic models for gas flow is better suited when the flow can be accepted as continuum (see continuum mechanics). Gas flows can not be treated as continuum, when their density drops or when they flow through micro or nano channels which have characteristic dimensions at the order of the mean free path of the gas molecules. Therefore, It might be necessary to model gas flows in microdevices in microscopic (molecular) level.

Kinetic theory of gases
Kinetic theory is a molecular model for gas flows. It recognizes the gas as a collection of many discrete molecules and, ideally, provides information on the position, velocity and the state of the molecules. This is done by following each molecule by using Newton's laws of motion and allowing binary collision of molecules with the other molecules and and the bounding walls. Macroscopic properties of gases, like density, pressure and temperature, can be best understood by utilizing this approach.

Let us take a cubic container with a volume of $$\ \displaystyle l^3 $$, which is full of $$\  \displaystyle N $$ simple gas (monatomic) molecules, and derive some of the macroscopic properties. We let the molecules to collide with each other and with the walls of the container. This concept is valid under the condition that collisions can be only of binary type and the molecules do not induce electromagnetic forces on each other. The mathematical expression for this conditions is as follows:


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The number density of the gas molecules in the container is:

$$\ \displaystyle n=\frac{N}{l^3} $$

Hence, the average volume for one molecule in this container is $$\  \displaystyle 1/n $$, so the mean molecular spacing  $$\  \displaystyle \delta $$ reads

$$\ \displaystyle \delta=n^{-1/3} $$

Gas molecules can be modeled by hard spheres, which has the effective molecular diameter $$\ \displaystyle (d) $$. Accordingly, the fraction of volume occupied by a molecule is in the order of $$\ \displaystyle (d/\delta)^3 $$. For very low number densities, mean molecular spacing $$\ \displaystyle (\delta) $$ becomes relatively large compared to the effective diameter of the molecule $$\  \displaystyle (d) $$. Hence, a molecule can freely move outside region of influence of other molecules and almost all collisions of molecules can be accepted to be a binary one or collision with a wall. This situation can be written as:

$$\ \displaystyle \delta>>d $$

and defines the circumstances in a dilute gas.

Density
The mass of each molecule can be calculated from the molar mass of the gas $$\displaystyle M [kg\ mol^{-1}]$$ and the Avogadro constant  $$\displaystyle N_A=6.02214179(30) \times 10^{23} [mol^{-1}]$$

$$\ \displaystyle m=\frac{M}{N_A} $$

Hence, the density of the gas reads:

$$\ \displaystyle \rho=\frac{m N}{l^3} $$

Pressure
Pressure is explained by kinetic theory as arising from the force exerted by fluid molecules impacting on the walls of the container. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the change of the momentum of the molecule ($$\Delta p$$) upon wall collision is:


 * $$\ \displaystyle \Delta p = p_{final,x} - p_{initial,x} = -m u -m u=-2 m u$$

where $$\ u $$ is the x-component of the initial velocity of the molecule.

The molecule impacts one specific side wall once every


 * $$\ \Delta t = \frac{2l}{u}$$

Hence the force exerted on the molecule by the wall is the rate of change of momentum per unit time:


 * $$\ F^* = \frac{\Delta p}{\Delta t} = -\frac{m u^2}{l}.$$

The force exerted by the molecule on the wall is in the negative direction:
 * $$\ F = \frac{m u^2}{l}.$$

Not all the molecules moves with the same speed. Say the speed of the molecules varies over the range $$\ u_{max}$$ and $$\ u_{min}$$, so the number of molecules moving over the range $$\ u+\Delta_u$$ is denoted by $$\ N_u$$. Using this distribution, it is possible to calculate the total force exerted by the molecules on the wall in the x-direction is the sum over all molecules:



\begin{array}{lll} F_{total, x} &= &\Sigma N_u \frac{m u^2}{l}\\ &&\\ &=& \frac{m}{l} N \frac{\Sigma N_u u^2}{N}\\ &&\\ &=& \frac{m}{l} N \overline{u^2} \end{array} $$ where $$\ \overline{u^2}$$ is the average of the square of x-component velocity of the molecules. As the pressure is the force per unit area, the x-component of the pressure reads

P_{x} = \frac{m}{l}N \overline{u^2}\frac{1}{l^2}=\frac{N m}{l^3} \overline{u^2} $$ Other components of the pressures can be similarly derived:



P_{y} = \frac{N m}{l^3} \overline{v^2} $$

P_{z} = \frac{N m}{l^3} \overline{w^2} $$

When the cubic container is small enough, the pressure on the walls of the container can be approximated by the average of them

\begin{array}{lll} P &=& \frac{1}{3}\left( P_x+P_y+P_z\right) \\ &&\\ &=& \frac{N m}{3 l^3}\left( \overline{u^2}+\overline{v^2}+\overline{w^2}\right)\\ &&\\ &=& \frac{N m}{3 l^3}\overline{c^2} \end{array} $$ where $$\ \displaystyle \overline{c^2} = \overline{u^2}+\overline{v^2}+\overline{w^2} $$ is the mean square velocity of the molecules of the simple gas.

Rearranging the terms shows that

Since $$\ N m$$ is the mass and $$\ l^3$$ is the volume, the pressure can also be written

$$\ \begin{array}{lll} P &=&\frac{1}{3} \rho \overline{c^2}\\ &&\\ &=& \frac{2}{3} E \end{array} $$

where $$\ E=\frac{1}{2}\rho \overline{c^2}$$ is the kinetic energy of the molecules per unit volume.

Temperature and kinetic energy
After defining the pressure by the kinetic approach, it is possible to define temperature also by utilizing the ideal gas law, which is:
 * $$\ P = \rho R_{\rm specific}T \ \ \text{or} \ \ P=n k_B T$$

where the specific gas constant is defined as
 * $$\ R_{\rm specific}=\frac{R}{M} \ \ \text{and} \ \ R=N_A k_B

\ \ \rightarrow \ \ R_{\rm specific}=\frac{k_B}{m} $$ in which $$\  R=8.314\,472(15)\ [J\ mol^{-1}\ K^{-1}]$$, $$\  M\ [kg\ mol^{-1}] $$, $$\  m\ [kg] $$ and  $$\displaystyle k_B=1.3806504(24) \times 10^{-23} [J K^{-1}]$$  are the gas constant, the molar mass of the gas molecules, mass of one molecule and  Boltzmann constant, respectively. It should be noted that ideal gas law is valid under ideal conditions, viz. at low pressure and at high temperatures, where $$\ \delta >> d $$, so that the molecular interaction is negligible.

Equating the pressure with the pressure derived by using kinetic approach
 * $$\ \frac{1}{3} \rho \overline{c^2}=\rho R_{\rm specific}T,$$

gives the temperature as:

However, this result is not justified as gas ceases to behave as a perfect gas before reaching to the absolute zero of temperature.

Other forms can be obtained by replacing the specific gas constant with $$\ R_{\rm specific}=\frac{k_B}{m}$$ as follows:


 * $$\ T=\frac{1}{3}\frac{1}{k_B} m \overline{c^2}$$

where $$\ m $$ is the mass of one gas molecule.

Mean free path
The distance traveled by a molecule between two successive collision is called the mean free path. Under the assumptions that the molecules are hard spheres and the probability distribution of the molecular velocities follows that of Maxwell distribution law, the mean free path can be derived as:


 * $$\ \lambda=\frac{1}{\sqrt{2}\pi d^2 n}$$

where $$\ d $$ and $$\ n $$ are the effective diameter of the gas molecule and the number density of the gas in the container.

Molecular speeds and the speed of sound
Although the macroscopic velocity under adiabatic conditions in the container is zero, the molecules are still moving around with the mean square velocity of $$\ \displaystyle  \overline{c^2}$$. The root mean square (rms) velocity of the molecules $$\ \displaystyle  c' $$ gives an idea of the speed of the molecules:

$$\ c'=\sqrt{\overline{c^2}}=\sqrt{\frac{3 P}{\rho}}=\sqrt{3 R_{\rm specific} T} $$

Two more statistically interesting speeds can be extracted from the Maxwell distribution of velocity:



\begin{array}{llll} \overline{c}&=&\sqrt{\frac{8}{\pi} R_{\rm specific} T} &\ \ \text{:average molecular speed}\\ c_p&=&\sqrt{2 R_{\rm specific} T} &\ \ \text{:most probable molecular speed} \end{array} $$

The speeds are ordered as follows:


 * $$ 0.886 \overline{c} = c_p < \overline{c} < c' = 1.085 \overline{c} $$

The speed of sound in an ideal gas is
 * $$\ c_s = \sqrt{\gamma \frac{p}{\rho}} $$

where $$\ \gamma=\frac{C_p}{C_v} $$ is the heat capacity ration and it takes the value 5/3 = 1.6667 for monoatomic and 7/5 = 1.4 for diatomic gas molecules. For ideal gas and air $$\ \gamma $$ is 1.4. Thus, in an ideal gas, the ratio between the speed of sound and rms velocity of the molecules is:

$$\ \frac{c_s}{c'}=\sqrt{\frac{\gamma}{3}} \approx 0.68 $$

which reveals that the speed of sound is slower than the all molecular speeds but it is at the same order of magnitude. This fact is not surprising while the waves are carried by molecular motion.

Viscosity
The viscosity of the fluid is responsible for the momentum transport in molecular level. The dynamic viscosity of an ideal gas, which is derived in the frame of kinetic theory, is:

$$\ \mu=\frac{1}{3}\rho \lambda \overline{c} $$

substituting the $$\ \lambda $$ and $$\ \rho $$ in the above equation results in

$$\ \mu=\frac{1}{3}\frac{m}{\sqrt{2}\pi d^2} \overline{c}=\frac{1}{3}\frac{m}{\sqrt{2}\pi d^2} \sqrt{\frac{8}{\pi} R_{\rm specific} T} $$

which states that the dynamic viscosity is only a function of the mass, the diameter and the average speed of the molecules, which is in turn a function of measured temperature. In other words, it is supposedly independent of pressure and density, which is in contradiction to experimental findings showing that viscosity is pressure dependent. In addition to that, this relation does not hold at very low pressures at which the mean free path reaches to the size of the container or flow apparatus and at very high pressures at which intermolecular forces can not be neglected. Furthermore, viscosity was observed to increase much faster than the $$\ \sqrt{T} $$ proportionality suggests. Nevertheless, this relation holds from a few mbars to several bars of pressure. A more accurate equation had been derived by using the intermolecular forces.

Liquid as a flowing medium
The molecules are too close to each other in a liquid, so that strong intermolecular interactions dominates the motion of the molecules. Moreover, the mean free path of the liquid molecules $$\ (\lambda) $$ are in the order of the mean molecular spacing $$\ (\delta) $$.

Definition of pressure
The pressure on a wall in liquids is a result of molecular collisions on a wall and the gravitation. In microfluidic systems, the effect of gravity is insignificant. However, the molecular collisions can not be easily modeled as was done for dilute gases by kinetic theory, because liquid molecules are densely packed and their motion highly influenced by the intermolecular interactions.

Viscosity
The derivation of viscosity using kinetic approach is not straightforward as the electromagnetic molecular interactions play dominant role in the liquids. One rough approximation was derived by Eyring and coworkers, which is:



\mu \approx \frac{N_A h \rho}{M} \exp\left(3.8 T_B / T\right) $$

where $$\ h $$ and $$\ T_B $$ are the planck constant and the boiling temperature of the liquid under one atmosphere, respectively. This equation, though being only an approximation, gives the observed decrease of viscosity with increasing temperature.