SimuLab 1: Nanoscale Structures at Phase Transition
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You will be able to:
a) Observe the nonideal Gas behavior;
b)Observe condensation as the Gas is cooled down ;
c) Explore phase equilibrium between Gas and Liquid and
study the surface tension;
d)Study supercooled Liquid;
e) Observe the formation of nanoscale structures during the
homogeneous nucleation;
f) Observe polycrystalite growth;
g) Understand the nanoscale structure of polycrystalites.
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1. Open the Universal Application. Press on Play Movie and select StatesofMatter.mov.
Select Averaging for Movie from the Options menu. Click Show Real Units. Press
Forward.
You see a two-dimensional system of 1024 atoms. All atoms are of identical
type A. You can think that they represent Argon. The hard core radius is rA
= 3.3, the distance of attraction is RAA = 12 and the attraction
energy is eAA = -1. The initial temperature of the heath bath is T0 = 2. The heat exchange
coefficient is a = 0.01.
While thermal motion in the gas is really chaotic, several examples may
be noticed here and there when atoms come close to each other and remain together
for some period of time. This indicates that attractive forces are indeed
there and this suggests that condensation is possible upon lowering overall
energy.
2. Watch the movie for 200 computer time units. This correspond to 65 picoseconds
of the real world. Record the values of the Temperature, Pressure, number
of particles and volume of the system from the Averaging for Movie window
and from the Real units Movie window.
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Q1.1: Are the conditions close to ambient ones? |
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Q1.2: Does the Gas obey the Ideal Gas equation: P
V = kb N T. Make calculation first in computer units
assuming kB = 1, and then for the real units , kB
= 1.38 ·10-23 J/K . |
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Now we will put the gas into the freezer with temperature T = 0.5 in computer
units.
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Q1.3: Is that temperature in real units above or
below the temperature of liquid nitrogen? |
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3. Reset the averaging window. Watch the temperature graph and the behavior
of the system. Press Forward.
4. After 200 time units press Stop. Record the temperature.
As the temperature goes down the gas start to condensate: large voids of low
density phase appear together with irregular patches of a dense phase. We
are now close to the critical point. For temperatures below the critical point
the liquid can coexist with the gas.
5. Switch the graph to Total energy, Potential Energy and Pressure.
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Q1.4: Why the Total energy goes down? |
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Q1.5: Why the Potential Energy and the pressure goes
down? |
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6. Reset Averaging for Movie. Press Forward and Stop after 20
time units. Record the values of the pressure and temperature.
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Q1.6: Does the Ideal gas law holds at this point?
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7. Switch the graph to temperature. Press Forward and watch the movie
till time 1200 units or 240 frames. Press Stop.
~The temperature almost reaches the thermostat value 0.5. You can see now
well defined phases: high density liquid and low density gas. Count how many
molecules are in the gaseous phase.
8. Switch the graph to Total Energy and Potential Energy.
~ Note that temperature almost reach the equilibrium one, the potential energy
still goes down. While the gas condenses into liquid the latent heat is taken
out.
9. Switch the graph to temperature vs. time. Press Forward and watch
the movie till frame 700 or 3500 time units.
~ Now the temperature completely reach the equilibrium, while the Total and
Potential energies keep going down. Count the number of molecules in the gaseous
phase.
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Q1.7: Does it changes significantly from the previous
measurement? |
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Q1.8: Why the potential energy goes down? |
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10. Reset Averages and average for 500 time units till time 4000 or 800 frames.
~ The pressure is negative (-2 ·10-4). How can we explain this? The pressure created by the gas, that still
remain in the container is positive and very small. The number of molecules
in the gas is Ng » 8 and the gas occupies the volume Vg which is approximately half
of the volume of the container Vg = 1/2 V. We can find the pressure
of Gas by applying Ideal gas Law
| Pg = |
Ng
Vg
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kb T » 0.5 ·10-4 |
. Thus, there is some extra force that acts on the walls of the container inwards.
This extra force must compensate the small positive pressure of gas and creates
a total negative pressure of 2.5 ¼10-4. This negative force is created by the two vertical surfaces of a liquid
band which tries to contract. The liquid tries to minimize its surface in order
to minimize its free energy. The force created by this effect is called surface
tension. How can we find this surface tension? The total force that acts on
the surface of the box inwards is 4 l¼P, where l = ÖV = 384 is the linear dimension of the box,
half of this force acts on the top wall and half on the bottom. Since the force
is created by the two surfaces of the liquid fs = l P = 384 ¼- 2 ¼10-4 = -0.09210 in computer units. Convert this number to Newtons.
11. Reset Averages. Press Forward. Now the temperature of the heat
bath is 0.3. Find the temperature of the system in Kelvins.
~ Copy the table below.
| Pressure |
Surface Tension |
Temperature |
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12. Watch the temperature graph for 500 units (100 frames). Press Pause
and record the values of the pressure and the temperature from the averaging
window. Press Reset Averages and press Forward.
13. Repeat the previous step for another 2 intervals of 100 units until time
5500 (frame 1100). Using the values of the pressure make a graph of the surface
tension vs. temperature.
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Q1.9: What happens with the surface tension as we
reduce the temperature? Why? |
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~ Watch the graph of the potential energy. The potential energy stops to decrease
and the temperature almost reach the temperature of the heat bath. Still the
system is disordered (there are no signal of crystallization yet). If you
look closely at the liquid you can notice that there are different arrangements
of particles. At some different places the particles form pentagons, small
patches of square lattice and dense patches of triangular lattice (See Fig. 1).
The size's arrangement are of the order of the nanometers. These different
patches have different potential energy depending on the number of neighbors
in each arrangement and different degree of disorder which can be quantified
by an entropy S. Different patches compete with each other in terms of free
energy. At low temperatures the term T S becomes very small, so the patch
with the lowest potential energy will win. The triangular patch as the maximal
number of neighbors in its attractive shell.
Figure 2.1: Picture of the system undergoing a phase transition to crystal.
At some different places the particles form pentagons, small patches of
square lattice and dense patches of triangular lattice. The different
patches are bordered by white.
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Q1.10: How many neighbors are in the attractive shell
of the densely patches disks of hard core repulsion 6.6 within theinteraction distance 12.
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Q1.11: What is the potential energy per particle
in the triangular lattice? |
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14. Find different triangular arrangement and watch their dynamic for another
100 frames. Keep measuring the average temperature from the averaging window.
Do not forget to reset averages before pressing forward.
~ Some triangular patches appear at different places. Some patches, as the
one on the top left of the liquid band reach a significant size. It is called
critical crystalline nucleus. As soon as the nucleus reaches a critical size
it cannot disappear and will only grow (For further discussion see Sect. 2.1.1).
15. Watch the movie for another 100 frames.
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Q1.12: How many critical nucleus appear in the system?
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~ The critical nucleus appear all over the places. This phenomena is called
homogeneous nucleation. Notice that the value of the temperature at which
that happens is T = 0.3 (see the averaging window).
~ Copy the table below:
| Frame number |
Temperature |
Potential Energy |
Total energy |
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16. Watch the movie till frame 1700 and record each 100 frames the values
of the temperature, potential and total energies as function of the frame
number.
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Q1.13: Why the temperature goes up with time? |
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17. Construct the temperature, potential energy, total energy and pressure
vs. frame number graphs.
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Q1.14: Explain the tendency of those graphs. |
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~ The system crystallizes into several distinct crystallites (4) with many
defects. This kind of substance is called polycrystalline. It is a characteristic
of the fast crystalline growth from a supercooled liquid far from equilibrium.
The size of those crystallites are of the order of several nanometers. The
nanoscale structure of the substance is crucial in nano-technology. That is
why it is extremely important to understand the effects of crystallization
conditions on the structure of the resulting solid state.
Figure 2.2: The system crystallizes into 4 distinct crystallites with many
defects. This solid is called polycristaline
Post-Lab In-Depth Discussions
As the surface of the crystalline nuclei increases, the rate of crystallization
increases as well. Since we still taking away heat from the system the total
energy decreases. The rate of cooling is proportional to the temperatures difference
between the system and the heat bath. As temperature increases, the crystallization
rate decreases and the heat exchange rate increases. At certain temperature
these two rates equilibrate each other and the system reaches the steady state.
At temperature T = 0.34, which is above the homogeneous nucleation temperature
of 0.31, and thus the liquid and crystal phases are still far from equilibrium.
The crystalline phase grows very fast. This means that the liquid at this temperature
is metastable. It is called supercooled liquid. We were able to supercool liquid
well below the equilibrium freezing point, when the crystal and the liquid are
at equilibrium, i.e. neither liquid or crystal are growing. However when
we cool the system to 0.31 the homogeneous crystallization happens (in water
that happens at -40 0 C. This is the lowest possible temperature that one can achieve
experimentally for a liquid.) .
Answers to the question
Simulab 1
A1.1: No, the temperature is about 2 C and the pressure 176 MPA » 1750 Atm.
A1.2: P V = 2149 while kb N T = 2048. Thus the gas is not ideal,
the deviation from ideal gas is 5%. Pressure is greater than one would expected
from Ideal gas law by 5%.
A1.3: It is below liquid nitrogen. The temperature of liquid nitrogen is 60
K.
A1.4: The Total Energy goes down because as we put our system in the freezer
the system gives away its heat to the freezer.
A1.5: The atoms attract each other and spend most of the time within the radius
of attraction of each other. The potential energy of each pair is negative.
As more and more atoms come within the radius of attraction, the potential energy
decreases. The pressure goes down due to two factors. Firstly, the temperature
goes down thus the particles collide with the walls with less speed. Secondly,
the particles attract to each other, thus, an extra force towards the bulk of
the container on the particles near the walls acts, thus reducing the pressure
on the walls.
A1.6: No it does not old.
A1.7: No, it is 8 molecules, previously it was 12.
A1.8: It is due to the surface tension. Atoms at the surface of the liquid have
few neighbors than in the bulk of the liquid and thus have a higher potential
energy. At thermal equilibrium the system tries to minimize its free energy
which at low temperature is almost equal to the potential energy. Thus it tries
to minimize its surface. This phenomenon is called surface tension.
Table 1
| Pressure |
Temperature |
Surface Tension |
| 2.00 10-4 |
0.5 |
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| 3.11 10-4 |
0.412 |
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| 3.99 10-4 |
0.34 |
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| 6.04 10-4 |
0.318 |
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A1.9: The surface tension is surface free energy (F = U -T S) per unit area. As we reduce the temperature the free energy decreases.
A1.10: 12
A1.11: -6
A1.12: 5
A1.13: The temperature goes up because the potential energy of interatomic interaction
decreases and is converted into kinetic energy of chaotic motion. The amount
of energy released during the crystallization is called latent heat.
A1.14: At time 8500 the temperature reaches its maximum because at this point
the crystalline surface is maximal, so is the crystallization rate (See "In-Depth Discussion").
As the amount of liquid phase decreases, the crystallization rate decreases
and the temperature finally reaches the temperature of the thermal bath, when
crystallization completely seizes.