### SimuLab 1: Nanoscale Structures at Phase Transition

 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.

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.

 Q1.1: Are the conditions close to ambient ones?

 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 .

Now we will put the gas into the freezer with temperature T = 0.5 in computer units.

 Q1.3: Is that temperature in real units above or below the temperature of liquid nitrogen?

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.

 Q1.4: Why the Total energy goes down?

 Q1.5: Why the Potential Energy and the pressure goes down?

6. Reset Averaging for Movie. Press Forward and Stop after 20 time units. Record the values of the pressure and temperature.

 Q1.6: Does the Ideal gas law holds at this point?

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.

 Q1.7: Does it changes significantly from the previous measurement?

 Q1.8: Why the potential energy goes down?

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 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

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.

 Q1.9: What happens with the surface tension as we reduce the temperature? Why?

~ 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.

 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.

 Q1.11: What is the potential energy per particle in the triangular lattice?

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.

 Q1.12: How many critical nucleus appear in the system?

~ 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

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.

 Q1.13: Why the temperature goes up with time?

17. Construct the temperature, potential energy, total energy and pressure vs. frame number graphs.

 Q1.14: Explain the tendency of those graphs.

~ 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.) .

#### 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 3.11 10-4 0.412 3.99 10-4 0.34 6.04 10-4 0.318

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.