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Lennard-Jones potentialOr why perfume does not spread across your room in 0.02 seconds
Lennard-Jones potentialOr why perfume does not spread across your room in 0.02 seconds
Derivation and explanation of the conceptIt is known that when molecules are infinitely apart, they do not interact with each other.
Graphics[{Circle[{5,0},0.5],Circle[{-5,0},0.5],{Dashed,Line[{{-4.5,0},{4.5,0}}]}}]
However, as the molecules approach closer to each other, they start interacting with each other:
Graphics[Circle[{2,0},0.25],Circle[{-2,0},0.25],{Dashed,Line[{{-1.75,0},{1.75,0}}]},Arrow[{{-1.75,0},{-1,0}}],Arrow[{{1.75,0},{1,0}}],Arrowheads[{-0.04,0.04}], Arrow[{{-2,-1},{2,-1}}],]
Where is attractive force and r is distance between the molecules/atoms
When the molecules approach too close to each other, they start to repel each other - repulsive forces dominate:
F
a
When the molecules approach too close to each other, they start to repel each other - repulsive forces dominate:
Graphics[{Circle[{0.25,0},0.25],Circle[{-0.25,0},0.25],Arrow[{{-0.5,0},{-1,0}}],Arrow[{{0.5,0},{1,0}}]}]
Where is repulsive force between the molecules.
The natural question is how those two forces (attractive and repulsive) complement each other into one uniform system.
It turns out that attractive force can be best described by London dispersion forces: forces, which are induced by instantaneous electron density shift in atoms/molecules, which causes neutral atoms/molecules to have partially electronegative and electropositive side.
This attractive energy can be described as:
F
r
The natural question is how those two forces (attractive and repulsive) complement each other into one uniform system.
It turns out that attractive force can be best described by London dispersion forces: forces, which are induced by instantaneous electron density shift in atoms/molecules, which causes neutral atoms/molecules to have partially electronegative and electropositive side.
This attractive energy can be described as:
U
a
3
2
α
1
α
2
I
1
I
2
I
1
I
2
1
6
r
The function can be simplified to:
U
a
6
σ
r
Here, ϵ is the potential energy, required to infinitely separate two atoms from equilibrium distance
r
0;
σ=
r
0
1/6
2
The attractive force is negligible at great distance, but becomes increasingly more significant as the get closer.
Below is a typical shape of attractive energy potential with increasing distance (Helium atoms are used for the example calculation):
Below is a typical shape of attractive energy potential with increasing distance (Helium atoms are used for the example calculation):
Plot-4*11000*,{r,0,}, AxesLabelStyle["r",Medium,Black], Style["",Medium,Black] , PlotLabelStyle["Potential energy, due to attractive London forces between the molecules at distance r", FontSize13, FontColorBlack ], PlotRange{0,-1000}
6
258.*
-12
10
r
-9
10
U
a
On the other hand, the repulsive force is described by Pauli exclusion principle - no two electrons can occupy the same orbitals in an atom. As the two atoms get too close, their electrons start to share the orbitals and therefore this repulsive force becomes highly dominant. It is described by:
U
r
12
σ
r
Which increases rapidly, as the atoms approach to each other.
Below is the shape of the potential energy due to repulsive forces for He atoms:
Below is the shape of the potential energy due to repulsive forces for He atoms:
Plot4*11000*,{r,0,4*}, AxesLabelStyle["r",Medium,Black], Style["",Medium,Black] , PlotLabelStyle["Potential energy, due to attractive London forces between the molecules at distance r", FontSize13, FontColorBlack ], PlotRange{0,10}
6
258.*
-12
10
r
-9
10
U
a
The total potential energy between the two molecules is:
U
total
12
σ
r
6
σ
r
Therefore, the total function of potential energy between the two atoms/molecules can be viewed:
σ={258.*,342.*,427.*,527.*,412.*,297.*,406.*};(*sigmavaluesofthesubstances*)ϵ={11000,128000,536000,454000,368000,34000,236000};(*epsilonvalues*)textValues={"He","Ar","","","","","Xe"};(*substancesused*)ManipulatePlot4ϵtype*-,{r,0,},(*creatingaplotforLennardJonespotentialfunction,whereσ〚type〛andϵ〚type〛aretheparameters,varyingwiththe"type"ofsubstance:*lookatthebottomofthecode*typevariesfrom1tothelengthofthelengthofσlist,andatthesametimethe"type"valuesfrom1to7areassignedtolookintheform"substance, σ value, ϵ value"*) AxesLabelStyle["r",Medium,Black], Style["",Medium,Black] , PlotLabelStyle["Overall potential energy between the molecules at distance r", FontSize13, FontColorBlack ], PlotRange{Automatic,{ϵ〚type〛/5,-ϵ〚type〛-2000}} , {type,1,"Substance:"},Thread[ Range@Length@σRow/@ Riffle[#,", "]&/@ Transpose[{textValues,"(σ)"σ,"(ϵ)"ϵ}]]
-12
10
-12
10
-12
10
-12
10
-12
10
-12
10
-12
10
Br
2
C
6
H
6
Cl
2
H
2
12
σ〚type〛
r
6
σ〚type〛
r
-9
10
U
a
The visualisation of interaction between the particles
The visualisation of interaction between the particles
You can explore how two Helium particles interact with each other in the following simulation: one particle can be adjusted at a fixed point, while the other interacts with the first one accordingly.
In this way it is made sure that the particles attract each other, however, repulsive force prevents them from getting too close.
Mantas Pastolis
mantas.pastolis.16@ucl.ac.uk
mantas.pastolis.16@ucl.ac.uk