Interaction energy between nonpolar particles

In summary, the conversation discusses an approach for calculating the total potential energy between two particles based on their electric fields and induced dipoles. The resulting potential energy is found to be proportional to the inverse square of the distance between the particles. The conversation also mentions the possibility of considering permanent dipoles in the calculation and the potential impact of dispersion forces.
  • #1
mastrofoffi
51
12
Homework Statement
I want to evaluate the potential energy of a configuration where nonpolar particle 1, of charge ##q## and polarizability ##\alpha_1## is at distance ##r## from nonpolar particle 2, of zero charge and polarizability ##\alpha_2##.

I am neglecting gravitational forces and dispersion forces, net charge can be thought as point-like and polarizabilites are considered scalar quantities. I'll be working in units where ##4\pi\epsilon_0\epsilon=\epsilon## for brevity.

I could not manage to find references to check my work.
Relevant Equations
Electric field generated by a point-charge: ## E_q(r) = \dfrac{q}{\epsilon r^2} ##
Electric field generated by a point-dipole: ## E_\mu(r) = \dfrac{\mu\sqrt{1+3\cos^2\theta}}{\epsilon r^3} ##
Dipole moment induced by electric field: ## \mu = \alpha E##
I tried an approach where I build the interactions one by one and then add them all to find the total potential energy.
The electric field of particle 1 induces a dipole moment on particle 2 given by ## \mu_2 = \alpha_2 E_q(2) ##(by ##E(2)## I mean the field evaluated at the position of particle 2); this induced dipole originates a reaction field which acts on particle 1
$$ E_\mu(1) = -\dfrac{\mu_2\sqrt{1+3\cos^2\theta}}{\epsilon r^3} = -\dfrac{2\mu_2}{\epsilon r^3} = -\dfrac{2\alpha_2E_q(2)}{\epsilon r^3} $$
where I used the fact that ##\theta## must be ##0## or ##\pi## because ##E_q(2)## acts along the line between the particles, hence the dipole is parallel to the field if ##q>0##, antiparallel if ##q<0##; this also explains the minus sign since this interaction will always be attractive.

Now I can evaluate the force that is acting on particle 1 as
$$ F(1) = qE_\mu(1) = -\dfrac{2q\alpha_2E_q(2)}{\epsilon r^3} = -\dfrac{2q^2\alpha_2}{\epsilon^2 r^5}$$
The potential energy of this configuration can then be found as the work needed to bring the charge from infinity up to a distance ##r## from the dipole, which is equal to the work needed for a charge ##q## to create a dipole ##\mu_2##:
$$ v'(r) = -\int_\infty^r \text{d}s F(s) = -\dfrac12\alpha_2 E_\mu^2(1) = -\dfrac{1}{\epsilon^2}\dfrac{\alpha_2 q^2}{2r^4} $$

I am pretty sure this charge-dipole contribution to the potential energy is correct; now is where I start having doubts.
The induced dipole on particle 2 induces in turn a dipole on particle 1, such that ##\mu_1 = \alpha_1E_\mu(1)##; this process should physically occur "simultaneously" to the induction of ##\mu_1## so that every infinitesimal increment ##\text{d}\mu_2## induces an infinitesimal ##\text{d}\mu_1##, but I think the result should not be affected by evaluating the 2 processes one after the other, since I only care for the equilibrium state and not for the transient.
The ##\mu_1## dipole is aligned to the ##\mu_2## dipole, thus the work needed to induce the dipole can be written as
$$ v''(r) = -\int_0^{\mu_1}\text{d}\mu E_\mu(1) = -\dfrac{1}{2}\alpha_1\int_0^{E_\mu(1)} \text{d}E E = -\dfrac{1}{2}\alpha_1 E_\mu^2(1)
= -\dfrac{2\mu_2^2\alpha_1}{\epsilon^2 r^6} $$
since the interaction energy between a permanent dipole and an electric field has the form ##v = -\vec{E}\cdot\vec{\mu}##.

At this point what I have is
$$ v(r) = v'(r) + v''(r) = -\dfrac{1}{\epsilon^2}\left[ \dfrac{\alpha_2 q^2}{2r^4} + \dfrac{2\mu_2^2\alpha_1}{r^6} \right] $$
Now I should be taking into account the fact that the induced dipole on particle 1 alters the field on particle 2, which adds another contribution the induced dipole of particle 2 and so on.. but the spatial decays add up pretty fast: the second term in ##v(r)## is already ##\sim r^{-10}##, so I don't think I want to go further unless, for some reason, it adds essential variations to the result.
Are my approach and result correct?
If not, what is wrong and why?
What if I wanted to consider 2 polar particles instead, i.e., each of them possesses a permanent dipole ##\mu_i^{(0)}##? Will the permanent dipole contributions just add on top on the non-polar interaction(I believe this should work fine)(also reminding that in this case I would have an orientational contribution arising in the polarizibilites), or should I rework it from the beginning(if so, why)?
I'm doing this just 'for fun', no constraints, so feel free to add whatever you think is relevant. Thank you.EDIT: I'm not neglecting dispersion forces because I think they are not relevant, I'm just temporarily pretending they don't exist.
 
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  • #2
I'm aware they exist and that they can be very important in some cases, but I'm not interested in them for now. I'm interested in the electrostatic interaction between two particles.
 

Related to Interaction energy between nonpolar particles

1. What is interaction energy between nonpolar particles?

Interaction energy between nonpolar particles refers to the attractive or repulsive force between two nonpolar molecules or particles. Nonpolar molecules do not have a permanent dipole moment and therefore do not interact through electrostatic forces. Instead, they interact through London dispersion forces, which arise due to temporary fluctuations in electron density within the molecules. This results in a weak, short-range attraction between nonpolar particles.

2. How is the interaction energy between nonpolar particles calculated?

The interaction energy between nonpolar particles is calculated using the Lennard-Jones potential, which is a mathematical model that describes the potential energy between two particles as a function of their distance. It takes into account both the attractive and repulsive forces between the particles and is typically represented by the equation E = 4ε[(σ/r)^12 - (σ/r)^6], where ε is the well depth and σ is the distance at which the potential energy is zero.

3. What factors influence the strength of interaction energy between nonpolar particles?

The strength of interaction energy between nonpolar particles is influenced by several factors, including the size and shape of the particles, the distance between them, and the polarizability of the particles. Larger particles with higher polarizability will have stronger interaction energy, while particles that are farther apart will have weaker interaction energy due to the inverse relationship between distance and force.

4. How does temperature affect the interaction energy between nonpolar particles?

Temperature has a direct effect on the interaction energy between nonpolar particles. As temperature increases, the particles gain more kinetic energy and are able to overcome the weak attractive forces between them. This results in a decrease in the interaction energy and a decrease in the overall attraction between the particles.

5. Why is understanding the interaction energy between nonpolar particles important?

Understanding the interaction energy between nonpolar particles is important in various fields, such as chemistry, biology, and materials science. It helps in predicting the behavior of molecules and particles, as well as the physical and chemical properties of substances. It also plays a crucial role in the formation of molecular structures and the stability of materials, as well as in the development of new drugs and materials with specific properties.

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