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Revision as of 21:18, 6 January 2009 by Dahlia (Talk | contribs) (Solving the Triangular System Exactly with nearest neighbor Interactions)

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My name is Dahlia and I am a first year grad student in chemistry. My specific program is called chemical physics. I just joined Eugene Shakhnovich's lab, where I plan to pursue theoretical biophysics. Top of Page

Fun facts on soft matter

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Final Project: FRUSTRATED SYSTEMS AND SOFT MATTER

My project concerns the very complex subject of frustration. I hope that I will be able to introduce the topic to you with some very simple spin systems, and then illustrate why this is important to soft matter!

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What is Frustration?

Frustration occurs in a myriad of different physical systems, ranging from polymer glasses, to proteins, to crystals. Although the mathematics and physics of very simple frustrated systems could be shown in various ways, I believe that the clearest way to present it is through the use of magnetic spins. All sorts of hard and soft-condensed matter systems have polarization, whether it be of magnetic spins, electron spin, charge on a molecule, etc. The way spin frustration is dealt with analytically and computationally--on a lattice--is the same technique that is used in many protein folding models, polymer melt models, and really any other system for which you can figure out how to discretize it on a lattice. For the purpose of clarity, here I illustrate some of the simplest techniques with magnetic spin examples. Thus the math we cover here is much more generally applicable than simply to spin systems. It is not difficult to extend the underlying principles of frustrated spin systems to other frustrated systems in soft matter which I will discuss subsequently. I'm assuming that everyone who reads this is familiar with the Ising model, but if you aren't there are lots of great statistical mechanics books that cover it in all its glory.

In its most general definition, when a system is in a frustrated state, it means that the minimum total energy of a system does not correspond to the sum of the minimum of all local interactions. Non-frustrated materials have a ground state that is characterized by a single potential well, which represents the uniform arrangement of perfectly ordered spins. By contrast, frustrated materials have a ground state that is an ‘energy landscape’ with many degenerate ground state configurations, separated by barriers of random height. This leads to low critical temperatures and strong dynamics at low temperatures. A simple example of a frustrated system is the following: In Figure 1, there is a triangular plaquette in which each spin must be opposite to one another. The blue and grey spins satisfy this easily, but the orange spin cannot. If its spin is down then it is not minimizing its interaction with the blue spin. If its spin is up the same happens with relation to the grey spin. The orange spin is thus frustrated. In other words, frustration occurs when a spin or multiple spins cannot find an orientation which fully satisfies all the interactions with its neighboring spins.

Trianglespins.jpg Figure 1

Before getting into more specifics, I first define some key features of frustrated systems. Two spins, Si and Sj with an interaction J have an interaction energy, E = -J (Si•Sj). If J > 0, the interaction is called ferromagnetic. Ferromagnetic materials have a minimum energy of –J, which occurs when all spins are parallel to one another. If J < 0, the interaction is called antiferromagnetic. Antiferromagnetic materials have a minimum energy of J, which occurs when all spins are anti-parallel to one another.

Ferroanti.jpg

The type of frustration that emerges from the competing interactions on triangular, face-centered cubic, and hexagonal-close-packed lattice structures with antiferromagnetic nearest-neighbor interactions is called geometrical frustration. Figure 1 also illustrates geometrical frustration. Later I will discuss an analogous case in soft matter: a self-avoiding, random walking polymer with geometrical frustration.

In addition to geometrical frustration, there is also frustration that occurs due to competing interactions. For example,suppose there is a one dimensional system of spins which have the condition that nearest neighbor (nn) interactions are ferromagnetic and next nearest neighbor (nnn) interactions are anti-ferromagnetic. There is no physical way for all interactions to be minimized in this system, and hence it is frustrated. Below is a picture of this type of frustration:

Competing.jpg

There is also frustration in glassy systems (which we will discuss later in glassy polymers). Top of Page

Solving the Triangular System Exactly with nearest neighbor Interactions

Taking only into account nearest neighbor interactions, the ground state spin state of the triangle in picture A can be solved. First let Si (i=1,2,3) of amplitude S make an angle Ox axis. In this case, taking the energy to be the sum of the spin interactions:
<math>E=J(\mathbf{S_1}\cdot\mathbf{S_2}+\mathbf{S_2}\cdot\mathbf{S_3}+\mathbf{S_3}\cdot\mathbf{S_1})=JS^2\left[\cos(\theta_1-\theta_2+\cos(\theta_2-\theta_3\cos(\theta_3-\theta_1)\right] </math>
<math>\frac{\partial E}{\partial\theta}=-JS^2\left[\sin(\theta_1-\theta_2)-\sin(\theta_3-\theta_1)\right]=0 </math>
<math>\frac{\partial E}{\partial\theta}=-JS^2\left[\sin(\theta_2-\theta_3)-\sin(\theta_1-\theta_2)\right]=0 </math>
<math>\frac{\partial E}{\partial\theta}=-JS^2\left[\sin(\theta_2-\theta_3)-\sin(\theta_1-\theta_2)\right]=0 </math>
<math>\theta_1-\theta_2=\theta_2-\theta_3=\theta_3-\theta_1=\frac{2*\pi}{3}=120^o</math>
From this we can determine the ground state of the antiferromagnetic triangular lattice, depicted in picture below:

Groundstate.jpg

This is probably the simplest example of a frustrated system that can be solved analytically. However, all frustrated systems that are superimposed on a lattice follow a general scheme. Interactions between particles are taken into account by a model such as the Ising models. This allows us to find the ground state configuration of frustrated systems as well as important thermodynamic properties such as the free energy, specific heat, and the critical points. We can also find out where the phase transition (between the disordered and ordered state) of our material will occur. Top of Page


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