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Fun facts on soft matter

Cells diffuse through 3-dimensional extracellular matrices.

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What is Frustration?



Trianglespins.jpg Figure 1







Solving the Triangular System Exactly with nearest neighbor Interactions




Solving frustrated Ising spin systems without disorder.

Systems without disorder have periodic interactions. A few of these systems are exactly solvable in 1 and 2 dimensions. A few such systems are: centered square lattice and generalized versions, the Kagomé lattice, an anisotropic centered honeycomb lattice. Like all frustrated systems, the ground states of these exactly solvable systems have very high, often infinite, degeneracy. Thermal fluctuations can preferentially select certain entropic states to others, thereby lowering levels of degeneracy. Another interesting property of these systems is that they have the property of the coexistence of order and disorder at equilibrium, which means that there will always be some spins that are disordered at all temperatures, even in an ordered phase.

The frustrated lattices can be described by using the 2D Ising model. Mathematically solving the Ising model involves first finding the partition function – a challenging problem. One commonly used method for finding the partition function involves mapping the Ising model onto a 16-vertex model (or a 32-vertext model). In this project I only describe the 16-vertex model and then show an example of how one can map the Ising model onto the 16-vertex model.

To understand the 16-vertex model, first imagine a lattice in which each vertex is connected by an edge (Figure 3). Each edge has two possible states, represented by arrows of different directions (up or down). Each vertex has 16 possible configurations, as shown in picture Figure 4. Each possible configuration has a different associated vertex weight (Boltzmann factor), <math>\omega_k = {\text{exp}}(\beta\epsilon_k)</math>, where <math>\epsilon_k</math> is the energy. The partition function is thus, <math>Z=\sum_c{\text{exp}}\left(-\beta\left[n_1\epsilon_1\right]\right)</math>

The sum is over all allowed configurations C of arrows on the lattice and <math>n_1</math> is the number of vertex arrangements of type j in a given configuration C. Z is a function of the Boltzmann weights: <math>Z=Z\left(\omega_1...\omega_{16}\right)</math>. In order to solve this exactly, the free-fermion condition has to be satisfied. In stipulating the free-fermion condition, the partition function is simplified. To satisfy the free-fermion condition, the following constraints are present,
<math>\omega_1=\omega_2</math>,<math>\omega_3=\omega_4</math>,<math>\omega_5=\omega_6</math>,<math>\omega_7=\omega_8</math>,<math>\omega_9=\omega_10</math>,<math>\omega_11=\omega_12</math>,<math>\omega_13=\omega_14</math>,<math>\omega_15=\omega_16</math> and the free energy is expressed as <math> f = \frac{-1}{4\pi\beta}\int_0^{2\pi}d\phi log(A(\phi)+Q(\phi)^\frac{1}{2})</math>
where: <math> A(\Phi)=a+c\cos(\Phi)</math>
<math> Q(\Phi)=y^2+z^2-x^2 - 2yx\cos(\Phi)+x^2\cos^2(\Phi)</math>
<math> a=.5(\omega_1^2+\omega_3^2+2\omega_1\omega_3+\omega_5^2+\omega_7^2+2\omega_5\omega_7)+2(\omega_9^2+\omega_{13}^2))</math>
<math> c=2[\omega_9(\omega_1+\omega_3)-\omega_{13}(\omega_5+\omega_7)] </math>
<math> z=.5[(\omega_1+\omega_3)^2-(\omega_5+\omega_7)^2]+2(\omega_9^2-\omega_{13}^2)]</math>
<math> x^2=z^2-.25[(\omega_1-\omega_3)^2-(\omega_5-\omega_7)^2]^2</math>

Phase transitions occur when one or more pairs of zeros of the expression <math>Q(\Phi)</math> close in on the real Φ axis and ‘pinch’ the path of integration in the expression of free energy. This occurs when <math>z^2=y^2</math>, or when <math>\omega_1+\omega_3+\omega_5+\omega_7+2\omega_9+2\omega_{13}=2{\text{max}}(\omega_1+\omega_3,\omega_5+\omega_7,2\omega_9,2\omega_{13}) </math>. This type of singularity in specific heat depends on whether <math>(\omega_1-\omega_3)^2-(\omega_5-\omega_7)^2\neq 0</math> (in which case there is a logarithmic singularity or <math>(\omega_1-\omega_3)^2-(\omega_5-\omega_7)^2=0</math> (in which case there is an inverse square root singularity). The mathematics required to figure this out is long and cumbersome and beyond the scope of this project.

As an example of how a 2D Ising model is mapped onto the 16-vertex model, we examine the case of the Kagomé lattice (see Figure 5). An Ising model defined on a Kagomé lattice is shown in Figure 6. In this picture, single lines represent nn interactions <math>(J_1)</math> and double lines represent nnn interactions <math>(J_2)</math>. The hamiltonian is:
<math>H=-J_1\sum_{<ij>}\sigma_1\sigma_2 - J_2\sum_{<<ig>>}\sigma_1\sigma_2</math>
where the first sume is over nn and the second sum is over nnn.

To transform the system into an exactly solvable free-fermion model, one decimates the central spin of each elementary cell of the lattice. The result is the "checkerboard" lattice (Figure 7), which is analogous to the lattice depicted in Figure 3. Each shaded box in the checkerboard lattice has an associated Boltzmann weight <math>W(\omega_1,\omega_2,\omega_3,\omega_4)</math>. The partition function of this checkerboard Ising model is,
<math>Z=\sum_{\sigma}\Pi W(\omega_1,\omega_2,\omega_3,\omega_4)</math>
Where <math>W(\omega_1,\omega_2,\omega_3,\omega_4)=2\cosh(K_1(\omega_1,\omega_2,\omega_3,\omega_4)){\text{exp}}[K_2(\omega_1\omega_4+\omega_2\omega_3)+K_1(\omega_1\omega_2+\omega_3\omega_4)] </math>
< Now impose another square lattice that crosses through the vertices of Figure 7 diagonally (Figure 8). Just as was done for the original 16-vertex model, on each bond there is an associated arrow with two possible directions (this type representing Ising spins). Each vertex has the same 16 possible arrow (spin) configurations as shown before in Figure 4. In sum, starting with an Ising model defined on a Kagomé lattice, we have recovered the 16-vertex model. By satisfying free-fermion conditions specific to the Kagomé lattice, free energy and critical conditions can be obtained, although specifics are again long and cumbersome and will not be written here.

Figure3.jpg Figure3
Figure4.jpg Figure4
Figure5.jpg Figure5
Figure6.jpg Figure6
Figure7.jpg Figure7
Figure8.jpg Figure8

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Complex Phases of Matter: Spin Ice



Water, however, is even more complex than this diagram demonstrates. There is a phase of water, or we could also term it more generally, a state of matter, called spin ice that reflects the frustration and disorder present in ice. Spin ices are frustrated ferromagnets. (How often do you think of ice as a ferromagnet?) In some ways we can think of the "ice-phase" as having a transition within itself in which it crosses the boundary of disorder-order. I am now going to go into more detail this very interesting description of ice. Note that more generally speaking, spin ice refers to the phase of matter that exists for any material made of up of a tetrahedron of ions (like water).


Spinice2.jpg Figure 2



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