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M.Sc. Project of Hanif Bayat Movahed

The Phase Transitions of Semiflexible Hard Sphere Chain Liquids

Supervisor: Prof. Don Sullivan

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Semiflexible Hard Sphere Model

1. Molecule : Chain: Hard sphere beads

2. Self avoiding beads (non intersecting monomers):

Just the two adjacent beads can penetrate each other.

3. Two non adjacent beads cannot intersect or penetrate each other.

4. Defining interaction potential energy between chains by considering just hard-sphere interactions.

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Different Phases of Liquid Crystal

• Liquid crystal: 1. Flow like liquids, 2. Light scattering like solids

• Multi chain systems can have three different phases for large enough bending stiffness (ε)

A. Isotropic, B. Nematic, C. SmecticPhase transition: Changing temperature or density

ISOTROPIC

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Energy Terms

Energy Function• One body and two body potential energy terms

One Body Potentials: 1. Short range potential function for bending energy 2. Hard Sphere Interaction (self avoiding)

Two Body Potential: Hard Sphere Potential (Onsager theory) 1. Infinity: If there is an overlap 2. Zero: Otherwise

Density function• The distances between adjacent beads are constant. • r i : Position of one bead , ωj : The orientation of bond j

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Helmholtz Free Energy

• Helmholtz free energy function can be calculated by

• Z is canonical partition function.

• Helmholtz free energy up to the second virial approximation (Onsager theory) for this model is

• fM Mayor function: the interaction between two chains 1 and 2. (Excluded volume)

• ρ0 replaced by ρ

• u1(R) (one body potential energy) includes both the external and intramolecular potential effects.

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Probability Distribution Function

• By considering: 1. Last page equation for Hemholtz free energy, 2. The normalization equation & 3. Minimization of Helmholtz free energy, the following “Self consistent” equation for probability distribution function is obtained:

• The important part: Calculation of the excluded volume between pairs of molecules

• Solving the above equation requires doing the following integral:

where

• Monte Carlo method for solving above integrals for uniform systems

• Barrett’s algorithm for calculation of the excluded volume

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Solving the EquationOverview I

• Uniform system like isotropic and nematic. ω is the set of orientations of the chain

• Unperturbed distribution function (chains are generated

proportional to this weight )

• Full distributed function (perturbed)• Self consistent equation for I ( f(ω) appear in both sides)

• Average with respect to the unperturbed distribution function

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Solving the Equation Overview II

• Finding I(ω) by using a self consistent equation

• Doing averages with respect to ω2 by using the Monte Carlo method

• Stored conformations (calculated just one time) are used for calculating the average values in evaluation of the following equation.

or

• This process is continued until within some tolerance

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ResultsParameters

η(volume fraction)=ρVmol, S2 (order parameter), pressure, Vmol are as follows:

• S2 is zero for isotropic and nonzero for nematic and smectic• Observing isotropic-nematic: calculation of S2 (order parameter) is enough• To observe nematic-smectic: S2 can’t be used- Another order parameter is required. Initial guess for iteration: To force equations to converge to the nematic solution and not trivial solution• The first point is η =1 and then η is decreased. Stiffness: The stiffness for our obtained results is near the rigid limit (βε=50). Number of Beads: 8 beads.Bond length-Diameter ratio: b/D=1(tangent case)

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Resultssome variables effects

Number of chains: Smoothness of the results (can be seen in the next slides)

Initial guess for chain orientation distribution:

• Different initial probability distributions

• The initial configuration should be near the nematic solution.

• Choosing the initial probability density can be interpreted as choosing initial value of I(ω).

Randomness in the chain generation: We tried to produce the same number of chains in different angular ranges by using the histogram technique

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ResultsGraphs (S2, η) I

Effects of number of chains in the results of S2 vs. η. • The jumps represent the phase transition points. • Left: 500 chains & Right: 5000 chains• The (x) line represents results of Jaffer et al. (Analytical approximation for excluded volume for semiflexible case).• Jaffer et al. and our results should be nearly the same at high stiffness (βε=50).

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Results Graphs (S2 , η) II

• The jumps represent the phase transition • Left: 11000 chains & Right: 14000 chains. • The (x) line represent the result of Jaffer et al.

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Results Graphs (S2 , η) III

The effects of random parameters and different initial guesses on the final results:

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ResultsGraphs (P, η)

• The effects of number of chains on results of p* vs. η • The phase transition occurs near the jump where both reduced pressure and the chemical potentials are equal (First order phase transition).

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Histograms Effects of number of chains on the histograms

Fluctuations of number of chain in each cosθ interval

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Extension to Smectic-A Phase

After solving the isotropic-nematic phase transition we focused on obtaining the smectic phase.

Bifurcation Analysis: Assume the smectic solution is a small perturbation around the nematic solution.

The probability distribution function depends on both orientation and position z:

A is the excluded area

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Fourier Series Representation

For arbitrary configuration of ω, Δf is not necessarily an even or odd function of z. In a lowest-order Fourier series representation:

q is 2π/d, where d is the period which should be near

By using z21=z2 - z1

The goal is to calculate φe(ω) and φo(ω)

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Solving the Δf Equation

It can proved that

where

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Smectic-A Final Equations

• After solving the last page coupled self consistent equations φe and φo can be inserted in

• The above equations have trivial solutions of zero (This was the main problem) . • The main goal is to find the proper q=2π/d that converge the above equations to a non trivial solution.

The Equilibrium Definition:

• If those equations reach equilibrium (non-zero constant value) for nearly all of the chains (2 parameters for each chain).• For example:98% reach non trivial equilibrium, 1% reach zero and 1% don’t reach any kind of equilibrium.

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Range of Smectic-A Results I

After plotting all the solutions of η vs. d/Rmean for different number of chains,

we obtained a similar feature to Mulder’s result for the completely aligned limit.

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Range of Smectic-A Results II

The results of the final equations for the normalized probability distribution are valid only

near the smectic-nematic phase transition.

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Main References

• "The nematic-isotropic phase transition in semi flexible fused hard-sphere chain fluids", J. Chem. Phys. 114, 3314 (2001) by K. M. Jaffer, S. B. Opps and D. E. Sullivan.

• “Simple Theories of Complex Fluids” , PhD Theses of Rene Van Roij, FOM-Institute for Atomic and Molecular Physics, Amsterdam, 1996 Based on 6 papers on him

• “The effects of shape on the interaction of colloidal particles”, L. Onsager , Ann. NY Acad. Sci. 51, 627 (1949). • Prof. Sullivan notes for describing the project.• Prof. Nickel notes and his previous program for producing the random semi flexible chains.Pictures: 1. http://www.elis.rug.ac.be/ELISgroups/lcd/lc/iso.gif , 2. chemistry.umeche.maine.edu/ CHY132/P3Q4.html , 3. http://friedel.dur.ac.uk/~dch0mrw/webpages/phases.gif, 4. http://www.favaca.org/img/develop/thank-you.gif