M.Sc. Project of Hanif Bayat Movahed

1. M.Sc. Project of Hanif Bayat Movahed The Phase Transitions of Semiflexible Hard Sphere Chain Liquids Supervisor: Prof. Don Sullivan

2. Semiflexible Hard Sphere Model • Molecule : Chain: Hard sphere beads • Self avoiding beads (non intersecting monomers): Just the two adjacent beads can penetrate each other. • Two non adjacent beads cannot intersect or penetrate each other. 4. Defining interaction potential energy between chains by considering just hard-sphere interactions.

3. 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. Smectic Phase transition: Changing temperature or density ISOTROPIC

4. 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

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

6. 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

7. 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

8. 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

9. 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)

10. 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

11. 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).

12. ResultsGraphs (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.

13. ResultsGraphs (S2 , η)III The effects of random parameters and different initial guesses on the final results:

14. 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).

15. HistogramsEffects of number of chains on the histogramsFluctuations of number of chain in each cosθ interval

16. Extension to Smectic-A Phase After solving the isotropic-nematic phase transition we focused on obtaining the smectic phase. The probability distribution function depends on both orientation and position z: A is the excluded area Bifurcation Analysis: Assume the smectic solution is a small perturbation around the nematic solution.

17. 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)

18. Solving the Δf Equation It can proved that where

19. 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.

20. 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.

21. 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.

22. 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