George Y. Panasyuk, Bioengineering, University of Pennsylvania

1. Computational physics for advanced undergraduate/graduate students Abstract George Y. Panasyuk, Bioengineering, University of Pennsylvania Nadina Gheorghiu, previous affiliation with Bryn Mawr College In trying to understand how regular structures develop in biological organisms, physics is expected to play an important role in changing biology to a more quantitative and predictive science. Through computational physics and bioengineering, models are compared to reality. For the undergraduate/graduate physics curriculum, laboratory experience is essential. At the same time, computational methods are increasingly needed in order to provide the understanding of experimental data. Too often, computational courses are offered to physics students in mathematics or computer science departments. Physics departments can offer computational physics courses and this should happen more often than it does now. Physics is all about the world around us that we should try to describe in various ways. We present here examples of computational physics approaches for undergraduate/graduate research.

2. Why computational physics? “All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.” James C. Maxwell Here are few examples of what computational physics can do Near-field tomography of a spherical scatterer, G.Y. Panasyuk, V. Markel, S. Carney, and J.C. Schotland, Appl. Phys. Lett. 89, 221116 (2006). Optical diffusion tomography of an absorbing phantom, Z. Wang, G.Y. Panasyuk, V. Markel, and J.C. Schotland, Opt. Lett.30, 3338 (2005). Space physics Liquid crystal display DNA N. Gheorghiu & G.Y. Panasyuk

3. The golden rule: Balancing analytical & computational work No doubt that the advance of computers revolutionized theoretical and computational physics in a life span of only one generation. Despite the fact that the major equations of modern physics, such as the Maxwell equations or Schrödinger equation where discovered many years ago, only a tiny number of their solutions have been obtained by the beginning of 70’s in the last century. Right after graduation of one of the authors of this report (G.Y.P.), only relatively simple, mostly 1-dimensional, problems were available for numerical solution. However, due to a very fast developing computer power and computational algorithms, nowadays we are able to solve the vast majority of realistic 2- or 3-dimensional problems of fundamental and applied physics, including, for example, inverse scattering problems in optical tomography, which characteristic complexity is up to 1012 simple float operations (addition or multiplication). This is why not only the development of some computational skills for physics students is absolutely necessary, but also students’ ability to write scientific programs using a high level computer language has to be as natural as to speak English. Emphasizing the importance of an ability to write state-of-art scientific programs, the authors have no intention to decrease the role of analytical methods, because a lot of problems are not amenable to even the current computer power! Only the right combination of computational and analytical methodstogether with a deep physical insight might bring success, let alone that analytical methods are able to better implement the underlying physics. In addition, teaching the same research topic at various, progressing levels is expected to be useful. N. Gheorghiu & G.Y. Panasyuk

4. Contents To illustrate these general ideas, in this poster we use a simple yet non- trivial example from liquid crystal physics which employs both analytical and computational methods. Our work has two parts that correspond to two levels of difficulty: • First, a rather intuitive approach is described at a level suitable for undergraduate research. Computational approaches and results are presented. • Subsequently, a more advanced analytical approach based on the Maxwell equations is introduced at a level suitable for graduate research. The authors’ intention is to show how a research topic, in this case within the field of computational physics, can be developed throughout all stages of academic instruction in a way that constantly relies on beauty and the excitement of the physics’ laws. N. Gheorghiu & G.Y. Panasyuk

5. Liquid crystal phase of matter n solid nematic liquid crystal ordinary liquid TS - NLC TNLC - I temperature As their name suggests, liquid crystals (LCs), in particular, nematic liquid crystals (NLCs) that we consider here, are intermediate phases between liquids and solids. They can flow like ordinary fluids, yet they also display an orientational long-range order, which is due to the anisotropic, rod-like shape of molecules. This intrinsic anisotropy of a NLC can be described by a unit vector n(r), called the director, aligned along the direction of preferred orientation of the molecules at a space point r. Thus, LCs are not ordinary fluids.The fine balance between rigidity and flexibility makes LCs perfect components of biological organisms. n(r,t) = uniaxial nematic director n2 = 1 Liquid crystals’ anisotropy N. Gheorghiu & G.Y. Panasyuk

6. Liquid crystals' elasticity There are three types of bulk deformations occurring in NLCs: splay, twist, and bend. Assuming no abrupt changes are present in the LC system, deformations can be described within the continuum theory (Frank-Zöcker-Oseen) by the elastic free energy density: where the elastic free energy is determined as N. Gheorghiu & G.Y. Panasyuk

7. Fundamental problems in liquid crystal research • There are two problems that naturally can arise in any LC study. • The first problem is to find the director field n(r) everywhere within the LC cell volume, which can be done by minimizing the free energy. • The second problem is to calculate light transmittance through the LC cell, which can be accomplished, in general, by solving the Maxwell equations. • Generally, these are complicated problems and the LC community spent tens of • years to develop effective methods to solve them. Despite the fact that in the vast • majority of cases only numerical solutions are available, we show here an • interesting yet non-trivial example of analytical solution to the problem of • calculating light transmittance through a LC layer with known, helical, director • structure. The resulting system of differential equations for the electric • components of the electromagnetic wave propagating through the LC layer will be • used to explain two possible numerical methods for solving the problem. By • comparison, the advantage of one of these numerical methods over the other will • be shown. N. Gheorghiu & G.Y. Panasyuk

8. Twisted nematic liquid crystal Monochromatic & unpolarized light Boundary conditions ? Helical director structure (1) Note: One can use the Euler-Lagrange equations for the free energy density to arrive at the same n(z) dependence. N. Gheorghiu & G.Y. Panasyuk

9. Ligth transmittance T Electric field components at the entrance of the (m+1)-th retarder layer (2) n s Divide the LC film into layers and consider each of them as a simple retarder which optical axis rotates from φ = 0 (first director layer) to φ = π/2 (last layer). N. Gheorghiu & G.Y. Panasyuk

10. The electric field in the wave incident on the (m+1)-th retarder layer (3) The electric field in the wave exiting from the (m+1)-th retarder layer (4) N. Gheorghiu & G.Y. Panasyuk

11. Numerical solution 1. Approximations (5) 2. Boundary conditions 3. Analytical solution (6) (7) N. Gheorghiu & G.Y. Panasyuk

12. The Euler and the Gauss-Seidel computational methods for solving differential equations (8) (9) (10) N. Gheorghiu & G.Y. Panasyuk

13. The advantage of the Gauss-Seidel method over the Euler method Despite the fact that the Gauss-Seidel method looks more complicated, it allows to use about 100 times coarser mesh to obtain the accuracy given by the Euler method (FORTRAN programs were used). Transmittance, T g N. Gheorghiu & G.Y. Panasyuk

14. Analytical solution to light propagation based on the Maxwell equations Ampère-Maxwell law (no current sources) (11) Faraday law For LCs (12) (13) LCs’ anisotropy Homogeneity in the xy plane N. Gheorghiu & G.Y. Panasyuk

15. Looking for a solution of (12) as results in: (14) Or, taking into account (1), (15) Introducing new field variables transforms (15) into (16) N. Gheorghiu & G.Y. Panasyuk

16. Choosing (17) one finds from (16): (18) (19) and its non-trivial solution results in a dispersion relation: (20) We are interested only in waves propagating in the z > 0 direction which gives For each and the resulting solution of (16) is N. Gheorghiu & G.Y. Panasyuk

17. (21) which are two elliptically polarized waves propagating in the z > 0 direction. The axes of the ellipse, x’ and y’, are rotated with respect to the xy coordinate system by an angle  = qz, which is exactly the director angle. is the ratio of the major to minor axes Optical domain: Helical pitch: (22) N. Gheorghiu & G.Y. Panasyuk

18. (23) Ordinary wave polarized approximately along the y’ axis, i.e. normal to the director n at any point inside the layer. Its speed is vo=c/no. (24) Extraordinary wave polarized approximately along the x’ axis, i.e. along the director n at any point inside the layer. Its speed is ve=c/ne. N. Gheorghiu & G.Y. Panasyuk

19. Finding again the light transmittance Generally, an arbitrary wave, here propagating along the z > 0 direction, can be written as a superposition of two waves: (25) where: (26) Because the incident wave is polarized along N. Gheorghiu & G.Y. Panasyuk

20. Thus, the x-component of the wave exiting from the LC slab (z = d)can be obtained from Eqs. 26 & 28 as: (29) Light transmittance (30) Note: The asymptotic result (  0) in Eq. 30 is the same as the one obtained from Eq. 7 considered at large values g=1/. For large values of the dimensionless parameter g=2d(n)/λ, the polarization of light propagating through the LC layer follows approximately the director n. the Mauguin limit N. Gheorghiu & G.Y. Panasyuk

21. Conclusions • We have considered light propagation and its transmission through an anisotropic • medium: a twisted nematic liquid crystal. We proposed a relatively simple derivation of • the formula applicable to light transmittance, based on accurate solving the Maxwell • equations preceded by its rather intuitive derivation. The computational part, based on an • intuitive description suitable to undergraduate research, showed an advantage of the • Gauss-Seidel method over the Euler method for the numerical integration of differential • equations. • 2. Subsequent, more elaborate derivations are appropriate to graduate research. The • analytical approach based on the accurate solving of the Maxwell equations is trying to • clarify for the student particular formulas introduced by books that are mostly oriented • on technical applications and frequently cite formulas without any derivations. At the • same time, monographs are usually dealing only with fundamental aspects of light • propagation and do not contain neither formulas for light transmittance through the liquid • crystal slab nor any details of their technical applications. Thus, our approach to a • particular case of the liquid crystal physics bridges the gap between the two. • 3. Computational physics is an opened field. Although here a particular physical system • was considered, the kind of analytical and computational tools presented here can be • adapted to other topics of both research and teaching interests, such as nanoscale • physics and biophysics. N. Gheorghiu & G.Y. Panasyuk