Anlytical model of colloidal particles in external periodic potentials

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(1)Analytical Model of Colloidal Particles in External Periodic Potentials. by. Juan Andrés León Gómez, BS. Thesis Presented to the Physics Department Universidad de los Andes in Partial Fulfillment of the Requirements for the Degree of. MAGISTER EN CIENCIAS - FISICA. Universidad de los Andes May 2005.

(2) Analytical Model of Colloidal Particles in External Periodic Potentials. Juan Andrés León Gómez, MS Universidad de los Andes, 2005. Supervisor: Gabriel Téllez Acosta. ABSTRACT A system of colloidal particles under the influence of periodic potentials is modelled as a one-component plasma to yield analytical results. The cases of one- and twodirectional periodicity of the potential are developed. In the latter case potential minima cells are present and the instances of one and two particles per cell are solved. RESUMEN Se modela un sistema de partı́culas coloidales bajo potenciales perioódicos como plasma de un componente para obtener resultados anaı́ticos. Se desarrollan los casos de periodicidad del potencial en una y dos direcciones. En el segundo caso. ii.

(3) hay celdas de mı́nimos de potencial y se solucionan las instancias de una y dos parı́culas por celda.. iii.

(4) Acknowledgments I would like to thank my supervisor Gabriel Téllez for both the brilliant insight in identifying a great Master’s Thesis subject, and all the help he provided me along the way, while at the same time allowing me freedom for independent work. I am also grateful with the Physics Department at Universidad de los Andes for the courses and financial support indispensable to complete this degree, and particularily Dr. Bernardo Gómez wo has always believed in me. I finally wish to thank the Drs. Plamen Nechev and Emmanuel Trizac who formed the evaluation committee.. The work that resulted in this document was partially funded by Colciencias and Fondo de Investigaciones, Facultad de Ciencias, Universidad de los Andes.. Juan Andrés León Gómez. Universidad de los Andes May 2005. iv.

(5) Contents Abstract. ii. Acknowledgments. iv. Chapter 1 Introduction. 1. 1.1. Colloidal Particle Systems . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. Description of the Problem . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.3. A Brief Introduction to the Proposed Model . . . . . . . . . . . . . .. 7. Chapter 2 The One-Component Plasma Framework. 9. 2.1. General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2. The Projector Method . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.3. Choice of Periodic Potential . . . . . . . . . . . . . . . . . . . . . . .. 14. Chapter 3 Potential in one direction only. 17. 3.1. The Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.2. Particle Densities and Correlations . . . . . . . . . . . . . . . . . . .. 18. 3.3. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. Chapter 4 Potential in two directions. 21. 4.1. Oblique Coordinate Generalization . . . . . . . . . . . . . . . . . . .. 22. 4.2. Orthogonalization Preliminaries . . . . . . . . . . . . . . . . . . . . .. 24. v.

(6) 4.3. The Almost Mathieu Equation . . . . . . . . . . . . . . . . . . . . .. 25. 4.4. Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . .. 27. 4.5. Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.5.1. One Particle per Cell . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.5.2. Two Particles per Cell . . . . . . . . . . . . . . . . . . . . . .. 31. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 4.6. Chapter 5 Conclusions. 39. Bibliography. 46. vi.

(7) Chapter 1. Introduction There has recently been a renewal of interest for the Soft Condensed Matter physicists in the problems of colloidal particles under the influence of external forces. With the the arrival of new experimental techniques, it has become possible to directly (i.e. optically) observe the behaviour of these multi-particle systems in a way as yet impossible for smaller scale bodies. This presents a unique opportunity, not just for the testing of general multiple-particle theories, but as a source of new, theoretically unknown phenomena. As we shall see in this chapter, this is a field where experiment leads theory, and the purpose of this work is no exception. After a review of the experimental results that have arisen interest in the problem of colloidal particles under the influence of external periodic potentials, we will propose a theoretical framework originally developed for very different purpose but quite similar in essence to the one of interest here. Chapter 2 will introduce our proposed model, while in chapters 3 and 4 it will be implemented to two particular cases of experimental interest: periodic potential wells in one and two directions.. 1.

(8) 1.1. Colloidal Particle Systems. Colloidal systems for the purposes of this work and most which it is based on, consist of particles of mesoscopic (µm) size suspended in a fluid. Under these conditions, stability of the system requires the presence of stabilizing charges, the liquid phase consisting of an electrolytic solution. We will concentrate on the timescale of the colloidal particles, which is much slower than that of the individual molecules of the solution, and describe the behaviour of these as a multiple same sized particle system. The particles’ surface charge and surrounding electrolytes induce an interaction potential between colloidal spheres of a screened Coulomb force [1],. φ(r) =. qe−κ(r−a) r(1 + κa). (1.1). with being the dielectric constant, a the effective radius of the charged colloidal particle, and κ the screening length, which depends on the density of charges in solution. In this reduced system of mesoscopic particles, the de Broglie wavelength, . λ=. h 2πmkT. 1/2. (1.2). where m is the mass of the particles, T the absolute temperature, and k the Boltzmann factor, is much smaller than other length scales in the problem, guaranteeing they can be safely described with classical physics.. 1.2. Description of the Problem. The experimental work that inspires this thesis comes mainly from the Bechinger group in Konstanz, Germany. In a series of publications ([2],[3],[4],[5]), it has given out many interesting results coming from the experimental set-up that we describe next: a thin layer of mesoscopic particles (about 3µm in diameter) dispersed in an. 2.

(9) Figure 1.1: A laser produced interference pattern is projected upon a colloidal particle system from above, pushing it into an effective two dimensional system. Light gradients induce effective forces in the direction of increasing intensity. Taken from [3] and [4]. electrolytic solution are observed by optical means from below, while powerful laser beams are shone on them from above. As described in [6], the electromagnetic momentum associated with light induces two useful effects on the sample. One, radiation pressure forces all particles to a thin layer, effectively reducing their degrees of freedom to two dimensions. And two, the focused light provides a light intensity gradient, which induces also an attraction to the places with the higher intensity, in effect working as a force field in space. In the experiments being discussed, the laser light is produced by two sources, producing an interference pattern. This pattern is the equivalent of a force gradient periodic in one direction. Additional pairs of source lights can produce periodicity in another direction, thus giving the possibility of simulating a lattice in two dimensions. Initially, experiments concentrated on the one-directional case, and results were striking, as interesting phase transition phenomena were discovered: with increasing light intensity (potential strength), the system passed through a series of. 3.

(10) Figure 1.2: With a one-directional potential, the colloidal system passes from an initially homogenous fluid (row 1) to a liquid ordered in direction with the potential valleys (row 2). Further increase induces a crystallization of the system, with fixed relative position of colloidal particles (row 3). At the most intense levels, there is a regression to the ordered liquid state (row 4). Taken from [2]. 4.

(11) stages. As intuitively expected, beginning from a radially symmetric ordering, the system turned to an oriented fluid in the direction of the potential valleys. With higher intensity came a phase transition into a crystal-like fixed location of the colloidal particles, and even more bizarre, with the highest levels of radiation came a retreat into the previous oriented fluid state. Since the effects of an isothermal increase in field strength can be thought of as thermodynamically equivalent to a temperature decrease at constant field strength, this was akin to a (highly counterintuitive) cold-indiced melting of a crystal, a process termed “reentrant melting”. Theoretical inquiries into these phenomena were soon carried out by Nelson, Frey and Radzihovsky with the help dislocation-unbinding theory of twodimensional solid state physics in [4],[7] and [8]. This framework examines from the crystalline phase the onset of disorder that marks the beginning of a phase transition. The authors were able to describe several different phases and boundaries betweem them were proposed. It was also shown that for specific values of the screening length of the colloid, this boundary could permit reentrancy with isothermic field variations (See figure (5.5)). This theory has remained unchallenged and experimenters could move on to other phenomena. With the availability of even stronger light intensities, it was possible to inquire into the phase transitions typical of two-directional modulation [5]. As with the previous case, the initial intuitive observation was that of particles having an increasing preference for potential minima, while still in a fluid state. At a certain intensity, there was again a novel transition to a crystalline order with low fluctuations, and for still higher values, there was an equivalent to the reentrant melting of the one-directional case. While the particles had a low probability of trascending the high-potential boundaries, they lost all inter-cell orientational order, at the same time keeping a relative fixed positions to their same-cell neighbours. And while the original experimental set-up consisted of periodic potentials. 5.

(12) Figure 1.3: System behaviour with cell forming potentials. On the left, the experimental results of [5] and on the right, computer simulations [9]. For both cases, there is a passage from a homogenous liquid to one with an increasing preference to potential minima, up to a point where barrier-hopping becomes rare. At yet higher potential strengths we find a crystallization of the system. And for certain conditions, there is a loss of inter-cell orientational ordering with the strongest potential intensities.. 6.

(13) forming a triangular lattice, soon generalizations to other geometries were carried out computationally by Reichhardt and Olson, with a particular interest in the ordering that particles in each minimum take in the “solid” phase [9]. As with the one-directional case, there have been explanation attempts, although from a quite different perspective. Agra, Trizac and van Wijland [10] have successfully applied an Ising analogy, and there is a general understanding that fluctuations, which are comparatively important in two-dimensional systems, are playing a crucial role. Yet up to date, all models that have been applied are “solid state” explanations, intrinsically unable to take account of the fluid phases. No consistent framework has been developed to comprehensively describe these directly as classical particle systems.. 1.3. A Brief Introduction to the Proposed Model. What we need and are going to develop in this thesis is a model with the following characteristics, 1. As is usually done with colloidal systems, we are going to concentrate on the mesoscopic particles. 2. We need to treat each colloidal particle as having an identical charge. 3. The system must be confined to two dimensions. 4. We must introduce the effect of fixed periodic potentials of varying strength. Before going into subject, we make a description of the model to be developed in this thesis, and which fulfills all requirements above. We use the simplest of all possible model for charged particles, termed the One-Component Plasma (OCP)[11]. For a recent review and applications, see [12] and [13]. 7.

(14) • A system to be described with classical statistical mechanics. • Only one kind of identical, point-like particles with an identical charge q. For a system with equally charged particles to be stable, an oppositely charged fixed background must be added. • We will restrict the system to two dimensions, and this condition will make the electrostatic potential to be extremely long range, a logarithmic function. This models a colloidal solution in the limit of long screening length. • The periodic potentials will be added through a modulation in the background charge [14]. As we shall see, this is one of a few systems in Statistical Mechanics that yields exact analytic results for certain conditions, as there is a restriction of the system to one temperature. We will be varying the potential strength at a fixed temperature, and while this is not entirely equivalent to a temperature variation for fixed potential, it is the same process that was done in the experiments. In order to produce analytical results, we are going to restrict ourselves to a particular shape of modulation, and will limit the cases to periodicity in one and two directions. For the two-directional case, potential minimum wells are formed and a crucial paramenter will be the particle density understood as the average amount of particles per cell. Due to time constraints, only the cases of one and two particles density were considered.. 8.

(15) Chapter 2. The One-Component Plasma Framework 2.1. General Procedure. Consider system of particles numbered 1 to N , all with an identical charge q. Gauss’ Law in two dimensions makes that instead of the familiar Coulomb’s inverse square law, we have a 1/r dependence of the electric interaction between the particles, that is consistent with a potential of the kind E(r) = q log(r/L). (2.1). L being an arbitrary length scale. We are setting it to one from now on. It is evident that such a system without additional restraints is unstable since it is all repulsive. We introduce a background charge of equal and opposite charge to all the particles −qn. The Hamiltonian for this particles+background system can be described in. 9.

(16) terms of three distinct contributions H = Hpp + Hpb + Hbb. (2.2). The first term comprising the energy of all particle-particle interactions, . Hpp =. q 2 log( ri − rj ). (2.3). i<j≤N. the second the interaction is that of each particle and the fixed background, Hpb = q 2. N . V ( ri ). (2.4). i=1. and depends of the particular geometry of the background. A third term of the interaction energy, of the charge density background interacting with itself, is a constant throughout this model. The Boltzmann factor can thus be expressed as −H/kT. e. 2. = C exp −q /kT. N i=1. . V ( ri ). . | ri − rj |q. 2 /kT. (2.5). i<j≤N. . The crucial feature of one-component plasma systems is that as we will see below, by setting the exponent q 2 /kT = 2, and with the definition of the Vandermonde Determinant . . |zj − zi |2 = . 2 . (rj eiθj − ri eiθi ). i>j. 10.

(17) =. =. ⎛ 1 1 ⎜ ⎜ ⎜ r2 eiθ2 r1 eiθ1 ⎜ det ⎜ .. .. ⎜ . . ⎜ ⎝ (r1 eiθ1 )N −1 (r2 eiθ2 )N −1 2 zi j−1 }i,j=1...N det{. ⎞2 ⎟ ⎟ ⎟ rN eiθN ⎟ ⎟ (2.6) .. ⎟ . ⎟ ⎠ iθ N −1 N (rN e ). .... 1. ... .. . .... the system can be reduced significantly. In a complex number notation we have e−H/kT = C. . e−V (ri ). i. . |zi − zj |q. 2 /kT. (2.7). i<j≤N. that is, with the said temperature, . 2 . e−H/kT = C det{e−V (ri ) zij−1 }i,j=1...N . (2.8). and this lays the groundwork for the present thesis. The earliest work on the OCP (see [13]) corresponded to uniform or radially symmetric charge, and the determinant, a Slater Determinant in those cases, could be expanded to a rather simple expression. This was possible due to the orthogonality of every function in the determinant with respect to any other of a different row as a consequence of the orthogonality with respect to integration by the angles θi . This however is true only as long as the V (r) functions have a radial symmetry, V (r). For more general potentials such as the periodic ones we need for this work, one could either try to work around this difficulty for the particular instance, or, as we shall do, use a general method that was developed later.. 11.

(18) 2.2. The Projector Method. In general, as long as the determinant remains invariant, one can choose another set ri ) such that of functions Ψj ( e−H/kT = C |det{Ψj ( ri )}i,j=1...N |2. (2.9). We are going to follow the method proposed by Cornu, Jancovici and Blum ([14] and [15]), which works in principle for potentials of an arbitrary shape. The idea is to produce the new basis functions Ψj (r) that are orthogonal. The expression above is then again a Slater determinant, and in terms of the projector Ψj (r1 )Ψ∗j (r2 ) r1 |P |r2 = 2. dr|Ψj (r)|. j. (2.10). the (truncated) particle densities are directly derived (see [16], Chapter 5): ρ(r) = r|P |r ρ(2) (r1 , r2 ) = −|r1 |P |r2 |2 . ρ(n) (r1 , r2 , ..., rn ) = (−)n+1. (2.11) ri1 |P |ri2 ...rin |P |ri1 . i1 ,i2 ,...,in. . In the last line, the summation includes all cycles (i1 , i2 , ...in ) that can be built with {1, 2, ..., n}. In practical terms for this thesis however, we are limiting ourselves to the density and 2-particle correlation function. Further advancement requires specific information of the V (r) function, and hence the fixed background charge. The starting point is the easiest possible configuration, that of a constant background ρ0 . Elementary-level physics leads this. 12.

(19) charge density to correspond to a potential of form 1 V0 (r) = V0 (r) = πρ0 r 2 2. (2.12). which can have an arbitrary constant we are omitting. As was already mentioned, this potential will automatically yield the density and correlations if replaced for in equation (2.8). The trick at this point is to have an exclusively periodic charge density ρ̃(r) superposed, such that the overall charge, and hence the basic requirement of OCP, does not change. This will correspond to a periodic potential φ(r), and V (r) = V0 (r) + φ(r). (2.13). Then we have to find an appropiate orthogonal basis of functions. Following ([15]), a replacement of the kind . − 12 πρ0 z−. z → ψk (z) = e j. . k πρ0. 2. (2.14). is made also absorbing the part of the potential depending on the constant background only, V0 (r). We end up with e−Vo (r) e−φ(r) ψk (z) ∝ Ψk (r) . k −φ( r )−πρ0 x− 2πρ. Ψk (r) = e. 2. 0. +iky. (2.15) (2.16). the plane waves upon which all further work will be based. In this new continuous basis, the projector will be r1 |P |r2 =. ∞ dk −∞. 2π. ∗. Ψ (r )Ψ (r ) ∞ k 1 k 2∗. 13. r Ψk (r)Ψk (r) −∞ d. (2.17).

(20) 2.3. Choice of Periodic Potential. It will be shown later in this document that a straightforward solution for the twodirectional potential case requires a charge density corresponding to the potential in equations (2.18) below. In order to keep the one directional case similar, an equivalent of it was adopted for that case as well: One Direction : e−2φ(r) = e−2φ(x) = 1 + λ cos . −2φ( r). Two Directions : e. . 2πx = 1 + λ cos Lx. . . 2πx Lx. . . + cos. 2πy Ly. . (2.18). Here Lx and Ly represent the distance between potential minima (wavelength) in each direction, whereas λ is a parameter that determines the amplitude of periodicity with the restrictions One Direction : |λ| ≤ 1 1 Two Directions : |λ| ≤ 2. (2.19) (2.20). brought forth by the non-negativity of the exponential function. It is through variations in λ that the effect of varying light intensity in the experiments will be modelled. In Chapter 4 we will see the directions of x and y need not necessarily be perpendicular.. 14.

(21) 2. 2. 1 0 -1 -1 1. 1. 1. 1. 0.5. 0. 0.5. -1 -1 1. 0 -0.5 -0.5. 0 φ. 0 -0.5 -0.5. 0. 0.5. Y. 0.5 1 -1. X. 1 -1. 2. 2. 1 0 -1 -1 1. 1. 1. 1. 0.5. 0. 0.5. -1 -1 1. 0 -0.5. 0 -0.5. -0.5. 0. -0.5. 0. 0.5. 0.5 1 -1. 1 -1. Figure 2.1: The one-directional potential for increasing values of λ {0.1, 0.4, 0.7, 0.99}.. 15. =.

(22) 1 0.5 0 -0.5 -1 -1 1 -0.5. 1 1 0.5 0 0.5 -0.5 -1 -1 1 0 -0.5 -0.5. 0. 1 0.5 0 -0.5. 0 0.5. 0.5. φ. 1-1. 1-1. Y X. 1 0.5 0 -0.5 -1 -1 1 -0.5. 1 1 0.5 0 0.5 -0.5 -1 -1 1 0 -0.5 -0.5. 0 0.5. 1 0.5 0 -0.5. 0 0.5. 1-1. 1-1. Figure 2.2: The two-directional potential for increasing values of λ {0.1, 0.3, 0.4, 0.49}.. 16. =.

(23) Chapter 3. Potential in one direction only This case has been studied before in [17], but we will revisit it in the context of colloidal particle systems. There had been no interest in the particle ordering and phase transitions the model might imply before.. 3.1. The Projector. The previously found Ψ functions are for this instance orthogonal when we take a potential function independent of y. Using the expression in equation (2.17), we get the projector for the one-directional case: −φ(x1 )−φ(x2 ). r1 |P |r2 = e. 1 2π. ∞ ik(y1 −y2 )−πρ0 [(x1 −k/2πρ0 )2 +(x2 −k/2πρ0 )2 ] e dk ∞ 2 −∞. −2φ(x)−2πρ0 (x−k/2πρ0 ) −∞ e. dx. (3.1). . This is exactly solvable as a Fourier series. All difficult integrals are either Gaussian, or of the form [18]: π cos(nπ)dx 0. π =√ 1 + a cos(x) 1 − a2. 17. √. 1 − a2 − 1 a. n. (3.2).

(24) .. 3.2. Particle Densities and Correlations. The particle density, . . 2πx ρ(r) = 1 + λ cos L. . 1 2π. ∞. 2. ∞. −∞. e−2πρ0 (x−k/2πρ0 ) dk. . −∞. . 1 + λ cos. 2πx L. . 2. e−2πρ0 (x−k/2πρ0 ) dx. (3.3). gives then . . ρ(r) = ρ0 1 + λ cos ∗. n. 2πx L. 1. . 1 − (λe−π/2ρ0 L2 )2. . (3.4). ⎛. ⎞n. ⎝. ⎠ e− 2ρ0 L2 cos. 1 − (λe−π/2ρ0 L2 )2 − 1 λe−π/2ρ0 L2. πn2. . 2πnx L. . . Similarly, the two-particle projector yields . r1 |P |r2 = −iρ0. . 2πx1 1 + λ cos L n. . . . 1 + λ cos. . 2πx2 − πρ0 (r1 −r2 )2 iπρ0 (x1 +x2 )(y1 −y2 ) 2 e e L. ⎛. 1 2. 1 − (λe−π/2ρ0 L )2. ⎝. 2. 1 − (λe−π/2ρ0 L )2 − 1 λe−π/2ρ0. L2. ⎞n. πn2. ⎠ e− 2L2 ρ0. . πn(y1 − y2 ) πn(x1 + x2 ) cosh L L πn(y1 − y2 ) πn(x1 + x2 ) sinh (3.5) + sin L L cos. from which one obtains the correlation function by taking its squared norm as in (2.11).. 18.

(25) 3.3. Numerical Results. Both the density and two-particle correlation function (for a particle at (x, y) = 0) are shown in figures (3.1) and (3.2). As can be immediately seen from the analytical solution, the particle density loses all y dependency. The corresponding figure shows the average ordering of particles across a potential well for varying intensities, and becomes evident that there is no important qualitative difference for increasing intensity besides a sharpening of preferences for the lower potential. Similarly, while there is a y-dependence in the correlation function, it is not of the periodic kind. A particle at the origin has a correlation with others that decreases exponentially in all directions while retaining the particle preference for potential minima. All this seems to correspond, for all potential strengths, to the oriented liquid phase of experiments. These results will be discussed further in the concluding chapter.. 19.

(26) 4 3 2. 0.8. 1 0.6. 0 -5. 0.4 0. ρ. 0.2. λ. 5 X. Figure 3.1: Particle density for the one-directional potential periodicity as a function of x-position and incresing potential strength for a wavelength L = 5 and a density ρ0 = 1.. 4 3 2. 1. 1 0 0 -1 0. 20 -1. ρ (2). Y. 1 X. Figure 3.2: The negative (for graphical clarity) 2-particle correlations for (r1 ) = (0, 0) for the one-directional potential periodicity, for an wavelength L = 0.5 and a density ρ0 = 1. ..

(27) Chapter 4. Potential in two directions This chapter, which constitues the central effort of the document, concerns the case of a superposition of two non-parallel periodic potentials. Its elaboration is more intrincate than the one-directional case as we need to take into account first a generalization for an arbitrary angle between the two modulation directions. Then, the added periodicity destroys the orthogonality of the Ψ functions and we must find another basis that is a superposition of the old one. And this basis will depend on the density of particles, since that determines how many of them will have to fit in a given potential minimum at a given time. And while the one particle per well case has been worked out before [15], although with a very different model in mind, the two-particle case is a novel contribution independent of its suitability for explaining the colloidal particle experiments. In this chapter, the two cases will be worked alongside for clarity.. 21.

(28) Figure 4.1: Coordinate transformation. 4.1. Oblique Coordinate Generalization. Consider a potential periodic along two different axes described by vectors a and b that are at an angle θ to each other such that φ(r + na + mb) = φ(r). (4.1). we are going to align b to the y-axis, and introduce dimensionless coordinates X and Y such that r = Xa + Y b x = Xa sin(θ) y = Y b + Xa cos(θ) dxdy = dXdY [ab sin(θ)]. 22. (4.2).

(29) In this system we have repeating cells of size ab sin(θ). We write the background density, which in this model corresponds to the charge density, as ρ0 =. d ab sin(θ). (4.3). where d is the amount of particles per cell. We are going to also separate k into an integer contribution n and a residue 0 ≤ t < 1 of the wavelength b: k = 2π. (t + n) b. (4.4). Our Ψ functions are now e−φ(r)−πd b cos(θ−π/2)X a. 2 +2π a b. cos(θ−π/2)X− πa cos(θ−π/2)(t+n)2 −2πi(t+n)X ab sin(θ−π/2) db. (4.5). which can be significantly reduced upon multiplication by the inocuous phase factor eπid b sin(θ−π/2)X a. 2 +πi a db. sin(θ−π/2)(t+n)2. = e−πid b cos(θ)(dX a. 2 + 1 (t+n)2 d. ). (4.6). finally obtaining −φ( r )−π ab e−i(θ−π/2). Ψt,n (r) = e. √. dX−. t+n √ d. 2. +2πi(t+n)Y. (4.7). which is applicable to a two-directional potential grid regardless of the periodicity wavelengths a and b as well as the angle θ between periodicity directions.. 23.

(30) 4.2. Orthogonalization Preliminaries. The purpose of the transformation to t and n becomes apparent upon checking orthogonality on Y: ∞ −∞. dY Ψt,n (r)Ψ∗t ,n (r) ∝. ∞ −∞. . . dY e−2φ(r)+2πi(n−n )Y +2πi(t−t )Y. (4.8). Periodicity of 1 in both axes of the potential ensures that save for the part depending on (t−t) , the system is translationally invariant by unities and orthogonal to the part periodic in (t − t ) for this latter having a necessarily larger wavelength, except when constant, i.e. (t − t )=0. ∞ −∞. dY. Ψt,n (r)Ψ∗t ,n (r). . ∝ δ(t − t ). 1 0. . dY e−2φ(r)+2πi(n−n )Y. (4.9). We have thus managed to discretize the basis for orthogonalization purposes. Now, our choice for periodic potentials gives us the purest periodic term e−2φ(r) in terms of cosine superposition. This will allow for a simple expression of the function products. We begin integrating the Y dependent part 1 0. . . e2πiY + e−2πiY dY 1 + λ cos(2πX) + λ 2. . e2πi(n−n )Y. (4.10). which gives δn,n (1 + λ cos(2πX)) +. λ δn,n +1 + δn,n −1 2. (4.11). Using this, we follow to the complete double integral. ∞ −∞. +. λ 2. drΨt,n (r)Ψ∗t,n (r) = ab sin(θ). δn,n +1 + δn,n −1. ∞ −∞. . dX δn,n (1 + λ cos(2πX)). −dπ a e−i(θ−π/2) (x− t+n )2 −dπ a ei(θ−π/2) (x− t+n )2 b d b d e. 24. (4.12).

(31) . For the sake of clarity we introduce a −i(θ−π/2) e b 2a sin(θ) γ = ν + ν∗ = b ν=. (4.13). and after standard Gaussian integrals we obtain ∞ −∞. drΨt,n (r)Ψ∗t,n (r). . . . . t+n 1 = ab sin(θ) √ 1 + λe−π/γd cos 2π δn,n d γd λ −πν ∗ /d+πν ∗ 2 /γd √ + e δn,n +1 + δn,n −1 (4.14) 2 γd. . It is immediately evident that our functions are not orthogonal, as nearest neighbour functions in n do not cancel. The next task will be to find another basis.. 4.3. The Almost Mathieu Equation. An infinite matrix Ann constructed with the inner product of all Ψ functions, ∞. r Ψt,n (r)Ψ∗t,n (r) −∞ d. will contain rows identical but for the cosine value in the. diagonal, which is n-dependent. We can through one row find the general eigenvalue equation for the diagonalization of this matrix. Our canonical basis is the infinite set of functions Ψt,n . We are going to construct another basis Ψ̃(r) from these, such that every function in it can be expressed as Ψ̃(r) =. . un Ψt,n (r). (4.15). n. and Ann Ψ̃ = Ψ̃. 25. (4.16).

(32) . The eigenvalue equation for coefficients, based on (4.14) must then be η(un+1 + un−1 ) + 2 cos. 2π(t + n) un = ζun d. (4.17). Where we have defined π d. . π − γd. ζ = λe. . ν ∗2 + γ1 −ν ∗ γ. . η=e √ γd −1 . ab sin(θ). −1. (4.18). Expression (4.17) is the Almost-Mathieu or Harper equation. Notice that the eigenvalue equation repeats itself every d steps in n. This suggests that we can reduce the problem to the diagonalization of a d sized matrix. If d is a fraction, then the periodicity is that of the most simplified integer numerator. Next we use the Bloch-Floquet theorem for periodic functions, which states that the solution must be of the form (see for instance [19]) un+d = un eidµ or un+d = un e−2πids. (4.19). and we construct the matrix ⎛. 2 cos 2π(t+1) d ⎜ ⎜ ⎜ η ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝. η. 0. .... 2 cos 2π(t+2) d. η. .... 0. η. 2 cos 2π(t+j) d. η. 0 .. .. .. .. η. ηe−2πids. 0. .... 26. 0 ... ... 0. ηe2πids. 0. η. .. η 2 cos 2π(t+d) d. ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ (4.20) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠.

(33) for which we must find eigenvalues and eigenvectors. Now, depending on the number of particles the problem changes.. 4.4. Eigenvalues and Eigenvectors. For the case d = 1, we have no matrix but a trivial equation . . η e2πis + e−2πis un + 2 cos(2πt)un = ζun. (4.21). we immediately retrieve the “eigenvalue”, and can arbitrarily choose the eigenvector coefficients un while obeying (4.19). ζ = 2η cos(2πs) + 2 cos(2πt) un = e−2πins . Ψ̃t,s (r) =. (4.22). Ψt,n (r)e−2πins. n. . We will leave it here for the time being in order to develop the simplification process jointly with the result of the two-particle per well case. For the d = 2 case, we have the matrix ⎛ ⎜ −2 cos ⎝. η+. 2πt 2. η. ηe−4πis. + ηe4πis. 2 cos. 2πt 2. ⎞ ⎟ ⎠. (4.23). which leads us to . ζ+ = 2 cos2. 2πt + η 2 cos2 2πs 2. 27. (4.24).

(34) . ζ− = −2 cos2. 2πt + η 2 cos2 2πs 2. (4.25). and, by replacing in the matrix and solving for the un , we obtain the following set simple eigenvectors or orthogonal funcions:. Ψ̃t,s,+ =. n. Ψ̃t,s,− =. n. . 2πt + η cos 2πs + (−1) cos 2. . 2πt − η cos 2πs + (−1) cos 2. . . cos2. n. . 2πt + η 2 cos2 2πs Ψt,n (r)e−2πins 2 . 2πt + η 2 cos2 2πs Ψt,n (r)e−2πins cos2 2. n. (4.26) It is quite striking to find such a symmetric expression for the two different kinds of Ψt,s functions, and it is this that will allow for a simple computation of the projectors.. 4.5. Projectors. At this stage, armed with the definition of the Ψt,n functions (4.7) and the way these are combined to make the orthogonal Ψ̃t,s , we proceed to replace in the definition of the projector r1 |P |r2 (2.10), in terms of the new parameters.. 4.5.1. One Particle per Cell. We have for the one-particle case:. r1 |P |r2 =. 1 0. 1. ds. 1. =. 0. 0. 1. ds. 0. dt dt . Ψ̃t,s (r1 )Ψ̃∗t,s (r2 ) drΨ̃t,s (r)Ψ̃∗t,s (r) nn. dr. 28. un u∗n Ψt,n (r1 )Ψ∗t,n (r2 ) ∗ r )Ψ∗t,n (r) nn un un Ψt,n (. (4.27).

(35) . First, since the funcions Ψ̃ are orthogonal, the normalization factor can be directly retrieved, as it is related to the (sole) eigenvalue. We retrieve back from equations (4.17) and (4.18), arriving at . drΨ̃t,s (r)Ψ̃∗t,s (r). π − γd. = ab sin(θ). 1 + λe. (η cos 2πs + cos 2πt) √ γd. (4.28). Next, we intend to eliminate the summations in the numerator. Since this part will also be useful for the case of d = 2, it will be done in general. We first introduce a change of variables with N = n − n. . =. . √ √ √ )2 −πν ∗ ( dX2 − t+n √ )+ 2πi(t+n)Y1 −(t+n )Y2 −πν( dX1 − t+n. . e−φ(r1 )−φ(r2 ) e2πis(n −n) e. nn. d. d. √ √ √ )2 −πν ∗ ( dX2 − t−n−N √ −πν( dX1 − t+n )+2πi(t+n)Y1 −2πi(t+n+N )Y2. e−φ(r1 )−φ(r2 ) e2πis(N ) e. d. d. nN. (4.29) Now, everything in terms of n is in form (t + n). This suggests, since the integral in (4.28) goes only from 0 to 1, to extend the integral to infinity absorbing n.. 1 0. ∞. f (t + n)dt =. n. −∞. f (t)dt. (4.30). This can be done because (for both d = 1 and d = 2) the part that is integrated over r is also only dependent on t in a way that is invariant through integer additions to t. Next we regroup everything that depends on N √ √ −φ(r1 )−φ(r2 )−πν( dX1 − √t )2 −πν ∗ ( dX2 − √t )2 +2πit(Y 1−Y 2). e. d. . d. √. 2 ∗ √ ( −πν ∗ Nd +2πi N [s−Y2 − iν d. e. N. 29. . dX2 − √t ) d. (4.31).

(36) and with the help of the Poisson relation . ∗ 2 − πν dN +2πiN α. e. . =. n. we obtain. . d − πd ∗ e ν ∗ ν N. d − πd∗ (α−N )2 e ν ν∗ N. (4.32). . 2 ∗ √ √ ( dX2 − √t −N ) s−Y2 − iν d. d. (4.33). and we can once more, since the integral in s is limited at zero and one, extend this integral by absorbing N . Like with t, the part integrated over r is invariant under integer additions to s. At this stage only the integrals are present and we have the reduced expression . d ν∗. ×. ∞ −∞. ∞. dt. ds. e−φ(r1 )−φ(r2 ) f (t, s). −∞ √ √ √ √ −πν( dX1 − √t )2 − πd (Y2 −s)2 −2πi( dY2 − ds)( dX2 − √t )+2πit(Y1 −Y2 ) ν∗. e. 2. d. (4.34). where f (t, s) is the integral over r and depends on the number of particles per cell. For the case d = 1, since we have already obtained it (4.28), we can follow through to. r1 |P |r2 = ×. 1 ab sin(θ). . γ −φ(r1 )−φ(r2 ) e ν∗. ∞ −∞. ∞. dt. −∞. ds. 2 − π (Y −s)2 −2πi(Y −s)(X −t)+2πit(Y −Y ) 2 2 2 1 2 ν∗. e−πν(X1 −t). −π γ. 1 + λe. 30. (η cos 2πs + cos 2πt). (4.35).

(37) 4.5.2. Two Particles per Cell. In this case we have a projector shaped like 1. r1 |P |r2 =. 0. 1. ds. 1. +. 0. 0. 1. ds. 0. dt dt . Ψ̃t,s,+ (r1 )Ψ̃∗t,s,+ (r2 ) drΨ̃t,s,+ (r)Ψ̃∗t,s,+ (r) Ψ̃t,s,− (r1 )Ψ̃∗t,s,− (r2 ). (4.36). drΨ̃t,s,− (r)Ψ̃∗t,s,− (r). Before we begin, a few words on the mathematical tools to be used. Let us define . . un,± = e−2πisn an,±. (4.37). In the one-particle case, we had an = 1 always: our eigenfunctions were normalized. In the two-particle case however this is not true. Besides of the (already known) eigenvalues, we need this normalization factor for the r integrals. This is where we begin. In order to perform the necessary calculation and for the rest of this section, it will be useful to keep in mind the following transformation: (−1)n cos πt = eiπn cos πt = cos π(t + n). (4.38). The procedure for integer coefficient n elimination in the projector is essentially the same as for the 1-particle case. Only now it is done for every term resulting from the computation of an a∗n ,. . . . . . η cos(2πs) + cos π(t + n) ± . × η ∗ cos(2πs) + cos π(t + n ) ±. 31. η 2 cos2 (2πs) + cos2 (πt). η ∗2 cos2 (2πs) + cos2 (πt). (4.39).

(38) It can be directly appreciated that upon replacing N = n − n, the system is again reducible in n through extension of the integrals to all s. All terms of the multiplication above are trivial except for those containing cos(t + n ) → cos π(t + n + N ) → cos πteiπN. (4.40). In this case, we have to notice that by grouping all terms containing N as we did in (4.31) we end up with . e. πN 2 ν ∗ +2πiN (s+1/2−Y2 −iν ∗ (X2 − 2t )) 2. .. (4.41). N. We can make the replacement s = s + 1/2 cos 2πs = − cos 2πs .. (4.42). Once all integer parameters have been absorbed, we rename s → s and will in effect have changed the sign of the second s-dependent cosine.. . ∗. [cos π(t)] × −η cos(2πs) + cos π(t) ±. . . η ∗2 cos2 (2πs). +. cos2 (πt). (4.43). A difference s−s of 1/2 will produce the same effect, and this is of immediate use since there is a pending problem with our orthogonalization: both eigenvalues (4.26) for functions obeying s − s = 1/2 evidently become equal, and one can with the replacement (4.42) calculate a non-vanishing inner product for the eigenvectors n. un,t u∗n,t−1/2 = ± cos πt. . η 2 cos2 (2πs) + cos2 (πt) +. 32. . . η 2∗ cos2 (2πs) + cos2 (πt).

(39) . For that particular value both eigenvectors are colineal. We will restrain our range of so that 0 ≤ s < 1/2 to avoid this. Our integrals won’t suffer since the symmetry of our system suggests 1/2 0. 1 2. f (s)ds =. 1 0. f (s)ds. (4.44). and we can, for operative purposes, express it that way for the dissolution of the n parameters. Again in a similar manner we obtain the indispensable normalization factor Ω± : . ×. . un,t u∗n,t = cos2 πt + ηη ∗ cos2 2πs +. n. η 2∗ cos2 (2πs). +. cos2 (πt). . η 2 cos2 (2πs) + cos2 (πt). . ± cos 2πs η η 2∗ cos2 (2πs) + cos2 (πt) +η. ∗. . . η 2 cos2 (2πs). +. cos2 (πt). (4.45). . The next step in the process, is to replace in the projector. We obtain r1 |P |r2 =. 1 2. . ×. d ν∗. ∞. ∞. ds. dt. A± B±. (4.46). −∞ −∞ √ √ √ √ t 2 −πν( dX1 − √t )2 − νπd ∗ (Y2 −s) −2πi( dY2 − ds)( dX2 − √ )+2πit(Y1 −Y2 ). ±. e. 2. d. where, once more with the help of (4.43) we have found A± as . ηη ∗ cos2 2πs + cos2 πt + (η − η ∗ ) cos 2πs cos πt + . . η 2 cos2 (2πs) + cos2 (πt). . × η 2∗ cos2 (2πs) + cos2 (πt) ± (η cos 2πs + cos πt) η 2∗ cos2 (2πs) + cos2 (πt) . ∗. ±(η cos 2πs + cos πt) 33. . η 2 cos2 (2πs). +. cos2 (πt).

(40) . π − γd. × 1 ∓ λe. . . η 2 cos2 (2πs). +. cos2 (πt). (4.47). and B± as 2π. . . 1 − λ− γd η 2 cos2 (2πs) + cos2 (πt) ab sin(θ) × Ω± γd. (4.48). . This is as far as we will go in the general case. For the particular instance where η=1, i.e. when the two potentials form squares regardless of their wavelength, the system can be significantly reduced by simple elimination. The result is r1 |P |r2 =. 2 ab. . γ −φ(r1 )−φ(r2 ) e ν∗. ∞ −∞. ∞. ds. −∞. 1 − λe− 2γ (cos(2πs) + cos(πt)) π. dt. −π γ. 1 − λ2 e. (cos2 (2πs) + cos2 (πt)). (4.50) √ √ √ √ t 2 −πν( 2X1 − √t )2 − ν2π ∗ (Y2 −s) −2πi( 2Y2 − 2s)( 2X2 − √ )+2πit(Y1 −Y2 ). ×e. 4.6. 2. 2. Numerical Results. Figures (4.2) and (4.3) describe de particle density for the cases of one and two particle per well, in the two extreme values of λ. For the one particle case, the main feature is the distribution resemblace for the two intensities, only differing by their scale relative to the average of 1. At a first glance, both cases seem consistent with a fluid of varying preference for the potential minima, but never really having regions of difficult transit. For the two-particle case, the immediate impression is that of a striking similarity to the one-particle case. Quantitatively, the difference in the two densities, besides of the doubling effect of incresing particle number, arises from. 34.

(41) 1.02 1 1.01 1 0.5 0.99 0.98 0 -1 1 -0.5 -0.5. 1.0005 1 0.9995 -1 1 -0.5 0 0.5 ρ. Y. 1 0.5 0 -0.5. 0 0.5. 1-1. 1-1. X. 1.04 1.02 1 0.98 -1 1 -0.5. 1 1.5 1 0.5 0.5 0 0 -1 1 -0.5 -0.5. 0 0.5. 1 0.5 0 -0.5. 0 0.5. 1-1. 1-1. Figure 4.2: Density for the one particle per cell, two-directional potential (above) for λ values of 0.01 and 0.499. The first column is the integral without the Exp{−φ} part, and the second column the real density. the numerical value of the integral (4.49) being somewhat larger in the two-particle case, while remaining qualitatively identical. These results indicate the presence, at all values, of a fluid system comparable to the one seen in Figure (1.3), above right. Certainly the incresing potential is not enough to stop particles from flowing across the entire system. Exploration of the two-particle correlation functions gives further proof to last statement, as again differences between the one- and two-particle per cell cases are hardly noticeable, and for both cases we observe the canonical landmark of the fluid state, an exponential decrease of correlations. For the systems under consid35.

(42) 2.005 2 1.995 -1 1 -0.5. 1 0.5 0 -0.5. 0 0.5 ρ. Y. 2.02 2 1.98 -1 1 -0.5. 1 0.5 0 -0.5. 0 0.5. 1-1. 1-1. X. 3 1 2 0.5 1 0 0 -1 1. 2.4 2.2 2 1.8 -1 1 -0.5. 0.5 0 -0.5. -0.5. 0. 1. -0.5. 0. 0.5. 0.5. 1-1. 1-1. Figure 4.3: Density for the two particle per cell, two-directional potential (above) for λ values of 0.01 and 0.499. The first column is the integral without the Exp{−φ} part, and the second column the real density.. 36.

(43) eration this decrease is strong enough to do away with all noticeable periodicity. Particles experience practically no effect from those in neighbouring cells, an effect even more noticeable in the d = 2 case. Further analysis in the conclusions.. 37.

(44) ρ (2). Y X. Figure 4.4: The negative (for graphical clarity) of 2-particle correlations for a particle at r = (0, 0), for the one (above) and two (below) particles per cell cases at λ = 0.49. The plots extends from −1 to 1 in both the X and Y directions.. ρ (2). Y X. 38.

(45) Chapter 5. Conclusions We have developed an exact model for colloidal particle behaviour under external periodic potentials, by modelling it as a one-component plasma. As results we have obtained expressions for the cases of a one-directional potential, and of cell forming, two-directional potentials. In the latter case particle density per well is an important parameter, and we have solved the system for one and two particles per potential minima. Analytical results and their graphical expression point to this model corresponding, for all physically sound potential intensity, to a fluid that is being modulated but never in a sufficient way to induce a qualitative change in the system. As we pointed out earlier, this corresponds to what is portrayed in figure (5.1). Since no approximations were made, one can infer that the failure to obtain a variety of phases in the colloidal particle systems described in Chapter 1 is probably a consequence of the differences between experiments and our model. A first noticeable problem comes from the shape of potentials. As can be seen in figures (2.1) and (2.2), even while reaching the highest physically sound potential intensity, the potenial barriers remain relatively unimpressive if crossed at a midpoint between cell corners. One could, physical considerations notwithstanding,. 39.

(46) Figure 5.1: Close-up of figure (1.3): probable behaviour of particles in our model. investigate the behaviour of the system for still higher values of λ considering that for certain cases the final expressions are analytical for unphysical values of that variable. In general, and we put the system in figure (5.2) as an example, the φ dependent part of the expressions start to produce negative values while both it and the rest of the functions continue to grow in proportion. The negativity allows for radical transformations of the density function, which qualitatively could be interpreted as further confinement of particles to the immediate area of potential minima. The shape of the density function in this vicinity even hints at a geometrically different ordering from that of the fluid states. The correlation functions for the same system indicate us how inside one of the formed rings, while particles do have an interaction, their positions still correlate in a quickly decreasing manner. There will not be a particle orientation as sharp. 40.

(47) 8 6 4 2 -1 1. 5 0 -5 0.5 -10 -15 -1 1 0 1. Y. -0.5. 0. -0.5. 0 X. 0 -0.5. -0.5 ρ. 1 0.5. 0.5. 0.5. 1-1. 1-1. Figure 5.2: Density for the two-directional, two-particle per well case, for the (un eπ/2 physical) almost largest possible value, λ = 2 − 0.1. On the left, the density function without the Exp{−φ} part. as the one shown in figure (5.3) below. Just a particle repulsion that extends to roughly one quarter to half of the ring. But in any case, one would need to take account of unphysical phenomena such as negative particle density, a direct consequence of a potential that is the logarithm of a negative function. If the negative regions can be interpreted as zero density, it might be useful to explore these density profiles in the future for the phenomena we expected to find in this thesis. A second limitation of our model may come from the extreme long range of electrical interactions and the point-like nature of the charges. In fact, previous theoretical work, [7], with one-directional potentials only, points to the reentrancy being present only for certain values of the colloidal screening constant κ. Our work perhaps corroborates and extends those results to two-directional cases. Finally, it has to be noted that our work for two-directional potentials here was restricted to one and two particles per cell. And while similar analytical calculations for higher particle numbers may be prohibitive due to the difficulty of obtaining nice symmetric eigenvectors for the Mathieu equation, they might show 41.

(48) 6 1. 4 2. 0.5. 0 -1. 0 -0.5 -0.5. 0 ρ. Y. 0.5 1 -1. X. Figure 5.3: For the (unphysical) λ of figure (5.2), ignoring the negative values of the density notice the resemblance to the system below, taken from figure (1.3).. 42.

(49) ρ (2). Y X. Figure 5.4: Above we have the negative correlation function for the (unphysical) λ of figure (5.3), for r = (0, 0) (left) and r = (−0.2, 0) (right). The axes extend roughly a formed cell, from −0.5 to 0.5 in both directions. Below,for a region extending from −0.75 to 0.75 we have the negative correlation function for r = (−0.15, 0) and r = (0.15, 0.15). For all plots, the peaks around |X| = |Y | = 0.5 are an artifice of the negative density.. ρ. (2). Y X. 43.

(50) Figure 5.5: In reference [7], it was proposed that, in one-directional potentials, the screening length of colloidal particle interaction determines whether the system experiences many phases or not. The thick arrow indicated a process of potential increase with parameters that allowed for reentrancy. Our system probably lies close to the horizontal axis, where we have a modulated liquid for all potential magnitudes.. 44.

(51) different behaviour. These cases also probably allow for explorations into higher λ values, if that line of work proves to be fruitful. These are all open possibilities for future work with the model that we inaugurate here.. 45.

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(54)