Eindhoven University of Technology MASTER. On the phase behaviour of semi-flexible rod-like particles. de Braaf, B.

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1 Eindhoven University of Technology MASTER On the phase behaviour of semi-flexible rod-like particles de Braaf, B. Award date: 2015 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 On the Phase Behaviour of Semi-Flexible Rod-Like Particles Bart de Braaf Thesis submitted for obtaining the degree of Master of Science in Applied Physics. TU/e TPS Supervisors: Paul van der Schoot Stefan Paquay May 20, 2015

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4 Abstract In this work we perform molecular dynamics simulations of semi-flexible rod-like particles for varying bending flexibilities and aspect ratios, from which we construct phase diagrams as function of density and pressure. We observe five distinct phases: isotropic, nematic, smectic-a, smectic-b and crystal. The isotropic-nematic and nematic-smectic-a phase transitions shift to higher densities for increasing flexibilities, consistent with literature. Additionally, for all but the highest aspect ratio, the smectic-a phase completely disappears above a certain flexibility, but is not replaced by a columnar phase, in contrast with predictions in the literature. We observe furthermore that for increasing densities the particles contract on account of their finite extensional compressibility, while at the isotropic-nematic phase transition there is a jump in their contour length, with the particles in the nematic phase being slightly longer. A similar jump occurs at the phase transition towards the smectic-b/crystal phase. These two observations agree with theoretical predictions we obtain by applying Onsager theory to extensible rods. We argue that this makes accurate measurements for the virial coefficents problematic as the length of the partilces is an annealed quantity in almost all computer simulations of rod-like particles. Keywords: Molecular dynamics, liquid crystals, semi-flexible particles, length changes

5 List of symbols Symbol explanation B 2 second virial coefficient c dimensionless density B 2 N/V D diameter of particles, so D = σ 0 d smectic layer thickness i i 2 = 1 k B Boltzmann constant k angle strength of harmonic potential for the angles k bond strength of harmonic potential for the bonds k eff k bond /N bond L 0 stretched contour length, reference length equal to L 0 = N bond l bond y L 0 /D aspect ratio of particles L instantaneous contour length L ete the end-to-end length L p persistence length l instantaneous length bonds l bond rest length of bonds m mass of beads ˆn director N number of particles/chain in the system N bond number of bonds in a chain P pressure r distance r c cutoff length of Lennard Jones potential S 2 nematic order parameter T temperature V volume of system v 0 volume of particle x relative contour length of a particle y relative length of single bonds α width of nematic orientation distribution β reciprocal temperature γ dimensionless parameter equal to βk bond lbond 2 /2 δ Dirac delta function ɛ dimensionless energy scale and strength of lennard jones potential Θ Heaviside step function θ angle between 3 consecutive beads θ 0 rest angle κ dimensionless parameter equal to βk eff L 2 0 /2 µ chemical potential Π osmotic pressure ρ packing entropy Continued on next page 2

6 Symbol σ σ 0 τ φ ψ 6 explanation orientation entropy dimensionless length and diameter of each bead smectic order parameter volume density bond order parameter 3

7 Contents 1 Introduction Rod-like particles Flexible rod like particles Our research Outline Method and analysis Method Single particle Stretch of the single bond Analysis Order parameters Correlation functions Previous work Simulation results Phase diagram Individual phase transitions Isotropic-nematic phase transition Nematic-smectic-A phase transition Individual phases The isotropic state The smectic layer thickness The higher-order phases Theory of compressible hard rods Onsager with a harmonic potential The isotropic phase The nematic phase Phase coexistence Onsager with a harmonic potential and flexibility Conclusions and outlook Conclusions Outlook Technology assessment Appendices A Extra analytical work A.1 Length of the chains A.2 Third virial coefficient

8 B Scripts B.1 Phase diagram B.1.1 LAMMPS input script for phase diagram B.1.2 LAMMPS restart script for phase diagram B.2 Stretch bond B.3 Onsager with a harmonic potential B.4 Onsager with a harmonic potential and flexibility

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10 Chapter 1: Introduction In the late 1800s Reinitzer [1] discovered a phase that did not behave like one of the ordinary phases: the gas, liquid or the crystal phase. This phase and others that have been discovered since, are called liquid crystals. These liquid crystals have characteristics of both liquids and crystals: they can flow like a simple liquid but can resist static deformations in some directions like crystals [2]. This behaviour is due to the aspherical shape of the particles. They are much longer than they are wide (rod-like) or much wider than the are long (disk-like). These examples are illustrated in figure 1.1. For the rest of this thesis we will focus on the rod-like particles. There are two types of liquid crystals: thermotropic, which means that energy drives the phase transitions, and lyotropics, which means that entropy drives the phase transitions [3]. This thesis focuses on lyotropic liquid crystals. Some materials that are used in experiments as lyotropics are: DNA [4], TMV [5], silica [6] and fd-virus [7]. Figure 1.1: A couple of possible shapes for particles. The red particle in the middle has the gas, liquid and solid phase. On the left in blue a rod-like particle and on the right in green a disk-like particle which also have liquid crystal phases. 1.1 Rod-like particles We now continue with the known phase behaviour of rod-like particles. The isotropic phase of the system is when the position and orientation of the particles are randomly distributed (see figure 1.2). Upon increasing the density the orientation will freeze in first, causing the particles to orient themselves along a common axis (the director). In this phase, however positional degrees of freedom are still randomly distributed. This phase is called the nematic phase (see figure 1.2). In the 1940s Onsager [8] analytically showed that the nematic phase is more stable than the isotropic phase for higher densities. He did this by calculating the second virial coefficient and making plausible that the third virial can be ignored in the limit that the aspect ratio of the particles goes to infinity. Furthermore, he showed that there is a competition between the orientation and packing entropy of the particles, the latter gaining the upper hand at high densities. If the density is increased even further the particles form layers along the director and within each layer there is a random distribution of the particles like a 2D liquid. This phase is called a smectic-a phase (see figure 1.2). Finally, after increasing the density even further, the crystal phase appears. The crystal phase also has layers along the director. However, inside the layers the particles are now 7

11 (a) (b) (c) (d) (e) (f ) Figure 1.2: Figures a, c and e obtained from experiments with silica rods [6], figures b,d and f obtained from molecular dynamics simulations, where the spherocylinders are moddelled as beadsspring chains. In figure a and b the isotropic phase is shown, where the particles are randomly distributed. In c and d the nematic phase is shown, where the particles are aligned along a common axis. In figures e and f the smectic phase is shown, where the particles are aligned along a common axis and the particles are in layers. ordered on a hexagonal lattice. Of the above described states the isotropic-nematic phase transition and the nematic-smectic-a transition can be analytically shown to exist [8, 9]. Bolhuis and Frenkel constructed a phase diagram based computer simulations, where they used sperocylinders that could not overlap for different particle lengths [10], shown in figure 1.3. In their results there are several different states: the isotropic, nematic, smectic-a, plastic crystal (rotator phase) and the crystal phase. In the plastic crystal the particles have a small aspect ratio and are positionally a crystal but can still rotate freely. In the crystal state there are two different ways of stacking, AAA and ABC. Here AAA means that the layers of the crystal are stacked on top of each other the particles also form columns, in the ABC stacking 8

12 between the layers is shifted. The density that they used is ρ = ρ/ρ cp with ρ cp the closed packed density of spherocylinders [10]. There are triple points within the phase diagram at aspect ratios of 3.2 and 3.7, with the aspect ratio of 3.2 the isotropic, smectic-a and crystal triple point. Below aspect ratio 3.2 there is no smectic-a phase. The second triple point at aspect ratio 3.7 is of the isotropic, nematic and smectic-a phases. Below this point there is no nematic phase. Figure 1.3: Figure obtained from the work of P. Bolhuis and D. Frenkel [10]. On the horizontal axis is the aspect ratio of the particles varying from L/D = 0 100, on the vertical axis is the dimensionless density ρ. There are several phases to be distinguished: I (isotropic), N (nematic), Sm (smectic-a), P (plastc crystal) and the crystal phase which is stacked in two ways, AAA and ABC. 1.2 Flexible rod like particles The phases discussed this far are found in systems where all particles have the same length and are rigid, meaning that they cannot bend, extend or otherwise change their shape. Furthermore they are considered to be hard, meaning that they cannot overlap. In experiments more phases have been discovered that cannot be explained with these hard rods. Two of these phases are the smectic-b and the columnar phase [4, 7, 11]. The smectic-b phase also has hexagonal order within the layers similar to that of the crystal state, but the smectic-b state has quasi-long range order, while the crystal state has long-range order [2]. The columnar phase is also hexagonally ordered perpendicular to the director and is also quasi-long ranged but there are no layers in the systems [2] (see figure 1.4). For the smectic-b phase no-one so far has shown theoretically that it can exist [13]. The columnar phase does not show up in second virial approximations or simulation for hard spherocylinders but it is shown that it exists for polydisperse systems [14]. The columnar phase was also observed for fd-viruses which are monodisperse by Grelet [11]. There are three options that can explain the columnar phase in Grelet experiments: charged rods [15, 16], chirality and flexible particles [17]. We focus on the flexible particles which due to their flexibility can 9

13 Figure 1.4: Figure obtained from the work of S. Naderi and P. van der Schoot [12]. The columnar phase, The particles are aligned along a common axis. There are no layers and the particles placed in columns. This columnar phase was created with an external potential they used it to study the dynamics within the columnar phase. create an effective polydispersivety. Theortical work already has shown that flexibility has an effect on the phase behaviour of flexible particles. Khokhlov and Semenov showed that due to flexibility the isotropic nematic phase transition shifts to higher densities [18, 19] and this is confirmed by computer simulations [20]. The flexibility is measured with the persistence length L p which is a typical length scale where orientational correlations along the contour length decay [21]. A similar shift in the density happens for the nematic-smectic-a phase transition if the particles are flexible. This was analytical shown by Van der Schoot [17] and Tkanchenko [22] and confirmed with computer simulations [23]. In experiments no-one has observed a smectic-a above L/L p 1, which is the ratio between the contour length of the particle and the persistence length. Because of this, Van der Schoot suggested that, due to the disappearance of the smectic-a phase other phases like the columnar phase can pop up. The flexibility of the particles could lead to the columnar phase observed in the work of Grelet. 1.3 Our research Our research focusses on finding the columnar phase by doing molecular dynamics computer simulations. We also constructed a phase diagram for flexible particles with different lengths. We attempt to replicate the system of Grelet, which consist of fd-viruses with flexibilities of L/L p = 0.09 and L/L p = 0.31 and an aspect ratio of 100. In our simulations we were only capable of aspect ratio of 10 due to computational cost. Aqueous dispersions of fd-virus have the following phases with increasing density: isotropic, nematic, smectic-a, smectic-b, columnar and the crystal phase. 10

14 1.4 Outline We continue this thesis in chapter 2 with a discussion of the simulation method used to find the different phases. In section we calculate a stretch that occurs in the bonds that we use to connect the beads. Also the analysis of those phases will be discussed, as will the earlier work of Naderi [24]. Our work is a follow upon this albeit with a slightly different simulation method. In chapter 3 we discuss our simulation results, varying the flexibility of the particles and their length. In this chapter we first discuss the general changes in the system. Secondly we focus on the the independent phase transitions and finally on some of the individual phases. In chapter 4 we use a theoretical model to explain length decrease of our particles with increasing density and the effects this has on the phase behaviour. In chapter 5 we present be a summary of our findings and suggestions for further research. 11

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16 Chapter 2: Method and analysis In this chapter we explain how we set up the molecular dynamics simulation (section 2.1) and analyse the results (section 2.2). In section 2.3 we discuss the work of S. Naderi to which this thesis is an extension albeit with a slightly different method. For the rest of this thesis we will use dimensionless parameters, and this means that all fundamental quantities including mass (m), length (σ 0 ), energy (ɛ) and the Boltzmann constant (k B ) are set to 1. All other quantities are scaled to those parameters. Furthermore we use the reciprocal temperature β = 1/T with T the temperature. Figure 2.1: A part of the system in its initial perfectly ordered state. The particles are perfectly aligned along the director, layers are formed along the director, within the layers the particles are hexagonally ordered and the layers are AAA stacked. On the left there is a layer visible showing the hexagonal lattice, on the right the layers along the director are visible. 2.1 Method For the molecular dynamics simulations we use LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [25]. We consider isobaric-isothermal (N P T with N the number of particles, P the pressure and T the temperature) ensembles of semi-flexible particles at various pressures P. For the NP T simulations we use the Nosé-Hoover thermo- and barostat [26] to control temperature and pressure. The system consists of 4608 semi flexible particles that are initially arranged in a crystal structure (see figure 2.1). They are stacked of equal perfect orientation in layers and within each layer they are hexagonally ordered. There are a total of 16 layers and they are AAA stacked. We use periodic boundary conditions to mimic an infinite size system. The barostat changes the volume anisotropically to make that we get the correct layer thickness in the smectic and crystal phases Single particle The particles are modelled as a chain of beads (see figure 2.2). We use three potentials to make our chain behave as semi flexible particles. First we use a truncated shifted Lennard- Jones potential [27] (equation (2.1)) to create the repulsion between the beads. To prevent 1 In appendix B.1 the LAMMPS input scripts can be found, which we used to run the molecular dynamics simulations. 13

17 non-physical repulsion between neighbouring beads, we exclude the nearest neighbour and the next-nearest neighbour from these interactions within the chain: Figure 2.2: The blue particles on the left is a capped cylinder used in many analytic and simulation studies on liquid crystals [8, 10]. The red particle on the right is a capped cylinder constructed out of beads which we are going to use in our simulations. The beads are connected with harmonic springs and a harmonic potential is also used to penalise bending within the chain. { ( ( 4 1 ) 12 ( r 1 ) ) 6 r if r r c = U LJ (r) = 0 otherwise, (2.1) with r the distance between two beads. Secondly we use a harmonic potential to create bonds between neighbouring beads within a chain: U bond (r) = 1 2 k bond (l l bond ) 2. (2.2) Here, l denotes the instantaneous length of the bond. The two parameter that can be tuned are k bond, the spring constant, which is set to 100. This should keep the length of the chains roughly equal to l bond. In section we discuss how much the bond length actually deviates from the rest lengthl bond. The second parameter l bond, is the rest length of the bond and is set to 0.5. The bond length of 0.5 was chosen to prevent interlocking of chains at high density, which is clearly impossible for spherocylinders [24]. The third potential is a angular potential that penalizes bending between three consecutive beads in the chain: U angle (θ) = 1 2 k angle (θ θ 0 ) 2. (2.3) θ 0 is the rest angle which is set to θ 0 = π and the k angle is the strength of the potential which we tune to get the desired flexibility of our chain: We calculate the persistence length of our particles by L p = βk angle l bond, which is valid in the limit N bond [24]. The ratio between length and persistence length becomes L 0 /L p = N bond l bond /βk angle l bond = N bond /βk angle with N bond the number of bonds in a chain. In our simulations we consider L 0 /L p = 0.05, 0.1, 0.2, 0.3, 0.4,

18 2.1.2 Stretch of the single bond The bonds have an energetic rest length l bond at which U(l) = 0. When thermal energy is added these bonds become longer than their rest length. We can calculate this stretching behaviour, assuming that the system exist of a single harmonic spring for which we can calculate the partition function. Without loss of generality we fix one end of the spring at the origin and the other end can move freely around it. This results in Z = ˆπ ˆ2π ˆ l 2 sin φ exp ( 12 ) βk bond (l l bond ) 2 dldθdφ. (2.4) With the use of y = l/l bond and γ = βk bond lbond 2 /2 > 0 we can rewrite and calculate the partition function: ( ( )) Z = 4πlb 3 exp ( γ) π (1 + 2γ) 1 + erf( γ) +. (2.5) 2γ Now we can calculate 2 the expectation value for y: 4γ 3 2 y = 4πl3 b Z ˆ 0 ( y 3 exp γ (y 1) 2) dy = 2 (1 + γ) exp ( γ) + πγ (3 + 2γ) ( 1 + erf( γ) ) ( ). 4γ 2 π(1+2γ)(1+erf( γ)) exp( γ) 2γ + 4γ 3 2 (2.6) The expectation value for y is a function of γ, for large γ equation (2.6) reduces to y = γ. (2.7) In equation (2.7) we can see that for a finite γ the bond is stretched, and only in the limit of γ to infinity the bond is not stretched lim y = 1. (2.8) γ Next we calculate for the general case l bond 0 the y 2 and use it to determine the standard deviation σ y y 2 = 4πl3 b Z ˆ 0 γ 4 exp ( γ(y 1) 2) dy = 2 γ (5 + 2γ) exp ( γ) + π (3 + 4γ (3 + γ)) ( 1 + erf γ ) ( ) 8γ 5 2 exp( γ) 2γ + σ y = 2 Mathematica 9 was used to solve these integrals. π(1+2γ)(1+erf γ) 4γ 3 2 (2.9) y 2 y 2. (2.10) 15

19 The final step is a comparison between our analytic calculations and a computer simulation 3. This computer simulation uses a Nosé-Hoover thermostat [26]. In this simulation we use 1000 dumbbells which consist of two beads connected by a spring. We varied γ by changing k bond, the rest length is kept constant at 0.5 and β = 1. The results in figure 2.3 show that the MD simulation behave like analytically predicted. 0.9 Stretch of two beads connected with a harmonic potential Simulation data Theoratical prediction l γ Figure 2.3: The stretch of the bond l as function of γ = βk bond lbond 2 /2 for a system of 1000 dumbbells consisting of two point masses connected with a harmonic bond. The simulation results are shown in black and the analytical prediction in red. For high γ the simulation and analytical results are almost a perfect match, as for the high values of γ the results go towards the rest length of the bond which was l bond = 0.5. The standard deviation does not match the analytical predictions, this is due to the Nose-Hoover not being ergodic for harmonic systems [28]. Now we can calculate the length of our particles that we are going to use for our simulations. We treat all the bonds within the chain independent so the total length becomes 4 L 0 = N bond l bond y, (2.11) where L 0 denotes the contour length of our particles. We use particles that are constructed out of 13, 15, 17 and 21 beads, so the contour length of the particles will be 6.46, 7.54, 8.62 and The diameter of our particles is D = 1, so the aspect ratios L 0 /D of the particles is equal to their length. 2.2 Analysis We are going to use two techniques to tell the different phases apart: order parameters and pair correlation functions. In our case the order parameters will have values between 0 and 3 In appendix B.2 the LAMMPS input scripts that we used 4 In appendix A.1 a more formal derivation of equation (2.11) is given. 16

20 1, 0 being not ordered at all and 1 being fully ordered. The pair correlation functions will show if there is a correlation between the positions of particles which will tell us if there is positional order within the system. For the analysis we make use of the centre of mass and average orientation of each particle Order parameters The first order parameter that we will use is the nematic order parameter [24] which tells us if the particles are aligned along a common axis: the director. If the value of the parameter is 0 then there is no alignment and the particles are randomly distributed. This is the isotropic phase. If the order parameter is 1 then the particles are perfectly aligned along the director. For the nematic order parameter we have to construct the orientational order tensor with components Q υω = 1 N ( 3 N 2êjυê jω 1 ) 2 δ υω, (2.12) j where υ and ω are the x,y and z directions, ê i is a unit vector along the main body axis of the particle, and the δ is the Kronecker delta. To get the nematic order parameter S 2 we calculate the eigenvalues and eigenvectors, and the director ˆn is then the eigenvector corresponding to the largest eigenvalue. The nematic order parameter only gives information about the alignment of particles, with small values of S 2 for the isotropic state but values around 1 for the nematic, smectic, columnar and crystal states. Therefore, we need a second order parameter τ, which tells us if the particles have formed layers along the director and how thick these layers are. To calculate τ the following function is maximised with respect to d. f(d) = 1 N N j ( exp 2πi r ) j ˆn d. (2.13) So τ = max f(d) and the corresponding d is the layer thickness, with r j the center of mass of each particle, i 2 = 1 and ˆn the director. The final order parameter that we use is the bond order parameter which indicates hexagonal ordering within the system [29], given by ψ 6υ = 1 N n N n exp (6iθ υj ), (2.14) j where N n are the nearest neighbours of particle υ and θ υj is the angle of the distance vector between particles υ and j and a reference vector. The nearest neighbours are selected by a cylindrical volume around the centre of mass of particle υ. The height of this volume is equal to he particle length and has a radius of 1.7. The modulus of ψ 6υ is the bond order per particle which we will use for one of the correlation function. If we first average the real part and the imaginary part of ψ 6υ and then calculate the modulus we get a global average of the system which is our order parameter ψ. With these 3 order parameters we can distinguish the different phases (see table 2.1). Based on the order parameters we cannot tell the difference between the Smectic-B and the crystal phase. Since they differ only in the long-range order within the layers, perpendicular to the director. The crystal will have long-range order, the smectic-b quasi-long range order 17

21 Table 2.1: The different phases and their perfect order parameters. The order will never be perfect, fluctuations within the system and the finite size of the system prevent perfect order. S 2 τ ψ Isotropic Nematic Smectic-A Smectic-B Columnar Crystal and the smectic-a will have short range order [30]. To check the range of the order we use correlation functions Correlation functions We will make use of two correlation functions, one pair correlation function and a bond order correlation function. The pair correlation function measures the positional order perpendicular to the director of the system. The pair correlation function perpendicular to the director will help with identifying the smectic-b and the crystal phase. It is calculated within the layers of the system and averaged over all layers. It can be calculated with g lay (r ) = 1 N 1 ρ i j i ( ) L δ [r r ij ˆn ] Θ 2 r ij ˆn, (2.15) with N the number of particles in the system, ρ = N/V the density, δ the Dirac delta function and Θ the Heaviside step function which we use as filter to take only the particles in the layer into account. The perpendicular pair correlation function shows if there is long range positional order within the system. In addition we can calculate the bond order correlation function which will show long range hexagonal order within the system. We calculate it within the layers of the system and average over the layers, it is calculated by [29]: g lay 6 (r ) = ψ 6 (r )ψ 6 (r r). (2.16) 2.3 Previous work This work expands on the work of S. Naderi [24] with a slightly different method. Naderi started with a high pressure and incrementally lowered it to the desired minimum, and then incrementally increased the pressure again to its initial value (see figure 2.4). The advantage of this method is that all interesting pressures are attained in one simulation. However there are some disadvantages. The first major disadvantage is that when we end up in the isotropic phase the simulation box becomes a cube, but in the nematic it becomes rectangular. We first explain why this happens and then we explain why this is a problem. This change of the simulation box happens because the boundaries of the box acts like a surface due to finite size effects, even if we invoke periodic boundary conditions. We can illustrate this with a simple model [31], in which we assume a nematic phase within a rectangular box with volume V = a a b and the director along the elongated direction of the box. Using Wulff 18

22 Pressure versus time 5 4 Pressure Ours Naderi Time Figure 2.4: A illustrative view of the pressure versus time during the simulations. Naderi (red line) did one simulation per length and flexibility, picked the pressures he was interested in and started at the highest pressure and incrementally lowered it. After reaching his minimum he increased the pressure again. We initialised the system at the interesting pressures and kept the pressure constant, so we did one simulation for each pressure (blue dots). theorem [32] we get ɛ ɛ = a b. (2.17) The surface tension is anisotropic therefore we use ɛ for that for the surface tension parallel to the director and ɛ perpendicular to the director. The dimension of ɛ is energy per unit area, and will be in the order of k B T due to the thermostat, while the unit area will depend on the size of the particles. In the parallel case it will be in the order of σ 0 L 0 and in the perpendicular case in the order of σ 2 0 with σ 0 the diameter of the particles and L 0 their length. Now we can rewrite equation (2.17) to σ 0 L 0 = a b. (2.18) Equation (2.18) tells us that the shape of the box will be stretched in the nematic state and the stretch will be in the order of the aspect ratio of the particles. If we now assume we have an isotropic state, ɛ = ɛ due to the random distribution of the particles, and this leads to a = b and the box will be a cube. This shape change of the box has a dramatic effect on the simulations. When the pressure drops and the isotropic phase is reached, and then the pressure is increased again to the nematic phase the box will stretch. If the time in the nematic state is too short, only a couple of layers are formed in the smectic phase, and the smectic order parameter and long-range positional correlations cannot be determined accurately. One way to avoid this issue with the isotropic phase is not entering it. However, for the shortest particles and the highly flexible 19

23 particles we were not certain if the nematic state was stable, it is possible that they should be isotropic. The second issue with this simulation technique is the time that the pressure is kept constant, which is too short. The system does not get enough time to relax to equilibrium before we change the pressure again. This leads to smectic phases like the smectic-c which we do not expect because of the size of the system [30]. For example, it is observed that some layers are tilted and others are not and some layers are more densely packed than others (see figure 2.5). Figure 2.5: An example of a smectic-a where some layers are tilted. From left to right layers 1, 3 and 6 are horizontal and layers 2, 4 and 5 are tilted. Also layer 1 appears more densely packed than layer 3. The third problem was that the smectic-a phase did not disappear for increasing flexibility, which is against expectation. The range of pressures where the smectic-a phase was found for the higher flexibilities was P = 0.3 even if different lengths were used. This is also because the system did not get enough time to relax. Due to these problems, we instead preform simulations at various but constant pressures. The phase that is reached at the end of the simulation is the equilibrium state for the given pressure and density. The disadvantages of this technique is that a lot more simulations have to be done which consumes a lot of time and creates a lot of data. Also the the changes in the shape of the box still occur and still lead to problems as we will see in chapter 3. 20

24 Chapter 3: Simulation results In this chapter we discuss the results of our molecular dynamics simulations. In section 3.1 we first discuss the phase diagram of our semi-flexible particles, where we focus on the effects of length and flexibility. In section 3.2 we focus on the isotropic-nematic and the nematicsmectic-a phase transition. Finally in section 3.3 we consider some individual aspects of the phase diagram: Firstly, we extract the virial coefficients from the isotropic phase. Secondly, the layer thickness of the smectic-a and the smectic-b/crystal phase are measured. Finally, we take a look at the higher order phases, that is, the smectic-b, columnar and the crystal phase. In particular, we will discuss our inability to find the columnar phase and why we cannot always distinguish between the semctic-b and crystal phase apart in our results Phase diagram for aspect ratio L 0 /D = 6.46 Flexibility versus density Isotropic Nematic Smectic A Smectic B Crystal Flexibility L 0 /L p Volume fraction φ =v 0 N/V Figure 3.1: Phase diagram of flexibility L 0 /L p versus the density φ for rods with aspect ratio L 0 /D = The isotropic phase are represented the green circles, the nematic phase by the red stars, the smectic-a phase by the blue diamonds, the smectic-b phase by the black squares and the crystal phase by the purple triangles. Firstly the isotropic-nematic phase transition shifts to higher densities with increasing flexibility, secondly the nematic-smectic-a phase transition also shifts to higher density for increasing flexibility and the smectic-a phase disappears for flexibilities above Phase diagram We present our phase diagrams in two different ways: flexibility L 0 /L p versus density φ and pressure P versus density φ. The density is a volume density φ = v 0 N/V, with N the number of chains in the system, V the volume of the system and v 0 = π 4 D2 L 0 + 4π 3 D3 the volume of a capped cylinder (see figure 2.2). L 0 is the length of the chain that we calculated in chapter 2 and D the diameter of the particles. In figure 3.1 the L 0 /L p φ phase diagram of aspect 21

25 ratio L 0 /D = 6.46 is shown and in figure 3.2 the P φ phase diagram, the lines in this figure connect the data points of a given flexibility. 5 4 Phase diagram for aspect ratio L 0 /D = 6.46 Pressure versus density Isotropic Nematic Smectic A Smectic B Crystal L0 /L p =0.5 [ Pressure T/σ 3 ] L0 /L p = Volume fraction φ =v 0 N/V Figure 3.2: Phase diagram of pressure versus the density φ for rods with aspect ratio L 0 /D = The isotropic phases is represented the green circles, the nematic phase by the red stars, the smectic- A phase by the blue diamonds, the smectic-b phase by the black squares and the crystal phase by the purple triangles. The lowest curve is the flexibility L 0 /L p = 0.05 and the highest curve flexibility L 0 /L P = 0.5. We can see the same behaviour as in figure 3.1, the isotropic-nematic phase transition shifts to higher densities, so does the nematic-smectic-a phase transition and the smectic-a disappears. There is significant density gap towards for the phase transition towards the smectic-b/crystal phase. No distinguishable phase gap was found for the isotropic nematic phase transition. In these phase diagrams we can distinguish, based on the criteria discussed in chapter 2 five different phases: The isotropic phase, the nematic phase, the smectic-a phase, the smectic-b phase and crystal phase. Due to finite size effects, which we elaborate later, we are not able to distinguish the smectic-b from the crystal phase. Previous works predict that both the isotropic-nematic [18 20] and the nematic-smectic-a transition [17, 23] shift to higher density with increasing flexibility, which is also what our phase diagrams show. The smectic-a phase completely disappears for flexibilities above L 0 /L p = 0.1. The nematic-smectic-b/crystal phase transition and the smectic-a-smectic-b/crystal phase transition are both first order, which is why there is a density gap between the different phases. In figure 3.2 the black lines that connect the data points are almost horizontal for the phase transition towards the smectic-b/crystal phase, indicating that we are close to the coexistence pressure. There is no distinguishable phase gap for the isotropic-nematic phase transition, so it is weakly first order or a second order phase transition. In section 3.2 we discuss the isotropic-nematic phase transition. In figure 3.3 the L 0 /L p φ phase diagram is shown and in figure 3.4 the phase diagram P φ is shown for an aspect ratio L 0 /D = Comparing figures 3.1 and 3.3 reveal some 22

26 Phase diagram for aspect ratio L 0 /D = Flexibility versus density Isotropic Nematic Smectic A Smectic B Crystal Flexibility L 0 /L p Volume fraction φ =v 0 N/V Figure 3.3: Phase diagram of flexibility L 0 /L p versus the density φ for rods with aspect ratio L 0 /D = The isotropic phase are represented the green circles, the nematic phase by the red stars, the smectic-a phase by the blue diamonds, the smectic-b phase by the black squares and the crystal phase by the purple triangles. Comparing this to figure 3.1 we can see that the isotropic-nematic phase transitions shifts to lower densities and there is still a smectic-a phase for flexibility L 0 /L p = 0.5. effects of an increasing aspect ratio. Firstly, there is still a smectic-a phase for the highest flexibility L 0 /L p = 0.5. Secondly, the phase transition shifts to lower densities for both the isotropic-nematic and the nematic-smectic-a phase transition for higher aspect ratios. This was also observed in the work of Bolhuis and Frenkel [10]. In figure 3.4 there is a distinguishable phase gap at the isotropic-nematic phase transition, so the phase transition seems first order. 3.2 Individual phase transitions In this section we focus on the isotropic-nematic and the nematic-smectic-a phase transitions Isotropic-nematic phase transition We start with the isotropic-nematic phase transition. In figure 3.5 the nematic order parameter S 2 (equation (2.12)) is shown as function of time, for an aspect ratio of L 0 /D = and flexibility L 0 /L p = 0.1. In this figure the system starts in a crystal phase and quickly drops to a nematic phase where a nucluation process take place and finally a weak first order phase transition takes place to the isotropic phase. In figure 3.6 a snapshot of the system from figure 3.5 is shownduring the last moments it was isotropic. However, in some cases the phase transition appears more like a second order transition. For example, aspect ratio L 0 /D = 6.46 and flexibility L 0 /L p = 0.5, which is a shorter and 23

27 Phase diagram for aspect ratio L 0 /D = Pressure versus density Isotropic Nematic Smectic A Smectic B Crystal L0 /L p =0.5 ] T/σ 3 0 [ Pressure L0 /L p = Volume fraction φ =v 0 N/V Figure 3.4: Phase diagram of pressure versus the density φ for rods with aspect ratio L 0 /D = The isotropic phase are represented the green circles, the nematic phase by the red stars, the smectic-a phase by the blue diamonds, the smectic-b phase by the black squares and the crystal phase by the purple triangles. The lowest curve is the flexibility L 0 /L p = 0.1 and the highest curve flexibility L 0 /L P = 0.5. Comparing this to figure 3.2, we can see that the isotropic-nematic phase transition has shifted to lower densities also the pressure is lower and there is a distinguishable phase gap for the isotropic nematic phase transition. more flexible particle than previous discussed. In figure 3.5 the order parameter as function of time is plotted, showing that the nematic order parameter drops and then rises again before it lowers to a value of 0.1. Based on this low value of the nematic order parameter we expect it to be an isotropic phase. In figure 3.6 a snapshot of the system for one time of is shown, from which we can see that the system is in, an isotropic on a global scale. However, on a local scale it seems nematic. The phase transition seems second order, this is further supported by the slow decay of the nematic order parameter S 2 in figure 3.5. We are not able to extend the simulation because shortly after this time step the system becomes unstable due to the barostat making the volume too thin, the effect discussed in section Nematic-smectic-A phase transition We now continue with the nematic-smectic-a phase transition. We take as an example for the nematic-smectic-a phase transition the particles with aspect ratio L 0 /D = 7.54 and flexibility L 0 /L p = The nematic-smectic-a phase transition in our case is second order because we see a continuous drop in the order parameter, but the order of the nematic-smectic-a phase 24

28 Nematic order Nematic order Order parameter Order parameter S2 0.8 S Time Time Figure 3.5: The nematic order parameter S2 versus time. On the left for aspect ratio L0 /D = and flexibility L0 /L0 = 0.1 and on the right for aspect ratio L0 /D = 6.46 and flexibility L0 /Lp = 0.5. Both simulations start in a crystal state. The figure on the left aspect ratio L0 /D = shows a quick drop into the the nematic phase where a nucleation process take place and finally the system drop into the isotropic phase. The figure on the right aspect ratio L0 /D = 6.46 shows a quick drop from the crystal state and a much slower decay towards the isotropic phase. (a) (b) Figure 3.6: Snapshots of the system at the phase transition from nematic to the isotropic phase. On the left for aspect ratio L0 /D = with flexibility L0 /Lp = 0.1 and on the right for aspect ratio L0 /D = 6.46 and flexibility L0 /Lp = 0.5. Both snapshots are at a time (see figure 3.5). From figure 3.5 we suspect both system to be in an isotropic phase, the snapshot on the left is but the on the right is on a global scale isotropic, locally nematic. transition is a matter of debate 1. In figure 3.7, we show four graphs of the smectic order parameter as function of time at four different pressures P = 1.52, 1.54, 1.56 and Due to the continuous drop in the order parameter, we also use a density profile along the director to tell the difference between the nematic and smectic-a phase. In figure 3.8 the density profiles of a smectic-a and nematic phase are shown. For the pressure of P = 1.52 the system is in a nematic phase, the order parameter is low (around 0.1) and the density profile (not shown) resembles that of the figure 3.8b. For the pressure of P = 1.58 the system is in the smectic-a phase as density shows a clear periodicity in that of figure 3.8a and the smectic order parameter is about τ = 0.4. In the case of pressure P = 1.54 and P = 1.56 we see that 1 Previous work has shown different results for the nematic-smectic-a transition. Bolhuis finds for smaller aspect ratios then ours a first order phase transition and for larger aspect ratio a second order transition [10]. Bladon found a weak first order transition for aspect ratios L0 /D = 6 and flexibilities ranging from L0 /Lp = [23]. In theoretical work of Chen they find a second order phase transition [33]. 25

29 the order parameter fluctuates between the low value of 0.1 and the higher value of 0.4, and a similar fluctuations between Smectic-A 3.8a and nematic 3.8b density profiles of the two states. Just as with the isotropic-nematic phase transition we reached some sort of a coexistence Smectic order Smectic order Order parameter τ Order parameter τ Time Time 1.0 Smectic order Smectic order Order parameter τ Order parameter τ Time Time Figure 3.7: The smectic order parameter τ versus time, for aspect ratio L 0 /D = 7.54 and flexibility L 0 /L p = There are four different pressures: P = 1.52 (top left), P = 1.54 (top right), P = 1.56 (bottom left) and P = 1.58 (bottom right). For P = 1.52 the smectic order parameter is low (τ 0.1). At this pressure, the system is nematic, while for P = 1.58 the system is in a smectic-a phase with τ = 0.4. For the intermediate pressures P = 1.54 and P = 1.56 the system fluctuates between nematic and smectic-a phases, which can be seen from the fluctuations in τ. 3.3 Individual phases In this section we discuss some individual aspects of the phase behaviour. Firstly we measure the virial coefficients for the isotropic phase. Secondly we take a look at the layer thickness of the smectic-a and smectic-b/crystal phase. Finally we discuss why we cannot tell the smectic-b apart from the crystal phase, and why we did not find a columnar phase. In chapter 2 we saw that the equilibrium length of our particles was longer than the rest length due to thermal effects. In the results presented here, there is on top of that, a compression with increasing density which we discuss in more detail in chapter 4. We mention it here because it will have major influences. 26

30 normalized density normalized density position along director (a) position along director (b) Figure 3.8: The density profile of the system along the director both the rod aspect ratio is L 0 /D = 7.54, flexibility L 0 /L p = 0.05 and at pressure 1.54, on the left a profile just above the nematic-smectic-a phase transition at time 8000, on the right just below the nematic-smectic-a phase transition at time The figure on the right is the density profile of a nematic, on the left the density profile of the smectic-a the peaks are still visible. If the density is increases the width of the peaks becomes narrower The isotropic state The second virial coefficient can be calculated analytically for hard rods [8] and for very long semi-flexible particles in the limit L L p D, and, furthermore, they claim to be equal [18]. Therefore, it is tempting to assume that it will be the equal for short semi-flexible particles as well. We can test this hypothesis by calculating the virial coefficients from our simulation data. We do this by fitting a polynomial to the pressure P vs density φ data points for aspect ratio L 0 /D = 6.46, where we include also higher virial coefficients, because for the short particles we have it is know that the higher virial coefficients are significant [8, 34]. We do not know how many virial coefficients should be taken in to account, so we do not know what order polynomial we should fit. We used two extra arguments to help use determine the order of the fit. Firstly, the second virial coefficient must be positive because the particles only repel each other. Secondly, the second virial coefficient must be of the order of the theoretical calculated value for the hard rods. Based on the above arguments, the most realistic order of the polynomial is 4. If a second order polynomial was used the fit was not accurate enough, for third order polynomial the second order virial coefficient became negative and for a higher order polynomials the second virial became much larger then the theoretical value. In figure 3.9 the results are shown, where the points are the same data as in figure 3.2 but in this case the lines are from the fit. In figure 3.10 we plotted the virial coefficients versus the flexibility. The virials are scaled because we used φ = v 0 N/V for the density instead of ρ = N/V. The standard virial expansion is which we rewrite to P T = 1 [ φ 1 + B 2 v 0 P = ρt [ 1 + B 2 ρ + B 3 ρ ], (3.1) v 0 φ + B 3 v 2 0 ] φ = 1 φ + B v 2φ 2 + B 3φ (3.2) If we use the second virial coefficient for a capped cylinder [8] to calculate our scaled second virial coefficient B , we see that the measured second coefficients are in that order, the 27

31 ] T/σ 3 0 [ Pressure data fit for virial coefficient in isotropic phase for aspect ratio L 0 /D = ideal gas virial approximation L 0 /L p 0.05 L 0 /L p 0.1 L 0 /L p 0.2 L 0 /L p 0.3 L 0 /L p 0.4 L 0 /L p Volume fraction φ =v 0 N/V Figure 3.9: Plotted are the data points for aspect ratio L 0 /D = 6.46 for the isotropic phase, the lines connecting the dots are 4th order polynomials that we fitted to the data points. The blue circles have flexibility L 0 /L p = 0.05, the red downward triangles L 0 /L p = 0.1, the green plus L 0 /L p = 0.2, the yellow upward triangles L 0 /L p = 0.3, the black squares L 0 /L p = 0.4 and the grey stars L 0 /L p = 0.5. We also added the ideal gas (brown line) and the prediction from the second virial approximation (black dashed line) upto the second order for hard rods, showing that we need higher virial coefficients to match our data. The data points also show that flexibility does not affect the isotropic phase much, the data points start to deviate from each other only for densities above φ = 0.3. lowest B 2 = 1.59 and the highest B 2 = Only the third and fourth virial coefficient are influenced by the flexibility. The third virial coefficient decreases with increasing flexibility and the fourth increases with increasing flexibility. As mentioned before the length of our particles changes and this has a major influence on the virial coefficents as we will show in chapter 4, where we will show that the second virial coefficient depends on the density The smectic layer thickness We measure the layer thickness d with use of equation (2.13), which we maximize to gain the smectic order parameter and the corresponding layer thickness. In figure 3.11 we plot the relative smectic layer thickness for aspect ratio L 0 /D = 8.62, at which the smectic-a phase disappears for flexibilities L 0 /L p = 0.3. As mentioned before, the contour length of the particles decreases with increasing density (see chapter 4 for more details), so we scale the smectic layer thickness with the measured average length of the particles, the scaled smectic layer is d/ L. In figure 3.11 we see that the smectic layer thickness jumps on the phase transition from the smectic-a phase to the smectic-b/crystal phase. The increase layer thickness happens because the density within the layers increases because the particles pack more optimally (see figure 3.13). This reduces 28